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Can Small Deviations from Rationality Make Significant

Differences to Economic Equilibria?

By GEORGE A. AKERLOF AND JANET L. YELLEN*

This paper concerns the robustness of eco-

nomic equilibria in famiUar economic mod-

els. It addresses the question whether a small

amount of nonmaximizing behavior by agents

is capable of causing changes in the equi-

librium of the system which are an order of

magnitude larger than the losses due to non-

maximization by the individual agents them-

selves.

For some, it would be reassuring to find

that the results of models based on maximiz-

ing behavior continue to hold as an ap-

proximation when the assumption of strict

maximization is relaxed just slightly. It is,

after all, difficult to believe that agents liter-

ally maximize all the time. People may suffer

from inertia. (Or, alternatively stated, they

may have some small transactions costs in

changing their actions.) They may also rely

on rules of thumb which produce acceptable

results on the average. So most economic

theory based on strict maximization is useful

when accompanied with the folk theorem

that the results of this theory are approxi-

mately correct if the deviations from opti-

mality (or the transactions cost of decision

making) are small.

On the other hand, if slight relaxation of

maximization results in rather different re-

sults from those of strict maximization, then

there is the possibility of explaining phenom-

ena that have been puzzling in the context of

economic theory based on strict maximizing,

such as the persistence of cartels and the

existence of the business cycle.

This paper constructs a number of exam-

ples which show that small deviations from

rationality can have first-order consequences

in microeconomic analysis. Our 1985 paper

shows that such deviations can account for

Keynesian business cycles. All of these ex-

amples assume that a fraction /i of the popu-

lation fails to maximize. The rest of the

population is totally rational. An equilibrium

of such an economy is termed near rational

if no nonmaximizer stands to gain a signifi-

cant amount by becoming a maximizer.

Technically, this means that the potential

losses due to nonmaximization are second-

order small (varying with the square of a

shift parameter). Nonetheless we show that

such behavior does commonly lead to

changes in equilibrium in familiar models

(relative to the equilibrium with full maximi-

zation) which are first-order, that is, that

vary approximately proportionately with the

shift parameter.

These results can be interpreted as a pre-

cise sense in which the conclusions usually

derived from models with strict maximizing

behavior are not very robust. We wish to

emphasize that this theory does pass Lucas'

test of a good model—that there are no

opportunities for large increases in profits,

or, in his terminology, that "there are no

$500 bills on the sidewalk." To be specific,

there is a fraction of the population who are

strict maximizers. If there are opportunities

for gain generally available (in Lucas' exam-

ple, to pick up $500 off the sidewalk), a

significant fraction of the population will

•University of Califomia, Berkeley, CA 94720. We

express our special gratitude to Robert Barro and Hajime

Miyazaki for greatly elucidating comments on an earlier

paper presented at the NBER-West Conference on Mac-

roeconomics, December 2, 1983, and to John Halti-

wanger and Michael Waldman for criticisms of an earlier

draft. We also thank Andrew Abel, Kenneth Arrow,

Sean Becketti, Ben Bemanke, Alan Blinder, Stephen

Blough, Roger Craine, Avinash Dixit, Richard Gilbert,

Paul Evans, Milton Friedman, Robert Hall, Bent Han-

sen, Robert Lucas, Thomas MaCurdy, John Roberts,

Joseph Stiglitz, Steven Stoft, Lawrence Summers, James

Tobin, and James Wilcox for valuable discussions which

have led directly or indirectly to this paper. Our re-

search was generously supported by National Science

Foundation grant no. SES 81-19150, administered by

the Institute for Business and Economic Research of the

University of California-Berkeley.

708

VOL. 75 NO. 4

AKERLOF AND YELLEN: DEVIATIONS EROM RATIONALITY

709

readily avail itself of these opportunities. Ad-

ditionally, the losses to those who are not

maximizing are small, in a well-defined sense.

Recent results in industrial organization

theory in the context of sequential games are

similar in spirit. In recent interesting papers

by Drew Fudenberg and Eric Maskin (1983),

David Kreps et al. (1982), Kreps and Robert

Wilson (1982), Paul Milgrom and John

Roberts (1982), and Roy Radner (1980), it

has been found that the presence of

nonmaximizing behavior, even in rather small

amounts, can significantly alter the equi-

librium obtained. Other papers which

examine the conditions under which non-

maximizing behavior can have a significant

impact on the characteristics of economic

equilibria include John Haltiwanger and

Michael Waldman (1985) and Thomas Rus-

sell and Richard Thaler (1985). John Conlisk

(1980) has examined conditions under which

rule-of-thumb behavior will coexist with

maximizing behavior in the long run.

Section I explains the basic logic of this

paper, which can be generally expressed in

terms of the envelope theorem. The next

sections present four examples of equilibria

that are near-rational, but that differ by a

first-order amount from the equilibrium that

would prevail with full rationality. Section II

presents an example of a pure exchange

economy. There is an initial (long-run) equi-

librium of this economy in which all agents

are exactly maximizing. Then each agent's

endowment of one good is increased by a

proportion e. A fraction fi of the population

has inertial behavior in its consumption of

that good while the rest of the population

strictly maximizes. It is shown that this be-

havior by nonmaximizers results in individ-

ual losses which are second-order in e, while

there is (almost always) a first-order change

(with respect to e) in the distribution of

income.

In the pure exchange economy, nonmaxi-

mization causes only a second-order move-

ment from the Pareto frontier. But if there

are externalities, nonmaximization which re-

sults in only second-order losses to the agents

can cause first-order changes in the econo-

my's deadweight loss. This proposition is

illustrated by two examples in Section III.

Section IV presents an example of cartel

formation. It is supposed initially that there

is a Coumot equilibrium of an oligopolistic

market. There is a posited reduction in out-

put by a fraction e by a proportion fi of the

firms, while the rest of the firms maximize

(according to the Cournot assumption that

the output produced by rival firms will re-

main unchanged.) With some qualification, it

is shown that such a reduction increases the

ohgopolists' profits by a first-order amount

while the expected gains from cheating on

the cartel are second-order in e.

I. Implications of the Envelope Theorem

There is a simple reason why deviations

from rationality, that involve only second-

order losses for nonmaximizing agents, nev-

ertheless can cause first-order changes in the

economy's equilibrium. The reasoning is

based on the envelope theorem and its gen-

eral equilibrium implications. The envelope

theorem states, in effect, that first-order

"errors" made by an agent in setting any de-

cision variable, such as quantities produced

or consumed, normally cause only second-

order losses in utility or profit. If a signifi-

cant fraction of the population makes such

errors and their mean value is nonzero, the

errors are likely to have general equilibrium

effects which are first-order.

More formally,^ consider the uncon-

strained maximization problem: maximize

f(x,a) where x is a choice variable and a is

a vector of parameters or variables exoge-

nous to the agent. Let x{a) denote the unique

maximizing choice of x, given a, and let

M{a) = f(x(a), a) denote the maximum

value of / for given a. According to the

envelope theorem:

(1) dM{a)/da=df{x(a),a)/da.

In words, the envelope theorem states that,

at the margin, the change in the objective

function caused by a change in a is identical

whether the agent adjusts x optimally or not

at all. Inertial behavior is virtually costless.

'See Hal Vadan (1978).

710

THE AMERICAN ECONOMIC REVIEW

SEPTEMBER 1985

The theorem is easily extended for con-

strained maximization problems.^

The proof of the envelope theorem is triv-

ial but instructive. Total differentiation of

M{a) yields

f(x,a)

(2)

dM{a) ^ df{x,a) dxja)

+ df{x(a),a)/da.

The first-order condition for an optimum

requires that df(x(a),a)/dx = O and thus

the envelope theorem follows. The first term

in (2), that is the indirect effect of a change

in a on / occurring via a change in optimal

X, is zero. Thus the effect of a change in a

on M is just the direct effect of a on / for

given X.

A graphical interpretation of the theorem

is given in Figure 1, which depicts / as a

function of x for two alternative values of a,

OQ. and a^. The optimal choices of x are XQ

and Xi, respectively, and the maximized val-

ues of the objective function / are M(aQ) =

/(^o. «o) and M(aj^) = f(x^,ai). An agent

who behaves inertially following a change in

a, leaving x fixed at XQ instead of changing

X to Xj will incur a loss, if, from his failure

to maximize equal to f(xi,ai)-f(xQ,a{)

which is obviously second-order small.

There is an intuitive explanation of this

result. The essential feature of optimal choice

is indifference at the margin. Take the case

of a consumer purchasing apples and banan-

as. At an optimum, the last dollar spent on

apples adds as much utility as the last dollar

spent on bananas. If the consumer is forced

to spend a small amount less than the opti-

mum on apples, the small amount extra he

or she can spend on bananas provides almost

perfect compensation.

More formally, writing the loss as a

Taylor-series expansion around Xj yields

FIGURE 1

tional not to the error made in setting x, but

to the square of the error. The loss can also

be expressed in terms of the change in a,

which can be thought of as a shock to the

system. Let e= a^- a^ denote the shock.

Then

(4)

where dx(a)/da normally differs from zero.

Thus, 3

(5)

(e) =..«•"((

where

dxda

If there is a shock of size e, an agent who

behaves inertially makes an error which is

ordinarily proportional to e, or first-order,

and incurs a loss which is proportion to e^,

or second-order. Behavior that leads to a loss

proportional to e^, where e is a shock param-

eter, is called near-rational.

Although the envelope theorem is itself

trivial, its implications are not. In many gen-

eral equilibrium models, first-order errors in

choice variables (for example quantities de-

manded or supplied) by a fraction of agents

will cause first-order changes in such endoge-

nous variables as relative prices. These en-

dogenous variables typically enter the objec-

(3) ^ = k

The loss from failure to maximize is propor-

Varian (pp. 268-69.)

•'This follows from the fact that dx(a)/da

(d^f/dd/d^fd^

VOL. 75 NO. 4

AKERLOFAND YELLEN: DEVIATIONS FROM RATIONALITY

711

tive functions of households and firms; they

are arguments in the a vector and any change

in these variables will accordingly have

welfare consequences. As long as the errors

do not just cancel out when averaged over

the population (a sensible assuinption in

many contexts), "small" departures from ra-

tionality can cause first-order changes in such

variables as prices, income distribution, or

deadweight loss.

To be more precise, let p denote an en-

dogenous variable (such as relative prices)

which enters the a vector in a particular

system under consideration. Assume that a

fraction /8 of the population fail to maximize

in a well-specified fashion. It is possible to

compute the equilibrium value of p corre-

sponding to altemative values of e and y3, so

that p can be written as p = p(E,P). Let

s{e,P) denote the difference between the

equilibrium value of p with a fraction /i of

nonmaximizers and the equilibrium value of

p with no nonmaximizers; that is, S(E,P) =

p(e,jS)-p(£,O). The function j(e,yS) gives

the systemic effect of nonmaximizing behav-

ior. Writing s(e,P) as a Taylor-series ap-

proximation around the point (0, ^8) yields

Methodology of the Examples. The methodol-

ogy employed in each of the next sections is

identical and can be outlined briefly. First, a

simple model is constructed and its equi-

librium is calculated under the assumption of

full maximization by all agents. Second, the

system is shocked in a specific way in amount

e. Third, the values of relevant endogenous

variables are computed in the equilibrium of

this model on the assumption that a fraction

P of agents fail to maximize in a well-

specified way. Finally, two questions are

posed: first, is the assumed behavior by non-

maximizers near-rational? In all cases, the

answer is yes and can be demonstrated by

showing that the loss from failure to maxi-

mize is second-order: SC(E) =• SC"{O)t^. Since

it is always the case in these examples that

i?(0) = 0, it is only necessary to demonstrate

that .S?"(0) = 0. This follows from the en-

velope theorem and in most cases proof will

therefore be omitted. Second, does the equi-

librium of the system corresponding to strict

maximization by all agents (j8 = 0) differ by

a first-order amount from the equilibrium of

the system when there are a fraction yS of

nonmaximizers? This involves showing that

(6)

Some Caveats. Some caveats are in order

concerning the appropriate interpretation of

our results. In each example we show that

the ratio of the systemic effect of nonmaxi-

mization to the individual loss from non-

maximization approaches infinity as e ap-

proaches zero. The potential for applying the

logic of the envelope theorem in any context

where maximizing behavior normally occurs

suggests that our theorem could have almost

unlimited applications. But for the theorem

to have practical relevance, it must be true

for finite values of e, corresponding to eco-

nomically noticeable shocks, and not just for

infinitesimal e, that the ratio R(e,P) is

"large." The behavior of i? as e increases

depends critically on the magnitudes of

.Sf"(«) and ds(E,P)/de in the neighborhood

of e = 0. If .Sf"(O) is extremely large, or if the

flrst partial of s is very small, the ratio of the

systemic effect to the individual loss will

diminish rapidly as e increases, limiting the

applicability of our inflnitesimal results. The

0,/9

As we demonstrate by example in Sections

II-IV, there are many models in which the

individual losses due to the assumed devia-

tions from maximization are approximately

proportional to e^ (second-order), but the

function s(e,l3) has a flrst partial derivative

that is nonvanishing. In such cases, the sys-

temic effect of nonmaximization is ap-

proximately proportional to e, or first-order;

the systemic effect of nonmaximization is an

order of magnitude larger than the individ-

ual losses. This implies that the ratio, de-

noted Rie,^), of the "systemic effect" of

nonmEiximizing behavior to the individual

loss approaches infinity as e approaches zero.

712

THE AMERICAN ECONOMIC REVIEW

SEPTEMBER 1985

simulations presented in Sections II and III

provide a simple way of evaluating the be-

havior of this ratio for finite e in some specific

models for reasonable parameter choices. In

these models the ratio of effects remains

large for "significant" e. Economic intuition

suggests, however, that there will be cir-

cumstances in which i?"(e) is quite large or

5.s(e,j8)/5e small."

Markets which are close to perfectly com-

petitive provide an obvious example in which

.Sf"(e) is large. Consider the case of a mo-

nopolistically competitive firm setting its

price. If the firm charges a price above the

optimum, sales and profits fall. In the limit-

ing case of perfect competition, sales plum-

met to zero and profits drop discontinuously.

A price below the optimum raises sales but

lowers profits. In the Umiting case of perfect

competition in which potential sales at the

market price are unlimited, price cuts lower

profits by an amount which is proportional

to the error in pricing. In effect, the function

relating lost profits to price errors has an

extremely large second derivative.

It is equally easy to envision cases in which

nonmaximizing behavior by a significant

fraction of the population would have only a

minor effect on the equilibrium of the system

so that ds(e,P)/de is small. Suppose, for

example, that a significant minority of ra-

tional market participants regards two goods

or assets as perfect substitutes at a given

price ratio; the opinions of such agents is apt

to dictate the market outcome even in the

presence of a substantial fraction of irra-

tional or nonmaximizing agents. Financial

markets provide the obvious example of a

context in which "efficient market" outcomes

can be achieved as long as some rational

arbitrageurs operate in the market. More

generally, the recent work of Haltiwanger

and Waldman shows that in any market

which can be characterized as one with "con-

gestion effects," it will be tme that sophis-

ticated or maximizing agents will have a

disproportionately large effect on the equi-

librium. In these circumstances, ds(e,P)/d£

is small. With the preceding words of warn-

ing, we now proceed to our first example.

II. Near-Rational Behavior in a

Pure Exchange Economy

The first example concerns the canonical

case of a pure exchange economy. A fraction

of individuals are assumed to demand one

good inertially, following an endowment

shock. This form of nonmaximizing behavior

is near-rational. The consequences of inertial

behavior for income distribution are first-

order, however. In comparison with the equi-

hbrium in which all agents maximize, it tums

out that some people are better off and some

worse off. There is a first-order movement

along the utility possibility frontier (but only

a second-order movement away from it).

The Initial Equilibrium. Consider an econ-

omy with equal numbers of consumers of

two types and two goods denoted by sub-

scripts 1 and 2. Consumers of groups 1 and 2

have utihty [/(xj, X2) and V(y^, y2), respec-

tively, and have initial endowments (3ci, jcj)

and (Ji, J2), respectively. Let p denote the

relative price of good 2. The equilibrium

value of p is determined by the condition

that excess demand for each good be zero.

Let the superscript o denote the values of

variables in the initial (long-run) equi-

hbrium, in which all agents maximize utility.

The Shock. It is then assumed that an en-

dowment shock occurs. Many different alter-

natives could be considered. We consider the

special case in which each individual's en-

dowment of good 2 increases by a fraction e.

The new endowment vectors are accordingly

(xi, ^2(1 + e)), (J^i, J'2(l + e))- A fraction fi

of both type-1 and type-2 consumers have

inertial consumption of good 2, their con-

sumption being x| and y2, respectively.

Two propositions will be demonstrated.

According to Proposition 1, the individual

•"In our examples, for given parameter values there

always exists a band of e values such that \R\ exceeds

any specified value within the band. However, as the

referees have pointed out to us, in most examples, it is

possible to select extreme parameter values which make

\R\ arbitrarily small for any (e,P) pair, «> 0.

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