Inframarginal Contributions to Development Economics

Increasing Returns and Inframarginai Economics - Vol. 3 0 ,

(2A)

that is, a small change in the activity undertaken by B will change the utility of A in the same direction; a marginal external diseconomy existing when, u*0. (3A) o This condition states that, while incremental changes in the extent of 5's activity, Yh have no affect on A's utility, the total effect of 5's action has increased A's utility. An infra-marginal diseconomy exists when (1) holds, and, for any given set of values for (Xx,X2,...,Xm), say, (C,,C2,...,Cm), then, u* = 0 , a n d \u*dYi < 0 .

(3B)

o

Thus, small changes inS's activity do not changer's level of satisfaction, but the total effect of 5's undertaking the activity in question is harmful to A. We are able to classify the effects of B's action, or potential action, on A's utility by evaluating the "partial derivative" ofA's utility function with respect to Y\ over all possible values for Y\. In order to introduce the further distinctions between relevant and irrelevant externalities, however, it is necessary to go beyond consideration of A's utility function. Whether or not a relevant externality exists depends upon the extent to which the activity involving the externality is carried out by the person empowered to take action, to make decisions. Since we wish to consider a single externality in isolation, we shall assume that B's utility function includes only variables (activities) that are within his control, including Y\. Hence, B's utility function takes the form, u»=u*(Xx,Y2,...,Ym).

(4)

Necessary conditions for utility maximisation by B are,

KK=f*lfrBt>

&

where 1} is used to designate the activity of B in consuming or utilising some numeraire commodity or service which is, by hypothesis, available on equal terms to A. The right-hand term represents the marginal rate of substitution in "production" or "exchange" confronted by B, the party taking action on Yu his production function being defined as,

/*=/*(^,y2,...,yj,

(6)

Externality

59

where inputs are included as activities along with outputs. In other words, the right-hand term represents the marginal cost of the activity, Yu to B. The equilibrium values for the Yi 's will be designated as Yt 's. An externality is defined as potentially relevant when the activity, to the extent that it is actually performed, generates any desire on the part of the externally benefited (damaged) party (A) to modify the behaviour of the party empowered to take action (B) through trade, persuasion, compromise, agreement, convention, collective action, etc. An externality which, to the extent that it is performed, exerts no such influence is defined as irrelevant. Note that, so long as (1) holds, an externality remains; utility functions remain interdependent. A potentially relevant marginal externality exists when,

*0.

uA

(7)

This is a potentially relevant marginal external economy when (7) is greater than zero, a diseconomy when (7) is less than zero. In either case, A is motivated, by 5's performance of the activity, to make some effort to modify this performance, to increase the resources devoted to the activity when (7) is positive, to decrease the quantity of resources devoted to the activity when (7) is negative. Infra-marginal externalities are, by definition, irrelevant for small changes in the scope of B's activity, Y\. However, when large or discrete changes are considered, A is motivated to change S's behaviour with respect to Y\ in all cases except that for which, u*\ _ = 0 , a n d

(8) K

M\y-V

7

u*(C1,C2,...,Cmjl)Zu*(Cl,C2,...,Cm,Yl)JoT&\l Y^%. When (8) holds, A has achieved an absolute maximum of utility with respect to changes over Y\, given any set of values for the X's. In more prosaic terms, A is satiated with respect to Y\3. In all other cases, where infra-marginal external economies or diseconomies exist, A will have 3

Note that, ui\

_ = 0 , is a necessary, but not a sufficient, condition for irrelevance.

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J. M. Buchanan, W. C. Stubblebine

some desire to modify 5's performance; the externality is potentially relevant. Whether or not this motivation will lead A to seek an expansion or contraction in the extent of 5's performance of the activity will depend on the location of the infra-marginal region relative to the absolute maximum for any given values of the JTs.4 Pareto relevance and irrelevance may now be introduced. The existence of a simple desire to modify the behaviour of another, defined as potential relevance, need not imply the ability to implement this desire. An externality is defined to be Pareto-relevant when the extent of the activity may be modified in such a way that the externally affected party, A, can be made better off without the acting party, B, being made worse off. That is to say, "gains from trade" characterise the Pareto-relevant externality, trade that takes the form of some change in the activity of B as his part of the bargain. A marginal externality is Pareto-relevant when5 (-)«£ /uXj > [ « * / « * - f*m

KK

>(r)[KK

]

~frJfrB\=?^

_ and when u{ /uXj < 0, and

(9)

u*/uXj >0.

In (9), Xj and Yj are used to designate, respectively, the activities of A and B in consuming or in utilising some numeraire commodity or service that, by hypothesis, is available on identical terms to each of them. As is indicated by the transposition of signs in (9), the conditions for Pareto relevance differ as between external diseconomies and economies. This is because the "direction" of change desired by A on the part of B is different in the two cases. In stating the conditions for Pareto relevance under ordinary two-person trade, this point is of no significance since trade in one good flows only in one direction. Hence, absolute values can be used.

4

In this analysis of the relevance of externalities, we have assumed that B will act in such a manner as to maximise his own utility subject to the constraints within which he must operate. If, for any reason, B does not attain the equilibrium position defined in (5) above, the classification of his activity for A may, of course, be modified. A potentially relevant externality may become irrelevant and vice versa. 5 We are indebted to Mr. M. McManus of the University of Birmingham for pointing out to us an error in an earlier formulation of this and the following similar conditions.

61

Externality

The condition, (9), states that^'s marginal rate of substitution between the activity, Yu and the numeraire activity must be greater than the "net" marginal rate of substitution between the activity and the numeraire activity for B. Otherwise, "gains from trade" would not exist between A and B. Note, however, that when B has achieved utility-maximising equilibrium, U»/!*%=/*/f°.

(10)

That is to say, the marginal rate of substitution in consumption or utilisation is equated to the marginal rate of substitution in production or exchange, i.e., to marginal cost. When (10) holds, the terms in the brackets in (9) mutually cancel. Thus, potentially relevant marginal externalities are also Pareto-relevant when B is in utility-maximising equilibrium. Some trade is possible. Pareto equilibrium is defined to be present when, (-)u$Juj

= W*/H£ - fyj fr. \> and when u^luj, < 0 , and (11)

KK =(-)[%1/M" -frjfy^\

when U

ZK

>0

-

Condition (11) demonstrates that marginal externalities may continue to exist, even in Pareto equilibrium, as here defined. This point may be shown by reference to the special case in which the activity in question may be undertaken at zero costs. Here Pareto equilibrium is attained when the marginal rates of substitution in consumption or utilisation for the two persons are precisely offsetting, that is, where their interests are strictly opposed, and not where the left-hand term vanishes. What vanishes in Pareto equilibrium are the Pareto-relevant externalities. It seems clear that, normally, economists have been referring only to what we have here called Pareto-relevant externalities when they have, implicitly or explicitly, stated that external effects are not present when a position on the Pareto optimality surface is attained.6

6

This applies to the authors of this paper. For recent discussion of external effects when we have clearly intended only what we here designate as Pareto-relevant, see James M. Buchanan, "Politics, Policy, and the Pigovian Margins", Economica, vol. XXVIX (1962),

62

J. M. Buchanan, W. C. Stubblebine

For completeness, we must also consider those potentially relevant infra-marginal externalities. Refer to the discussion of these as summarised in (8) above. The question is now to determine whether or not, A, the externally affected party, can reach some mutually satisfactory agreement with B, the acting party, that will involve some discrete (non-marginal) change in the scope of the activity, Y\. If, over some range, any range, of the activity, which we shall designate by AYX, the rate of substitution between Y\ and Xj for A exceeds the "net" rate of substitution for B, the externality is Pareto-relevant. The associated changes in the utilisation of the numeraire commodity must be equal for the two parties. Thus, for external economies, we have AuA

I AuA /

AY, J AXj

, N AuB lAuB > (-) AY / AYj X

AfB lAfB AY, / AYj

(12)

and the same with the sign in parenthesis transposed for external diseconomies. The difference to be noted between (12) and (9) is that, with infra-marginal externalities, potential relevance need not imply Pareto relevance. The bracketed terms in (12) need not sum to zero when B is in his private utility-maximising equilibrium. We have remained in a two-person world, with one person affected by the single activity of a second. However, the analysis can readily be modified to incorporate the effects of this activity on a multi-person group. That is to say, 5's activity, Yu may be allowed to affect several parties simultaneously, several's, so to speak. In each case, the activity can then be evaluated in terms of its effects on the utility of each person. Nothing in the construction need be changed. The only stage in the analysis requiring modification explicitly to take account of the possibilities of multi-person groups being externally affected is that which involves the condition for Pareto relevance and Pareto equilibrium. For a multi-person group i

[uBYJuBYj - f»/f*

+ u$JuXj ]

and uY> juAXj > ( - ) [ « * / « * - f*j'f^

+ u^luXj

, when u{ juXj < 0, (9B) _ , when u* lux > 0.

In (9B), Y1 represents the "private" equilibrium value for Yu determined by B, after the ideal Pigovian tax is imposed. As before, the bracketed terms represent the "net" marginal evaluation of the activity for the acting party, B, and these sum to zero when equilibrium is reached. So long as the left-hand term in the inequality remains non-zero, a Pareto-relevant marginal externality remains, despite the fact that the full "Pigovian solution" is attained. The apparent paradox here is not difficult to explain. Since, as postulated, A is not incurring any cost in securing the change in B's behaviour, and, since there remains, by hypothesis, a marginal diseconomy, further "trade" can be worked out between the two parties. Specifically, Pareto equilibrium is reached when,

(r)u$lux

= u§lul - ffl'fl,+uYlJux~\

< / < =(-)[ t. In other words, each type a persons has higher trading efficiency than each type b person. Also, we assume that k2 = k. The assumptions can be interpreted as follows. Country 2 is a developed country that has greater trading efficiencies. Country 1 is a less developed country where some residents (who might be urban or coastal residents) have the same trading efficiency as in the developed country, but the rest of the population has a lower trading efficiency. The production conditions are the same for all individuals in the same country, but different between the countries. Hence, the production functions for a type i consumer-producer are X +X

J J=L%'

yj+ysj=Ljy'

x2+xs2=If2x,

y2+ys2=rL2y,

j=*>b'

(2a)

(2b)

where x., yst are respective quantities of the two goods sold by a type i person; Lis is the amount of labor allocated to the production of good s (= x, y) by a type / person, and Lix + Liy = B > 1. For simplicity, we assume that B = 2. It is assumed that r, c > 1. This system of production

84

X. Yang, D. Zhang

functions and endowment constraint displays economies of specialization in producing good x and constant returns to specialization in producing good y. But an individual in country 2 has a higher productivity in producing good y than an individual in country 1. Economies of specialization are individual specific and activity specific, that is, they are localized increasing returns, which are compatible with the Walrasian regime.5 If all individuals allocate the same amount of labor to the production of each good, then an individual in country 1 has the same average labor productivity of good x as an individual in country 2. But the average and marginal labor productivity of good y for an individual in country 2 is higher. This is similar to the situation in a Ricardian model with exogenous comparative advantage. Country l's productivities are not higher than country 2 in producing all goods, but may have exogenous comparative advantage in producing good x. But if an individual in country 2 allocates much more labor to the production of x than an individual in country 1, her productivity is higher than that of the latter. Similarly, if an individual in country 1 allocates more labor to the production of good x than an individual in country 2, her productivity of good x will be higher. This is referred to as endogenous comparative advantage, since individuals' decisions on labor allocation determine 5

Marshall was aware of the incompatibility between his marvellous marginal analysis of demand and supply and classical economic thinking about the implications of specialization and division of labor. As a result he drew the distinction between internal economies of scale and external economies of scale. The latter is supposed to relate to economies of division of labor in the market place. However, as Allyn Young (1928) argued, "the view of the nature of the processes of industrial progress which is implied in the distinction between internal and external economies is necessarily a partial view. Certain aspects of those processes are illuminated, while, for that very reason, certain other aspects, important in relation to other problems, are obscured." As Young and Yang (2001) have pointed out, economies of specialization are diseconomies of a person's scope of production activities and economies of division of labor consist of economies of specialization of all individuals and economies of diversity of different occupations. They are different from (internal or external) economies of scale, though related to local economies of scale of an individual labor in producing a good. Hence, it seemed to Young that the concept of external economies of scale is a misrepresentation of the classical concept of economies of specialization and division of labor.

Economic Development, International Trade, and Income Distribution 85 difference in productivity between them. But an individual in country 1 has no endogenous comparative advantage in producing good y since her marginal and average productivity of y is always 1, lower than r, independent of her labor allocation. However, country 1 has exogenous comparative advantage in producing x and country 2 has exogenous comparative advantage in producing y. The decision problem for a type i individual involves deciding on what and how much to produce for self-consumption, to sell and to buy from the market. In other words, the individual chooses six variables xnx° ,xf ,yt,y° ,yf > 0 . Hence, there are 26 = 64 possible corner and interior solutions. As shown by Wen (1998), for such a model, an individual never simultaneously sells and buys the same good, never simultaneously produces and buys the same good, and never sells more than one good. We refer to each individual's choice of what to produce, buy and sell that is consistent with the Wen theorem as a configuration. There are three configurations from which the individuals can choose: (1) self sufficiency. Configuration A, where an individual produces both goods for self-consumption. This configuration is defined by x,,yt >0,

x' = x," = yt' = yf =0,i

= a,b,2.

(2) specialization in producing good x. Configuration (x/y), where an individual produces only x, sells x in exchange for y, is defined by Xi,x;,yf>o,

x?=yi=y;=o.

(3) specialization in producing good y. Configuration (y/x), where an individual produces only y, sells y in exchange for x, is defined by

yny:,x?>Q,

yf=Xt=x,'=Q.

The combination of all individual's configurations constitutes a market structure, or structure for short. Given the configurations listed above, fourteen structures may occur in equilibrium. Structure AAA, as shown in panel (1) of Figure 1, is an autarky structure where individuals in both countries choose self-sufficiency (configuration A). Here, the first two letters denote the configurations chosen by type a and type b persons, respectively in country 1 and the

86

X. Yang, D. Zhang

00- g)fc^ fcw0' ^ 'g) g) g) country 1 country 2 (1) Structure AAA

country 1 country 2 (7) Structure XXP

country 1 country 2 (2) Structure PAY

country 1 country 2 (8) Structure XXD

Figure 1: Configurations and Structures

country 1 country 2 (3) Structure XAP

country 1 country 2 (9) Structure XXY

Economic Development, International Trade, and Income Distribution

87

third letter denotes the configuration chosen by individuals in country 2. In each panel upper-left circles represent configurations chosen by type a individuals, lower-left circles represent configurations chosen by type b individuals, and circles on the right hand side represent configurations chosen by individuals in country 2. Structure PAY, shown in panel (2) of Figure 1, means that some type a persons choose configuration (x/y) and others choose autarky, all type b persons choose autarky, and individuals in country 2 choose configuration (y/x). Letter P stands for a type of person partially involved in the division of labor: some of them completely specialize and others choose autarky. This structure involves three types of dual structure. In a type I dual structure ex ante identical individuals are divided between specialization and autarky. In a type II dual structure, different types of individuals in the same country are divided between specialization and autarky. In a type III dual structure, one country is completely involved in the division of labor, while some residents in the other country are self-sufficient. Structure XAP, shown in panel (3) of Figure 1, means that all type a persons choose configuration (x/y), all type b persons choose autarky, some individuals in country 2 choose (y/x) and others choose autarky. Structure DAY, shown in panel (4) of Figure 1, means that some type a persons choose (x/y) and others choose (y/x), all type b persons choose autarky, and individuals in country 2 choose (y/x). Letter D stands for division of individuals of a certain type between (x/y) and (y/x). Structure XAY, panel (5) of Figure 1, means that all type a persons choose (x/y), all type b persons choose autarky, and individuals in country 2 choose configuration (y/x). Structure XAD implies that all type a persons choose (x/y), all type b persons choose autarky, some individuals in country 2 choose (y/x) and others choose (x/y). Structure XPY, panel (6), means that all type a persons choose (x/y), some type b persons choose autarky and others choose (x/y), and individuals in country 2 choose configuration (y/x).

88

X. Yang, D. Zhang

Structure XXP, panel (7), means that all type a and type b persons choose (x/y), some individuals in country 2 choose configuration (y/x), and others choose autarky. Structure XXD, panel (8), means that all type a and type b persons choose (x/y), some individuals in country 2 choose configuration (y/x), and others choose (x/y). Structure XXY, panel (9), implies all type a and type b persons choose (x/y) and all individuals in country 2 choose configuration (y/x). Structure XDY means that all type a persons choose (x/y), some type b individuals choose (x/y) and others choose (y/x), all individuals in country 2 choose configuration (y/x). Structure YDY means that all type a persons choose (y/x), some type b individuals choose (x/y) and others choose (y/x), all individuals in country 2 choose configuration (y/x). Structure DXY means that some type a persons and all type b persons choose (x/y), other type a persons choose (y/x), all individuals in country 2 choose configuration (y/x). Structure DYY means that some type a persons and all type b persons choose (y/x), other type a persons choose (x/y), all individuals in country 2 choose configuration (y/x). It can be shown that other structures cannot occur in equilibrium. 3. General Equilibrium and its Inframarginal Comparative Statics According to Sun, Yang and Zhou (1998), a general equilibrium exists and is Pareto optimal for the kind of models in this paper under the assumptions that the set of individuals is a continuum and preferences are strictly increasing and rational; both local increasing returns and constant returns are allowed in production and transactions. Also, the set of equilibrium allocations is equivalent to the set of core allocations. An equilibrium in this model is defined as a relative price of the two goods and all individuals' labor allocations and trade plans, such that (i) Each individual maximizes her utility, that is, the consumption bundle generated by her labor allocation and trade plan maximizes utility function (1) for given p.

Economic Development, International Trade, and Income Distribution

89

(ii) All markets clear. For simplicity, we assume that J3= 0.5. Let the number (measure) of type i individuals choosing configuration (x/y) be Mix, choosing (y/x) be Miy, and choosing A be MA. Since the interior solution is never optimal in this model of endogenous specialization and there are many structures based on corner solutions, we cannot use standard marginal analysis to solve for a general equilibrium. We adopt a two step approach in solving for a general equilibrium. In the first step, we consider a structure. Each individual's utility maximizing decision is solved for the given structure. Utility equalization condition between individuals choosing different configurations in the same country and market clearing condition are used to solve for the relative price of traded goods and numbers (measure) of individuals choosing different configurations. The relative price and numbers and associated resource allocation are referred to as a corner equilibrium for this structure. According to the definition, a general equilibrium is a corner equilibrium in which all individuals have no incentive to deviate, under the corner equilibrium relative price, from their chosen configurations. Hence, in the second step, we can plug the corner equilibrium relative price into the indirect utility function for each configuration, then compare corner equilibrium values of utility across all configurations. The comparisons are called a total cost-benefit analysis which yields the conditions under which the corner equilibrium utility in each constituent configuration of this structure is not smaller than any alternative configuration. This system of inequalities can thus be used to identify a subspace of parameter space within which this corner equilibrium is a general equilibrium. With the existence theorem of general equilibrium proved by Zhou, Sun and Yang (1998), we can completely partition the parameter space into subspaces, within each of which the corner equilibrium in a structure is a general equilibrium. As parameter values shift between the subspaces, the general equilibrium will discontinuously jump between structures. The discontinuous jumps of structure and all endogenous

90

X. Yang, D. Zhang

variables are called inframarginal comparative statics of general equilibrium. We now take the first step of the inframarginal analysis. As an example, we consider structure XAP. Assume that in this structure, M2y individuals choose configuration (y/x) and M2A individuals choose autarky in country 2, where M2y + M2A = M2. Ma individuals choose configuration (x/y) and Mb individuals choose configuration A in country 1. Since all individuals in country 2 are ex ante identical in all aspects, the maximum utilities in configurations A and (y/x) must be the same in country 2 in equilibrium. Marginal analysis of the decision problem for an individual in country 2 choosing autarky yields her maximum utility in configuration A: U2A = (r2c+1y)0'5, where y- ccl(c + l) e+1 . Marginal analysis of the decision problem for an individual in country 2 choosing configuration (y/x) yields the demand function xd2 = rip, the supply function^ = r, and indirect utility function: U2y = r(k/p)05. The utility equalization condition U2y = U2A yields p = pjpy = kr/2°ny. Similarly, the marginal analysis of the decision problem of a type a individual choosing configuration (x/y) yields the demand function, yd _ jc-\ ip ^ m e SUppiy function xsa = 2C_1 , and indirect utility function: Uax =2c~l(kp)05 . The marginal analysis of the decision problem for a type b individual choosing configuration A in country 1 yields her maximum utility in autarky UbA = (2c+lyf5. Inserting the corner equilibrium relative price into the market clearing condition for good x, Msx x" - M2y %d2 , yields the number of individuals selling good y, M2y = kMJAy, Mm = Ma, and MbA = Mh. Indirect utility functions for individuals choosing various configurations in the two countries are listed in Table 1.

Configurations Type a person Type b person Country 2

Table 1: Indirect Utility Functions Indirect utility functions (x/y) (y/x) t/„ = 2c-\kpf5 U!Ly = (k/pf5 $ UbK = ¥-\tpf Uby = (t/pf5 c 5 U2x = 2 ~\kpf U2y = r(k/pf5

A U^ = (2c+lrf5 UbA = (T+lrf5 U2A = (l^ryf5

Economic Development, International Trade, and Income Distribution

91

Following this procedure, we can solve for corner equilibrium in each structure. The solutions of all corner equilibria are summarized in Table 2. We can then take the second step to carry out total cost-benefit analysis for each corner equilibrium and to identify the parameter subspace within which the corner equilibrium is a general equilibrium. Consider the corner equilibrium in structure XAP as an example again. In this structure Ma individuals choose configuration (x/y) and Mh individuals choose autarky in country 1, and M2y individuals choose configuration (y/x) and M2A individuals choose autarky in country 2. For a type a person in country 1, equilibrium requires that her utility in configuration (x/y) is not smaller than in configurations (y/x) and A under the corner equilibrium relative price in structure XAY. Also equilibrium requires that all type b persons in country 1 prefer autarky to configurations (x/y) and (y/x) and that all individuals in country 2 are indifferent between configurations (y/x) and A and receive a utility level that is not lower than in configuration (x/y). In addition, this structure occurs in equilibrium only if M2ye (0, M2). All the conditions imply Um > Uay, Uax > UaA, UhA > UbM UbA > Uby, U2A=U2y>U2x,

M2ye(0,M2),

where indirect utility functions in different configurations and corner equilibrium relative price are given in Tables 1 and 2. The conditions define a parameter subspace: kt < 16/Vr, k e (4yVr, Min{4f, 4M2y/Ma}), where y = cc/(c + l) c+1 . Within this parameter subspace, the corner equilibrium in structure XAP is the general equilibrium. Following this procedure, we can carry out total cost-benefit analysis for each structure. The total cost-benefit analysis in the second step and marginal analysis of each corner equilibrium in the first step yields inframarginal comparative statics of general equilibrium, summarized in Table 3. From this Table, we can see that the parameter subspaces for structure AAD, XAA, and all structures in which country 1 exports y and country 2 exports x to occur in general equilibrium are empty.

92

Structure AAA PAY XAP DAY XAY XAD XPY XXP XXD XXY XDY YDY DXY DYY

X. Yang, D. Zhang Table 2: Comer Equilibria Numbers of individuals choosing various configurations A4A = Ma, MbA = Mb, M2A = M2 r23~c/k Mix = M2kr/4y. M^ = M a - M2kr/4y, MbA = Afb, M2A = M2 krlyl^1 MK = Ma, MbA = Mb, M2y = MJdAy, M2A= M2- (MJdAy) 21M ax = (Ma+ rM2)/2, May= (M a - rM2)/2, MbA = Mb,M2v = M2 •Mx = Ma, MbA = Mb, M2v = M2 M2r2'^/Afa ^ = M , MbA = Mb, M2y = (Ma + M2)/2 r2'" c Mx = Ma, Mbx = (foM2/4x) - Ma, M2v = M2 yl^lk krly2M M« = Ma, M,x = Mb, M2y = *(Ma + Mb)/4r, M 2 A =M 2 -WM a +M b )/4^ Max = M,, Mbx = Mb, M2y = (Ma + Mb + M2)I2, a1-* M2x = (M 2 - M - M)/2 Mx = Ma, Mbx = Mb, M2v = M2 M2r2i-c/(M!i+ Mb) 21Mx = Ma, Mbx = (Mb - Ma + rM2)/2, Mbv = (Ma + Mb - rM2)/2, M2v = M2 May = Ma, Mbx = (Ma + Mb + rM2)/2, 21Mbv = (Mb - Ma - rM2)/2, M2v = M2 2l-c Mbx = Mb, Max = ( M - Mb + rM2)/2, Mav = (Ma + Mh - rM2)/2, M2v = M2 2l-c Mbx = Mb, M« = ( M + Mb + rM2)/2, Mav = (Ma - Mb - rM2)/2, M2v = M2

Relative price of xto y

When a dual structure occurs in equilibrium, individuals in autarky appear as underemployed or underdeveloped in the sense that they receive none of the gains from trade and income differential between them and other individuals who are involved in the division of labor increases as a result of a shift of equilibrium from AAA to a dual structure. These self-sufficient individuals cannot find a job to work for the market. Figure 1 illustrates the equilibrium structures.6 We say the 6

The coexistence of two equilibria distinguishes the model of endogenous specialization from a standard Arrow-Debreu equilibrium model in which the number of multiple equilibria is always odd.

w

i« -^ £

A

x •*

s.

ft

13

^

^

11

T

^J->

I*

V

us

-

S3

Economic Development, International Trade, and Income Distribution

V V

*

5tv

•

^

V

fs

Si

X

"V.

5?*!'

v 1

x

V -?

ST

93

Table 3: {Continued) Mi/M2 >r

MJ M2< 1

XAP

MJM2 < Vr

AAA

MJ M2 e

ihr)

MJM2 > Vr

k< 4yM2IMa XAP

; < AyMJrMi

AyMJrMi

k> 4rM2/Ma XAY

XAY

XPY

4yMJrM2 PAY

4j-Ma/f-M2 XAY

MJ Mi >r

X

XPY

PAY

X

Economic Development, International Trade, and Income Distribution

95

level of division of labor increases if occurrence of letter A or P decreases and/or the occurrence of letter D, X, and/or Y increases in the equilibrium structure. We can identify five levels of division of labor. The lowest one is associated with structure AAA where all three types of individuals are self-sufficient. The second level is associated with structures PAY and XAP in each of which two types of individuals are self-sufficient. The third level is associated with structures DAY, XAY, XAD in each of which all of individuals of a certain type are selfsufficient. The fourth level is associated with structures XPY and XXP in each of which only part of the population of a certain type are selfsufficient. The fifth level is associated with structures XXD, XXY, XDY, YDY, DXY, DYY, in each of which all individuals are specialized. In this table y= ccl(c + l) c+1 . X stands for configuration (x/y) chosen by individuals of a certain type, Y stands for (y/x) chosen by individuals of a certain type, A stands for autarky, P stands for the partial division of labor where individuals of a certain type are divided between autarky and specialization, D stands for the division of individuals of a certain type between (x/y) and (y/x). Structure XAP involves type I and type II dual structures, structure XXP involves type I and type III dual structures. Structures DAY, XAY, XAD involve type II and type III dual structures. Structures PAY and XPY involve all three types of dual structure. With the definition of division of labor in mind, we can now closely examine Table 3. As trading efficiencies are improved, the equilibrium level of division of labor increases. If the equilibrium level of division of labor increases from level 1 to level 2, then dual structure emerges, or duality of economic structure increases. If the equilibrium level of division of labor increases from 2 to 3, duality decreases as those individuals who are left behind catch up. If the equilibrium level of division of labor increases from 3 to 4, duality increases again as some of individuals in a group that was in autarky before choose specialization. As the equilibrium level of division of labor increases from 4 to 5, duality decreases again as those who were left behind catch up. But it is also possible for improvements in trading efficiencies to generate a jump of equilibrium from level 2 to level 4 in the division of labor. Such a

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jump will reduce duality. A jump from level 3 in the division of labor to level 5 will reduce duality as well. This analysis implies that as different groups of the same type or different types of individuals gradually choose a higher level of specialization one by one, the degree of duality fluctuates. If we use the difference between the highest and the lowest per capita real income i.e. the difference in per capital real income between type a and type b individuals in country 1 as a measure of income inequality, its increase will raise inequality of income distribution and its decrease will reduce inequality. Hence, as different individuals gradually become involved in the network of the division of labor as a result of improved trading efficiencies, the equilibrium degree of inequality fluctuates. It does not monotonically increase or decrease. The relationship between inequality and per capita real income may not be an inverted U-curve. Inserting the corner equilibrium relative price in Table 2 into indirect utility functions in Table 1, we can compare the difference of per capita real incomes (equilibrium levels of utility) of two types of individuals in country 1, considering the parameter subspaces that demarcate the structures. The comparisons confirm our analysis. We first consider the path of evolution in the division of labor: AAA-»XAP->XAY-»XPY —»XXY. The difference in per capita real income between type a and type b individuals in country 1 is 0 in structure AAA. It is positive in structure XAP. It decreases as the equilibrium jumps from XAP to XAY. It then increases as the equilibrium jumps from XAY to XPY. It decreases again as the equilibrium jumps from XPY to XXY. We can find other evolutionary paths of the division of labor with a fluctuating degree of inequality of income distribution. Also, comparisons of ratios of the highest to the lowest real income between corner equilibria generate similar results. All of the results on the evolution of the division of labor, dual structure and trade pattern are summarized in the following proposition, illustrated in Figure 1. Proposition 1: As trading efficiency increases from a very low to a very high level, the equilibrium level of the domestic and international

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division of labor increases from complete autarky in both countries to complete division of labor in both countries. As different individuals are gradually involved in the division of labor, the duality of economic structure fluctuates, generating fluctuation of the equilibrium degree of inequality of income distribution in the less developed country. This degree is neither monotonically increasing nor monotonically decreasing. The relationship between the inequality and per capita real income might not be of an inverted U-curve. The inframarginal comparative statics of general equilibrium in Table 3 can be used to establish two corollaries. The first is that evolution in the division of labor generated by improvements in trading efficiencies will raise equilibrium aggregate productivity. In order to establish the above statement, we consider the aggregate PPF for individual 1 (from country 1) and individual 2 (from country 2). As shown in Figure 2 where c = 2, the PPF for individual 1 is curve BC, that for individual 2 is curve AC. In autarky, the two persons' optimum decisions for taste parameter j6 G (0, 1) are functions of /?. Let f3 change from 0 to 1; we can calculate equilibrium values of Y = yx + y2 and X=X\ + x2 as functions of J3. The values of X and Y for different values of /? constitute curve EGD in Figure 2. The equilibrium aggregate production schedule in structure AAA is a point on the curve, dependent on the value of j5. But the aggregate PPF for the two individuals is the curve EFD. Since in a structure with complete division of labor the equilibrium production schedule is point F which is on the aggregate PPF, the aggregate productivity in a structure with the complete division of labor is higher than in structure AAA. The difference between EFD and EGD can be considered as economies of the division of labor. Following the same reasoning, we can prove that the equilibrium aggregate productivity in a dual structure is lower than the PPF. Hence, Proposition 1 implies that as trading efficiencies are improved, the equilibrium level of division of labor and equilibrium aggregate productivity increase side by side.

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2+2a

2&A

C 2

0

B 4

8

^x

Figure 2: Economies of Division of Labor Based on Endogenous and Exogenous Comparative Advantage

The second corollary is that deterioration of a country's terms of trade and increase of gains received by this country from trade may concur. Suppose that the initial values of parameters satisfy (Ma + Mh)/M2 e (Vr, r), MJM2 e (1, r), kt > I6f/r. Within this parameter subspace, suppose that the initial value of/ satisfies /'< 4){M!i+ Mb)/M2r. Table 3 indicates that structure XPY occurs in equilibrium where country l's terms of trade are p' = y23~c/t'. Now assume that / increases to the value /", so that t"> 4XM a + Mb)/M2. According to Table 3, the equilibrium jumps to structure XXY where country l's terms of trade arep" = M2r21_7(Ma+ Mb). It can then be shown that country l's terms of trade deteriorate as a result of the change in / within the parameter subspace. But this shift of the equilibrium from XPY to XXY increases utility of each individual in country 1 if k'lt' < k"M2r/4f{Ma+ Mh). This has established the claim that there exists some parameter subspace within which the deterioration of a country's terms of trade may concur with an increase of gains that this country receives from trade. There are other parameter subspaces within which changes in parameters may

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generate concurrence of the deterioration of one country's terms of trade and an increase in its gains from trade.7 This corollary generates the following policy implications. In the transitional stage of economic development and globalization, the terms of trade are against the less developed country that has relatively low trading efficiency: some residents in the less developed country receive autarky utility and most gains from trade go to the developed country. There are two policies to change this inferior position. One is to impose a tariff to improve the terms of trade and the other is to improve trading efficiency to expand the network of trade. The former is to increase the share of gains received by the less developed country from a shrunk pie because of the deadweight caused by the tariff. The latter is to provide a greater share of gains from trade by enlarging the pie. The expanded network of division of labor can generate productivity gains. As long as productivity improvements outpace the deterioration of the terms of trade, the less developed country can receive more gains from trade not only because of productivity gains, but also because of more equal division of gains from trade between the countries as all individuals are involved in the international and domestic division of labor. 4. Effects of Trade on Income Distribution in the Developed Country Since we assume that there is only one type of individuals in the developed country that has greater trading efficiency, the model in the previous section cannot explore effects of economic development and international trade on income distribution in the developed country. In this section we extend the model in the previous section by specifying two types of individuals in the developed country. The new model is exactly the same as the previous one except that there are two types of

7

Empirical and theoretical research on economic development and terms of trade can be found, for instance in Morgan (1970), Kohli and Werner (1998), Sen (1998).

Table 4: Equilibrium Structure

0 s > t > 0. For simplicity, we assume that M]a = Mu, = M2a = M2\, = 1/4. Following the method in the previous section, the inframarginal comparative statics of this new model can be solved as in Table 4, where y = ccl (1 + c)c+1, X denotes configuration (x/y), Y denotes (y/x), P denotes a division of individuals of a certain type between specialization and autarky, D denotes a division of individuals of a certain type between (x/y) and (y/x). The first of four letters denoting a structure represents the configuration chosen by type la persons, the second represents that chosen by type lb persons, the third represents that chosen by type 2a persons and the fourth represents that chosen by type 2b persons. Figure 3 gives an intuitive illumination of the evolution of the division of labor and economic development as a result of improvements in trading efficiencies. In this figure, an arrow -» denotes the direction of the evolution of the division of labor, the numbers 1 and 2 next to the arrows denote the two countries and the signs + and - denote increase and decrease, respectively, of the inequality of income distribution in a country. A question mark implies that either an increase or decrease of inequality is possible. From the results we can see that although evolutionary paths of division of labor AAAA—»AADA—»XAYA-» XAYD—»XXYY generates an inverted U-curve of inequality in the less developed country, path AAAA->XAPA->XPYA->XAYP->XXYY generates fluctuation of inequality. Other paths with ? may generate such fluctuation, too. Again, inframarginal comparative statics of general equilibrium confirm Proposition 1 even if income distribution in the developed country is considered. This yields Proposition 2.

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1:0 2: AAAA 1 2:0

AADA

-XAYD

1:.2:XXYY

XAPA

1:2-A

-XPYA

1:+ 2:1

-XXYP

1:2:-

Figure 3: Evolution in Division of Labor

Proposition 2: As trading efficiency is improved, the equilibrium level of division of labor and aggregate productivity increase. The degree of inequality of income distribution in the less developed country and the developed country fluctuates as duality of economic structure fluctuates. The degree of inequality is nonmonotonic and it might not be an inverted U-curve. An interesting feature of our inframarginal analysis of general equilibrium is that economic development and structural changes can take place in the absence of changes of tastes and production functions. All development and trade phenomena can be generated by changes in trading efficiencies that are neither on the demand side nor on the supply side. All recent general equilibrium models of "high development economics" share this feature (see Fujita and Krugman 1995; Krugman and Venables, 1995; Kelly, 1997; Shi and Yang, 1995; Yang, 1991, and Yang and Rice, 1994, for instance). Hence, they explore the limitation of the analysis that explains trade and development phenomena by changes in demand and/or supply sides. 5. Conclusion and Policy Implication In this paper, we have proposed a theory of ratcheting inequality and equality of income distribution using inframarginal analysis of a model of endogenous specialization. According to our view, there are two resources concerning the source of the income inequality. One is the

Economic Development, International Trade, and Income Distribution 103

difference in transaction condition between groups; the other is the relative number of people of different types. This theory accommodates the empirical evidence for the coexistence of positive and negative correlation between inequality and economic development which relates to trade. Because of the trade off between economies of the division of labor and transaction costs, the equilibrium level of the division of labor increases as trading efficiencies are improved. If trading efficiencies are different between countries or between different groups of individuals in the same country, different individuals will be involved in the division of labor sequentially. If some individuals have a higher level of specialization (or commercialization) than other individuals, a dual structure will increase inequality of income distribution. As those poor individuals increase their levels of specialization, the dual structure disappears and inequality decreases. But as the level of division of labor increases further, while some individuals' levels of specialization increase before others' do, a new dual structure will occur, which will disappear as latecomers catch up. This ratcheting process generates fluctuation of the equilibrium degree of inequality as the division of labor evolves. Therefore, there is neither a monotonic relationship between inequality of income distribution and economic development nor an inverted U-curve. In other words, inequality (equality) of income distribution is irrelevant to economic development. We should pay more attention to improvements in trading efficiencies that will enhance aggregate productivity and all individuals' welfare than to the relationship between inequality and economic development (or trade). Our model has policy implications. The government can reduce inequality of income distribution by implementing policies that affect trading efficient parameters. The government may increase trading effciency of a poor population by removing institutional obstacles of free migration from inland or rural areas to coast or urban areas, or by constructing a transportation infrastructure in poor areas. A logical extension of this paper is to add more goods and/or more countries and introduce a tariff into the Ricardian model. More structures may occur in equilibrium and the ratcheting process will be more pronounced. If political economics of income distribution and rent

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seeking is considered, negative effects of unfair income distribution on economic development may be analyzed.

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CHAPTER 6 PURSUIT OF RELATIVE UTILITY AND DIVISION OF LABOR*

Jianguo Wang a and Xiaokai Yang b * ^National University of Singapore

Monash

University

1. Introduction Adam Smith (1776) argued that the development of the division of labor is due to humankind's inclination for exchange. By contrast, Emile Durkheim (1933) argued that the division of labor may be the result of individuals' pursuit of relative economic standing. According to Durkheim, the division of labor is a result of the struggle for existence; different occupations can coexist without being obliged mutually to destroy one another because they pursue different objects. Frank (1984, 1985a,b) also argues that people's pursuit of relative economic standing generates a positive effect on specialization because individuals intend to open new fields to avoid competition and to rank themselves at higher positions. Henri Saint-Simon, Joseph Fourier, Robert Owen, and Karl Marx (e.g., 1844) and many modern management scientists believe that individuals normally dislike specialization but prefer diverse activities. However, the existing theories in the literature of this field have not explored the relationship between changes in preference for relative utility and the division of labor even in a descriptive stage, not to mention in the * Reprinted from Journal of Comparative Economics, 23 (1), Jianguo Wang and Xiaokai Yang, "Pursuit of Relative Utility and Division of Labor," 20-37, 1996, with permission from Elsevier. * We are grateful to two anonymous referees and to Yew-Kwang Ng, Robert Frank, Jeff Borland, Ben Heijdra, and Murray Kemp for helpful comments. Any remaining errors are ours. Ill

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context of general equilibrium. In this paper, we shall develop two general equilibrium models to explore the relationship between the degree of individuals' desire for relative utility and the development of the division of labor. Unfortunately, the neoclassical microeconomic framework cannot be used to achieve this for the following reason. In the neoclassical microeconomic framework in which a production function or technology, either with constant returns to scale or with increasing returns to scale, is employed to describe the relationship between output and inputs, total factor productivity is independent of the level of specialization and of the internal organization of a firm. This paper shall combine Yang's (1990) framework with consumerproducers, economies of specialization, and transaction costs with Wang's (1993) approach to modeling individuals' behavior pursuing relative utility to formalize classical thoughts about the relationship between pursuit of relative position and the division of labor. Yang's framework can be used to investigate the relationship between total factor productivity, the level of specialization, and trade dependence. Wang's (1993) approach, which is built on the literature of pursuit of relative economic standing (see, for example, Basmann (1988), Frank (1984, 1985a,b, 1989), Mezias (1988), Miner (1990), Ng and Wang (1993), Stadt et al. (1985), Schoeck(1969), Veblen (1899), and Veenhoven (1991)), can be used to explore equilibrium implications of pursuit of relative utility. The blend of the two analytical vehicles will enable us to inquire into the relationship between individuals' pursuit of relative utility and the development of the division of labor. Two key concepts in this paper, division of labor and relative utility, should be given here first. Roughly speaking, an economic organizational pattern is said to involve the division of labor if it allocates the labor of different individuals to different activities. Hence, the levels of specialization of individuals and the number of professional activities are two sides of the level of division of labor. Relative utility is the utility ratio between individuals, e.g., the ratio of person i's utility to person/s utility. Note that relativity of utility and relative utility are two distinct concepts. Relativity of utility means that formation of preferences is relative. We have the relativity of utility so long as individuals pursue relative

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economic standing, which includes relative utility but may include something else such as relative income. The main finding of our analysis is that, given the existing structure of individuals' preferences and the degree of increasing returns to specialization, (i) a stronger desire for relative economic standing, e.g., envy, will promote the level of division of labor and productivity if it reduces the differential between preferences for goods or if it weakens an individual's preference for variety of her productive activities; and (ii) the consumption of relative economic standing may or may not generate positive effects on the division of labor, depending on specific conditions. In the first model, with two goods and M consumer-producers (section 2), each person as a consumer prefers diverse consumption and as a producer prefers specialized production. An increase in the division of labor increases, of course, the number of transactions. Since there is a cost occurring from each transaction, i.e., transaction cost, a higher level of the division of labor must generate a higher level of transaction costs. The trade-off between economies of specialization and transaction costs implies that the level of division of labor is determined by transaction efficiency. If transaction efficiency is low, then individuals choose autarky because transaction costs outweigh economies of specialization generated by the division of labor. If transaction efficiency is sufficiently improved, then individuals choose the division of labor because transaction costs are outweighed by economies of specialization. Each individual's utility is a function of the quantities of goods consumed as well as a ratio of her utility level to others' utility levels. The behavior pursuing relative utility is equivalent to behavior with envy. This implies that a person's utility can be expressed as a function of the quantities of all goods consumed and others' utility levels, which have negative effects on her utility level if she pursues relative utility. Each individual maximizes her utility function given others' utility levels under a well-specified and enforced private property system so that stealing is impossible. The equilibrium relative prices of traded goods, relative numbers of different specialists, and equilibrium level of division of labor are then functions of the parameters that characterize pursuit of relative utility. The comparative statics of general equilibrium yield the

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equilibrium implications of individuals' pursuit of relative utility. Since, in the context of preference interdependence, an increase in the desire for relative utility implies a relative decrease in the desire for direct consumption, the increase in the desire for relative position will affect the structure of preferences for goods. The first equilibrium model with two goods is developed in section 2 to show that if an increase in the desire for relative position increases the differential between preferences for various goods, then the equilibrium is more plausibly to be associated with a lower level of division of labor. On the other hand, if an increase in the desire for relative position reduces the differential between preferences for various goods, then a higher level of division of labor is more plausibly the equilibrium. In the second model, section 3, the number of goods is assumed to be N instead of 2, and each individual is assumed to prefer a variety of her productive activities in addition to direct consumption and relative position. An increase in the desire for relative position will change the structure of preferences. If this increase weakens the preference for a variety of activities, then the equilibrium level of division of labor will be increased by the increased desire for relative position. 2. A Model with Two Consumer Goods and M Consumer-Producers 2.1. Specification of the model Consider an economy with M ex ante identical consumer-producers and two consumer goods. The self-provided amounts of the two goods are x and y, respectively. The amounts of the goods sold at the market are xs and ys, respectively. The amounts of the goods purchased in the market are xd and yd, respectively. The fraction 1 - k of a unit of goods purchased disappears in transit because of transaction costs. Hence, a person receives k when she buys one unit of goods. Each person is equipped with a system of production functions and an endowment constraint. xt + xf = Laxi,

yt + yf - Layi

(production function)

Pursuit of Relative

Lxj + Lyi = 1

Utility

115

(endowment constraint), (1)

where Lri is person i's labor share in producing good r. We define the share as person z's level of specialization in producing good r. The parameter a represents the degree of economies of specialization. This system of production functions displays economies of specialization if a > 1. This implies that, for a > 1, labor productivity of a good increases as a person's level of specialization in producing this good increases. Note that economies of specialization are individual specific or increasing returns are localized. rt + r/ is the quantity of good r(r = x, y) that is produced by person /. Due to transaction costs, krf is the quantity that is received by person i from the purchase of rf, and rt + ty is the quantity of good r consumed by person i. Hence, each consumer-producer's utility function is specified as1 f Y U^ix^kxfYiy^kyfY n^T >

(2a)

where J is a set of all individuals except person i, a and/? are respective preference parameters for the two goods, and y is the parameter that represents the preference for relative utility. We assume a + f3 + y -1 for simplicity or for normalization. However, this constant-returns- to-scale assumption is not essential. What is essential is that these preference parameters are interdependent. Even if we assume a + jB + y = g where g may be larger or smaller than 1, our main results will remain the same. 1

The strategic interactions are not taken into account in this model mainly because they do not make much difference since no uncertainty is considered and with a large population the strategic effect is negligible in the models. It may be argued that the utility function in (2a) involves circularity and information problems since, in a model of relative utility, Uj itself depends on Ut. Moreover, it might also be argued that individual;' cannot observe Uj. However, as shown in Wang (1993), the existence of envy is equivalent to the existence of pursuit of relative utility. If we start from a model allowing for real incomes as the source of envy, it can be converted into a model with utilities as the source of envy. Moreover, ex ante identical individuals who can only take others' utility levels as given when they make their optimum decision have no information problems for their decision making. Thus, individual / may know Uj via individual j's real income. Also, (2a) is not an acceptable formation if Mis a variable. However, since Mis a constant here, (2a) is acceptable. See Wang (1993) for the case of a variable M. In fact, relative utility may be more observable than relative income. For example, if we use one's smile frequency as one of her happiness indicators, obviously, her smile frequency is easier to be seen than her income.

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J. Wang, X. Yang

Obviously, our utility function is an application of the Cobb-Douglas log-linear hypothesis, and its functional form is very similar to what Khinchin (1957) discusses. However, we do not use the logarithm of the objective function mainly because its algebra is nearly the same, and the results will be the same. The utility function characterizing the pursuit of relative utility can be rearranged as N -r-|V(l-(M-l)/)

U,=

a

{xi+kx?) {yi+ky?y

(2b)

where M is the population size. This utility function characterizes an individual's envy, which can be derived from behavior pursuing relative position, characterized by the utility function in (2a). Let/? be the price of good y in terms of x, then person i's decision problem is - i/(i-(w-i)r)

a

{xi+kxt) {yi+kyfY\YlU U, = lx,+kxfYlv,+kyfY \\U..j xt+xf -If.,xi' L„.+L = 1

ys=La.

v. + J i

(utilityfunction)

Si

xf + pyf = xf + pyf

(production function)

yi

(endowment constraint) (budget constraint). (3)

2.2. Individuals' decision problems and corner solutions The model is similar to Yang's except that the pursuit of relative utility is allowed. Therefore, our analysis focuses mainly on the relationship between the pursuit of relative utility and the development of division of labor and skips those relationships analyzed by Yang. Due to the great number of ex ante identical individuals and individual-specific economies of specialization or localized increasing returns, no individual has monopoly power to manipulate market supply and demand and, therefore, to influence prices of traded goods. Thus, the regime here is a special Walrasian one: the numbers of individuals selling different goods play the same role as prices do in the conventional Walrasian regime.

Pursuit ofRelative Utility

117

According to Yang (1990), an individual's optimum decision is always a corner solution in the models of this kind. Facing many corner solutions, she must decide which one is optimal. Since the utility levels are discontinuous from one corner solution to another, the conventional marginal analysis works only within a corner solution but not for choosing the optimum decision out of many corner solutions. Yang (1990) has solved this problem by applying the Kuhn-Tucker theorem. He first identifies a set of candidates for the optimum solution, rules out the interior solution and many corner solutions from this set, and then identifies an optimum decision by comparing utility levels across many corner solutions. This total benefit-cost analysis across corner solutions in addition to the marginal analysis of each corner solution is referred to by Buchanan and Stubblebine (1962) as inframarginal analysis. To follow Yang's approach, it is necessary to adopt some important concepts from him. A pattern of economic organization has a higher level of division of labor than another pattern if all individuals' levels of specialization in the former pattern are higher than in the latter. A pattern of economic organization is said to be autarky if every individual produces her own consumption. A combination of zero and nonzero variables that is compatible with the Kuhn-Tucker conditions and the budget constraint is called a configuration. A combination of configurations that is compatible with the market-clearing condition is called a market structure or simply a structure. If n goods are traded, n - 1 relative prices and n - 1 relative numbers of individuals choosing different configurations are determined by n - 1 independent market clearing conditions and n - 1 utility equalization conditions across n configurations. The n - 1 relative number and the n - 1 relative prices define a corner equilibrium in a given structure. All the corner equilibria constitute a set of candidates for the general equilibrium. General equilibrium is defined as a fixed point that satisfies the following conditions, (i) Each individual uses inframarginal analysis, which means total benefit-cost across corner solutions in addition to marginal analysis for each corner, to maximize her utility with respect to configurations and quantities of each good produced, consumed, and traded for a given set of relative prices of traded goods and a given set of the number of individuals selling different goods; and (ii) the

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J. Wang, X. Yang

set of relative prices of traded goods and the set of number of individuals selling different goods clear the markets for traded goods and equalize utility for all individuals. According to this definition, the general equilibrium is the corner equilibrium that yields the highest per-capita real income because it satisfies the two conditions for general equilibrium and other corner equilibria do not satisfy condition (i). We will see in the following sections that computation of the equilibria is rather complicated and computational cost is quite high. In reality, equilibria are approached through some iterative improvements. However, this dynamic process must be described by a dynamic model (see Yang and Ng (forthcoming)) who show in a dynamic model that individuals can search for the efficient pattern of division of labor through the price system over time if pricing efficiency is not too low). In our static model, this dynamic iterative process is not described, and we can see only the outcome of the process. For a person's optimum decision, Yang (1990, p. 206) has proven the following lemma. Lemma: According to the necessary condition for an optimum decision, an individual does not produce and purchase a good at the same time, and sells at most one good. According to this lemma, there are only three configurations in the present model. Following an approach for solving for the equilibrium based on corner solutions developed by Yang (1990), we can solve for the equilibrium in two steps. First, we can solve for a corner solution for each configuration and a corner equilibrium for each structure. Then the general equilibrium is identified from the set of corner equilibria. A. Autarkic configuration Configuration autarky or simply A is a profile of zero and nonzero variables defined by xs=ys - xd=yd = 0, x,y, Lx, Ly > 0. In configuration A, each consumer-producer self-provides all goods she needs. Let xs = ys - xd = yd = 0 in (3), and person /'s decision problem for configuration A becomes

Pursuit ofRelative Utility

P maxC/, = L"*{\-L.T XI

XI

V

)

(

m

V JeJ

Yr'

119

i/(i-(Af-i)r)

(4a) J

The first-order condition dUldLxi = 0 yields the corner solution for structure A as

P

a a+p

n

' x =

L* =1-L* =x a + p' v

a

a + ft

(

y

v

P

(4b)

« + /?

where subscript / is dropped when no confusion is caused. Inserting (4b) into (4a) and using the factt/. = [/. for all i andy yields the per-capita real income in configuration A, UA, given by a a +p

U,

P

\"P

(4c)

a+p

B. Configuration(x\y) Configuration(x|y\ is defined by x, xs,yd> 0, Lx -1, xd - y = ys - Ly = 0 . A person choosing this configuration sells good x and buys good y. Inserting x, xs, yd > 0, Lx = 1 , xd = y = ys = Ly = 0 into (3), person f s decision problem for(x| jA is ,

maxC/,= (\-x!)a(kx;/pY

v y

-|i/(HM~1)r)

YlUj eJ

vJ

(5a)

• )

The first-order condition dUxi jdx\ = 0 yields the optimum decisions xs

- •

P

a + j3

y

d

—.

P

P{cc + PY

r

u

a ^af kp p(a + p) a+p

where Ux(p) is the indirect utility function for(x|_y).

,

(5b)

120

J. Wang, X. Yang

C. Configuration (y\x) Configuration (y\ x) is defined by y, ys, xd > 0, Ly = 1, yd = x = xs = Lx = 0. A person choosing (y\x) sells good y and buys good x. Inserting y, ys, xd >0 , Ly = 1, yd -x = xs = LX = 0 into (3), person z's decision problem for (y\x\ is max £/,

(1-j/fteO 0 \Uuj)

•

( 6a )

The first-order conditionrfC/^./<s^f = 0 yields the optimum decisions a a + /3

r=

,

pa pa ' a + (3

„ ( U =

fi Y kpa a+p {a + P)

(6b)

whereUy(p)is the indirect utility function for(j|x). 2.3. Corner equilibrium and general equilibrium and its comparative statics A. Market structures and corner equilibria There are two structures in the model: autarky denoted by A and the division of labor denoted by D. Due to the absence of trade, the comer solution for autarky is a comer equilibrium. The comer equilibrium for structure D is determined by two conditions: the market clearing condition, which is established by equilibrating market demand and supply, and the utility equalization condition, which comes from the assumption of free entry and utility-maximization behavior. We can solve for the relative price p and the relative number of individuals choosing (x| y) and yy\ x) from the market-clearing and utility-equalization conditions. The utility-equalization condition Ux(p) = Uy(p) yields p* = £(^-«)/(1->') . The market-clearing condition Mxxs - Myxd yields M^ - ap/p ,2 where Mx and My are the respective numbers of 2

Mmust be an integer. Yang and Ng (1993) have shown that if the integer condition is not satisfied, the equilibrium may not exist.

121

Pursuit of Relative Utility

individuals selling good x and good y and M^ = Mx/My . The market-clearing condition for goody is not independent of that forx due to Walras' law. Inserting the corner equilibrium level ofp into indirect utility function Ux(p) or U (p) yields the corner equilibrium utility level for structure D as UD =

\a

P a+P) \a+P a

V

fc2aft/(a+fi)

(V)

B. General equilibrium and its comparative statics Since the general equilibrium is the corner equilibrium that generates maximum per-capita real income, which can be identified by comparing the utility levels across structures, hence, having compared UD with UA, we have

uD>uA

iff

r

\

(a-\)(a+p)l7P

a k>k° = a + (3

P

\

a +p

(a-\)(a+p)l2a

(8)

Expression (8) yields the following proposition. Proposition 1: Equilibrium is the division of labor if the coefficient of transaction efficiency k> k° and is autarky ifk < k°. Figure 1 provides an intuitive illustration of Proposition 1. Two individuals in Figure la are used to illustrate autarky. In fact there are more individuals than shown in Figure 1. Circles represent configurations. The lines represent flows of goods self-provided. In autarky there is no market demand and supply. Productivity is low because of a low level of specialization. However, transaction costs do not exist because people do not do business with each other. Figure lb represents the division of labor, structure D. The lines represent the flows of goods. For structure D, there is a local business community composed of two persons. There are two markets, one for x and the other for j / .

122

J. Wang, X. Yang

a

b

A yC 3 33f Xs

>V )

Figure 1: (a) Autarky and (b) the Division of Labor

There are two distinct professional sectors. The per-capita output of each good is higher than in autarky because of a higher level of specialization for each person. But transaction costs are higher than in autarky too, because each person must undertake two transactions to obtain the necessary consumption. Moreover, the degree of interdependence between individuals, the degree of production concentration of each good, and the degree of integration of society are higher for structure D than in autarky. The most important implication of the inframarginal analysis and framework with consumer-producers, economies specialization, and transaction costs is to predict many macroeconomic phenomena on the basis of a sound microfoundation; see Yang and Ng (1993, Chaps. 17 and 18). For the static simple model in this paper, it is predicted that aggregate demand and supply, the extent of the market will shrink as transaction efficiency k is reduced due to, say, oil crisis. According to our theory, demand and supply and the division of labor, and the division of labor and the extent of market are two sides of the same coin. Hence, macroeconomic phenomena that relate to aggregate demand can be explained by a change in the parameter of pursuing relative position through its effect on the equilibrium level of division of labor. Unfortunately, due to the limitation of space, we need another paper to investigate the macroeconomic implications of the model.

Pursuit of Relative

123

Utility

The effects of the degree of individuals' desire for relative utility, y, on the development of the division of labor can be seen from the effects of a change in y on k° via its effect on a and fl . If the effect of an increase in y on k° is negative (positive), the requirement in transaction efficiency for the division of labor to be the equilibrium is lower (higher). A total differentiation of k° with respect to y yields dink0 da

d\nk°

d\nk° d/3 —, (9) dy da dy dfi dy wheredfijdy = -(dajdy)-1 due toa + fi + y-l. The division of labor is more plausibly the equilibrium for a stronger desire for relative utility3 if d In k°/dy < 0 and less plausibly to be equilibrium if J In k°jdy > 0. A differentiation of k° in (8) yields 51nA:° a-\ '.ta-S^-Zb. » \ >0 2 da fS a + j3 a a + fi =

+

dink0

and^-^-O

dfi

if£a>0,y

V(10)

'

which implies that (i) dlnk°jdy 0 , and/or d0/dy / ? , da/dy0 ; (ii) dlnk° Idy>0 , if a fi , da/dy>0, and/or dfijdy < 0 ; (iii) dlnk°jdy-0 ifa = fl, no matter what sign dajdy and dfijdy are; and (iv) for the case where both dajdy

(endowment constraint)

heH

prx\r = ]T Pgx?g

(budget constraint),

(16)

where pr is the price of good r, and the decision variables are xir,xl,xfg,Ljr,Lih,xih,and n. Having inserted all constraints into [/and differentiating[/, with respect toLir,xsir,xfg, individual optimal decisions are given by

Lir=n{\-Lc)/N, X* =Kl(n-\)

= Llln,

x°r=(n-l)Li/n xih

=[(l-Lc)/Nj

128

J. Wang, X. Yang 4 =(Pr/Pg)xl/(n-l)

= {Pr/Pg)Ll/",

VgeG,

(17)

where Lc = c(n -1) . Inserting these optimal solutions into Uj , and noticing pr = pg in a symmetric model, Ui can be expressed as a function of n \an

U,=

ka[n-\)

'l-c(n-l)'

N

aa(N-n)

N

x(N-n + iy.

(18)

Maximizing Ui with respect to n yields the optimal number of traded goods, n , for an individual selling good r dhxU,L = -a(]nn*+\) dn

aaNc — -+ • l-c(«*-l)

N~n'+\ +a\nk = F^0.

(19)

It is not difficult to identify the sign of dn*/dy from (19). Since the larger optimal number of traded goods, n , means the higher level of division of labor, we can use the sign of dn*jdy to investigate the effect of a stronger desire for relative utility on the level of division of labor. Because n cannot be solved explicitly, we must apply the implicit function theorem to (19) so that dn* __ dF/dy dy dF/dn'

(20)

where y = \- aN - a . The second-order condition for interior optimal solution n is dF/dn* = d2 In Ul /dn2 < 0 . Since we assume a, a, and^ are interdependent and the sum of them is a constant, we have F = F[a(y), o(y)]. Umce,dF/dy = (dF/da)(da/dy) + (dF/da) (da/dy). Differentiating F and using F = 0 in (19) yields dF/da = a/a (N - n +1) > 0. Also dF/da < 0 can be derived from (19). Therefore, dF — 0, dy

da if

da . dy dy

(21b)

Pursuit of Relative

129

Utility

Expressions (21a) and (21b) together with the second-order condition for the interior n , implies that dn*

n

.. da

< 0 if

dy This leads us to

n

da

0>

.

(22)

dy

Proposition 3:5 A stronger desire for relative utility has positive effects on the level of division of labor n {or the number of traded goods) if it weakens the preference for variety of each person's productive activities and increases the preference for direct consumption of goods. Proposition 3 is intuitive. If a stronger desire for relative utility weakens the preference for the variety of activities, a decrease in a, individuals will prefer more specialization so that the equilibrium level of division of labor is promoted. Figure 2, where it is assumed that m = N=A, provides an intuitive illustration of the evolution of division of labor when an increase in the degree of the desire for relative utility reduces the degree of the desire for variety of a person's productive activities and increases the desire for direct consumption of goods. The lines signify goods flows. The small arrows indicate direction of goods flows. The numbers beside the lines signify goods involved. A circle with number / signifies a person selling good i. Figure 2a denotes autarky, where each person self-provides four goods, i.e., n - 1 = 0, because of a low degree of the desire for relative utility that leads to a high degree of the desire 5 Compared to Proposition 2, no difference in preferences across consumer goods is assumed for the general model from which Proposition 3 is drawn. That is to say, an individual has the same preference for all consumer goods. Hence, the effects of the size of preference differentials between different consumer goods on the development of division of labor cannot be investigated in this general model. Otherwise, the conclusions from this more general model must be more ambiguous. This assumption is made since our main purpose is to focus on the effects of the size of preference differential between all consumer goods as a whole and the variety of her productive activities on the development of division of labor (in fact, here our suggestion is based on, as many economists have argued, that "individuals could also have preferences for variety in their activities"). Also if we assume preferences are different across N goods, the model is not tractable. Of course, in reality, the preferences are not symmetrical and solving for the equilibria is a dynamic process of iterative improvement.

130

J. Wang, X. Yang

'$>,'$>.

'$'$

Figure 2: The evolution of division of labor, (a) autarky, n = 1, four communities; (b) partial specialization, n = 2, two communities; (c) complete specialization, n = 4, the integrated market.

for variety of productive activities. Figure 2b denotes partial specialization, where each person sells one good, buys one good, trades two goods, and self-provides three goods, i.e., n = 2, because of a higher degree of the desire for relative utility that leads to a lower degree of the desire for variety of activities. Figure 2c denotes extreme specialization, where each person sells and self-provides one good, buys three goods, and trades four goods, i.e., n = 4, because of a very high degree of the desire for relative utility that leads to a very low degree of the desire for variety of activities. Note that here there are multiple equilibria but all of them yield the same utility and symmetric comparative statics. Since our model is symmetrical, only the number of traded goods is relevant, and patterns of

Pursuit of Relative

Utility

131

trade do not make any differences. For instance, for an equilibrium structure in Figure 2b, it is indeterminate if goods 1 and 2 or goods 2 and 3 are traded. Also, it is indeterminate who is specialized in which profession. An infinitesimal differential in parameters of tastes, production, and transactions across goods and individuals will rule out the multiplicity of equilibria. Since there exists a positive relationship between the level of division of labor and productivity and trade dependence as shown in Yang (1990), Yang and Borland (1991), and Borland and Yang (1992), the following corollary can be derived from Proposition 3. Corollary: A stronger desire for relative utility has positive effects on labor productivity and trade dependence if it decreases the desire for a variety of each person's productive activities and increases the desire for quantities of consumption goods. It can be shown that as long as economies of specialization are individual specific, that is, increasing returns to specialization are localized, and individuals pursue their relative utility under perfect competition, the equilibrium is Pareto optimal. For detailed proof of this proposition, see Yang and Ng (1993) and Wang (1993).6 4. Concluding Remarks We have explored the relationship between the pursuit of relative utility and the development of division of labor in two equilibrium models in this paper. It is shown that the equilibrium level of division of labor depends on some critical level of transaction efficiency, which is in turn determined by preference parameters for consumption, for a variety of activities, and for relative utility. An increase in the degree of the desire for relative utility will change the structure of preferences for consumption 6

According to Debreu's theory of value, the existence of global increasing returns and/or externalities distorts market. However, Yang and Ng (1993) and Wang (1993) show that (i) if increasing returns to specialization instead of to scale and are localized to each of many individuals who have no monopoly power (this is just the case in this paper), a Walrasian regime prevails; and (ii) relative utility effects do not distort market under perfect competition but relative consumption effects and relative income effects do.

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and for a variety of activities. This increase will generate positive effects on the development of division of labor if it reduces the differential between preferences for different goods or if it reduces the desire for a variety of each person's productive activities. Our finding also means that the consumption of relative position may or may not generate positive or negative effects, depending on specific conditions. Thus, some of the intuitions and conclusions of Marx, Durkheim, Veblen, and other economists on the relationship between pursuit of relative position and division of labor are not correct without qualification.

References Basmann, Robert L., Molina, David J., and Slottje, Daniel J., "A Note on Measuring Veblen's Theory of Conspicuous Consumption." Rev. Econom. Statist. 70, 3:531-535, Aug. 1988. Borland, Jeff, and Yang Xiaokai, "A New Approach to Economic Organization and Growth." Amer. Econom. Rev. 82, 2:460-482, May 1992. Buchanan, James, and Stubblebine, Wm G., "Externality." Economica 29, 116:371-384, Nov. 1962. Durkheim, Emile, The Division of Labour in Society. New York: The Free Press, 1933. Frank, Robert H., "Are Workers Paid Their Marginal Products?" Amer. Econom. Rev. 74, 4:549-571, Sept. 1984. Frank, Robert H., "The Demand for Unobservable and Other Non-Positional Goods." Amer. Econom. Rev. 75, 1:101-116, Mar. 1985a. Frank, Robert H., Choosing The Right Pond: Human Behaviour and The Quest for Status. New York: Oxford Univ. Press, 1985b. Frank, Robert H., "Frames of Reference and The Quality of'Life." Amer. Econom. Rev. 79, 2:80-85, May 1989. Khinchin, Aleksandr I., Mathematical Foundations of Information Theory. New York: Dover, 1957.

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Marx, Karl, In Dirk J. Struik, Ed.; Martin Milligan, Trans., Economic and Philosophic Manuscripts of 1844. New York: International Publishers, 1971. Mezias, Stephen J., "Aspiration Level Effects: An Empirical Investigation." J. Econom. Behav. Organiz. 10, 4:389^100, Dec. 1988. Miner, Frederick C , Jr., "Jealousy on the Job." Personnel. J. 69, 4:88-95, Apr. 1990. Ng, Yew-Kwang, and Wang, Jianguo, "Relative Income, Aspiration, Environment Quality, Individual and Political Myopia: Why May the Rat-Race for Material Growth Be Welfare-Reducing." Math. SocialSci. 26, 1:3-23, July 1993. Schoeck, Helmut, Envy: A Theory of Social Behaviour. New York: Harcourt, Brace & World, 1969. Smith, Adam, Reprint, In E. Cannan, Ed. An Inquiry into The Nature and Causes of The Wealth of Nations. Chicago: Univ. of Chicago Press, 1976. Stadt, Huib van de, Kapteyn, Arie, and Geer, Sara van de, "The Relativity of Utility: Evidence from Panel Data." Rev. Econom. Statist. 67, 2:179-187, May 1985. Veblen, Thorstein, The Theory of The Leisure Class. New York: Macmillan Co., 1899. Veenhoven, Ruut, "Is Happiness Relative?" Social Indicators Res. 24, 1:1-34, February 1991. Wang, Jianguo, "Pursuit of Relative Economic Standing," Ph.D. Dissertation, Department of Economics, Monash University, 1993. Yang, Xiaokai, "Development, Structure Change, and Urbanisation." J. Develop. Econom. 34, 1-2:199-222, Nov. 1990. Yang, Xiaokai, and Borland, Jeff, "A Microeconomic Mechanism for Economic Growth," J. Polit. Econom. 99, 3:460^182, July 1991. Yang, Xiaokai, and Ng, Yew-Kwang, Specialization and Economic Organisation: A New Classical Microeconomic Framework, Contributions to Economic Analysis Series. Amsterdam: North Holland, 1993. Yang, Xiaokai, and Ng, Yew-Kwang, "Specialization, Information, and Growth: A Sequential Equilibrium Analysis."/. Econom. Behav. Organiz., forthcoming.

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Part 4

Urbanization

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CHAPTER 7 DEVELOPMENT, STRUCTURAL CHANGES AND URBANIZATION*

Xiaokai Yang* Monash

1.

University

Introduction

As far as the problem of production was concerned, Adam Smith (1776) and Allyn Young (1928) emphasized the productivity implications of economic organization (the division of labor). Neoclassical microeconomics cannot explore such implications for the following reason. For production functions with constant returns to scale, an agent's productivity of a good is not greater when he produces only this good than when he produces many goods. On the other hand, production functions with increasing returns to scale cannot be used to characterize the level of specialization within a firm. The concept of economies of scale presupposes a complete separation of pure consumers from pure producers. 'Scale' relates to a firm which is a pure producer, but is irrelevant to a pure consumer. The separation of pure producers from pure consumers is a basis of Debreu's theoretical framework and neoclassical microeconomics. This artificial separation has perhaps misled economic theory. In autarky, there is neither a pure consumer nor a pure producer; each individual is a

Reprinted from Journal of Development Economics, 34 (1-2), Xiaokai Yang, "Development, Structural Changes and Urbanization," 199-222, 1990, with permission from Elsevier. * The author is grateful to Gene Grossman, Barry Nelebuff, Edwin Mills. Yew-Kwang Ng, Roslyn Anstie, and two referees for their comments. Thanks also go to the Ford Foundation and the Open Society Fund for financial support. The author is responsible for the remaining errors. 137

138

X. Yang

producer/consumer. The division of labor will increase the portion of a person's production that is not consumed by himself (i.e. the portion sold to other people) and increase the portion of a person's consumption that is not produced by himself (i.e. the portion purchased from other people). We can view this change as an increase in the degree of separation between production and consumption though each person is a producer/consumer even in the division of labor. The degree of such separation depends on the level of division of labor (or inversely on the degree of self-sufficiency). As to endogenizing the level of division of labor and thereby the degree of such separation, Debreu's framework is irrelevant since in his framework, pure consumers are completely separated from pure producers and the degree of the separation of consumption from production cannot be defined. In order to capture the ideas of Smith and Young, this paper specifies production functions for each producer/consumer such that an individual's productivity increases with the level of specialization and the aggregate transformation curve for the whole economy depends positively on the level of division of labor. There are several implications of this method of specifying production functions for the theory of equilibrium. First, each individual is a producer/consumer. He must decide how many goods are self-provided, i.e. what is the level of specialization. Hence, the model in this paper can be used to endogenize the level of division of labor. According to conventional microeconomics, pure consumers cannot choose the level of specialization since they must buy all goods from firms. Second, a Cobb-Douglas utility function is specified for each producer/consumer. Consequently each individual as a consumer prefers diverse consumption and as a producer prefers specialized production. This implies that the division of labor will incur great transaction costs. Therefore, there is a trade-off between economies of specialization and transaction costs. In other words, our method of specifying production functions makes the level of division of labor crucial for productivity; while transaction efficiency is critical for the determination of the level of division of labor. Because of increasing returns to specialization, the production possibility frontier (PPF) is associated with extreme

Development,

Structural Changes and Urbanization

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specialization. Extreme specialization will, however, incur prohibitively great transaction costs since people prefer diverse consumption. Hence, the welfare frontier may differ from the PPF.1 A natural conjecture is that a competitive equilibrium will balance the trade-off between economies of specialization and transaction costs, and that improvement of transaction efficiency will move the equilibrium closer to the PPF, resulting in an increase in the division of labor. The major purpose of this paper is to prove this conjecture. In other words, Smith's conjecture of the 'invisible hand' and his insights into increasing returns to specialization will be reconciled in this paper. A crucial assumption leading to this result is that labor is specific for each person who is able to produce all goods. This assumption combined with the assumption of free entry ensures the first welfare theorem even if there exist increasing returns to specialization. Intuitively, we can see that if increasing returns to specialization are specific for each individual, economies of scale are limited. Since production functions are specified for each producer/consumer, prices are determined by the numbers of individuals selling different goods. This number cannot be manipulated by any individual because of the assumption of free entry and the assumption that each person is able to produce all goods. Hence, nobody is able to manipulate prices. Nevertheless, if labor can be divided in fine detail and the population size is very large, economies of division of labor may be very large. Hence, a competitive market may be compatible with the substantial economies of the division of labor. Therefore, we can use this method to develop the concepts of equilibrium and Pareto optimum in relation not only to the resource allocation for a given level of division of labor, but also to the determination of the level of division of labor and productivity. The implications of this method for other fields of economic analysis are important. The level of division of labor based on increasing returns to specialization is intimately related to the extent of market, trade dependence, trade pattern, market structure, and economic structure, so these can be made endogenous in our model. Productivity is related to 1 According to conventional microeconomics, the welfare frontier coincides with the PPF.

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the level of division of labor which depends on transaction efficiency, which is in turn affected by urbanization, government policies, and institutional arrangements. Hence, our model can be used to investigate the impacts of urbanization, government policies, and institutional arrangements on the equilibrium level of division of labor (related to the market structure, trade dependence, and so on) and productivity. This paper shows that the equilibrium trade volume depends positively on the absolute degree of increasing returns to specialization in production and transaction, and negatively on the average distance between a pair of neighbors. The trade pattern is determined by the relative degree of increasing returns to specialization in producing different goods and relative preference for different goods. In addition, it is shown that increases in diversification of the economic structure, concentration of production, integration of the economy, specialization, and the output share of roundabout productive activities are different versions of the evolution of division of labor resulting from improvements of transaction efficiency which are in turn caused by urbanization, liberalization policies, or changes in institutional arrangements. Many economists have proposed similar ideas. Nevertheless few among them have been successful in formalizing them. On the other hand, the formal models proposed by mainstream economists are often inconsistent with these ideas. This may be due to the difficulty of formalizing the ideas of Smith and Young. For example, many economists [see, e.g. Helpman and Krugman (1985) and Herberg and Tawada (1982)] point to problems which are considered to make an equilibrium model with increasing returns to specialization and transaction costs unmanageable. Such problems include the issue of corner solutions based on increasing returns to specialization, the problem of infinite combinations of individual corner solutions in solving for equilibrium, notorious complications in dealing with indexes of variables in models with transaction costs, and the problem of existence of equilibrium. Formalizing the notion of increasing returns to scale is much easier than formalizing the notion of increasing returns to specialization. This might explain why it is hard to find microeconomic equilibrium models that formalize the theory of production proposed by Smith and Young.

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In this paper, we try to formalize the essence of the ideas of Smith and Young as well as to keep an equilibrium model tractable. Our techniques for achieving these two goals are to specify a specific transaction technology and to devise a multiple-step approach to handling the issue of combinations of corner solutions. In order to get around the problem of the existence of equilibrium, we propose a specific model. Using this model, we can prove the existence of equilibrium although it is impossible to reach a general conclusion on this issue for a model with increasing returns to specialization. Fortunately, these measures are not only effective in keeping the model tractable, but also useful in working out the meaningful comparative statics of equilibrium. Many interesting economic phenomena which are not addressed in the conventional theory can be explained by the kind of model in this paper. This paper is organized as follows. Section 2 sets out a model with three goods. Sections 3-7 develop a multiple-step approach to handling the model with increasing returns to specialization. Section 3 solves for the corner solutions for the individual decision problem. Section 4 solves for all candidates for equilibrium in various market structures. Section 5 solves for the restricted Pareto optimum in each market structure and the full Pareto optimum. Section 6 investigates the relationship between equilibria and the Pareto optima. Section 7 solves for an equilibrium and investigates its comparative statics. Some simple conclusions are summarized in the final section. 2.

A Model with Three Goods

Let us first consider an economy with M consumers/producers and three consumer goods. The self-provided amounts of these goods are x, y, and z, respectively. By self-provided we shall mean that quantity of a good produced by an individual for his own consumption. The amounts of these goods sold at the market are xs, /, and z\ respectively. The amounts of these goods purchased in the market are xd, y\ and z , respectively. An 'iceberg' type of transaction technology is characterized by the coefficient k. Fraction i of a shipment disappears in

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transportation. Thus, (1 - k)xd, (1 - k)yd , and (1 - k)zd are the amounts a person receives from the purchases of three goods, respectively. Furthermore, we assume that (1 - k) depends on the quantity of labor used in transaction. (1 - k) can be viewed as transaction service. Such services are categorized into self-provided ones and traded ones. Let 1 - k = T + Td where T is the self-provided quantity of transaction service and T1 is the quantity purchased of transaction service. The more transaction service T + Td, the greater portion of a purchase is received by its buyer. Here, T + Td < 1. Signifying the quantity sold of transaction service by Ts , transaction technology and production functions are thus given by x + xs=Lax,

y + ys=Lby,

z + z"=Lcz,

T + TS=L'T,

(2.1a)

where x + x5, y + ys, z + zs, and T + TS are the output levels of four goods and service, respectively. Ls is the amount of labor used in producing good (or service) s where s - x,y,z,T . (2.1) is assumed to be identical for all individuals. In such iso-elasticity production functions parameters a, b, c, and t characterize the returns to specialization. If a, b, c,t>\, then there are increasing returns to specialization. Adopting the concept of localized technology proposed by Sah and Stiglitz (1986), we assume that the total quantity of labor available for an individual is specific for him. Let this quantity be one; there is an endowment constraint of the specific labor for an individual: Lx+Ly+Lz+LT=l,

0G(xly) = r"G(ylx) = U(y/x),

(4.3)

where G's depend on a, b, c, a, ft, y, t, and s. The intersection of U(x/y) and U(y/x) determines a corner equilibrium value of r. Inserting the value of r back into the utility function, (4.3) gives a corner equilibrium utility U*, which is real income as well as the real returns to labor in this market because of the assumption that each person has one unit of labor.

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For the category of non-basic markets, there is the following proposition: Proposition 2: There does not exist a corner equilibrium for any non-basic market. For a non-basic market, there are m structures and n traded goods, and m > n. According to Appendix A, in any optimal structure an individual sells at most one type of good. This implies that the types of structures selling the same type of good exceed one if m > n. The necessary conditions for a corner equilibrium lead to m - 1 conditions of utility equalization. These m - \ equations contain n - 1 relative prices. Noting the fact that these m - 1 equations are log-linear, nonhomogeneous, and independent of one another, this system has no consistent solution since m > n. Therefore, Proposition 2 has been established. Noting the following two points, we can prove the existence of a corner equilibrium for a given basic market. (A) For a given basic market with n traded goods and n component structures, the individual supplies of all traded goods are constants and the individual demands for all traded goods are functions of relative prices. The indirect utility functions are determined only by relative prices of the traded goods. Letting the n indirect utility functions equal one another, we obtain (n - 1) log-linear equations containing (n - 1) relative prices. Again, noting that these equations are non-homogeneous and independent of one another, we can solve for a vector of log-relative prices. The relative prices are positive. (B) Given the relative prices solved in (A), (n - 1) independent market clearing conditions can be transformed into a system of equations containing (n - 1) relative numbers of individuals selling the different goods. This system looks like a system of linear equations associated with an input-output system. For example, in the market with three traded goods, such a system is

Development, Structural Changes and Urbanization

-filia + -y(a + r)

1

b^M^y s

V

151

(4.4a)

Mzxpzxz

If the number of goods traded by individuals selling good z is less than that traded by the whole market, e.g. structure (z/x) trades two goods and structures (x/y, z) and (y/x, z) trade three goods, then (4.4a) becomes MyxPyxys

1

fx^

(4.4b)

where My= MJMj is the relative number of individuals selling good i to those selling good j . Myx and Mzx are unknown. It is easy to see that the solution vector (Myx, Mzx)' for this kind of system of linear equations is positive. Considering the positiveness of equilibrium relative prices shown in (A), the positiveness of the solution of (4.4) guarantees the existence of corner equilibrium in a basic market. This leads us to Proposition 3:

There exists a corner equilibrium for any basic market.

There are multiple corner equilibria. All these corner equilibria comprise a set of the candidates for equilibrium. These candidates satisfy the following conditions (i) All excess demands for goods are zero for uniform positive relative prices of goods and a uniform positive price of labor (the real returns to labor). (ii) Individuals maximize their utilities for given prices and for a given basic market. Note that we have not yet imposed full maximization by individuals at the moment since this can be done only when all candidates for equilibrium are enumerated. By enumerating all corner equilibria, we will find the Pareto optimum corner equilibrium and prove that it is a full equilibrium in the next two sections.

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X. Yang

The Pareto Optimum Corner Equilibrium

This section first proves that a corner equilibrium is Pareto efficient for a given basic market. Then, we prove that all non-basic markets are neither equilibria nor Pareto optimal. Finally the relationship between the real return to labor in a corner equilibrium and the preference and technology parameters will be investigated. If the market is P as shown in Figure 1(a), i.e., some individuals choose structure (x/y) and other individuals choose structure (y/x), then we can derive the necessary conditions for a restricted Pareto efficient allocation from the problem in Appendix B. By 'restricted', we shall mean that the market is given. From the necessary conditions of that problem, it can be shown that the corner equilibrium found in sections 3 and 4 is the restricted Pareto optimum for the market. Following this procedure, we can show that each corner equilibrium is the restricted Pareto optimum for a given basic market. Moreover, Appendix B has proven Proposition 4: Each corner equilibrium is Pareto efficient for a given basic market and all non-basic markets cannot satisfy the necessary conditions for the Pareto optimum. Combining this with Proposition 2, we conclude that neither equilibrium, nor the Pareto optimum is associated with a non-basic market. This leads us to Proposition 5: Both candidates for an equilibrium and for the Pareto optimum market are associated with some basic market. By comparing real returns to labor in different basic markets, we can find the Pareto optimum corner equilibrium: it is the corner equilibrium with the maximum real return to labor. This Pareto optimum corner equilibrium depends upon t, s and the differences among (a,a), (b, /?), (c, y). Given the number of traded goods, it can be shown that (i) the corner equilibrium including as traded goods those with larger preference

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parameters is Pareto superior to that including as traded goods those with smaller preference parameters; and (ii) the corner equilibrium including as traded goods those with a high return to specialization is Pareto superior to that including as traded goods those with a small return to specialization, (i) and (ii) lead us to Proposition 6: The corner equilibrium including as traded goods those with a large preference parameter has a greater real return to labor, and the corner equilibrium including as traded goods those with higher returns to specialization has a greater real return to labor. Proof: See Appendix C. Assume that a = b = c = t and a = /? = y, then differences in the real returns to labor in various markets depends only upon the number of traded goods and service. The composition of traded goods has no effect on the real returns to labor. With this assumption, there are five possible market configurations: (1) Autarky. We refer to it as market A. This market consists of structure (x, y, z). (2) Partial division of labor in production. We refer to it as market P. This market consists of structure (x/y) and (y/x). (3) Complete division of labor in production. We refer to it as market C. This market consists of structure (x/yz), (y/xz) and (z/xy). (4) Partial division of labor in production and transaction. We refer to it as market PT. This market consists of structure (x/yT), (y/xT), and (Tlxy). (5) Complete division of labor in production and transaction. We refer to it as market CT. This market consists of structure (xlyzT), (ylxzT), (z/xyT), and (T/xyz). Having solved the corner equilibria in these five markets and compared the real returns to labor, we obtain

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Proposition 7: (1) For s (the average distance between a pair of neighbors) < 0.54, the real returns to labor in markets P and PT cannot be the maximum. (i) Market A has the maximum real return to labor if a < 0.41 0.251og(l-1.414s). (ii) Market C has the maximum real return to labor z/0.41 - 0.251og (1 - 1.4145) < a < 2.77 - 1.091og (1 - 1.732*). (Hi) Market CT has the maximum real return to labor if 2.11 - 1.091og (1 - 1.7325) < a. (2) For 0.54 < s < 0.58, the real return to labor in market P cannot be the maximum. (i) Market A has the maximum real return to labor if a < 0.41 0.25 log (1-1.4145). (ii) Market C has the maximum real return to labor ifOAl - 0.251og (1 - 1.414s) 0.8U/dys = Ogives the necessary condition for the optimum ys . Inserting this condition into 8U/dLx and differentiating the resulting first-order derivative with respect to Lx again yields d2U/8L2x>0

if 8U/dLx=0.

(A.3a)

This implies that the optimum value of Lxis either zero or one if xd > 0. Lx = 1 conflicts the assumptions^ 0, implied by the assumption xd > 0, and requiring Ly > 0. Hence, (A. 3 a) means the optimum value of Lx is zero if xd> 0. (A.3b) In other words, an individual will not produce and purchase a good at the same time. This is just the first part of the above proposition. Next, the second part of the proposition is proven. Without loss of generality, we assume ys, zs > 0 . Because / cannot be negative for i = x, y, z, T and because i" = 0 if id > 0 due to (A.2), it can be shown that yd = zd = 0 if ys , zs > 0 . dU/dys = dll/dzs = 0 gives the necessary conditions for the optimum ys and zs. Inserting these conditions into dU/dLy and differentiating the resulting first-order derivatives with respect to Ly again yields d2U/dL2y>0

if dU/dLy=0.

(A.4a)

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This implies that the optimum value of Ly is either zero or one. Ly = 0 conflicts the assumption ys > 0, and Ly = 1 conflicts the assumption z > 0. Hence, (A.4a) means ys and zs cannot be positive at the same time.

(A.4b)

In other words, an individual will not sell two goods at the same time. This is just the second part of the above proposition. Appendix B: Corner Equilibrium and the Restricted Pareto Optimum If market is P in Figure 1(a), i.e., some individuals choose structure (x/y), signified by subscript 1 and other individuals choose structure (y/x), signified by subscript 2, then we can obtain the necessary conditions for the Pareto efficient allocation from the problem below: d

max. z

i

Ux=x?\T(\-K)y?Yzl, i

(B.l)

Z

JV*;? > 2>>'2. >.. --• T S.t.

A | ~T J l | — Ltx,

Z | — ^1

Ljj

l^x J

y2+y°2=L»y,z2=(l-LT-Lyy,

,

T = LT,

MjXf = M2x( or rx{ = x(, M2y\=Mxydx

or ys2=ryf,

U2 = yp2 U(l - K)xd2 \a z\>u. where u is a constant. The first-order conditions are MRSI=MRSIJ\ MRSlx = MRTz\, MRSl=MRT$, MRS\y=MRFfy, MRSyx=MRTyx, 2 MRTL=MRTIT , yx yx

(B.2) MRSl=MRT>, MRS^MRT^,

(B.3) (B.4)

where MRS* =(dUk/dxJ)/(dUk/dxi) is the marginal rate of substitution between good i and j for individuals choosing structure k; MRT* = (dxJdL)/(dXj/dL) is the marginal rate of transformation between good i and j for individuals choosing structure k.

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These are just the necessary conditions for corner equilibrium. From the corner equilibrium for market P, solved in subsection LB andI.C., we can derive that MRS^/T = Py/px=

MRS%T or MRS\X =

MRS^.

This is just (B.2). From this equilibrium we can obtain (B.3) and (B.4) too. Actually, the relative number of individuals, r, is also a decision variable in the problem of Pareto efficient allocation. However, from the problem (B.l), we can solve for the unique xf, x{ , ys2, and yf; they determine a unique r through the market clearing condition since the number of traded goods equals the number of structures in the basic market. For a non-basic market, e.g., that consists of structures (z/xy), (x/yz), (z/xy), and (x/y), we have a similar problem of Pareto optimal allocation as (B.l) which maximizes U(z/xy) subject to all individual production functions, transaction technologies, and balance between consumption and production giving that other utilities are not smaller than some constants. However, the relative numbers of individuals in structures (x/yz), and (x/y) are flexible. Name these two numbers as M\ and M2, respectively. For the relevant Lagrange function associated with this restricted optimization problem, LA, we can show dLA/dMt \M _o < 0 if AdU (z/xy)/dx + BdU (z/xy)/by < 0, 8LA/8M2 \M=0 0, (B.5b)

where we use the facts thatM = ^TM(. , M is the total number of individuals, and Mt is the number of individuals in structure /. LA is the Lagrange function. (B.5) implies that for any value ofAdU(z/xy)/dx + B8U(z/xy)/dy, the Pareto optimum requires either Mi = 0 or M2 = 0. In other words, this non-basic market cannot satisfy the necessary conditions for the Pareto optimum. Applying the above procedure to each market, we can show Proposition 4 in section 5.

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Appendix C: A Proof to Proposition 6 Assume, for example, that the market is P, as shown in Figure 1(a); it is not difficult to show (i) the comer equilibrium including as traded goods those with larger preference parameters is Pareto superior to that including as traded goods those with smaller preference parameters; and (ii) the comer equilibrium including as traded goods those with great return to specialization is Pareto superior to that including as traded goods those with small return to specialization. (i) Assuming a = (5 and a = b = c = /, we have utilities for structure (x/y) and (y/x): log[/(x/j) = a(21ogx + l o g r - l o g r ) + ^logz + «log(l-icr), (C.la) logU(y/x) = a(2logx + logT + logr) + ylogz + a l o g ( l - K ) , (C.lb) where x = [2a/( 3 a + j ' ) ] / 2 , z = y/(3a + y)

andT = [a/(3a + y)J.

The condition of utility equalization becomes E = logU(x/y) - logU(y/x)

= 2alogr = 0.

(C.2)

(i) will be established, if it can be shown that d\ogU*/da\a=y

=dlogU*/da\a=r+(d\ogU*/dr*)(dr*/da)\a__r>0,(C3)

where U* is real income and r* = M*xJM*y is relative number of the individuals choosing different structures in the comer equilibrium. From (C.2) it can be derived that dr'lda\a__f = -(dE/da)/(8E/dr%__r where 8E/da\

=

dlogU'/da\

= 0,

(C.4a)

= 2 log r = 0 because r = 1 if a = y, and >0.

(C.4b)

(C.4) ensures that (C.3) holds. (C.3) implies that if individuals prefer x and y to z, the comer equilibrium with x and y as traded goods will have a greater real return to labor than that with x and z or with y and z as traded goods.

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(ii) Assuming a = p = y and a = b = t, (C.l) holds if x = [2a/(3a + c)]72, z = [cl{3a + c)]c, and 71 = [2a/(3a + c)]°. Also, there is the condition of utility equalization (C.2). (ii) will be established, if it can be shown that d]ogU*/da\aae=dlogU'ma=c+(d\ogU'/drtXdr'/da^g=c>0.

(C.5)

From (C.2) with x = [2a/(3a + c)]72, z = [c/(3a + c)]c, and T = \2aLI (3a + c)]a, it can be derived that. dlogU*/da\a=c >0, and dr*jda\a=c =-(dE/da)/(dE/dr*)\a=c where 8E/da\ '

=0, (C.6)

=0.

\a=c

(C.6) ensures (C.5) to hold. (C.5) implies that if the returns to specialization in producing x and y are greater than that in producing z, then the corner equilibrium with x and y as traded goods will have a greater real return to labor than that with x and z or with y and z as traded goods. Proposition 6 can be established by using (i) and (ii).

References Chenery, M., 1979, Structural change and development policy (Oxford University Press, Oxford). Ethier, W.J., 1986, The theory of international trade, Discussion paper no. 1 (International Economics Research Center, University of Pennsylvania, Philadelphia, PA). Fei, J. and G. Ranis, 1964, Development of the labor surplus economy (Irwin, Homewood, IL). Helpman, E. and P. Krugman, 1985, Market structure and foreign trade (The MIT Press, Cambridge, MA). Herberg, H., M.C. Kemp and M. Tawada, 1982, Further implications of variable returns to scale in general equilibrium theory, International Economic Review 9, 261-272.

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Kuznets, S., 1966, Modern economic growth (Yale University Press, New Haven, CT). Lewis, W., 1955, The theory of economic growth (Allen and Unwin, London). Sah, R. and J. Stiglitz, 1986, On leaning to learn, localized learning, and technological progress, Mimeo. (Department of Economics, Princeton University, Princeton, NJ). Smith, A., 1976, An inquiry into the nature and causes of the wealth of nations, edited by E. Cannan (The University of Chicago Press, Chicago, IL). Yang, X., 1988, A microeconomic approach to modeling the division of labor based on increasing returns to specialization, Ph.D. dissertation (Department of Economics, Princeton University, Princeton, NJ). Young, A.A., 1928, Increasing returns and economic progress, The Economic Journal 1, 527-542.

CHAPTER 8 AN EQUILIBRIUM MODEL ENDOGENIZING THE EMERGENCE OF A DUAL STRUCTURE BETWEEN THE URBAN AND RURAL SECTORS*

Xiaokai Yang and Robert Rice* Monash University

1. Introduction The intimate relationship between the emergence of cities and the trade-off between economies of specialization and transaction costs is noted by Mills and Hamilton [9] and others. Some formal models have been developed to capture different aspects of the relationship in isolation. Kendrick [6] examines the trade-off between economies of scale and transaction costs in a planning model of plant site location, but the relationship between the trade-off and urbanization is not explored. A general transaction technology is introduced into Debreu's general equilibrium framework with constant returns to scale by Hahn [4], Karman [5], Kurz [8], and others. These models cannot formalize the trade-off between economies of specialization and transaction costs because of the assumption of constant returns to scale. Schweizer [11] develops some equilibrium models to address the trade-off between

* Reprinted from Journal of Urban Economics, 35 (3), Xiaokai Yang and Robert Rice, "An Equilibrium Model Endogenizing the Emergence of a Dual Structure between the Urban and Rural Sectors," 346-368, 1994, with permission from Elsevier. * The authors are grateful to two anonymous referees, Jan Brueckner, Jeff Borland, Pak Wai Liu, Yew-Kwang Ng, and Victor Bottini for helpful comments. Thanks also go to Edwin Mills and Gene Grossman who encouraged the first author's research that relates to the paper. We are responsible for any remaining errors.

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X. Yang, R. Rice

economies of scale and transaction costs. Krugman [7] explores the implications of the trade-off between economies of scale and transaction costs for geographical concentration of economic activities.1 However, economies of scale differ from economies of specialization, a sort of diseconomies of scope, despite the connection between the two concepts.2 The paper develops a general equilibrium model which derives the emergence of a dual structure between the urban and rural sectors within Yang's framework of consumer-producers, economies of specialization, and transaction costs. In Yang [12], this framework is used to formalize a trade-off between economies of specialization and transaction costs. The labor productivity of goods and transaction services depends on the distance between neighbors. If this distance is large, then transaction costs generated by the division of labor outweigh economies of specialization, so that individuals will choose a low level of specialization. A decrease in this distance, which may be associated with urbanization, can increase the equilibrium level of specialization and labor productivity by reducing transaction costs.3 In that model, as urbanization develops, the distance between all neighbors decreases by the same amount. Hence, Yang's model cannot endogenize the emergence of a dual structure between the urban and rural sectors. In this paper, we extend the model, introducing a differential in transaction cost parameters between the agricultural and manufacturing sectors. A dual structure between the urban and rural sectors may emerge with the level of urbanization, depending on the transaction cost parameters. In the model to be considered there are four consumer goods and many ex ante identical consumer-producers. Each good is essential in 1

In Krugman [7], economies of scale are realized within a firm; in this paper, economies of specialization result from the division of labor between individuals. 2 The distinctions between economies of scale and specialization and between economies of specialization and diseconomies of scope are identified by Yang and Ng [15]. Scale can be specified as the output level of a firm, while level of specialization is specified as a person's labor share in producing a certain good, which decreases with the scope of his activities. Scope can be specified for a firm, but level of specialization must be specified for each person as well as for each activity. This implies that the scope of a firm and the levels of specialization of many different specialists in the firm may be great at the same time although each person's level of specialization decreases with his scope of activities. 3 Specialization in production at the level of the individual differentiates Yang's model [12] from the general equilibrium models in location theory presented in Schweizer [11].

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consumption. Each individual, as a consumer, prefers diverse consumption and, as a producer, prefers specialized production due to economies of specialization in the production of each good. Assume that goods 1, 2, and 3 are manufactured goods and their production is not land intensive; the producers of these goods can either reside in a single city or be dispersed in the countryside. We refer to them as C-type producers. Good 4 is food and its production is land intensive, so that its producers have to reside dispersedly in the countryside. We refer to them as R-type producers. The assumption that good 4 is land intensive and goods 1, 2, and 3 are not implies that the transportation distance and thereby the transaction cost coefficient of a unit of good is much smaller among C-type producers than it is among R-type producers and between R-type and C-type producers. We draw the distinction between the quantities of the same good that are self-provided, sold, and purchased. This distinction leads to corner solutions of consumer-producers. Using the Kuhn-Tucker approach to corner solutions, we can show that each individual sells at most one good and does not simultaneously buy and sell nor simultaneously buy and self-provide the same goods. The combinations of individuals' corner solutions that are consistent with this condition generate many candidates for the general equilibrium which satisfy the market-clearing conditions but not all conditions for full utility maximization of individuals. Following the approach developed in Yang [12], we can identify four types of market structures as candidates for the general equilibrium. The first is autarky where all people reside dispersedly and no city exists. The second is the partial division of labor where each R-type producer trades good 4 for good 1, each C-type producer trades good 1 for good 4, and the level of specialization is the same for all individuals. All people reside dispersedly in the countryside and no city exists. Each and every individual self-provides three goods and trades two goods. The third is associated with a higher level of division of labor where each C-type producer resides in the city and the level of specialization is lower for all R-type producers than for all C-type producers. The city emerges from the division of labor. The fourth is a complete division of labor where everybody trades all four goods in a dual structure between the urban and rural sectors.

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Applying the approach developed in Yang [12], we prove that the equilibrium is autarky if a variable transaction cost coefficient for a unit of good and/or a fixed cost for each transaction are sufficiently large, since in this case transaction costs outweigh economies of specialization generated by the division of labor. As the transaction cost coefficients are slightly lowered, the general equilibrium shifts to the partial division of labor without cities where no division of labor exists between C-type persons. If the transaction cost coefficients are further lowered, the general equilibrium entails a city where C-type specialists have the division of labor. For this general equilibrium, a dual structure between the urban and rural sectors emerges from the partial division of labor. If the transaction cost coefficients are sufficiently small, the general equilibrium entails the complete division of labor with a city since economies of specialization outweigh the transaction costs generated by the complete division of labor. As the transaction cost coefficients are reduced, the division of labor evolves, resulting in an increase in the ratio of urban to rural population and per capita real income. Section 2 of this paper describes the model which has consumerproducers, economies of specialization, and transaction costs. Section 3 investigates individuals' decision problems and equilibrium. In section 4 the comparative statics of the equilibrium are used to explore the implications of the model for the emergence of a dual structure between the urban and rural sectors from the division of labor. 2. A Model with Consumer-Producers, Economies of Specialization, and Transaction Costs Consider an economy with M consumer-producers and four consumer goods where Mis large. Each consumer-producer's self-provided amount of good i is Xj. The amounts of good i sold and purchased in the market are xf and xf , respectively. Assume that goods 1, 2, and 3 are manufactured goods and that not much land is needed in producing them; then the producers of these goods can be either concentrated in a city or dispersed in many localities. However, good 4 is a land-intensive agricultural good. Its producers can only reside dispersedly in the rural

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area. The production functions for the four goods, which are the same for each consumer-producer, are given by xt+x^=L%

Lt+L2+L3+L4=l, a>\,

L,e[0,l],

1 = 1,2,3,4,

(1)

where x, + xf is the output level of good i and Li is a person's labor share in producing good i, which we term his level of specialization in producing good i. Each consumer-producer is endowed with one unit of labor. Parameter a represents the degree of economies of specialization. There are several distinctive features to this system of production functions and endowment constraint. First, this system is specified for an individual rather than for a firm. Consequently, each individual is a consumer-producer. Demand (or supply) depends not only on the utility (or production) function, but also on the production (or utility) function. Specifying production functions as identical for all individuals makes the distinction between our production function and the traditional production function clearer. The model in this paper does not involve comparative advantage in the conventional sense. According to neoclassiccal microeconomics, such a model with identical technology and preferences for all individuals is too trivial to be interesting. However, this paper will show that such a model is capable of not only generating gains from trade, but also endogenizing the level of division of labor and trade dependence.4 Second, it is assumed that the available time input is specific for each individual and for each activity. Therefore, the time available to an individual is fixed and there is an endowment constraint in this system. This point ensures that the economies of specialization are limited and are individual specific as well as good specific. Third, the production of any two goods is separable. This implies that an individual can choose extreme specialization. A comparison between this functional form and alternative specifications will make this point clearer. Suppose the production function is specified as x°x* = L , where 4

The models developed by Dixit and Stiglitz [1], Ethier [2], Krugman [7], and Grossman and Helpman [3] can also generate gains to trade between ex ante identical agents. However, our model here is based on the concept of economies of specialization, which is a sort of diseconomies of scope and differs from the concept of economies of scale specified in those models.

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a + b < 1 for increasing returns to scale and L is the total amount of labor allocated to the production of goods /' andy. This production function is not well defined forx,. = Oorxy = 0, implying that individuals cannot choose extreme specialization. Suppose alternatively that the production function is specified as xf + x* = L , where a, b < 1 for increasing returns to scale and L is the total amount of labor allocated to the production of goods i and/. With this functional form, the labor allocated to produce good i cannot be distinguishable from that allocated to produce goodj. Thus, this production function will not be able to capture the stylized fact that learning by doing which generates economies of specialization is not only specific for an individual, but also specific for a good. There are two types of transaction costs. The first is a variable transaction cost, characterized by the coefficient K e (0, 1). The second is a fixed transaction cost, characterized by the coefficient c e (0, 1). We first examine the variable transaction cost. Fraction 1 - K of a shipment disappears in transit because of transaction costs. Thus, Kxf is the amount a person receives from the purchase of good i. The individuals producing good 4 reside in the rural area. We refer to them as R-type persons. Here, a city is defined as the area where many individuals have the division of labor and reside in close proximity to one another. If some individuals specialize in producing good 1,2, or 3, they can reside either in a city or in the rural area. We refer to these individuals as C-type persons. If they reside within a city, then the distance between each pair of C-type neighbors will be short. This short distance between a pair of C-type trade partners saves on transaction costs if C-type persons choose to have the division of labor among themselves.5 If there is no division of labor among producers of goods 1, 2, and 3, nobody will choose to reside in a city since living closely together has no benefit. A C-type person cannot reside closely to a R-type person because each R-type person has to occupy a sufficiently large area for agricultural activities. Thus, if C-type persons choose to reside in a city, then the distance between a pair of R-type and C-type neighbors is much greater than that between a pair of

5

The distance between a person and his trade partner may be greater than the distance between neighbors. The former may increase with the number trade partners, which increases in turn with the level of division of labor.

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C-type neighbors. Hence, the distance between a pair of R-type and C-type trade partners is much greater than that between a pair of C-type trade partners who live in the same city. Let the transaction efficiency coefficient between C-type trade partners be k and that between a pair of R-type and C-type partners be s. Then, s is much smaller than k if C-type persons reside in a city. Here, the transaction efficiency parameter is related to the technical conditions of transactions and has nothing to do with the concept of efficiency in the sense of welfare analysis. To summarize, we have K = k between C-type trade partners, and K = s between R-type and C-type partners,

(2)

with k being much larger than s if all C-type persons reside in a city. Later we will see that ex ante identical individuals can choose between becoming R-type persons and becoming C-type persons as they choose different levels of specialization and different professions. An individual's consumption of good i is xi + Kxf . The utility function is identical for all individuals:

V = Y\(xi+Kxf).

(3a)

Here, 1 - K is the coefficient of the first type of transaction cost. The second type of transaction cost is a fixed cost in terms of utility loss, incurred in transactions.6 The fixed transaction cost is proportional to the number of goods purchased by each individual and is independent of the quantities of traded goods. For each good purchased, a fraction c of utility V disappears, so that an individual enjoys only the fraction (1 - nc) of utility V when he buys n goods. This fixed cost coefficient can be interpreted as the cost of creating a new market or of finding the price of a particular good; for example, the investment cost of facilities and instruments that are necessary to implement trade of a particular good or utility loss in discovering the price of the good. Hence, the utility level that is enjoyed by an individual is 6

Introducing only the variable transaction cost without the fixed transaction cost means that the condition for the market structures with an intermediate level of division of labor to be the equilibrium will not be satisfied, so that the equilibrium will be at a corner, either autarky or complete division of labor. This point will be shown in Proposition 1.

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U = (l-nc)V,

(3b)

where n is the number of goods purchased. We assume that there is free entry for all individuals into any sector and the population size Mis large. These assumptions imply that ex ante identical individuals treat prices parametrically. 3. Configurations and Market Structures In this section we investigate the relationship between individuals' decisions of their levels and patterns of specialization and the level and pattern of division of labor and urbanization for society as a whole. It is assumed that each consumer-producer maximizes his utility with respect to the quantities of goods produced, traded, and consumed by him for a given set of relative prices of traded goods subject to the budget constraint, production functions, and endowment constraint. Since the quantities self-provided, produced, and traded of the same good are distinguished from one another, an individual's decision is always a corner solution which involves zero values of some decision variables. Combinations of zero and nonzero values of 12 decision variables xnxf,xf (i = 1, 2, 3, 4) generate 212 = 4096 profiles of variables and thereby one interior solution and 4095 corner solutions. We call a combination of zero and nonzero variables a configuration. Yang [12] has proven that for the models of the kind in this paper, a consumer-producer does not simultaneously buy and sell nor simultaneously buy and self-provide the same goods; also, he sells one good at most (see Proposition 1 and its proof in Yang [12]). Using this proposition, the interior solution and many corner solutions can be ruled out from the set of candidates for the optimal decision. Since preference and production parameters are identical for all goods and for all individuals and the transaction cost coefficient is the same for all C-type persons, the symmetry between C-type persons who sell different goods can be used to rule out more corner solutions from consideration. Having taken this and the budget constraint into account, the candidates for the optimal decision are reduced to only 29 corner solutions. A combination of configurations in which demand for any traded good is matched by supply of that good is called a market structure or simply a

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structure. Free entry and competition among identical utility maximizers will establish utility equalization and market-clearing conditions for each structure. The utility equalization and market-clearing conditions determine a set of relative prices for the traded goods and the relative numbers of individuals selling different goods. The set defines a corner equilibrium for each structure. Yang [12] has proven that the corner equilibrium that generates the highest utility level is the general equilibrium, whereas corner equilibria that do not generate the highest utility level cannot be the general equilibrium because individuals have incentives to deviate from them.7 In this section, individuals' corner solutions and the corner equilibrium are solved for each structure, and the general equilibrium is identified by comparing all corner equilibrium levels of utility. Having taken Proposition 1 in Yang [12] into account, there are four categories of configurations. (i) Configuration autarky or A. An individual who chooses this configuration produces four goods for his own consumption. (ii) Configuration (ilj). An individual who chooses this configuration sells good i, buys goody, and self-provides other goods, where i,j = 1,2, 3, 4; i ^j; i before the slash denotes a good sold, and j after the slash denotes a good purchased. There are 12 configurations of this category: (1/2), (1/3), (1/4), (2/1), (2/3), (2/4), (3/1), (3/2), (3/4), (4/1), (4/2), (4/3). (iii) Configuration (i/jr). An individual who chooses this configuration sells good i, buys goods j and r, and self-provides the other goods, where i,j, r = 1,2,3,4, and i £j ^ r. There are 12 configurations of this category: (1/23), (1/24), (1/34), (2/13), (2/14), (2/34), (3/12), (3/14), (3/24), (4/12), (4/13), (4/23). 7

Yang and Ng [15, pp. 161-165] have shown how a decentralized market sorts out the efficient pattern and level of division of labor from many corner equilibria. Since the concept of corner equilibrium is a vehicle for solving for general equilibrium so long as the concept of corner solution is a vehicle for a person to solve for the optimum decision, multiple corner equilibria do not imply multiple general equilibria as long as each person's multiple corner solutions do not imply multiple optimum decisions. Yang and Ng [15, Chap. 6, Proposition 6.5] have shown that the general equilibrium is Pareto optimal. The assumption that economies of specialization are limited to each consumer-producer is crucial for this result.

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(iv) Configuration (i/jrt). An individual who chooses this configuration sells and self-provides good i and buys goods j , r, and t, where i,j, r, t = 1, 2, 3, 4, and i ^ j ± r ^ t. There are 4 configurations of this category: (1/234), (2/134), (3/124), (4/123). Combinations of the configurations that are compatible with the market-clearing conditions (i.e., the demand for any traded good is matched by the supply of that good) generate a myriad of structures. Since all preference, production, and transaction parameters are identical for all individuals of the same type, many structures with different trade compositions generate the same per capita real income (utility). Hence, there is indeterminacy of trade composition, and only the number of traded goods matters. Thus, the distinct structures can be characterized by the numbers of traded goods for the two types of persons. Let (nm) denote a structure in which a C-type person trades n goods and a R-type person trades m goods; there are 10 distinct market structures that generate different utility levels. 1. Autarky, depicted in Figure la, which consists of configuration A and is referred to as structure A. 2. Structure (22), depicted in Figure lb, in which each C-type person and each R-type person trade two goods, respectively. This structure consists of configurations (1/4) and (4/1). In this structure, R-type persons exchange good 4 for good 1 with C-type persons. The structures consisting of (z'/4) and (4//), where z = 2 or 3, are symmetric with this structure and generate the same corner equilibrium per capita real income. Hence, we do not draw the distinction between the three structures. This structure is referred to as balanced partial division of labor without cities since the two types of persons have the same low level of specialization. 3. Structure (23), in which each C-type person trades two goods and each R-type person trades three goods. This structure consists of configurations (z'/4) and (4/if), where i,j = 1, 2, 3, and i £j. 4. Structure (24), in which each C-type person trades two goods and each R-type person trades four goods. This structure consists of configurations (z'/4) and (4/123), where z = 1, 2, 3.

Dual Structure between Urban and Rural Sectors

j

177

4

Figure 1: Emergence of a dual structure between urban and rural sectors, (a) Autarky, (b) Structure (22), without city and with balanced partial division of labor, (c) Structure (32), with a city and unbalanced division of labor, (d) Structure (33), with a city and balanced partial division of labor, (e) Structure (43), where urban residents are completely specialized and rural residents are partially specialized, (f) Structure (44), with complete division of labor.

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5.

Structure (32), depicted in Figure lc, in which each C-type person trades three goods and each R-type person trades two goods. This structure consists of configurations (i/4j) and (4//), where i,j = 1, 2, 3, and i ± j . This structure is referred to as a low level of unbalanced partial division of labor with a city since all C-type persons have a higher level of specialization than R-type persons. 6. Structure (33), depicted in Figure Id, in which each C-type person and each R-type person trade three goods. This structure consists of configurations (z'/4/) and (4/y), where i,j = 1, 2, 3, and i ^j. This structure is referred to as balanced partial division of labor with a city since two types of persons have the same level of specialization in a dual structure between the urban and rural sectors. 7. Structure (34), in which each C-type person trades three goods and each R-type person trades four goods. This structure consists of configurations (i/4j) and (4/123), where i,j = 1, 2, 3, and i i^j. 8. Structure (42), in which each C-type person trades four goods and each R-type person trades two goods. This structure consists of configurations (i/4jr) and (4/z), where i,j, r= 1,2,3, and / £j ^ r. 9. Structure (43), depicted in Figure 1 e, in which each C-type person trades four goods and each R-type person trades three goods. This structure consists of configurations (i/4jr) and (4/z)'), where i,j, r = 1, 2, 3, and i £j ^ r. This structure is referred to as a high level of unbalanced division of labor with a city since all urban residents have a higher level of specialization than rural residents and all individuals have a higher level of specialization than in structure (32). 10. Structure (44), depicted in Figure If, in which each C-type person and each R-type person trades four goods. This structure consists of configurations (z'/4/>) and (4/123), where i,j, r= 1,2,3, and z ^ j ^ r. This structure is referred to as complete division of labor. There are some structures that are symmetric to and generate the same corner equilibrium per capita real income as each of the structures listed above. We do not distinguish between these structures and the corresponding ones that we have listed.

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Letting the number of individuals selling good i be M., a product of M, and individual supply (demand), given by the individual optimal decisions, yields the market supply of (or demand for) each traded good. Equilibrating the market demand to the market supply yields the market-clearing conditions for each market structure. Furthermore, free entry will ensure utility equalization through individuals' behavior of maximizing utilities. The market-clearing and utility equalization conditions determine the relative prices of traded goods and the relative numbers of individuals choosing different configurations. The set of relative prices and relative numbers defines a comer equilibrium in each structure. The concept of comer equilibrium for a structure is an analogue of comer solution for an individual configuration. A comer equilibrium is not a general equilibrium unless each individual's utility is maximized with respect to comer solutions because a comer solution is not an optimal decision unless it maximizes an individual's utility. Let us investigate these 10 structures one by one. (1) In structure A (autarky) there is only one configuration, A. The individual decision problem for this configuration is max U - xxx2x3xA = L°La2L° (1 - L, - L2 - L3 )a s.t. x(. =Lj, i = \, 2,3,4 i , + L2 + L3 + L4 = 1 The optimum decision is /,={,

1 = 1,2,3,4,

(4a)

(production function) (endowment constraint). U(A) = 2-*°,

(4b)

where U(A) is the per capita real income in autarky. (2) In structure (22), there are two different configurations, (1/4) and (4/1). The decision problem for (1/4) is max U = (1 - cn)xxx2x3sxf = (1 - c){Lax - x[ )La2La3 sx{ /p4l s.t. x, + xf = L°, xi -Lat, i = 2,3 Lx + L2 + L3 = 1 x{ = p^xf

(5a)

(production function) (endowment constraint) (budget constraint),

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X. Yang, R. Rice

where p„ is the price of good 4 in terms of good 1. Note that all transactions here are between R-type persons and C-type persons. Hence, K = s. Moreover, the number of goods purchased, n, is one for this configuration. The optimum decision is A = i . A = j , i = 2,3, xf = x, = L>/2, xf = xf/pn

(5b)

Ul(p4l) = s(l-c)2-*"-ip-4}, where£/,Cp41) is the indirect utility function for configuration (1/4). The decision problem for configuration (4/1) is symmetric to (5). The optimum decision is

U4(pj =

s(l-c)2-*^ptl,

where f/4(/>41) is the indirect utility function for configuration (4/1). The market-clearing and utility equalization conditions for structure (22) are MlX;=MAx?(pJ

(7a)

t/ 1 (p 41 ) = C/4(/>41).

(7b)

Due to Walras' law, the market-clearing condition for good 4 is not independent of (7a). Equation (7a) ensures a special Walras regime. Prices are determined by market demand and supply, which are determined by the numbers of persons selling any specific good. Each person is able to produce all goods, and thereby the numbers of persons selling different goods cannot be manipulated by any individual. Combined with the fact that economies of specialization are limited to each individual and to each good, this then implies that prices are parameters to individuals. Here the relative numbers of persons selling different goods play the same role as the relative prices in the conventional Walras regime. Inserting the individual optimum decisions into the relevant variables in (7), we find the corner equilibrium in structure (22), p41=M4/M1=l,

C/(22)= 5 (l-c)2-^-2,

(8)

where U(22) is the per capita real income in structure (22). Repeating this procedure, we have solved for the corner equilibria in other structures. The per capita real income, relative prices, relative numbers of different type

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181

persons, and level of specialization for individuals choosing different configurations are listed in Table 1. In the table, ^ . Mi /M4 , i = 1,2,3, is the ratio of urban population size to rural population size for those structures involving a city, which is defined as the degree of urbanization, andL; is the level of specialization for a person selling good /. We have multiple corner equilibria. All of these corner equilibria are candidates for the general equilibrium. These candidates satisfy the following conditions: (i) Excess demands for all traded goods are zero for the uniform positive relative prices of the traded goods, the relative numbers of individuals choosing different configurations, and the uniform positive price of labor (real return to labor), and (ii) individuals maximize their utility for a given set of relative prices and for a given structure. Note that we have not yet imposed full maximization of utility by individuals across structures. Now we allow individuals to choose configurations across structures. Then each individual will choose a configuration that generates the highest real income, taking a set of relative prices of traded goods as given. By enumerating all candidates for the general equilibrium, we will find the corner equilibrium that generates the highest per capita real income and prove that it is the general equilibrium. The details of this proof are in Yang [12] or in Yang and Ng [15, Chap. 6, Proposition 6.5]. From the individual decision problems in this section, we can see the effects of our method of specifying production functions for individual demand and supply, indirect utility functions, and the theory of equilibrium. First, individual demand depends on individual supply since each individual is a consumer-producer. Demand and supply functions and indirect utility functions are not continuous across structures. Second, an individual not only solves for the efficient resource allocation for a given level of specialization, but also chooses a configuration in order to find the efficient level of specialization. The indirect utility function differs from configuration to configuration despite ex ante identical utility functions for all individuals. A corner equilibrium in this paper is equivalent to a general equilibrium in neoclassical microeconomics. A corner equilibrium determines an efficient allocation of resources for a given level of division of labor. However, for our model, the general equilibrium will determine the efficient level of division of labor. Many economic phenomena can be explained by changes in organizational

Table 1: Corner Equilibria in 10 Market St Structure type

Per capita real income

Corner equilibrium relative prices

A (Fig. la) 2-8° Structure (22) (Fig-lb) s(l-c)2- 6 "- 2

p4i =

1

Structure (23) , 4 / 3 ( 1 _ c f 3 ( l _ 2 c f

^

^

x 3„-l2-(20o+4)/3

S

™ic)(32)

5

[Ml-^)(l-2c)f

X3

(

g

}

=

x 31-«, 2

[(l_c)A(1_2c)]>/3 2

("-')/3

A, = P« = [k(l-2c)/(l-c)f2(^ p4l = pa= (k/sf

x33(«-l)2-8«

Structure (24) ^3/2 (j _ C)V« ^ _ 3cy/4 x2-(9«+7)/2

{

X31*-')

IJ(-I)2-7-I

Structure(33) ^ / 3 ( l _ 2 c )

=

M

Al

= Pa = PG = .y-I/2 ["(! _ c y ( ! _ 3 c ) j ' / 4 21.5(l-«)

=

(

Table 1: (Continued) Structure type Structure (42)

Per capita real income

^[(l- C )(l-3c)]'

/2

x2-3a-5

Structure (34)

c)f

x2 (»-')A:

s^k^(l-2cfA

P*l = PA2 = Pn

(skf(l-2cf 2/3

= [(l-3 C )/(l-2 C )*f x 2 l/4 x l/2 2 2(»-l)33(l~»)/4

x39(„-l)/4

x(l-3c)

A i = P 4 2=P43 = [ ( 1 - 3 c ) / ( 1 3

x(l-3c) 1 / 4 2-o«-' Structure (43) (Fig.le)

Corner equilibrium relative prices

(M

P« = Pt2 = Pil, 2 8

3°-'2- - °/3

3

= H [(l-3c)/5(l-2c)]

V3

=

x 31-a 2 -l+8«/3

Structure (44) ( ^ ) 3 / 2 ( l - 3 c ) 2 - " (Fig. If)

P« = Pa = Pn = {klsT =

Note. ^ ; MJMA, / = 1,2,3, is the ratio of urban population size to rural population siz defined as the degree of urbanization, and L-L is the level of specialization for a person s

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patterns of division of labor even if utility and production functions are ex ante identical for all individuals. Therefore, we refer to this approach as organization oriented while neoclassical microeconomics is resource allocation oriented. 4. The General Equilibrium and its Comparative Statics By comparing per capita real income in all structures, we can identify the corner equilibrium with the maximum per capita real income. We refer to this corner equilibrium as the Pareto-optimum corner equilibrium. The Pareto-optimum corner equilibrium is the general equilibrium. A comparison among the per capita real incomes in the 10 structures yields Proposition 1: (1) For c > 1, the general equilibrium is structure A; (2) For c e ( \ , 1), the general equilibrium is structure A if a < a\ and is structure (22) if a > a\\ (3) For c G ( y , | ) , where c is sufficiently close to 1/3, the general equilibrium is structure A if a < a2, structure (22) if a e (a 2 , a3), structure (32) if a e (a 3 , a4), structure (33) if a > a4; (4) For c G (0, j ), where c is sufficiently close to 0, the general equilibrium is structure Aif[a< a5 andsk > b] or [a < a6 andsk< b], structure (43) if a G (a5, a7) andsk> b, structure (44) if [a > aq andsk > b] or [a > a6 andsk < b]. Here a is the degree of economies of specialization, s is the transaction efficiency parameter between C-type and R-type persons and between R-type persons, k is the transaction efficiency parameter between C-type persons when they reside in the same city, and c is the fixed transaction

Dual Structure between Urban and Rural Sectors

185

cost coefficient.8 The critical values of a that make the general equilibrium shift between structures are listed below. or, sl-0.721ns,

a2 = l - 0 . 3 5 1 n [ s ( l - c ) ] ,

a3 s0.05 + 0.091n[(l-c)/ifc(l-2c)], aA = 1 + 0.52 In [(l - c)/(l - 2c)] - 0.35 In s, a5 =0.52 -0.28 In (sk), a6=l-0.27h\(sk),

an = 4.08 - 0.12 In (sk),

6 = 1/4603119211, a2 < a3 < a4,

if c e (|,^) and c is sufficiently close to ~,

a5 b, c e (0,1J, and c is sufficiently close to 0,

a5>a1,

if sk 1. Also, per capita real incomes in all structures except in structures A and (22) are non-positive if c>\ . Hence, the set of candidates for the general equilibrium is composed of only structures A and (22). A comparison of per capita real income between the two structures indicates that (7(A) > 0(22) iff a < ax. If c e (y ,y), then the per capita real incomes in structures (24), (42), (43), (34), and (44) are non-positive. A comparison of the per capita real incomes in structures (22), (23), and (32) indicates that 0(23) < either 0(22) or 0(32). Thus, the set of candidates for the general equilibrium consists of structures A, (22), (32), and (33). A comparison between [/(A), 0(22), 0(32), and 0(33) establishes the statements that A generates the highest per capita real income if a < a2 structure (22) generates the highest per capita real income if a e (az, a3), structure (32) generates the highest per capita real income if a e (a3, a4), and structure (33) generates the highest per capita real income if a > a4, where the values of a, are given in Proposition 1. A comparison between a2, a3, and a4 shows that a2 < a3 < a4 if c is sufficiently close to y from the right-hand side. Either or both of the two inequalities may not hold if c G (y, j) and c is sufficiently close to \ . For a3 > a4, we have 0(33) > 0(32) > 0(22) if a > a3, and 0(32) < 0(22) if a < a3. This implies that structure (32) cannot be the equilibrium. The equilibrium will jump from (22) over (32) to (33) as transaction efficiency improves. If c is sufficiently close to zero, then it can be shown that structures (22), (32), (23), (33), (24), and (42) cannot generate the highest per capita real income. This statement can be established in several steps. Suppose c = 0. First, it can be shown that 0(24) < 0(42), so that structure (24) cannot

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generate the highest per capita real income. Then, it can be established that (7(23) < (7(32) if a > 1 and k> s, which is assumed in this paper. Therefore, structure (23) cannot be the equilibrium. Furthermore, comparisons among U(32), (7(42), and U(22) indicate that (7(42) > U(32) if a > 1 - 0.48 In k and (7(22) > (7(32) if a < 1 - 0.48 In k. This implies that structure (32) cannot be the equilibrium. A comparison establishes that U(33) < (7(43) if a > 1. Also, (7(34) < t/(43) if a > 1 and k > s. This means that structures (33) and (34) cannot be the equilibrium. Now, the set of candidates for equilibrium is narrowed down to structures A, (22), (42), (43), and (44). Comparisons among (7(A), (7(22), and (7(43) yield that (7(43) > (7(22) if a > ag = 0.88 - 0.1 In s - 0.39 In k and (7(22) > (/(A) if a >a9 = 1 - 0.72 In s. It is straightforward that ag ag, £7(22) < C/(A) and U(43) if a e (a 8 , a9), and (7(22) < t/(A) if a < as. Therefore, structure (22) cannot be the equilibrium. Similarly, it can be shown that (7(42) < C/(43) if a >an = 1 - 0.29 \n(sk); U(42) < (7(43) and (/(A) if a G (a10, ^ l), where a10 = -1.31 - 0.25 \n(sk); and (7(42) < (/(A) if a b, then A is the equilibrium when a a7. Ifsk < b, then (7(43) < (/(A) or (7(44), so that A is the equilibrium if a < a6 and (44) is the equilibrium if a > a6. If c e (o,j) and c is sufficiently close to \ from the left-hand side, then more structures with an intermediate level of division of labor may occur in equilibrium. Here, the values of b and all at for i < 8 are given in Proposition 1.

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9.

E. Mills and B. Hamilton, "Urban Economics," Scott, Foresman, Glenview, IL (1984).

10. Adam Smith, "An Inquiry into the Nature and Causes of the Wealth of Nations" (E. Cannan, Ed.), University of Chicago Press, Chicago (1976). 11. U. Schweizer, General equilibrium in space, in "Location Theory." (J. J. Gabszewicz et al., Eds.), Harwood, London (1986). 12. X. Yang, Development, structural changes, and urbanization, Journal of Development Economics, 34, 199-222 (1990). 13. X. Yang and J. Borland, A microeconomic mechanism for economic growth, Journal of Political Economy, 99, 460-482 (1991). 14. X. Yang and G. Hogbin, The optimum hierarchy, China Economic Review, 2, 125-40(1990). 15. X. Yang and Y. Ng, "Specialization and Economic Organization," Elsevier Science, Amsterdam (1993). 16. X. Yang and H. Shi, Specialization and product diversity, American Economic Review, 82, 392-398 (1992).

CHAPTER 9 AGGLOMERATION ECONOMIES, DIVISION OF LABOUR AND THE URBAN LAND-RENT ESCALATION: A GENERAL EQUILIBRIUM ANALYSIS OF URBANISATION*

Guang-Zhen Sun a and Xiaokai Yang b * "Max Planck Institute and University of Macau

h

Monash

University

1. Introduction The purpose of the paper is to explain several important phenomena of urbanization, including the urban land-rent escalation, decreases in the relative per capita consumption of land in the urban and rural area, increases in the population size of the urban area compared to the rural area, and the absolute increase of diversity of occupations in the urban area as well as relative to that in the rural area, as different aspects of evolution in the division of labour. The intimate relationship between cities and the division of labour has indeed long been recognized by Xenopnon (Gordon 1975), William Petty (1682), Alfred Marshall (1890, Ch. 9-10), Mills (1972), Scott (1988) and others. However, as noted by

Reprinted from Australian Economic Papers, 41 (2), Guang-Zhen Sun and Xiaokai Yang, "Agglomeration Economies, Division of Labour and the Urban Land-Rent Escalation: A General Equilibrium Analysis of Urbanisation," 164-184, 2002, with permission from Blackwell. * We are grateful to Laurent Calvet, Don Snodgrass, Guoqiang Tian, our colleagues at Max Planck Institute and Monash University, seminar participants at Duke University and Harvard University and the anonymous referee for helpful comments on previous versions. Sun gratefully acknowledges the financial support from the Research Committee of the University of Macau. The usual disclaimer applies.

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Stigler (1976, pp. 1209-1210), there was no formal theory of division of labour and specialisation in the mainstream economics: "The last of Smith's regrettable failures is one for which he is overwhelmingly famous — the division of labour (A)lmost no one used or now uses the theory of division of labour, for the excellent reason that there is scarcely such a theory.... Smith gave the division of labour an immensely convincing presentation — it seems to me as persuasive a case for the power of specialization today as it appeared to Smith. Yet there is no evidence, so far as I know, of any serious advance in the theory of the subject since his time, and specialization is not an integral part of the modern theory of production, which may well be an explanation for the fact that the modern theory of economies of scale is little more than a set of alternative possibilities." In the recent two decades, there emerged a growing literature on specialisation and the division of labour, including Becker (1981), Rosen (1978, 1983), Baumgardner (1988), Kim (1989), Locay (1990), Yang (1990), Becker and Murphy (1992) and Yang and Ng (1993) among others, with accentuation of the endogenisation of individuals' levels of specialisation. In this paper, we address the progressive division of labour and urbanisation process using the framework developed by Yang (1990) and Yang and Ng (1993) which is featured by the concept of consumer-producers (rather than consumers and firms) and increasing returns to labour specialisation. Although considerable progress has been made in modelling urbanisation and agglomeration, particularly within the New Economic Geography, there have been few analyses that address the emergence and growth of cities resulting from the evolution in division of labour in a general equilibrium setting. Remarkable exceptions are Yang and Rice (1994) and Fujita and Krugman (1995).1 Taking evolution of division of labour as evolution in the number of traded goods and in individuals' levels of specialisation, Yang and Rice's story runs as follows. Due to the trade off between economies of specialisation and transaction costs, 1

Other related works in one way or another are Fujita (1985), Fujita and Mori (1997), Hochman (1997), Krugman and Venables (1996) and Yang (1990).

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as a unit transaction cost coefficient falls, each individual reduces her number of self-provided goods and increases her level of specialisation, so that the equilibrium level of division of labour evolves. In addition, the production of agricultural goods is land intensive and hence farmers must have their residence dispersed, while producers of manufactured goods can freely choose between dispersed residences and concentralised residences. If the partial division of labour between the production of the agricultural good and the production of one manufactured good occurs in equilibrium, then each manufacturer resides nearby a farmer, no city exists. If a high level of division of labour between manufacturers as well as between farmers and manufactures emerges from sufficiently improved transaction conditions, all manufactures will reside together in cities to reduce transaction costs between them. Their model can predict the following phenomena as different aspects of the evolution in division of labour and urbanisation. Productivity of all goods and per capital real income increase, the trade dependence, the extent of the market, individuals' levels of specialisation, and the degree of diversity of economic structure increase. However, this model cannot predict some important phenomena of urbanisation such as increases in the land rent differential between the urban and rural areas and decreases in relative per capita consumption of land of urban and rural residents, as the consumption of land does not enter the utility functions of the agents in their model. These two phenomena have been confirmed by many empirical works, of which Colwell and Munneke (1997) is the latest one among others.2 Similarly, in Fujita and Krugman (1995), there are also tradeoffs between global economies of scale, utility benefit of consumption variety of manufactured goods, and transport costs. An increase in population size or in transaction efficiency will enlarge the scope for trading off one against others among the conflicting forces, thereby increasing productivity, per capita real income, and consumption variety. In addition, there is a trade off between transaction costs that can be 2

von Thunen (1826), Beckmann (1969), Ben-Akiva et al. (1989) and others have studied the land price differentials without taking account of the equilibrium structure of division of labour.

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saved by concentrated residences of manufacturers in a city and transaction costs between rural farmers and urban manufacturers that is increased by the concentrated urban residences. The increase in the number of manufactured goods, an aspect of division of labour, will move the efficient balance of the latter trade off toward a more concentrated residential pattern of manufacturers, making a city more likely to emerge. The benefit of concentrated residences of manufactures caused by an increase in the number of manufactured goods is called economies of agglomeration. In fact, their model is a modern version of von Thunen's (1966) isolated state, in which the driving force of city formation is the circular linkage between economies of agglomeration and concentration of production. They conducted a very interesting comparative static analysis demonstrating that a decrease in the unit transaction cost, a larger population size, and a higher productivity of the agricultural sector will make a city more likely to emerge from a greater number of manufactured goods and will increase the population share of urban residents. But the city in Fujita and Krugman (1995) is a dimensionless single point and hence the residential land market could not be addressed. As in Yang and Rice (1994), the consumption of residential land does not enter the utility functions of agents. Furthermore, the degree of market integration is not endogenised because all agents are always connected by an integrated market. Increases of individuals' specialisation could not be addressed in Fujita and Krugman either. Both the Yang and Rice model and Fujita and Krugman model cannot predict the increases in relative land rent and population density in the urban and rural areas since economies of agglomeration in their models come from the concentration of manufacturers' residences (refereed to as type I economies of agglomeration hereafter). But in reality, we can observe economies of agglomeration that are generated by geographical concentration of transactions (type II economies of agglomeration hereafter). The geographical concentration of transactions will generate transaction cost advantage to residing in a city since urban residents can considerably benefit from informational spillovers, and save on travelling faraway as most transactions are executed at the city.

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The competition for the urban land that is a central market place where transactions are executed will then bid up urban land rent. Hence, type II economies of agglomeration may be the main driving force behind land rent differential between the urban and rural areas.3 In the current paper, a general equilibrium model with endogenous urban-rural occupation structure and the division of labour is developed to endogenise the following phenomena. Concomitantly with the urbanisation resulting from the expansion of the network of division of labour that is in turn driven by improvements in transaction efficiency, the land rent in the urban area increases absolutely as well as relative to that in the rural area, the number of occupations in the urban area increases absolutely as well as relative to that in the rural area, per capita consumption of land in the city decreases and the per capita consumption of rural residents increases, the number of traded goods for each individual as well as for the society as a whole increases, and the extent of endogenous comparative advantages between individuals of different occupations increases. Our story runs as follows. Each agent is a consumer-producer, and consumes both consumption goods and land for residence. The trade off between economies of specialisation and transaction costs implies that the equilibrium level of division of labour increases as a unit transaction cost coefficient of goods falls. A larger network size of division of labour will generate a larger number of transactions per person, so that the concentrated pattern can save on travelling cost per person by shrinking a large transaction network into a concentrated area (the city). Those residents at the central market place can trade all goods without travelling faraway and benefit from intra-city informational spillovers (Quigley 1998). Hence, competition for residing in the city will bid up the land rent of the urban area. Free migration between the urban and rural areas and between different professions will equalise per capita real income of all individuals such that transaction advantage of urban residents are offset by a higher land-rent for residence and smaller per 3

Our distinction between the two types of economies of agglomeration echoes Lindsey et al. (1995), McCann (1995) and Nakamura (1985) who argue that the concepts of economies of agglomeration and externalities of urbanisation need to be refined.

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capita consumption of land. As transaction conditions are improved, a larger network size of division of labour will be chosen and type II economies of agglomeration increase, so that all concurrent phenomena mentioned above take place as different aspects of the evolution in division of labour. In short, stripping the process of urbanisation and concentration of residence down to its bare essentials, it is the complicated interplay among the division of labour, geographical patterns of transactions and residence, and the trading efficiency that gives rise to urban land-rent escalation and the structural rural-urban shift in terms of residents, occupations, and concentration of economic activities.4 In an interesting study by Brueckner and Zenou (1999) of the augmented Harris-Todaro model with a land market, the urban land-rent escalation also provides a force against the rural-urban migration. The Brueckner and Zenou model is nevertheless a partial equilibrium model in which the product prices are exogenously given. Moreover, in their model, the evolution of the diversity of occupations, a remarkable feature concomitant with the urbanisation, is not addressed. Neither is the division of labour and specialisation. By combining Henderson's (1974) modern classic piece and Krugman (1991), one of founding models of the new economic geography, Tabuchi (1998) recently contributes a general equilibrium analysis of urban agglomeration economies due to product variety and agglomeration diseconomies due to intra-city congestion in a two-city system framework. It demonstrates dispersion may take place when the interregional transportation cost is sufficiently low. The economies of agglomeration in Tabuchi's model come from the concentration of manufacturing, whereas in our model what matters is the type II economies of agglomeration coming from the concentration of transactions (more discussion on this point will be given in the end of section 5, below). More importantly, the urban land rent escalation and the evolution of the diversity of urban occupations, most remarkable features among others concomitant with the 4

As far as we know, Scott (1988) is the first one who spells out, in a descriptive way, the interdependence among the structure of division of labour, location patterns of economic activities and transaction conditions.

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urbanisation, are not addressed in Tabuchi (1998). Neither is the diversity of the traded goods for consumption and individuals' specialisation. The paper proceeds as follows. Section 2 is devoted to an analysis of the interdependence among the geographical pattern of transactions, trading efficiency and the network size of the division of labour. The model is specified in section 3. Section 4 solves for the general equilibrium urban-rural occupation structure and the network of division of labour. A comparative statics is conducted in the section 5. The final section concludes. 2. Economies of Transaction Agglomeration and Division of Labour In the preceding section, we highlight that type II economies of agglomeration, which are associated with the network effects of the division of labour and the concentrated location pattern of transactions, may be the most important driving force of the land-rent differentials between the urban and rural areas. This section is motivated to further spell out the mechanism of the intrinsic relation among the network size of the division of labour, the trading efficiency and the economics of geographical concentration of transactions. Presumably, if the geographical pattern of individuals' residences is fixed and each pair of trade partners trade in the geographical mid-point of their residences, total travel distance and related cost will increase more than proportionally as the network of transactions required by a particular level of division of labour is enlarged. But if all individuals conduct their transactions at a central place, the large network of transactions can be geographically shrunk and be concentrated in that central place to significantly reduce the total travel distance of all individuals. To be sure, the economies of transaction agglomeration differ from economies of scale. Some economists call them positive externalities of cities. Indeed, they are generated by interactions between the positive network effects of division of labour and the effects of the geographical concentration of transactions. On the one hand, the extent of the division of labour is determined by trading efficiency, which itself depends on the geographical pattern of transactions; but, the effect of the

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geographical pattern of transactions on trading efficiency is in turn determined by the level of division of labour, on the other. Hence, trading efficiency, geographical pattern of transactions, and level of division of labour are interdependent and should be simultaneously determined in a general equilibrium model, as is shown in the following sections. One important insight among others that could be drawn from such a perspective is that the crucial determinant of the land rent of a city is the size of network of division of labour that is associated with the city as its center of transactions. The above analysis regarding how geographical concentration of transactions can improve the trading efficiency through the network effects of the division of labour could be illustrated by considering a simple numerical example. An Example. Consider a local community with n goods (n = 2, 3,..., 7) traded among agents, as shown in Figure 1. The distance between each pair of neighbours is assumed to be 1. For simplicity, we restrict ourselves to the symmetric case in which each agent is assumed to sell only one good to exchange for the other (« - 1) traded goods. The number of types of specialists producing some particular goods, n, is a proxy for the degree of the division of labour. Abstracting from irrelevant complication, suppose that exogenous transaction costs are proportional to the travel distance of individuals in conducting the transactions required by the division of labour and that one unit of travel distance costs $1. In panel (a), each pair of individuals trades at the geographical middle point of their residences, which is represented by a small circle. In panel (b), all individuals go to the centre of the local community, represented by the small circle, which is the residence of an individual, to trade with each other. Now, consider the case n = 7. In panel (a), each of the six individuals residing at the periphery of the community has a farthest trade partner. She travels to the centre, which is the middle point between her and the partner, to trade with the partner. She can trade with the person at the centre by the way of the trip. The travel distance for the return trip cost $2. She has other two neighbouring trade partners. It costs her $1 to trade with each of them. The distance between her and each of the

A General Equilibrium Analysis of Urbanisation »=2

-e

•

n=3

«=4

n=5

n=6

n=l

(a) Dispersed location of transactions

(b) Concentrated location of transactions

Figure 1: Dispersed vs. Concentrated Location Patterns of Transactions

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G.-Z. Sun, X. Yang

other two trade partners is V3. A return trip to the middle point between her and each of them is thus V3. Hence, it cost her $2V3 to trade with the two trade partners. Her transaction costs with six trade partners then total $(2 + 2 + 2A/3) = $7.46. For the person at the centre, transaction costs are zero, since all other individuals will trade with her at the centre as they stop by there to trade with their farthest partners. In the geographical pattern of panel (b), all individuals not residing at the centre bring their goods there to trade. Total transaction cost for each of them is $2. A comparison of transaction costs between panels (a) and (b) indicates that geographical concentration of transactions can save on transaction costs if the size of the network of division of labour is sufficiently large. Williamson refers to the pattern of transactions in panel (b) as the pattern of the wheel, and that in panel (a) as the pattern of all channels. If the purpose of travelling is to obtain information about products, prices, and partners, then increasing returns to the geographical concentration of transactions will be more significant. We may define a central market place as a geographical location where many trade partners conduct transactions. This definition implies a corresponding geographical concentration of transactions. Geographically dispersed bilateral transactions are not associated with the market according to this definition. In fact, it can be shown that the market is not needed if the level of division of labour is low. Consider, for instance, the case n = 2. It can be shown that if the geographical pattern of transactions is such that each pair of trade partners trade at the mid-point between their residences, the transaction cost to each of them is only $ 1. But if all individuals go to the central market place to trade, as shown in Figure 14.2, panel (b), for the case with n = 7, then each individual's transaction cost is $2. This illustrates that for a low level of division of labour, geographically concentrated transactions will generate unnecessary transaction costs. Thus, the capacity of a geographically concentrated pattern of transactions to save on transaction costs depends on the level of division of labour. In other words, trading efficiency is dependent not only on the geographical pattern of transactions, but also on the level of division of labour. But the level of division of labour is itself determined by trading

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efficiency. This interdependence among the level of division of labour, the geographical pattern of transactions, and trading efficiency implies that the three variables are simultaneously determined in a general equilibrium environment. It is analogous to the interdependence between the prices and quantities of goods that are consumed and produced in the standard general equilibrium model, where the optimum quantities of goods consumed and produced are dependent on prices, while the equilibrium prices are themselves determined by individual agents' decisions on the optimum quantities. 3. The Model Consider an economy with M ex ante identical agents in terms of all characteristics.5 Each agent is a consumer-producer, being endowed with a unit of labour. The sizes of residential land in the urban and rural areas are assumed to be A and B, respectively.6 Each individual can freely choose any occupation configuration and residence location between rural and urban areas. There are m consumption goods. The number of goods that are actually traded, however, is to be endogenised by the complicated interplay among the location pattern of transactions, the division of labour and the trading efficiency. Traded goods produced in the urban and rural areas are referred to as A-type and B-type goods, respectively. For each individual, the job choice is associated with the choice between working in the urban area producing A-type goods and working in the rural area producing B-type goods. The utility function of each individual is given by u(z, R), where z represents the amount of the composite consumption good and R the 5

The ex ante identity assumption allows us to formalise Smith's notion of endogenous comparative advantages, which, different from Ricardo's exogenous comparative advantages, means that differences in productivity among individuals are the outcome rather than cause of the division of labour (1776/1976, Chapter 2, p.28). Indeed, as argued forcefully by Smith (1776, Ch.1-3), the labour heterogeneity is not a necessity for the division of labour to emerge. Unfortunately, this point seems to have been simply neglected by many contemporary economists. 6 The assumption of exogenously given lot sizes simplifies the algebra significantly. We leave the endogenisation of the sizes to future research.

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consumption of residential land, i.e., the lot size of the house. Specifically, we assume thatw(z,i?) = zi?andzis composted in such a way from different consumption goods that, z = Y\ ™ 1 xf, where xf is the amount of good/the individual consumes. Thus, the utility function is f m

\

Y\xf R

(1)

V i'=i

where xct - xt + ktxf is the amount of good i consumed, xt is the amount of good i self-provided, xf is the amount of good i purchased from the market, /c, is an iceberg transaction efficiency coefficient (fraction 1 - kt of a unit of good purchased disappears in transit) of good i, and R the amount of land consumed. The production function is identical for all goods, no matter whether they are produced in the city or in the rural area, with a constant marginal return and a fixed set-up cost, Xf + xf -I-a

(2)

where /. is the amount of labour employed in producing good /, which is referred to as the individuals level of specialisation in producing good /, xf is the amount of good i sold to the market, and a is the fixed learning cost, which generates economies of specialisation in production. As shown by Houthakker (1956), Becker (1981), Rosen (1983), the fixed learning cost implies that the division of labour can save on total learning cost for society as a whole by avoiding duplicated learning. The aggregation economies of specialisation for all individuals can generate economies of division of labour which are sometimes referred to as positive network effects. A city, in the urban economics literature, is usually defined as an agglomeration of transactions and manufacturing.7 As Glaeser (1998, p. 140) puts it, 'Conceptually, a city is just a dense agglomeration of people and firms.' Yang and Rice (1994, p.350) hold a similar view, 'a city is defined as the area where many individuals have the division of 7

Nakamura (1985, p. 108) typifies the idea that the benefit from agglomeration of economic activities constitutes the basic driving force of the city formation and urbanisation, 'Agglomeration economies are the most important in explaining modern cities.'

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labour and reside in close proximity to one another'. For residents in the urban area, they benefit not only from the low transport cost as they exchange their produce for other goods that they need in the concentrated urban area but also from the informational spillovers that is much more pronounced in the urban area than in rural areas.8 Those who reside in the rural area pay a higher commuting cost, as they have to travel from the dispersed rural area to the city for their transaction activities, and they suffer from the less spillover of information compared to urban residents. In short, the transaction efficiency for the urban residents is higher than that for rural residents. For simplicity, the ratio of the transaction efficiency coefficient of urban residents, being assumed the same for all traded goods, denoted as kA, and that of rural residents for all traded goods, denoted as kB, is assumed to be a constant greater than one, kAlkB=X>\

(3)

Although individuals might have incentives to reside in the urban area for the facility in transaction, competition for residence in the urban area bids up the residential land rent in the city which in turn means less space for residence in the urban area. The tradeoff between economies of agglomeration and the disutility from less consumption of land due to the land rent escalation in the urban area leads to an equilibrium residence structure and an equilibrium location pattern of transactions, while the efficient trade off between economies of specialisation and transaction costs determines the equilibrium network size of division of labour. Here, the location pattern of residences, the geographical pattern of transactions, the consumption pattern of land, relative land rent between urban and rural areas, and the network size of division of labour are interdependent. Therefore, the concept of general equilibrium is a Glaeser (1998, p. 140) argues that, 'All of the benefits of cities come ultimately from reduced transport costs for goods, people and ideas' and further points out that even now 'the positive impact of agglomeration that comes from reducing the costs of moving people and ideas appear to be as important as ever.' Quigley (1998) emphasises that agglomeration economies are associated with the benefits from the diversity of economic activities in cities. Fujita and Thisse (1996) provide a rather comprehensive overview of economies and diseconomies of agglomeration of economic activities.

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powerful vehicle for figuring out a mechanism that simultaneously determine all of the interdependent variables, as to be shown below in section 4. Lastly, a public ownership of the residential land in both urban and rural areas is assumed.9 Total rent of residential land within the urban (rural) area is equally shared among urban (rural) residents. Hence, each individuals income consists of a component from selling her professional produce and a component from the equally-shared land rent. 4. Equilibrium Residence Structure and Division of Labour What we consider in this paper is a framework with consumerproducers, economies of specialisation in production, and transaction costs in which each individual is allowed to decide upon her specialisation pattern, i.e., to choose the numbers of goods self-provided and purchased from the market. That is, each individual may choose the zero value of some decision variables. The invisible hand coordinates the decentralised decision of individuals on their specialisation pattern and resource allocation and leads to the general equilibrium in which the residence pattern and the network size of division of labour for the society are endogenised. This generates the high degree of complexity of modelling as well as high explanatory power of the model. When the zero value of each decision variable is allowed, there are 23m possible corner and interior solutions for the problem with m decision variables. Fortunately, as proved by Wen (1998), in this kind of model, each individual sells at most one good and does not self-provide and purchase the same good. Although Wen's (1998) proof is somehow complicated mathematically, the economic intuition is rather straightforward: the agent would otherwise bear some unnecessary transaction costs. 9

This assumption, which was also made in Sun and Yang (1998) for simplifying the algebra, indeed can be to some extent justified. 'By institutional facts, half of urban land is in the public domain and the other half is controlled by planning and zoning' (Henderson, 1996, p.32). Note the assumption of a public land ownership is also adopted in Fujita and Thisse's (1986, section 5), in which a Stackelberg strategy location model is developed with firms as leaders and households as followers, as well as in Helsley and Strange (1990).

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As is in section 3, traded goods produced in the urban (rural) area are denoted as A-type goods (B-type goods). Then, producers of A-type goods (B-type goods) have incentives to reside in the urban (rural) area to save on commuting cost between the residence and working location. Therefore, for each consumer-producer, the decisions on residence location and job choice become independent: one chooses to reside in the urban (rural) area if and only if she chooses to produce A-type (Btype) goods. To make the algebra tractable, in the remainder of this paper, we assume that no goods are of both type A and type B. Let nA (nB) be the number of A-type (B-type) traded goods, and n = nA+nB be the number of all traded goods. Clearly, the values of nA, nB, and n are not independent. Given a pair of (nA, nB), each agent can optimise on her resource allocation for the given structure of division of labour with nA occupation configurations in the urban area and nB occupations in the rural area. Each pair of (nA, nB) thus could be used to describe an occupation structure between urban and rural areas. But in a general equilibrium setting, the equilibrium structure of division of labour, which is fundamentally important for understanding the urbanisation process and residence patterns, is endogenously determined. In the following, we will use a two-step approach to solve for the equilibrium pattern of residence and social division of labour. First, we define a value profile of nA, nB, and n as a structure. All individuals' utility maximising decisions are solved for given values of nA, nB, and n. Then market clearing conditions and utility equalisation conditions are used to solve for the equilibrium for a given urban-rural occupation structure (nA, nB). Secondly, taking account of possible value profiles of nA, nB, and n, of which each profile corresponds to a rural-urban occupation structure, we solve for the general equilibrium, which, as is shown below is actually the structure associated with the highest per capita real income.

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a) An equilibrium analysis for a given rural-urban occupation structure The symmetry of the model implies that we need to consider only two types of decision problems. For a urban resident producing a A-type good, the decision problem is Max: uA ^{xA)(kAxdA)"^(kAxdB)^xJA" s.t. xA + xA = lA — a xjA = ljA - a

R

A

(4)

(production function for the good sold)

(production function for a non-traded good)

lA+(m —ri)ljA< 1 (endowment constraint for working time) PAXA (HA

~ 1) + PB4nB

+ r R

A A= PAXA + EA

(budget constraint)

where nA(nB) is the number of A-type (B-type) goods, n = nA+nB is the number of all traded goods, xA is the quantity of the A-type good the individual self-provides, xA is her amount of the A-type good sold to the market, and lA is her quantity of labour allocated to the production of this good, defined as her level of specialisation in producing this good, xA is her amount of a A-type good purchased from the market, xB is her amount of a B-type good purchased from the market, xjA is her amount of a non-traded good produced and consumed, ljA is her quantity of labour allocated to the production of a non-traded good. The symmetry implies that xdA is the same for the (nA - 1) A-type goods purchased, xB is the same for nB B-type goods purchased, and ljA is the same for m - n of non-traded goods. pA and pB are the respective prices of A-type and B-type goods. rA (rB ) is the land rent in the urban (rural) area. RA is the lot size of each urban resident's house. EA= rAAIMA is the land rent distributed to each urban resident, where MA (MB) is the number of urban (rural) residents. The decision variables are xA, xA, lA, an XJA, UA, XA 5 XB d RA10

Given the rather complicated interplay among the geographic pattern of transactions, the network size of the division of labour and the trading efficiency, it is natural for variables characterising specialisation patterns, time allocation, land rents, prices of

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The solution of the decision problem for given values of nA, nB, and n in (4) is called an individual's resource allocation for a given rural-urban occupation structure. Standard marginal analysis is applicable to such resource allocation problems. It yields the demand and supply functions and indirect utility function as follows npA [l - (m - n + \)a\ -{m-n

+ \)EA

{m + \)pA pA(l + na) + EA (m + \)pA

(5b)

pA\-(m-n + X)a\ + E,A xdA = ^—-. —-± (m + l)pA PA[l-(m-n

+l a

)]

+

EA

{m + \)pB pA[l-(m-n + Y)a] + EA {m + \)rA f

K

A

r

APl

P.A

^" B

EA+ pA[\-(m-n

+ \)a}}

m+\

\PBJ

(5a)

(5c)

(5d) ^

(5f)

The optimum decision for a rural resident is symmetric to (5), that is, it can be obtained by exchanging subscripts A and B and letting kB=k. Due to the free rural-urban migration, the equilibrium for structure (nA, nB) can be obtained from the utility equalisation between urban and rural residents and the market clearing conditions for residential lands and goods. The land market clearing conditions are MARA=A

(6a)

MBRB=B

(6b)

The market clearing condition for A-type goods yields goods and so on to be incorporated. Appendix A contains a glossary of all the variables, categorised into parameters and endogenous variables, for the readers' reference.

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npA[l-(m-n

+ l)a]-(m-n

+ l)EA MA

{m + \)pA

nA

PA[\-(m-n -

+ \)a] + EA MA. -(nA-l)-\ (m + \)pA

pB[l-(m-n + l)a] + EB ;{m + \)p ; MB A

(7) The market clearing condition for B-type goods is not independent of (6) and (7) duo to Walras' law. Free rural-urban migration implies the equalisation of utilities for residents in urban and rural areas, from which one obtains,

^ + ^ [ 1 - ( " » - w + l)g]| EB+pB[l-(m-n + \)a]\

\"-B J

r

\

(8)

\PBJ

i.e.

X

„_, \EA+pA[\-(m-n

+ \)a\

E

B +pB[i-(m~n

+ l)a],

r

\

(8)'

PB

vny (6)-(8), the population equation MA+ MB = M, and the definitions EA = rAAIMA and EB = rBBIMB yield an equilibrium for a given structure defined by nA, nB, and n (let /? s = 1 , i.e., the price of B-type goods is used as the numeraire), PA =

h=lB=—

B 0 A 1-0

l-(m-n

_[l-(m-n

m

X«

+ \)a

—+ «

+ l)a]0M mA

[\-{m-n + \)a](\- 8)M mB M^^-^OM

(9a) (9b) (9c)

(9d) (9e)

A General Equilibrium Analysis of Urbanisation

MB=^-

(9f)

= (\-0)M

RA^AMB^A(l~0) RB BMA BO u* = k/~l

l-(m-n m

where / =

-r

6 j v 1-0

+ Y)a

,(\-0)„

213

(9g) A_ \ + f M f (l-0)n+l

(9h)

6= nAln is the urban share of traded goods.

A general equilibrium of residence pattern and division of labour in this paper is comprised of two components. The first component consists of a set of relative prices of traded goods and land, and a set of numbers of individuals choosing different configurations of occupation and residence, and a location pattern of transactions that satisfy the market clearing and utility equalisation conditions. The second is an equilibrium rural-urban occupation structure defined by (nA, nB), ox{n,6), which is endogenised by the interaction among economies of the division of labour, transaction costs and the residence pattern. In the next subsection we will take account of the second component and solve for the general equilibrium, based on which a comparative statics analysis is to be conducted in section 5. b) The general equilibrium residence structure and social division of labour In the preceding subsection, both the number of all traded goods and that of A-type traded goods are fixed. However, both the emergence and evolution of the urban system result from the expansion of the network of division of labour, which is in turn associated with the structural shift of transaction location patterns and the increase of the diversity of occupations (see, e.g., Quigley 1998). In this subsection a general equilibrium analysis is done to solve for the urban-rural structure and the total number of occupations.

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Note that in our model occupation structures are Pareto rankable in terms of utility due to the utility equalisation within any structure, which in turn results from the ex ante identity of agents. In fact, the general equilibrium is in the structure that yields the maximum per capita real income, as shown by the following Lemma.11 Lemma: The occupation structure with the highest per capita real income is a general equilibrium. Any Pareto inefficient structure is not a general equilibrium. This Lemma indeed establishes the claim that a decentralised market can fully utilise economies of agglomeration (which look like externality) and the network effect of division of labour by choosing the Pareto efficient pattern of individuals' residences, the efficient location pattern of transactions, and the efficient network size of division of labour. It also rules out multiple equilibria with different per capita real incomes and shows that there is no coordination failure in a static general equilibrium model with network effects of division of labour and location pattern.12 The very function of the market is to coordinate individuals' decisions in choosing their specialisation patterns and location patterns of residence and transactions in order to fully utilise network effects of the division of labour and economies of agglomeration.13 11

Sun, Yang and Zhou (2001) have shown that both the existence of general equilibrium and the First Welfare Theorem still hold for a broad class of models in the framework with consumer-producers, transaction costs and increasing returns to specialisation in production. One can modify their argument regarding the First Welfare Theorem by considering the augmented model (with a land market incorporated) presented in this paper, and prove the following Lemma. For an alternative proof of the Lemma, see Sun and Yang (1998). 12 But because of indeterminacy about who specialises in which activity, which n out of the m goods are traded, and which nA out n traded goods are produced in the urban area in equilibrium, we have multiple equilibria that generate the same per capita real income. 13 The two-step approach adopted in this section allows us to sort out the general equilibrium, in which nobody has incentives to deviate from her current occupation (location) and resource allocation. It could be shown that if the representative A-type agent is allowed to maximise her utility with respect to numbers of A-type goods and Btype goods she purchases from the market at the first step of our algorithm, and so is the

A General Equilibrium Analysis of Urbanisation

215

With the above Lemma, we can solve for the general equilibrium by maximising the per capita real income given in (9) with respect to nA and nB. Since nA + nB = n, this is equivalent to maximising per capita real income in (9h) with respect to n and 9 = nAln, for which the first order condition leads to (see Appendix B for details), A-i = * • - * A 1-0

(10)

and In* +

— = -#ln/l \-{m-n + Y)a

(11)

Note that (10) implies \h\X\dn =

1 0(1-0)

d0

(12)

that is, 0 = nAln increases with n. In other words, the number of traded goods produced in the urban area increases more than proportionally as the network size of division of labour n increases. (10) and (11) uniquely determine the general equilibrium values of n and 0, which characterise the size and pattern of the network of division of labour and the location pattern of individuals' residences and transactions.14 Consequently, the general equilibrium values of the number of A-type goods and B-type goods (nA and nB), the urbanisation extent (MA or MA/M), the prices of traded goods (pA and pB, note the latter is used as the numeraire), the residential land rents in the urban B-type representative, then their optimal decisions would not match with each other, and hence such an one-step approach does not lead to an occupation equilibrium. Individuals may have coordination failure in deciding their occupation and trade patterns in such an asymmetric dual structure. But that does not mean the coordination failure of the 'invisible hand' in sorting out the equilibrium structure of occupations and division of labour, as is further discussed below. 14 A careful reader may be concerned with the possibility of multiple equilibrium structures characterised by («*,#*) satisfying (10) and (11). As to be shown in section 5, for any («*,#*) satisfying (10) and (11), bothw* and 0* are increasing with k. But from (12), n* and 6* are increasing with each other. Thus the equilibrium occupation structure is unique.

216

G.-Z. Sun, X. Yang

and rural areas (rA and rB), the per capita lot size of residence for urban and rural areas (RA and RB), and the specialisation level of individuals (IA and lB) can be obtained by using (10)—(11) and the equilibrium solution for a given structure derived in the preceding subsection, as below (the subscript" " is suppressed). nA=0n

(13a)

nB = {l-0)n

(13b)

K t=0M

(13c)

PB=1

PAZ =

(13d)

_[\-(m i-n + \)a]i9M 'A

mA

[l-(m- n + \)a](\-0)M 'B

mB B A

0 1-0

(13h)

K

RB=

.

(13i)

(l-0)M

\-(m-n

lA=lB=—i

(13f) (13g)

RA=— A 0M B

(13e)

+ \)a L- + a

,,_.. (13j)

m Note that not only the number of traded goods produced in the urban area increases more than proportionally as the network size of division of labour n increases, but more interestingly, both the absolute urban land-rent (rA) and the relative urban-rural land-rent {rA/rB) increase even faster than the urbanisation speed (0 = nA/n-MAIM) . That is, concomitant with the urbanisation process and the expansion of the network of social division of labour, is the even faster escalation of the land rent in urban areas.

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217

5. Comparative Statics In this section we conduct a comparative statics with respect to the transaction efficiency to investigate the implication of the institutional arrangements and infrastructure for the urbanisation and the network size of the social division of labour. Intuitively, as the transaction efficiency is improved, the unit transaction cost involved in each transaction is decreased, and therefore individuals are allowed a larger scope for trading off economies of specialisation against transaction costs. That means, more goods would be traded through the market. One may naturally expect more traded goods to be produced in the urban area because of the better facility for transaction in the urban area compared with that in the rural area. But the more trade dependent, the more attractive the urban area as residence location to individuals. That means, as the transaction condition is improved, more and more people would migrate from the rural area to the city, thereby further bidding up the urban land-rent. Consequently, the lot size of residence in the urban area decreases. The balance of the tradeoff between economies of the transaction agglomeration and the disutilities from the smaller per capita consumption of land in the urban area leads to a new equilibrium for the improved transaction condition. The following comparative static analysis confirms this conjecture and provides some important insights into other phenomena associated with urbanisation, among which is the increase of specialisation levels of individuals. Our comparative statics starts with an analysis of changes of the occupation diversity and the urban-rural structure of traded goods, i.e., n (=nA+nB) and 0{= nAln) , in response to the improvement of transaction condition, i.e, the increase of the transaction efficiency coefficient k. As shown by (10) and (12) in section 4, the A-type share of traded goods, 0, increases (more than proportionally) with the number of all traded goods, n. Differentiating both sides of (10) with respect to k yields,

f-rai-tfUn*]^ ak

dk

(.4)

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G.-Z. Sun, X, Yang

Inserting (12) into the total differentiation of (11), together with the second order condition for maximising the per capita real income (9h) with respect to 0 and n that requires the negative definiteness of the Hessian matrix (D2u*(n,6) < 0), leads to (see Appendix C for proof) ^=ldk

k

-2

^ [l-(m-n

l

-

>0

(15)

-0(l-0)(ln^ + l)a]

Combining (14) and (15), we have, dk (15) and (16) summarise the comparative statics of general equilibrium network size of division of labour and the equilibrium location pattern of residence and transactions. That is, not only that the number of traded goods for each individual as well as for society as a whole, n, increases with transaction efficiency, but the number of traded goods produced in the urban area increases more than proportionally as improvements in transaction conditions enlarge the network size of division of labour, n. Using (9), the comparative statics of general equilibrium values of endogenous variables can be easily worked out, as shown below. ^ = —*- = — increases with transaction efficiency k nB

MB

1-6

nA = nO increases with transaction efficiency k nB = n(\ - 0) ambiguous — = — = — increases with transaction efficiency k r,

1-0

A

_ [ \m *—"J— increases with transaction efficiency k A

mA \\-{m-n+\)a\a-e)M

rB = ^-^

mB

..

ambiguous

-^- = ——- decreases with transaction efficiencyJ k R„

BO

A General Equilibrium Analysis of Urbanisation

219

RA = — decreases with transaction efficiency k RB =

increases with transaction efficiency k

h=h=

~n+ 'a + a increases with transaction efficiency k

u (per capita real income) increases with transaction efficiency k Note that the envelope theorem is used to derive the positive effect of A: on u from (9h) and (10). Besides the implication of the improvement in transaction facility for the urban land-rent escalation, the structural rural-urban shift of production of traded goods and the urbanisation, it can be further shown that the following concurrent phenomena are also different aspects of the evolution in division of labour driven by improvements in transaction conditions. The diversity of occupations, the differentials between different occupations, and the number of markets for different goods, increase. The number of transactions for each individual, the trade dependence, which is defined by trade volume to real income, interdependence among individuals of different occupations, the extent of the market, which is defined by per capita aggregate demand from the market, and the degree of commercialisation, which is defined by the ratio of income from the market to total real income, increase. The extent of endogenous comparative advantage, which is defined as the difference in productivity between sellers and buyers of traded goods, increases. Production concentration, which is defined as the reciprocal of the number of producers of each traded goods, the per capita real income and productivity of each good increase also. In summary, we have Proposition: Urban land rent increases absolutely as well as relative to the land price in the rural area in the urbanisation process which in turn is a result of the expansion of the network of division of labour driven by improvement in transaction efficiency. At the same time the per capita land consumption decreases for urban residents and increases for rural residents, and the number of occupation in urban areas increases

220

G.-Z. Sun, X. Yang

absolutely as well as relative to that in the rural area. Improvement in transaction conditions also generates the following concurrent phenomena. The number of traded goods for each individual as well as for the society as a whole increases. The number of traded goods produced in the urban area increases absolutely as well as relative to that in the rural area. The population ratio of urban and rural residents increases. Each individual's level of specialisation and the extent of endogenous comparative advantage between individuals of different occupations increase. The number of markets, the number of transactions for each individual, the trade dependence and the interdependence among individuals of different occupations increase. The extent of the market, the geographical concentration of both production and transaction increase. Per capita real income and productivity of each good increase. It seems worthwhile to point out that a distinguishing feature of type II economies of agglomeration in this model is that even if it is indeterminate in equilibrium which «B out of n traded goods are produced in the rural area, a high level of division of labour will make the concentration of transactions and residences in the urban area to emerge from ex ante identical individuals with identical production technologies. The assumptions made in Yang and Rice (1994) and Fujita and Krugman (1995) that land-intensive agricultural production must be dispersed in the rural area is not necessary for the existence of type II economies of agglomeration that mainly result from the agglomeration of transactions and residences. Furthermore, a variety of manufactured goods that is not land intensive may possibly be produced in the rural area, like what have happened in most developed countries after 1970s that is sometimes termed as suburbanisation (see, e.g., Glaeser 1998, p. 145 and Tabuchi 1998, p.334), because of the trade off between economies of agglomeration and high land rent in the city. This has a flavour of the theory of complexity: some phenomena that do not exist for each individual element of a system emerge from a complex structure of a collection of numerous ex ante identical individual elements. This is just like different DNA structure of the same molecules generating many species of animals. This suggests that a hierarchical system of cities

A General Equilibrium Analysis of Urbanisation

221

might be developed based upon type II economies of agglomeration alone. 6. Concluding Remarks A general equilibrium model with consumer-producers, economies of specialisation, and transaction costs is developed in this paper to address the urban land rent, trade concentration in urban area and other issues associated with the urbanisation process. Urbanisation results from the expansion of the network of the social division of labour driven by improvements in transaction conditions. It is shown that as urbanisation develops, the urban land rent increases absolutely as well as relative to the land rent in the rural area, the land consumption decreases for urban residents and increases for rural residents, and the number of occupations in urban areas increases absolutely as well as relative to that in the rural areas. Improvement in transaction conditions also generates the following concurrent phenomena as different aspects of the evolution in division of labour. The number of traded goods for each individual as well as for society as a whole increases. The number of traded goods produced in the urban area increases absolutely as well as relative to that in the rural area. The population ratio of urban and rural residents increases. Each individual's level of specialisation and the extent of endogenous comparative advantage between individuals of different occupations increase. The number of markets, the number of transactions for each individual, the trade dependence and the interdependence among individuals of different occupations increase. The extent of the market, the degrees of commercialisation, the geographical concentration of both production and transactions increase. Per capita real income and productivity of each good increase. We have shown that the invisible hand can fully exploit the network effect of division of labour and economies of agglomeration (which look like externality of urbanisation) by efficiently coordinating individuals' decisions in choosing their individual networks of transactions and their location patterns of residence and transactions. Interpreting agglomeration economies as increasing returns, Ades and Glaeser's (1999) empirical investigation of the (positive) relationship

222

G.-Z. Sun, X. Yang

among extent-of-market, the division of labour and economic growth effectively provides a support to our insight that the interplay between geographical patterns of transactions and trading efficiency leads to urbanisation and increase of productivity. Furthermore, the theory developed in this paper has some important policy implications for urbanisation and economic development. Concomitant with the urbanisation is usually the diversity of economic activities in the city that is conducive to the economic development and hence enhances the welfare of the economy. As is demonstrated in China in the post-Mao era when the government interventions with economic activities attenuates and thus the transaction efficiency is improved, the population share of urban residents increases from 17.9 percent in 1978 to 30.4 percent in 1998 (PRCY 1999, p. 783). Potentially, the theory developed in this paper also has important implications for the real estate business practice. Once elabourated, it may hopefully provide a vehicle to predict the potential for price increase of the urban land. The potential is critically dependent on the equilibrium network size of the division of labour that is in turn determined by transaction conditions. It should be pointed out that our model is simple and limited in some aspects. One possible way to expand the model is to relax the assumption of fixed land sizes in urban and rural areas, and then the city size in terms of the size of residential land would be endogenised. That will significantly enrich the urbanisation story and allow for more insights into the geographic structure of the dual economy. As such, the presumption taken in this paper that the land rent within the city (and the rural area) is the same could also be relaxed. Indeed, as far as transaction costs are concerned, only the distinction between rural and urban areas is made, but within the urban (rural) area the distance from the central market place is assumed irrelevant to the transaction costs in our model. Thus, the distance-related transaction costs could be addressed in the extended model. Another extension in to modify the model to allow for the endogenisation of the diversity of consumptions by introducing a CES utility function. By doing so, increase in the diversity of consumption product, a notable feature with the urbanisation process, would be endogenous rather than exogenously given as is in this paper.

A General Equilibrium Analysis of Urbanisation

223

That is actually an appealing attempt but the mathematics may become quite complicated. Thirdly, no intermediate goods are allowed in the current model, and therefore a very important aspect of the division of labour, i.e. the roundaboutness of production and its connection to urbanisation process is ignored. One possible way to undertake this task is to integrate the present model with Sun and Lio's (1998) model of industrialisation that centres around the co-evolution of the division of labour and roundaboutness in production to address the interplay between urbanisation and industrialisation as an important aspect of the progressive division of labour. We attempt to conduct further analyses in sequels to this paper.

Appendix A Parameters: kA transaction efficiency coefficient for urban residents kB = k transaction efficiency coefficient for rural residents X = kA/kB urban-rural transaction efficiency ratio a set-up cost in all production functions M population size of the economy A{B) size of residential land in the urban (rural) area Endogenous variables: z composite consumption goods consumption of residential land of the urban (rural) resident /, labour employed to produce good i lA(lB) specialisation level of the urban (rural) resident xA(xB) amount of the traded good self-provided by the urban (rural) resident xsA (xsB) amount of a traded goods sold by the urban (rural) resident RA{RB)

224

G. -Z. Sun, X. Yang

nA{nB) number of traded goods produced in the urban (rural) area n = nA + nB number of all the traded goods 9= njn urban share of traded goods m number of all necessity consumption goods MA(MB) population of urban (rural) residents PA{PB) prices of traded goods produced in the urban (rural) area rA(rB) land rent in the urban (rural) area Appendix B Maximisation of the utility (9h) with respect to the A-type share of traded goods, # yields In

X"~ —

e

1

1

+—: 9

e1

(Al)

\-e

Note that the left hand side of (A 1) is an increasing function while the right hand side is a decreasing function of — ^ — - , so that (Al) hold if < r-'A(i-8). and only if =1, i.e. BO A--*.-' A 1-9

(A2)

K

Inserting (A2) into the first order condition for maximising (9h) with respect to the number of the traded goods, n ,yields ma ln& + = -9\nA 1 - (m - n + \)a

(A3)

Appendix C Insertion of (A2) and (A3) into the second order derivatives of the utility (9h) with respect to the number of traded goods (n) and the A-type share of traded goods (9) yields

A General Equilibrium Analysis of Urbanisation

d2u dO1

n(n + 2) (n + l)2

=

d2u* dOdn

225

1 0(1-0)

n(n + 2) In X (n + l)2

(A5)

ma2

6(l-6)(lnX)2 = — (A6) T +— dn2 [\-(m-n + \)a\ (n + l)2 The negative definiteness of the Hessian matrix D2u* (n, 6) requires that (A4) and (A6) be negative and15