The fundamental principle of intertemporal optimization

Economics Letters 0165-1765/93/$06.00

41 (1993) 273-280 0 1993 Elsevier

273 Science

Publishers

B.V. All rights

reserved

The fundamental principle of intertemporal optimization Consumer Hiroaki

behavior under recursive preferences

Hayakawa*

Department of Economics.

Suezo

University, 56-l

Kita-machi,

Toji-in, Kita-ku, Kyoto 603, Japan

Ishizawa

Department of Economics, Received Accepted

Ritsumeikan

Nagoya Gakuin University, 1350 Kamishinano-cho,

Seto-Shi, Aichi-Ken

480-12, Japan

1 February 1993 1 March 1993

Abstract This paper shows that a simple principle, expressible by a single equation, runs through intertemporal consumer behavior: the effect of any given variations of consumption on the utility must be balanced against the effect of such variations on subsequent stocks of assets. The former is measured by the directional marginal utility and the latter by the directional marginal value of assets; the implicit utility price of assets, necessary to convert the latter into utility units, is then determined in reference to the maximized value of the utility functional. It is this equation that is implied by Pontryagin’s maximum principle.

1. Introduction Pontryagin’s maximum principle is well known [Pontryagin et al. (1962), Arrow and Kurz (1970)]. Yet, the precise economic meaning of this principle has not been made explicit partly because the marginal quantities required for economic analysis - such as marginal utility, the marginal rate of substitution, and the marginal rate of transformation - are difficult to define in continuous time modeling of intertemporal behavior and partly because an effective way of characterizing the intertemporal optimality itself has not been proposed. In this paper we show that a simple principle runs through intertemporally motivated consumer behavior, which can be summarized in a single equation in two well-defined marginal quantities-directional marginal utility and directional marginal value of assets - that are linked by the implicit price of assets expressed in utility units. We then show that this equation is equivalent to what Pontryagin’s maximum principle entails, thereby clarifying the economic meaning of this principle as applied to intertemporal consumer problems. Here we deal with an infinitely-lived consumer possessed with recursive time preferences of the Uzawa-Epstein kind. * Correspondence

to: Hiroaki

Hayakawa,

4-5

Satsukigaoka,

Kani-Shi,

Gifu-Ken

509-02,

Japan.

H. Hayakawa and S. Ishizawa I Economics

274

2. The fundamental

equation

of intertemporal

Letters 41 (1993) 273-280

optimization

Consider an infinitely-lived consumer who tries to optimize intertemporally in continuous time. His preferences vary endogenously after the Uzawa-Epstein kind [Uzawa (1968) and Epstein (1987a, b)], and his asset accumulation is constrained by a dynamic budget equation governing how the stock of his assets changes over time as a result of his consumption and saving decisions. There are are n goods, xi, x2,. . . , x,. The consumption space, X, therefore consists of n-tuples of their time paths (i.e. X consists of n-vectors of piecewise continuous functions, x = (x1, x2,. . .) xn), with each function being a mapping from the non-negative real line representing time into the non-negative real line representing the instantaneous rate of consumption, where the image of function xi at time t is denoted x’(t)). While each element of X is an n-tuple of paths of II individual goods, we shall call it a path. In what follows, a dot over a variable denotes its time rate of change, and a subscript to a function denotes its partial derivative (‘i’ in the case of good i and ‘a’ in the case assets). Let a consumer have his preferences defined on this consumption space, and suppose that his preferences are specified after the Uzawa-Epstein kind as follows: u(x) = IX 44t))

exp[ - 1)’ G(7))

0

dr ] dt

,

where u( . ) and 6(. ) are real-valued and twice continuously differentiable functions of x(t). Suppose that real assets are represented by a single variable, a(t), and that its dynamics governed by the following transition equation: b = f@(r) 3x(t)> 3

(1)

are

(2)

where f( . ) is held to be twice continuously differentiable with respect to both a(t) and x(t); x(t) is a vector of control variables at time t, which is identified here with a vector of II goods. 1 The problem we consider, therefore, is formally stated as z

max X(l)

I 0

G(t))

exp [ - IO’G(T))

d7] dt

r=o

subject

to 4 =f(a(t),

with initial

x(t))

stock of assets,

u(O) given

at a,, .

The decision is made at time 0. At every point in time a consumer can choose between consuming now for immediate gratification and saving for postponed gratification. If his choice is optimal, the marginal utility of consumption should match the marginal utility of saving. To explicate this rule, consider small variations of n goods in arbitrary directions in a neighborhood of time T along the optimal path x”. Such variations affect the value of the utility functional. This effect can be measured by computing the rate at which the value of the utility functional changes as the variations are made arbitrarily small. Technically, this rate is given by a directional derivative, which we call the directional marginal utility. The same variations also affect subsequent stocks of assets. Hence, computing the rate at which any subsequent stock of assets changes as the variations are made arbitrarily small, we get an analogous quantity, to be called the ’ When multiple assets are involved, the stock of real assets, a, here should be interpreted as the aggregation of individual assets, and the consumption path x should be interpreted as containing these assets. As an example, see Blanchard and Fischer (1989) for their treatment of Sidrauski’s (1967) model involving two assets, money and capital.

H. Hayakawa

and S. Ishizawa

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Letters

41 (1993) 273-280

275

directional marginal value of assets. This directional quantity measures the rate at which the time-t stock of assets changes due to arbitrarily small variations of II goods around time T. And, this derivative is defined for any point in time t after T. To relate these two directional derivatives, the directional marginal value of assets must be translated into its utility equivalent. This requires an implicit price (or shadow price) of assets. The price at time t is naturally determined by computing the extent to which the future life-time utility changes if the stock of assets at time t is increased by one unit and by discounting this value back to time 0. The optimization rule then amounts to requiring that the directional marginal utility equal the implicit utility value of the directional marginal value of assets. With this understanding, we now derive it explicitly. At an arbitrary point in time T, a consumer has a certain stock of assets, aT. Starting with this stock of assets, consider maximizing 8(x(~)) d7

1

dt

(3)

subject to constraint (2), where r.~ is the right-hand tail of x after time T, and denote the optimal paths of n goods and the stock of assets by x* and a*. Let V(T; aT) be the maximized value of this utility functional, i.e.

(4) r=T

subject

to b(t) = f(a(t)),

x(t)) for every

t2 T

and a(T) = aT given. If the time horizon the sum of two terms, V(T; aT):

is segmented into two intervals, [0, T] and [T, cc), U(x*) can be written as one the integration of (1) from 0 to T and the other the discounted value of

U(x*) = V(0; a,,) = i,‘u(x*(7))D(O

, 7; x*) d7 + D(0, T; x*)V(T;

a*(T))

,

(5)

where D(0, t;x*) iexp

[ - 1: S+*(T))

.

dr]

Suppose that the optimal path is altered in a small interval of time [T - As, T) in the direction of h.2 Such an alteration affects U(x*) in two separate ways. First, it affects U(x*) directly

* Around

the optimal

path I*, consider

h(t) = [h’(t), h’(t), for every

t in the non-negative

a difference

path h given by

, h”(t)] real line. And,

define

x(t) + h(t)Ac , if t is in [T-As, f(f) = otherwise, [ x(t) >

a modified

path f as

T),

where AC is a small real number and As is a small positive real number. Altering the optimal time [T - As, T) in the direction of h amounts to considering such a modified path.

path in a small interval

of

H. Hayakawa and S. Ishizawa I Economics

276

Letters 41 (1993) 273-280

through changes in consumption around time T, by the amount of the directional marginal utility, denoted SU(T; x*, h). 3 We call this effect the direct consumption effect at time T. Second, the subsequent stock of assets for t > T is affected simultaneously. This effect is represented by the directional marginal value of time-t assets denoted 8a(t, T; x*, h). 4 This derivative gives an extra amount of assets which the consumer can expect to accumulate (or decumulate) by time t provided he returns to the optimal consumption path, x*, after time t. If he allocates this extra amount of assets to consumption at time t, U(x*) can be raised in the amount of 6U: 6lJ = D(0, t; x*)

dV(t; a*(t) I

da(t)

iSa(t, T; x*, h)

= v(t)6a(t, T; x*, h) , where

dV(t; a*(t)

v(t) i D(0, t; x*)

da(t)

1’

v(t) is the implicit price of assets at time t in utility units; it corresponds to the marginal utility of income in static theory. We call v(t) the marginal utility of assets at time t. Since 6U represents the indirect effect on the utility functional through asset accumulation, we call it the indirect consumption effect at time t. If x* is an optimal consumption path, an alteration around time T along x* should not change requires that the direct consumption effect at time T, the value of U(x*). This, therefore, 6U(T; x*, h), be balanced against the indirect consumption effect for any time t; that is, 0 = 6U(T; x, h) + v(t)Sa(t, T; x*, h)

for every T and every t 2 T.

(7)

As shown below, this equation exhausts all optimality information, and we call it the fundamental equation of intertemporal optimization. To explicate the meaning of this condition, first observe that the directional marginal utility can

’ Taking a modified path i as defined in footnote 2, we define direction h, 6U(T; I*, h), by the following limit:

the directional

marginal

utility

at time T along

path x* in

U(i) - U(X*) SU(T, h; x*) A

lim (AS.dr,-(,,. 0,

AsAc

provided that this limit exists. See Hayakawa 4 For the optimal path x* (which is continuous), (2); it satisfies IL*(t) = a,, + ,,’ A+*(r),

i

x*(r)) dr

’ and lshizawa (1992) for details. we assume that there exists a unique

$*(t) to differential

equation

,

where a,, is the initial stock of assets (at time 0) which is given. Consider we assume that a continuous function 4(t) satisfies

4(t)= a,,+

solution

a modified

path i as defined

in footnote

2. For f,

s,,’

fC&CtL +I) dt

This function can be viewed as a solution to differential equation (2) with right-hand derivatives taken into account at points of discontinuity of P. Then the directional marginal value of time-t assets in direction h along path x*, 6a(t, T, h; x), is defined by the following limit: Wr, provided

lim T, h; x) Z (,r,<)_(,j.

that this limit exists.

0)

44) - ti*(t) AsAc

See Hayakawa

’

and Ishizawa

(1992)

for details.

H. Hayakawa

and S. Ishizawa

be expressed uniquely as an inner product MU(T; x*), ’ and a vector of time-T values 6U(T; x*, h) = (MWT; where

x*), h(T))

MU(T;

x*) A [MU’(T;

MU’(T;

x*) &(x*(T)

I Economics

Letters 41 (1993) 273-280

277

of an n-vector of the marginal of variations, h(T), i.e.

utilities

T,

at time

,

(8)

x*), MU2(T;

x*), . . . , MU”(T;

- G;(x*(T))U(

+*)]D(O,

x*)]

T; x*) ,

i = 1,2,. . . , n .

Likewise, the directional marginal value of assets can be expressed uniquely as an inner product of an n-vector of the marginal values of assets with respect to n individual goods, MA(t, T; x), and a vector of variations, h(T), i.e. &(t,

T; x*, h) = (MA@, T; x*), h(T))

where

MA@, T;

x*)

5

(MA’(t,

MA’@, T; x*) LE(t,

T;

(9) MA’@,

x*),

T; x*)h(a*(t),

T; x*), . . . , MA”@, T;

x*(t))

E(t, T; x*) bexp [ 1; f,(a*(r),x*(r)

,

dr]

See Hayakawa and Ishizawa (1992) for expressions (8) and (9) We now show that the fundamental equation (7) produces two familiar First, letting t = T in (7), we obtain 0 = 6U(T; x*, h) + v(T)Sa(T, = (MU(T;x*),

h(T))

optimality

T; x*, h)

+ v(T)(MA(T,

x*) + v(T)L(u*(T),

x*(T))

conditions.

(10) T;x*)),

h(T))

,

where MA(T, T; x*) = V,f(a*(T), x(T)), where V,f(u*(T), x(T)) is a f(u*(t), x(t)) with respect to x at time T. Since the vector h(T) is arbitrary, and only if MU’(T;

x*))

= 0,

i = 1,2,.

. . , n, for every

gradient vector of this equality holds if

T( z-0).

(11)

This is a snap-shot condition satisfied along the optimal consumption-asset path (x*, a*) at any moment of time T. It implies that the direct gains in utilities from a small variational increase in consumption of good i around time T, MU’(T; x*), must be offset by the indirect negative effect caused thereby in the form of an asset decumulation, f;(u*(T), x*(T)), converted to utility units by the marginal utility of assets at time T, v(T). The fundamental equation (7) also determines the dynamics of the marginal utility of assets, v(t). To see it, equate (7) and (10) ’ to obtain v(T)Su(T, from which,

together

T; x*, h) = v(t)ih(t,

T; x*, h)

for every t 2 T ,

with (9), follows

’ These marginal utilities have also been considered by Wan (1970) and Epstein (1987a, ’ Since the first term on the right of Eq. (7) 1sindependent of time t, it holds that v(t’)da(t’, for two points

(12)

T; x*, h) = v(t)Sa(t,

in time (ZT),

t and

b).

T; x*, h)

t’. Equation

(12) is obtained

from

this by setting

t’ = T.

278

H. Hayakawa

and S. Ishizawa

v(T) = v(t) exp [ 1; f,@*(r),

I Economics

Letters 41 (1993) 273-280

for every t 2 T

x*(r) dr]

(13)

T set at 0, this demonstrates that the product of the marginal utility of time t-asset, v(t) and at level v(0); that is, the marginal utility of assets exp[& _&r*(r), x*(r) d 71 remains constant declines exponentially at the rate of f,(a*(t), x*(t)):

With

fi(t> -4) = -f,@*(t), x*(t)) , While converse

with v(0) =

wo; %I> aa from

conditions (11) and (14) necessarily also holds. ’ The optimality condition

follow from (7) therefore

(14). We now show that conditions (11) and (14) are exactly entails. First, let y(t) be a variable that satisfies Y(l) = +qx(t))y(t)

,

(6)

.

0

y(O)

the fundamental equation is equivalent to conditions

what Pontryagin’s

(7), the (11) and

maximum

principle

(15)

= 1

Since the equation yields the solution consumer can be re-expressed as

y(t) = exp[ - Ji 8(x(~)) dr], the optimization

problem

of the

32

max U(x) = subject

I

+(r))Y(t)

dt

to Y(:) = -6(x(t))y(t) 4t)

= f(u(t),

,

x(t)) 9

y(0) = 1 and u(O) = a, We assume that this problem has a solution has a solution, denoted x*. Given path x*, the differential equations for the state variables (y(t), u(t)), together with their initial values, determine their time paths, denoted (y*(t), u*(t)). With the Hamiltonian specified as H(x(t),

a(t), y(t), h(t), v(t)) = +(t))Y(t)

Pontryagin’s maximum conditions for all t 20: [4x(t))

-

principle

states

triplet

(x*(t),

+ v(t)f(n(t)>

y*(t), u*(t))

x(t)) >

satisfies

the

(16) following

WFW))ly(t) + W.fMt>, 4)) = 0 , i = 1, . . . , n ,

Jw = -[4x(t)> - ww))] fi(t>= - v(t)f,@(t>,x(t)> where A(t) and v(t) respectively. Let U(,x), defined

that

- A(t)+(t))y(t)

are

co-state

,

(18) (19)

2

variables

associated

in (3), be differentiated

’ The converse can be seen as follows. condition (ll), imply (7).

(17)

First, condition

with

with respect

(14) implies

the

state

variables

y(t)

and

u(t),

to t to obtain

(13). Then,

conditions

(13), (8), and (9), together

with

H. Hayakawa and S. lshizawa I Economics

dKx)ldf

= -[4x(t)) - K4W(t))l

which shows that U(,x) satisfies solution to that equation, ’ i.e.

the differential

>

(20)

equation

(18). We assume

that

A@) = U(,X) . With

(21) substituted [u;(x(t))

279

Letters 41 (1993) 273-280

this is a unique

(21) into (17),

conditions

- u(tx)si(x(t))lY(t)

(17)-(19)

can be rewritten

+ v(t)h(a(t)~ x(t)) = O 9

fi(t) = - +)f,@(% x(t)) .

as

(22) (23)

Since y(t) = D(0, t; x*) along the optimal path x*, condition (22) is identical to condition (11) [see (S)], whereas condition (23) is nothing but condition (14). Hence, by virtue of the equivalence established above, conditions (22) and (23) are equivalent to the fundamental equation (7). Thus, what Pontryagin’s maximum principle entails for the optimization problem (A) is identical to our fundamental equation of intertemporal optimization; the principle amounts to stating that the directional marginal utility must equal the implicit utility value of the directional marginal value of time-t assets, where the implicit utility price of time-t assets declines at the rate of f,(a(t), x(t)).

3. Conclusion It has been demonstrated that a simple principle runs through intertemporal consumer behavior. The effect of any given variations of consumption on the utility of a decision-maker must be balanced against the effect of such variations on subsequent stocks of assets. With these effects formally captured by the directional marginal utility and the directional marginal value of assets, a rational consumer plans in such a way that the former is equated with the implicit utility value of the latter while the implicit utility price of assets is determined with reference to the maximized value of the utility functional. The optimization principle, therefore, can be expressed by a single equation involving these two directional marginal quantities and the implicit price of assets. It is precisely this equation that is implied by Pontryagin’s maximum principle when applied to consumer behavior.

References Arrow, K.J. and M. Kurz, 1970, Public investment, the rate of return, and optimal fiscal policy (Johns Hopkins Press, Baltimore, MD). Blanchard, O.J. and S. Fischer, 1989, Lectures on macroeconomics (The MIT Press, Cambridge, MA). Epstein, L.G., 1987a, A simple dynamic general equilibrium model, Journal of Economic Theory 41, 68-95. Epstein, L.G., 1987b, The global stability of efficient intertemporal allocations, Econometrica 55, 329-355. Hayakawa, H. and S. Ishizawa, 1992, Foundation of intertemporal consumer choice, Unpublished manuscript. Pontryagin, L.S., V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchencko, 1962, The mathematical theory of optimal processes, translated by K.N. Trirogoff (Interscience, New York). Sidrauski, M., 1967, Rational choice and patterns of growth in a monetary economy, American Economic Review 57, 534-544. ’ This is equivalent

to assuming

the transversality

condition:

lim,_,

h(t)y(t) = 0

280

H. Hayakawa

and S. lshizawa

I Economics

Letters 41 (1993) 273-280

Uzawa, H., 1968, Time preferences, the consumption function, and optimum asset holdings, in: J.N. Wolfe, ed., Value, capital, and growth: Papers in honour of Sir John Hicks (University of Edinburgh Press, Edinburgh) 485-504. Wan, H.Y., Jr., 1970, Optimal saving programs under intertemporally dependent preferences, International Economic Review 11, 521-547.