Stochastic methods in economics and finance (Advanced Textbooks in Economics)

ADVANCED TEXTBOOKS IN ECONOMICS VOLUME 17 Editors: C.J.BLISS M. D. INTRILIGATOR Advisory Editors: W.A.BROCK D. W.JO...

Author: A. G. Malliaris | W. A. Brock

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ADVANCED TEXTBOOKS IN ECONOMICS VOLUME

17

Editors: C.J.BLISS M. D. INTRILIGATOR

Advisory Editors:

W.A.BROCK D. W.JORGENSON A.P.KIRMAN J.-J. LAFFONT

J.-F. RICHARD

ELSEVIER SCJENCE Amsterdam Lausanne •

•

New York Oxford •

•

Shannon

•

Singapore

•

Tokyo

S TOCHAS TIC METHODS IN ECONOMICS AND FINANCE

A. G. MALLIARIS Loyola University ofChicago

with a Foreword and Contributions by W.A.Brock University of Wisconsin, Madison

ELSEVIER SCIENCE Amsterdam Lausanne •

•

New York

•

Oxford Shannon Singapore Tokyo •

•

•

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 21 1,

1000 AE Amsterdam. The Netherlands

� 1982 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science. and the following tcnns and conditions apply to its use·

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Theorem 4.3. (Fatou). For a non-negative sequence of random variables {Xn} on en, !F, P),

f lim inf X, dP �lim inf f X, dP. n n

n

Theorem 4.4. (Dominated convergence). Let {X,} be a sequence of random variables and Y an integrable random variable all on (U, .tF, P) such that I Xn I � Y w.p.l, for all n . If Xn -+ X w.pJ, then X and X,1 are integrable and

f X,. dP -+ f XdP.

n

n

Results from probability

11

Proofs of these theorems may be found in Ash (1972, pp. 44-50). With the above background on integration we now state some fundamental defmitions. Let X be a random variable on (n, !!F, P) and let k > 0. We say that E(Xk) is the kth moment of X and that E ((X - E (X)l) is the kth central moment. When k I , E(X) is usually called the mean of X and when k = 2, the second central moment is called the variance of X, written as var X and denoted by a2 , t.e. =

•

a2

=

var X= E((X- E (X))2 ),

(4 .9)

provided E(X) < oo. The positive square root a is called the standard deviation. Note that if k > 0 and E(XK} < oo , then E(x2) < oo for 0 < Q < k. Also, if X1, , Xn arc independent random variables on (n, .¥, P) such that E (X;) < oo for all i = 1 , 2, ... , n, then E(X1 , , Xn ) < oo and also •••

•••

E (X1 , ... , Xn ) = E(X1 )

···

E(Xn );

(4 .1 0)

furthermore

n var (X1 + ... + X,) = _L var X;. t=l

(4.11)

For two random variables, X and Y, each having a finite expectation, the co variance of X and Y is defined by cov (X, Y)

=

E((X- E (X)) ( Y - E(Y)})

=

E (XY)- E(X) E ( Y).

(4.12)

From (4.1 0) observe that if X and Y are independent, then cov ( X, Y) = 0; however, the converse is not true. Consider the random variables X and Y and suppose that their variances, de noted by a,i and a}, respectively, are nonzero and finite. We define the co"ela tion coefficient between X and Y, denoted by p (X, Y) as p (X, Y) = cov (X, Y)faxay.

(4.13)

We close this section by stating a useful fact. Let X be a random variable on (n, Y, P) with distribution function F. Assume that g: R -+ R is a measurable function and let Y = g (X). Then E(Y) = f g (x} d F (x). R

(4.14}

12

Stochastic methods in economics and finance

In various applications it becomes easier to compute the expectation of a ran dom variable by integrating over R instead of over n. In such cases we may use (4.14} by letting g(x) = x. S.

Conditional probability

Recall from elementary probability that the conditional probability of a set A given a set B, denoted by P(A 1 B), is given by

P(A I B) = P(A nB)/P(B),

(5.1)

provided P(B) -:1= 0. A conditional probability is associated with events in a given subset of the space n. Intuitively ,a conditional probability represents a re�valua tion of the probability of A occurring in light of the information that B has al ready occurred. In this section we study conditional probability in a general con text for a space (!2, !F, P), where the conditional probability of a set A is defined with respect to a a-field � in .'F. The notation used is P [A I f/J ] and the intuitive interpretation is analogous to (5 .l ), in the sense that the conditional probability of the set A is being evaluated in light of the information available in the a-field c!J , with •!J contained in .OF. At this point a digression is necessary to present two definitions and the Radon-Nikodym theorem which will establish the existence of conditional probability. Consider two measures v and P on a measurable space (!2, §). We say that the measure v is absolutely continuous with respect to the measure P, or equivalently that v is dominated by P, if for each A E .
�IJ2 • Then

w.p.l.

For a proof see Tucker ( 1967, p. 2 1 2).

X and Y be random variables on (n, .¥, P), and suppose Y and XY are integrable and X is �IJ-measurablc. Then Theorem 5.5. Let

E[XY I � ] = XE( Y I �1 ]

for ·.tl a a-field in ."F.

w.p.l

For a proof sec Tucker (1 967, pp. 2 1 3-2 1 4). Finally we state Theorem 5.6. (Conditional form of Jensen's inequality). Let

h be a convex

16


X a random variable on (n, !IF, P) with t'!J a a-field in .'F. Sup X lz (X) arc both integrable. Then lz(E[X I � ]) � E [h(X) I �] w.p.l.

function on R and pose that and

For a proof see Tucker (1967, p. 2 1 7). We close thls section by stating two important facts. Suppose that is a random variable on .�, and let C!J be a a-field in !F. In our discussion thus far we have explained the meaning of conditional proba bility whlch readily extends to the case of a random variable since 1 I t§] = for We also explained the mean I (§ ], with = ing of conditional expectation, writ ten as I 1J ] . For both concepts of con ditional probability and conditional expectation we would like to associate a probability distribution function. In this analysis we follow Billingsley (1979, p. 390) and we state

X

P[A

(n, P)

A {w: X(w)eA'}

A ' e�1 • E [X

P[X E A

X on (n, !F, P) with conditional probability P[A I �4] = P[X eA' I rg ) P[w: X(w)eA' I � ] for A IE 9f 1 there exists a function p (A'' w) defined for A ' E !Jf 1 and wE n satisfying two properties: ( I ) for each w E n, p(A ', w) is, as a function of A', a probability measure on 9f 1 , and (2) for each A 1 e 9f 1 , p (A', w) is, as a function of w, a version of P[XeA I �1 ]. The probability measure p ( · , w) is called the conditional distribution of X given
'

·,

R

6. Martingales and applications

Having presented the elements of conditional probability we now study martin gale theory and some of its applications in economics and finance. Martingale theory uses conditional probability extensively so the usefulness of the previous section will shortly become apparent. Let us also mention that martingale theory,


17

like various other areas of probability theory, has its origins in notions of gambling in the following sense: let X1 , X2 , be a sequence of random variables defmed on a common probability space Within the gambling context these random variables denote the gambler's total winnings after 1 , 2 , ... , n, ... trials in a succession of games. The gambler's expected fortune after trial n + 1 , given that he has completed n trials already, is denoted as E[Xn + 1 1 X1 , , Xn ]. As in the previous section E [Xn + 1 I X1 , , Xn ] denotes the expectation of Xn+ 1 condi tioned upon the a-field generated by the random variables X1 , , Xn . Intuitively, the a-field generated by x. ' .. , xn contains all the past information of the gam bler's fortune up to and including the nth trial. If a game is fair then the gambler after the (n + 1 )st trial will expect on the average to be neither wealthier nor poorer than he was before this trial, i.e. •••

(n, �' P).

•••

•••

•••

.

(6.1) Stated differently, eq. (6.1 ) , called the martingale property, states that in a fair game the gambler's fortune on the next play is on the average his current fortune and is not otherwise affected by the previous history. If instead of = in (6.1) we put � or �' then the game is favorable or un favorable, respectively. With this motivation we state the defmition of a martin gale.

6.1

Definitions

Let X1 , X2 , be a sequence of random variables defined on a common proba bility space (U, !:!', and let !F1 , !F2 , be a sequence of a-fields all belonging to !F. The sequence {(Xn , .'Fn), n = 1 , 2 , ... } is a martingale if for each n it satis fies the four conditions below: •••

P)

•••

( 1 ) . in (4*) we put �, then {(X, , .'F, ), n = 1 , 2, ... } is called a supermartingale. Using (6.4) we state that {(X, , -�,), n = I , 2 , ... } is a submartingale if and only if I xn+ l d P � I X, dP A A

for A E .OF, and

A E .-F,

{(Xn , .� ), 11 = -,

I x,+ 1 dP� f x, d P. A A 6.2.

(6.5) 1 , 2 , ... } is a supermartingalc if and only if for

(6.6)

hxamplcs

With the notions stated above we now give some examples of a probabilistic na ture following Billingsley ( 1 979, p. 408) and Doob ( 1 97 1 ), before we illustrate the application of the martingale concept in economics and finance. (1) Let (n, .'¥', P) be a probability space, let v be a finite measure on .�, and let -�1 , -�2 , . be a nondecreasing sequence of a-fields in .'F. Suppose that v is dominated by P when both arc restricted to .� , i.e. suppose that A E -�, and ..


20

P(A) = 0 imply v(A) = 0. By theorem 5.1 there exists a Radon-Nikodym deriva tive of v with respect to P, when both are restricted to � , and denote it by Xn . Note that Xn is measurable � , integrable with respect to P and by (5 .2) it satis fies for A E � v(A) =

f Xn dP.

(6.7)

A

But A E g;;, implies A E -�1+ 1 so that v(A) = fA xn + 1 dP. Therefore (6.4) holds and {(Xn , -�), n = 1 , 2 , ... } is a martingale. (2) Another example is this; suppose Y is an integrable random variable on (n, � P) and { .� } is a non decreasing sequence of a-fields in .(Xn). To prove (2) we need only to show again that E[cp(Xn+ l ) I � ] � cf>(Xn). Since Xn is a submartingale E[Xn + t 1 .¥,, ] � X, . By hypothesis ct> is nondecreas ing so cp{E [Xn+ 1 1 .� ]} � cf>(Xn)· Now apply again Jensen's inequality to con clude Proof.

Theorem 6.2. (Kolmogorov's inequality). Let X1 , X2 , gale on . 0. Then,

(n,

•••

,

Xn be a submartin

Proof. Let A > 0 be given and define A 1 , ... , An, A as follows:

Note that the A k 's as defined above are disjoint. Also note that since X 1 , X2 , , X, is a submartingale one inductively may obtain E [Xn +k I .� ] � Xn for k � 1 Then n f Xn dP = L f X, cJP k=l Ak A n = L f E [Xn l � ] d P k = l Ak n � L f Xk dP k=l A k n � A L P(A k ) AP(A), k=l •••

.

.

=

Results from probability with A

31

k E .�k = a(X1 , , Xk). Therefore, •••

AP(A)

=

AP,w: �ax X; (w) � A l

l

j

t �n

� J X, dP � J X� d P � E( I Xn I ). A

This concludes the proof.

!�

Next we present the notion of an

upcrossing which is fundamental

for the Up

crossing Theorem. This upcrossing theorem is used by probabilists in the proof of the important result known as the martingale convergence theorem. However,

we present the notion of upcrossing here because it may be useful to researchers in finance and economics also. Let

[a, In

be an interval with

a < (3, and let X 1 ,

The number o f u pcrossings of [a, (3] by X1

•••

, Xn

be random variables.

(w), ... , Xn (w) is the number of times the sequence passes from below a to above {3 (see fig. 6.1 ) .

0

0

0

0

0

0

Figure 6 . 1 .

In the figure above there arc two upcrossings, for

to the strings of consecutive

follows:

yl

= 0,

2 �k �n + l 0

yk

=

1 's above

0

1 1

if

if

if

if

yk - l yk yk - 1 yk-1 J

=

1

=

0

=

1

=

0

and

and

and

and

the graph for

xk - l � (3, xk- l > a, xk - l < {3, xk - 1 � a.

16. These correspond the variable Y defined as

n =

32

Stochastic methods in economics and finance According to the definition an upcrossing corresponds in

1

0 on either side. Now define

an unbroken string of 's with a if

Yk =

1

and

Y2 , Y3 ,

•••

, Yn

to

Yk + 1 = 0,

otherwise, then the number of upcrossings is

Let g;-0

2, . . , n + I .

=

{0 , n} and !fie = a(X1 , , Xk). Then Yk is measurable � - 1 , k = 1 , •••

and U is measurable.

X1 , Xn be a submartingale (n, !F, P). Then the number of upcrossings, U of [a, J3], satisfies

Theorem 6.3.

(Martingale upcrossings).

E ( I Xn I) + I a l

E(U) �

J3 - o:

For a proof see Billingsley

Let

•••

,

on

.

(1979, p. 415) or Ash (1972, p. 291).

The fmal result establishes convergence of martingales.

Theorem 6.4.

(Martingale convergence).

Let

X 1 X2 , ,

•••

be a submartingale on

(fl, fF, P) and assume that K = supn E( I Xn I ) < oo . Then Xn � X w.pl, where X is a random variable such that E( I X I) � K. For a proof see Billingsley

(1979, p. 416).

7. Stochastic processes

stochastic process is a collection o f random variables {Xt, tE T} on the same probability space (n, �, P). Note that X,(w) = X (t, w) has as its domain the k product space T x n and as its target space R or R . The points of the index or parameter set T are thought of as representing time. If T is countable, especially if T {0, I , 2, 3, ... } N, i.e. the set of non-negative integers, then the process is called a discrete parameter process. If T = R or T = [a, b 1 for a and b real numbers or T = [0, oo) i.e. if Tis uncountable, then we have a continuous parameter proA

=

=

,

33


cess. Although the index set T can be rather arbitrary, in this section and the rest of this book the most often used index set is T = [0, oo). n denotes the random or sample space and for fiXed w E n, Xt (w) = X ( w) for t E T, is called a sample path or sample function corresponding to w. Other terms used in various texts to describe Xt(w) for a fiXed w are realization or trajectory of the process. For ·,

fiXed w E n the usual notation of the process is Xt; however, X(t) is also used in this text as well as in the literature. Obviously for a fiXed

t E T, Xt(w) = X(t,

)

is a random variable. The space in which all the possible values of Xt lie is called ·

state space. Usually the state space is the real line R and Xt is called a real

the

valued stochastic process or just a stochastic process. It is also possible for the

state space to be

k R , in

which case we say that

Xt

is a k-vector stochastic pro

cess. The various martingales presented in the previous sections are good examples

of stochastic processes. Additional examples will be discussed in this section; in

particular we briefly plan to discuss the

Markov process, and

the

Brownian motion or Wiener process, the Poisson process. However, before we do so we state

some basic notions in the form of definitions and theorems of stochastic processes.

7. 1.

Basic notions

An important feature of a stochastic process

the random variables of the

{Xt , t E T} is the relationship among This relaprocess, say X, , .. . , Xtn for t 1 , , tn

E T.

tionship is specified by the joint distribution function of these variables given by

Px for

t

•

•

...•

x

t n

(H) = P [w

•

:

•••

(7

(Xt. (w), ... , Xtn (w)) eH]

.I)

HE {jf n . It must be pointed out a t the outset that a system o ffinite-dimen

sional distributions of the form of (7 . I ) does not completely determine the prop erties of the process in the case of an arbitrary index set

T.

However, the fust

step in the general theory of stochastic processes is to construct processes for given fmite-dimensional distributions.

Suppose that we are given a stochastic process having (7

.I) as a fmite-dimen sional system. Note that (7 .I) necessarily satisfies two consistency properties. The first property is the condition ofsymmetry. Let p be a permutation of (I , 2, ... , n) and define fp :

R

n

-+ R n by

(7.2)

The random vector

(Xt

•

,

... , X, ) n

=

f.p (Xtp l , .. . , X,pn )

(7.3)


34

must have distribution Px 1

l

• . .• • x

from the left-hand side of (7 .3) and (7 . 1 ) 1, from the right-hand side of (7 .3). This leads to the

and also Px 1 • •• • • x 1 1; 1 pi pn condition of symmetry written as px,

t

.

- px , . .. . . x , r.p... . x 1 piZ t1 pl

l

(7.4)

·

The second consistency property is called the is written as pX t

t

•

... , X t (H) = pX t n

•

• ... , X t • X t n+ 1 11

condition of compatability; it

(H x R I )

(7.5)

for HE .'Jf " . The conclusion of the above analysis is this: given a stochastic process {X1 , t E T} then its finite-dimensional distributions satisfy properties (7 .4) and (7 .5). Naturally, the mathematical question arises: Does the converse hold true? That is to say, given finite-dimensional distributions having properties (7 .4) and (7 .5), does there exist a stochastic process having these finite-dimensional distribu tions? The question is answered affirmatively by the famous Kolmogorov theorem. Theorem 7 .1. (Kolmogorov). Given a system of finite-dimensional distributions satisfying the symmetry and compatability consistency conditions, then there exists a probability space (n, � P) and a stochastic process {X1, tE T} defmed on this space, such that the process has the given finite-dimensional distributions as its distributions.

For a proof see Billingsley ( 1 979, section 36). Kolmogorov's existence theorem puts the theory of stochastic processes on a firm foundation. Next, we proceed to state some useful definitions. Consider two stochastic processes {X,, tE T} and { Y,, tE T}, both defined on the same probability space (n, fF, P). These two processes are said to be stochastically equivalent if for every tE T, P[w: X1 (w) -:/=- Y,(w)] = 0. Alterna tively, the two processes are equivalent if for every t E T, X1(w) = Y1 (w) w.p.l. If two processes are equivalent we say that one is a version of the other and we conclude that their finite-dimensional distributions coincide. However, it is not always the case that equivalent processes have sample paths with the same prop erties. This can be illustrated with a simple example which also substantiates the remark made earlier that finite-dimensional distributions do not completely de termine all the properties of the process. The example considers two processes


35

{X,, t � 0} and { Y,, t � 0 } on the same probability space (n,.¥, P). Define first X, for t � 0 as X,(w) = 0

(7.6)

for all W E n,

and secondly define Y, for t � 0 as Y,(w)

=

{�

if V(w) = t,

(7.7)

if V(w) * t,

where V is a positive random variable on (n, !!F, P) with a continuous distribu tion given by P[w: V(w) =x) = 0 for eachx > 0. For X1 and Y, defined as above, P[w: X1 (w) * Y1(w)) = 0 for each t � 0 and therefore they arc stochastically equivalent. Also, X1 and Y1 have the same finite-dimensional distribution which, for t 1 , , tn say, are given by •••

pX t

I

' . . . , Xt

tr

(H) = Py t

I

' ... , y t

n

(H) =

{

1

if II contains the origin of R " ,

0 otherwise,

for HE ;% n . However, note that the equivalence of the two processes is not suf ficient to guarantee the same sample paths for x, and Y,. For w E n, from (7 .6) we know that X1 (w) = 0 wlillc from (7.7) Y1 (w) = 0 with a discontinuity at t V(w), where it obtains the value I. Thus, the sample paths of these two pro cesses arc not the same. To correct irregularities of this form probabilists have in troduced the concept of a separable process. Consider a process {X1, t E [ 0, oo)} defined on a complete probability space (n, :F, P). This process is separable if there is a countable, dense subset of T = [0, oo) denoted by S = {t1 , t2 , } such that for every interval (a, b) C [0, oo ) and every closed set A C R it holds that =

•••

P[w: X1 (w)E A for all I E (a, b) () S) = P[w: X,(w)E A for all ! E (a, b)) .

(7.8)

Observe that the definition requires that the probability of the set where [X1 E A for all t E (a, b) n S] and [X1 E A for all t E (a, b)] differ is zero. The motivation of this definition is to make a countable set of time points serve to characterize the sample paths of a process. The important question is: Given a


36

stochastic process having a system of consistent finite-dimensional distributions does there exist a separable version with the same distributions? Fortunately the answer is yes and we are therefore allowed to consider separable versions of a given process. For a detailed statement of the existence theorem and its proof see Billingsley (1 979, section 38) or Tucker (1967, section 8 .2). We next present some simple facts about three important processes. 7.2.

The Wiener or Brownian motion process

A Wiener process or a Brownian motion process {z t , t E [0, oo)} is a stochastic process on a probability space (!1, .� P) with the following properties: ( I ) z0 (w) 0 w .pJ, i.e. by convention we assume that the process starts at 0. (2) lf O � t0 � t 1 � •• • � tn are time points then for HE � � , =

P[zt - z,.

,

1- 1

.

E H; for

i � n] = iX . = n 2 n+2 n +2

(8.9)

Eq. (8.9) suggests that we should not stop because our expected reward after one more toss, Xn + 1 , conditioned on X, , is greater than our reward without any further tossing. But, suppose that we act wisely and have another tossing. Even tually, a tail will occur and our final reward will be 0. Thus, acting wisely at each stage does not imply the best long-run policy. This suggests that for the reward function (8.8) no optimal stopping rule exists. However, stopping rules do exist. Consider, for example, the class of stopping rules {Tk }, k 1 , 2, ... , where Tk stops after the kth toss no matter what sequence of heads and tails has appeared. For such stopping rules we have =

E(X k ) r

=

I 2k

-

·

k2k + k+J

(

I

-

)

1 - ·0 2k

=

k k+I

From (8 .1 0) as k � oo we obtain using (8 .1) that V = exist but no optimal rule exists. 8. 1.

(8.10)

--

l.

Thus, stopping rules

Mathematical results

Having motivated the concept of optimal stopping we proceed to establish some basic mathematical results and then give some illustrations from the theory of job search and stochastic capital theory. Our analysis draws heavily from the two classic books of Chow, Robbins and Siegmund ( 1 97 1 ), and DeGroot ( 1 970). Let (D., fF, P) be a probability space and let {,:;;;, , n = 1 , 2, .. . } be a sequence of increasing a-fields belonging to .¥. Let Y1 , Y2 , be random variables having •••

46


a known joint probability distribution function F and defined on (!1, .�, P). We assume that we can observe sequentially Y1 , Y2 , • •• and we denote by X 1 , X2 , •• • the sequence of rewards. If we stop at the n th stage, Xn = [,, (Y1 , Y2 , ••• , Y,). It is assumed that X 1 , X2 , ••• are measurable with respect to .?'1 , .tF"2 , •• • , i.e. X, is measurable � for n = I , 2 , . .. . A stopping rule or stopping variable or stopping time is a random variable T = T(w) defined on (!1, .�, P) with target space the positive integers 1 , 2 , . . whlch satisfies two conditions. First, .

P[w: T(w) < oo ) = 1 ,

(8. 1 1 )

and secondly

{w: T(w) n} e .�1 =

(8.12)

for each n .

Eq. (8. 1 1 ) says that the stopping rule takes a finite value w.p.l and (8.12) says that the decision to stop at time n depends only on past information included in the a-field .�1 • Put differently, (8.12) indicates that no future information is available to influence the decision to stop at time n. In general � is quite arbitrary, although in some applications we may take .� = a(Y1 , Y2 , ... , Y, ). The collection of all sets A E § such that A n {r = n} e �� for all integers n is a a-field in.¥ and is denoted by .�. We note that T and X7 are measurable .� . For any stopping rule T, the reward at time T is denoted by X7 whlch is a ran dom variable of the form on

{T n}, =

otherwise,

(8.13)

for 11 = I , 2, . . As in (8 .1) the value of the reward sequence, V, is the supremum of E(X7) for T e C. Note that l 0 denotes the fiXed cost of every observation. Eq. (8.1 7) says that our reward at period n is the difference between the largest value ob served among the random variables Y1 , , Yn and the cost of such observations or sampling. We then have the following theorem. :::

•••

Theorem 8.2. Suppose that Y1 , Y2 , is a sequence of independent and iden tically distributed random variables on with a distribution function F and let Xn , n = l , 2, ... be a sequence of rewards as in (8.17). If •••

(D, .07, P)

E(Y� ) < oo

(8. 1 8)

for n = 1 , 2, ... , then there exists a stopping rule which maximizes E(X7) and which has the form: stop as soon as some observed value Y � V and continue if


48

Y < V, where V is the value of the reward sequence obtained as a unique solu tion of the equation 00

J

v

( Y - V) d F(Y) = c.

(8.19)

For a proof see DeGroot (1 970, p. 352).

8.2. Job search Before we state additional mathematical theorems on optimal stopping we move to apply the results stated so far to the theory of job search. In this application we follow Lippman and McCall (1 976a). Consider an individual who is seeking employment and who searches daily, until he accepts a job and who receives exactly one job offer every day. The cost of generating each offer is c > 0. There are two possibilities: sampling with recall, i.e. when all offers are retained, and sampling without recall, i.e. when offers are made and not accepted are lost. Notationally, the random variables Yn represent the job offers at periods n = I , 2 , .. and we assume that the job searcher knows the parameters of the wage distribution F from which his wage offers Y, are ran domly generated. To keep the analysis simple we assume the participant in the job search to be risk neutral and seeking to maximize his expected net benefit. The searcher's decision is when to stop searching and accept an offer. Note that his sequence of rewards has the simple form .

Xn = max { Y1 ,

• • •,

Yn }

-

nc

as in (8.17), for n = 1 , 2, ... , in the case of sampling with recall, and the form

Xn

=

Yn

-

nc

in the case of sampling without recall. This last equation simply says that in the case of sampling without recall the searcher's reward is the difference between the current offer and his total cost of the search. Below we discuss the case of sampling with recall for independent and identically distributed Y1 , , Y, . This job search problem can be analyzed using the results from the theory of optimal stopping. We proceed to discuss the existence and nature of optimal stopping and then we obtain some additional insights from the analysis. By theorem 8 .l , to establish the existence of an optimal stopping rule for the •••


49

job search problem, we need to establish the two conditions in (8 . 1 5 ) and (8.16). This is accomplished in the following lemma in which independence of Y1 , Y2 , is not needed.

•••

Lemma 8.2. Let Y1 , Y2 , be a sequence of identically distributed random variables having a distribution function F. Let c > 0 be a given number and define •••

Z

= sup X, , ,

(8.20)

where X, is as in eq. (8.1 7). If the mean ofF exists, then lim X, --+ - oo as n � oo , w.p.l. If the variance of F is finite then E( I Z I ) < oo . For a proof see DeGroot ( 1 970, pp. 350-352). This lemma is useful because it helps us establish the existence of an optimal rule for the job search problem. Note that this lemma is a purely mathematical result which states sufficient conditions for (8.15) and (8 . 1 6) to hold. More specifically, if the mean and variance of the common distribution F exist then 2 (8.15) and (8.16) hold. Thus, assuming that E ( Y ) < oo , n = 1 , 2, ... , we can use lemma 8.2 and theorem 8.1 to conclude the existence of an optimal rule. The nature of the optimal rule in the job search model is described by theorem 8.2. More specifically, for any wage offer Y, the optimal stopping rule for the job searcher is of the nature or form ,

accept job if continue search if

y�

V,

Y
0 is, the higher the reservation wage V and the longer the duration of search will be. Studying the equation H(V) c, which is (8.19), we obtain a simple economic interpretation, i.e. the value V is chosen so that equality holds between the expected marginal return from one more observation, H(V), and the marginal cost of obtaining one more job offer, c. In this application the job searcher behaves myopically by com paring his wage from accepting a job with the expected wage from exactly one more job offer. We conclude by remarking that in the infinite time horizon case with F( ) known there is no difference in the analysis between sampling with recall and sampling without recall. In the latter case search always continues until the reser vation wage is exceeded by the last offer. Note, however, that if the time horizon is finite, or if F( · ) is not known, these two assumptions cause the results to be different, as is shown in Lippman and McCall (1 976a). =

·

8.3. Additional mathematical results We continue the analysis on optimal stopping by illustrating the role of martin gale theory in problems of optimal stopping. Let X1 , X2 , ••• be a sequence of random variables on (n, !F, P), denoting re wards and let F1 , .fF2 , ••• be an increasing sequence of a-fields in .¥. We assume, as earlier, that X, is measurable with respect to �" n I , 2, ... , and that X, = fn ( Y1 , .•• , Y, ). The pair {(X, , �� ), n = 1 , 2, ... } is called a stochastic sequence. If E( I X, I ) < oo, n = I , 2, .. ., then we say that {(Xn , .�), n = I , 2, . .. } is an in tegrable stochastic sequence. If we interpret a stochastic sequence as a martin gale it is natural to inquire whether E(X1 ) = E(X7), with a stopping time. The reader will recall that the martingale property may be viewed as representing the notion of a fair gamble and therefore asking whether E(X1 ) = E(X7) means that we are inquiring whether the property of fairness is preserved under any stop ping time r. The results that are stated next explore this question and are obtain ed from Chow, Robbins and Siegmund (1971) and DeGroot (1 970). =

T

Theorem 8.3. Suppose that {(Xn , !F,,), n = I , 2, ... } is a submartingale on (il, �, P), r is a stopping time and n is a positive integer. (1) If P[r � n]

(2) If P[r < oo]

=

=

1 , then E [X, 1 .�7 ] � X7• I and

E(X7) < oo and also

(8.24)

52


f lim inf 1Z {T > tt x,: dP = 0, }

(8.25)

then for each n, (8.26) For a proof of this theorem see Chow, Robbins and Siegmund (1971 , p. 2 1 ). From eq. (8.24) we conclude that the unconditional expectations satisfy the relation (8.27) provided P[r � n] = 1 . For supermartingales the conclusion is as in (8.27) with the inequalities reversed and for martingales we have (8 .27) with equality signs. Concerning the condition in (8.25), we note that it is satisfied for a sequence of random variables X 1 , X2 , which are uniformly integrable. The defmition is this: a sequence of random variables {X" , n = I , 2 , ... } on (!1, !F, P) is wzi fomzly integrable if, as a -+ oo, then •••

lim sup n

f I X11 I d P { I X11 1 > a}

-+

0.

(8.28)

Note that (8 .28) implies that sup11 E( I X11 I ) < oo , which allows us to conclude in particular that E(X7) < oo for a stopping time, and therefore (8.26) holds. The last result of this subsection is the monotone case theorem. For a stochastic sequence {(X11 , �,), n = 1 , 2, . . } assumed to be integrable let

T

.

(8.29) for n = 1 , 2, ... . If

A 1 C A 2 C ... CA, C ...

(8.30)

and 00

u

n= I

A

n

=n

(8.31)

both hold, we say that the monotone case holds. In this case the next theorem tells us the nature of the optimal stopping rule.


53

Theorem 8.4. (Monotone case). Suppose that {(X, p .�1 ), n � 1 , 2, ... } is a sto chastic sequence wl1ich satisfies the monotone case. Let the stopping variable s be defined by

s = first n ;;;?; l such that xtl ;;;?; E [XPI+ i :�, J ' provided E(X-;) < oo. Then if lim inf "

f x,: {s > " }

�

I

0

holds, we obtain that

for all T such that E(XT ) < oo and lim inf n

f X,� = 0. {T > n }

For a proof see Chow, Robbins and Siegmund ( 1 97 1 , p. 55).

8.4. Stochastic capital theory We conclude this section by presenting a brief analysis of stochastic capital theory following Brock, Rothschild and Stiglitz (1 979). Introductory lectures on capital theory often begin by analyzing the follow ing problem: you have a tree which will be worth X(t) if cut down at time t, where t = 0, 1 , 2, ... . If the discount rate is r, when should the tree be cut down? Also, what is the present value of such a tree? The answers to these questions are straightforward. Suppose we choose the cutting date T to maximize e-rT X(T). Note that at t < T a tree is worth crt e-n Most of capital theory can be built on this simple foundation. It is our purpose to analyze how these simple questions of timing and evaluation change when the tree's growth is stochastic rather than deterministic. Suppose a tree will be worth X(t, w) if cut down at time t, where X(t, w) is a stochastic process. When should it be cut down? What is its present value? We ask these questions because, as in the certainty case, one can usc such an analysis to answer many other questions of valuation and timing. Before we can analyze the problem of when to cut down a tree which grows stochastically, we must specify both the stochastic process which governs the

X(T).


54

tree's growth and the valuation principle used. Here we analyze a discrete time model because for such models analysis of some problems can be done both more easily and in greater generality. We assume the tree's value follows a discrete time real-valued Markov process which we write as , X1• To complete the specification of our problem, we must describe what the person who owns the tree is trying to maximize. The simplest assumption is that he is maximizing expected present discounted value. Thus, if we let C be the set of stopping times for X1, then the problem is to find r E C to maximize e- rr. An apparently more general approach would involve the maximization of expected utility, is a strictly e-rr). This is only apparently more general because if increasing function, w)) is a Markov process with essentially the same properties as X w) and there is no analytical difference between maximizing E(e- rr X(r)) and maximizing Of course, the interpretation of the stochastic properties of is thought of as being depends on whether measured in dollars or in utiles. If the discount rate is

X1 , X2 ,

•••

EX(r)

E(U[X(r)]

U(X(t,

(t,

(3

=

1

1

+r

U(X)

E(U[X(r)] e-n). X(t)

X(t)

'

(8.32)

E{f37 X7).

our problem is to choose a stopping time r to maximize Let us choose the simplest possible specification for the X1 process, i.e. suppose that (8.33) where €1 are independent and identically distributed random variables with ex pected value p.. It is clear that in thi� case the optimal rule will be of a particu larly s!mple form: pick a tree size X and cut the tree down the ·first time that

x, � x.

If the process xt is deterministic, x,+ I = x, + p., it is easy to fmd the optimal cutting size, which we denote by X. X must satisfy

X = (3(X + p.),

(8.34)

since the l.h.s. of (8.34) is the value of cutting a tree down now and the r.h.s. is the present discounted value of the tree next period. If X satisfies (8.34), then the tree owner is indifferent as to whether he sells the tree now or keeps it for a period. For small the value of the r.h.s. (keeping the tree for a period) exceeds the value of the l.h.s. (harvesting the tree now). Note that if X is a solution to (8.34), then p.(X = (1 - (3)/(3 r, or X = J.L/r, i.e. the growth rate equals the inter est rate. How is tllis analysis changed if the tree's growth is uncertain? The answer

X,

=


55

depends on whether the stochastic process Xt is strictly increasing or not. Let X be the optimal cutting size for the random process in (8 .33). Then if € is positive, which implies that X, is an increasing process, uncertainty has no effect on the cutting size. Theorem 8.5. If

(8.35)

P[Xt + 1 > Xr] = 1 , then X = X .

We give a heuristic argument rather than a rigorous proof. This result is implied by the more general theorem 8.7 below. Let V(X) be the value of having a tree oi size X assuming it will be cut down when it reaches the optimal size. Then if X is the cutting size it must be that A

V(X) = X

for X � X

V(X) > X

for X < X.

(8.36)

and A

(8.37)

i,

Also at the tree owner must be indifferent be!ween cutting the tree down now and letting it grow for a period. That is to say, X must satisfy

X = {3 E(max {X + E, V(X + €)}).

(8.38)

However, since € is non-negative, V(X + e) = X + € and (8.38) becomes X = f3 E(X + €) = {3(X + p.) which is the same as (8 .34). Since the solution to (8.34) is unique, X = X. If € is negative, tltis argument does not go through. A

-

Theorem 8.6. If P[E < 0] > 0, then X > X. This theorem says that trees will be harvested when they are larger under uncer tainty rather than under certainty. Again we give a heuristic rather than a rigorous proof. Suppose for simplicity that € has a density function f( ·) with support on 1 , + 1 ]. Then (8.38) becomes

[-


56

X= A

( 0 ( J (i 0

(3 f V(i -1

>P

-1

+ e) [(e) d e +

+ e) [(e) d e +

1

J (i + e)[(e) d e

0 1

J (i + e) [(e) de

0

= f3(X + p.). Thus,

i>

1

J.l(3 -

(3

=

11

r

)

)

= x.

We note here two implications of these simple propositions: that uncertainty can increase the value of a tree and that strictly increasing processes behave dif ferently from processes which can decrease. Theorem 8.6 implies that uncertainty can in some cases increase the value of a tree. Suppose you have a tree of size X. Then if its future growth is certain, its value is just X. If its evolution is uncertain and if its size may decrease, then you will not cut it down; its value exceeds X. Writing (X) as the value of a tree of size X in the deterministic case and ve (X) as the value of a tree when the incre ments in the tree's growth are the (nondegenerate) random variable €, we have shown that if e were negative

Vd

(8.39)

for some X. (Continuity implies that (8.39) holds for X E [X - o, �) for some o .) It is natural to ask whether (8.39) holds under more general circumstances. Another implication of theorems 8.5 and 8.6 is that strictly increasing pro cesses are different from processes which may decrease. We show that this is gen erally true by showing in the next theorem below that theorem 8.5 holds for a very wide class of increasing processes. Theorem 8.7. Let X, be a Markov process such that

P[Xt+ t � X, ] = 1 and that

E [.BX,+ 1 - X, I X, � X, ] � O, E [.8Xt+ 1 - X, I x, � X, ] � 0,

57


t where xt is a nonincreasing sequence. Let yt = [3 Xt. lf xt � Xt , then Yt, yt+ I ' Yt+ 2 , ... is a supermartingale. This proposition implies that uncertainty does not affect the time at which the tree is cut down. By theorem 8.3 we know that if T is any stopping time, then E [ YT I xt � Xt ] � Yt ; it is optimal to stop when the tree's height first exceeds xt . To establish this theorem we must show that for all t

However, since

it suffices to observe that E(.Bxt+T+ 1

-

xt+T 1 xr � xt 1 = E(E(.Bxt+ T+ 1

= E(E[.8xt+T+ I

-

xt+T 1 xt+T J 1 xt � Xt) xt+T I xt+T

;;;. xt +T] I xt :>: Xt) � 0. -

9. Miscellaneous applications and exercises ( 1 ) Suppose that {An} is a countable sequence of sets belonging to the a field !F. Show that n, A, E :':!'. 1hls fact establishes that a a-field is closed under the formation of countable intersections. Next, suppose that A E ff and BE.¥', where .� is a a-field. Show that A - B E .�. (2) Let be the set of all positive integers and let .91 be the class of subsets A of such that A or A c is fmite. Is sl a a-field? Explain. (3) Let R 1 be the set of real numbers and let !!A 1 be the a-field of Borel sets. The class Bf 1 contains all the open sets and all the closed sets on the real line. This shows that the class fA 1 of Borel sets is sufficiently large. However, the reader must be warned: there do exist sets in R 1 not belonging to For such an example see Billingsley (1979, section 3). (4) Suppose that n = { 1 , 2, 3 , ... }, i.e. n is the class of positive integers and !F is the a-field consisting of all subsets of n. Define J.l. (A) as the number of points, i.e. integers, in A and �-t(A) = oo if A is an infinite set. l11en 11 is a measure on�; it is called the counting measure. Next, consjder (!2, .�) as above and let p 1 , p2 , ... be non-negative numbers corresponding to the set of positive integers such that 'J:,;P; = I , i = 1 , 2, ... . Define

Z Z

:'11 1

•


58

�-t(A) = L P;· x;EA Then 11 is a probability measure. (5) Probability as a special kind of a measure satisfies several useful proper ties. Here are some such properties. Let (n. !F, P) be a probability space. (a) If A E !F and B E .� and A C B then P(A) � P(B). This is the monoto

nicity property. (b) If A E .�then P(Ac) 1 - P(A). (c) If {A, } is a countable sequence of sets in !F then =

( )

P U , A, � :r , P(A,).

This is called the countable subadditivity property

or Boole 's inequality. (d) If {A, } is an increasing sequence of sets in !F having A as its limit then P(A,) increases to P(A). For proofs of these statements see Billingsley (1979, section 2) and Tucker ( 1 967, pp. 6 -7). (6) Suppose X and Y are random variables defined on the same space (il, !F, P). Then (a) eX is a random variable for c e R; (b) X + Y is a random variable provided X (w) + Y(w) =1= oo - oo for each w; (c) X Y is a random variable provided X(w) Y (w) =I= 0 • oo for each w; and (d) XJY is a random variable provided X(w)/Y(w) =1= oo I oo for each w. be a random variable on (il, /F, P) with finite expectation E (X) (7) Let and finite variance var X. Then for k such that 0 < k < oo we have

X

P[w : I X(w) - E(X(w)) l � k ] �

var X

k

2

•

This is called Chebyshev's inequality and it indicates that a random variable with small variance is likely to take its values close to its mean. (8) Let X be a random variable om (il, .'F, P) with finite expectation and sup pose that is a convex real-valued function such that E((X)) < oo. Then (E(X)) � E ((X)). This is called Jensen's inequality and it has found several applica tions in economic theory. If a consumer prefers risk then his utility function is convex, and Jensen's inequality says that the expected value of his random utility function is greater or equal to the utility of the expected random variable X. In this case the consumer wH: pay to participate in a fair gamble. Suppose that is a concave real-valued function with E(X) < oo and E((X)) < oo. Then (E(X)) � E ((X)), which is also called Jensen's inequality, and in economics it may be


59

used to describe a risk-averse consumer who will avoid a fair gamble by purchasing insurance. Friedman and Savage (1 948) have suggested a utility function com posed of both concave and convex segments. For further applications of Jensen's inequality in economics see Rothschild and Stiglitz (1 970). (9) Let en,� P) be a probability space with A E !F such that P(A) > 0. Let {Bn } be a finite or countable sequence of disjoint events such that P(Un Bn) = 1 and P(Bn) > 0 for all n. Then for every k,

P[Bk I A ] = P[A I Bk ] P[Bk ] I :r n P[A I B, ] P[Bn ]. This is called Bayes ' theorem. Using conditional probability this theorem may be stated as follows:

P[B I A )

=

f

B

P[A I .19] dP I f P[A I 9i ] dP n

for B E .'M, where EM is a a-field in .'F. ( 1 0) Theorems 4.2, 4.3 and 4.4 can be extended for conditional expecta tion. They are as follows. (a) Conditional form of monotone convergence theorem. Let {X, } be an in creasing sequence of non-negative random variables on (n, .� P) and let Xn (w) -+ X(w) w.p.l and assume that E(X) < oo . Then for rJ a a-field in !F, w.p.l as n � oo . (b) Conditional form of Fatou's lemma. If {Xn } is a sequence of non-negative random variables on (n, .�, P) with finite expectations and if E(lim infn Xn) < oo as n -+ oo , then E [lim inf n Xn I _ The property of dependence ofsolutions on parameters is established by the next theorem. Theorem 7.1. Let x(p, t) be the solution of (7.1) and suppose that f(p, t, x) and a(p, t, x) satisfy for all p the conditions of existence and uniqueness, i.e. (6.4) and (6.5). Also, assume the following: (7.2) (2) for every N> 0,

I x I � N, t E [0, T], as p

lim sup ( lf(p, t, x) - f(p0, t,x) I + I X

-+

Po then

a(p, t,x) - a(p0 , t, x) I ) = 0;

(3) there is a constant K, independent of p such that for t € [0, T]

Then, as p -+ Po , sup t

p

l x(t,p) - x(t,p0 ) l -+ 0.

(7.3)

For a proof of this theorem see Gihman and Skorohod {1972, pp. 54-55). As a special case of theorem 7.1 suppose that the functions f and a are inde pendent of the parameter p, i.e. let

dx(p, t) = f(t, x) d t + a(t, x)dz(t),

(7.4)

with initial condition x(p, 0) = c(p) for t E [0, T]. Here we maintain the depen dence of initial condition on the parameter p. As a corollary of theorem 7 .l we now obtain the property of the stochastically continuous dependence of solu tions on initial data, i.e. for eq. (7 .4) assuming existence and uniqueness of its solution, (7 .2) implies (7 .3).


98

We now define mean-square-differentiability for a stochastic process. Let x (t) be a stochastic process with t E [0, T] and let u E [0, T] be a specific point. We say that x(t) is mean-square-differentiable at t = u, with random variable y(u) as its derivative, if the second moments of x(t) and y {u) exist and satisfy for h > 0 EI

[x(u + h) - x(u)]/h - y(u) 1 2

�

0

as h � 0.

We apply the above definition to a stochastic process depending upon a para meter and given by t

x(s, t, c) = c + J f(v,x (s, v, c)) dv +

t

s

J a(v, x(s, v, c)) dz (v), s

{7.5 )

where v E [s, T], 0 � s < t � Tand x(s, s, c) = c E R. A question naturally arises: Under what assumptions is x (s, t, c) of (7.5) mean-square-differentiable with re spect to initial data c? The answer is given by the next theorem. Theorem 7.2. Suppose that /and a of (7.5) are continuouswith respect to (t,x)

and they have bounded first and second partial derivatives with respect to the argument x. Then for given t = u, u E [s, T], the solution x (s, u , c) is twice mean square-differentiable with respect to c and

a

u

& x(s, u, c) = 1 + J fx (v, x(s, v, c)) s u

a x (s, v, c) d v ac

a + J ax (v, x(s, v, c)) - x(s, v, c) dz(v). ac s For a proof of this theorem and its generalization see Gihman and Skorohod (1972, pp. 59-62}. Next we recall the defmition of a Markov process and estab lish the fact that solutions of stochastic differential equations are Markov pro cesses. A real-valued process x(t) for t E [0, T] defined on the probability space (n, .tF, P) is called a Markov process if for 0 � s � t � T and for every set B e &I, with 9'1. denoting the Borel a-field of R, the following equation holds w .p.l:

P[x(t) E B 1 .� ] = P[x(t} E B I a(x(s))].

Stochastic calculus

99

Here .'f_; is the a-field generated by x (s) for s E [0, T] and a(x(s)) denotes the a-field generated by the single random variable x(s). In words, the definition says that for a Markov process the past and the future are statistically independent when the present is known. For a Markov process x(t) we obtain from the theory of conditional probability the existence of a transition probability function, de noted by P(s, x, t, B), which yields the probability that x (t) E B, given that x (s) = s, for 0 � s � t � T. Theorem 7.3. Consider the stochastic differential equation (6 . I ) with initial con dition as in (6.2) and suppose that a unique solution x(t), t E [0, T] exists. Then x (t) is a Markov process whose initial probability distribution at t = 0 is c and whose transition probability is given by

P(s, x, t, B) = P[x(t) E B I x (s) = x ]. For a proof see Arnold (1 974, p. 147). A special class of Markov processes is the class of diffusion processes. A real-valued Markov process x(t), t E [0, T], with almost certainly continuous sample paths, is called a diffusion process if its transition probability P(s, x, t, B) satisfies the following three conditions for every s E [0, T],x E R and E > 0: th

(1) lim

I

f

t-s ly-x I > «=

-

P(s, x, t , dy) = 0;

(7.6)

(2) there exists a real-valued function f(s, x) such that th

lim

I

f

t-s ly-x 1 .-

-

€

(y-x) P(s,x, t, dy) = f(s,x);

(7.7)

(3) there exists a real-valued function h (s, x) such that tls

lim

1t-s

f (y-x)2 P(s,x, t, dy) = h(s,x). ly-x l .- t:

(7.8)

Note that f is called the drift coefficient and h is called the diffusion coeffi cient and they are obtained from conditions (2) and (3). We remark that condi tion ( 1 ) says that large changes in x (t) over a short period of time are improbable. Theorem 7 .4. Consider the stochastic differential equation (6.1 ) with initial con dition as in (6.2) and suppose that a unique solution x (t), t E [0, T] exists. Sup-

100


pose also that the functions f and a are continuous with respect to t. Then the solution x(t) is a diffusion process with drift coefficient f(t, x) and diffusion coefficient h(t,x) = a2 (t,x}. For a proof qf this result see Arnold (1 974, p. 1 5 3). The usefulness of this last theorem can be explained as follows. Note from the definition of a diffusion process that the transition probability function P(s, x, t, B) is crucial in defining the drift coefficient f(t, x) and the diffusion coeffi. cient h(t, x). A question arises. Suppose that a diffusion process is given with coefficients f and h. Can we obtain P(s, x, t, B) from f and h? The answer is yes. Actually the decisive property of a diffusion process is that the transition proba bility P(s, x, t, B) is uniquely determined under certain assumptions by the co efficients f and h. This fact is surprising because f and h involve only the first and second moments which, in general, are not sufficient to define a distribu tion. Therefore given a stochastic differential equation that satisfies the hypo theses of theorem 7 .4 we know that its solution is a diffusion process which under certain assumptions could yield a transition probability without actually having to find an explicit solution. How is this done? We answer the question by making certain definitions and by stating relevant theorems. To each diffusion process with coefficients f = (fi), i = 1 ; 2, , d and h = (h;i), i,j = 1 , 2 , . , d we assign the second-order differential operator .

� =

...

.

�fi(s,x) z

a

a

-

X·'

+ 4 � � h;j (s, x) z

1

a

a

2 a

X·I X · I

(7.9)

gg can formally be written for every twice partially differentiable function g(x). For real-valued f and h (1 9) becomes .

!'2

= f(s ' x)

dx

d

-

d2 . + 2! lz (s x) dx2 '

(7.10)

Denote by Es,x expectation conditioned upon x at time s. The following theorem is important in our analysis. Theorem 7.5. Suppose that x(t) for t E [0, T) is a d-dimensional diffusion pro cess with continuous coefficients f(s, x) and h(s, x) and that conditions (1 ), (2)

and (3) in (7 .6), (7 .7) and (7 .8) hold uniformly in s E [0, T]. Let g(x) denote a continuous, bounded, real-valued function such that for s < t, t fixed,x E R d , u(s,x) = Es ,x g (x (t)) =

f g(y)P(s,x, t, dy). Rd

(7 .1 1 )

Stochastic calculus

101

Suppose u has continuous bounded partials

au . 32 u . 3x; ' 3x;3xi '

1 0 on (0, oo ] and (8.3) holding. From theorem 7.4 we know that the solution of (8.2), x(t), is a diffusion process taking on values in the interval [0, oo ]. Furthermore, the endpoints of the interval [0, oo ] are ab sorbing states, i.e. if x (t) 0, then x(u) = 0 for u > t and similarly if x (t) = oo , then x(u) = oo for u > t. In general, a stationary distribution will always exist in the sense that x will either ( l ) be absorbed at one of the boundaries or (2) it will have a finite density function on the interval (0, oo ) or (3) it will have a discrete probability mix of (1) and (2). Possibility (2) is the nontrivial case which is of interest for (8.2) under the stated assumptions. Note, however, that in case (2) the boundaries are in accessible in the sense that, as e -+ 0, =

(9. 1 )

P[x(t) � e] -+ 0, and also P [x(t) � 1/d -+ 0.

(9.2)

A necessary and sufficient condition for (9 .l) and (9 .2) to hold is that eqs. (9 .3) -(9.5) are satisfied, where X

f

/2 (u)du = oo ,

(9.3)

f

/2 (u) du = oo ,

(9.4)

f

/1 (u) du 0, a > 0, b > 0 and p > 0. Discover an optimum solution. Note that this problem and one of its generaliizations are discussed in the next chapter. IS.

Further remarks and references

Most economists are familiar with the analysis of discrete time stochastic models from studying econometrics. In this chapter we usc briefly the discrete time case in order to motivate the analysis of the continuous time case. Problems related to discrete time stochastic models, beyond the introductory econometrics level such as in lntriligator ( 1978), arc presented in some detail in Chow (1 975), Aoki ( 1 976), and Bertsekas and Shreve ( 1978). Also, note that Sargent ( 1 979) has a chapter on linear stochastic difference equations. The modeling of uncertainty in continuous time is presented in Astrom (1970), Balakrishnan (1973), Fried man (1975), Soong (1973), Tsokos and Padgett (1 974), and Gihman and Skoro hod (1969, 1 972). Ito's stochastic differential equation appeared in Ito (1946, 1 950) and was later studied in some detail in Ito (195 1 b). In his 1946 paper Ito discovered that certain mathematical questions, raised by Kolmogorov and later by Feller about partial differential equations related to diffusion processes, could be studied by solving stochastic differential equations. Later in his 1 95 1 Memoir, Ito expanded the Picard iteration method of ordinary differential equations to establish theo rems of existence and uniqueness of Ito stochastic differential equations. It is worth remarking that Ito's stochastic differential equation is one among many stochastic equations. Syski ( 1967) considers a basic system of random dif ferential equations of the form

dx

dt = f(x(t), y(t), t) for t E T, with initial condition x(t0) = x0 , where {, x and y can be vectors of appropriate dimensions, and he classifies such equations into three basic types: random differential equations with (1) random initial conditions, (2) random inhomogeneous parts, and (3) random coefficients. The Ito stochastic differen-

Stochastic calculus

1 33

tial equation is a special class of random differential equations which, however, is important for several reasons. First, the conditional mean and the conditional variance as functions are sufficient statistics for Ito equations. Thus, for Ito equa tions the calculations of the conditional mean and the conditional variance func tions completely determine the whole process. This is analogous to the statistical fact that the mean and variance are sufficient statistics in normal distribution theory. Secondly, the Ito equation exhibits a nonanticipating property which is use ful in modeling uncertainty. In other words, if the true source of uncertainty in a system is dz and we want the differential equations not to be clairvoyant, then the evolution of the state variable x in the next instant should depend only on uncertainty evolving in that instant. An Ito equation is consistent with this lack of clairvoyance. Thirdly, the Ito equation has solutions which when they exist have nice prop erties. Section 7 describes these properties and it suffices to say here that the Mar kov property and the diffusion property are useful properties with a well de veloped theory available about them. Finally, although economists are just dis covering the usefulness of the Ito equation, it is worth noting that this equation has found important applications in the engineering literature, and particularly in the areas of control, filtering and communication theory. The discussion of section 3 concentrates on Ito's integral . Basic references are Ito ( 1 944, 1 95 l b}. Doob (1 953) presents in some detail Ito's original ideas with some extensions. See also Doob ( 1 966) for an explanation of the connec tion between Wiener's integral and Ito's generalization. Recently, Astrom (1 970) and Arnold ( 1 974) have presented simplified discussions of Ito's integral. A de tailed account about stochastic integrals is given in McKean ( 1 969). A differ ent approach to stochastic integration and in general to stochastic calculus may be found in McShane (1 974). Wong and Zakai ( 1965} discuss the convergence of ordinary integrals to stochastic integrals. We note that section 3 is a brief intro duction to stochastic integration having as its purpose to motivate and supply a definition for the classical Ito stochastic integral. For the reader who is interested in pursuing his study on stochastic integrals we suggest Metivier and Pellaumail ( 1 980), McKean { 1 969) and Kussmaul (1977). These books develop the theory according to the Ito prototype. The idea of defining stochastic integrals with re spect to square integrable martingales, suggested in Doob ( 1 953), was extended in Kunita and Watanabe (1 967). Further extensions of the stochastic integral with respect to a special class of Banach-valued processes is presented in Meyer (1 976) and with respect to Hilbert-valued processes in Kunita ( 1 970). Naturally, the various extensions of the concept of a stochastic integral have implications on Ito's lemma and the study of stochastic differential equations. Metivier and


1 34

Pcllaumail ( 1980) provide detailed references of such extensions. In this book we usc Ito's integral because we chose to analyze Ito's stochastic differential equations. The reader, however, should be informed that an alterna tive approach to stochastic integration has been proposed by Stratonovich (1966). To illustrate the difference between the Ito and the Stratonovich integral we con sider the simple special case t

J z(u)dz(u),

(15.1)

s

where z(t) is a Wiener process with unit variance. In this special case, for a par tition of the form of (3.8) the analysis of section 3 in particular lemma 3.4 in this chapter showed that Ito's integral denoted by A is defined such that as e � 0, then

E I A - � z(t;) [z(t;+ 1 ) - z (t;)] l

1 2 � o.

( 1 5 .2)

Recall that (15 .2) comes from the two equations in (3.1 1 ) of this chapter. Stra tonovich (1966, p. 363) defines his stochastic integral, in our special case, as

E

S-

]

[z(t;) + z(t;+ 1 ) 2 [z(t;+ 1 ) - z(t;)l fL

2 � o,

( 1 5.3)

where S denotes the Stratonovich stochastic integral. Note that the Stratonovich stochastic integral is a particular linear combination of the integrals A and B de fined in (3 .1 1 ). More specifically,

S - -I A + -I B

-2 2 ,

where A is Ito's stochastic integral, and A and JJ arc as in (3.1 1 ). The Ito and Stra tonovich integrals are not greatly different. Actually, Stratonovich has developed formulae representing one integral in items of the other, and these formulae are not complicated. See Stratonovich (I 966, p. 365). However, because of the na ture of stochastic calculus these two integrals have different properties. The Ito integral and the Ito differential equation maintain the intuitive idea of a state model. Also, the Ito integral has the useful properties that it is a martingale and it preserves the interpretations that the expectation of dx in (2. 1 2) is [dt and the conditional variance of dx is d t. The main disadvantage of the Ito integral is that it does not preserve the differentiation rules of ordinary calculus as Ito's lemma demonstrates. The Stratonovich integral preserves many computational

a2

Stochastic calculus

135

rules of ordinary calculus but it does not have the just stated advantages of the Ito integral. For further details about the relation between these two integrals see Meyer (1976). So far, all economics and finance applications of stochastic calculus have used the Ito integral because the associated Ito differential equations provide a mean ingful modeling of uncertainty , as explained earlier. Consequently, Ito's lemma is important for computing stochastic differentials of composite random functions. Put differently, Ito's lemma is a formula of a change of a variable for processes that arc stochastic integrals with respect to a Wiener process. Note that Ito's lemma first appeared in Ito ( 195 I a) and later in Ito (1961 ). We repeat here what we said earlier, namely that Ito's lemma and Ito's integral were discovered by Ito as he was working on partial differential equations related to diffusion pro cesses. This original research led also to the development of stochastic differen tial equations. It was Ito, and later Doob, Gihman and Skorohod among others, who established the field of stochastic differential equations. At this point we will discuss briefly one method for solving stochastic differ ential equations of first order as presented in Gihman and Skorohod (1972, pp. 33-39). Consider the equation on [0, T], dx(t) = f( t , x(t)) d t + a(t,x(t)) dz(t)

( 1 5 .4)

and note that it can be written in integral form as

x (t) = x(O) +

I

t

J f(s,x(s)) d s + J a(s, x (s)) dz (s),

0

( I 5.5)

0

where x(O) is the initial condition. The idea of the technique is to discover ap propriate transformations of ( 1 5 .4) so that the right-hand side of ( 1 5 .5) has a more convenient form, i.e. the unknown function does not appear on the right hand side of ( I S .5). We illustrate this technique with an example and refer the reader to Gihman and Skorohod ( 1 972, pp. 33 -39) for the detailed mathematics and for more examples. Consider the equation dx( t) = a0x(t) d t + a0x(t) dz,

(1 5.6)

with a0 and a0 constants. To solve this equation consider the substitution y = log x and usc Ito's lemma to obtain:


136

dy

=

1 1 dx + X 2

-

( - -I ) (dx) = - (a0xdt + a0xdz) X2

I

2

X

1 1 2 2 - - a0 x d t 2 X2

Integrating this last equation we get:

y(t) - y(O) =

t

t

J (a0 - � a� ) d t + J a0 dz

0

,

0

which can be written as:

y(t) = y(O) + (a0 - � a�) t + a0z(t). Recalling the substitution y = log x, i.e. x that the solution of ( l 5 .6) is:

x(t)

�

eY( t )

�

x (O) exp

( 1 5 .7)

= e Y , for x(O) = eY (O ) , we conclude

{ (a0 - a; ) t + a0z(t)} .

Thus, finding the appropriate substitution and using integration can lead to find ing the solution of stochastic differential equations. In sections 8 and 9 we discussed the problem of stochastic stability and we distinguished between stability of a point equilibrium and stability in the sense of convergence in distribution to a steady state distribution. Mathematicians have studied stability for a point equilibrium where noise has disappeared. while the area of stability in the sense of convergence in distribution to a steady state dis tribution independent of initial conditions has not received much attention. Even so, both areas remain quite open for future research. One of the first papers in the stochastic stability of point equilibrium is the work of the Russian mathema ticians Kats and Krasovskii ( 1 960) where they extend the deterministic Liapunov ( 1 949) method. Note that Antosiewicz ( 1 958), Borg ( 1 949), Hahn ( 1 963, 1967), Cesari (1 963), J lartman ( 1 964), Krasovskii ( 1 965), La Salle and Lcfschetz (1961), La Salle ( 1 964), Massera (1 949, 1956), and Yoshizawa ( 1 966} are among the basic references for Liapunov deterministic method to stability. In the United States, Bucy (I 965) and Wonham (I 966a. 1 966b ), among others, contributed to the solution of problems of stochastic stability in the spirit of the Liapunov method. Kushner ( 1 967a) gives a complete account of the major stochastic sta bility results following the Liapunov method. The reader may find Kushner's ( 1 972) stochastic stability survey useful for an introduction to the subject. AI-

Stochastic calculus

137

though general results about stochastic stability are not abundant, some progress has been achieved in the stability of linear stochastic systems. Some of these results, with many references, appear in Kozin (1972). Our distinction between stochastic stability of a point equilibrium and sta bility in the sense of convergence in distribution independent of initial condi tions should not imply that these are the only two concepts of stochastic stability. Within the area of stochastic stability of point equilibrium several definitions and theorems exist. Our section 8 gives only one notion of stochastic stability of point equilibrium. Now we mention two more definitions of stochastic stability of a point equilibrium to illustrate the scope of this important area of mathe matical research. For example, if we let x(t) denote the solution of a stochastic differential equation such as (8 . 1 ) on [0, ), with x = 0 being the equilibrium solution, we then say that tlze 0-equilibrium is stable in the mean if the expecta tion exists, and given € > 0 there exists an 11 > 0 such that I x I = I x(O, w) I < 11 implies oo

(

E s� p I x(t) I

)
0. Several other definitions are available in the mathematical literature. See Kozin ( 1972) and Kushner ( 1 97 1 ). A simple illustration reported in Kozin ( 1 972, pp. 1 42 - 1 92) may be ap propriate at this point to give the reader an indication of the mathematical curi osities that arise in stochastic stability. Consider the Ito equation dx(t) = ax(t) d t + ax (t) d z , where a and a are constants and z is a Wiener process with unit variance; suppose that the initial condition is x(O) x0• The solution of this equation, w .p.l, is =

x(t) ; x(O) exp

[(a - a; ) t + uz (t)}

and the nth moments of this solution are given by

138

�(�a - 2 ) nt


E(xn (t)) = xn (0) exp

02

02112

+ -2

��-j "

From this last eq�ation Kozin ( 1 97 2 , p. 1 92) concludes that there is exponen tial stability of the nth moment provided

a

,a

Thus, for n = I , < 0 implies that the first moment is exponentially stable, but note that higher moments are unstable. For n = 2 < - a 2 guarantees the expo nential stability of the first and second moments, but higher moments are un stable. Recall that the Liapunov- Kushner method of section 8 applied to the same equation showed stability of x(t), i.e. of the sample paths w.p .l provided < - a2 /2 but nothing was said there about the stability of the moments. It is hard to give an economic interpretation to an economic model described by an Ito equation whose point equilibrium is stable but whose higher moments are unstable. Therefore we need to distinguish between the sample path behavior and the moment behavior and actually choose to give priority regarding stability characteristics to one or the other behavior. The area of stochastic stability in the sense of convergence in distribution to a steady state distribution remains an open field for research. It is difficult to establish the existence of an equilibrium distribution in a general setting, let alone prove its stability. For special cases we have existence theorems for an equilibrium distribution. The basic results in this area arc reported in the book by Mandl ( 1968) and some of the original work first appeared in Feller ( 1 954), Tanaka (1957) and Khasminskii ( 1 962). Merton (I 975a) has a stochastic stability result in a special case of a continuous growth model which he obtains by assuming constant savings functions being maximized over the set of constant savings func tions. This is a very special stability result. The problem of stochastic stability in a continuous stochastic optimal growth, i.e. the problem of proving that opti mal stochastic processes converge in distribution to a steady state distribution, remains open. Brock and Mirman (I 972) proved this result for the discrete sto chastic growth model. Concerning the topic of stochastic control, the problem stated in eqs. (I 0 . 1 ) and (I 0.2) is a stochastic version o f the deterministic control problem studied by Arrow and Kurz ( 1 970, pp. 27 -5 1 ) We chose to study (1 0 . 1 ) and ( 1 0.2) be cause most economists are familiar with the deterministic control problem analy zed by Arrow and Kurz ( 1 970) and the present stochastic version will enable the reader to compare the familiar with the new mathematical issues and results of the stochastic extension.

a

,

.

Stochastic calculus

139

The analysis of section I 0 uses Bellman's principle of optimality as it first ap peared in Bellman (I 957), and subsequently popularized in books such as Drey fus (1 965), Hadley (1 964) and Mangasarian ( 1 969), along with stochastic analy sis to derive stochastic conditions of optimality. Aoki (1 967), Astrom (1 970), Kushner ( 1 97 1 ) and Bertsekas (1 976) have presentations of stochastic control at the introductory level. However, unlike the area of deterministic optimal control where many books are available, such as Anderson and Moore ( 1 97 1 ), Athans and Falb (1 966), Berkovitz (1 974), Bryson and Ho ( I 979), Kwakernaak and Sivan (1 972), Strauss ( 1 968), Pontryagin et al. (1 962), Hestenes (1 966), and Lee and Markus (1 967), the literature on stochastic optimal control is not yet very large. At an advanced level the reader is encouraged to consult the book by Fleming and Rishel ( 1 975) and the papers of Benes ( 1 97 1 ), Bismut (1973, 1 976), Davis ( 1 973), Fleming ( 1969, 1 97 1 }, Kushner (1965, 1967b, 1975), Rishel (1 970) and Won ham (I 970). We note that Fleming and Rishel ( 1 975, ch. 5) provide a rigorous analysis for the stochastic optimal control problem which supplements our heuristic approach. In what they call the verification theorem, Fleming and Rishell ( 1 97 5 , p. 1 59) give suf ficient conditions for an optimum, supposing that a well-behaved solution exists for the Hamilton -Jacobi- Bellman nonlinear partial differential equation with the appropriate boundary conditions. In section 1 2 we present a generalized Ito formula and a maximum principle for jump processes. Our approach is intuitive and its aim is to familiarize the reader with some of the mathematics used in Merton (1971 ). A rigorous analysis would require an analysis of the exact mathematical properties of the term g(c, x)dq (t) of eq. ( 1 2 . 1 ) and the meaning of the jump process integrals. Kushner (1967, p. 1 8) and Doob ( 1 953, p. 284) discuss some of these issues. For a more detailed treatment see Dellacherie ( 1 974). It is appropriate to note that as economists apply the techniques of stochastic control theory to economic models a need will soon develop for a stability analysis of such stochastic models. Recent economic research in the deterministic case by Araujo and Scheinkman (1 977), Benveniste and Scheinkman (1977, 1979), Scheinkman ( 1 976, 1 978), Brock and Scheinkman ( 1 977, 1 976), Brock ( 1 976, 1 977), Cass and Shell (I 976), McKenzie ( 1 976 ), Magill ( I 977a), Samuel son (1 972), and Levhari and Liviatan ( 1 972), among others, has demonstrated how a wide class of economic problems arise from deterministic optimal control whose stability properties are crucial for correctly specifying such models. The availability of mathematical results concerning the stability of deterministic con trol systems such as Gal'perin and Krasovskii (1 963}, Hale ( I 969), Hartman (196 1 ), Hartman and Olech ( 1962), Lefschetz ( 1 965), Mangasarian ( 1 963, 1 966}, Markus and Yamabe (1 960), Rockafellar (1 973, 1976), and Roxin (1965, 1 966), have helped economic researchers. On the other hand, the nonavailability of

140


enough mathematical results on the stability of systems of stochastic differential equations arising from a stochastic control context may delay economic research in this area. For a sample of such economic problems see Brock and Magill ( 1979) and Magill (I 977b). Many applications of stochastic methods in economics and finance deal with stochastic control in discrete time as opposed to the continuous time case dis cussed in this chapter. We chose to include in the next two chapters discrete time stochastic applications to enrich the reader's education, although we have not discussed explicitly discrete stochastic methods in this chapter. There are many similarities, however, and the various applications will illustrate them in what follows. It suffices to mention here that Kushner ( 1 9 7 1 ) establishes various similarities and relationships between discrete and continuous time stochastic models. Discrete time stochastic problems are usually less complicated than the corresponding continuous time problems. The latter may be approximated by the discrete time procedure using time intervals of length Since h may be made arbitrarily small, relationships between discrete and continuous time models can be established. In finance, Merton ( 1 978) approximates the continuous time model by using only elementary probability methods to derive the continuous time theorems. In so doing he uncovers the economic assumptions imbedded in the continuous time mathematical theory.

lz.

CHAPTER 3

APPLICATIONS IN ECONOMICS

For the person who thinks in mathemat ics, and does not simply translate his ver bal thoughts or his images into mathemat ics, mathematics is a language of discovery as well as a language of verification. II.A. Simon (1977, p. xv)

I . Introduction In this chapter we present several examples to illustrate the use of stochastic methods in economic analysis. Some applications use specific results from the previous chapters, while some other applications introduce new techniques.

2. Neoclassical economic growth under uncertainty The research of Bourguignon (1 974) and Merton (1 975a) has extended the neo classical model of growth developed by Solow ( 1 956) to incorporate uncertainty. Such an extension uses Ito's lemma as a tool for introducing uncertainty into the deterministic model. Consider a homogeneous production function F(K, L ) of degree 1 , where K denotes units of capital input and L denotes units of labor input. From homo geneity we obtain that F(K/L, 1 ) = f(K/L) = f(k ), where k = KfL. For equilib rium to obtain, investment must equal saving, i.e . .

dK dt

K = - = sF(K , L), O < s < I ,

142


s is the marginal propensity to save. The Solow neoclassical differential equation ofgrowth for the certainty case is obtained as follows (a dot above a where

variable denotes its time derivative): k=

dk

dt

=

( ) = dr L d

K

KL - LK L2

=

k. L

-

L

L.

K = sf(k) - nk, L

(2. 1 )

where LjL = n, i.e. it is assumed that L (t) = L (O)e" 1 ,

(2.2)

L (O) > O, O < n < I .

The existence, uniqueness and global asymptotic stability properties of the steady state solution of the neoclassical differential equation in (2.1) are presented in Burmeister and Dobell ( 1 970, pp. 23- 30). Suppose now that instead of LjL = n, labor growth is described by the sto chastic differential equation, d L = nL dt + oLdz.

(2.3)

The stochastic part is dz, where z = z (t, w) = z (t ) is a Wiener process defined on some probability space (n, !F, P). In the engineering literature dz is usually cal led white noise. The drift of the process, n , is the expected rate of labor growth per unit of time and the variance of the process per unit o f time is o2 Note that dL/L = n d + adz says that over a short period of time the proportionate rate of change of the labor force is normally distributed with mean n d t and variance o2 dt. The new specification of the growth of labor n i (2.3) alters the neoclassical differential equation of growth. To compute the stochastic neoclassical differen tial equation ofgrowth we make use of Ito's lemma. We are given that •

t

,

dK = sF(K L)dt, dL = nL d t + oLdz. Letting k = KJL , Ito's lemma yields ak ak ak dk = - d t + - dL + - d K + at aL aK +

�[:��

(dK)2 + 2

a�:L

(dK} (dL) +

:;_� J (dL)2

Applications in economics

= -

=

� (nL d t L

+

(�

Ka2 L 2 d t aL dz) + _!_ sF(K, L)dt + ]_ L 2 L

[sf(k) - (n-a2 )k ]dt - kadz.

)

143

(2.4)

Note that if a = 0 for all k E [0, oo), then eq. (2.4) yields as a special case the certainty differential equation of neoclassical growth in (2.1}. Comparing eqs. (2. 1 ) and (2.4) we see that because of the new specification o f the labor growth in (2.3}, Ito's lemma has enabled us to obtain random fluctuations in the changes of the output per labor unit ratio dk. These random fluctuations are due, of course, to the random fluctuations of the labor growth. Thus, uncertainty with respect to labor growth via Ito's lemma is translated into uncertainty with re spect to output per labor unit, which is consistent with our intuition that fluc tuations in an input are expected to cause fluctuations in output.

3. Growth in an open economy under uncertainty Let F(K, L) be a homogeneous production function of degree such an economy be saving = sF(K, L )dt + pK dz,

1

and let saving in

(3.1)

where , as before, s is the marginal propensity to save and pKdz indicates a ran dom inflow of capital from the rest of the world. Observe that we do not explain the causes of p K dz because our purpose in this application is to illustrate ItO 's lemma and how uncertainty in the rest of the world causes uncertainty in the domestic economy. Here we assume that there are ad hoc random inflows of capital but we hasten to add that one possible explanation of pK dz may be the random fluctuations in the differential between domestic interest rates and the average interest rate prevailing in the rest of the world. Next, as in the previous application, we assume that the behavior of labor growth is given by dL = nL d t + aL dz. The model consists of the labor growth equation and the equilibrium condi tion equation dK = sF(K, L)d t + pKdz.

(3.2)


144

Note that (3.2) says that over a short period of time the proportionate rate of change of the capital stock is normally distributed with mean sF(K, L)dt and variance p2 d t. Thus, in this application we have two sources of uncertainty, i.e. uncertainty due to fluctuations in the labor force and uncertainty due to fluctu . ations in the inflow of foreign capital. Both such fluctuations are expected to influence the changes in the domestic output per labor unit and the precise for mulation is obtained by computing dk. To obtain dk, k = K/L, we use Ito's lem ma which yields

dk = +

K 1 [nLdt + aLdz) + [sF(K, L)dt + pKdz] L L2

�[

-2

C2) pKoL d t + ::; o2 L2 dr]

= - .£. nL dt - .!_ aLdz + sF(K, L) dt + p � d z L L L2

1 L

/}

2 pKaL dt +

-LK a2 L 2 dt 3

= (sf(k) - (n-a2 + pa)k ]d t - (a-p)kdz.

(3.3)

Obviously, if p = 0 and there is no inflow of foreign capital, the last equation reduces to eq. (2.4).

4.

Growth under uncertainty: Properties of solutions

Consider the stochastic differential �equation of economic growth derived in sec tion 2 of this chapter:

dk = [sf(k) - (n-a2 )k]dt - akdz,

(4.1 )

with initial random condition k(O, w) = k(O) = k0 > 0. Suppose that (4.1) has a unique solution k (t, w) = k (t) for t e (0, oo) What are the properties of such a solution? The answer is provided by the following theorems. .

Theorem 4.1 (The Markov property). Suppose that the stochastic differential eq. (4. 1 ), with initial random condition k0 > 0, has a unique solution. Then its


145

unique solution k(t), t E [0, ) is a Markov process whose initial probability distribution at t = 0 is k0 and whose transition probability is P(s, k, t, B) = P[k(t) E B I k(s) = k]. The proof follows immediately from theorem 7.3 of the previous chapter. This theorem is a useful result, particularly for economic policy considera tions. Suppose that for an economy the process of capital per worker is described by the Markov process k (t). Given that the economy has capital per worker k at time s, the economic policy makers may be interested in knowing the probability that at some future time t the capital per worker will fall within the interval oo

We now establish that k(t) is a diffusion process. Theorem 4.2 (The diffusion property). Suppose that the stochastic differential eq. (4.1), with initial condition k0 > 0, has a unique solution. Then its unique solution k(t), t E (0, oo), is a diffusion process with drift coefficient [sf(k(t)) (n -a2 )k(t)] and diffusion coefficient a2 k2 (t). The proof of this theorem follows from theorem 7.4 of the previous chapter if we note that eq. (4.1) is autonomous stochastic differential equation and therefore the continuity with respect to t is vacuously satisfied. The economic significance of this result may be made clear by a comparison of various systems. For example, in the deterministic model of economic growth we obtain a differential equation whose present state determines its future evolu tion. The Markov property of the stochastic differential equation of growth is richer in content because the current state completely determines the probability of occupying various states at all future times. The diffusion property goes even further: it describes changes in the process of capital accumulation per worker during a small unit of time, say D.t, as the sum of two factors. The first factor, i.e. the drift coefficient, is the macroeconomic average velocity of the random motion of capital accumulation when k (s) = k. The second factor, i.e. the dif fusion coefficient, measures the local magnitude of the fluctuation of k(t) k (s) about the average value, which is caused by collisions of the process of capital accumulation with economic and noneconomic variables undergoing a random movement. A straightforward generalization of eq. (4. 1) is when s and a are functions of k, written as s(k) and a(k), instead of being nonrandom and non-negative con stants. Assuming this to be the case we may rewrite (4.1) as an

146


dk [s(k)[(k)- (n-a2 (k))k)dt - a(k)kdz. =

(4.2)

Similar analysis as the one just completed can show that if a unique solution of (4.2) exists then it will satisfy the Markov and the diffusion properties. S.

Growth under uncertainty: Stationary distribution

The tools of stochastic calculus can be used to further study neoclassical growth under uncertainty. In this section we are interested in the existence of a station which is assumed to be the solu ary distribution of the stochastic process tion of eq. (2.4). As discussed in section 9 of the previous chapter, the stochastic process for is completely characterized by the drift and diffusion coefficients. Using the results of section 9 of Chapter 2, the stationary distribution for k, de noted by is given by

k(t)

k 1r(k), (n- a2 )y dy 7T(k) = a2mk2 exp �2 Jk s[(y)a2y2 J 2 k sf(y) mk 2 exp a2 f y2 dy =

-

nI

°

2

[

.

-

--

J

(5 . 1 )

.

Note that (5. 1 ) for the special case when the production function is given by a Cobb - Douglas function, 0 < a < 1 , becomes

i.e.[(k) k0, 2s 7T (k) mk- 2nfo 2 exp IL(l-a)a 2 k-(1 - o:)lJ . In eqs. (5 .1) and (5 2 ) m is determined so that J;' 7T{y)dy =

=

(5 2 )

-

.

.

=

I . For further

analysis of (5 . 1 ) and (5 .2) see Merton ( I975a). The remainder of this section follows Merton (1 975a) in comparing the ex pected stationary value of per capita output with the steady state certainty value. To achieve this goal we need a brief analysis and a technical lemma. Consider (2.4) and suppose that it has a stationary distribution denoted by 7T. Let be a twice continuously differentiable function and use Ito's lemma to compute i.e.

g(k)

dg(k), dg(k) g'(k)dk + � g" (k)(dk)2 = (g'(k)[s[(k) - (n- a2 )k] + � g"(k)a2 k2 ) d t - g'(k)a2 k2 dz. =


147

The following lemma is useful.

g(k}

Lemma 5.1 . Suppose is twice continuously differentiable and that (2.4) has a stationary distribution rr. Also, assume that

(g'(k}a2 k2 rr(k}) =

lim

k-o

lim

k-�

(g'(k)o2k2 rr(k)] = 0.

Then (5.3} For a proof see Merton (1975a, p. 392). as a special case to compute the expected We use lemma 5 . I with stationary value of per capita output, i.e. We use (5 .3) as follows:

g(k) = k

E(f(k)}.

E(l [sf(k} - (n-a2 }k] + 0} = 0.

(5.4)

•

From (5.4) we obtain

E(sf(k)) = (n- a2 )E(k) and thus

E(f(k)) = IZ-02 s E(k).

(5.5)

The result in (5 .5) is of particular interest because it allows for a comparison = with the certainty case. For example, let < a < I . From growth theory we know that the certainty estimate of the steady state per capita output

f(k} ko:, 0

IS

(5.6) How do (5.5) and (5.6) compare? Obviously

1 - o:) 2 n-a s o:/( E(f(k}) = E(k) > ( n ) s -

·

(5.7)

Eq. (5.7) illustrates that the certainty estimate is biased since (5.5) is larger than


148

(5 .6) and that therefore care must be taken in using the certainty analysis, even as an approximation of stochastic analysis.

6. The stochastic Ramsey problem

(I

optimal saving

In this section we follow Merton 975a) in determining the policy function under uncertainty. The problem is to a saving policy T - such that we

t)

s*(k,

fmd

T

J u(c)dt

maximize E0

(6. 1 )

0

subject to

dk = (sf(k) - (n-a2 )k)dt - akdz and k(t) � 0 for each t w .p. l , and in particular k(1) � 0. Here, u is a strictly concave, von Neumann-Morgenstern utility function of per capita consumption c for the representative consumer. Note that (6.2) c (l -s)[(k). =

To solve this stochastic maximization problem we use Bellman's Optimality Principle, as in section 1 0 of Chapter 2 . Let

J(k (t), t, 1)

=

m�x Et

T

J [(1-s)[(k)]dt. t

u

(6.3)

The Hamilton-Jacobi-Bellman equation for (6.3) is given by

{ u [(I-s)f(k)] + 2 + _!_ a2 k 2 } . s

0 = max

aJ ar

-

aJ sf(k) - (n-a2 )k ] + +ak [

a J 2 a k2

(6.4)

The first-order condition to be satisfied by the optimal policy

0 = u' [(1-s*)f(k)](-f(k)) +

�� [(k),

s* from (6.4) is


149

which becomes

aJ . u , [(l-s*)f(k)] = ak u'

(6.5)

s*

s*,

Note that means du/dc. To solve for in principle, one solves (6.5) for and and then substitutes this solution into (6.4) as a function of which becomes a partial differential equation for Once (6.4) is solved then its solution is substituted back into (6.5) to determine s* as a function of and The nonlinearity of the Hamilton-Jacobi-Bellman equation causes difficul ties in finding a closed form solution. One way of overcoming this difficulty is by letting oo, in which case the partial differential equation is reduced to an ordinary differential equation. This is done next. Observe that is a time-homogeneous process and that is not a function of time; thus from (6.3) we deduce that

k, T-t

aJjak,

J.

k

T-t.

T-+

u

k

aJ = - E, {u[(l-s*(k, T- t))f(k(T-t)))}. (6.6) at Suppose that an optimal policy exists, f is a well-behaved production function, and n -a2 > 0, then as T-+ lim s*(k, T-t) = s*(k, ) = s*(k), there will exist a stationary distribution for k associated with the optimal policy s*(k) and denoted by n*. Let T in (6.6) to obtain . a' (6.7) hm 3i = -E*(u[(l-s*)f(k)]) = --B, oo

oo

{ }

-+

oo

E*

where is the expectation operator over the stationary distribution n* and B is the level of expected utility of per capita consumption in the Ramsey optimal stationary distribution. Use (6.7) in (6.4) to write as oo,

T-+ aJ 1 a2J 0 = u [(l-s*)f(k)] -B + - [s*[(k) - (n-a2)k] + - -- a2k2 • ak 2 ak2 Next we differentiate (6.5) with respect to k:

::� = u"[(l-s")f(k)) ( (1-s")((k) - �s: f(k) ) .

(6.8)

(6.9)

150


Finally, substitute (6.5) and (6.9) into (6.8) and rearrange terms to conclude d S* 0 = (--2I a2 k 2ju· ) -- + (fiu ' "

dk

- -2 a2 k 2 u "( )s * + I

(6.10) Note that (6.1 0) is a first-order differential equation for s* with boundary con dition as t � oo,

lim E0 {J(k (t), t)} = 0. In (6.10), if namely

a=

0, the classical Ramsey rule of the certainty case is yielded,

u's*f- u 'nk + u - B = 0, which is usually rewritten as

B-u , s*f-- nk = -u, where B is the bliss level of utility associated with maximum steady state con sumption and k s*f- nk along the optimal certainty path. =

7. Bismut on optimal growth In this section we give an application of Bismut's approach to optimal stochastic control presented in section 1 1 of Chapter 2. Here we follow Bismut ( 1 975). Consider a one-sector optimal growth model with the usual notation i.e. k is capital per worker, f(k) is a well-behaved production function, s is the marginal propensity to save, u is a concave utility function and p is the discount rate. The problem is to maximize the expected discounted intertemporal utility assuming u ' (0) = oo,

m!x

E0

00

J e-Pt u ((l -s)f(k))d t

0

subject to the constraints

(7 . 1 )

App/icatiorrs itt economics

151

(7.2)

dk = sf(k)dt + a(k, sf(k))dz, k(O) = k0 > 0. The transformed Hamiltonian function, i.e. :II

( 1 1 .5)

of Chapter

2,

is written as

(7.3)

= u((l -s)f(k)) + psf(k) + H a(k, s[(k)).

Maximize .f(' in terms of s, where 0 < s < 1 , to get

- u ' (c)f + pf + Ha1[ = 0, which becomes, after dividing by [,

(7.4)

u'(c) = p + Ha1 .

Note that c is per capita consumption and a1 is the partial derivative of a with respect to investment where investment equals sf(k ). Next, we write eq. (1 1 .7) of Chapter 2 as it applies to our case . We have dp = - [( 1-s)f'(k)u'(c) + psf'(k) - pp + H(ak

+ sf'(k)a1 )] dt + (7 .5)

where ak denotes the partial derivative of a with respect to k. Eq. (7 .5) may be rewritten as dp

=

{- (p + Ha1 )['(k) - Hak + pp} dt + /ldz + d1l1.

(7.6)

Eqs. (7 .4) and (7 .6) uncover important economic reasoning. Eq. (7 .4) indicates that the consumer will consume up to the point where the marginal utility of his consumption is equal to the expected marginal value of capital in terms of utili ty, minus the marginal risk of investment valued at its cost. If, for example, the consumer is a risk-averter with - H > 0, this will tend to make the consumer con sume more than he would with the same p when no risk is involved. Let R denote the cost of capital. The consumer pays p + Ha1 to the producer and the cost of the producer is Rk - /Ia. The equation of profits is (p

+ Ha1 )f(k) + Ha(k, J) - Rk,

(7.7)


152

which once maximized in

k gives (7.8)

Then the profit rate in terms of the marginal value of capital

(

)

H . H a f, (k) +-a R r== 1 +p p p k I

p, denoted by r, is (7.9)

Now look for a moment at (7 .6) and rewrite it as

dp

,

dz dt

-= {-(p +Ha1)[ (k) -Hak + pp } +H dt

Multiply both sides of (7 .1 0) by

+ dM dt

·

(7.10)

(1/p) and take its expected value to conclude (7 . 1 1 )

Use (7

.1 1) to rewrite (7 .9) as 1 ' r + E dp dt P

( )

P=

which is the neoclassical relation between interest rate the expected inflation rate. Finally, compare and from (7 .9) to obtain

(7. 1 2)

p, rate of return r, and

f' (k) r r - (Hfp)ak f'(k) 1 + (11/p)a1 ' where the instantaneous risk premium f'(k) - r is equal to (Hfp) (ak + ra1) 1 + (Hfp)a1 =

Having presented several applications of stochastic calculus to economic growth we next discuss the concept of rational expectations which recently has received attention from several economists. Rational expectations use the tech niques of stochastic analysis and they are incorporated in several applications in this chapter and the next.

Applications in economics 8.

153

The rational expectations hypothesis

In this section we present the

rational expectations hypothesis postulated by

Muth ( 1 9 6 1 ) which has found many applications in stochastic economic and financial models. Theorists realize that to make stochastic models complete, an expectations hypothesis is needed. What kind of infonnation is used by agents and how it is put together to frame an estimate of future conditions is impor tant because the character of dynamic processes is sensitive to the way expecta tions are influenced by the actual course of events. Muth ( 1 9 6 1 ) suggests that expectations, since they are informed predictions of future events, are essentially the same as the predictions of the relevant economic theory. This hypothesis may be restated: the subjective probability distributions of outcomes tend to be distributed, for the same information set, about the objective probability distri butions of outcomes. This hypothesis asserts that: ( 1 ) infonnation is scarce and the economic system generally does not waste it; (2) the way expectations are formed depends specifically on the structure of the relevant system describing the economy; and (3) a prediction based on general information will have no substantial effect on the operation of the economic system. The hypothesis does not assert that predictions of economic agents are perfect or that their expecta tions are all the same . From a theoretical standpoint there are good reasons for assuming rational expectations because: ( 1 ) it is a hypothesis applicable to all dynamic problems, and expectations in different markets would not have to be treated in different ways; (2) if expectations were not moderately rational there would be opportu nities to make profits; and (3) rational expectations is a hypothesis that can be modified with its analytical methods remaining applicable in systems with in complete or incorrect information . As an illustration of the ideas above, we present Muth 's

(I 96 1 )

model of

price fluctuations in an isolated market, with a flXed production lag, of a com modity which cannot be stored. The model is given by: {8. 1 ) demand, C(t) -{3p(t) (8.2) P(t) 'YPe(t) + (t) supply , (8.3) market equilibrium, P(t) C(t) where P(t) represents the number of units produced in a period lasting as long as the production lag, C(t) is the amount consumed , p (t) is the market price in the tth period, pe(t) is the market price expected to prevail during the tth period on the basis of infonnation available through the (t- 1 )st period, and finally (t) is = = =

u

u

an error term. All the variables used here are deviations from equilibrium values.

1 54


Put (8. 1 ) and (8.2) in (8.3) to get p(t) = -('Y/�)pe(t) - (1/(3)u (t).

(8.4)

Note that u (t) is unknown at the time the production decisions are made but it is known and relevant at the time the commodity is purchased in the market. Suppose that the errors have no serial correlation and that Eu (t) = 0. Then the prediction of the model in (8 .4) is (8.5) In (8.5) E p(t) denotes the prediction of the model or the theory and it is objec tive, while pe(t) denotes the subjective prediction of the firms. If the prediction of the model were different from the expectations of the firms, there would be opportunities for profit. Note that we do not use parentheses with the expecta tion operation to simplify the notation. The rationality assumption given by (8.6) states that such profit opportunities could no longer exist. If ('Y/�) * - 1 in (8.5), then the rationality assumption implies that (8.7) i.e. the expected price equals the equilibrium price. Let us now introduce more realism into our illustration by allowing for ef fects in demand and alternative costs in supply. We assume that part of the shock variable may be predicted on the basis of prior information. From (8.4), taking conditional expectation, we write Ep(t) = - ("t/f3)pe(t) - ( l /(3)Eu (t),

(8.8)

and using the rationality assumption in (8.6) we obtain (8.9) which yields

pe(t) = - ( 1 /((3 + 'Y )] Eu (t).

(8. 1 0)


155

If the shock is observable, then the conditional expected value may be found directly. If the shock is not observable, it must be estimated from the past histo ry of variables that can be measured. In this latter case we shall write the u's as a linear combination of the past history of normally and independently distributed random variables x (t) with zero mean and variance o2 , i.e. 00

i=O

u (t) = 1: w(i)x(t-i),

(8. 1 1 )

Ex (i) = 0,

(8.12)

Ex (i)x (j) =

{ 02 0

if i = j, if i =Fi.

{8.13)

The price will be a linear function of the same independent disturbances and will be written as

00

i=O

(8.14)

p (t) = 1: W(i)x(t-i). Similarly, from (8.12) 00

i= 1

pe(t) = W{O)Ex (t) + 1: W(i)x(t-i) =

:E i= 1 00

W(i)x(t-i).

(8.15)

Putting (8.14) and (8.15) in (8 . 1 ) and (8.2), respectively, and solving using (8.3) we have

W(O)x(t) +

(

)

oo 1 oo "/ 1 + - 1: W(z)x(t-i) = --1: w(z)x(t-i). (3

I

(3

0

(8.16)

Eq. (8.16) is an identity in the x's and therefore the relation between W(i) and w(i) is as follows: W{O) = W(i) = -

1 w(O), (3 1

f3 + 'Y

w (i) for i = 1 , 2, 3 ... .

(8. 1 7) (8.18)


156

Note that (8 . 1 7) and (8 . 1 8) give the parameters of the relation between the price function and the expected price function in terms of the past history of indepen dent shocks. The next step is that o f writing the expected price in terms of the history of observable variables, i.e. 00

pe(t) = � V(j)p(t-j). i

(8.19)

=I

Use (8.14), (8. 1 5) and (8.19) to obtain

pe(t)

= =

=

00

00

� W(i)x(t-z) = � V(j)p(t-j)

i= 1

[

j= I

]

-� V(j) � W(i-j)x (t-i- j)

1=

1

[ _f

-�

I= 1

J= 1

t=O -

J

V(j) W(i-j) x (t-i).

(8.20)

As before, we again conclude from (8.20) that the coefficients must satisfy

W(i) =

t

i= 1

V(j) W(i-j)

(8.21)

since the equality in (8 .20) must hold for all shocks. In (8 .2 1 ) we have a system of equations with a triangular structure which may be solved successively for V1 ,

v2 , . . .

As a particular illustration suppose that in (8.1 1 ) w(i) = 1 for all i = O, 1 , 2, ... , which means that an exogenous shock, say a technological change, affects all future conditions of supply. Then using (8.17) and (8.18) eq. (8.2 1 ) yields

pe(t)

=

- 00 ( {3 )j {3

'Y

-�

J=

1

'Y

+ ')'

p(t-j),

(8.22)

which expresses the expected price as a geometrically weighted moving average of past prices.


157

9 . Investment under uncertainty In this application we follow Lucas and Prescott ( 197 1 ) who introduced an un certain future into an adjustment-cost type model of the firm to study the time series behavior of investment, output and prices. Their paper is rich in methodol ogical ideas and techniques and in this section we plan just to describe the model and state an existence theorem. Consider an industry consisting of many small firms each producing a single output, q, by using one single input capital, k , , under constant returns to scale. With an appropriate choice of units we may use k, to also denote production at full capacity and denote the production function as (9.1) Gross investment, denoted by x, , is related to capacity in a nonlinear way: (9.2) where h is assumed to be bounded, increasing, h (0) > 0, continuously differen tiable, strictly concave, and that there exists a l) , 0 < l) < 1 , such that l) h -r { 1 ). The last assumption means that 5k, is the investment rate that is needed to maintain the capital stock k,. The assumption of strict concavity is made be cause it gives rise to adjustment costs of investment and the model thus reflects gradual changes in capital stock as opposed to immediate passage to a long-run equilibrium level. Let p1 denote the product price and r the cost of capital with r > 0. Using the standard discount factor, /3, where f3 1 /( l + r), then ex post the present value of the firm, V, is given by =

=

(9.3) We use the notation kt , xt and pt interchangeably for industry and firm variabies. The objective of the firm is the maximization of the mean value of (9 .3) with the stochastic behavior of P, somehow specified. However, because we have omitted variable factors of production in (9 . I ) the firm will choose to produce at full capacity and the only nontrivial decision of the firm is the choice of an investment level. This investment level is decided, as usual, by comparing the known cost of a unit of investment to an expected marginal return. To simplify the firm's investment decision we place the burden of evaluating the income


158

stream changes due to a given investment to the traders in the firm's securities. We denote by w: the undiscounted value per unit of capital expected to prevail next period . The firm's problem can now be stated as x>o

max [ - x + (3k1h (xfkt )w*], r

(9.4)

w1k1 P,k, - x + (3k1h(x/k1)w:

(9.5)

- 1 + (3h ' (xfk1)w: :s;;, 0 with equality, if x > 0.

(9.6)

where x is the cost of investment and (3k t+ 1 w: is the next period value resulting from x. Using (9.2) we obtain (9.4). The maximization problem of (9.4) is sub ject to the two constraints =

and

Since (9.5) and (9.6) are solved jointly for x, and w: as functions of k1,p1 and w1, we can write the investment function as

x, = k1g(w,

�

p1),g' > 0.

(9.7)

The industry demand function is assumed to be subject to random shifts and is written as

P, = D(q1, u1) ,

(9.8)

where {u 1 } is a Markov process with a transition function p ( , · ) defined on R 2 • For given u 1 , D is a continuous, strictly decreasing function of q 1, p 1 = D(O, u 1) < oo, and with •

q

J D(z, u1)dz

(9.9)

0

bounded uniformly in u 1 and q. We also assume that D is continuous and in creasing in u 1 so that an increase in u 1 causes a shift to the right of the demand function. For given (k0 , u0) an anticipated price process is defined to 1be a sequence {pt } of functions of (u 1 , u 2 , .•• , u t ) , or functions with domain R • Similarly, an investment-output plan is defmed as a sequence {q1 , x1 } of functions on R 1 • We restrict the sequences {pt }, {q t } and {xt } to belong to L i.e. to be elements of +,

159

Applications itr economics

the class L with non-negative terms for all (t, u 1 , u2 , ••• , u1). Here L denotes the set of all sequences x {x1}, t 0, 1 , 2 , . . . , where x0 is a number and for t � 1 , x1 is a bounded measurable function on R1, bounded i n the sense that norm is finite, i.e. =

II x II = sup t

(u 1

1

sup •••

, 11

=

1)E E

1 I x (u 1 , '

.••

, u1 ) I

< oo.

Therefore , for any sequences {p1}, {q1} and {x1}, elements of L •, the present value V in (9 .3) is a well-defmed random variable with a finite mean. The objec tive of the firm then becomes the maximization of the mean value of V with respect to the investment-output policy, given an anticipated price sequence. To link the anticipated price sequence to the actual price sequence we assume that expectations of the firms are rational or that the anticipated price at time t is the same function of (u 1 , , u 1 ) as the actual price. We are now ready to de fmc an industry equilibrium for fiXed initial state (k, u) as an element {q� , x� , p� } of L + x L + x L + such that (9 .8) is satisfied for all (t, u 1 , , u1) and such that •••

•••

(9.10) for all {q,, x 1} E L + x L + satisfying (9 . 1 ) and (9 .2). Note that the expectation of the tth term in (9 .1 0) is taken with respect to the joint distribution of (u 1 , ,u 1). Having defmed industry equilibrium the question naturally arises of whether a unique equilibrium exists. Lucas and Prescott (1971) first show that a competi tive equilibrium leads the industry to maximize a certain "consumer surplus" ex pression, and then they show that the latter maxinlUm problem can be solved using the techniques of dynamic programming. Define the function s(q, u), q � O , u E R by •••

s (q, u) =

q

J D(z. u)dz, 0

(9. 1 1 )

so that for given u , s (q, u) is a continuously differentiable, increasing, strictly concave, positive and bounded function of q , and for given q, s is increasing in u . Note that s(q t , u t ) is the area under the industry's demand curve at an output of q1 and with the state of demand u1• Let the discounted consumer surplus, S, for the industry be 1

S = E ( 1�0 J1 [s (q1, u1) - x1 ) ) .

(9. 1 2)

160


We are interested in using the connection between the maximization of S and competitive equilibrium in order to determine the properties of the latter. Asso ciated with the maximization of S is the functional equation

v(k, u) = x;;o.o sup (s(k, u) -x + (3fv [kh(x/k),z )p(dz, u) ) . We now state a basic result of Lucas and Prescott (I 971 ).

(9.13)

v v0,

Theorem 9.1 . The functional eq. (9 .1 3) has a unique, bounded solution on the right-hand side of (9.13) is attained by a unique (0, oo) x R ; and for all and is the unique industry equilibrium , given In terms of given by

(k, u) x(k, u),

x(k, u).

k0

x1 x(k1, u1), kt+1 k1h[x(k1, u1)/k1), q, = k, , P1 = D(q1 , u1)

(9.14)

=

{9.15)

=

for t = 0,

I , 2 , ... , and all realizations of the process {u,}.

Proof. See Lucas and Prescott ( 1 97 1 , pp. 666-67 1 ). With the question of existence and uniqueness being settled Lucas and Prescott study the long-run equilibrium assuming ( 1 ) independent errors and (2) serially dependent errors. Below we state the results for the frrst case and refer the reader to the Lucas and Prescott paper for the results of the second case. Suppose that the shifts and are independent for s =1= t, i.e. the transition will function It follows in this case that does not depend on not depend on and from (9 .14) and (9 . 1 5) it follows that the time path of the capital stock will be deterministic, given by

p(z . u) u

u1

us

u.

x(k, u)

(9.16)

x(k1) x(k) ok,

v(k1, u).

where is the unique investment rate obtaining We defme a capital stock /cC > 0 to be a stationary solution of (9 .16) if and only if it is a solution to = since = l.

h (o)

{u1} there k0 > 0 and

Theorem 9.2. Under the hypothesis of independence of the process are two possibilities for the behavior of the optimal stock First, if if

k1 •


f D(O, u)p (du) > o + [rfh ' (o)] holds, then plicitly by

161

(9.17)

k , will converge monotonically to the stationary value kc , given im·

f D(k, u)p(du) = o

+ [rfh' (o)] .

Or, secondly, if (9. 1 7) fails to hold, or if k0 = 0, then k1 will converge monoto· nically to zero.

Proof. See Lucas and Prescott (197 1 , pp. 671 -673). 10. Competitive processes, the transversality condition and convergence

The methods of Bismut (1973, 1975), briefly presented earlier, have been used by Brock and Magill (1 979) in an attempt to develop a general approach to the continuous time stochastic processes that arise in dynamic economics. In this section we follow Brock and Magill {1979) who show that under a concavity assumption, to be specified, a competitive process which satisfies a transversality condition is optimal under a discounted catching.up criterion. Let (!2, F, P) denote a complete probability space, � a a·field on n, and P a probability measure on .� Let I = [0, oo) denote the non·negative time interval and (/, Jt, p) the complete measure space of Lebesgue measurable sets ..II, with Lebesgue measure p. Let (n x /, :1{', P x p) denote the associated complete pro duct measure space with complete measure P x p and o·field .Yl' ::> y; x dt. Let (Rn , vfln ), with n � 1 , denote the measurable space formed from the n�imen· sional real Euclidean space Rn with a-field of Lebesgue measurable sets. tt " . Let

be an .tf·measurable function (random process) induced by the following sto· chastic control problem. Find an .)f-measurable control v(w, t) E U c Rs, s � I , such that for o > 0 sup J Je- 6 vEU n I

t u(w, t, k(w, t), v(w, t))dtdP(w), t

k(w, t) = k0 + Jt(w, k(w, ) v(w, T))dT + 0 T,

T ,

{10. 1 )


162 T

+

J o(w, 0

T,

k(w, T), v(w, T)) d z ( w, T),

(1 0.2)

where u

E Rl ;

f = (/1 , , f" ) E R" ; 0 = •.•

E R"m and k0 E K c R"

is a nonrandom initial condition; u(·, k, v),f(·, k, v), o(·, k, z,) are .W'�measur able random processes for all (k, v) in k x U c R" x Rs and u (w, ),/(w, . ), o(w, ), are continuous on I x K x U for almost all w, while z (w, t) E Rm , m � 1 , is a Brownian motion process. Let •

•

.� = .�(z (w, t)), T E [0, t]) denote the smallest complete o-ficld on n relative to which the random varia abies {z (w, T ), T E [0, t)} are measurable. We require that k(w, t) be .'F,-mea sureable for all t E /, so that /(·) and o( · ) are nonanticipating with respect to the family of a-fields {.�t' t E I}. To ensure the existence of a unique random process k(w, t) as a solution of {10.2) we assume that the Lipschitz and growth conditions of Chapter 2 are satisfied, i.e. we make the following assumption. Assumption 1 0.1 . Lipschitz and growth conditions: there exist positive con stants, et and /3, such that (i) II /(w, t, k, v) - !(w, t, k , v) I I + II o (w , t, k, v) - o (w, t, k, v) I I � a I I k for all (k, v), (k, v) E K x

-

k II

U, for almost all (w, t) E n x I, and

(ii) 11/{w, t,k, v) W + II o(w, t,k, v)l l 2 � {3(1 + II k 11 2 ) for all (k, v) E K x U, for almost all (w, t) E n x I. Here, as in previous sections, double bars denote vector norms. Recall from Chapter 2 that assumption 1 0.1 is sufficient for the existence and uniqueness of a solution of a stochastic differential equation. We will exhibit a sufficient condition for a random process to be a solution of the problem (1 0.1) and (1 0 .2) in tenns of a certain price support property,

163


the nature of which is most clearly revealed by restr�cting this stochastic control in the manner of Bismut ( 1 973, p. 393) and Rockafellar (1970, p. 1 88) as fol lows. Consider the new integrand L (w, t , k , k , a) =

sup u(w, t, k, v) 1/(w, t, k, v) = k, a(w, t, k, v) = a vEV oo if there is no v E U such that -

f(w, t, k, v) = k, a(w, t, k, v) = a.

Note that L (w, t, · ) is upper semicontinuous for all (w, t) E n x I and L (w, t, k(w, t), k(w, t), a(w, t)) is Jf-measurable whenever k(w, t), k (w, t) and a(w, t) are Jf-measurable. We impose indirect concavity and boundedness conditions on the functions u (w, t, ), f(w, t, · ) and a(w, t,• ), and a convexity condition on the domain K x U by the following assumption. •

Assumption 10.2. Concavity-boundedness: L (w, t, · ) is concave in {k, ic, a) for all (k, k, a) E R n R n X R nm for all (w, t) E n X I and there exists 'Y E R, I 'Y I such that L ( ) < 'Y for all (w, t, k, ic, a) E n X I X R n X R n X R n m .
* under which it maximizes profit almost surely, at almost every instant. For -(p- op) denotes the vector of unit rental costs, -rr denotes the matrix of unit risk costs induced by the disturbance matrix a, while ( 1 , p) is the vector of unit output prices, so that . ,, k + (p-op) k + tr(1ro ) L + p ,..:.. is the (imputed) profit which is maximized almost surely, at almost every instant, by a competitive random process. We also give a geometric interpretation which is this: the random process (p- bp, p , rr) e fP * genera�es supporting hyperplanes to the epigraph of -L(w, t, k, k, a) at the point (k, k, a) for almost all (w, t) E n X I. The hyperplanes parallel to a given supporting hyperplane indicate hyper planes of constant profit, so that the supporting hyperplanes are precisely the hyperplanes of maximum profit at each instant. Note that under assumption 1 0.2 a random process ( k, k, a) E f? is competi tive if and only if

(p {w, t) - Op (w, !), p(w, !), 7r (W, !)) E - oL(w, t, k(w, t), k(w, t)a(w, t))

(10.10)

for almost all (w, t) E n X /, where oL denotes the subdiff('rential of L (w, t, . ) . Eq. (10.10) is a generalization of the standard Euler-Lagrange equation. The Fenchel conjugate of -L (w, t, k, k, a) with respect to (k, a) will be cal led the generalized Hamiltonian

G(w, t, k,p, rr) =

sup rzm n (k ,o)E R x R

+ L (w, t, k, k, a)}.

{p' k + tr(rra') + (10. 1 1 )


166

G(w, t, k,

(w, t)

Observe that E p, 1r) is concave in k and convex in (p. 1r) for aJI " m n n n X I and is defined for all p , 7T) E R X R X R . Under assumption 1 0 .2 if ) is differentiable, a random process (k, k, a) E f1' is competitive if and only if

(k,

G(w, t,

·

I

t

k(w, t) = k0 + J GP (w, T) d T + J Grr(w, T)dz (w, T), 0 0

( 1 0 . 1 2)

and also

I

p(w, t) = Po + f [ - Gk (w, T) + op(w, T)]dT + 0 t

+

J 1r(w, T)dz {w, T).

( I 0 . 1 3)

0

stochastic Hamiltonian equa

Eqs. ( 1 0 . 1 2) and ( 1 0 . 1 3), which will be called the are a generalization of the standard Hamiltonian canonical equations for a discounted stochastic variational problem. Assume that for all (x, u, E R" x R " x R n m that

tions,

s)

L(w, t,x, s) = L (x, s) for all (w, t) E n X /, so that L is nonrandom and time independent. When ( 1 0.5) is fmite we deftn e the current value function W(k}: u,

u,

R"

W(k(t)) =

�up

(k ,k,o) E .?

E,

�

R

00

J e-o (r-r) L (k(w, T), k(w, T), a(w, T)} dT, t

( 1 0 . 1 4)

k

t,

k(t) K.

where E, denotes the conditional expectation given at time and where replaces as the initial condition in (10.3). is a concave function for all k E Note that under assumption 10.2, In establishing convergence properties, the following class of Mc!(enzie competi tive processes is of special importance. A random process (k, k, a ) e q> is Mc if it is competitive and if the dual random price process t) supports the value function

k0

W (k}

Kenzie competitive p(w, W(k(w, t)) - p(w, t)'k(w , t) � W(k) - p(w, t)' k for all k for almost all (w, t) E Q X /. " ER ,

( 1 0 . 1 5)


167

If (k, k, a) fi' is McKenzie competitive then in (10.8) is determined by the condition p0 E a W{k0). We are now ready to state The9rem 10.1 . (Transversality condition). A competitive random process (k, k, o) E Jf"with dual price process CP - f>p, p, 1r) E #*, which satisfies the transversality condition lim sup E0e- 6 T p(w, 1)'k(w, T) � O E

Po

T-+oo

is optimal in the class %of random processes for which lim inf E0e- 6 p(w, T)' k (w, T) � 0. T-oo

T

(10.16)

Sec Brock and Magill ( 1979). The sample paths of a McKenzie competitive process starting from nonrandom initial conditions have a remarkable convergence property. Consider a point k0 E K and a McKenzie competitive process emanating from this point. Under as sumptions, which include a strict concavity assumption on the basic integrand L , a McKenzie competitive process emanating from any other point k0 E K con verges almost surely to the first process. This result, which has its origin in the dual relationship between the prices and quantities of a McKenzie competitive process, may be stated as follows. Theorem 10.2 (Almost sure convergence). Let assumption 10.2 be satisfied and let the function L be time independent and nonrandom as in (10.14). If two McKenzie competitive random processes (1 0.17) (k, k, a) E :� and (k, k, a ) E f¥>, with associated dual price processes (10.18) (p- fJp, p , 1r) E .�* and ("p-fJp, p, 1f) E .� *, starting from the nonrandom initial conditions

Proof.

satisfy the following conditions:

168


{i) there exists a compact convex subset M

c

R" x Rn

such that for all t

E

I

(k(w, t), p(w, t)) = (k(w, t; k0), p(w, t; Po)) E M, (k(w, t), p(w, t) = (k(w, t; k0), p(w, t; Po) E ft1

for almost all W E n; (ii) there exists > 0 such that the function 11

V(k-k, p-p) = - (p-p) ' (k- k)

(10.19)

satisfies

fi2

V(k-k, p-p)� - 11 11 (k-k , p-p) 11 2

(10.20)

for all (k-k,p-p) e Y = {(k-k, p-p) l (k,p), (k,p) e ft1}; {iii) the value function is (a) strictly concave, {b) differentiable, and {c) strictly concave and differentiable, for all k in the interior of K , K = { k I (k,p) E M }; then (i), (ii) and (iii) (a) imply k(w, t) - k(w. t) -+ 0 w.p.l. as t -+ (i), (ii) and (iii) (b) imply p(w, t) - p(w, t) -+ 0 w .p.l. as t -+ (i), (ii) and (iii) (c) imply (k(w, t) - k(w, t),p(w, t) - p (w, t)) -+ 0 w.p.l. as t -+ oo Proof Sec Brock and Magill (1979, pp. 852-855). oo;

oo;

.

II.

Rational expectations equilibrium

In this section we follow Brock and Magill (1979) to show how the concept of a competitive process, in conjunction with the stochastic Hamiltonian equations (10.12) and (10.13), provides a useful framework for the analysis of rational expectations equilibrium. We examine in particular a rational expectations equi-

169


librium for a competitive industry in which a fixed finite number of firms behave according to a stochastic adjustment cost theory by creating an extended inte grand problem analogous to that of Lucas and Prescott (1971 ) . Consider therefore an industry composed of N � 1 firms, each producing the same industry good with the aid of n � 1 capital goods. All firms have identical expectations regarding the industry product's price process, which is an �Yf-mea surable , nonanticipating process

r(w, t): (n x I,.Yt ) -+ (R+ , J t).

(1 1 .1 )

The instantaneous flow of profit of the i th fmn is the difference between its reve where and and its costs C; nue in = v ) denote the capital stocks and investment rates of the ith firm, and where f; arc the standard strictly concave production and and -C; adjustment cost functions. If l) > 0 denotes the nonrandom interest rate, then each firm seeks to maximize its expected discounted profit by selecting an $" measurable, nonanticipating investment processes

r(w, t)/ (k; (w, t)) v; (v;1, , (v1) (k;)

(vi (w, t)),

k ; = (kit, ... , ki")

•••

i v (w, t): (n x 1, .1(') -+ (R" , JI"),

such that sup Eo

00

f e-

O

T

i=

1,

..., N,

[r(w, T)[i (k ; (w, T)) - c; (vi (w, T))]dT'

v1(w ,t) 0 t t 1 k; (w, t) = k� + J vi (w, T)d T + J a; (k1 (w, T))dz (w, T), 0

0

a; (k ; )dz; = H1i , ag

(H1ik1 + ag )dz1i, l

�

i=

(1 1 .2)

(I 1 .3)

are n x n and n x I matrices with constant coefficients and T ) is an m-dimensionaJ Brownian motion process. This model is a simple stochastic version of the basic Lucas 1967) and Mortensen ( 1973) adjustment cost model, with the standard additional neoclassical assumption that the invest ment and output process of the ith firm have no direct external effects on the investment and output processes of the kth firm, for i if= On the product market the total market supply, given by where _

z' (w,

(

k.

1 (w, t) = Q5 :E ! (k1(w, t)), N

i= 1

170


depends in a complex way through the maximizing behavior of firms on the price process {1 1 . 1 ). On the demand side of the product market we make the simplifying assumption that the total market demand depends only on the cur rent market price Q0 (w, t) = 1/1- 1 (r(w, t)), r � 0,

where 1/I(Q) > 0, 1/1' (Q) < 0 and Q � 0. A rational expectations equilibrium for the product market of the industry is an .1t'-measurable, nonanticipating random process {1 1 .1) such that (1 1 .4) QD (w, t) = Qs (w , t) for almost all (w, t) E n X I. The firms' expectations are rational in that the anticipated price process coin cides almost surely with the actual price process generated on the market by their maximizing behavior. Consider the integra] of the demand function 'lt(Q) =

Q

f 1/J(y)dy,

Q � O,

0

so that 'It ' (Q) = 1/I(Q), 'It " (Q) = 1/J '(Q) < 0 and Q � 0. We call the problem of fmding N £-measurable, nonanticipating investment processes (u1 (w, t), . . , UN (w, t)): (Q X /,.tf) -+ (R n N ,.Jt"N) .

such that

(1 1 .5)

where (k 1 (w, t), . . , kN (w , t) satisfy (1 1 .2) and ( 1 1 .3), almost surely, the ex tended integrand problem. .

171


Theorem I I . I .

(Rational expectations equilibrium). If the generalized Hamil tonian of the extended integrand problem (1 1 .5) is differentiable, if (k (w, t),p (w, t)) = (k1 (w, t), .. . , kN (w, t),p1 (w , t), . . , j5N(w, t) is a competitive process for (1 1 .5) which satisfies the transversality condition (1 1 .6) limT-+oo sup E0 e- 5 T p(w, T)' k(w, T) � 0, and if for any alternative random process k(w� t) with k0 = k0 (1 1 .7) lim inf E0 e-5 Tj5(w , T)' k(w, T) � 0, T-+oo then the Jf-measurable, nonanticipating random process

.

(1 1 .8) is a rational expectations equilibrium for the product market of the industry. Proof Since the generalized Hamiltonian for the extended integrand problem is differentiable, (k(w, t), p(w, t)) is competitive if and only if, writing (I 0.12) and (10.13) in shorthand form, d k; = h ;(p; ) d t + ai(k,.)dz; dp ; = (Bp i 'It' � f; (k i) f; - � 1Tiia ii. ] d t + 7T ;d z ; '

- (

i= 1

) . k

1

i= 1

where Jz'. = (C'. ;) -1 . Eqs. {1 1.8) and (1 1 .9) imply v

k

1

.

i = 1 , . . ,N ,

,i =

(I 1 .9)

.

1, .. , N. {1 1.10)

Eqs. ( 1 1 .6), {11.7) and ( 1 1 .8) are sufficient conditions for each firm to maxi mize expected discounted profit by theorem 10.1 . Eq. (1 1 .8) implies that (1 1 .4) is satisfied and the proof is complete.

Stochastic methods in economics and /iiUlnce

172

1 2. Linear quadratic objective function

In this section and the following one we present the analysis of particular sto chastic control problems which have found applicability in economics and finance. Consider the one-dimensional state and control problem - W(x(t)) = min Et v

J e-P (s- r ) {a(x(s))2 + b (v(s))2 }ds

s=t

subject to x (t) known and dx (t) = v(t)d t + ox(t)dz (t). Note that o > 0, a > 0, b > 0 and we try as a solution

p

> 0.

Since the objective function is convex

W(x) = -Px2 ,

( 12.1)

which yields W = -2Px

and W = -2P. (12.2) We now use the Hamilton-Jacobi-Bellman equation for the case with discount ing, i.e. X

XX

to write in our specific case -pPx2 = max { -ax2 -bv2 2 Pxv v

-

+

� ( -2Po2 x2 )}.

(12.3)

From (12.3) we obtain that -2bv-2Px = 0, or that V0

=

-Pxjb.

Substitute (12.4) into (12.3); then -pPx2

=

{ -ax2 - b ( - Pxb }2 - 2Px (- bPx } - Po2 x2 } ,

(12.4)


173

which after simple algebraic manipulations is reduced to

1 - P2 + b

(p -o2 )P - a = 0.

(1 2.5)

Choose the largest root because of the convexity of the objective function and denote it by P+. A candidate solution that is optimum is given by

dx(t)= [- !

J

the solution of which is x (t) = x0 exp

13.

(1 2.6)

P. x (r) d t + ox(t)dz (r),

{ (- ! P. - 0; )

t}

t + oz ( )

.

(12.7)

State valuation functions of exponential form

Jn the previous section we saw that the state valuation function of the linear quadratic example turned out to be quadratic in the state variable . This is a gen eral principle in linear quadratic problems. We tum now to a type of problem where the state valuation function turns out to be of exponential form. In exam ining this type of problem we hope that the reader will pick up the common thread of technique that is used in searching for closed form solutions for the state valuation functions when they exist. We also hope that this will illustrate a technique of showing that a state valuation function of a particular closed form cannot satisfy the Hamilton-Jacobi-Bellman equation as well. In this way the search for closed form solutions to the Hamilton-Jacobi-Bellman partial differential equations is systematic instead of random guesswork. It is well to learn how to prove that solutions of a particular form do not exist as well as formulating hypotheses upon the objective and the constraints of the problem so that a closed form solution of a particular type does exist. Consider the problem J(x(t), t,N) = max subject to

E,

{1

D(s)u (c(s), s)ds + B(x(N), N) }

dx (s) = (b(s)x(sf<s> - c(s))d s + o(x (s), s)dz(s).

(13.1) (1 3.2)


174

Here D(s), c(s), u (c(s), s), x (s), B(x(N), N), b (s), (j(s), a(x (s), s), and d z (s) de note discount rate at time s, consumption at time s , utility of consumption at time s, state variable at time s which in economic applications is usually capital, bequest function of the state variable at time N, efficiency multiplier of produc tion function at time s , output elasticity at time s , standard deviation function at time s , and Wiener process that is standardized at each moment of time, respec tively. Notice that the problem in ( 13.1) and ( 1 3 2) is quite general. We shall see how general we can make it, and still get a closed form solution for the state valuation function J. The stochastic maximum principle applied to {13.1) and ( 1 3 2 ) gives .

.

0

where

= maximum {D(t)u (c, t) +�(J)}, c>O

!f (J) =

(13.3)

1

lim E { [J(x (t + tl t), t + tl t, N) - J(x (t), t, N)]/ tl t }.

�t-+0

(13.4)

The operator .!fJ( ) is just the conditional expectation of the instantaneous rate of change of J and is a general case of the operator defined in eq. (7.9) of Chapter 2. See also eq. (4.15) of Chapter 4. Let us calculate the partial differential equation given in eq. (13.3) for the constant relative risk-aversion class of utility functions given by ·

u (c(s), s) = (k (s) (c(s))1 - a(s))/(1 -a(s)).

(13.5)

c.

Here k (s) is just a constant independent of Assume that the optimal consump tion given by eq. (1 3.3) is positive. Then we can write (1 3.6)

In eq. (13.6) it is understood that everything is a function of time, and sub scripts denote the obvious partial differentiation. Insert the solution for c from eq. (13.6) into eq. (13.3) to get the partial differential equation (13.7)

Eq. (1 3.7) is obtained by the following steps. First, evaluate the operator in eq. (13.4) to get


175

{1 3.8) Eq. (13.8) is obtained either by a direct application of Ito's lemma, or by ex panding J into a formal Taylor series discarding all terms of higher order than � t, and taking conditional expectation of the remaining conditional on informa tion received at time t. Secondly, insert the optimal value for c from eq. ( 1 3 .6) into eq. (13.8). Insert all of this into eq. (13.3) and rearrange terms to get eq. (13.7). Notice that everything in eq. (13.7) is a function of time, but we have suppressed notation of this dependence in order to simplify the expression. Now look carefully at eq. (1 3.7), especially the last three terms. If we tried as a candidate for J a function of the form

J(x, t, N) g(t, N)xe ( t) + [(t, N) =

(I 3.9)

then we should attempt to determine the unknown exponent e(t) by equating exponents and testing for consistency by examining the possibility of identical exponents on the x variable of all four terms of eq. (1 3.7). The exponents of x in eq. (1 3.7) for each of the four terms reading from left to right are:

(e- 1 ) (a-1 )Ia = e + exponent on (xe log x) = (e- 1) + (3 =

(e-2) + exponent on x from a2 •

(13.10)

Eq. (13.10) follows immediately by inspecting the partial derivatives of J, which we calculate by using eq. (13 .9), and list below in equations

(13. 1 1 ) J

X

=

egxe - l . J '

XX

=

e(e-1)gxe- 2

'

(13.12)

We see immediately from eqs. (I 3.1 1 ) and (13.12) that in our search for the most general problem that will give us a solution for the state valuation function of the form (1 3.9), we will have to assume the conditions listed below in order to get the same exponent for the last three equalities of eq. (13.10). First assume that

(13.13) Assumption ( 1 3 . 1 3) is needed to get rid of the term xe log x that obstructs the validity of eq. (13 . 1 1 ). Secondly, assumption

Stochastic rnetlrods in economics and finance

1 76

3.14) on the variance function is needed so that the last equality of eq. (13.1 0) is equal to e. Finally, assumption (13.1 5) {3 (s) 1 is needed for identity with the other terms of eq. (13.10). Thus, assumptions (13.13)- (13.15) allow us to assert that it is possible to find a function of the form in eq. (13.9) that will satisfy cq. (13.7). To solve for the unknown exponent e, which by eq. (13.13) must be con stant in time, we solve (e-1} (a- 1 )/a = e. (13.16) Thus, (13.17) e = 1 -a. We see now that a must be independent of time in order to obtain a solution of the form of (13.9). We have now solved for the exponent, e, in eq. (13.9). What remains is to solve for the functions g and f. In order to solve for the functions and/, write eq. (13.7) using eq. (13.10) as follows: (I

=

g

+ � e(e-1 )ghxe .

(13.18) Now, eq. (13.18) must hold for all x and therefore/, = 0 must hold. Hence,[ is independent of time. Now cancel xe off both sides of (13.18) to get 0 = (eg)(a- 1 )fa ka- 1 a(a-1 )- 1 + g, + beg + � e(e-l )glz = 0. Notice that (13.19) can be written in the form

(13.19) (13.20)

where g =.gr


177

We see that (1 3.20) is a differential equation that looks formidable to solve. However, such a differential equation can be transformed. Experiment with transformations of the form (13.21) y =g'Y to discover that if-y 1/cx, then eq. (13.20) can be transformed into the form =

(13.22)

Cancel y a - 1 off of both sides of ( 1 3 .22) to get (13.23) 0 = a0 + a 1 y + cxy, which is a differential equation linear in y, y. Tltis is a standard fonn which can be solved regardless of whether the coefficients a0 and a 1 are dependent upon time or not. Here the coefficients a0 and a1 are defmed in the obvious manner by eq. (13.19). A boundary condition is needed before eq. (13.23) can be solved. This is obtained from the restriction (1 3.24) J(x,N,N) = B(x,N). Of course, we will not be able to solve eqs. ( 1 3 .23) and (13.24) for arbitrary be quest functions. To see how things go put (1 3.25) B(x,N) := O. Look now at eq. (1 3.9). From eqs. (13.9) and (13.25) we infer immediately that f(t,N) = 0. This is so because f,(t,N) = 0 for all t and f(N,N) = 0. Fur thermore, eqs. (13.25) and (1 3.19) imply g(N, N) = 0 and y(N, N) 0. (13.26) Thus, we have our boundary condition on eq. (1 3.23) and it can be solved for an explicit solutiony(t,N), from whichg can be calculated from eq. (13.21). What about the case of more general bequest functions than that given in eq. (13.25)? We see immediately by inspection of eq. (1 3.24) that if there is any hope of a solution for J of the form given by eq. (1 3.9), then the exponent on x in the function B(x, N) must be the same as in eq. (1 3.9). Thus, a more gener al class of bequest functions where x enters with the same exponent as that of =

e

1 78


the utility function may be treated without much extra effort. Thus, closed form solutions exist in this case as well. Now, retrace through our derivation using the Hamilton-Jacobi-Bellman equation, eq. ( 1 3 .7), and recall how much had to be assumed on the structure of the utility function, the production function, and the bequest function in order to get a closed form solution for J. We saw from eqs. ( 1 3 . 1 3) and (13.15) that the exponent on the utility function had to be independent of time and the ex ponent on the production function had to be unity at all points in time. Further more, the variance had to be proportional to the square of the state variable, but the constant of proportionality could vary in time . Also, all other coefficients could vary in time. It is worthwhile to note that other sources of randomness may be introduced into problem ( 1 3 . 1 ) and ( 1 3 .2) above and beyond the randomness in the change of the state variable, provided that these sources of randomness are independent of the Wiener process dz; that is to say the coefficients k , b and h may be gener ated by Ito equations as well. We may summarize this application by noticing that it is fairly illustrative of the method of searching for closed form solutions when they exist and deter mining assumptions that are necessary to put on the problem in order to get the existence of a closed form solution. Furthermore, we have approached the prob lem of searching for a closed form solution in such a way that illustrates the maximum amount of generality that one can have and still get the existence of a closed form solution. Once the solution has been found, then the optimal con trols may be solved for explicitly, and the optimal law of motion that describes the system may be written down in closed form as well. Quadratic objectives with linear dynamics or exponential objectives with linear dynamics are by far the most common examples where closed form solutions to the HJB equation are available.

14. Money, prices and inflation In this section we follow Gertler (1 979) to present a rational expectations macro economic model to illustrate the convergence of the state variables to a stable distribution over time. We begin by stating the deterministic system of structural relationships:

(14.1) In (M/P) = c In Y - mi,

c , m > 0,

(1 4.2)


rrd = i\ (In Y - a In K ) + 1T *, i\. a > 0 , 0�¢� 1,

1 79

(14.3) (14.4)

1T = 1T*,

(14.5)

i = r + 1T*.

(14.6)

Equation (14.1) is a reduced form IS function relating the logarithm of real output positively to the logarithm of the capital stock and negatively to the real interest rate. It is assumed that the capital stock is constant. Eq. (14.2) is the LM function relating the logarithm of real money balances positively to the logarithm of real income and negatively to the nominal interest rate. Eq. (14.3) relates the desired rate of inflation, 1T d , to the aggregate excess effective demand and the anticipated inflation rate 1T *. Note that aggregate excess effective demand is the difference between output and the natural full employment level of output. Eq. (14.4) describes the price velocity constraint where actual current inflation rr is the sum of the ad hoc inertia inflation if and a convex combination of 1T d and if. Eq. (14.5) denotes myopic perfect foresight and it is the equivalent of rational expectations in the deterministic context. Finally, eq. ( 1 4.6) is a definitional identity where the nominal interest rate i is the sum of the real interest rate r and the anticipated inflation rate 1T *. Next we introduce uncertainty into the detem1inistic system by postulating In Y d t = b 1 ln Kd t - b2 rd t + dz.

(14.7)

Note that this last equation generalizes eq. (14.1) since the random tenn dz is being added. In (14.7), dz equals sy(t)dt, where s is a parameter and y(t) is a normally distributed random variable with mean zero, unit variance, and the y (t)'s are serially uncorrelated. The introduction of uncertainty also affects eq. ( 14.5). Assuming rational ex pectations instead of (14.5) we now postulate 1r *(t) = lim E[1r(t) 1 .�U ] , U-+t

(14.8)

where .OF is the a-field incorporating all information available at time u . Such information includes the structure of the model and both the initial values and the past behavior of the state variables. Using eqs. (14.2), (14.3) and (14.7) we obtain: u


180

ttdt = ¢Xw In (M/P)dt + ¢(Xwm + l)tt*dt + ( 1 -¢)7Tdt + ¢'A [(wmb.Jb2 ) - a] I n Kdt + [(¢Xmw)/b2 ) dz,

(14.9)

where for convenience we have w = b 2 /(m + cb 2 ). Eq. (14.9) describes the dy namic random behavior of actual inflation as a function of various parameters, of the logarithm of real money supply, the anticipated inflation rate, the inertia inflation of the system, the logarithm of the capital stock and uncertainty. From the assumption of rational expectations in (14.8) and (14.9) we conclude that

tt* = [mw/(1 - mcS w) In (M/P) + [ 1/(1-mcS w)] ii - [ cS/(1 -mcS w)]a In K + [ b1mcS wfb 2 (1-mcS w)] In K.

(14.10)

In this last equation we let cS = ¢'A (I -¢). Note that cS is the coefficient on excess demand in the price adjustment equation. Substituting (14.10) into (14.9) we have

ttdt = [ cS w/(1-mcS w)] In (MJP)dt + [1/(1-mcSw)) irdt - [cS/(1 -mcSw)]a In Kdt + [b1mcSwfb2 (1-mcSw)] In Kdt (14. 1 1 ) + (¢"A.mwfb2 )dz. At this point we assume that the money growth rate is deterministic and fixed at p.. Using Ito's lemma we obtain d In

(M/P) = pdt - ttdt + (¢Xmwjb2)2 dt,

(14.12)

where ttd t is as in (14.1 1 ). Finally, we write dir = {3 * (ttdt - 1i'dt),

(14.13)

dh = Alzdt + f' dt + dv,

(14.14)

which is an assumption about the evolution of the system's inertia inflation ii. We are now ready to study the problem of convergence. Substituting ttdt from (14. 1 1 ) into (14.12) and (14.13) we obtain the system of linear stochastic differential equations

where

181


( (:P)) ' ( (= + n

h=

-l>wf(l -ml>w) -l/(1 - ml>w)

A -

f

{3*l> wf(l-ml>w) {3 *ml>w/(1 -m�w)

JJ.

•

[l>/(1 - ml>w)]a In K - [b1ml>wfb 2 ( 1 - ml>w)] In K

- [{3*l>/(l - ml>w)]a In K + [{3*b1 ml>wfb2(I - ml>w)] In K

( dv=

-(¢"Amw/b 2 )dz {3*(¢"Am wfb2)dz

1

,

= eA h (t) + J e A 0

)'

).

The solution of (14.14) is given by h (t)

)

{1 -s)

1

fds J e A (r- s) dv(s) ds. +

0

( 14.15)

Take the expected value of h (t) in this last equation to yield Eh(t) = e A h(O) + J e A (t- s> fds. t

(14.16)

0

Assume that A is negative-definite. Then from (14.16) we conclude that the mean, Eh (t), converges to a stable path. The necessary and sufficient conditions for stability are {3*m < 1

(14.17)

l> < l/mw.

(14.18)

and


1 82

Eq. {14.17) restricts the adjustment speed of the price inertia and requires that the demand for real balances cannot be too sensitive to the nominal interest rate. Eq. (14.18) restricts the size of 8 , the coefficient on excess demand in the price adjustment mechanism. Gertler (1979, p. 232) also shows that if A is negative-definite the variance covariance matrix of h (t) will converge to a stable value. The above discussion concludes our illustration and we refer the reader to Gertler (1 979) for further analysis of specific aspects. 1 5 . An N-sector discrete growth model In this application we follow Brock ( 1 919) to present an n-process discrete opti mal growth model which generalizes the Brock and Mirman ( I 972, 1 973) model. This section and sections 1 1 - 1 6 of Chapter 4 attempt to put together ideas from the modern theory of finance and the literature on stochastic growth mod els. Here we develop the growth theoretic part of an intertemporal general equilibrium theory of capital asset pricing. Basically , what is done is to modify the stochastic growth model of Brock and Mirman (1972) in order to put a non trivial investment decision into the asset pricing model of Lucas ( 1 978). The fmance side of the theory is presented in sections 1 1 - 1 6 of Chapter 4 and they derive their inspiration from Merton (1 973b). However, Merton's ( 1 973b) inter temporal capital asset pricing model (ICAPM) is not a general equilibrium theory in the sense of Arrow-Debreu, that is to say, the technological sources of un certainty are not related to the equilibrium prices of the risky assets in Merton (1973b). To make Merton's ICAPM a general equilibrium model, first the Brock and Mirman (1972) stochastic growth model is modified, and secondly, Lucas' ( I 978) asset pricing model is extended to include a nontrivial investment deci sion. This is done in such a way as to preserve the empirical tractibility of the Merton formulation and at the same time determine endogenously the risk prices derived by Ross (1 976) in his arbitrage theory of capital asset pricing. The model is given by

1 (3'� 1 t 00

maximize E 1

=

such that ct+ 1 + x t+ 1 -x, = x, =

N

.�

1= 1

xit , xit

u

(c,)

(15.1)

N

�1 [g;(xit ' r, ) - 8 f;, l,

� 0,

i

=

1 , 2, ... , N,

t=

l , 2, . . . ,

( 1 5 .2) ( 1 5.3)


c, � 0,

t

=

1 , 2, ... ,

i 1 , 2, ... ,N,r1 historically given, =

1 83

(15.4) (15.5)

where E 1 , (3, u , c, . x,, g; , x;,. r, and f>; denote mathematical expectation condi tioned at time 1 , discount factor on future utility, utility function of consump tion, consumption at date t, capital stock at date t, production function of pro cess i, capital allocated to process i at date t, random shock which is common to all processes i, and depreciation rate for capital installed in process i, respectively . The space of {c,};: 1 , {x,};: 1 over which the maximum is being taken in (15.1) needs to be specified. Obviously, decisions at date t should be based only upon information at date t. In order to make the choice space precise some for malism is needed which is developed in what follows. The environment will be represented by a sequence {r,};: 1 of real vector valued random variables which will be assumed to be independently and identi cally distributed. The common distribution of r, is given by a measure p: Yi(Rm) [0, 1 ], where .'-Jf(R'" ) is the Borel a-field of R'" . In view of a well-known one to-one correspondence (sec, for example, Loeve, 1 977), we can adequately rep resent the environment as a measure space (Q, .o.F, v) , where n is the set of all sequences of real m vectors, .dF is the a-field generated by cylinder sets of the form n;: 1 A ,, where A, E �J(Rm), t I , 2, ... , and A1 = Rm for all but a finite number of values of t. Also v , the stochastic law of the environment, is simply the product probability induced by J..L , given the assumption of independence. The random variables r, may be viewed as the tth coordinate function on n , i.e. for any W {w, };: I E Q , r1(w) is defined b y r1(w) = W1• We shall refer to w as a possible state of the environment, or an environment sequence, and shall refer to w1 as the environment at date t. In what follows, .tF, is the a-field guaranteed by partial histories up to period t (i.e. the smallest a-field generated by cylinder sets of the form n;=l A T , where AT is in .� (R'") for all t, and A7 Rm for all T > t). The a-field -�t contains all of the informa tion about the environment which is available at date t . I n order to express precisely the fact that decisions c1,x, only depend upon information that is available at the time the decisions are made, we simply re quire that c1, x, be measurable with respect to .o.Ft . Formally the maximization in (15 .1) is taken over all stochastic processes {c,}�= 1 , {x, };: 1 that satisfy {15.2)-{15.5) and such that for each t = I , 2, . .. , c, x 1 are measurable .;x;, units of capital have evaporated at the end of t. Thus, net new output is K;(x;, , r1 ) - f>;xit from process i. The total output available to be divided into consumption and capital stock at date t + 1 is given by N

.�1 [g;(x;, , r1) - f>;x;,J + x1

t=

N

=

=

t= I

.� [g; (xi , r,) + (1- f> ;)X;r ] t N

(15.6)

-� f;(x;, , r,) = y '+ 1 '

l= 1

where

(15.7) denotes the total amount of output emerging from process i at the end of period t. The output y'+ 1 is divided into consumption and capital stock at the begin ning of date t + 1 and so on it goes. Note that we assume that it is costless to install capital into each process i and it is costless to allocate capital across processes at the beginning of each date t. The objective is to maximize the expected value of the discounted sum of utilities over all consumption paths and capital allocations that satisfy (15 .2)

(1 5.4).

In order to obtain sharp results we place restrictive assumptions on this prob lem. We collect the basic working assumptions below. A.l . The functions u ( ), /;( ) are all concave, increasing, and are twice contin uously differentiable. •

·

A.2. The stochastic process { r1} �= 1 is independently and identically distributed. Each r1 : (il, .t;f, p) � R m , where (f2, .�, JJ) is a probability space. Here n is the space of elementary events, 9f is the sigma field of measurable sets with respect


185

to and is a probability measure defined on subsets B n, B E Yf. Further more, the range of r1, r, (fl), is compact. A.3. For each {x;1 }� r the problem in (15.1) has a unique optimal solution (unique up to a set of realizations of {r1} of measure zero). Notice that A.3 is implied by A.l and strict concavity of and {!;}� Rather than try to find the weakest possible assumptions sufficient for uniqueness of solutions to (15 .l ) it seemed simpler to assume it. Furthermore, since we are not interested in the study of existence of optimal solutions in this application we have simply assumed that also. Since the case N = 1 has been dealt with by Brock and Mirman (1972, 1973) and Mirman and Zilcha (1975, 1976, 1977), we shall be brief where possible. By A.3 we see that to each output level y1 the optimum c,,x,,x;1, giveny, may be written (15.8) c, = g(y,); x, = h(y,); x;, = h;(y,). J.l,

c

J.1

1 ,

1

u

,

1 .

The optimum policy functions g(· ), h(·) and h;(·) do not depend upon t be cause the problem is time stationary. Another useful optimum policy function may be obtained. Given x1 and r1 , A.3 implies that the optimal allocation {x;,}� and next period's optimal capi tal stock x 1 are unique. Furthermore, these may be written in the form (I 5.9) X;r = a;(x1, r,_ ) and (15.10) x = H(x,, r,). Eqs. (15 .9) and (15 .1 0) contain r, _ and r1, respectively, because the allocation decision is made after r,_ 1 is known but before r, is revealed, whlle the capital consumption decision is made after y is revealed, i.e. after r, is known. Equation (15.10) looks very much like the optimal stochastic process studied by Brock-Mirman and Mirman-Zilcha. It was shown in Brock and Mirman (1972, 1 973) for the case N = 1 that the stochastic difference equation (15.10) converges in distribution to a unique limit distribution independent of initial conditions. We shall show below that the same result may be obtained for our N process model by following the argument of Mirman and Zilcha (1975). Some lemmas are needed. 1

t+

1

t+ 1

1

t+ 1


1 86

Assume A.l . Let U(y . ) denote the maximum value of the objec tive in (15.1) given initial resource stocky1 • Then U(y1 ) is concave, nondecreas ing in y1 and, for each y1 > 0, the derivative U' (y . ) exists and is nonincreasing iny Proof. Mirman and Zilcha (1975) prove that Lemma 1 5 . 1 . • .

for the case N = 1 . The same argument may be used here. The details are left to the reader. Note that g(y ) in the last equation is nondecreasing since u " (c) < 0 and U' (y) is nonincreasing in y owing to the concavity of U( ). Lemma 15.2. Suppose that A.2 holds and u (c) � 0 for all c. Furthermore, as sume that along optima 1 E 1 (3 - 1 U(y1) � 0 as t � oo. 1

•

t ;:

if {c,};: 1 , {x1 };: 1 , {xu}� 1 , 1 , 2 ; ... , is optimal then the following conditions must be satisfied. For each i, t (15.1 1) u ' (c,) � (3E1 { u' (c,+ 1 ) t; (x;, , r1 ) } , (15.12) u' (c,)x;, = (3E, { u ' (ct+ 1 ) J;(x;p r,)x; r } and (15.13) lim E1 { (3, _ 1 u' (c,) x,) = 0. Proof. The proof of (15.1 1) and (15 .12) is an obvious application of calculus to {15.1) with due respect to the constraints c, � 0 and x, � 0. An argument anal ogous to that of Benveniste and Scheinkman ( I 977) establishes ( I 5 .13). By con cavity off; , i = 1, 2, ... N, U( ·) and by lemma 15.1 we have for any constant 'Y, Then

,_00

O < -y < l

187


(15.14) But since U is nondecreasing1 -iny1 and each!; is increasing in X;, the l.h.s. of(I 5.14) 1 is bounded above by E1 /3 U(y1) which goes to zero as t � oo Since u ' � 0 and y1 � 0, the r.h.s. of (15.14) must go to zero as well. But by (15.12) as t � oo .

I

El il'- u'(c,l

[f

!i(x;, ' -

" '• - I

)(x;, t-

1

)] EliJ'- (f =

'

u'(c, - I ) xi, t- 1

)

= E1 /3 1 - 2 u '(c, _ 1 )x, _ 1 � o,

as

was to be shown. Lemma 15.3. Assume that u'(c) > 0, u"(c) < 0 and u'(O) = oo Furthermore, assume that fj(O, r) = O,fj'(x, r)> 0 and fj'(x, r) < 0 for all values ofr. Also sup pose that there is a set of r-values with positive probability such that fj is strictly concave in x. Then the function h(y) is continuous in y, increasing in y, and h(O) = 0. Proof. See Brock (I 979). Now by A.3 and (15.8)-(15.10) it follows thaty,+ 1 may be written ( 15.15) Y r + I = F(x,,r,). Following Mirman and Zilcha (1975) define (15.16) F(x) = min F(x, r), F(x) = max F(x, r), where R is the range of the random variable (U, PA, J.l) Rm which is compact by A.2. The following lemma shows that F and F are well defined. Lemma I 5.4. Assume the hypotheses of lemma 1 5.3 and suppose that each f;(x, r) is continuous in r for each x. Then F(x, r) is continuous in r. .

rER

rER

r:

�


188

Proof This is straightforward because

Yt+ 1 = � /j(xit• r, ) = � /j (fli (x, )x,, r, ) F(x,, r,). I I Since fli (x 1) is continuous in x1 > 0 and each f; (x, r) is continuous in r we con clude that F(x, r) is continuous in x and r. This concludes the proof. =

Let x, x be any two ftxed points of the functions

H(x) = h(F(x)); H(x) = h(F(x)),

(15. 17)

respectively. Then Lemma 1 5.5. Any two fixed points of the pair of functions deftned in ( 1 5. 1 7) must satisfy x < x . Proof See Brock ( 1 979). Finally, applying arguments similar to Brock and Mirman following:

( 1972) we obtain the

Theorem I 5.1 . There is a distribution F(x) of the optimum aggregate capital stock x such that F,(x) � F(x) uniformly for all x. Furthermore, F(x)does not depend on the initial conditions (x 1 , r 1 ) . Here F1 ( x) = P[x, < x].

Proof See Brock ( 1979). Theorem 1 5.1 shows that the distribution of optimum aggregate capital stock at date t, F1 (x), converges pointwise to a limit distribution F(x). Theorem 15.1 is important because we will usc the optimal growth model to construct equilibrium asset prices and risk prices. Since these prices will be time stationary functions of x,, and since x1 converges in distribution to F, we will be able to use the mean ergodic theorem and stationary time series methods to make statistical inferences about these prices on the basis of time series observations. More will be said about this in chapter 4. 1 6.

Competitive frrm under price uncertainty

In this application we follow Sandmo ( 1 97 1) to illustrate the use of stochastic techniques in the theory of the competitive ftrm under price uncertainty. These


189

techniques arc rather elementary and make usc of concepts introduced in Chap ter 1 . Consider a competitive firm in the short run whose output decisions arc dom inated by a concern to maximize the expected utility of profits. The sales price, p, is a non-negative random variable whose distribution is subjectively deter mined by the firm's beliefs . The density function of the sales price is f(p) and we denote the expected value of the sales price by i.e. E(p) = We assume that the firm is a price taker in the sense that it is unable to influence the sales price distribution. Let u denote the von Neumann- Morgenstern utility function of the firm and 7t(x) the profits function, where x is output. We assume that u is a bounded, concave, continuous and differentiable function such that u'(1r) > 0 and u "(1r) < 0. ( 16.1) Thus, the firm is assumed to be risk averse. The total cost function of the firm, F(x), consists of total variable cost, C(x), and fixed cost B. We write ( 16.2) F(x) = C(x) + B, where C(O) = 0 and C'(x) > 0. ( 16.3) In the usual way we define the firm's profit function by (16.4) 1r(x) px - C(x) - B, and the firm's objective to maximize the expected utility of profits can be written as (16.5) E(u(px - C(x) - B)). The necessary and sufficient conditions for a maximum of (16.5) are obtained by differentiating (16.5) with respect to x; they are (16.6) E(u '(1r) (p - C'(x))) :::: 0 and (16.7) E(u "(1r)( p - C'(x))2 - u'(1r)C"(x)) < O. p,

=

J.l..

190


Suppose that eqs. ( 1 6.6) and ( 16.7) determine a positive, finite and unique solu tion to the maximization problem ( 1 6.5). For our analysis the basic question is: How does the optimal output under uncertainty compare with the well-known competitive solution under certainty, where price is equated with marginal cost? To provide an answer we proceed as follows. Rewrite ( 16.6) as

( 1 6.8)

E(u'(1r)p) = E(u'(1r)C'(x)) and subtract E(u'(7T)J.L) on each side of ( 1 6 8) to get .

E(u '(1r) (p - p.)) = E (u'(1r) [C'(x) - p.]).

(16.9)

Note that taking the expectation of ( 1 6.4) we obtain

E(1r) = E(p)x - C(x) - B = p.x - C(x) - B. Therefore, 1r(x) - E(1r) = px - p.x = (p - p.)x, or equivalently 1r(x) = E(1r) + (p - p.)x. If p � p., then from the last sentence we obtain that 1r(x) � E(1r) and therefore u' (1r) � u'(E{1r)). In general,. for all p we have that

u'(1r)(p - p.) � u '(E(1r)) (p - Jl).

(16.10)

Take expectations on both sides of ( 16.1 0) to get

E(u'(1r) (p - p.)) � u'(E(1r)) E(p - p.) = 0.

( 1 6. 1 1)

Observe that the zero in the right-hand side o f ( 16. 1 1) comes from the fact that u'(E(1r)) is a constant and E(p) = p.. Combine the result in ( 16.1 1), namely that E(u'(1r) (p - p.)) � 0 with eq. (16.9) to conclude that

E(u'(1r) [C'(x) - p.]) � 0,

(16. 1 2)

which finally implies

C'(x) � p.

(16.13)

because u'(1r) > 0 by (16.1). Our result in (16.13) says that optimal output for a competitive firm under price uncertainty is characterized by marginal cost being less than the expected price. If we characterize the certainty output as that quan tity where C'(x) = p., then we may conclude that under price uncertainty, out-

191


put is smaller than the certainty output. This result is a generalization of McCall's ( 1967) theorem for the special case of a constant absolute risk aversion utility function. Next, suppose that x* denotes the positive, finite and unique optimum out put which is the solution to {16.6) and satisfies (16.7). Then x* will give a global utility maximum provided (16.14) E(u(px* - C(x*) - B)) � u(-B), where -B is the level of profit when x 0. Consider the left-hand side of{l6.14) and approximate it by a Taylor series around the point p p. to rewrite ( 16.14) as =

=

E(u(p.x* - C(x) - B) + u'(p.x* - C(x*) - B)x*(p - p.) + & u"(p.x* - C(x*) - B)x*2 (p - p.)2) � u (-B).

(16.15) Note that higher-order terms in the Taylor series have been neglected and also note that by definition the second term on the left-hand side of {16 .15) is zero. Rearranging the remaining terms in ( 16.15) and dividing through by u '(px* C(x*) - B) so as to make the expressions invariant under linear transformations of the utility function, we then obtain u(px* - C(x*) - B) - u( -B) I u"(p.x* - C(x*) - B) X*2 E(p - IJ.)2 �- u '(p.x* - C(x*) - B) 2 u' (p.x* - C(x*) -- B)

..:... � .._ _ ..:... _ .._ _ ....:. .._ ....:. .._ _ __:. ;._ _

·

(16.16) Observe that the factor -u "/u' on the right-hand side of (16.16) is the risk aver sion function evaluated at the expected level of profit for optimum output x*. The factor x*2 E(p - p.)2 denotes the variance of sales. Each of these two factors is positive and therefore from ( I 6.16) we conclude that p.x* - C(x*) - B > - B, (16.17) given that the utility function is strictly increasing. Finally from ( 16.1 7) we ob tain C(x*) < p., x*

(16.18)

192


which says that at the optimum level of output of a competitive firm under un certainty, the expected price is greater than average cost. This fact further im plies that the firm requires strictly positive expected profit in order to operate in a competitive environment under price uncertainty. Therefore, price uncer tainty leads to a modification of the standard results of the microeconomic theo ry of the competitive finn in an environment of certainty. 17. Stabilization in the presence of stochastic disturbances In this section we follow Brainard ( 1 967) and Turnovsky (1977) to illustrate the effects of uncertainty in stabilization policy. To fully demonstrate the role of uncertainty we usc the simple case of one target, denoted by y, and one instru ment, denoted by x, related linearly as follows:

(17.1)

y = aX + U .

Here we assume that y and x are scalars while a and u are random variables having expectations and variances denoted by E(a) = a , E(u) = u , var(a) = a� , and var(u) = a�, respectively. Suppose that the policy-maker has chosen a target value y*. To fix the ideas involved, suppose that ( 1 7 . I ) is a reduced form equation between GNP, y, and the money supply, x, subject to additive, u, and multiplicative, a, disturbances. Given y*, the stabilization problem is to choose x so that the policy-maker will maximize his expected utility. Brainard (1967) uses a quadratic utility function, U, of the form u=

-

(y - y*)2 .

(1 7.2)

The problem then becomes max E(U) = max E(-(y - y*)2 ) X

X

(17.3)

subject to ( 1 7. I ) Substituting ( 1 7. 1 ) to ( 1 7.3) and taking expectations we obtain

(17.4)

where p denotes the correlation coefficient between a and u. Put (1 7.4) into ( 1 7.3), differentiate with respect to x, and set this derivative equal to zero to obtain the optimal value of x; it is given by

193


( 17.5) a0 = au =

0, In ( 1 7 .5) xu denotes the optimal value under uncertainty. Note that if which means that certainty prevails, then the certainty optimal value of the in strument variable, denoted xc, is given by y* - u (17 .6) X = -c

a

Comparing (17 .5) and ( 1 7.6) we observe that the difference between the values of xu and xc depends on If 0 then xu = xc . This shows that for an addi tive random disturbance the values of xu and .xc are the same; this is called the certainty equivalence result. Therefore, to understand the role of uncertainty we must study the role of the multiplicative disturbances arising from the random variable a. To do so, suppose that 0; then ( 1 7.5) becomes a0 •

aa =

au =

( I 7.7)

from which we obtain that ( 17.8)

This last equation means that the policy-maker is more conservative when uncer tainty prevails. What are the implications of such a conservative policy? Inserting xu from ( 17.7) into ( 17.1) we solve for the target value achieved from xu in the special case under discussion, i.e. when � 0 and u u the result is a

y

=

aa(y* - u) +U. a2 + a 2 a

=

=

;

(1 7.9)

Without loss ofgenerality we assume in ( 1 7.9) thaty* > ii; this assumption simply says that the target value y* is greater than the expectation of the additive dis turbance term. From ( 1 7.9) we are now able to discover the implications of un certainty in economic policy. Observe that


194

u) + u) - y*

a (y* E(y) - y* = E l 7i 2 a 2 + aa

\

=

=

a 2 (y*

_

u)

a�(u - y*) < a2

+

(a2

+

2

a

�) (u - y*)

+a

o.

a

( 1 7. 1 0)

From ( 17.1 0), under the assumption that y* > u, we conclude that E (y) < y*, which means that the target variable will on the average undershoot its desired value. If we assume that a� = 0 and a� =I= 0, it can be shown in a similar way that E(y) = y*, so that y will fluctuate randomly about its target value. Thus even in the simplest possible case with one target on one instrument variable, the results of stochastic analysis are more general and richer than those of the certainty analysis.

18. Stochastic capital theory in continuous time In this section we conduct an analysis similar to that of section 8 of Chapter 1 for discrete time except that now we work with the case of continuous time. Consider the Markov process {X1 } �= 0 with t e [0, oo ). Most of the time we sup pose that {X1 } �=o is given by the Ito stochastic differential equation

a

dX = [(X, t)d t + (X, t)dz,

(18.1)

-y (t, X, T) =

(18.2)

where dz is normal with mean zero and variance dt. Consider the problem sup

t< T < T

E [e- X7 I X(t) = X]. n

Here the supremum is taken over the set of measurable stopping times, i.e. the events { T � s } depend only upon {Xr } for r < s . The existence of optimal stopping times and of critical boundaries becomes a very technical matter in continuous time as the analysis of section 1 3 of Chapter 2 has shown. Hence, we shall proceed heuristically making use of some ideas from section 1 3 of Chapter 2. Actually the basic ideas are simple, intuitive and quite pretty when unencumbered by technicalities. We define the continuation region C(t, T) for problem (1 8.2) by


C(t, T) = { (s, X) I -y(s, X, T) > e-nX, t � s � T}

195

(1 8.3)

Note that solutions X(s, T) to (18.4) -y(s, X, T) = X(s, T)e- rs play the role that the critical numbers {X,} played in the discrete time case Just as in the discrete time case we would like the critical numbers to exist and to be unique so that X(s, T) is a function and not a point to set mapping. We want to describe the boundary of C(t, T) by a function. We will be particularly interested in C(O, T) C(T) and C(O, ) = C. Fortunately a theorem by Miroshnichenko (1975, p. 388) gives us what we need under mild assumptions. In order to motivate Miroshnichenko's theorem we proceed heuristically as follows. Put =

oo

R (s, X) = e- rsx.

(18.5)

E

Since (t, X) C(T) therefore -y(t, X, T) > R (t, X). Now sample paths are con tinuous. Therefore if llt is small enough, it will always be worthwhile to con tinue on from (t + 11t, X(t + At)). Since the value at (t, X) is the maximum of the value of stopping before t + At and continuing on after t + llt, it follows that (t, X) C(T) implies e

-y(t, X, T) = E[-y(t + llt, X(t + llt), T) I X(t) = X].

(18.6)

In order to shorten notation, put (1 8.7)

for any function H(t, X). Let A(t, T) = {{s, X) I LR(s, X) > 0, t � s � T}.

Then we claim that A (t, D C C(t, n. To prove our claim let

(s, X) e -y(t, T) = { (s, X) I -y(s, X, T) = R(s, X)}.

Hence,

Stochastic methods in economcs i and finance

196

R (s, X) =

sup

s X is =

202

Stochastic methods in economics andfinance

H(X, Y) = Y

(X) <J>( Y)

--

= Y exp

(

y

-

J

X

)

g(u)d u .

It follows from (1 8.20) that g satisfies a first-order differential equation

g'(X) rfb - (a/b)g(X) - g 2 (X).

(18.28)

=

Since g is a first-order differential equation, only one boundary condition is re quired to identify a particular solution. Thus, in principle, to calculate H(X, Y) in a particular case, use a boundary condition of the form (18.22) to identify a particular solution of (1 8.28) and integrate along that solution from X to Y to get - log(H(X, Y)/Y). This observation suggests a way of getting comparative statics results. Sup pose g and g are two solutions to (1 8.28) corresponding to different parameters. For simplicity we call the tree which grows according to the process which de termines g( · ) the g tree and that which grows according to the process which determines g( · ) the g tree. Suppose also that

g(X) < g(X) Then exp

( J g(u)du ) > ( y

-

X

(1 8.29)

for all X.

exp

y

-

J

X

K lu )d u

)

and

H(X, Y) > h(X, Y).

This means that for any cutting size Y the g tree is worth more than the g tree. If X* is the optimal cutting size for the g tree and X* is the optimal cutting size for the g tree, then in an obvious notation

V(X) = H(X, X*) �H(X, X*) > fl(X, X*) = V(X), so that the g tree is worth more than the g tree. It also follows that X* > X*. The optimal cutting size for the g tree, X*, must maximize X/<J>(X). First-order conditions for maximization imply X * must satisfy (X*) = X*¢' (X*) or that g(X*) =

1/X*.

(18.30)

Second-order conditions are that g(X) intersect 1/X from below. Thus, if g(X) < g(X), it must be that if g(X*) = X*-1 , g(X*) < X*-1 ; at X* the value of the g tree is still increasing. Thus, X* > X*.


203

To use this method of analysis it is necessary to translate the boundary con ditions on f given by (18.22a, b, c) into boundary conditions on g. Since solu tions to (18.20) are of the form (18.24), in general .ellX • g(X) = A Xe'J...'J...X + Bp.p.X

Ae X

+

Be

The solution 'YQ (X) corresponding to a reflecting barrier at Q for [, satisfies 'YQ(Q) = 0 and is given by 'YQ (X) =

1

exp[(X - p..)(X - Q)] -

I

(18.3 1)

1 - exp[(X - p..)(X - Q)] - X p..

The solution cxQ (X) corresponding to an absorbing barrier at Q = 0 for /is given by ex

Q (X) = _

X exp[(X - p..) X] - p.. exp[(X - p..) X] -

The solution to ( 18.20) which is bounded at the corresponding g = f' /f is given by f3(X) = X.

(18.32)

1

-oo

is of the form f(X) = A e'J... X ;

( 18.33)

We now have enough information to draw a phase diagram for g. Eq. (1 8.28) can be rewritten as

g' = (X - g)(g - p),

(1 8.34)

where X and p.. are the roots of the characteristic equation off( · ) and are given by {18.25) and (18.26). For g > X, g'(X) < 0; for X > g > p, g'(X) > 0, and for g < p, g' (X) < 0, so the phase diagram looks as in fig. 18.1. In this figure we have drawn cxQ (X), 'YQ (X) and f3(X) solutions corresponding to an absorbing barrier at Q, a reflecting barrier at Q, and bounded behavior at - oo, respectively. Since solutions to (18.28) cannot cross, a (X) > (3(X) > 'YQ (X). It is easy to Q calculate from (18.3 1 ) and (18.32) that

{18.35)

204


and lim o:Q(X) = lim (3Q(X) Q -.. Q -.. - oo - oo

=

(18.36)

(3(X) = X.

Also, if Q > Q, it is easy to see that O!Q (X) > o:0 (X) > (3Q (X) > (3Q (X). We may sum up these observations in two propositions. Proposition 18.1. As Q � -oo, trees with reflecting barriers at Q become indis tinguishable from trees with absorbing barriers at Q or trees which are simply bounded at oo. Proposition 18.2. Trees with reflecting barriers are worth more than trees with absorbing barriers. The value and the optimal cutting size of a tree with a reflect ing barrier is an increasing function of the barrier. For trees with absorbing bar riers, the opposite is true. g(X)

\ (t(X) A

/

Figure

1 8. 1 .


205

The phase diagram makes it easy to do comparative statics, to analyze the ef fects of changes in parameters on value and cutting size. The parameters of our problem are a, b and r. Differentiating (1 8. 28) we see that dg'(X)/dr = 1/b > O and dg'(X)/da = - (1/b)g(X) which is negative since we are only interested in positive g(X). This proves the rather obvious Proposition 1 8.3. Increasing the infinitesimal mean growth rate of a tree or de creasing the interest rate increases the value and the optimal cutting time of a tree. The effects of increasing2 the variance are only slightly more complicated. Note that dg'(X)/d b = - (I /b )[r - ag(X )] which is negative whenever g(X) < r/a. It is straightforward to calculate that A < rfa. If g( ) corresponds to a reflecting barrier at g(X) � A for all X, it follows that increasing the variance increases the cutting size and value of the tree. Proposition 1 8.4. If a tree's growth process is a diffusion with constant coeffi cients which is bounded at infinity or which has reflecting barriers, then increas ing the variance increases the value and cutting size of the tree. The effects of increasing the variance on the absorbing case arc more complex and the reader is referred to Brock, Rothschild and Stiglitz (1 979). We conclude this analysis of the stationary case by observing that under uncertainty the opti mal cutting size is greater than it would be under certainty. ·

-oo,

19. Miscellaneous applications and exercises

(1 )

Consider the stochastic differential equation of economic growth derived in section 2, dk = [sf(k) - (n-a2 )k ]dt - akdz,

(19.1)

with initial random condition k(O) = k0 > 0. Find a set of sufficient conditions for the existence of a unique solution k(t), t [0, oo) and use theorem 6.1 of Chapter 2 to establish the existence and uniqueness of k(t). (2) Equation (19 . 1 ) may be generalized by assuming that s and a are no longer constants but instead are functions of k, written as s(k) and a(k). With the new functions s(k) and a(k) the generalized stochastic differential equation of growth becomes E

206


dk

=

[s(k)f(k) - (n-- o2 (k))k] d t - o(k)kdz,

(19.2)

with initial condition k (O) = k0 > 0. Find a set of sufficient conditions for the existence of a unique solution k (t), t E [0, oo) and use theorem 6.1 of Chapter 2 to establish the existence and uniqueness of k (t). (3) Suppose that eq. (19 .2) has a unique solution k (t). Discover sufficient conditions such that k (t) is bounded w.p.l , i.e. such that

P[w: k(t, w) < oo ] = I . (4) Assume that the coefficients of ( 1 9 .2) vanish for the equilibrium solu tion k * , where k * is a nonrandom, nonzero constant; that is, assume that

[s(k*)f(k*) - (n-o2 (k*))k *] = o(k*)k*

=

0.

Also, assume that there exists an T] > 0 such that o(k)k > 0 for 0 < I k-k * I < TJ . Is the equilibrium solution k* stable? (5) Money, growth and uncertainty: Tobin ( 1 965) was one of the first econ omists to introduce money in an economic growth model. The Tobin model has been studied by several authors. In this application we will first present briefly the deterministic Tobin model following Hadjimichalakis (1971) and then pro ceed to propose an extension by introducing uncertainty. We begin the model by describing its equations: homogeneous production functions of degree I

Y = F(K, L),

( 1 9.3)

p

( 1 9.4)

("1)

perfect foresight

-p = q ,

investment function

l = K +- -

saving function

s;s

labor growth

L (t) = L (O) en , L (O) > 0,

(19.7)

money supply growth

lv/(t) = Jltt(O) e8 1 ,

(1 9.8)

.

[

d dt

y+

p

'

:t (�) l t

/v/(0) > 0.

(19.5) (1 9.6)

Note that PJP denotes actual inflation and q denotes expected inflation. It is assumed that the two are equal under the assumption of perfect foresight .


207

Under equilibrium, saving must equal investment. Thus,

which yields

K=s

[ Y + dt (A1P�)� - d( (Afp) · d

d

This last equation is called Tobin 's fundamental equation. On the basis of the above, the differential equation of money and growth, where we define m = M/P 1/L, i.e. m is per capita real money balances, is given by

k

=

s[(k) - (I - s) (8 - q)m - nk.

( 19.9)

Note that if 8 = 0 = q, then we obtain as a special case the Solow equation (2.1 ). Suppose that uncertainty is now introduced in Tobin's monetary growth model by postulating randomness in the growth of the money supply and de scribed by the stochastic differential equation

dA1 = 8JV!

dt + J.Livldz.

(19. 10)

Usc eq. ( 1 9 . I 0) and the appropriately modified model of cqs. ( I 9 .3)-( 1 9 .7) to study the effects of uncertainty in the money supply growth. (6) Random demand functions: Let a consumer solve max U(X,

Y)

(19.1 1 )

subject to

(19.12) Let

solve (19 .I I ) and processes:

(19 .1 2).

Now, let iH, p 1 and p 2 be random and follow the Ito

{19. 13)

208


dp1 = JJ.p d t + aP. dzP , . 1 dp 2 = JJ.p d t + aP dzP . 2

.

2

2

(19.14) {19.15)

Assume that each instantaneous mean and variance of (19.13)-(19.15) is a function of (l'l1, p1 , p2 ). Assume that { zM }, {zp , } , {zP, } are Wiener processes that satisfy the formal rules dzM · dzP = pd t; dzM · dzP = pd t; dzp · dzP = pd t, 2 1 , , dzP ·dzP = dt; d zP dzP = dt; d zM ·dzM = dt. , . . Use Ito's lemma to write down the stochastic differential equations of the demands X = g(M, p1 , p2 ), and Y = h (M,p 1 , p2) for the case 2

U(X, Y) = AXa yP,

•

0 < a < 1 , 0 < {3 < 1 , 0 < a + {3 � l .

This is a random world when the individual cannot store commodities, i.e. it is a stochastic world without stock variables just as standard static demand theory is a deterministic world with no stock variables. (7) Edgeworth's random boxes: This is an Edgeworth Box with random en dowment. Let consumer i = 1 , 2 solve max X� y{J I

I

where e 1 = (a, 0), e2 = (0, b ),p · e; is a scalar product. Note that a and {3 do not vary across consumers. Use problem (6) to solve for the demand functions Determine the relative price p 1 /p 2 by supply equals demand: h 1 + h 2 = a, g1 + g2 = b.

Now let da = P.ad t + aad za, d b = p.bdt + abdzb ,


209

determine the endowments, a and b, of the two consumers. Use Ito's lemma to find the stochastic process that equilibrium relative prices p 1 /p2 must follow. (8) Marshall with a2 : Consider the following algebraic version of the Marshal Han Cross: p

= D(q) = Aq - a = S(q) = BqiJ, A > 0, B > 0, a > 0, 13 > 0.

Find the stochastic processes generating equilibrium p (t) and q(t) when

= JlA d t + aA dzA , dB = Jln d t + a8 dz8 ,

dA

where we assume dzA dz8 = 0 for simplicity. Does the old proposition about random supply and demand leading to high price variation and low quantity variation when supply and demand are inelastic hold up in this new framework? (9) Monopoly vs. competition in exploring for oil: A monopolist solves max

E0

00

J

0

e-Pt (R(q) - I)dt

{19.16)

subject to

dE(t) = -q(t)dt + b(E(t))da(t),

(19.17)

where

da(t) = 1 da(t) = 0

with probability A.(/(t))dt, with probability

1 - A.{/(t))dt.

Here R (q) = D(q)q = total revenue, and /(t) = investment in enlarging the stock of oil, E(t). Investment takes the form of allocating money /(t) to "dig" where with probability A.(I(t))dt of "find" of size b (E(t)) will be discovered. Write down the Hamilton-Jacobi-Bellman equation for {19.16) and (19.17). Let competition solve the same problem with R (q) replaced by CS(q) where

CS(q) =

q

f D(y)dy,

qo

and q0 is a small positive lower limit. Note that provided that

q0

could be taken to be zero


210 q

J

D(y )dy for q > 0, is finite.

Write down the Hamilton-Jacobi-Bellman equation for competition. Point out the difference from monopoly. Also, attempt to say if monopoly or competition will invest more in finding new wells. The above model may be looked upon as a proxy for the amount of innova tion in the sense of divesting profits into projects to enlarge the stock of salable resources. One would expect that monopoly would invest less than competition. Any hints from this model? Finally , try to find a form of D(q), b (E(t)), 'A(l(t)) that permits a closed form solution for the Hamilton-Jacobi-Bellman equation. ( 1 0) Exhaustible resource problem : Consider the problem max E0

00

J e- rt R (q)dt

0

subject to dE =

-

qdt + a0Edz,

for the special case where

As in the previous problem R (q) is total revenue for the firm, q is quantity of output and E(t) is the stock of the resource at time t with E(O) = E0 > 0. Solve this problem for a closed form solution for the cases: (i) a0 = 0 and (ii) a0 > 0, and compare your solutions. (1 1) Write the quantity theory of money equation MV = PO,

where M is money supply, V is velocity of money, P is price level, and 0 is real output. Let

and


211

where, E1dz � = dt, E1dz � = d t , E1dz 1 dz 2 = pd t. Let V be constant over time and nonrandom. Write out the formula for dPfP in percentage terms. ( 1 2) Consider the quadratic problem presented in section 1 2:

- JV(x (t)) = mJn E,

00

J

e-P(s- t ) { a(x (s))2 + b (v(s))2 } ds.

s=t

Solve this problem for each of the following laws of motion:

(i)

dx(s) = v(s)dt + a0 x(s)dz(s).

where a0 is a constant independent of time and of x; (ii)

dx(s) = [v(s) + Q0] dt + a0x(s)dz(s),

where Q0 is a nonrandom constant; and (iii) d x(s) = [ax(s) + v (s)] dt + a0x(s)dz (s), where a is a nonrandom constant. Furthermore, in each case check that your solution satisfies the transversality condition, lim E 0 e-P'q (t) ·x(t) = 0. l-+00

{13) Exchange rate in a two-country stochastic monetary model: We follow Lau (1977) to formulate a simple international monetary model. Let there be

two countries: country I , say England. and country 2, say France. Let there be two goods: good 1 cloth, and good 2, wheat . This is not a production model, so we assume that both goods are perishable and that at the beginning of each peri od England is endowed with a constant amount of cloth, y 1 , and France is en dowed with a constant amount of wheat, y 2 • We use q 1 to denote the price of good I , cloth, and q 2 to denote the price of good 2, wheat. We choose q 1 = I , i.e. cloth is the numeraire. Let c;i be the real consumption by country i of good j and Jet M;i be country i's holdings of money j. Denote by S; the discount rate for country i. Let P1 be pounds per unit of cloth and P2 francs per unit of cloth and let E be the exchange rate. By purchasing power parity, P1 = EP2 or E = PtfP2 , i.e. E is pounds per franc. We assume that the money stocks, il1f follow the stochastic differential equations. ,

Stochastic met/rods in economics and finance

212

and

where J.L1 , J.L2 , a 1 and a2 are constants, and z 1 and z 2 are normalized Wiener pro cesses such that

With the above information we now formulate the problem: England solves max E0

00

Je

0

-

t> ,

t

[u1 (c 11 , c 12 ) + vi (Mu/PI , M12 /P2 )] dt

subject to

,

1 } ;: 1 , { cf2 } ;: {Mf1 } ;:

for { cf

max E0

00

1

1

2 } ;: , while France solves

and {Mf

1

J e - 6 2 t [u2 (c21 , c22 ) + v2 (M2. /P1 , M22/P2 )l d t

0

subject to

for {c�1 } �= 1 , {c� };: 1 , {M�1 };: 1 and {M� };: 1 • Then {P 1 } �= 1 and {P };: 1 is 2 2 2 a monetary perfect foresight equilibrium if for every t E [0, oo) all of the follow ing hold:

Observe that we postulate separable utility functions u; and V; and also note that the superscript d denotes quantities demanded. Make any additional assumptions that are needed to study in this model the behavior of the exchange rate E . (14) Stochastic search theory: Consider a representative individual who searches for a higher wage. This search takes the fonn of allocating time to in fluence the probability of arrival of a Poisson event. The more time that he allo-

Applicatio11s in economics

213

cates to search activity the more frequently this Poisson event will arrive. If the Poisson event arrives the searcher's wage increases by a jump. The searcher faces a stochastic differential equation that determines the evaluation of his wage. This equation consists of an exogenous rise in the nominal wage plus a Brownian motion term plus a Poisson term. Only the Poisson term is influenced by the searcher's search activity. Let us get into the details. Consider the following model: max E0

00

J e-Pt y(t)dt =J(W(O), 0) 0

subject to

d W(t) = rW(t)dt + aW(t)dz(t) + g(W(t))dq(t), W0 given, where W(t), y(t), {z(t)};: 0 and {q(t)}�= o are nominal wage , flow income, discount rate on future income, standardized Wiener process, and Poisson pro cess, respectively. The probability that the Poisson event occurs, i.e. q(t + �t) q(t) = 1 , is given by X(21 )�t + o(�t) and y(t) = W(t) (1-21 (t)), where 21 (t) is the percentage of time devoted to wage augmentation activity, i.e. search activity. Furthermore, E, denotes expectation conditioned at t. The numbers r and a do not depend on t or W. Here g( W(t)) is the amount of the wage jump if the Pois p,

son event occurs. The idea is that the amount of a better job offer should depend upon current wage. Form

c/>(21, W, t) = e- Pt (l-2.)W +11 +lwrW + � lwwa2 W2 + X[J(W +g(W), t) -J(W, t)]. Assume an interior solution for 21 : a c�> dX - = 0; 0 = -e-P' W + - [J(W + g(W), t)-J(W, t)). d21 a 21

(19.18)

(19.19)

Notice that eqs. (19 . 1 8) and (19 .19) are the standard partial differential equa tions of stochastic control. Make any assumptions that seem economically reasonable and proceed to study the effects of search on nominal wages.

214

Stochastic methods in economics and fir�ar�ce

20. Further remarks and references

We would like to note that there is some arbitrariness on the part of the author in the selection of applications and in distinguishing them either as economics or finance applications. This chapter and the next are not intended to be exhaustive nor is it possible to have an empty intersection. An important book of readings and exercises which supplements this chapter is Diamond and Rothschild ( 1 978). The use of continuous time stochastic calculus in macroeconomic growth under uncertainty first appeared in the papers of Bourguignon (1 974), Merton {1 975a) and Bismut ( 1 975). These papers build on several earlier papers on economic growth under uncertainty such as Brock and Mirman ( 1972), Levhari and Srinivasin { 1 969), Mirman ( 1 973), Stigum (1 972), Leland ( 1 974) and Mir man and Zilcha (1975). among others. The main unresolved issue in continuous time economic growth under uncertainty is the stochastic stability of the sta tionary distribution. Stochastic point equilibrium is discussed in Malliaris ( 1 978). For discrete time growth models under uncertainty several results are presented in Brock and Majumdar ( 1 979) and section 1 5 of this chapter. Two recent con tributions on optimal saving under uncertainty are Foldes ( 1 978a, 1978b ). Our analysis of growth in an open economy under uncertainty is very limited. However, there has been great interest recently in introducing uncertainty in in ternational economics with several papers, such as Batra ( 1 975), Mayer ( 1 976), Baron and Forsythe ( 1 979), and the recent book by Helpman and Razin (1 979). Sec also the international monetary model under uncertainty n i exercise 1 3 of section 1 9 due to Lau (I 977). In section 8 we presented the concept of rational expectations to illustrate the use of stochastic methods and more importantly to familiarize the reader with the defmitional aspects of this concept. Rational expectations as a concept has found numerous applications and for a survey article we suggest Shiller (1 978) and Kantor ( 1 979). We use rational expectations in sections 9 and 1 1 . The cost of adjustment type of investment theory developed by Lucas (1967b, 1967), Gould (1 968) and Treadway ( 1 969, 1 970), among others, was extended under uncertainty in the paper of Lucas and Prescott {1971 ). Elements of the Lucas and Prescott ( 197 1 ) paper are presented in section 9 to illustrate the use of stochastic methods in microeconomic theory. Another microeconomic application is found in section 1 6. Some basic references in the analysis of the firm under uncertainty, other than Sandmo ( 1 97 1 ), are Mills ( 1959), Nelson ( 1 96 1 ), Pratt (1 964), McCall (1 967), Stigum (1 969a, l 969b), Baron ( 1 970, 1 97 1 ), Zabel ( 1 970), Leland ( 1 972). Ishii (1 977), Wu ( 1979) and Perrakis ( 1980), just to mention a few . Note that there is a considerable bibliography on problems of general equilibrium under uncertainty. We have not presented any


215

applications in this area primarily because the techniques are not similar to the ones presented in this book. Techniques of general equilibrium under uncertainty usually involve arguments of a topological nature and/or arguments of functional analysis. A representative paper in general equilibrium under uncertainty is Bewley (1 978). However, we remark that stochastic stability techniques have been used in general equilibrium in some specific fonnulations such as Turnovsky and Weintraub (1971). In sections 1 0 and 1 1 we attempt to develop a general approach to continuous time stochastic processes that arise in dynamic economics from maximizing be havior of agents, as in Brock and Magill (1979). The analysis considers a class of stochastic discounted infinite horizon maximum problems that arise in economics and uses Bismut's ( 1 973) approach in solving these problems. It is shown that the idea of a competitive path, introduced in the continuous time deterministic case in Magill (l 977b ), generalizes in a natural way in the case of uncertainty to a competitive process. Theorem 1 0.1 shows that under a concavity assumption on the basic integrand of the problem a competitive process which satisfies a transversality condition is optimal under a discounted catching-up criterion. Next, we consider the sample path properties of a competitive process. If for almost every realization of a competitive process the associated dual price pro cess generates a path of subgradients for the value function, we call the process McKenzie competitive, since it was McKenzie ( 1976) who first recognized the importance of this property in the detenninistic case. Theorem 1 0.2 shows that two McKenzie competitive processes starting from distinct nonrandom initial conditions converge almost surely if the processes are bounded almost surely and if a certain curvature condition is satisfied by the Hamiltonian of the system. The problem of finding sufficient conditions for the existence of a McKenzie competitive process remains an open problem .. Business cycles and macroeconomic stabilization methods in an environment where uncertainty prevails are areas of research in which stochastic calculus tech niques are quite appropriate . However, the research has just begun; we note that Lucas' ( 1 975) paper represents a methodological advance in business cycle theory. In the Lucas (I 975) paper, discrete time stochastic techniques arc used to show that random monetary shocks and an accelerator effect interact to generate serially correlated cyclical movements in real output and procyclical movements in prices, in the ratio of investment to output, and in nominal interest rates. Also, note the Slutsky ( 1 937) paper in which it is shown that a weighted sum of independent and identically distributed random variables with mean zero and finite variance leads to approximately regular cyclical motion. Slutzky's (1 937) ideas have not yet been fully utilized. Magill (I 977a) has a brief analysis in which he shows that the introduction of uncertainty imbeds the short-run study

216


of the business cycle into the long-run process of optimal capital accumulation. Tinbergen's (1952) classic work on static stabilization has been extended to al low for uncertainty by Brainard (1 967) and in section 1 7 we give a simple illus tration. See also Poole (1 970), Chow ( 1970, 1 973) and Tumovsky (I 973). In section 14 we illustrate the use of stochastic calculus techniques in a macroeconomic model with rational expectations, following Gertler ( 1 979). The Gertler { 1 979) paper explores the consequences for price dynamics of imperfect price flexibility and it demonstrates that the same condition which ensures sta bility in the deterministic system also ensures that the distribution of the state variables converges to a stable path in the stochastic case. In section 1 4 we illus trated the stochastic case; for a comparison between the deterministic and the stochastic case see Gertler (1 979). Section 1 8 follows Brock, Rothschild and Stiglitz (1 979) to illustrate various methods of continuous optimal stopping to stochastic capital theory. This sec tion treats the time independent case. Similar results continue to hold in the gen eral case when the instantaneous mean and the instantaneous variance of the dif fusion process are functions of the tree's current size. For details see the Brock, Rothschild and Stiglitz ( 1 979) paper. See also the paper by Miller and Voltaire ( 1 980) which treats the repeated or sequential stochastic tree problem, i.e. the problem of deciding when to harvest and replant trees given the knowledge of the process of each tree's growth history. Miller and Voltaire (I 980) show that the results for the nonrepeated and for the repeated cases are qualitatively similar. The reader who is interested in further applications of optimal stopping methods in economics should consult Samuelson and McKean ( 1965), Boyce ( 1 970) and Jovanovic (1 979a).

CHAPTER 4

APPLICATIONS IN FINANCE

There is no need to enlarge upon the im portance of a realistic theory explaining how individuals choose among alternate courses of action when the consequences of their actions are incompletely known to them. Arrow ( 1 97 1 , p. 1 ).

1 . Introduction In this chapter we present several applications of stochastic methods in finance to illustrate the techniques discussed in Chapters 1 and 2 . We also include some applications which use additional techniques to familiarize the reader with a suf ficient sample of stochastic methods applied in modem finance.

2. Stochastic rate of inflation In this application we illustrate the use of Ito's lemma in determining the solu tion of the behavior of prices and real return of an asset when inflation is described by an Ito process. The analysis follows Fischer ( 1 975). Suppose that the rate of inflation is stochastic and the price level is describable by the process dP

p

= fl dt + sdz.

(2. 1 )

Stochastic: methods in economics and finance

218

The stochastic part is d z with z being a Wiener process. The drift of the process, n, is the expected rate of inflation per unit of time. It is defined by . n _ hm -

ll - 0

E,

}_ Il

{

P(t + Jz) - P(t) p (t)

}

(2.2)

,

where E, is the expectation operator conditioned on the value of P(t). The vari ance of the process per unit of time is defined by 1 s2 = hm E1 h-o h •

-

{t

P(t +

lz) - P(t)

P(t)

]}

2

-

n/1

(2.3)

A discrete time difference equation which satisfies II and s2 as defined in (2.2) and (2.3) is

P(t + h) - P(t) = 11/z + sy(t) (h) tf2 ' P(t)

(2.4)

where y(t) is a normal random variable with zero mean and unit variance which is not temporally correlated. The limit as h 0 of (2 .4) then describes a Wiener process for the variable sy(t) (h) 11 2 and eq. (2 . 1 ) can be written as --+

dP p

= fl d t + sy(t) (h) 11 2 = n d t + sdz,

where dz = y(t) (h) 1 12 . Note that (2 . 1 ) says that over a short time interval the proportionate change in the price level is normal with mean n d t and variance s2 d t. Rewrite (2.1) as dP = Plldt + Psdz and let

y (t) = P(O) exp

(2.5)

[(

n

-

;) /

s

t+s

]

(2.6)

dz

We use Ito's lemma to show that y (t) satisfies eq. (2.5). Let

F(t , z ) = P(O) exp

[ ( s; ) / n-

t +S

]

dz

Applications in finance

219

and compute oF/ot, o F/oz and o 2 F/oz2 as below:

{ ( n ; } I ] (n - ; } [ (n - ;} I ] { ( ;} I ]

oF or

P(O) ex

oF oz

P(O) exp

::{

=

P(O) ex

-

n-

•

•

t +s

dz

•

t+s

dz

s = sy(t),

•

t+s

dz

2 s = s2y (t).

= y ( t)

( n ;} , -

•

Thus, applying Ito's formula and by making the necessary substitutions we end up with dy

oF

= at d t + = y (t)

oF I o2 F (dz)2 dz + 2 oz2 oz

( ;} II -

•

d t + sy(t) dz + 4 s2 y(t) (dz)'

s2 = y(t) I ld t - y(t) 2 d t + sy(t) d z + i s2 y (t) dt

= y(t) n d t + y (t)sdz. Thus, (2.6) satisfies eq. (2.5), which is what we wanted to show. We continue with a further application. Consider the two Ito processes

dP

p = lid! + sdz

and

dQ

Q=

rd

t.

(2.7)

We usc Ito's lemma to compute the stochastic process describing the variable q = u (P, Q) = QJP. Recall that Ito's formula for this case is:

oq or

oq oP

oq oQ

dq = - d t + - d P + - d Q


220

Computing the various terms and using the multiplication rules in (4.12) in Chap ter 2, the result is: dq

= =

Finally, dq q

-

=

� dP � d Q +

- pl

Q

+

� (!�) (Ps)2 dt

(OPdt + Psdz) +

p

1

(rQdt) +

p

Q

s2 d t.

(r - n + s2 ) d t - sdz,

which describes the proportional rate having a nominal return as in (2 .7).

of change of the real return of an asset

3. The Black-Scholes option pricing model In this section we follow Black and Scholes ( 1 973) and Merton (1 973a) to de velop an option pricing model. Consider an asset, a stock option for example, denoted by A , the price of which at time t can be written as

W(t) = F(S, t),

(3.1)

where F is a twice continuously differentiable function. Here S(t) is the price of some other asset, denoted by B, for example the stock upon which the option is written. The price of B is assumed to follow the stochastic differential equation dS(t) = f(S(t), t) d t + 17 (S(t), t) dz (t),

(3.2)

S(O) = S0 given. Consider an investor who builds up a portfolio of three assets, A, B and a riskless asset denoted by C. We assume that C earns the competitive rate of re turn r(t). The nominal value of the portfolio is

P(t) = N1 (t)S(t) + N2 (t) W(t) + Q(t),

(3.3)

where N1 denotes the number of shares of B, N2 the number of shares of A, and Q is the number of dollars invested in the riskless asset C. Assume that B pays no dividends or other distributions. By Ito's lemma we compute


221

(3.4) =

where

a b

=

=

adt + bdz, F1 + Fsf + � Fss TJ2 = aw W,

(3.5)

Fs TJ = aw W.

(3.6)

Here we follow Black and Scholes (1973) and assume as a simplifying special case that f(S, t) = aS and that TJ(S, t) = aS, where a and a are constants. Next we write the dynamics for S(t) in this special case of (3.2) in percentage terms as

dS

= ad t + adz.

S

(3 7) .

Now for a portfolio strategy where N1 and N2 are adjusted slowly relative to the change in S, W and t we may assume that dN1 = dN2 = 0 and proceed to study the change in the nominal value of the portfolio, dP, as follows:

N1 (dS) + N2 (dW) + dQ = (ad t + adz)N1 S + (aw d t + ow dz)N2 W + rQdt.

dP

=

(3.8)

(3.9) Design the proportions wl and w2 so that the position is riskless for all t � 0: var,

(�) =

var, (W1 adz +

W2 owdz) = 0,

where var1 denotes variance conditioned on choose (W1 , W2 ) = ( W1 , W2 ) so that

(3.10)

(S(t), W(t), Q(t)). In other words, (3.1 1 )

Then from (3.9)

( ) dP

-

E, p = [aW1 + aw W2 + r( 1 - W1 - W2 )] d t = r(t) dt -

-

-

(3.12)


222

since the portfolio is riskless. Eqs. (3.1 I ) and (3 . 1 2) yield the famous Black Scholes-Merton equations:

W1

=-

°w

(3. 1 3)

and (3. ) 4) which simplify to -

-

(3. 1 5 )

--

Eq. (3. 1 5) says that the net rate of return per unit of risk must be the same for the two assets. For a further special case

a(S, t) = a0 ; a(S, t) = a0 ; r(t) = r0 ,

(3.16)

where a0 , a0 and r0 are constants that are independent of (S, t), by using eq. (3.15) and making the necessary substitutions from (3.5) and (3.6) we obtain the partial differential equation

� a� S2 F88 (S, t) + r0SF8 (S, t) - r0F(S, t) + F, (S, t) = 0.

(3. 1 7)

Its boundary condition is determined by the specifications of the asset. For the case of an option which can be exercised only at the expiration date T with exercise price £, the boundary condition is

F(0, r) = 0,

r=

T-t,

F(S, T) = max [O , S- E].

(3.18)

Let W(S, r; E, r0, a�) denote the solution F, subject to the boundary con dition. This solution is given by Black -Scholes (I 97 3) and Merton ( 1973a) as (3.19) where

Applications in {i11ance

1

223

(3.20)

(v) = 2 7T)l/2 (

-

i.e. the cumulative normal distribution, with

1

(3. 2 1 )

4. Consumption and portfolio rules Another useful application o f stochastic calculus techniques is found i n Merton (1971 ). Assume that in an economy all assets are of a limited liability type, that there exist continuously trading perfect markets with no transaction costs for all assets, and that prices per share P;(t) arc generated by Ito processes, i.e.

(4. 1 ) where ex; is the instantaneous conditional ex pected percentage change in price per unit of time and oi is the instantaneous conditional variance per unit of time. In the particular case where the geometric Brownian motion hypothesis is as sumed to hold for asset prices, ex.; and O; will be constants and prices will be sta tionarily and log-normally distributed. To derive the correct budget equation it is necessary to examine the discrete time formulation of the model and then to take limits to obtain the continuous time form. Consider a period model with period length h , where all income is generated by capital gains. Suppose that wealth, W(t), and P;(t) are known at the beginning of period t. The notation used is,

N;(t) = number of shares of asset i purchased during period t, i.e. between t and t + h, where h > 0� and C(t) = amount of consumption per unit of time during t. The model assumes that the individual comes into period t with wealth invested in assets so that

W(t)

=

n

L N;(t - h)P;(t). I

(4.2)


224

Note that we write N1 (t -lz) because this is the number of shares purchased for the portfolio during the period t-h to t, evaluated at current prices P1(t). The decisions about the amount of consumption for period t, C(t), and the new port folio, N1(t), are simultaneously made at known current prices: n

- C(t) h =

(4.3)

[N;(t) -N;(t- h)]P;(t).

� 1

(4.2) and (4.3) by h to eliminate backward differences and thus

Increment eqs. obtain

n

- C(t + h)h =

�

l n

= � 1

+ and

[N;(t + h) - N1 (t)] P1(t + h) [N1(t + h) - N1(t)] [P1(t + h) -P1 (t)] n

� I

[N;(t + h) -N;(t)]P1(t)

(4.4)

"

W(t + lz) = :E N;(t)P;(t + h).

(4.5)

I

Take limits in (4.4) and n

(4.5) as h � 0 to conclude that n

- C(t) d t = � dN;(t) dP1(t) + � dN1 (t)P;(t) 1 and, similarly,

W(t) =

J

n

� I

N;(t)P;(t).

Use Ito's lemma to differentiate

d W(t) =

(4.6)

"

� 1

N1(t) dP;(t) +

(4.7) W(t) to obtain n

� 1

dN1(t)P1(t) +

"

� 1

dN;(t) dP1(t).

(4.8)

The last two terms, �� dN1P1 + �;' dN1 dP1, in (4.8) are the net value of addi tions to wealth from sources other than capital gains. If dy(t) denotes the in stantaneous flow of noncapital gains., i.e. wage income, then

dy - C(t) d t =

"

� I

dN1P1 +

which yields the budget equation

"

� 1

dN1 dP1,

225

Applications in finance n

dW= � J

N;(t)dP; + dy - C(t) dt.

(4.9)

Define the new variable w;(t) = N;(t)P;(t)/W(t) and usc (4. 1 ) to obtain, dW

=

n

� 1

W; Wa;dt - Cdt + dy +

n

� I

W; Wa; dz;.

(4.10)

Merton (1971) assumes that dy = 0, i.e. all income is derived from capital gains and also that an = 0, i.e. the nth asset is risk free. Therefore , letting an r, =

d W=

7

n-J

W;(a;- r) Wdt + (rW- C) d t +

t

n- 1

W;a;Wdz;

becomes the budget constraint. The problem of choosing optimal portfolio and consumption rules for the individual who lives T years can now be formulated. It is the following max subject to

Eo [/ u(C(t), t) dt + B(W(T), T)J

W(t) � 0 dW=

n- 1

7

for all

t

E

(4.1 1 )

(4.12}

[0, T] w.p.l,

W;(a;- r) Wdt + (r W-C) dt +

Here u and B are strictly concave in C and

n-l

7

W; a;Wdz;.

(4. 1 3)

W.

To derive the optimal rules we use the technique of stochastic dynamic pro gramming presented in Chapter 2. Define (4.14) and also define

<J> (w, C, W, P, t) = u(C, t) + Y [J), where !f

=

]

:r { f w, ; w - c

a

aw

(4.15)

+

�

a

,

P

,

a

�

;


226

+

,

n

� � P.1 Ww-I o1.1. 1

I

a2 3P; 3 W

---

(4. 1 6)

Under the assumptions of the problem there exists a set of optimal rules w* and C* satisfying

0 = { max {¢ (C, w; W, P, t)} C, w } = ¢ (C*, w* ; W, P, t) for t E [0, T) .

(4 .17 )

In the usual fashion of maximization under constraint we define the Lagrangian L = ¢ + "A [ I - �� W;] and obtain the first-order conditions:

(4.18) + � 1; w II

I

0

=

ok;

P; hi,

L>-. (C*, w*) = l

-

k = I , ... , n, ,

� wr I

(4.19} (4.20}

Merton solves for C* and w* and inserts the solutions to (4.17) and (4. 1 5) to obtain a complicated partial differential equation. See Merton (197 1 , p. 383). If this partial differential equation is solved for J, then its solution, after appro priate substitutions, could yield the optimal consumption C* and portfolio rules w*. 5. Hyperbolic absolute risk-aversion functions

In this application we specialize the analysis of Merton ( 1971) as presented in the previous section for the class of hyperbolic absolute risk aversion (HARA} utility functions of the form u (C, t) = e - P t v(C),

where

( 1 --y

I - -y

v(C) =


{3C

--

'Y

+ 11

Note that the absolute risk (v" jv') is given by v

"

A (C) = - , = v

)

'Y

(5. 1 )

•

aversion

----

c

1 --y

227

11 + -

denoted by A (C) and defined to be A (C) =

> 0,

(5.2)

{3

provided -y =1- I ; (3 > 0; ({3C/(l - -y)) + 11 > 0; and TJ = 1 if -y = - oo. This family of utility functions is rich because by suitable adjustments of the parameters one can have a utility function with absolute, or relative, risk aver sion that is increasing, decreasing or constant. Without loss of generality assume that there are two assets, one risk-free and one risky. The return of the risk-free asset is r and the price of risky asset is log normally distributed satisfying dP - = a d t + odz. p

(5 .3)

The optimality equation for this special case is

0=

{1 - -r? 'Y

� +

e- p r 11

[ ]

eP'Jw 'Y /('Y- 1 >

{3

J

0 - 'Y) fi + r W lw tJ

1

2

lw

-

ww

+ 1,

(a -r?

(5.4)

2 a o2

subject to J ( W, T) = 0. For simplicity we assume that the individual has a zero bequest function. A solution of the partial differential equation in (5 4) is given by Merton ( 1 973c, p. 2 1 3) .

[:

+

; { 1 -cxp [-r(T-1)]�

')'

,

p -- f> v

(5.5)


228

S

-r)2/2S a1 (W, t) -t))]/{3r.

r

S

S

where = 1 -')', v = + (a and is assumed positive. If < 0, and � in (5 .5) will hold only when 0 � therefore "Y > 1 , the solution J ("Y- 1 ) 71 [ 1 -exp(-r (T The explicit solutions for optimal consumption and portfolio rules are given below:

C* (t)

=

6

and

w* (t)

-j;{I - { �

[p-')'v) [w(t) +

= a-r2 + Sa

ex

1

{ 1 -exp

p

71(a-r)

W(t) {3ra2

')'V

[r(t- T))

(t -

T�}

{ 1 -exp

�

-

S 71

{3

-

[r(t- T)]}.

W(t)

(5.6)

(5.7 )

The basic observation obtained from the above two equations is that the de mand functions are linear in wealth. It can be shown that the HARA family is the only class of concave utility functions which imply linear solutions.

6. Portfolio jump processes We follow Merton ( 1 97 1 ) to discuss an application of the maximum principle for jump processes in a portfolio problem. Consider a two-asset case with a com mon stock whose price is log-normally distributed and a risky bond which pays an instantaneous rate of interest r when not in default, but in the event of de fault the price becomes zero. The process which generates the bond's price is as sumed to be given by d P = rPd

t

-

Pd q ,

(6. 1 )

q(t)

with being an independent Poisson process. The new budget equation that replaces (4. 1 3) is d

W = {w W(a-r) + r W - C} dt + waWdz - (l - w) Wdq.

(6.2)

Note that (6.2) is an example of mixed Wiener and Poisson dynamics. An appli cation of the generalized Ito formula and the maximum principle for jump pro cesses in section 1 2 of Chapter 2 yields the optimality equation


0

=

229

u(C*, t} + J, (W, t) + A. [J(w* W, t) - J (W, t)] + lw (W, t)[(w * (a-r)+r) W - C*] + � lww (W,t)a2 w*2 W2 , (6.3)

where the optimal assumption implicit equations

C* and portfolio w * rules are determined by the

0

= uc(C*, t) - Jw (W, t)

(6.4)

0

= A.Jw (w * W, t) + lw (W, t) (a-r) + lww (W, t) a2 w * W.

(6.5)

and

There is one additional novelty in this Merton problem that is not present in the pure Brownian motion case. That is, for a HARA utility function you must not only conjecture a solution from J(W, t) = g(t) wa and solve for the exponent by the method of equating exponents, you must also conjecture a form for the demand function w W = d W + e, where the term d is dependent of wealth. This is a natural conjecture to make for the form of the demand function for the risky asset given the separation theorem which says that for utility functions of the hyperbolic absolute risk-aversion class that the proportion of wealth held in the risky asset should be independent of the investor's wealth level and independent of his age . It is natural to conjecture the same sort of separation theorem for the Poisson case as well. At any rate, the philosophy is to try it and see if it works. Furthermore, conjecture that the term e = 0. This is natural because if wealth is zero there would be no demand for the risky asset. Next equate exponents on wealth in the partial differential equation for the state valuation function, J, to solve for the unknown exponent on wealth. I t will turn out to be the same exponent as that on consumption in the utility function. Cancel all terms involving wealth off the partial differential equation (6.3) for the state valuation function. That will give an ordinary differential equation for the unknown function g(t). Furthermore , examine the necessary condition (6.5) to determine the unknown proportion d. Cancel off all terms involving wealth and all terms involving the unknown function g(t} in the necessary condition (6.5} to get the relationship

d=

A. a-r + o2 (1 --y) a2 ( 1 --y)

tJY- J

·

(6.6)

This last is the same as Merton's (80') in Merton ( 1 97 1 , p. 397) with w = d. Thus, conjecturing a linear demand function for the risky asset worked.


230

The moral of this exercise is that for the Poisson case it was necessary to con jecture a form of the demand function for the risky asset in order to obtain a closed form solution. But the conjecture of the appropriate form for the demand function was motivated by the form of the demand function that was derived in the pure Brownian motion case. In other words, in the pure Brownian motion case when the utility function was hyperbolic absolute risk averse then the demand function for the risky asset was linear and the proportion of the investor's port folio held in the risky asset turned out to be independent of the investor's wealth level and of his or her age. This independence is called the separation theorem. The name separation theorem derives from the fact that the consumption deci sion and the portfolio diversification decision turn out to be determined inde pendently of each other in this particular case. Merton's (

1 97 1 ) paper contains several other examples of closed form solution

determinations for Poisson processes. Furthermore, it contains closed fo rm solu tion determinations for more general processes as well.

7.

The demand

for index bonds

Consider, as Fischer

( 1975)

does, a household with three assets in its portfolio:

a real bond, a risky asset and a nominal bond. Assume that the portfolio can be adjusted instantaneously and costlessly . We also assume that the rate of inflation is stochastically describable by the process dP - = fl d t + sdz.

(7 . 1 )

p

The real bond pays a real return of r1 and a nominal return o f r 1 plus the realized

rate of inflation. Note that d

Q1 -Q.

dP = r1 d t + - = (r1 + fl ) d t + s dz = R 1 d t + s 1 dz 1

7 )

( .2

p

is the equation describing the nominal return on the "index bond. The nominal return on equity is d

2

Q Q, = R2 d t + s2 dz2 , where R2 is the expected nominal return on equity per unit of time and

(7 .3) s�

is the

variance of the nominal return per unit time. Using the results of application

2,


231

if we let d Q3 /Q3 = R 3 d t describe the deterministic nominal return of the nomi nal bond, then the real return on the nominal bond is (7.4) Let w 1 , w2 and w3 be the proportions of the portfolio held in real bonds, equity and nominal bonds, respectively. Obviously, w1 w2 w3 I . The flow budget constraint, giving the change in nominal wealth, W, is similar to (4.1 0): 3 2 d W = .r1 W; R; Wdt - PCdt + .r1 w; s; Wd z; , (7.5) +

+

=

where C is the rate of consumption. Uncertainty about the change in nominal wealth arises from holdings of real bonds and equity. Since w3 = 1 - w1 - w2 , may rewrite eq. (7 .5) as 2 2 d W = .r w;(R; - R3) Wd t (R3 W - PC) d t + .r W; S; Wdz; (7.6) We are now in a position to formulate the household's choice problem: (7.7) max E0 J u [C(t), t] d t {C,w;} subject to (7 .6) and we

+

I

.

I

00

o

W(O) = W0 ,

where u is a strictly concave utility function in C, and E0 is the expectation con ditional on P(O). The first-order necessary conditions of optimality are: 0 = uc(C, t) - Plw, (7.8) (7.9) (7.10) where, as before, J(W, P, t) = max E, J u(C, s) ds {C,w,. } 00

t


232

and p is the instantaneous coefficient of correlation between the Wiener processes dz 1 and dz2 and I p I < 1 . It is now possible to solve for asset demands from the two equations (7 .9) and (7 .1 0) and the fact that L W; = 1 , to obtain

(7 . I I ) (7 .12)

From (7. 1 1 )-(7.13) Fischer ( 1 975) studies the complete properties of the demand functions for the three assets. In particular consider the demand func tion for index bonds in (7 . 1 1 ) Observe that the coefficient -JwlJww W is the inverse of the degree of relative risk aversion of the household. If we make the simplifying assumption that p = 0 in (7 .1 1 ), then (7 . l l ) says that the demand for index bonds depends on (i) the degree of relative risk aversion, (ii) the dif ference between expected nominal returns on the two types of bonds, R 1 R3 , and (iii) the variance of inflation , s � . But how about the term lwpPflww W in (7 .1 1 )? This term can be related to the degree of relative risk ave·rsion as follows: .

-

(7 .J 4)

To obtain (7 .14) differentiate (7 .8) first with respect to P to get

(7 . 1 5 ) and with respect to

W to get

(7 . 1 6) Finally, note that since consumption is a function of real wealth

ac aP

w ac

= - P aw ·

(7 .17)


233

Combining (7 .1 5)-(7 .17) we get (7 .14). There are many other valuable insights that this analysis uncovers. We mention just one more. Consider the yield differentials in terms of real returns when 1 = 0, i.e. when the household has no index bonds in its portfolio, given by w

(7 .18) w2

Suppose that = I , i.e. the net quantities of real and nominal bonds are zero. If there is positive covariance between equity returns and inflation then from (7 .18) we obtain that r1 -r3 > 0, which means that index bonds will have to pay a higher return than the expected real yield on nominal bonds. In other words, if equity is a hedge against inflation, then index bonds do not command a pre mium over nominal bonds. Conversely, if equity is not a hedge against inflation then index bonds will command a premium over nominal bonds. 8. Term structure in an efficient market

Methods of stochastic calculus similar to the ones used by Black and Scholes (1973) and Merton (197 1 ), which were presented in earlier sections, have been used by Vasicek (1 977) to give an explicit characterization of the term structure of interest rates in an efficient market. Following Vasicek (1977) we describe this model below. Let P(t, s) denote the price at time t of a discount bond maturing at time s, with t � s. The bond is assumed to have a maturity value, P(s, s), of one unit, i.e. (8.1)

P(s, s) = 1 .

The yield to maturity, R(t, T), is the internal rate of return at time t on a bond with maturity date s = t + T, given by R(t, D = -

1 logP(t, t + T), T> O. T

(8.2)

From (8.2) the rates R(t, T) considered as a function of Twill be referred to as the term structure at time t. We use (8.2) to define the spot rate as the instan taneous borrowing and lending rate, r(t), given by r(t) = R(t, 0) = lim R(t, n. r-o

(8.3)

234


It is assumed that r(t) is a continuous function of time described by a stochastic differential equation of the form

dr =f(r, t)dt + p(r, t)dz,

(8.4)

where, as usual, z(t) is a Wiener process with unit variance. It is assumed that the price of a discount bond, P(t, s), is determined by the assessment, at time t, of the development of the spot rate process (8.4) over the term of the bond, and thus we write

P(t, s) = P(t . s, r(t)).

(8.5)

Eq. (8.5) shows that the spot rate is the only state variable for the whole term structure, which implies that the instantaneous returns on bonds of different maturities are perfectly correlated. Finally, we assume that there are no trans actions costs, information is available to all investors simultaneously, and that investors act rationally; that is to say, we assume that the market is efficient. This last assumption implies that no profitable riskless arbitrage is possible. From eqs. (8.4) and (8.5) by using Ito's lemma we obtain the stochastic dif ferential equation

dP = Pp(t, s, r(t)) dt - Pa(t,s, r(t)} dz, which describes the bond price changes. In (8.6} the functions J.l and as follows:

p(t' s, r) -

1 P(t, s, r)

a are defined (8.7)

a(t. s,r) = - --- p 0 P(t, s , r). P(t,s,r) r 1

(8 .6)

a

(8.8)

Consider now the quantity q(t, r(t)) given by

q (t, r) =

JJ(t, s, r) -r , r

Stochastic methods in economics and finance (Advanced Textbooks in Economics)

Stochastic methods in economics and finance

Economics - Stochastic Modeling in Economics and Finance

Stochastic Modeling in Economics and Finance

Stochastic methods in finance

Numerical Methods in Economics

Differential Equations, Stability and Chaos in Dynamic Economics (Advanced Textbooks in Economics)

Stochastic Modeling in Economics and Finance (Applied Optimization)

Stochastic Modeling in Economics and Finance (Applied Optimization)

Stochastic Modeling in Economics and Finance (Applied Optimization)

Understanding Macroeconomics Theory (Advanced Texts in Economics and Finance)

Mathematical Economics and Finance

History of Economic Theory (Advanced Textbooks in Economics)

Mathematical Finance (Routledge Advanced Texts in Economics and Financea)

Lectures on Microeconomic Theory (Advanced Textbooks in Economics)

Numerical Methods in Finance and Economics: A MATLAB-Based Introduction

Soft Computing in Economics and Finance

Numerical Methods in Finance and Economics A MATLAB based Introduction

Numerical Methods in Finance and Economics: A MATLAB-Based Introduction

Handbook of Empirical Economics and Finance (STATISTICS: Textbooks and Monographs)

Stochastic Dominance and Applications to Finance, Risk and Economics

Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management (Advanced Textbooks in Economics)

Applied computational economics and finance

Applied Computational Economics and Finance

Applied Computational Economics and Finance

Applied Computational Economics and Finance

Applied Computational Economics and Finance

Optimization in Economics and Finance: Some Advances in Non-Linear, Dynamic, Multi-Criteria and Stochastic Models (Dynamic Modeling and Econometrics in Economics and Finance)

Optimization in Economics and Finance: Some Advances in Non-Linear, Dynamic, Multi-Criteria and Stochastic Models (Dynamic Modeling and Econometrics in Economics and Finance)

Stochastic methods in economics and finance (Advanced Textbooks in Economics)