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o. Assume throughout that the monopolist's objective runction is concave in q and I. (a) Derive the first·order conditions for the monopolist's choices.
C
12.C.8 Consider a homogeneous-good J·firm Cournot model in which the demand function x(p) is downward sloping but otherwise arbitrary. The firms all have an identical Cost function c(q) that is increasing in q and convex. Denote by Q the aggregate output of the J firms, and let Q-, = L. •• i q•. (a) Show that firm j's best response can be written as b(Q _ j).
(b) Compare the monopolist's choices with those of a benevolent social planner who can control both q and 1 (a "first-best" comparison).
(b) Show that h(Q _ j) need not be unique (i.e., that it is in general a correspondence, not a function).
(e) Compare the monopolist's choices with those of a benevolent social planner who can cantrall but not q (a "second·best" comparison). Suppose that the planner chooses 1 and then the monopolist chooses q.
(e) Show that if Q_j > Q- i , q, E h(Q_j), and qjE b(Q -i)' then (4 j + Q-i) ~ (qj + Q_j). Deduce from this that h(') can jump only upward and that b'(Q _i) ~ -I whenever this derivative is defined.
12.B.10· Consider a monopolist that can choose both its product's price p and its quality q. The demand for its product is given by x(p, q), which is increasing in q and decreasing in p. Given the price chosen by the monopolist, does the monopolist choose the socially efficient quality level?
(d) Use you result in (e) to prove that a symmetric pure strategy Nash equilibrium exists in this model. (e) Show that multiple equilibria are possible. (f) Give sufficient conditions (they are very weak) for the symmetric equilibrium to be the only equilibrium in pure strategies.
12.C.IA In text. 12.C.2C Extend the argument of Proposition 12.C.1 to show that under the assumptions made in the text [in particular, the assumption that there is a price p < co such that x(p) = 0 for all p ~ p], both firms setting their price equal to c with certainty is the unique Nash equilibrium of the Rertrand duopoly model even when we allow for mixed strategies.
(a) Derive the Nash equilibrium of this model. Under what conditions does it involve only one firm producing? Which will this be' (b) When the equilibrium involves both firms producing. how do equilibrium outputs and profits vary when firm I's cost changes?
(a) Show that both firms naming prices equal to the smallest multiple of A that is strictly greater than c is a pure strategy equilibrium of this game. Argue that it does not involve either firm playing a weakly dominated strategy.
(e) Now consider the general case of J firms. Show that the ratio of industry profits divided by industry revenue in any (pure strategy) Nash equilibrium is exactly H 1£, where £ is the elasticity of the market demand curve at the equilibrium price and H, the Herfindahl index of concelltration, is equal to the sum of the firms' squared market shares L.i(q;lQ*)'. (Note: This result depends on the assumption of constant returns to scale.)
12.C.4" Consider altering the Bertrand duopoly model to a case in which each firm j's cost per unit is ('j and C 1 < C2' (a) What are the pure strategy Nash equilibria of this game?
-
12.C.9" Consider a two·firm Cournot model with constant returns to scale but in which firms' costs may differ. Let c j denote firm j's cost per unit of output produced, and assume that c, > c,. Assume also that the inverse demand function is p(q) = a - bq, with a > c,.
12.C.3" Note that the unique Nash equilibrium of the Rertrand duopoly model has each firm playing a weakly dominated strategy. Consider an alteration of the model in which prices must be named in some discrete unit of account (e.g., pennies) of size A.
(b) Argue that as A _ 0, this equilibrium converges to both firms charging prices equal to c.
431
,-------------------------------------------------------------~
12.C.10· Consider a J·firm Cournot model in which firms' costs differ. Let Cj(qi) = ajc(qj) denote firm j's cost function, and asSUme that c(·) is strictly increasing and convex. Assume that 17 1 >'" > (XJ.
432
CHAPTER
12:
MARKET
POWER
~~~--------------------------------------------------------(a) Show that if more than one firm is making positive sales in a Nash equilibrium of this model, then we cannot have productive efficiency; that is, the equilibrium aggregate output Q* is produced inefticiently.
--
(c) Provide an example in which wclfare decreases when a firm becomes more productive (i.e .. when ':X J falls for somej). [Him: Consider an improvement in cost for IIrm I in the model of Exercise 12.C.9.] Why can this happen?
12_C.16" Derive the Nash equilibrium prices and profits in the circular city model with J firms when travel costs are quadratic, as in Exercise 12.C.IS. Restrict attention to the case in which v is large enough that the possibility of non purchase can be ignored. What happens as J grows large! As I falls?
12.C.11 (' Consider a capacity-constrained duopoly pricing game. Firm j's capacity is 'Ij for j = 1,2, and it has a constant cost per unit of output of (' 0 and that there exists a price p such that x(f» = 'I, + 'I,. Suppose also that x(p) is concave. Let p(.) = x - '(.) denote the inverse demand function. Given a pair of prices charged. sales are dt:termined as follows: consumers try to buy at the low·priccd firm first. If demand exceeds this firm's capacity, consumers are servcd in order of their valuations. starting with high· valuation consumers. If prices are the same, demand is split evenly unless one firm's demand exceeds its capacity, in which case the extra demand spills over to the other firm. Formally, the firms' sales arc given by the functions x,(p" p,) and x,(p"p,) satisfying [x i(·) gives the amount firm i sells taking account of its capacity limitation in fullilling demand]
12,C.17" Consider the linear city model in which the two firms may have different constant unit production costs c, > 0 and c, > O. Without loss of generality, take c, OS; c, and suppose that I' is large enough that non purchase can be ignored. Determine the Nash equilibrium prices and sales levcls for equilibria in which both firms make strictly positive sales. How do local changes in c, affect the equilibrium prices and profits of firms I and 2? For what values of c, and c, docs the equilibrium involve one firm making no sales? 12.C.IH" (The SI"ckl"herli leaderxhip model) There are two firms in a market. Firm I is the -'leader" and picks its quantity lirst. Firm 2, the "follower," observes firm I's choi I' + 31 in the linear city model discussed in Example 12.C.2, a Jirm j's best response to any price of its rival p _i always results in all consumers purchasing
(e) It takes K periods to respond to a deviation.
frol11 one of the two firms. 12.C.l4
C
12.0.2" In text.
Consider the linear city model discussed in Example 11.C.2.
(a) Derivc the bcst·rcsponsc functions when equilibrium in this case is = p! = c + l.
l' E
I'r
(b) Repeat (a) for the case in which v E (c
+
(c
+
2f, c
I' -
+ 1')/2
(a) Under what conditions can the symmetric joint monopoly outputs (q" q,) = (q~/2, q~/2)
l/, c + 21).
l' < f + t. the unique Nash equilibrium involves prices of and some consumers not purchasing from either firm.
(e) Show that when (v
12.D.3" Consider an infinitely repeated Coumot duopoly with discount factor b < I, unit costs of c > 0, and inverse demand function p(q) = a - bq, with a > c and b > O.
+ 31). Show that the unique Nash
(d) Show that when v E (c + I, C + 11), the unique symmetric equilibrium is 112. Are there asymmetric equilibria in this case?
be sustained with strategies that call for (q~12, q~ 12) to be played if no one has yet deviated
pi = pi =
and for the single-period Cournot (Nash) equilibrium to be played otherwise?
pt
(b) Derive the minimal level of o. In stage 2. the demand function for firm j as a function of the price vector p = (p, •. .. PJ) of the J active firms is Xj(p) = '[i' -/I(Jp/LI P.)]. Analyze the welfare properties as the size (:;l) and the substitution (fl) parameters change. 12.G.l" Consider the linear inverse demand Cournot duopoly model and the linear city dilTerentiated-price duopoly model with differing unit costs that you examined in Exercises 12.C9 and 12.CI7. Find the derivative, with respect to a change in firm J's unit cost, affirm 2's equilibrium quantity in the Cournot model and equilibrium price in the linear city model. In which model is this change in firm 2's behavior beneficial to firm I? 12.AA.IA In text.
What happens to this loss as K ~ 0" 12.E,4" Consider a two-stage model of entry in which all potential entrants have a cost per unit of (" (in additional to an entry cost of K) and in which, whatever number of firms enter, a perfect cartel is formed. What is the socially optimal number of firms for a planner who cannot control this cartel behavior? What are the welfare consequences if the planner cannot
12.AA.2C Prove Proposition 12.AAA. [I/illl: Consider a strategy profile of the following form: the players arc to play an outcome path involving some pair (qt' q,) in period I and «Ii, qj) in every period thereafter. If either player deviates, this outcome path is restarted.] 12.BB.1A In text.
control entry? I2.E.SC Consider a two-stage entry model with a market that looks like the market in Exercise 12.CI6. The entry cost is K. Compare the equilibrium number of firms to the number that a planner would pick who can control (a) entry and pricing and (b) only entry. I2.E.6" Compare a one-stage and a two-stage model of entry with Cournot competition [all potential entrants arc identical and production costs arc c(q) = cq]. Argue that any (SPNE) equilibrium outcome of the two-stage game is also an outcome of the one-stage game. Show by example thal the reverse is not true. Argue that we cannot, however, have more firms active in the one-stage game than in the two-stage game. 12,E.7" Consider a one-stage entry model in which firms announce prices and all potential firms have average costs of AC('l) (including their. fixed setup costs) with a minimum average
=
12.1IB,2" Show that if the incumbent in the entry deterrence model discussed in Appendix II is indifTercnt between deterring entry and accommodating it, social wclfare is strictly greater if he chooses deterrence. Discuss generally why we might not be too surprised if entry deterrence could in somc cases raise social welfare. I2,BII.3 C Consider the linear city model of Exercise 12.C2 with v> (. + 31. Suppose that firm I enters the market first and can choose to set up either one plant at onc end of the city or two plants, one at each end. Each plant costs F. Then firm E decides whether to enter (for simplicity, restrict it to building one plant) and at which end it wants to locate its plant. Determine the equilibrium of this model. How is it affected by the underlying parameter values? Compare the welfare of this outcome with the welfare if there Were no entrant. Compare with the case where there is an entrant but firm I is allowed to build only one plant.
435
C
Adverse Selection, Signaling,
HAP
T
E
R
SEC T ION
13
13.A Introduction One of the implicit assumptions of the fundamental welfare theorems is that the characteristics of all commodities are observable to all market participants. Without this condition, distinct markets cannot exist for goods having differing characteristics, and so the complete markets assumption cannot hold. In reality, however, this kind of information is orten asymmetrically held by market participants. Consider the following three examples: (i) When a firm hires a worker, the firm may know less than the worker does about the worker's innate ability. (ii) When an automobile insurance company insures an individual, the individual may know more than the company about her inherent driving skill and hence about her probability of having an accident. (iii) In the used-car market, the seller of a car may have much better information about her car's quality than a prospective buyer does.
436
I N FOR MAT ION A l A S Y M MET R I E SAN 0
A 0 V E R S ESE LEe T ION
will be low. Moreover, this fact may even further exacerbate the adverse selection problem: If the price that can be received by selling a used car is very low, only sellers with really bad cars will ofTer them for sale. As a result, we may see little trade in markets in which adverse selection is present, even if a great deal of trade would occur were information symmetrically held by all market participants. We also introduce and study in Section 13.B an important concept for the analysis of market intervention in settings of asymmetric information: the notion of a coIISrmilled Pareto oprimal allocarioll. These are allocations that cannot be Pareto improved upon by a central authority who, like market participants, cannot observe individuals' privately held information. A Pareto-improving market intervention can be achieved by such an authority only when the equilibrium allocation fails to be a constrained Pareto optimum. In general, the central authority's inability to observe individuals' privately held information leads to a more stringent test for Paretoimproving market intervention. I n Sections 13.C and 13.0, we study how market behavior may adapt in response to these informational asymmetries. In Section \3.C, we consider the possibility that informed individuals may find ways to sigllill information about their unobservable knowledge through observable actions. For example, a seller of a used car could ofTer to allow a prospective buyer to take the car to a mechanic. Because sellers who have good cars are more likely to be willing to take such an action, this offer can serve as a signal of quality. In Section 13.0, we consider the possibility that uninformed parties may develop mechanisms to distinguish, or screen, informed individuals who have dilTcring information. For example, an insurance company may ofTer two policies: one with no deductible at a high premium and another with a significant deductible at a much lower premium. Potential insureds then self-selecr, with high-ability drivers choosing the policy with a deductible and low-ability drivers choosing the no-deductible policy. In both sections, we consider the welfare characteristics of the resulting market equilibria and the potential for Pareto-improving market intervention. For expositional purposes, we present all the analysis that follows in terms of the labor market example (i). We should nevertheless emphasize the wide range of settings and fields within economics in which these issues arise. Some of these examples are developed in the exercises at the end of the chapter.
and Screening
A number of questions immediately arise about these settings of asymmetric illjomlllrioll: How do we characterize market equilibria in the presence of asymmetric information? What are the properties of these equilibria? Are there possibilities for welfare-improving market intervention? In this chapter, we study these questions, which have been among the most active areas of research in microeconomic theory during the last twenty years. We begin, in Section 13.B, by introducing asymmetric information into a simple competitive market model. We see that in the presence of asymmetric information, market equilibria often fail to be Pareto optimal. The tendency for inefficiency in these settings can be strikingly exacerbated by the phenomenon known as adverse selecrioll. Adverse selection arises when an informed individual's trading decisions depend on her privately held information in a manner that adversely affects uninformed market participants. In the used-car market, for example, an individual is more likely to decide to sell her car when she knows that it is not very good. When adverse selection is present, uninformed traders will be wary of any informed trader who wishes to trade with them, and their willingness to pay for the product offered
1 3 • B:
437
----------------------------------------------------~----~~~
!
13.B Informational Asymmetries and Adverse Selection Consider the following simple labor market model adapted from Akerlof's (1970) pioneering work: I there arc many identical potential firms that can hire workers. Each produces the same output using an identical constant returns to scale teChnology in which labor is the only input. The firms arc risk neutral, seek to maximize their expccted prolits, and act as price takers. For simplicity, we take the price of the firms' output to equal I (in units of a numeraire good). Workers difTer in the number of units or output they produce if hired by a firm, I. Akcrlof (1970) used the example of a used-car market in which only the setler of a used car knows if the car is a "lemon." for this reason, (his type: of model is sometimes referred to as a It'mon.~ model.
I
1
438
C HAP T E R
1 3:
A 0 V E R 5 ESE L E C T ION
I
S I G N A LIN G,
AND
S CREE NI N G
SECTION
---------------------------------------------------------------which we denote by 0.' We let [Q, 0] c R denote the set of possible worker productivity levels, where 0 ~ Q< Ii < 00. The proportion of workers with productivity of 0 or less is given by the distribution function F(O), and we assume that F(') is nondegenerate, so that there are at least two types of workers. The total number (or, more precisely, measure) of workers is N. Workers seek to maximize the amount that they earn from their labor (in units of the numeraire good). A worker can choose to work either at a firm or at home, and we suppose that a worker of type 0 can earn r(O) on her own through home production. Thus, r(O) is the opportunity cost to a worker of type 0 of accepting employment; she will accept employment at a firm if and only if she receives a wage of at least r(O) (for convenience, we assume that she accepts if she is indifferen!).' As a point of comparison, consider first the competitive equilibrium arising in this model when workers' productivity levels are publicly observable. Because the labor of each different type of worker is a distinct good, there is a distinct equilibrium wage 11"(0) for each type O. Given the competitive, constant returns nature of the firms, in a competitive equilibrium we have IV'(O) = 0 for all 0 (recall that the price of their output is I), and the set of workers accepting employment in a firm is
13.8:
INFORMATIONAL
ASYMMETRIES
e(l\') =
~
{II: r(O)
Consider, next, the demand for labor as a function of IV. If a firm believes that the average productivity of workers who accept employment is fl, its demand for labor is given by if" < if I'
I\'
= I\'
if I' >
(13.B.3)
1\'.
Now, if worker types in set e' are accepting employment offers in a competitive equilibrium, and if firms' beliefs about the productivity of potential employees correctly renect the actual average productivity of the workers hired in this equilibrium, then we must have I' = £[0 I 0 E eo]. Hence, (I3.B.3) implies that the demand for labor can equal its supply in an equilibrium with a positive level of employment if and only if I\' = £[0 10 E eo]. This leads to the notion of a competitive equilibrium presented in Definition 13.B.1.
(I3.B.I)
(This is simply the total revenue generated by the workers'labor.)5 Aggregate surplus is therefore maximized by setting 1(0) = I for those () with r(O) ~ 0 and 1(0) = 0 otherwise (we again resolve indifference in favor of working at a firm). Put simply,
Definition 13.B.1: In the competitive labor market model with unobservable worker productivity levels, a competitive equilibrium is a wage rate w' and a set e' of worker types who accept employment such that
2. A worker's productivity could be random without requiring any change in the analysis th r and some with 0 < r. In this setting, the Pareto optimal allocation of labor has workers with 0 ~ r accepting employment at a firm and those with 0 < r not doing so. Now consider the competitive equilibrium. When r(O) = r for all 0, the set of workers who are willing to accept employment at a .given wage, e(w), is either [Q, 0] (if w ~ r) or 0 (if w < rl. Thus, £[010 E e(wl] = £[0] for all wand so by (I3.B.5) the equilibrium wage rate must be w' = £[0]. If £[0] ~ r, then al/ workers accept employment at a firm; if £[/1] < r, then none do. Which type of equilibrium arises depends on the relative fractions of good and bad workers. For example, if there is a high fraction of low-productivity workers then, because firms cannot distinguish good workers from bad, they will be unwilling to hire any workers at a wage rate that is sunicicnt to have them accept employment (i.e., a wage of at least r). On the other hand, if there arc very few low-productivity workers, then the average productivity of the workforce will be above r, and so the firms will be willing to hire workers at a wage that they arc willing to accept. In one case, too many workers arc employed relative to the Pareto optimal allocation, and in the other too few. The cause of this failure of the competitive allocation to be Pareto optimal is simple to see: because firms are unable to distinguish among workers of differing productivities, the market is unable to allocate workers efficiently between firms and home production"
--
SECTION
13.8:
INfORMATIONAL
ASYMMETRIES
AND
ADVERSE
SELECTION
441
I. . - - - - - - - - - - 45'
!
I I I I
£[/1]
~ !
£[Olr(O):5"]
I
Figure 13.B.1
I I I I
r@
A compelitive equilibrium wilh adverse selection.
'I'
not so; indeed, the market may fail completely despite the fact that every worker type should work at a lirm. To see the power of adverse selection, suppose that r(O)!> 0 for all 0 E [Q, 0] and that r(') is a strictly increasing function. The first of these assumptions implies that the Pareto optimal labor allocation has every worker type employed by a firm. The second assumption says that workers who are more productive at a firm arc also more productive at home. It is this assumption that generates adverse selection: Because the payolT of home production is greater for more capable workers, only less capable workers accept employment at any given wage "' [i.e., those with r(O) !> "l The expected value of worker productivity in condition (13.B.5) now depends on the wage rate. As the wage rate increases, more productive workers become willing to accept employment at a firm, and the average productivity of those workers accepting employment rises. For simplicity, from this point on, we assume that F(') has an associated density function [(.), with [(0) > for all 0 E [Q, 0]. This insures that the average productivity of those workers willing to accept employment, E[O 1 1'(0) !> w], varies continuously with the wage rate on the set"' E [r@,oo]. To determine the equilibrium wage, we use conditions (13.B.4) and (11B.5). Together they imply that the competitive equilibrium wage IV' must satisfy
°
Adverse Seieclioll and Markel Ullravelillg
A particularly striking breakdown in efficiency can arise when r(O) varies with O. In this case, the average productivity of those workers who arc willing to accept employment in a firm depends on the wage, and a phenomenon known as adverse select;oll may arise. Adverse selection is said to occur when an informed individual's trading decision depends on her unobservable characteristics in a manner that adversely alTects the uninformed agents in the market. In the present context, adverse selection arises when only relatively less capable workers are willing to accept a firm's employment offer at any given wage. Adverse selection can have a striking effect on market equilibrium. For example, it may seem from our discussion of the case in which r(O) = r for all 0 that problems arise for the Pareto optimality of competitive equilibrium in the presence of asymmetric information only if there are some workers who should work for a firm and some who should not (since when either 0 < r or Q> r the competitive equilibrium outcome is Pareto optimal). In fact, because of adverse selection, this is
w' = £[01 r(O) !> 11"].
(13.8.6)
We can use Figure I3.B.1 to study the determination of the equilibrium wage w·. There we graph the values of £[0 1r(O) !> w] as a function of w. This function gives the expected value of Ii for workers who would choose to work for a firm when the prevailing wage is II'. It is increasing in the level IV for wages between r(O) and r(ii), has a minimum value of Q when II' = r(q), and attains a maximum value ~f £[0] for IV ~ 1'(1)).7 The competitive equilibrium wage 11" is found by locating the wage rate at which this function crosses the 45-degree line; at this point, condition (13.B.6) is satisfied. The set of workers accepting employment at a firm is then e' = {o: r(O)!> w £[0] = ... marginally reduces her supply of tabor 10 a firm here, Ihe firm is made worse olT, in contrast with the situation in a competitive market with perfect information, where the wage exactly equals a worker's marg.inal productivity.
7. The figure does nol depici Ihis funclion for wages below r@. Because £[0] > r(q) in this model. no wage below r(q) can be an equilibrium wage under our assumption that £[/118(",) = 0] = £[0]. 8. For another diagrammatic determination of equilibrium, see Exercise 118.1.
J
442
CHAPTER
13:
ADVERSE
SELECTION,
SIGNALING.
AND
SECTION
SCREENING
13.B:
INFORMATIONAL
ASYMMETRIES
AND
ADVERSE
SELECTION
443
--------------------------------------------------------------------- ,--------------------------------------------------------------------45" 45'
I\"*
£[0]
,, ,,
£[/1]
:----r, , ,, ,,
= 0
£[Olr(O),; .. ]
£[Ulr(O),; w]
Complete market failure.
,, ,, ,,
r(lI) = ~
r(li)
,,
.
r(Q)
t
.
"'~ w~ w~ .'
We can see immediately from Figure I3.B.1 that the market equilibrium need not be ellicient. The problem is that to get the best workers to accept employment at a firm, we need the wage to be at least r(O). But in the case depicted, firms cannot break evcn at this wage because their inability to distinguish among different types of workers leaves them receiving only an expected output of £[0] < r(O) from each worker that they hire. The presence of enough low-productivity workers therefore forces thc wage down below r(ih, which in turn drives the best workers out of the market. But once the best workers are driven out of the market, the average productivity of the workforce falls, thereby further lowering the wage that firms are willing to pay. As a result, once the best workers are driven out of the market, the next-best may follow; the good may then be driven out by the mediocre. How far can this process go? Potentially very far. To see this, consider the case depicted in Figure 13.B.2, where we have r@ = and r(O) < 0 for all other O. There the equilibrium wage rate is w' = ~, and only type workers accept employment in the equilibrium. Because of adverse selection, essentially no workers are hired by firms (more precisely, a set of measure zero) even though the social optimum calls for all
q
q
to be hired,9 Example 13.8.1: To see an explicit example in which the market completely unravels let r(O) = 0:0, where 2 < 1, and let 0 be distributed uniformly on [0, 2]. Thus, r(q) = q (since (J = 0), and r(O) < 0 for 0 > 0. In this case, £[0 \ r(O) ,;; w] = (w/20:). For 0: > !, £[0 \ r(-O) ,;; 0] = 0 and £[0 \ r(O) ,;; w] < w for all w > 0, as in Figure 13.B.2.10 The competitive equilibrium defined in Definition 13.8.1 need not be unique. Figure 13.B.3, for example, depicts a case in whieh there are three equilibria with strictly positive employment levels. Multiple competitive equilibria can arise because there is virtually no restriction on the slope of the function £[0\ r(O) ,;; w). At any wage II'. this slope depends on the density of workers who are just indifferent about accepting employment and so it can vary greatly if this density varies.
9. In this equilibrium, every agent receives the same payoff as if the market were abolished: every firm earns zero and a worker of type 0 earns rIO) for all 0 (including 0 = q). 10. This example is essentially the one developed in Akeriof (1970). His example corresponds to the case): = ~.
Flgur. 13.B.2 (left)
Figure 13.B.3 (right)
Multiple eompetitiv, equilibria .
Note that the equilibria in Figure I3.B.3 can be Pareto ranked. Firms earn zero profits in any equilibrium, and workers are better off if the wage rate is higher (those workers who do not accept employment are indifferent; all other workers are strictly better off). Thus, the equilibrium with the highest wage Pareto dominates all the others. The low-wage, Pareto-dominated equilibria arise because of a coordination failure: the wage is too low because firms expect that the productivity of workers accepting employment is poor and, at the same time, only bad workers accept employment precisely because the wage is low.
A
GllI1lC- Theoret ic
Approach
The notion of competitive equilibrium that we have employed above is that used by Akerlof (1970). We might ask whether these competitive equilibria can be viewed as the outcome of a richer model in which firms could change their offered wages but choose not to in equilibrium. The situation depicted in Figure 13.B.3 might give you some concern in this regard. For example, consider the equilibrium with wage rate \\'!. In this equilibrium, a firm that experimented with small changes in its wage offer would find that a small increase in its wage, say to the level w' > \I'! depicted in the figure, would raise its profits because it would then attract workers with an average productivity of £[0\ r(O) S \\"] > 11". Hence, it seems unlikely that a model in which firms could change their offered wages would ever lead to this equilibrium outcome. Similarly, at the equilibrium involving wage II'f, a firm that understood the structure of the market would realize that it could earn a strictly positive profit by raising its offered wage to ",'. To be more formal about this idea, consider the following game-theoretic model: The underlying structure of the market [e.g., the distribution of worker productivities F(') and the reservation wage function r(')] is assumed to be common knowledge. Market behavior is captured in the following two-stage game: In stage 1, two firms simultaneously announce their wage offers (the restriction to two firms is without loss of generality). Then, in stage 2, workers decide whether to work for a firm and, if so, which one. (We suppose that if they are indifferent among some set of firms, then thcy randomize among them with equal probabilities.)" Proposition 13.B.1 characterizes the subgame perfect Nash equilibria (SPNEs) of this game for the adverse selection model in which r(') is strictly increasing with 1'(0) S () for all 0 E [Q, 0] and F(') has an associated density f(·) with f(O) > 0 for all () E [Q, 0]. Proposition 13.B.1: Let W' denote the set of competitive equilibrium wages for the adverse selection labor market model, and let w· = Max {w: WE W·}. (i) If w· > r(Q) and there is an r. > 0 such that E[O \ r(O) ,;; w'] > w' for all w' E (w* - r., w*), then there is a unique pure strategy SPNE of the two-stage game-theoretic mode\. In this SPNE, employed workers receive It. Note that if there is a single type of worker with productivity 0, this model is simply the labor market version of the Bertrand model of Section 12.C and has an equilibrium wage equal to 0, the competitive wage.
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CHAPTER
13:
ADVERSE
SELECTION.
SIGNALING,
AND
SCREENING
a wage of w', and workers with types in the set 0(w') = (II; r(lI) :s w'} accept employment in firms. (ii) If w' = rIO), then there are multiple pure strategy SPNEs. However, in every pur~ strategy SPNE each agent's payoff exactly equals her payoff in the highest-wage competitive equilibrium. Proof: To begin, note that in any SPNE a worker of type II must follow the strategy of accepting employment only at one of the highest-wage firms, and of doing so if and only if its wage is at least r(O)." Using this fact, we can determine the equilibrium behavior of the firms. We do so for each of the two cases in turn. (i) 11" > rIO); Note, first, that in any SPNE both firms must earn exactly zero. To see this, supp;se that there is an SPNE in which a total of M workers arc hired at a wage IV and in which the aggregate profits of the two firms are
n=
M(£[Olr(O):s IV] - IV) > O.
Note that n > 0 implies that M > 0, which in turn implies that IV ~ r@. In this case, the (weakly) less-profitable firm, say firm j, must be earning no more than n/2. But firm j can earn profits of at least M(£[O I r(O) :s IV + IX] - IV - IX) by instead offering wage Ii' + ex for IX> O. Sinoe £[Olr(O):s 11'] is continuous in 11', these profits can be made arbitrarily close to by choosing IX small enough. Thus, firm j would be better off deviating, which yields a contradiction; we must therefore have :s O. Because neither firm can have strictly negative profits in an SPNE (a firm can always offer a wage of zero), we conclude that both firms must be earning exactly zero in any SPNE. From this fact, we know that if IV is the highest wage rate offered by either of the two firms in an SPNE, then either IV E W' (i.e., it must be a competitive equilibrium wage rate) or Ii' < r(O) (it must be so low that no workers accept employment). But suppose that IV < w,-= Max {II'; WE W'}. Then either firm can earn strictly positive expected profits by deviating and offering any wage rate 11" E (II" - e, 11"). We conclude that the highest wage rate offered must equal IV' in any SPNE. Finally, we argue that both firms naming 11" as their wage, plus the strategies for workers described above, constitute an SPNE. With these strategies, both firms earn zero. Neither firm can earn a positive profit by unilaterally lowering its wage because it gets no workers if it does so. To complete the argument, we show that £[0 I rIO) :s IV] < IV at every IV > 11", so that no unilateral deviation to a higher wage can yield a firm positive profits either. By hypothesis, IV" is the highest competitive wage. Hence, there is no IV> w' at which £[0 I rIO) :s 11'] = IV. Therefore, because £[0 I rIO) :s 11'] is continuous in IV, £[0 I riO) :s 11'] - IV must have the same sign for all 1\' > IV'. But we cannot have £[0 I rIO) :s IV] > II' for all II' > 11'" because, as W .... 00, £[(11 rIO) :s 11'] .... £[0], which, under our assumptions, is finite. We must therefore have £[0 I rIO) :s 11'] < II' at all IV> 11". This completes the argument for case (i). The assumption that there exists an e > 0 such that £[111 r(lI) :s 11"] > 11" for all IV' E (IV' - t, IV') rules out pathological cases such as that depicted in Figure 13.BA.
n
n
(ii) IV' = rIO); In this case, £[Olr(O):s IV] < '" for all IV> IV', so that any firm attracting workers at a wage in excess of "," incurs losses. Moreover, a firm must 12. Recall that we assume that a worker accepts employment whenever she is indilTerent.
---
SECTION
13.8:
INFORMATIONAL
ASYMMETRIES
AND
ADVERSE
SELECTION
445
,-----------------------------------------------------------, i
E[O)
~!
i, E[Olr(O) ~ .. ) ,,i!~ ,, ,,, ,, ,,
r@
r(ii)
earn exaclly zero by announcing any IV S; 11'". Hence, the set of wage offers (w" "'2) that can arise in an SPNE is {(w" w2 ): Wj:S 11'" for j = I, 2}. In everyone of these SPNEs, all agents earn exactly what they earn at the competitive equilibrium involving wage rate IV"; both firms earn zero, and a worker of type 0 earns rIO) for all liE [q, /1]. • One difference between this game-theoretical model and the notion of competitive equilibrium specified in Definition I3.B.1 involves the level of firms' sophistication. In the competitive equilibria of Definition I3.B.I, firms can be fairly unsophisticated. They need know only the average productivity level of the workers who acoept employment at the going equilibrium wage; they need not have any idea of the underlying market mechanism. In contrast, in the game-theoretic model, firms understand the entire structure of the market, including the full relationship that exists bet ween the wage rate and the quality of employed workers. The game-theoretic model tells us that if sophisticated firms have the ability to make wage offers, then we break the coordination problem described above. If the wage is too low, some firm will find it in its interest to offer a higher wage and attract better workers; the highest-wage competitive outcome must then arise.')
COllstrained Pareto Optima alld Market intervention We have seen that the presence of asymmetric information often results in market equilibria that fail to be Pareto optimal. As a consequence, a central authority who knows all agents' private information (e.g., worker types in the models above), and can engage in lump-sum transfers among agents in the economy, can achieve a Pareto improvement over these outcomes. In practice, however, a central authority may be no more able to observe agents' private information than are market participants. Without this information, the authority will face additional constraints in trying to achieve a Pareto improvement. For example, arranging lump-sum transfers among workers of different types will be impossible because the authority cannot observe workers' types directly. For Pareto-improving market intervention to be possible in this case, a more stringent test must therefore be passed. An allocation that cannot be Pareto improved by an J 3. See Exercise 13.8.6, however, for an example of a model of adverse selection in which, for some parameter values, the highest-wage competitive equilibrium is not an SPNE of our gametheoretic model.
Figure 13.8.4
A pathologicat exampte.
446
CHAPTER
'3:
ADVERSE
SELECTION.
SIGNALING.
AND
SCREENING
5 E C T ION
, 3 • B:
I N FOR MAT ION A L A . V M MET A IE.
AND
A D V E A S ESE l E C T ION
447
---------------------------------------------------------------------- ------------------------------------------------------------------authority who is unable to observe agents' private information is known as a constrained (or second-best) Pareto optimum. Because it is more difficult to generate a Pareto improvement in the absence of an ability to observe agents' types, a constrained Pareto optimal allocation need not be (fully) Pareto optimal [however, a (full) Pareto optimum is necessarily a constrained Pareto optimum]. Here, as an example, we shall study whether Pareto-improving market intervention is possible in the context of our adverse selection model (where r(') is strictly increasing with r(O) S 0 for all 0 E [Q, Ii] and F(') has an associated density f(·) with f(O) > 0 for all 0 E [~, 0]) when the central authority cannot observe worker types. That is, we study whether the competitive equilibria of this adverse selection model are constrained Pareto optima. In general, the formal analysis of this problem uses tools that we develop in Section 14.C in our study of principal-agent models with hidden information (see, in particular, the discussion of monopolistic screening). As these techniques have yet to be introduced, we shall not analyze this problem fully here. (Once you have studied Section 14.C, however, refer back to the discussion in small type at the end of this section.) Nevertheless, we can convey much of the analysis here. By way of motivation, note first that in examining whether a Pareto improvement relative to a market equilibrium is possible, we might as well simply think of intervention schemes in which the authority runs the firms herself and tries to achieve a Pareto improvement for the workers (the firms' owners will then earn exactly what they were earning in the equilibrium, namely zero profits). Second, because the authority cannot distinguish directly among different types of workers, any differences in lump-sum transfers to or from a worker can depend only on whether the worker is employed (the workers otherwise appear identical). Thus, intuitively, there should be no loss of generality in restricting attention to interventions in which the authority runs the firms herself, offers a wage of w. to those accepting employment, an unemployment benefit of w. to those who do not [these workers also receive r(O)]. leaves the workers free to choose whether to accept employment in a firm, and balances her budget. (In the small-type discussion at the end of this section, we show formally that this is the case.) Given this background, can the competitive equilibria of our adverse selection model be Pareto-improved upon in this way? Consider, first, dominated competitive equilibria, that is, competitive equilibria that are Pareto dominated by some other competitive equilibrium (e.g., the equilibrium with wage rate wf shown in Figure 13.B.3). A central authority who is unable to observe worker types can always implement the best (highest-wage) competitive equilibrium outcome. She need only set w, = w', the highest competitive equilibrium wage, and w. = O. All workers in set 0( w') then accept employment in a firm and, since w' = E[O 1r(/I) S w'], the authority exactly balances her budget." Thus, the outcome in such an equilibrium is not a constrained Pareto optimum. In this case, the planner is essentially able to step in and solve the coordination failure that is keeping the market at the low-wage equilibrium. 14. An equivalent but less heavy·handed intervention would have the authority simply require any operating firm to pay a wage rate equal to w*. Firms will be willing to remain
operational because they break even at this wage rate, and a Pareto improvement results.
What about the highest-wage competitive eqUilibrium (i.e., the SPNE outcome in the game-theoretic model of Proposition 13.B.1)? As Proposition 13.B.2 shows, any such equilibrium is a constrained Pareto optimum in this model. proposition 13.B.2: In the adverse selection labor market model (where r(') is strictly increasing with rIO) S /I for all /I E [0, Ii] and F(') has an associated density (.) with (0) > 0 for all 0 E [Q.O]). the- highest-wage competitive equilibrium is a constrained Pareto optimum. Proof: If all workers are employed in the highest wage competitive equilibrium then the outcome is fully (and, hence, constrained) Pareto optimal. So suppose some are not employed. Note, first, that for any wage w. and unemployment benefit w. offered by the central authority the set of worker types accepting employment has the form [Q, 0] for some 0 [it is {O: .... + r(O) S IV.}]. Suppose, then,that the authority attempts to implement an outcome in which worker types 0 s ~ for bE [Q, Ii] accept employment. To do so, she must choose .... and w. so that
"'. + r(O) = w•. In addition, to balance her budget, IV, and IV. must also satisfy" IV,F(O) + IV.(I - F(O» =
f
Of(O) dO.
(13.B.7)
(I3.B.8)
Substituting into (13.B.7) from (13.B.8), we find that. given the choice of 0, the values of w .. and WI' must be ",.(0) =
and
"',(0) =
r r
Of(O) dO - r(O)F(O)
(l3.B.9)
Of(O) dO + r(O)(1 - F(O)),
(13.B.10)
or, equivalently, IV.(O) = F(O)(E[OIO
s
/I] -
r(9»
w,(O) = F(O)(E[O lOs 9] - riO»~
(13.B.11)
+ r(O).
(l3.B.12) Now, let 0* denote the highest worker type who accepts employment in the highest-wage competitive equilibrium. We know that r(O') = E[/il 0 SO']. Hence, from conditions (13.B.II) and (I3.B.12), we see that w.(O') = 0 and w.(O') = r(O'). Thus, the outcome when the authority sets 0 = 0' is exactly the same as in the highest-wage competitive equilibrium. We now examine whether a Pareto improvement can be achieved by setting 6 ¥ 0*. Note that for any 0 E [Q, 0] with 6 ¥ 0', type Q workers are worse off than in the equilibrium if .... (0) < r(O') [r(O') is their wage in the equilibrium] and type workers are worse off if "',(Ii) < O. Consider 0 < 0* first. Since r(O*) > r(O), condition (l3.B.lO) implies that
o
w.(O)
s
r
Of(O) dO
+ r(O')(1
- F({J)),
15. The authority will never wish to run a budget surplus. If "". and w .. lead to a budget surplus, then setting w = "' .. +,; and wf' = WI' + (; for some I: > 0 is budget feasible and is Pareto superior. (Note that the set of workers accepting employment would be unchanged.) lI
448
CHAPTER
13:
ADVERSE
SELECTION,
SIGNALING,
AND
SCREENING
--------------------------------------------------------------------------and so
11',(0) - ,(0*) :s; F(O)(E[OI 0 :s; 0] - ,(0*») = F(O)(E[OIO:s;
0] -
E[OIO:s; 0*])
< O. Thus, type 0 workers must be made worse off by any such intervention. Now cO;lsider 0 > 0*. We know that E[O I,(0) :s; 11'] < II' for all II' > 11'* (see the proof of Proposition 13.8.1). Thus, since ,(0*) = 11'* and ,(-) is strictly increasing, we have £[0 I ,(0) :s; ,(0)] < ,(ti) for all Ii> 0*. Moreover,
E[OI'(O):s; ,(0)]
= E[OIO:s; 0],
and so £[0 I 0 :s; 0] - ,(0) < 0 for all 0> 0*. But condition (l3.B.II) then implies that 11',(6) < 0 for all 0 > 0*, and so type ii workers are made worse off by any such intervention. _ Hence, when a central authority cannot observe worker types, her options may be severely limited. Indeed, in the adverse selection model just considered, the authority is unable to create a Pareto improvement as long as the highestwage competitive equilibrium (the SPNE outcome of the game-theoretic model of Proposition 13.B.I) is the market outcome. " More generally, whether Paretoimproving market intervention is possible in situations of asymmetric information depends on the specifics of the market under study (and as we have already seen, possibly on which equilibria result). Exercises 13.B.8 and 13.B.9 provide two examples of models in which the highest-wage competitive equilibrium may fail to be a constrained Pareto optimum. Although it is impossible to Pareto improve a constrained Pareto optimal allocation, market inlcrvcntion could still be justified in the pursuit of distributional aims. For example, if social welfare is given by the sum of weighted worker utilities
r
[1(0)0
SECTION
13.8:
INFORMATIONAL
ASYMMETRIES
- I(O»,(O)]i.(O) dF(O),
(13.B.13)
where i.(0) > 0 for all 0, then social welfare may be increased even though some worker types end up worse off. In the applied literature, for example, it is common to see aggregate surplus used as the social welfare function, which is equivalent to the choice of i.(O) = N for all 0." When society has this social welfare function, social welfare can be raised relative to the competitive equilibrium in Figure 13.B.1 (which, by Proposition 13.B.2, is a constrained Pareto optimum) simply by mandating that all workers must work for a firm and that all firms must
16. Proposition 13.8.2 Can also be readily generalized to allow r(O) > 0 for some O. (See Exercise I3.B.IO.) 17. Note that when types cannot be observed. aggregate surplus is no longer a valid weHare measure for any social welfare function because, unlike the case of perfect information, lump~sum
transfers across worker types are infeasible. (See Section 1O.E for a discussion of the need for lump-sum transfers to justify aggregate surplus as a welfare measure for any social welfare function,)
ADVERSE
pay workers a wage of £(0). Although workers of type ii are made worse off by this intervention, welfare as measured by aggregate surplus increases.'· An interesting interpretation of the choice of aggregate surplus as a social welfare function is in tcrms of an unborn worker's ex ante expected utility. In particular, imagine that each worker originally has a probability /(0) of ending up a type 0 worker. If this unborn worker is risk neutral, then her ex ante expected utility is exactly equal to expression (13.B.13) with i.(O) = I for all O. Thus, maximization of aggregate surplus is equivalent to maximization of this unborn worker's expected utility. We might then say that an allocation is an ex ante COII.wrailJeti Pareto optimum in this model ir, in the absence of an ability to observe worker types, it is impossible to devise a market intervention that raises aggregate surplus. We see, therefore. thai whether an allocation is a constrained optimum (and. thus, whether a planned intervention leads to a Pareto improvement) can depend on the point at which the welrare evaluation is conducted (i.e., before the workers know their types, or after)'· Lei us now use the techniques of Section 14.C 10 show formally that we can restrict attention in searching for a Pareto improvement to interventions of the type considered above. We shall look for a Pareto improvement for the workers keeping the profits of the firms' owners nonnegative. For notational simplicity, we shall treat the firms as a single aggregate firm. By the revelation principle (see Section 14.C), we know that we can restrict attention to direct rcvcl;.ttion mechanisms in which every worker type tells the truth, Here a direct revelation mechanism assigns, for each worker type () E [q, 0], a payment from the authority to the worker of ",(Ii) E R, a lax t(O) paid by the firm to the authority, and an employment decision /(Ii) E 10, I:. The sci of feasible mechanisms here are those that satisfy the illdividual ratiollality c0I1s1raillf for the firm.
r
[/(0)0 - t(O)] dF(O)
(13.B.14)
the hwly£'[ hulcmce i'olldilion for the central authority,
f
[t(O) - ...(0)] dF(O)
(13.B.15)
and the lrur/Hellillf} (or ;'Icentivt! compatibilil)', or seltse/(-ftiml) constraints that say that for
all 0 and
iJ 1\'(0) + (I - 1(0»,.(0)
(I3.B.l7)
I R. Moreover. bec 0, c.. (e, 0) > 0, c.(e,O) < 0 for all e > 0, and c.. (e, 0) < 0 (subscripts denote partial derivatives). Thus, both the cost and the marginal cost of education are assumed to be lower for high-ability workers; for example, the work required to obtain a degree might be easier for a high-ability individual. Letting U(IV, e 10) denote the utility of a type 0 worker who chooses education level e and receives wage IV, we take U(IV, e I0) to equal her wage less any educational costs incurred: U(IV, e 10) = w - c(e, 0). As in Section 13.B, a worker of type 0 can earn r(O) by working at home.
Random mOve of
Flgure13.C.l
worker Iype.
The ex.tensive form of the education signaling
,.A,,""------------ nature determines
game. , ; : : . - - - - - - - - - - - - - - - " " : I l.....I ' - - - - Worker chooses
education level contingent on her Iype (really a continuous choice),
Conditional on seeing a level of t. say".
firms make wage offers simultaneously (really a continuous choice),
Worker decides which olTer 10 accept. if any. }
J
452
CHAPTER
13:
ADVERSE
SELECTION,
SIGNALING,
AND
SECTION
SCREENING
--------------------------------------------------------------------
13.C:
SIGNALING
453
~---------------------------------------------------------------
Note that, in contrast with the model of Section 13.B, here we explicitly model only a single worker of unknown type; the model with many workers can be thought of as simply having many of these single-worker games going on simultaneously, with the fraction of high-ability workers in the market being).. In discussing the equilibria of this game, we often speak of the "high-ability workers" and "low-ability workers," having the many-workers case in mind. The equilibrium concept we employ is that of a weak perfect Bayesian equilibrium (see Definition 9.C.3), but with an added condition. Put formally, we require that, in the game tree depicted in Figure 13.C.l, the firms' beliefs have the property that, for each possible choice of e, there exists a number /lie) E [0, 1] such that: (i) firm I's belief that the worker is of type 011 after seeing her choose e is /lie) and (ii) after the worker has chosen e, firm 2's belief that the worker is of type 011 and that firm I has chosen wage offer w is precisely /l(e)ur(w I e), where ur(w I e) is firm I's equilibrium probability of choosing wage offer IV after observing education level e. This extra condition adds an element of commonality to the firms' beliefs about the type of worker who has chosen e, and requires that the firms' beliefs about each others' wage offers following e are consistent with the equilibrium strategies both on and off the equilibrium path. We refer to a weak perfect Bayesian equilibrium satisfying this extra condition on beliefs as a perfect Bayesian equilibrium (PBE). Fortunately, this PBE notion can more easily, and equivalently, be stated as follows: A set of strategies and a belief function /,(e) E [0, 1] giving the firms' common probability assessment that the worker is of high ability after observing education level e is a PBE if
011 ------------------------
Flgur. 13.C.2 (left)
Indifference curves for high- and low·ability workers: the single-crossing
-----------------------
()I.
property. Flgur. 13.C.3 (rlghl)
o Knowing this fact, we turn to the issue of the worker's equilibrium strategy, her choice of an education level contingent on her type. As a first step in this analysis, it is useful to examine the worker's preferences over (wage rate, education level) pairs. Figure IlC.2 depicts an indifference curve for each of the two types of workers (with wages measured on the vertical axis and education levels measured on the horizontal axis). Note that these indilTerence curves cross only once and that, where they do, the indifference curve of the high-ability worker has a smaller slope. This property of preferences, known as the single-crossing property, plays an important role in the analysis of signaling models and in models of asymmetric information more generally. It arises here because the worker's marginal rate of substitution between wages and education at any given (IV, e) pair is (dIVide)" = e.(e, 0), which is decreasing in 0 because (" ... (e, II) < O. We can also graph a function giving the equilibrium wage offer that results for each education level, which we denote by wee). Note that since in any PBE wee) = /,(e)1I1I + (I - /,(e»Ol for the equilibrium belief function /lie), the equilibrium wage offer resulting from any choice of e must lie in the interval [0,.,0,,]. A possible wage offer function w(e) is shown in Figure 13.C.3. We are now ready to determine the equilibrium education choices for the two types of workers. It is useful to consider separately two different types of equilibria that might arise: separating equilibria, in which the two types of workers choose different education levels, and pooling equilibria, in which the two types choose the same education level.
(i) The worker's strategy is optimal given the firm's strategies. (ii) The belief function /lie) is derived from the worker's strategy using Bayes' rule where possible. (iii) The firms' wage offers following each choice e constitute a Nash equilibrium of the simultaneous-move wage offer game in which the probability that the worker is of high ability is /l(e).20 In the context of the model studied here, this notion of a PBE is equivalent to the sequential equilibrium concept discussed in Section 9.C. We also restrict our attention throughout to pure strategy equilibria. We begin our analysis at the end of the game. Suppose that after seeing some education level e, the firms attach a probability of /,(e) that the worker is type 0Il' lf so, the expected productivity of the worker is /l(e)OI/ + (1 - /l(e»Ol' In a simultaneous-move wage offer game, the firms' (pure strategy) Nash equilibrium wage offers equal the worker's expected productivity (this game is very much like the Bertrand pricing game discussed in Section 12.C). Thus, in any (pure strategy) PBE, we must have both firms offering a wage exactly equal to the worker's expected productivity, /l(e)OIl + (I - /,(e»Ol'
Separating Equilihria To analyze separating equilibria, let e*(tI) be the worker's equilibrium education choice as a function of her type, and let 1V*(e) be the firms' equilibrium wage offer as a function of the worker's education level. We first establish two useful lemmas. Lemma 13.C.1: In any separating perfect Bayesian equilibrium, w*(e*(OH» = 0H and w*(e*(OLl) = 0L: that is, each worker type receives a wage equal to her productivity level. Proof: In any PBE, beliefs on the equilibrium path must be correctly derived from the equilibrium strategies using Bayes' rule. Here this implies that upon seeing education level e*(Ol)' rlrms must assign probability one to the worker being type 0,.. Likewise, upon seeing education level e*(OIl)' firms must assign probability one
20. Thus, the extra condition we add imposes equilibrium-like play in parts of the tree off the equilibrium path. See Section 9.C for a discussion of the need to augment the weak perfect Bayesian equilibrium concept to achieve this end.
J
A wage schedule.
454
CHAPTER
13:
ADVERSE
SELECTION,
SIGNALING,
AND
--- --
SCREENING
Type 0" II'
Type Ot
qJ ,// i
,, , ,,
. I
/
0" ""(to)
.--
I
(/,.
._._._._._.--;-----------
I
I
I I
I
,,
I
e
11
11
e'(O,.)
e'(O,,)
(h
and 0",
Figure 13.C.4 (left)
Low-ability worker's outcome in a
Lemma 13.C.2: In any separating perfect Bayesian equilibrium, e'(Otl = 0; that is, a low-ability worker chooses to get no education. Proof: Suppose not, that i~, that when the worker is type OL' she chooses some strictly positive education level e ::. O. According to Lemma 13.C.I, by doing so, the worker receives a wage equal to 0L' However, she would receive a wage of at least 0,_ if she instead chose e = O. Since choosing e = 0 would have save her the cost of education, she would be strictly better off by doing so, which is a contradiction to the assumption that > 0 is her equilibrium education level. _
e
Lemma 13.C.2 implies that, in any separating equilibrium, type O,.'s indifference curve through her equilibrium level of education and wage must look as depicted in Figure 13.CA. Using Figure 13.CA, we can construct a separating equilibrium as follows: Let c'(O,,) = ii, let e'(Od = 0, and let the schedule ",'(e) be as drawn in Figure 13.C.5. The firms' equilibrium beliefs following education choice e are JI'(e) = (w'(e) - Od/(Oll - Od. Note that they satisfy I,'(e) E [0, I] for all e;::o; 0, since \\,'(e) E [OL' 0,,]. To verify that this is indeed a PBE, note that we are completely free to let firms have any beliefs when e is neither 0 nor On the other hand, we must have JI(O) = 0 and Il(e) = 1. The wage offers drawn, which have ",'(0) = 0,- and ""(e) = 0", renect exactly these beliefs. What about the worker's strategy? It is not hard to see that, given the wage function IV'(e), the worker is maximizing her utility by choosing e = 0 when she is type 0L and by choosing e = when she is type 0". This can be seen in Figure 13.C.S by noting that, for each type that she may be, the worker's indifference curve is at its highest-possible level along the schedule ""(e). Thus, strategies [e'(O), ""(e)] and the associated beliefs JI(e) of the firms do in fact constitute a PBE. Note that this is not the only PBE involving these education choices by the two types of workers. Because we have so much freedom to choose the firms' beliefs off the equilibrium path, many wage schedules can arise that support these education
e.
0,.
/~r'\" I
"'(e)
L/ I --·---r : -~:'~------i--------t----I I I I I
I I , , ,
I
o
to the worker being type 0". The resulting wages are then exactly respectively. _
/
'-.
----------.----------,
"'(e'(O,.)) = 01.
,,
e
A separating equilibrium with the
"'(e)
------------~----------
separating equilibrium.
Figure 13.C.S (right)
A separating equilibrium: Type is inferred from education level.
SIGNALING
Figure 13.C.6 (Iell)
(/"
. . . . . ·*'
I
13.C:
Type O.
'I"~---------
8" ------------ ,----------
,, ,, ,, ,
SECTION
"'(0,.)
11
"
r'(O,,)
.'«(/,,)
11
choices. Figure 13.C.6 depicts another one; in this PBE, firms believe that the worker is certain to be of high quality if c ;::0; i' and is certain to be of low quality if e < e. The resulting wage schedule has ""(e) = 0" if e ;::0; ii and ""(e) = OL if e < ii. In these separating equilibria, high-ability workers arc willing to get otherwise useless education simply because it allows them to distinguish themselves from low-ability workers and receive higher wages. The fundamental reason that education can serve as a signal here is that the marginal cost of education depends on a worker's type. Because the marginal cost of education is higher for a low-ability worker [since c,.• (e, 0) < 0], a type 0" worker may find it worthwhile to get some positive level of education e' > 0 to raise her wage by some amount "'"' > 0, whereas a type OL worker may be unwilling to get this same level of education in return for the same wage increase. As a result, firms can reasonably come to regard education level as a signal of worker quality. The education level for the high-ability type observed above is not the only one that can arise in a separating equilibrium in this model. Indeed, many education levels for the high-ability type arc possible. In particular, any education level between i! and el in Figure 13.C.7 can be the equilibrium education level of the high-ability workers. A wage schedule that supports education level e'(O,,) = e, is depicted in the figure. Note that the education level of the high-ability worker cannot be below in a separating equilibrium because, if it were, the low-ability worker would deviate and pretend to be of high ability by choosing the high-ability education level. On the other hand, the education level of the high-ability worker cannot be above €, because, ifit were, the high-ability worker would prefer to get no education, even if this resulted in her being thought to be of low ability. Note that these various separating equilibria can be Pareto ranked. In all of them, firms cam zero profits. and a low-ability worker's utility is 0,.. However, a high-ability worker does strictly better in equilibria in which she gets a lower level of education. Thus, separating equilibria in which the high-ability worker gets education level e (e.g., the equilibria depicted in Figures 13.C.S and 13.C.6) Pareto dominate all the others. The Pareto-dominated equilibria are sustained because of the high-ability worker's fear that if she chooses a lower level of education than that prescribed in the equilibrium firms will believe that she is not a 'high-ability worker. These beliefs can be maintained because in equilibrium they are never disconfirmed.
e
same education choices as in Figure 13.C.S but different
off-equilibriumpath beliefs. Figure 13.C.7 (right)
A separating equilibrium with an
education choice
> e by high-ability workers.
e"(OH)
455
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SCREENING
.
w
Figure 13.C.8
~TypeO.
E[O) --- --------------------
0, ------------------------
(a)
SEC T ION
1 3 • C:
S I G N A LIN G
457
----- ,------------------------------------------------------------------------
0, -------------------------
(b)
\I'
Separating equilibria may be Pareto dominated by the no·signaling OUlcom (a) A. separating e equlhbnum that is nOI Pareto domInated by the no,s'gnaling
w
Type 0.
0" ------ ------~~~~:-~~
£[11]
,,--.~
I I I
/:
I
V~
outcome.
-----~--------------- I
I
(b) A separating equilibrium that is Pareto dominated by the nO'signaling
0,
.-.,,/
.. '(e)
\
------r---------------
I
I I
t"
I I I
e'
"
outcome.
"~(Oil; j
Figure 13.C,10 (right)
= L, H
The only remaining issue therefore concerns what levels of education can arise in a pooling equilibrium, It turns out that any education level between 0 and the level e' depicted in Figure 13.C.9 can be sustained. Figure 13.C.1O shows an equilibrium supporting education level e', Given the wage schedule depicted, each type of worker maximizes her payoff by choosing education level e', This wage schedule is consistent with Bayesian updating on the equilibrium path because it gives a wage offer of £[0] when education level e' is observed, Education levels between 0 and e' can be supported in a similar manner. Education levels greater than e' cannot be sustained because a low-ability worker would rather set e = 0 than e > e' even if this results in a wage payment of OL' Note that a pooling equilibrium in which both types of worker get no education Pareto dominates any pooling equilibrium with a positive education leveL Once again, the Pareto-dominated pooling equilibria are sustained by the worker's fear that a deviation will lead firms to have an unfavorable impression of her ability. Note also that a pooling equilibrium in which both types of worker obtain no education results in exactly the same outcome as that which arises in the absence of an ability to signal. Thus, pooling equilibria are (weakly) Pareto dominated by the no-signaling outcome,
It is of interest to compare welfare in these equilibria with that arising when worker types arc unobservable but no opportunity for signaling is available. When education is not available as a signal (so workers also incur no education costs), we arc back in the situation studied in Section I3.B. In both cases, firms earn expected profits of zero. However, low-ability workers are striclly worse off when signaling is possible. I n both cases they incur no education costs, but when signaling is possible they receive a wage of OL rather than £(0). What about high-ability workers~ The somewhat surprising answer is that high-ability workers may be either better or worse off when signaling is possible. In Figure 13.C.8(a), the high-ability workers are better off because of the increase in their wages arising through signaling. However, in Figure I3.C.8(b), even though high-ability workers seek to take advantage of the signaling mechanism to distinguish themselves, they are worse off than when signaling is impossible! Although this may seem paradoxical (if high-ability workers choose to signal, how can they be worse olP), its cause lies in the fact that in a separating signaling equilibrium firms' expectations are such that the wage-education outcome from the no-signaling situation, (w, e) = (£[0],0), is no longer available to the high-ability workers; if they get no education in the separating signaling equilibrium, they are thought to be of low ability and offered a wage of 0L' Thus, they can be worse off when signaling is possible, even though they are choosing to signal. Note that because the set of separating equilibria is completely unaffected by the fraction i. of high-ability workers, as this fraction grows it becomes more likely that the high-ability workers are made worse off by the possibility of signaling [compare Figures 13.C.8(a) and 13.C.8(b)]. In fact, as this fraction gets close to I, nearly every worker is getting costly education just to avoid being thought to be one of the handful of bad workers!
Multiple Equilibria alld Equilibrium Refinemelll The multiplicity of equilibria observed here is somewhat disconcerting, As we have seen, we can have separating equilibria in which firms learn the worker's type, but we can also have pooling equilibria where they do not; and within each type of equilibrium, many different equilibrium levels of education can arise, In large part, this multiplicity stems from the great freedom that we have to choose beliefs off the equilibrium path, Recently, a great deal of research has investigated the implications of pulling "reasonable" restrictions on such beliefs along the lines we discussed in Section 9.D, To see a simple example of this kind of reasoning, consider the separating equilibrium depicted in Figure 13,C.7, To sustain e l as the equilibrium education level of high-ability workers, firms must believe that any worker with an education level below e I has a positive probability of being of type OL' But consider any education level eE (e, el)' A type OL worker could never be made better off choosing such an education level than she is getting education level e = 0 regardless of what
Poolillg Equilibria Consider now pooling equilibria, in which the two types of workers choose the same level of education, e'(OL) = e'(O/l) = eO. Since the firms' beliefs must be correctly derived from the equilibrium strategies and Bayes' rule when possible, their beliefs when they see education level e' must assign probability). to the worker being type 0/1' Thus, in any pooling equilibrium, we must have w'(e') = ).0/1 + (I - i.)OL = £[0]. I
J
(Ienl The highest-possible education level in a pooling equilibrium.
Figure 13.C.9
A pooling equilibrium.
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firms believe about her as a result. Hence, any belief by firms upon seeing education level e > e other than !l(e) = I seems unreasonable, But if this is so, then we must have w(e) = 9", and so the high-ability worker would deviate to e. In fact, by this logic, the only education level that can be chosen by type 0" workers in a separating
---
SECTtON
"'II'
wage of If so, low-ability workers would choose e = 0 and high-ability workers would choose e = ell' This alternative outcome involves firms incurring losses on low-ability workers and making profits on high-ability workers. However, as long as the firms break even on average, they are no worse off than before and a Pareto improvement has been achieved. The key to this Pareto improvement is that the central authority introduces cross-subsidization, where high-ability workers are paid less than their productivity level while low-ability workers are paid more than theirs, an outcome that cannot occur in a separating signaling equilibrium. (Note that the outcome when signaling is banned is an extreme case of cross-subsidization.)
equilibrium involving reasonable beliefs is e. In Appendix A we discuss in greater detail the use of these types of reasonablebeliefs refinements. One refinement proposed by Cho and Kreps (1987), known as the illluirive criterion, extends the idea discussed in the previous paragraph to rule out not only the dominated separating equilibria but also all pooling equilibria. Thus, If we accept the Cho and Kreps (1987) argument, we predict a unique outcome to this two-type signaling game: the best separating equilibrium outcome, which is shown in Figures 13.C.S and 13.C.6.
Exercise l3,C.3: In the signaling model discussed in Section I3.C with r(O,,) = r(O,,) = 0, construct an example in which a central authority who does not observe worker types can achieve a Pareto improvement over the best separating equilibrium through a policy that involves cross-subsidization, but cannot achieve a Pareto improvement by simply banning the signaling activity. [Hint: Consider first a case with linear inditTerence curves.J
Secolld-Best Market Intervention I n contrast with the market outcome predicted by the game-theoretic model studied in Section I3.B (the highest-wage competitive equilibrium), in the presence of signaling a central authority who cannot observe worker types may be able to achieve a Pareto improvement relative to the market outcome. To see this in the simplest manner, suppose that the Cho and Kreps (1987) argument predicting the best separating equilibrium outcome is correct. We have already seen that the best separating equilibrium can be Pareto dominated by the outcome that arises when signaling is impossible. When it is, a Pareto improvement can be achieved simply by banning the signaling activity, In fact, it may be possible to achieve a Pareto improvement even when the no-signaling outcome does not Pareto dominate the best separating equilibrium. To see how, consider Figure l3.C.1 I. In the figure, the best separating equilibrium has low-ability workers at point (OL' 0) and high-ability workers at point (01/, e). Note that the high-ability workers would be worse off if signaling were banned, since the point (£[OJ, O) gives them less than their equilibrium level of utility. Nevertheless, note that if we gave the low- and high-ability workers outcomes of (IVL,O) and ("'", ell), respectively, both types would be better off. The central authority can achieve this outcome by mandating that workers with education levels below ell receive a wage of"'L and that workers with education levels of at least ell receive a
,, ,I
13.C:
The case with r(OIl) = r(O"l = 0 studied above, in which the market outcome in the absence of signaling is Pareto optimal. illustrates how the use of costly signaling can reduce welfare. Yet, when the market outcome in the absence of signaling is not efficient, signaling's ability to reveal information about worker types may instead create a Pareto improvement by leading to a more efficient allocation of labor. To see this point, suppose that we have r = ,«(lL) = '(011)' with 0L < r < 0" and £[0] O. Contract (wI. + e, I,J will attract all type 01. workers. and contract (IV II + e, 11/) will attract all type 01/ workers. [Note that since type 0, initially prefers contract (w" I,) to (WI' tl)' we have w, - C(I" 0,) ~ WI - C(II' 0,), and so (Wi + e) - C(ti' Oil ~ (wj + e) - c(tjo 0,).] Since e can be chosen to be arbitrarily small. this deviation yields this firm profits arbitrarily close to n, and so the firm has a profitable deviation. Thus. we must have n :5 O. Because no firm can incur a loss in any equilibrium (it could always earn zero by oITering no contracts), both firms must in fact earn a profit of zero. _
w
Type (I, :..,....-- Indifference Curve
13,0:
~
til
lemma 13,0.2: No pooling equilibria exist.
for type 0, with perfttt observability.
Proof: Suppose that there is a pooling equilibrium contract (WP.I P). By Lemma 13.0.1, it lies on the pooled break-even line, as shown in Figure 13.0.3. Suppose that firm j is oITering contract (w p • I P ). Then firm k ¥ j has a deviation that yields it a strictly positive profit: It ofTers a single contract (1\', i) that lies somewhere in the shaded region in Figure IlD.3 and has Ii· < 0". This contract attracts all the type 0" workers and none of the type {lL workers, who prefer (w p• I P) over (IV, i). Moreover, since II- < 0", firm k makes strictly positive profits from this contract when the high-ability workers accept it. _
Flgur. 13.0.2 (right)
Break-even lines.
We now consider the possibilities for separating equilibria. Lemma 13.0.3 shows that all contracts accepted in a separating equilibrium must yield zero profits. Lemma 13.0.3: If (WL' ttl and (WH' tH ) are the contracts signed by the low- and high-ability workers in a separating equilibrium. then both contracts yield zero profits; that is, w L = OL and w H = 0H' Proof: Suppose first that 11'1. < 0,.. Then either firm could earn strictly positive profits by instead oITering only contract (w,., rd, where 01. > wL > IVI.' All low-ability workers would accept this contract: moreover, the deviating firm earns strictly positive profits from any worker (of low or high ability) who accepts it. Since Lemma 13.0.1 implies that no such deviation can exist in an equilibrium, we must have II"L ~ OL in any separating equilibrium. Suppose, instead, that 11'11 < 011' as in Figure 13.0.4. If we have a separating Flgur. 13.0.3 (left)
Type OL
No pooling equilibria Type 0"
exist. Figure 13.0.4 (right)
Lemma 13,0.1: In any equilibrium, whether pooling or separating, both firms must earn zero profits.
The high·ability contract in a
Proof: Let (wI.' t,J and (IV II • til) be the contracts chosen by the low- and high-ability workers, respectively (these could be the same contract), and suppose that the two firms' aggregate profits are n > O. Then one firm must be making no more than n/2. Consider a deviation by this firm in which it alTers contracts (WI. + e, tl.) and
°L ----------------------
1
separating equilibrium cannot have wli < 0Il'
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SCREENING
equilibrium, then the type Ii, contract (w"ILl must lie in the hatched region of the figure (by Lemma 13.0.1, it must also have 11', > lid. To see this, note that since type 011 workers choose contract (11'1/, 11/), contract (11'" Id must lie on or below the type 011 indifference curve through (11'1/, I,,), and since type 0, workers choose (11'" I,) ovcr (11'", I,,), contract (w" Id must lie on or above the type 0, indifference curve through (11'11' III)' Suppose that firm j is offering the low-ability contract (11',,1 1.1. Then firm k # j could earn strictly positive profits by deviating and offering only a contract lying in the shaded region of the figure with a wage strictly less than 011' sllch as (Ii', i). This contract, which has 11'/1 < 011' will be accepted by all the type 011 workers and by none of the type 0, workers [since firm j will still be offering contract (11'1., I,J]. SO we must have 11'11 ~ 0" in any separating equilibrium. Since, by Lemma 13.0.1, firms break even in any equilibrium, we must in fact have 11'1. = III. and 11'11 = 011' •
---- ...--Oc (~,;,,;)------------------
Proposition 13.0.2 summarizes the discussion so far. Proposition 13.0.2: In any subgame perfect Nash equilibrium of the screening game, low-ability workers accept contract (OL' 0), and high-ability workers accept contract (OH' iH), where iH satisfies OH - e(iH. Ot! = OL - e(O, Ot!. Proposition 13.0.2 does not complete our analysis, however. Although we have established what any equilibrium must look like, we have not established that one exists. In fact, we now show that one may nOI exist. Suppose that both firms are offering the two contracts identified in Proposition 13.0.2 and illustrated in Figure 13.0.7(a). Does either firm have an incenlive to deviate? No firm can earn strictly positive profits by deviating in a manner that attracts either only high-ability or only low-ability workers (just try to find such a deviation). But what about a deviation that attracts 0/1 workers? Consider a deviation in which the deviating firm attracts all workers 10 a single pooling contract. In Figure 13.0.7(a), a contract can attract both types of workers if and only if it lies in the shaded region. There is no profitable deviation of this type if, as depicted in the figure, Ihis shaded area lies completely above the pooled break-even line. However, when some of the shaded area lies strictly below the pooled break-even line, as in Figure 13.D.7(b), a profitable deviation to a pooling contract such as (I", i) exists. In this case, 110 eql/i1ibrium exists. Even when no single pooling contract breaks the separating equilibrium, it is possible that a profitable deviation involving a pair of contracts may do so. For example, a firm can attract both types of workers by offering the contracts (IV" Id and ("'1/,1/1) depicted in Figure 13.0.8. When it does so, type 0, workers accept contract (1"/., I,) and type 0/1 workers accept (WI/,1I/)' If this pair of contracts yields the firm a positive profit, then this deviation breaks the separating contracts identified
Proof: Consider Figure 13.0.6. By Lemmas 13.0.3 and 13.0.4, we know that = (0,,0) and that 11'11 = 01/' In addition, if the type 0, workers are willing to acccpt contract (0/.,0), III must be at least as large as the level ill depicted in the
(11'/., I,J
Figure 13.0.5 (leH)
The low-ability workers must recci\c
contract (Oc,O) in an! separating equilibrium (Ii',i)
I I I
I
I I I
I
o
ill
(b)
(0/1, i/l)' •
Wc can now derive the high-ability workers' contract.
Of. (;~.t~)-----r------------
o
(al
posilive profit by also offering, in addition to its current contracts, a contract lying in the shaded region of the figure with "'/I < 0/1, such as (w, I). This contract attracts all the high-ability workers and does not change the choice of the low-ability workers. Thus, in any separating equilibrium, the high-ability contract must be
Lemma 13.0.5: In any separating equilibrium, the high-ability workers accept contract (OH' i H ), where iH satisfies OH - c(iH , Ii L ) = OL - c(O, Ot!.
I
o
figure. Note that low-ability workers are indifferent between contracts (OL' 0) and (0/1' i/l), and so 0/1 - e(il/' Od = 0, - e(O, 0,). Suppose, then, that the high-ability contract (0/1' III) has 1/1 > il/' as in the figure. Then either firm can earn a strictly
I'roof: By Lemma 13.0.3,11'1. = 0, in any separating equilibrium. Suppose that the low-ahility workers' contract is instead some point (0/., li.1 with Ii. > 0, as in Figure 13.0.5. (Although it is not important for the proof, the high-ability contract must then lie on the segment of the high-ability break-even line lying in the hatched region of the figure, as shown.) If so, then a firm can make strictly positive profits by offering only a contract lying in the shaded region of the figure, such as (w, I). All low-ability workers accept this contract, and the contract yields the firm strictly positive profits from any worker (of low or high ability) who accepts it. •
I
SCREENING
465
Figure 13.0.7
Lemma 13.0.4: In any separating equilibrium, the low-ability workers accept contract (OL'O); that is, they receive the same contract as when no informational imperfections are present in the market.
Typ< Ii,.
13.0:
£[0] -----
Lemma 13.0.4 identifics the contract that must be accepted by low-ability workers in any separating equilibrium.
\I'
SECTION
Flgur. 13.0.6 (rig hi)
The high·ability workers must receivc
contract (Oil' ill) in any separating equilibrium.
J
An equilibrium may not exist. (a) No pooling contract breaks the separating equilibrium. (b) The pooling contracl (Ii>, i) breaks the separating equilibrium.
466
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AND
--- --
SCREENING
Type 0.
W
Flgur. 13.0.8 ~----------------------
(wL.td
in Proposition I3.D.2 and no equilibrium exists. More generally, an equilibrium exists only ir there is no such profitable deviation.
Welfare Properties of Screening Equilibria Restricting attention to cases in which an equilibrium does exist, the screening equilibrium has welrare properties parallel to those or the signaling model's best separating equilibrium [with r(Od = r(O.) = 0). First, as in the earlier model, asymmetric inrormation leads to Pareto inefficient outcomes. Here high-ability workers end up signing contracts that make them engage in completely unproductive and disutility-producing tasks merely to distinguish themselves rrom their less able counterparts. As in the signaling model, the low-ability workers are always worse ofT here when screening is possible than when it is not. One difference rrom the signaling model, however, is that in cases where an equilibrium exists, screening must make the high-ability workers better off; it is precisely in those cases where it would not that a move to a pooling contract breaks the separating equilibrium [see Figure 13.D.7(b)]. Indeed, when an equilibrium does exist, it is a constrained Pareto optimal outcome; ir no firm has a deviation that can attract both types or workers and yield it a positive profit, then a central authority who is unable to observe worker types cannot achieve a Pareto improvement either."
A profitable deviation uSlOg a pair of
contracts may eXist that breaks the separating equilibrium.
A P PEN 0 I X
A:
A E A SON A B L E • BEL I E F 8
A E FIN E MEN T SIN
S I G N A LIN 0
money. But if (w', t') is withdrawn as a result, then low·ability workers will accept (w, i) and this deviation ends up being unprofitable. Hellwig (1986) examines sequential equilibria and their refinements in a game that explicitly allows for such withdrawals. By introducing such reactions, these papers establish the existence of pure strategy equilibria. Introducing reactions of this sort does not simply eliminate the nonexistence problem. however, but also yields somewhat different predictions regarding the character· istics of market equilibria and their welfare properties. For example, when firms can make multiple offers as we have allowed here, cross·subsidization can arise in Wilson equilibria. Indeed, Miyazaki (1977) shows that in the case in which multiple offers are possible, a Wilson equilibrium always exists and is necessarily a constrained Pareto optimum. In the screening model examined above. we took the view that the uninformed firms made employment offers to the informed workers. Vet we could equally well imagine a model in which informed workers instead make contract offers to the firms. For example, each worker might propose a task level at which she is willing to work, and firms might then offer a wage for that task level. Note, however, that this alternative model exactly parallels the signaling model in Section 13.C and. as we have seen, yields quite different predictions. For example. the signaling model has numerous equilibria, but here we have at most a single equilibrium. This is somewhat disturbing. Given that our models are inevitably simplifications of actllal market processes, if market outcomes are really very sensitive to issues such as this our models may provide us with little predictive ability. One approach to this problem is offered by Maskin and Tirole (1992). They note that contracts like those we have allowed firms to offer in the screening model discussed in this section are still somewhat restricted. In particular, we could imagine a firm offering a worker a contract that involved an ex post (after signing) choice among a set of wage-task pairs (you will sec more about contracts of this type in Section 14.C). Similarly, in considering the counterpart model in which workers make offers. we could allow a worker to propose such a contract. Maskin and Tirole (1992) show that with this enrichment or the allowed contracts (and a weak additional assumption) the sets of sequential equilibria of the two models coincide (there may be mUltiple equilibria in both cases).
APPENDIX A: REASONABLE·BELIEFS REFINEMENTS IN SIGNALING GAMES
What can be said about the potential nonexistence of equilibrium in this model? Two paths have been followed in the literature. One approach is to establish existence of equilibria in the larger strategy space that allows for mixed strategies; on this, see Dasgupta and Maskin (1986). The other is to take the position that the lack of equilibria indicates that, in some important way. the model is incompletely specified. The aspect the literature has emphasized in this regard is the lack of any dynamic reactions to new contract offers [see Wilson (1977), Riley (1979), and Hellwig (1986»). Wilson (1977), for example, uses a definition of equilibrium that captures the idea that firms are able to withdraw unprofitable contracts from the market. A set of contracts is a Wilson equilibrium if no firm has a profitable deviation that remains profitable once existing contracts that lose money after the deviation are withdrawn. This extra requirement may make deviations less attractive. In the deviation considered in Figure 13.0.3, for example, once contract (w, i) is introduced, the original contract (w', t') loses 25. Actually, there is a small gap: An equilibrium may exist when there is another pair of Contracts that would give higher utility to both types of workers and that would yield the firm deviating to it exactly zero profits. In this case, the equilibrium is not a constrained Pareto optimum.
In this appendix, we describe several commonly used reasonable-beliefs refinements or the perrect Bayesian and sequential equilibrium concepts for signaling games, and we apply them to the education signaling model discussed in Section 13.c. Excellent sources for rurther details and discussion are Cho and Kreps (1987) and Fudenberg and Tirole (1992). Consider the rollowing class or signaling games: There are I players plus nature. The first move or the game is nature's, who picks a "type" for player I, E 0 = {O" ... , ON}' The probability ortype is [(0), and this is common knowledge among the players. However, only player I observes O. The second move is player I's, who picks an action a rrom set A aner observing O. Then, aner seeing player I's action choice (but not her type), each player i = 2, ... , I simultaneously chooses an action s, rrom set S,. We define S = S2 X . . . x St. Ir player I is of type 0, her utility rrom choosing action a and having players 2, ... , I choose s = (S2" .. ,St) is ",(a, s, 0). Player ii'I receives payoff ",(a, s, 0) in this event. A perrect Bayesian
e
a
0 A M ES
467
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SCREENING
equilibrium (PBE) in the sense used in Section 13.C is a profile of strategies
(a(O), s2(a), ... ,s,(a)), combined with a common belief function 1'(0 Ia) for players 2, ... ,I that assigns a probability 1'(0 I a) to type 0 of player I conditional on
---- --
observing action a E A, such that (i) Player l's strategy is optimal given the strategies of players 2, ... , I. (ii) The belief function }leO I a) is derived from player l's strategy using Bayes' rule where possible. (iii) The strategies of players 2, ... , 1 specify actions following each choice a E A that constitute a Nash equilibrium of the simultaneous-move game in which the probability that player I is of type II is 11(11 I a) for all II E 0. In the context of the model under study here, this notion of a PBE is equivalent to the sequential equilibrium notion. The education signaling model in Section 13.C falls into this category of signaling games if we do not explicitly model the worker's choice between the firms' offers and instead simply incorporate into the payoff functions the implications of her optimal choice (she chooses from among the firms offering the highest wage if this wage is positive and refuses both firms' offers otherwise). In that model, 1 = 3,0 = {II,., II/I}, the set A = :r: r;e, O} contains the possible education choices of the worker, and the set Si = : II': \\' E IR) contains the possible wage offers by firm i.
Domillatioll-Based Refillemellts of Beliefs The simplest reasonable-belief refinement of the PBE notion arises from the idea (discussed in Section 9.0) that reasonable beliefs should not assign positive probability to a player taking an action that is strictly dominated for her. In a signaling game, this problem can arise when players 2, ... ,I (the firms in the education signaling model) assign a probability }leO I a) > 0 to player I (the worker) being of type 0 after observing action a, even though action a is a strictly dominated choice for player I when she is of type O. Formally, we say that action a E A is a strictly dominated choice for type 0 if there is an action a' E A such that Min IIt(a', s', 0) > Max " t (a, s, 0). 2"
REASONABLE.BELIEFS
REFINEMENTS
IN
SIGNALING
Unfortunately, in the education signaling model discussed in Section l3.C, this refinement does not narrow down our predictions at all. The set 0(e) equals {OL' Ou} for all education levels e because either worker type will find e to be her optimal choice if the wage offered in response to e is sufficiently in excess of the wage offered at other education levels. Thus, no beliefs are ruled out, and all PBEs of the signaling game pass this test. If we want to narrow down our predictions for this model, we need to go beyond the use of refinements based only on notions of strict dominance. 28 Recall the argument we made in Section 13.C for eliminating all separating equilibria but the best one. We argued that since, in Figure 13.C.7, a worker of type 0,. would be better off choosing e = 0 than she would choosing an education level above i' for any beliefs and resulling equilibrium wage Ihal mighl follow Ihese Iwo edllcalion lel'e/s, no reasonable belief should assign a positive probability to a worker of type II,. choosing any e > e. This is close to an argument that education levels e > e are dominated choices for a type OL worker, but with the critical difference reflected in the italicized phrase: Only equilibrium responses of the firms are considered, rather than all conceivable responses. That is, we take a backwardinduction-like view that the worker should only concern herself with possible equilibrium reactions to her education choices. To be more formal about this idea, for any nonempty set 0 c 0, let S*(0, a) c S, x ... x S, denote the set of possible equilibrium responses that can arise after action a is observed for some beliefs satisfying the property that 1'(11 I a) > 0 only if II E 0. The set S*(0, a) contains the set of equilibrium responses by players 2, ... ,I that can follow action choice a for some beliefs that assign positive probability only to types in 0. When 0 = 0, the set of all conceivable types of player I, this construction allows for all possible beliefs. 29 We can now say that action a E A is strictly dominated for type 0 in this stronger sense if there exists an action a' with Min
u,(a', s', 0)
> Max
u,(a, s, 0).
(13.AA.2)
nS·(9.4I'
Using this stronger notion of dominance, we can define the set 0*(a) = {O: there is no a'
E
A satisfying (13.AA.2)j,
containing those types of player I for whom action a is not strictly dominated in the sense of (\ 3.AA.2). We can now say that a PBE has reasonable beliefs if for all a E A with 0*(a) i' 0, I/(a, II) > 0 only if 0 E 0*(a). Using this reasonable-beliefs refinement significantly reduces the set of possible outcomes in the educational signaling model, sometimes even to a unique prediction. In that model, S*(0, e) = [OL' 011] for all education choices e because, for any belief I' E [0, I], the resulting Nash equilibrium wage must lie between OL and 0/1' As a
(13.AA.I)
For each action a E A, it is useful to define the set 0(a) = [0: there is no a' E A satisfying (13.AA.I»).
This is the set of types of player I for whom action a is not a strictly dominated choice. We can then say that a PBE has reasonable beliefs if, for all a E A with 0(a) i' 0, onlyif
A:
s'£S·(9.o',
Sf,S
II(Ola»O
APPENDIX
2S. We could, in principle, go further with this identification of strictly dominated strategies for player I by also eliminating any strictly dominated strategies for players 2, ... ,I, then looking to see whether we have any more strictly dominated actions for any of player I's types, and so on. However, in the educational signaling model, this does not help us because the firms have no strictly dominated strategies. 29. Note that when there is only one player responding (so I = 2), the set S'(0, a) is exactly the set of responses that are not strictly dominated for player 2 conditional on following action a. Note also that in this case a strategy s,ta) is weakly dominated for player 2 if, for any a e A, it involves play of some Sf S'(0, a).
OE0(a)
and we consider a PBE to be a sensible prediction only if it has reasonable beliefs."
26. Note that a strategy a(O) is strictly dominated for player I if and only if it involves play of a strictly dominated action for some type O. 27. Doing this is equivalent to first eliminating each type (J's dominated actions from the game and then identifying the PBEs of this simplified game.
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~~~---------------------------------------------------Type
(I,.
e can survive because, as we argued in Section D.C. the high-ability worker will do better by deviating to an education level slightly in excess of e. Furthermore, we can also eliminate any pooling equilibrium in which the equilibrium outcome is worse for a high-ability worker than outcome (Ii", e), such as in the equilibrium depicted in Figure 13.AA.I, since any such equilibrium must involvc unreasonable beliefs: If 1,(0" 1e) = I for all e > thcn a type Ii" worker could do better deviating to an education level just above where she would receive a wage of 0". In fact, when the high-ability worker prefers outcome (0", e) to (£[0],0), this argument rules out all pooling equilibria, and so we get the unique prediction of the best separating equilibrium.
e, e
E(/UiliiJl'iulIl Domination and the Intuitive Criterion We now consider a further strengthening of the notion of dominance, known as 0 for 0 > 0 and r(0) < 0 for 0 < O. Let the densilY of workers of type 0 be f(O), with f(O) > 0 for all 0 E [Q, 0]. Show that a competitive equilibrium with unobservable worker types necessarily involves a Pareto inefficient outcome.
Although the use of either equilibrium domination or the intuitive criterion yields a unique prediction in the education signaling model when there are two types of workers, they do not accomplish this when there are three or more possible worker types (see Exercise 13.AA.l). 'Stronger refinements such as Banks and Sobel's (1987) notions of divilliry and universal diviniry, Cho and Kreps' (1987) related notion called D I, and Kohlberg and Mertens' (1986) srabiliry do yield the unique prediction of the best separating equilibrium in these games with many worker types. See Cho and Kreps (1987) and Fudenberg and Tirole (1992) for further details.
13.B.3" Consider a positive selection version of the model discussed in Section 13.B in which r(') is a continuous, strictly decreasing function of O. Let the density of workers of type Ii be f(O), wilh f(O) > 0 for all 0 E [Q, 0].
(a) Show that the more capable workers are the ones choosing to work at any given wage. (b) Show that if reO) > 0 for all 0, then the resulting competitive equilibrium is Pareto efficient.
(e) Suppose that there exists a (; such Ihat reO) < 0 for 0 > () and reO) > 0 for 0 < 0. Show that any competitive equilibrium with strictly positive employment necessarily involves too
ml/eh employment relative to the Pareto optimal allocation of workers. 13.B.4" Suppose two individuals, I and 2, are considering a trade at price p of an asset that they bOlh use only as a store of wealth. Ms. I is currently the owner. Each individual i has a privately observed signal of the asset's worth YI' In addition, each cares only about the expected value of the asset one year from now. Assume that a trade at price p takes place only if both parties think they are being made strictly better off. Prove that the probability of trade occurring is zero. [Hilll: Study the following trading game: The two individuals simullaneously say either "trade" or "no trade," and a trade at price p takes place only if they bOlh say" trade."]
REFERENCES Akcrlof. G. (1970). The market for lemons: Quality uncertainly and the market mechanism. Quarterly JOU"'QI of Economics 89: 488-500. Banks. J.. and J. Sobel. (1987). Equilibrium selection in signaling games. £wnomt'lr;ca 55: 647-62. Cho. I·K .. and D. M. Kreps. (1987). Signaling games and stable equilibria. Quarurly Journal of Economics 102: 179-221. Dasgupta. P., and E. Maskin. (1986). The existence of equilibrium in discontinuous economic games. Rt'lliew of Economic Studies 46: 1-41. fudcnbcrg. D .. and 1. Tirole. (1992). Game Thear),. Cambridge, Mass.: M IT Press. Hellwig, M. (1986). Some recent developments in the theory of competition in markets with adverse selection. (University or Bonn. mimeographed). Holmstrom, B., and R. B. Myerson. (1983). Efficient and durable decision rules with incomplete information. Econometrica 51: 1799-819. Kohlberg. E.. and J.-F. Mertens. (1986). On the strategic stability of equilibria. ECotlOmetrica 54: 1003-38. Maskin, E., and J. Tirole. (1992). The principal.agcnt relationship with an informed principal,ll: Common values. Econometrica 60: 1-42. Miyazaki. H. (1977). The rat race and internal labor markets. Bell JOUr/wi of Economics 8: 394-418. Riley, 1. (1979). Informational equilibrium. Econometrica 47: 331-59. Rothschild, M .. and 1. E. Stiglitz. (1976). Equilibrium in competitive insurance mark.ets: An essay in the economics of imperfect information. Quarterly Journal of Economics 80: 629-49. Spence. A. M. (1973). Job mark.l signaling. Quarl.,ly Journal of Economics 87: 355-74. Spence, A. M. (1974). Markel Signaling. Cambridge. Mass.: Harvard University Press. Wilson, C. (1977). A model of insurance markets with incomplete information. Journal of Economic Theory 16: 167-207. Wilson, C. (1980). The nature or equilibrium in markets with adverse selection. Bell Journal of Economics 11: 108-30.
I3.B.5" Reconsider the case where reO) = r for all 0, but now assume thai when the wage is such thai no workers are accepting employment firms believe that any worker who might accept would be of the lowest quality, that is, £[0 Ie = 0] = Q. Maintain the assumption that all workers accept employment when indifferent. (a) Argue that when £[0] :2: r > Q, there are now two competitive equilibria: one with e" = [Q,O] and one with w" = Q and e" = 0. Also show that when Q :2: r Ihe unique competitive equilibrium is w" = £[0] and e" = [Q,O], and when r > £[0] the unique competitive equilibrium is w" = Q and e" = 0.
1\""
= £[0] and
(b) Show that when £[0] > r and there are two equilibria, the full-employment equilibrium Pareto dominates the no-employment one. (e) Argue that when £[0] :2: r the unique SPNE of the game-theoretic model in which two firms simultaneously make wage offers is the competitive equilibrium when this equilibrium is unique, and is the full-employment (highest-wage) competitive equilibrium when the competitive equilibrium is not unique and £[0] > r. What happens when £[0] = r? What about the case where £[0] < r? (d) Argue that the highest-wage competitive equilibrium is a constrained Pareto optimum.
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13.B.6C [Based On Wilson (1980)] Consider the following change in the adverse selection model of Section 13.B. Now there are N firms. each of which wants to hire at most I worker. The N firms differ in their productivity: In a firm of type y a worker of type 0 produces yO units of output. The parameter y is distributed with density function g(') on [0,00], and Y(i') > 0 for all y E [0, 00].
---
E X E R CIS E S
13.C2C Reconsider the two-type signaling model with r(Od = r(8 H ) = 0, assuming a worker's productivity is 0(1 + I'e) with I' > O. Identify the separating and pooling perfect Bayesian equilibria, and relate them to the perfect information competitive outcome. 13.C3" In text.
(a) Let :(11', II) denote the aggregate demand for labor when the wage is II' and the average productivity of workers accepting employment at that wage is II. Derive an expression for this function in terms of the density function g(').
I3.C4" Reconsider the signaling model discussed in Section I3.C, now assuming that worker types are drawn from the interval [Q,O] with a density function f(O) that is strictly positive everywhere on this interval. Let the cost function be c(e, 0) = (e' /0). Derive the (unique) perfect
(b) Let 11(\\') = £[Olr(O),;; w], and define the aggregate demalld fUlletiall for labor by z*(\\') = :(\\',11(\\')). Show that :*(11') is strictly increasing in II' at wage IV if and only if the
Bayesian equilibrium.
elasticity of It with respect to w exceeds 1 at wage \\. (assume that all relevant functions arc uifferentiable).
13.CS" Assume a single firm and a single consumer. The firm's product may be either high or low qualily and is of high quality with probability i .. The Consumer cannot observe quality before purchase and is risk neutral. The consumer's valuation of a high-quality product is v,,; her valuation of a low-quality product is VL' The costs of production for high (H) and low (L) quality are "'*. Can you use these facts to give a simple proof of Proposition 13.8.2?
13.C6' Consider a market for loans to finance investment projects. All investment projects require an outlay of I dollar. There are two types of projects: good and bad. A good project > 0 and a probability (I - Po) of yielding has a probability of Po of yielding profits of profits of zero. For a bad project. the relative probabilities are P. and (I - P.), respectively, where Po > P" The fraction of projects that are good is i, E (0, I). Entrepreneurs go to banks to borrow the cash to make the initial outlay (assume for now that they borrow the entire amount). A loan contract specifies an amount R that is supposed to be repaid to the bank. Entrepreneurs know the type of project they have, but the banks do not. In the event that a project yields profits of zero, the entrepreneur defaults on her loan contract. and the bank receives nothing. Banks are competitive and risk neutral. The risk-free rate of interest (the rate the banks pay to borrow funds) is r. Assume that
n
I3.B.S" Consider the following alteration to the adverse selection model in Section I3.B. Imagine that when workers engage in home production, they use product x. Suppose that the amount consumed is related to a worker's type, with the relation given by the increasing function x(O). Show that if a central authority can observe purchases of good x but not worker types, then there is a market intervention that results in a Pareto improvement even if the market is at the highest·wage competitive equilibrium. 13.8.9" Consider a model of positive seleetioll in which r(') is strictly decreasing and there are two types of workers, 0" and 01., with 00 > 0" > 01• > O. Let i. = Prob (0 = 0,,) E (0,1). Assume that r(O,,) < 0" and that r(Od> 0L' Show that the highest-wage competitive equilibrium need not be a constrained Pareto optimum. [Hint: Consider introducing a small unemployment benefit for a case in which £[0] = r(Od. Can you use the result in Exercise 13.8.7 to give an exact condition for when a competitive equilibrium involving full employment is a constrained Pareto optimum?]
Pc; n
on
-
(I + r) > 0 > p. n
-
(I
+ r).
(a) Find the equilibrium level of R and the set of projects financed. How does this depend i., n, and r'~
{'G' PH'
(b) Now suppose that the entrepreneur can offer to contribute some fraction x of the I dollar initial outlay from her own funds (x E [0, I]). The entrepreneur is liquidity constrained, howe,.r. so that the effective cost of doing so is (I + I')x, where I' > r. (i) What is an entrepreneur's payoff as a function of her project type. her loan-repayment amount R, and her contribution x? (ii) Describe the best (from a welfare perspective) separating perfect Bayesian equifibrium of a game in which the entrepreneur first makes an offer that specifies the level of x she is willing to put into a project, banks then respond by making offers specifying the fevel of R they would require, and finally the entrepreneur accepts a bank's offer or decides not to go ahead with the project. How does the amount contributed by entrepreneurs with good projects change with small changes in P" PG, A, n, and r?
13.B.10" Show that Proposition 13.8.2 continues to hold when r(O) > 0 for some O. l3.CI" Consider a game in which, first, nature draws a worker's type from some continuous distribution on [Q, 0]. Once the worker observes her type, she can choose whether to submit to a cost less test that reveals her ability perfectly. Finally, after observing whether the worker has taken the test and its outcome if she has, two firms bid for the worker's services. Prove that in any subgame perfect Nash equilibrium of this model all worker types submit to the test, and firms offer a wage no greater than Q to any worker not doing so.
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----------------------------------------------------------------------(iii) How do the two types of entrepreneurs do in the separating equilibrium of (b)(ii) compared with the equilibrium in (o)?
The Principal-Agent Problem
\3.0.1" Extend the screening model to a case in which tasks are productive. Assume that a type 0 worker produces O( I + 1'1) units of output when her task level is r where II > O. Identify the subgame perfect Nash equilibria of this model.
14
I3.D.2" Consider the following model of the insurance market. There are two types of individuals: high risk and low risk. Each starts with initial wealth W but has a chance that "n accident (c.g., a firc) will reduce her wealth by L. The probability of this happening is PL for low· risk types and PH for high-risk types, where PH > Pl.. Both types are expected utility m"ximizers with a Bernoulli utility function over wealth of u(w), with u'(w) > 0 and u"(I\') < 0 at all
1\'.
There arc two risk-neutral insurance companies. An insurance policy consists of a
premium payment M made by the insured individual to her insurance firm and a paymcnt R from the insurance company to the insured individual in the event of a loss. (0) Suppose that individuals are prohibited from buying more than one insurancc policy.
Arguc that a policy can be thought of as specifying the wcalth levels of the insured individual in the two states "no loss'" and "loss."
i
I 14,A Introduction In Chapter 13, we considered situations in which asymmetries of information exist between individuals at the time of contracting. In this chapter, we shift our attention to asymmetries of information that develop subsequem to the signing of a contract. Even when informational asymmetries do not exist at the time of contracting, the parties to a contract often anticipate that asymmetries will develop sometime after the contract is signed. For example, after an owner of a firm hires a manager, the owner may be unable to observe how much effort the manager puts into the job. Similarly, the manager will often end up having better information than the owner about the opportunities available to the firm. Anticipating the development of such informational asymmetries. the contracting parties seek to design a contract that mitigates the difficulties they cause. These problems arc endemic to situations in which one individual hires another to take some action for him as his "agent." For this reason. this contract design problem has come to be known as the principal-agent problem. The literature has traditionally distinguished between two types of informational problems that can arise in these settings: those resulting from hidden actions and those resulting from hidden in/ormation. The hidden action case. also known as moral hazard. is illustrated by the owner's inability to observe how hard his manager is working; the manager's coming to possess superior information about the firm's opportunities, on the other hand. is an example of hidden information.' Although many economic situations (and some of the literature) contain elements of both types of problems. it is useful to begin by studying each in isolation. In Section 14.B, we introduce and study a model of hidden actions. Section 14.C analyzes
(b) Assumc that the insuranc~ companies simultaneously olTer policies; as in Section 13.0, they can each olTer any finitc number of policics. What are the subgamc perfcct Nash equilibrium outcomes of the model'! Docs an equilibrium necessarily exist'!
I3.D.3 c Consider the following extension of the model you developed in Exercise 13.0.1. Suppose that therc is a fixed task level T that all workers face. The monetary equivalent cost of accepting cmployment at this task level is c > 0, which is independent of worker type. However, now a worker's actual output is observable and verifiable. and so contracts can base I;ompensation on the worker's ex post observed output level. (0) What is the subgame perfect Nash equilibrium outcome of this model?
(b) Now suppose that the output realization is random. It can be either good (qG) or bad ('I.). Thc probability th"t it is good is PH for a high-ability worker and PI. for a low-ability worker (Pu > Pt.>. If workers are risk-neutral expected utility maximizers with a Bernoulli utility fUllction over wealth of u(w)
= "',
what is the subgame perfect Nash equilibrium
outcomc? (\:) \Vltat ir workers are strictly risk averse with II"(\\,) < 0 at all w?
13.\).4" Reconsider the scrcening model in Section 13.0, but assume that (i) there is an infinite nllmber of firms that could potentially enter the industry and (ii) firms can each offer at most one contract. [The implication of (i) is that, in any SPNE, no firm can have a profitable entry opportunity.] Characterize the equilibria for this casc. I3.AA.l c Consider the extension of the signaling model discussed in Section \3.C to thc OiISC of three types. Assumc all thrce types have rIO) = O. Provide an example in which more than olle perfect Bayesian equilihrium satisfies the intuitive critcrion.
1. The literature's use of the term moral hazard is not entirely uniform. The term originates in the insurance literature. which first focused attention on two types of informational imperfections: the "moral hazard" that arises when an insurance company cannot observe whether the insured exerts effort to prevent a loss and the "adverse selection" (see Section 13. B) that occurs when the
insured knows more than the company at the time he purchases a policy about his likelihood of an accident. Some authors use moral hazard to refer to either of the hidden action or hidden
information variants of the principal-agent problem [see, for example, Hart and Holmstrom (1987)]. Here, however, we use the term in the original sense.
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---------------------------------------------------------------------a hidden information model. Then, in Section 14.0, we provide a brief discussion of hybrid models that contain both of these features. We shall see that the presence of postcontractual asymmetric information often leads to welfare losses for the contracting parties relative to what would be achievable in the absence of these informational imperfections. It is important to emphasize the broad range of economic relationships that fit into the general framework of the principal-agent problem. The owner-manager relationship is only one example; others include insurance companies and insured individuals (the insurance company cannot observe how much care is exercised by the insured), manufacturers and their distributors (the manufacturer may not be able to observe the market conditions faced by the distributor), a firm and its workforce (the firm may have more information than its workers about the true state of demand for its products and therefore about the value of the workers' product), and banks and borrowers (the bank may have difficulty observing whether the borrower uses the loaned funds for the purpose for which the loan was granted). As would be expected given this diversity of examples, the principal-agent framework has found application in a broad range 'of applied fields in economics. Our discussion will focus on the owner-manager problem. The analysis in this chapter, particularly that in Section 14.C, is closely related to that in two other chapters. First, the techniques developed in Section 14.C can be applicd to the analysis of screening problems in which, in contrast with the case studied in Section 13.0, only one uninformed party screens informed individuals. We discuss the analysis of this monopolistic screening problem in small type at the end of Section 14.C. Second, the principal-agent problem is actually a special case of "mechanism design," the topic of Chapter 23. Thus, the material here constitutes a first pass at this more general issue. Mastery of the fundamentals of the principalagent problem, particularly the material in Section 14.C, will be helpful when you study Chapter 23. A good source for further reading on topics of this chapter is Hart and Holmstrom
--
(1987).
14.B Hidden Actions (Moral Hazard) Imagine that the owner of a firm (the principal) wishes to hire a manager (the agent) for a one-time project. The project's profits are affected, at least in part, by the manager's actions. If these actions were observable, the contracting problem between the owner and the manager would be relatively straightforward; the contract would simply specify the exact actions to be taken by the manager and the compensation (wage payment) that the owner is to provide in return. 2 When the manager's actions are not observable, however, the contract can no longer specify them in an effective manner, because there is simply no way to verify whether the manager has fulfilled his obligations. In this circumstance, the owner must design the manager's compensation scheme in a way that indirectly gives him the incentive to take the correct
SEC T ION
1 4 • 8:
HID DEN
ACT ION S
t
M 0 RA L
actions (those that would be contracted for if his actions were observable). In this section, we study this contract design problem. To be more specific,let It denote the project's (observable) profits, and let e denote the manager's action choice. The set of possible actions is denoted by E. We interpret e as measuring managerial effort. In the simplest case that is widely studied in the literature, e is a one-dimensional measure of how "hard" the manager works, and so E c R More generally, however, managerial effort can have many dimensionshow hard the manager works to reduce costs, how much time he spends soliciting customers, and so on-and so e could be a vector with each of its elements measuring managerial effort in a distinct activity. In this case, E c AM for some M.l In our discussion, we shall refer to e as the manager's effort choice or effort level. For the nonobservability of managerial effort to have any consequence, the manager's effort must not be perfectly deducible from observation of It. Hence, to make things interesting (and realistic), we assume that although the project's profits are affected bye, they are not fully determined by it. In particular, we assume that the firm's profit can take values in ['.!' x] and that it is stochastically related to e in a manner described by the conditional density function f(lt I e), with f(lt Ie) > 0 for all e E E and all 11 E ['.!, xl Thus, any potential realization of 11 can arise following any given effort choice by the manager. In the discussion that follows, we restrict our attention to the case in which the manager has only two possible effort choices, ell and eL (see Appendix A for a discussion of the case in which the manager has many possible actions), and we make assumptions implying that ell is a "high-effort" choice that leads to a higher profit level for the firm than eL but entails greater difficulty for the manager. This fact will mean that there is a conflict between the interests of the owner and those of the manager. More specifically, we assume that the distribution of It conditional on ell first-order stochastically dominates the distribution conditional on e/.; that is, the distribution functions F(lt Ied and F(1I I ell) satisfy F(lt I ell) ::; F(lt I ed at alllt E ['.!, x], with strict inequality on some open set n c ['.!, x] (see Section 6.0). This implies that the level of expected profits when the manager chooses ell is larger than that from eL: J1If(1I I ell) d1l > J1If(1I I eLl d1l. The manager is an expected utility maximizer with a Bernoulli utility function U(IV, e) over his wage IV and effort level e. This function satisfies u.. (w, e) > 0 and u.... (w, e) ~ 0 at all (w, e) (subscripts here denote partial derivatives) and u(\\', ell) < II(IV, "L) at all w; that is, the manager prefers more income to less, is weakly risk averse over income lotteries, and dislikes a high level of effort! In what follows, we focus on a special case of this utility function that has attracted much of the
1 In fact, more general interpretations are possible. For example. e could include non-effortrelated managerial decisions such as what kind of inputs are purchased or the strategies that are adopted for appealing to buyers. We stick to the effort interpretation largely because it helps wilh intuition. 4. Note that in the multidimensional-effort case, it need not be that eH has higher effort in every dimension; the only important thing for our analysis is that it leads to higher profits and entails a larger managerial disutility than does iL.
2. Note that this requires not only that the manager's actions be observable to the owner but also that they be observable to any court that might be called upon to enforce the contract.
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attention in the literature: u(w, e) = v(w) - g(e).' For this case, our assumptions on u(w, e) imply that v'(w) > 0, v"(w) ~ 0, and g(eH) > geed. The owner receives the project's profits less any wage payments made to the manager. We assume that the owner is risk neutral and therefore that his objective is to maximize his expected return. The idea behind this simplifying assumption is that the owner may hold a well-diversified portfolio that allows him to diversify away the risk from this project. (Exercise 14.B.2 asks you to consider the case of a risk-averse owner.)
--- --
f
(11 - W(lI» [(111 e) dll
S.t.
f
f
v(w(n»[(nl e) dn -gee)
-[(lIle)
+ yv'(w{lI))[(nle)
,,(w:> -
f
481
= 0,
(14.B.3)
gee) = ii.
(14.B.4)
n[(l1l e) dll - v-'(ii
+ gee)).
(14.B.5)
The first term in (14.8.5) represents the gross profit when the manager puts forth effort e; the second term represents the wages that must be paid to compensate the manager for this effort [derived from condition (14.B.4)]. Whether ell or eL is optimal depends on the incremental increase in expected profits from ell over eL compared with the monetary cost of the incremental disutility it causes the manager. This is summarized in Proposition 14.B.1.
to offer the manager? Second, what is the best choice of e? Given that the contract specifies effort level e, choosing w(n) to maximize S(n - w(nll[(nl e) dn = (S lI[(nl e) dn) - (S w(n)[(nle) dn) is equivalent to minimizing the expected value of the owner's compensation costs, S w(n)[(l1l e) dll, so (14.B.l) tells us that the optimal compensation scheme in this case solves
S.t.
I
or
~ ii.
Proposition 14.B.1: In the principal-agent model with observable managerial effort, an optimal contract specifies that the manager choose the effort e* that maximizes n f (nl e) dn - v- '(D + gee»~] and pays the manager a fixed wage w* = v- '(D + g(e*)). This is the uniquely optimal contract if v"(w) < 0 at all w.
(14.B.2)
w(n)[(nle)dll
H A Z A R0
Note that since Ilk,,) > g(e,,), the manager's wage will be higher if the contract calls for effort ell than if it calls for eL . On the other hand, when the manager is risk neutral, say with v(w) = w, condition (14.B.3) is necessarily satisfied for any compensation function. In this case, because there is no need for insurance, a fixed wage scheme is merely one of many possible optimal compensation schemes. Any compensation function w(n) that gives the manager an expected wage payment equal to ii + gee) [the level derived from condition (14.B.4) when v(\\') = IV] is also optimal. Now consider the optimal choice of e. The owner optimally specifies the effort level e E :e,., ell} that maximizes his expected profits less wage payments,
It is convenient to think of this problem in two stages. First, for each choice of
f
(M 0 R A L
w:
e that might be specified in the contract, what is the best compensation scheme w(n)
Min
ACT ION S
If the manager is strictly risk averse [so that v'(w) is strictly decreasing in w], the implication of condition (14.B.3) is that the optimal compensation scheme W{lI) is a constant; that is, the owner should provide the manager with a fixed wage payment. This finding is just a risk-sharing result: Given that the contract explicitly dictates the manager's effort choice and that there is no problem with providing incentives, the risk-neutral owner should fully insure the risk-averse manager against any risk in his income stream (in a manner similar to that in Example 6.C.1). Hence, such given the contract's specification of e, the owner offers a fixed wage payment that the manager receives exactly his reservation utility level:
(l4.B.I)
V(W(lI)) [(111 e) dn - I/(e)
HID 0 E N
condition 6
I
The Optimal Contract when EjJort is Observable
Max
1 4. B:
V'(W{lI» = y.
It is uscful to begin our analysis by looking at the optimal contracting problem when effort is observable. Suppose that the owner chooses a contract to offer the manager that the manager can then either accept or rcject. A contract here specifies the manager's effort e E {e,., ell} and his wage payment as a function of observed profits W(lI). We assume that a competitive market.for managers dictates that the owner must provide the manager with an expected utility level of at least ii if he is to accept the owner's contract offer ('I is the manager's reser"",ioll lIIility le,'e/). If the manager rejects the owner's contract offer, the owner receives a payoff of zero. We assume throughout that the owner finds it worthwhile to make the manager an offer that he will accept. The optimal contract for the owner then solves the following problem (for notational simplicity, we suppress the lower and upper limits of integration ~ and x):
,'EI"/.t'"I. ".. _I
SEC T ION
0
~ 'I.
6. The first-order condition ror w{tt) is derived by taking the derivative with respect to the manager's wage at each level or 1t separately. To see this point, consider a discrete version or the model in which there is a finite number or possible profit levels (7[1 •...• 1t N ) and associated wage levels (Wi •...• wN ). The first-order condition (14.8.3) is analogous to the condition onc gels in the discrete model by examining the first-order conditions ror each W II , n = I..... N (note that we allow the wage payment to be negative). To be rigorous. we should add that when we have a continuum of possible levels or 7[, an optimal compensation scheme need only satisry condition
The constraint in (14.B.2) always binds at a solution to this problem; otherwise, the owner could lower the manager's wages while still getting him to accept the contract. Letting y denote the multiplier on this constraint, at a solution to problem (14.B.2) the manager's wage W(lI) at each level of n E [~, x] must satisfy the first-order
(14.B.3) aL a seL of profiL levels LhaL is of full measure.
5. Exercise 14.B.1 considers one implication or relaxing this assumption.
.1
------"
482
CHAPTER
14:
THE
PRINCIPAL·AGENT
PROBLEM
-------------------------------------------------------------------------The Optimal Colltract whell Effort is Not Observable
-
The optimal contract described in Proposition 14.B.1 accomplishes two goals: it specifics an efficient effort choice by the manager, and it fully insures him against income risk. When effort is not observable, however, these two goals often come into conflict because the only way to get the manager to work hard is to relate his pay to the realization of profits, which is random. When these goals come into conflict, the nonobservability of effort leads to inefficiencies. To highlight this point, we first study the case in which the manager is risk neutral. We show that in this case, where the risk-bearing concern is absent, the owner can still achievc the same outcome as when effort is observable. We then study the optimal contract when the manager is risk averse. In this case, whenever the first-best (full ohscrvability) contract would involve the high-efTort level, eflicient risk bearing and ellieient incentive provision come into conflict, and the presence of nonobservable actions leads to a welfare loss.
14.B:
HIDDEN
ACTIONS
(MORAL
The manager is willing to accept this contract as long as it gives him an expected utility of at least ii, that is, as long as
f
rrf(rr Ie') drr - ex - g(e') ;;:: ii.
(14.B.8)
Let (I.' be the level of ex at which (14.B.8) holds with equality. Note that the owner's payoff if the compensation scheme is w(rr) = rr - IX' is exactly 0:' (the manager gets all of" except for the fixed payment (I.'). Rearranging (14.B.8), we see that 0:' = rrf(rr Ie') drr - g(e') - U. Hence, with compensation scheme "~rr) = rr - (I.', both the owner and the manager get exactly the same payoff as when effort is observable. _
J
The basic idea behind Proposition 14.B.2 is straightforward. If the manager is risk neutral, the problem of risk sharing disappears. Efficient incentives can be provided without incurring any risk-bearing losses by having the manager receive the full marginal returns from his effort.
A ri ....{-II('lIt,.al HllIIUlfWr
A risk-averse manager
Surpose that 1'(\\') = \\'. Applying Proposition 14.B.I, the optimal efTort level (" when elTort is observable solves Max ('eh',.(',d
f
l£f(l£ I e) dl£ - y(e) - U.
When the manager is strictly risk averse over income lotteries, matters become more complicated. Now incentives for high efTort can be provided only at the cost of having the manager face risk. To characterize the optimal contract in these circumstances, we again consider the contract design problem in two steps: first, we characterize the optimal incentive scheme for each efTort level that the owner might want the manager to select; second, we consider which efTort level the owner should induce. The optimal incentive scheme for implementing a specific effort level e minimizes the owner's expected wage payment subject to two constraints. As before, the manager must receive an expected utility of at least II if he is to accept the contract. When the manager's effort is unobservable, however, the owner also faces a second constraint: The manager must actually desire to choose effort e when facing the incentive scheme. Formally, the optimal incentive scheme for implementing e must therefore solve
(14.B.6)
The owner's profit in this case is the value of expression (l4.B.6), and the manager receives an expected utility of exactly II. Now consider the owner's payoff when the manager's efTort is not observable. In Proposition 14.B.2, we establish that the owner can still achieve his full-information payofT. Proposition 14.B.2: In the principal-agent model with unobservable managerial effort and a risk-neutral manager, an optimal contract generates the same effort choice and expected utilities for the manager and the owner as when effort is observable. Proof: We show explicitly that there is a contract the owner can ofTer that gives him the same payoff that he receives under full information. This contract must therefore be an optimal contract for the owner because the owner can never do better when effort is not observable than when it is (when efTort is observable, the owner is always free to offer the optimal nonobservability contract and simply leave the choice of an effort level up to the manager). Suppose that the owner offers a compensation schedule of the form w(l£) = 1£ - (I., where 1 is some constant. This compensation schedule can be interpreted as "selling the project to the manager" because it gives the manager the full return rr except for the fixed payment ~ (the "sales price"). If the manager accepts this contract, he chooses e to maximize his expected utility,
f
SECTION
I\'(rr) I(rr I e) I at alinE Ii. But ify = O,condition (14.B.IO) then implies that "'(II'(n» :$ 0 at any such n (recall that JJ : O. On the other hand, if I' = 0 in the solution to problem (14.B.9) then, by condition (14.B.10), the optimal compensation schedule gives a fixed wage payment for every profit realization. But we know that this would lead the manager to choose eL rather than ell' violating constraint (iil/) of problem (14.B.9). Hence, J1 > O. • 7. Although problem (l4.B.9) may not appear to be a convcx programming problem. a simple
transformation of the problem shows lhat (14.B.1O) is both a necessary and a sufficient condition for a solution. To see this. reformulate (I4.B.9) as a problem of choosing the manager's level of utility for each profit outcome n. say ii(n). Lelling 4>(') = .. -1(.), the objective function becomes
reformulated problem (see Section M.K of the Mathematical Appendix). The first-order condition
for this problem is
"~(wen»~ = v(n).
[(nle..) < I f(nlel/)
'v
if
f(nle,J --~~-~ > I. I(nlel/)
X. A more direct argument for constmint (i) being binding goes as follows: Suppose that w(n) is a solution to (14.B.9) in which constraint (i) is not binding. Consider a change in the compensation function that lowers the wage paid at each level of 7t in such a way that the resulting decrease in utility is equal at all n. that is, to a new function "~'In) with [v(w(n)) - v(,,'(n))] = t.v > 0 at all 7r E [~. ill This change does not affect the satisfaction of the incentive constraint (ii H ) since if the manager wa~ willing to pick ell when faced with w(n), he will do so when faced with \v(n). Furthermore, because constraint (i) is not binding, the manager will still accept this new contract if j,r is small enough. Lastly. the owner's expected wage payments will be lower than under w(n). This yields a contradiction.
which is convex in ii(n). and the constraints are then all linear in ti(n).
Thus. (Kuhn-Tucker) first·order conditions are both necessary and sufficient for a maximum of this
Defining wen) by
if
This relationship is fairly intuitive. The optimal compensation scheme pays more than IV for outcomes that are statistically relatively more likely to occur under ell than under (',. in the sense of having a likelihood ratio [f(nl e,J/f(n I el/)] less than I. Similarly, it ofTers less compensation for outcomes that are relatively more likely when "" is chosen. We should stress, however, that while this condition evokes a statistical interpretation, there is no actual statistical inference going on here; the owner kllows whal level of effort will be chosen given the compensation schedule he ofTers. Rather, the compensation package has this form because of its illcenril'e effects. That is, by structuring compensation in this way, it provides the manager with an incentive for choosing el/ instead of "t. This point leads to what may at first seem a somewhat surprising implication: in an optimal incentive scheme, compensation is not necessarily monotonically increasing in profits. As is clear from examination of condition (14.8.10), for the optimal compensation scheme to be monotonically increasing, it must be that the likelihood ratio [/(nle..)/I(nle,,)] is decreasing in n; that is, as n increases, the likelihood of getting profit levelrr if effort is ell relative to the likelihood if effort is eL must increase. This property, known as the monotolle likelihood ratio property [see Milgrom (1981)], is lIot implied by first-order stochastic dominance. Figures 14.B.l(a) and (b), for example, depict a case in which the distribution of n conditional on ell stochastically dominates the distribution of n conditional on e L but the monotone likelihood ratio property does not hold. In Ihis example, increases in effort serve to convert low profit realizations into intermediate ones but have no effect on the likelihood of very high profit realizations. Condition (14.8.10) tells us that in this case. we should have higher wages at intermediate levels of profit than at very high ones because it is the likelihood of intermediate profit levels that is sensitive to increases in effort. The optimal compensation function for this example is shown in Figure 14.8.I(c).
or
S4>(l'(nli/(n I",,) dn.
w(n) > ,(,
wIn}
01. and Prob (0,,) = J. E (0, I). (Exercise 14.CI asks you to consider the case of an arbitrary finite number of states.) A contract must try to accomplish two objectives here: first, as in Section 14.B, the risk-neutral owner should insure the manager against fluctuations in his income: second, although there is no problem here in insuring that the manager puts in elTort (because the contract can explicitly state the elTort level required), a contract that maximizes the surplus available in the relationship (and hence, the owner's payoff) must make the level of managerial effort responsive to the disutility incurred by the manager, that is, to the state O. To fix ideas, we first illustrate how these goals are accomplished when is observable: we then turn to an analysis of the problems that arise when II is observed only by the manager.
--- --
1 ... C:
HID 0 E N
IN FOR MAT ION
(A NOM 0 N 0 POL 1ST I
These conditions indicate how the two objectives of insuring the manager and making elTort scnsitive to the state are handled. First, rearranging and combining conditions (14.C2) and (14.C3), we sec that
rr'(en = (I,.(e1, 0,)
for i = L, H.
(14.C7)
This condition says that the optimal level of elTort in state 01 equates the marginal benefit of elTort in terms of increased profit with its marginal disutility cost. The pair (11'1. is illustrated in Figure 14.CI (note that the wage is depicted on the vertical axis and the elTort level on the horizontal axis). As shown, the manager is beller off as we move to the northwest (higher wages and less elTort), and the owner is beller 01T as we move toward the southeast. Because the manager receives utility level II in state 0" the owner seeks to find the most profitable point on the manager's state 0, indilTerence curve with utility level U. This is a point of tangency between the manager's indilTerence curve and one of the owner's isoprofit curves. At this point, the marginal benefit to additional effort in terms of increased profit is exactly equal to the marginal cost borne by the manager. The owner's profit level in state 0, is n1 = rr(e1) - 1'-'(11) - l/(e1, 0,). As shown in Figure 14.C1. this profit is exactly equal to the distance from the origin to the point at which the owner's isoprofit curve through point (\\'1, hits the vertical
en
°
(14.CI)
Max w/ •• '/2!O
>\·/I.""~O
i. v(w" - g(e", 0,,»
+ (I
- i.)v(w l •
-
g(e l., Otl);:> II.
en
In any solution [(11';, en. (wr" er,)] to problem (14.CI) the reservation utility constraint must bind; otherwise, the owner could lower the level of wages olTered and still have the manager accept the contract. In addition, letting y ;:> denote the multiplier on this constraint, the solution must satisfy the following first-order conditions: (14.C2) -i. + }·i.v'(wr, - g(er" 0,,)) = o.
°
. '( ell.)
I.n
-
. '(. yJ.V W II -
491
°°
If () is observable, a contract can directly specify the elTort level and remuneration of the manager contingent on each realization of (note that these variables fully determine the economic outcomes for the two parties). Thus, a complete information contract consists of two wage-elTort pairs: (w", ell) E JR x JR+ for state 0" and (11"/., erJ E JR x JR+ for state 01.' The owner optimally chooses these pairs to solve the following problem:
+ y( I
R E E N I N G)
so the manager's marginal utility of income is equalized across states. This is the usual condition for a risk-neutral party optimally insuring a risk-averse individual. Condition (14.C6) implies that ",r, - y(er" 0Il) = wt - y(et, 01,), which in turn implies that 1'(II'r, - y(er" 011)) = 1'(11'; - y(et, 01.)); that is, the manager's utility is equalized across states. Given the rescrvation utility constraint in (14.CI), the managcr therefore has utility level Ii in each state. Now consider the optimal elTort Icvels in the two states. Since 1/,.(0, 0) = and rr'(O) > 0, conditions (14.C4) and (14.CS) must hold with equality and e1 > for i = 1,2. Combining condition (14.C2) with (14.C4), and condition (14.C3) with (14.CS), we see that the optimallevcl of elTort in state 0" e1, satisfies
TIle State (] is Observable
-( I - i.)
esc
(14.C6)
°
s.t.
SEC T ION
- i.) D'(W! -
gee!, 0tl) = 0 .
~ 0, 9 (* ell. 0)) II ge (* ell' 0II ){ = 0
(I - i.)rr'(erJ - y(1 - i.)v'(II'! - gee!, 0tl)g,(e!,
OIJ{ ~~'
Figure 14.C.1
The optimal wage-effort pair for state OJ when states are observable. n(e) - w
(14.C3)
if
eil > 0.
if et > O.
(14.CA)
(14.CS)
Profits of { 0 Owner in Slalc 0,
m:)
12. As with the case of hidden actions studied in Section 14.B, nonobservabililY causes no welfare loss In the case of managerial risk neutralily. As there. a "sellout" contract that races the manager with the full marginal returns from his actions can generate the first-best outcome. (See
Exercise 14.C.2.)
J
= n7':
492
CHAPTER
14:
THE
PRINCIPAL·AGENT
v(.' - y(e, Od)
PROBLEM
="
SECTION
v( .. - y(r,
0.» = U
.(e) - w =
.( 0 (i,e" the owner lowers the wage payments in both states by c), This new contract still satisfies constr'!int (i) as long as c is chosen small enough, In addition, the incentive compatibility constraints are still satisfied because this change just subtracts a constant, c, from each side of these constraints, But if this new contract satisfies all the constraints, the original contract could not have been optimal because the owner now has higher profits, which is a contradiction. _ Lemma 14.C.3: In any optimal contract: (i) e L :5 et; that is, the manager's effort level in state OL is no more than the level that would arise if 0 were observable, (ii) eH = e;,; that is, the manager's effort level in state OH is exactly equal to the level that would arise if 0 were observable,
Stale { ()
"
/,
I;~olil
and, as is evident in Figure I 4.C.S, the truth-telling constraints are still satisfied, Thus, > e;' cannot be optimal. a contract with Now consider part (ii), Given any wage -elTort pair (Ii'/" "/,) with ,"- :5 such as that shown in Figure 14,C.6. the owner's problem is to find the location for (1\'11'"11) in the shaded region that maximizes his profit in state 11 11 , The solution occurs at a point of tangency between the manager's state 1111 indifTerence curve through point (Ii',.. ell) and an isoprofit curve for the owner. This tangency occurs at point (lVII' cr,) in the figure, and necessarily involves efTort level er, because all points of tangency between the manager's state 0" indifTerence curves and the owner's isoprofit curves occur at efTort level er, [they are characterized by condition (14.C7) for i = If]. Note that this point of tangency occurs strictly to the right of efTort level e,. because
"I.
Ftgur. 14.C.4 (teft)
I n a feasible contract alTering (wL' eel for state OL' the pair ("'H,eH) must lie in the shaded region, Figure 14.C,5 (right)
An optimal contract has eL. :s; ei.
el. :s ei.
g(l". 0,,)
(iv) WL - g(IL'
ad
~ WI/ - g(I",
ad·
This problem has exactly the same structure as (14.C.8) but with the principal's (here the firm's) profit being a function of the state. As noted above, the analysis of this problem follows exactly the same lines as our analysis of problem (14.C.8). This class of models has seen wide application in the literature (although often with a continuum of types assumed). Maskin and Riley (1984b), for example, apply this model to the study of monopolistic price discrimination. In their model, a consumer of type 0 has utility t~x. 0) - T when he consumes x units of a monopolist'S good and makes a total payment of T to the monopolist, and can earn a reservation utility level of V(0. 0) = 0 by not purchasing from the monopolist. The monopolist has a constant unit cost of production equal to c > 0 21. The model studied in Section 13.D with ltj(t) = 0, corresponds to the limiting case where
I' -0.
eC
MOD E L 5
501
,---------------------------------------------------------------------
Exercises 14.C.7 to 14.C.9 ask you to study some examples of monopolistic screening models.
14,D Hidden Actions and Hidden Information: Hybrid Models Although the hidden action - hidden information dichotomization serves as a useful starting point for understanding principal-agent models, many real-world situations (and some of the literature as well) involve elements of both problems. To consider an example of such a model, suppose that we augment the simple hidden information model considered in Section 14.C in the following manner: let the level of efTort e now be unobservable, and let profits be a stochastic function of efTort, described by conditional density function f(n I e). I n essence, what we now have is a hidden action model, but one in which the owner also docs not know something about the disutility of the manager (which is captured in the state variable 0). Formal analysis of this model is beyond the scope of this chapter. but the basic thrust of the revelation principle extends to the analysis of these types of hybrid problems. In particular, as Myerson (1982) shows, the owner can now restrict attention to contracts of the following form: (i) After the state (/ is realized, the manager announces which state has occurred. (ii) The contract specifies, for each possible announcement 0 E e, the efTort level dO) that the manager should take and a compensation scheme 1\'(11 I 0). (iii) In every state II, the manager is willing to be both Irulirful in stage (i) and obedient following stage (ii) [i.e., he finds it optimal to choose effort level e(O) in state OJ, This contract can be thought of as a revelation game, but one in which the outcome of the manager's announcement about the state is a hidden action-style contract, that is, a compensation scheme and a "recommended action." The requirement of "obedience" amounts to an incentive constraint that is like that in the hidden action
22. The regulator's objective function can be generalized to allow a weighted average of consumer and producer surplus, with greater weight on consumers. In this case, the runction 7r i (·) will depend on 0;.
502
APPENDIX
A:
MULTIPLE
EFFORT
LEVELS
IN
THE
HIDDEN
ACTION
MODEL
503
-------------------------------------------------------------------------- ,---------------------------------------------------------------CHAPTER
14:
THE
PRINCIPAL.AGENT
PROBLEM
model considered in Section 14.B; the "truthfulness" constraints are generalizations of those considered in our hidden information model. See Myerson (1982) for details. One special case of this hybrid model deserves particular mention because its analysis reduces to that of the pure hidden information model considered in Section 14.C. In particular, suppose that effort is unobservable but that the relationship between effort and profits is determillistic, given by the function n(e). In that case, for any particular announcement 8, it is possible to induce any wage-effort pair that is desired, say (w, e), by use of a simple "forcing" compensation scheme: Just reward the manager with a wage payment of w if profits are n(e), and give him a wage payment of - OCJ otherwise. Thus, the combination of the observability of n and the one-to-one relationship between nand e effectively allows the contract to specify e. The analysis of this model is therefore identical to that of the hidden information model considered in Section 14.C, where wage-effort pairs could be specified directly as functions of the manager's announcement. To see this point in a slightly different way, note first that because of the ability to write forcing contracts, in this model an optimal contract can be thought of as specifying, for each announcement 6, a wage-profit pair (w(O), n(O)). Now, for any required profit level n, the effort level necessary to achieve a profit of n is e such that nee) = n. Let the function e(n) describe this etTort level. We can now think of the manager as having a disutility function defined directly over the profit level which is given by y(n, 0) = g(e(n), II). But this model looks just like a model with observable etTort where the effort variable is n, the disutility function over this etTort is y(n, II), and the profit function is n(7I) = 71. Thus, the analysis of this model is identical to that in a pure hidden information model. A similar point applies to a closely related hybrid model in which, instead of the manager's disutility of effort, it is the relation between profit and effort that depends on the state. In particular, suppose that the disutility of effort is given by the function g(e) and profits are given by the function 7I(e, Ii), where 71.(') > 0, 71 .. (') < 0, 71,(') > 0, and n.. (·) > 0. Effort is not observable, but profits are. The idea is that the manager knows more than the owner does about the true profit opportunities facing the firm (e.g., the marginal productivity of effort). Again, we can think of a contract as specifying, for each announcement by the manager, a wage-profit pair (implicitly using forcing contracts). In this context, the effort needed to achieve any given level of profit 71 in state II is given by some function 1'(71, Ii), and the disutility associated with this effort is then g(n, 0) = g(';(7I, II». But this model is also equivalent to our basic hidden information model with observable etTort: just let the etTort variable be n, the disutility of this etTort be g(n, 0), and the profit function be n(n) = 71. Again, our results from Section 14.C apply.
Figure 14.AA.1
Density functions for E = {eL , eM' ell}: effort choice eAt may not be
implementable.
Profit Realization
to the more general specification initially introduced in Section 14.B in which E is the feasible set of effort choices. As in Section 14.B, we can break up the principal's (the owner's) problem into several parts: (a) What are the effort levels e that it is possible to induce'! (b) What is the optimal contract for inducing each specific etTort level e E E'! (c) Which etTort level e E E is optimal'! In a multiple-action setting, each of these three parts becomes somewhat more complicated. For example, with just two actions, part (a) was trivial: eL could be induced with a fixed wage contract, and ell could always be induced by giving incentives that were sufficiently high at outcomes that were more likely to arise when ell is chosen. With more than two actions, however, this may not be so. For example, consider the three-action case in which E = {e L, eM' ell} and the conditional density functions are those depicted in Figure 14.AA. I. As is suggested by the figure, it may be impossible to design incentives such that eM is chosen because for any w(n) the agent may prefer either eL or ell to eM' (Exercise 14.B.4 provides an example along these lines.) Part (b) also becomes more involved. The optimal contract for implementing effort choice e solves Min w(lt)
f
(14.AA.I)
\I'(n)f(nle)dn
s.t. (i)
f
v(w(n)) f(n I e) dn - g(e)
(ii) e solves
~.~~
f
~ ii
v(w(n»I(n\e) dn - g(e).
Ifwe have K possible actions in set E, the incentive constraints in problem (14.AA.I) [constraints (ii)] consist of (K - I) constraints that must be satisfied. In this case, with a change of variables in which we maximize over the level of utility that the manager gets conditional on n, say v(7I), we have a problem with K linear constraints and a convex objective function [see Grossman and Hart (1983) and footnote 7 for more on this]. However, if E is a continuous set of possible actions, say E = [0, e] c IR, then we have an infinity of incentive constraints. One trick sometimes used in this case to
APPENDIX A: MULTIPLE EFFORT LEVELS IN THE HIDDEN ACTION MODEL
In this appendix, we discuss additional issues that arise when the effort choice in the hidden action (moral hazard) model discussed in Section 14.B is more complex than the simple two-effort-choice specification e E {e L, ell} analyzed there. Here, we return
d
504
CHAPTER
1.:
PRINCIPAl.AGENT
THE
PROBLEM
----------------------------------------------------------------------------------simplify problem (14.AA.l) is to replace constraint (ii) with a ./irst-order condition (this is sometimes called the ./irst·order approach). For example, if e is a one. dimensional measure of effort, then the manager's first-order condition is
f
~:P:P:E:N~O~':X~B~'~S~O~"~U~T~'~O~N__~O_'
,--
Using Lemma 14.C.1 we can restate problem (14.C.8) as Max ..... //.1'/1
v(w(n» !e(n I e) dn - g'(e) = 0,
OBLEM WITH HIDDEN INFORMATION ___ 505 __T_H_E__P_R_'_H_C_'_P_A_l_._A_G_E_N__T __P_A________________________________
~
).[n(el/) - wl/]
(14.AA.2)
1
• I.
+ I'
[!.(n1e)] f(n Ie)
The condition that ratio Ue(n I 11
+ 'l -
1/
= O.
(14.IlB.2)
+ , = o.
(14.BB.3)
- 1.
(14.1313.4)
(I 4. 1lB.5)
Step 4: Steps 1 to 3 imply that 4>, = O. Suppose not: i.e., that ,. > O. Then constraint (iv) must be binding. We shall now derive a contradiction. First, substitute for 4'" in conditions (14.BB.4) and (14.BB.5) using the fact that 4>11 = 4>,. + i.from condition (14.BB.2). Then, using the fact that (e,., ell)>> O. we can write cond,t,ons (14.1l1l.4) and (14.BB.5) as
+ 4>1.[y,.(e ll . 0,.)
-II,.(eIl' 0Il)] = 0
and
+ (I + d[g,(e/.. 011) -
g,(e" 0,,)] =
o.
But 0 and Ye(O. 0,) = a for i = L. H. Similarly for condition (14.B8.5) and e,.
~ v·'(ii)
lVI/ -
g(el/,
Step 2: Adding conditions (14.BB.2) and (14.BB.3) implies that i' = I. Hence. constmint (i) must bind at an optimal solution.
w,J
(iii)
w, - gte"~ 0Il) "'II -
Step I: Condition (14.BB.2) implies that 11 > O. Thus, constraint (iii) must bind (hold with equality) at an optimal solution.
(I - i.)[n'(e,.) - 1I,(e/.. 011)] WI. -
(14.B8.I)
along with the complementary slackness conditions for constraints (i), (iii). and (iv) [conditions (M.K.7)]. Let us break up the analysis of these conditions into several steps.
0
s.t. (i)
~
if 1', > 0
i.[rr'(ell) - r/,(ell. 0Il)]
+ (I
OLl
if 1'" > 0
Recall problem (14.C.8): ""/1. ell
WI/ -
(iv)
-(I - i.)
APPENDIX B: A FORMAL SOLUTION OF THE PRINCIPAL-AGENT PROBLEM WITH HIDDEN INFORMATION
i.[n(el/) - w,,]
~ V-'(ol)
(iii)
-i.
Finally, to answer part (c), we need to compute the optimal contract from part (b) for each action that part (a) reveals is implementable and then compare their relative profits for the principal. With more than two effort choices, two features of the two-effort-choice case fail to generalize. First, nonobservability can lead to an upward distortion in effort. (Exercise 14.B.4 provides an example.) Second, at the optimal contract under nonobservability we can get boch an inefficient effort choice and inefficiencies resulting from managerial risk bearing.
Max
OLl
IV,]
Letting (i', 4>", 4>d ~ 0 be the multipliers on constraints (i), (iii), and (iv), respectively, the Kuhn- Tucker conditions for this problem can be written (see Section M.K of the Mathematical Appendix)
(14.AA.3)
.
).)[n(ed -
s.t. (i) w, - g(e"
where f.(n I e) = af(n I e)IDe. If we replace constraint (ii) with (14.AA.2) and solve the resulting problem, we can derive a condition for w(n) that parallels condition (14.8.10):
v'(w(;i) =
+ (I -
0, ""',.t'/ ~ 0
rr'(ed - y,.(e" 011) >
a > n'(ell ) -
g,(e ll , 011),
which implies 1'1/ > e, since n(e) - gte, 011) is concave in e. But if el/ > e, and constraint (iii) binds (which it does from Step I), then constraint (iv) must be slack
....
506
CHAPTER
14;
THE
PRINCIPAL-AGENT
PROBLEM
-------------------------------------------------------------------------------------------because we then have
l
EX E R CIS E S
~----------
Maskin. E.. and J. Riley. (l984b). Monopoly with incomplete information. Rand Journal of Economics IS:
("'" - wel =
=
,. = 0, we know from (14.BB.2) that 4>11 two values into conditions (14.88.4) and (14.B8.5) we have
= J..
Substituting these
1['(e,,) - Y,(e", 0,,) = 0
(14.B8.6)
EXERCISES
(14.B8.7)
14.B.I" Consider the two-elTort-level hidden aelion model discussed in Section 14.B with the gene«11 utility function u(w, e) for Ihe agent Must the reservalion utility constraint be binding
and
[1['(e,.) - y •. (e L• 0,.>]
i.
+ l·~-;: [g.(eL' 0,,) -
y.(e L • 0,.>] =
o.
in an optim uk.).
14.C.6C Reconsider the labor market screening model in Exercise 13.0.1, but now suppose that there is a single employer. Characterize the solution to this firm's screening problem (assume that both types of workers have a reservation utility level of 0). Compare the task levels in this solution with those in the equilibrium of the competitive screening model (assuming an equilibrium exists) that you derived in Exercise 13.0.1.
(a) Suppose that the owner wants to implement effort choice e" and that both Rand C are observable. Derive the first-order condition for the optimal compensation scheme I\'(R. C). How does it depend on Rand C?
14.C.7" (1. Tirole) Assume that there are two types of consumers for a firm's product. OH and II,.. The proportion of type 0,. consumers is i.. A type Us utility when consuming amount x of the good and paying a total of T for it is u(x, T) = Ov(x) - T. where
(b) How would your answer to (a) change if the manager could always unobservably reduce the revenues of the firm (in a way that is of no direct benefit to him)?
1_(1 -x)' vex) = ----2---. The firm is the sole producer of this good, and its cost of production per unit is c > O.
(e) What if, in addition, costs are now unobservable by a court (so that compensation can be made contingent only on revenues)?
(a) Consider a nondiscriminating monopolist. Derive his optimal pricing policy. Show that he serves both classes of consumers if either OL or i. is "large enough."
14.B.7C Consider a two-period model that involves two repetitions of the two-effort-level hidden action model studied in Section 14.8. There is no discounting by either the firm or the manager. The manager's expected utility over the two periods is the sum of his two single-period expected utilities £[v(w) - g(e)), where v'(·) > 0 and v"(·) < O. Suppose that a contract can be signed ex ante that gives payoffs in each period as a function of performance up until then. Will period 2 wages depend on period I profits in the optimal contract?
(b) Consider a monopolist who can distinguish the two types (by some characteristic) but can only charge a simple price p; to each type 0,. Characterize his optimal prices.
(e) Suppose the monopolist cannot distinguish the types. Derive the optimal two-part tariff (a pricing policy consisting of a lump-sum charge F plus a linear price per unit purchased of p) under the assumption that the monopolist serves both types. Interpret. When will the monopolist serve both types? (d) Compute the fully optimal nonlinear tariff. How do the quantities purchased by the two types compare with the levels in (a) to (e)?
14.B.SC Amend the two-effort-choice hidden action model discussed in Section 14.B as follows: Suppose the principal can, for a cost of c, observe an extra signal y of the agent's effort. Profits n and the signal )' have a joint distribution fen, y Ie) conditional on e. The decision to investigate the value of )' can be made after observing It. A contract now specifies a wage schedule wen) in the event of no investigation, a wage schedule I1'(It, }') if an investigation occurs, and a probability pen) of investigation conditional
14.C.S" Air Shangri-la is the only airline allowed to fly between the islands of Shangri-la and Nirvana. There are two types of passengers, tourist and business. Business travelers are willing to pay more than tourists. The airline, however, cannot tell directly whether a ticket purchaser is a tourist or a business traveler. The two types do differ, though, in how much they are willing to pay to avoid having to purchase their tickets in advance. (Passengers do not like to commit themselves in advance to traveling at a particular time.) More specifically, the utility levels of each of the two types net of the price of the ticket, P, for any given amount of time W prior to the flight that the ticket is purchased are given by
on n. Characterize the optimal contract for implementing effort level ell'
14.C.I C Analyze the extension of the hidden information model discussed in Section 14.C where there are an arbitrary finite number of states (0" ... , ON) where 0;+, > 0; for all i. 14.C.2" Consider the hidden information model in Section 14.C, but now let the manager be risk neutral with utility function v(w) = IV. Show that the owner can do as well when 0 is unobservable as when it is observable. In particular, show that he can accomplish this with a contract that offers the manager a compensation scheme of the form w(It) = It - a and allows him to choose any effort level he wants. Graph this function and the manager's choices in (w, e)-space. What revelation mechanism would give this same outcome?
BIISilless: Tourisc:
v - O.p - W, v - OfP - W,
where 0 < O. < 0,.. (Note that for any given level of W, the business traveler is willing to pay more for his ticket. Also, the business traveler is willing to pay more for any given reduction in W.)
..
509
510
CHAPTER
t.:
THE
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PROBLEM
The proporlion of travelers who are tourists is .I.. Assume that the cost of transporting a passenger is e. Assume in (a) to (d) that Air Shangri·la wants to carry both types of passengers.
-
PAR
T
F
0
U
R
General Equilibrium
(0) Draw the indifference curves of the two types in (P, W)-space. Draw the airline's
isoprofit curves. Now formulate the optimal (profit·maximizing) price discrimination problem mathematically that Air Shangri·la would want to solve. [Hinr: Impose nonnegativity of prices as a constraint since, if it charged a negative price, it would sell an infinite number of tickets at this price.] (b) Show that in the optimal solution, tourists are indifferent between buying a ticket and not going at all. (c) Show that in the optimal solution. business travelers never buy their ticket prior to the night and are just indifferent between doing this and buying when tourists buy. (d) Describe fully the optimal price discrimination scheme under the assumption that they sell to both types. How does it depend on the underlying parameters i., 0•• Or, and ,,? (0) Under what circumstances will Air Shangri·la choose to serve only business travelers?
P"rt IV is devoted to an examination of competitive market economies from a
gelleral equilibrium perspective. Our use of the term "general equilibrium" refers both
14.C.9" Consider a risk·averse individual who is an expected utility maximizer with a Bernoulli utility function over wealth u(·). The individual has initial wealth Wand faces a probability IJ of suffering a loss of size L, where W> L > O. An insurance contract may be described by a pair (c" e,), where e, is the amOunt of wealth the individual has in the event of no loss and c, is the amount the individual has if a loss is suffered. That is, in the event no loss occurs the individual pays the insurance company an amount (W - e,), whereas if a loss occurs the individual receives a payment [e, - (W - L)] from the company.
to a methodological point of view and to a substantive theory. Methodologically, the general equilibrium approach has two central features. First, it views the economy as a closed and interrelated system in which we must simultaneously determine the equilibrium values of all variables of interest. Thus, when we evaluate the effects of a perturbation in the economic environment, the equilibrium levels of the entire set of endogenous variables in the economy needs to be recomputed. This stands in contrast to the partial equilibrium approach, where the impact on endogenous variables not directly related to the problem at hand is explicitly or implicitly disregarded. A second central feature of the general equilibrium approach is that it aims at reducing the set of variables taken as exogenous to a small number of physical realities (e.g., the set of economic agents, the available technologies, the preferences and physical endowments of goods of various agents). From a substantive viewpoint, general equilibrium theory has a more specific meaning: It is a theory or the determination of equilibrium prices and quantities in a system of perfectly competitive markets. This theory is often referred to as the Walrasian theory of markets [from L. Walras (1874)], and it is the object of our study in Part IV. The Walrasian theory of markets is very ambitious. It attempts no less than to predict the complete vector of final consumptions and productions using only the fundamentals of the economy (the list of commodities, the state of technology, prererences and endowments), the institutional assumption that a price is quoted for every commodity (including those that will not be traded at equilibrium), and the behavioral assumption of price taking by consumers and firms. Strictly speaking, we introduced a particular case of the general equilibrium model in Chapter 10. There, we carried out an equilibrium and welfare analysis of perfectly competitive markets under the assumption that consumers had quasilinear preferences. In that setting, consumer demand functions do not display wealth effects (except for a single commodity, called the numeraire); as a consequence, the analysis of a single market (or small group of markets) could be pursued in a manner understandable as traditional partial equilibrium analysis. A good deal of what we do in Part IV
(0) Suppose that the individual's only source of insurance is a risk·neutral monopolist (i.e., the monopolist seeks to maximize its expected profits). Characterize the contract the monopolist will offer the individual in the case in which the individual's probability of loss, IJ. is observable.
(b) Suppose, instead, that 0 is not observable by the insurance company (the individual knows 0). The parameter 0 can take one of two values {OL,OH}' where 0" > OL > 0 and Prob (Od = i.. Characterize the optimal contract offers of the monopolist. Can one speak of one type of insured individual being "rationed" in his purchases of insurance (i.e., he would want to purchase more insurance if allowed to at fair odds)? Intuitively. why does this rationing occur? [Him: It might be helpful to draw a picture in (c" e,)·space. To do so. start by locating the individual's endowment point. that is, what he gets ifhe does not purchase any insurance.] (c) Compare your solution in (b) with your answer to Exercise I3.D.2. 14.AA.I" Show that [f,(lt Ie)ll(lt Ie)] is increasing in It for all e E [a, b] c R if and only if for any e', e" E [a. b]. with e" > e', [f(ltle")II(ltle')] is increasing in It.
"L.
l4.AA.2" Consider a hidden action model with e E [0, i] and two outcomes It,, and with "L' The probability of ltH given effort level e is I(lt" Ie). Give sufficient conditions for the first·order approach to be valid. Characterize the optimal contract when these conditions arc satisfied.
"If >
l4.B8.I" Try solving problem (14.B8.I) by first solving it while ignoring constraint (iv) and then arguing that the solution you derive to this "relaxed" problem is actually the solution to problem (14.B8.I).
511
...
512
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PART
EQUILIBRIUM
IV:
GENERAL
EQUILIBRIUM
513
--------------------------------------------------------------------------- --------------------------------------------------------------------------------can be viewed as an attempt to extend the ideas of Chapter 10 to a world in which wealth effects are significant. The primary motivation for this is the increase in realism it brings. To make practical use of equilibrium analysis for studying the performance of an entire economy, or for evaluating policy interventions that affect large numbers of markets simultaneously, wealth effects, a primary source of linkages across markets, cannot be neglected, and therefore the general equilibrium approach is essential. Although knowledge of the material discussed in Chapter 10 is not a strict prerequisite for Part IV, we nonetheless strongly recommend that you study it, especially Sections 10.B to 10.0. It constitutes an introduction to the main issues and provides a simple and analytically very useful example. We will see in the different chapters of Part IV that quite a number of the important results established in Chaptcr 10 for the quasilinear situation carryover to the case of general preferences. But many others do not. To understand why this may be so, recall from Chapters 4 and 10 that a group of consumers with quasilinear preferences (with respect to the same numeraire) admits the existence of a (normative) representative consumer. This is a powerful restriction on the behavior of aggregate demand that will not be available to us in the more general settings that we study here. It is important to note that, relative to the analysis carried out in Part III, we incur a cost for accomplishing the task that general equilibrium sets itself to do: the assumptions of price-taking behavior and universal price quoting-that is, the existence of markets for every relevant commodity (with the implication of symmetric information)-are present in nearly all the theory studied in Part IV. Thus, in many respects, we are not going as deep as we did in Part III in the microanalysis of markets. of market failure, and of the strategic interdependence of market actors. The trade-off in conceptual structure between Parts III and IV reflects, in a sense, the current state of the frontier of microeconomic research. The content of Part IV is organized into six chapters. Chapter 15 presents a preliminary discussion. Its main purpose is to illustrate the issues that concern general equilibrium theory by means of three simple examples: the tlVo-consumer Edgeworth box economy; the one-consumer, one-firm economy, and the sm O. In short, only the relative prices pUp! are determined in an equilibrium.
xr,
Flgur. 15.B.4 (top right)
Optimal consumption for consumer I at prices p.
Flgur. 15.B.5 (bottom)
xr
Consumer I's offer curve. This implies that the consumer's offer curve lies within the upper contour set of w, and that, if inditTerence curves are smooth, the offer curve must be tangent to the consumer's indifference curve at the endowment point. Figure 15.B.6 represents the demanded bundles of the two consumers at some arbitrary price vector p. Note that the demands expressed by the two consumers are not compatible. The total demand for good 2 exceeds its total supply in the economy ';'" whereas the total demand for good I is strictly less than its endowment w,. Put somewhat ditTerently, consumer I is a net demander of good 2 in the sense that he wants to consume more than his endowment of that commodity. Although consumer 2 is willing to be a net supplier of that good (he wants to consume less than his endowment), he is not willing to supply enough to satisfy consumer I's needs. Good 2 is therefore in excess demand in the situation depicted in the figure. In contrast, good I is in excess supply. At a market equilibrium where consumers take prices as given, markets should clear. That is, the consumers should be able to fulfill their desired purchases and
Example \S.B.I: Suppose that each consumer i has the Cobb-Douglas utility function
=
=
II,(X,;, Xli) xiix~i-'. In addition, endowments are WI = (I, 2) and w, (2, I). At prices p = (PI' p,), consumer I's wealth is (p, + 2p,) and therefore his demands lie
on the offer curve (recall the derivation in Example 3.0.1): OC,(p) = (Il(PI
+ 2p , ), (I PI
d
- 1l)(PI p,
,»).
+ 2P
figure 15.B.6
A price vector with excess demand for good 2 and excess supply for good I.
520
Ci1A":',"ER
15
GENERAL
EQUILIBRIUM
xi
2
~~~______r-__________~'02
THEORY:
_______
EXAMPLES
SEC 1 ION
Indifference Curves Ihrough w ~-+~~ -+'O~C~,~________~02
~~
1 5 . B:
PUR E
______-r___
E X C HAN C. E:
T H £
E 0 GE W 0 A TH
BOX
~02
__
x_~ ~:_. >. ,-
,',1,----------,-I
SOME
____ :'__-I_ __ } ,', p'
Figure 15.B.9
Multiple Walrasian equilibria.
0,
o,,-,_~.~_
x~ I (b)
(a)
~-I----------~~
The Edgeworth box, simple as it is, is remarkably powerful. There are virtually no phenomena or properties of general equilibrium exchange economies that cannot be depicted in it. Consider, for example, the issue of the uniqueness of Walrasian equilibrium. In Chapter 10, we saw that if there is a numeraire commodity relalive to which preferences admit a quasilinear representation, then (with strict convexity of preferences) the equilibrium consumption allocation and relative prices are unique. In Figure 15.B.7, we also have uniqueness (see Exercise 15.B.2 for a more explicit discussion). Yet, as the Edgeworth box in Figure 15.B.9 shows, this property does not generalize. In that figure, preferences (which are entirely nonpathological) are such that the offer curves change curvature and interlace several times. In particular, they intersect for prices such that p,/p, is equal to t, I, and 2. For the sake of completeness, we present an analytical example with the features of the figure.
Figure 15.B.7 ('op)
(a) A Walrasian equilibrium. (b) The consumer's offer curves intersect at the Walrasian equilibrium allocation.
" + x". Note that the utility functions are quasilinear (which, in particular, facilitates the computation of demand), but with respect to different numeraires. The endowments are w, = (2, r) and (1), = (r, 2), where r is chosen to guarantee that the equilibrium prices turn out to be round numbers. Precisely, r = 21/9 > O.ln Exercise 15.B.5, you are asked to compute the offer curves of the two consumers. They are:
Observe that the demands for the first and the second good are, respectively, decreasing and increasing with p,. This is how we have drawn OC, in Figure 15.B.7(b). Similarly, OC,(p) = (a(2p, + p,)/p" (I - 0()(2p, + p,)/p,). To determine the Walrasian equilibrium prices, note that at these prices the total amount of good I consumed by the two consumers must equal 3 (= W II + (1) ,,). Thus, a(pi
+ 2p!l + 0(2pi + p!) = pi
2"10 -
3. OC,(p" p,)
pi
Solving this equation yields
!1=_a_ p!
I - a
_
= (2 + r(~) (~)"IO, (~) -1/ )>> 0 9
and
(15.B.2) OC,(p" p,) =
Observe that at any prices (pi, p!) satisfying condition (15.B.2), the market for good 2 clears as well (you should verify this). This is a general feature of an Edgeworth box economy: To determine equilibrium prices we need only determine prices at which one of the markets clears; the other market will necessarily clear at these prices. This point can be seen graphically in the Edgeworth box: Because both consumers' demanded bundles lie on the same budget line, if the amounts of commodity I demanded are compatible, then so must be those for commodity 2. (See also Exercise 15.B.1.) •
(( P,p,)
-I/O
,2 + r
(p,) p; - (p,)"IO) p, »0.
NOle that, as illustrated in Figure 15.B.9, and in contrast with Example 15.B.I, consumer I's demand for good I (and symmetrically for consumer 2) may be increasing in p ,. To compute the equilibria it is sufficient to solve the equation that equates the total demand of the second good to its total supply, or
(p )"10 = 2 + r. ~ - ~ (~p)-'/9 + 2 + r (p)
d
521
522
CHAPTER
15:
GENERAL
EQUILIBRIUM
THEORY:
SOME
SECTION
EXAMPLES
PURE
EXCHANGE:
THE
EDGEWORTH
BOX
523
Ahr-----~~~------_.O,
THE
conditions hold:
these goods.'o Output is sold in world markets. Factors. on the other hand. are immobile and must be used for production within the country. The central question for our analysis concerns the equilibrium in the factor markets; that is. we wish to determine the equilibrium factor prices W = (w, •...• wd and the allocation of the economy's factor endowments among the J firms." Given output prices p = (p, •...• PJ) and input prices w = (w, •...• wL ). a profitmaximizing production plan for firm j solves Max
15.0:
I ..... L.
I
pj~(z;l
(15.0.5)
j
s.t.
LZj = i.
zn
How does the equilibrium factor allocation (zt .. ... compare with what this planner does? Recall from Section 5.E that whenever we have a collection of J price-taking firms, their profit-maximizing behavior is compatible with the behavior we would observe if the firms were to maximize their profits jointly taking the prices of outputs and factors as given. That is. the factor demands (zt •. .. , z1) solve
Because of the concavity of firms' production functions. first-order conditions are both necessary and sufficient for the characterization of optimal factor demands. Therefore. the L(J + I) variables formed by the factor allocation (=r ..... =1) E R~J and the factor prices w' = (IVt •...• wl) constitute an equilibrium if and only if they satisfy the following L(J + 1) equations (we assume an interior solution here):
Max
(15.0.6)
(:I •..•• :J)~O
for j = I •...• J and { = I ....• L
(15.0.1)
Since Lj zj = i (by the equilibrium property of market clearing). the factor demands must also solve problem (15.0.6) subject to the further constraint that Lj Zj = i. But this implies that the factor demands (zt ....• z1) in fact solve problem (15.0.5): if we must have Lj Zj = i. then the total cost w' -(LI z) is given. and so the joint profit-maximizing problem (15.0.6) reduces to the revenue-maximizing problem (15.0.5).
(zt.···. z1)
and for ( = I•...• L.
(15.0.2)
The equilibrium output levels are then qj = ~(zj) for every j. Equilibrium conditions for outputs and factor prices can alternatively be stated using the firms' cost functions cl(w. qj) for j = I•...• J. Output levels (qt • ...• qj) » 0 and factor prices 11'* »0 constitute an equilibrium if and only if the following
One benefit of the property just established is that it can be used to obtain the equilibrium factor allocation without a previous explicit computation of the equilibrium factor prices; we simply need to solve problem (l5.D.5) directly. It also provides a useful way of viewing the equilibrium factor prices. To sce this. consider again the joint profit-maximization problem (15.D.6). Wc can approach this problem in an equivalent manner by first deriving an aggregate
10. See Exen:isc 15.D.4 ror an endogenous determination (up to a scalar multiple) or the pri~cs I' ~ (PI"
.. PJ)'
II. Note that once the factor prices and allocations are determined. each consumer's demands G.ln be readily determined from his demand function given the exogenou5 prices (PI" .. • PJ) and
12. Note thai maximization or economy-wide revenue rrom production would be the goal of any planner who wanted to maximize consumer welrare: it allows ror the maximal purchases of consumption goods. at the fixed world prices.
Ihe wealth derived from factor input sales and profit distribulions. Recall that the currenl model is completed by assuming that this demand is met in the world markets.
rl
MODEL
531
5:.:..::
~HAPTEK
i!i:
GENERAL
EQUILIBRIUM
THEORY:
SOME
.:)tCTION
EXAMPLES
15.0.
tHE
", -.~
f(:",:,,)
=I
FIgure 15.0.1
2)(2
PRODUCTION
__________________
MODEL
533
~o,
0,
(a) A unit isoquant. (b) The unit cost (b)
(a)
function. (a) Figure 15.0.2
production function for dollars: f(:) =
Max
p.!.(Z.)
S.t.
+ ... + pJlJ(ZJ)
(h) (a)
An inefficient factor allocation.
(b)
The Pareto set of factor allocations.
represent the possible allocations of the factor endowments between the two firms in an Edgeworth box of size i, by The factors used by firm I are measured from the southwest corner; those used by firm 2 are measured from the northeast corner. We also represent the isoquants of the two firms in this Edgeworth box. Figure 15.D.2(a) depicts an inemcient allocation z of the inputs between the two firms: Any allocation in the interior of the hatched region generates more output of hOlh goods than docs :. Figure 15.D.2(b), on the other hand, depicts the Pareto set of factor allocations, that is, the set of factor allocations at which it is not possible, with the given total factor endowments, to produce more of one good without producing less of the other. The Pareto set (endpoints excluded) must lie all above or all below or be coincident with the diagonal of the Edgeworth box. If it ever cuts the diagonal then because of constant returns, the isoquants of the two firms must in fact be tangent all along the diagonal, and so the diagonal must be the Pareto set (see also Exercise 15.B.7). Moreover, you should convince yourself of the correctness of the following claims.
=,.
Liz)=:,
The aggregate factor demands must then solve Max.~o({(z) - w'z). For every I, the first-order condition for this problem is IV, = Df(z)/elz, . Moreover, at an equilibrium, the aggregate usage of factor ( must be exactly :/ Hence, the equilibrium factor price of factor ( must be W, = (if(:)/ill, ; that is, rhe prke of factor { mu.W be exactly equal to its aggregate margi/wl productivit}' (in rerms of revenue). Since f(·) is concave, this observation by itself generates some interesting comparative statics. For example, a change in the endowment of a single input must change the equilibrium price of the input in the opposite direction. Let us now be more specific and take J = L = 2, so that the economy under study produces two outputs from two primary factors, We also assume that the production functions I,(z", ZIt), 1'(Z12' ZIt) are homogeneous of degree one (so the technologies exhibit constant returns to scale; see Section 5.B). This model is known as the 2 x 2 production model. In applications, factor I is often thought of as labor and factor 2 as capital. For every vector of factor prices W = (w" w,), we denote by cj(w) the minimum cost of producing one unit of good j and by OJ(w) = (o,iw), o'l(w» the input combination (assumed unique) at which this minimum cost is reached. Recall again from Proposition 5.C.2 that Vcj(w) = (olj(w), O'j(w», Figure IS.D.I(a) depicts the unit isoquant of firm j,
Exercise 15.0.1: Suppose that the Pareto set of the 2 x 2 production model does not coincide with the diagonal of the Edgeworth box. (a) Show that in this case, the factor intensity (the ratio of a firm's use of factor I relative to factor 2) of one of the firms exceeds that of the other at every point along the Pareto set. (b) Show that in this case, any ray from the origin of either of the firms can intersect the Pareto set at most once. Conclude that the factor intensities of the two firms and the supporting relative factor prices change monotonically as we move along the Pareto set from one origin to the other.
{(zlj' Z,j) E R~: Jj(z'j' z,;l = I}, along with the cost-minimizing input combination (o,j(w), o,iw». In Figure IS.D.I(b), we draw a level curve of the unit cost function, {(WI' w,): ciw" w,) = c}. This curve is downward sloping because as w, increases, w, must fall in order to keep the minimized costs of producing one unit of good j unchanged. Moreover, the set {(WI' w,): Cj(w" WI) 2: c} is convex because of the concavity of the cost function Cj(w) in IV. Note that the vector VCj(w), which is normal to the level curve at W = (w" w,), is exactly (o,j(w), O'j(w», As we move along the curve toward higher w, and lower
In Figure 15.0.3, we depict the set of nonnegative output pairs (q"q,) that can be produced using the economy's available factor inputs. This set is known as the production possihilil)' SCI. Output pairs on the frontier of this set arise from factor allocations lying in the Pareto set of Figure IS.D.2(b). (Exercise IS.o.2 asks you to prove that the production possibility set is convex, as shown in Figure 15.0.3.)
w" the ratio O'j(w)/o'j(w) falls. Consider, first, the efficient factor allocations for this model. In Figure IS.D.2, we
g
J
534
CHAPTER
'5:
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15.0:
THE
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535
q, Flgur. 15.0.3 (left)
The production possibility set. Flgur. 15.0.4 (right)
The equilibrium [actor q,
prices and factor
Figure 15.D.5
intensities in an interior equilibrium.
The equilibrium factor allocation.
factor intensity condition, there is at most a single pair of factor prices that can arise liS the equilibrium factor prices of an inlerior equilibrium. t s Once Ihe equilibrium factor prices 11'* are known, Ihe equilibrium output levels can be found graphically by delermining the unique point (z!, z;! in the Edgeworth box of factor allocations at which both firms have the factor inlensities associated wilh faclor prices "", that is,
With the purpose of examining more closely the determinants of the equilibrium factor allocation (z!. z;! and the corresponding equilibrium factor prices 11'* = ("'t. w!). we now assume that the fuctor intensities of the two firms bear a systematic relation to one another. In particular. we assume that in the production of good I. there is. relative to good 2. a greater need for the first factor. In Definition IS.D.I we make precise the meaning of" greater need ". is relatively more intensive in factor 1
Definition 15.0.1: The production of good than is the production of good 2 if
and
a22 (w)
at all factor prices w = (w,. w 2). To determine the equilibrium factor prices. suppose that we have an illterior equilibrium in which the production levels of the two goods are strictly positive (otherwise. we say that the equilibrium is specialized). Given our constant returns assumption, a necessary condition for (IV!. "';! to be the factor prices in an interior equilibrium is that it satisfies the system of equations and
C,(W,. 11',) = p,.
a,,(IV')
The construction is depicted in Figure 15.0.5. An important consequence of this discussion is that in the 2 x 2 production model, if Ihe factor intensity condition holds, then as long as the economy does not specialize in Ihe production of a single good [and Iherefore (15.0.7) holds], the equilibrium factor prices depend only on the technologies of the two firms and on the oil/put prices p. Thus. the levels of the endowments matter only to the extent that Ihey delermine whether the economy specializes. This result is known in the international trade literature as the faclOr price equalizatioll theorem. The theorem provides conditions (which include the presence of tradable consumption goods, idenlical produclion technologies in each country. and price-taking behavior) under which the prices of nontradable factors are equalized across nonspecialized countries.
a,,(w) > a'2(w)
a2 ,(w)
zI, = ~~(IV') z!,
(15.0.7)
That is, at an interior equilibrium, prices must be equal to unit cost. This gives us two equations for the two unknown factor prices "'I and "".'3 Figure 15.0.4 depicts the two unit cost functions in (15.0.7). By expression (15.D.7). a necessary condition for (w,. 'v,) to be the factor prices of an interior equilibrium is that these curves cross at (w" w,). Moreover. the factor intensity assumption implies that whenever the two curves cross. the curve for firm 2 must be Oatter (less negatively sloped) than that for firm I [recall that VCj(w) = (a,j(w), a'j(w))], From this, it follows that the two curves can cross at most once. I. Hence. under the
We now present two comparative statics exercises. We first ask: How does a change in the price of one of the outputs, say Pt. affect the equilibrium factor prices and factor allocations? Figure 15.D.6(a), which depicts the induced change in Figure IS.D.4. identifies the change in factor prices. The increase in p, shifts firm I's curve
15. Note, however. that although ('\',. \\ 0 and (1\""2 < 0, as we wanted. -
du'
PropositIon 15.0.1: (Stolper-Samuelson Theorem) In the 2 x 2 production model with the factor intensity assumption. if Pi increases. then the equilibrium price of the factor more intensively used in the production of good i increases, while the price of the other factor decreases (assuming interior equilibria both before and after the price change).'·
wr
We have just seen that if p, increases, then /"'! increases. Therefore, both firms must move to a less intensive use of factor I. Figure 15.D.6(b) depicts the resulting change in thc equilibrium allocation of factors. As can be seen, the factor allocation moves to a new point in the Pareto set at which the output of good I has risen and that of good 2 has fallen. For the second comparative statics exercise, suppose that the total availability of factor I increases from i , to i'" What is the effect of this on equilibrium factor prices and output levels? Because neither the output prices nor the technologies have changed, the factor input prices remain unaltered -, xr for some i. By (16.C2), we must have P' x, ~ w, for all i, and by (16.CI) P' x, > w, for some i. Hence,
L, P'x, > L,
w, = p'w
+ L p.y;, j
2. The terminology" Xi is maximal for ;::i in set B" means that x, is a prderence-maximizing choice for consumer; in the set B; that is, ."1 E B and i ;2:/ for all E B.
x x;
(16.CI)
That is, anything that is strictly preferred by consumer i to xr must be unaffordable to her. The significance of the local nonsatiation condition for the purpose at hand is that with it (16.CI) implies an additional property:
Lxi = W + L yii
Wi'
x;
Morcover, because yj is profit maximizing for firm j at price vector p,
=
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16.D The Second Fundamental Theorem of Welfare Economics The second fundamental welfare theorem gives conditions under which a Pareto optimum allocation can be supported as a price equilibrium with transfers. It is a converse of the first welfare theorem in the sense that it tells us that, under its assumptions, we can achieve any desired Pareto optimal allocation as a market-based equilibrium using an appropriate lump-sum wealth distribution scheme. The second welf~re theorem is more delicate than the first, and its validity requires additional assumptions. To see this, reconsider some of the examples discussed in Ch-'X,~, then p'x, ~ w:'). '
Figure 16.C.1 A price equilibrium with transrers that is not a Pareto optimum.
we have p'w
+ Lj P'yj
~ p'w
+ Lj P·Yj· Thus,
L P'X
i
> p'w +
L P·Yj·
(16.C.3)
But then (x, y) cannot be feasible. Indeed, Li Xi = W + Lj Yj implies Li P' Xi = P'W + P' J'j' which contradicts (16.C.3). We conclude that the equilibrium allocation (x*, y*) must be Pareto optimal. _
L
The central idea in the proof of Proposition 16.C.1 can be put as follows: At any feasible allocation (x, Y), the total cost of the consumption bundles (x I' ... ,x,), evaluated at prices P, must be equal to the social wealth at those prices, P' W + L} P' Yj' Moreover, because preferences are locally nonsatiated, if (x, Y) Pareto dominates (x*, yO) then the total cost of consumption bundles (Xl' ... ' x,) at prices p, and therefore the social wealth at those prices, must exceed the total cost of the equilibrium consumption allocation p'(L x1j = p'w + Lj p' yj. But by the profitmaximization of Definition 16.B.4, there are no technologically feasible production levels that attain a value of social wealth at prices P in excess of P' W + Lj P' yj. The importance of the nonsatiation assumption for the result can be seen in Figure 16.C.I, which depicts an Edgeworth box where local nonsatiation fails for consumer I (note that consumer I's indifference "curve" is thick) and where the allocation x>, a price equilibrium for the price vector P = (PI' pz) (you should verify this), is not Pareto optimal. Consumer I is indifferent about a move to allocation x, and consumer 2, having strongly monotone preferences, is strictly better otT. (See Exercise 16.C.3 for a first welfare theorem compatible with satiation.) Two points about Proposition 16.C.1 should be noted. First, although the result may appear to follow from very weak hypotheses, our theoretical structure already incorporates two strong assumptions: ulliversal price quolillg of commodities (market completeness) and price taking by economic agents. In Part III, we studied a number of circumstances (externalities, market power, and asymmetric information) in which these conditions are not satisfied and market equilibria fail to be Pareto optimal. Second, the first welfare theorem is entirely silent about the desirability of the equilibrium allocation from a distributional standpoint. In Section 16.0, we study the second fundamental theorem of welfare economics. That result, a partial converse to the first welfare theorem, gives us conditions under which any desired distributional aims can be achieved through the use of competitive (price-taking) markets.
Definition 16.~.1~ Given an economy specified by ({(Xi' ;::i)}!-1' P~-lf-1' W) an allocalion (~ • V ). and a price vector p = (P" ... ,PL) # 0 constitute a price quasieqUlhbnum with transfers if there is an assignment of wealth levels (W,' ... , w,) with LiWi = p·w + LiP'Vi" such that
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(i) For every
i.
AND
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yt maximizes profits in If; that is. P'Yj ~ p·yt for all Yj Elf·
(ii) For every i. if x;>-;xi then p'x; ~ (iii)
BASIC
is,
always make. we must have p' x; =
Xi
2= p'
=~
Y=
THEOREM
OF
WELFARE
x~,
ECONOMICS
553
that
v. = {~x, E RL
X,
E V" ... ,
x, v,} E
Wi
~ lj = {~YjER":)"
E
Y" .. . ,J'JE YJ}'
Thus, V is the set of aggregate consumption bundles that could be split into I individual consumptions, each preferred by its corresponding consumer to x,*. The set l' is simply the aggregate production scI. Note that the set Y + {(v}, which geometrically is the aggregate production set with its origin shifted to (V, is the set of aggregate bundles producible with the given technology and endowments and usable, in principle, for consumption. St('l'l: Ev('ry s('t v; iscOlw('.t. Suppose that x, >-, x: and x; >-, x:' Take 0 $ (% !> I. We want to prove that <Xx; + (I - Cl)X;~, Because preferences are complete, we can assume without loss of generality that Therefore, by convexity of preferences, we have ax, + (I - a),,; ~,x;, which by transitivity yields the desired conclusion: ax, + (I - (X)x; >-, x,* [recall part (iii) of Proposition I.B.I).
x:.
Jor ever)' i. This means that we could just as well not
mention the w,'s explicitly and replace part (ii) of Definition 16.D.1 by
xr then p'
lUNOAMt.hTAl
and
j
Note also that when consumers' preferences are locally nonsatiated, part (ii) of Definition 16.D.1 implies p'x! ~ w, for every i.l In addition, from part (iii), we get L,P'x; = P'w + Lj r' y; = L, II',. Therefore, Imder Ihe assumption of focally /J(J/Isaliat.d preferences, which we
If ."(/ >i
,;,fCONO
We begin by defining, for every i, the set V. of consumptions prererred to = {x, E X,: x, >-, x~} c RL. Then define V
Part (ii) of Definition 16.0.1 is implied by the preference maximization condition of the definition of a price equilibrium with transfers [part (ii) of Definition 16.B.4]: If xi is prererence maximizing in the set {Xi E X,: P' X, ~ w,j, then no x, >-, xi with p' Xi < Wi can exist. Hence, any price equilibrium with transfers is a price quasiequilibrium with transfers. However, as we discuss later in this section, the converse is not true.
(ii')
THL
V.
Wi'
'LA = w + L yr ;
16.0:
x;.
That is, allocation (x·, y.) and price vector p constitute a price quasiequilibrium with transfers if and only if conditions (i), (ii'), and (iii) hold' Moreover, with locally nonsatiated is expenditure minimizing on the set preferences, condition (ii') is equivalent to saying that {x, E X: x, ;::,xn (see Exercise 16.D.I). Thus, our discussion later in Ihis section of the conditions under which a price quasiequilibrium with transfers is a price equilibrium with transfers can be interpreted in the locally non satiated case as providing conditions under which expenditure minimization on the set {x, e X,: x,;::, xi I implies preference maximization on the set {x,eX,: P'x,~p'xn = {x,eX,: P'x,!> w,l . •
x,;::, x;.
SI~p 2: Tire sels Valid Y + {w} are cO/wex. This is just a general, and easyto-prove, mathematical fact: The sum of any two (and therefore any number of) convex sets is convex.
xr
Slep 3: V f"\ (Y + {w}) = 0. This is a consequence of the Pareto optimality of (x·, y.). If there were a vector both in Vand in Y + {w}, then this would mean that with the given endowments and technologies it would be possible to produce an aggregate vector that could be used to give every consumer i a consumption bundle that is preferred to
x:.
Proposition 16.0.1 states a version of the second fundamental welfare theorem.
St~p
Tlrer~ is P = (PI' ... ' 1',) '" 0 alld a lIumber r Stlclr Ilral p' z ;>: r for every p'Z S r for every Z E Y + {w}. This follows directly from the separating hyperplane theorem (see Section M. G. the Mathematical Appendix). It is illustrated in Figure 16.0.1.
Proposition 16.0.1: (Second Fundamental Theorem of Welfare Economics) Consider an economy specified by ({(X;, ~;)li-" {Y;}t-" iii). and suppose that every Y; is convex and every preference relation ~; is convex [i.e., the set {X; E X;: X; ~j Xj} is convex for every x; E X;] and locally nonsatiated. Then, for every Pareto optimal allocation (x*, y*). there is a price vector p = (p" ... ,pd '" 0 such that (x·, y*. p) is a price quasiequilibrium with transfers.
4:
=E V IIl1d
Figure 16.0.1
The separation argument in the proof of the second welfare
Proof: In its essence, the proof is just an application of the separating hyperplane theorem for convex sets (see Section M.G. of the Mathematical Appendix). To facilitate comprehension, we organize the proof into a number of small steps.
theorem.
3. To see this, observe that if preferences are locally nonsalialed and p' x7 < Wi) then close to Xi with Xi >-ix7 and p' Xi < Wj. contradicting condition (ii) of Definition 16.D.1. 4. A similar observation applies, incidentally, 10 the definition of price equilibrium with transfers
x7 there is an
(Definition 16.B.4). If preferences are locally nonsatiated, we get an equivalent definition by not referring explicitly to the w:s and replacing part (ii) of the definition by (ii"): If x, >-, x~ then P'x j > P'x7- Thus. in this locally nonsatiated case, condition (ii") says that x~ is preference maximizing on {XI E Xi: p·.'t l S p' x7}.
x,
=
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Budget
Seep 5: If Xi ?::" Xi for every j then p' (L, x,) ;:: r. Suppose that x, :::, xi for every
A price quasi-equilibrium that
x,
is not a price equilibrium.
Step 6: p' (Li xi) = P' (w + Lj yj) = r. Because of step 5, we have P' O::i xi) ~ r. On the other hand, Li xi = Lj yj + W E Y + (wi, and therefore P'(Li xi) s: r. Thus, p' (Li = r. Since Li xi = w + Lj yj, we also have P' (w + Lj yj) = r.
}-IX{. Beca use of steps 5 and 6, we ha ve P{Xi +
'~i xi) ~
I'
= p.(X!
+
'~I xi).
Proof: The idea of the proof is indicated in Figure 16.0.3 (where we take p'xi = Wi only because this is the leading case; the fact plays no role in the proof). Suppose that, contrary to the assertion of the proposition, there is an Xi >- I xi with p' Xi = Wi' By the cheaper consumption assumption, there exists an x; E XI such that P' x; < lVi. Then for all IX E [0, I), we have rzx i + (I - rz)x; E Xi and P·(tXXl + (1 - IX)X;) < WI" But if rt is close enough to I, the continuity of ?::'i implies that lXX, + (I - IX)X; >- i xi, which constitutes a contradiction because we have then found a consumption bundle that is preferred to xi and costs less than lVi. • Note that in the example of Figure 16.0.2, we have WI = 0 in the price quasicquilibrium supporting allocation x·, and so there is no cheaper consumption for (p, w.)." As a consequence of Proposition 16.0.2, we have Proposition 16.0.3.
Step 9: The lI'ealth levels IVI = p'xi for i = I, ... , I support (x·, y., p) as a price 411asic(luilihrium lVith transfers. Conditions (i) and (ii) of Definition 16.0.1 follow from steps 7 and 8; condition (iii) follows from the feasibility of the Pareto optimal allocation (x·, y.) . • In Exercise 16.0.2, you are asked to show that the local nonsatiation condition is required in Proposition 16.0.1. When will a price quasiequilibrium with transfers be a price equilibrium with transfers? The example in Figure 15.B.IO(a), reproduced in Figure 16.0.2, indicates that there is indeed a problem. Figure 16.0.2 depicts the quasiequilibrium associated with the Pareto optimal allocation labeled The unique price vector (normalizing PI = I) that supports x· as a quasiequilibrium allocation is p = (1,0); the associated wealth levels are WI = p·xt = (I,O)'(O,x!,) = 0 and W2 = P·x!. However, although the consumption bundle xt satisfies part (ii) of Definition 16.0.1 (indeed, p' x I 2: 0 = WI for any XI ~ 0), it is not consumer I's preference-maximizing bundle in her budget set {(x",x21)ER~:(1,0)'(x",x21)S:0} = {(XII'X21)ER~:
x·.
Xli
=
555
Figure 16.0.2 (lett)
Line
i. By local nonsatiation, for each consumer j there is a consumption bundle Xi arbitrarily close to Xi such that >-, Xi' and therefore Xi E V;. Hence, Li Xi E V, and so p'(L :li) 2: r, which, taking the limit as Xi -+ Xi' gives P'(L, Xi) ~ r. s
ECONOMICS
Proposition 16.0.3: Suppose that for every i, Xi is convex, 0 E Xi' and :::i is continuous. Then any price quasiequilibrium with transfers that has (w" ... , wd »0 is a price equilibrium with transfers.
O}.
An important feature of the example just discussed, however, is that consumer I's wealth level at the quasiequilibrium is zero. As we shall see, this is key to the failure of the quasiequilibrium to be an equilibrium. Our next result provides a sufficient condition under which the condition MXi>-IXi implies P'X i ~ lV i" is equivalent to the preference maximization condition" Xi >-i xi implies p' Xi > Wi'"
6. If, as in all our applications. ~j is locally nonsatiated and Wj = p·xi. then Proposition 16.0.2 olTers 5ufHcienl conditions for the eq uivalence of the statements" x7 minimizes expenditure relative to p in the s~l {''(j E Xj: X,~, x;}" and" xi is maximal for 2:, in the budget set lXiE X,: P·X . ~ p·xj}." 7. A similar argument can be used to show that if X, is convex and the Walrasian demand function xj(p, \\'j} is well defined, then there is a cheaper consumption for (P. w,) if and only if there is an x; arbitrarily close to xl(p. Wi) with p' x; < lVi . In the Appendix A, of Chapter 3 the latter concept was called the locall.v cheaper consumption condition. 8. Note also that Proposition 16.0.2 generalizes the result in Proposition 3.E.t(ii), which assumed local nonsatiation. Wi = p'x7 > O. and Xj = R';.
5. Geometrically. what we have done here is show that the set 1:1 {Xi E Xj: Xi;::j x7} is contained in the closure of V (see Section M.F of the Mathernalieal Appendix for this concepl), which, in turn. is contained in the half-space (v E ilL: p' v ~ r).
m
Ftgure 16.0.3 (right)
Suppose there exists a "cheaper consumption" (an x; E Xi such that p'x; < WI)' Then if the preferred set does intersect the budget set (p' XI :S WI for some xi>jx:), it follows that the preferred set does intersect the interior of the budget set (p·x; < \Vi for some
Xi >-IXn.
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Consider the implications of Proposition 16.0.3 for a pure exchange economy in which W »0 and every consumer has X, = R~ and continuous, locally nonsatiated preferences. In such an economy, by free disposal and profit maximization, we must have p ~ 0 and P ~ 0 at any price quasiequilibrium 9 Thus, under these assumptions, any price quasiequilibrium with transfers in which x~ »0 for all i is a price equilibrium with transfers (since then = p' x,' > 0 for all iJ. But there is more. Suppose that, in addition, preferences are strongly monotone. Then we must have p » 0 in any price quasiequilibrium with transfers. To see this, note that p ~ 0, p ~ 0, and w » 0 imply that L, w, = p' 'v > 0 and therefore that w, > 0 for some i. But by Proposition 16.0.2, this consumer must then be maximizing her preferences in her budget set {x, E R~: P' x, :0;; w,}, which, by strong monotonicity of preferences, cannot occur if prices arc not strictly positive. Once we know that we must have p »0, we can conclude that u,(x i ) for some i. But (u; •. . "ui) E U only if there is a feasible allocation (x', y') such that ui(xi) ;;:: ui for all i. It follows then that (x'. y) Pareto dominates (x. y). Conversely. if (x. y) is not a Pareto optimum. then it is Pareto dominated by some feasible (x', y'). which means that Ui{Xi) ~ u,(x i ) for all i and ui(xi) > "i(Xi) for some i. Hence. (u,(x,), ...• u,(x,» f UP . • We also note that if every Xi and every lj is convex. and if the utility functions IIi ( .) arc concave, then the utility possibility set U is convex (see Exercise 16.E.2).1l One such utility possibility set is represented in Figure 16,E.2. Suppose now that society's distributional principles can be summarized in a social welfare /ullctioll W(u" ... , u,) assigning social utility values to the various possible vectors of utilities for the J consumers. We concentrate here on a particularly simple class of social welfare functions: those that take the linear form W(u, •...• u,) =
such that
L i.;Ui
ui(x,) for i = I•... , J}. 12. However, nol every poinl in the boundary must be Pareto optimal. Go back, for example, to Figure 16.C.1: The ulility values associated with x· belong to the boundary of the utility possibility set because it is impossible to make both consumers better off. Yet. x· is not a Pareto optimum. 13. II can be shown that under a mild technical strengthening of the strict convexity assumption on preferences (essentially the same condition used to guarantee differentiability of the Walrasian demand function in Appendix A of Chapter 3), there are in the family of utility functions ",(.) that represent ;::j some utility functions that are not only quasiconcave but also concave.
II. Two faclS eSiablished in Chapter 17 lend plausibility to this claim. First, in Section 17.1, WI! show that convexity is not required for the (approximate) existence or a Walrasian equilibrium In a l'lrgc economy. Second. in Section I7.C, we argue that the second welfare theorem can be rcphnlscd as an assertion of the existence or a Walrasian eqUilibrium ror economies in which endowments are distributed in a particular manner, and it can therefore be seen as implied by the conditions guaranteeing the general existence of Walrasian equilibria.
I
Figure 16.E.2 (right)
A convex utility possibilily sel.
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",
1 I. F!
FIR S T - 0 R DE R
CON 0 I T ION S
FOR
P A A E TOO P TIM A LIT y
561
x,
Figure 16.E.4 Ftgur. 16.E.3
Maximizing the utili! y or a representative consumer.
Maximizing a linear social welfare function.
for some constants i. = p., ..... i.,).'4 Lelling 1/ = (u, •...• U,). we can also write W(I/) = i. '1/. Because social welfare should be nondecreasing in the consumer's utility levels. we assume that i. :2: O. Armed with a linear social welfare function, we can select points in the utility possibility set U that maximize our measure of social welfare by solving Max w.u
i.·I/.
(16.E.I)
Figure 16.E.3 depicts the solution to problem (16. E.I). As the figure suggests. we ha ve the result presented in Proposition 16.E.2. Proposition 16.E.2: If u· = (u1, . ..• u7) is a solution to the social welfare maximization problem (16.E.1) with i.» O. then u· E UP; that is, u· is the utility vector of a Pareto optimal allocation. Moreover. if the utility possibility set U is convex. then for any ii = (ii, • ...• ii,) E UP, there is a vector of welfare weights i. = (i." ... , i.,):2: O. i. #< 0, such that i.·ii:2: ).·u for all UE U. that is. such that ii is a solution to the social welfare maximization problem (16.E.1). Proof: The first part is immediate: if u· were not Pareto optimal, then there would exist a /I E U with u:2: u' and u #< u*; and so because ).» 0, we would have i.'u>i.'fl*, For the second part, note that if ii E UP. then ii is in the boundary of U. By the supporting hyperplane theorem (see Section M.G of the Mathematical Appendix), there exists a i. #< 0 such that i.. ii :2: i.' u for all u E U. Moreover, since the set U has been constructed so that U - R'+ c U, we must have ). :2: 0 (indeed, if )" < O. then by choosing a u E U with u, < 0 large enough in absolute value, we would have i.·/1 > i.' ,i). • Proposition 16.E.2 tells us that for economies with convex utility possibility sets, there is a close relation between Pareto optima and linear social welfare optima: Every linear social welfare optimum with weights i. » 0 is Pareto optimal, and every Pareto optimal allocation (and hence. every Walrasian equilibrium) is a social welfare optimum for some welfare weights V." ... , i.,) :2: 0.15 14. See Chapter 22 for a discussion of more general types of social welfare functions. 15. The necessity of allowing for some i' l to equal zero in the second part of this statement parallels the similar feature encountered in the characterization of efficient production vectors in
Proposition 5.F.2.
As usual. in the absence of convexity of the set U. we cannot be assured that a Pareto optimum can be supported as a maximum of a linear social welfare function. The point ,i in Figure 16.E.1 provides an example where it cannot. fly using the social welfare weights associated with a particular Pareto optimal allocation (perhaps a Walrasian equilibrium), we can view the latter as the welrare optimum in a certain single-consumer. single-firm economy. To sec this, let (x·, y.) be a P.ucto optimal allocation and suppose that i. = 0.1 .... ' i.,)>> 0 is a vector of welfare weights supporting U at (ul(xn .... ,II,(xi)). Define then a utility runction u).(.X) on aggregate consumption vectors in
X=
L,
X;
C
R " by u,(.> 0 for all YjE !;IL. The meaning of the last condition is that if fit)~ = O. so that Yj is in the transformation frontier of Y;, then any attempt to produce more of some output or use less of some input makes the value of Fj (') positive and pushes us out of Y; (in other words, YJ is production ellicient. in the sense discussed in Section 5.F. in the production set y;).17 Note that. for the moment, no convexity assumptions have been made on preferences or production sets. The problem of identifying the Pareto optimal allocations for this economy can be reduced to the selection of allocations
Max
".F:
Xli:
iJu 0, --- 1', { i
OXfi
Problem (16.F.I) states the Pareto optimality problem as one of trying to maximize the well-being of consumer I subject to meeting certain required utility levels for the other consumers in the economy [constraints (I)] and the resource and technological limitations on what is feasible [constraints (2) and (3). respectively]. By solving problem (16.F.I) for varying required levels of utility for these other consumers ('1, •...• iii)' we can identify all the Pareto optimal allocations for this economy. Indeed. you should pause to convince yourself of this by solving Exercise 16.F.1.
oF.
:0;
0
= 0
Pt - Yj ---L = 0 oYtj
.
If Xli > 0
for all i,
t,
(16.F.2)
for all j,
t.
(16.F.3)
As is well known from Kuhn-Tucker theory (see Section M.K of the Mathematical Appendix), the value of the multiplier Pt at an optimal solution is exactly equal to the increase in consumer J's utility derived from a relaxation of the corresponding constraint, that is. from a marginal increase in the available social endowment wt of good t. Thus. the multiplier Pt can be interpreted as the marginal value or "shadow price" (in terms of consumer I's utility) of good t. The multiplier Ii" on the other hand, equals the marginal change in consumer J's utility if we decrease the utility requirement Ui that must be met for consumer i"f. I. Condition (16.F.2) therefore says that, at an optimal interior allocation, the increase in the utility of any consumer i from receiving an additional unit of good t. weighted (if i "f. I) by the amount that relaxing consumer i's utility constraint is worth in terms of raising consumer I's utility, should be equal to the marginal value PI of good t. Similarly, the multiplier Yj can be interpreted as the marginal benefit from relaxing the jth production constraint or. equivalently, the marginal cost from tightening it.
Exercise 16.F_I: Show that any allocation that is a solution to problem (16.F.I) is Pareto optimal and that any Pareto optimal allocation for this economy must be a solution to problem (16.F.I) for some choice of utility levels (ii, •...• iii)' [Him: Use the fact that preferences are strongly monotone.] Because utility functions are normalized to take nonnegative values. from now on we consider only required utility levels that satisfy U, ~ 0 for all i. The point of Exercise 16.F.I can be seen by examining the utility possibility set U in Figure 16.F.1. If we fix a required nonnegative utility level for consumer 2. we can locate a point on the frontier of the utility possibility set U by maximizing
18. Recall that for expositional ease we are not imposing any boundary constraints on the vectors h We note also that the assumption of strictly positive gradients of the functions uc( .} and fj( .} implies that the constraint qualification for the necessity of the Kuhn-Tucker conditions is satisfied. (See Section M.K of the Mathematical Appendix for the specifics of first-order conditions for optimization problems under constraints.)
17. For expositional convenience. we have taken every FJ(') to be defined on the entire R'·. A consequence of this (and the assumption that VF;(y/)» 0 for all YJ} is that every commodity is both an input and an output of the production process. Because this is unrealistic, we emphasize that no more than expositional ease is involved here.
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Hence. yPJFj/vYt;) is the marginal cost of increasing Ytl and thereby effectively tightening the constraint on the net outputs of the other goods. Condition (16.F.3) says. then. that at an optimum this marginal cost is equated. for every j. to the marginal benefit Jlt of good t. If we suppose that we have an interior solution (i.e .• x,» 0 for all i). then conditions (16.F.2) and (16.F.3) imply that three types of ratio conditions must hold (see Exercise 16.F.3):
(Ju;/Dx" iJu;/iJx n
for all i. i'. f. I'.
(16.FA)
t.1'.
(16.F.5)
for all i.j.t.I'.
(16.F.6)
for allj./.
cJFj"/iJY{"j"
iJll;/(:X~, =J!I/~Ytj
cJU;/(lx{",
vFI/vYf"1
Condition (16.F.4) says that in any Pareto optimal allocation. all consumers marginal rates of substitution between every pair of goods must be equalized [sec Figures 15.B.II(b) and 15.B.12 for an illustration in the two-good. two-consumer case]; condition (16.F.5) says that all firms' marginal rates of transformation between every pair of goods must be equalized [see Figure IS.D.2(b) for an illustration in the two-good. two-firm case]; and condition (16.0.6) says that every consumer's marginal rate of substitution must equal every firm's marginal rate of transformation for all pairs of goods [see Figure 15.C.2 for an illustration in the case of the one-consumer. one-firm model with two goods]. Conditions (l6.F.4) to (16.F.6) correspond to three types of efficiency embodied in a Pareto optimal allocation (see Exercise 16.F.4).
(16.F.7) •.•
XLi) ;;,
(2) L,xl/:!;;x,
u,
Fltri
PARETO
(16.F.9)
s.t. }', 5 fCY, •...• y,).
To explore the relationship of the first-order conditions (16.F.2) and (16.F.3) to the first and second welfare theorems. we make the further. and substantive. assumption that every 11,(') is a quasiconcave function (hence. preferences are convex) and that every fj(') is a Convex function (hence. production sets are convex). The virtue of this assumption is that with it we do not have to worry about second-order conditions; in all thc maximization problems to be considered. the first-order nceessary conditions arc automatically sufficient. In this differentiable. convex framework. conditions (16.F.2) and (16.FJ) can be uscd to establish a version of the two welfare theorems. To see this. note first that (x*, .1'*, p) is a price equilibrium with transfers (with associated wealth levels 1\', = p' for i = I •...• I) if and only if the first-order conditions for the budgetconstrained utility maximization problems
xr
and the profit maximization problems Max
(=
s.t. Fj(Yj)
#~"
I, ...• L.
,"Xli
(16.F.8) (n ..... .,))
{=
2•...• L
j = I •... • J.
The first-order conditions for this problem lead to condition (16.F.5).
P'Yj :!;;
0
are satisficd. Denoting by 7, and PI the respective multipliers for the constraints of these problems. the first-order conditions [evaluated at (x'. Y')] can be written as follows:
i = 2, ...• 1
(ii) Elficienr productioll across tec/tllologies. The aggregate production vector should be elficienr in the sense discussed in Section 5.F. That is, it should be impossible to reassign production plans across individual production sets so as to produce, in the aggregate, more of a particular output (or use less of it as an input) without producing less of another. Focusing. in particular. on the first good. this means that given required total productions (Yl' ... ,yc! of the other goods. we want to solve
(2) Fj(y):!;; 0
u«O, + y, ..... '0,. + h)
Max
_~,Pt
The first-order conditions for this problem lead to condition (I6.F.4).
s.t. (I) Lj Y'j;;' }',
CONDITIONS
(iii) Optimal agf/regllle production levels. We also must have picked aggregate production levels that generate a desirable assortment of commodities available for consumption. Keeping the utility requirements (", •...• ti,) fixed. let u(.i, •...• xc! and J(Y, .. ' .. h) denote. respectively. the value functions for problems (I6.F.7) and (16.F.8). Then we want to solve
(i) Oprimal al/oeation oj available goods aeros., consumers. Given some aggregate amounts (.i" ...• xc> of goods available for consumption purposes, we want to distribute them to maximize consumer I's well-being while meeting the utility requirements (u ...• ",) for " consumers 2•... , I. That is. we want to solve
s.t. (I) u,(x Ii"
fIRST·ORDER
16.f:
The first-order conditions of this problem lead to condition (16.F.6).
vu,./cJx". ou,.fJx{",.
~!"I/VYt j = J!j"~~Y{j: cJFj/ 0
: 0 is feasible if
Example 16.G.2: Occupational Choice Suppose that every individual could, in principle, work either as a classics scholar or as an economics professor. But not all individuals are equally good at both things. A way to capture the different comparative advantage is to assume that for every individual i, there is an Il i ;:>: 0 measuring how many "effective hours of economics professorial services" it takes to produce "an effective hour of classical scholarship." A relatively low Ili indicates comparative advantage in classical scholarship. Suppose also that every individual i has an amount of professorial hours that she can supply; we assume that I professorial hour can produce I effective hour of economics professorial services or 1/':1. i effective hours of classical scholarship by individual i. There is a single consumption good on which the individual i can spend her earnings. It is important to be able to imbed this problem in our formal structure because we certainly want to be able to analyze how, for example, competitive labor markets will perform when individuals have occupational choices as well as choices about how mllcil labor to supply. This is how it can be done (it is not the only possible way): suppose we list consumption and effective hours supplied as a three-dimensional vector (c i , t d , t'i)' where CI is individual j's consumption and t" ~ 0 and t" ~ 0 are the effective hours spent working as a classics scholar and as an economics professor, respectively. Because the latter two quantities are supplies-that is, services offered by the individual to the market-we follow the convention of measuring them as negative numbers. We can then define the consumption set of individual j as XI = (c" t", 1.1 ):
II.G:
q ~ f(:),
,
LXI'
+:
=
w.,
and
q=
x,.
«x;., ...
z'»
It is Pareto optimal if there is no other feasible allocation ,. O.
(16.G.l)
(ii) For any i, xi is maximal for I '-'
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Koopmans. T. (1957). Three Essays on the Stale of Economic Science. New York: McGraw-Hili. Lange, O. (1942). The foundation or welfare economics. Econometrica 10: 21S-228. Quinzii, M. (1992). Increasing Returns and Efficiency. New York: Oltrord University Press. Samuelson. P. (1947). FOUildarions of Economic AnalysiS. Cambridge. Mass.: Harvard University Press.
/(Y+ {w}) n R~
EXERCISES Figure 16.AA.l The set or reasible
16.C.t" Show that ir a consumption set Xi c RL is nonempty. closed. and bounded and the prererence relation 0 for some i and then argue that p » 0.]
REFERENCES
16.0.4C Consider a two·good exchange economy with r identical consumers. The consumption set is R!. the individual endowments are co e R! •• and the prererences are continuous and strongly mono lone but not necessarily convex. Argue that the symmetric allocation in which
Albis. M. (1953). Traile d'ecOIlOmie pure. Paris: Publications du CNRS. Arrow, K., and F. Hahn. (1971). Genrral Competitive Analysis. San Francisco: Holdcn·Day. Dcbrcu. G. (1959). Tllt'orr IIf Value, New York: Wiley.
L
>-, xr then
xr" is equivalent to the condition: "x~ is expenditure minimizing for the price vector Xj: xr }." p in the set
p' Xi ~ P'
Proof: Note that U = U' - IR~ where E
xr
p'X; ~ Wi'"
The case with several production sets is more delicate. and it is here that the irreversibility assumption comes to the rescue. Very informally. we can dcrive. as in the preceding paragraph. the boundedness of feasible aggregate productions and feasible individual consumptions. Now. the only way that unboundedness would be possible at the individual production level while remaining bounded in the aggregate is if. so to speak. the unboundedness in one individual production plan was to be canceled by the unboundedness of another. However, this would imply that the collection of all technologies in the economy (i.e.• the aggregate production set) allows the reversal of some technologies (see Exercise 16.AA.3 for more details). Incidentally, it can also be shown that irreversibility, with the other assumptions. yields the closed ness of Y, so we do not actually need to assume this separately. _
U' = {(u,(x,), ... , U/(X/)): (x. Y)
xr
16.C.2' Suppose that the prererence relation 0 for some i and then apply Proposition 16.D.2.]
Now suppose there inc two consumers and that their preferences are identical to those above. One owns all of the land and the other owns all of the labor. In this society, arbitrary lump-sum taxes arc not possible. It is the law that any deficit incurred by a public enterprise must he covered by a tax on the value of land. (d) In appropriate notation, write the transfer from the landowner as a function of the government's planned production of education.
(0) Find a marginal cost price equilibrium for this economy where transfers have to be compatible with the transfer function specified in (d). Is it Pareto optimal?
16.E.2K Show that the utility possibility set U of an economy with convex production and consumption sets and with concave utility functions is convex.
16.AA.1 A Show that if every Xi and every lj is closed, then the set A of feasible allocations is closed.
16.F.l" In text. 16.F.2A Derive the first-order conditions (16.F.2) and (I6.FJ) of the maximization problem (16.F.I).
16.AA.2K Show that( Y + (w}) ('\ R~ is compact if the following four assumptions are satisfied: (i) Y is closed, (ii) l' is convex, (iii) 0 E Y, and (iv) if v E Y ('\ R~ then v = O. Exhibit graphically four examples showing that each of the four assumptions is indispensable.
16.F.3 A Derive conditions (16.F.4), (16.F.5), and (I6.F.6) from the first-order conditions (16.F.2) and (16.F.3).
16.AA.3" Suppose that Y = Y, + Y, c R'; satisfies the assumptions given in Exercise 16.AA.2 and that 0 E 1'" 0 E l',. Argue that if the irreversibility assumption holds for Y then (.I', E Y,: y, + y, + w ~ 0 for some y, E Y,} is bounded.
16.F.4 A Derive the first-order conditions (16.F.4), (16.F.5), and (16.F.6) from problems (16.F.7), (16.F.8), and (16.F.9), respectively. 16.G.1A Prove Proposition 16.G.1 using the first-order conditions (16.F.2) and (16.F.3). I6.G.2A In text. 16.G.3" Exhibit graphically a one-consumer, one-firm economy with two inputs and one output where at the (unique) marginal cost price equilibrium, cost is 1101 minimized. [/lillt: Choose the production function to violate quasi concavity.] 16.G.4" Show that under the general conditions of Section 16.G if there is a single consumer (perhaps a normative representative consumer) with convex preferences, then there exists at least one marginal cost price equilibrium that is an optimum. 16.G.S" In a certain economy there are two commodities. education (e) and food (f), produced by using labor (L) and land (D according to the production functions e = (Min (L, T})'
and
j=(LD'/'
.....
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the three sections is the role of two sufficient conditions: the weak axiom of revealed preferellce ill Ille aggregale (a way of saying that wealth effects do not cancel in the aggregate the positive influence of the substitution effects), and the property of gruss slIhslilllliulI (a way of saying that there are not strong complementarities among the goods in the economy). In Section 17.1, we return to the role of convexity in guaranteeing the existence of Walrasian equilibrium. We qualify this role by showing that nonconvexities that are "small" relative to the aggregate economy (e.g., the indivisibility represented by a car) are not an obstacle to the (near) existence of equilibria, even if they are "large" from the standpoint of an individual agent. This chapter is of interest from both methodological and substantive points of view. From a substantive standpoint, it deals with an important theory: that of Walrasian equilibrium. Methodologically, the qucstions that we ask (e.g., does an equilibrium exis!"! Are the equilibria typically isolated? Is the equilibrium unique? Is it stable? What arc the effects of shocks') and the techniques that we use are questions and techniques that arc of relevance to any theory of equilibrium.
of Equilibrium
17.A Introduction III this chapter. we study the theoretical predictive power of the Walrasian equilibrium model. Thus. in contrast with Chapter 16. our outlook here is positive rather than
17.B Equilibrium: Definitions and Basic Equations
Ilormative.
The concept of a private ownership economy was described in Section 16.B. In such an economy. there are I consumers and J firms. Every consumer i is specified by a consumption set Xi c R/.• a preference relation ,.illlll presented in Section 16.B. We then introduce the notion of an ayyregllle ncess 0 such that Z/(P) > -5 for every commodity If pn _ p, where p # 0 and PI = 0 for some t, then
Max {z,(pn) .... , zdpn)} _
DEFINITIONS
AND
BASIC
t and all
defined for some p» [because we may have 11j(p) = 00 for some j). Nevertheless. an equilibrium price vector is still characterized by i(p) = 0. When production sets are not strictly convex, matters become more complicated because the correspondences Yj(p) may no longer be single-valued. Indeed, a production situation of considerable theoretical and practical importance-and one lhat we certainly do not want to rule out by assumption-is the case of constant returns to scale. With constant returns, however, production sets are neither strictly convex nor bounded above (except for the trivial case in which no positive amount of any good can be produced). In principle, we could still view the equilibria as the zeros of a "production inclusive excess demand correspolldellce," defined as in (17.B.3) for a subset of strictly positive prices.· Correspondences, however, do not make good equational systems (e.g., they cannot be differentiated). It is therefore usually much more convenient in such cases to capture the equilibria as the solutions of an extended system of equations involving the production and the consumption sides of the economy. We illustrate this idea in the small type discussion that follows.
p.
00.
Proof: With the exception of property (v), all these properties are direct conseq uenees of the definition and the parallel properties of demand functions.' The bound in (iv) follows from the nonnegativity of demand (i.e., the fact that X, = R~), which implies that a consumer's total net supply to the market of any good t can be no greater than his initial endowment. You are asked to prove property (v) in Exercise 17.B.2. The intuition for it is this: As some prices go to zero, a consumer whose wealth tends to a strictly positive limit [note that, because P-(L, w.l > 0, there must be at least one such consumer] and with strongly monotone preferences will demand an increasingly large amount of some of the commodities whose prices go to zero (but perhaps not of all such commodities: relative prices still matter). _
To sc.::c how an extended system of equations can be constructed, consider the case in which
production is of the linear activity type (this case is reviewed in Appendix A of Chapter 5). Say that, in addition to the dispos,,1 technologies, we have J basic activities ii" ••. , ilJ E R /·.
°
Finally, note that because of Walras' law, to verify that a price vector I' » clears all markets [i.e., has :/(1') = for all t] it suflices to check that it clears all markecs hll/ 0" 0 for all f}.
We shall proceed in five steps. In the first two, we construct a certain correspondence /(.) from 6 to 6. In the third, we argue that any fixed point of /(.), that is, any 1'* with 1'* E /(1'*), has z(p*) = O. The fourth step proves that /(.) is convex valued and upper hemicontinuous (or, equivalently, that it has a closed graph). Finally, the fifth step applies Kakutani's fixed-point theorem to show that a 1'* with 1'* E f( 1'*) necessarily exists. For notational clarity, in defining the sct f(p) c 6, we denote the vectors that are clements of /(1') by the symbol q.
~
:(p).q'
for all q'E6}.
where s is the bound in excess supply given by condition (iv).'o In summary, for p' c10sc cnough to Boundary 6, thc maximal demand corresponds to some of the commoditics whose price is close to zero. Therefore, we conclude that, for large n, any q' E /(p') will put nonzero weight only on commodities whose prices approach zero. But this guarantees p'q = 0, and so q E f( ,,).
= (q E 6: q, = 0 if z,(p) < Max {z,(,,), ... , zdp)}}.
5/ OJ.
Because 1'1 = 0 for some f, we have f(p) '" 0. Note also that with this construction, no price from Boundary 6 can be a fixed point; that is, I' E Boundary 6 and p E f(p) cannot occur because P'P > 0 while P'q = 0 for all q E f(p).
9. Noto also Ihat ror 'my p E /'J., the set I( p) is always a race or the simplex /'J.; that is, it is One or the subsets or /'J. spanned by a finite subset or unit coordinates. For P E Boundary /'J., I(p) is the race or /'J. spanned by Ihe zero coordinates or p. For p E Interior /'J., I(p) is the race spanned by the coordinates corresponding to commodities with maximal excess demand. 10. In words. the last chain of inequalities says that the expenditure on commodity {is bounded because it has to be financed by. and thererore cannot be larger than, the bounded value of excess supplies.
51 o. Proof: Because of homogeneity of degree zero we can restrict our search for an equilibrium to the unit simplex 11 = {p E IR\: 2:1 PI = I}. Define on 11 the function z+(·) by zt(p) = Max {ZI(p),O}. Note that z+(·) is continuous and that z+(p)'z(p) = 0 implies z(p):;; O. Denote alp) = 2:1 [PI + z;(p)]. We have alp) ~ I for all p. Define a continuous function f(·) from the closed, convex set 11 into itself by
f(p) = [l/a(p)](p
E
+ z+(p)). The second welfare theorem of Section 16.0 can be seen as a particular case of the current existence result. To see this, suppose that ., = (x" ... , x,,) is a Pareto optimal allocation of a pure exchange economy satisfying the assumptions leading to Proposition 17.C.1. Then, by Proposition 17.C.t, a Walrasian equilibrium price vector" and allocation .i = (. 0 such that if p' "" P. pi = PL = 1. and lip' - pil < c, then z(p') "" O. Moreover, if the economy is regular. then the number of normalized equilibrium price vectors is finite.
15. For any number" '" 0, sign" =
... v .... A,
5 I:. C T
16. This result was first shown by Dierker (1972). 17. For advanced treatments on the topic of this section, refer to Balasko (1988) or Mas-Colell (InS).
-I according to whether" > 0 or " < O.
1.
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f(·; q) if q' is close to q. Hence, the notion that the regularity of a system f(·; q) = 0 is typical, or generic, could be captured by demanding that for almost every q, f(·; q) = 0 be regular; in other words, that nonregular systems have probability zero of occurring (with respect to say, a nondegenerate normal distribution on RS).'9 It stands to reason that some condition will be required on the dependence of f(·; q) on q for this to hold. At the very least, f(·; q) has to actually depend on q. The important mathematical theorem to be presented next tells us that little beyond this is needed. 20
f'(u): Perlurbed S~stem
I(d
17.D:
Figure 17.0.4 The regular case is typical.
Proposition 17.0.3: (The Transversa/iCy Theorem) If the M x (N + S) matrix Of(v; q) has rank M whenever f(v; q) = 0 then for almost every q, the M x N matrix OJ(v; q) has rank M whenever f(v; q) = O.
description of the solution set. In particular, if M > N, the system should be overdetermilled and have no solution; if M = N, the system should be exactly determined with the solutions locally isolated; and if M < N, the system should be underdetermilled and the solutions not locally isolated. Clearly, all these statements are not always true (you can see this just by considering examples with linear equations). So, what does it mean to be in the "normal case"'? The implicit function theorem provides an answer: one needs the equations (which we assume are differentiable) to be independent (that is, truly distinct) at the solutions. Definition 17.0.3 captures this notion.
Heuristically, the assumption of the transversality theorem requires that there be enough variation in our universe. If Df(v; q) has rank M whenever f(v; q) = 0, then from any solution it is always possible to (differentially) alter the values of the function f in any prescribed direction by adjusting the v and q variables. The conclusion of the theorem is that, if this can always be done, then whenever we are initially at a nonregular situation an arbitrary random displacement in q breaks us away from nonregularity. In fanciful language, if our universe is nondegenerate, then so will be almost every world in it. Note one of the strengths of the theorem: the matrix Df(v; q) has M rows and N + S columns. Hence, if S is large, so that there are many perturbation parameters, then the assumption of the theorem is likely to be satisfied; after all, we only need to find M linearly independent columns. On the other hand, D../(v; q) has M rows but only N columns. It is thus harder to guarantee in advance that at a solution D.f(v; q) has M linearly independent columns. But the theorem tells us that this is so for almost every q. Observe that if M > N (more equations than unknowns), then the M x N matrix D.f(v; q) cannot have rank M. Hence, the theorem tells us that in this case, generically (i.e., for almost every q). f(v; q) = 0 has no solution.
Definition 17.0.3: The system of M equations in N unknowns f(v) = 0 is regular if rank Of(v) = M whenever f(v) = O. For a regular system, the implicit function theorem (see Section M.E of the Mathematical Appendix) yields the existence of the right number of degrees of freedom. Ir M < N, we can choose M variables corresponding to M linearly independent columns of Df(v) and we can express the values oC these M variables that solve the M equations f(v) = 0 as a Cunction oC the N - M remaining variables (see Exercise 17.0.2). Ir M = N, equilibria must be locally isolated for the same reasons as discussed earlier in this section for the system zIp) = O. And iC M > N, then rank Df(v):s; N < M Cor all v; in this case, Definition 17.0.3 simply says that, as a matter of definition, the equation system f(v) = 0 is regular if and only if the system admits no solution. It remains to be argued that the regular case is the "normal" one. Figure 17.0.4 suggests how this can be approached. In the figure, the one-i:Quation, one-unknown system f(v) = 0 is not regular [because of the tangency point of the graph of f(·) and the horizontal axis]. But clearly this phenomenon is not robust: if we slightly perturb the equation in an arbitrary manner [say that the shocked system is /,(. )], we get a regular system. On the other hand, the regularity of a system that is already regular is preserved for any small perturbation.'· This intuitive idea of a perturbation can be formalized as follows. Suppose there are some parameters q = (q" ... , qs) such that, for every q, we have a system of equations f(v; q) = 0, as above. The set of possible parameter values is RS (or an open region of RS ). We can then justifiably say that f(·; q') is a perturbation of
Let us now specialize our discussion to the case of a system of L - I excess demand equations in L - I unknowns, i(p) = O. We have seen by example that nonregular economies are possible. We wish to argue that they are not typical. To 19. More formally. we could say that in a system defined by finitely many parameters (taking values in. say, an open set) a property is generiC in the first sense if it holds for a set of parameters of full measure (i.e.• the complement of the set for which it holds has measure zero). The property is gelwrk ;11 the .~econd .~ense if it holds in an open set of full measure. A full measure set is dense but it need not be open. Hence. the second sense is stronger than the first. Yet in many applications (all of ours in fact). the property under consideration holds in an open set, and so genericity in the first sense automatically yields genericity in the second sense. In some applications there is no finite number of parameters and no notion of measure to appeal to. In those cases we could say that a property is generic in the third sense if the property holds in an open and dense set. When no measure is available, this still provides a sensible way to capture the idea that the property is typical; bUI it should be nOled that with finitely many parameters a set may be open, dense and have arbitrarily small (positive) measure. In this entire section we deal with genericity in the first sense, and we simply call it genericity. 20. for this theorem, we assume that f(v; q) is as many times differentiable in its two arguments as is necessary.
18. The perturbation should control the values and Ihe derivatives of the funclion. In technical language. it should be a C' perturbation.
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do so, we could resort to a wide variety of perturbation parameters influencing preferences or endowments (or, in a more general setting, technologies). A natural set of parameters are the initial endowments themselves:
L 0 CAL
U N I QUE N E S SAN 0
THE
I N 0 EXT H E 0 REM
Because of the index theorem, this picture, in which the number of equilibria changes discontinuously from 3 to 1 at some points in the space of endowments is typical of the multiple-equilibrium case. A very extensive analysis of this equilibrium set has been carried out by Balasko (1988).
We can write the dependence of the economy's excess demand function on endowments explicitly as zIp; w). We then have Proposition 17.0.4.
We conclude the discussion of genericity with two observations: First, the generic local determinateness of the theory extends to cases with externalities, taxes, or other "imperfections"
Proposition 17.0.4: For any p and w, rank D.,i(p; w) = L - 1.
leading to Ihe failure or the first welfare theorem. (See Exercise 17.0.6.) This should be clear from the generality of the malhematical techniques which. in essence. rely only on the ability 10 express the equilibria of the theory as the zeros of a natural system of equations with Ihe
Proof: It suffices to consider the endowments of a single consumer, say consumer I, and to show that the (L - 1) x L matrix D."z(p; w) has rank L - 1 [this implies that rank D.,i(p; w) = L - 1]. To show this, we can either compute D.. ,z(p; w) explicitly (Exercise 17.0.3) or simply note that any perturbation of w,' say dw" that leaves the wealth of consumer 1 at prices p unaltered will not change demand and therefore will change excess demand by exactly -dw,. Specifically, if p'dw , = 0 then, denoting dw , = (dw ll , .•• , dWL_I.,), we have D.. ,i(p; w) dWI = D..,z,(p; w) dw , = -d such that zO(ji) = 0 and (ii) signIDio(ji)1 = (_I)c-I. For example, zO(p) could be generaled from a single-consumer Cobb-Douglas economy (Exercise 17.0.8). The idea is that fO( p) is both simple and familiar to us and that, as a consequence, we can use it to learn aboul the properties of the unfamiliar t( pl. Consider the following one· parameter family (in technical language, a homotopy) of excess
See Exercises 17.0.4 to 17.0.6 for variations on the theme of Proposition 17.0.4. In Figure 17.0.5, we represent the equilibrium set E = {(w" w 2 , PI): Z(PI' 1; w) = O} of an Edgeworth box economy with total endowment w = w, + w 2 • The set E is the graph of the correspondence that assigns equilibrium prices to economies w = (w" W2)'
demand functions:
P,
zip, I) = I:(p)
___""--;-----'7 0,
Figure 17.0.5 The equilibrium scI.
+ (I
- I)io(p)
for 0 :5 I :5 I.
Thc syslcm i(p, I) = 0 has L - I equations and L unknowns: (P,'··.' PC-I,I). Typically, Iherefore, the solution set £ = ((p,I): z(p, I) = OJ has one and only one degree of freedom al any of its points (that is, it looks locally like a segment). Moreover, since this solution set cannot escape to infinite or zero prices (because of the boundary conditions on excess demand) and is closed [because or the continuity of t(p, I)], it follows that the general situation is well represented in Figure 17.0.6. In Figure 17.0.6, we depict £ as formed, so to speak, by a finite number of circle·like and segmenl-like components, with the endpoints of the segments at Ihe I = 0 and I = I boundaries. Since Ihere arc two endpoints per segment, there is an even number of such endpoints. By construclion, (> is the only endpoint at the I = 0 boundary." Therefore, there must be an odd number of endpoints at the I = 1 boundary; that is, there is an odd number of solutions to :(p) = z(P. I) = O. Suppose now that we follow a segment from end to end. Whal
21. To be quite explicit, this means that the set of endowments that yield nonregular economies
is a subset or RLI that has (LI-dimensional) Lebesgue measure zero, or, equivalently. probability zero for. say, a nondegenerate LI·dimensional normal distribution.
22. More generally, if z(p: I) is an .rbilrary homolopy. then the typical situalion is well reprcscnled by any or the Figures 17.0.I(a), (b), or (c).
597
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Index: +
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EQUILIBRIUM
+)
I= I
FIgure 17.0.6
The equilibrium set under a homotopy.
is Iho rdalion belween Ihe indices al the two ends'? A moment's reflection (keeping the il11plicit function theorem in mind) reveals that as long as we move in a given
diroction relative to I (i.e., forward or backward), the index, (_I)L -, sign IDpi(p, 1)1, does not change. and that the index changes sign precisely when we reverse direction.2) Now, a segment that begins ~tnd ends at the same boundary must reverse direction an odd number or times; heneo. tho indices at the two endpoints have opposite signs. You can verify this in Figure 17.0.6. Therefore, the sum of the indices at I = I equals the index of the lone equilibrium of i(') connected by a segment to the equilibrium p oUo(.) at the boundary I = O. It is represented by in Figure 17.0.6. The segment that connects;; to p. in £ reverses directions an even number of times (possibly none); therefore, we conclude that the index of this equilibrium at I = I equals the index of p for to(.), which, by construction, is + I. Hence, the sum of the indices al I = I is + I, as Proposition 17.D.2 asserts to be true in complete generality.
"oz,(p) L...
,
,,*
ANYTHING
aI',
_ 0 1',-
GOES:
THE
SONNENSCHEIN-MANTEL-DEBREU
for all ( and I' [or Dz(p)p
" "z,(p) L... 1', - - = -Zt(p) , apt
= 0]
(17.E.I)
(17.E.2)
for all ( and p [or p'Dz(p) = -z(p)]
These arc the excess demand counterparts of expressions (2.E.I) and (2.E.4) for demand functions. They follow, respectively, from the homogeneity of degree zero and the Walras' law properties of excess demand. More interestingly, from z(p) = L; (x;(p, P'w;) - w;) we also get
17.E Anything Goes: The Sonnenschein-Mantel-Debreu Theorem
(17.E.3)
We have seen that under a number of general assumptions (of which the most substantial concerns convexity), an equilibrium exists and the number of equilibria is typically finite. Those are important properties, but we would like to know if we could say more, especially for predictive or comparative-statics purposes (see Section 17.G). We may well suspect by now (especially if the message of Chapter 4 on the difficulties of demand aggregation has been well understood) that the answer is likely to be negative; that is, that, in general, we will not be able to impose further reslrictions on excess demand than those in Proposition 17.8.2, and therefore that no further general restrictions on the nature of Walrasian equilibria than those already studied can be hoped for. Special assumptions will have to be made to derive stronger implications (such as uniqueness; see Section 17.F). I n this section, we confirm this and bring home the negative message in a particularly strong manner. The theme, culminating in Propositions 17.E.3 and 17.E.4, is: AII)'llrillg satisjying tire jew properties tlrat we have already shown must Iwld,
call
17.E:
The analysis that follows develops the logic of this conclusion through a series of intermediate results that have independent interest. Some readers may wish, in a first reading of this section, to skip these results and examine directly the statements of Propositions 17.E.3 and 17.E.4 and the accompanying discussion of their interpretations. To be specific, we concentrate the analysis, as usual, on exchange economies formalizcd by means of excess demand equations. Focusing on exchange economies makes sense because, as we know from Chapter 5, aggregation effects are unproblematic in production. The source of the aggregation problem rests squarely with the wealth effects of the consumption side. We begin by posing a relatively simple but nonetheless quite important question: To what extent can we derive restrictions on the behavior of excess demand at a given price p. In particular, we ask for possible restrictions on the L x L matrix of price effects DZ(p).24 Suppose that z(p) is a differentiable aggregate excess demand function. In Exercise 17.E.I, you are asked to show that
Five + Equilibria Pall = I
~----..
+ 1=0
SECTION
where, as usual, S;(p, P 'w;) is the substitution matrix (see Exercise I7.E.2). Expression (I7.E.3) is very instructive. It tells us that if it were not for the wealth effects, Dz( 1') would inherit the negative semidefiniteness (n.s.d.) property of the substitution matrices. How much havoc can the wealth effects cause? Notice that the matrix
i!!~, p'w,l z,,(p) ow,
D., ',(I', p·w.)Z,(p)T
=
...
iJxI,(p, P'w,) ZU> 0 and then choose a utility function that has OJ + z(p) as the
23. To see Ihis, think of the case where L = 2. Applying the implicit function Iheorem to i,(p,. t) = O. verify Ihen that a reversal of direction occurs precisely where OZ,(p"I)/iJp, = O.
demanded point
1
THEOREM
599
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direction of price change." Thus, we should expect that if I < L then there are some negative semidefiniteness restrictions left on Dz( pl. That this is the case is formalized in Proposition 17.E.1.
:L z,(p) =
0, at most I of the I
+
THE
= L (lip, )e'(p,a')
,
SONNENSCHEIN-MANTEL-DE8REU
THEOREM
601
= L, e' a' = A.
and so we would have accomplished our objective. Can we find these L consumers? The answer is "yes." Begin by choosing a collection of
I vectors,
endowments (w\ ... . , w,) yielding strictly positive consumptions when excess demands arc =,(p) = _p,(a')T; that is, x, = w,- p,(a')T» 0 for every i. Observe then that, for cvery i = I. ... , L, thc candidate individual excess demand satisfies Wal",s' law
can be linearly independent. Since I < L, it follows that we can find a nonzero vector dp E RL such that p·dp = 0 and z,(p)·dp = 0 for all i.ln words: dp is a nonproportional price change that is compensated (i.e., there is no change in real wealth) for every consumer. But then from (17.EJ) we obtain
L dp'S,(p, P'w,) dp 5: O.
GOES:
Dz(p) = - L D.. x,(p, P'w,)Z,(p)T
(P,ZI(P), ... ,z,(p)} c IRL
dp'Dz(p) dp =
ANYTHING
and
Proposition 17.E.1: Suppose that 1< L. Then for any equilibrium price vector p there is some direction of price change dp ¥ 0 such that p'dp = 0 (hence, dp is not proportional to p) and dp'Dz(p) dp 5: O. Proof: Because z( p) =
17.E:
P'z,(p)
= -p,p'a' = 0
(because Ap = 0),
and. also. that the candidate wealth effect vector satisfies the necessary condition of Proposition 2.E.3 p·D..,x,(p, P'w,)
•
=(llp,)p'e' = I.
Figure 17.E.1 should thcn be persuasive enough in convincing us that wc can assign prdacnccs to i = I, ...• L in such a way that the chosen consumption at p is Xi. the wealth efTect vector at p is proportional to f' (and therefore must equal (1Ip,Je')." and the indifference map has a kink at x,. The figure illustrates the complete construction for the case /. = 2" In Exercise 17.E.3, you are asked to write an explicit utility function . •
Parallel reasoning should make us expect that if I ;e: L (i.e., if there are at least as many consumers as commodities), then there may not be any restriction left on Dz(p) beyond (17.E.I) and (17.E.2). After all, thc direction of an individual wealth effect vector at a given price is quitc arbitrary (and can be chosen independently of the substitution effects of the corresponding individual); and with I ;e: L wealth effect vectors to be specified, there is considerable room to maneuver. Proposition 17.E.2 confirms this suspicion.
\"li
Figure 17.E.1
Decomposition of excess demand and
price effects at a price vector p (for L = 2).
Proposition 17.E.2: Given a price vector p, let z E RL be an arbitrary vector and A an arbitrary L x L matrix satisfying p'z = 0, Ap = 0 and p'A = -z. Then there is a collection of L consumers generating an aggregate excess demand function z(·) such that zIp) = z and Dz(p) = A. Proof: To keep the argument simple, we restrict ourselves to a search for consumers that at their demanded vectors have a null substitution matrix, SlIp, P'w,) = 0, that is, whose indifference sets exhibit a vertex at the chosen poin!." We can always formally rewrite the given L x L matrix A as A = Le'a',
,
where e' is the (th unit column vector (i.e., all the entries of e' are 0 except the (th entry, which equals I) and a' is the Ith row of A [i.e., at = (an, .. . , a,d)' Suppose now that we could specify L consumers, i = I, ... , L, with the property that, for every i, consumer i has, at the price vector p, an excess demand vector z,(p) = _p,(a')T, a wealth effect vector Dw,x,(p, P'w,) = (1/p,)e', and a substitution matrix S,(p, P'w,) = 0 (where 1 L l L 0 , ••. ,a and e , •.. ,e are as defined above). Then we would have both zIp)
= LZ,(P) = -LP,(a')T = _ATp = , ,
-p'A
27. Indeed. if Dxj(p, P'w j ) = (t/e i , then 1 = p'Dxj(p, P'w i ) = "iP'e l = rlfpj. Hence. (t, = lip;. 28. At no extra cost, we could actually accomplish a bit more. We could also require the substitution matrices of the consumers i = I•...• L to be any arbitrary collection of L x L matrices S, satisfying the properties: Sj is symmetric. negative semidefinite, p,Sj = 0, and SiP = O. The spccil1cation of consumers generating excess demand z( p) and excess demand effects D:(p) at p would proceed in a manner similar (0 the proof just given except that the argument would now be applied to A - L S,. By using matrices S, of maximal rank (i.e.• of rank L - I), we could insure that the resulting L consumers display smooth indifference sets at their chosen consumptions.
=z
25. For example. it cannot hurt in any direction of price change that is orthogonal to the weallh effects vector D..,xj(p. P"W j ) or to the excess demand vector Zj(p). A more precise argument is given
in Proposition 17.E.1. 26. The term "vertex" refers to what is usually called a "kink" in the case L
= 2.
i
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17.E:
ANYTHING
GOES:
THE
SONNENSCHEIN-MANTEL-OEBREU
Up to now, we have studied the possibility of restrictions on the behavior of excess demand at a single price vector. Although the results of Propositions 17.E.! and !7.E.2 are already quite useful, we can go further. The essence of the negative point being made is, unfortunately, much more general. Consider an arbitrary function z(p), and let us for the moment sidestep boundary issues by having zIp) be defined on a domain where relative prices are bounded away from zero; that is, for a small constant £ > 0, we consider only price vectors p with pt!Pt. ;:: £ for every ( and t'. We could then ask: "Can z(·) coincide with the excess demand function of an economy for every p in its domai~?" Of course, in its domain, z(·) must fulfill three obvious necessary conditions: it must be continuous, it must be homogeneous of degrec zero, and it must satisfy Walras' law. But for any z{·) satisfying these three conditions, it turns out that the answer is, again, .. yes.""
{p
E R~:
ptfPt· ;::
£
Ftgure 17.E.2 preferences (in the case L = 2) for the offer curve of an excess
demand function :J') such that ZI( p) = 0 h", no solution with 1/& < PI/P, < 1:.
for every ( and t'} U!.
and with values in RL. Assume that, in addition, z(·) is homogeneous of degree zero and satisfies Walras' law. Then there is an economy of L consumers whose aggregate excess demand function cOincides with zIp) in the domain p'.'O
= tZI(P)
+r
the initial endowment point. We then see in the figure that no matter how complicated the
Strictly speaking, Proposition 17.E.3 docs not yet settle our original question, "0111 we assert ill1ything more about the equilibria of an economy than what we have derived in Sections 17.e and 17.D?" The problem is that Proposition 17.E.3 characterizes the behavior or excess demand away rrom the boundary, whereas it is the power or the boundary conditions that yields some or the restrictions we have already established: existence, (generic) finiteness, oddness. the index rormula.)( To argue that we cannot hope ror more restrictions than these on the equilibrium set, we need to guarantee that ir a candidate equilibrium set satisfies them, then the construction or the "explaining" economy will not add new equilibria. The result presented in Proposition 17.E.4, whose proor we omit, provides therefore the final answer to our question."
[accordingly, zHp) = -(PI/P,)Z:(p)] [accordingly, z~(p)
p.
olT(:r curve! m~ly otherwise be, we can always fit an indifference map so that for any p E ~ we generale precisely the demands flJ j + :i(p). •
and :;(p) = !ZI(P) - r
+ :I( p) is the intersection point of the offer curve with the budget line perpendicular to
The olTer curve is continuous and. because Zi(p) = 0 has no solution in p,. it does not touch
Proof: AI the end of this section, we olTer (in small·type) a brief discussion of the general proof of this result. Here, we limit ourselves to the comparatively simple ease where L = 2. Suppose then Ihat L = 2 and that an & > 0 and a function z(·) satisfying the assumption of the proposition are given to us. The continuity and homogeneity of degree zero of z(·) imply the existence of a number r> 0 such that IZI(p)1 < r for every peP,. We now specify two functions Zl(.) and z'(·) with domain P, and values in A', which are also continuous and homogeneous of degree zero, and satisfy Walras' law. In particular, we let z:(p)
603
Construction of
Proposition 17.E.3: Suppose that z(·) is a continuous function defined on
p. =
THEOREM
= -(PI/P,)Z;(p)].
Note that zIp) = Zl(p) + z'(p) for every peP,. We shall show that for i = 1,2 the function Zl(.) coincides in the domain P, with the excess demand function of a consumer. To this elTect, we usc the following properties of Zl(.): continuity, homogeneity of degree zero, satisfaction of Walras law, and the fact that there is no peP, such that Zl(p) = O. In Exercise 17.E.4, you are asked to show by example that this last requirement is needed. Choose a WI » 0 such that WI + Zl(p) » 0 for every pep'. In Figure 17.E.2, we represent the offer curve OCI associated with Zl(.) in the domain p,. In the figure, for every pep',
PropOSition 17.E.4: For any N ;:: 1, suppose that we assign to each n = 1, ... ,N a price vector pn, normalized to IIpnll = 1, and an L x L matrix An of rank L - 1, satisfying Anpn = 0 and pn'A n = O. Suppose thaI. in addition, the index formula Ln (_1)L -1 sign IAnl = + 1 holds.)) If L = 2, assume also that positive and negative index equilibria alternate. Then there is an economy with L consumers such that the aggregate excess demand z( .) has the properties:
29. The question was posed by Sonnenschein (1973). He conjectured that the answer was that. indeed, On the domain where PI ~ t for all It the three properties were not only necessary but also suil1cicnl; that is. we could always find such an economy. He also proved that this is so for the
(i) zIp) = 0 for Ilpll = 1 if and only if p = pn for some n. (ii) Dz(pn) = An for every n.
two·eommodilY case. The problem was then solved by Mantel (1974) for any number of commodities. Mantel made use of 2L consumers. Shortly afterwards, Debreu (1974) gave a different and very simple proof requiring the indispensable minimum of L consumers. This was topped by Mantel (1976), who refined his earlier proof to show that L homothetic consumers (with no
31. Note, for example, that although a candidate function z(·) defined on
p, may
not have any
solution. we can still successfully generate it from an economy. What happens. of course, is that the equilibria of the economy (which must exist) are all outside of p.:.
restrictions in their initial endowments) would do.
30. NOle. in particular, that this result implies that for any / ~ L, there is an economy of I consumers Ihat generates z(·) on p,. We need only add to the L eonsumers identified by the
32. For this and more general results, see Mas-Colell (1977). 33. Here. A" is the L - I x L - 1 matrix obtained by deleting one row and corresponding column from A.
proposition J - L consumers who have no endowments (or. alternatively. whose most preferred consumption bundle at all price vectors in p.: is their endowment vector).
1
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-
EQUILIBRIUM
Proposition 17.E.4 tells us that for any finite collection of price vectors {pl •... , pN} and matrices of price effects {DZ(pl), ... , DZ(pN)}, we can find an economy with L consumers for which these price vectors are equilibrium price vectors and {DZ(pl), ... , DZ(pN)}. are the corresponding price effects at these equilibria. The result implies that to derive further restrictions on Walrasian equilibria we will need to make additional (and, as we shall see, strong) assumptions. This is the subject of the next three sections. An excellent survey for further reading on the topic of this section is Shafer and Sonnenschein (1982).
SECTION
11.E:
ANYTHING
uOES:
THE
SONNEHSCHEIN-MANTEL-OEBAEU
p Revealed Preferred to p' Weak Axiom Satisfied
Figure 17 .E.4
We should point out that the initial endowments of the consumers obtained by means of Propositions 17.E.2, 17.E.3 or 17.E.4 are not a priori limited in any way. If there are constraints on permissible initial endowments. the nonnegativity conditions on consumption come into play and there may, in fact, be other restrictions on the function z(·). For example, you are asked in Exercise 17.E.5 to verify that the excess demand vectors z( p) and z( p') represented in Figure 17.E.3 cannot be decomposed into individual excess demand functions generated by rational preferences if the amount of any commodity that any consumer may possess as an initial endowment is prescribed to be at most I and if consumptions must be nonnegative.
Revealed preference for excess demand.
z'(p) #< z'(p')
and
P'z'(p')!> 0
(17.E.4)
(see Figure 17.E.4). We say that p is indirectly re""altd preferred 10 p' if there is a finite chain p' ..... p' such that p' = p, p" = p', and p' is directly revealed preferred to p" I for all II !> N - I. The SA then says:
=,
For "1'"'.1' p "lid p'. if p is (direc·tly or indirectly) revealed preferred preJ O.
(17.E.4')
Suppose that .,(.) is an arbitrary real-valued function of p such that .,(p) > 0 for all p € P,. The hasic observation of the proof is then the following: if z'(·) is a proportionally one-to-on" l'xC('ss clt'mand /utlcliml that 5alisjies the SA, then lhe same propert;~s are trut of the function ,,(. )z'(·). Indeed, for any p and p' the revealed preference inequalities (17.E.4') hold for z'(·) if 1II1 O. This is precisely what we will now do. For every normalized p € P,. denote T, = {z € RL: p'Z = O} and for every i = I, ... , L, let :'( p) € Tp be the point that minimizes the Euclidean distance liz - e'li (or, equivalently, maximizes the concave "utility function" -liz - ill) for z € Tp , where e' is the ith unit vector (the column vector whose ith entry is I with zeros elsewhere). Geometrically, z'( p) is the perpendicular projection of e' on the budget hyperplane Tp; that is, z'(p) = e' - p, P. where p, is the ith component of the vector p (recall that i :s; L). Then z'(·) is proportionally one-to·one (see Exercise 17.E.6) and satisfies the SA (since it is derived from utility maximization; see also Exercise 17.E.7). Now let r > 0 be a large-enough number for us to have z(p) + rp » 0 for every normalized p € P, [such an r exists by the continuity of z( . ) and the fact that the set of normalized price vectors in P, is compact and includes only strictly positive price vectors]. For every i = I •...• L and every normalized p € P" define o,(p) = z,(p) + rp, > O. where z,(p) is the ith component
Proof of Proposition '7.E.3 continued: Although a complete proof of the proposition for the case of any number of commodities would take us too far afield. the essentials of the proof by Debreu (1974) are actually not too difficult to convey. We shall attempt to do so. We note that. when carefully examined, the proof can be seen as a generalization of the argument for the L = 2 case presented earlier. In Section 3.1. we saw that the strong axiom of revealed preference (SA) for demand functions is equivalent to the existence of rationalizing preferences. The same is true for excess demand functions: If an excess demand function z'(·) satisfies the SA (we will give a precise definition in a moment), then z'(-) can be generated from rational preferences." It is thus reasonable to redefine our problem as: Given a function z(') that. on the domain P" is continuous. homogeneous of degree zero, and satisfies Walras' law (for short, we refer to these functions as excess demand functions). can we find L excess demand functions z'(·). each satisfying the SA. such that L, z'(p) = z(p} for every p E P,? Before proceeding, let us define the SA for an excess demand function z'(·). The definition is just a natural adaptation of the definition for demand functions. We say that p is directly reveald preferred to p' if
34. We refer to the proof of Proposition 3.1.1 for the justification of this claim.
.L
605
,
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p,
.~
/ -- O.
36. The sel Y can be thought as an aggregate produclion sel. The restriction that Y be of constant returns is made merely for convenience of exposition. It allows us. for example. not to worry about the distribution of profits to consumers (since profits are zero in any equilibrium). Note also that the constant returns model includes pure exchange as a special case (where y= -R';).
35. Reviews for this topic are Kehoe (1985) and (1991). and Mas-Colell (1991).
L
EQUILIBRIA
607
608
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EQUILIBRIUM
SECTION
17.F:
UNIQUENESS
OF
EOUILIBRIA
609
Proposition 17.F.2: Suppose that the excess demand function z(·) is such that, for any constant returns convex technology Y, the economy formed by z(·) and Y has a unique (normalized) equilibrium price vector. Then z(·) satisfies the weak axiom. Conversely, if z(·) satisfies the weak axiom then, lor any constant returns convex technology Y, the set 01 equilibrium price vectors is convex (and so, il the set of normalized price equilibria is finite, there can be at most one normalized price equilibrium). Figure 17.F.l
A violation of the weak axiom implies
multiplicily of equilibria for some Y.
In words, the definition says that if p is revealed preferred to p', then p' cannot be revealed preferred to p [i.e., z( p) cannot be affordable under p']. It is the same definition used in Sections I.G and 2.F, but now applied to excess demand functions.'" The axiom is always satisfied by the excess demand function of a single individual, but it is a strong condition for lI!JqreYlIre excess demand (see Section 4.C for a discussion of this point). We lirst note that, given:('), the WA is a necessary condition for us to be assured of a unique equilibrium for every possible convex, constant returns technology Y that :(.) is coupled with. To see this, suppose that the WA was violated; that is, suppose that for some p and p' we have z(p) # z(p'), p'z(p') ,.:; 0, and p'·z(p),.:; o. Then we claim that both p and p' are equilibrium prices for the convex, constant returns production set given by
Y'
I
= lYE R .: P'y,.:;
0 and P"y,.:; O}.
Figure 17.F.1 depicts this production set for the case L = 2. Note that we have :( p) E Y' and p' Y ,.:; 0 for every Y E Y·. Thus, by Proposition 17.F.I, p is an
Proof: The first part has already been shown. To verify the convexity of the set of equilibrium prices, suppose that p and p' are equilibrium price vectors for the constant returns convex technology Y; that is, z(p) E Y, z(p') E Y, and, for any )'E Y, P'y,.:; 0 and p"Y ,.:; O. Let p" = I1.p + (I - alp' for 11. E [0, I). Note, first, that p" •.\" = ap' Y + (I - a)p"J''':; 0 for any)'E r. To show that p" is an equilibrium, we therefore need only establish that :(1''') E Y. Because 0 = p",z(p") = I1.p,z(p") + (I - l1.)p'·z(p"), we have that either p·z(p"),.:; 0 or P"z(p")":; O. Suppose that the first possibility holds, so that P':(p")":; 0 [a parallel argument applies if, instead, p'·z(p"),.:; 0]. Since :(1') E Y we have p"·z(p),.:; O. But with p"':(p),,:; 0 and p·z(p")":; O. a contradiction to the WA can be avoided only if :(p") = :(p). Hence z(p") E y.lO •
We arc therefore led to focus attention on conditions on preferences and endowments of the I consumers guaranteeing that the aggregate excess demand function z( p) fulfills the WA. To begin with a relatively simple case, suppose that all the endowment vectors w, arc proportional among themselves; that is, that w, = (1,(ii, where w is the vector of total endowments and (1, ~ 0 are shares with (1, = I. I n such an economy, the distribution of wealth across consumers is independent of prices. Normalizing prices to P'w = I, the wealth of consumer I is (x, and =,(p) = x,(p, 11.,) - w,. The aggregate demand behavior ofa population of consumers with fixed wealth levels was studied in Section 4.C. We repeat our qualitative conclusion from there: if individual wealth levels remain fixed, the satisfaction of the WA by aggregate demand (or excess demand), although restrictive, is not implausible.'o
:L
equilibrium price vector. The same is true for p'. Since z(p) # z(p'), we conclude that the equilibrium is not unique for the economy formed by z(·) and the production set yo. What about sufficiency? The weak axiom is not quite a sufficient condition for uniqueness, but Proposition 17.F.2 shows that it does guarantee that for any convex, constant returns Y, rile set of equilibrium price vecrors ;s convex. Although this convexity property is certainly not the same as uniqueness, it has an immediate uniqueness implication: if an economy has only a finite number of (normalized) price equilibria (a generic situation according to Section 17.0),38 the equilibrium must be unique.
A proportionality assumption on initial endowments is not very tenable in a general equilibrium context. It is important, therefore. to ask which new effects are at work (relative to those studied in Section 4.C) when the distribution of endowments docs not satisfy this hypothesis. Unfortunately, it turns out that nonproportionality of endowments can reduce the likelihood of satisfaction of the weak axiom by aggregate excess demand. To see this, consider the relatively simple situation in which preferences arc homothetic. Recall from Sections 4.C and 4.0 that, when endowments arc proportional, this case is extremely well behaved; not only is the WA satisfied, but the model even admits a representative consumer. Yet, as we proceed to discuss
37. A formal, and inessential, difference is that we now define the revealed preference relation on the budget sets (i.e., on price vectors) directly rather than on the choices (i.e., on commodity
39. Observe thai we have cSlablished thai eilher :(p") = z(p) or :(p") = z(p'). Since this is true for any" E [0, I]. and since Ihe function z(') is conlinuous, this implies Ihal z(p) = z(p') for any
vectors).
two equilibrium price vectors p and p'; that is, if the WA holds for =('), then every Walrasian equilibrium for the given endowments must have the same aggregate consumption vector and. hence. the same Llggrcgate production vector.
38. Although our discussion in Section 17.D focused on the case of exchange economies, its conclusions regarding generic local uniqueness and finiteness of the equilibrium set can be extended to the present production context.
40. On this point. consull also Ihe references given in Chapter 4, especially Hildenbrand (t994).
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-------------------------------------------------------------------------------------------------below (in small type), even with homothetic preferences, the WA can easily be violated when endowments are not proportional.·'
Example 17.F.l: This is an example of a failure of the WA compatible with homotheticity and even with the property of gross substitution, which we will discuss shortly. Consider a
In Section 2.F we olfered a dilferential version of the WA for the case of demand functions. 1n a parallel fashion we can also do so for excess demand functions. It can be shown that a sufficient dilferential condition for the WA is
for only the first two goods; that is. he has an excess demand function Z,(P) = Z,(P" p,) that does not depend on p, and P. and. further. is such that z,,(p) = z.. (p) = 0 for all p. Similarly. consumer 2 has preferences and endowments for only the last two goods." We claim that if there is a price vector p' at which the excess demand of the two consumers is nonzero [i.e., z,(p') '" 0 and z,(p') '" 0]. then the aggregate excess demand cannot satisfy the WA. To sec this, choose (p" p,) and (p,. p.) arbitrarily, except that P,ZII(P') + P,Zll(P') < 0 and P,Zll(P') + P.z.,(p') < O. For a> O. take q = (p'" Pl' ap,. afi.) and q' = (ap" ap" Pl' p~). Then if a> 0 is sufticiently large. we have q,z(q') < 0 and q'·z(q) < 0 (Exercise 17.F.2). •
four·commodity economy with two consumers. Consumer I has preferences and endowments
dp'Dz(p) dp < 0 whenever dp'z(p) = 0 (i.e., whenever the price change is compensated) and dp is not proportional /0 p (i.e .• relative prices change).
(17.F.I)
Allowing for the first inequality to be weak, expression (l7.F.I) constitutes also a necessary condition. 42
Under the homotheticity assumption, we have
See Exercise 17.F.3 for yet another example.
I Dw-',(p. P'w,) = - - x,(p, p·w,). p.Wj
I
Denoting S,
= S,(p, P'w,), x, = . PI and Pk = Pk for k "f. t, we have Zk(P') > Zk(P) for k oF t.
41. To reinforce this point. it is also worth mentioning that. in fact. jf we are free to choose initial endowments, then the class of homothetic preferences imposes no restrictions on aggregate
If, as is the case here, we are dealing with the aggregate excess demand of an economy. then the fact that z(·) is also homogeneous of degree zero has the consequence that with gross substitution we also have z{(p') < zAp) whenever p' and pare related as in Definition 17.F.2. To see this, let p = a.p, where a. = PI/p{, Note that PI = PI and p, > p; for k "f. t. Then the homogeneity of degree zero of z(·)
demand. Indeed. as we noted in Section 17.E, the basic conclusion of Proposition 17.E.2 can still be obtained with the further restriction that preferences be homothetic. See Mantel (1976) and the su rvey of Shafer and Sonnenschein (1982). 42. Suppose that dp'z(p) = (p' - p),z(p) = O. Definition I7.F.1 implies then that dp'dz = (p' - p)'(z(p') - zIp)) s; O. Going to the dilferentiallimitand using the chain rule, it follows that dp·Oz(p)dp s; 0 whenever dp'z(p) = O. 41 But this cannot happen if the ,'(,(P, p·w . ) are collinear among themselves or if the collinear among themselves. See Exercise l7.F.1.
WI
are
44. Thus, this example can also be seen as a case of positive association between endowments and demands.
1.
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z,
E (. T ION
( will respond positively to an increase in P•. But if response.'6 •
The offer Curve of a
"
Wli
UNIOUENESS
bl~
= 0, there will be no
Proof: It suffices that we show that z(p) = z( p') cannot occur whenever P and p' are two price vectors that are not collinear. By homogeneity of degree zero, we can assume that p' ~ p and p, = PI for some I. Now consider altering the price vector p' to obtain the price vector p in L - I steps, lowering (or keeping unaltered) the price of every commodity k '" I one at a time. By gross substitution, the excess demand of good I cannot decrease in any step, and, because P '" p', it will actually increase in at least one step. Hence, =/(P) > ZI(P') . • One might hope 10 establish uniqueness in economics with production by applying the as property to the production inclusive excess demand :(.). However, the direct use of the GS property in a production context is limited. Imagine. for example. a situation in which inputs
slIhslirllliolt. 45
Figure 17.F.2 represents the offer curve of a gross substitute excess demand function L = 2. As the relative price of good I increases, the excess demand for good I decreases and the excess demand for good 2 increases. An important characteristic ofthe gross substitute property, which follows directly from its definition, is that it is additive across excess demand Junctions. In particular, if the individual excess demand functions satisfy it, then the aggregate function does also.
and Olltputs ure distinct goods. If the price of an input increases, the demand for every other
input may decrease. not increase as the as property would require, simply because the optimal level of output decreases. Indirectly. though. the gross substitute concept may still be quite helpful. Recall. in particular. that at th. end of Section 17.B. we argu.d that it is always possible to
reduce a production economy to an exchange economy in which, in effect. consumers
c,change factor inputs and then engage in horne production using a freely availabl. constant returns technology. The aggregate excess demand in this derived exchange economy for factor inputs combines elements of both consumplion and production and may well satisfy the as property."
Example 17.1'.2: Consider a utility function of the form u,(x,) = L' UI/(x,,), [f -[xIiUi,(XIi)/ui,(xl/)] < I for all ( and Xli' then the resulting excess demand function z,(p) has the gross substitute property for any initial endowments (Exercise 17.F.5). This condition is satisfied by U,(X,) = (L, ~IiXt,)lJp for 0 < p < I (Exercise l7.F.5). The limits of these preferences as p -+ I and p .... 0 are preferences representable, respectively, by linear functions and by Cobb-Douglas utility functions (recall Exercise 3.C.6). As far as the gross substitution property is concerned, Cobb-Douglas preferences constitute a borderline case. Indeed, the excess demand function for good ( is then ZIi(P) = rt.1i(P·W,)/PI - wli . [f W" > 0, the excess demand for good
What is the relationship between gross substitution and Ihe w.ak axiom? Clearly, the \V A docs not imply the as property (the latter can be violated ev.n in quasilinear, one·consumer economies). The converse relationship is not as obvious. but it is nevertheless true that the GS property does not imply the WA. [n fact, Example 17.F.I. which viola led the \VA. could perfectly well satisfy as" There is. however. one connection that is important. The gross substitute property implies that If z(p) = 0 and ;(p') ¢' 0, then p,z(p') > O.
(17.F.3)
We shall not prove condition (17.FJ) here. For the case in which L = 2, you are asked for a proof in Exercise 17.F.7. To understand (17.F.3). note that if p is the price vector of an
45. It is worth mentioning that functions satisfying the GS property arise naturally in many economic contexts. For example. if A is an (L - I) x (L - I) input~output matrix and (.' E IR'.-'.
then (. - (I - A). satisfies the (w.ak) GS property as a function of a E R'.-' (see Appendix A of Chapter 5 for the interpretation of th.s. concepts). More generally, the equation system g(a) - a associated with the fixed-point problem [i.e., find a such that y(') = aJ of an ;naeas;nrl function r/: R~ - R~ [i. •.• g(.) ~ y(.') whenever a ~ a'J satisfies it (perhaps. again. in its weak version).
46. See also Grandmont (1992) for an interesting result wh.r. a Cobb-Douglas positive representalive consumer. and therefore GS excess demand, is derived from a requirement that at any given price, the choice behavior is widely dispersed (in a certain precise sense) across consumers. Grandmonl's is an example of a model in which the individual excess demand functions may not
Note that in these cases there is no homogeneity of degree zero or Walras' law~-conditions specific to general equilibrium applications-to complement the GS property. This is significant because exploration of the implications of the OS property without homogeneity of degree zero or Walras'
satisfy the gross substitute property but th. aggr.gat. function does. 47. See Mas-Colell (1991) and Exercis. 17.F.6 for further elaborations on this point. 48. Therefore, in view of Proposition 17.F.I. we know that in a constant returns economy the fulfillment of the GS property by the excess demand of the consumers does not imply the uniqueness
law.
of equilibrium.
as property. See
EQUILIBRIA
Proposition 17.F.3: An aggregate excess demand function z(') that satisfies the gross substitute property has at most one exchange equilibrium; that is, z(p) = 0 has at most one (normalized) solution.
gross substitute excess demand function.
tells us that 0 = zl(ii) - Z/(P) = zl(ii) - ZI(P') + Z/(P') - Z/(P), However, gross substitution implies that z/(ii) - z,(p') > 0 (change sequentially each price P; for k '" I to Pl' applying the GS property at each step), and so ZI(P') - Z/(P) < O. The differential version of gross substitution is clear enough: At every p, it must be that iJz.(p)/c1PI > 0 for k '" I; that is, the L x L matrix Dz(p) has positive off-diagonal entries. In addition, when z(·) is an aggregate excess demand function, homogeneity of degree zero implies that Dz(p)p = 0, and so c:z,(p)/c1p, < 0 for all I = I, ... , L: the diagonal entries of Dz(p) are all negative. If in these definitions the inequalities arc weak, one speaks of \\'(,lIk IJ'OSS
these conditions add substantially to the power of the
OF
III tile special case oj excllange economies if the gross substitute property holds for aggregate excess demand then equilibrium is unique.
Figure 17.F.2
'(pI
;::
Exercise 17.F.16 for an
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price vector p for each consumer i, we have x, ;t,w, for all i. However, by the assumption of the proposition and the first welfare theorem, (w" ...• wc) is a Pareto optimal allocation and so we must have x, -,wc for all i. But then we can conclude that x, = w, for all i, because otherwise, by the strict convexity of preferences. the allocation Ox, + !w" ... , !x, + !w,) would be Pareto superior to (w" . ..• w,) . •
Il1dex Al1alysis and Ul1iqueness ( ... and NOl1ul1iqueness) =1 - -__ :(p)
Figure 17.F.3
The index theorem (Proposition 17.D.2) provides a device to test for uniqueness in any given model. The idea is that if merely from the general maintained assumptions of the model we can attach a definite sign to the determinant of the Jacobian matrix of the equilibrium equations at any solution point, then the equilibrium must be unique. After all, the index theorem implies that sign uniformity across equilibria is impossible if there is multiplicity. As a matter of fact, we could have proceeded by means of this index methodology for many of our previous uniqueness results. Take, for example, an exchange economy. In both the WA and the GS cases, whenever z(p) = 0, the matrix Dz(p) is necessarily negative semidefinite [see the small-type discussion of expression (17.F.I) and Proposition 17.F.4]. Moreover, if an equilibrium is regular (i.e., if rank Dz(p) = L - I), the negative scmidefiniteness of Dz(p) can be shown to imply that the index of the equilibrium is necessarily + I (see Exercise 17.F.II). Hence, we can conclude that in both the WA and GS cases, any regular economy must have a unique (normalized) equilibrium price vector. Although the index methodology provides a good research tool. it is often the case that, as here, uniqueness conditions lend themselves to direct proofs. It is a notable fact that some of the more subtle uses of index analysis are not to establish uniqueness but rather to establish nonuniqueness [the first usage of this type was made by Varian (1977)]. This is illustrated in Example 17.F.3.
The revealed preference property of gross substitution.
(e,change) equilibrium and p' is not, then, sinoe :(p); 0, we have p',z(p); 0, and therefore any nonequilibrium p' is revealed preferred to p. Henoe, the requirement in (17.F.3) that ".:( p') > 0 amounts to a restricted version of the WA asserting that no equilibrium price veclor p C'In be revealed preferred to a nonequilibrium prioe vector p'. Geometrically, it says that the range of the excess demand function, {:(p'): p' »O} c RL (Le., the offer curve), lies entirely above the hyperplane through the origin with normal vector p (see Figure 17.F.3). In par;dlcl to Proposition 17.F.2, condition (17.F.3) implies the convexity of the equilibrium price set of Ihe exchange economy, that is, of (pe R'++: z(p); O} c R" (in Exercise 17.F.B, you are asked 10 show this). Interestingly, condition (17.F.3) is satisfied not only in the WA and the GS cases but also in the no-trade case, to be reviewed shortly. In Ihe differentiable case, there is a parallel way to explore the connection between the WA and gross substitution. Let z(p) = O. The sufficient differential condition (17.F.I) for the WA tells us that dp'D:(p)dp < 0 for any dp not proportional to p. Suppose now that instead of the WA, we require that Dz(p) has the gross substitute sign pattern. Because z(p) = 0, we have p' D:(p) = 0 and Dz(p)p = 0 [recall (17.E.I) and (17.E.2)]. Using these two properties it can then be shown that again we obtain dp' Dz(p) dp < 0 for any dp not proportional to p (sec Section M.D of the Mathematical Appendix). Henoe, we can conclude that aC all exc/ulIIgc "'{lIjlibrjllll/ prke veccor, the GS property yields every local restriction implied by the WA. This is summarized in Proposition 12.F.4.
Example 17,F_3: Suppose we have two one-consumer countries. i = 1,2. Countries are symmetrically positioned relative to the home (H) and the foreign (F) good. To be specific, let each country have one unit of the home good as an endowment and none of the foreign good, and utility functions u,(xu;, xF/) = Xu; - X:, for -I < p < O. Merely from symmetry considerations, it follows that there is a symmetric equilibrium p = (1.1). But we may be interested in knowing whether there are asymmetric equilibria. One way to proceed is as follows: compute the index of the symmetric equilibrium; a sufficicnt (but not necessary) condition for the existence of an asymmetric equilibrium is that this index be negative (i.e., _1).49 If we carry out the computation for the present example (you are asked to do so in Exercise 17.F.I3), we see that the index is negative if at prices p = (I, I) the wealth effects in each country are so biased toward the home good that an increase in the price of the good of country I. say, actually increases the demand for this good in country I by more than it decreases the demand from country 2. •
Proposition 17.F.4: If z(') is an aggregate excess demand function, zIp) = 0, and Oz(p) has the gross substitute sign pattern, then we also have dp-Oz(p) dp < 0 whenever dp ,",0 is not proportional to p.
Uniijlleness as an Implication of Pareto Optimality We now present a result that is not of great significance in itself but that is nonetheless interesting because it highlights a uniqueness implication of Pareto optimality. For simplicity, we restrict ourselves again to an exchange economy (see Exercise 17.F.9 for a generalization allowing for production). Proposition 17.F.S: Suppose that the initial endowment allocation (w" .. . , wc) constitutes a Walrasian equilibrium allocation for an exchange economy with strictly convex and strongly monotone consumer preferences (Le., no-trade is an equilibrium). Then this is the unique equilibrium allocation.
49. In this, as typically in any example, the excess demand function fails to be differentiable at
prices at which demand just "hits" the boundary. Typically (we could say "generically"), these prioes will not be equilibrium prioes and the validity of the index theorem is not alTected by these
Proof: Let an allocation x = (XI' ... , xc) and price vector p constitute a Walrasian cquilibrium when consumers' endowments are (w" . .. , w,). Since w, is affordable at
nondilTerentiabilities.
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17.G Comparative Statics Analysis
Proof: Let Ihe firsl consumer have endowmenls wilh Ihe prescribed amounlS of Ihe firsl L - I commodilies, and give 10 Ihis consumer arbitrary preferences, with the single reslriclion Ihal 0"" t ,(p; ,;,,) be nonsingular (it suffices for Ihis thallhe demand function of ~onsumer I salis?es a slricl normalilY condilion; again see Exercise 17.G.I). Since 0 ..,1(;;; WI) = 0~,1,(;;; w,), expression (17.G.I) lells us Ihal we are looking for an addilional collection of L consumers such Ihal Ihe resulling (L + I)-consumer economy has z(p; ';',) = 0 and
z(p; q) = (ZI(P; q), ... , Z,._,(P; q)).
Here, q E IRN is a vector of N parameters influencing preferences or endowments (or both). Throughout, we normalize P,. = I. Suppose the value of the parameters is given initially by the vector ii and that p is an equilibrium price vector for ii; that is, z(p; ii) = O. We wish to analyze the effect of a shock in the exogenous parameters q on the endogenous variable p solving the system. A first difficulty for doing so is the possibility of multiplicity of equilibrium: the system of L - I equations in L - I unknowns i('; q) = 0 may have more than one solution for the relevant values of q, and thus we may need to decide which equilibrium to single out after a shock. If the change in the values of the parameters from ii is small, then a familiar approach to this problem is available. It consists of focusing on the local effects on p, that is, on the solutions that remain near p. Assuming the differentiability of i(p; q), we may determine those eITects by applying the implicit function theorem (see Section M.E of the Mathematical Appendix). Indeed, if the system z(·; ii) = 0 is regular at the solution p, that is, if the (L - I) x (L - I) matrix Dpz(p; ij) has rank L - l,so then for a neighborhood of (p; ij) we can express the equilibrium price vector as a function p(q) = (p,(q), ... , PL-,(q» whose (L - I) x N derivative matrix at ii is
ii)] - , D.i( p; ii).
(I7.G.2) NOlc Ihal Ihe (L - I) x (L - I) matrix defined in (17.G,2) is nonsingular. Thus, we have reduced our problem 10 Ihe following: can we find L consumers whose aggregale excess demand al p is -:,(;;; ,;',) and whose aggregate (L - I) x (L - I) matrix of price effects is A = - D,o,: ,(P;';', )8-' - O,Z,(;;; ';',)1 II follows from Proposition 17.E.2 Ihal the answer 10 Ihis queslion is "yes" (nole Ihallhe reslriclions Ihal Proposilion 17.E.2 imposes on Ihe L x L matrix A place no restriclion on Ihe malrix obtained by deleling one row and one column of A) . • Proposilion 17.G.1 shows that any firsl-order effecl is possible. As in Section 17.E (recall Figurc 17.E.3), il is also the case here thaI if there are prior reslriclions on initial endowments and if consumplion musl be nonnegalive, Ihen Ihere are again comparative statics restrictions of a global character. [See Brown and Matzkin (1993) for a recent investigation of this point]
(17.G.I)
There are a number of comparative static effects that, ideally. we would like to have and that seem economically intuitive: For example, that if the endowment of one good increases, then its equilibrium price decreases. Nevertheless, strong conditions are required for them to hold. By now this should not surprise us: We already know that wealth effects and/or the lack of sufficient substitutability can undermine intuitive comparative static effects. The latest instance we have seen of this occurring has been precisely Proposition 17.G.1. The analysis of uniqueness in Section 17.E may lead us to suspect that good comparative statics effects can hold if aggregate excess demand satisfies either weak-axiom-like conditions (recall Definition 17.F.I) or gross substitution properties (sec Definition 17.F.2). This is in fact so. We consider first the implications of a weak-axiom-like restriction on aggregate excess demand.
What can we say about the first-order eITects Dp(ij)? Expression (17.G.I) and Proposition 17.E.2 [which told us that the matrix of price effects Dpz(p; ij) is unrestricted when I 2: L] strongly suggest that, without further assumptions, the "anything goes" principle applies to the comparative statics of equilibrium in the same manner that in Section 17.E it applied to the closely related issue of the effects of price changes on excess demand. We now elaborate on this point in the context of a specific example. = (w,', ... , WL_I.') Let the list of parameters under consideration be the vector of initial endowments of the first consumer for the first L - I commodities. All of the remaining endowments are kept fixed. As before we assume that i('; dJ,) = 0 is regular at the solution p. It can be shown (see Exercise 17.G.I) that if the demand function of the first consumer satisfies a strict normality condition, then rank Dp(6J,) = L - I, where p(.) is the locally defined solution function with p(J,,) = p. Proposition 17.G.I tells us that if there are enough consumers then this is all that we can say.
w,
50. In a slighl abuse of nOlalion. we leI D,z(i;; ii) stand for Ihe matrix oblained from D,z(p; Ii) by deleling Ihe lasl row and column.
COM PAR A T I v EST A TIC S
Proposition 17.G.1: Given any price vector p, endowments for the first consumer of the first L - 1 commodities J" = (w", ... , wL -",), and a (L - I) x (L - 1) nonsingular matrix 8, there is an exchange economy formed by L + 1 consumers in which the first consumer has the prescribed endowments of the first L - 1 commodities, i(ft; J,,) = 0, i(', J,,) = 0 is regular at p and Dp(J,,) = 8.
Comparative statics is the analytical methodology that concerns itself with the study of how the equilibria ofa system are affected by changes (often described as "shocks") in various environmental parameters. In this section, we examine the comparative static properties of Walrasian equilibria. To be concrete, we consider an exchange economy formalized by a system of aggregate excess demand equations for the first L - I commodities:
Dp@ = - [Dpi( p;
11. 0:
Proposition 17.G.2: Suppose that i(p; ij) = 0, where i(') is differentiable. If Dqi(p; ij) is negative definite," then (Dqi(p; ij) dq)'(Dp(ij) dq) 2: 0 for any dq,
i
1
(17.G.3)
51. This condition is independent or which particular commodity has been labeled as L (see
Section M.D of Ihe Malhemalical Appendix).
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first L - I goods (relative to the price of the Lth good) decrease. 53 In particular. suppose again that consumer I's initial endowment of some good decreases. By labelling commodities appropriately. we can let this good be commodity LUnder the assumption of normal demand for consumer I. a decrease in WLl. at the fixed price vector P. will decrease the excess demand for the first L - I goods. Therefore. the prices of the first L - I goods decrease and so we now reach the conclusion that we could not obtain by means of Proposition 17.G.2: if the endowments of a single good decrease then its price (relative to the price of any other good) increases. This suggests. incidentally. that the assumptions of Proposition 17.G.3 are strictly stronger than those of Proposition 17.G.2. Indeed. as we saw in Proposition 17.F.4. if z(p; ij) = 0 and the L x L matrix D,z(p; ij) satisfies the gross substitute property. then dp'D,z(p; ij) dp < 0 whenever dp '" 0 is not proportional to p. In particular. by letting drc = 0 we have that the matrix D,i(p; q) is negative definite.
Proof: The inverse of a negative definite matrix is negative definite. Therefore [D,i(p; q)] -, is negative definite (see Section M.D of the Mathematical Appendix). Hence. by (17.G.I) we have (D,z(p; ij) dq) . (Dp(ij) dq)
17.G:
= - D.i(p; ij) dq· [D,z(p; ijl] -, D.i(p; q) dq
which is precisely (17.GJ). _ The weak axiom implies the negative semidefiniteness of D,i(P; q) whenever 0 [see expression (17.F.I) and the remark following it]. Therefore. the assumption of Proposition 17.G.2 amounts to a small strengthening of this implication. Its conclusion says that for any infinitesimal shock dq in q. the induced shock to excess demand at prices fixed at P. D.i(p; q) dq. and the induced shock in equilibrium prices. D.p(ij) dq. move "in the same direction" (more precisely. as vectors in R / -' they form an acute angle). For example. a shock that at fixed prices alTects only the aggregate excess demand of the first good. 52 say by decreasing it. will necessarily decrease the equilibrium price of this good. Note that this docs 110/ say that if (I)" increases then the equilibrium priee of good I decreases. Under an assumption of normal demand. this change in w" does indeed decrease the excess demand for good I at ;; but it also alTects the excess demand for all other goods (sec Exercise 17.G.2). We next consider in Proposition 17.G.3 the implications of gross substitution (or. more precisely. of gross substitution holding locally at (p; q)).
:( p; ti) =
Expression (17.G.I) allows us to explicitly compute the effects of an infinitesimal shock. In fact. it also offers a practical computational method to estimate the local effects of small (but pcrh"ps not infinitesimal) shocks. Suppose that the value of the vector of parameters after the shock is ci and, for IE [0, I]. consider a continuous function i(' ,I) that, as I ranges from I ; 0 to I = I. distorts :(.; ti) into i('; iiI. An example of such a function, called a homotopy. is :('./) = (I - I):(';q)
+ If(·;ii).
Denote the solution set by E = {(I, pI: f(p, t) = O}. Then we may attempt to determine p(q) by following a segment in the solution set that starts at (0, p)." If ij is close to q. and the initial situation (1 is regular, then we are in the simple case of Figure 17.G.I(a): there is a unique segment that connects (0. p) to some (I. p)." Naturally. we then put P(ii) - p. If q is not close to ij but nevertheless i(' ,I) is a regular excess demand function for every I [this will be the case if. for example, z(· ./) satisfies. for every I. any of the uniqueness conditions covered in Section 17.F], then this procedure will still succeed in going from I = 0 to I = I and. therefore, in determining an equilibrium for ii.'· Unfortunately. if the shock is large, we can easily find ourselves in situations such as Figures 17.G.I(b) and 17.G.I(c), where at some t' the economy i(·. n is not regular and at (I'. Po') there is no natural
Proposition 17.G.3: Suppose that i(;;; ij) = 0, where i(·; .) is differentiable. If the L x L matrix Dpz(;;; ij) has negative diagonal entries and positive off-diagonal entries, then [Dpi(;;; ij»)" 1 has all its entries negative. Proof: Because of the homogeneity of degree zero of excess demand (recall Exercise 17.E.I). we have D,z(p;q)p=O. and so D,z(p;q)p«O. where p=(p, ..... p.-,). Denote by I the (L - I) x (L - I) identity matrix and take an r > 0 large enough for the matrix A = (l/r)D,z(p; q) + I to have all its entries positive. Then D,i(p; q) = -r[l- A]. and therefore D,l(p; q)p« 0 yields (I - A)p» 0; that is. the positive matrix A. viewed formally as an input-output matrix. is productive (see Appendix A of Chapter 5; the fact that the diagonal entries of A are not zero is inessential). Hence, as we showed in the proof of Proposition 5.AA.I. the matrix [I - A)"' exists and has all its entries positive. From [D,i(p; q)J"' = -(I/r)[1 - A)"' we have our conclusion. _
53. This conclusion holds for nonlocal shocks as well. To see this let Dz(p; q) have the gross subs,itu'e sign pattern throughout its domain and suppose that l(p; ij)« l(p: ij) for all p. For IE [0.1]. define zIp; I) = Ii(p; ii) + (I - I)i(p; ij). Denote by P(/) the solution to l(p; I) = O. Note 'hat D.i( P(/); I) dl = 1(P(/); ii) - 2(P(/); ij)« 0 for all I and therefore. by Proposition 17.G.3. Dp(/) dl « 0 for all I. But then. for any ( = I, .... L - I. we have
PI (q-) - PI (-) q = f.'[ilPI(t)]d dI I < 0 . 0
[t follows from Proposition 17.GJ and expression (17.G.I) that, given gross subSlitution. if D,z(p; q) dq «0. that is. if the excess demand for all of the first L - I goods decreases as a consequence of the shock (and therefore the excess demand for the Llh good increases), then Dp(q) dq «0. That is. the equilibrium prices of the
In Exercise 17.G.3 you can find a more direct approach to the global theory. See also Milgrom and Shannon (1994) for much more on the latter approach. 54. In practice. "following" a segment involves the application of appropriate numerical techniques; see Garcia-Zangwill (1981), Kehoe (1991). and references therein. 55. Moreover, if the shock is sufficiently small. the p so obtained is independent of the particular homotopy used. 56. However. if there are multiple equilibria at ii. then which equilibrium we find may now depend on the homotopy.
52. What this means is 'hat the excess demand of good 2 to L - I is not changed. By Walras' law. the excess demand of good L must change.
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o o (b)
(a)
Figure 17.G.l
OF
EOUILIBRIUM
however, that those are just two examples. Indeed, one of the difficulties in this area is the plethora of plausible disequilibrium models. Although there is a single way to be in equilibrium, there are many different ways to be in disequilibrium.
Price Ttitollllemenl
,.
We consider an exchange economy formalized by means of an excess demand function
o
z(·). Suppose that we have an initial p that is not an equilibrium price vector, so
(e)
that z(p) '" O. For example, the economy may have undergone a shock and p may be the pres hock equilibrium price vector. Then the demand-and-supply principle suggests that prices will adjust upward for goods in excess demand and downward for those in excess supply. This is what was proposed by Walras; in a ditTerential equation version put forward by Samuelson (1947), it takes the specific form
Comparative statics in the large: the general case.
conlinuation of the palh as I increases." To obtain an equilibrium i> for ij there is then no real alternative but to appeal to general algorithms for the solution of the system of equations :(.: ij) = O. It is a sobering thought that which solution we come up with at ij may be dictated more by our numerical technology than by our initial position (p; ij). This is most unsatisfactory, 'lnd it is a manifestation of a serious shortcoming-the lack of a theory of equilibrium
dp,
- = c,z,(p) de
for every (,
(17.H.I)
where dp,/til is the rate of change of the price for the fth good and c, > 0 is a constant atTecting the speed of adjustment. Simple as (17.H.I) is, its interpretation is fraught with difficulties. Which economic agent is in charge of prices'! For that malter, why must the "law of one price" hold out of equilibrium (i.e., why must identical goods have identical prices out of equilibrium)"! What sort of time does "I" represent? It cannot possibly be realtime hecause, as the model stands, a disequilibrium p is not compatible with feasibility (i.e., not all consumption plans can be simultaneously realized). Perhaps the most sensible answer to all these questions is that (17.H.I) is best thought of not as modeling the actual evolution of a demand-and-supply driven economy, but rather as a tentative trial-and-error process taking place in fictional time and run by an abstract market agent bent on finding the equilibrium level of prices (or, more modestly, bent on restoring equilibrium after a disturbance).'· The hope is that, in spite of its idealized nature, the analysis of (l7.H.I) will provide further insights into the properties of equilibria. Even perhaps some help in distinguishing good from poorly behaved equilibria. The analysis is at its most suggestive in the two-commodity case. For this case, Figure 17.H.l represents the excess demand of the first good as a function of the relative price pdp,. The actual dynamic trajectory of relative prices depends both on the initial levels of absolute prices and on the differential price changes prescribed by (17.H.I).60 But note that, whatever the initial levels of absolute prices, p,(I)/p,(t) increases at e if and only if z,(p,(t)!p,(I), I) > O. In Figure I7.H.I we see the following two features of the adjustment equations (l7.H.I).
selection.
17.H Tatonnement Stability We have, so far, carried out an extensive analysis of equilibrium equations. A characteristic feature that distinguishes economics from other scientific fields is that, for us, the equations of equilibrium constitute the center of our discipline. Other sciences, such as physics or even ecology, put comparatively more emphasis on the determination of dynamic laws of change. In contrast, up to now, we have hardly mentioned dynamics. The reason, informally speaking, is that economists are good (or so we hope) at recognizing a state of equilibrium but are poor at predicting precisely how an economy in disequilibrium will evolve. Certainly there are intuitive dynamic principles: if demand is larger than supply then the price will increase, if price is larger than marginal cost then production will expand, if industry profits are positive and there are no barriers to entry, then new firms will enter, and so on. The difficulty is in translating these informal principles into precise dynamic laws. 58 The most famous attempt at this translation was made by Walras (1874), and the modern version of his ideas have come to be known as the theory of ealonnemenl slabililY. In this section, we review two tatonnement-style models, one of pure price adjustment and the other of pure quantity adjustment. We should emphasize,
(a) Call an equilibrium (p" p,) locally slable if, whenever the initial price vector is sufficiently close to it, the dynamic trajectory causes relative prices to converge to the equilibrium relative prices p,/p, (the equilibrium is locally tolally unstable if any
57. Note that by reversing the direction of change of I we can continue to move along the segments in these two figures (this is actually quite a general facl). If p is the only solution at I = 0, as in 17.G.I(b), then the segment necessarily ends with a (I, p). Thus, in some sense we have succeeded in finding an equilibrium for ii that is associated with our initial p. But the association is very weak: it may depend on the particular homotopy and it requires the parameter-reversal procedure. If, as in Figure I7.G.I(c~ ;Hs not the only equilibrium at I = 0, then the procedure may simply not work: the segment that starts at (0, p) goes back to I = O. 58. Refer to Hahn (1982) for a general review.
59. This is, in essence, the idea of Walras (li'onnement means "groping" in French), who took inspiration from the functioning of the auctioneer·directed markets of the Paris stock exchange.
The idea was made completely explicit by Barone (t908) and by Lange (1938), who went so far as to propose the tatonnement procedure as an actual computing device for a centrally planned economy.
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=,
17.H:
TATONNEMENT
STABILITY
623
L=3 I: Locally Stable. Index +1 2: Locally TOlally Unslable. Index +1 3: Saddle. Index -I p,/p,
Flgur.
=, (P,/Pl. I)
17.H.l
Tatonnement
Irajeclories for L = 2.
disturbance leads the relative prices to diverge from pdp,). Then a (regular) equilibrium PI/P, is locally stable or locally totally unstable according to tlae sign of tlae slope of excess demand at Ihe equilibrium, that is, according to the index of the equilibrium (recall Definition 17.0.2). If excess demand slopes downward at p,/p, (as in Figure 17.H.l), then a slight displacement of PI/P, above pdp, will generate excess supply for good I (and excess demand for good 2), and therefore the relative price will move back toward the equilibrium level p,/p,. The effect is the reverse if excess demand slopes upward at pdp,. (b) There is sySlem stability, that is, for any initial position (PI(O), p,(O)), tlae
Flgur. 17.H.2 An example of tatonnement
trajeclories for L = 3.
price of a good goes to zero the excess demand for the good becomes positive (thus, in particular, the trajectories point inward near the boundary). However, properties (a) and (b) are both violated: There are (regular) equilibria that are neither locally stable nor locally totally unstable (they are "saddle points," such as the equilibrium labeled 3 in the figure), and from some initial positions prices may not converge to any equilibrium."' In a more positive spirit, we now argue that for the cases where we have succeeded in proving the uniqueness of Walrasian equilibrium, we are also able to establish the convergence of any price trajectory to this equilibrium (this property is called global stability).62 The next proposition covers, in particular, the weak axiom, the gross substitute, and the no-trade cases studied in Section 17.F.63 These three cases have in common that they satisfy the weak axiom when we restrict ourselves to comparisons between equilibrium and nonequilibrium prices [see the discussion of condition (17.F.3) in Section 17.F]. That is, for the unique (normalized) equilibrium price vector p* arising in these cases we have: "If z(p*) = 0 then pO. z(p) > 0 for any p not proportional to p*."
corresponding trajectory of relalive prices PI(t)/P,(t) converges to some equilibrium arbitrarily closely as t -+ ct:). For regular, two-commodity, economies, properties (a) and (b) give a complete picture of the dynamics. It is very satisfactory picture that accounts for the persistency of tatonnement stability analysis: a theory yielding properties (a) and (b) must be saying something with economic content. Unfortunately, as soon as L> 2 neither the local conclusions (a) nor the global conclusions (b) of the two-commodity case generalize. This should not surprise us, since the price dynamics in (17.H.l) are entirely driven by the excess demand function, and we know (Propositions 17.E.2 and 17.E.3) that the latter is not restricted in any way (beyond the boundary conditions). Consider an example for L = 3 and c, = c, = c, = I. In Figure I7.H.2 we represent the normalized set of prices S = {p » 0: (p,)' + (p,)' + (P3)' = I}. This normalization has the virtue that, for any excess demand function z(p), the dynamic flow p(/) generated by the differential equation dptfdt = z(p), ( = 1,2,3, remains in S [i.e., if P(O) E S then p(t) E S for all t]. This is a consequence of Walras' law:
Proposition 17.H.l: Suppose that z(p*) = 0 and p*,z(p) > 0 for every p not proportional to p*. Then the relative prices of any solution trajectory of the differential equation (17.H.l) converge to the relative prices of pO.
d(p,(t)' + p,(t)' + pit)') dt = 2p,(t)z,(p(t» + 2p,(t)z,(p(t)) + 2p,(t)Z3(P(t» = O.
Proof: Consider the (Euclidean) distance function f(p) = Lt (l/c{)(Pt - pi)'. For any trajectory p(t) let us then focus on the distance f(p(t)} at points I along the trajectory. We have
Thus, the dynamics of p can be represented by trajectories in S, the direction vector of the trajectory at any P(/) being the direction of the excess demand vector z( p(t». We conclude, therefore, that the only restrictions on the trajectories imposed by the general theory are those derived from the boundary behavior of excess demand. In Figure 17.H.2 we represent a possible field of trajectories. In the figure, when the
61. We should warn againsl deriving any comfort when prices converge 10 a limil cycle. Recall that this price tatonnement is not happening in real time. The dynamic analysis has a hope of telling us something significant only ir it converges. 62. Warning: uniqueness by itself does nol imply slabilily-excepl ror L = 2. You should Iry
60. NOle Ihal allhough Ihe change in Pf all prescribed by (I7.H.I) depends only on Ihe relative prices p,/p, for { = I, 2, Ihe change in Ihe price ralio p,/p, al I depends bolh on Ihe curren I price
10
ratio and on the current absolute levels of PI and Pl"
1
draw a counlerexample in Ihe slyle of Figure 17.H.2. 63. For a proof specific 10 Ihe gross Subslilule case. see Exercise 17.H.1.
624
C HAP T E R
1 7:
THE
P 0 5 I T I VET H E 0 R Y
0 F
E 0 U IL
I8
~ E. C T I
R I lJ ;.,
<J
h
• 1 • H:
TAT 0 NNE MEN T
S T It. B I l i T Y
6"::~
succeeds in restoring equilibrium afler a small dislurbance. Thus we see the contrast: for lalonnemenl stability. we impose few informalional reslriclions on the adjustment process [to determine Ihe change in p we only need 10 know f(p); in particular. no knowledge of Ihe derivalives of 1(') is required]. bUI convergence is guaranleed only in special circumslances. For Ihe NeWlon method. local convergence always oblains. bul to delermine Ihe direclions of price change al any p we need 10 know alllhe excess demands f(p) and all the price effecls D:(p). See Smale (1976) and Saari and Simon (1978) for classic conlributions to Ihis Iype of
- p" z(p(r» !> 0,
Newton price dynamics.
where the last inequality is strict if and only if p(r) is not proportional to p'. We conclude that the price vector p(r) monotonically approaches the price vector p' [in fact, since the same argument applies to (1.p', p(r) must be monotonically approaching any (1.p']. This does not mean that p(r) reaches a vicinity of p'. Typically it will not: the rate of approach of p(r) to p' will go to zero before p(r) gets near p'. But the rate of approach can go to zero only if p(r) becomes nearly proportional to p' as r ..... 00, in which case the relative prices do converge.· 4 •
Qlllllltity Tatollllemelll In Ihe analysis so far. prices could be out of equilibrium but quantities. that is to say Ihe amounls demanded and supplied. are always at their equilibrium (i.e.• utility and profit·maximizing) values. We now briefly consider a model in which quantities rather Ihan prices may be in disequilibrium.· 7 This is besl done in a production context. To be very concrele. suppose that there is a single production set y.6. At any moment of lime. we assume that there is given a single. fixed production vector Y E Y. Prices. however. are always in equilibrium in the sense that the general equilibrium syslem of the economy. conditional on y. generales some equilibrium price system p 0, or. if the relevant inverse exists,
as {liven:
Definillon 17.H.1: We say that the differentiable trajectory y(t) E Y is admissible if p(y(t))·(dy(t)/dt) ~ 0 for every t. with equality only if y(t) is profit maximizing for p(y(t)) (in which case we could say that we are at a long·run equilibrium). A difference belween the price and the quantity tatonnement approaches that adds appeal to the second is Ihal feasibility is now insured at any I and that, as a result. we can interpret the dynamics as happening in real time. 69 •7o Will an admissible Irajectory necessarily take us to long. run equilibrium? We cannot really explore this matter here in any detail. As usual, the answer is "only
~·I:·
I
~.
~
Ii
'!!!. =
-;.[Di(pJr'i(p)
(17.H.2)
67. We could also look al Ihe general case where both could be in disequilibrium; sec. for example. Mas·Colell (1986). 68. There is no difficulty in considering several. Also. Y can be interpreted as an individual or
dt
This adjuslmenl equalion is known as Newlon's method and is a slandard lechnique of numerical analysis. If Df(p') is nonsingular. so Ihal [Dt(p·)r' exisls. then (l7.H.2) always
as an aggregate production set.
69. Nonelheless. it is importanl 10 realize thaI. even then. Ihis is not a fully dynamic model: The optimization problems of the consumers remain static and free of expectational feedbacks and firms follow naive. short· run rules of adjustment (in a more positive spirit one might call this adap,ive. rather than naive, behavior). For an extensive analysis of market adjustment procedures in real
64. Conlinuous rcal·valued functions Ihallake decreasing values along any dynamic trajectory and the value zero only at stationary points are known as Lyapunov functions.
time. see Fisher (1983). 70. The quantity dynamics of Definition 17.H.1 are reminiscenl of Marshall (1920) and arc
65. How could we prelend to know much about speeds of adjustments? 66. NOle Ihat Ihis fils nicely wilh Proposilion 17.H.1 because Ihe revealed.preference·like property poslulaled Ihere implies the negalive (.. mijdefiniteness of Dt(p) al Ihe equilibrium price vector p•.
ortcn referred to as Marshallian dynamics, especially in a partial equilibrium context. ]n contrast. the price dynamics are frequently called Walrasian dynamics.
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SEC T ION
Proof: Consider u(y(l) + w) where u(·) and ware respectively the utility function and the endowments of the consumer. The unique equilibrium production vector is the single production vector y that. maximizes u(y + w) on Y; recall the oneconsumer, one-firm example of Section IS.C. The argument is much simpler if we assume that u(·) is differentiable. We claim that utility must then be increasing along any admissible trajectory. Indeed,
+ w)
V ( () uyr
+w
) dy(l) '-d-I-
dyer) = !I(I)p(y(r»'d/
NON CON V E X I TIE S
627
As we have mentioned repeatedly, especially in Chapters 10 and 12, a central justification of the price-taking hypothesis is the assumption that every economic agent constitutes an insignificant part of the whole economy. Literally speaking, however, this cannot be satisfied in the model of this chapter because, formally, we allow for no more than a finite number I of consumers. (This is particularly true of our examples, where we typically have I = 2.) A straightforward reinterpretation is possible, however. We illustrate it for the case of a pure exchange economy. Suppose we consider economies whose consumers have characteristics (preferences and endowments) that fall into I given types, with r consumers of each type (a generalization to unequal numbers per type is possible; see Exercise 17.1.1). That is, the set of consumers is formed by r replicas of a basic reference set of consumers. Furthermore, an allocation denoted by (Xl' ... ' x,) is understood now to specify that each consumer of type j consumes XI (so the totality of consumers of type j consume rx i ). We observe then that the analysis and results presented up to this point are not modified by this reinterpretation; they simply do not depend in any way on the parameter r. In this way, we can conclude informally that the theory so far covers cases with an arbitrarily large number, even an infinity, of consumers; in particular, we see that any equilibrium of our earlier model is an equilibrium of the r-replica economy (for any integer r ~ I). There is, however, an important qualification. The ability to interpret the model and results in a manner that is fully independent of the number of consumers depends crucially on the convexity assumption on preferences. Without this assumption, it is not justified to neglect allocations that assign different consumption bundles to different consumers of the same type. Consider, for example, the Edgeworth box of Figure 17.1.1. If there is only one consumer of each type, then no equilibrium exists; but if we have two of each type, then there is an equilibrium. To see this, give W2 to the convex consumers, let one of the two nonconvex consumers receive the bundle XI' and let the other receive the different bundle Thus, in the nonconvex case, the
Proposition 17.H.2: If there is a single strictly convex consumer, then any admissible trajectory converges to the (unique) equilibrium.
--d-I-- =
L A R GEE CON 0 M I E SAN D
17.I Large Economies and Nonconvexities
under special circumstances." A limited, but important example (it covers the shortrun/long-run model of Section IO.F) is described in Proposition 17.H.2.
du(y(l)
1 7 • I:
> 0,
with equality only at equilibrium. Here we have used the fact that at a short-run (interior) equilibrium, the price vector p(y(I» weighted by the marginal utility of wealth !I(t) must be equal to the vector of marginal utilities of the consumer. Now, since utility is increasing, we must necessarily reach the production vector y at which utility is maximized in the feasible production set (i.e., the equilibrium). This is illustrated in Figure 17.H.3. (We are sidestepping minor technicalities: to proceed completely rigorously, we should argue that the dynamics cannot be so sluggish that we never reach the equilibrium. To do so we would need, strictly speaking, to strengthen slightly the concept of an admissible trajectory). •
x;.
Note that the single consumer of Proposition 17.H.2 could be a (positive) representative consumer standing for a population of consumers. Figure 17.1.1
Good 2 1,(I)p(r(l))
= Vu(r(t) + w)
Figure 17.H.3
Equilibrium with
An example of
nonconvex preferences in economies of changing size.
quantity tatonnemcnt.
Indifference Curves of the (Representative) } Consumer
o,L---------\---t-----"-+-__
y" R~+ Good I i
1
THE
POSIT .... £
THEORY
OF
EQUILIBRIUM
SEC T ION
I r
I = r
(the sum has r terms)
+ ... + Z,,: =" E z,(p), ... , z"
A Ii
t:.
u
E C 0 HOM i E SA'" 0
,,0,.. l.. V
J"oj
~ 1. ), I TIE S
IJL:;;1
In the previous reasoning, the convexification or aggregate excess demand, with its existence implication, depends on our ability to prescribe very carefully which of several indifferent consumptions each consumer has to choose. Only in this way can we make sure that the "ggreg"tc consumption will be precisely right. Whatever we may think about the possible processes that may lead consumers to select among indifferent optimal choices in the right proportions, there can be little doubt that it would be better if we did not have to worry about this; that is, ir, given any price, practically every consumer had a single optimal choice. It is therefore or interest to point out that, while not a necessity, this is a most plausible occurrence if the number or consumers is large. Indeed, if the dist,ihution of individual prefere/lces ;s di.'ipas('d across the populCltion (so that, in particular. no two consumers arc exactly identical B), r!Jeli ('f('1f if til I,there need not be any inclusion relationship between E(,") and E(,') (except ir ," = m,' ror some integer m > I, in which case E(,') c E(,")). 72. See Starr (t969) ror a classic contribution to this topic. 73. See the comment after the proof of Proposition t 7.C.2 regarding demand correspondences, and also Exercise t 7.C.1.
l
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EQUILIBRIUM
APPENDIX
We comment briefly on economies with production. Suppose that the consumption side of the economy is generated, as before, as the r-replica of a basic reference set of (possibly nonconvex) consumers. There are also J production sets l). Each lJ is closed, contains the origin, and satisfies free disposal (these are all standard assumptions). In addition, we assume that there is an upper bound (a capacity bound perhaps) on every lJ; that is, there is a number s such that )'/J ~ s for all ( and 11 e lJ. The production sets may be nonconvex. It is then possible to argue that the economy will possess a near equilibrium if r is large relative to the bound s (i.e., if the size of the consumption side of the economy is large relative to the maximal size of a single firm). On the average, the production side of the economy is also being convexified, so to speak (see the small-type discussion of Section S.E for a related point)." Note that the bounded ness property of the production sets is important. Suppose, for example, that every firm has the technology represented in Figure IS.C.3. Then no matter how many consumers there are, the potential profits of every firm are infinite (as long as p, > 0). Thus, there is no reasonable sense in which a near equilibrium exists. For the averaging·out effect to work the nonconvexity in production has to be of bounded size (see Exercise 17.1.2).
A:
CHARACTERIZING
EQUILIBRIUM
THROUGH
WELFARE
EQUATIONS
",
o,~--------------~~------~~~--
I
i
I
P(,)'(w, - "(s)) = g,(s) p,(s) . p,(s)
proportional to SEA (see Exercise I7.AA.I). In words, SEA stands for the values of utility distribution parameters and the determined allocation distributes "welfare" in accordance with the "shares" s = (s" ... ,sIlo Figure I7,AA.1 illustrates the construction. An arbitrary SEA will typically not correspond to an equilibrium. How can we recognize those sEA that do? To answer this question we can resort to the second welfare theorem. From Propositions 16.D.1 (and the discussion in small type following Proposition 16.D.3), we know that, under our assumptions, associated with x(,,) there is a price vector pes) E RL that supports the allocation in the sense that, for every i, x; >-; x,(s) implies pes)' x; > p(s) , x;(s). Therefore, (x(s'), p(s'» constitutes a Walrasian equilibrium if and only if S· E A solves the system of equations
APPENDIX A: CHARACTERIZING EQUILIBRIUM THROUGH WElFARE EQUATIONS
We have seen, beginning in Section 17.B, that if our economy satisfies sufficiently nicc properties (e.g., strict convexity of preferences) then we can resort, for the purposes of the analysis, to formalizing our theory by means of highly reduced systems of equilibrium equations. In the text of this chapter, we have focused on excess demand equations, But this is not the only possibility. In this appendix, we briefly illustrate a second approach that builds on the welfare properties of equilibria. We again concentrate on a pure exchange economy in which each consumer i = I, ... ,J has the consumption set R~ and continuous, strongly monotone, and strictly convex preferences. We also assume that WI ~ 0 for all i and 1:, w, » o. We know from Chapter 16 that a Walrasian equilibrium of this economy is a Pareto optimum (Proposition 16.C.l). Therefore, to identify an equilibrium, we can as well restrict ourselves to Pareto optimal allocations. To this effect, suppose we fix continuous utility functions u,(·) for the J consumers with u,(O) = O. Then to every vector s = (s I' ... , Sl) in the simplex A = {s' E R~ : 1:, s; = I} we can associate a unique Pareto optimal allocation xes) E R~I such that (UI(XI(S», ... , UI(XI(S))) is
g;(S')
= p(s')·[w; -
x,(s')]
=0
for every i = I, ... , J.
(I7.AA.I)
The Edgeworth box example of Figure 17.AA.2 explains the point that we are currently making. This Pareto-based equation system was first put forward by Negishi (1960), and was the approach taken by Arrow and Hahn (1971) in their proof of existence of equilibrium. It can be quite useful when the number of consumers (say, the number of countries in an international trade model) is small relative to the number of commodities. In contrast, if the number of consumers is large relative to the number of commodities, then an approach via excess demand functions will be superior. A limitation of the Negishi approach is that it is very dependent on the fact that an equilibrium must be a Pareto optimum. The excess demand approach is more easily adaptable to situations where this is not so (for example, because of tax distortions; see Exercise 17.C.3).7.
77. Observe that the average is with respect to , (the size of the economy in terms of the number of consumers). not with respect to J. Ir. as , increases. J is made to vary and is kept in some approximate fix.ed proportion with r, then rrom the qualitative point or view it does not matter how we measure size (this is a possible way to interpret, in the current context, the discussion or Section 5.E). But for the validity of the convexifying etrecl there is no need 10 vary J with r. The number J may be kept fixed and, thus, J could well be small relative to r (in which case the "averaged" economy is praclically one of pure exchange) or it could be large; il could even be Ihat J = co. The last case corresponds to a model with free entry, where the equilibrium-or the near equilibrium-
determines. endogenously, Ihe set of active firms. Typically, with free entry the sel of the aClive firms increases as the number of consumers, measured by r, grows (this point has also been discussed
78. The syslems of equations (17.B.2) and (17.AA.I) can be formally conlrasted as follows. In both of Ihem. at any poinl of Ihe domain of the equal ions, consumers and firms salisfy the utility maximizalion conditions for some prices and distribution of weallh. In (l7.B.2) this dislribution of weallh is always Ihe one induced by the inilial endowments, bUI feasibility (i.e., the equalilY of demand and supply) is insured only at the solulion. In (l7.AA.I) it is the other way around: feasibililY is always satisfied, but Ihe agreement of the wealth distribution with that induced by Ihe
in Section IO.F in a partial equilibrium context; there is not much more to add here).
initial endowments is insured only at the solution.
1
(ten) Construction of the welfare-theoretic equation system: first step.
Figure 17.AA.l
Figur. 17.AA.2 (rig hi)
Construction of the welfare-theoretic equation syslem: second step.
631
632
A P PEN 0 I X
Definition 17.BB.3: An allocation (x', y') and a price system P disposal quasiequilibrium if
if Xi
then (x·, p) cannot be a Walrasian quasiequilibrium because consuming nothing costs zero and is
preferred by the first consumer to any other consumption. 8!. Because lj is convex and closed, - R~ C lj implies lj -
R~ c lj (Exercise 5.8.5). X, for every i. With this assumption.
Proposition 17.BB.2 yields the existence of a true equilibrium. not just of a quasiequilibrium. Wj ~ .i, can be interpreted (keeping in mind the possibility of free disposal) simply as saying that consumer; could survive Economically, however, the latter assumption is considerably stronger:
without entering the markets of the economy, while the market a strictly positive amount of every good.
Wj
»X, says thac the consumer can supply to
">-; Xi
then p'x; ~ P'w;
+ L 0i/P' Y,'. /
(iii') Li xi ~ L; W;
have p »0. By profit maximization (using the free-disposal technology) and the possibility of inaction, we have p'(x1 + x! - W, - w,) ~ O. Since p'x! $ p'W" this yields p'x1 ~ p'W, > O. But
Wi»
0 constitute a free-
(i) for every i, P'Y, ~ P'Y,' for all Y,E If· (ii') For every i, P'x!' ~p'w;+ LjOijP'Y,', and
79. Recall, however, the important qualification of Section 17.1, and see also the discussion at the end of this appendix. 80. In Figure 17.8B.2, the second consumer has conventional strongly monotone preferences; but for the first consumer both commodities are bads and. thereCore. he is satiated at the origin. Also (I), »0 and w 2 » O. Suppose that x· = (xT. xf) and price vector p#-O constitute a Walrasian quasiequilibrium. Because the preferences of the second consumer 3fC strongly monotone, we must
82. A stronger condition would require that
~
I
I
1
+ L/ Y,'
and p' (Li xi - Li Wi - L/ lj') = O.
Thus. all we have done is replace in Definition 17.BB.I of a quasiequilibrium the exact feasibility condition 'Ti xi = L Wi + Lj yt" by (iii') above. That is. we allow the excess supply of some goods provided that they are free. In Exercise 17.BB.4 you arc asked to show that if one production set. say 1I. satisfies the free-disposal property and if (xf' .... x7. yf ....• yr. p) is a free-disposal quasiequilibrium. then there is l'* ~ yf such that (xf" ... x7. l,·. y! ..... yJ. p) is a Walrasian quasiequilibrium. Therefore. to establish Proposition 17.B8.2, it is enough for us to show that a free-diposal quasiequilibrium exists. We proceed to formalize the free-disposal quasiequilibrium notion as a kind of noncooperative equilibrium for a certain game among I + J + I players. The I and J players are the consumers and the firms. respectively. and their strategies are demand-supply vectors. The extra player is a fictitious market agent (a "grand coordinator") having as his strategy the prices of the L different goods. Since the set A of feasible allocations is bounded, there is r > 0 such that whenever (xl •. ··.x,.y, •...• yJ)E A we have Ix,d < rand IY'jl < r for all i,j, and t. Because we need to have compactness of strategy sets to establish existence, we begin by
635
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EOUILIBRIUM
B:
GENERAL
APPR"" .. "
Tu
IHE
E.XISTENCl
(d,
WAlRASIAN
EQUILloHlljJol
-------------------------------------------------------------------------------------Denote by x,(x, Y, p) c X, the set of consumption bundles xi so defined. Firm j: Chooses productions yj € ~ that are profit maximizing for P on ~. (Firm j's payoff function is simply its profit.) Denote by )~(x, Y, p) c ~ the set of production plans yj so defined. Market Agent: Chooses prices q €!:J. so as to solve
replacing every X, and every lj by a truncated version:
X, = {x, € X,: IXl;!
:5 r for all t}.
~ = {YJ€ lj: IYol :5 r for all
tl·
t.
Note that A c X, x ... X X, x y, x ... x Because (.£" ... ,.£" ... ,0, ... ,0) € A, it follows that .£, € X, for every i, and 0 € ~ for every j. In particular, all the strategy sets are nonempty. Lemma 17.BB.1 shows that in our search for a free-disposal quasiequilibrium we can limit ourselves to the truncated economy.
Max
Lemma 17.BB.1: If all X; and If are convex and (x", y", p) is a free-disposal quasiequilibrium in the truncated economy, that is, if (x", y", p) satisfies Definition 17.BB.3 of free-disposal quasiequilibrium with the consumption and production sets replaced by their truncated versions, then (x", y", p) is also a free-disposal quasiequilibrium for the original untruncated economy.
ilEA
Lemma 17.BB.2: Suppose that (x·, Y", p) is such that xi € x;(x", Y·, p) for all i, v," € Yj(x·, y", p) for all j, and p € p(x·, Y", pl. Then (x·, y., p) is a free-disposal quasiequilibrium for the truncated economy. Proof of Lemma 17.BB.2: We note first that P' yj ;::: 0 for every j (because 0 € 5'). By the definition of x,(·) and jij('), conditions (i) and (ii') of Definition 17.B8.3 are then automatically satisfied. Hence, the only property that remains to be established is (iii'), that is,
Lxi - LllI, - Lyt:5 0
~ O/jP' yj}'
We have P'x~:5 w/(p,Y") = P'w, The strategy sets are:
P'(LX~ i
E
~/.:
PI ;::: 0 for all t and LI Pr
= I}.
Given a strategy profile (x, Y, 1') = (X,' ... ' X" Y,' ... ' YJ' p), the payoff functions and best-responses of the different agents are: Consumer i:
Chooses consumption vectors xi € X, such that (I) p'xi :5 w,(p, y) and (2) xi ?:;,x7 for all xi' E X, satisfying P' XI < w/(p, y). (Consumer i's payoff function can be thought of as giving a payoff 1 if he chooses a consumption vector satisfying this condition, and 0 otherwise.)
(17.BB.l)
J
Only the behavior of the market agent needs comment. Given the total excess demand vector, the market agent chooses prices so as to maximize the value of this vector. Hence, he puts the whole weight of prices (which, recall, have been normalized to lie in the unit simplex) into the commodities with maximal excess demand. As we have already observed when doing the same thing in the proof of Proposition 17.C.1, this is in accord with economic logic: if the objective is to eliminate the excess demand of some commodities, try raising their prices as much as possible. Lemma 17.BB.2 says that an equilibrium of this noncooperative game yields a frec-disposal quasiequilibrium for the truncated economy.
We are now ready to set up a simultaneous-move noncooperative game. To do so we need to specify the players' strategy sets and payoff functions. To simplify notation we assign to every consumer i, price vector P and production profile Y = (YI'···' YJ), a limited liability amount of wealth
For consumer i: X, For firm j: ~ For the market agent: !:J. = {p
i
i
Denote by p(x, y, p) the set of price vectors q so defined.
Proof of Lemma 17.BB.l: Consider a consumer i (the reasoning is similar for a firm). Because (x·, y.) € A, we have Ix1,1 < r for all t; that is, the consumption bundle of consumer i is interior to the truncation bound. Suppose now that x~ fails to satisfy condition (ii') of Definition 17.BB.3 in the nontruncated economy, that is, that there is an x, € X, such that x, >-,x~, and P' x, < P'W, + LJ O/jP' yt- Denote x7 = (I - (I/n»xr + (l/n)x,. For all n we have P' xi < P'W, + Lj O'JP' yt and, by the convexity of preferences, xi ?:;,x7- Also, we can choose an n large enough to have Ixi,l < r for all t. By local nonsatiation there must then be an xi € X, such that xi >-,xi and p'xi < P'W, + Lj O'JP' yt- But then xi € X, and xi >-,x7 ?:;,X~, and so in the truncated economy xr fails to satisfy condition (ii') of Definition 17.BB.3. Thus, (x·, y., p) must not be a free-disposal quasiequilibrium in the truncated economy. This contradiction establishes the result. _
w,(p, y) = P'w, + Max{o,
(LX,-LW,-LYJ)·q.
I !
I
L
and
+ L J8/jp·yt
for all i and therefore
t):5 o.
LW' - L Y j
j
This implies L, xi - L, w, - LJ yt :5 0 because otherwise the value of the solution to problem (17.BB.l) would be positive and so P (which as we have just seen has "'(L, xi - L, w, - LJ ytl:5 0) could not be a maximizing solution vector, that is, a member of p(x", y", pl. It follows that (x·, Y·) € A and so, X?, < r for all i and t. From this we get that the budget equations are satisfied with equality (i.e., ". xi = p'W, + Lj OIjP' yt for all i) because otherwise local nonsatiation yields that for some consumer i there is a preferred consumption strictly interior to consumer i's budget set in the truncated economy, implying x~ ¢ X,(X·, y., pl. We therefore conclude that we also have P'(L, xr - LI W, - LJ ytl = O. This completes the proof. _
638
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N OW, as we discussed in Appendix A to Chapter 8 (see the proof of Proposition
8.0.3 presented there), under appropriate conditions on the best-response correspondences, this noncooperative game has an equilibrium. Lemma 17.88.3: Suppose that the correspondences x;!"), ~(.), and p(.) are nonempty, convex valued, and upper hemicontinuous. Then there is (x', y', p) such that x7 E x,(x', yO, p) for all i, Yj' E ~(x', yO, p) for all j, and p E p(x', yO, pl. Proof of Lemma 17.BB.3: We are simply looking for a fixed point of the correspondence '1'(.) from X, x ... X X, X YI X •• '. x >J x A to itself defined by 'I'(x.y,p) = x,(x,y,p)
x··· x x,(x,y,p) x y,(x, y, p) x··· x yAx,y,p) x p(x, y, p).
The correspondence '1'(.) is nonempty, convex valued, and upper hemicontinuous. The existence of a fixed point follows directly from Kakutani's fixed point theorem (sec Section M.I of the Mathematical Appendix). _ Lemmas 17.BB.4 to 17.BB.6 verify that the best-response correspondence of this noncooperative game is nonempty, convex valued, and upper hemicontinuous· ' Lemma 17.88.4: For all strategy profiles (x, y, p), the sets xi(x, y, p), pIx, y, p) are nonempty.
~(x,
y, p), and
Proof of Lemma 17.BB.4: For Yj(x, y, p) and pIx, y, p) the claim is clear enough since we arc maximizing a continuous (in fact, linear) function on, respectively, the noncmpty, compact sets ~ and A. For Xi(X, y, p), recall that the continuity of ;:::i implies the existence of a continuous utility representation "i(') for ;:::, .• 4 Let be a maximizer of the continuous function "i(X,} on the nonempty compact budget set (Xi E Xi: p' Xi::;; Wi(p, yO)}. Then x; E Xi (X, y, pl. The budget set is nonempty because Xi E Xi and ,Xi::;; Wi' With P
x;
Lemma 17.88.5: For all strategy profiles the sets xi(x, y, p), Yj(x, y, p), and pIx, y, p) are convex. Proof of Lemma 17.BB.5: We establish the claim for Xi(X, y, pl. You are asked to complete the proof in Exercise 17.BB.6. Suppose that Xi' x; E Xi (x, y, p) and consider Xi. = aXi + (I - a)x;, for any a E [0,1]. Note first that P·Xi. ::;; W,(P, y). In addition, by the convexity of preferences we cannot have Xi :>iXi. and x; :>, x,. (Exercise 17.BB.5). So suppose that Xi. ;:::iXi' Consider now any xi e X, with P'x, < wi(p, y). Then since x, E Xi(X, y, p) we have Xi;::: i xi, and so Xi. ;:::ixi. We conclude that Xi. e Xi(X, y, pl. A similar conclusion follows if Xi. ;:::i x;. Hence, xi(x, y, p) is a convex set. _ Lemma 17.88.6: The correspondences
xi ('), ~(.), and p(.) are upper hemicontinuous.
Proof of Lemma 17.BB.6: Again, we limit ourselves to Xi(·)' Exercise 17.BB.7 asks you to complete the proof for M') and p('). 83. For the firms and the market game this result is covered by Proposition 8.DJ, but for the consumers we need a special argument (as defined, the payoff runctions of the consumers arc not continuous).
84. This was proved in Proposition lC.1 for monotone preferences on Rt As we pointed out there. however. the conclusion actually depends only on the continuity of the preference relation.
B:
GENERAL
APPRQACH
TQ
THE
EXISTENCE
OF
WALRASIAN
x;
Let p" - t p, y" - t y, x" - t x, and x;" -+ as n - t 00, and suppose that x;" E x,(x", y", pO). We need to show that x; e XI(P, x, y). From p"'x;" ::;; W,(P", yA) we get p'x; ::;; W,(pA, yA). Consider now any xi e with x;' :>iX;. Then, by the continuity of preferences, xi :>, x;" for n large enough. Hence, p". xi :?: wi(p", yO). Going to the limit we get p' x7
Xl
x;
closed-graph property that we have replaced preference maximization by the weaker objective of expenditure minimization in the definition of the objectives of the consumer. _
The combination of Lemmas I7.BB.4 to 17.BB.6 establishes that the given best-response correspondences satisfy the properties required in lemma 17.BB.3 for the existence of a fixed-point, which completes the proof of Proposition 17.BB.2. _ The assumptions on preferences and technologies can be weakened in an important respect. Our existence argument requires only that the best·response correspondence .i,(x, y, p) and rj(x, y, p) be convex valued and upper hemicontinuous. Beyond this, the proof imposes no restrictions whatsoever on the dependence of consumers' and firms' choices on the ~state" variables (x. y. pl. Thus we could allow consumers' tastes, or firms' technologies, to depend on prices (money illusion?), on the choices of other consumers or firms (a form of externalities), or even on own consumption (e.g., tastes could depend on a current reference point-a source
I).·"··
of incompleteness or non transitivity of preferences already illustrated in Chapter The following is an example of the sort of generality that can be accommodated: Suppose that consumer preferences are given to us by means of utility functions -,('; x, y, p) defined on X, but dependent, in principle, on the state of the economy. If for every (x, y, p) the conditions of Proposition 17.BB.2 are satisfied, and the parametric dependence on (x, y, p) is continuous, then a Walrasian quasiequilibrium still exists. The proof does not need any change. We can make a similar point with respect to the possibility that firms' technologies depend on external effects, with, then, an added theoretical payoff. It allows us to see that equilibrium exists if the technology of the firm is convex: il does nol mailer iflhe ~aggregale"lechnology oflhe economy is convex. See Exercise 17.B8.8 for more on this. The existence proof we have given in this appendix is an example of a "large space" proof. The fixed'point argument (in our case phrased as a Nash equilibrium existence argument) has been developed in a disaggregated domain where all the equilibrating variables have been listed separately. The advantage of proceeding this way is that the argument remains very flexible and allows us to incorporate the weakest possible conditions without extra effort (as the last paragraph has illustrated). The disadvantage, of course, is that the fixed point may be 85. Suppose, for example, that the utility function of a consumer is given to us in the form
that is, the evaluation of possible consumptions depends on the current consumption. Without loss of generality we can normalize u,(x,: x,) = 0 for every x,. Define the induced weak and strict preference relations '=/ and >-1 on Xi by, respectively, .. xj ~,x/ if ",(xi; x,) ~ 0'" and .. xi >-iXi if uj(x/; x,) > 0." Then the relations ~, and >-, contain all the relevant information for equilibrium analysis. Note, however, that it is perfectly possible for It, not to be complete and for neither It; nor r; to be transitive. See Shafer (1974) and Gale and Mas-Colell (1975) for more u;(·: x;);
on this.
86. Another example of dependence on the overall consumption vector of the economy arises if, for example. we are considering equilibrium at a given point in time. Then current consumptions in the economy (e.g., purchases of physical or financial assets) will typically affect future prices; these, in turn, will innuence current preferences via expectations.
EQUILIBRtUM
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EOUILIBRIUM
hard to compute and cumbersome to analyze. Usually, as we have seen in Section 17.C and in Appendix A of this chapter, it is possible to work with more aggregated, reduced systems. In fact, the general point duly made, it is worthwhile to observe that this is so even under the assumptions of Proposition 17.BB.2.17 We elaborate briefly on this. We can prove Proposition 17.BB.2 by selling up a two-player game instead of an I + J + I one'· The first player is an aggregate consumer-firm that has L, X, - IL' w,} - LI ~ as its strategy set; the second is, as before, a market agent having 6. as its strategy set. Given p E 6., the first agent responds with the set of vectors z expressible as z = 1:. x, - L, w, - LI YI' where Jj is profit maximizing in ~ for every j, and x, E X, is such that (I) p' x, :5: P'W, + LI 0'1 p' Yj and (2) x, :::,x; whenever p'x; < p'W, + LI 0'1 p' YI' As before, the market agent responds with the set of q E 6. that maximize Z'q on 6.. Once this two-person game has been set up, the proof proceeds as for Proposition 17.BB.2. You should check this in Exercise 17.BB.9. If for any p E 6. the preference-maximizing choices of consumers, x,( p), and the profitmaximizing choices of firms, )j(p), were single valued, we could go one step further and consider a game with a single player (the market agent). Given p, we would then let the best response of the market agent be the set of price VectOrsqE 6. that maximizes [1:, x,{p) - L, w, - LI y)(p»)-q on Ii. In essence, this is what we did in the proof of Proposition I7.CI.
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North·Holland. Shoven. 1.. and J. Whalley. (1992). Applring Gtn"al £quilibrium. New York: Cambridge University Press. Sonnenschein, H. (1973)_ Do Walras' identity and continuity characterize the class of community excess demand runctions'! Journal of Economic Theory 6: 345-54. Smale. S. (1976). A convergent process of price adjustment and global Newton methods. Journal of MllIlrematical Economics 3: 107-20. Starr. R. (1969). Quasi-equilibria in markets with non--convex preferences. Econometrica 37: 25-38. V.uian, H. (1977). Non-Walrasian equilibria. Econometrica 45: 573-90. Walras, L. (1874). Elements d'Economie Politique Pure. Lausanne: Corbaz. [Translated as: EI('mems of Pure E(·(tn(tmic_~. Homewood, III.: Irwin, 1954.]
aggregation problem. Journal of Economic Theory 57: 1-35.
Hahn, F. (1982). Stability. Chap. 16 in Handbook of Mathematical Economics, vol. II. edited by K. Arrow. and M. Intriligator. Amsterdam: North-Holland. 87. But it is not so for the generalizations described in the previous paragraph. 88. This was the approach taken in Debreu (1959).
EXERCISES
17.B.IA Show that for a pure exchange economy with J = 1 and Y, = -R~, "J'j:5: 0, p'rt = 0, and p ~ 0" ir and only ir "yj E YI and p' yj ~ p' Yt for all Yt E YI ."
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------------------------------------------------------------------------------------------------17.0.2" Prove property (v) of Proposition 17.0.2. The proof of Proposition 17.B.2 in the text contains a hint. Recall also the following technical fact: any bounded sequence in RL has a convergent subsequence. 17.B.3" Suppose that z(·) is an aggregate excess demand function satisfying conditions (i) to (v) of Proposition 17.8.2. Let p. - P with some, but not all, of the components of P being zero.
17,C.3" Consider an exchange economy in which every consumer I has continuous, strongly monotone, strictly convex preferences, and w, » O. The peculiarity of the equilibrium problem to be considered is that the consumer will now pay a type of tax on his gross consumption; moreover, this tax can differ across commodities and consumers. We will also assume that total tax receipts are rebated equally across consumers and in a lump-sum fashion. Specifically, for every i there is a vector of given tax rates Ii = (t 11_ .•. tu) ~ 0 and for every price vector P » 0 the budget set of consumer i is 1
(a) Show that as n becomes large, the maximal excess demand is always obtained for some commodity whose price goes to zero. (b) Argue (if possible by example) that a commodity whose price goes to zero may actually remain in excess supply for all n. [Hint: Relative prices mailer.] 17.B.4" Suppose that there are J firms whose production sets Y ... , >J C RL are closed, " strictly convex, and bounded above. Suppose also that a strictly positive consumption bundle is producible using the initial endowments and the economy's aggregate production set Y = Li Ij (i.e., there is an x » 0 such that x e {LI w,} + Y). Show that the production inclusive aggregate excess demand function i(p) in (l7.B.3) satisfies properties (i) to (v) of Proposition 17.B.2. 17.B.S' Suppose that there are J firms. Each firm produces a single output under conditions of constant returns. The unit cost function of firm j is cJ(p), which we assume to be dilTerentiable. The consumption side of the economy is expressed by an aggregate excess demand function z(p). Write down an equation system similar to (l7.B.4)-(I7.B.5) for the equilibria of this economy. 17.8.6 C [Rader (1972)] Suppose that there is a single production set Yand that Y is a closed, convex cone satisfying free disposal. Consider the following exchange equilibrium problem. Given prices P = (P,' ... , pd, every consumer i chooses a vector V, e RL so as to maximize ?:;:, on the set {x,e Xi: P'V,:S; P'w" and x, ~ V, + y for some ye Y}. The price vector p and the choices v· = (vr, ... , vr> are in equilibrium ifL' v~ = L, w,. Show that, under the standard assumptions on preferences and consumption sets, the price vector and the individual consumptions constitute a Walrasian equilibrium for the economy with production. Interpret. 17.C.I' Verify that the correspondence f(·) introduced in the proof of Proposition l7.C.1 is convex· valued.
17.C.2C Show that a convex-valued correspondence z( -) defined on R~ + and satisfying the conditions (i) to (v) listed below (parallel to the corresponding conditions in Proposition 17.C.I) admits a solution; that is, there is a p with Oe z(p). (i) (ii) (iii) (iv) (v)
z(·) is upper-hemicontinuous.
{z;, ... , zi.} -
00.
[Hilll: If you try to replicate exactly the proof of Proposition l7.C.l you will run into difficulties with the upper-hemicontinuity condition. A possible three-step approach goes as follows: (I) Show that fo 0 small enough the solutions must be contained in 11, {p e 11: PI ~ dar all (2) argue then that for r > 0 large enough, one has z(p) c [-r, r]L for every p E 11,; finally, (3) carry out a fixed-point argument in the domain 11, x [-r, rlL. For an easier result, you could limit yourself to prove the convex-valued parallel to Proposition l7.e2. The suggested domain for the fixed-point argument is then 11 x [-r, rl'.
n;
R~: ~ (I + IIi)PIXIi :s; Wi}'
An equilibrium wilh laxes is then a price vector p »0 and an allocation (xr, ...• xl) with L. Wi such that every i maximizes preferences in Bi(p, P'W, + (II/XL" I"PIX"»'
LX~ =
(a) Illustrate the notion of an equilibrium with taxes in an Edgeworth box. Verify that an equilibrium with taxes need not be a Pareto optimum. (b) Apply Proposition 17.C.l to show that an equilibrium with taxes exists. (c) As formulated here, the taxes are on gross consumptions. If they were imposed instead on net consumptions, that is, on amounts purchased or sold, then (assuming the same rate for buying or selling) the budget set would be B,(p,
r,)
=
{x;eR~:p.(X'-W;) + ~l/lpli(x" -w,,)I:s; r,},
where the r, are the lump-sum rebates. In what way does this budget set differ from that described previously for the case of taxes on gross consumptions? Represent graphically. Notice the kinks. (d) Write down a budget set for the situation similar to (c) except that the tax rates for amounts bought or sold may be different. (c) (More advanced) How would you approach the existence issue for the modification described in (c)? 17.C.4 A Consider a pure exchange economy. The only novelty is that a progressive tax system is instituted according to the following rule: individual wealth is no longer p'W,; instead, anyone with wealth above the mean of the population must contribute half of the excess over the mean into a fund, and those below the mean receive a contribution from the fund in proportion to their deficiency below the mean. (a) For a two-consumer society with endowments w, = (1,2) and after-tax wealths of the two consumers as a function of prices.
Wz
= (2, I), write the
(b) If the consumer preferences are continuous, strictly convex, and strongly monotone, will the excess demand functions satisfy the conditions required for existence in Proposition I7.CI given that wealth is being redistributed in this way?
z(·) is homogeneous of degree zero.
For every p and z e z(p) we have p' z = 0 (Walras' law). There is 5 E R such that Zl> -5 for any z e z(p) and p. If p. _ p ",. 0, z· e z(p·) and PI = 0 for some t, then Max
Bi(p, Wi) = {Xi E
=
17.C.S" Consider a population of / consumers. Every consumer i has consumption set R~ and continuous, strictly convex preferences ?:;:i' Suppose, in addition, that every i has a household technology>; c RL satisfying 0 E t;. We can then define the induced preferences ?:;:~ on by Xi ?:;:~ xi if and only ifror any Yi e t; with xi + Yi ~ 0 there is y, e lj with x, + y, ~ 0 and Xi + Yi ?:;:,x; + yi (i.e., whatever can be done from xi, something at least as good can be obtained from Xi)'
R'.
(a) Show that induced preferences are rational, that is, complete and transitive. (b) Show that if t; is convex then induced preferences
?:;:~
are convex.
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(e) Suppose that goods are of two kinds: marketed goods and nonmarketed household goods. Initial preferences ;::;, care only about household goods, and initial endowments W, have nonzero entries only for marketed goods. Use the concept of induoed preferences to set up the equilibrium problem as one that is formally a problem of pure exchange among marketed goods. Discuss.
-
E X ERe I 5 E 5
17.0.S' Show by explicit computation that the index of the equilibrium of a one-consumer Cobb- Douglas pure exchange economy is + I. 17.E.I' Derive expressions (17.E.I) and (I7.E.2). 17.E.2' Derive expression (17.E.3).
17.C.6B Let L = 2. Consider conditions (il, (iii), and (iv) of Proposition 17.B.2 Exhibit four examples such that in each of the examples only one condition fails and yet the system of equations z(p) = 0 has no solution. Why is condition (ii) not included in the list?
17.E.3" Provide explicit utility functions rationalizing at a given price vector p the individual excess demands Z,( p) and matrices of price effects Dz,(p) constructed in the proof of Proposition 17.E.2.
17.0.IB Consider an exchange economy with two commodities and two consumers. Both consumers have homothetic preferences of the constant elasticity variety. Moreover, the elasticity of substitution is the same for both consumers and is small (i.e., goods are close to perfect complements). Specifically,
17.E.4" Consider the two-commodity case. Give an example of a function z(p) defined on = {(PI' P2)>> 0: E < (p,lp2) < (lIE)}, and with values in R2, that is continuous, is homogeneous of degree zero, satisfies Walras' law, and cannot be generated from a rational preference relation. Represent graphically the offer curve associated with this function. NOle that it goes through the initial endowment point and compare with the construction used in Figure 17.E.2.
u,(x", x,,) = (2x~,
+ X~,)"P
and
u,(x", x,,) = (Xf2
+ 2x~2)''''
r,
and p = -4. The endowments are w, = (1,0) and 002 = (0,1). Compute the excess demand function of this economy and verify that there are multiple equilibria.
17.E.SA Show that the choices represented in Figure 17.E.3 cannot be generated from consumers wilh endowment vectors bounded above by (1,1) and nonnegative consumption.
17.0.2' Apply the implicit function theorem to show that if J(v) = 0 is a system of M equations in N unknowns and if at jj we haye J(V) = 0 and rank DJ(jj) = M, then in a neighborhood of jj the solution set of J(') = 0 can be parameterized by means of N - M parameters.
I7.E.6 A Show that the excess demand function Z,(P) = e' - P,P, defined for IIpll = I. is proportionally one-to-one in the sense used in the general proof of Proposition 17.E.3 (at the end of Section 17.E).
17.0.3' Carry out explicitly the computations for Proposition 17.0.4.
17.E.7" Show directly that Ihe excess demand function z,(p) = e' - P,P used in the general proof of Proposition 17.E.3 salisfies the strong axiom of revealed preference.
17.0.4c Consider a two-commodity, two-consumer exchange economy satisfying the appropriate differentiability conditions on utility and demand functions. There is a total endowment vector 6i» O. Show that for almost eyery W,« 6i the economy defined by the initial endowments w, and W2 = 6i - w, has a finite number of equilibria. This differs from the situation in Proposition 17.0.2 in that total endowments are kept fixed. [Hint: You should use the properties of the Slutsky matrix.] 17.0.5" Consider a two-commodity, two-consumer exchange economy satisfying the appropriate differentiability conditions on utility and demand functions. Set the equilibrium problem as an equation system in the consumption variables x, e R~ and X2 e R~, the price variables p e R~, and the reciprocals of the marginal utilities of wealth )., e Rand ).2 e R (neglect the possibility of boundary equilibria). The parameters of the system are the initial endowments (00" (2) e R". Prove without further aggregation that (after deleting one equation and one unknown) the system satisfies the full rank condition of the transyersality theorem. 17.0.68 The setup is identical to Exercise 17.D.S except that an externality is allowed: The (differentiable) utility function of consumer I may depend on the consumption of COnsumer 2; that is, it has the form u,(x"x2) where X, is consumer i's consumption bundle [but we still have U2(X 2 )]. Equilibrium is defined as usual, with the proviso that consumer I takes consumer 2's consumption as given. Show that, generically on initial endowments (00" ( 2 ) E R", the number of equilibria is finite. 17.0.7 8 Suppose the agents of an overall exchange economy are distributed across N islands with no communication among them. Each island economy has three equilibria. (a) Argue that the number of equilibria in the overall economy is 3N • (b) Suppose now that the islands' economies are identical and that there is a possibility of communication across the islands: free and costless transportation of commodities. Show that then the number of equilibria is 3.
17.F.lc Show that expression (I7.F.2) gives rise to a negative semidefinite matrix of price effects. D:( pl. if initial endowments are proportional among themselves or if consumptions are proportional among themselves. 17.F.2' Complete the requested verification of Example 17.F.1. 17.F.3" There are four goods and two consumers. The endowments of the consumers are "'I = (W,I.W21'0.0) and 002 = (W , 2. Wu. 0,0). Consumer I spends all his wealth on good 3 while consumer 2 does the same on good 4. Specify some values of w, and W 2 for which the corresponding excess demand of this economy does not satisfy the weak axiom of revealed preference.
17.F.4A Suppose that there are L goods but that for every consumer there is a good such that at any price the consumer spends all his wealth on that good (perhaps goods are distinguished by their location). Show that the aggregate excess demand will satisfy the (weak) gross substitute property. 17.F.Sc Complete the missing steps of Example 17.F.2. 17.F.6 c Consider a two-consumption-good, two-factor model with constant returns and no joint production. In fact. suppose that the production functions for the two consumption goods are Cobb-Douglas. Consumers have holdings of factors and have preferences only for the two consumption goods. The economy is a closed economy (at equilibrium. consumption must equal production). Suppose that the two goods are normal and gross substitutes in the demand JUII(·tion of the consumers. Define an induced exchange economy for factors of production by assuming that at any vector of factor prices the two goods are priced at average cost and the final demand for them is met. Show that the resulting aggregate excess demand for factors of production has the gross substitute property and, consequently. that there is a unique equilibrium for the overall economy.
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------------------------------------------------------------------------------------------------example, be the system of excess demands corresponding to a subgroup of markets with the prices of commodities outside the group kept fixed.
17.F.7A Prove expression (17.F.3) for L = 2. 17.F.SA Show that expression (J 7.F.3) implies that the set of solutions to z(p) = 0 is convex. 17.F.9 S Consider an economy with a single constant returns production set Y. Preferences are continuous, strictly convex, and strongly monotone. Suppose that the feasible consumptions (Xl' ... ' x,) are associated with a Walrasian equilibrium. Assume, moreover, that no trade is required to allain these consumptions if Yis freely available to all consumers; that is XI - W, E Y for all i. Show then that those are the only possible equilibrium consumptions. 17.F.10A Show that expression (17.F.3) implies that Dz(p) is negative semidefinite at an equilibrium p.
=
=
17.F.J1 B Show that if z(p) 0, rank Dz(p) L - I, and Dz(p) is negative semidefinite, then, for any (, the (L - I) x (L - I) matrix obtained from Dz(p) by deleting the (th row and column has a determinant of sign (_I)L-'. [Hint: From Section M.D ofthe Mathematical Appendix you know that rank Dz(p) = L - I implies that the (L - I) x (L - I) matrix under study is nonsingular. Consider then Dz(p) - d.] 17.F.12" Show that if z(p) = 0 and Dz(p) has the gross substitute sign pallern, then the (L - I) x (L - I) matrix obtained from Dz(p) by deleting the Ith row and column has a
lIegalive dominanl diagonal (see Section M.D of the Mathematical Appendix for this concept)
and is therefore negative definite. 17.F.13 A Provide the missing computation for Example 17.F.3. B
17.F.14 Consider a firm that produces good lout of goods 1= 2, ... , L by means of a production function /(v" .. .• vd. Assume that /(.) is concave, increasing. and twice continuously differentiable. We say that I and I' are complements at the input combination v = (v" . .. , vd if 0' /(v)/ov, ov,. > O. (a) Verify that for the Cobb-Douglas production function f(v" ... , VL) = vi' x ... x ~, + ... + ~L !> I, any two inputs are complements at any •.
.r',
(b) Suppose that f(') is of the constant returns type. Show that at any. and for any there is an (' that is a complement to t at v.
t
(c) Suppose now that f(·) is strictly concave and that any two inputs are complements at any •. Let ",(p" ... , pel be the input demand functions. Show thaI, for any t, a.tlap, > 0, < 0, and < 0 for t' # I.
av,/ap,
a.tlaPr
(d) Discuss the implications of (a) to (c) for uniqueness theorems that rely on the gross substitute property.
(a) We say that g(.) satisfies the strong gross subslilule properly (SGS) if for some" > 0 every coordinate of the function "g(p) + p is strictly increasing in p and (IXU(P) + p) E [0. rJ" for every p E [0, r]H. Show that if g(p) has the SGS property then it also has the GS property. (b) Show by example that the GS property does not imply the SGS property. Establish, however, that if g(') is continuously differentiable and the GS property is satisfied then the SGS property holds.
From now on we assume that g(') satisfies the SGS property. (e) Show that there is an equilibrium, that is, a p with g(p) = O. Illustrate graphically for the case N = I. [Hinl: Quote the Tarski fixed point theorem from Section M.I of the Mathematical Appendix, or, if you prefer, assume continuity and apply Brouwer's fixed point theorem.] (d) Give an example for N = 2 where the equilibrium is not unique.
=
=
(e) Suppose that g(p) g(p') O. Show that there must be an equilibrium p' such that p. :2: P and p' :2: p'. Similarly, there is an equilibrium p- such that p- :S P and p- :S p'. [Hint: Apply the argument in (e) to the domain [Max {p" p;}. r] x ... x [Max {PH' pj,}, r].]
(f) Argue (you can assume continuity here) that the equilibrium set satisfies a strong and very special property, namely, that it has a maximal and a minimal equilibrium. That is. there are pm.. and pm;. such that g(pm .. ) = g(pm;") = 0 and pm;" S p!> p-' whenever g(p) = O. (g) Assume now that g(.) is also differentiable. Suppose that we know that at equilibrium, that is, whenever g(p) = 0, the matrix Dg(p) has a negative dominanl diagonal; that is, Dg(p)v« 0 for a .» O. Argue (perhaps non rigorously) that the equilibrium must then be
unique. (h) Suppose that g(') is the usual excess demand system for the first N goods of an economy with N + I goods in which the last price has been fixed to equal I and the overall (N + I)-good excess demand system satisfies the gross substitute property. Apply (g) to show that the equilibrium is unique. 17.F.17A [Becker (1962), Grandmont (1992)] Suppose that L = 2 and you have a continuum of consumers. All consumers have the same initial endowments; they arc not rational, however. Given a budget set, they choose at random from consumption bundles on the budget line using a uniform distribution among the nonnegative consumptions. Let z(p) be the average excess demand (= expected value of a single consumer's choice). Show that z(·) can be generated from preference maximization of a Cobb-Douglas utility function (thus the economy admits a positive representative consumer in the sense of Section 4.0).
17.F.lS" Consider a one-consumer economy with prOduction and strictly convex preferences. There is a system of ad valorem taxes I = (I" ... , Id creating a wedge between consumer and producer prices; that is, PI = (I + I,)q, where P, and q, are, respectively, the consumer and producer price for good t. Tax receipts arc turned back in lump-sum fashion. Write the definition of (distorted) equilibrium. Show that the equilibrium is unique if the production sector is of the Leontief type (a single primary factor, no joint production, constant returns) and all goods are normal in consumption. can you argue by example the nondispensability of the last normality condition? If this is simpler, you can limit your discussion to the case of two commodities (one input and one output).
17.G.l" Suppose that in an exchange economy (and with the normalization PL = I) we are given equilibrium prices p(w,) as a differentiable function defined as an open domain of the endowments of the first L - I goods of the first consumer, W, = (w", ... , w L -,.,). All the remaining endowments are kept fixed. Suppose that the demand function of the first consumer is strictly normal in the sense that Dw,x,(p, w,) » 0 through the relevant domain of (p, w,). Show then that for any';', and;; = P(';',), we have rank D.. i,(;;;';',) = L - 1 and rank Dp(';',) = L - I, where il(p; w,) is the excess demand function ~f the first consumer for the first L - 1 goods.
17.F.16C Suppose that g(p) = (g,(p), ...• gH(P» is defined in the domain [0, r]H and that g(O, . .. ,0) » (0, ... ,0), g(r, ... , r) « (0. ...• 0). Note that we do not assume Walras' law, homogeneity of degree zero, or, for that maller, continuity. The function g(') could, for
it;;;
17.G.2" The setting is as in Exercise 17.G.1 or as in Proposition 17.G.2. Suppose that cD,) = O. Show that there are economies with D,£(;;; cD,) an (L - I) x (L - I) negative definite matrix but where op,(,;")/aw,, > O. [Hint: Use Proposition 17.G.I and the arguments employed in its proof.]
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------------------------------------------------------------------------------------------------I7.G.3C The setting is a. in Exercise 17.F.16. except that now we have two functions g(p) ERN and (J(p) eRN. Each of these functions satisfies the conditions of Exercise 17.F.16 (in particular the SDS property). In addition. we assume that b(-) is an upward shift of g('); that i•• (J(p) ~ g(p) for every pE [O.r]N. Prove that if (pml •• p .... ) and (~ml •• P"''') are the minimal and maximal equilibrium price vectors (see Exercise 17.F.16) for g(') and b('), respectively. then pmln ~ pmln and pm.. ~ pm... [You can assume that g(') and !I(') are continuous; if this makes things simpler. assume also that both functions have a unique solution.] Represent graphically for the case N = I. 17.H.l c Suppose that the system of excess demand functions z(p) satisfies the gross substitute property. Consider the tatonnement price dynamics dp, = Z/(P) dt
For any price vector p let o/t(p)
for every
t.
(0)
= Max {z,(p)/p, •...• zL(p)/pd.
(I) Argue that if pCt) is a solution for the above tatonnement dynamics (i.e .•
dp,(t)/dt - ZI(pCt» for every t and t) and z(pCO» ¢ 0 then o/t(pCt» should be decreasing through time. [Him: If ZI(pCt»/PI(t) = o/t(pCr)) then PI(t)/PI'(t) cannot decrease at t for any t'. Hence. Z/(pCt» cannot increase. whereas P, surely increases.]
(b) Argue that p(t) converges to an equilibrium price as t dynamics (0) Walras' law implies that LI pJ(t) = constant.]
00 •
[Hint: Recall that for the
17.H.2" There is an output good and a numeraire. The price of the output good is p. The data of our problem are given by two functions: The consumption side of the economy provides an excess demand function z(p) for the output good. and the production side an increasing inverse output supply function pCz). Both functions are differentiable. In addition. their graphs cross at (1.1). which is the equilibrium we will concentrate on in this exercise. Given this selling we can define two one-variable dynamics: (i) In Walras price dynamics we assume that at p the price increases or decreases according to the sign of the difference between excess demand and (direct) supply at p. (ii) In Marshall quantity dynamics we assume that at z production increases or decreases according to the sign of the difference between the demand price (i.e .• the inverse excess demand) and the supply price (i.e.• p(z» at z. (a) Write the above formally and interpret economically. (b) Suppose that the technology is nearly of the constant returns type. Show then that around the equilibrium (1.1) the system is always Walrasian stable but that Marshallian stability depends on the slope of the excess demand function (in what way?). (c) Write general price and quantity dynamics where prices move II la Walras and quantities la Marshall. Draw a (P. z) phase diagram and argue that in the typical case dynamic trajectories will spiral around the equilibrium.
a
(d) Go back to the technology specification of (b). Show that the system in (c) is locally stable if and only if the equilibrium is Marshallian stable.
17.1.1A Argue that the replica procedure described at the beginning of Section 11.1 does effectively include the case where the numbers of consumers of different types are not the same (a"ume, for simplicity. that the proportions of the different types are rational numbers). [Hint: Redefine the size of the original economy.]
v'.
17.1.2A Consider for a one-input. one-output problem the production function q = where " is thc amount of input. Show that the corresponding production set Y is additive but that the smallest cone containing it. yo. is not closed. Discuss in what sense the nonconvexity in Y is large. Argue that, whatever the number of consumers, there is no useful sense in which an equilibrium (nearly) exists. 17.1.3" There arc three commodities: the first is a high-quality good. the second is a low·quality good, and the third is labor. The first and second goods can be produced from labor according to the production functions f,(v) = Min {v. I} and fiv) = Min {v', I} for 0 < /1 < I. The economy has one unit of labor in the aggregate. Labor has no utility value. There are two equally sizcd classes of agents, with a very large number of each. "Rich" and "poor" have identical endowments, but the rich own all the shares in the firms of the economy. The rich spend all their wealth on the high-quality good; the poor must buy either one quality or the other-they cannot buy both. The utility function of the poor is U(XI' x,) = x, + lx" defined for (x,. x,) not both positive. (a) Which standard hypothesis of the general model docs this economy fail to satisfy? (h) Show that there can be no equilibria other than one in which both qualities of product arc produced. (e) Show that an equilibrium exists. 17.AA.IA Consider an exchange economy in which the preferences of consumers are monotone, strictly convex, and represented by the utility functions (u,(·) •... , u,(·». Show that for any (-" ......,) »0 there can be at most one Pareto optimal allocation x = (x" ... , x,) such that (III (x;) • ... , u,(x,)) is proportional to (5, •. . .• 5,). 17.AA.2" Consider the welfare-theoretic approach to the equilibrium equations described in Appendix A (the Negishi approach). The existence of a solution to the system of equations q(s) = 0 defined there follows from a fixed-point argument similar to the one carried out in Proposition 17.C.2. Assume that you are in an exchange economy with continuous, strictly convex and strongly monotone preferences, and that w, »0 for every i. Assume also that yes) turns out to be a function rather than a correspondence (a sufficient condition for this is that preferences be representable by differentiable utility functions and that at every Pareto optimal allocation at least one consumer gets a strictly positive consumption of every good). (a) Show that yes) is continuous. (b) Show that yes) satisfies a sort of Walras' law:
"L' y,(s) = 0, for every 5."
(e) Show that if s, = 0 then g,(s) > O. [Hint: If 5, = 0 then u,(x,(s» = 0 and so pCs)' x,es) = 0.] (d) Complete the existence proof. (Note that g(s) is also defined for 5 with zero components. This makes mailers simpler.)
(e) Consider the simplest price and quantity dynamics in the limit case where there are constant returns and excess demand is also a constant function. Draw the phase diagram. Suppose now that the quantity dynamics is modified by making the quantity responses depend not only on price and cost but also on the "expectation of sales. that is. on the excess demand. Will this have a stabilizing or a destabilizing effect?
17.AA.3" Suppose that. in an exchange economy, consumption sets are R~ and preferences are representable by concave. increasing utility functions u,(·). Let f1 = p. E R~: L, i., = I} be a simplex of utility weights. Suggest an equation system for Walrasian equilibrium that proceeds by associating with every i. a linear social welfare function.
17.H.3A For L = 3 draw an example similar to Figure 17.H.2 but in which there is a single equilibrium that. moreover. is locally totally unstable. Could you make it a saddle?
17_BB.1A Give a graphical example (for L = 2) of a Walrasian quasiequilibrium with strictly positive prices that is not an equilibrium for an economy in which:
fl
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-------------------------------------------------------------------------------------------------(i) (ii) (iii) (iv)
For For For For
every j, Ij = -R~. every i, XI is nonempty, closed, convex and satisfies XI + R~ C XI' every i, preferences arc continuous, convex, and strongly monotone. every i, WI e XI'
Why does this example not contradict any result given in the text (see the small-type discussion after the proof of Proposition 17.BB.I)? 17.BB.2" Consider an economy in which every consumer desires only a subset of goods and has holdings of only some goods. For the commodities desired, however, the preferences of the consumer are strongly monotone (they' are also continuous) on the corresponding nonnegative orthant. Suppose in addition that Li WI » 0 and that the economy satisfies the following jlldemmposabililY condition: It is not possible to divide consumers into two (nonempty) groups so that the consumers of one of the groups do not desire any of the commodities owned by the consumers of the other group.
Show then that any Walrasian quasiequilibrium is an equilibrium. 17.BB.3c Consider an Edgeworth box where preferences are continuous, strictly convex and locally nonsatiated (but not necessarily monotone). Suppose also that frcc disposal of commodities is not possible. Argue that, nonetheless, the offer curves must cross and, therefore, Ihat an eq uilibrium exists. Show that at equilibrium the two prices cannot be negative. In fact, at least one price must be positive (this is harder to show). 17.BB.4A Prove that if (x', y', p) is a free-disposal quasiequilibrium and Y, satisfies free disposal, then we can get a true quasiequilibrium by changing only the production of firm I. 17.BB.5 A Provide the missing step in the proof of Lemma 17.BB.5 (that is, show that the convexity of preferences implies that XI >-1 X .. and xj >-,x .. cannot both occur for Xi.
=
IXXI
+ (I
-
IX)X;).
17.BB_6 A Complete the proof of Lemma 17.BB.S by verifying the convexity of YJ(x, y, p) and of p(", y, pl. 17.BB.7A Complete the proof of Lemma 17.BB.6 by verifying the upper hemicontinuity of the correspondences jiJ(') and ;;(.). 17.BB.S" [Existence with production externalities; see Chipman (1970) for more on this topic.] There are L goods. Good L is labor and it is the single factor of production. Consumers have consumption set R~, continuous, strongly monotone, and strictly convex preferences, and endowments only of labor. Good t = I, ... , L - I is produced in sector t, which is composed of )1 identical firms. The production function of a firm in sector ( is it("t) = IXt"~' for 0 < PI :s; I. The peculiarity of the model is that the productivity coefficient IXt will not be a constant but will depend on the aggregate use of labor in sector t. Precisely, IXt
=
Yt(t "n)",
Yt> 0 and Pt
~ O.
(a) Define the notion of Walrasian equilibrium. Assume in doing so that individual firms neglect the effect on IXt of their use of labor. To save on notation, suppose also that profit shares are equal across consumers. (b) Prove the existence of a Walrasian equilibrium for the current model (make the standard additional assumptions that you find necessary). [Him: The general proof of Appendix B needs very few adaptations.]
EX ERe I S E S
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-------------------------------------------------------------------------------------------(c) Derive and represent the aggregate production set of each sector. Which conditions on the parameters (Jt, Yt, Pt guarantee that the aggregate production set of sector t exhibits increasing. constant, or decreasing returns to scale? (d) Note that the existence conditions of (b) may be satisfied while the aggregate production set is not convex. What would happen if the externality of sector ( were internalized by putting all the firms of the sector under joint management? (e) Suppose that L = 2, (JI = 1 and individual preferences are quasilinear in labor; that is, they admit a utility function ",(x,,) + Xli' Discuss, both analytically and graphically, the bias of the equilibrium level of production relative to the social optimum.
17.88.9" Carry out the existence argument for the two-player-game approach described at the end of Appendix B.
C
Some Foundations for
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T
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18.A Introduction
AND
Definition 18.B.1: A coalition Sc:I improves upon, or blocks, the feasible allocation x· = (xf, ... , xn e R\I if for every i E S we can find a consumption Xi ~ 0 with the properties: (i)
Xi">-i Xi
for every i e S.
(ii) Los Xi e Y + {LoS W;}. Definition \S.B.\ says that a coalition S can improve upon a feasible allocation
x' if there is some way that, by using only their endowments L i d WI and the publicly available technology Y, the coalition can produce an aggregate commodity bundle that can then be distributed to the members of S so as to make each of them better off. I. The constant returns assumption is important. With general production sels the difficulty is Ihat we cannot avoid being explicit aboul ownership shares. However, these have been defined to be profll shares, which makes our conceptual apparatus dependent on the very notion of prices whose emergence we are currently trying to explain. Thus we stick here to the case of constant returns. This is not a serious restriction: recall from Section S.B (Proposition S.B.2) that it is always possible to reduce generailechnologies to the constant returns case by reinterpreting the ownership shares as endowments of an additional "managerial" input.
18.B Core and Equilibria The theory to be reviewed in this section was proposed by Edgeworth (IS81). His aim was to explain how the presence of many interacting competitors would lead to 652
CORE
the emergence of a system of prices taken as given by economic agents, and consequently to a Walrasian equilibrium outcome. Edgeworth's work had no immediate impact. The modern versions of his theory follow the rediscovery of his solution concept (known now as the core) in the theory of cooperative games. Appendix A contains a brief introduction to the theory of cooperative games; this section, however, is self-contained. For further, and very accessible, reading on the material of this section, we refer to Hildenbrand and Kirman (19SS). The theory of the core is distinguished by its parsimony. Its conceptual apparatus does not appeal to any specific trading mechanism nor does it assume any particular institutional setup. Informally, the notion of competition that the theory explores is one in which traders are well informed of the characteristics (endowments and preferences) of other traders, and in which the members of any group of traders can bind themselves to any mutually advantageous agreement. The simplest example is a buyer and a seller exchanging a good for money, but we can also have more complex arrangements involving many individuals and goods. Formally, we consider an economy with I consumers. Every consumer i has consumption set R\, and endowment vector WI ~ 0, and a continuous, strictly convex, strongly monotone preference relation ;::" There is also a publicly available constant returns convex technology Y c: RL.' For example, we could have Y = - R\, that is, a pure exchange economy. All of these assumptions are maintained for the rest of the section. As usual, we say that an allocation x = (x" ... , XI) E R\I is feasible if:L XI = Y + LIW; for some ye Y. With a slight abuse of notation, we let the symbol I stand for both the number of consumers and the set of consumers. Any nonempty subset of consumers Sc:I is then called a coalition. Central to the concept of the core is the identification of circumstances under which a coalition of consumers can reach an agreement that makes every member of the coalition better off. Definition IS.B.! provides a formal statement of these circumstances.
Competitive Equilibria
Up to this point of Part IV, the existence of markets in which prices arc quoted and taken as given by economic agents has been assumed. In this chapter, we discuss four topics that, in essence, have two features in common: The first is that they all try to single out and characterize the Walrasian allocations from considerations more basic than those stated in its definition. The second is that they all emphasize the role of a large number of traders in accomplishing this task. In Section IS.B we introduce the concept of the core, which can be viewed as embodying a notion of unrestricted competition. We then present the important core equivalence theorem. Section IS.C examines a more restricted concept of competition: that taking place through well·specified trading mechanisms. The analysis of this section amounts to a reexamination in the general equilibrium context of the models of noncooperative competition that were presented in Section 12.F. The motivation of the remaining two sections is more normative. In Section IS.D we show how informational limitations on the part of a policy authority (constrained to use policy tools relying on self-selection, or envy freeness) may make the Walrasian allocations the only implementable Pareto optimal allocations. In Section IS.E the objective is to characterize the Walrasian allocations, among the Pareto optimal ones, in terms of their distributional properties. In particular, we ask to what extent it can be asserted that at the Walrasian allocation everyone gets her "marginal contribution" to the collective economic well-being of society. A number of the ideas of this chapter (especially those related to the core, but also some in Section IS.E) have come to economics from the cooperative theory of games. This therefore seems a good place to present a brief introduction to this theory; we do it in Appendix A.
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------------------------------------------------- ------------------------------------------------x"
~-+--------------~------------~~
Flgur. ".8.2 ~:---+-2:;,
Figure 18.B.l
~2
Definition 18.B.2: We say that the feasible allocation x' = (x! ..... xi) E R~I has the core property if there is no coalition 01 consumers ScI that can improve upon x*. The core is the set 01 allocations that have the core property. We can see in the Edgeworth box of Figure 18.B.1 that for the case of two consumers the core coincides with the contracl curve. With two consumers there are only three possible coalitions: {I, 2}, {I}, and {2}. Any allocation that is not a Pareto optimum will be blocked by coalition {I, 2}.2 Any allocation in the Pareto set that is not in the contract curve will be blocked by either {I} or {2}. With more than two consumers there are other potential blocking coalitions, but the fact that the coalition of the whole is always one of them means that all allocations in lite core are ParelO optimal. We also observe in Figure 18.B.I that the Walrasian equilibrium allocations, which belong to the contract curve, have the core property. Proposition 18.B.1 tells us that this is true with complete generality. The proposition amounts to an extension of the first welfare theorem. Indeed, in the current terminology, the first welfare theorem simply says that a Walrasian equilibrium cannot be blocked by the coalition of the whole.' The following result, Proposition 18.B.I, shows that it also cannot be blocked by any other coalition. Proposition 18.B.1: Any Walrasian equilibrium allocation has the core property. Proof: We simply duplicate the proof of the first welfare theorem (Proposition J6.C.I). We present it for the exchange case. See Exercise 18.B.1 for the case of a general constant returns technology. Let x' = (x!, ... , be a Walrasian allocation with corresponding equilibrium
xn
2. With continuity and strong monotonicity of preferences. if a feasible allocation is Pareto dominated. then it is Pareto dominated by • reasible allocation that strictly improves the utility of ('t'('ry consumer. To accomplish this we simply transfer a very sman amount of any good from the consumer that is made better off to every other consumer. If the amount transrerred is sufficiently small then. by the continuity or prererences, the transrerring consumer is still beller olT. while. by strong monotonicity. every other consumer is made strictly better otT. 3. Keep in mind the point made in rootnote 2.
O,~--------------------------~~r_~
The core equals the contract curve in the two-consumer casco
price vector p ~ O. Consider an arbitrary coalition ScI and suppose that the consumptions (x.} •• s are such that >-. for every i E S. Then p'X, > p·w. for every i E S and therefore P·(L •• s x.) > P·(L •• s w;). But then L •• s Xi :s: L.d w, cannot hold and so condition (ii) of Definition 18.B.1 is not satisfied (recall that we are in the pure exchange case). Hence coalition S cannot block the allocation x' . •
x. x:
The converse of Proposition 18.B.1 is, of course, not true. In the two-consumer economy or Figure 18.B.1 every allocation in the contract curve is in the core, but only one is a Walrasian allocation. The core equivalence theorem, of which we will soon give a version, argues that the converse does hold (approximately) if consumers are numerous. Quite remarkably, it turns out that as we increase the size of the economy the non-Walrasian allocations gradually drop from the core until, in the limit, only the Walrasian allocations are left. The basic intuition for this result can perhaps be grasped by examining the Edgeworth box in Figure 18.B.2. Take an alloca tion such as x where consumer I receives a very desirable consumption within the contract curve. Consumer 2 cannot do anything about this: She could not end up better by going alone. But suppose now that the preferences and endowments in the figure represent not individual consumers but types of consumers and that the economy is actually composed of four consumers, two of each type. Consider again the allocation x, interpreted now as a symmetric allocation, that is, with each consumer of type 1 receiving x I and each consumer of type 2 receiving x 2 • Then matters are quite different because a new possibility arises: The two members of type 2 can form a coalition with one member of type I. In Figure 18.B.2, we see that the allocation x can indeed be blocked by giving x; to the one consumer of type 1 in the coalition and x; to the two consumers of type 2 [note that - 2(x; - W2) = (x; - W I )].4 4. Observe that all this has the flavor or Bertrand competition, as reviewed in Section 12C. Indeed. we can look at what happens with this three-member coalition as the rollowing: One or the consumers or type 1 bids away the transactions or the COnsumers or type 2 with the other consumer or type I. Although this is a topic we shall not get into, we remark that, in fact, there are strong parallels between Bertrand price competition and core competition. Note, in particular. that core competition is as shortsighted as Bertrand competition. By undercutting the other consumer or her type. the consumer or type I is only initiating a process or blocking and counterblocking (mutual underbidding in the Bertrand selling) that eventually leads to a result (perhaps th. Walrasian allocation) where she will be worse off than at the initial position.
An allocation in the contract curve that can be blocked with two replicas.
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--------------------------------------------------------------------------------------------The ability to do this depends. of course. on the way we have drawn the indifference curves. Nonetheless. as we will see. we are always able to form a blocking coalition of this sort if we have sufficiently many consumers of each type. The version of the core equivalence theorem that we will present is in essence the original of Edgeworth. as generalized by Debreu and Scarf (1963). It builds on the intuition we have just discussed. To begin, let the set H = {I, ... , H} stand for a set of types of consumers, with each type 10 having preferences ;::, and endowments w,. For every integer n > 0, we then define the N-repliCll economy as an economy composed of N consumers of each type. for a total number of consumers IN = N H. We refer to the allocations in which consumers of the same type get the same consumption bundles as eqlwl-treatment "l1ocClliolls. Proposition 18.B.2 shows that any allocation in the core must be an equal-treatment allocation. (We hasten to add that this is true for the current replica structure, where there arc equal numbers of consumers of each type. It does not hold in general; see Exercise IS.B.2.) Proposition 18.B.2: Denoting by hn the nth individual of type h, suppose that the allocation x· = (XT1' ... • xrnl' .. • xtN" .. . xj." • ... xl.tn • ... . X;"N) E R~HN I
belongs to the core of the N-replica economy. Then x' has the equal-treatment property. that is, all consumers of the same type get the same consumption bundle: for all 1 :!> m. n:!> Nand 1 :!> h:!> H. Proof: Suppose that the feasible allocation x = (X" •...• X"N) E R/;."N does not have the equal-treatment property because. say. x, .. 1- x .. for some III 1- n. We show that x does not have the core property. In particular, we claim that x can be improved upon by any coalition of H members formed by choosing from every type a worst-treated individual among the consumers of that type. Suppose without loss of generality that, for every h. consumer h I is one such worse-off individual. that is. x" for all hand n. Define now the average consumption for each type: .", = (liN) L. x,•. By the strict convexity of preferences we have (recall that consumers of type I are not treated identically)
x,. ;::.
.x, ;::, x"
for all h
and
X,>-, X " ,
(IS.B.I)
We claim that the coalition S={II •...• hl, ...• HI}. formed by H members, can attain by itself the consumptions (x""" .x,,) E R~H. Therefore. by (IS.B.l). the original nonequal-treatment allocation can be blocked by S.' To check the feasibility of (.x I' •.•• XII) E R/;." for S, note that, because of the feasibility of x = (x", ...• x/IN) E 1R/+"N. there is Y E Y such that L. Ln x,. = y + N(Lh w,), and therefore
5. Recall that preferences are strongly monotone and continuous, so that if S can achieve an allocation that does strictly better than x· for some of its members. and at least as well as x· for all of them, then it can also achieve an allocation that does strictly better for all of its members.
S l ~ ; I 0 H
1
a . 8:
C 0 A E
AND
E
u U I LIB
RI A
657
-------------------------------------------------------------------------------------------But by the constant returns assumption on Y. (I/N)y E Yand so we conclude that (x l' ...• XII) E R';" is feasible for coalition S. • Proposition IS.8.2 allows us to regard the core allocations as vectors of fixed size LH. irrespective of the replica that we are concerned with. As a matter of terminology, we call a vector (x" ...• XII) E R~" a Iype allocation and, for any replica N. interpret it as the equal-treatment allocation to consumers where each consumer of type h gets x,. A type allocation (x l ' . . . , XII) E IR~" is feasible if L, x. = Y + L. W, for some )' E Y. Note that for any replica N the corresponding equal-treatment allocation is feasible because
and Ny E Y by the constant returns assumption on Y. By Proposition IS.8.2 the core allocations of a replica economy can be viewed as feasible type allocations. Define by eN C R~H the set of feasible type allocations for which the equal-treatment allocations induced in the N-replica have the core property. Note that eN docs depend on N. Nonetheless, we always have eN. I C eN because a type allocation blocked in the N-replica will be blocked also in the (N + I )-replica by a coalition having exactly the same composition as the one that blocked in the N-replica. Thus. as a subset of RLII the core can only get smaller when N - 00. At the same time, we know from Proposition IS.B.I that the core cannot vanish because the Walrasian equilibrium allocations belong to eN for all N. More precisely, the set of Walrasian type allocations is independent of N (see Exercise IS.8.3) and contained in all eN' The core equivalence theorem (which. in the current replica context, is the formal term for the combination of Propositions 18.B.I, IS.8.2 and the forthcoming Proposition IS.B.3) asserts that the Walrasian equilibrium allocations are the only surviving allocations in the core when N - 00.
Ri
Proposition 18.B.3: If the feasible type allocation x· = (x~, ...• xii) E H has the core property for all N = 1,2, ... , that is, x· E eN for all N. then x· is a Walrasian equilibrium allocation. Proof: To make the proof as intuitive as possible we restrict ourselves to a special case: a pure exchange economy in which. for every h. ;::, admits a continuously differentiable utility representation u,(') [with Vu,(x,)>> 0 for all x,]. In addition. the initial endowments vector w, is preferred to any consumption x, that is not strictly positive. This guarantees that any core allocation is interior. We emphasize that these simplifying assumptions are not required for the validity of the result. Suppose that x = (X, •... ' XII)E RLII is a feasible type allocation that is not a Walrasian equilibrium allocation. Our aim is to show that if N is large enough then x can be blocked. We may as well assume that X is Pareto optimal (otherwise the coalition of the whole blocks and we are done) and that x, » 0 (otherwise a consumer of type /0 alone could block). Because of Pareto optimality we can apply the second welfare theorem (Proposition 16.D.I) and conclude that X is a price equilibrium with transfers with respect to some p = (P,' ...• pLl.lf X is not Walrasian then there must be some 10, say /0 = I. with P'(X, - w,) > O. Informally. type 1 receives a positive net transfer from the rest of the economy and is thus relatively favored (interpretatively. think
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of type I as the most favored). We shall show that, as long as N is large enough, it would pay for the members of all the other types in the economy to form a coalition with N - I consumers of type I (i.e., to throw out one consumer of type I). More precisely, if a member of type I is eliminated then to attain feasibility the rest of the economy must absorb her net trade x, That, of course, presents no difficulty for the positive entries (those commodities for which the rest of the economy is a net contributor to this consumer of type I), but it is not so simple for the negative ones (the commodities where the rest of the economy is the net beneficiary). The most straightforward methodology is to simply distribute the gains and losses equally. In summary, our coalition is formed by (N - I) + N(H - I) members and, for every type I,. every member of type h gets
+ N(H
AND
We saw in Proposition 18.B.I that the half of the core equivalence theorem that asserts that Walrasian allocations have the core property generalizes the first welfare theorem. In its essence. the half asserting that. provided the economy is large, core allocations are Walrasian constitutes a version of the second welfare theorem. To understand this it may be useful to go back to the general (non replica) setup and formulate the property of a core allocation being Walrasian in terms of the existence of a price support for a certain set. For simplicity. we restrict ourselves to the pure exchange case. Given a core allocation x = (xt, ... , 4) e R~' then. in analogy with the construction used in the proof of the second welfare theorem (Proposition 16.0.1) we can define the setsy
I
_ I)
CORE
and, therefore, will also be individually favorable (recall Section 3.1 for similar arguments)" •
W,.
x; = x, + (N
1I.B:
V. =
_ I) (x, - w,).
{XI: XI >-IXrj U {WI} c: RL
Note that (N - I)x',
+ Nx~ + ... + Nx;' =
+ Nx, + ... + NXII + (x, x, + x, - W, I)w, + Nw, + ... + Nw, .
(N - I)x,
- w,)
We have L, WI E V. But there is more: Ihe CO" property for x· implies Ihal LI WI belongs Ihe houlldary of V. To see this, note that ifL' w, is in the interior of Vthen there is: e V such that:« L, w,; that is, there is x' = (xi •.. ·, X,) with x; E V. for every i and LI x; LI w,. Hence. x' is feasible, x' ". (w, •. .. ,w,), and, for every i, either x; >-1 xr or x; = WI' It follows that the set of consumers S = Ii: x;". W,} is nonempty. that x; >-,xr for every i E S, and that
= Nw, + ... + NWII = (N -
10
Hence, the proposed consumptions are feasible for the proposed coalition. Note also that the consumptions are nonnegative if N is large enough. For every h. every consumer of type h in the coalition moves from .x, to x;. Is this an improvement or a loss"! The answer is that if N is large enough then it is an unambiguous gain. To sec this. observe that p'(x, - w,) > 0 implies Vu,(x.)·(x, - w,) > 0 for every h because p and Vu,(x.) are proportional. As we can then see in Figure 18.B.3 (or. analytically, from Taylor's formula; see Exercise 18.B.4) there is Ii> 0 with the property that, for every h, u.(x. + IX(X, > u,(x,) whenever 0 < IX < Ii. Hence, for any N with (I/[(N - I) + N(H - I)]) < Ii the coalition will actually be blocking. Intuitively, we have done the following. The coalition needs to absorb x, - W,. Evaluated at the marginal shadow prices of the economy, this is a favorable "project" for the coalition since p'(x, - w,) > O. If the coalition is numerous then we can make sure that every member will have to absorb only a very small piece of the project. Hence the individual portions of the project will all be ~at the margin"
x, + (x, - "',)
I~«I~-I~=I~-I~=I~
to p'x;
change of a consumer
of type h in the blocking coalilion.
x; ~ .', +
I
(r -1)+ r(H -I)
".
(x, - w,)
ifS
hil
,_s
i.S
xr.
Figure 18.B.3
for p, > 0
I.'
Thus S is a blocking coalition. The next claim is that if P = (P""" pd ". 0 supports Vat LI w" that is, p': :2: P'(LI Wi) for all Z E V, then P must be a Walrasian price vector for x· = (xt, ... , xf). To verify this, note first that, for every i. we have xi >i xf for some xj arbitrarily close to Therefore, xi + L. .. w. E Vand so p-(x; + L •• I co,):2: P'(WI + L ... w.). Going to the limit (i.e., letting x; ~ xn this yields p'xr ~ P'W, for all i. Because LI xr s L.WI, we must therefore have p' xr = P'W, for all i.ln addition. whenever x;>-.xr we have P'(x; + L •• IW.):2: P'(w, + L •• I w.) and so p' x; :2: p·w,. If we exploit the continuity and strong monotonicity of preferences as we did in Section 16.0 (or in Appendix B of Chapter 17). we can strengthen the last conclusion
The consumption
VU.(x,) ~ ~,P
=:«
i~S
w,»
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>
p·W;.
The key difference from the case of the second welfare theorem (studied in Section 16.0) is that V c: R" does nol need 10 be convex and that therefore a nonzero peRI;. supporting V at L, w, may not exist. The reason for the lack of convexity is that the individual sets V, c: RL need not be convex: V. is the union of the preferred set at xr, which is convex, and the initial endowment vector which will typically be outside this preferred set and therefore disconnected from it. However, if Ihe (possibly nonconvex) sets V. c: RL being added are Ilumerous. then Ihe sum L. V. c: RL is "almost" convex. Thus, the existence of (almost) supporting prices for core allocations can be seen as yet another instance of the convexifying effects of aggregation. We end by mentioning an elegant approach to core theory pioneered by Aumann (1964) and Vind (1964). It consists of looking at a model where there is an actual continuum of consumers and where we replace all the summations by integrals. The beauty of the approach is that all the approximate results then hold exactly. The core equivalence theorem, for example,
w,.
6. See Anderson (1978) for a different line of proof 'hat makes minimal assumptions on the economy.
·· ... PTER
15
SOME
FOUNDATIONS
f")R
COMPETI1IVl;,
EQUIL,BRIA
--------------------------------------------------------------------------------------------takes the form: An allocation belongs to the core if and only if it is a Walrasian equilibrium allocation.
Definition lS.C.l: The profile of actions a* = (aT, ... , ail EA, x ... equilibrium if. for every i.
0 but the limit of pte, e) as e goes to zero remains bounded away from zero.'3 • Example IS.C.3: Tradillg Poses. This example belongs to a family proposed by Shapley and Shubik (1977). It is not particularly realistic but it has at least three 12. When every firm produces ,b'/J,the profils or one firm are ,b'/)). > I/y. Bul I/y is an upper bound ror Ihe profils or any firm Ihal deviates rrom Ihe suggeSled produclion by producing more. Hence an output level or rb'll ror every firm constitutes an equilibrium. 13. The complemenlarily makes it impossible ror .p to be continuously dillerenliable al Ihe origin. Thererore, p(' ) rails 10 be continuous. This is Ihe crucial aspecl ror Ihe example. NOle Ihal discontinuity at the origin is a natural occurrence: it will arise. ror example. whenever the indifference map or.p(·) is homothetic (bul not linear). See Harl (1980) ror more on Ihis issue.
663
ob~
... HAPTER
lei:
SOME
..
uv .... OAYlu ...
t-Gft
COMPETITIVE
fQUILIBRIA
S E to T I 0
~
\
a . LI.
T 11 E l l Mil &
l O R E 0 1ST R I 8 UTI 0 N
ootl
-------------------------------------------------- -------------------------------------------------------------------------.>
Figure lB.C.4
o
An effective budget set for the trading post Example IS.C.3.
x"
virtues: it constitutes a complete general equilibrium model. all of the participants interact strategically (in the two previous examples. consumers adjust passively). and it is analytically simple to manipulate. There are L goods and J consumers. Consumer i has endowment w, € R~. The Lth commodity. to be called "money." is treated asymmetrically. For each of the first L - I goods there is a trading post exchanging money for the good. At each trading post ( :5 L - I. each consumer i can place nonnegative bids a" = (ai" a;,) € IR~. The interpretation is that an amount ai i of good I is placed at the offer side of the trading post to be exchanged for money. Similarly. an amount a; i of money is placed in the demand side to be exchanged for good I. Accordingly. the bids are also constrained by aii :5 Wt; and LIS L-' (Iii :5 WLi' Given the bids of consumer i in the trading posts (:5 L - I and prices (p, •...• PL-,.I) the mechanism is completed by the trading rule: gt(a 1h
···,
ai i
aL-l,i; PI"'" Pl.-I' I) = -
-
, QIi
PI
for all t < L - I. The trade for the money good is derived from the budget constraint of the consumer. Given a vector a = (a" •...• aL-I."" .• a" •. ..• aL-I.,) of bids for all consumers. the clearing prices in terms of money are determined as the ratio of the amount of money offered to the amount of good offered:
ria;i
PI(a)=-~.
Lian
t
= I •...• L-I.
(IS.C.I)
Note that PI(a) is well defined and continuous except when there are no offers at the trading post I [i.e .• except when aii = 0 for all i]'" A typical effective budget set for agent i is convex and. provided that L .. i (Ii. "# 0 and L ... "# 0 for alii:::; L - I. it has an upper boundary containing no straight segments (you are asked to formally verify this in Exercise IS.C.I). This reRects the fact that as a consumer increases her bid in one side of a market the terms of trade turn against her. Figure IS.C.4 gives an illustration for the case L = 2.
a;.
t4. For the special. but important. case in which there is a single trading post (i.e .• L = 2). we can go a bit farther. When L, ail > 0 and L. a,. = O. the relative price of money is still well defined: it is zero. The essential difficulty in defining relative prices arises when L, aii = 0 and L.a,. = O.
It follows from expression (IS.C.I) that approximate price taking will prevail in any trading post that is thick in the sense that the aggregate positions taken on the two sides of the market are large relative to the size of the initial endowments of any consumer. A necessary condition for thickness is that there be many consumers. But this is not sufficient: it is possible even in a large economy to have equilibrium where some market is thin and. as a consequence. a trading equilibrium may be far from a Walrasian equilibrium. In fact. any trading equilibrium for a model where a trading post I is closed (i.e .• the trading post does not exist) will remain an equilibrium if the trading post is open but stays inactive. That is. if we put a" = (a;'" ail) = 0 for all i. Economically. this is related to Example 18.C.2: it takes at least two agents (here a buyer and a seller) to activate a market. Mathematically. the difficulty is again the impossibility of assigning prices continuously when at; = 0 for all i. Up to now. in this and previous examples. all of the instances of trading equilibria not approaching a Walrasian outcome when individual competitors are small have been related to failures of continuity of market equilibrium prices. But the current example also lends itself to illustration of the individual spanning problem. Indeed. even if markets are thick and therefore prices. from the individual point of view. are almost fixed. it remains true that the trading post structure imposes the restriction that goods can only he exchanged for money on hand (in macroeconomics this restriction is called the cash-in-advance. or the Clower. constraint). Money obtained by selling goods cannot be applied to buy goods. Therefore. for a given individual the Walrasian budget set will be (almost) attainable only if the initial endowments of money are sufficient. that is. only if at the solution of the individual optimization problem the constraint Lf " L-' WL' is not binding. But there is no general reason why this should be so. Suppose. to take an extreme case. that WL' = O. Then consumer i simply cannot buy goods at all. _
ar, : :;
18.D The Limits to Redistribution In Section 16.0 we saw that. under appropriate convexity conditions and provided that wealth can be transferred in a lump-sum manner. Pareto optimal allocations can be supported by means of prices. However, as we also pointed out there. a necessary condition for lump-sum payments to be possible is the ability of the policy authority to tell who is who-that is. to be able to precisely identify the characteristics (preferences and endowments) of every consumer in the economy. In this section, we shall explore the implications of assuming that this cannot be done to any extent; that is. we shall postulate that individual characteristics are private and become public only if revealed by economic agents through their choices. We will then see that under very general conditions the second welfare theorem fails dramatically: the only Pareto optimal allocations that can be supported involve no transfers. that is. they are precisely the Walrasian allocations. Thus. if no personal information of any sort is available to the policy authority. then there may be a real conAict between equity and efficiency: if transfers have to be implemented we must give up Pareto optimality. The nature of this trade-off is further explored in Sections 22.B and 22.C. We place ourselves in an exchange economy with J consumers. Each consumer i has the consumption set R~, the endowment vector w, ~ O. and the continuous. monotone. and strictly quasiconcave utility function u,(·).
666
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11:
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FOR
COMPETITIVE
EQUILIBRIA
SECTION
-------------------------------------------------------------------------------------Good 2
Good 2
11.0:
- '"')' w,
X~?:IX;.
o
(a)
(b)
LIMITS
TO
REDIITRI8UTION
Good I B
Figure 18.0.1
~
0 for all (
~
L - I and h.
If we have a sequence of finite economies (Ii, ... ,I';,) such that I' = :[.1. -+ 00 and (1/1')/. -+~. for every h, then we can properly regard (Il., ... , ~II) as the continuum limit of the sequence of increasingly large finite economies. Exercise IB.E.I: Show that the function v('): of degree one.
R~ -+
IR is concave and homogeneous
The function v(·) is a sort of production function whose output is social utility and whose inputs are the individual consumers themselves. Further, in the limit, every individual of type h becomes an input of infinitesimal size. For the time being, we concentrate our discussion on the continuum limit. We assume also that v(') is differentiable. 2 • Definition lB.E.l: Given a continuum population ~ = (~1" .. '~H) E R': a feasible allocation 22 (xf, ... , xk) is a marginal product, or no·surplus, allocation if
•
Uh(X h )
av(~)
=-a~h
for all h.
(18.E.3)
In words: at a no-surplus allocation everyone is getting exactly what she contributes at the margin. Proposition lB.E.l: For any continuum population ii = (ii, .. . ,iiH) »0 a feasible allocation (xf, ... , xk) » 0 is a marginal product allocation if and only if it is a Walrasian equilibrium allocation. Proof: If x· = (xt, . .. , x~) is a marginal product allocation then, using Euler's formula (see Section M.B of the Mathematical Appendix), we have
( _) v~
,,_ av(ii)
= 7:~'
a~,
,,_
= 7: ~,u,
(.)
x, .
Hence, x· solves problem (IB.E.2) for ~ = ii. Suppose now that x· = (xt, ... , x~) is a feasible allocation that gives rise to social utility vIii); that is, it constitutes a solution to problem (IB.E.2) for Il = ii. Denote by Pt, ( = I, ... ,L, the values of the multipliers of the first-order conditions associated with the constraints:[, ii,(xt. - (010) ~ 0, ( = I, ... , L, in the optimization problem (IB.E.2); see Section M.K of the Mathematical Appendix. By the quasilinear form of u,(') we have PL
for all (
:s L
=I
and
PI
= Vlt/I,(xt"
... , xl-I .• )
(IB.E.4)
- I and all I ~ h ~ H.
21. This could be derived from more primitive assumptions. 20. Because utility functions are concave the maximum utility can be reached while treating consumers of the same type equally.
P A INC I P L E
671
--------------------------------------------------------
22. We assume thai consumers of the samelype are treated equally. Feasibilily means therefore that
r,,, Jl"xt
:s;
LII P.W ...
672
CHAPTER
,.:
SOME
FOUNDATIONS
FOR
COMPETITIVE
EQUILIBRIA
--------------------------------------------------------------------------------------------It follows from (IS.E.4) that the vector of multipliers p = (P .. ... ,pd is the vector of Walrasian equilibrium prices of this quasilinear economy (recall the analysis of Section 10.0). In addition, by the envelope theorem (see Section M.L of the Mathematical Appendix), applied to problem (IS.E.2). we have (Exercise IS.E.2): ov(il) - = u. (*) x. 0/1.
+ p. (w. -
*) x •.
(IS.E.5)
Therefore. we conclude that x* is Walrasian if and only if x* solves problem (18.E.2) for /1 = /' and (18.E.3) is satisfied. that is. if and only if x* is a marginal product allocation. _ Expression (18.E.5) IS mtUltlve. The lert-hand side measures how much the maximum sum of utilities increases if we add onc cxtra individual of type II. The right-hand side tells us that there are two effects. On the one hand. the extra consumer of type It receives from the rest of the economy the consumption bundle and so she directly adds her utility u.(x:) to the social utility sum. On the other. while she contributes her endowment vector w •. Hence the net change for the receiving How much is this worth to the rest of the economy'! rest of the economy is w. The vector of social shadow prices is precisely p = (p, • ...• pd. and so the total change for the rest of the economy comes to p·(w. - x:>, Note that the Walrasian allocations are thus characterized by this second effect being null: the utility of the consumer equals her entire marginal contribution to social utility. In Exercise IS.E.4 you are asked to verify that the smoothness assumption on utility functions is essential to the validity of Proposition IS.E.!. Let us now consider a finite economy (I", ..• Ill) » O. We can define the marginal contribution of an individual of type h as
x:.
x:.
x:.
A.v(I ...... Ill) = v(I, •...• I ••...• Ill) - v(l, •...• I. - I •...• I,,).
Typically. there does not exist a feasible allocation (xT •...• x7,) with u.(x:l = A.v(/", .. , Ill) for all It. To see this. note that by the concavity of v(.) we have A.v ~ ov/O/1. [both expressions evaluated at (/, •... ,III)]. Except for degenerate cases. this inequality will be strict. Moreover. L. I.(ov/o/1.) = v(/, •...• Ill) by Euler's formula (see Section M.B of the Mathematical Appendix). and thus we conclude that L1I.(&.v) > v(I, •...• Ill); that is. it is impossible to give to each consumer the full extent of her marginal contribution while maintaining feasibility. [n contrast with the continuum case, individuals are not now of negligible size: their whole contribution is not entirely at the margin. [n particular. you should note that in a finite economy the Walrasian allocation is typically not a marginal product allocation. [t follows from expression (IS.E.5) that an allocation (xT •...• x7t) that solves problem (18.E.2) for (/1 , •...• /11l) = (I ...... 1/1) is a Walrasian equilibrium allocation if and only if
illl u.(x:) = - - (/, •...• I,,).
a/I.
But we have just argued that normally &.V(/" ...• Ill) > ov(l" . .. , IIl)/il/I•. [n words: At the Walrasian equilibrium consumers are compensated according to prices determined by the marginal unit of their endowments. But they lose the extra social surplus provided by the inframarginal units. This is yet another indication that the concept of Walrasian equilibrium stands on firmer ground in large economics.
- - - - ..... _ - - -
APPENDIX
A:
COOPERATIVE
GAME
THEORY
O/,.}
-----------------------------------------------------------------------------We have just seen that in the context of economies with finitely many consumers it is not possible to feasibly distribute the gains of trade while adhering literally to the marginal productivity principle. The cooperative theory of games provides a possibility for a sort of reconciliation between feasibility and the marginal productivity principle. It is known as the Shapley value. In Appendix A. devoted to cooperative game theory. we offer a detailed presentation of this solution concept. For an economy with profile (I, •... • Ill) the Shapley value is a certain utility vector (Sit, •...• Shll) E R"that satisfies L,l,Sh, = v(/, •...• I H ). For every type h.the utility Sh, can be viewed as an al'eraye oj marginal utilities d,v(/; •. ..• I~). The average is taken over profiles (I',' ...• /~) S (I" ...• (11 ), where the probability weight given to (I; •...• /~) equals 1//, interpreted as the probability assigned to sample size I; + ... + I~, times the probability of gelling thc profile (I; •...• 1;,) when independently sampling I; + ... + I~ consumers out of the original population with I consumers and profile (I, •...• I H ). See Appendix A for more on this formula. An allocation that yields the Shapley value (let us call it a Shapley alloration) is not related in any particular way to the Walrasian equilibrium allocation (or for that mailer to the core). Except by chance. they will be different allocations. Yet. remarkably. we also have a convergence of these concepts in economies with many consumers: the Walrasian and the Shapley allocations are then close to each other. This result is known as the value equivalence llieorelli. A rigorous proof of this theorem is too advanced to be given here [see Aumann (1975) and his references]. but the basic intuition is relatively straightforward. There are two key facts. First. if the entries of (I'" ...• /~) are large. then subtracting a consumer of type Ii amounts to very lillIe. and so
d,v(I', •...• I~) '" ov(/;, ...• I~)/o/". Second, if the entries of (I, •. ..• I H ) are large then, by the law of large numbers. most profiles (I; •.... I~) constitute a good sample of (I, •... • Ill) and are therefore almost proportional to (I, ... .• I,,). Using the homogeneity of degree one of v(') (hence the homogeneity of degree zero of "1'/ v(1) for any partition of I into two coalitions S. TJ.
680
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FOUNDATIONS
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COMPETITIVE
A P P E '" v I X
EQUILIBRIA
A:
COO PER A T I V EGA MET H E 0 R Y
players i, h E S, utility differences are preserved in a manner similar to the two-player case: ShieS, v) - S/,,(S\{h}. v) = Sh.(S, v) - Sh.(S\{i}, v)
",
for all ScI, i, h E S,
L ShieS, v) = v(S)
(IS.AA.2)
for all ScI,
iES
1~{2H-I----4'=-.J.._-~.----
Figure 1B.AA.8
o
u,
{(u,. u,):", + u, =
Egalitarian division for two-player games.
'111. 2})}
Expressions (IS.AA.2) determine the numbers ShieS, v), i E S, uniquely. This is clear for Sh,({i}, v). From here we can then proceed inductively. Suppose that we have dcfined S/,,(S, v) for all ScI, S 'I' I, i E S. We show that there is one and only one way to define Sh,(f, v), i E I. To this effect, note that (tS.AA.2) allows us to express every Sh,(I, v) as a function of Sh,(f, v) and of already determined numbers:
Shi(f, v) = Sh,(f, v) + Sh,(f\{ I}, v) - Sh,(1\{i}, v) Then to determine Sh,(f, ") use :[;,' Sh i(1. v)
a reasonable, or "fair," way to divide the gains from cooperation, taking as a given the strategic realities captured by the characteristic form.29 We study only the TV case, for which the theory is particularly simple and well established. The central concept is then a certain solution called the Shapley value. 30 Suppose that individual utilities are measured in dollars and that, so to speak, society has decided that dollars of utility of different participants are of comparable social worth. The criterion offairness to which value theory adheres is egalitarianism: the aim is to distribute the gains from trade equally. To see what the egalitarian principle could mean in the current TV context let us begin with a two-player game (I, v) = ({ I, 2}, v). Then the gains (or losses, if superadditivity fails) from cooperation are
v(l) - v({I}) - v({2}). Therefore, the obvious egalitarian solution, which we denote (Sh,(I, v), Sh 2 (1, v», is (see Figure IS.AA.S)
Sh;(I, v) = v({i}) + !
REFERENCES
18.B.4' Use Taylor's formula to complete Ihe proof of Proposition IS.B.3.
Anderson. R. (1978). An elementary core equivalence theorem. Econometrica 46: 83-87. Aumann. R. (1964). Markets with a continuum of traders. Econometrica 32: 39-50. Aumann. R. (1975). Values of markets with a continuum of traders. Econometrica 43: 611-46. Champsaur. P., and G. laroque. (1981). Fair allocations in large economies. Journal of Economic Theory
IX-B.S" Consider an economy composed of 21 + I consumerS. or these, I each own One right shoe and I + I each own a left shoe. Shoes are indivisible. Everyone has the same utility function. which is Min {R. L}. where Rand L are. respectively. the quantities of right and lerl shoes consumed.
25: 269-82. Debreu. G .• and H. Scarf. (1963). A limit theorem on the Core of an economy. inlr'nOlionat Economic Review ~: 235-46. Edgeworth. F. Y. (1881). Mathematical Psychics. London: Kegan Paul. Foley. D. (1967). Resource allocation and the public sector. Yale Economic Essays 7: 45-98.
Gabszewicz. J. J.• and J. P. Vial. (1972). Oligopoly·. la Cournot· in a general equilibrium analysis. Journal of EClmomic Theory 4: 381-400. Hart, O. (1980). Perfect competicion and optimal product differentiation. Journal of Economic Theory 11: 165-99. Hildenbrand. W. and A. Kirman. (1988). Equilibrium Anal),sis. New York: Norlh·Holland. Mas·Colell, A. (1982). The Cournotian roundations or Walrasian equilibrium: an exposition or recent theory. Chap. 7 in Advanct's in Economic Theory, edited by W. Hildenbrand. New York: Cambridge University Press. Moulin, H. (1988). Ax/o,.., of Cooperative Game Theory. New York: Cambridge University Press. Myerson, R. (1991). Game Theory: Analysis of Conflict. Cambridge, Mass.: Harvard UniversilY Press. Novshek, W.. and H. Sonnenschein. (1978). Cournot and Walras equilibrium. Journal of Economic Theory
19: 223-66. Roberts. K. (1980). The limit points of monopolislic competition. Journal of Economic Theory 22: 256-278. Osborne, M. and A. Rubinstein. (1994). A Course in Game Theory. Cambridge, Mass.: MIT Press. Ostray, J. (1980). The no-surplus condition as a characterization of perfectly competitive equilibrium. Journal of Economic Theory 22: 65-9\. Owen. G. (1982). Game Theory, 2nd ed. New York: Academic Press. Schmeidler, D. and K. Vind. (1972). Fair net trades. Econometrica 40: 637-47. Shapley. L.. and M. Shubik. (1977). Trade using a commodity as a means of payment. Journal of Political Economy 85: 937-68. Shubik, M. (1959). Edgeworth's market games. In Contributions to the Theory of Games. IV. edited by R. D. Luce. and A. W. Tucker. Princeton, NJ.: Princeton University Press. Shubik, M. (1984). Game Theory in the Social Sciences. Cambridge, Mass.: MIT Press. Thomson, W., and H. Varian. (1985). Theories of justice based on symmetry. Chap. 4 in Social Goals and Social Organizations, edited by L. Hurwicz, D. Schmeidler, and H. Sonnenschein. New Vorlc: Oxford University Press. Varian. H. (1976). Two problems in the theory of fairness. Journal of Public Economics S: 249-60. Vind, K. (1964). Edgeworth allocations in an exchange economy with many traders./nlt'rnalional Economic Review 5: 165-77.
(a) Show that any allocalion of shoes that is matched (i.e .• every individual consumes the same nllmhcr of shoes of each kind) is a Parelo optimum, and conversely. (b) Which Parelo oplima are in the COre of this economy? (This time, in Ihe definition of the COre allow for weak dominance in blocking.) (e) Let P. and Pi. be the respective prices of the two kinds of shocs. Find the Walrasian equilibria of this economy.
(d) Comment on Ihe relationship between the core and the Walrasian equilibria in this economy. IS.C.I'" ESlablish the properties of effective budget sets claimed in the discussion of Example I X.C.3. You can restrict yourself to the case L = 2. IS.O.I" Consider an Edgeworth box wilh continuous, strictly COnvex and monotone preferences. Show that every feasible allocalion where both consumers are at least as well off as al their initial endowments is self.selective. IS.E.I" In texl. \S.E.2' Use the envelope theorem (see Section M.L of the Mathemalical Appendix) to derive expression (IS.E.5). IS.E.3" By considering an example with L-shaped preferences for two non.numeraire goods (hence, the utility function cannol be differentiable), argue that it is possible that at a Walrasian allocalion with a continuum of traders every trader gets less than her marginal conlribution. 18.AA.I" A collection of coalitions S" ... , SH C I is a generalized partition if we can assign a weIght b, E [0. I] to every I S n S N such that, for every player i E I. we have LI" i.S.1 b, = 1. ExhIbit examples of generalized partitions, with the corresponding weights. We say thai a TU-game (I, v) is balanced if for every generalized partition we have L. I\, ..(S.) S ..(I). where b, are the corresponding partition weights. Show that the game has a nonempty Core if and only if it is balanced. [Hint: Appeal to the duality theorem of linear programming (see Section M.M of the Mathematical Appendix).] IS.AA.2' In texl. A
EXERCISES
I8.AA.3 Show that the proportional allocation of Example IS.AA.6 is the only allocation in the core if average product is constant.
18.B.IA Show that Walrasian allocations are in the core for the model with a constant returns technology described in Section 18.B.
18.AA.4C Show that if the Shapley value is defined by formula (lS.AA.4)-or, equivalently. by (18.AA.3 )-then the preservation of differences expression (IS.AA.2) is satisfied.
685
686
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18:
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FOR
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EOUILIBRIA
18.AA.5" We say that a game (I, v) is a unanimity game if there is a nonempty Sci such that v(T) = v(S) if SeT and v(T) = 0 otherwise. Show then that under the efficiency, symmetry, and dummy axioms we arc led to distribute v(S) equally across the members of S. 18.AA.6" Show that any TU-game (I, v) can be expressed as a linear combination of unanimity games. Then use the Exercise IS.AA.5 and the linearity axiom to show that there is a unique solution satisfying the efficiency, symmetry, dummy, and linearity axioms. Connect your discussion wilh the Shapley value.
CHAPTER
General Equilibrium Under
19
Uncertainty
18.AA.7c Show that the production game described in Example IS.AA.S is convex. 18.AA.8" In the context of the production example of Example IS.AA.S, give an example of a two-input production function that is convex (as a function) but for which, nonetheless, the core is empty (thus. the induced game cannot be convex). 18.AA.9" Consider the game with four players defined by v({i}) = 0, v({12}) = v({34}) = 0, = v({ 14}) = v({23}) = v({24}) = I, v({ijk}) = I for all three-player coalitions {ijk}, and ,,({I234}) = 2.
L{{ 13})
(a) Show that this is the game that you would get from the utility production technology Min {=" =,l, where z, and z, are the amounts of two factors, if the factor endowments of the four consumers are w, = w, - (1,0) and wJ = W. = (0,1). ,
(b) Show that the core of this game contains all points of the form (a. a, I - a, I E [0, I].
~)
for
(e) Show that if v({ 134}) is increased to 2, holding all other coalition values constant. there is then only one point in the core. Compare the welfare of player I at this point to what she would get at all the points in the core before the increase in v({I34}). (d) Compute the Shapley value of the game [before the modification in (e)) without using the brUle-force enumeration technique. [Hint: Use symmetry considerations and other axiomatically based simplifications to go part of the way to the answer.] (e) How does the Shapley value change under the modification of part (e)? Discuss the difference between the changes in the Shapley value and in the core. 18.AA.IO" Consider a firm constituted by two divisions. The firm must provide overhead in Ihe form of space. (x" x,), to each of them. The cost of aggregate amounts of space is given by C(x, + x,) = (x, + x,)', 0 < y < I. (a) Suppose that, whatever the usage of space (x" x,), the total cost must be exactly allocated between the two divisions. Propose a cost allocation system based on the Shapley value to accomplish this. (b) Compute the marginal cost imposed on each of the two divisions [according to the cost allocalion system identified in (a)) whenever a division increases its usage of space. (c) Suppose now that the profits accruing to the two divisions arc Il,X, and Il,X" respectively (we assume that 11, > 0 and 11, > 0), and that each division uses space to the point where marginal profits equal own marginal costs [as determined in (b)). Will this lead to an efficienl (that is, profit-maximizing) choice of overhead? (d) Is there any distribution rule ""(x,, x,), ""(x,, x,), with ""(x,, x,) + ""(x,, x,) = C(x, + .' ..• Ps) e RLS at I = I. every consumer i formulates a consumption. or trading. plan (z 1/ ••••• ZSi) e RS for contingent commodities at I = O. as well as a set of spot market consumption plans (x 1/ •••• , x,,) E RLS for the different states that may occur at t = I. Of course, these plans must satisfy a budget constraint. Let U,(') be a utility function for ;:::,. Then the problem of consumer i can be expressed formally as Max
(19.D.1)
(xl •• ··· • .xs.)ER~.s
(:u •. ·· ••s.)ERS
s.t.
(i)
L, q'Z,i
~
O.
(ii) p, x" ~ p,W"
+ p"Z"
for every s.
Restriction (i) is the budget constraint corresponding to trade at I = O. The family of restrictions (ii) are the budget constraints for the different spot markets. Note that the value of wealth at a state s is composed of two parts: the market value of the initial endowments. p, ·W". and the market value of the amounts z,' of good 1 bought or sold forward at t = O. Observe that we are not imposing any restriction on the sign or the magnitude of z". If z" < - W"i then one says that at t = 0 consumer i is selling good 1 shorl. This is because he is selling at t = 0, contingent on state s occurring. more than he has at I = I if s occurs. Hence, if s occurs he will actually have to buy in the spot market the extra amount of the first good required for the fulfillment of his commitments. The possibility of selling short is. however. indirectly 6. In principle, expectations could differ across consumers. but under the assumption of correct expeclations (soon 10 be introduced) they wiU nol.
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limited by the fact that consumption, and therefore ex post wealth, must be nonnegative for every s.' To define an appropriate notion of sequential trade we shall impose a key condition: Consumers' expectations must be self-fuljilled, or rational; that is, we require that consumers' expectations of the prices that will clear the spot markets for the different states s do actually clear them once date t = I has arrived and a state s is revealed. Definition 19.0.1: A collection formed by a price vector q = (q, •.. . ,qs) E contingent first good commodities at t = O. a spot price vector
W for
for every s. and. for every consumer i. consumption plans zi = (zT;•...• z$;) E RS at t = 0 and xi = (xT; . .... x$;) E IRLS at t = 1 constitutes a Radner equilibrium [see Radner (1982)] if:
1 9 • 0:
SEQ U E N T I • L
Proposition 19.0.1: We have: (i) If the allocation x· E R lSI and the contingent commodities price vector (P" ...• PsI E R~S+ constitute an Arrow-Oebreu equilibrium. then there are prices q E R~ + for contingent first good commodities and consumption plans for these commodities z· = (zT ..... zrl E RSI such that the consumptions plans x'. z·. the prices q. and the spot prices (p, • ...• PsI constitute a Radner equilibrium. (ii) Conversely. if the consumption plans x· E nlSI. z· E R SI and prices q E A~ +. (p, •...• PsI E n~s+ constitute a Radner equilibrium. then there are multipliers (II, •. ..• lIS) E n~ + such that the allocation x' and the contingent commodities price vector (II,P, •. .•• jlsPs) E R~s+ constitute an Arrow-Oebreu equilibrium. (The multiplier is interpreted as the value. at t = O. of a dollar at t = 1 and state s.)
I'.
Proof: (i) It is natural to let q, = P" for every s. With this we claim that. for every consumer i. the budget set of the Arrow-Oebreu problem. BtO = {(Xli' ... ' xS/) E R,!:
L, p,'(x"
- w,,) SO}.
is identical to the budget set of the Radner problem, (i) For every i. the consumption plans
zi. xi solve problem (19.0.1).
Br =
{(x 1/ ••••• x.,,)
E
IR'! : there are (z li • . . . • zs.> such that L. q,z" S 0 and P,(X,,- w,,) S P"Z" for every s}.
see this. suppose that X, = (Xli •...• X5') E Bto. For every s. denote (I/PI')p,·(x" - w,,). Then L,q,z" = L,PI'Z" = L,P,'(X" - w,,) s 0 and P.,·(x" - W,.> = P"Z" for every s. Hence, X, E Bf. Conversely. suppose that x,=(x" •...• xs,)EBr; that is. for some (z" •...• zs,) we have L,q,z" SO and p,(x" - w,,) S P"Z" for every s. Summing over s gives L,P,·(X" - w,,) S LJPtsZ:wi = Lsqszsi S O. Hence, Xi e Bto. We conclude that our Arrow-Oebreu equilibrium allocation is also a Radner equilibrium allocation supported by q = (Pi' •...• PIS) E RS• the spot prices (P, ....• PsI, and the contingent trades (zr, ..... zt,) E RS defined by =.;. = (I/p,,)p,·(x:' - w,,). Note that the contingent markets clear since. for every s. To
At a Radner equilibrium, trade takes place through time and, in contrast to the Arrow-Oebreu setting. economic agents face a sequence of budget sets, one at each date-state (more generally, at every date-event). We can see from an examination of problem (19.0.1) that all the budget constraints are homogeneous of degree zero with respect to prices. This means that the budget sets remain unaltered if the price of one physical commodity in each date-state (that is, one price for every budget set) is arbitrarily normalized to equal I. It is natural to choose the first commodity and to put p" = I for every s, so that a unit of the s contingent commodity then pays off I dollar in state S.8 Note that this still leaves one degree of freedom, that corresponding to the forward trades at date 0 (so we could put q, = I. or perhaps L, q, = I). In Proposition 19.0.1, which is the key result of this section, we show that for this model the set of Arrow-Oebreu equilibrium allocations (induced by the arrangement of one-shot trade in LS contingent commodities) and the set of Radner equilibrium allocations (induced by contingent trade in only one commodity, sequentially followed by spot trade) are identical. 7. Observe also that we have taken the wealth at I = 0 to be zero (that is. there are no initial endowments or the contingent commodities). This is simply a convention. Suppose. ror example. that we regard
Wlli~
the amount of good t available at t
:a=
1 in state s. as the amount of the s
contingent commodity that i owns at I = 0 (to avoid double counting. the initial endowment or commodity I in the spot market' at I = I should ,imultaneously be put to zero). The budget constraints are then: (i) L,q,(:~ - "'I") ~ 0 and (ii) p,·x,. ~ LI~I P"",, + Ph:;' ror every s. But letting
Z:i = tli +
W bl •
we see that these are exactly the constraints of (l9.D.1).
8. It rollows rrom the possibility or making this normalization that. without loss or generality. we could as well suppose that our contingent commodity pays directly in dollars (see Exercise 19.D.1 ror more on this).
TRA0 E
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---------------------~~~~~~~~-=
Z" =
L.=!
=
(l/p,,)P,·[L,(X:' -
w,,» SO.
(ii) Choose I', so that jl,P" = q,. Then we can rewrite the Radner budget set of every consumer i as
Bf
=
{(x " •...• xs.> E ALS : there are (z" •...• zs,) such that L,q,z" S 0 and jl,p,(x" - W,.> s q,z" for every s}.
But from this we can proceed as we did in part (i) and rewrite the constraints. and therefore the budget set. in the Arrow-Oebreu form:
Bf = BtO =
{(XI/ •...• Xs,)E
ALS: L,jl,P,·(x" - w,,) SO}.
x:
Hence, the consumption plan is also preference maximizing in the budget set Bt"· Since this is true for every consumer i. we conclude that the price vector LS (II. P,.···. lISPS) E A clears the markets for the LS contingent commodities. _ Example 19.0.1: Consider a two-good. two-state. two-consumer pure exchange economy. Suppose that the two states are equally likely and that every consumer has the same. state-independent. Bernoulli utility function u(x,,). The consumers differ only in their initial endowments. The aggregate endowment vectors in the two states
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Good 2
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p 0, Initial Endowment in State 2
Good I Good 2
are the same; however, endowments are distributed so that consumer I gets everything in state I and consumer 2 gets everything in state 2. (See Figure 19.0.1.) By the symmetry of the problem, at an Arrow-Debreu equilibrium each consumer gets, in each state, half of the total endowment of each good. In Figure 19.0.1, we indicate how these consumptions will be reached by means of contingent trade in the first commodity and spot markets. The spot prices will be the same in the two states. The first consumer will sell an amount ex of the first good contingent on the occurrence of the first state and will in exchange buy an amount {J of the same good contingent on the second state. (You are asked to provide the details in Exercise
19.0.2.) • It is important to emphasize that, although the concept of Radner equilibrium cuts down the number of contingent commodities required to attain optimality (from LS to S), this reduction is not obtained free of charge. With the smaller number of forward contracts, the correct anticipation of future spot prices becomes crucial. Up to this point we have discussed the sequential implementation of an Arrow-Debreu equilibrium when there are two dates,9that is, for the date-eventtree of Figure 19.B.1. Except for notational complications, the same ideas carryover to a tree such as that in Figure 19.B.2 where there are T + I periods and information is released gradually. (See the small· type discussion at the end of Section \9.B for basic concepts and notation.) We would then have spot markets at every admissible date-event pair IE (i.e., those IE where E e.9';, the information partition at r). With H the set of basic physical commodities, we denote the spot prices by P,. e RH. At every rE we could also have trade for the contingent delivery of physical good I at each of the sucoessor date-events to tE. Denote by q,.(t + I, E') the price at tE of one unit of good I delivered at r + I if event E' is revealed (of course, we require E' e .9';., and E' c: E). The problem of the consumer consists of forming utility-maximizing plans by choosing, at every admissible rE, a vector of consumption of goods X,E' e R~ and, for every sucoessor (r + I, E'), a contingent trade Z,E,(I + I, E') of good I deliverable at (I + I, E'). Overall, the budget constraint to be satisfied at tE is
+
ASSET
One can then proceed to define a corresponding concept of Radner equilibrium and to show that the Arrow-Debreu equilibrium allocations for the model with H(T + I)S contingent commodity markets'· at I = 0 are the same as the Radner equilibrium allocations obtained from a model with sequential trade in which, at each date-event, consumers trade only current goods and contingent claims for delivery of good I at successor nodes. Exercises 19.D.3 and 19. D.4 discuss this topic further.
Good I
P,.· x,"
1I.E:
I
q,.(r
+ I, E')z,.,(1 + I, E')
!> P,E'W'E;
+ PUEZ,- ..... ,(t, E)
1£'.,Y',.,:E'cEI
where E - is the event at the date
I -
I predecessor to event E at
I.
9. To be as simple as possible, we have also assumed that there is no consumption at 1=0.
Figure 19.0,1 Reaching the Arrow-Debreu equilibrium by means of contingent trade in the first good only.
19.E Asset Markets Thc S contingent commodities studied in the previous section serve the purpose of transfcrring wealth across the states of the world that will be revealed in the future. Thcy arc. however, only theoretical constructs that rarely have exact counterparts in reality. Nevertheless, in reality there are assels, or securities, that to some extent perform the wealth-transferring role that we have assigned to the contingent commodities. It is therefore important 10 develop a theoretical structure that allows us to study the functioning of these asset markets. We accomplish the task in this section by extending the formal notion of a contingent commodity and then generalizing the theory of Radner equilibrium to the extended environment.'1 We begin again with the simplest situation, in which we have two dates, I = 0 and t = I, and all the information is revealed at t = I. Further, for notational simplicity we assume that consumption takes place only at t = 1. We view an asset, or, more precisely, a unit of an asset, as a title to receive either physical goods or dollars at t = I in amounts that may depend on which state occurs. '2 The payoffs of an asset are known as its returns. If the returns are in physical goods, the asset is called real (a durable piece of machinery or a futures contract for the delivery of copper would be examples). If they are in paper money, they are called financial (a government bond, for example). Mixed cases are also possible, Here we deal only with the real case and, moreover, to save on notation we assume that the returns of assets are only in amounts of physical good 1.13 It is then convenient to normalize the spot price of that good to be I in every state, so that, in effect, we are using it as numeraire. Definition 19.E,1: A unit of an asset, or security, is a title to receive an amount f. of good 1 at date t = 1 if state s occurs. An asset is therefore characterized by its feturn vectof f = (f" . .. ,fS) € R S.
10. A contingent commodity is a promise to deliver a unit of physical commodity h at date t if state" occurs. Recall from Section 19.B that the consumption sets have to be defined imbedding in them the inrormation measurability restrictions. thai is, making sure that at date t no consumption is dependent on inrormation not yet available. II. See Radner (1982) and Kreps (1979) [complemented by Marimon (1987)] for treatmenls in the spirit of this section. 12. As usual "title to receive" means "duty to deliver" ir the amount is negative. Although negative returns present no particular difficulty. we will avoid them. 11 This assumption also has an important simplifying reature: At any given slate the returns of all assets are in units of the same physical good. Therefore, the relative spot prices of the various physical goods in any given state do not affect the relative returns or the different assets in that state.
MARKETS
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70t
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UNCERTAINTY
Example 19.E.l: Examples of assets include the following: (i) r = (1, ... , I). This asset promises the future noncontingent delivery of one unit of good 1. Its real-world counterparts are the markets for commodity futures. In the special case where there is a single consumption good (i.e., L = I). we call this asset the safe (or riskless) asset. It is important to realize that with more than one physical good a futures contract is not riskless: its return in terms of purchasing power depends on the spot prices of all the goods l 4
Dellnltlon 19.E.2: A collection formed by a price vector q = (q, • ... ,qK) E RK for assets traded at t = O. a spot price vector P. = (p, •• ...• PL.) E RL for every s. and. for every consumer i. portfolio plans zi = (zr;..... z.tl E RK at t = 0 and consumption plans xi = (Xri . ...• x~;) E R LS at t = 1 constit~tes a Radne, equilibrium if: (i) For every i. the consumption plans Max
zi. xi solve the problem
U,{X'i' .... XSi)
(Xli • ... JI's,)ER~S
(ii) r = (0•...• 0, I. 0•...• 0). This asset pays one unit of good I if and only if a certain state occurs. These were the assets considered in Section 19.0. In the current theoretical setting they are often called Arrow securities. (iii) r = (1,2. 1.2•...• 1.2). This asset pays one unit unconditionally and. in addition. another unit in even-labeled states. Example 19.E.2: Options. This is an example of a so-called derivative asset. that is. of an asset whose returns are somehow derived from the returns of another asset. Suppose there is a primary asset with return vector r E RS• Then a (European) call option on the primary asset at the strike price c E R is itself an asset. A unit of this asset gives the option to buy. after the state is revealed (but beforc the rcturns are paid). a unit of the primary asset at price c (the price c is in units of the "numeraire." that is. of good I). What is the return vector rIc) of the option? In a given state s. the option will be exercised if and only if r, > c (we neglect the case r, = c). Hence
(Z" •...• z~,)ER ..
s.t.
(a)
L. q.·Z.i ~ 0
(b) p, X'i !> P,W'i + L. P,.z./,.
for every s.
(ii) LiZki ~ 0 and LiX:i ~ L;W.; for every k and s. In the budget set of Definition 19.E.2, the wealth of consumer i at state s is the sum of the spot value of his initial endowment and the spot value of the return of his portfolio. Note that. without loss of generality. we can put Ph = I for all s. From now on we will do so. It is convenient at this point to introduce the concept the retllrn matrix R. This is an S x K matrix whose kth column is the return vector of the kth asset. Hence. its generic sk entry is r... the return of asset k in state s. With this notation. the budget constraint of consumer i becomes
B,j p, q,
B).1 "R~
, fo,"om' portfolio ,,' R'
~ ho~
q",
rIc) = (Max {O. rl - c} •... , Max {O, rs - c}).
Pt'(XII:- WII») [rtt ...... ruj_
For a primary asset with returns r = (4,3,2. I) specific examples are r(3.5) =
(.5. 0 • 0
(
p,'(xs, -
0).
r(2.5) = (1.5.
0.5. 0
r( 1.5) = (2.5.
1.5. 0.5, 0)._
0).
We proceed to extend the analysis of Section 19.0 by assuming that there is a given set of assets. known as an asset structure, and that these assets can be freely traded at date t = O. We postpone to the next section a discussion of the important issue of the origin of the particular set of assets. Each asset k is characterized by a vector of returns r. E RS• The number of assets is K. As before. we assume that there are no initial endowments of assets and that short sales are possible. The price vector for the assets traded at t = 0 is denoted q = (ql •... , qK)' A vector of trades in these assets. denoted by z = (z I' ••• , ZK) E R K, is called a portfolio. The next step is to generalize the definition of a Radner equilibrium to the current environment. In Definition 19.E.2. U;(·) is a utility function for the preferences ~; of consumer i over consumption plans xs;) R~s.
(XII .... '
~
. WS,)
'.
0 for every k. Also, without loss of generality. we assume thai no row of the relurn malrix R has all of its entries equal to zero," Given an arbitrage-free assel price veclor q E RK. consider Ihe convex set V= (UE
R':. = Rz for some ZE R" wilh
q'Z =
·w.,
r.. z:,.
for every k = I •...• K.
Ol·
That is, the vector of expected marginal utilities of the K assets must be proportional to the vector of asset prices. 18 With this we have attained our result. since by taking
The arbitrage freeness of q implies Ihal V () (R~ \(Oll = 0. Since both V and R~ \(Ol are convex sets and the origin belongs 10 V. we can apply the separating hyperplane theorem (see Section M.G of the Mathematical Appendix) to oblain a nonzero vector II' = (11', •...• lis) such that 11". $ 0 for any UE V and 1"'. ~ 0 lor any .E Note Ihal il must be Ihalll' ~ O. Moreover,because.E V implies -UE V. it follows Ihatll'·. = ofor any uE V. Figure 19.E.I(a) depicts this construction for Ihe Iwo-slale case.
w..
Figure 19.E.l
Vo
V = : 1': I' = R:. q': = O. :
E R I: Then we cannot summarize the individual decision problem by means of an indirect utility of the asset portfolio. The relative prices expected in the second period 2 ' also matter. This substantially complicates the formulation of a notion of constrained Pareto optimality. Be that as it may, there appears not to be a useful generalization of the "constrained Pareto optimal" concept in which we could assert the constrained Pareto optimality of Radner equilibrium allocations. Example 19.F.2, due to Hart (1975), makes the point.ln it we have an economy with several Radner equilibria where two of them are Pareto ordered. That is, we have a Radner equilibrium that is Pareto dominated by another Radner equilibrium. To the extent that it seems natural to allow a welfare authority, at the very least, to select equilibria, it follows that the first equilibrium is not constrained Pareto optimal.)O Example 19.F.2: Pareto Ordered Equilibria. Let 1= 2, L = 2, and S = 2. There are no assets (K = 0). The two consumers have, as endowments. one unit of every good in every state. The utility functions are of the form 7lIlUi(Xlli' X21i) + 7l2iUi(X12i, x 2,,). Note that although the probability assessments arc different for the two consumers (these probabilities will be specified in a moment), the spot economies are identical in the two states. Suppose that this spot economy has several distinct equilibria (e.g., it could be the exchange economy in Figure 15.B.9). Let p', p" E R2 be the Walrasian prices for two of these equilibria and let Vi(P) be the spot market utility associated with Ui(', .) and the spot price vector P E R2. Suppose that v,(p') > v,(p"). By Pareto optimality in the spot market, V2(P') < V2(P·). We now define two Radner equilibria. The first has equilibrium prices (p" P2) = (p', p") E R4 and the second has (p" P2) = (p., p') E R4. Because there is no possibility of transferring wealth across states, these are indeed Radner equilibrium prices and, moreover, they are so for any probability estimates 7li' However, the expected utility of these Radner equilibria for the different consumers depends on the 7l i . We can sec now that if consumer I believes that the first state is more likely than the second, that is, he has It" > }, then he will prefer the first equilibrium to the second. Indeed, 7t1l > 1 and v,(p') > v,(p") imply 7lIlV,(P') + 7l"v,(p") > 7tIlV,(P") + 7l 2l V,(p'). Similarly, if the second consumer believes that the second state is more likely than the first, that is, he has 7122 > 1, then he will also prefer the first equilibrium to the second: 7l'2 >! and v,(p') < v2(p") imply 1t'2V2(P') + 7lnv2(P") > 7l"V2(P") + 7tnv,(p'). Thus, the Radner equilibrium with prices (p', pH) Pareto dominates the one with prices (p", p') . • The consensus emerging in the literature seems to be that failures of restricted Pareto optimality (for natural meanings of this concept) are not only possible but even typical [Geanakoplos and Polemarchakis (1986)]. In Exercise 19.F.3 you are asked to develop a related optimality paradox: it is possible for the set of assets to expand and for everybody to be worse off at the new equilibrium! We shan not pursue the constrained optimality analysis
sec T ION
1 8 • G:
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in any greater depth. At some point the analysis runs into the difficulty that it is hard to proceed sensibly without tackling the difficult problem of the determination of the asset structure.
We could also analyze the positive issues studied in Chapter 17 within an incomplete market sellmg. For eXIStence, there is a new set of complexities related to the fact that unbounded short sales are possible. In some contexts this may lead to existence failures (see Exercise 19.F.4).'1 New subtleties also arise for the issue of the determinacy of equilibria (i.e., the number and local uniqueness of equilibria). As we have seen in Section 17.0, with a complete asset structure we have generic finiteness. But with incomplete markets the nature
of the assets (e.g., whether real or financial) mailers, as may the size of S.
19.G Firm Behavior in General Equilibrium Models under Uncertainty In the previous sections we have concentrated on the study of exchange economies. For once, this has not been just for simplicity. The consideration of production and firms is genuinely more difficult in a context of possibly incomplete markets. The rca son relates to the issue of the objectives of the firm.12 As before, we consider a setting with two periods, t = 0 and t = I, and S possible statcs at I = I. There are L physical commodities traded in the spot markets of period t = I and K assets traded at t = O. There is no consumption at I = O. The returns of the assets are in physical amounts of the good I (which we call the numeraire). The S x K return matrix is denoted R. We introduce into our model a firm that produces a random amount ofnumeraire at date t = I (perhaps by means of inputs used at time t = O. but we do not formalize this part explicitly). We let (a ... , as) denote the state-contingent levels of produc" tion of the firm. There are also shares ()i ~ 0, with Li (), = I, giving the proportion of the firm that belongs to consumer i. We take, for the rest of this section (except in the small-type paragraphs at the end) the natural point of view that the firm is an asset with return vector a = (a" ... , as) whose shares are tradeable in the financial markets at t = 0.)) Suppose now that the firm can actually choose, within a range, its (random) productIOn plan. Say, therefore, that there is a set A c RS of possible choices of return 31. Unbounded short sales are at the origin of a discontinuity in the dependence on asset returns of the space of attainable wealth transfers across states. No matter how close asset returns (in dollar terms) may be t~ displaying a linear dependence. consumers can plan to attain. by using trades of very large magnatude, any wealth transfer in the subspace spanned by the asset returns. But when :et~rns bcco~e exactly linearly dependent, this attainable subspace suddenly drops in dimension. As 1O~lcated. thl~ can lead to an existence failure in some COntexts. The model we have analyzed in t~IS chapter IS not. however. one of those. If. as here, in every state all assets have returns in a Single good. whic~. moreover. is the same across assets, then the discontinuity does not arise.
32. The ciass,c paper on Ihis topic is Diamond (1967). For a more recent survey see Merton 29. Or the relative prices of goods between the second and third period. if we are considering more than two dates. 30. That is. the first equilibrium is not Pareto optimal relative to any set of constrained feasible allocations that includes all Radner equilibrium allocations.
(1982).
33. A minor difference with the setting so far is that the firm does really produce the vector (a l • . . . • "s), and. therer~re the total endowment of this asset is not zero. In fact, by putting L, 0, = 1 we have normahzed thiS total endowment to be I.
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-------------------------------------------------------------------------------------------Bo; =
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UNDER
{(xu,···, x s;) E R~: there is a portfolio Z,E RK such that p,(x" - rob) S L Phr.. z" for every 50 and q-z, S O,v(a. q)},
,
(19.G.1)
Figure 19.G.l
An example of possible production choices of the firO).
vectors (a" ... , as) E A of the firm. See Figure 19.0.1 for the case where S = 2. We assume that the return vector a E A is chosen before the financial markets of period I = 0 open. Thus, the decision is made by the initial shareholders (since shares may be sold in period I = 0, the shareholders at the end of period I = 0 may be a different set). Which production plan should these initial owners choose? It turns out that the answer is very simple if A can be spanned by the existing assets and is very dillicult if it cannot. Definition 19,G.1: A set A c R S of random variables is spanned by a given asset structure if every a E A is in the range of the return matrix R of the asset structure, that is, if every a E A can be expressed as a linear combination of the available asset returns.
If we assume, first, that A is spanned by R and, second, that we are dealing with a small project (i.e., all the possible productions a E A are small relative to the size of t he economy; e.g., a,flIL; ro,;!1 is small for all s), then we are (almost) justified in taking the equilibrium spot prices P = (PI'" . ,Ps) E RLS and asset prices q = (ql' ... ' qd E R" as constants independent of the particular production plan chosen by the firm." For the asset priccs q E R" the markel value v(a, q) of any production plan a E A can be computed by arbitrage: if a = L, ex,r, then v(a, q) = L, 'J.,q,. In Exercise 19.0.1 you are asked to show that if the firm is added as a new asset to the given list of assets, and each production plan a E A is priced at its arbitrage value v(a), then any budget-feasible consumption plan of any consumer can actually be reached without purchasing any shares of the firm (the fact can be deduced from Proposition 19.E.3). Thus, for fixed asset prices q E R" and spot prices p = (p I' .•. , Ps) E R LS, the budget constraint of consumer i is"
It follows from the form of this budget constraint that at constant prices every consumer-owner (i.e., any i with 0, > 0) faced with the choice between two production plans a, a' E A, will prefer the one with higher market value, Indeed, if v(a, q) ~ v(a', q) then B•. ; c Bo" Thus, the objective of market value maximization will be the ullanimous desire of the firm's initial owners.'" If A is not spannable by the given asset structure we run into at least two serious difficulties. The first difficulty has to do with price quoting and is common to any commodity innovation problem. Without spanning, the value of a production plan a E A cannot be computed from current asset prices simply by arbitrage, The value IS not, so to speak, implicitly quoted in the economy. Therefore, it would need to be anticipated by the agents of the economy from their understanding of the workings of the overall economy-no mean task. The second difficulty, more specific to the financial context, has to do with price laklllY. Due to the possibility of unlimited short sales there is a discontinuity in the
plausibility of the price-taking assumption. With spanning we can argue, as we did that if the project is small then the effect of production decisions on asset prices, on spot prices at I = I, is also small. But if a new asset a E A, no matter how small, IS not generated by the current asset structure, then its availability increases the span of avatlable wealth transfers by one whole extra dimension. The impact is therefore substantial, and may well have a dramatic effect on prices. 31 There is then no reason for owne~s' preferences over different production plans to be dictated merely by the tocrease to wealth at the prices prior to the introduction of the firm (see Exercise 19.0.2). These two difficulties, to repeat, are serious. There is no easy way out.
0;
A variation of the above model entirely eliminates the asset role of the firm at t = O. Let us assume that the firm's shares cannot be traded at I = 0.>8 If owners at t = 0 choose " E A, this simply means that their endowments at I = I are modified by the random variable Ilia that, recall, pays in good I (i.e., the new endowment of consumer i becomes (w" + (O,a.. 0, ... , 0» E RL for every state sl. If a E A can be spanned, then we are as in the previous model. It does not matter whether shares of the firm can be sold or not at r = O. In either case consumers can take positions in the asset markets that will guarantee that the resulting final consumptions at t = I are the same (Exercise 19.G.3). If a E A cannot be spanned, matters are different. The good news is that, because no new tradeable asset is created at r = 0, the price-taking discontinuity problem disappears. The bad news IS that there IS now another difficulty: Because there is no market for the shares at I = 0, the value of the asset cannot be computed as a deterministic amount at I = O. It is rather a
34. Both assumptions are important for this conclusion. Suppose for a moment that there are
zero 10lal endowments of the asset. Then, since the asset is redundant, Proposition 19.E.3 (see also Exercise 19.E.4) implies that at the Radner equilibrium the new asset is absorbed without any change in prices. What we are now assuming is that this remains approximately true if the total endowment
of the asset is small (i.e., if the project is small). 35. Note that the value of the initial endowments at t = 0 is the value or the shares of the firm 0,1"(
It is a good exercise to verify that (20.B.I) satisfies the stationarity property and that the property can be violated by utility functions of the form V(e) = L, a;u(e,), that is, with a time-dependent discount factor (Exercise 20.B.2). The property of stationarity should nol be confused with the statement asserting that if the consumption streams c and e' coincide in the first T - I periods and a consumer chooses one of these streams at t = 0, then she will not change her mind at T. This "property" is tautologically true: at both dates we are comparing Vee) and V(e').· The stationarity experiment compares V(e) and V(e') at I = 0, but at period T it compares the utility values of the future streams shifted to t = 0, that is, V(e T ) and V(C'T). Thus, stationarity says that in the context of the form (20.B.2), the preferences over the future are independent of the age of the decision maker. Time stationarity is not essential to the analysis of this chapter (except for Sections 20.E and 20.F on dynamics), but it saves substantially on the use of subindices.
(3) Additive separability. Two implications of the additive form of the utility function are that at any date T we have, first, that the induced ordering on consumption streams that begin at T + I is independent of the consumption stream followed from 0 to T, and, second, that the ordering on consumption streams from o to T is independent of whatever (fixed) consumption expectation we may have from T + I onward (see Exercise 20.B.3). In turn, these two separability properties imply additivity; that is, if the preference ordering over consumption streams satisfies these separability properties, then it can be represented by a utility function of the form Vee) = L, u,(c,) [this is not easy to prove, see Blackorby, Primont and Russell (1978)]. How restrictive is the assumption of additive separability? We can make two arguments in its favor: the first is technical convenience; the second is a vague sense that what happens far in the future or in the past should be irrelevant to the relative welfare appreciation of current consumption alternatives. Against it we have obvious counter-examples: Past consumption creates habits and addictions, the appreciation of a particularly wonderful dish may depend on how many times it has been consumed in the last week, and so on. There is, however, a very natural way to accommodate these phenomena within an additively separable framework. We could, for example, allow for the form V(e) = L. U,(C'_I' c,). Here the utility at period t depends not only on consumption at date I but also on consumption at date t - I (or, more generally, on consumption at several past dates). We can formulate this in a slightly different way. Define a vector Z, of "habit" variables and a household producrioll recilllology that uses an input vector C'-I at t - I to jointly produce an output vector C _ I of consumption goods at I - I and a vector z, = C._I of "habit" ' variables at I. Then, formally, u. depends only on time t variables and total utility is L. u,(z" c,). In summary: additive separability is less restrictive than it appears if we allow for household production and a suitable number (typically larger than I) of current variables. (4) Lellglh of period. The plausibility of the separability assumption, which makes the enjoyment of current consumption independent of the consumption in other periods, depends on the length of the period. Because even the most perishable consumption goods have elements of durability in them (in the form, for example, of a flow of "services" after the act of consumption), the assumption is quite strained if the length of the elementary period is very short. What determines the length of the period? To the extent that our model is geared to competitive theory, this period is institutionally determined: it should be an interval of time for which prices can be taken as constant. On a related point, note that the value of fJ also depends, implicitly, on the length of the period. The shorter the period, the closer should be to 1.
a
(5) Recursive uriliry. With the form (20.B.I) for the utility function, we have = u(c o) + aV(c ' ) for any consumption stream C = (co, C I , ••• ,e... ..). If we think of u = u(co) as current utility and of V = V(e ' ) as future utility, we see that the marginal rate of substitution of current for future utility equals and is therefore independent of the levels of current and future utility. The recursive utility model [due to Koopmans (1960)] is a useful generalization of (20.B.I) that combines two features: it allows this rate to be variable but, as in the additively separable case, it has the property that the ordering of future consumption streams is independent of the consumption stream followed in the past. Vee)
2. Hence, the completeness of the preference relation on consumption streams is guaranteed. 3. Ramsey (t928) called the assumption a 'weakness of the imagination." 4. This property is often called lime consistency. Time inconsistency is possible if tastes change through time (recall the example of Ulysses and the Sirens in Section 1.8!), but, as we have just argued, it must necessarily hold if the preference ordering over consumption streams (co . ...• c" . .. ) does not change as lime passes. In line with the entire treatment of Part IV, we maintain the assumption of unchanging tastes throughout the chapter.
a
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The recursive model goes as follows. Denote current utility by II ~ 0 and future utility by V ~ O. Then we are given a current utility function II(C,) and an aggregator function G(II, V) that combines current and future utility into overall utility. For example, in the separable additive case we have G(II, V) = II + cS V. More generally we could also have, for example, G(u, V) = II' + cSV', 0 < " S I. In this case, the indifference curves in the (II, V) plane arc not straight lines. The utility of a consumption stream c = (co, ... , c" . .. ) could then be computed recursively from (20.8.3) V(c) = G(u(c o), V(e')) = G(u(co), G(u(e,), Vee'))) = .... For (20.B.3) to make sense we must be able to argue that the influence of V(e T ) on V(c) will become negligible as T ~ 00 [so that Vee) can be approximately determined by taking a large T and letting V(CT) have an arbitrary value]. This amounts to an assumption of time impatience. In applications, it will typically not be necessary to compute V(c) explicitly. See Exercise 20.B.4 for more on recursive utility.
(6) Altruism. The expression V(c) = u(co) + b V(c') suggests a multigeneration interpretation of the single-consumer problem (20.B.I). Indeed, if generations live a single period and we think of generation 0 as enjoying her consumption according to u(co), but caring also about the utility V(c') of the next generation according to f, V(c'), then V(c) = u(co) + bV(c') is her overall utility. If every generation is similarly altruistic, then we conclude, by recursive substitution, that the objective function of generation 0 is precisely (20.B.I). The entire "dynasty" behaves as a single individual. With this we also have another justification for b < I. The inequality means then that the members of the current generation care for their children, but not quite as much as for themselves. See Barro (1989) for more on these points.
20.C Intertemporal Production and Efficiency Assume that there is an infinite sequence of dates t = 0, I, .... In each period t, there are L commodities. If it facilitates reading, you can take L = 2 and interpret the commodities as labor services and a generalized consumption-investment good (see Example 20.C.1). One of the great advantages of vector notation, however, is that in some cases-and this is one-there is no novelty involved in the general case. Thus, while you think you are understanding the simple problem, you are at the same time understanding the most general one. We shall adopt the convention that goods are nondurables. This is a convention because, in order to make a good durable, it suffices to specify a storage technology whose role is, so to speak, to transport the commodity through time. lfwe were exogenously endowed with some amount of resources (e.g., some initial capital and some amount of labor every period), we would ask what we could do with them. To give an answer, we need to specify the production technology. We already know from Chapter 5 how to do this formally by means of the concept of a production set (or a production transformation function, or a production function). With minimal loss of generality, we will restrict our technologies to be of the following form: the production possibilities at time t are entirely determined by the production decisions at the most recent past, that is, at time t - I. If we keep in mind that we can always define new intermediate goods (such as different vintages of a machine),
SEC T 10"
2 0 • C:
I N T E R T E M P 0 R ALP ROD U C T I 0"
A" 0
E F Fie lEN C Y
737
------------------------------------------------------and also that we can always define periods to be very long, we see that the restriction is minor. Thus, the technological possibilities at t will be formally specified by a production set Y c R2L whose generic entries, or production plans, are written y = (Y., y.). The indices b and a are mnemonic for "before" and "after." The interpretation is that the production plans in Y cover two periods (the "initial" and the "last" period) with y. E IRL and y. E IRL being, respectively, the production plans for the initial and the last periods. As usual, negative entries represent inputs and positive entries represent outputs. We impose some assumptions on Y that are familiar from Section 5.B: (i) Y is dosed and convex. (ii) Y n IR~/ = {O} (no free lunch). (iii) Y - IR~L C Y (free disposal). An assumption specific to the temporal setting is the requirement that inputs not be used later than outputs are produced (i.e., production takes time). This is captured by (iv) If)" = (Yo,Y.)E Y then (y.,O)E Y (possibility oJtrllllcation). In words, (iv) says that, whatever the production plans for the initial period, not producing in the last period is a possibility. A simple case is when y.. ~ 0 for every r E Y, that is, when all inputs are used in the initial period. Then (iv) is implied by the free-disposal property (iii). Example 20.C.I: Ramsey-Solow Model.' Assume that there are only two commodities: A consumption-investment good and labor. It will be convenient to describe the technology by a production function F(k, I). To any amounts of capital investment k ;:>: 0 and of labor input I ~ 0, applied in the initial period, the production function assigns the total amount F(k, I) of consumption-investment good available at the last period. Then
Y = {( -k, -I, x, 0): k
~ 0,
I
~ 0, x S;
F(k, I)} -
R~.
Note that labor is a primary factor; that is, it cannot be produced. _ Example 20.C.2: Cost-oJ-Adjustmem Model. Suppose that there are three goods: capacity, a consumption good, and labor. With the amounts k and I of invested capacity and labor at the initial period, one gets F(k, I) units of consumption good output at the last period. This output can be transformed into invested capacity at the last period at a cost of k' + y(k' - k) units of consumption good for k' units of capacity, where y(.) is a convex function satisfying y(k' - k) = 0 for k' < k and i'(k' - k) > 0 for k' > k. The term y(k' - k) represents the cost of adjusting capacity upward in a given period relative to the previous period. (Note the marginal cost of doing so increases with invested capacity of the period.) Formally, the production set Y is Y = {( - k, 0, -I, k', x, 0): k : 0, k' ;:>: 0, x
:$;
F(k, I) - k' - y(k' - k)} -
I\l~. _
5. See Ramsey (1928) and Solow (1956). The same model was atso inlroduced in Swan (1956).
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--------------------------------------------------------------------------------Example 20.C.3: Two-Sec/or Model. We could make a more general distinction between an investment and a consumption good than the one embodied in Examples 20.C.1 and 20.C.2. Indeed. we could let the production set be 1
I 1
y
= {( -k, 0, -I, k', x, 0): k?; O,I?; 0, k'?; 0, x $
G(k, I, k')} - R~,
where k, k' are, respectively, the investments in the initial and the last periods. Note that the investment and the consumption good need not be perfectly substitutable [they are produced in two separate sectors, so to speak; see Uzawa (\964)]. If they are [i.e., if the transformation function .G(k, I, k') has the form F(k, I) - k'] then this example is equivalent to the Ramsey-Solow model of Example 20.C.t. If it has the form G(k.I, k') = F(k, I) - k' - y(k' - k) then we have the cost·of-adjustment model of Example 20.C.2. • Example 20.C.4: (N + I)-Sec/or Model. As in Example 20.C.3, we have a consumption good and labor, but we now interpret k and k' as N-dimensional vectors. For simplicity of exposition, in Example 20.C.3 we have taken G(k,I, k') to be defined for any k ?; 0, k' ?; O. In general, however, this could lead to the production of negative amounts of consumption good. To avoid this it is convenient to complete the specification by means of an admissible domain A of (k,I, k') combinations. Then
y=
I( -k, 0,
-I, k', x, 0): (k,I, k') E A and x $ G(k, I, k')} -
R~(H+2J.
•
Once we have specified our technology, we can define what constitutes a path of production plans. Definition 20.C.1: The list (Yo' y" ... , y" ... ) is a production path, or trajectory, or program, il y, EYe RIL lor every t. Note that along a production path (Yo,' .• , y" •• . ) there is overlap in the time indices over which the production plans y,_, and y, are defined. Indeed, both Y•. ,_' E RL and Y.. E RL represent plans, made respectively at dates / - I and /, for input use or output production at date /. Thus, we have. at every /, a net input-output vector equal to Y•. ,_, + Y., E RL (at / = 0, we put Y•. _I = 0; this convention is kept throughout the chapter)." The negative entries or'this vector stand for amounts of inputs that have to be injected from the outside at period / if the path is to be realized, that is, amounts of input required at period / for the operation of y,_1 and y, in excess of the amounts provided as outputs by the operation of y,_1 and y,. Similarly, the positive entries represent the amounts of goods left over after input use and thus available for final consumption at time /. The situation is entirely analogous to the description of the production side of an economy in Chapter 5. If we think of the technology at every / as being run by a distinct firm (or as an aggregate of distinct firms) and of p, as an infinite sequence with nonzero entries (equal to y,) only in the / and / + I places, then L, p, is the aggregate production path; and it is also precisely the sequence that assigns the net input-output vector Y•. ,_, + Yo, e RL to period t. If we had a finite horizon, the current setting would thus be a particular case of the description of production in 6. A minor point of notation: when there is any possibility of confusion or ambiguity in the reading of indices. we insert commas; for example. we write Y•. ,_I instead of Y.'_I"
SEC.TION
20.C:
'NTERTEMPORAL
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-----------------------------------------------------------------------------Chapter 5. With an infinite horizon there is a difference: we now have a countable infinity of commodities and of firms instead of only a finite number. As we shall see. this is not a minor difference. It will. however. be most helpful to arrange our discussion around the exploration of the analogy with the finite horizon case by asking the same questions we posed in Section 5.F regarding the relationship between efficient production plans and price equilibria. Definition 20.C.2: The production path (Yo, ...• y" ... ) is efficient il there is no other production path (Yo, ... , y;, ... ) such that Y•. '-1
+ Yb' $
y~.'-I
+ Yb'
lor all t,
and equality does not hold lor at least one t. I n words: the path (Yo,"" y.. ... ) is efficient if there is no way that we can produce at least as much final consumption in every period using at most the same amount of inputs in every period (with at least one inequality strict). The definition is exactly parallel to Definition 5.F. I. What constitutes a price vector in the current intertemporal context? It is natural to define it as a sequence (Po, p" . .. ,p" . .. ), where p, e RL. For the moment we shall not ask where this sequence comes from. We assume that it is somehow given and that it is available to any possible production unit. The prices should be thought of as present-value prices. We shall discuss further the nature of these prices in the nex t section. Given a path (Yo, ... ,y" ... ) and a price sequence (Po, . .. ,p" ... ), the profit level associated with the production plan at / is
P'·Ybl
+ P,+l"Ys,'
We now pursue the implications of profit maximization on the production plans made period by period. Definition 20.C.3: The production path (Yo" .. , y" ... ) is myopically, or short-run, profit maximizing for the price sequence (Po' ... ,p" ... ) il lor every t we have P,'Yb, + P'+I·Y.,?; P,'Yb, + P'+I'Y~,
lor all y;e Y.
Prices (Po,' .. ,p" . .. j capable of sustaining a path (y" . ..• y ... .. ) as myopically profit-maximizing are often called Malinvaud prices for the path [because of Malinvaud (1953»).' Does the first welfare theorem hold for myopic profit maximization? That is, if (Yo .... ,Y.. ... ) is myopically profit maximizing with respect to strictly positive prices, does it follow that (Yo, ... , y ..... ) is efficient? In a finite-horizon economy this conclusion holds true because of Proposition 5.F.I, but a little thought reveals that in the infinite-horizon context it need not. The intuition for a negative answer rests on the phenomenon of capical overaccumulation. Suppose that prices increase through 7. Observe that we do not require that L. P,"(Y•. I_I + Ybl) < 00. In principle. a production path may have an infinite present value. We saw in Sections S.E and S.F. where we had a finite number of commodities and firms that individual. decentralized profit maximization and overall profit maximization amounted to the same thing. Because of the possibility of an infinite present value. the existence of a countable number of commodities and production sets makes this a more delicate matter in the current context. See Exercises 20.C.2 to 20.C.S for a discussion.
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---------------------------------------------------------------------------------------time fast enough. Then it may very well happen that at every single period it always pays to invest everything at hand. Along such a path, consumption never takes place-hardly an efficient outcome. Example 20.C.5: With L = I, let Y = ( -k, k'): k ~ 0, k' $ k} C R2. This is just a trivial storage technology. Consider the path where Y, = (- 1,1) for all t; that is, we always carry forward one unit of good. Then Y•. _I + Y.o = - I and Y•. ,_I + Y., = 0 for alit> O. This is not efficient; just consider the path Y; = (0,0) for alit, which has Y~.'_I + = 0 for all t ~ O. But for the stationary price sequence where p, = I for all t, (Yo, ... , y" ... ) is myopically profit maximizing. _
Proposition 20.C,1: Suppose that the production path (Yo" .. , Yr" .. ) is myopically profit maximizing with respect to the price sequence (Po' ... , Pr , ••. ) » O. Suppose also that the production path and the price sequence satisfy the transversality condition Pr+,'Y., -+ O. Then the path (Yo.·'" yr' ... ) is efficient. Proof: Suppose that the path (y~ •...• y;, ... ) is such that Y•. ,_I + Y., $ Y;.'_I + Yb' for all t, with equality not holding for at least one t. Then there is e > 0 such that if we take a T sufficiently large for some strict inequality to correspond to a date previous to T, we must have T
I
.=0
T
p"(Y~.'-1 + Yb') >
I
p,'(Y•. '-1
+ Y.,) + e.
1=0
In fact, if T is very large then PT+ I • Y.T is very small (because of the transversality condition) and therefore T
I
I
T
T-l
PT' YOT
+
I (p,+ .=0
I'
y~,
+ P"Yb') >
I (p,+ .=0
I ' Y.,
+ p,' Yo,).
We must thus have either p, + 1 'y~, + p,' Y" > p,+ I 'Y., + p,' Y., for some t ::; T - I or PT' YOT > PT+ 1 'Y.T + PT' Y.T' In either case we obtain a violation of the myopic profit-maximization assumption [recall that by the possibility of truncation we have (YbT' 0) E Y]. Therefore, no such path (y~, ... , y;, ... ) can exist. Note that the essence of the argument is very simple. The key fact is that if the transversality condition holds, then for T large enough we can approximate the overall profits of the truncated path (Yo, ... , YT) by the sum of the net values of period-by-period input-output realizations (up to period T). It does not matter whether we match the inputs and the outputs per period or per firm (that is, "per production plan "). If the horizon is far enough away, either method will come down to Profits = Total Revenue - Total Cost. _
AND
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741
(ii) If the answer to (i) is yes, can we conclude that the pair (Yo, ... , y" ... ). (Po, ... , p" ... ) satisfies the transversality condition? Ti,e allswer to (ii) is "not necessarily." In Section 20.E we will see, by means of an example, that the transversality condition is definitely not a necessary property of Malinvaud prices. The answer to (i) is .. Essentially yes." We illustrate the matter by means of two examples and then conclude this section by a small-type discussion of the general situation.
Example 20,e.6: Ramsey-Solow Model Continued. In this model, we can summarize a path by the sequence (k" I" c,) of total capital usage, labor usage, and amount available for consumption. From now on we assume that k,+ 1 + c,+ I = F(k" I,) and that the sequence I, of labor inputs is exogenously given. Then it is enough to specify the capital path (k o, .•. , k" .• .). Denoting by (q" w,) the prices of the two commodities at t, we have that profits at tare q,+,F(k"I,) - q,k, - w,l, and, therefore, the necessary and sufficient conditions for short-run profit maximization at ( are
~
q,+
p"(Y'.'-1 +y.,).
By rearranging terms-a standard trick in dynamic economics-this can be rewritten as (recall the convention Y•. _I = Y~._I = 0)
PRODUCTION
(i) Is there a system of Malinvaud prices (Po, ... , p" .. .) for (Yo, ... , y" . .. ), that is, a sequence (Po,""p" ... ) with respect to which (Yo, ... ,Y" ... ) is myopically profit maximizing?
T
p,'(Y;.'-1 +yb,»PT+I'Y.T+
INTERTEMPORAL
Proposition 20.e. I tells us that a modified version of the first welfare theorem holds in the dynamic production setting. Let us now ask about the second welfare theorem: Given an efficient path (Yo, ... , y" ... ), can it be price supported? In Proposition 5.F.2 we gave a positive answer to this question which applies to the finite-horizon case. In the current infinite-horizon situation we could decompose the question into two parts:
Y',
Efficiency will obtain if, in addition to myopic profit maximization, the (present) value of the production path becomes insignificant as t ... 00. Precisely, efficiency obtains if the (present) value of the period t production plan for period t + I goes to zero, that is, if p,+ I' Y., ... 0 as t -+ 00. This is the so-called transversality condition. Note that the condition is violated in the storage illustration of Example 20.C.5.
20.C:
--------------------------------------------------------------------------------------
= V,F(k"I,)
and
I
~ = V2 F(k" I,).
q,.l
Notc that, up to a normalization (we could put qo = I), these first-order con· ditions determine supporting prices for any feasible capital path (see Exercise 20.e.6). The transversality condition says that q,+ ,F(k" I,) -+ O. If the sequence of productions F(k" I,) is bounded, then it suffices that q, -+ O. In view of Proposition 2a.e.I. we can conclude that a set of sufficient conditions for efficiency of a feasible and bounded capital path (k o, ..• , k" ... ) is that there exist a sequence of output prices (qo, ... , q" ... ) such that
~ = V,F(k" I,)
for alit
(20.C.I)
q,+1
and
q,
-+
a
(equivalently, Ijq,
-+ 00).
(20.C.2)
Because of the possibility of capital overaccumulation, (2a.e.I). which is necessary, is not alone sufficient for efficiency. On the other hand, (20.e.2) is not necessary (see Section 20.E). Cass (1972) obtained a weakened version of (20.e.2) that, with (2a.c.I),
-
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20.0:
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is both necessary and sufficient.· The condition is
"" I
L-= ,-0 q,
(20.C.2')
00.
-
Example 20.C.7: Cost of Adjustment Model continued. In the cost or adjustment model. a production plan at time t - I involves the variables k,_ .. 1,_1' k,. c,. We associate with these variables the prices q,_I' q,. s,. Profits are then
w,_"
s,(F(k,.,.I,.,) - k, -y(k,...,. k,_,))
+ q,k, -
Production Possibility Set . Y () ({Y•. ,_,) X RL)
q,_,k,., - w,.,I,.,.
(Y •. Y•. ,_I + YIN,
thus contradicting efficiency.
(a)
(b)
We construct the desired price sequence (Po •...• P, •.. ·) by induction. Put Po = q", (i.e.• the relative prices at r = 0 are the M RTs between goods at the initial part of the production plan Yo EYe R'L). Suppose now that the prices (Po •...• PT) have already been determined, and that every y, up to I = T - 1 is myopically profit maximizing for these prices. Because of the first-order conditions for profit maximization at T - I. we have that PT = aq•. T ' I for some a > O. We know that q•. T ' I = {Jq .. for some {J> O. Then PT = a{Jq'T' Therefore. if we put p,.. 1 = a/iq.", we have that (PT' PT.,) = (a/iq'T' a{Jq.T) is proportional to qT = (q'T' q.T). which means that YT is profit maximizing for (PT. PT. I)' Hence we have extended our sequence to (Po • ...• PT. I) and we can keep going. Note that, as in Examples 20.C.6 and 20.C.7. the construction of the supporting short-run prices does not make full use of the efficiency. What is used is that the production path is "short-run efficient" (that is. the production path cannot be shown inefficient by changes in the production plans at a finite number of dates). The above observations can be made into a perfectly rigorous argument for the existence of Malinvaud prices in the smooth case. The proof for the nonsmooth case is more complex. It must combine an appeal to the separating hyperplane theorem (to get prices for truncated horizons) with a limit operation as the horizon goes to infinity. With a minor technical condition (call IlOnlighrness in the literature). this limit operation can be carried out.
20.D Equilibrium: The One-Consumer Case In this section. we bring the consumption and the production sides together and begin the study or equilibrium in the intertemporal setting. We shall start with the one-consumer case. As we will see in Section 20.G. the relevance or this case goes beyond the domain or applicability or the representative consumer theory or Chapter 4. An economy is specified by a short-term production technology Y c R2L, a utility Junct ion u( . ) defined on R~. a discount factor lJ < I, and, finally, a (bounded) sequence or initial endowments (W o," .• w" .. .), w, e R~. We assume that Y satisfies hypotheses (i) to (iv) or Section 20.C and that u(·) is strictly concave. differentiable. and has strictly positive marginal utilities throughout its
domain. 8. Some additional, very minor, regularity conditions on the production runction F(') are required ror the validity or this equivalence.
Prices are given to us as sequences (Po •. ..• p" . .. ) with p, e R~. As in Chapter 19 we can interpret these prices either as the prices or a complete system of rorward
Figure 2O.C.l (ten)
Smooth production scI. Flgur. 20.C.2 (rtgh')
A production path that is inefficient at T.
744
CHAPTER
20:
EQUILIBRIUM
AND
markets occurring simultaneously at t = 0 or as the correctly anticipated (present value) prices of a sequence of spot markets. We will consider only bounded price sequences. In fact, most of the time we will have IIp,lI _ 0. 9 Given a production path (Yo,"" y.. ... ), y, E Y, the induced stream of consumptions (co, ... , c,' ... ) is given by C,
+ P'+I-Y.'
Fixing T and rearranging the terms of we get
L
(It,+p,'w,)-
L, '" T p,'C, = L, '" T p'(Y•. ,_, + Y., + w,)
L
p,·C,=PT+'·Y.T
(20.0.1)
20. D:
E QUI L' • R , U M:
THE
0 N E - CON SUM E RCA S E
there is a forward market for every commodity at every date, or, in another, that assets (e.g., money) are available that are capable of transferring purchasing power Ihrough time (see Exercise 20.0.1 for more on this). Secondly, observe that the strict monotonicity of u(·) implies that if we have reached utility maximization then, a fortiori, total wealth (denoted w) must be finite; that is,
Lit, + LP,'w, < 00.
w=
for every t.
T , ~"
Moreover, at the equilibrium consumptions the budget constraint of (20.0.4) must hold with equality. An important consequence of the last observation is that at equilibrium the transversality condition is satisfied. Formally, we have Proposition 20.D.1. Proposition 20.0.1: Suppose that the (bounded) production path (yt,···, vi, . .. ) and the (bounded) price sequence (Po, ... ,p" ... ) constitute a Walrasian equilibrium. Then the transversality condition P,+,' 0 holds.
Y:, -
Proof: Denote
c: = Y:.,_, + y:' + w,. By expression (20.0.1) we have L (It,+p,'w,)- L p,·C,=PT+'·Y.T· 1$
T
IS T
Since cach of the sums in the left-hand side converges to w < conclude that PH' • Y:T -+ O. •
00
as T
-+ 00.
we
IS T
IS T
Expression (20.0.1) is an important identity. It tells us that the transversality condition is equivalent to the overall value of consumption not being strictly inferior to wealth (i.e., there is no escape of purchasing power at infinity). The definition of a Walrasian equilibrium is now as in the previous chapters. One only has to make sure that a few infinite sums make sense. Definition 20.0.1: The (bounded) production path (y~, ... , y~" .. ), y~ E Y, and the (bounded) price sequence p = (Po' ... ,p" ... ) constitute a Walrasian (or competitive) equilibrium if:
Y:.,_, Y6, + w, ~ 0
(i) c~ = + (ii) For every t,
1t,
for all t.
(20.0.2)
= P,'Yb, + p,+,'V:, ~ P,'Yb + p,+,·Y.
(20.0.3)
for all Y = (Vb' Y.) E Y. (iii) The consumption sequence (ct, ... ,cr . .. ) ~ 0 solves the problem Max
L, o'u(c,) s.t.
(20.0.4)
L,P,'c, ~ L,1t, + L,p,·w,.
Condition (i) is the feasibility requirement. Condition (ii) is the short-run, or myopic, profit-maximization condition already considered in Section 20.C (Definition 20.C.3). The form of the budget constraint in part (iii) deserves comment. Note first that there is a single budget constraint. As in Chapter 19, this amounts to an assumption of completeness, which means, in one interpretation, that at time t = 0 9. Keep in mind that prices are to be thought
or as
measured in current-value terms.
745
---------------------------------------------------------------------------------------
+ Y'u + 00"
= Y.... _I
If c, ~ 0 for every t, then we say that the production path (Yo, ... ,y" . .. ) is feasible: Given the initial endowment stream the production path is capable of sustaining nonnegative consumptions at every period. To keep the exposition manageable from now on we restrict all our production paths and consumption streams to be bounded. Delicate points come up in the general case, which are better avoided in a first approach. Alternatively, we could simply assume that our technology is such that any feasible production path is bounded. Given a production path (Yo, . .. ,Y... .. ) and a price sequence (Po, ... ,P.. · .. ), the induced stream of profits (Ito, ... , It" ... ) is given by
n, = P,·Y'n
s [C
TIME
-----------------------------------------------------------------------------------
Another implication of Definition 20.0.1 by
w
T. Because /J < I, there is f. > 0 such that if T is large enough then there is an improvement of utility of more than 2e in going from (1'0' ... , "" ••• J to (co, ... , c;, ... J. Since w < 00, the amount I:t> riP,' (c, - c;)1 can be made arbitrarily small. Hence, for large T the stream (c;;' . .. , c;, . ..) is almost budget feasible. It follows that it can be made budget feasible by a small sacrifice of consumption in the first period resulting in a utility loss not larger than e. Overall, it still results in an improvement. But this yields a contradiction because only the consumption in a finite number of periods has been altered in the process. _
20.0:
EOUILtBRIUM:
THE
ONE-CONSUMER
(ii) Suppose instead that Wo = I and w, = 0 for t > O. There is, however, a linear production technology transforming every unit of input at t into IX > 0 units of output at I + I. Because of the boundary behavior of the utility function, consumption will be positive in every period, and therefore the technology will be in operation at every period. The linearity of the technologies then has the important implication that the equilibrium price sequence is completely determined by the technology. Putting Po = I, we must have P, = I/a'. Wealth is IV = PoWo = I, and therefore the equilibrium consumptions must be c: = [.5'(1 - t5)]/p, = (exb)'(1 - b). Note that, as long as I ~ a < I/b, both the price and the consumption sequences are bounded. Observe also the interesting fact that for this example we have been able to compute the equilibrium without explicitly solving for the sequence of capital investments. (iii) We are as in (ii) except that we now have a general technology F(k) transforming every unit k" of investment at t into F(k,) units of output at r + I. This output can then be used indistinctly for consumption or investment purposes at I + I. That is, c,., = F(k,) - k,. I' The logarithmic form of the utility function allows for a shortcut to the computation of equilibrium prices. Indeed, say that (Po," . 'PI' ... ) are equilibrium prices and (c~, ... ,c~, ... ), (k~, ... ,k:, . .. ) equilibrium paths of consumption and capital investment. Then we know that at any T a constant fraction 0 of remaining wealth is invested. That is,
PT.,k~., = b( L
p,c:)
= t5PT+,F(k:J.
'2: T+ 1
Example 20.0.1: In this example we illustrate the use of conditions (20.0.6) for the computation of equilibrium prices. Suppose that we are in a one-commodity world with utility function L, /J' In c,. Given a price sequence (Po,' .. ,p" ... ) and wealth IV, the first-order conditions for utility maximization (20.0.6) are
b'
).p, = c,
for all I,
and
L p,c, = IV. ,
Hence, W = L, p,C, = (1/).) L, b' = (1/).)[1/(1 -.5)] and so p,c, = .5'/i. = .5'(1 - b)1V for all r. Note that this implies a conslant rale of savings because PTcdeL, " T p,c,) = I - 6, for all T (Exercise 20.0.4).'0 We now discuss three possible production scenarios. (i) The economy is of the exchange type; that is, there is no possibility of production and we are given an initial endowment sequence (wo. ... , WI' ... ) » O. Then the equilibrium must involve = w, for every I, and therefore, normalizing to L, p,W, = I, the equilibrium prices should be
c:
0'(1 - b) P,=----
w,
for every t.
10. Logarithmic utility functions facilitate computation and are very important in applications. However, they are not continuous at the boundary (In 1:, _ - 00 as c, - 0) and therefore violate one of our maintained assumptions. This does not affect the current analysis but should be kept in mind.
CASE
747
----------------------------------------------------------------------------------
Therefore, we must have k:., = of(k:J for every I. With ko = Wo = I given, this allows us to iteratively compute the sequence of equilibrium capital investments. The sequence of prices is then obtained from the profitmaximization conditions P,. ,F'(k:J- p, = o.• Since a Walrasian equilibrium is myopically profit maximizing and satisfies the transversality condition (Proposition 20.0.1), we know from Proposition 20.C.1 that it is production efficient (assuming p, » 0 for all I). Can we strengthen this to the claim that the full first welfare theorem holds? We will now verify that we can. In the current one-consumer problem, Pareto optimality simply means that the equilibrium solves the utility-maximization problem under the technological and endowment constraints: Max
L b'u(c,), S.t. (", =
Yu,l-l
(20.0.7)
+ Ybt + w,;,:: 0
y,
and
Proposition 20.0.3: Any Walrasian equilibrium path planning problem (20.D.7).
(V~,
E
Y for all I.
... , V?, . .. ) solves the
Proof: Oenote by B the budget set determined by the Walrasian equilibrium price sequence (Po,· .. , p" ... ) and wealth IV = L,1!, + L, p,·w" where 1[,
= p,'Y:'
+ P,+l'Y:,,+l
748
CHAPTER
EQUILIIIRIUM
20:
AND
TIME
SEC T ION
--------------------------------------------------------------------------------------------for all I. That is.
have
B = {(co•...• c; •.. .): c; ~ 0 for all t and
L, p,'c; !> w}.
By the definition of Walrasian equilibrium. the utility of the stream (c~ •...• c: •... ) defined by = Y:.,_1 + + £0, is maximal in this budget set. It suffices. therefore, to show that any feasible path (y~•. ..• y~, ... ), that is, any path for which y~ E Y and c~ = Y;.,_1 + Y;' + co, ~ 0 for all I, must yield a consumption stream in B. To see this note that, for any T.
c:
Y:,
L
p,'c;=
1ST
L (P,'Yb,+P,+I'Y;,)+Pr'Ybr+ L rST-l
c; »
E QUI L III R I U M:
THE
0 N E • CON. U MER
CAS E
749
0 for all t and. moreover, for it to be legitimate to determine the sign of
L, b'(u(C;) -
u(cn) = ~T(U(CT) - u(cm + b T' '(u(c~+ ,) - u(4+ ,))
by signing the first·order term b T Vu(cf)'(cr - cf) + bT+' Vu(cf+ ,)'(CT" - cf, ,) = PT'(Y;T - rtT) + PT' ,'(Y~r - Y:T) = PTe Y;T
+ PT+ I" Y~T - PTe Y:T - PT+ I" Y:T > O.
But this conclusion contradicts the assumption that
p,·W,.
(y~,
... ,
Y: .... )solves (20.0.7). •
1sT
By the possibility of truncation of production plans, we have (Ybr, 0) E Y. Therefore, by short-run profit maximization, p,' Y,r :$; 1[r and p,' + P, + I • :$; 1[, for all t:$; T- I. Hence,
y"
L
L
p,'c;:$;
1ST
which implies
20. D:
-----------------------------------------------------------------------------
L, p,'c;
1ST
1[,+
L
p,'w,:$;w
y:,
for all T,
1ST
:$; w. •
Let us now ask for the converse of Proposition 20.0.3 (i.e .• for the second welfare theorem question; see chapter 16): Is any solution (Yo, ... , y" ... ) to the planning problem (20.0.7) a Walrasian equilibrium? In essence. the answer is "yes," but the precise theorems are somewhat technical because. to obtain a well-behaved price system (i.e .• a price system as we understand it: a sequence of nonzero prices). one needs some regularity condition on the path. We give an example of one such result." Proposition 20.0.4: Suppose that the (bounded) path (Y6,"" vi, ... ) solves the planning problem (20.0.7) and that it yields strictly positive consumption (in the sense that, for some t > O. en = Yi•. '-l + Yib' + COft > t for all t and t). Then the path is a Walrasian equilibrium with respect to some price sequence (Po.···.P" ... ). Proof: We provide only a sketch of the proof. A possible candidate for an equilibrium price system is suggested by expression (20.0.6): p, = b'VU(Cn
for all t,
y:.,_,
where c: = + Y:' + w,. Because (c~, ...• c: •... ) is bounded above and bounded away from the boundary (uniformly in t) we have L, IIp,1I < 00, which implies the transversality condition. In turn. by expression (20.0.1) this yields L, Poc: = L, (n, + p,'w,) = w < 00. Therefore. by Proposition 20.0.2, the utility·maximization condition holds. I! remains to establish that short·run profit maximization also holds. To that effect suppose that this is not so. that is. that for some T there is y' E Y with PT·Y~
+ PT+t 'y~ > PT'Y:T + PT+ I'Y:T =
nT'
Let (y; •. ..• y;, ... ) be the path with Yr = y' and y; = Y: for any t >F T. Let (co •... , c; •. .. ) be the associated consumption stream. Because of the convexity of Y and the strict positivity for us to property of (c~ •. ..• c: •. .. ) we can assume that YT = Y' is sufficiently close to
yr·
II. A general treatment would involve, as in Sections IS.C or 16.D,lhe application ora suitable
version (here infinite·dimensional) of the separating hyperplane theorem. The next result gets around this by exploiting the differentiability of It(·). I! is thus parallel to the discussion in Section 16.F.
The close connection between the solutions of the equilibrium and the planning problem (20.0.7) has three important implications for. respectively. the existence. uniqueness. and computation of equilibria. The first implication is that it reduces the question of the exiSlence of an equilibrium to the possibility of solving a single optimization problem. albeit an infinite-dimensional one. Proposition 20.0.5: Suppose that there is a uniform bound on the consumption streams generated by all the feasible paths. Then the planning problem (20.0.7) attains a maximum; that is, there is a feasible path that yields utility at least as large as the utility corresponding to any other feasible paths. The proof. which is purely technical and which we skip. involves simply establishing that, in a suitable infinite-dimensional sense. the objective function of problem (20.0.7) is continuous and the constraint set is compact. The second implication is that it allows us to assert the uniqueness of equilibrium. Proposition 20.0.6: The planning problem (20.0.7) has at most one consumption stream solution. Proof: The proof consists of the usual argument showing that the maximum of a strictly concave function in a convex set is unique. Suppose that (yo •...• y, •. .. ) and (yo •...• y; •... ) are feasible paths with L, .5'u(c,) = L, .5'u(c;) = Y. where (co •...• c, •. .. ) and (co •. ..• c; •.. .) are the consumption streams associated with the two production paths. Consider y~ = lY, + !y;. Then the path (y~, ...• y~ •. .. ) is feasible and at every t the consumption level is c~ = !c, + !c;. Hence. L, .5'u(c~) ~ Y. with the inequality strict if c, l' c; for some t. Thus. if c, '# c; for some t. the paths (yo, ... , y, •.. .). (y~ •...• y; •... ) could not both solve (20.0.7). • The third implication is that Proposition 20.0.3 provides a workable approach to the computation of the equilibrium. We devote the rest of this section to elaborating on this point.
The Computation of Equilibrium and Euler Equations It will be convenient to pursue the discussion of computational issues in the slightly restricted setting of Example 20.C.4, the (N + I)-sector model. To recall. we have N capital goods, labor, and a consumption good. We fix the endowments of labor to a constant level through time. A function G(k. k'). gives the total amount of consumption good obtainable at any t if the investment in capital goods at t - I is
-
750
c HAP T E R
2 0:
E QUI LIB R I U"
AND
T
I .. E
SEC T ION
20.
D: E QUI LIB R I
U .. :
THE
0
NE· CON. U .. E RCA. E
751
-------------------------------------------------------------------------------------------given by the vector k E RH. the investment at t is required to be k' E R~. and the labor usage at t - I and t is fixed at the level exogenously given by the initial endowments. We denote by A c RH X RH the region of pairs (k. k') E R2H compatible For notational with nonnegative consumption [i.e .• A = Ilk. k') E R2H: G(k. k') ~ convenience. we write u(G(k. k'» as u(k. k'). We assume that A is convex and that 11(', .) is strictly concave. Also. at t = 0 there is some already installed capital invcstment ko and this is the only initial endowment of capital in the economy. In this economy the planning problem (20.0.7) becomes'2
On
(20.0.8) s.t. (k,_" k,) E A for every t. and ko =
ko ·
From now on we assume that (20.0.8) has a (bounded) solution. Because of the strict concavity of u(· • . ) this solution is unique. For every / ~ I the vector of variables k, E RH enters the objective function of (20.0.8) only through the two-term sum o'u(k,_,. k,) + 0'+ 'u(k,. k, + d. Therefore, diffcrentiating with respect to these N variables. we obtain the following necessary conditions for an interior path (k o•...• k, •. .. ) to be a solution of the problem (20.0.8): 13 for every n :> Nand /
~
I.
In vector notation. for every t
~
I.
(20.0.9)
Conditions (20.0.9) are known as the Euler equations of the problem (20.0.8). Example 20.0.2: Consider the Ramsey-Solow technology of Example 20.C.1 (with I, = I for all t). Then. u(k. k') = u(F(k) - k') and A = Ilk. k'): k' $; F(k)}. Therefore, the Euler equations take the form -II'(F(k,_d- k,)
+ ou'(F(k,) -
k,+,)F'(k,) = O.
for all/2'o I
or
~=F'(k,)
for all t
~
I.
ou (c,+ ,)
In words: the marginal utilities of consuming at t or of investing and postponing consumption one period are the same. _ Example 20.0.3: Consider the cost-of-adjustment technology of Example 20.C.2 (except that as in Example 20.0.2 we fix " = I for all t and drop labor as an explicitly considered commodity) and suppose we have an overall firm that tries to maximize the infinite discounted sum of profits by means of a suitable investment policy in capacity. Output can be sold at a constant unitary price that. with a constant rate 12. By convention we put u(k_ 1• ko) == O. 13. The expression "interior path" means that (kit k,. I) is in the interior of A for all t. For the interpretation of the expression to come, recall also that k. and k~ stand, respectively. for the nth and Ihe (N + 1I)lh argumenl or u(k, k').
••
of interest. gives a present value price of 0'. Thus the problem becomes that of maximizing L, o'[F(k'_I) - k, - y(k, - k,_,)]. The Euler equations are then -I-y'(k,- k,_,)
+ o[F'(k,) +y'(k,+,
-
k,n =0
for all t ~ 1.
In words: the marginal cost of a unit of investment in capacity at t equals the discounted value of the marginal product of capacity at t plus the marginal saving in the cost of capacity expansion at t + I. Note that. iterating from t = I. we get I
+ y'(k,
- k o) =
L li'(F'(k,) -
,,,I
I).
In words: At the optimum. the cost of investing in an extra unit of capacity at t = I equals the discounted sum of the marginal products of a maintained increase of a unit of capacity. '4 See Exercise 20.0.5 for more detail.'· _ Suppose that a path (k o•..•• k" ... ) satisfies the Euler necessary equations (20.0.9). From their own definition, and the concavity of u(· •. ). it follows that the Euler equations are also sufficient to guarantee that the trajectory cannot be improved upon by a trajectory involving changes in a single k,. In fact, the same is true if the changes are limited to any finite number of periods (see Exercise 20.0.6). Thus. we can say that the Euler equations are necessary and sufficient for short-run optimization. The question is then: Do the Euler equations (or. equivalently. short-run optimization) imply long· run optimization? We shall see that. under a regularity property on the path (related. in a manner we shall not make explicit. to the transversality condition 16). they do. We say that the path (k o•. ..• k, •.. .) is strictly interior if it stays strictly away from the boundary of the admissible region A. [More precisely. the path is strictly interior if there is £ > 0 such that for every t there is an £ neighborhood of (k,. k,+ I) entirely contained in A.] Proposition 20.0.7: Suppose that the path (*0' ... ' k" ... ) is bounded, is strictly interior, and satisfies the Euler equations (20.0.9). Then it solves the optimization problem (20.0.8). Proof: The basic argument is familiar. Ir 0'0'" .. k, •. .. ) does not solve (20.0.8). then there is a feasible trajectory (/ ko• then k, is unbounded. The only value of k, generating a bounded k, is k, = ko' Therefore. t/I(k o) = ko for any ko' It is instructive to see what happens if we try k, ~ ko. Then, the path induced by the ditTerence equation is feasible and. in fact, we have a constant level of consumption C, = 2k,_, - k, = 2ko - k,. Thus, for k, > k o• we have here an example of a path that is compatible with the Euler equations but that is not optimal. because at k, = ko we get a higher level of constant consumption." _
It may be helpful at this stage to introduce the concept of the value function V(k) and the policy fUllctioll t/I(k). Given an initial condition ko = k. the maximum value attained by (20.D.8) is denoted V(k). and if (k o• k l ••••• k, •... ) is the (unique) trajectory solving (20.D.8) with ko = k. then we put t/I(k) = k,. That is. t/I(k) e IRN is the vector of optimal levels of investment. hence of capital. at t = I when the levels of capital at t = 0 are given by k. What accounts for the importance of the policy function is the observation that if the path (k o•...• k" ...) solves (20.D.8) for ko = ko then. for any T, the path (k T • ••• ,k T +, •. ..) solves (20.D.8) for ko = k,.. Thus, if (k o, ...• k" . .. ) solves (20.D.8) we must have
k,+, = t/I(k,) for every t.
2 0 • D:
CA. E
753
---------------------------------------------------------
The dynamic programming approach exploits the recursivity of the optimum problem (20.D.8). namely. the fact that
V(k) =
(20.D.IO)
Max
u(k. k')
+ 0 V(k'),
(20.D.11)
,'""Uh(i,l')EA
and we see that the optimal path can be computed from knowledge of ko and the policy function t/I(.). But how do we determine t/I(')1 We now describe two approaches to the computation of t/I(.). The first exploits the Euler equations; the second rests on the method of dynamic programming. The Euler equations (20.D.9) suggest an iterative procedure for the computation of t/I(k). Fix ko = k and consider the equations corresponding to k,. With ko given, we have N equations in the 2N unknowns k, e IRN and k2 eRN. There are therefore N degrees of freedom. Suppose that we try to fix k, arbitrarily [equivalently. we try to fix - V2u(ko. k,), the marginal costs of investment at t = I] and then use the N Euler equations at t = 1 to solve for the remaining k2 unknowns [equivalently, we adjust the commitments for investment at t = 2 so that the discounted marginal payofT of investment at t = I. bV,u(k" k 2 ). equals the preestablished marginal cost of investment at t = I, i.e. - V2u(ko, k,)]. Suppose that such a solution k2 is found [by the strict concavity of u(·), if there is one solution then it has to be unique]. We can then repeat the process. The N Euler equations for period 2 are now exactly determined: Both kl and k2 are given. but we still have the N variables k) corresponding to t = 3 with which we can try to satisfy the N equations of period 2. Suppose that we reiterate in this fashion. There are three possibilities. The first is that the process breaks down somewhere, that is, that given k, _, and k, there is no solution k,+, [or, more precisely, no solution with (k,. k,+ ,) e A]; the second is that we generate a sequence that is unbounded (or nonstrictly interior); the third is that we generate a bounded (and strictly interior) sequence (k o, k" ... ,k" ... ). In the third case, by Proposition 20.D.7 we have obtained an optimum, and since by Proposition 20.D.6 the optimum is unique, we can conclude that given ko, tile third possibility (the trajectory startillg at ko and kl is strictly interior alld bounded) can occur for at most one value of k l . If it occurs, tltis value of kl is precisely t/I(k o). Thus, the computational method is: Solve the ditTerence equation induced by the Euler
and obtains t/I(k) as the vector k' that solves (20.D.II). This, of course, only transforms the problem into one of computing the value function V(·). However, it turns out that. first, under some general conditions [e.g.• if V(·) is bounded] the value function is the only function that solves (20.D.11) when viewed as a functional equation, that is, V(·) is the only function for which (20.D.lI) is true for every k. and, second, that there are some well-known and quite effective algorithms for solving equations such as (20.D.II) for the unknown function V(·). (Sec Section M.M. of the Mathematical Appendix.) We end this section by pointing out two implications of the definition of the value function (sec Exercise 20.D.8): (i) The value fUllction V(k) is concave. (ii) For every perturbacion parameter z e IRN with (k
V(k + z)
~
+ z, t/I(k» e A we have
u(k + z, t/I(k» + oV(t/I(k)).
(20.D.12)
Suppose that N = I and (k, t/I(k)) is interior to A. For later reference we point out that from (i), (ii), and V(k) = u(k. t/I(k)) + 0 V(t/I(k» we obtain
V'(k) and, if V(·) is twice-differentiable,
= V,u(k. t/I(k»
VH(k) ~ V:,u(k, t/I(k)). (See Figure 20.0.1 and Exercise 20.D.9. 18 ) 17. Hence. when k, > k., the Euler equations lead to capital overaccumulation. We note. without further elaboration, that given a path satisfying the Euler equal ions we could use the equations themselves to determine a myopically supporting price sequence. However, jf k I > ko Ihis sequence will violate the transvcrsality condition. 18. The expression V;~f(') denotes the ij second partial derivative of the real-value function f(·).
1
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V(k +:)
Figure 20.0.1
Along an optimal P"h the value function is majorized by the
utilities of singleperiod adjustments.
20.E Stationary Paths, Interest Rates, and Golden Rules In this section, we concentrate on the study of steady states. This study constitutes a first step towards the analysis of the dynamics of equilibrium paths. We refer to Bliss (1975), Gale (1973), or Weizsiicker (1971) for further analysis of steady-state theory. We begin with a production set Y c I1\llL satisfying the properties considered in Section 20.C. Recall that a production path is a sequence (yo, ... , y" ... ) with y, E Y for every r. DefinItion 20.E.1: A production path (yo, ... , y" ... ) is stationary, or a steady state, if there is a production plan y = (Yb' Y.) E Y such that y, = I' for all t> O. Abusing terminology slightly, we refer to the "stationary path (y, ... , y, .. .)" as simply the "stationary path y." The first important observation is that stationary paths rhat are also efficient are supportable by proportional prices. '9 This is shown in Proposition 20.E.!. ProposItion 20.E.1: Suppose that Ii E Y defines a stationary and efficient path. Then, there is a price vector Po E I1\lL and an ex > 0 such that the path is myopically profit maximizing for the price sequence (Po' expo' ... , ex'PD' ... ). Proof: A complete proof is too delicate an affair, but the basic intuition may be grasped from the case in which production sets have smooth boundaries. For this case we can, in fact, show that every (myopically) supporting price sequence must be proportional. By the efficiency of the path (y, . .. , )" ... ), the vector y must lie at the boundary of Y. Let q = (qo, q,) be the unique (up to normalization) vector perpendicular to Y at y. Also, by the small type discussion at the end of Section 20.C, there exists a price sequence (Po, ... , P..... ) that myopically supports this efficient path. Because }' E Y is short-run profit maximizing at every r we must have (P .. P.. ,) = ).,(qo, q ,) for some ;., > O. Therefore, p, = ).,qo and p,+, = ;.,q, for all r. In particular, p, = ;., _, q, and p, = ;., qo. Combining, we obtain p, = (}.,/;., _ ,) p, and
+,
+,
+,
t 9. To prevent possible misunderstanding, we warn that establishing the inefficiency of a given stationary path will typically require the consideration of nonstationary paths.
I·
'I
S TAT' 0 N A A Y
PAT H S,
'N TEA EST
A ATE S ,
AND
+,
= ()., + ,/;., ) p,. From this we get ).,/)., _, = )., + ,/)., for all t 2: 1. Hence, denoting this quotient by ex, we have p,+, = exp, = ex 2 p,_, = ... = ex'+'po' The factor ex has a simple interpretation. Indeed, r = (I - ex)/ex [so that p, = (I + r)p,+ ,] can be viewed as a rate oj interest implicit in the price sequence (see Exercise 20.E.I). Proposition 20.E.I is a sort of second welfare theorem result for stationary paths. We could also pose the parallel first welfare theorem question. Namely, suppose that (Y, ... , }', ... ) is a stationary path myopically supported by a proportional price sequence with rate of interest r. If r> 0, then P, = (1/(1 + r»'po -+ 0 and therefore the transversality condition p, y. -+ 0 is satisfied. We conclude from Proposition 20.C.1 that the path is efficient. If r ~ 0, the transversality condition is not satisfied (p, docs not go to zero), but this does not automatically imply inefficiency because the transversality condition is sufficient but not necessary for efficiency. Suppose that r < 0 and, to make things simple, let us be in the smooth case again. Consider the stationary candidate paths defined by the constant production plan y, = (Y. + r.e,),. - u), where e = (1, ...• 1) E RL. This candidate path uses fewer inputs (or produces more outputs) at t = 0 and generates exactly the same net input-output vector at every other t. Therefore, if for some £ > 0, the candidate path is in fact a feasible path; that is. if y, E Y, then the stationary path y is not efficient (it overaccumulates). But if Y has a smooth boundary at y, the feasibility of y, for some r. > 0 can be tested by checking whether y, - y = £(e, - e) lies below the hyperplane determined by the supporting prices (Po, [1/(1 + r)]po). Evaluating. we have £(1 - 1/(1 + r»po·e < 0, because r < O. Conclusion: For'£ small enough, the stationary path)' is dominated by the stationary path y,. We record these facts for later reference in Proposition 20.E.2. p,
__~=---u(k +:, ",(k)) + 6V(I{t(k))
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Proposition 20.E.2: Suppose that the stationary path (y, ... , y . .. ), YE Y, is myopically supported by proportional prices with rate of interest r, then the path is efficient if r > 0 and inefficient if r < O. We have not yet dealt with the case r = 0, which as we shall see, is very important. 20 We will later verify in a more specific setup that efficiency obtains in this case. Let us now bring in the consumption side of the economy and consider stationary equilibrium pat Its. Assuming differentiability and interiority, a stationary path (y, ... , y, ... ) that is also an equilibrium can be supported only (up to a normalization) by the price sequence p, = IJ' Vu(c). where c = Yb + y.; recall Proposition 20.D.4 and expression (20.D.6). That is. a slationar.v equilibrium is supported by a price sequence clIlbodyillf} (I proportiollalil,l' Jacror equal 10 rite discount Jactor .5, or, equivalently, with rate of interest r = (I - .5)/.5. Definition 20.E.2: A stationary production path that is myopically supported by proportional prices p, = rx'Po with (X = .5 is called a modified golden rule path. A stationary production path myopically supported by constant prices p, = Po is called a golden rule parh. 20. No'e that 0 is 'he ra'e of growth implicit in the pa,h (Y, ... , y, .. .). In a more general treatment we could allow for a constant returns technology and for the production path to be proportional (but not necessarily stationary). Then Proposition 20.E.2 remains valid with 0 replaced by the corresponding rate of growth.
756
c
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E
Q
U , L' B R , U II
.. NOT' II E
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----------------------------------------------------------------------------------- ---------------------------------------------------------------------Depending on the technology and on the discount factor b, there may be a single or there may be several modified golden rule paths (see the small-type discussion at the end of this section). But in any case we have just seen that a statiollary equilibrium path is lIecessarily a modified goldell rule path. Thus, we have the important implication that the calldidates for statiollary eqllilibrium paths (y, . .. ,y, ... ) are
Qu'put
completely determined by the technology and the discount faclOr and are independent <J{ the lIIility fllnctioll u(·).
To pursue the analysis it will be much more convenient to reduce the level of ahstraction. Consider an extremely simple case, the Ramsey-Solow model technology of Example 20.C.1. We study trajectories with I, = I for all t (imagine that there is available one unit of labor at every point in time). We can then identify a production path with the sequence of capital investments (k o, ... , k" ... ). Given (k o, ... , k" .. .), denote r, = V,F(k" 1) - I. Thus, r, is the lIel (i.e., after rcplacing capital) marginal productivity of capital. Suppose that k, > 0 and that the sequence of output prices (qo, ... ,q" .. .) and wages (w o, ... , H'" ... ) myopically price supports the given path. Then, by the first-order condition for profit maximization, we have q,+,(1 + r,) - q, = O. Hence r, is the output rate of interest at time t implicit in the output price sequence (qo, ... ,q" ... ). Let us now focus on the stationary paths of this example. Any k ;,: 0 fixed through time constitutes a sleady stale. With any such steady state we can associate a constant surplus level c(k) = F(k, 1) - k and a rate of interest r(k) = V,F(k, I) - I, also constant through time. 1 ' Therefore, the supporting price-wage sequence is with H'o
V1F(k,l)
q. = V,F(k, 1>"
Denote by w(k) the real wage wo/qo so determined. It is instructive to analyze how the steady-state levels of consumption e(k), the rate of interest r(k), and the real wage w(k) depend on k. Let k be the level of capital at which the steady-state consumption level is maximized [i.e., k solves Max F(k, 1) - k). Note that k is characterized by r(k) = V, F(k, 1) - 1 = O. Thus k is precisely the goldell rule steady state. The construction is illustrated in Figure 20.E.I, where we also represent the modified golden rule k. [characterized by r(kd ) = V,F(k., 1) - 1 = (I - 0)/0). Observe that if k < k then r(k) > O. As we saw in Proposition 20.E.2, r(k) > 0 implies that the steady state k is etl1cicnt (thus, in particular, the modified golden rule is etl1cient: it gives less consumption than the golden rule but it also uses less capital). Similarly, if k > k then r(k) < 0 and we have inefficiency of the steady state k. What about k?" We now argue that the golden rule steady state k is efficient. A graphic proof will be quickest. Suppose we try to dominate the constant path k by starting with ko < k, so that consumption at I = 0 is raised. Since the surplus at t = 1 must be at least
21. Thus. c(k) is the amount of good constantly available through time and usable as a flow
for consumption purposes. 22. Recall that the associated price sequence is constant and that the transversality condition is therefore violated.
Figure 2O.E.l
I"pu,
-k I
k.
t
The production technology of the Ramsey-Solow model and the golden rule.
I
Golden Mod,fied _ Rule Golden Rule J k,
='0 -
( O. We will not state or demonstrate this theorem precisely, but the main idea of its proof is quite accessible. We devote the next few paragraphs to it. Suppose for a moment that for a given u(', .) our candidate 1/1(') is such that oJ!(k) solves, for every k, the following "complete impatience" problem: Max
u(k, k').
(20.F.2)
t':
This would be the problem of a decision maker who did not care about the future. While this is not quite the problem that we want to solve, it approximates it if we take" > 0 to be very low. Then the decision maker cares very little about the future and therefore its optimal action k' will, by continuity, be very close to I/I(k). Hence, in an approximate sense, we are done if we can find a u(', .) such that oJ!(k) solves (20.F.2) for every k. In order for a oJ!(k) > 0 to solve (20.F.2), u(k, .) cannot be everywhere decreasing in its second argument (the optimal decision would then be k' = 0). [n the simplest version of the Ramsey-Solow model (Example 20.C.I), the returns of k', the investment in the current period, accrue only in the next period, and therefore the utility function u(k, k') is decreasing in k'. But in the current, more general, two-sector model there is no reason that forces this conclusion. Suppose, for example, that there are two consumption goods. The first is the usual consumption-i~vestment good, while the second is a pure consumption good not perfectly substitutable with the first. Say that with an amount k of investment at time t - lone gets, jointly, k units of the consumption-investment good at time t and k units of the second consumption good at time t - I. Accordingly, with k' units of the consumption-investment good invested at lone gets, jointly, k' units of the consumption-investment good at t + 1 and k' units of the second consumption good at t. Thus, if k and k' are the amounts of investment at 1 - I and I, respectively, then the bundle of consumption goods available at 1 is (k - k', k'). Hence, the utility function u(',') has the form u(k, k') = u(k - k', k'), where a(·, .) is a utility function for bundles of the two consumption goods. Therefore, our problem is reduced to the following: Given oJ!(k) can we find a(·, .) such that oJ!(k) solves Max,. u(k - k', k') for all k in some range? The problem is represented in Figure 20T5. 24 We see from the figure that the problem has formally become one of finding a concave utility function with a prespecified Engel curve at some given prices (in our case, the two prices are equal). Such a utility function can always be obtained. It is a well-known, and most plausible fact that the concavity of a(·) imposes no restrictions on the shape that a single Engel curve may exhibit (see Exercise 20.F.I). The news is not uniformly bad, however. In principle, as we have seen, everything may be possible; yet there are interesting and useful sufficient conditions implying a
24. We also assume that of;(k) < k for all k.
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---------------------------------------------------------------------------------=-~~~~--------------------------------Second Hence, if the discount factor t5 is close to 1, it is a plausible conclusion that 1!/I'(k)1 < 1 Consumption Good
for all k. In technical language: !/I(') is a contraction, and this implies global convergcnce to a unique steady state.'· In Exercise 20.F.2 you are invited to draw the policy functions and the arrOw diagrams for this case. A particular instance of a contraction is exhibited in Figure 20.F.1. ~_ _
(k - y,(k). y,(k))
Figure 20.F.S
Construction of an
arbitrary policy function in the completely impatient casco
45 First Consumption Good
well-behaved dynamic behavior. We discuss two types of conditions: a low discollllr of time and ('ross derivatives of uniform posit;l'£, sign.
Low Discoullr of Time One of the most general results of dynamic economics is the turnpike rheorem, which, informally, asserts that if rhe one-period utility function is strictly concave and tile decision maker is very patient, then rhere is a single modified golden rule sready scare
rhar, moreover, attracts the optimal trajectories from any initial position. I n the context of the two-sector model studied in this section, we can give some intuition for the turnpike theorem. Suppose that the value function V(k), which is concave, is twice-differentiable." At the end of Section 20.D, we saw that since by definition,
V(k
Cross Derivative of Uniform PositilJe Sign
+ z) 2: u(k + z, !/I(k» + W(t/I(k»
for all z and k (with equality for z = 0), we must have
V'(k) = V,u(k, t/I(k»
and
V"(k) 2: V?,u(k, !/I(k»
for all k.
Also for all k, t/I(k) solves the first-order condition
V2 u(k, t/I(k»
+ t5 V'(!/I(k)) = O.
(20.F.3)
Differentiating this first-order condition, we have (all the derivatives are evaluated at k, !/I(k) and assumed to be nonzero)
V 2 u(') t/I'(')
Because
= - Vi,"(';'+ t5V"(')
Vl 11l(') :;:; 0 and SV;'Il(') :;:; b V"(-) :;:; 0, it follows that 1\11'(')1 :;:; \Vi,u(
The lurnpike Iheorem is valid for any number of goods. The precise slatement and the proof of the Iheorem are subtle and technical [see McKenzie (1987) for a brief survey], but the main logic is simply conveyed. Consider the extreme case where there is complete patience, that is, "only the long-run matters." A difficulty is that it is not clear what this means for arbitrary paths; but at least for paths that are not too "wild," say for those that from some time become cyclical, it is natural to assume that it means that the paths are evaluated by taking the average utility over the cycle. Observe now that for any cyclical nonconstant path. the strict concavity of the utility function implies that the constant path equal to the mean level of capital over the cycle yields a higher utility. It may take some time 10 carry out a transition from the cycle to the constanl path (e.g., it may be necessary to build up capital) bul, as long as this can be done in a finite number of periods, the cost of the transition will not show up in the long run. Hence the cyclical nonconslant path cannot be optimal for a completely patienl optimizer. By continuity, all this remains valid if 0 is very close 10 I. We can conclude. therefore, that if a path tends to a nonconstant cycle then we can always implement a finite Iransition to a suitable constant "long-run average," for a relatively large long-run gain of utility and a relatively low short-run cost. In fact, this conclusion remains valid whenever a path does not stabilize in the long-run. It follows that the optimal path must be asymptotically almost constant, which can only be the case if the path reaches and remains in a neighborhood of a modified golden rule steady state (recall from Section 20.E that those are the only constant paths that can be equilibria, and therefore optimal)."
~i~u~~~ u(. )\. I
We shall concern ourselves here with the particular case of the two-sector model studied so far where V,u(k, k') > 0 and V,u(k, k') < 0 for all (k, k'). By a cross derivative of uniform positive sign we mean that V12 u(k, k') > 0, again at all points of tlte domain. In words: An increase in investment requirements at one date leads 10 a situation of increased productivity (in terms of current utility) of the capital installed the previous date. Examples are the classical Ramsey-Solow model u(F(k) - k') and the cost-of-adjustment model u(F(k) - k' -y(k' - k)) (see Exercise 20.F.3). We shall argue that under tltis cross derivative condition tIle policy function is increasing (as in Figures 20.F.1 or 20.F.2), and therefore the optimal path converges to a stationary path. To prove the claim, it is useful to express \II(k) as the k' solution to Max
u(k, k')
+ bV
(20.FA)
(II',VI
s.t. V:;:; V(k'),
By the concavity of 11(') we have (see Sections M.e and M.D of the Mathematical Appendix)
26. We nole thai y,(.) need not be monotone and Ihe convergence may be cyclical. although the cycles will dampen through lime.
25. For a (very advanced) discussion of this assumption. see Santos (1991).
i
~L
27. Also. with ~ close 10 1, the modified golden rule willlypically retain the uniqueness property of the golden rule.
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765 -------------------------------------------------------------------- ----------------------------------------------------------------------------Indifference Curve for Ihe Ulilily + W, k fixed
u(k, k')
v
SEC T ION
-",-",//
k:I-_ _ _. ....L... /~=------ !/I,(')
>k
k;'
r-A
r----~~/-"..."...------!/I(. ) / / / ~ Transilory Shock //
_.",&----V(k')
,,/
"" ",,""
!/I(k)
S EVE R A l e 0 N SUM E R S
Permanenl Shock / /
J /
I'(o/I(k))
E QUI l I • R I U M:
k'
Indifference Curve for Ihe Ulilily , u(i. k') + J V, k fixed
i:
J. . . . . . . .
2 0 • G:
/
/"
,, I, ,,I
~5'
I
f
Figure 2O.F.6
k'
where V(·) is the value function. For fixed k, problem (20.F.4) is represented in Figure 20.F.6. The marginal rate of substitution (M RS) between current investment k' and future utility V at s = (I/I(k), V(I/I(k))) is O/b)V2u(k. I/I(k» < O. Suppose now that we take k > k. Then the indifference map in Figure 20.F.6 changes. Because V'211(k. I/I(k)) > O. the MRS at s is altered in the manner displayed in the figure. that is. the indifference curve becomes flatter. But we can see then that necessarily I/I(k) > I/I(k), as we wanted to show. The cross derivative condition does not. by itself. imply the existence of a single modified golden rule. Thus, we could be in Figure 20.F.2 rather than in Figure 20.F.!. Note, however. that in many cases of interest it may be possible to show directly that the modified golden rule is unique. Thus. in both the classical Ramsey-Solow model of Example 20.C.1 and in the cost-of-adjustment model [with )1'(0) = 0] of Example 20.C.2. the modified golden rule is characterized by F'(k) = lib. Hence it is unique and. because the policy function is increasing. we conclude that every optimal path converges to it. We also point out that if the cross derivative is of uniform negative sign. then. by the same arguments. 1/1(') is decreasing. While this allows for cycles. the dynamics are still relatively simple. In particular. the non monotonic shape associated with the possibility of chaotic paths (Figure 20.F.4) cannot rise. See Deneckere and Pelikan (1986) for more on these points. Figure 20.F.6 is also helpful in illuminaling Ihe dislinclion bel ween Irallsilory and permanelll shocks. One of Ihe importanl uses of dynamic analysis in general. and of global convergence turnpike results in particular. is in the examination of how an economy at long-run rest reacts 10 a perlurbalion of the dala al lime I = I. In an exlremely crude c1assificalion, Ihese perturbalions can be of Iwo Iypes: (i) Transitory shocks affeci the environment of Ihe economy only al I = I; Ihal is, Ihey aller ko or. more generally. u(ko• . ). Ihe ulilily function at I = I. Then Figure 20.F.6 allows us to see how the equilibrium path will be displaced. The (k'. V) indifference curve of u(k o• k') + {, V changes, bul the constraint function V(k') remains unaltered. Hence. afler the (transilory) shock, Ihe new k', corresponds 10 the solulion of the optimum problem depicled in Figure
With the uniform positive sign cross derivative condition, the policy funclion is increasing.
20.F.6 bUI with Ihe new indifference map. From
t =
2 on we simply follow Ihe old policy
function.
(ii) Permanelll shocks move the economy to a new utility function u(k. k') constant over time. Then Ihe entire policy funclion changes to a new ';('). In terms of Figure 20.F.6 Ihere would be a change in both the indifference curves and the constraint. The new kf is now harder 10 determinc and to compare with the preshock k, or. for the same shock at period I. wilh k;'; bul il can oftcn be done. We pursue Ihe matter through Example 20.F.1. Example 20.F.I: Consider the separable utilily u(k. k') = y(k) + h(k'). This could be the inveslmenl problem of a firm: g(k) is Ihe maximal revenue obtainable wilh k. and -h(k') is Ihe cost of investment. Then Vl,u(k. k') = 0 at all (k. k'). Our previous analysis of Figure 20.F.6, lells us Ihal in this case o/t ... ,y" ... ), and (COl' ••• , C,I' •.• )
+ V1,u(·) + oVi,u(·) + Vf,u('))
o(Vi,u(')
(20.G.6)
If there are no externalities [i.e., if V1,u(') = Vf,u(') = 0] then the cO'lcavity of u(·, ,) implies that expression (20.G.6) is larger than I in absolute value (you should verify this in Exercise 20.G.5). Thus, in agreement with the discussion of Section 20.0, we are not then able to find a non·steady-state solution of the Euler equations. But if the externality effects are significant enough, inspection of expression (20.G.6) tells us immediately that dk1+ ,/dk, can perfectly well be less than 1in absolute value. The same is true for dk1+ ,/ok,_" and therefore we can conclude that robust examples with a continuum of equilibria are possible.
2Q,H Overlapping Generations In the previous sections we have studied economies that, formally, have an overlapping structure of firms but only one (or, in Section 20.G, several), infinitely long-lived, consumer. We pointed out in Section 20.B that in the presence of suitable forms of altruism it may be possible to interpret an infinitely long-lived agent as a dynasty. We will now describe a model where this cannot be done, and where, as a consequence, the consumption side of the economy consists of an infinite succession of consumers in an essential manner. To make things interesting, these consumers, to be called generations, will overlap, so that intergenerational trade is possible. The model originates in Allais (1947) and Samuelson (1958) and has become a workhorse of macroeconomics, monetary theory, and public finance. The literature on it is very extensive; see Geanakoplos (1987) or Woodford (1984) for an overview, Here we will limit ourselves to discussing a simple case with the purpose of highlighting, first, the extent to which the model can be analyzed with the Walrasian equilibrium
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-------------------------------------------------------------------------------------methodology and, second, the departures from the broad lessons of the previous sections. We shall classify these departures into two categories: issues relating to optimality and issues relating to the multiplicity of equilibria. We begin by describing an economy that, except for the infinity of generations, is as simple as possible. We have an infinite succession of dates t = 0, I, ... and in every period a single consumption good. For every I there is a generation born at time I, living for two periods, and having utility function u(c." COl) where c., and c.. are, respectively, the consumption of the Ith generation when young (i.e., in period C), and its consumption when old (i.e., in period 1 + I); the indices b and a are mnemonic symbols for "before" and "after." Note that the utility functions of the dilTerent generations over consumption in their lifespan are identical. We assume that 11(', .) is quasiconcave, differentiable and strictly increasing. Every generation 1 is endowed when young with a unit of a primary factor (e.g., labor). This primary factor does not enter the utility function and can be used to produce consumption goods contemporaneously by means of some production function J(:)." Say that J(I) = I. Under the competitive price-taking assumption, total profits at C, in terms of period-I good, will be f. = I - 1'(1) and, correspondingly, labor payments will be I - t. Thus, we may as well directly assume that the initial endowments of generation 1 0 units of consumption good. Now let (Po, ... , p" ... ) be an infinite sequence of (anticipated) prices. We do not require that it be bounded. For the budget constraint of the different generations we take P,C.,
+ P.. ,Ca, $; (I
- t)p,
for t > 0
(20.H.I)
and POC.O
+ P'CaO
$;
(1- t)po
+ t(~
p,) + M.
(20.H.2)
These budget constraints deserve comment. For 1 > 0, (20.H.I) is easy to interpret. The value of the initial endowments, available at I, is (I - t)p,. Part of this amount is spent at time 1 and the rest, (I - t)p, - P,c." is saved for consumption at t + I. The saving instrument could be the title to the technology, which would thus be bought from the old by the young at 1 and then sold at 1 + I to the new young (after collecting the period t + I return). The price paid for the asset is the amount saved, that is, (I - 0) p, - P,c.,. The direct return at 1 + I is tp, + I and so, if the asset market is to be in equilibrium, the selling price at t + I should be (I - t)p, - P,C., - cp,+ ,. In summary, in agreement with the budget constraint (20.H.l) this leaves (I - ")p, - P,C., to be spent at 1 + I. The constraint (20.H.2) for 1 = 0 is more interesting. Its right-hand side is the value of the asset to generation O. Note that asset market equilibrium requires that
33. The assumption thal production is contemporary with input usage fits well with the lenglh
or I he period being long.
20. H:
0 Y E R lAP PIN G
G ENE RAT ION.
771
----------------------------------------------------------------------------this value should be at least the Jundamental value, that is, t(l:, p,).)' Indeed, the value of the asset at t = 0 equals the profit return EPO plus the price paid by the young of generation I. At any T, the price paid by the young of generation Tshould not be inferior to the direct return tPT+ I' In turn, at T - 1 it should not be inferior to the direct return plus the value at T; that is, it should be at least t(PT + PT+ I)' Iterating, we get the lower bound t(p, + ... + PT+ ,) for the price paid by generation I, which, going to the limit and adding tpo, gives tel:, p,) as a lower bound for the value to generation O. Thus, in terms of expression (20.H.2) a necessary condition for equilibrium is M 0). We did not do so in Sections 20.D or 20.G because with a finite number of consumers, bubbles are impossible at equilibrium. The equality of demand and supply implies that the (finite) value of total endowments plus total profits equals the value of total consumption, and therefore at equilibrium no individual value of consumption can be larger than the corresponding individual value of endowments and profit wealth (you should verify this in Exercise 20.H.I). We will see shortly that under some circumstances bubbles can occur at equilibrium with infinitely many consumers. It would therefore not be legitimate to eliminate them by definition. The definition of a Walrasial1 equilibrium is now the natural one presented in Definition 20.H.1. Defin1tion 20.H.1: A sequence of prices (Po • ...• P, • ... ). an M 0, there is a single price sequence (with Po = I) that can be continued indefinitely, and therefore a single equilibrium path.
34. Strictly speaking, we are saying that if the consumption good prices are given by (Po.···. P,.···) and the asset prices present no arbitrage opportunity, then the price of the asset should be at least as large as its fundamental value.
772
SEC T ION
CHAPTER
20:
EOUILIBRIUM
AND
TIME
~~~~~~~----------------------
Suppose first that s > O. We say then that the asset is real (it has "real" returns). At an equilibrium the wealth of generation 0, (1 - s)Po + seL p,) + M, must be finite (how could this generation be in equilibrium otherwise?). Therefore, if s > 0, it follows that L, p, < OOB An important implication of this is that the aggregate (Le., added over all generations) wealth oj society, which is precisely L, P.. isjinite. In Proposition 20.H.1 we now show that, as a consequence, the first welfare theorem applies for the model with, > O.
Offer Curve
Proposition 20.H.1: Any Walrasian equilibrium (Po,." ,p" .. .), {{c~,. c:,l},=o, with L, Pt < 00 is a Pareto optimum; that is, there are no other feasible consumptions {{Cb" ca,)};':,o such that u{cb" Ca,) :? U{Cb,. c:,) for all t :? 0, with strict inequality for some t.
"'' -:)1-,
f',', , \'" "",_-+, '{'" I ___
Proof: We repeat the standard argument. Suppose that {(c... c.,)}~o Pareto dominates {(e:.. c:,J},":.o· From feasibility, we have c:, + C:.'_I = I and c., + C•. '_I :; 1 for every I. Therefore, L, p,(c:' + C:.,_I) = L, p, and L, p,(c., + C•. '_I) :; L, p,. Because L, p, < 00, we can rearrange terms and get
Figure 20.H.l
~""l--
C. 2
L'--C-'.-,---c...L,,~l"'-'--t+-~-C-o-'n~s-umplion in
first Period of Life
Overlapping generations: construction of
the equilibrium (case t > 0).
L (Pt c., + p,+ I C.,) :; L (p,c:' + p,+ , C:,) = L, p, < 00. Because the utility function is increasing and (c:" e:') maximizes utility in the p,c:, + p, + IC:, for every t, with at budget set we conclude that P,c., + PH I least one strict inequality. Therefore, L, (p" C.I + p,+ IC.,) > L, (p,c:, + p,+ IC:')Contradiction. _
C., ;::
Consumption in Second Period of Life 45' =
0 V E R LAP PIN G
Pareto Optimality
Consumption in Second Period of Life
c.,
2 0 _ H:
~
•
No-Trade Sleady-Stale COl
C.o
Consumption in first Period of Life
G ENE RAT ION S
773
------------------------------------------------------------------------------
Proposition 20.H.1 is important but it is not the end of the story. Suppose now that the asset is purely nominal (i.e., , = 0; for example, the asset could be fiat money, or ownership of a constant returns technology). Then it is possible to have equilibria that are /lot optimal. I n fact, it is easy to see that we can sustain autarchy (i.e., no trade) as an equilibrium_ Just put M = 0 (no bubble, worthless fiat money) and choose (Po, ... , p". _.) so that, for every t, the relative prices p,/p, + I equal the marginal rate of substitution of u(', .) at (I, 0), denoted by p. This no-trade stationary state (also called tlte nonmonetary steady state) where every generation consumes (1,0) is represented in Figure 20.H.2. As it is drawn (with p < I), we can also see that the no-trade outcome is strictly Pareto dominated by the steady state (y, 1 - y) [or, more precisely, by the consumption path in which generation 0 consumes (I, 1 -)') and every other generation consumes (y, 1 - y)]. What is going on is simple: in this example the open-ended ness of the horizon makes it possible for the members of every generation I to pass an extra amount of good to the older generation at t and, at the same time, be more then compensated by the amount passed to them at 1+ 1 by the next generation_ Note that, in agreement with Proposition 20.H.I, the lack of optimality of this no-trade equilibrium entails p,/p, + I = P< I for all t; that is, prices increase through time. It is also possible in the purely nominal case for an equilibrium with M > 0 not to be Pareto optimal. Note first if {(e:,. c:,)},"';,o, (Po"", P..... J and M constitute an
Figure 20_H.2
Overlapping generations: construction of
equilibria (case t = 0)
It corresponds to the stationary consumptions (y, 1 - y) and the price sequence p, = 0:', where 0: = (I - s - y)/(l - y) < L Note that the iterates that ~gin at a ~alue c.o '" 1 - y unavoidably "leave the picture," that is, bec~me unfe~s~~le. In Flg~re 20.H.2, where s = 0, there is a continuum of equlhbna: any tnltlal conditIOn
C.o :; 1 - y can be continued indefinitely. .. . It is plausible from Figures 20.H.1 and 20.H.2 that the existence of an eqUlllbnum can be guaranteed under general conditions- This is indeed the case [see Wilson
35. You can also verify this graphicaJly by examining Figure 20.H.l.
(1981)].
1
774
CHAPTER
20:
EQUILIBRIUM
AND
TIME
equilibrium, then we have (recall that cto = I) p,+ ,c:,
= p,(1
- ct,)
= p,c:,,_, = ... = p,e:o = M
for every t.
-
c.,
/
Monetary Sleady State
1-",
Consumption in First Period of Life
0 V E R LAP PIN G
ProposItion 20.H.2: Suppose that at an equilibrium we have LtP t
Y we have a nonstationary equilibrium trajectory with trade (hence M > 0) which is also strictly Pareto dominated by the steady state (y, I - y). Nonetheless, it is still true that for any equilibrium with COl> Y we have Mfp, -+ 0; that is, in real terms the value of the asset becomes vanishingly small with time. For CO, = y, matters are quite different. We have a steady-state equilibrium (called the monetary steady state) in which the price sequence p, is constant and therefore the real value of money remains constant and positive. This monetary steady state is the analog of the golden rule of Section 20.E and, as was the case there, we have that. in spite of L, p, < 00 being violated. the monetary steady state is Pareto optimal. We will not give a rigorous proof of this. The basic argument is contained in Figure 20.H.3. There we represent the indifference curve through (y, I - y) and check that any attempt at increasing the utility of generation 0 by putting c.. < y leads to an unfeasible chain of compensations; that is, it cannot be done. The discussion just carried out of the examples in Figures 20.H.2 and 20.H.3 suggests and confirms the following claim, which we leave without proof: In the purely nominal case, of all equilibrium paths the Pareto optimal ones are those, and only those, that exhibit a bubble whose real value is bounded away from zero throughout time. It is certainly interesting that a bubble can serve the function of guaranteeing the optimality of the equilibria of an economy, but one should keep in mind that this happens only because an asset is needed to transfer wealth through time. If a real asset exists then this asset can do the job. If one does not exist then the economy, so to speak, needs to invent an asset. To close the circle, we point out that if there is a real asset then not only is a bubble not needed but, in fact, it cannot occur.
c.,
SEC T ION
G ENE RAT ION S
775
-------------------------------------------~~~~~~~~
We have already seen, in Figure 20.H.2, a model with a purely nominal asset (i.e., c = 0) and very nicely shaped preferences (the offer curve is of the gross substitute type) for which there is a continuum of equilibria. Of those, one is the Pareto optimal monetary steady state and the rest are nonoptimal equilibria where the real value of money goes to zero asymptotically. The existence of this sort of indeterminacy is clearly related to the ability to fix with some arbitrariness the real value of money (the "bubble") at t = 0, that is Mfpo. It cannot occur if bubbles are impossible, as, for example, in the model with a real asset (i.e., £ > 0) where, in addition, we know that the equilibrium is Pareto optimal. One may be led by the above observation to suspect that the failure of Pareto optimality is a precondition for the presence of a robust indeterminacy (i.e., of a continuum of equilibria not associated with any obvious coincidence in the basic data of the economy). This suspicion may be reinforced by the discussion of Section 20.G, .where we sa~ that the Pareto optimality of equilibria was key to our ability to claim the generic determinacy of equilibria in models with a finite number of consumers. Unfortunately, with overlapping generations the number of consumers is infinite in a fundamental way,'" and this complicates matters. Whereas with a rear asse~ the Pareto optimality of equilibria is guaranteed and the type of indeterminacy of Figure 20.H.2 disappears, it is nevertheless possible to construct nonpathological examples with a continuum of equilibria. The simplest example is illustrated in Figure 20.H.4. The figure describes a real-asset model with the steady state (y, I - }'). Suppose that, in a procedure we have resorted to repeatedly, we tried to construct an equilibrium with c. o slightly different from I - y. Then, normalizing to Po = I, we would need to use p, to clear the market of period 0, P2 to do the same for period I, and so on. In the leading case of Figure 20.H.I, we have seen that this eventually becomes unfeasible. A change in p, that takes care of a disequilibrium at t - I creates an even larger disequilibrium at t, which then has to be compensated by a change of a larger magnitude in p, + I In an explOSive process that finally becomes impossible. But in Figure 20.H.4, the utility function is such that, at the relative prices of the steady state, a change in the pnce of the second-period good has a larger impact on the demand for the first-period good than on the demand for the second-period good. Hence. the successive adjustments necessitated by an initial disturbance from c = 1 _ " dampen with each iteration and can be pursued indefinitely. We conclud~ that a~ equilibrium exists with the new initial condition. As a matter of terminology, the
, 36. By this v k, and by g(k' - k) = 0 for k' S k. Say as much as you can about the policy. In particular, determine the steady·state trajectory of investment. 20.D.6 8 Verify the claim made in the proof of Proposition 20.0.7 that the Euler equations (20.0.9) are the first-order necessary and sufficient conditions for short-run optimization. In other words: they are necessary and sufficient for the nonexistence of an improving trajectory differing from the given one at only a finite number of dates. 20.D.7 A With reference to Example 20.0.4, show that, for the functional forms given, the Euler equations are as indicated in the example: k,., = 3k, - 2k,., for every I. Also verify that the solution to this difference equation given in the text is indeed a solution, that is, that it satisfies the equation. 20.D.8 A Verify that the value function V(k) does satisfy the properties (i) and (ii) claimed for it at the end of Section 20.0. 20.D.9 A Argue that the properties (i) and (ii) of the value function referred to in Exercise 20.0.8 yield the two consequences, concerning V'(k) and V"(k), claimed at the end of Section 20.0. 20.E.IA Discuss in what sense the term r defined after the proof of Proposition 20.E.1 can be interpreted as the rate of interest implicit in the proportional price sequence. 20.E.2" Suppose that the production set Y c RL is of the constant return type and consider production paths that are proportiollal (but not necessarily stationary), that is, paths (Yo .. ..• .1'..... ) that satisfy .1', = (I + II)Y,., for aliI and some II. (a) Argue that the conclusion of Proposition 20.E.1 remains valid for proportional paths. (b) State and prove the result parallel to Proposition 20.E.2 for proportional paths. 20.E.3" Suppose that in the Ramsey-Solow model k solves Max (F(k, I) - k) (see Figure 20.E.2). Show that if k, S k - • for aliI, then the path determined by (k o,' .. , k" ... ) is efficient. [Hilll: Compute prices and verify the transversality condition.] 20.EAA Prove the three neoclassical properties stated at the end of the regular type part of Section 20.E. 20.E.S A Carry out the requested verification of expression (20.E. I).
E X E R CIS E S
785
------------------------------------------------------------------20.E.6
A
Carry out the verification requested in the discussion of Figure 20.E.3. A 20.E.7 In the Ramsey-Solow model. two dilTerent steady states are associated with different rates of interest. This is not so in the example illustrated in Figure 20.E.3, at first sight very similar. The key difference is that in the Ramsey-Solow model the consumption and investment goods are perfect substitutes in production. Clarify this by proving, in the context of the example underlying Figure 20.E.3, that if the two goods are perfect substitutes then r(k) # r(k) whenever k # i:. [Hilll: Their being perfect substitutes means that G(k, k' + a) = G(k, k') - a for any a < F(k. k'j.] A 20.E.8 Consider the proportional production paths with rate of growth equal to II > 0 (recall Exercise 20.E.2) in the context of a Ramsey-Solow technology of constant returns. Show that among these paths the one that maximizes surplus (at I = I, or, equivalently, normalized surplus or surplus" per capita") is characterized by having the rate of interest equal to II. This path is also called the lIoldell rule sleady slale palh. A 20.E.9 Argue that, for the one-Consumer model of Section 20.0, the golden rule path cannot arise as part of a competitive equilibrium. [Hint: The key fact is that J < I.] c 20.F.l Consider two arbitrary functions y,(w) and y,(w) that are defined for w > 0, take nonnegalive values, and satisfy y,(w) + y,(w) = w for all w. Suppose also that they are twice continuously ditTerentiable. Show thaI for any a > 0 there is a utilily function for lwo commodities. u(x" x,), that is increasing and concave on the domain {(x"x,): x, + x, S a} and is such that (y,(w), y,(w)) coincides wilh lhe Engel curve functions for prices p, = I, p, = I and wealth w < a. [Hinl: Let u(x" x,) ~ (x, + x,)'/2 - £[(x, - y,(x, + x,))' + (x, - y,(x, + x,))'] and take £ to be small enough. Verify then that Vu(x" x,) is strictly positive and D'u(x" x,) is negative definite for any (x,. x,) such that 0 < x, + x, SiX. and that the Engel Curve is as required.] 20.F.2A Suppose lhat. for k E R., the policy function ' .. , "'I) E R ,
where "'i takes the value 1,0, or -I according to whether agent i prefers alternative x to alternative y, is indifferent between them, or prefers alternative y to alternative x, respectively.' Definition 21.B.1: A social welfare functional (or social welfare aggregator) is a rule F(~" ... , "',) that assigns a social preference, that is,F(a., ... , "',) E {-1, O. 1}. to every possible profile of individual preferences (IX ••...• a,) E { -1. O. 1}'. All the social welfare functionals to be considered respect individual preferences in the weak sense of Definition 2I.B.2. Definition 21.B.2: The social welfare functional F(", ••.. .• "',) is Paretian, or has the Pareto property. if it respects unanimity of strict preference on the part of the agents. that is. if F(l • ...• 1) = 1 and F( -1 •...• -1) = - I. Example 2I.B.I: Paretian social welfare functionals between two alternatives abound. Let (fJ, •...• fJ,) E IR'+ be a vector of nonnegative numbers, not all zero. Then we
2 •• 8:
ASP Eel A l e A S E:
SOC I ALP REF ERE NeE SOY E R
TWO
could define F(IX" ... , ",,)
= sign 2., Pi(X',
where. recall, for any a E R, sign a equals I, 0, or -I according to whether a > 0,
a = 0, or a < 0, respectively. An important particular case is majority voting, where we take p, = I for every i. Then F(IX, •... , ",,) = I if and only if the number of agents that prefer alternative x to alternative y is larger than the number of agents that prefer y to x. Similarly, F(a, •...• a,) = -I if and only if those that prefer y to x are more numerous than those that prefer x to y. Finally, in case of equality of these two numbers, we have F(IX" ...• IX,) = 0, that is. social indifference. a Example 2I.B.2: DictatorslJip. We say that a social welfare functional is dictatorial if there is an agent h, called a dictator, such that, for any profile ("'I" .. ,"',), "'. = I implies F(cx" ... , "',j = I and, similarly, "'. = -I implies F("" , •.• , "',j = -I. That is. the strict preference of the dictator prevails as the social preference. A dictatorial social welfare functional is Paretian in the sense of Definition 2I.B.2. For the social welfare functionals of Example 21.B.1, we have dictatorship whenever "'. > 0 for some agent It and "'i = for i ¥ It. since then F(a" ... , a,) = "' •. a
°
The majority voting social welfare functional plays a leading benchmark role in social choice theory. In addition to being Paretian it has three important properties, which we proceed to state formally. The first (symmetry among agents) says that the social welfare functional treats all agents on the same footing. The second (neutrality between alternatives) says that, similarly, the social welfare functional does not a priori distinguish either of the two alternatives. The third (positive responsiveness) says, more strongly than the Paretian property of Definition 2I.B.2, that the social welfare functional is sensitive to individual preferences. Definition 21.B.3: The social welfare functional F("' ...... "',) is symmetric among agents (or anonymous) if the names of the agents do not matter. that is. if a permutation of preferences across agents does not alter the social preference. Precisely. let 1I:{1, ... . /} -+ {1 ....• 1} be an onto function (i.e., a function with the property that for any i there is h such that 1I(h) = i). Then for any profile (a, •...• IX,) we have F(cx •. ... ,,,,,) = F(IX'I')'" . ,a,I'I)' Definition 21.B.4: The social welfare functional F("', •.... :x,) is neutral between alternatives if F(a, •...• a,) = -F(-IX, •... , -:x,) for every profile (:x, •...• a,). that is. if the social preference is reversed when we reverse the preferences of all agents.
rormally the principle involved. Note. in particular. that this specification precludes the usc of any
Definition 21.B.S: The social welfare functional F(~, •. ..• :x,) is positively responsive if. whenever (IX, •...• 7,) ~ (ex; .... • ~;). ("', ....• IX,) ¥ ("'; .... , IX;). and F(",; • ...• cx;) ~ O. we have F(a, • ...• (1.,) = + 1. That is. if x is socially preferred or indifferent to y and some agents raise their consideration of x. then x becomes socially preferred.
"cardinal" or "intensity" information between the two alternatives because this intensity can only be calibrated (perhaps using lotteries) by appealing to some third alternative. A rortiori, the specification also precludes the comparison of feelings of pleasure or pain across individuals. In Chapter 22, we discuss in some detail matters pertaining to the issue of interpersonal comparability of utilities.
It is simple to verify that majority voting satisfies the three properties of symmetry among agents, neutrality between alternatives, and positive responsiveness (sec Exercise 21.B.I). As it turns out, these properties entirely characterize majority voting. The result given in Proposition 21.B.1 is due to May (1952).
I. In the whole of this chapter we make the restriction that only the agents' rankings between the two alternatives matter for the social decision between them. In Section 21.C we will state
A L T ERN AT, V E S
791
792
CHAPTER
.,:
SOCIAL
CHOICE
THEORY
Proposition 21.B.1: (May's Theorem) A social welfare functional F(IX" ... ,lXd is a majority voting social welfare functional if and only If it is symmetric among agents, neutral between alternatives, and positive responsive. Proof: We have already argued that majority voting satisfies the three properties. To establish sufficiency note first that the symmetry property among agents means that the social preference depends only on the total number of agents that prefer alternative x to y, the total number that are indifferent, and the total number that prefer y to x. Given (IX., ... ,IX,), denote n+(IX., ... ,IX,) = #(i:IX,= I),andn-(IX., ... ,IX,)= #(i:IX,= _1).2 Then symmetry among agents allows us to express F(IX., ... , IX,) in the form F(IX., ... , IX,) = G(n+(IX., ... , IX,), n-(IX., ... , IX,». Now suppose that (IX., ... ,IX,) is such that n+(IX ...... IX,) = n-(IX., ... ,IX,). Then 11+( -IX I, ... , -IX,) = n-(IX., ... , IX,) = n+(IX., ... , IX,) = n-( -IX ..... , -IX,), and so
--
SECTION
".C:
THE
GENERAL
CASE:
ARROW'S
IMPOSSIBILITY
there are I agents, indexed by i = I, ...• J. Every agent i has a rational preference relation ::::, defined on X. The strict preference and the indifference relation derived from ::::, are denoted by >-, and -" respectively.3 In addition, it will often be convenient to assume that no two distinct alternatives are indifferent in an individual preference relation ::::,. It is therefore important, for clarity of exposition, to have a symbol for the set of all possible rational preference relations on X and for the set of all possi ble preference relations on X having the property that no two distinct alternatives are indifferent. We denote these sets, respectively, by fit and 9. Observe that 9 c: .'11." In parallel to Section 21.B, we can define a social welfare functional as a rule that assigns social preferences to profiles of individual preferences (::::" ... , ::::,) E fit'. Definition 21.C.1 below generalizes Definition 21.B.I in two respects: it allows for any number of alternatives and it permits the aggregation problem to be limited to some given domain .O=IX;+ •. Therefore, by the positive responsiveness property, we must have F(IX., ... ,IX,) = I. In turn, if n-(IX., ... ,IX,) > n+(IX ..... ,IX,) then n+(-IX ..... ,-IX,» n-( -(X., ... , -IX,) and so F( -IX ..... , -IX,) = I. Therefore, by neutrality among alternatives: F(IX., ... ,IX,)= -F(-IX., ... , -IX,) = -I. We conclude that F(IX., ... , IX,) is indeed a majority voting social welfare functional. _ In Exercise 2I.B.2, you are asked to find examples dilTerent from majority voting that satisfy any two of the three properties of Proposition 21.8.1.
2LC The General Case: Arrow's Impossibility Theorem We now proceed to study the problem of aggregating individual preferences over any number of alternatives. We denote the set of alternatives by X, and assume that
Definition 21.C.1: A social welfare functional (or social welfare aggregator) defined on a given subset .j is irrcnexive (x >jX cannot occur) and transitive (x >j'y and y >jX implies x >iX), Similarly, -j is fI:nexive (x -jX for all x EX). transitive (x - j y and)' -jZ implies x -,x) and symmetric (x - j y implies Y - j x). 4. Formally, the preference relation ~j belongs to ~ if it is reflexive (x ~I x for every x EX). transitive (x ;::i y and )' ~j z implies x ~I x) and IOtal (if x "# y then either x ~i y or y ~i x, but not
bOlh). Such preference relations are often referred to as slricl preferences (although Slricl-lolal preferem'('''' would be less ambiguous) or even as linear orders, because these are the properties of the usual "larger than or equal to" order in the real line. 5. In particular, there are no individual utility levels and, therefore. there is no meaningful sense
in which any conceivable information on individual utility levels could be compared and matched up. We refer again to Chapter 22 (especially Section 22.D) for an analysis of the problem that 2. Recall the notation # A = cardinality of the set A = number of clemen Is in the set A.
focuses on the information used in the aggregation process.
THEOREM
793
794
C HAP T E R
2,:
• 0 CI AL
C HOI C E
THE 0 R Y
----------------------------------------------------------------------~
Definition 21.C.2: The social welfare functional F:sI -+ fJl is Paretian if. for any pair of alternatives {x. y} c X and any preference profile (1::;, •...• 1::;,) e sI. we have that x is socially preferred to y. that Is. x Fp(I::;, •...• 1::;,) y. whenever x >-1 y for every i. In Example 2l.C.l we describe an interesting class of Paretian social welfare functionals. Example 21.C.l: The Borda Count. Suppose that the number of alternatives is finite. Given a preference relation 1::;, e fJl we assign a number of points c,(x) to every alternative x e X as follows. Suppose for a moment that in the preference relation 1::;, no two alternatives are indifferent. Then we put c,(x) = n if x is the nth ranked alternative in the ordering of 1::;,. If indifference is possible in 1::;, then c,(x) is the average rank of the alternatives indifferent to x. 6 Finally. for any profile (1::;"" .• 1::;,) E fJl' we determine a social ordering by adding up points. That is. we let F(I::;, •...• I::;,)efJl be the preference relation defined by xF(I::; ...... I::;,)y if L, c,(x) ~ L, c,(y). This preference relation is complete and transitive [it is represented by the utility function -c(x) = - L,C,(X)]. Moreover. it is Paretian since if x >-, y for every i then c,(x) < c,(y) for every i. and so L, c,(x) < L, c,(y) . • We next state an important restriction on social welfare functionals first suggested by Arrow (1963). The restriction says that the social preferences between any two alternatives depend only on the individual preferences between the same two alternatives. There are three possible lines of justification for this assumption. The first is strictly normative and has considerable appeal: it argues that in settling on a social ranking between x and y. the presence or absence of alternatives other than x and y should not matter. They arc irrelevant to the issue at hand. The second is one of practicality. The assumption enormously facilitates the task of making social decisions because it helps to separate problems. The determination of the social ranking on a subset of alternatives does not need any information on individual preferences over alternatives outside this subset. The third relates to incentives and belongs to the subject matter of Chapter 23 (see also Proposition 2I.E.2). Pairwise independence is intimately connected with the issue of providing the right inducements for the truthful revelation of individual preferences. Definition 21.C.3: The social welfare functional F:sI .... tJt defined on the domain sI satiSfies the pairwise independence condition (or the independence of irrelevant alternatives condition) if the social preference between any two alternatives {x. y} c X depends only on the profile of individual preferences over the same alternatives. Formally'. for any pair of alternatives {x. y} eX. and for any pair of preference profiles (1::;, ....• 1::;,) E d and (1::;., •...• 1::;;) E d with the property that. for every i. Xl::;iY xl::;;y and yl::;iX yl::;;x. 6. Thus if X = {x. y. z} and x~, y -, z then e,(x) = 1. and e,(y) = e,(z) = 2.5. 7. The expressions Ihat follow are a bil cumbersome. We emphasize Iherefore thaI Ihey do nothing more than to capture formally the statement just made. An equivalent formulation would be: for any {x.y} c: X. if ~,!{x.y} = ~;!{x.y} for all i. then F(~, ..... ~r)!{x.y} = F(;:'" ... , ;:;)1 {x. y}. Here;: !{x, y} 51ands for Ihe restriction of Ihe preference ordering;: 10 Ibe set {x, y}.
SECT'ON
'1.C:
THE
GENERAL
CASE:
ARROW'S
'MPOSSIB'L'TY
THEOREM
795
--------------------------------------------------------------------------we have that xF(I::;, •...• 1::;,) y
xF(I::;; ..... 1::;;) y
yF(~, •...• ~,)
yF(~·, •...• I::;;)x.
and
x
Example 21.C.l: conrinued. Alas. the Borda count does not satisfy the pairwise independence condition. The reason is simple: the rank of an alternative depends on the placement of every other alternative. Suppose. for example. that there are two agents and three alternatives {x. y. z}. For the preferences x >-, z >-, y.
y>-,x>-,z we have that x is socially preferred to y [indeed. e(x) = 3 and ely) = 4]. But for the preferences
x >-; y >-', z. y>-,z>-,x we have that .v is socially preferred to x [indeed. now c(x) = 4 and ely) = 3]. Yet the relative ordering of x and y has not changed for either of the two agents. For another illustration, this time with three agents and four alternatives {x. y. t. w}. consider
z >-, X
>-, J' >-, lV.
z >-,x >-,y>-, w. y
>- , z >-, w >- J x.
Here. y is socially preferred to x [c(x) = 8 and ely) = 7]. But suppose now that alternatives z and w move to the bottom for all agents (which because of the Pareto property is a way of saying that the two alternatives are eliminated from the alternative set): x >-', y >-', z >-', IV.
>-2 y >-, z >-', w. y >-3X >-3 z >-', IV. x
(21.C.l)
Then x is socially preferred to y [e(x) = 4. c(y) = 5]. Thus the presence or absence of alternatives z and w matters to the social preference between x and y. Another modification would take alternative x to the bottom for agent 3:
x >-;}' >-; z >-';
lV,
>-:;y >-:;z >-:; lI', }' >-; z >-; IV >-; x. X
Now .r is socially preferred to x [which. relative to the outcome with (21.C.l). is a nice result from the point of view of agent 3]. • The previous discussion of Example 21.C.l teaches us that the pairwise independence condition is a substantial restriction. However. there is a way to proceed that will automatically guarantee that it is satisfied. It consists of determining the social prefcrence between any given two alternatives by applying an aggregation rule that uses only the information about the ordering of rhese two alternatives in
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individual preferences. We saw in Section 21.B that, for any pair of alternatives, there are many such rules. Can we proceed in this pairwise fashion and still end up with social preferences that are rational. that is. complete and transitive? Example 21.C.2 shows that this turns out to be a real difficulty. Example 21.C.2: The Condorcet Paradox." Suppose that we were to try majority voting among any two alternatives (see Section 21.B for an analysis of majority voting). Does this determine a social welfare functional? We shall see in the next section that the answer is positive in some restricted domains d c !Jtl • But in general we run into the following problem. known as the Condorcet paradox. Let us have three alternatives {x. y. z} and three agents. The preferences of the three agents are
x >-. y >-1 z.
----
--
SECTION
21.C:
THE
GENERAL
CASE:
ARROW'S
IMPOSSIBILITY
Definition 21_C.4: Given F('), we say that a subset of agents ScI is: (i) Decisive for x over y if whenever every agent in S prefers x to y and every agent not in S prefers y to x, x is socially preferred to y. (ii) Decisive if, for any pair {x, y} eX, S is decisive for x over y. (iii) Completely decisive for x over y if whenever every agent in S prefers x to y, x is socially preferred to y. The proof will proceed by a detailed investigation of the structure of the family of decisive sets. We do this in a number of small steps. Steps 1 to 3 show that if a subset of agents is decisive for some pair of alternatives then it is decisive for all pairs. Steps 4 to 6 establish some algebraic properties of the family of decisive sets. Steps 7 and 8 use these to show that there is a smallest decisive set formed by a single agent. Steps 9 and 10 prove that this agent is a dictator.
Z>-2X>-2Y. Y >- 3Z>- 3x.
Then pairwise majority voting tells us that x must be socially preferred to Y (since x has a majority against y and. a fortiori. y does not have a majority against x). Similarly. y must be socially preferred to z (two voters prefer y to z) and z must be socially preferred to x (two voters prefer z to x). But this cyclic pattern violates the transitivity requirement on social preferences. _ The next proposition is Arrow's impossibility theorem. the central result of this chapter. It essentially tells us that the Condorcet paradox is not due to any of the strong properties of majority voting (which. we may recall from Proposition 2I.B.l. are symmetry among agents, neutrality between alternatives, and positive responsiveness). The paradox goes to the heart of the matter: with pairwise independence there is no social welfare functonal defined on !Jtl that satisfies a minimal form of symmetry among agents (no dictatorship) and a minimal form of positive responsiveness (the Pareto property). Proposition 21.C.1: (Arrow's Impossibility Theorem) Suppose that the number of alternatives is at least three and that the domain of admissible individual profiles. denoted d, is either d = 91 1 or d = iJl l • Then every social welfare functional F:.flI -+ !Jt that is Paretian and satisfies the pairwise independence condition is dictatorial in the following sense: There is an agent h such that, for any {x, y} eX and any profile (;::; I' . . . , ;::;/) ed, we have that x is socially preferred to y. that is, x Fp (;::;, •••. , ;::;d y, whenever x >-hY' Proof: We present here the classical proof of this result. For another approach to the demonstration we refer to Section 22.D. It is convenient from now on to view I not only as the number but also as the set of agents. For the entire proof we refer to a fixed social welfare functional F: d -+ !Jt satisfying the Pareto and the pairwise independence conditions. We begin with some definitions. In what follows, when we refer to pairs of alternatives we always mean distinct alternatives.
8. This example was already discussed in Section 1.8.
Step I: If for some {x. y} c X. ScI is decisive for x over y. then. for any alternative z # x. S is decisive for x over z. Simiiarly,Jor any z # y. S is decisive for z over y. We show that if S is decisive for x over y then it is decisive for x over any z # x. The reasoning for Z over y is identical (you are asked to carry it out in Exercise 21.C.l). If z = y there is nothing to prove. So we assume that z # y. Consider a profile of preferences (;::; ••...• ;::;1) e d where
for every i e S and for every i e I \S. Then. because S is decisive for x over y. we have that x is socially preferred to y, that is. xF,(;::; ••...• ;::;/)Y' In addition. since y;::;/z for every iel. and F(') satisfies the Pareto property it follov.s that y F,(;::; ...... ;::;1) z. Therefore. by the transitivity of the social preference relation. we conclude that x F,- x (i.e., such Ihal y ~ x bUI nol x ~ y). Thus, for any integer M we can find a
chain
x' >- x' >-. ,,>- x". where x· E X' for every m =
I .... , M. If M is larger than the number
of alternatives in X', then there must be some repetition in this chain. Say that x·' = x'" for m > m', By quasi transitivity. x"" >- x'" = x"", which is impossible because >- is irrenexive by definition. Hence, ;:: must be acyclic. An example or an acyclic but not quasitransitive relation will be given
in Example 2I.D.2. The relation >- derived from a rational prererence relation ~ is transitive (Proposition I.B.J). An example of a quasitransitive. but not rational. prererence relation is given
in Example 2t.D.t.
2 1 • D:
S0 ME
P D S SIB I LIT Y
RES U L T S:
RES T RIC TED
{v, w} c {x, y, z} we have that v is socially at least as good as w if either V;::I w, or v = y, w = x and v >- 2 w. In Exercise 21.0.3 you should verify that the social preferences so defined are acyclic but not necessarily quasitransitive. _
Single-Peaked Preferences We proceed now to present the most important class of restricted domain conditions: single-peakedness. We will then see that, in this restricted domain, nondictatorial aggregation is possible. In fact, with a small qualification, we will see that on this domain pairwise majority voting gives rise on this domain to a social welfare functional. Definition 21.0.2: A binary relation
x
y
then
z >- y
and If Y
> z - y.
In words: There is an alternative x that represents a peak of satisfaction and, moreover, satisfaction increases as we approach this peak (so that, in particular, there cannot be any other peak of satisfaction). Example 21.0.4: Suppose that X = [a,b] c R and ~ is the "greater than or equal to" ordering of the real numbers. Then a continuous preference relation;:: on X is single peaked with respect to ~ if and only if it is strictly convex, that is, if and only if, for every WE X, we have ay + (I - a)z >- W whenever y;:: W, Z ;:: w, y '" z, and a E (0,1). (Recall Oefinition 3.B.5 and also that, as a matter of definition, preference relations generated from strictly quasiconcave utility functions are strictly convex.) This fact accounts to a large extent for the importance of single-peakedness in economic applications. The sufficiency of strict convexity is actually quite simple to verify. (You are asked to prove necessity in Exercise 21.0.4.) Indeed, suppose that x is a maximal element for;::, and that, say, x > z > y. Then x ;:: y, y ;:: y, x '" y, and Z = ax + (I - a)y for some a E (0, I). Thus, z >- y by strict convexity. In Figures 2I.D.I and 21.0.2, we depict utility functions for two preference relations on X = [0, I]. The preference relation in Figure 21.0.1 is single peaked with respect to ;::" but that in Figure 21.D.2 is nol. _ Definition 21.0.4: Given a linear order : c 9t the collection of all rational preference relations that are single peaked with respect to
D 0 M A INS
801
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---------------------------------------------------------------Ftgure 21.D.1 (left)
Utility
Utility
Preferences are sing). peaked with resP e ar~ and any pair {x, y} c X, we put x i(::::" ... , ::::, )y, to be read as "x is socially at least as good as y", if the number of agents that strictly prefer x to y is larger or equal to the number of agents that strictly prefer y to x, that is, if #(ie/:x>-IY};;:: #{ie/:y>-,x}. Note that, from the definition, it follows that for any pair {x, y} we must have either x i(::::" ... , ::::,) y or y i(::::" ... , ::::,) x. Thus, pairwise majority voting induces a complete social preference relation (this holds on any possible domain of preferences). In Exercise 21.0.5 you are asked to show in a direct manner that the preferences of the Condorcet paradox (Example 2I.C.2) are not single peaked with respect to any possible linear order on the alternatives. In fact, they cannot be because, as we now show, with single-peaked preferences we are always assured that the social preferences induced by pairwise majority voting have maximal elements, that is, that there are alternatives that cannot be defeated by any other alternatives under majority voting. Let (::::,' ... , ::::,) e ar~ be a fixed profile of preferences. For every i e I we denote by XI e X the maximal alternative for ::::, (we will say that Xi is "i's peak"). Definition 21.0.5: Agent h e I is a median agent for the profile (::::1' ... , ::::,) e ar~ if #{ie/:x;;;::xh};;::2I
and
I # {'le/:xh;;::xi } ~2'
A median agent always exists. The determination of a median agent is illustrated in Figure 2I.D.3. If there are no ties in peaks and # I is odd, then Definition 21.D.5 simply says that a number (I - 1)/2 of the agents have peaks strictly smaller than x. and another number (I - I )/2 strictly larger. In this case the median agent is unique. Proposition 21.0.1: Suppose that ~ is a linear order on X and consider a profile of preferences (::::1"" , ::::,) where, for eve!y i, ::::; is single peaked with respect.to ;;::. Let h e I be a median agent. Then Xh F(:::: l' ••• , ::::,) y for every y e X. That IS, the peak xh of the median agent cannot be defeated by majority voting by any other alternative. Any alternative having this property is called a Condorcet winner. Therefore, a Condorcet winner exists whenever the preferences of all agents are singlepeaked with respect to the same linear order.
Flgur. 21.0.3
Agent 5 is the Median Agent 3 VOlers
3 VOlers
Proof: Take any y e X and suppose that x. > y (the argument is the same for y > x.). We need to show that y does not defeat x, that is, that #{ie/:x.>-,y};;:: #(ie/:y>-,x.}.
Consider the set of agents Sci that have peaks larger than or equal to x., that is, S = {i e I: XI ;;:: x.}. Then x, ~ x. > y for every i e S. Hence, by single-peakedness of :::: I with respect to ~, we get x. >-, y for every j e S. On the other hand, because agent h is a median agent we have that #S ~ 1/2 and so # {i e I: y >-,x.} ~ #(/\S) ~ 1/2~ #S:;, #{ie/:x.>-,y}._ Proposition 21.0.1 guarantees that the preference relation i(::::" ... , ::::,) is acyclic. It may, however, not be transitive. In Exercise 21.0.6 you are asked to find an example of nontransitivity. Transitivity obtains in the special case where I is odd and, for every i, the preference relation ::::, belongs to the class 9"~ c ar~ formed by the rational preference relations:::: that are single peaked with respect to ~ and have the property that no two distinct alternatives are indifferent for ::::. Note that, if I is odd and preferences are in this class, then, for any pair of alternatives, there is always a strict majority for one of them against the other. Hence, in this case, a Condorcet winner necessarily defeats any other alternative. Proposition 21.0.2: Suppose that I is odd and that ;;:: Is a linear order on X. Then pairwise majority voting generates a well-defined social welfare functional F: .'Ji'~ -+ .'11. That is, on the domain of preferences that are single-peaked with respect to ;;:: and, moreover, have the property that no two distinct alternatives are indifferent, we can conclude that the social relation F(::::1' ... , ::::,) generated by pairwise majority voting is complete and transitive. Proof: We already know that i(::::" ... , ::::,) is complete. It remains to show that it is transitive. For this purpose, suppose that x i(::::" ... , ::::,)y and y i(::::" ... , ::::,) z. Under our assumptions (recall that I is odd and that no individual indifference is allowed) this means that x defeats y and y defeats z. Consider the set X' = {x, y, z}. If preferences are restricted to this set then, relative to X', preferences still belong to the class .'Ji'~, and therefore there is an alternative in X' that is not defeated by any
Determination of a median for a single-peaked family.
803
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-------------------------------------------------------------~
21.0:
SOME
POSSIBILITY
RESULTS:
RESTRICTED
DOMAINS
605
---------------------------------------------------------------x = R'
Figure 21.0.4
Indifference curves for the preferences of Example 21.0.5.
other alternative in X'. This alternative can be neither y (defeated by x) nor z (defeated by y). Hence, it has to be x and we conclude that x transitivity. _
FC??:." . .. , ~,) z, as
required by
In applications, the linear order on alternatives arises typically as the natural order, as real numbers, of the values of a one-dimensional parameter. Then, as we have seen, singlepeakedness follows from the strict quasiconcavity of utility functions, a restriction quite often satisfied in economics. It is an unfortunate fact that the power of quasiconcavity is confined to one-dimensional problems. We illustrate the issues involved in more general cases by discussing two examples. Example 21.0.5: Suppose that the space of alternatives is the unit square, that is, X = [0, I]'. The generic entries of X arc denoted x = (x" x,). There are three agents I = {t, 2, 3}. The preferences of the agents are expressed by the utility functions on X:
o example is that the cone spanned by the nonnegative combinations of the gradient vectors of the three utility functions equals the entire R' (see Figure 21.0.4). Exercises 21.0.7 and 21.0.8 provide further elaboration on this issue. The reason why in two (or more) dimensions, quasiconcavity does not particularly help is that, in contrast with the one-dimensional case, there is no sensible way to assign a "median" to a set of points in the plane. This will become clear in the next, classical, Example 21.0.6 which we now describe. Example 21.0.6: Euclidean Preferences. Suppose that the set of alternatives is R'. Agents have preferenees 0, is preferred by agents I and 2 to x. You should verify the claims made in (i), (ii), and (iii). _ The situation illustrated in Example 21.0.5 is not a peculiarity. The key property of the
10. The preferences
or this example are not strictly convex. This is immaterial. Without changing
the nature of the example we could modify them slightly so as to make the indifference curve map strictly convex.
A(y,z)
= {XE R": IIx -
yll < IIx-zlI}.
See Figure 21.0.6 for a representation. Geometrically, the boundary of A(y, z) is the hyperplane perpendicular to the segment connecting y and z and passing through its midpoint. We will consider the idealized limit situation where there is a continuum of agents with Euclidean preferences and the population is described by a density function g(x) defined on R', the set of possible peaks. Then given two distinct alternatives y, z E R", the fraction of the total population that prefers y to z, denoted m,(y, z), is simply the integral of g(') over the region A(y, z) c R'. When will there exist a Condorcet winner? Suppose there is an x· E R' with the property that any hyperplane through x· divides R' into two half-spaces each having a total mass of ! according to the density g('). This point could be called a median for the density g('); it coincides with the usual coneept of a median in the case n = I. A median in this sense is a Condorcet winner. It cannot be defeated by any other alternative because if y ¥- x· then A(x·, y) is larger than a half-space through x· and, therefore, m,(x·, y) ~ t. Conversely, if x·
11. For an example in the same spirit where the two roles arc kept separate, see Grandmont (1978) and Exercise 21.0.9.
Flgur. 21.0.5 (left)
Euclidean preferences in R'. Flgur. 21.D.6 (right)
The region of Euclidean preferences that prefer y to z.
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THEORY
-------------------------------------------------------------------~ A(x"
+ "I,x")
=>
m•. ix"
+ "I, X") > i Figure 21.D.7
If x" is not a median then it is not a Condorcet winner.
Figure 21.D.8
---
-------~-------(a)
----
--B(b)
is not a median then there is a direction q E R' such that the mass of the half-space {z E R': q'Z > q·x·J is larger than 1. Thus, by continuity, if < > 0 is small then the mass of the translated half-space A(x" + f.q, x") is larger than t. Hence x· + -,w. then the preference relatIOn >-, defi~~ by y>-iw >-iz >-ix takes {y.w} to the top from
",
As a first step in the study of policy decision problems, this section is concerned with the description of the set of options available to a policy maker. The following section will consider the objectives of the policy maker. I The starting point of the analysis is a nonempty set of alternatives X and a collection of I agents. In contrast with Chapter 21, where we used preference relations, we will now assume that agents' tastes are given to us in the form of utility functions II,: X -+ R. One may wonder what is the exact meaning of the utility values II.eX): DO they have cardinal or ordinal significance? Are they comparable across individuals? These questions will be considered in Section 22.0. For current purposes there is no need to answer them. It is a traditional, and firm, principle of welfare economics that policy making should not be paternalistic. At a minimum, this means that alternatives that cannot be distinguished from the standpoint of agents' tastes should not be distinguished by the policy maker either. We are therefore led to the idea that only the agents' utility values for the different alternative should matter and therefore that the relevant constraint set for the policy maker is the utility possibility set [introduced by Samuelson (1947)], which we now define.
U = {(u" ... , ud
SEC T ION
BARGAINING
Figure 22.B.1 (leH)
A utility possibility set.
u
Flgur. 22.B.2 (right)
",
",
Example 22.B.I corresponds to a first-best situation. A first-best problem is one in which the constraints defining X are only those imposed by technology and resources. The policy maker cannot produce from a void and, therefore, must respect these constraints, but otherwise she can appeal to any conceivable policy instrument. If, as is often the case, there are other restrictions on the usable instruments, we say that we have a second-best problem. The restrictions can be of many sorts: legal, institutional, or, more fundamentally, informational. The last type were amply illustrated in Chapters 13 and 14 (and will be seen again in Chapter 23). We should warn, however, that the conceptual distinction between first-best and second-best problems is not sharp. In a sense, adverse selection or agency restrictions are as primitive as technologies and endowments.
Example 22.8.2: Ramsey Taxation. Consider a quasilinear economy with three goods, of which the third is the numeraire. The numeraire good can be freely transferred across consumers (more formally, one of the policy instruments available to the policy maker is the lump-sum redistribution of wealth). The first two goods are produced from the numeraire at a constant marginal cost equal to I. Consumers face market prices that are eq·.al to marginal cost plus a commodity tax whose level is fixed by the policy maker. Tax proceeds are returned to the economy in lump-sum form. Finally, the amounts consumed are those determined by the demand functions of the different consumers. We know from the second welfare theorem (Section 16.0) that any utility vector in the first-best UPS can be reached with the above instruments (it suffices to set the tax rates at a zero level and distribute wealth appropriately). But suppose that we now have an unavoidable distortion-the policy maker is constrained to raise a total amount R of tax receipts. This has then become a second-best problem. To determine the corresponding second-best UPS, note first that, since the numeraire is freely transferable across consumers, the boundary of this set is still linear, as in the first-best case (i.e., as in Figure 22.8.2). Hence, to place this boundary it suffices to find the level of prices P" P2 that maximizes V(P" Pi), the indirect utility function of a representative consumer (which, up to an increasing transformation, equals the
A utility possibility set: transferable utility.
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AXIOMATIC
aggregate consumer surplus; see Section 4.D and Chapter 10 for these concepts). J Denote by XI(PI' p,) and X,(PI' p,) the aggregate demand functions. Then we must solve the problem Max
--- --
BARGAININQ
v( PI' p,) (PI - I)x,(p" p,)
S.t.
+ (p,
(PI - 1) oxI(p" p,) OP1
_ dxl(p,) .),(PI - 1) - - = (1 -1.)X,(p.) dp,
and
Denoting by tl = (PI - I)lpl the tax rate on good I, we can write this condition in elasticity form as 0:
t1 = - -
for some
0:
> O.
PO S SIB I LIT Y
+ (p,
_ 1) ox,(p" P1) = O. OP1
(22.B.2)
Example 22.B.4: Few Policy Instruments. In Examples 22.B.2 and 22.B.3 we have assumed that the unrestricted transfer of numeraire across consumers is one of the instruments available to the policy maker. Because of this, in those two examples the UPS had a "full" frontier, that is, a frontier that is an (1- I)-dimensional surface. In addition, quasilinearity insured that this surface was flat (and therefore that the UPS was convex). We now explore the implications of limiting the extent to which the numeraire is transferable. We assume that we have two goods and that the utility functions of I consumers are quasilinear with respect to the first good (which is untaxed). Arbitrary transfers of numeraire are not permitted, however. The policy maker now has a single instrument: a commodity tax (or subsidy) on the second good. Again, this good can be produced at unit marginal cost. The policy maker's surplus (or deficit) is given back to the consumers according to some fixed rule (hence, no arbitrary transfers of numeraire are permitted). Say, to be specific, that this rule is that the surplus-deficit is absorbed by the first consumer. Then the (second-best) UPS is [denoting by Vi(P1) the indirect utility f.. nction of consumer i]
.( p, - - 1) dx,(p,) - - - = (1 - j.') X, (-) p, . dp,
j.
and
UTI L , T Y
Note that except in the separable case, where ox,(p" p,)lop, = 0, we have p, i' 1; that is, even if the initial distortion involves only the first market, second-best efficiency requires creating a compensatory distortion in the second market [this point was emphasized by Lipsey and Lancaster (1956)]. This is an intuitive result: suppose that we were to put P1 = 1; then the last (infinitesimal) unit demanded of the second good makes a contribution P1 - 1 = 0 to the total surplus (recall that p, will equal the marginal utility for good 2). Therefore, a small tax on good 2 is desirable because its effect is to divert some demand toward good I, where the contribution to total surplus of the last unit demanded is PI - 1 > O. •
There is i. < 0, such that
0:
22. B:
assume that PI is fixed at some level PI > I. s The policy instruments are any transfer of numeraire across agents and the level of a commodity tax on the second good. The net revenue in the two markets is given back to consumers in a lump-sum form. The solution P1 of the surplus-maximization problem is then characterized by the first-order conditions (see Exercise 22.B.3)
- I)x,(p" p,) ;e: R.
Suppose, to take the simplest case, that the utility functions of the different consumers are additively separable. This means that the two demand functions can be written as x ,(PI) and x,(p,). Then the first-order conditions satisfied by a solution (PI' p,) of the maximization problem are (carry out the calculation in Exercise 22.B.2):
tl =-£I(PI)
SEC T ION
(22.B.I)
£1(P1)
Expression (22.8.1) is known as the Ramsey taxation formula [because of Ramsey (1927)]. An implication of it is that if the demand for good 1 is uniformly less elastic than that for good 2, then the optimal tax rate for good I is higher. This makes sense: For example, if the demand for good I is totally inelastic then there is no deadweight loss from taxation of this good (see Section IO.C) and therefore we could reach the first-best optimum by taxing only this good.' • Example 22.B.3: Compensatory Distortion. The basic economy is as in Example 22.8.2, except that we do not necessarily assume that the utility functions of the consumers are additively separable. The distortion is now of a different type. We
U = {u
E
R': u :S (V,(P1)
+ (p,
- 1) Li Xi(P1), v1(p,), ... , V,(P1» for some P1 > O}.
Two points are worth observing. The first is that U does not need to be convex (you should show this in Exercise 22.B.4; recall from Proposition 3.D.3 that the indirect utility functions are quasi-convex. An example is represented in Figure 22.B.3. The second is that U is defined by means of a single parameter, P2' and therefore its Pareto frontier (which, naturally,lies in R') is one-dimensional. See Figure 22.B.4 for a case with I = 3. This feature is entirely typical. As long as the instruments available to the policymaker are fewer than I - 1 in number, the frontier of the UPS cannot be (I - 1)-dimensional. Note that when there is free transferability of numeraire across
3. Because total surplus equals consumer surplus plus the fixed amount of tax revenues R. by maximizing consumer surplus we maximize total surplus. We nole also thai the assumption that the amount R must be raised through commodity taxation is somewhat artificial in a context where lump-sum redistribution is possible. We make the assumption, in this and the next example, merely 10 be pedagogical. Alternalively, we could rule oul Ihe possibilily of lump-sum Iransfers. In this case the exercise carried out in this example (and the next) determines the first-order conditions ror Ihe problem of maximizing the sum of individual utilities (the "purely utilitarian social welfare function" in the terminology of Section 22.C) 4. We should warn that the formulas in (22.B.I) constitute only first-order conditions. As we shall see in Ihe forthcoming examples, second-best problems are frequently nonconvex and therefore the satisraction of first-order conditions does not guarantee that we have determined a true maximum.
5. More generally. we could think of the market for good I as being beyond the control of the policy maker and giving rise, perhaps because of a monopolistic structure. to a price higher than marginal cost.
1
SET S
821
822
CHAPTER
22:
ELEMENTS
OF
WELFARE
ECONOMICS
AND
AXIOMATIC
BAROAININO
SECTION
~
22 •• :
UTILITY
POSSIBIL'TY
SETS
823
---------------------------------------------------------------------------------",
Figure 22.B.3 (left)
",
A nonconvex second-best utility possibility set (Example 22.B.4).
Flgur. 22.B.6
Figure 22.B.4 (rlghl)
",
",
"J
A second-best utility possibility set for a case with few instruments: low-dimensional Pareto frontier (Example 22.B.4).
the I consumers, this automatically gives us the necessary minimum of I - I instruments. _
",
A nonconvex utility possibility set for a first·best problem wilh externalities (Example 22.8.5).
Chapter 6). then the (expected, or ex ante) UPS is convex since it is just the set of convex combinations of the utility vectors in the UPS associated with deterministic policies. There is no general theoretical reason to prevent the policy making from randomizing. On the other hand, the practical admissibility of stochastic policies cannot be decided on a priori grounds either. We conclude this section with a final example [borrowed from Atkinson (1973)] that highlights the contrast between first-best and second-best problems. Example 22.B.6: Unproductive Taxation. Suppose that there are two commodities and two consumers. We call the first commodity "labor", or leisure, and the second the "consumption good." There is a total of one unit of labor which is entirely owned by the first consumer. The consumption good can be produced by the first consumer from labor at a constant marginal cost of I (there is also free disposal). The first consumer has a utility function u,(x, I> X2') and the second has "2(X22)' In Figure 22.B.7 we illustrate the construction of the first-best Pareto frontier for this model. Suppose that u l is given. Then, subject to attaining the level of utility ", for consumer I, we want to give to consumer 2 as much utility as possible. If consumer I gets (x 11' X2') then the labor supply is I - XII and the amount of consumption good available for consumer 2 is I - XI' - X2" Thus, we should first determine (X'I' X2') by minimizing x,, + X2' subject to u,(x, I> X2') ~ "" and then let U2 = u2(1 - XII - .x2d· We now study the second-best problem where consumer I cannot be forced to supply labor. The only available policy instrument for providing consumption good
Example 22.B.5: First-best N onconvexities. In Example 22.B.4 the possible nonconvexity of the UPS is due to the second-best nature of this set.lflump-sum transfers of numeraire were allowed, then the corresponding first-best UPS would be convex. Yet a first-best UPS may also be nonconvex. Two familiar sources of nonconvexities in first-best problems are indivisibilities and externalities. As for the first, suppose that there are two locations and two agents with identical locational tastes (in particular, they both prefer the same location). There are only two possible assignments of individuals to locations and therefore the UPS will be as in Figure 22.B.5. As for externalities, suppose that there is a single good and that the utility functions of two consumers are u,(x,) = x, and u,(x .. x,) = x,/x,. Then the UPS is as in Figure 22.B.6 (see Appendix A of Chapter II for more on nonconvexities due to externalities). _ Examples 22.B.4 and 22.B.5 have provided instances where the UPS is nonconvex. There is a procedure that permits one, in principle, to convexify the UPS. It consists of allowing the policy maker to randomize over her set of feasible policies. If random outcomes are evaluated by the different agents according to their expected utility (see
Figure 22.B.7
",
Construction of the first·best Pareto frontier for Example 22.8.6.
Figure 22.B.5
A nonconvex utility possibility set for a first-best locational problem (Example
22.B.5).
u
",
"
x" Labor Supply
824
C HAP T E R
2 2:
E L E MEN T 8
0 F
W ELF ARE
E CON 0 M I C 8
AND
A X 10M A TIC
• A RQ AI NINQ
---........
.,
Flgur. 22.B.8 (le"1
Construction of Ih, second-best Pareto frontier for Example 22.B.6. Flgur. 22.B.8 (rlghll
First-best and second-best utility possibilily sels for the unproductive taxalio n example (Example 22.B.6).
x"
Labor Supply
to consumer 2 is a linear tax t(1 - XI') on whatever amount of labour the first consumer decides to supply given the tax rate. The construction of the secondbest frontier is illustrated in Figure 22.B.8. For t ~ O. consumer I will choose X'I so as to maximize u,(x". (I - t)(I - XII»' Observe that this is as if she had chosen the point in her offer curve corresponding to the price vector (I. 1/(1 - t». Denote this point by xl(t) = (xl,(t). X21 (t». The utility of consumer 2 is then u 2 (t(1 - XII(t)))· The first-best and second-best UPS are displayed in Figure 22.B.9. 6 In the second-best case the figure also depicts the locus of utility pairs Q c RI obtained as t ranges from 0 to I. that is.
Q = {(UI(XI(t)). uit(1 - XI I (t)))) e
R2: O:S; I:S;
I}.
Note that Q does not coincide with the Pareto set of the second-best UPS because it exhibits a characteristic nonmonotonicity. The economic intuition underlying it is clear: if t is low. consumer 2 will get very little of the consumption good; but if I is very high. the situation is not much better. Consumer 2 will now get a large fraction of the labor supplied by consumer I. but for precisely this reason not much labor will be supplied by consumer I. • We can distill yet another lesson from Example 22.B.6. We see in Figure 22.B.9 that it is quite possible for the first-best and second-best Pareto frontiers to have some points in common; that is. there may well be second-best Pareto optima that are first-best Pareto optima. Yet Figure 22.B.9 tells us that it would be quite silly to select a point in the second-best Pareto frontier merely according to the criterion of proximity to the first-best frontier. The resulting selection may be distributionally 7 very biased. The investigation of more sensible selection criteria will be the purpose of Section 22.C.
6. Again, the second-best frontier mayor may not be convex.
7. We may add Ihal il may also be uninteresting from Ihe point of view of policy: in Figure 22.B.9 the only second-best policy that yields a first-best result is t = 0, that is, no policy al all!
--
SECTION
22.C:
SOCIAL
WELFI.RE
FUNCTIONS
AND
SOCIAL
22,C Social Welfare Functions and Social Optima In Section 22.8 we described the constraint set of the policy maker. or social planner. The next question is which particular policy is to be selected. The application of the Pareto principle eliminates any policy that leads to utility vectors not in the Pareto frontier. Yet this still leaves considerable room for choice." which. by necessity. must now involve trading off the utility of some agent against that of others. In this section we assume that the policy maker has an explicit and consistent criterion to carry off this task. Specifically. we assume that this criterion is given by a social welfare function W(u) = W(u ...... u,) that aggregates individuals' utilities into social utilities. We can imagine that W(u) reflects the distributional value judgments underlying the decisions of the policy maker." In Section 22.E (and subsequent ones) we will discuss a somewhat different approach. one that puts more emphasis on the bargaining. or arbitration. aspects of the determination of the final policy selection. In the current section, we refrain from questioning the assumption of interpersonal comparability of Ittility, which is implicit in our use of levels of individual utility as arguments in the aggregator function W(u l ••••• u,). Section 22.0. which links with the analysis of Chapter 21, is devoted to investigating this matter. Thus, for a given social welfare function W(·) and utility possibility set U c R', the policy maker's problem is Max
W(u l ••••• u,)
(22.C.1)
s.t.(ul.···.u,)eU. A vector of utilities. or the underlying policies. solving problem (22.C.1) is called a social optimum. If the problem has a second-best nature. and we want to emphasize this fact. then we may refer to a constrained social optimum. We now present and discuss some of the interesting properties that a social welfare function (SWF) may. or may not. satisfy. (i) N onpaternalism. This first property is already implicit in the concept itself of a SWF. It prescribes that in the expression of social preferences only the individual utilities matter: Two alternatives that are considered indifferent by every agent should also be socially indifferent. The planner does not have direct preferences on the final alternatives. (ii) Paretiall property. Granted the previous property. the Paretian property is an uncontroversial complement to it. It simply says that W(·) is increasing; that is, if It; W(u). We also say that W(· ) is strictly Paretian if it is strictly increasing; that is. if u; W(u). If W(·) is strictly Paretian then a solution to (22.C.I) is necessarily a Pareto optimum. 8. Only exceptionally will the Pareto frontier consist of a single point. Recall also that, as we saw in Example 22.B.3, in second-best situations with few instruments, the requirement of Pareto optimality may not succeed in ruling out many policies.
9. This approach 10 welfare economics was firsllaken by Bergson (1938) and Samuelson (1947).
OPTIMA
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---
BARGAINING
Figure 22.C.l (left)
K /
/
A symmetric social welfare function.
/
/
/
".C:
",
",
SOCIAL
WELFARE
FUNCTIONS
AND
SOCIAL
/~.'~
The optimum of a symmetric, strictly concave social welfare function on a
Invarianl When Reflecled on Diagonal
",
Symmetric and Convex
",
(iii) Symmetry. The symmetry property asserts that in evaluating social welfare all agents are on the same footing. Formally, W(·) is symmetric if W(u) = W(u') whenever the entries of the vector u [e.g., u = (2,4,5)] constitute a permutation of the entries of the vector u' [e.g., u' = (4, 5, 2)]. In other words, the names of the agents are of no consequence, only the frequencies of the different utility values matter. The indifference curves of a symmetric W(·) are represented in Figure 22.C.1 for a two-agent case. Geometrically, each indifference curve is symmetric with respect to the diagonal. Note also that, because of this, if the indifference surfaces are smooth then the marginal rates of substitution at any u = (Ul> •.. ,u,) with identical coordinates are all equal to I. (iv) Concavity. Finally, a most important property is the concavity of W(·). We saw in Chapter 6 that, in the context of uncertainty, the (strict) concavity of a utility function implies an aversion to risk. Similarly, in the current welfare-theoretic context it can be interpreted as an aversion to inequality condition. A straightforward way to see this is to simply note that if W(·) is concave and W(u) = W(u'), then W(tu + tu') ~ W(u) [with the inequality strict if u ". u' and W(·) is strictly concave]. Another is to observe that if the UPS is convex and symmetric, then the utility vector that assigns the same utility value to every agent is a social optimum of any symmetric and concave SWF (see Figure 22.C.2 and Exercise 22.C.l).10 Thus, with convex UPSs and concave, symmetric SWFs some inequality is called for only if, as will typically be the case, the UPS is not symmetric. It is to be emphasized that in general, and especially for second-best problems, the UPS may not be convex. This means that even if W(·) is concave the identification of social optima is not an easy task. A utility vector that satisfies the first-order conditions of problem (22.C.1) may not satisfy the second-order conditions or, if it does, it still may not constitute a global maximum. We can gain further insights by discussing some important instances of social welfare functions. 10. The set U c R' is symmetric if" e U implies "' e U for any"' e RL that differs from" only by a permutation of its entries. The interpretation of the symmetry property of a UPS is that there is no bias in the ability to produce utility for different agents. In other words, from the point of view of their possible contributions to social welfare. all agents arc identical.
symmetric and con"x utility possibility set ~ egalitarian.
(a)
"I
(b)
",
(c)
OPTIMA
Figure 22.C.3
Figure 22.C.2 (right)
/'
/
/
--
SECTION
",
Example 22.C.I: UriliIarian. A SWF W(II) is pllrely utilitarian if it has the form W(II) = L; U; [or, in the nonsymmetric situation, W(u) = LI PiU,]. In this case, the indifference hypersurfaces of W(·) are hyperplanes. They are represented in Figure 22.C.3(a). Note that W(·) is strictly Paretian. In the purely utilitarian case, increases or decreases in individual utilities translate into identical changes in social utility. The use of the purely utilitarian principle goes back to the very birth of economics as a theoretical discipline. In Exercise 22.C.2 you are asked to develop an interpretation of the purely utilitarian SWF as the expected utility of a single individual "behind the veil of ignorance." Another line of defense, based also on expected utility theory, has been offered by Harsanyi (1955); see Exercise 22.C.3. Because only the total amount of utility matters, the purely utilitarian SWF is neutral towards the inequality in the distribution of utility. It is important not to read into this statement more than it says. In particular, it does not say "distribution of wealth." For example, if there is a fixed amount of wealth to be distributed among individuals and these have strictly concave utility functions for wealth, then the purely utilitarian social optimum will be unique and distribute wealth so as to equalize the marginal utility of wealth across consumers. If, say, the utility functions are identical across individuals then this will choose as the unique social optimum the vector in the Pareto frontier that assigns the same utility to every agent (see Exercise 22.C.1 for generalizations). _ Example 22.C.2: Maximin. A SWF is of maximin or Rawlsian type [because of Rawls (1971)) if it has the form W(u) = Min {u l , ... , u,} [or, in the nonsymmetric case, W(u) = Min {P,u" ... , p,u,}]. In other words, social utility equals the utility value of the worst-off individual. It follows that the social planning problem becomes one of maximizing thl utility of the worst-off individual." The (L-shaped) indifference curves of the maximin SWF are represented in Figure 22.C.3(b). II. One could refine this criterion by adopting a lexical, or serial, maximin decision rule. First maximize the utility of the worst-ofT, then choose among the solutions of this first problem by maximizing the utility of the next worst-off, and so on. With this. the objectives of the policy maker can still be expressed by a le:dm;n social welfare ordering of utility vectors, but the ordering is not conlinuous and cannot be represenled by a SWF (compare with Example 3.CI). Even so, the refinement is natural and important. For example. we are then guaranteed that the social optimum is a Pareto optimum. You 3re asked to show all this in Exercise 22.C.4. Note that the maximin SWF is Paretian but not strictly Paretian. This makes for some difficulties. In Figure 22.C.4 the
Social welfare functions. (a) Purely utilitarian. (b) Maximin or Rawlsian. (c) Generalized utilitarian.
827
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",
",
/ /
45'
/
FIgure 22.C.4
// / /
--- --
/
FIgure 22.C.S (rIght)
Maximin 0plimum Ulililarian 0plimum
Q
Q~------~~--~
"I
. O' . I FII"·be,, pllmum POlo
~
"I
It is reasonably intuitive that this concave SWF will have strong egalitarian implications. In fact, the preference for equality is quite extreme. Suppose, in effect, that U E RI is an arbitrary UPS and that u E U has all its coordinates equal. Then u fails to be the Rawlsian social optimum only if u is not Pareto optimal. Hence, if there is a u = (u., ... , u / ) in the Pareto frontier of U with all its coordinates equal, then u is a maximin optimum. Note, in contrast, that for a purely utilitarian SWF we reached the social optimum at complete equality only in the case where U is convex and symmetric. In Figure 22.C.4, which continues the analysis of Example 22.6.6, we depict a situation where maximin optimization leads to the selection of a policy (a tax level) that does not yield complete equality. Nonetheless, even in this case, the purely utilitarian social optimum is significantly more unequal than the maximin optimum. _
Example 22.C.3: Generalized Utilitarian. A SWF is generalized utilitarian if it has the form W(u) = LI g(u,) [or, in the nonsymmetric case, W(u) = LI g,(u / )), where g(.) is an increasing, concave function. The generalized utilitarian SWF is strictly Paretian and could be regarded as an instance of the purely utilitarian case where the individual utility functions u,(·) have been replaced by g(u,(·». This is not, however, a conceptually useful point of view. The point is precisely that, given the individual utility functions, there is a deliberate social decision to attach decreasing social weight to successive units of individual utility. The social indifference curves for this case are represented in Figure 22.C.3(c). We can also verify in Figure 22.C.4 and 22.C.5 that the equality implications of the generalized utilitarian SWF are intermediate between those of the purely utilitarian and of the maximin SWFs. _ Example 22.C.4: COnstalll Elasticity. An instance of generalized utilitarian functions that is very useful in applications is provided by the family defined by social utility functions g(.) whose marginal utilities have constant elasticity. This is a family in which attitudes towards inequality can be adjusted by means of a single parameter p '2! O. point at the boundary of U with equal coordinates is a maximin optimum but not a Pareto optimum. In the figure we have selected as "maximin optimum'" the leximin optimum (which. by definition, is a maximin optimum itself).
Range of generali~ utilitarian optima for Example 22.B.6 and the constant elasticity SWF of Example 22.C.4 (p E [0, <Xl]).
22.C:
SOCIAL
WELFARE
FUNCTIONS
AND
SOCIAL
For the rest of the example, individual utility values are restricted to be nonnegative. Then, for any p '2! 0, we let
(t.~)
A maximin oplimum for Example 22.8.6.
/
SECTION
BARGAINING
gp(u,) = (I - p)u! -p
and
if p'# I, if p = I.
Note that, as claimed, the elasticity of g~(u,> is constant because we have u,g7(u,>/g~(u,) = -p for all values u,. Taking into account that, for p '# I, h(W) = [1/(1 - p)] WI/(I-P) is an increasing transformation of W, we can represent the generalized utilitarian social preferences in a particularly convenient manner as w,,(u) =
(L, u! -0)11'-0
for p'# I,
w,,(u) =
L, In u,
for p = I.
and
Thus, we obtain the CES functions that are well known from demand and production theories (see Exercises 3.C.6 and 5.C.IO, respectively). Note that for p = 0 we get Wo(u) = L, u" the purely utilitarian case, and as p -+ 00 we get w,,(u) -+ Min {u l , . · . , U/}, the maximin case. (See Exercise 22.C.5.) In Figure 22.C.5 we depict the range of solutions to Example 22.B.6 as we vary p. We see that as the aversion to inequality increases (that is, as p -+ 00) the optimal tax rate increases. Note, however, that even for very high p we do not approach complete equality. On the other hand, none of these second-best solutions corresponds to the point in the Pareto frontier that is also Pareto optimal for the first-best problem. The latter distributes utility so unequally that the equity considerations underlying any symmetric and concave SWF leads us to sacrifice some first-best efficiency for an equity gain. _
The Compensation Principle We could ask ourselves to what an extent we can do welfare economics without social welfare functions. If the purpose of the SWF is the determination of optimal points in a given Pareto frontier, then resorting to them seems indispensable. This is the usage of social welfare functions that we have emphasized up to now; but in practice this is not the only usage. Often, the policy problem is given to us as one of choosing among several different utility possibility sets; these may correspond, for example, to the UPS associated with different levels of a basic policy variable. 12 If we have a social welfare function W(·), then the choice among two utility possibility sets U and U' should be determined by comparing the social utility of the optimum in U with that of the optimum in U'. However, even if there is no explicit social welfare function one may attempt to say something meaningful about this problem using revealed preference-like ide \s. This is the approach underlying the compensation principle (already encountered in Sections 4.0 and to.E). Let us first take the simplest case: that in which we have two utility possibility sets such that U c U'. Then one is very tempted to conclude that U' should be preferred to U. This would certainly be the case if the points that would be chosen 12. Formally. we can reduce this problem to the previous one by considering the overall UPS formed by the union of the UPSs over which we have to choose. But this may not be the most convenient thing to do because it loses the sequential presentation of the problem (first choose among UPS. then choose the utility vector).
OPTIMA
829
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(,;HAPTER
22:
ELEMENTS
OF
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ECONOMICS
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AXIOMATIC
BARGAININQ
---......
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22.0:
IN VARIANCE
PROPERTIES
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831
Figure 22.C.7 Flgur. 22.C.&
",
U' passes the weak compensation test over (U, u).
",
within each of V and V' were the optima of a social welfare function. But even if no social welfare function is available the set V' might still be considered superior to V according to the following strong compensation test: For any possible U E V there is a u' E V' such that uj ;,: u, for every i. That is, wherever we are in V it is possible to move to V' and compensate agents in a manner that insures that every agent is made (weakly) better off by the change to V'. If the compensation is actually made, so that every agent will indeed be made better off by a switch from V to V', there is no doubt that the switch should be recommended. But if compensation will not occur, matters are not so clear: By choosing V' over V based only on a potential compensation we are neglecting quite drastically any distributional implication of the policy change. In fact, it is even possible that the change leads to a purely egalitarian worsening (see Exercise 22.C.6). Recall from Section 10.0 that in the quasilinear case we always have V c V' or V' c V. This is because the boundaries of these sets are hyperplanes determined by the unit vector (hence parallel). In addition, this property also guarantees that the strong compensation criterion (which in Sections 3.0 and 10.E we called simply the compensation criterion) coincides with the choice we would make using a purely utilitarian social welfare function. In this quasilinear case, therefore, the strong compensation criterion does not neglect distributional issues to a larger extent than do purely egalitarian social welfare functions. Matters are more delicate when we compare two utility possibility sets V and V' which are such that one is not included in the other, that is, whose frontiers cross (see Figure 22.C.6). Suppose that we know that the outcome with utility possibility set V is the vector u E V, and that we are considering a move to V'.'3 If u E V', and we were to allocate utility optimally in V' according to a social welfare function, then the move to V' would be advisable. More generally, whenever u E V', the move from (V, u) to V' passes the following weak compensation test: There is au' E V' such that u; ;:: u, for every i. That is, given that we know that the outcome at V is u, we could move to V' and compensate every agent in a manner that makes every agent (weakly) better off. In Figure 22.C.6, V' passes the test with respect to (V, u) but not with respect to (V, u). Again, if the compensation is actually paid, then the weak compensation criterion 13. For e~ample. the original U could correspond to some underlying economy and u could be Ihe ulility values of a market equilibrium.
carries weight. If it is not paid, then it is subject to two serious criticisms. The first is the same as before (it disregards distributional consequences). The second is that it may lead to paradoxes. As in Figure 22.C.7, it is possible to have two utility possibility sets U and U', with respective outcomes u E U and u' E U', such that U' passes the weak compensation test over (U, u) and V passes the weak compensation test over (U', u').ln Exercise 22.C.7 you are asked to provide a more explicit example of this possibility in an economic context. Further elaborations are contained in Exercise 22.C.8.
22.D Invariance Properties of Social Welfare Functions In this section, we probe deeper into the meaning of the comparisons of individual utilities implicit in the definition of a social welfare function. The significance of the matter derives from the fact that whereas a policy maker may be able to identify individual cardinal utility functions (from revealed risk behavior, say), it may actually do so but only up to a choice of origins and units. Fixing these parameters unavoidably involves making value judgments about the social weight of the different agents. It is therefore worth examining the extent to which such judgments may be avoided. Thus, following an approach to the problem taken by d'Aspremont and Gevers (1977), Roberts (1980), and Sen (1977), we explore such questions as: What are the implications for social decisions of requiring that social preferences be independent of the units, or the origins, of individual utility functions?" To answer these types of questions, we need to contemplate the dependence of social preferences on profiles of individual utility functions. Thus, the social welfare functionals introduced in Chapter 21 provide a natural starting point for our analysis. However, we mudify their definition slightly by specifying that individual characteristics arc given to us in the form of individual utility functions u,(') rather than as individual preference relations. From now on we are given a set of alternatives X. We denote by 'fI the set of all possible utility functions on X, and by iJt the set of all possible rational (i.e., complete and transitive) preference relations on X.
I !
J
14. In addition to the previous references, you can consult Moulin (1988) for a succinct presentation of the material of this section.
A paradox: V' passes the weak com· pensation test over (V, u), and V passes the weak com· pensation test over (U', u').
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Definition 22.0.1: Given a set X of alternatives. a social welfare functional F: Cfil ..... 9t is a rule that assigns a rational preference relation F(ii, • ...• iii) among the alternatives in the domain X to every possible profile of individual utility functions (u,(·) ....• UI('» defined on X. The strict preference relation derived from F(u, •. ..• u,) is denoted Fp(ii, • ...• u,) .. • As in Chapter 21. we will concern ourselves only with social welfare functionals that are Paretian. Definition 22.0.2: The social welfare functional F: Cfil ..... 9t satisfies the (weak) Pareto property. or is Paretian. if. for any profile (u, • ...• iii) e Cfil and any pair x. veX. we have that u,{x);;o: ii,{V) for all i implies x F(ii, •.. .• iii) V. and also that u,{x) > u,{V) for all i implies x Fp(ii, •. ..• ii,) V. The first issue to explore is the relationship between these social welfare functionals and the social welfare functions of Section 22.C. A social welfare function W(·) assigns a social utility value to profiles (Ul •.••• UI) e RI of individual utility values. whereas a social welfare functional assigns social preferences to profiles (u ,....• u,) of individual utility functions (or. in Section 21.C, of individual preference relations). From a social welfare function W(·) we can generate a social welfare functional simply by letting F(UI' ...• u,) be the preference relation in X induced by the utility function u(x) = W(ul(x) •...• UI(X». The converse may not be possible. however. In order to be able to "factor" a social welfare functional through a social welfare function, the following necessary condition must. at the very least. be satisfied. Suppose that the profile of utility functions changes, but that the profiles of utility values for two given alternatives remain unaltered; then the social ordering among these alternatives should not change (since the value given by the social welfare function to each alternative has not changed). That is. the social ordering among two given alternatives should depend only on the profiles of individual utility values for these alternatives. Apart from being formulated in terms of utilities, this property is analogous to the pairwise independence condition for social welfare functionals (Definition 21.C.3). We keep the same term and state the condition formally in Definition 22.0.3. Definition 22.0.3: The social welfare functional F: Cfil ..... 9t satisfies the pairwise independence condition if. whenever x. veX are two alternatives and (u, • .... UI) eCfi I • (ii; • ...• iii) e Cfil are two utility function profiles with ii,{x) = D;{x) and u,(V) = ii;(V) for all i. we have xF(ii, ..... iil)V
xF(u; ..... ul)v.
The necessary pairwise independence condition is almost sufficient: In Proposition 22.0.1 we now see that if the number of alternatives is greater than 2, and the Pareto and pairwise independence conditions are satisfied. then we can derive from the social welfare functional a social preference relation defined on profiles (u l •...• UI) e 9t1 of utility values.'6 A standard continuity condition then allows us to represent this
BARGAINING
833
---...... ----------------------------------------------------------SECTION
22.0:
IN VARIANCE
PROPERTIES
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WELFARE
preference relation by means of a function W(u, •...• u,). thereby yielding a social welfare function. proposition 22.0.1: Suppose that there are at least three alternatives in X and that the Paretian social welfare functional F: Cfi' ..... 9t satisfies the pairwise independence condition. Then there is a rational preference relation;::; defined on HI [that is. on profiles (u, . ...• uJl e HI of individual utility values] that generates F(·). In other words. for every profile of utility functions (u, •... • iii) e Cfil and for every pair of alternatives x. veX we have x F(ii, . ...• uJl V
(u,(x) •...• iii (x)) ;::; (u,(V) •. ··• iil(v)).
Proof: The desired conclusion dictates directly how ;::; should be constructed. Consider any pair of utility profiles u = (u, •. ..• u,) e R' and u' = (u; •. ..• ui) e R'. Then we let u;::; u' if x F(u ...... u,)y for some pair x.yeX and a profile (u, •. ..• u,) e Cfi' with Ui(X) = Ui and Ui(Y) = u; for every i. We argue first that the conclusion u ;::; u'. is independent of the particular two alternatives and the profile of utility functions chosen. Independence of the utility functions chosen is an immediate consequence of the statement of the pairwise independence condition. Proving independence of the pair chosen is a bit more delicate. It sunices to show that if we have concluded that u;::; u' by means of a pair x. Y then. for any third alternative z (recall that by assumption there are third alternatives). we obtain the same conclusion using the pairs x, z or z. y." We carry out the argument for x. z (in Exercise 22.0.2 you are asked to do the same for z. y). To this effect. take a profile of utility functions (UI •...• UI) e Cfi' with Ui(X) = Ui. Ui(Y) = u;. and ui(z) = u; for every i. Because we have concluded that u;::; u' using the pair x, y. we must have x F(u, •... , il,) y. By the Pareto property. we also have Y F(u, •. .. , UI) z. Hence. by the transitivity of F(u, •. ..• UI)' we obtain x F(u, •.. . , u,) z. which is the property we wanted. It remains to prove that ;::; is complete and transitive. Completeness follows simply from the fact that the preference relation F(u, •...• UI) is complete for any (u l •...• ,i,) e Cfil. As for transitivity. let u;::; u';::; u·. where u, u'. u· e R'. Take three alternatives x. Y. z e X and a profile of utility functions (u, ..... u,) e Cfil with Ui(X) = Ui. ui(Y) = u;. and ui(z) = u, for every i. Since u;::; u' and u' ;::; UN. it must be that x F(u, •. .. , u,) y and y F(u, ... .• UI) z. Because of the transitivity of F(u, •. ..• U/). this implies x F(u , •...• u,) z. and so u ;::; UN. Hence. ;::; is transitive. _ By the Pareto condition, the social preference relation;::; obtained in Proposition 22.0.1 is monotone. You are asked to show this formally in Exercise 22.0.3.
Exercise 22.0.3: Show that if the social welfare functional F: Cfi' ..... 9t satisfies the Pareto property. then a social preference relation;::; on utility profiles for which the 17. Indeed. suppose Ihat we initially used the pair (x. y). Consider any other pair (v. w). If v = x u'. Hence, let the chain of (v. w). There y) - (x. z) -
15. Tha' is. x Fi", ..... "lb' if x F(", ..... u,)y but not y F(u, ..... "I)X. 16. In Exercise 22.D.1 you can find examples showing that the Pareto condition and Ihe restriction on the number of alternatives cannot be dispensed with for the result of Proposition
or w = y then we have just claimed that we get the same ordering between u and t' # x and w # y. If. in addition, v # y. then we reach the same ordering by replacements: (x. y) - (v. y) _ (v. w). Similarly. if w # x we can use (x. y) - (x. w) remains the case (v. w) = (y. x). Here we use a third alternative. z. and the chain (x.
22.D.1.
(y,z) - (y.x).
FUNCTIONS
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conclusion of Proposition 22.0.1 holds must be monotone in the sense that if u' then u' ;:: u, and if u' » u then u' :> u.
~
u
The social preference relation;:: on R' obtained in Proposition 22.0.1 need not be continuous or representable by a utility function. Consider, for example, a lexical dictatorship (say that there are two agents and let u:> u' if U I > u; or if U I = u; and U2 > u;) and recall from Example 3.C.1 that this type of ordering is not representable by a utility function. Nonetheless, we want to focus on social welfare functions and so from now on we will simply assume that we deal only with social welfare functionals that, in addition to the assumptions of Proposition 22.0.1, yield a continuous social preference relation ;:: on R'. As in Section 3.C, such a social preference relation can then be represented by a utility function: in fact, a continuous one. This is then our social welfare function W(u" ... , u,). Note that any increasing, continuous transformation of W(·) is also an admissible social welfare function. In summary, we have seen that the existence of a social welfare function generating a given social welfare functional amounts, with some minor qualifications, to the satisfaction of the pairwise independence condition by the social welfare functional. Therefore, we will concern ourselves from now on with a social welfare functional F: Oft' -+ {It that can be generated from an increasing and continuous social welfare function W: R' -+ R, or equivalently, from a monotone and continuous rational preference relation;:: on R'. We will discover that, in this context, natural utility invariance requirements on the social welfare functional have quite drastic effects on the form that we can choose for W(·) and, therefore, on the social welfare functional itself. Definition 22.0.4: We say that the social welfare functional F: Oft' -+ {It is invariant to cammon cardinal transformations If F(ii" ... , ii,) = F(ii;, ... , iii) whenever the profiles of utility functions (ii" ... , ii,) and (ii;, ... , iii) differ only by a common change of origin and units, that Is, whenever there are numbers (J > 0 and IX such that ii,{x) = (Jii;{x) + IX for all i and x eX. If the invariance Is only with respect to common changes of origin (i.e., we require (J = 1) or of units (i.e., we require IX = 0), then we say that F(') is invariant to common changes of origin or of units, respectively. It is hard to quarrel with the requirement of in variance with respect to common cardinal transformations. Even if the policy maker has the ability to compare the utilities of different agents, the notion of an absolute unit or an absolute zero is difficult to comprehend. We begin by analyzing the implications of invariance with respect to common changes of origin. Suppose that the social welfare functional is generated from the social welfare function W(·). We claim that the invariance with respect to common changes of origin can hold only if W(u) = W(u') implies W(u + lXe) = W(u' + lXe) for all profiles of utility values u e R', u' e R' and IX e R, where e = (I, ... , I) is the unit vector. Indeed, let W(u) = W(u') and W(u + lXe) < W(u' + lXe). Consider a pair x, y e X and profile (17" ..• , 17,) e U' with u,(x) = and u,(y) = for every i. Then x F(u I' ••• , 17,) y. However, x F(u;, .•• , 17;) y does not hold when 17;0 = 17,(') + IX, can tradicting the invariance to common changes of origin. Geometrically, the assertion that W(u) = W(u') implies W(u + lXe) = W(u' + lXe) says that the indifference curves of W(·) are parallel with respect to e-they are
u,
u;
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Figure 22.0.1
Indifference map of a social welfare function invariant to identical
changes of utility origins.
obtained from each other by translations along the e direction (see Figure 22.0.1). In Proposition 23.0.2 [due to Roberts (1980)], we show that this property has an important implication: up to an increasing transformation, the social welfare function can be written as a sum of a purely utilitarian social welfare function and a dispersion term. Proposition 22.0.2: Suppose that the social welfare functional F: Oft' -+ {It is generated from a continuous and increasing social welfare function. Suppose also that F(') is invariant to common changes of origins. Then the social welfare functional can be generated from a social welfare function of the form W(U" ... , Ut) = 0 - g(u, - 0, ... , u, - 0).
(22.0.1)
where 0 = (1//) LjUj. Moreover, if F(') is also independent of common changes of units, that is, fully invariant to common cardinal transformations, then g(.) is homogeneous of degree one on its domain: {s e R': LjSj = OJ. Proof: By assumption the social welfare functional F: Oft' -+ {It can be generated by a continuous and monotone preference relation;:: on R'. Moreover the invariance to identical changes of units implies that if u - u' then u + lXe - u' + lXe for any IX e R. We now construct a particular utility function W(·) for ;::. Because of continuity and monotonicity of;:: there is, for every u e R', a single number IX such that u - lXe. Let W(u) denote this number. That is, W(u) is defined by u - W(u)e. (See Figure 22.0.2 for a depiction.) Because of the monotonicity of preferences, W(·) is a legitimate utility representation for ;::.1. The fi.st part of the proof will be concluded if we show that W(u) - ii depends only on the vector of deviations (u, - 17, ... , u, - 17) = u - ae, that is, that if u - iie = u' - a'e then W(u) - 17 = W(u') - a'. But this is true because u - W(u)e and the invariance to common changes of origin imply that if u - ae = u' - a' e then u'
= u + (ii' -
ii)e - W(u)e
+ (U'
- a)e
= [W(u) + (a' -
a)]e
18. Up to here this is identical to the parallel construction in consumption theory carried out in Proposition 3.CI. We refer to the proof of the latter for details.
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origin case.
and therefore, W(u') = W(u) + (ii' - ii) as we wanted. The construction is illustrated in Figure 22.0.2.'9 To prove the second part, suppose that F(') is also invariant to common changes of units. Because F(') is generated from W(·), this can only happen if for every u - u' and {J > 0 we have {Ju - {Ju'. But then u - W(u)e implies {Ju - {JW(u)e, and so W({Ju) = {J W(u) for any u e R' and {J > O. That is, W(·) is homogeneous of degree one, and since g(.) coincides with - W(·) on the domain where ii = 0, we conclude that g(.) is also homogeneous of degree one. _ Going further, if the policy maker is not empowered with the ability to compare the absolute levels of utility across consumers, then the social welfare functional must satisfy more demanding invariance notions. Delinltlon 22.0.5: The social welfare lunctional F: 'fI' -+ [Jt does not allow interpersonal comparisons of utility il F(ii, . ... , ii,) = F(ii; •. ... iii) whenever there are numbers p; > 0 and r1.; such that ii,{x) = p;ii/(x) + r1.; lor all i and x. If the invariance is only with respect to independent changes 01 origin (Le .. we require P; = 1 lor all i). or only with respect to independent changes of units (Le., we require that r1.; = 0 for all i), then we say that F(') is invariant to independent changes of origins or of units. respectively. We have then Proposition 22.0.3. 20 19. We can gain some intuition on the form of this utility function by noticing its similarity to the quasilinear representations in consumer theory. Here we can write any vector u e R' as u = lie + (u - lie) and indifference sets can he obtained by parallel displacements in the direction e. In consumer theory we can write any vector x E RL as x = (x,. 0, ...• 0) + (0. Xl •••• I xd and indifference sels are parallel in the direction (1.0, ...• 0). Similarly. lhe conclusion in both cases is
thatthere is a utility function that is linearly additive in the first lerm (i.e., in the direction in which indifference sets are parallel), 20. See d'Aspremont and aevers (1977) for more results of this type.
0 F
soc I A L
W ELF ARE
proposition 22.0.3: Suppose that the social welfare functional F: 'fI' -+ [Jt can be generated from an increasing. continuous social welfare lunction. If F(·) is invariant to independent changes of origins. then F(') can be generated from a social welfare function W(·) of the purely utilitarian (but possibly nonsymmetric) form. That is, there are constants b; ~ 0, not all zero. such that W(u" ...• u,) =
Flgur. 22.0.2 Construction of the social welfare fUnetion of form (22.D.\) for the invariant to identical changes of
PRO PER TIE S
L b;u;
for all i.
(22.D.2)
Moreover. if F(') is also invariant to independent changes of units [Le., if F(') does not allow lor interpersonal comparisons of utility], then F is dictatorial: There is an agent h such that. for every pair x, VEX, iih(x) >iih(y) implies x Fp(ii, •... , ii,) y. Proof: Suppose that ;::; is the continuous preference relation on R' that generates the given F(·). For a representation of the form (22.0.2) to exist, we require that the indifference sets of ;::; be parallel hyperplanes. Since we already know from Proposition 22.0.2 that those sets are all parallel in the direction e, it suffices to show that they must be hyperplanes, that is, that if we take two u, u· e R' such that u - u'. then for u" = !U + !u· we also have u· - u - u'. The invariance of F(') with respect to independent changes of origins means. in terms of ;::;. that for any r1. e H' we have u + a;::; u" + r1. if and only if u ;::; UN. Take r1. = 1(u' - u). Then u + r1. = u" and u" + r1. = u'. Hence. u;::; u· if and only if UN;::; u'. If u;:: u" then u·;::; u' and so u· - u. If u' > u then u' > u· which contradicts u - u'. We conclude that u· - u - u'. as we wanted. Once we know that indifference sets are parallel hyperplanes. the same construction as in the Proof of Proposition 22.D.2 will give us a W(·) of the form (22.0.2). In addition, the Pareto property yields b, ~ 0 for all i. Finally, suppose that F(') is also invariant to independent changes of units. Then dictatorship follows simply. Choose an agent h with b. > O. Take u, u' e R' with u, > u~. Then, by invariance to independent changes of units. we have that L, b,u, > L, b,u; if and only if b,u, + e L, .. , bl", > b.u. + eLI .. , blu; for any e > O. Therefore, since b,u, > b,u. we get. by choosing e > 0 small enough, that LI blu, > L, b,u;. Thus. agent h is a dictator (show, in Exercise 22.0.4, that in fact b, = 0 for all i >F II). _ We point out that for the dictatorship conclusion of Proposition 22.0.3, it is not necessary that F(') be generated from a social welfare function. It suffices that it be generated from a social preference relation on R'. Proposition 22.0.3 (extended in the manner indicated in the last paragraph) has as a corollary the Arrow impossibility theorem of Chapter 21 (Proposition 2I.C.I), which is, in this manner, obtained by a very different methodology. Indeed, suppose that F(') is a social welfare functional defined. as was done in Chapter 21, on profiles of preference relations (;::;" ... , ;::;,) e !Jl'. Then we can construct a social welfare functional 1"(.) oefined on profiles of utility functions (ii" .... ii,)e'fl' by letting F'(ii" ... , ii,) = F(;::;" .•.• ;::;,). where ;::;, is the preference relation induced by the utility function ii,(·). In Exercise 22.0.5 you are asked to verify, first, that F'(') inherits the Paretian and pairwise independence conditions from F('), second, that F'(') does not allow for interpersonal comparisons of utility and, third. that a dictator for F'(') is a dictator for F(·).
FUN C T , 0 N S
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Other invariance properties of social welfare functionals have been found to be of interest. We mention two. We say that the social welfare functional F: 'fI1 .... {It is invariant to common ordinal transformations if F(ii" ... , ii,) = F(ii;, ... , iii) whenever there is an increasing function y(-) such that ii,(x) = y(ii;(x» for every x e X and all i. The interpretation of this invariance is that although the social planner has no notion of individual utility scales she can, nonetheless, recognize that one individual is better off than another (but the question "by how much?" is meaningless). An example is provided by the social welfare functional induced by the symmetric Rawlsian social welfare function W(u) = Min {u" ... , U/}, With this SWF, the ordering Over policies depends only on the ability to determine the worse-off individual (see Exercise 22.0.8 for further elaboration). We say that a social welfare function W(·) generating a given social welfare functional F: '1l' .... R is independent of irrelevant individuals if, when we split the set of agents into any two groups, the social preference among utility vectors in one of the groups is independent of the level at which we fix the utilities of the agents in the other group (we should add that, if so desired, the condition can be formulated directly in terms of the social welfare functional). This is a sensible requirement It says that the distributional judgments concerning the inhabitants of, say, California, should be independent of the individual welfare levels of the inhabitants of, say, Massachusetts. As in the formally similar situation in consumer theory (Exercise 3.G.4), a social welfare function for I > 2 agents that is continuous, increasing, and independent of irrelevant individuals has, up to an increasing transformation, the addilively separable form W(u) = L, g,(u,); that is, W(u) is generalized utilitarian, possible nonsymmetric. Moreover, under weak conditions it is also true that the only social welfare functions that, up to increasing transformations, both admit an additively separable form and are invariant to common changes of origin are the utilitarian W(u) = L, b,u" Thus, from an invariance viewpoint we can arrive at the utilitarian form for a social welfare function by two roads: one, Proposition 22.0.3, is based on invariance to independent changes of origins; the other, just mentioned, is based on independence of irrelevant individuals and invariance to common changes of origins. See Maskin (1978) for more on this. Example 22.0_1: Fix an alternative x* and define a social welfare functional F(') by associating to every profile of individual utility functions (ii., ... , ii,) the social preference relation generated by a utility function Vex) = L, g,(u,(x) - ii~x*». Then, informally, this social welfare functional is both invariant to independent changes of origins and independent of irrelevant individuals, but it is neither utilitarian nor dictatorial. Note, however, that this functional cannot be generated from a social welfare function because it is not pairwise independent the social preference among two alternatives may depend on the ueiliey of ehe lhird afternative x* . •
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bargaining games (such as those considered in Appendix A of Chapter 9) by adopting an axiomatic point of view. Thus, the approach is more related to ideas of cooperative game theory (as reviewed in Appendix A of Chapter 18).2' For current purposes, the description of a bargaining problem among 1 agents is composed of two elements: a utility possibility set U c H' and a threat, or status-quo, point u* E U. The set U represents the allocations of utility that can be settled on if there is cooperation among the different agents. The point u* is the outcome that will occur if there is a breakdown of cooperation. Note that cooperation requires the unanimous participation of all agents, in which case, to repeat, the available utility options are given by U c R/. If one agent does not participate, then the only possible outcome is the vector 11*. This setup is completely general with two agents and, because of this, the two-agent case is our central reference case in this section. With more than two agents, the assumption is a bit extreme, since we may want to allow for the possibility of partial cooperation. We take up this possibility in Section 22.F. Throughout this section we assume that U c R' is convex and closed and that it satisfies thcfree disposal property U - R I. C U (i.e. if u' ::;; u and u E Uthen u' E U). As in Delinition 22.B.1, U c R' could be generated from a set of underlying 22 alternatives X, which could well include lotteries over deterministic outcomes. For simplicity we also assume that u* is interior to U and that {u E U: u ~ u*} is bounded.
Definition 22.E.l: A bargaining solution is a rule that assigns a solution vector f(V, u*) E V to every bargaining problem (V, u*).21 We devote the rest of this section to a discussion of some of the properties one may want to impose on f(·) and to a presentation of four examples of bargaining solutions: the egalitarian, the utilitarian, the Nash and the Kalai-Smorodinsky solutions. We should emphasize, however, that a strong assumption has already been built into the formalization of our problem: we are implicitly assuming that the solution depends on the set X of feasible alternatives only through the resulting utility values. Definition 22.E.2: The bargaining solution f(·) is independent of utility origins (IUO). or invariant Co independent changes of origins, if for any Il = (Il, ... ,Il,) E H' we have for every i f,(V', u* + Il) = f,(V, u*) + Il, whenever U' = {(u.
+ Il., ...
,u,
+ <X,): u E V}.
The IUO property says that the bargaining solution does not depend on absolute scales of utility. From now on we assume that this property holds. Note that we therefore always have feU, 11*) = feu - {II'}, 0) + II'. This allows us to normalize our problems to 11* = O. From now on we do so and simply write f(U) for f(U, 0).
22_E The Axiomatic Bargaining Approach In this section, we briefly review an alternative approach to the determination of reasonable social compromises. The role of a planner endowed with her own preferences is now replaced by that of an (implicit) arbitrator who tries to distribute the gains from trade or, more generally, from cooperation in a manner that reflects "fairly" the bargaining strength of the different agents. The origin of the theory is game-theoretic. However, it sidesteps the construction of explicit noncooperative
21. For general introductions to the material of this section, see Roth (1979). Moulin (1988). and Thomson (1995). 22. In principle. the underlying set X and Ihe corresponding utility functions on X could be dilTerent for dilTerent U ell'. For the Iheory that follows all that matters is the utility sel U. 23. Thus, a bargaining solution is a choice rule in the sense of Chapter 1. If an underlying alternalive sel X is kepi fixed and. Iherefore, the form of U, generated as in Definition 22.B.I, depends only on the utility functions. we can also regard the bargaining solution as a choice of function in the sense of Definition 2t.E.1.
APPROACH
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u,
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u,
Figure 22.E.2
The property o[ independence of
Symmelric Sel Flgur. 22.E.1
Threal Poinl
It should not be forgotten, however, that a change in the threat point (which will now show up as a change in U) will affect the point settled on. Definition 22.E.3: The bargaining solution f(·) is independent of utility units (IUU). or invariant to independent changes of units. if for any P= (Pt •...• P,) e R' with Pi> 0 for all i. we have fi(U') = PJi(U)
whenever U' = {(PtUt •....
for every i
P,u,): ue U}.24
With independence of utility origins (implicitly assumed in Definition 22.E.3), independence of utility units tells us that, although the bargaining solution uses cardinal information On preferences, it does not in any way involve interpersonal comparisons of utilities. Definition 22.E.4: The bargaining solution f(') satisfies the Pareto property (P). or is Paretian. if. for every U. flU) is a (weak) Pareto optimum. that Is. there is no u e U such that u, > fAU) for every i. Definition 22.E.S: The bargaining solution f(') satiSfies the property of symmetry (S) if whenever U c R' is a symmetric set (Le .. U remains unaltered under permutations of the axes;2S see Figure 22.E.l). we have that all the entries of flU) are equal. The interpretation of the symmetry property is straightforward: if, as reflected in U, all agents are identical, then the gains from cooperation are split equally.
Definition 22.E.6: The bargaining solution f(') satisfies the property of individual rationality (IR) if flU) ~ o. In words: the cooperative solution does not give any agent less than the threat point (recall also that, after normalization, we consider only sets U with 0 e U). It is a sensible property: if some agent got less than zero, then she would do better by opting out and bringing about the breakdown of negotiation. The next property is more substantial. 24. Geomelrically. U' is obtained from U by stretching the different axes by the rescaling factors ((J, ..... {J,). 25. More precisely. if U E U then u' e U for any u' differing from u only by a permutation of ils entries.
'"
The symmetry property [or bargaining solutions.
\'
841
----------------------------------------------------------------------
Threat Point
Definition 22.E.7: The bargaining solution satisfies the property of independence of irrelevant alternatives (IIA) if, whenever U' c U and flU) e U', it follows that flU') = flU).
The I1A condition says that if flU) is the "reasonable" outcome in U and we consider a U' that is smaller than U but retains the feasibility of f(U), that is, we only eliminate from U "irrelevant alternatives," then flU) remains the reasonable outcome (sec Figure 22.E.2). This line of justification would be quite persuasive if we could replace" reasonable" by "best." Indeed, if f( U) has been obtained as the unique maximizer on U of some social welfare function W(u), then the IIA condition is clearly satisfied [if f (U) maximizes W(·) on U then it also maximizes W(·) on U' c U]. We note that while the converse is not true, it is nonetheless the case that, in practice, the interesting examples where I1A is satisfied involve the maximization of some SWF. We proceed to present four examples of bargaining solutions. To avoid repetition, we put on record that all of them satisfy the Paretian, symmetry, and individual rationality properties (as well as, by the formulation itself, the independence of utility origins). You are asked to verify this in Exercise 22.E.I. In Exercise 22.E.2 you are asked to construct examples violating some of these conditions. Example 22.E.1: Egalitarian Solution. At the egalitarian solution fe('), the gains from cooperation are split equally among the agents. That is, for every bargaining problem U c IR t , f..(U) is the vector in the frontier of U with all its coordinates equal. Figure 22.E.3 depicts the case I = 2. Note also that. as illustrated in the figure, every f..(U) maximizes the Rawlsian social welfare function Min {u" ... , ut } on U. The egalitarian solution satisfies the IIA property (verify this). Clearly, for this olution. utility units are comparable across agents. and so the lUU property is not satisfied. 20 • Example 22.E.2: Utilitariall Solution. For every U we now let J.(U) be a maximizer of L, u, on U n IR~. If U is strictly convex, then this point is uniquely defined and. therefore. on the domain of strictly convex bargaining problems the IIA property is satisfied. As witt the previous example. the solution violates the IUU condition. Figure 22.E.4 illustrates the utilitarian solution in the case I = 2. • 26. Do not forget that the utility values arc nol absolute values but rather utility differences from the threal point. It is because of this that changes of origins do not matter.
irrelevant alternatives [or bargaining
solutions.
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.ARaAININa
---....... u, /
/
45'
//
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",
Flgur. 22.E.3 (teft) The egalitarian solution for bargaining problems.
---.-_ _f.",.(U)/
Flgur. 22.E.6 Flgur. 22.E.4 (right)
"'-
u,
Threat Point
u,
length [b, f.,(U)] = length [I.(U), aJ
U
, ,
I t
i
I
I
", ~,
-----11 1"2 = constant
Figure 22.E.5
The Nash solution for bargaining problems.
'"
Threat Point
Example 22.E.3: Nash Solution. For this solution, we take a position intermediate between the two previous examples by requiring that f.(V) be the point in V n R/+ that maximizes the product oC utilities u l x ..• x u" or, equivalently, that maximizes Li In U i (this corresponds to the case p = 1 in Example 22.C.4). In Figure 22.E.5, we provide an illustration Cor I = 2.ln this case, the Nash solution has a simple geometry: !.( V) is the boundary point oC V through which we can draw a tangent line with the property that its midpoint in the positive orthant is precisely the given boundary point f.( V); see Exercise 22.EJ. As with the egalitarian and the utilitarian examples, the Nash solution satisfies the IIA property (because it is defined by the maximization oC a strictly concave Cunction). Interestingly, and in contrast to those solutions, the condition of independence of ucility units (IUU) holds for the Nash solution. To see this, note that Li In Ui 0 it satisfies "IU, = ... = '1,U, = Y. that is, ~, = ,.( I/u,) for every i. Consider now any u' E U. We have L, q,u;!> L, 'I'"' and therefore L,( I/u,)u; !> L, (1/11,)11,. Since (I/u, .... , I/u,) is the gradient of the concave function Li In u, at (,i" ... , ,i,), this implies L, In u; !> L, In Il, (see Section M.C of the Mathematical Appendix). Hence ,i maximizes L, In u, on U, that is, u= f.,(U).". See Figure 22.E.6 for an illustration of the argument. In Exercise 22.E.3 you should show the converse-that the Nash solution is simultaneously utilitarian and egalitarian for appropriate choice of units.
• The Nash solution was proposed by Nash (1950), who also established the notable fact that it is the only solution that satisfies all the conditions so far. Proposition 22.E.l: The Nash solution is the only bargaining solution that is independent of utility origins and units, Paretian, symmetric, and independent of irrelevant alternatives. 2 • Proof: We have already shown in the discussion of Example 22,E.4 that the Nash solution satisfies the properties claimed. To establish the converse, suppose we have a candidate solution f(·) satisfying all the properties. By the independence of utility origins, we can assume, as we have done so far, that f(·) is defined on sets where the threat point has been normalized to the origin. Given now an arbitrary V, let Ii = f.(V) and consider the sets
U' = {II E IR':
L II,/Ii, S; I}
and
27. To rcpc,ll in more geometric terms: the hyperplane with normal ('110 •..• ",) passing through ii leaves U belo,,"' it (because of the utilitarian property). Thus, it suffices to show that the set (/I: L, In II,';; L, In Ii,} lies above the hyperplane. BUI note that this follows from the fact that, because of the egalitarian property. (~" ... , ~,) is proportional to (I/Ii, •... , 1/';,), which is the gradient of the concave function Li In at U. 28. Note that we do not assume individual rationality explicitly: tt turns out to be implied by the other conditions.
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(left) The N~sh solution is determmed uniquely from . the independell({h: nth) < n(i)})
represents how much agent i contributes when she joins the group of her predecessors in the ordering. This is the amount the predecessors would agree to pay i if she had
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all the negotiating power, that is, if she could make a take-it-or-leave-it offer. 31 Note that L, g•.• U) = v(l) for all permutations It. The agents do not come to us ordered. They all stand on the same footing. We may account for this by giving every agent the same chance of being in any position, thereby making all positions equally likely. Equivalently, we could take the (equal weighting) average of agent i contributions over all permutations It (there are I! of these). This is precisely the Shapley value solution. Definition 22.F.6: The Shapley value solution f. ---
BARGAINING
II X IRe I • I •
Is it plausible to you? If you wanted to escape from it, how would you do it? What does this all say about the independenoe axiom as applied to social decisions? Suppose now that there are I agents and that in addition to the social utility function U(·) we are also given I individual preference relations ~ defined on the same set of lotteries X. We assume that they are also represented by utility functions of the expected utility form
EXERCISES
for 1-1, .... 1.
22.B.I A Give sufficient conditions for the convexity of the first·best utility possibility set in the context of the exchange economies of Example 22.8.1. 22.B.2A Derive the first·order conditions stated in Example 22.B.2. 22.B.3 A Derive the first·order conditions (22.B.2) of Example 22.B.3. 22.B.4" Show as explicitly as you can that the utility possibility set of Example 22.B.4 may not be convex. 22.C.l A Suppose that the utility possibility set U c R' is symmetric and convex. Show that the social optimum of an increasing, symmetric, strictly concave social welfare function W(·) assigns the same utility values to every agent. [Note: A set U is symmetric if u e U implies u' E U for any u' obtained from u by a permutation of its entries.] Observe that the same conelusion obtains if W(·) is allowed to be just concave, as in the utilitarian case, but U is required to be strictly convex. 22.C.2A Suppose that we contemplate a decision maker in an original position (or ex·ante, or hehind the veil oj ignorance) before the occurrenoe of a state of the world that will determine
which of I possible identities the decision maker will have. There is a finite set X, of possible final outcomes in identity i. Denote X = X, x ... X X,. (a) Appeal to the theory of state·dependent utility presented in Section 6.E to justify a utility function on X of the form
U(x" ... , x,) = ",(x,)
+ ... + u,(x,).
Interpret and discuss the implications of this utility function for the usage of a purely utilitarian social welfare function. (b) Suppose that X, = ... = X, and the preference relation on X defined by the utility function in (a) is symmetric. What does this imply for the form of the utility function? Discuss and interpret. 22.C.3" We have N final social outcomes and we consider a set of alternatives X that is the set oftotteries over these outcomes. An alternative can be represented by the list of probabilities assigned to the different final outcomes, that is, p = (p" . .. , p.) where p. :?; 0 for every nand P,
+ ... + P. = I.
We assume that we are given a social preferenoe relation ~ on X that is continuous and conforms to the independence axiom. Thus, it can be represented by a utility function of the expected utility form U(p) = u,p,
+ ... + ".P •.
From now on we assume that this social utility function U(·) defined on X is given. (0) Suppose that there are two final outcomes and that they are specified by which of two individuals will reoeive a oertain indivisible object. Suppose also that social preferenoes are symmetric in the sense that there is social indifferenoe between the lottery that gives the object to individual 1 for sure and the lottery that gives the object to individual 2 for sure. Show that all the lotteries must then be socially indifferent. Discuss and interpret this conclusion.
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We say that the social utility function U(·) is Parelian if we have U(p) > U(p') whenever U,(p) > U,(p') for every i. (b) Consider a case with N - 3 and I = 2 and illustrate, in the 2-dimensional simplex of lotteries, how the indifferenoe map of the utility functions of the two agents and of the social utility function fit together when the social utility function is Paretian. (c) Exhibit a case where the Paretian condition determines uniquely the social indifferenoe map (recall that we are always assuming the independenoe axiom fot social preferences!). Argue. however. that in general the Paretian condition does not determine uniquely the social indifferenoe map. In fact. exhibit an example where any social utility function is Paretian. (d) Argue (you can restrict yourself to N - 3 and I - 2) that if the social utility function U(p) is Paretian then it can be written in the form U(p) - Il,U,(p)
+ ... + Il,U,(p)
where Il, :?; 0 for every i and Il, oF 0 for some I. What does this conclusion say for the usage of a purely utilitarian social welfare function? Interpret the Il, weights, as well as the fact that they need not be equal across individuals. 22.C.4A The 'eximin ordering, or preference relation, on R' has been mentioned in footnote 11 of this chapter when discussing the Rawlsian SWF. It is formally defined as follows. Given a vector u = (u, •. ..• u,) let u' e A' be the vector that is the nondecreaslng rearrange"",nl or u. That is, the entries of u' are in nondecreasing order and its numerical values (mUltiplicities included) are the same as for u. We then say that the vector u is at least as good as the vector Q in the leximin order if u' is at least as good as Q' in the lexicographic ordering introduced in Example 3.C.1. (a) Interpet the definition ofthe leximin as a refinement ofthe Rawlsian preference relation. (b) Show that the leximin ordering cannot be represented by a utility function. It is enough to show this for I = 2. (c) (Harder) Show that the social optimum of a leximin ordering is a Pareto optimum. You can limit yourself to the case I = 3. 22.C.s B Consider the constant elasticity family of social welfare functions (Example 22.C.4). Argue that w,,(u) .... Min {u, •...• u,} as p .... 00. 22.C.6 A Suppose that U and U' are utility possibility sets and that we associate with them Pareto optimal utility outcomes ii e U and u' e U', respectively. Show graphically that: (8) It is possibk for U' to pass the strong compensation test over U and yet for the outcome with U' to be worse than the outcome with U, as measured by the purely utilitarian SWF.
(b) If the utility possibility sets are derived from a quasilinear economy and U' passes the weak compensation test over U. then it also passes the strong compensation test and, moreover, the outcome for U' is a utilitarian improvement over the outcome for U. Is this conclusion valid if we evaluate social welfare by a nonutilitarian SWF? 22.C.7 B Construct an explicit example of two Edgeworth box economies, differing only in their distributions of the initial endowments. such that the utility possibility set of each one
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passes the weak compensation test over the utility possibility set of the other, when the utility outcome in the laller is chosen to correspond to one of its competitive equilibria.
22.0.3 is valid under the weakened assumption that F is generated from a social preference relation on R I.)
22.C.8 A Suppose we have two utility possibility sets U, U' with respective outcomes U E U and u' E U'. We say that (U', u') passes the Kaldor compensation test over (U, u) if U' passes the weak compensation test over (U, u) and U does not pass the weak compensation test over
12.0.68 This exercise is concerned with social welfare functions satisfying expression (22.0.1).
(U',u').
(a) For I = 2, represent graphically a situation where Kaldor comparability is possible and one where it is not. (b) Observe that Kaldor comparability is asymmetric. Oefine your terms.
22.0.18 In this exercise we verify the indispensability of the assumptions of Proposition 22.0. I.
= lx, y}. The social
(a) Suppose there are three agents and only two alternatives, that is, X welfare functional is given by if and only if
ii,(x)
~
ii,(y) for every i
Y F(ii,. ii,. ii,)x
if and only if
ii,(y)
~
u,(x) for at least one i.
and Check that the social preference relation is always complete, that the social welfare functional cannot be represented by means of a social welfare function, and that only the condition on the number of alternatives fails from Proposition 22.0.1.
X
(e) Argue that if in (22.0. I) the function g(') is homogeneous of degree one and differentiable. then it must be linear (and so we arc back to the utilitarian case). 12.0.78 Consider the constant elasticity family of social welfare functions studied in Example
F(ii,. ii,. ii,)y
(b) Now we have three agents and three alternatives, that is, X = welfare functional is given by
(c) Show that the symmetric Rawlsian social welfare function W(u) = Min lu" •.. , u/l can be wrillen in the form (22.0. I). What about nonsymmetric Rawlsian social welfare functions? [Him: Check the condition of invariance to common changes of origins.] (d) Give other examples satisfying (22.0.1), in particular, examples with g(.) ~ 0 and intermediate between the utilitarian and the Rawlsian cases. Interpret them.
(e) Show that Kaldor comparability may not be transitive.
X
(a) Show that the nonsymmetric utilitarian function W(u) = L b,u , can be wrillen in the form (22.0.1). (b) Show that if W(·) is symmetric and g(O) = 0 then g(') ~ O.
lx, y, z}.
The social
F,(ii" ... , iii) y F,(ii ... , iii) z "
for every (ii" ... , iii) E fli. Show that, again, no representation by means of a social welfare function is possible and that, of the assumptions of Proposition 22.0.1, only the Paretian property fails to be satislied. (c) Exhibit an example in which the only condition of Proposition 22.0.1 that fails to be satisfied is pairwise independence. 22.0.2A Carry out the verification requested in the second paragraph of the proof of Proposition 22.0. I. 22.0.3 A In text. 22.0.4A A social welfare functional F is lexically dictatorial if there is a list of n > 0 agents h" ... , h. such that the strict preference of h, prevails socially, the strict preference of h, prevails among the alternatives for which hi is indifferent, and so on. (a) Show that if F is lexically dictatorial then F is Paretian, is pairwise independent, and does not allow for interpersonal comparisons of utility. (b) Under what conditions can a social welfare functional that is lexically dictatorial be generated from a social welfare function? (c) Show that if a dictatorial social welfare functional is generated from a social welfare function W(u) = L b,u then b, = 0 for every i distinct from the dictator. "
22.0'sc Complete the proof of Arrow's impossibility theorem along the lines suggested in the last paragraph prior to the small-type text at the end of Section 22.0. (Assume that Proposition
22.C.4. (a) Show that the social welfare functionals derived from SWFs in this family are invariant to common changes of units. (b) Show that the only members of this family which arc also invariant to common changes of origins, and therefore admit a representation in the form (22.0.1), are the purely utilitarian (i.e., p = 0) and the Rawlsian (i.e., p = 00). 22.0.8 8 This is an exercise on the property of invariance to common ordinal transformation. (a) Show that the symmetric, Rawlsian social welfare function satisfies the property. (b) Show that the anti-Rawlsian function W(u) = Max lui, ... , u/l also satislies it. (c) Show that the property is satislied for dictatorial social welfare functionals. (d) (Harder) Suppose that I = 2 and W(u) = W(u') for two vectors u, u' E R2, with
u; < u, < u, < ul' Assume also that W(·) is increasing. Show that the induced social welfare functional cannot be invariant to identical ordinal transformations. From this, argue informally (you can do it graphically) that for I = 2 a continuous, increasing social welfare function that is also invariant to identical ordinal transformations must be either dictatorial, Rawlsian, or anti-Rawlsian. 22.E.IA Verify that the bargaining solutions in Examples 22.E.I to 22.E.4 are independent of utility origins, Paretian, symmetric, and individually rational. It is enough if you do so for I = 2. 22.E.2A State nonsymmetric versions of the four bargaining solutions studied in Section 22.E (egalitarian. utilitarian, Nash, and Kalai-Smorodinsky). Motivate them. 22.E.3 8 This is an exercise on the Nash solution. (a) Verify that for I = 2, [.(U) is the boundary point of U through which we can draw a tangent line with th. property that its midpoint in the positive orthant is precisely the given boundary point J.( U). (b) Verify that if U c RI is a bargaining problem then there are rescaling units for the individual utilities with the property that the Nash solution becomes simultaneously egalitarian and utilitarian. 22.E.4A Verify that the Kalai-Smorodinsky solution satisfies the property of independence of utility units but violates the property of independence of irrelevant alternatives. You can restrict yourself to I = 2.
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22.E.5 B This is an exercise on Ihe monolonicily property. (0) Show Ihat Ihe egalitarian solulion is the only bargaining solution that is independent of ulilily origins, Paretian, symmetric and monotonic. [Hint: Consider first a family of symmetric ulility possibility sels with linear boundaries. Notice then that for any two sets U, U' we always have U,... U' c U and U ,... U' c U'.]
(b) (Harder) Suppose that f(') is a bargaining solution that is independent of utility origins, Paretian, and strongly monotonic [if U c U' then flU) !> flU') and, in addition, if flU) is interior to U' then f(U)« flU')]. Show that there is a curve in R' starting at the origin and strictly increasing such that, for every U, f(U) is the intersection point of the boundary of U with this curve. You can restrict yourself to the case I = 2. 22.E.6c Let I = 2. A bargaining solution fO is partially monotone if when U c U' and u'(U) = u'(U'), that is, U' expands U only in the direction of agent j ~ i, we have fi( U') :l: fi( U) for j ~ i. Argue that the Kalai-Smorodinsky solution is characterized by the following properties: independence of utility origins and units, Pareto, symmetry, and partial monotonicity. [Hint: use sets U such that U' c U and II'(U) = II'(U'), II'(U) = II'(U')]. 22.E. 7 A Consider a family of bargaining solutions f'(') such that, for every set of agents I, f'(') is independent of utility origins and is generated by maximizing the social welfare function L. g(u,) on normalized bargaining problems U c R', where g(') is increasing, strictly concave, and independent of the particular I considered. Show that the family f' is consistent. 22.E.Sc Show by example that the Kalai-Smorodinsky solution is not consistent. It is enough to consider three agents and its subgroups of two agents. 22.E.9 A This exercise is aimed at showing the independence of the assumptions of Proposition 22.E.1. To this effect, give five examples such that for each of the five assumptions of Proposition 22.E.1 there is one of the examples that violates this assumption but satisfies the
remaining four.
EXERCIIEI
22.F.1 A Show that in the transferable utility case any bargaining solution that is invariant to independent changes of origin, symmetric, and Paretian divides the gains from cooperation equally among the agents. 22.F.2A Show that the Shapley value cooperative solution presented in Section 22.F satisfies the following properties: invariance to independent changes of utility origins, in variance to common changes of utility units, Paretian, symmetry, and the dummy axiom. 22.F.3 A Suppose that for a given set of agents I we take two characteristic forms v and v' and consider their sum v + v'; that is, v + v' is the characteristic form where (v + v'XS) = v(S) + v'(S) for every ScI. (0) Verify that the Shapley value is linear in the characteristic form; that is, /.,(v f,,(v) + f,,(v') for all v, v' and i.
+ v') =
(b) Interpret the linearity property as a postulate that agents are indifferent to the timing of resolution of uncertainty when we randomize among bargaining situations. 22.F.4 c The linearity property of the previous exercise can be restated in a perhaps more intuitive form. We say that a characteristic form v(') is a IInanimity game if for some Tel we have that v(S) = v(T) if T c S, and v(S) = 0 otherwise (thus, the bargaining situations of Section 22.E correspond to T = I). (0) Show that the independence of utility origins and invariance to common changes of utility units, Pareto, symmetry, and dummy axiom properties imply that, for a unanimity game v('), any cooperative solution f(·) assigns the values [,(v) = (1/nv(1) if JET, and J.(v) = 0 otherwise.
(b) We say that the cooperative solution f(·) is weakly linear if for any v and v' differing only by a unanimity game [i.e., there is Tel and «E R such that .'(S) - v(S) + « if T c S, and v'(S) v(S) otherwise] we have that f,(v') f,(v) + «IT if JET, and ft.u) - ft.v) otherwise. Show that if, in addition to the properties listed in (a), the cooperative solution f(') is weakly linear, then it is fully linear, that is, f(v + v') f(v) + f(v') for any two characteristic forms v and v'.
=
=
=
22.E.l0A Give an example of a utilitarian bargaining solution (Example 22.E.2) that violates the property of independence of irrelevant alternatives. [Hint: It suffices to consider I = 2. Also, the violation should involve sets U that are convex but not striclly convex.] 22.E.l1 c Go back to the infinite horizon Rubinstein's bargaining model discussed in the Appendix A to Chapter 9 (specifically, Example 9.AA.2). The only modification is that the two agents are risk averse on the amount of money they get. That is, each has an increasing, concave, differentiable utility function 1I,(m,) on the nonnegative amounts of money that they receive. The factor of discount ~ < I is the same for the two agents. Also 11,(0) = O. The total amount of money is m. (a) Write down the equations for a subgame perfect Nash equilibrium (SPNE) in stationary strategies. Argue that there is a single configuration of utility payoffs that can be obtained as payoffs of a SPNE in stationary strategies. (b) Consider the utility possibility set
U = {(II,(m,), 1I,(m,» E R': m, + m, = m} - R~. Show that if ~ is close to 1 then the payoffs of a SPNE in stationary strategies are nearly equal to the Nash bargaining solution payoffs. (c) (Harder) Argue that every payoff configuration of a SPNE can be obtained as the payoff configuration of a SPNE in stationary strategies. Thus, the uniqueness result presented in Example 9.AA.2 extends to the case in which the agents have strictly concave, possibly different, utility functions for money.
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(c) Show that the Shapley value is the only cooperative solution that satisfies the following prope.'ties: independence of utility origins and in variance to common changes of utility units, Parellan, symmetry, dummy axiom, and linearity. 22.F.sc In this exercise we describe another cooperative solution for a game in characteristic form: the nucleolus. For simplicity we do it for the particular case in which 1= 3, v(1) = v(2) = v(3) = 0, and 0 !> v(S) !> v(l), for any group S of two agents. Given a ulility vector II = (II" II" II,) :l: 0 and an ScI the excess of S at II is e(u, S) = v(S) - L,.s u,. We define the first maximllm excess as m,(II) = Max {_(II, S): 1 < #S < 3}. Choose a two-agent coalition S such that m,(u) = e(u, S). Then we define the second maximum excess as m,(II) = Max {e(II, S'): I < #S' < 3 and S' ~ S}. We say that an exactly feasible [i.e., II, = v(l)] utility profile II = (II" II" 1I,):l: 0 is in thr nucleolus if for any other such profile II' we have either m,(II) < m,(u') or m,(u) = m,(u') and m,(II) !> m,(u').
L,.,
(a) Show that if u = (u" u" u,) is in the nucleolus then either the three excesses for two-agent coalitions are identical or two are identical and the third is larger. (b) Show that there is one and only one utility profile in the nucleolus. [Hint: Argue first that there is a two-agent coalition S such that e(u, S) = m,(u) for every profile in the nucleolus.] From now on we refer to this profile as the nucleolus solution. (c) Argue that the nucleolus solution is symmetric.
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(d) Suppose that agent I is a dummy. Then ", = 0 at the nucleolus solution. (e) Suppose that tv(J):$ v(S) for any coalition S of two agents. Show then that at the nucleolus profile the three excesses for two-agent coalitions are identical.
Incentives and Mechanism
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(f) Compute and compare the Shapley value and the nucleolus for the characteristic form: v(l) = v(2) = v(3) = 0, v({i, 2}) = v{(l, 3}) = 4, v({2, 3}) = 5, v(J) - 6. (g) Show that if the core is nonempty (see Appendix A to Chapter 18 for the definition of the core in this context) then the nucleolus utility profile belongs to the core.
Design
22.F.6" Consider a regulated firm that produces an output by means of a cost function c(q). Assuming a quasilinear economy, the consumer surplus generated by q is Seq). (a) Suppose that c(q) is strictly concave (i.e., strictly increasing returns to scale). Show that at the first·best price the firm will not cover costs. Conversely, for any q suppose that the price p(q) is determined so that the cost is covered; that is, p(q) = c(q)/q. Show that if q is then determined so as to have p(q) = S'(q), we will not reach the first-best optimum. l11ustrate graphically. (b) Suppose that the quantity produced, q, has to be determined under the constraint that with p = S'(q) we have pq :2: c(q). Solve this second-best weIrare problem. l11ustrate graphically.
(e) Interpret the units of output as "projects." For any production decision q, what is the cost allocation suggested by the Shapley value?
22.F.7 c This exercise is similar to Exercise 22.F.6, except that the firm now produces two outputs under the separable cost functions c,(q,), c,(q,). The surplus S,(q,) + S,(q,) is also separable. (a) The second-best problem [first studied by Boiteux (1956)J is now richer than in Exercise 22.F.6. Suppose that the quantities q" q, have to be determined so that with p, = S'(q,) and p, = S'(q,) we have p,q, + p,q, :2: c,(q,) + c,(q,) (equivalently, at the chosen prices demand must be served and cost covered). Derive first-order conditions for this problem. Make them as similar as possible to the Ramsey formula of Example 22.B.2. (b) (Harder) Interpret the units of outputs as projects. Suppose that these units are very small, so that a given production decision (q" q,)>> 0 represents the implementation of many projects of each of the two types. Can you guess, given (q" q,), what is an approximate value for the cost allocation suggested by the Shapley value? [Hint: For most orderings of projects, any particular project will have preceding it an almost peneet sample of all the projects.J (e) Suppose that for the productions (9" 9,), the Shapley value cost allocation assigns cost per unit of c, and c, (note that "projects" of the same type receive the same cost imputation). Suppose also that c, = oS,(q,)/oq, and c, = oS,(ii,)/oq,. Interpret. Argue that, in general, these productions will not correspond to either the first-best or the second-best optima of the problem.
23.A Introduction In Chapter 21, we studied how individual preferences might be aggregated into social preferences and ultimately into a collective decision. Howe~er, an .im~~rtant !eature of many settings in which collective decisions must be made tS that mdlVlduals actual preferences are not publicly observable. As a result, in one way or another, individuals must be relied upon to reveal this information. In this chapter, we study how this information can be elicited, and the extent to which the information revelation problem constrains the ways in which social decisions can respond to individual preferences. This topic is known as the mechanism design problem.
Mechanism design has many important applications throughout economics. The design of voting procedures, the writing of contracts among parties who will come to have private information, and the construction of procedures for deciding upon public projects or environmental standards are all examples. I The chapter is organized as follows. In Section 23.B, we introduce the mechanism design problem. We begin by illustrating the difficulties introduced by the need to elicit agents' preferences. We also define and discuss the concepts of social choice functions (already introduced in Section 21.E), ex post efficiency, mechanisms, implementation, direct revelation mechanisms, and trutliful implementation. In Section 23.C, we identify the circumstances under which a social choice function can be implemented in dominant strategy equilibria when agents' preferences are private informdtion. Our analysis begins with a formal statement and proof of the revelation principle, a result that tells us that we can restrict attention to direct revelation mechanisms that induce agents to truthfully reveal their preferences. Using this fact, we then study the constraints that the information revelation
I. Simple examples of the last two applications were encountered in Sections 14.C and J I.E, respectively. 857
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problem puts on the set of implementable social choice functions. We first present the important Gibbard-Satterthwaite theorem, which provides a very negative conclusion for cases in which individual preferences can take unrestricted forms. In the rest of the section, we go on to study the special case of quasilinear environments, discussing in detail Groves-Clarke mechanisms. In Section 23.D, we study implementation in Bayesian Nash equilibria. We begin by discussing the expected externality mechanism as an example of how the weaker Bayesian implementation concept can allow us to implement a wider range of social choice functions than is possible with dominant strategy implementation. We go on to provide a characterization of Bayesian implementable social choice functions for the case in which agents have quasilinear preferences that are linear in their type. As an application of this result, we prove the remarkable revenue equivalence rheorem for auctions. In Section 23.E, we consider the possibility that participation in a mechanism may be voluntary and study how the need to satisfy the resulting parricipation ('onsrraillls limits the set of implementable social choice functions. Here we prove the important Myerson-Satterthwaite theorem, which shows that, under very general conditions. it is impossible to achieve ex post efficiency in bilateral trade settings when agents have private information and trade is voluntary. In Section 23.F, we discuss the welfare comparison of mechanisms. defining the notions of ex ante and interim incentive efficiency, and providing several illustrations of the computation of welfare optimal Bayesian mechanisms. Appendices A and B are devoted to, first, a discussion of the issue of multiple equilibria in mechanism design and. second, the issue of mechanism design when agents know each others' types but the mechanism designer does not (so-called complete information environments). References for further reading are provided at the start of the various sections. We would be remiss. however. not to mention here two early seminal articles: Mirrlees (1971) and Hurwicz (1972).
23.B The Mechanism Design Problem In this section. we provide an introduction to the mechanism design problem that we study in detail in the rest of the chapter. To begin. consider a setting with I agents. indexed by i = I •...• I. These agents must make a collective choice from some set X of possible alternatives. Prior to the choice. however. each agent i privately observes his preferences over the alternatives in X. Formally. we model this by supposing that agent i privately observes a parameter. or signal. Ii, that determines his preferences. We will often refer to Ii, as agent fs rype. The set of possible types for agent i is denoted 0,. Each agent i is assumed to be an expected utility maximizer, whose Bernoulli utility function when he is of type 0, is u,(x. 0,). The ordinal preference relation over pairs of alternatives in X that is associated with utility function u,(x,O,) is denoted ;t,(O,). Agent i's set of possible preference relations over X is therefore given by
fJP, = {;t,: ;t, = ;t,(O,) for some 0, e 0,}. Note that because 0, is observed only by agent i. in the language of Section 8.E
--
SECTION
22.1:
THE
MECHANIIM
DEIION
we are in a setting characterized by incomplete information. As in Section 8.E, we suppose that agents' types are drawn from a commonly known prior distribution. In particular. denoting a profile of the agents' types by 9 = (0 1 , . . . ,0,), the probability density over the possible realizations of 9 e 0 1 x ... X 0, is .p(.). The probability density .p(.) as well as the sets 0 ..... ,0, and the utility functions u;(·.O,) are assumed to be common knowledge among the agents. but the specific value of each agent fs type is observed only by i. 2 Because the agents' preferences depend on the realizations of 0 = (0 1 , •••• 0,). the agents may want the collective decision to depend on O. To capture this dependence formally. we introduce in Definition 23.B.I the notion of a social choice function. a concept already discussed in Section 21.E.3
Dellnltlon 23.B.1: A social choice function is a function f: 0 1 x ... x 0 t - X that. for each possible profile of the agents' types (0 1, ••• • 0,). assigns a collective choice f(Ol.···,O,)eX·
One desirable feature for a social choice function to satisfy is the property of ex post efficiency described in Definition 23.B.2. Dellnltlon 23.B.2: The social choice function f: 0 1 x ... X 0, - X is ex post efficient (or Paretian) if for no profile 0 = (0 1, •.•• 0,) is there an x e X such that Ui(X, Oil ui(f(O), 0i) for some i. Definition 23.B.2 says that a social welfare function is ex post efficient if it selects, for every profile 0 = (0 1, •••• 0,). an alternative f(O) e X that is Pareto optimal given the agents' utility functions UI(·'OI) •...• u,(·.O,). The problem faced by the agents is that the Ois are not publicly observable, and so for the social choice f(OI, .... 0,) to be chosen when the agents' types are (0 1, •••• 0,). each agent i must be relied upon to disclose his type 9,. However. for a given social choice function f(·). an agent may not find it to be in his best interest to reveal this information truthfully. We illustrate this information revelation problem in Examples 23.B.1 through 23.B.4. which range from very abstract to more applied settings.
2. The formulation here is restrictive in one sense: in some settings of interest. agents' preferences over outcomes depend not only on their own observed signals bUI also on signals observed by others (e.g .• agent ts preferences over whether 10 hold a picnic indoors may depend on agent j's knowtedge of likely weather condilions). Through most of this chapler. we focus on the case in which an agent's 'lreferences depend only on his own signal, known as the private values case. We generalize our analysis in Section 23.F. 3. In Section 21.E an agent's type was equivalent to his ordinal preferences over X, and so a social choice function was defined there simply 35 a mapping from Uti x ... X 91, to X. Moreover, it was assumed there that for all i we have {II, = 11, the set of all possible ordinal prderence orderings on X. 4. Two points should be noted about this definition. First. it restricts attention to delerministic social choice funclions. This is largely for exposilional purposes; ahhough much of Ihe chapter considers deterministic social choice functions. in Sections 23.0 to 23.F we allow social choice functions that assign lotreries over X. Second, as in Section 21.E, we limit our attention to singte·vatued choice functions.
PROBLEM
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DESIOH
Example 23.B.I: An Abstract Social Choice Selling. In the most abstract case, we are given a set X and, for each agent i, a set fJl l of possible rational preference orderings on X. To consider a very simple example, suppose that X = {x, y, z} and that J = 2. Suppose also that agent I has one possible type, so that 0, = {B,}, and that agent 2 has two possible types, so that 0, = {O'"Oi}. The agents' possible preference orderings fJl l = (;::I(B I )} and fJl 2 = (;::,(8'2), ;::2(02)} are given by
;::1 (B,)
----...
--
SEC T ION
23. B:
THE
ME C HAN I I M
DEli 0 N
z
y
y
Figure 23.B.1
In the social choice runction that selccts a Walrasian equilibrium ror each prererence profile, agent 2 has an incentive to claim to be type when he is really type 8;.
y x
x
[A higher positioned alternative is strictly preferred to a lower positioned one; so, for example, x :>,(B,) y :>,(B I ) z.] Now suppose that the agents wish to implement the ex post efficient social choice function f(·) with
Ao"0:,) = y
and
f(B" 0i) = x.
8,
0,
are (;:: ,(ii,), ;::2 (Oi))], an allocation that he strictly prefers to f(O"Oll when his preferences are ;::2 (0:,). _
If so, then agent 2 must be relied upon to truthfully reveal his preferences. But it is apparent that he will not find it in his interest to do so: When O2 = 0;, agent 2 will wish to lie and claim that his type is 0'2' In abstract social choice settings, a case of central interest arises when fJl l is, for each agent i, equal to fJI, the set of all possible rational preference relations on X. In this case, an agent has many possible false claims that he can make and, intuitively, it may be very difficult for a social choice function always to induce the agents to reveal their preferences truthfully. We will see a formal illustration of this point in Section 23.C when we present the Gibbard-Satterthwaite theorem. _
Example 23.B.3: A Public Project. Consider a situation in which J agents must decide whether to undertake a public project, such as building a bridge, whose cost must be funded by the agents themselves. An outcome is a vector x = (k, I" ... , I,), where k E (0, I} is the decision whether to build the bridge (k = I if the bridge is built, and k = 0 if not), and II E R is a monetary transfer to (or from, if II < 0) agent i. The cost of the project is c;:: 0 and so the set of feasible alternatives for the J agents is
Example 23.B.2: A Pure Exchange Economy. Consider a pure exchange economy with L goods and J consumers in which agent i has consumption set R~ and endowment vector WI = (W'h"" w u )>> o(see Chapter 15). The set of alternatives is
The constraint Lltl ~ -ck reflects the fact that there is no source of outside funding for the agents (so that we must have c + LI I, :s 0 if k = I, and LI II :S 0 if k = 0). We assume that type O;'s Bernoulli utility function has the quasilinear form
X = {(k, I" ... , I,): k E
to, I}, liE R for all i, and L II ~ -ck}. I
X
=
«x" ... ,XI): xIER~
and LXf/ ~ LW(i for I
(=
I, ... L}.
I
In this setting it may be natural to suppose that fJI;, each consumer i's set of possible preference relations over alternatives in X, is a subset of fJl E , the set of individualistic (i.e., depending on XI only), monotone, and convex preference relations on X. To consider a simple example, suppose that J = 2, that consumer I has only one possible type, so that 0, = (B,} and fJI, = (;::, (B,)}, and that for consumer 2 we have;1l2 = iJI E • Imagine then that we try to implement a social choice function that, for each pair (;::, (0,), ;::, (0,)), chooses a Walrasian equilibrium allocation (note that this social choice function is ex post efficient). As Figure 23.B.I illustrates, consumer 2 will not generally find it optimal to reveal his preferences truthfully. In the figure, f(BI,Oll is the unique Walrasian equilibrium when preferences are (;::, (ii,), ;::2 (0'2)) [it is the unique intersection ofthe consumers' offer curves OC, and OC:, occurring at a point other than the endowment point]. However, by claiming that he has type Oi, which has as its offer curve OC;, consumer 2 can obtain the allocation f(OI' 0;) [the unique Walrasian equilibrium allocation when preferences
861
-E-4--------------------------~~
;::2 (0:')
x
PRO B L E II
uI(x, 01) = Olk + (ml + II), where lii l is agent i's initial endowment of the numeraire ("money") and 0, E R. We can then interpret 01 as agent i's willingness to pay for the bridge. In this context, the social choice function f(O) = (k(O), 1,(0), ... ,1,(0» is ex post efficient if, for all 0,
k(O)=t
ifLO,;:: c, I
(23.B.I)
otherwise,
and
L 1,(0) =
-ck(O).
(23.B.2)
I
Suppose that the agents wish to implement a social choice function that satisfies (23.B.I) and (23.B.2) and in which an egalitarian contribution rule is fOllowed, t~at is, in which 11(0) = -(cj I)k(O). To consider a simple example, suppose that 0, = to,} for i '# I (so that all agents other than agent I have preferences that are known) and
862
CHAPTER
23:
'NCENT'VES
AND
MECHAN'SM
DES,GN
IICTION
L,,., e,
L,,., e, L,,..e, -
e,
~I = (c - '0"L e, + &) + tit. - ~I =
(C(l-I
I) _
L
,,..
e, +&) + m•.
Example 23.B.4: Allocation of a Single Unit of an Indivisible Private Good. Consider a setting in which there is a single unit of an indivisible private good to be allocated to one of 1 agents. Monetary transfers can also be made. An outcome here may be represented by a vector x = (y" ... , Y/O t., ... , t,), where y, = I if agent i gets the good, y, = 0 if agent i does not get the good, and t, is the monetary transfer received by agent i. The set of feasible alternatives is then
= {(y"
e
e
(l
But, for & > 0 small enough, this is less than m" which is agent I's utility if he instead claims that 8. = 0, a claim that results in the bridge not being built. Thus, agent I will prefer not to tell the truth. Intuitively, under this allocation rule, when agent I causes the bridge to be built he has a positive externality on the other agents (in the aggregate). Because he fails to internalize this effect, he has an incentive to understate his benefit from the project. _
x
DIIIGN
eo
Agent l's utility in this case is
+ m. -
MICHANII ..
e.
e,
8.
TNI
Two special cases that have received a great deal of attention in the literature deserve mention. The first is the case of bilateral trade. In this case we have I ... 2; agent I is interpreted as the initial owner of the good (the "seller"), and agent 2 is the potential purchaser of the good (the "buyer"). When ~ > there are certain to be gains from trade regardless of the realizations of 8. and 82; when ~, > 2 there are certain to be no gains from trade; finally, if ~2 < and ~, < 2 then there may or may not be gains from trade, depending on the realization of 8. The second special case is the auction setting. Here, one agent, whom we shall designate as agent 0, is interpreted as the seller of the good (the" auctioneer") and is assumed to derive no value from it (more generally, the seller might have a known value 80 = different from zero). The other agents, I, ... , I. are potential buyers (the "bidders").5 To illustrate the problem with information revelation in this example. consider an auction setting with two buyers = 2). In the previous examples, we simplified the discussion of information revelation by assuming that only one agent has more than one possible type. We now suppose instead that both buyers' (privately observed) valuations 8, are drawn independently from the uniform distribution on [0, I] and that this fact is common knowledge among the agents. Consider the social choice function f(8) = (Yo(8), y.(8), Y2(0), 10 (8), t.(8), t 2(8» in which
0, = [0, (0). Suppose also that C> > c(1 - 1)/1. These inequalities imply, first, that with this social choice function agent I's type is critical for whether the it is; if 8, < C it is not), and that the bridge is built (if 8, ~ C sum of the utilities of agents 2, ... , I is strictly greater if the bridge is built under c(1 - 1)/1> OJ. this egalitarian contribution rule than ifit is not built [since Let us examine agent I's incentives for truthfully revealing his type when e, = c - L,o" + & for & > O. If agent I reveals his true preferences, the bridge will be built because
L,,., e,
21 •• :
... ,y"t., ... ,t,):y,E{O, I} and t,EIR for all i, LY'
,
e
i)
=
if 0,
82 ;
= 0 if 8. < O2
(23.8.3)
if 8, < 8 2 ;
= 0 if 8. ~ 8 2
(23.B.4)
yo(8) = 0
for all 8
(23.8.5)
t.(8) = -8.y,(8)
(23.B.6)
t 2 (8) = - 8 2y,(8)
(23.8.7)
t o(8) = -(t.(8)
= I, and LI,;5; OJ.
e,y, + (Iii, + I,),
where 1ft, is once again agent i's initial endowment of the numeraire ("money"). Here IIi E IR can be viewed as agent i's valuation of the good, and we take the set of possible valuations for agent i to be 0, = [Q" 0,] c R. In this situation, a social choice function flO) = (y.(O), ... , y,(O), 1.(8), ... ,1,(0)) is ex post efficient if it always allocates the good to the agent who has the highest valuation (or to one of them if there are several) and if it involves no waste of the numeraire; that is, if for all 0 = (0., ... ,8,) E 0. X ••• x 0/0 y,(O)(O, - Max{O., ... , O,}) = 0
~
h(O) = I Y2(8) = I
+ t 2(8)).
(23.B.8)
In this social choice function. the seller gives the good to the buyer with the highest valuation (to buyer 1 if there is a tie) and this buyer gives the seller a payment equal to his valuation (the other, low-valuation buyer makes no transfer payment to the seller). Note that f(·) is not only ex post efficient but also is very attractive for the seller: if f(·) can be implemented, the seller will capture all of the consumption benefits that are generated by the good. Suppose we try to implement this social choice function. Assume that the buyers are expected utility maximizers. We now ask: If buyer 2 always announces his true value, will buyer I find it optimal to do the same? For each value of 8., buyer l's problem is to choose the valuation to announce, say ~" so as to solve
We suppose that type e;,s Bernoulli utility function takes the quasilinear form u,(x,
PlloallM
863
------------------------------------------------------------------------
Max i,
(II. - ~.) Prob (8 2 ~ ~.)
or Max i,
for all i
and
(8. - ~.)~ •.
S. Note that, for ease of notation, we take there to be I
,
.~
+ 1 agents
in the auction setting.
864
CHAPTER
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INCENTIYES
AND
MECHANISM
DESIGN
The solution to this problem has buyer 1 set 01 = Bd2. We see then that if buyer 2 always tells the truth, truth telling is not optimal for buyer I. A similar point applies to buyer 2. Intuitively, for this social choice function, a buyer has an incentive to understate his valuation so as to lower the transfer he must make in the event that he has the highest announced valuation and gets the good. The cost to him of doing this is that he gets the good less often, but this is a cost worth incurring to at least some degree." Thus, we again see that there may be a problem in implementing certain social choice functions in settings in which information is privately held. (For a similar point in the bilateral trade context, see Exercise 23.B.2.) Although buyers have an incentive to lie given the social choice function described in (23.B.3) to (23.B.8), this is not true of all social choice functions in this auction setting. To see this point, suppose we try to implement the social choice function j(.) that has the same allocation rule as that above [i.e., in which the functions y,(.) for i = 0, 1,2 are the same as those described in (23.8.3) to (23.B.5)] but instead has transfer functions 1,(0) = -O,y,(O) I,(B) = -OIY,(O) 10 (0) = -(1,(0)
+ t,(B».
In this social choice function, instead of buyer i paying the seller an amount equal to his own valuation 0, if he wins the object, he now pays OJ, where j ¢' i; that is, he pays an amount equal to the second-highest valuation. Consider buyer I's incentives for truth telling now. If buyer 2 announces his valuation to be 0, $ BI , buyer I can receive a utility of (0, - 0,) ;:: 0 by truthfully announcing that his valuation is B,. For any other announcement, buyer I's resulting utility is either the same (if he announces a valuation of at least 0,) or zero (if he announces a valuation below 0,). So if 0, $ 0" announcing the truth is weakly best for buyer I. On the other hand, if buyer 2's announced valuation is (), > B" then buyer l's utility is 0 if he reveals his true valuation. However, buyer 1 can receive only a negative utility by making a false claim that gets him the good (a claim that his valuation is at least (),). We conclude that truth telling is optimal for buyer I regardless of what buyer 2 announces. Formally, in the language of the theory of games, truth telling is a weakly dominant strategy for buyer I (see Section 8.B). A similar conclusion follows for buyer 2. Thus, this social choice function is implementable even though the buyers' valuations are private information: it suffices to simply ask each buyer to report his type, and then to choose j(O).' • Examples 23.B.I to 23.B.4 suggest that when agents' types are privately observed the information revelation problem may constrain the set of social choice functions that can be successfully implemented. With these examples as motivation, we can now pose the central question that is our focus in this chapter: What social choice junctiolls can be implemented when agellls' types are privale information?
6. This /rade-otT is similar 10 that faced by a monopotist (see Section t2.B): when the monopolist raises his price, he lowers his sales but makes more on his remaining sales. 7. For other examples of implementable social choice [unctions, see Exercise 23.8.1.
-- ----
SECTION
23.8:
THE
MECHAN"M
DESIGN
To answer this question, we need in principle to begin by thinking of all the possible ways in which a social choice function might be implemented. In the above examples we have implicitly imagined a very simple scenario in which each agent i is asked to directly reveal 0, and then, given the announcements (0" ... ,0,), the alternative j(O" ... , 0,) E X is chosen. But this is not the only way a social choice function might be implemented. In particular, a given social choice function might be indirectly implemented by having the agents interact through some type of institution in which there are rules governing the actions the agents may take and hoW these actions translate into a social outcome. To illustrate this point, Examples 23.B.5 and 23.B.6 study two commonly used auction institutions. Example 23.B.5: Firsl-Price Sealed-Bid Auclion. Consider again the auction setting introduced in Example 23.B.4. In a first-price sealed-bid auction each potential buyer i is allowed to submit a sealed bid, b,;:: O. The bids are then opened and the buyer with the highest bid gets the good and pays an amount equal to his bid to the seller.· To be specific, consider again the case where there are two potential buyers (I = 2) and each 0, is independently drawn from the uniform distribution on [0, I]. We will look for an equilibrium in which each buyer's strategy M') takes the form MO,) = CI.,O, for CI., E [0, I]. Suppose that buyer 2's strategy has this form, and consider buyer I's problem. For each B, he wants to solve Max bl
(0, - b,) Prob(b,(B,)
$
b,).
:
Since buyer 2's highest possible bid is CI., (he submits a bid of CI., when 0, = I), it is evident that buyer I should never bid more than cx,. Moreover, since B, is uniformly distributed on [0, I] and b,(B,) $ b, if and only if B, $ (b,/cx,), we can write buyer I's problem as Max
(B, - b,)(bdCl.,)·
bIE{O.1I2J
The solution to this problem is b,(B,) =
{W' CI.,
if tB, $ ietB, >
CI." CI.,.
By similar reasoning, iftB, $ CI." if!B,>CI.,.
to,
Letting CI., = CI., = t, we see that the strategies b,(O,) = for i = 1,2 constitute a Bayesian Nash equilibrium for this auction. Thus, there is a Bayesian Nash equilibrium of this first-price sealed-bid auction that indirectly yields the outcomes specified by the social ~hoice function j(O) = (Yo(B), y,(B), y,(O), t o(9), 1,(0), t,(O»
8. If there are several highest bids, we suppose that the lowest numbered of these bidders gets the good. We could equally well randomize among tho highest bidders if there are more than one, but 'his would require that we expand 'he set of alternatives to A(X), the set of all lotteries over X. In fact. we do precisely this when we sludy auctions in Sections 23.0 and 23.F.
PROBLEM
865
866
C H A , TEA
23:
INC E N T lYE 8
AND
M E C HAN IBM
DE' I Q N
--------------------------~~~~----------------~ in which
y,(O) = I
• E C T ION
23. 8:
THE
M E C HAN
II M
DE' I Q N
a Bayesian game of incomplete information. That is, leUing ii,(sl •...• s" 0,) = u,(g(s, •... ,s,), 0,). the game
(23.B.9)
Yl(/I) = I
if /I, < 0l;
(23.B.l0)
[I, {S,}, {ii,(')},e, x .. · x e"t/>(')]
yo(/I) = 0
for all 0
(23.B.11)
is exactly the type of Bayesian game studied in Section R.E. Note that a mechanism could in principle be a complex dynamic procedure. in which case the elements of the strategy sets S, would consist of contingent plans of action (see Chapter 7)." For the auction setting, the first-price sealed-bid auction is the mechanism in which S, = IR+ for all i and, given the bids (b ..... , b,) E R~. the outcome function g(b" ... , b,) = ({y,(b" ... , b, {t,(b, •.... b, )}I-,) is such that
t,(O) = -!O,y/(O)
(23.B.12)
tl(O) = -tOlY'(O) totO) = -(t,(8)
(23.B.l3)
+ t,(/I».
(23.B.14)
nt-"
•
y,(b, ..... b.>
Example 23.8.6: Second-Price Sealed-Bid Auction" Once again, consider the auction setting described in Example 23.B.4. In a second-price sealed-bid auction each potential buyer i is allowed to submit a sealed bid, b, ~ O. The bids are then ~pened and the buyer with the highest bid gets the good, but now he pays the seller an amount equal to the second-highest bid.'o By reasoning that parallels that at the end of Example 23.B.4, the strategy b,(O.) .= 0, for all /I, E [0, I] is a weakly dominant strategy for each buyer i (see ExerCise 23.B.3). Thus, when I = 2 the second-price sealed-bid auction implements the social choice function flO) = (yo(O). y,(O), Yl(O). t o(8). t,(O), tl(O» in which y,(O) = I
if 0, ~ 0l;
=
0 if 8, < Ol
y,(O) = I
if 0, < 0l;
=
0 if /I, ~ /l l
YolO) = 0
for all /I
=I
if and only if i
= Min{j: b) =
Max{b, ..... bdl.
t,(b, •...• b,) = -b,y,(b l •••• , b,).
In the second-price sealed-bid auction, on the other hand. we have the same strategy sets and functions y,('), but instead t,(b ...... b,) = -Max{b):J I< i}y,(b ..... ,b,). A strategy for agent i in the game of incomplete information created by a mechanism r is a function s,: e, -+ S, giving agent i's choice from S, for each possible type in e, that he might have. Loosely put, we say that a mechanism implements social choice function f(·) if there is an equilibrium of the game induced by the mechanism that yields the same outcomes as f(·) for each possible profile of types 8 = (0" ...• 0,). This is stated formally in Definition 23.B.4. Deflnlllon 23.B.4: The mechanism r = (5, ..... 5,. g(.)) implements social choice function f (.) if there is an equilibrium strategy profile (sH·) ..... s1 (.)) of the game induced by r such that g(s~(O,) •...• 51(/1,)) = f(/I, •...• /I,) for all (/I, .... . 0.) E e, x ... x e,.
1,(0) = -/lly,(/I)
Il(O) = -O'Yl(O) 10(0) = -(1,(/1)
Note, however, that we have not specified in Definition 23.B.4 exactly what we mean by an Mequilibrium". This is because, as we have seen in Part II, there is no single equilibrium concept that is universally agreed upon as the appropriate solution concept for games. As a result, the mechanism design literature has investigated the implementation question for a variety of solution concepts. In Sections 23.C and 23.D we focus on two central solution concepts: dominant strategy equilibrium and Bayesian Nash equilibrium. ll Note also that the notion of implementation that we have adopted in Definition 23.B.4 is in one sense a weak one: in particular, the mechanism r may have more (/'an one equilibrium, but Definition 23.B.4 requires only that one of them induce outcomes in accord with f(·). Implicitly, then, Definition 23.B.4 assumes that. if multiple equilibria exist, the agents will play the equilibrium that the mechanism designer wants. Throughout the chapter we shall stick to this notion of implementation. Appendix A is devoted to a further discussion of this issue.
+ tl(/I» . •
Examples 23.B.5 and 23.B.6 illustrate that, as a general matter, we need to consider not only the possibility of directly implementing social choice functions by asking ag~nts .to ~evea~ their .types but also their indirect implementation through the design of institutions In whIch the agents interact. The formal representation of such an institution is known as a mechanism. Definition 23.B.3: A mechanism r = (5, ..... 5,. g(.)) is a collection of I strategy sets (5, ..... 5,) and an outcome function g: 5, x ... X 51 -+ X. A mechanism can be viewed as an institution with rules governing the procedure for making the collective choice. The allowed actions of each agent i are summarized by the strategy set S,' and the rule for how agents' actions get turned into a social choice is given by the outcome function g('). For~alIy, th.e mechanism r combined with possible types (e" ... , e/)' probabIlIty denSity t/>(.), and Bernoulli utility functions (u,(·), ... , ul (·» defines
I 1. Note also that we are representing the game created by a mechanism using its normal form. For all the analysis Ihat follows in Ihe text this will be sufficient. In Appendix B. however. we consider a case where the extensive form representation is used. 12. Appendix B considers several other equilibrium concepts in the special context of camp/ere information settings in which the players observe each others' types.
9. This auclion is also called a Vickrey aucr;on. afler Vickrey (1961). 10. If there is more than One highest bid. we again select the lowest-numbered of these bidders.
J
, A
0 8 LEM
867
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MECHANISM
DESIGN
The identification of all social choice functions that are implementable may seem like a daunting task because, in principle, it appears that we need to consider all possible mechanisms-a very large set. Fortunately, an important result known as the revelation principle (to be formally stated and proven in Sections 23.C and 23.D) tells us that we can often restrict attention to the very simple type of mechanisms that we were implicitly considering at the outset, that is, mechanisms in which each agent is asked to reveal his type, and given the announcements (~" ... ,~,), the alternative chosen is f(~" ... , ~,) E X.13 These are known as direct revelation mechanisms, and formally constitute a special case of the mechanisms of Definition 23.B.3.
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SECTION
STRATEGY
IMPLEMENTATION
Because of the revelation principle. when we explore in Sections 23.C and 23.D the constraints that incomplete information about types puts on the set of implementable social choice functions, we will be able to restrict our analysis to identifying those social choice functions that can be truthfully implemented. Finally, we note that, in some applications, participation in the mechanism may be voluntary, and so a social choice function must not only induce truthful revelation of information but must also satisfy certain participation (or individual rationality) constraints if it is to be successfully implemented. In Sections 23.C and 23.D, however, we shall abstract from issues of participation to focus exclusively on the information revelation problem. We introduce participation constraints in Section 23.E.
23.C Dominant Strategy Implementation
Definition 23.B.6: The social choice function f (.) is truthfully implementable (or incentive compatible) if the direct revelation mechanism r = (El" ... ,El" f(·J) has an equilibrium (sf(·) .... , s1 (. J) in which s1 (Oi) = 0i for all 0i E Eli and all i = 1•... ,I; that is. if truth telling by each agent i constitutes an equilibrium of r = (El, ....• El" f(·J).
In this section, we study implementation in dominant strategies. '4 Throughout we follow the notation introduced in Section 23.B: The vector of agents' types 0= (0" . .. ,0,) is drawn from the set El = El, x ... x El, according to a probability density t/J('), and agent i's Bernoulli utility function over the alternatives in X given his type 0, is u,(x, 0,). We also adopt the notational convention of writing 0_, = (0 1, ••• ,0,_"0,+,, ... ,0,), 0 = (0" 0_/), and El_/ = El, x ... x El/_ I x Eli+' X • " x El,. A mechanism r = (S" ... ,'Slo g(.)) is a collection of I sets S" . .. ,S" each S/ containing agent i's possible actions (or plans of action), and an outcome function g: S -+ X, where S = S, x ... X S,. As discussed in Section 23.B, a mechanism r = (S, .... , Slo g(')) combined with possible types (El ..... , El,), density t/J('), and Bernoulli utility functions (u,(·), ... , u,(·» defines a Bayesian game of incomplete information (see Section 8.E). We will also often write L, = (5" ... ,S,_"5,+,, ... ,s,), 5 = (5/,5_/), and S_/ = S, X . . . X S,_, X S/+' x .. · X S,. Recall from Section 8.B that a strategy is a weakly dominant strategy for a player in a game if it gives him at least as large a payoff as any of his other possible strategies for every possible strategy that his rivals might play. In the present incomplete information environment, strategy 5/: El, ... S, is a weakly dominant strategy for agent i in mechanism r = (S" ... ,S" g(.» if, for all 0, E El/ and all possible strategies for agentsj '" i, L,(·) = [s,(·), ... ,5,_,('),5,+,('), ... ,51 (,)],1$
To offer a hint as to why we may be able to restrict attention to direct revelation mechanisms that induce truth telling, we briefly verify that the social choice functions that are implemented indirectly through the first-price and second-price sealed-bid auctions of Examples 23.B.5 and 23.B.6 can also be truthfully implemented using a direct revelation mechanism. In fact, for the second-price sealed-bid auction of Example 23.B.6 we have already seen this fact, because the social choice function implemented by the second-price auction is exactly the social choice function that we studied at the cnd of Example 23.B.4 in which truth telling is a weakly dominant strategy for both buyers. Example 23.B.7 considers the first-price sealed-bid auction. Example 23.B.7: Truthful Implementation of the Social Choice Function Implemented by the First-Price Sealed-Bid Auction. When facing the direct revelation mechanism (El" ... , El,,f(·)) with f(O) = (Yo(O), y,(O), Y2(0), totO), t,(O), t 2(0)) satisfying (23.B.9) to (23.B.14), buyer I's optimal announcement ~, when he has type 0, solves Max ;,
DOMINANT
The first-order condition for this problem gives ~, = 0,. So truth telling is buyer l's optimal strategy given that buyer 2 always tells the truth. A similar conclusion follows for buyer 2. Thus, the social choice function implemented by the first-price sealed-bid auction (in a Bayesian Nash equilibrium) can also be truthfully implemented (in a Bayesian Nash equilibrium) through a direct revelation mechanism. That is, the social choice function (23.B.9) to (23.B.l4) is incentive compatible. _
Definition 23.B.S: A direct revelation mechanism is a mechanism in which Si = Eli for all i and g(O) = f(O) for all 0 EEl, x ... x El,. Moreover, as we shall see, the revelation principle also tells us that we can further restrict our attention to direct revelation mechanisms in which truth telling is an optimal strategy for each agent. This fact motivates the notion of truthful implementation that we introduce in Definition 23.B.6 (we are again purposely vague in the definition about the eqUilibrium concept we wish to employ).
23.C:
(0, - t~,) Prob(02 ~ ~,)
E, .[U,(g(5,(O,), 5_,(0 _,»,0,)10,] ;;:: E,Au,(g(§" L,(O _/»,0/)10/]
for all §, E S/. (23.C.I)
or Max ;,
869
-----------------------------------------------------------------------
Condition (23. ':.1) holding for all
(0, - t~,)/l,.
L
,(-l and 0, is equivalent to the condition that,
14. Good sources for fu"her reading on the subject of this section are Dasgupta, Hammond and Maskin (1979) and Green and Larront (1979). IS. The expecta'ion in (23.C.1) is taken over realizations of E (,L,.
13. Some early versions of the revelation principle were derived by Gibbard (1973). Green and Larront (1977). Myerson (1979). and Dasgupta. Hammond and Maskin (1979).
e_,
1
870
c HAP T E R
2 3:
INC E N T lYE.
AND
III E C HAN I • III
0 E• IQ N
----------------------------------------------------------------------for all O,e e .. u,(g(s,(O,), L,), 0,) 2: u,(g(J.. L,), 0,)
(23.C.2)
for aIlJ,eS, and all s_,eS_,.'6This leads to Definition 23.C.1. DefinItIon 23.C.1: The strategy profile s·(·) = (st( .), ... ,s1(')) is a dominant strategy equilibrium of mechanism r = (S" ... ,SI' g(.)) if, for all i and all 01eel' u;(g(s7(01)' L;). 0;) 2: u;(g(si. s _1).0;)
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• E C T ION
23. C:
DO III I N A III T
• T R AT E Q Y
whether a particular J(.) is truthfully implementable in the sense introduced in Definition 23.C.3.
DefinItion 23.C.3: The social choice function f(') is truthfully implementable in dominant strategies (or dominant strategy incentive compatible. or strategy-proof. or straightforward) if s1(01) = 01 for all O;e e; and i = 1, ... ,I is a dominant strategy equilibrium of the direct revelation mechanism r = (e ,. ... ,e l • f( That is. if for all i and all 0; e e;.
'».
for all sieS;and all s_;eS_;.
ul(f(O;. 0_1). 0;) 2: UI(f(OI. 0_;). ( 1)
We now specialize Definition 23.B.4 to the notion of dominant strategy equilibrium. DefinItion 23.C.2: The mechanism r = (S" ...• SI' g(.)) implements the social choice function f (.) in dominant strategies if there exists a dominant strategy equilibrium of r. s·(·) = (st(·) •... ,si(·)). such thatg(s·(O)) = flO) for allOee. The concept of dominant strategy implementation is of special interest because if we can find a mechanism r = (S" ... ,SI' g('» that implements J(.) in dominant strategies, then this mechanism implements 1(') in a very strong and robust way. This is true in several senses. First, we can feel fairly confident that a rational agent who has a (weakly) dominant strategy will indeed play it." Unlike the equilibrium strategies in Nash-related equilibrium concepts, a player need not correctly forecast his opponents' play to justify his play of a dominant strategy. Second, although we have assumed that the agents know the probability density t/>(.) over realizations of the types (0" ... ,01 ), and hence can deduce the correct conditional probability distribution over realizations of 0_ .. if r implements J(.) in dominant strategies this implementation will be robust even if agents have incorrect, and perhaps even contradictory, beliefs about this distribution. In particular, agent i's beliefs '8 regarding the distribution of 0 _, do not affect the dominance of his strategy Third, it follows that if r implements J(') in dominant strategies then it does so regardless of the probability density t/>( '). Thus, the same mechanism can be used to implement 1(') for any t/>(.). One advantage of this is that if the mechanism designer is an outsider (say, the "government"), he need not know t/>(.) to successfully implement J(. ). As we noted in Section 23.B, to identify whether a particular social choice function Fortunately, it turns out that for dominant strategy implementation it suffices to ask
16. Condilion (23.C.2) follows from (23.C.1) simply by selling ,_,(0_,) = ,_, for all O_,E 0 _,. To see that (23.C.2) implies (23.C.1). consider the case where S_, is a finite set. Then. for any".
Thus. (23.C.2) implies (23.C.I). 17. We leave aside the question of what might happen if an agent has several weakly dominant strategies. This is Ihe issue of multiple equilibria Ihat we discuss in Appendix A. Even so. we at least mention one conclusion from that discussion: The problem of multiple equilibria is relatively small when we arc dealing with dominant strategy equilibrium. 18. In fact. the implementation of 1(' ) using r is also robusl to substantial relaxations of the hypothesis that agents maximize expected utility.
(23.C.3)
lor all (J;e e; and all O_le e_;. The ability to restrict our inquiry, without loss of generality, to the question of whether 1(') is truthfully implementable is a consequence of what is known as the revelation principle Jor dominant strategies. Proposition 23.C.1: (The Revelation Principle for Dominant Strategies) Suppose that there exists a mechanism r = (S" ...• SI' g(.)) that implements the social choice function f(·) in dominant strategies. Then f(') is truthfully implementable in dominant strategies. Proof: If r = (S" ... , SJo g(.» implements 1(') in dominant strategies, then there exists a profile of strategies s·(·) = (sr(·), ... , s1 (.» such that g(s·(O» = J(O) for all and, for all i and all 0, e e"
°
u,(g(s:(O,), L,), 0,) 2: u,(g(J.. L,), 0,)
for all §, e S, and all i and all 0, e e ..
sr (.).
J(.) is implementable. we need, in principle, to consider all possible mechanisms.
1111 P L ElliE N TAT ION
L,
u,(g(sr(O,), s!,(O_,», 0,) 2: u,(g(snb,), s!,(O_,», 0,)
for all (J, e e, and all all i and all 0, e e"
°_, °
(23.C.4)
e S_,. Condition (23.C.4) implies, in particular, that for all
e
e_,. Since g(s·(O»
(23.C.S)
= J(O) for all 0, (23.C.S) means that, for
u,(J(O.. 0_,), 0,) 2: u,(J(b.. 8_,), 0,)
for all (J, e e, and all _I e e _,. But, this is precisely condition (23.C.3), the condition for 1(') to be truthfully implementable in dominant strategies. • The intuitive idea behind the revelation principle for dominant strategies can be put as follows: Suppose that the indirect mechanism r = (S" . . , ,Sh g(.» implements J(.) in dominant strategies, and that in this indirect mechanism each agent i finds playing s~ (0,) when his type is 0, better than playing any other 5, e S, for any choices e S_, by agentsj ¥ i. Now consider altering this mechanism simply by introducing a mediator who says to each agent i: "You tell me your type, and when you say your type is 0" I will play s~(O,) for you." Clearly, if sr(O,) is agent i's optimal choice for each 0, r- e; in the initial mechanism r for any strategies chosen by the other agents. then agent i will find telling the truth to be a dominant strategy in this new scheme. But this means that we have found a way to truthfully implement 1(')' The implication of the revelation principle is that to identify the set of social choice functions that are implementable in dominant strategies, we need only identify those that are truthfully implementable. In principle, for any 1('), this is just a matter of checking the inequalities (23.C.3).
s_,
871
872
--------------------------------------------------------------CHAPTER
23:
INCENTIVEI
AND
MECHANI8M
DEIIGN
SEC T ION
2 3 . C:
0 C MIN A H T
& T RAT E G Y
IMP L E MEN TAT
ION
The inequalities (23.C.3), which are necessary and sufficient for a social choice function f(·) to be truthfully implementable in dominant strategies, can be usefully thought of in terms of a certain weak preference reversal property. In particular, consider any agent i and any pair of possible types for j, and 8';. If truth telling is a dominant strategy for agent i, then for any O_lee_ 1 we must have
e;
uM(O;, 0_,), 0;)
~
uM(Oj, 0_,), 0;)
uM(O;', 0_,), OJ)
~
u,(f(Oj, 0_,), OJ).
/.(0;,8_,) must lie
in shaded set ~.(O:)
°_,)
L,(x, 0,) =
{z e X: u,(x, 0,)
0; 0,
Figure 23.C.2 depicts a change in some agent i's type from to in an exchange setting in which agent i's preferences satisfy the single-crossing properry that we discussed in Sections 13.C and 14.C. In the figure. we denote agent i's allocation in outcome f(O,. O2 ) by J,(O" O2 ), According to Proposition 23.C.2. h(O;'. 0_,) must lie in the shaded region of the figure if truth telling is to be a dominant strategy for agent i. Thus. the characterization in Proposition 23.C.2 can be seen as a multiperson extension of the truth-telling constraints that we encountered in Section 14.C (here they must hold for every possible 0 _, e 0 _,). In the remainder of this section we explore in more detail the characteristics of social choice functions that can be truthfully implemented in dominant strategies.
~ u,(z. 0,»).
Using this lower contour set we get the characterization of the set of social choice functions that can be truthfully implemented in dominant strategies that is given in Proposition 23.C.2.
The Gibbard-Satterthwaite Theorem
Propoaltlon 23.C.2: The social choice function f(') is truthfully implementable in dominant strategies if and only if for all i. all 0_; e u,(f(O). 0,) for all i (recall that no two alternatives can be indifferent). Because f(0) = X. there exists a O' E 0 such that flO') = y. Now choose a vector of types 0" E 0 such that, for all i, u,(y, 0;') > u;(f(O),O~) > u,(z, Oil for all z >F- flO). y. (Remember that all preferences in 9 are possible.) Since L,(y, 0;) c L,(y, Oil for all i, monotonicity implies that flO") = y. But. since L,(f(O), 0,) c L,(f(O). IJ~) for all i. monotonicity also implies that f(O") = flO): a contradiction because y >F- flO). Hence. f(') must be ex post efficient.
Step 3: A social clooice function f(·) tloar is monotonic and ex post efficient is necessarily dictatorial. Step 3 follows directly from Proposition 21.E.1. Together, steps I to 3 establish the result. _ It should be noted that the conclusion of Proposition 23.C.3 does not follow if
X contains two clements. For example. in this case, a majority voting social choice function (sec Section 21.E) is both nondictatorial and truthfully implementable in dominant strategies (Exercise 23.C.2). Note also that when .iff, = .'1' for all i, any ex post efficient social choice function muse have f(0) = X (verify this in Exercise 23.C.3). Thus, the Gibbard-Satterthwaite theorem tells us that when 91, = ? for all i, and X contains more than two elements. the only ex post efficient social choice functions that are truthfully implementable in dominant strategies are dictatorial social choice functions. Given this negative conclusion, if we are to have any hope of implementing desirable social choice functions, we must either weaken the demands of our implementation concept by accepting implementation by means of less robust equilibrium notions (such as Bayesian Nash equilibria) or we must focus on more restricted environments. In the remainder of this section, we follow the latter course, studying the possibilities for implementing desirable social choice functions in dominant strategies when preferences take a quasilinear form. Section 23.D explores the former possibility: It studies implementation in Bayesian Nash equilibria. Proposition 23.C.3 is readily extended in two ways. First. the resuh's conclusion still follows whenever dt; contains [j' (the set of all rational preference relations having the property that no two alternatives are indifferent). and so it extends to environments in which individual indifference is possible. This is stated formally in Corollary 23.C.1. Corollary 23.C.1: Suppose that X is finite and contains at least three elements. that iJ' c: iii, for all i, and that ((0) = X. Then the sociat choice function f(') is truthfully implementable in dominant strategies if and only if it is dictatorial. Proof: It is again immediate that a dictalorial social choice function is truthfully implementable. We now show that under the stated hypotheses f(·) must be dictatorial if it is truthfully implementable. An implication of Proposition 23.C.3 is that there must be an agent h such that /(0) E {x EX: u,(. ",(z, 0;') > ",(x, Oil for all x;. (f(IJ'), .}; and (iii) "/(" OJ) > "1(f(IJ'), OJ) > ",(x, OJ) for all x;. (f(O'), z}. Consider the profile of types (O~, 0'" ... ,0;). By Proposition 23.C.2, we must have flO') E L,(f(Oj, 0'" ... , 0;), O~), and so it must be that I(~, 0'" .. ,' 11;) ~ f(lJ'). The same argument can be applied iteratively for all i '" Ito show that f(~, ... , 1Ii-" 0;) = f(O')· Next, note that (by Proposition 23.C.2) we must have f(O~, ... , OJ_I' 11;) E LI(f(O"), OJ). Hence, f(O") E (t, flO')}. But (by Proposition 23.C.2) we must also have f(O") E LI(j(O;, ... , 0; -,,0;),0;), and since ",(z, 0;) > ",(f(O'), 0',) this means we cannot have flO") = z. Hence, flO") = flO'). But, since ",(z, OJ) > ",(f(O'), 0;,), this contradicts agent J being a dictator whenever ;::,(0,) E 9' for all i . •
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SEC T ION
23•
c:
0 0 .. I NAN T
S T RAT E G Y
I .. P L E .. E N TAT ION
877
of alternatives is therefore'o
x=
{(k,'" ... ,I,): k e K,I,E R for all i, and
L I, ~ OJ.
Note that this environment encompasses the cases studied in Examples 23.B.3 and 23.B.4: Example 23.el: A Public Projecl. We can fit a generalized version of the public project setting of Example 23.8.3 into the framework outlined above. To do so, let K contain the possible levels of a public project (e.g., if K = (0, I), then either the project is "not done" or "done") and denote by elk) the cost of project level k E K. Suppose that v,(k, 0,) is agent i's gross benefit from project level k and that, in the absence of any other transfers, projects will be financed through equal contribution [i.e., each agent i will pay the amount c(k)/I).l' Then, we can write agent i's nel benefit from project level k when his type is 0, as v,(k, 0,) = v,(k, 0,) - (c(k)//). The t,'s are now transfers over and above the payments e(k)/I . •
As our second extension, we can derive a related dictatorship result for social choice
functions whose image fee) is smaller than X. We first offer Definition 23.C.6. Deflnttlon 23.C.6: The social choice function f(') is dictatorial on set X c X if there exists an age"nt i such that. for all a = (a" ... ,0,) E e, flO) E (x E X: u,(x, 0,) '" u,(y, 0,) for all YEX}.
Example n.e2: AI/ocalion of a Single Unil of an Indivisible Private Good. Consider the environment described in Example 23.B.4 in which an indivisible unit of a private good is to be allocated to one of I agents. Here the "project choice" k = (y, •... , y,) represents the allocation of the private good and K = {(y" ...• Y/): y, E {O, I} for all i and :L.v, = I}. Agent i's valuation function takes the form v,(k, 0,) = (), y,. •
This weaker notion of dictatorship requires only that f(·) select one of the dictator's most preferred ahernatives in X, rather than in X. Corollary 23.C.2: Suppose that X is finite, that the number of elements in fee) is at least three, and that 9' c!il, for all i = 1, ... ,I. Then f(·) is truthfully implementable In dominant strategies if and only if it is dictatorial on the set fee).
A social choice function in this quasilinear environment takes the form f(·) = (k(-), 1 1(-), ••. ,1,(') where, for all 0 e 0. k(O) e K and L,I,(O) ~ O. Note that if the social choice function f(·) is ex post efficient then, for all 0 e 0, k(O) must satisfy
Proof: It is immediate that f(·) is truthfully implementable if it is dictatorial on the sct I(e), and so we now show that under the stated hypotheses f(·) must be dictatorial on sct f(9). If f: e .... X is truthfully implementable in dominant strategies when the sct of alternatives is X, then the social choice function /: 9 .... f(9) which has /(0) = f(8) for all 0 E 8 is truthfully implementable in dominant strategies when the sct of alternatives is f(8). By Corollary 23.C.I, /(.) must be dictatorial. Hence, f(·) is dictatorial on the set 1(9). •
,
L
/
L
v,(k(O), 0,) ~
i-1
v,(k, ()I)
for all k e K.
(23.C.7)
1-1
We begin with a result that identifies a class of social choice functions that satisfy (23.C.7) and that are truthfully implementable in dominant strategies.
The implication flowing from Corollary 23.c.2 is therefore this: When !ii, c 9' for all i, the set of social choice functions which have an image that contains at least three elements and which are truthfully implementable in dominant strategies is exactly the sct of social choice functions that can be implemented (indirectly) by restricting the sct of possible choioes to some subset X c X and assigning a single agent i to choose frcom within this set.
Proposition 23.C.4: Let k*(') be a function satisfying (23.C.7). The social choice function f(') = (k*('), t , (·), ... , til')) is truthfully implementable in dominant strategies if, for all i = 1, ... , I, t,(O) =
[.L. ,,,,,
vj(k*(O), OJ)]
+ hi(O-i)'
(23.C.8)
where h,{-) is an arbitrary function of O-i'
Quasilinear Environments: Groves-Clarke Mechanisms
Proof: If truth is not a dominant strategy for some agent i, then there exist 0" and 0_, such that
In this subsection we focus on the special, but much studied, class of environments in which agents have quasilinear preferences. In particular. an alternative is now a vector x = (k,I" ... ,1/), where k is an element of a finite sct K, to be called the "project choice," and I, e R is a transfer of a numeraire commodity ("money") to agent i. Agent i's utility function takes the quasilinear form u,(x,O,) = v,(k, 0,)
v,(k'(b" 0 _,),0,)
b,.
+ 1,(0,.0_,) > v,(k'(O" 6_,), 0,) + 1,(0" 0_,).
20. Observe that X is not a compact set. This explains what might appear as a small paradox: in this setting, there arc no dictatorial social choice runctions because any agent i. when allowed to pick his best ahernative in X, faces no bound on how much money he can extract from the other
+ (m, + I,),
where m, is agent i's endowment of the numeraire. We assume that we are dealing with a closed system in which the I agents have no outside source of financing. The sei
agents.
21. NOlhing we do depends on this choice tor the "base" method of contribution.
,
J
s
878
c HAP TEA
2 3:
INC E N T lYE SAN D
M E C HAN IS M
Substituting from (23.C.S) for I,(~" 0_ 1) and 1,(0" 0_ 1), this implies that I
L v;(k*(~"
I
L vj(k*(O), OJ),
0_ 1), OJ) >
j O. type 01 will strictly prefer to falsely report that he is type 0, when the other agents' types are 0_,. To sec this. note first that k·(o;. 0_,) = k·(O,. 0_,) since setting k = k·(O,. 0_,) maximizes v.(k. 0;) + LI'" vl(k. 01)' Thus. truth teIling being a dominant strategy requires that v,(k·(O,. 0_,). 01)
+ 1,(01. 0_,) ~ v,(k·(O,.
°
_,).0,)
+ 1,(0" 0_,).
or. substituting. from (23.C.9) and (23.C.IO).
But by the logic of part (i). h,(D:.
t
+ h,(o;. 0_,)
t
+ h,(b,.
~
h,(O,. 0_.).
°_,) = h,(O,. °_,) because k·(o;. °_,) = k·(O,. °_,). This gives
°_,)
~ h,(O,.
°_,).
(23.C.ll)
By hypothesis we have h(O,.O_,) > h(b,.o_,). and so (23.C.ll) must be violated for smaIl enough t > O. This completes the proor. _ Thus. when all possible functions v,(·) can arise for some 8, EO,. the only social choice functions satisfying (23.C.7) that are truthfully implementable in dominant strategies are those in the Groves class. Groves mechanisms and budgel balance Up to this point. we have studied whether we can implement in dominant strategies a social choice function that always results in an efficient choice of k [one satisfying (23.C.7)]. But ex post efficiency also requires that none of the numeraire be wasted. that is. that we satisfy the budgel balance condilion: LI,(8)=0 forall8E0.
,
(23.C.12)
We now briefly explore when fully ex post efficient social choice functions [those satisfying bOlh (23.C.7) and (23.C.l2)] can be truthfully implemented in dominant strategies. Unfortunately. in many cases it is impossible to truthfully implement fully ex post efficient social choioe functions in dominant strategies. For example. the result [due to Green and Laffont (1979)] in Proposition 23.C.6. whose proof we omit, shows that if the set of possible types for each agent is sufficiently rich. then no social choice functions that are truthfully implementable in dominant strategies are ex post efficient.2> Proposition 23.C.6: Suppose that for each agent i = 1 •...• I. {vk .8j ): 8j E OJ} = 1'"; that is. every possible valuation function from K to R arises for some 8j E OJ. Then there is no social choice function ((.) = (k·(·). t,(·) •...• t,(·» that is truthfully implementable in dominant strategies and is ex post effiCient. that is. that satisfies (23.C.7) and (23.C.12). Thus. under the hypotheses of Proposition 23.C.6. the presence of private information means that the 1 agents must either accept some waste of the numeraire
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SEC T ION
DO MIN ANT
& T RAT E G Y
IMP L E MEN TAT ION
[I.e.• have L, 1,(8) < 0 for some O. as in the Clarke mechanism] or give up on always having an efficient project selection [i.e .• have a project selection k(8) that does not satisfy (23.C.7) for some OJ. One special case in which a more positive result does obtain arises when there is at least one agent whose preferences are known. For notational purposes. let this agent be denoted "agent 0". and let there still be 1 agents, denoted i = I •... • 1. whose preferences are private information (so that we are now lelling there be 1 + I agents in total). The simplest case of this phenomenon. of course. occurs when agent 0 has no preferences over the project choice k. that is. when his preferences are u,(x) = "'0 + 10, We saw one example of this kind in Example 23.B.4 when we considered auction sellings (agent 0 is then the seller). Another example arises in the case of a public project when the project affects only a subset of the agents in the economy (so that agent 0 represents all of the other agents in the economy). When there is such an agent. ex post efficiency of the social choice function still requires that (23.C.7) be satisfied; but now ex post efficiency is compatible with any transfer functions c,(·) •...• 1,(') for the 1 agents with private information, as long as we set coCO) = - L, .. o 1,(0) for all 8. That is. in this (I + I)-agent selling. the Groves mechanisms identified in Proposition 23.C.4 (in which only agents i = I •... • 1 announce their types) are ex post efficient as long as we set the transfer of agent 0 to be coCO) = - L" ot,(O) for all O. In essence. the presence of an "outside" agent 0 who has no private information allows us to break the budget balance condition for those agents who do have privately observed types. We should offer. however. one immediate caveat to this seemingly positive result: Up to this point. we have not worried about whether agents will find it in their interest to participate in the mechanism. As we will see in Section 23.E. when participation is voluntary. it may be that no ex post efficient social choice function is implementable in dominant strategies even when such an outside agent exists. The differentiable case
It is common in applications to encounter cases in which K = R. the v. 2 see Laffont and Maskin (1980) and Exercise 23.C.10]. By (23.C.13). for all 0= (0,,0,), we have
E. ,[ui(g(si(Oi)' S~i(O_i»' Oil I0;] ~ E.,[U;(g(§i' S~i(O-i»' 0ill0;]
Definition 23.0.2: The mechanism r = (5" ... ,Sj. g(.» implements the social choice function f(') in Bayesian Nash equilibrium if there is a BayesIan Nash equilibrium of r, s·(·) = (si(·) • ... ,sr such that g(s'(O» = flO) for all E e.
00,
Thus, for all
°
ok
(.».
()',,(O)
_iJ'_v-,-,,(k_O-,:(O~)._O-,-,,)_ ilk_'(_O) o_k_·(_O) o~
il~
+ ov,(k·(O). 0,) o_'_kO_(O_)
00,
ft
OO,il~
(23.C.16)
Delinltlon 23.0.3: The social choice lunction f(') is truthfully implementable in Bayesian Nash equilibrium (or Bayesian incentive compatible) il Sf(Oi) = 0i for al\ O,E i and i = 1•... , J is a Bayesian Nash equilibrium 01 the direct revelation mechanism r = (e" .... e" f(·)). That is. illor all i = 1•... , J and all
and
il',,(O)
00,00,
_o'_v",,(k-:-·~(O-,-,)._O~,) _ok_'(_O) _ok_·(_O) + ov,(kO(O), 0,) _o'_k'_(_O) ok' 00, 00, ok 00,00,'
e
(23.C.17)
O,E
O'V,(kO(O). 0,) ok'
+ o'v,(k·(O). 0,)] ok'(O) ilk'(O) ok'
00,
00,
ei•
(23.D.l)
If we have budget balance. then ,,(0) = -,,(8) for all 0, and so we must have 0',,(8)/00,00, = -0',,(0)/00,00,. But this would imply, by adding (23.C.16) and (23.C.17), and using (23.C.15). that
[
°
As with implementation in dominant strategies (see Section 23.C). we will see that a social choice function is Bayesian implementable if and only if it is truthfully implementable in the sense given in Definition 23.0.3.
00,
= (0" 0,),
00,00,
23.D Bayesian Implementation
for al\ 5i E Si'
ilv,(k·(O). 0,) ok'(O)
00,
IMPLEMENTATION
e
and
0,,(0)
BAYESIAN
Definition 23.0.1: The strategy profile s·(·) = (si(·) •.... si(·» Is a Bayesian Nash equilibrium of mechanism r = (5" ...• 5,. g('» if, for all i and all O,E i •
ov,(k·(O). 0,) ok·(O) ok
23.0:
In this section. we study implementation in Bayesian Nash equilibrium. 17 Throughout we follow the notation introduced in Section 23.B: The vector of agents' types 0=(0, •...• 0,) is drawn from set e=e, x· .. xe, according to probability density 0, we
and
Proposition 23.0.2 shows that to identify all Bayesian incentive compatible social choice functions in the linear setting. we can proceed as follows: First identify which functions k( .) lead every agent i's expected benefit function v,(') to be nondecreasing. Then. for each such function. identify the expected transfer functions I, (.)..... I, (.) that satisfy condition (23.0.12) of the proposition. Substituting for U,(·). these are precisely the expected transfer functions that satisfy. for i = I, ...• I.
1,(0,) = 1M,)
+ ~,v,(~,) - O,v,(O,) +
f" v,(s) ds ~,
for some constant l,{~,). Finally. choose any set of transfer functions (tl(O) •. ..• t,(O» such that E, ,[t,(O,. 0_,)) = 1,(0,) for all 0,. In general. there are many such functions t,(· •. ); one. for example. is simply t,(O,. = 1,(0,)." We now illustrate one implication of this characterization result for the auction setting introduced in Example 23.B.4. Some further implications of Proposition 23.0.2 are derived in Sections 23.E and 23.F.
°_,)
Auctions: tile revenue equivalence theorem Thus, - (0.)
v"
~ U,(O,) - U,(II,) ~ -(0)
b,- 0,
v"
•
(23.0.13)
Expression (23.0.13) immediately implies that v,(') must be nondecreasing (recall that we have taken b, > 0,). In addition, letting 8, -+ 0, in (23.D.13) implies that for all 0, we have
Ui(O,) = v,(O,) and so
U,(O,) =
U,{~,)
+
f"
v,(s) ds
for all 0,.
!,
(ii) Sufficiency. Consider any 0, and 0, and suppose without loss of generality that 0,. If (23.0.11) and (23.0.12) hold, then
0, >
u;(o,) - U,(O,) =
', f" f'·
_ v,(s) ds
;:: _ v,(8,) ds I.
= (0, - O,)v,(8,).
32. Observe Ihat the agent's preferences here over his expected benefit jj, and expected transfer I, satisfy the single-crossing property that played a prominent role in Sections I3.C and 14.C.
889
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Let us consider again the auction setting introduced in Example 23.B.4: Agent 0 is the seller of an indivisible object from which he derives no value. and agents 1•... , 1 are potential buyers. 34 It will be convenient. however, to generalize the set of possible alternatives relative to those considered in Example 23.B.4 by allowing for a random assignment of the object. Thus. we now take y,(O) to be buyer i's probability of getting the object when the vector of announced types is = (0" ...• 0,). Buyer i's expected utility when the profile of types for the 1 buyers is 0= (0" ... ,0,) is then 0, y,(O) + tote). Note that buyer i is risk neutral with respect to lotteries both over transfers and over the allocation of the good. This setting corresponds in the framework studied in Proposition 23.0.2 to the case where we take k=(yl .... ,y,). K={(yl .... ,y,):y,E[O.I] for all i = 1, ... ,1 and L, y, ~ I}, and v,{k) = Yo' Thus, to apply Proposition 23.0.2 we can write VitO,) = y,(O,), where y,(8,) = E._.[y,(O" 0_,)] is the probability that i gets the object conditional on announcing his type to be 8, when agents j .;. i announce their types truthfully, and U,(O,) = 0d,(Od + 1,(0,).
°
33. However. if we wish the social choice function f(') = (k('), ,,(.)•...• 1,('» to satisfy some further properties, such as budget balance. only a subset (possibly an empty one) of the transfer functions generating the expected transfer functions (t.(8.). ...• t,(8,» may have these properties. 34. We note that our assumption that the seller in an auction setting derives no value rrom the
object is not necessary for the revenue equivalence tbeorem. (As we shall see, tbe result characterizes the expected revenues generated for the seller in different auctions. and so is valid for any utility function that the seUer might bave.) In the absence of tbis assumption, however, tbe seller in an auction will generally care about more than just the expected revenue he receives.
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We can now establish a remarkable result, known as the revenue equivalence theorem. 3 >
--- --
(.v,(O,lO, -
t
MO,) dO,) -
dO,J - U,(Q.).
I -,(0.> ,(O'»)J( ,., .n j(III») dO, ... dll,J , (23.0.16)
By inspection of (23.0.16). we see that any two Bayesian incentive compatible social choice functions that generate the same functions (y,(O) •...• y,(II» and the same values of (U,(Q,) •...• U,(Q,» generate the same expected revenue for the seller. _ As an example of the application of Proposition 23.0.3, consider the equilibria of the first-price and second-price sealed-bid auctions that we identified in Examples 23.8.5 and 23.B.6 (where the buyers' valuations were independently drawn from the uniform distribution on [0. I]). for these equilibria. the conditions of the revenue equivalence theorem are satisfied: in both auctions the buyer with the highest valuation always gets the good and a buyer with a zero valuation has an expected utility of zero. Thus, the revenue equivalence theorem tells us that the seller receives exactly the same level of expected revenue in these equilibria of the two auctions (you can confirm this fact in Exercise 23.0.3). More generally. it can be shown that in any symmetric auction setting (i.e .• one where the buyers' valuations are independently drawn from identical distributions). the conditions of the revenue equivalence theorem will be met for any Bayesian Nash equilibrium of the first-price sealed-bid auction and the (dominant strategy) equilibrium of the second-price sealed-bid auction (see Exercise 23.0.4 for a consideration of symmetric equilibria in these settings). We can conclude from Proposition 23.0.3. therefore. that in any such setting the first-price and second-price sealed-bid auctions generate exactly the same revenue for the seller.
U,(Q,).
Moreover. integration by parts implies that
.v,(s) dS),(O,) dO, =
j(Oj»)dll, ...
I
,.L, U,(Q,).
.v,(s) dS),(O,) dO,
.v,(s) dS) ,(0,) dO,J -
n
[ y,(O, •...• 0,)(0, [ f.O,' ... fO' 0, ,=,
r
[r r r(r (r (r =
E[ -1.(0,)] =
(23.0.15)
E[ -1,(0)) = E•.[ - 1,(0,))
(.v,(0;) 11, - U,(Q,) -
P A A TIC I PAT ION
Thus. the seller's expected revenue is equal to
Proof: By the revelation principle. we know that the social choice function that is (indirectly) implemented by the equilibrium of any auction procedure must be Bayesian incentive compatible. Thus, we can establish the result by showing that if two Bayesian incentive compatible social choice functions in this auction setting have the same functions (y,(O) •...• y,(O)) and the same values of (U,(Q,), ... , U,(Q,» then they generate the same expected revenue for the seller. To show this. we derive an expression for the seller's expected revenue from an arbitrary Bayesian incentive compatible mechanism. Note. first. that the seller's expected revenue is equal to :Lf., E[ -1,(0». Now.
r
23. E:
or. equivalently.
6' ... f6' y,(O, •...• 0,)(0, - I - ,(0,»)( [ f,,~, ,(0,) ,.
Proposition 23.0.3: (The Revenue Equivalence Theorem) Consider an auction setting with I risk-neutral buyers. in which buyer i's valuation Is drawn from an interval [Q;.6;] with Q; ;I< 6; and a strictly positive density 11>;(·) > 0, and in which buyers' Iypes are statistically independent. Suppose that a given pair of Bayesian Nash equilibria of two different auction procedures are such that for every buyer i: (i) For each possible realization of (0 ...• 0,). buyer i has an identical probability " of getting the good in the two auctions; and (ii) Buyer i has the same expected utility level in the two auctions when his valuation for the object is at its lowest possible level. Then these equilibria of the two auctions generate the same expected revenue for the seller.
=
5 ECT, 0 N
23.E Participation Constraints
.v,(O,),(O,) dO,)
In Sections 23. B to 23.0. we have studied the constraints that the presence of private information puts on the set of implementable social choice functions. Our analysis up to this point. however, has assumed implicitly that each agent i has no choice but to participate in any mechanism chosen by the mechanism designer. That is. agent i's discretion was limited to choosing his optimal actions within those allowed by the mechanism. In many applications. however. agents' participation in the mechanism is voluntary. As a result. the social choice function that is to be implemented by a mechanism must not only be incentive compatible but must also satisfy certain participation (or individual rationality) constraints if it is to be successfully implemented. In this section. we provide a brief discussion of these additional
Substituting. we see that (23.0.14)
35. Versions of the revenue equivalence theorem have been derived by many authors; see McAfee and McMillan (1987) and Milgrom (1987) for references as well as for a further discussion of the result.
.l
CON 5 T R A I N T 5
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--------------------------------------------------------------constraints on the set of implementable social choice functions. By way of motivating our study, Example 23.E.I provides a simple illustration of how the presence of participation constraints may limit the set of social choice functions that can be successfully implemented. Example 23.E.l: Participation Constraints in Public Project Choice. Consider the following simple example of public project choice (recall our initial discussion of public project choice in Example 23.S.3). A decision must be made whether to do a given project or not, so that K = {O, I}. There are two agents, I and 2. For each agent i, 0, = {~, 8}, so that each agent either has a valuation of~, or a valuation of iI. We shall assume that 8 > 2~ > O. The cost of the project is c E (2~, 8). Suppose that we want to implement a social choice function having an ex post efficient project choice; that is, one that has k·(III' (1 2) = 1 if either 0 1 or 112 is equal to 8, and P(OI' (1 2) = 0 if II, = 112 =~. In the absence of the need to insure voluntary participation, we know from Section 23.C that we can implement some such social choice function in dominant strategies using a Groves scheme. Suppose, however, that each agent has the option of withdrawing from the mechanism at any time (perhaps by withdrawing from the group), and that, if he does, he will not enjoy the benefits of the project if it is done, but will also avoid paying any monetary transfers. Can we implement a social choice function that achieves voluntary participation and that has an ex post efficient project choice?l6 The answer is "no." To see this, note that if agent I can withdraw at any time, then to insure his participation it must be that I,(~, 8) ~ -~. That is, it must be that whenever his valuation for the project is ~, he pays no more than ~ toward the cost of the project. Now consider what agent l's transfer must be when both agents announce that they have valuation 8: If truth telling is to be a dominant strategy, then t, (0, 0) must satisfy
Ok·(O, 0)
+ 1, (0, 0) ~
Ok·(~, 0)
+ t,(~, 0),
or, substituting for k·(O, 0) and k·(~, ii),
0+ t,(O, 8) ~ 8 + t,(~, 8). Since t,(~, 0) ~ -~, this implies that 1, (0, 8) ~ -~. Thus, we conclude that agent I must not make a contribution toward the cost of the project that exceeds ~ when (II" ( 2 ) = (0,8). Moreover, by symmetry, we have exactly the same constraint for agent 2's transfer when (II" (1 2 ) = (8, 8), namely, t , (8, 8) ~ -~. Hence, t,(O, 0) + t 2 (O, 0) ~ -2~. But if this is so, then because 2~ < c, the feasibility condition t,(O, 8) + t 2(8, 8) S -c cannot be satisfied. We conclude,therefore, that it is impossible to implement a social choice function with an ex post efficient project choice when the agents can withdraw from the mechanism at any time. Note also that the presence of an "outside agent" (say "agent 0") who does not care about the project decision does not help at all here when that agent can also withdraw from the mechanism at any time. This is because, to insure this agent's participation, his transfer to(lI .. (1 2 ) must be nonnegative for every realization of
36. Note that any social choice funclion thai fails to have both agents participate is necessarily e., post inefficient because one of the agents is excluded from the benefits of Ihe project.
--
IECTION
U.E:
PARTICIPATION
(0 (1 2 ), In particular, w.,e ..must h~v: t o(8, 82 ~ 0, and so we must fail to satisfy the " feasibility condition to(II, II) + t,(II, II) + t 2(O, II) s -c. • As a general matter, we can distinguish among three stages at which participation constraints may be relevant in any particular application. First, as in Example 23.E.I, an agent i may be able to withdraw from the mechanism at the ex post slage that arises after the agents have announced their types and an outcome in X has been chosen. Formally, suppose that agent i can receive a utility of ii,(II,) by withdrawing from the mechanism when his type is 11,.37 Then, to insure agent i's participation, we must satisfy the ex post participation (or individual rationality) constraints 3 ' (23.E.I) In other circumstances, agent i may only be able to withdraw from the mechanism at the interim stage that arises after the agents have each learned their type but before they have chosen their actions in the mechanism. Letting U,(O.lfl = E•. ,[u,(f(II" 11_,), II,) IlIa denote agent i's interim expected utility from social choice function f( . ) when his type is II" agent i will participate in a mechanism that implements social choice function f(') when he is of type II, if and only if U,(O,lfl is not less than ii,(O,). Thus, interim participation (or individual rationality) constraints for agent i require that for all 0,.
(23.E.2)
In still other cases, agent i might only be able to refuse to participate at the ex ante stage that arises before the agents learn their types. Letting U,(f) = E.,[U,(O;l fl] = E[u,(f(II.. II _,),0,)] denote agent i's ex ante expected utility from a mechanism that implements social choice function f('), the ex ante participation (or individual rationality) constraint for agent i is U,(f) ~ E.,[ii,(II,)].
(23.E.3)
Participation constraints are of the ex ante variety when the agents can agree to be bound by the mechanism prior to learning their types. When, instead, agents know their types prior to the time at which they can agree to be bound by the mechanism, we face interim participation constraints. 39 Finally, if there is no way to bind the
37. We assume that agent i's utility from withdrawal depends only on his own Iype. 38. We assume throughout that it is always optimal to insure that each agent is always willing to participate. In fact, however, there is no loss of generality from assuming this: When agents can "not participate: any outcome that can arise when some subset I' of the I agents does not participate, say x', should be included in the set X. Because we can always have the mechanism select x' in the circumstanoes when this subset of agents would have refused to participate, if the set X is defined appropriately we can always replicate the outcome of any mechanism that causes non participation with a mechanism in which all agents are always willing to participate. 39. Recall that the assumption in a Bayesian game that types are drawn from a commOn prior density is often merely a modeling device for how agents form beliefs about each others' types (see Section 8.E). That is, for analytical purposes we may be representing a setting in which agents' types are already determined but are only privately observed by assuming that there has been a prior random draw of types from a commonly known distribution; but there may not actually be any such prior stage at which the agents could possibly interact.
CONITIIAINTI
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DEIIGN
---------------------------------------------------------------------~
agents to the assigned outcomes of the mechanism against their will, then we face ex post participation constraints.'· Note that if f(·) satisfies (23.E.I), then it satisfies (23.E.2); and, in turn, if it satisfies (23.E.2), then it satisfies (23.EJ). However, the reverse is not true. Thus, the constraints imposed by voluntary participation are most severe when agents can withdraw at the ex post stage, and least severe when they can withdraw only at the ex ante stage. In summary, when agents' types are privately observed, the set of social choice functions that can be successfully implemented are those that satisfy not only the conditions identified in Sections 23.C and 23.0 for incentive compatibility (in, respectively, either a dominant strategy or Bayesian sense, depending on the equilibrium concept we employ) but also any participation constraints that are relevant in the environment under study. In the remainder of this section, we illustrate further the limitations on the set of implementable social choice functions that may be caused by participation constraints by studying the important Myerson-Salterthwaite theorem [due to Myerson and Satterthwaite (1983)].
SECTION
23.£:
PARTlelPATION
news: Whenever gains from trade are possible, but are not certain,'1 there is no ex post efficient social choice function that is both Bayesian incentive compatible and satisfies these interim participation constraints. Thus, under the conditions of the theorem, the presence of both private information and voluntary participation implies that it is impossible to achieve ex post efficiency. (For an illustration of the result for specific functional forms, see Exercise 23.E.7.)
Proposition 23.E.1: (The Myerson-Satterthwaite Theorem) Consider a bilateral trade setting in which the buyer and seller are risk neutral, the valuations 0, and O2 are independently drawn from the intervals [~" 8,] c R and [~2' 62] c R with strictly positive densities, and (Q" 0,) ("\ (~2' 62) '" 0. Then there is no Bayesian incentive compatible social choice function that is ex post efficient and gives every buyer type and every seiler type nonnegative expected gains from participation.
Proof: The argument consists of two steps:
Slep I: In any Bayesian incentive compatible and interim individually rational social choice function f(·) = [y,(.), y,{- ),t,(' ),t,(-)) in which y,(O" 0,) + y,(O" 0,) = I and t,(O" 0,) + t,(O" 0,) = 0, we musl have
f fi, i,
The Myerson-Satterthwaite Theorem
Consider again the bilateral trade setting introduced in Example 23.B.4. Agent I is the seller of an indivisible object and has a valuation for the object that lies in the interval 0 1 = [~I' 61] c R; agent 2 is the buyer and has a valuation that lies in 0, = [~" 6,] c R. The two valuations are statistically independent, and 0, has distribution function 0 for iii]. We let y,(O) denote the probability that agent i receives the good all 0i E given types 9 = (0 1,9,), and so agent i's expected utility given 0 is O,y,(O) + t,(O) (we normalize mi = 0 for all i). The expected externality mechanism studied in Section 23.0 shows that in this setting we can Bayesian implement an ex post efficient social choice function (or what, in this environment, we might call a "trading rule"). A problem arises with the expected externality mechanism, however, when trade is voluntary. In this case, every type of buyer and seller must have nonnegative expected gains from trade if he is to participate. In particular, if a seller of type 01 is to participate in a mechanism that implements social choice function f('), that is, if participation in the mechanism is to be individually rational for this type of seller, it must be that UI(O.!f> ~ 0" because this seller can achieve an expected utility of 0, by not participating in the mechanism and simply consuming the good. Likewise, a buyer of type 0, can always earn zero by refusing to participate, and so we must have U,(O,If> ~ O. Unfortunately, these interim participation constraints are not satisfied in the expected externality mechanism (you are asked to verify this in Exercise 23.E.I). The Myerson-Satterthwaite Theorem tells us the following disappointing piece of
"
"
.
.
y,(O"0,)
[(
I - ,(0,») (0, + ,(0,»)] - 4>,(0,)4>,(0,) dO, dO, ~ O.
0, - - - - - 4>,(0,)
4>,(0,)
~~
To see this, note first that the same argument that leads to (23.0.15) can be applied here to give [throughout the proof we suppress the argument f in U,(0.lf) and simply write U,(O,)]:
mi'
fi, fi, y,(O,. 0,)(0, -
£[ -i,(O,)) = [
"
I - ,(0'»)4>,(0,)4>,(0,) dO, dO,] -
"
.
4>,(0,)
.
Also, because (23.D.12) implies that
V,(Q,)
=
VItO,) -
i, ii, f ~I
U,(~,). (23.E.5)
y,(O" 0,)4>,(0,) dO, dO"
!l
condition (23.0.15) also implies that
-
[ fi, fi, y,(O"O,) (,(0,») 0, + - - 4>,(0,)4>,(0,) dO, dO, ] - V,(O,).
£[ -1,(0,)] =
" .
"
4>,(0,)
.
(23.E.6)
Then, since J',(O,. 0,) = I - y,(O,. 0,) we have
-
£[-1,(0,))=
-
[ f" f" (0 , + ,(0.1) - - 4>,(O,)tJ>,(O,)dO,dO, ] 4>,(0,) ,,~,
[ f fi, i' ~,
"
»)
y,(O,.o,) ( II, + - '(0 - ' 4>,(0,)4>,(0,) dO, dll, 4>,(0,)
e,) '"
]-
U,(ii,).
41. That is, whenever (~" 8,)" (q" 0 (or equivalently, 8, > ~, and 8, > ~,), so that for some realizations of 0 = (0 1.8 2 ) there are gains from trade but for others there are not.
40. For example, if the mechanism can lead an agent into bankruptcy, the provisions of bankruptcy law provide an elfective lower bound on ex post utilities.
J
CONSTRAINT.
895
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23:
INCENTIVES
.. NO
"ECH .. NIS ..
--- --
DESIGN
But dO, dO,J [ f•i,, 'fi,, (0, + ,(0'»),(0,),(0,) ,(0,)
= [
fi, [0,,(0,) + ,(0,)] dO,J !,
= [0,,(0,)]::
=0,. Thus. £[ -1,(0,)]
0, - [
=
fi, fi, y,(O" 0,)(0, + ,(0'»),(0,),(0,) dO, dO,J ,(0,) "
.
U,(O,).
h
.
~~
Now. the fact that 1,(0" 0,) + 1,(0,,0,) = 0 implies that £[ -1,(0,,0,)] + £[ -1,(0,,0,)] = So. adding (23.E.5) and (23.E.7) we see that
o.
[U,(O,) - 0,] + U,(q,) =
f"i, fi," }"(O,, 0,)[(0, -
I - ,(0,») _ (0,
+ ,(O'»)J,(O,),(O,) dO, dO,.
,(0,)
,(0,)
But individual rationality implies that U,(O,) ~ 0, and U,(q,) ~ 0, which establishes (23.E.4). Seep 2: Cundilion (23.£.4) cannol be salisfied },,(O,. 0,) = 0 whenever 0, < 0,.
if
y,(O" 0,) = I whenever 0, > 0, and
Suppose it were. Then the left-hand side of (23.E.4) could be written as
f f";'I.' .• i
,
"
"
oI
[(0, - I - ,(0,) - 0,),(0,) - ,(O,)J,(O,) dO, dO, ,(0,) =
=
f"[( f"[(
J";,,.,.',I ,(0,) dO,
"
I - ,(0,) ) 0, - - - - - 0, ,(0,) ,(0,) "
!,
I - ,(0,) . - ) ,(Mm{O"O,}) . O,-----Mm{O"O,} ,(O,)dO, ,(0,)
= -
f
~ [I
J
- ,(0,)],(0,) dO,
+ f~ . [(0, - 0,),(0,) + (,(0,) -
i,
= -
f"
I)] dO,
'1
~l
I
[I - ,(0,)],(0,) dO,
+ [(0, - 0,)(,(0,) - 1)],:
23.F:
OPTI .... L
... VEII .. N
whether trade will occur and at what price.·' By the revelation principle, we know that the social choice function that is indirectly implemented in a Bayesian Nash equilibrium·) of such a mechanism must be Bayesian incentive compatible. Moreover, since participation is voluntary, this social choice function f(·) must satisfy the interim individual rationality constraints that UI (8 1 1f) ~ 8 1 for all 8 1 and U,(O,1 f) ;;>: 0 for all 0,. Thus, the Myerson-Satterthwaite theorem tells us that, under its assumptions, no voluntary trading institution can have a Bayesian Nash equilibrium that leads to an ex post efficient outcome for all realizations of the buyer's and seller's valuations.
:
1
23,F Optimal Bayesian Mechanisms In Sections 23.B to 23.E we have been concerned with the identification of implementable social choice functions in environments characterized by incomplete information about agents' preferences. In this section, we shift our focus to the welfare evaluaton of implementable social choice functions. We begin by developing several welfare criteria that extend the notion of Pareto efficiency that we have used throughout the book in the context of economies with complete information to these incomplete information settings. With these welfare notions in hand, we then discuss several examples that illustrate the characterization of optimal social choice functions (and, by implication, the optimal direct revelation mechanisms that implement them). We restrict our focus throughout this section to implementation in Bayesian Nash equilibria, discussed in detail in Section 23.0. Unless otherwise noted, we also adopt the assumptions and notation of Section 23.0. Good sources for further reading on the subject of this section are Holmstrom and Myerson (1983), Myerson (1991), and Fudenberg and Tirole (1991). For economies in which agents' preferences are known with certainty, the concept of Pareto efficiency (or Pareto optimality) provides a minimal test that any welfare optimal outcome x E X should pass: There should be no other feasible outcome:i E X with the property that some agents are strictly better off with outcome :i than with outcome x, and no agent is worse off. The extension of this welfare test to social choice functions in settings of incomplete information should read something like the following: The social choice function 1(') is efficient if it is feasible and if there is no other feasible social choice function that makes some agents strictly better off, and no agents worse off.
completes the argument. •
SECTION
q,
and
q,
< 0,. This contradicts (23.E.4) and
Recalling the revelation principle for Bayesian Nash equilibrium (Proposition 23.0.1), the implication of the Myerson-Satterthwaite theorem can be put as follows: Consider any voluntary trading institution that regulates trade between the buyer and the seller. This includes, for example, any bargaining process in which the parties can make offers and counteroffers to each other, as well as any arbitration mechanism in which the parties tell a third party their types and this third party then decides
To operationalize this idea, however, we need to be more specific about two things: First, what exactly do we mean by a social choice function being ~feasible',? Second,
42. Strictly speaking, for a direct application of Proposition 23.E.I, the date of delivery and consumption of the good must be fixed (so the bargaining processes studied in Appendix A of Chapter 9 would not count). But through a suitable reinterpretation Proposition 23.E.l can be applied to settings in which trade may take place over real time, where not only delivery of the good matters but also the rime of delivery (sec Exercise 23.E.4 for details). 43. And, hence, in any perfect Bayesian or sequential equilibrium (sec Section 9.C).
.. ECH .. NIIN.
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precisely what do we mean when we say that no other feasible social choice function .. makes some agents strictly better off, and no agent worse off"? Let us consider the first of these issues. The identification of the set of feasible social choice functions when agents' preferences are private information has been discussed extensively in Sections 23.0 and 23.E. Suppose that we define the set FBtc = {f: El .... X: f(·) is Bayesian incentive compatible}.
(23.F.I)
The elements of set Fatc in any particular application are the social choice functions that satisfy condition (23.0.1), the condition that assures that there is a Bayesian Nash equilibrium of the direct revelation mechanism r = (0 , .. , Elto f(·» in which " truth telling is each agent's equilibrium strategy. Likewise, following the discussion in Section 23.E, we can also define the set
FI /( = {f: El .... X: f(·) is individually rational}.
(23.F.2)
The set FIR contains those social choice functions that satisfy whichever of the three types of individual rationality (or participation) constraints (23.E.I)-(23.E.3) are relevant in the application being studied. If no individual rationality constraints are relevant (i.e., if agents' participation is not voluntary), then we simply have FtR = {f: El .... X}, the set of all possible social choice functions. The content of our discussion in Sections 23.0 and 23.E is therefore that the set of feasible social choice functions in environments in which agents' types are private information is precisely F* = FBIC n F,/(. Following Myerson (1991), we call this the incentive feasible set to emphasize that it is the set of feasible social choice functions when, because of incomplete information, incentive compatibility conditions must be satisfied. Now consider the second issue: What do we mean when we say that no other feasible social choice function would "make some agents strictly better off, and no agents worse off"? The critical issue here has to do with the liming of our welfare analysis. In particular, is the welfare analysis occurring before the agents (privately) learn their types, or after? The former amounts to a welfare analysis conducted at what we called in Section 23.E the ex ante stage (the point in time at which agents have not yet learned their types); the latter corresponds to what we called in Section 23.E the interim stage (the point in time after each agent has learned his type, but before the agents' types are publicly revealed). To formally define the different welfare criteria that arise in these two cases, let us once again denote by U.(0.lf) agent i's expected utility from social choice function f(·) conditional on being of type 0 Also let U,(f) = E.,[ U,(O;lf)] denote agent i's ex ante expected utility from social "choice function f(·). We can now state Definitions 23.F.1 and 23.F.2. Definition 23.F.1: Given any set of feasible social choice functions F, the social choice function f(') E F is ex ante efficient in F if there is no I (.) E F having the property that Vi(/) ~ V;(f) for all i = 1, ... , I, and Vi(/) > Vi(f) for some i. Definition 23.F.2: Given any set of feasible social choice functions F, the social choice function f(·) E F is interim efficient in F if there is no 1(·) E F having the property that Vi(Oil/) ~ Vi(Oilf) for all OiE Eli and all i= 1, . .. ,1, and Vi(Oill) > Vi(Oilf) for some i and 0iE Eli' The motivation for the ex ante efficiency test is straightforward: If agents have not yet learned their types, then when comparing two feasible social choice functions
---
IECTION
U.F:
O,TIMAL
.AYEIIAN
MECHANIIMI
899
---------------------------------------------------------------------we should evaluate each agent's well-being using his expected utility over all of his possible types. However, when our welfare analysis occurs after agents have (privately) learned their types, things are a bit trickier. Although the agents each know their types, we-as outsiders-do not know them. Thus, the appropriate notion for us to adopt in saying that one social choice function !(.) welfare dominates another social choice function f(·) is that !(.) makes every possible type of every agent at least as well off as does f('), and makes some type of some agent strictly better off. This leads to the concept of interim efficiency given in Definition 23.F.2. Proposition 23.F.1 compares these two notions of efficiency. Proposition 23.F.1: Given any set of feasible social choice functions F, if the social choice function f(') E F is ex ante efficient in F. then It is also Interim efficient in F. Proof: Suppose that f(·) is ex ante efficient in F but is not interim efficient in F. Then there exists an !(')EF such that U,(O.!!)~ U,(O.!f) for all O,EEl, and all i = I, ... , I, and U,(O.!!) > V,(O.! f) for some i and 0, EEl,. But since, for all i, U.(f) = E.,[U,(O.!f)] and V,(!) = E.,[U,(O,I!)], it follows that V/(!) ~ V/(f) for all j = I, ... , I, and V,(!) > V,(f) for some i, contradicting the hypothesis that f(·) is ex ante efficient in F . • The ex ante efficiency concept is more demanding than is interim efficiency (and so fewer social choice functions f(·) pass the ex ante efficiency test) because a social choice function !(.) can raise every agent's ex ante expected utility relative to the social choice function f(') even though !(.) may lead some type of some agent i to have a lower expected utility than he does with f(·). Putting together the elements developed above, we conclude that when agents' types are already determined at the time we are conducting our welfare analysis, the proper notion of efficiency of a social choice function in an environment with incomplete information is interim efficiency in F·, the set of Bayesian incentive. compatible and individually rational social choice functions.·· On the other hand, if our analysis is conducted prior to agents learning their types, then the proper notion of efficiency is ex ante efficiency in F*.·5 These two notions are often called simply ex ante incentive efficiency and interim incentive efficiency [the terminology is due to Holmstrom and Myerson (1983)], where the modifier "incentive" is meant to convey the point that the set F* is being used.·' These two welfare notions differ from the ex post efficiency criterion introduced in Definition 23.B.2. To see their relationship to it more clearly, Definition 23.F.3 44. These cases often correspond to situations in which our assumption that the agents' types are drawn from a known prior distribution is being used merely as a device to model agents' beliefs about each others' types, as described in Section 8.E, rather than as a description of any actual prior lime at which the agents could interact or our welfare analysis might have been done. 45. This case often arises in contracting problems when, It the time of contracting, the agents anlicipate thatlhey will later come to acquire privale information aboutthcir types. Then the natural welfare standard to use in comparing different contracts (i.e, different mechanisms) is the ex ante criterion. The principal-agent model studied in Section 14.C and Example 23.F.1 below is an example along Ihese lines. 46. However, since the relevant individual rationality constraints Vlry from one application to another, it is usually clearer to describe precisely the sct F within which efficiency is being evaluated.
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develops the ex post efficiency notion in a manner that parallels Definitions 23.F.I and 23.F.2. Dellnltlon 23.F.3: Given any set of feasible social choice functions F. the social choice function f (.) e F is ex post efficient in F if there is no 1(·) e F having the property that Uj(/(O).Oj)~Uj(f(O).Oj) for all i=I ..... 1 and all Oee. and u,(i(O), OJ) > uj(f(O), OJ) for some i and 0 e e. The ex post efficiency test in Definition 23.F.3 conducts its welfare evaluation at the ex post stage at which all agents' information has been publicly revealed. Using this definition. we see that a social choice function f(·) is ex post efficient in the sense of Definition 23.B.2 if and only if it is ex post efficient in the sense of Definition 23.F.3 when we take F = {f: e ... X}. Note that the criterion of ex post efficiency in (f: e -+ X }, or more generally. of ex post efficiency in when individual rationality constraints are present. ignores issues of incentive compatibility. As a result, it is appropriate as a welfare criterion only if agents' types are in fact publicly observable. Because F* c Fill' allocations that are ex ante or interim incentive efficient need not be ex post efficient in this sense. Indeed. the Myerson-Satterthwaite theorem (Proposition 23.E.1) provides an illustration of this phenomenon for the bilateral trade setting: under its assumptions. no element of F * is ex post efficient. Examples 23.F.1 to 23.F.3 provide further illustrations. (For one way in which the notion of ex post efficiency is nevertheless still of interest in settings with privately observed types. see Exercise 23.F.1.) Note also that even in settings in which agents' types are public information, the use of ex post efficiency in F,. as our welfare criterion is appropriate only when agents' types are already determined. When our welfare analysis instead occurs prior to agents learning their types, the appropriate notion is instead the stronger criterion that f(·) be ex ante efficient in These two notions are sometimes called ex post classical efficiency and ex ante classical efficiency [again, the terminology is due to Holmstrom and Myerson (1983)] to indicate that no incentive constraints are involved in defining the feasible set of social choice functions.
F,.
F,.,
In the remainder of this section we study three examples in which we characterize welfare optimal social choice functions. In Examples 23.F.I and 23.F.2. it is supposed that one agent who receives no private information chooses a mechanism to maximize his expected utility subject to both incentive compatibility constraints and interim individual rationality constraints for the other agents. These two examples therefore amount to a characterization of one particular interim incentive efficient mechanism. In Example 23.F.3, we provide a full characterization of the sets of interim and ex ante incentive efficient social choice functions for a simple setting of bilateral trade with adverse selection. Example 23.F.l: A Principal-A gem Problem with Hidden Information. In Section 14.C we studied principal-agent problems with hidden information for the case in which the agent has two possible types. Here we consider the case where the agent may have a continuum of types. Recall from Section 14.C that in the principal-agent problem with hidden information. the principal faces the problem of designing an optimal (i.e .• payoff maximizing) contract for an agent who will come to possess private information. In doing so, the principal faces both incentive constraints and
--- ---
SEC T ION
23. F:
0 P TIM ALB AYE. I A N i l E C HAN I • II •
a reservation utility constraint for the agent. Recall also from Section 14.C that, in the limiting case in which the agent is infinitely risk averse, the agent must be guaranteed his reservation utility for each possible type he may come to have, and so this contracting problem is identical to the contracting problem that would arise if the agent already knew his type at the time of contracting. Here we shall set things up directly in these terms, assuming that the agent already possesses this information when contracting occurs. With this formulation. the principal's optimal contract can be viewed as implementing one particular interim incentive efficient social choice function. (When the agent actually does not know his type at the time of contracting and is infinitely risk averse. then this social choice function is also ex ante incentive efficient.) To introduce our notation. we suppose that the agent (individual I) may take some observable action e e R. (his "effort" or "task "level) and receives a monetary payment from the principal of t,. The agent's type is drawn from the interval [~, 6]. where ~ < 6 < O. according to the distribution function (') which has an associated density function (.) that is strictly positive on [Q,6]. We assume that this distribution satisfies the property that [0 - «I - (0»/(0))] is nondecreasing in IJ.47 The agent's Bernoulli utility function when his type is IJ is u,(e. 11.0) = I, + IJg(e). where g(.) is a differentiable function with g(O) = O. g(e) > 0 for e > O. g'(O) = O. g'(e) > 0 for e > O. and g"(') > 0; that is. Og(e) represents the agent's disutility of effort (recall that 0 < 0). with higher effort levels leading to an increasing level of disutility to the agent. Note that a larger (i.e., less negative) level of 0 lowers. at any level of e. both the agent's total level of dis utility and his marginal dis utility from any increase in e. As noted above, we suppose that the agent must be guaranteed an expected utility level of at least ii for each possible type he may have. The principal (individual 0) has no private information. His Bernoulli utility function is uo(e. (0 ) = v(e) + 10, where 10 is his net transfer and v(.) is a differentiable function satisfying v'(·) > 0 and v"(·) < O. A contract between the principal and the agent can be viewed as specifying a mechanism in the sense we have used throughout this chapter. By the revelation principle for Bayesian Nash equilibrium (Proposition 23.D.I). the equilibrium outcome induced by such a contract, formally a social choice function that maps each possible agent type into effort and transfer levels, can always be duplicated using a direct revelation mechanism that induces truth telling. Thus, the principal can confine his search for an optimal contract to the set of Bayesian incentive compatible social choice functions f(·) = (e('), 10 (' ),1 , ('» that give the agent an expected utility of at least ii for every possible value of O. In what follows. we shall (without loss of generality) restrict attention in our search for the principal's optimal contract to contracts that have to(lJ) = - t,(IJ) for all IJ (i.e., that involve no waste ofnumeraire). The principal's problem can therefore be stated as Max
E[v(e(O» - t,(IJ)]
f(·)"'(~{·I.II(·»
S.t. f(·) is Bayesian incentive compatible and individually rational. 47. For a discussion of how the analysis changes when this assumption is not satisfied, see Fudenberg and Tirol. (1991).
901
902
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DEIIGN
SECTION
The present model falls into the class of models with linear utility studied in Section 23.0 [specifically, in the notation of Proposition 23.0.2, k = e, vl{k) = g{e), and VI{O) = g{e{O» here]. Letting VI{O) = II{O) + Og{e{O» denote the agent's utility if his type is 0 and he tells the truth, Proposition 23.0.2 can be used to restate the principal's problem in terms of choosing the functions e{') and VI (.) to solve E[v{e{O»
Max
+ Og{e{O» -
UI{O)]
(23.FJ)
r
s.t. (i) e{') is nondecreasing (ii) UI{O) = UI @ +
g(e(s» ds for all 0
u for all O~
(iii) UI(O);::
Constraints (i) and (ii) are the necessary and sufficient conditions for the principal's contract to be Bayesian incentive compatible, adapted from Proposition 23.0.2 [constraint (i) follows because g(.) is increasing in e], while constraint (iii) is the agent's individual rationality constraint. Note first that if constraint (ii) is satisfied, then constraint (iii) will be satisfied if and only if UI{Q) ~ u. As a result, we can replace constraint (iii) with (iii')
UI{Q)
~
U.
Next, substituting for UI(O) in the objective function from constraint (ii), and then integrating by parts in a fashion similar to the steps leading to (23.0.14), problem (23.FJ) can be restated as
[f
Max
S.t.
{v{e(o))
+ g{e{O»(0 - I ~(:O»)}4>{O)dOJ - VI @
(23.Fo4)
(i) e{') is nondecreasing (iii') UI{Q) ~ u.
It is now immediate from (23.Fo4) that in any solution we must in fact have = U. Thus, we can write the principal's problem as one of choosing e(') to solve
UI (Q)
Max n')
[f
{v{e(o»
+ g(e(O»
(0 - I
~(~~0»)}4>(0) dO] -
u
(23.F.5)
S.t. (i) e{') is nondecreasing. Suppose for the moment that we can ignore constraint (i). Then the optimal function e(') must satisfy the first·order condition·· v'(e{O»
+ g'(e(l!» ( 0 -
1- cI>{O») -- = 0 4>{I!)
for alll!.
U.F:
OPTI .... L
."'EII .. N MECHANISMS
903
----------------------------------------------------------------------
(23.F.6)
But note that, under our assumption that [I! - {(I - cl>{1!))/4>(0))] is nondecreasing in 0, the implicit function theorem applied to (23.F.6) tells us that any solution e{') to this relaxed problem must in fact be nondecreasing. Thus, (23.F.6) characterizes the solution to the principal's actual problem (see Section M.K of the Mathematical Appendix). The optimal VI {') [and, hence, ll{')] is then calculated from constraint (ii) of (23.FJ) using this optimal e{') and the fact that UI{Q) = ii.
It is interesting to compare this solution with the optimal contract for the case in which the agent's type is observable. This contract solves
Max
E[v{e{I!» - tl{I!)]
s.t. tl{l!)
+ Og(e(O)) ~ u for
all O.
Hence, the optimal task level in this complete information contract is the level e*(O) that satisfies, for all 0, v'(e*(O» + g'(e*{O))1! = O. Note that e*(O) is the level that arises in any ex post (classically) efficient social choice function. In contrast, the principal's optimal e(') when I! is private information is such that v'{e(l!» + g'(e(I!»I!{> 0 at alll!.< 0, = 0 at 0 = O. We see then that e(l!) < eO(O) for all I! < 0, and e(0) = eO{O). This is a version of the same result that we saw for the two· type case in Section 14.C. In the optimal contract, the type of agent with the lowest disutility from effort (here type 0; in Section 14.C, type 0,,) takes an ex post efficient action, while all other types have their effort levels distorted downward. The reason is also the same: doing so helps reduce the amount the agent's utility exceeds his reservation utility for types 0 > Q (his so·called information rents). To see this point heuristically, suppose that starting with some function e(') we lower e(iI) by an amount de < 0 for some type ~ E (Q, 0) and lower this type's transfer to keep his utility unchanged!' The decrease in the transfer paid to type a is ag'{e(O» de, while the direct effect on the principal is v'{e{~» de. At the same time, according to constraint (ii), this change in e{~) lowers the utility level, and hence the transfer, that must be given to all types 0 > ~ by exactly g'{e{~» de. The expected value of this reduction in the transfers paid to these types is -(I - cI>{O»g'(e{O»de. If the original e{') is an optimum, the sum of the first two changes in the principal's profits (those for type ~) weighted by the density of type ii, [v'(e(a» + ag'(e(a))] 4>(0) de, plus the reduction in payments to types 0 > a, (I - cI>(a»g'(e(a» de, must equal zero. This gives exactly (23.F.6). • Example 23.F.2: Optimal Auctions. We consider again the auction setting introduced in Example 23.Bo4. Here we determine the optimal auction for the seller of an indivisible object (agent 0) when there are I buyers, indexed by i = I, ... , I. Each buyer has a Bernoulli utility function 0, y,(I!) + t,{I!), where y,{O) is the probability that agent i gets the good when the agents' types are I! = (Ol' ... ' 0,). In addition, each buyer i's type is independently drawn according to the distribution function ,(') on [Q" 0;] c: Ii! with Q, -F 0, and associated density 4>,(') that is strictly positive on [Q" 0;]. We assume also that, for i = I •...• 1, 1- cI>,(O,)
0,----4>,{O,)
is nondecreasing in 0,. ~o 49. We say "heuristically" because to do this rigorously we need to perform this reduction in
48. It can be shown that under our assumptions, the optimal contract is interior, that is, has e(0) > 0 for (almost) all O.
e over an interval of types and then take limits.
50. For a discussion of the case in which this assumption is not met, see Myerson (1981).
904
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--
DES'GN
A social choice function in this environment is a function f(·) = (YO('), •.. ,y,(.), 10 (') ••••• 1,(')) having the properties that. for all fi e e. Yi(fi) e [0, I] for all i, :[,,,0 y,(fi) = I - yo(fi), and lo(fi) = - L,o'o 1.(fi).51 The seller wishes to choose the Bayesian incentive compatible social choice function that maximizes his expected revenue E,[lo(fi)) = - E.[L.o'o I,(fi)] but faces the interim individual rationality constraints that V,(fi,) = fi,Y.(fi,) + 1,(0,)
ij·
f
[y,(fi,)O, - V,(fi;)]
»)][ n' ]
cIl,{fi ... , 0,) ( 0i -I-- O. For small enough £ > 0, this alternative social choice function satisfies ali of the constraints of problem (23.F.17) (note that it satisfies ill < I because, by step 2, y~ < I; check the other constraints too). Moreover, it yields a larger value of the objective function of (23.F.17) than (Y!, I!, y~, I~)-a contradiction. This establishes step 3. Slep 4: satisfied.
tH
I_
I_ =
BAYEIIAN
= y~ = I). But we have
already noted above that no such social choice function is incentive feasible (i.e., is an element of F·). Slep 3: equality).
OPTIMAL
Figure 23.F.1 (len)
Step 2: Any solulion 10 problem (23.F.Il) has YII < I; Ihal is, in any inlerim incenlive efficienl social choice function, trade does nOI occur with certainly when Ihe good is of high qualilY. (y!,I!, y~,I~) is ex post (classically) efficient (i.e., it has Y!
40y_ + ",_
23. F:
UI = 24
+ IH
- 24YH'
U2 = 8
+ 26YH
- I H·
In Figure 23.F.3, for an arbitrary point 0 > OL and OL + OH > O. The agents' valuations are statistically independent with Prob (0, = 8Ll = ). E (0, I) for I = 1,2. In the expected externality mechanism, each agent i announces his valuation and agent;os transfer when the announced types are (8" 8,) has the form t,(8" 8_,) = E._.[O_;k*(O;,IL;)] + h,(0_,),wherek*(8 I ,O,1 = OifO I = 0, = 8L ,and k*(8 .. 8,) = I otherwise. As we saw in Section 23.D, in one Bayesian Nash equilibrium of this mechanism, truth telling is each agent's equilibrium strategy. But this truth-telling equilibrium is not the only Bayesian Nash equilibrium. In particular, there is an equilibrium in which both agents always claim that 0H is their type. To see this, consider agent j's optimal 56_ The "strong" terminology is not standard; in the literature it is not uncommon, for example. to see the strong implementation concept simply referred to as .. implementation ....
912
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------------------------------------------------------------------------------strategy if agent - j will always announce 0H' Whichever announcement agent j makes, the project is done. Thus, regardless of his actual type, agent j's direct benefit (i.e., O,k·(O" 62 » is not affected by his announcement (it is OL if he is of type OL' and 0H if he is of type 0H)' It follows that agent j's optimal strategy is to make an announcement that maximizes his expected transfer. Now, agent j's expected transfer if he announces 0" is ().OL + (1 - )')OH) + h,(O,,), whereas if he announces OL his expected transfer is (I - )')OH + h,(OH)' Hence, agent j will prefer to announce 0" regardless of his type if agent - j is doing the same. It follows that both agents always announcing 0", and the project consequently always being done, constitutes a second Bayesian Nash equilibrium of this mechanism. _
A P PEN D I X
B:
IMP L E MEN TAT tON
IN
EN V I RON MEN T'
WIT H
COM P LET E
observe 0, there is still an implementation problem: Because no outsider (such as a court) will observe 0, the agents cannot write an enforceable ex ante agreement saying that they will choose outcome f(O) when agents' preferences are O. Rather, they can only agree to participate in some mechanism in which equilibrium play yields f(O) if 0 is realized. 57 Note that a complete information setting can be viewed as a special case of the general environment considered throughout this chapter, in which the probability density 0,; =0 if 0, SO,.
(e) Prove that if [(.) satisfies IPM, and if 91, c i? for all i. then J(.) is monotonic.
+ O,)y,(O,. 0,).
1,(0,.0,) = -!(O,
(d) Prove that if [(.) is monotonic, and 91, = i? for all i. then J(') satisfies IPM.
+ O,)y,(O,. 0,).
Suppose that the seller truthfully reveals his type for all 0, worthwhile to reveal his type? Interpret.
E
[0. I]. Will the buyer find it
23.B.3 8 Show that b,(O,) = 0, for all 0, E [0. I] is a weakly dominant strategy for each agent i in the second-price sealed-bid auction. 23.B.4 c Consider a bilateral trade setting (see Example 23.B.4) in which both the seller's and the buyer's types are drawn independently from the uniform distribution on [0. I]. (a) Consider the double auclion mechanism in which the seller (agent I) and buyer (agent 2) each submit a sealed bid. bi ~ O. If b, ~ b,. the seller keeps the good and no monetary transfer is made; while if b, > b" the buyer gets the good and pays the seller the amount ! O. Show that the continuously differentiable soeial choice function f(-) - (k(·),I,(·), ... , 1,('» is truthfully implementable in dominant strategies if and only if, for all i = I, ... ,I,
23.0.4 c Consider a first-price sealed-bid auction with I symmetric buyers. Each buyer's valuation is independently drawn from the interval [~, 0] according to the strictly positive density 1/>(').
k(O) is nondecreasing in 0,
and
(a) Show that the buyer's equilibrium bid function is nondecreasing in his type. 1,(0,,0 _,) = 1,(0,.0 _.) _ -
f."
(b) Argue that in any symmetric equilibrium (b·(·), . .. ,b*('» there can be no interval of types (0',0"),0' '" 0", such that b'(O) is the same for all 0 E (0', 0"). Conclude that b'(') must therefore be strictly increasing.
ov.(k(s, 0 -.), s) ok(s, 0 -,) ds.
!,
ok
Os
23.C.IO" (8. Holmstrom) Consider the quasilinear environment studied in Section 23.C. Let
(c) Argue, using the revenue equivalence theorem, that any symmetric equilibrium of such an auction must yield the seller the same expected revenue as in the (dominant strategy) equilibrium of the second-price sealed-bid auction.
k'(') denote any project decision rule that satisfies (23.C.7). Also define the function V'(O) = L:, v,(P(O), 0,).
(a) Prove that there exists an ex post efficient soeial choice function [i.e., one that satisfies condition (23.C.7) and the budget balance condition (23.C. I 2)] that is truthfully implementable in dominant strategies if and only if the function V·(·) can be written as V'(O) = L:, 1'1(0 _,) for some functions V,(·)" .. , 1'1(.) having the property that V,(.) depends only on 0_, for all i.
=
23.0.Sc For the same assumptions as in Exercise 23.0.4. consider a sealed-bid all-pay auction in which every buyer submits a bid, the highest bidder receives the good, and every buyer pays the seller the amount of his bid regardless of whelher he wins. Argue that any symmetric equilibrium of this auction also yields the seller the same expected revenue as the sealed-bid second-price auction. [Hint: Follow similar steps as in Exercise 23.0.4.]
=
(b) Use the result in part (a) to show that when 1=3, K R, 9, R. for all i, and v,(k, 0,) = O,k - (!)k' for all i an ex posteffieient social choice function exists that is truthfully implementable in dominant strategies. (This result extends to any I > 2.)
23,0,6c Suppose that I symmetric individuals wish to acquire the single remaining ticket to a concert. The ticket office opens at 9 a.m. on Monday. Each individual must decide what time to go to get on line: the first individual to get on line will get the ticket. An individual who waits I hours incurs a (monetary equivalent) disutility of {JI. Suppose also that an individual showing up after the first one can go home immediately and so incurs no waiting cost. Individual i's value of receiving the ticket is 0., and each individual's 0, is independently drawn from a uniform distribution on [0, I]. What is the expected value of the number of hours that the first individual in line will wait? [Hint: Note the analogy to a first-price sealed-bid auction and use the revenue equivalence theorem.] How does this vary when {J doubles? When I doubles?
(c) Now suppose that the v,(k, 0,) functions are such that V·(·) is an I-times continuously differentiable function. Argue that a necessary condition for an ex post efficient social choice function to exist is that, at all 0, o'V'(O) =
o.
00, ... 00, (In fact, this is a sufficient condition as well.) (d) Usc the result in (c) to verify that, under the assumptions made in the small type discussion at the end of Section 23.C, when I = 2 no ex post effieient social choice function is truthfully implementable in dominant strategies.
23.E.I" Consider again a bilateral trade selling in which each 0, (i = 1,2) is independently drawn from a uniform distribution on [0, I]. Suppose now that by refusing to participate in the mechanism a seller with valuation 0, receives expected utility 0, (he simply consumes the good). whereas a buyer with valuation 6, receives expected utility 0 (he simply consumes his endowment of the numeraire, which we have normalized to equal 0). Show that in the expected externality mechanism there is a type of buyer or seller who will strictly prefer not to participate.
23.C.II A Consider a quasilinear environment, but now suppose that each agent i has a Bernoulli utility function of the form u,(v,(k, 0,) + m, + I,) with u;(') > O. That is, preferences over certain outcomes take a quasilinear form, but risk preferences arc unrestricted. Verify that Proposition 23.C.4 is unaffected by this change. 23.0.1" [Based on an example in Myerson (1991)] A buyer and a seller arc bargaining over the sale of an indivisible good. The buyer's valuation is O. K 10. The seller's valuation takes one of two values: 0, e {O, 9}. Let I be the period in which trade occurs (I = 1,2, ... ) and let p be the price agreed. Both the buyer and the seller have discount factor lJ < I.
23.E.2A Argue that when the assumptions of Proposition 23.E.I hold in the bilaterai trade selling: (a) There is no social choice function 1(') that is dominant strategy incentive compatible and interim individually rational (i.e., that gives each agent i nonnegative gains from participation conditional on his type 0" for all 0,).
(a) What is the set X of alternatives in this selling? (b) Suppose that in a Bayesian Nash equilibrium of this bargaining process, trade occurs
...
921
922
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-----------------------------------------------------------------------------(b) There is no social choice function f(·) that is Bayesian incentive compatible and ex post individually rational [i.e., that gives each agent nonnegative gains from participation for every pair of types (9" 9,». 23.E.3" Show by means of an example that when the buyer and seller in a bilateral trade selling both have a discrete set of possible valuations, social choice functions may exist that are Bayesian incentive compatible, ex post efficient, and individually rational. [Hint: It suffices to let each have two possible types.] Conclude that the assumption of a strictly positive density is required for the Myerson-Sallerthwaite theorem. 23.E.4" A seller (i = I) and a buyer (i = 2) are bargaining over the sale of an indivisible good. Trade can occur at discrete periods t = 1,2, .... Both the buyer and the seller have discount factor ~ < I. The buyer's and seller's valuations are drawn independently with positive densities from [~" 0,] and [~" 0,], respectively. Assume that (Q" 0,)" (~" 0,) # 0. Note that in this selling ex post efficiency requires that trade occur in period I whenever 0, > 0" and that trade not occur whenever 0, > 0,. Use the Myerson-Sallerthwaite theorem to show that, in this selling with discounting, no voluntary trading process can achieve ex post efficiency. 23.E.S" Suppose there is a conrinuum of buyers and sellers (with quasilinear preferences). Each seller initially has one unit of an indivisible good and each buyer initially has none. A seller's valuation for consumption of the good is 0, e [Q" ii,], which is independently and identically drawn from distribution CI>,(') with associated strictly positive density 4>,('). A buyer's valuation f~om consumption of the good is 0, E [Q" 0,], which is independently and identically drawn from distribution CI>,(') with associated strictly positive density 4>,c-). (_) Characterize the trading rule in an ex post efficient social choice function. Which buyers and sellers end up with a unit of the good? (b) Exhibit a social choice function that has the trading rule you identified in (a), is Bayesian incentive compatible, and is individually rational. [Hint: Think of a "competitive" mechanism.] Conclude that the inefficiency identified in the Myerson-Sallerthwaite theorem goes away as the number of buyers and sellers grows large. [For a formal examination showing that, with a finite number of traders, the efficiency loss goes to zero as the number of traders grows large, see Gresik and Sallerthwaite (1989).] 23.E.6" Consider a bilateral trading selling in which both agents initially own one unit of a good. Each agent i's (i = 1,2) valuation per unit consumed of the good is 0,. Assume that 0, is independently drawn from a uniform distribution on [0, 1]. (a) Characterize the trading rule in an ex post efficient social choice function. (b) Consider the following mechanism: Each agent submits a bid; the highest bidder buys the other agent's unit of the good and pays him the amount of his bid. Derive a symmetric Bayesian Nash equilibrium of this mechanism. [Hine: Look for one in which an agent's bid is a linear function of his type.] (0) What is the social choice function that is implemented by this mechanism? Verify that it is Bayesian incentive compatible. Is it ex post efficient? Is it individually rational [which here requires that U,(O,) ~ 9, for all 9, and i = 1,2]? Intuitively, why is there a difference from the conclusion of the Myerson-Satterthwaite theorem? [See Cramton, Gibbons, and Klemperer (1987) for a formal analysis of these "partnership division" problems.]
23.E.7" Consider a bilateral trade setting in which the buyer's and seller's valuations are drawn independently from the uniform distribution on [0, I]. (0) Show that if f(·) is a Bayesian incentive compatible and interim individually rational
EXEIICIIEI923
----------------------------------------------------------------------------------social choice function that is ex post efficient, the sum of the buyer's and seller's expected utilities under f(·) cannot be less than 5/6. (b) Show that, in fact, there is no social choice function (whether Bayesian incentive compatible and interim individually rational or not) in which the sum of the buyer's and seller's expected utilities exceeds 2/3. 23.F.l c Consider the quasilinear setting studied in Sections 23.C and 23.D. Show that if the social choice function f(·) e F' is ex post classically efficient in FlO then it is both ex ante and interim incentive efficient in P. [From this fact, we see that if an ex post classically efficient social choice function can be implemented in a setting with privately observed types (i.e., if it is incentive feasible), then no other incentive feasible social choice function can welfare dominate it. Note, however, that there may be other ex ante or interim incentive efficient social choice functions that are not ex post efficient; for example, you can verify that in Example 23.F.1 there is an ex post classically efficient social choice function that is incentive feasible, but the particular interim incentive efficient social choice function derived in the example is not ex post efficient.] 23.F.2" [Based on Maskin and Riley (1984)] A monopolist seller produces a good with constant returns to scale at a cost of c > 0 per unit. The monopolist sells to a consumer whose preference for the product the monopolist cannot observe. A consumer of type 9 > 0 derives a utility of Ov(. 0 and VO(.) < O. The set of possible COnsumer types is [q,O] with > Q > 0, and the distribution of types is 4>('), with an associated strictly positive density function 4>(') > O. Assume that [9 - «I - 4>(0))/4>(9))] is nondecreasing in O. Characterize the monopolist's optimal selling mechanism to this consumer, assuming that a consumer of type 0 can always choose not to buy at all, thereby deriving a utility of O.
°
23.F.3 c An auction with a reserve price is an auction in which there is a minimum allowable bid. Suppose that in the auction selling of Example 23.F.2 the I buyers are symmetric and that Q = O. Argue that a second'price sealed·bid auction with a reserve price is an optimal auction in this case. What is the optimal reserve price? Can you think of a modified second·price sealed·bid auction that is optimal in the general (nonsymmetric) case? 23.F.4" Derive the optimal y,{') functions in the auction selling of Example 23.F.2 when the seller's valuation for the object is 00 > O. 23.F.5 B Suppose that a monopolist seller who has two potential buyers has a total of one divisible unit to sell; that is, production costs are zero up to one unit, and infinite beyond that. The demand function of buyer i is the decreasing function x,(p) for i = 1,2. The monopolist can name distinct prices for the two buyers. (a) Characterize the monopolist's optimal prices. (b) Relate your answer in (0) to the optimal auction derived in Example 23.F.2. [For more on this, see Bulow and Roberts (1989).] 23.F.6 c [Based on Baron and Myerson (1982)]. Consider the optimal regulatory scheme for a regulator of a monopolist who has known demand function x(p), with x'(p) < 0, and a privately observed constant marginal cost of production 9. The regulator can set the monopolist's price and can make a transfer from or to the monopolist, so the set of outcomes is X = {(p, I): p > 0 and Ie R}. The regulator must guarantee the monopolist a nonnegative profit regardless of his production costs to prevent the monopolist from shutting down. The monopolist's marginal cost 0 is drawn from [~, 0] with 8 > ~ > 0 according to the distribution function ('), which has an associated strictly positive density function 4>(') > O. Assume that
924
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MECHANISM
DESION
$(O)/IIJ(O) is nondecreasing in O. Denote a type-O monopolist's profit from outcome (p,l) by "(p, I, 0) = (p - O)x(p)
+ I.
-
EXERCISEI925
-------------------------------------------------------------------------------------------normalized valuation function, that is, a function such that v.(k o) = O. Suppose that kO(.) and the Groves transfers are calculated using these announcements. Does each agent have a unique (weakly) dominant strategy in this normalized Groves mechanism?
(a) Adapl the characterization in Proposition 23.0.2 to this application.
23.BB.IA Consider the dynamic mechanism in Example 23.BB.1.
(b) Suppose that the regulator wants to design a direct revelation regulatory scheme
(a) For each possible preference profile, write down its normal form and identify its Nash equilibria.
(p('), 1(')) that maximizes the expected value of a weighted sum of consumer and producer
surplus,
f-(I,~
xis) ds
(b) For each possible preference profile, identify this mechanism's subgame perfect Nash equilibria.
+ ",,(P(O), 1(0), 0),
where" < I. Characterize the regulator's optimal regulatory scheme. What if " :!: 1?
23.BB.28 Is a social choice function that is implementable in dominant strategies necessarily implementable in Nash equilibrium? What if we are interested in strong implementation instead?
23.F.7c [Based on Dana and Spier (1994)] Two firms, j = 1,2. compete for the right to produce in a given market. A social planner designs an optimal auction of production rights to maximize the expected value of social welfare as measured by
w=
l>J + S + (;' J
23.BB-3c Consider a setting of public project choice (see Example 23.8.3) in which K = {O, II· Let 0, denote agent i's benefit if the project is done (i.e., if k = 1); normalize the value from k = 0 to equal zero. Assume that El, = R. In this setting, the only mechanisms that involve an ex post efficient project choice are Groves mechanisms. Let kO(.) denote the project choice rule in such a mechanism. Also, suppose that I ~ 3. The transfers in a Groves mechanism are characterized by two properties:
I) ~:rJ' J
where I; denotes the transfer from firm j to the planner, S is consumer surplus, "; is the gross (pretransfer) profit of firmj, and;' > I is the shadow cost of public funds. The auction specifies transfers for each of the firms and a market structure; that is, it either awards neither firm production rights, awards only one firm production rights (thereby making that firm an unregulated monopolist), or gives production rights to both firms (thereby making them compete as unregulated duopolists). Each firm j privately observes its fixed cost of production OJ. The fixed cost levels 0, and 0, are independently distributed on [g,O] with continuously differentiable density function (') is increasing in O. The firms have common marginal cost c < 1 and produce a homogeneous product for which the market inverse demand function is p(x) = I - x (this is publicly known). If both firms are awarded production rights, they interact as Cournot competitors (see Section 12.C). Characterize the planner's optimal auction of production rights.
°
(i) if kO(O" _I) = kO(O;, 0_,), then I.{O" 0_,) = 1,(0;, 0_,); (ii) if kO(O" 0_,) = I and kO(O;, 0_,) = 0, then 1,(0,,0_,) - 1,(0;,0_,) = 2.J'" OJ. Which, if any, of these two properties must be satisfied by any Nash implementable social choice function that involves an ex post efficient project choice?
23.F.8 A Show that any ex post classically efficient social choice function in Example 23.F.3 has YL = y" = I. 23.F_9 8 Show that in the model of Example 23.F.3: (a) No feasible social choice function is ex post efficient. (b) In any feasible social choice function, y" S YL and I" S I L • (c) In any feasible social choice function, the expected gains from trade of a low-quality seller are at least as large as the expected gains from trade of a high-quality seller; that is, It - 20j't ~ I" - 40y". 23.F.10· Characterize the sets of interim and ex ante incentive efficient social choice functions in the model of Example 23.F.3 when trade is not voluntary for the seller (but it is voluntary for the buyer). 23.AA.1 • Reconsider Exercise 23.C.8. Exhibit a mechanism r = (S., . .. , S" g(')) that is not a direct revelation mechanism that truthfully implements f(') in dominant strategies and for which each agent has a unique (weakly) dominant strategy. 23.AA.2 8 Let K = {ko, k., ... , kN I be the set of possible projects and suppose that, for each agent i, {v,L 0,): Ole El,l = 'V, that is, that every possible valuation function from K to R arises for some 0, eEl,. Do players in a Groves mechanism have a unique (weakly) dominant strategy? Consider instead a mechanism in which each agent i is allowed to announce a
.....
Mathematical Appendix
SECTION
M.A:
MATRIX
NOTATION
FOR
particular, that if M = 1 (so that [(x) E R) then DI(x) is a 1 x N matrix; in fact VI(x) = [D [(x)]'. To avoid ambiguity, in some cases we write DJ(x) to indicate explicitly the variables with respect to which the function [(.) is being differentiated. For example, with this notation, if I: RNH .... RM is a function whose arguments are the vectors x E RN and y E RX, the matrix D.[(x, y) is the M x N matrix whose mn th entry is iJIm(x, y)/iJx •. Finally, for a real-valued differentiable function I: RN .... R, the Hessian matrix D2[(X) is the derivative matrix of the vector-valued gradient function V[(x); i.e., D2 [(x) = D[V[(x)). In the remainder of this section, we consider differentiable functions and examine how two well-known rules of calculus-the chain rule and the product rule---{;ome out in matrix notation. The Chain Rule
Suppose that g: RS .... RN and I: RN .... RAt are differentiable functions. The composite function [(g(')) is also differentiable. Consider any point x E RS. The chain rule allows us to evaluate the M x S derivative matrix of the composite function with respect to x, DJ(g(x» by matrix multiplication of the N x S derivative matrix of g('), Dg(x), and the M x N derivative matrix of I(') evaluated at g(x), that is, D I(y), where y = g(x). Specifically,
This appendix contains a quick and unsystematic review of some of the mathematical concepts and techniques used in the text. The formal results are quoted as "Theorems" and they are fairly rigorously stated. It seems useful in a technical appendix such as this to provide motivational remarks, examples, and general ideas for some proofs. This we often do under the label of the "Proof" of the mathematical theorem under discussion. Nonetheless, no rigor of any sort is intended here. Perhaps the heading "Discussion of Theorem" would be more accurate. It goes without saying that this appendix is no substitute for a more extensive and systematic, book-length, treatment. Good references for some or most of the material covered in this appendix, as well as for further background reading, are Simon and Blume (1993), Sydsaeter and Hammond (1994), Novshek (1993), Dixit (1990), Chang (1984), and Intriligator (1971).
D.[(g(x)) = D[(g(x» Dg(x).
(M.A.I)
The Product Rule
Here we simply provide a few illustrations. (i) Suppose that [: RN .... R has the form [(x) = g(x)h(x), where both g(.) and h(') are real-valued functions of the N variables x = (x" . .. ,XN) (so that g: RN .... R and h: RN .... R). Then the product rule of calculus tells us that D[(x) = g(x) Dh(x)
+ h(x) Dg(x).
(M.A.2)
which, transposing, can also be written as
M.A Matrix Notation for Derivatives
VI(x) = g(x) Vh(x)
+ h(x) Vg(x).
(ii) Suppose that J: RN .... R has the form I(x) = g(x)·h(x) where both g(.) and h(') are vector·valued functions which map the N variables x = (Xl' ... ' x N ) into RM. Then D[(x) = g(x)'Dh(x) + h(x)·Dg(x). (M.A.3)
We begin by reviewing some matters of notation. The first and most important is that formally and mathematically a "vector" in RN is a column. This applies to any vector; it does not matter, for example, if the vector represents quantities or prices. It applies also to the gradient vector VI(x) E RN of a function at a point x; this is the vector whose nth entry is the partial derivative with respect to the nth variable of the real-valued function I: RN .... R, evaluated at the point x ERN. Expositionally, however, because rows take less space to display, we typically describe vectors horizontally in the text, as in x = (Xl' ... ' XN)' But the rule has no exception: all vectors are mathematically columns. The inner product of two N vectors X E RN and y E RN is written as X' Y = L. x.y•. If we view these vectors as N x 1 matrices, we see that X' Y = X T y, where T is the matrix transposition operator. An expression such as "x·" can always be read as "XT"; for example, the expression X' A, where A is an N x M matrix, is the same as
Note that h(x)' Dg(x) = [h(X)]T Dg(x) is a I x N matrix, as is the other term in the right-hand side. Thus, the vector-valued case (M.AJ) implies the scalar-valued formula (M.A.2). (iii) Suppose that I: R .... RM has the form [(x) = a(x)g(x), where a(') is a real-valued function of one variable (i.e., a: R .... R) and g: R .... RM. Then D [(x) = a(x) Dg(x) + a'(x)g(x). (M.A.4) (iv) Suppose that [: RN .... RM has the form [(x) = h(x)g(x) where h: RN .... R and g: RN .... RM. Then DI(x) = h(x) Dg(x)
+ g(x) Dh(x).
(M.A.5)
Note that g(x) is an M-element vector (i.e., an M x 1 matrix) and Dh(x) is a 1 x N matrix. Hence, g(x) Dh(x) is an M x N matrix (of rank 1). Observe also that (M.A.4) follows as a special case of (M.A.5).
xTA.
If I: !\IN .... RM is a vector-valued differentiable function, then at any x ERN we denote by D[(x) the M x N matrix whose mnth entry is iJIm(x)/iJx•. Note, in 926
.....
DERIVATIVES
927
928
MATHEMATICAL
APPENDIX
M.B Homogeneous Functions and Euler's Formula
-
SECTION
M.B:
HOMOGENEOUS
FUNCTIONS
AND
EULER'S
FORMULA
929
---------------------------------------------------------------------------------X,
In this section. we consider functions of N variables, f(x l ••••• x N). defined for all nonnegative values (Xl ••••• x N ):2: O. Definition M.B.1: A function f(x" ... ,xN) is homogeneous of degree r (for r = ... , -1.0,1, ... ) if for every I> 0 we have f(IX" ... ,IXN ) = (,f(x, • ...• x N ).
As an example. f(x i • x,) = XI/X, is homogeneous of degree zero and f(x ,• x,) = (XIX,)I" is homogeneous of degree one. Note that if f(x" ...• x N) is homogeneous of degree zero and we restrict the domain to have XI > 0 then, by taking I = I/x i • we can write the function f(·) as
j(-)
= t' Flgur. M.B.1
f(-) = I X,
f(l. x,/x l ••••• XN/X I) = f(x l ••••• x N). Similarly, if the function is homogeneous of degree one then and the slope of the level set containing point IX for
f(l. x,/x I•...• XN/X I ) = (l/xI)f(x , •...• x N ).
I
An illustration of this fact is provided in Figure M.B.1. Suppose that f(·) is homogeneous of some degree r and that h(·) is an increasing function of one variable. Then the function hU(x" ... , XN» is called homothetie. Note that the family of level sets of hU('» coincides with the family of level sets of f(·). Therefore. for any homothetic function it is also true that the slopes of the level sets are unchanged along rays through the origin. A key property of homogeneous functions is given in Theorem M.B.2.
> O. By the definition of homogeneity (Definition M.S.\) we have f(IX , •...• IX N ) -I'f(x , •... , XN) = O.
Differentiating this expression with respect to x. gives t
> 0 at IX is
_ Of..~X]!~1 = _ I' - , 0 f(x)/ox I 0f(x)/ox, iJf(IX)/OX 2 1'-' of(x)/ox, = - iJf(x)/ox,'
Theorem M.B.1: If f(x" . .. , xN) is homogeneous of degree r (for r = ...• -1.0,1 •... ), then for any n = 1, ... ,N the partial derivative function of (x, , ...• xN)/ox n is homogeneous of degree r - 1. Proof: Fix a
I
Of(IX" .... tXN) ,of(xI.· ... XN) 0 -t = • ox, ox.
Theorem M.B.2: (Euler's Formula) Suppose that f(x" ... ,XN) is homogeneous of degreer(for somer = ... , -1, 0, 1, ... ) and differentiable. Then at any (x" . .. , xN ) we have
so that
of(IX " ... ,IX N ) = t,-I of(X " ...• XN). OX, OX, By Definition M.B.I, we conclude that of(X " ...• xN)/ox, is homogeneous of degree r - I. •
f!.L
of (x, , ... ,xN )
n=l
iJxn
-
-
f(-
xn = r
Xl.···
- )
I
XN •
or, in matrix notation, Vf(x)·x = rf(x).
For example. for the homogeneous of degree one function f(x ,• x,) = (XIX,)I/'. we have of(X I. x,)/ox i = f(X,/XI)I/', which is indeed homogeneous of degree zero in accordance with Theorem M.B.1. Note that if f(·) is a homogeneous function of any degree then f(x ...• XN) = " is, a radial f(x;, ... , x;") implies f(tx , •... , tXN) = f(tx;, ... , tx;") for any t > 0; that expansion of a level set of f(·) gives a new level set of f(·). I This has an interesting implication: the slopes of the level sets of f(·) are unchanged along any ray through the origin. For example, suppose that N = 2. Then, assuming that of(x)/ox, "" 0, the slope of the level set containing point x = (XI' x,) at X is -(of(x)/ox,)/(of(x)/ox,),
Proof: By definition we have
f(lx t .... ,IX N) - t'!(x, .... , XN)
= O.
Differentiating this expression with respect to t gives
Evaluating at
I
= I, we obtain Euler's formula . •
For a function that is homogeneous of degree zero, Euler's formula says that t. A level set offunction 1(,) is a set of the form {x E
R~: I(x)
= k} for some k. A radial
expansion of this set is the set of points obtained by mUltiplying each vector X in this level set by some positive scalar t > O.
....
The level sels of a homogeneous funclion.
930
MATHEMATICAL
-
APPENDIX
As an example. note that for the function f(x i • x,) = XI/X,. we have of(x I> x,)/ox i = l/x2 and of(x I' x,)/ox, = -(xl/(x,)'). and so ~ of(xl.···.XN) _ '~I aX. x.
I -
XI
= .i; XI
-
-
0
(x,)' x, = .
• E C TI 0 N
M.
c:
CON CAY E
AND
~ of(XI' •••• XN) - - f(L.
X,,-
AX,.
(M.C.2) for any collection of vectors XI E A, ...• x that 0: 1 + ... + O:K = I.
- )
X1"",XN'
For example. when f(x l• x,) = (XIX,)I/'. we have of(X I' X,)/OXI of(.x l • X2 )/OX 2 = HXI/X2)1/'. and so
= t(X,/XI)I/'
K
E
A and numbers 0:, ~ O•..• , 0:" ~ 0 such
and Let us consider again the one-variable case. We could view each number 0:, in condition (M.C.2l as the "probability" that x' occurs. Then condition (M.C.2) says that the value of the expectation is not smalier than the expected value. Indeed, a concave function f: R .... R is characterized by the condition that
~o.f'...2(-'xl,,-''_'....c'._X"-,,N) __ I (X,)I/' _ I (XI)I/' _ X - ax. • 2 -XI XI + -2 -X, X2
L.. .~,
= (X IX2)1/2 =
f(X,.
(M.C.3)
x,).
for any distribution function F: R .... [0, I). Condition (M.C.3) is known as J,nsen's inequality.
M.C Concave and Quasiconcave Functions In this section. we consider functions of N variables f(x, •...• x N) defined on a domain A that is a convex subset of RN (such as A = RN or A = R~ = {x E IR N: x ~ O}).' We denote X = (Xl •...• x N ).
The properties of convexily and slriel convexily for a function f(·) are defined analogously but with the inequality in (M.CI) reversed. In particular, for a strictly convex function f('), a straight line connecting any two points in its graph should lie entirely above its graph, as shown in Figure M.C.2. Note also that f(·) is concave if and only if - f(·) is convex. Theorem M.C'! provides a useful alternative characterization of concavity and strict concavity.
Definition M.C.l: The function f: A .... R. defined on the convex set A eRN, is con-
cave if f(rJ.X'
+ (1
- o:)x) ~ o:f(x')
+ (1
- o:)f(x)
(M.C.1)
for all x and x' E A and all 0: E [0, 1]. If the inequality is strict for all x' 0: E (0,1), then we say that the function is strictly concave.
~
x and all
Theorem M.C.l: The (continuously differentiable) function f: A .... R is concave if and only if
Figure M.CI(a) illustrates a strictly concave function of one variable. For this case, condition (M.C.I) says that the straight line connecting any two points in the graph of f(·) lies entirely below this graph. 3 In Figure M.C.I(b). we show a function
fIx
R
,f(·
Theorem M.C.3's characterization of quasiconcave functions is illustrated in Figure M.CS. The content of the theorem's condition (M.CS) is that for any quasiconcave function f(·) and any pair of points x and x' with fIx') ~ fIx), the gradient vector VfIx) and the vector (x' - x) must form an acute angle.
6. See Section M.E for a discussion of the properties of such matrices.
d
935
936
MATHEMATICAL
APPENDIX
Theorem M.D.1: Suppose that M is an N x N matrix. (i) The matrix M is negative definite if and only if the symmetric matrix M + MT is negative definite. (ii) If M is symmetric. then M is negative definite if and only if all of the characteristic values of M are negative. (iii) The matrix M is negative definite if and only if M- 1 is negative definite. (iv) If the matrix M is negative definite. then for all diagonal N x N matrices K with positive diagonal entries the matrix KM is stable.'
--- -
• E C T ION
Proof: Part (i) simply follows from the observation ihat z'(M + MT)Z = 2z'Mz for every z E RH. The logic of part (ii) is the following. Any symmetric matrix M can be diagonalized in a simple manner: There is an N x N matrix of full rank e having e T = e - I and such that CM C T is a diagonal matrix with the diagonal entries equal to the characteristic values of M. But then z· Mz = (Cz)· eMCT(Cz). and for every Z E RH there is a z such that i = Cz. Thus. the matrix M is negative definite if and only if the diagonal matrix CMC T is. But it is straightforward to verify that a diagonal matrix is negative definite if and only if everyone of its diagonal entries is negative. Part (iii): Suppose that M - I is negative definite and let z ¥- O. Then z· M z = (Z'MZ)T = z'MTz = (MTz)'M-I(MTz) < O. Part (iv): It is known that a matrix A is stable if and only if there is a symmetric positive definite matrix E such that EA is negative definite. Thus, in our case, we can take A = KM and E = K- . •
M. 0:
.. A T RIC E . :
NEG A T I V E
( • E .. II 0 E FIN I TEN E ••
AND
0 THE R
Proof: (i) The necessity part is simple. Note that by the definition of negative definiteness we have that every ,M, is negative definite. Thus, by Theorem M.D.!, the characteristic values of ,M, are negative. The determinant of a square matrix is equal to the product of its characteristic values. Hence, I,M,I has the sign of (-IY. The sufficiency part requires some computation. which we shall not carry out. It is very easy to verify for the case N = 2 [if the conclusion of (i) holds for a 2 x 2 symmetric matrix. then the determinant is positive and both diagonal entries are negative; the combination of these two facts is well known to imply the negativity of the two characteristic values]. For (ii), we simply note the requirement to consider all permutations. For example. if M is a matrix with all its entries equal to zero except the N N entry, which is positive. then M satisfies the nonnegative version of (i) but it is not negative semidefinite according to Definition M.D.1. Notice that in part (iii) we only claim necessity of the determinantal condition. [n fact, for nonsymmetric matrices the condition is not sufficient. • Example M.D.I: Consider a real-valued function of two variables, I(x" x,). In what follows. we let subscripts denote partial derivatives; for example, 112(X,. x,) = iJ' I(x I' x,)/iJx, iJx,. Theorem M.C.2 tells us that 1(') is strictly concave if
Xl)]
D'/(x" x,) = [/"(X,, x,) 112(X" I,I(X I • X,) I,,(x l , X,)
is negative definite for all (XI' x,), According to Theorem M.D.2, this is true if and only if
'
For positive definite matrices, we can simply reverse the words "positive" and "negative" wherever they appear in Theorem M.D.1. Our next result (Theorem M.D.2) provides a determinantal test for negative definiteness or negative semidefiniteness of a matrix M. Given any T x S matrix M, we denote by ,M the r x S submatrix of M where only the first t ~ Trows are retained. Analogously, we let M, be the T x s submatrix of M where the first s ~ S columns are retained, and we let ,M, be the t x s submatrix of M where only the first r oS T rows and s oS S columns are retained. Also. if M is an N x N matrix, then for any permutation It of the indices {I, ...• N} we denote by M' the matrix in which rows and columns are correspondingly permuted.
and
/ll(X I • x,) 112(X" X')I > 0, I"(x,, x,)
IIII (x " x,)
or equivalently, if and only if
and
111(X" x,)/,,(x l , x,) - U12(X" x,)]' > O. Theorem M.C.2 also tells us that 1(') is concave if and only if D' I(x" x,) is negative semidefinite for all (XI' x,). Theorem M.D.2 tells us that this is the case if and only if
Theorem M.D.2: Let M be an N x N matrix. (i) Suppose that M is symmetric. Then M is negative definite if and only if (-l),I,M,1 > 0 for every, = 1..... N. (ii) Suppose that M is symmetric. Then M is negative semidefinite If and only if ( -1 )'J,M;I 2: 0 for every' = 1•...• N and for every permutation It of the indices {1 ....• N}. (iii) Suppose that M is negative definite (not necessarily symmetric). Then ( -1 )'J,M;I > 0 for every, = 1....• N and for every permutation It of the indices {1 .... • N}.8
and
/II(X" x,) l , x,)
I111(x
ltiXI,Xl)I2:0. 1,,(xI' x,)
and, permuting the rows and columns of D'/(x l , x,).
1/,,(x l • x,)1
~
0
and
Xl)l . .
I,,(x I , x,) 111(x" O. '" 111(X" x,)
I112(X I, x,)
Thus. 1(') is concave if and only if 111(x I , x,)
7. A matrix M is stable irall or its characteristic values have negative real pans. This terminology is motivated by the ract that in this case the solution or the system or differential equations dx(r)/dl = MX(I) will converge to zero as I _ 00 ror any initial position x(O). 8. A matrix M such that - M satisfies the condition in (iii) is called a P matrix. The reason is that the detenninant of any submatrix obtained by deleting some rows (and corresponding columns)
:s 0,
I"(x,, x,) oS 0, and
•
is positive.
.....
PRO PER TIE'
937
938
MATHEMATICAL
-
SECTION
APPENDIX
-------------------------------------------------------------------------A similar test is available for positive definite and semidefinite matrices: The results for these matrices parallel conditions (i) to (iii) of Theorem M.D.2, but omit the factor (_1)'.9 Theorem M.D.3: Let M be an N x N symmetric matrix and let B be an N x S matrix with S ~ N and rank equal to S.
(-1)'!
,M:
(,B")T
= O}
tSEMI)DEFINITENESS
AND
OTHER
and (performing the appropriate permutations) f"(x,, x,)
f"(x,, x,)
f,(x" x,)
f12(X" x,)
fl1(X" x,)
f,(x" x,) ~ O.
f,(x" x,)
f,(x" x,)
0
-
To characterize matrices that are positive definite or positive semidefinite on the subspace {t e IJl:N: Bz = O}, we need only alter Theorem M.D.3 by replacing the term ( _ I)' with ( _I )s.
0
for r = S + 1, ... , N. (ii) M is negative semidefinite on {z eRN: Bz z ERN with Bz = 0 and z ,;. 0) if and only if
NEGATIVE
2f,(x" x,)f,(x" x,)fdx" x,) - [f,(x" x,)]' f"(x,, x,) - [f,(x" x,)]'fl1 (x" x,) ~ O.
,M, ,B! > 0
(,B)T
MATRICES:
Computing these two determinants gives us the necessary and sufficient condition
(i) M is negative definite on {z eRN: Bz = O} (i.e., z· Mz < 0 for any z eRN with Bz = 0 and z ,;. 0) if and only if
(-1)'!
M.D:
(I.e., z'Mz ~ 0 for any
Theorem M.D.4: Suppose that M is an N x N matrix and that for some p » 0 we have Mp = 0 and MTp = O. Denote Tp = {z eRN: p'z = O} and let if be the (N - 1) x (N - 1) matrix obtained from M by deleting one row and the corresponding column.
,B"I ~0 0
for r = S + 1, ... , N and and every permutation n, where ,8" is the matrix formed by permuting only the rows of the matrix ,8 according to the permutation n (,M; is, as before, a matrix formed by permuting both the rows and columns of ,M,).
(i) If rank M = N - 1, then rank Nt = N - 1. (ii) If z· Mz < 0 for all z e Tp with z,;. 0 (i.e., if M is negative definite on Tp), then z· Mz < 0 for any z eRN not proportional to p. (iii) The matrix M is negative definite on Tp if and only if Nt is negative definite.
Proof: We will not prove this result. Note that it is parallel to parts (i) and (ii) of Theorem M.D.2 with the bordered matrix here playing a role similar to the matrix there. _
Proof: (i) Suppose that rank M < N - I, that is, Mz = 0 for some i e RN -, with i ,;. O. Complete to a vector z e RN by letting the value of the missing coordinate be zero. Then we have that, first, z is linearly independent of p (recall that p» 0) and, second, Mz = 0 and Mp = O. Thus, rank M < N - I, which contradicts the
z
Example M.D.2: Suppose we have a function of two variables, f(x l , x,). We assume that Vf(x) ,;. 0 for every x. Theorem M.C.4 tells us that f(·) is strictly quasi concave if the Hessian matrix D' f(x l , x,) is negative definite in the subspace (z e R': Vf(x)' z = O} for every x = (x" x,). By Theorem M.D.3 the latter is true if and only if fl1(X" x,)
f12(X" x,)
fleX"~ x,)
f"(x,, x,)
f"(x,, x,)
f,(x" x,) > 0,
fleX"~ x,)
f,(x" x,)
hypothesis. (ii) Take a z e RN not proportional to p. For IX, = (P'z)/(P'p) and z* we have z* e Tp and z' ,;. O. Because MTp = Mp = 0, we have then
(iii) This is similar to part (ii). In fact, part (ii) directly implies that M is negative definite if M is negative definite on Tp (because for any ze RN -', z'Mi = z'Mz, where z has been completed from z by placing a zero in the missing coordinate, and if z ,;. 0 this z is by construction not proportional to pl. For the converse, let n denote the row and column dropped from M to obtain M. If for every z' e Tp with z'';' 0 we let z = z' - (z;/P.)p, then z. = 0 and z';' 0 [if z were equal to zero, then we would have z' = (z;/P.)p in contradiction to z'·p = 0]. Moreover, z"Mz' = z·M. = z'Mz < O. _
0
2f,(x" x,)f,(x" X,)f12(X" x,) - [flex"~ x,)]'fdx" x,) - [f,(x" x,)]' fl1(X" x,) > O.
= XIX, we get 2x,x, > 0 confirming that the function is strictly quasiconcave. By Theorem M.C.4, f(·) is quasiconcave if and only if the Hessian matrix D' f(x" x,) is negative semidefinite in the subspace (z e R': Vf(x)' z = O} for every x = (x" x,). By Theorem M.D.3 this is true if and only if
If we apply this test to f(x" x,)
fl1(X" x,)
fdx" x,)
f,(x" x,)
f"(x,, x,)
f,(x" x,) ~ 0,
f,(x l , x,)
f,(xI' x,)
IX,p,
z'Mz = (z· + IX,p)'M(z' + IX,p) = z"Mz' < O.
or equivalently, if and only if
f"(x,, x,)
=z-
Definition M.D.2: The N x N matrix M with generic entry aji has a dominant diagonal if there is (p" ... ,PN) »0 such that, for every i = 1, ... , N, !pjajA > Li*; !Pia;J Dellnltlon M.D.3: The N x N matrix M has the gross substitute sign pattern if every nondiagonal entry is positive. Theorem M.D.S: Suppose that M is an N x N matrix.
0
(i) If M has a dominant diagonal, then It Is nonsingular. (ii) Suppose that M is symmetric. If M has a negative and dominant diagonal then it is negative definite.
9. Recall that M is positive (semi)definite if and only if - M is negative (semi)definite. Moreover,
hM,1 = (-I)'I,M,I·
...
PROPERTIES
939
940
MATHEMATICAL
APPENDIX
(iii) If M has the gross substitute sign pattern and if for some p » 0 we have Mp « 0 and MTp « 0, then M is negative definite. (iv) If M has the gross substitute sign pattern and we have Mp = MTp = 0 for some p » 0, then Nt is negative definite, where Nt Is any (N - 1) x (N - 1) matrix obtained from M by deleting a row and the corresponding column. (v) Suppose that all the entries of M are nonnegative and that Mz« z for some z » 0 (i.e., M is a productive input-output matrix). Then the matrix (1- M)-' exists. In fact, (/- M)-' = L~:cf M*.
-
SECTION
IMPLICIT
FUNCTION
THEOREM
941
x
R
~(ij)
-------7!T--- ~(.)
x' --------=x x' ------- I I I I I I
I
:
: I I I I I
: I I I I I
Figure M.E.1
A locally solvable equation. (a) Solutions or f(x; q) = 0 near (;C, q). (b) The graph or ~(- ).
q' ij q'
f(·;q')
j{-; ij) f(·;q')
(a)
(b)
Suppose that x = (x, •.... xN) E A and ii = (ii, •.. ·• iiM) E B satisfy equations (M.E.I). That is. I.(x. ii) = 0 for every n. We are then interested in the possibility of solving for x = (x, •...• x N ) as a function of q = (q, •...• qM) locally around ii and .ii. Formally. we say that a set A' is an open neighborhood of a point x E IRN if A' = {x' E IR N: IIx' - xII < £} for some scalar £ > O. An open neighborhood B' of a point q E IRM is defined in the same way. Definition M.E.1: Suppose that x = (x" ... ,xN) E A and q = (q" ... ,qM) E B sa~sf! the equations (M.E.1). We say that we can locally solve equations (M.E.1) at (x, q) for x = (x, . ... ,XN) as a function of q = (q" ...• qM) if there a.re open neighborhoods A' c A and B' c B, of x and q, respectively, and N untquely determined "implicit" functions '1,( .), ... , '1N(') from B' to A' such that 'n('1,(q). ... ,'1N(q); q) = 0
for every q
E
B' and every n,
and for every n. In Figure M.E.I we represent. for the case where N = M = I, a situation in which the system of equations can be locally solved around a given solution. The implicit function theorem gives a sufficient condition for the existence of such implicit functions and tells us the first-order comparative statics effects of q on x at a solution.
M.E The Implicit Function Theorem
Theorem M.E.1: (Implicit Function Theorem) Suppose that every equation I n (·) is continuously differentiable with respect to its N + M variables and that we consider a solution x = (x" ... , XN) at parameter values q = (q" ...• qM)' that is, satisfying 'nIx; q) = 0 for every n. If the Jacobian matrix of the system (M.E.1) ~It~ respect to the endogenous variables, evaluated at (x, q). is nonsingular, that IS, If
The setting for the implicit function theorem (1FT) is as follows. We have a system of N equations depending on N endogenous variables x = (x, •...• x N) and M parameters q = (q, •...• qM): XN;
THE
I.~---------~~---------------
Proof: (i) Assume, for simplicity. that p = (I •...• I). Suppose. by way of contradiction. that Mz = 0 for z # O. Choose a coordinate n such that Iz,l ~ Iz,.1 for every other coordinate n'. Then la•• z.1 > L, •• la.,z.1 ~ L,.,la.jzjl. where 0/) is the generic entry of M. Hence. we cannot have Lj a,)z) = O. and so M z # O. Contradiction. (ii) If M has a negative dominant diagonal then so does the matrix M - aI, for any value a ~ O. Hence. by (i) we have (_I)NIM - all # O. Now if a is very large it is clear that (-ltIM - all> 0 (since (_I)NIM - all = (-I)NaNI(Mja) - II and 1-/1=(-1)''). Moreover. since (-I)NIM-aII is continuous in a and (_I)NIM - all # 0 for all a ~ O. this tells us that (-I)NIM - all> 0 for all a ~ O. Hence. (-I )NIMI > O. By the same argument. (-IYI,M,I > 0 for all r. So. if M is also symmetric then by part (i) of Theorem M.D.2 it is negative definite. (iii) The stated conditions imply that M + MT has a negative and dominant diagonal [in particular. note that Mp« 0 and AfT p «0 implies that p.(2a.. ) < - L) •• Pj(aj • + a.) for all n. where 0/) is the generic entry of M]. Because, by the gross substitute property. 0/) > 0 for i # j.this gives us Ip.(2a.. )1 > IL) .. p)(a", + 0,,)1 for all n. Hence. the conclusion follows from part (ii) of this theorem and part (i) of Theorem M.D.!. (iv) If M satisfies the condition of (iv). then the fact that M has the gross substitute sign pattern implies that M does as well and that Mp «0 and MT p «0. Hence. M satisfies the conditions of (iii) and is therefore negative definite. (v) This result was already proved in the Appendix to Chapter 5 (see the proof of Proposition 5.AA.I).
f,(x, •...•
M.E:
q, •...• qM) = 0 (M.E.I)
Of,(x, q)
Of,(x, q)
ox,
oXN
fN(x, •...• xN;q, •...• qM) = 0
# 0,
The domain of the endogenous variables is A c RN and the domain of the parameters is Be RM.'O
OfN(x, q)
OfN(x, q)
ox,
oXN
(M.E.2)
then the system can be locally solved at (x, q) by implicitly defined lunctions 'In: B' _ A' that are continuously differentiable. Moreover, the first-order effects
10. In wha, follows. we ,ake A and B to be open sets (sec Section M.F) so as to avoid boundary problems.
..
942
MATHEMATICAL
- -
APPENDIX
of q on x at (x, 17) are given by Dq'l(q) = -[Dxf(x; qlr'Dqf(x; 17).
(M.E.3)
Proof: A proof of the existence of the implicit functions 'I.: B' -+ A' is too technical for this appendix, but its common-sense logic is easy to grasp. Expression (M.E.2), a full rank condition, tells us that we can move the values of the system of equations in any direction by appropriate changes of the endogenous variables. Therefore, if there is a shock to the parameters and the values of the equation system are pushed away from zero, then we can adjust the endogenous variables so as to restore the "equilibrium." Now, given a system of implicit functions '1(q) = ('1,(q), ... ,'1N(q)) defined on some neighborhood of (x, Ii), the first-order comparative static effects iJ'I.(ii)/iJqm are readily determined. Let f(x; q) = (/, (x; q), ... ,fN(X; q)). Since we have
f('1(q); q) = 0
CONTINUOUS
FUNCTIONS
AND
COMPACT
Definition M.E.2: Given open sets A c ~ and 8 c RM , the (continuously differentiable) system 01 equations f(·;
