JOURNAL OF ECONOMIC THEORY 46, 409413 ( 1988)

On a Criterion of Infinite-Horizon Efficiency

ABHIJIT SENGUPTA *

Department of Economics, University of We.stern Ontario, London, Ontario N6A 5C-7. Canada

Received December 19, 1986; revised September 14. 1987

We prove a property of the class of programs covered by Cass’ pioneering criterion of inefficiency of infinite-horizon programs, A notion that we call gross inefficiency is introduced. It is, in general, a much stronger property than inefficiency. We then show that a program in the class studied by Cass is inefficient if and only if it is grossly inefficient. Journal of Economic Literafure Classification Numbers: 022. 1 I 1. ,( 1988 Academic Press. lnc

1. INTRODUCTION

It was in 1953 that Malinvaud [2] posed the problem of characterizing efficiency in an infinite-horizon model, and observed that the standard criterion of efficiency for models with a finite horizon-namely, profit maximization-is an inadequate guarantee of efficiency when the horizon is infinite. Malinvaud also provided, in a rather general setting, a sufficient condition. It soon became apparent, however, that the condition proposed by Malinvaud was overly strong and not a necessary condition for efficiency. It was not until 1972 that a breakthrough was made by Cass [ 1 ] when he obtained a complete characterization of efficiency for a particular class of programs.

The purpose of this note is to bring out a property of the class of programs considered by Cass. We introduce a notion that we call gross inefficiency which is based on the concept of inefficiency of finite-horizon programs and is, in general, a much stronger property than inefficiency of infinite-horizon programs. We then show that a program in the class studied by Cass is inefficient if and only if it is grossly inefficient.

* I am deeply indebted to Amitava Bose for introducing me to the subject of this note and for many illuminating discussions. I have also benetited from comments and suggestions by Bob Rosenthal and an anonymous referee. Financial support from the U.S. National Science Foundation under Grant SES 83 17924 is gratefully acknowledged.

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CopyrIght ,,‘I 1988 by Academic Press, Inc. All rights of reproductmn in any form reserved

410 ABHIJIT SENGUPTA

2. CASS' CRITERION

The iiterature analyzes a one-good model with a technology given by a production function f: R + + R + , which is increasing, with f(0) = 0. The production possibilities consist of input k and output y =f(k) for k 2 0. The initial stock of input, k > 0, is assumed to be historically given.

A program is a sequence (k, I’, c) with (k,, y,, , , c,+ I) satisfying

k, = k; O<k,d y,, tb 1; f(k,)=y,+l, ta0;

c,+1= y r+1 -kt+l, t30.

(We interpret c, as consumption associated with the program.) A program (k’, y’, c’) dominates a program (k, y, c) if c: 2 c,, for all

t>,l, and ci>c, for some t. A program is inef$cient if it is dominated by another program; otherwise

it is efficient. Associated with any program is a price sequence (p) which maximizes

the intertemporal profits of the program. It is called a competitive price sequence and is given by

j-Jo= 1, P,+ ,/P( = l/f ‘(k,).

The class of programs considered by Cass [l] is assumed to satisfy:

(C.1) There exist real numbers k and k such that

O<k,<k,dE<oo, t 2 0.

(C.2) f is twice continuously differentiable and there exists a real number m such that

0 < -f “(k,) <m < 00, t 3 0.

We will sometimes refer to a program (k, y, c) which satisfies (C.l) and (C.2) as a Cass program.

THEOREM A (Cass). Under (C.l) and (C.2), a program (k, y, c) is inefficient if and only if

,g (VP,) < 00.

INFINITE-HORIZON EFFICIENCY 411

3. A PROPERTY OF CASS PROGRAMS

In the simplest terms, a program is inefficient if it is possible to realize a positive “gain” at zero “cost.” Any characterization of inefficiency therefore must identify appropriate representations of the notions of “gain” and “Cost.” The representation of a gain is obvious-namely, a consumption bonus. Now, consider a given program (k, y, c) and a contemplated consumption bonus in some period (taken, without loss of generality, to be period 1) of E,, 0 < E, < k,. Th$ sequence of minimum decrements in k, required to maintain the consumption streams generated by (k, y, c) in future periods is given by

& r+l =f(k,)-f(k,-&,I, fb 1. (1)

Proposition 0 characterizes what constitutes a gain at no cost in the con- text of an infinite horizon.

PROPOSITION 0. A program (k, y, c) is inefficient if and only if0 < E, <k, for all t > 1.

Proof: This follows from trivial modifications of the proof in Cass [l, pp. 204-205-j. 1

What the proposition says is that in the context of an infinite-horizon model no cost occurs so long as it is not necessary to reduce k, to non- positive levels. Thus if a consumption bonus today can be sustained at all, a gain is made at zero cost.

Note that while the concept of a “gain” is identical in finite and intinite- horizon models, there is an essential difference in the concept of “cost.” In models with a finite horizon, with an end-of-the-horizon terminal capital requirement (say, kT), a consumption bonus today that entails reducing k, would not constitute a “gain at zero cost.” In other words, a reduction in future capital (E, > 0) always corresponds to a cost in finite horizon; in infinite horizon a cost occurs only when E, 2 k, for some t in the future.

The finite-horizon analogue of “a gain at zero cost” in the context of an infinite horizon would be the notion that a current consumption gain is achievable without any eventual loss of future capital, i.e., E, -+ 0 as t + co. Of course, that would be too strong a notion of inefficiency, and not at all appropriate for an infinite horizon.

DEFINITION. Call a program (k, y, c) grossly inefficient if 0 <E, <k, for all t>l, and E,+O as t-+co.

We will now show that for the class of programs considered by Cass

412 ABHIJIT SENGUPTA

an inefficient program is always grossly inefficient in the sense thatathere exists a consumption gain E, > 0 such that EJE, --t 0 as t + W.

PROPOSITION 1. Under (C.l) and (C.2), a program (k, y, c) is inefficient if and only if it is grossly inefficient.

ProoJ: It is obvious that if (li, ~1, c) is grossly inefficient then it is ineflicient. Now suppose (k, y, c) is inefficient. By Taylor expansion of (1 ),

c:, + , = 6, f ‘(k,) + #[ -f “(S,)],

for some X, E [k, - E,, k,]. Multiplication by p,+ , = p,/f ‘(k,) gives

Pt+1&,+1 = PAI 1 + t&,C-f”(Xl)llf’(kr)}.

By (C.2), there exist m, n > 0 such that -f “(k,) <m < CQ, and f ‘(k,) 2 n, so that

Pr+lE,+I 6P,%(l+qq24.

Taking reciprocals and defining ti = m/2n yields

l/P ,+,&,+I 3(l/P,E,)Cl -fi&,/(l +%)I

2 (l/P,&,)--m(l/P,). (2)

Summing (2) from 1 to (t - 1) and cancelling repeated terms, we get

r-1 l/P,b3(l/P,&,)-fi c (l/P,).

A=1 (3)

Now, by Theorem A,

CX:> f (i/p,j=s, say, 5 = 0

so that (3) can be written as

(4)

Multiplying (4) by p,~, and then taking reciprocals, we have

&r/E, d ~/lP,C(l/P,)-~~~,1~.

Then it follows that for &I < l/tiiSp, ,

E,IE, < llm*p,, m*>O.

INFINITE-HORIZON EFFICIENCY 413

By Theorem A, C;LE0 (l/p,) < m, so that l/p, + 0 as t -+ cc1. Therefore, e,+O as t--tsm. 1

This is a rather surprising result which shows that the notion of inefficiency captured by Theorem A is much stronger (namely, gross inefftciency) than is appropriate for an infinite horizon, and, in fact, is conceptually closer to finite-horizon inefhciency.

In the context of finite-horizon models the ideas of “cost” and “gain” associated with an efficient program had long been characterized through the concept of rates of transformation at the margin, or equivalently, in terms of competitive prices. When the horizon is infinite, however, com- petitive prices do not yield a natural measure of cost. Approximations can be secured through regularity conditions on technology, but then the regularity conditions imposed by Cass have the effect of making inefficiency equivalent to gross inefficiency. Any criterion of efficiency in terms of com- petitive prices therefore has an inherent limitation in an infinite-horizon model.

REFERENCES

1. D. CASS. On capital overaccumulation in the aggregative model of economic growth: A complete characterization, J. Econ. Theorv 4 (1972), 2OCb223.

2. E. MALINVAUD, Capital accumulation and efficient allocation of resources, Econome~ricu 21 (1953). 233-268.