American Economic Association
Rational Choice Under an Imperfect Ability To Choose Author(s): André de Palma, Gordon M. Myers, Yorgos Y. Papageorgiou Source: The American Economic Review, Vol. 84, No. 3 (Jun., 1994), pp. 419-440 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/2118060 Accessed: 14/12/2010 13:15 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=aea. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
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RationalChoice Under an ImperfectAbilityTo Choose By ANDRE DE PALMA, GORDON M. MYERS, AND YORGOS Y. PAPAGEORGIOU*
We consideran individualwho lacks the information-processing capacityrequiredfor a directcomparisonof all feasibleallocations.Insteadof findingat once a best allocation,the individualmyopicallyadjustshis currentallocation towardhigherutility.The individualmakes adjustmenterrorsinverselyproportional to his abilityto choose. We comparethe stationarystate of thisprocess with the standardmodel. We see how an imperfectabilityto choose modifies both positive and normativepredictionsof the standardmodel and how the standardmodel can be obtainedfrom our moregeneralone as the specialcase to perfectability.(JEL D 11) corresponding
The standardmodel of consumerbehavior asserts that individualshave unlimited information-processingcapacity, which allows them to solve their choice problemin a strictly optimal manner irrespectiveof the difficultyof the problem.However, existing evidenceindicatesthat individualshave only limited information-processing capacity.' In
consequence,when decidingamongalternative courses of action, individualsuse simple, local and myopic choice procedures which adapt choice behaviorto their capacity limitations (see e.g., Allen Newell and Herbert A. Simon, 1972). The quality of choice procedures used reflects the ability to choose. Since such proceduresare imperfect, they cause random errors when the choice problem is of sufficient complexity. within our framework, individual Thus, * De Palma: Department COMIN, Universite choice behavior can only be determinedup Geneve, 102 BoulevardCarlVogt, CH-1211Geneve 4, to a probabilitydistributionwhich serves as Switzerland;Myers: Departmentof Economics,Unia "blackbox"to summarizecomplexbehavversityof Waterloo,200 UniversityAve. West, Waterloo, ON N2L 3G1, Canada;Papageorgiou:BSB 329, ioral aspectsof the individual.Such ambiguMcMasterUniversity,1280 Main Street West, Hamility in choice predictionsis deeply rooted in ton, ON L8S4K1, Canada.We thankRichardArnott, the real world. It arises because individuals Tom Muller,the participantsof a McMastereconomics not know their preferences perfectly may workshop, and three anonymous referees for their comments.G.M.M.and Y.Y.P. acknowledgeSSHRCC well, and it differs from that generated by grants410-91-1851and 410-91-0906,respectively. an uncertainfuture in the context of ratio'This observationhas given rise to informationnal choice (von Neumannand Morgenstern, processingtheories of choice, which are dominantin 1944). Of course, variance may well arise the behavioralsciences (see e.g., James R. Bettman, throughuncertaintyabout the future. But it 1979;Daniel Kahnemanet al., 1982).In those theories, individualsacquireinformationfromvarioussourcesin may also arise as an inherent property their environment,which they perceive,interpret,and of choice behavior even when the future evaluate drawingupon past experienceand upon the context in which they obtained it. Thus information- can be known (James G. March, 1978 processingtheories are not just aboutwhat individuals pp. 598-99). know, but also about how individualsuse what they We consider the choice problem of an know. In economics, such processing-capacitylimitaindividualwho does not have the informationshavebeen expressedas a "competence-difficulty" tion-processingcapacity required for a digap (Ronald A. Heiner, 1983), whereby the comperect comparisonof all feasible allocations. tence of individualsto solve a choice problemdoes not Instead of finding at once a best allocation matchthe difficultyof that problem. 419
420
THEAMERICANECONOMICREVIEW
as in the standardmodel of consumer behavior,the individualadjustsmyopicallyhis current consumption in order to improve utility.Beginningwith some initialcommodity stock, the individualspends at a constant rate which defines periods correspondingto one unit of expenditure. The individual's commoditystock changes between periods following the rates at which he consumes the various commoditieson the one hand, and his purchasingdecisions on the other. At the beginning of each period, the individual spends the unit of money currently availableon a single commodity.The problem now is to find a commoditywhich gives the highest marginalutility. Thus the individual reduces the unmanageableproblem of comparingall feasible consumptionbundles to a manageableproblemof comparing the utility incrementsderived by adding in turn an amount of unit value to each current commoditystock. In principle,all that is necessaryto solve this problemis knowledge about marginal abilities at a point, rather than about the utility function itself. Followingthe information-processing approach to choice, we allow for the possibility that the individualmakes errorsin comparing marginal utilities. This creates a difference between perceived and true marginal utilities, which decreases as the abilityto choose increases.For a particular degree of ability, the above choice procedure correspondsto a particularstationary state in consumption. As the ability to choose increases,the stationarylevel of utility also increases. At the limit, where the ability to choose becomes perfect, the stationary state coincides with the solution of the standard consumer choice problem. Thus, for different degrees of ability to choose, our model predicts a dispersionof choice decisions even in cases where rational individualshave identical preferences, endowments,and access to information-a phenomenon which has been observed experimentally(R. J. Herrnstein,1991).In the context of the standard consumer choice model, such dispersion must signify irrational behavior. When the individual can choose to improve his ability, he faces a trade-off be-
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tween the level of ability and the income left for consumption. Under these conditions, a maximumfeasible level of utility can be reached,providedthe individualdoes not make errorsin improvinghis ability.In the presence of such errors,our framework leads to a failure of both fundamentalwelfare theorems which does not arise from externalities or a lack of information,but ratherbecause the individualdoes not have the information-processingcapacity necessary to equate the marginalcost of improving abilitywith the correspondingmarginal benefit. In such cases, paternalismmay be justified. We next examinethe possibilitythat random errorsare biased.A naturalinterpretation for such errors can be found in the context of image advertisement. In most economic models, such advertisementhas been treated as a good that yields either information or utility. We, on the other hand, focus on the manipulativecharacteristics of image advertisementwhich can be either good (if error bias encourages consumptionthat increasesutility) or bad (if it encouragesconsumptionthat decreasesutility). The impact of such advertisementbecomes smaller for higher ability to choose until, when the abilityto choose is perfect, manipulative advertisement is rendered completelyineffective.It follows that, when individualsare perfect instrumentsof choice and have fixed preferences, as in standard economictheory,there is no conceptualbasis for manipulativeadvertisement:individuals either benefit from it or eventuallydiscard it. In contrast, an imperfect ability to choose does not negate existing economic approachesto advertisement.For example, image advertisementcan enter the utility function as a complement to the good advertised (Gary Becker and Kevin M. Murphy,1993) and, at the same time, bias the perceptionof true utilitywhen the individual has an imperfect ability to choose. Our approachprovidesa rationalefor laws against false advertisement,for the regulation of advertisementof commodities for which errors may become dangerous, and for advertisementaimed specificallyat lowabilitygroupssuch as young children.
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VOL.84 NO. 3
Throughoutthe paper, we provide examples about how an imperfect ability to choose modifies both the positive and normative implications of the standard consumer model.2 In the final section we provide a model of socially optimal product differentiationwith fixed prices in which normative implications vary systematically with the degree of ability to choose. When the ability to choose is perfect, we derive the standardsolution.The optimalproducts become increasinglysimilarfor lower ability until, below a certain level, there remains only one optimalproduct.Besides providing a formalization of the familiar idea that there can be too much productvariety,this exampleprovidesanothercase in which the implications of variable ability to choose cannot be dismissedas insignificant. I. MyopicAdjustments
If an individualcan evaluate and rank all feasible consumptionbundles, the description of the choice procedure adopted becomes trivial:the individualsimplyranksall feasible consumptionbundles accordingto his preferences and chooses a most preferred one. Under these circumstances,the choice problem can be expressed in the familiar,compactform: (1)
max U(Q1
Qn)
subjectto n EPiQ~=Y = PIQ 1
where U( ) is the ordinalutilityfunction,pi and Qi are the price and the consumption of the ith commodity,and Y is income. If, however, the individualdoes not have the
2See also George A. Akerlof and Janet L. Yellen (1985)for a discussionabout how even small errorsin consumptiondecisionscan have large effects on equilibria.
421
ability for making the global comparisons requiredto solve a problem (1), then he is bound to adjusthis choice behavioraccordingly. For example, if we confronted you with a complete displayof all the consumption choices you made last month, together with a reasonable(for you) alternativedisplay of consumptionchoices, we doubt that you would be prepared to undertake the global comparisonsrequired for expressing preferencebetween those two consumption bundles. You would be prepared,however, to discuss the advantagesof choosing this brand of cereal over that brand, or the excessive amountsyou spent last month on entertainment. More generally, excepting some well-definedcases in which the problem of choosing among a small number of alternatives is independent of everything else, consumptionchoices rarely, if at all, concern entire consumptionbundles. Typically consumptionchoicesare aboutelements of the consumption bundle and about change. In this sense, consumption-bundle consumption choices are typically local ratherthan global.3 One main goal of our paper is to model a sequence of consumptionchoices that describes local consumption-bundle adjustments. We imagine that this sequence occurs during a time intervalof length T. At the beginningof the time interval,the individual receives a fixed income Y, which it spends on the n commodities.To fix ideas, let T representa month and let Y represent the salaryreceived at the beginningof the month. There is no borrowing or saving across months. We assume that spending occurs at a uniform rate y = Y/ T which determines the partitioningof the month into periods of length At correspondingto one unit of expenditure: (2)
yAt=1.
3This is closer to a Marshallianpoint of view (Alfred Marshall, 1920 [Vol. 3, Chapter III]). It is interestingto note that John R. Hicks,a protagonistin the establishmentof ordinalutilitytheory,later recognised the descriptiveaccuracyof the old consumer theory(Hicks,1976pp. 137-38).
THEAMERICANECONOMICREVIEW
422
At the beginningof each period [t, t + At] (t = O, At, 2At ..., (Y - 1)At) the individual spends the availableunit on a single commodity. We thus replace the global choice of one among an infinite number of commodity combinations in the feasible consumption set with a sequence of simpler, local decisions about spending one unit of moneyon one out of n discretealternatives. Let Qi,t represent the stock of commod-
ity i at the beginning of the period [t, t + At]. During that period, this stock is consumed at a uniformrate qi t determinedat the beginningof the period. The continuous decrease of the stock during that period is given by (3)
Qi,t(T)
=
Qi,t(
-
C)
and ci is a conwhere Omax{Aijtv+8e1t;jiii}.
In the context of problem (1), this choice procedure portrays an individualwho applies a gradientprocess to climb myopically
denotes the probabilitythat the individual will spend the unit of money on commodity i at t. We shall firstexaminethe simplecase in which error variance remains the same over time. If those errorsare independently and identicallydistributed6and if the preference orderingof the individualis invariant under uniformexpansionsof the choice set then, accordingto theorem 6 in John I. Yellott (1977), the random errors are double-exponential.7Under these circumstances, we can write the random errors as 1
where 1/,u is the dispersion parameterof ECO,and where E is double-exponential with a zero location parameter and unit 6
5Followinginformation-processing theories,we can imaginethat the individualuses informationfrom the environmentas an input to an individual-specific processing technologyand produces decisions. The processing technologywill reflect the individual'sinvestment for improvinghis processingcapacitythrough,for example,education.However,it will also reflectinherent characteristicsof the individual,such as perceptiveness, intelligence, quality of memory, and so forth, which make some individualsbetter instrumentsof choice than others. In this framework,errorscan arise from incompleteinformation,the distortionof information during processing,or wrong deductions.The last possibilityhas been called a "lackof logicalomniscience"by BartonL. Lipman(1992).Betterprocessing technologyimpliessmallererrorson average.
P1i,, Pr{Ai,,v + i,
> max{Ai,tv + 8jt; Ii}}
at the beginningt of each period in [0,T] spendyAt on a commodityi such that (7)
423
A justificationfor stochasticindependencecan be made if a commodityis understood to represent a distinct class of differentiatedproducts sold at the same price by a large number of competing firms. There is no restrictionon the entryor exit of firms,or on the introductionof new productsand discontinuation of old ones. We imaginethat the individualdraws a randomcombinationof productsof unit value belongingto the same commodity.Since the choice environmentis in continuousflux,the individualcan make errorsin predictingthe consumptioneffect of the particulardrawon utility;and since everydrawconsistsof a randomcombination,those errors can be independent. 7A uniformexpansionof a set can be obtained by replicatingevery element in the set the same number of times.
424
THEAMERICANECONOMICREVIEW
dispersion parameter. Then, following Daniel McFadden(1974), the marginalallocation probabilities (8) are given by the multinomiallogit model: (10)
pi,t
exp( ,.tAtv)
E exp(,uxAj tv ) j=1
Since errorsbecome smallerfor largerI,, this parameter can be interpreted as the abilityof the individualto choose, where a larger ,u signifies higher ability. We also have ( 11)
limO PDi,t=n
lim Pi t if AIiv t> max{A,j,Itv
(1
0
i}
otherwise.
When there is no abilityto choose, discrete choices are equiprobable, irrespective of differences in the true marginal value of alternatives;and when the abilityto choose is perfect, the best choice is made with certainty. More generally, the individual's abilityto choose is reflectedby the distribution of marginal allocation probabilities around alternativesof higher marginalutility. As the ability to choose increases, the distributionof marginalallocationprobabilities tightens aroundbetter alternativesuntil, at the limit, the individualadjusts only towardthe best alternative.
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Therefore expected changes in the value of the commoditystock between intervalsobey (12)
ApiQi t = (yPiPt - pjqj t)At.
Since the individualaims to improve his utilitylevel, yPDitin (12) can be interpreted as the value of the currentlydesired consumption rate on commodity i. Thus (12) implies that expected changes in the value of the commoditystock are driven by the difference between the values of currently desired and experiencedconsumptionrates. ThroughIPit, (12) depends on the entire distributionof commoditystocks. Thus the dynamicsof (12) may be complicated,and they may or may not lead the individualto stationarychoice behavior,such that no further changes in the value of the commodity stock are observed between intervals.8We do not study these dynamicshere. Instead, we confine our analysisto stationarychoice behavior which corresponds to the timeinvariantsystem (13)
y jAt = pjqjAt.
Summingover one interval,(13) implies (14)
YPi= Tpiqi.
Summing(14) over the n commodities,we obtain the stationarybudget constraint n
(15)
y= Epiqi i=l1
since EiPi = 1. Furthermore, substituting (10) into (13), we have
III. StationaryChoiceBehavior
Changes in the value of the commodity stock between periods are driven by the flows of expenditureand consumption.We have alreadyarguedthat choice, and hence expenditure,can be determinedonly up to a probabilitydistribution.Using (2), the expected expenditureon commodityi during the period equals yAtPi't, while the corresponding
consumption
equals
piqi1tAt.
1plii (16) Aiv -Alv=-In pl pql
8
>1
A stationarysolutionexists (see e.g., proposition4 in Victor Ginsburghet al. [1985]).Sufficientconditions for the stabilityof stationarystates for systemsmore general than ours are given in proposition5 of Ginsburghet al. (1985).
VOL.84 NO. 3
DE PALMAETAL.: IMPERFECTABILITY TO CHOOSE
IV. Relationship with Maximizing Behavior
Let us begin by defininga simple expression for true utility increments at the stationary state. We consider the decision of spendingthe unit of money on commodityi at the beginningof a period and its consequences for future periods along the stationary path. Denote the initial change in the stock of commodity i by AQ(?), and denote the portionof AQi() remainingafter m periods by AQ(m). Taking into account
AQi(?)=1/Pi and (3), we have (17)
AQ(m)=-(1-CjAt)m
Pi
m = 0,1,2,....
Using (4), the correspondencechange in the consumptionflow is (18)
Aqfm)= =(1-ciAt)m Pi
m=0,1,2,... We requirethat the true utilityincrement takes fully into account the future consequences on utility of the initial spending
425
diate step follows from (18). From now on, we shall assume that (20) holds as an equality. Introducing(20) in (16), we obtain (21)
v1 v1
t
P1
A
vi _I=_n
Pi
piqi p1q1
I> 1
where vi /pi is the marginalutility per unit of expenditure.Equations(15) and (21) can be used to determine the stationary consumption rates (q, ...., q4) where, from now
on, an overbarwill denote variablesat stationary state. According to our interpretation of (12), stationary consumption rates also represent desired consumption rates: since, in this case, the aspirations of the individualmatch its experiences,there is no need for further change in the commodity stock. Solving(15) and (21), we have (22)
4i = i( Y,P;A)
where p is the vector of prices. Using (13) and (22), the correspondingstationaryutility level can be written as
decision:
(23) vq,... (19)
Aiv=
E
)=v
Y
m=O V(yP;
We also assume that the function v(q1,..., qn),which determinesthe true current utilitygeneratedby the n consumption rates, is differentiable, strictly increasing, and strictlyquasi-concave.If we expandv(*) in Taylor series around qi and retain only linear terms, we can express true utility incrementsas (20)
Aiv-
E
Aqim)vj
m=O
=
CiV
I I
Ygn
A1v(m).
00
E (1-c1At)
Pi m=0 Vi
piAt
where vi dv /dqi and where the interme-
yA)
When there is no ability to choose, (21) implies piii = pq. for all i and j, which is consistent with '(1). That is, under complete inabilityto determinewhat constitutes a good choice, the individual allocates an equal amountof money per unit of time to all commodities.On the other hand, when the abilityto choose is perfect, (21) implies that, for all i and j, at least one of the following conditions must hold: (a) vi /Pi = viJ/pi; (b) qi = 0; (c) qi = 0; (d) both Qi = 0
and ij = 0. It follows immediatelythat the stationaryconsumptionrates corresponding to a perfect ability to choose satisfy the necessary conditions of maximizing v(*) subject to Yi2piqi< y, which is consistent
with (11). Furthermore,it can be shownthat the stationaryconsumptionrates tend to the
THEAMERICANECONOMICREVIEW
426
solution of the constrained maximization problem as A -: oo, which implies the conti-
nuity of the stationarysolutionsfor ,u> 0. V. SatisficingBehavior
We now turn to a main result of our paper, namely,that the behaviorof the individual under an imperfectabilityto choose is satisficingin the sense that its stationary choice behavior corresponds to a level of utility that is below the maximumpossible (Simon, 1955; Leibenstein, 1976 Ch. 5). In particular,we shall prove that d3 / dA > 0 for 0 < t -0.
2 RHS
At x1 = 1/2 we have Pl = 1/2. Hence, for the same location, FOC = 0, and SOC = 1-
A0. It follows that xl = 1/2 correspondsto a local maximumof W(*) for A < 1/0 and to a local minimumfor A > 1/0. We prove below that the maximumis global. Therefore x4 = 1/2 if and only if ,u < 1/0. For > 1/0, x also corresponds to a global maximumof W( ). Total differentiationof (B10) further yields that x4 is strictly decreasingand convexin A for A > 1/0. Hence the minimum occurs at ix = o. In order to
evaluatethis minimumnotice that limPD= 1 and lim ix
=0 for ,
-+
oo. These, in con-
junctionwith the FOC, imply lim,l 4x = 1/4. It remains to prove that the result is global. Using x1
1/2 - z, after some cal-
2
FIGURE
Bi.
SOLUTIONS TO EQUATION
(B13)
(B13) that correspondsto a local minimum of W(*), the only other solutionmust correspond to a global maximum. 2. Anderson and de Palma (1988)
In order to determine optimal location policies analogous to our optimal design policies, Andersonand de Palma(1988) employed the consumer-surplusfunction (B14) CS(x1,x2)
culations,we can write (B10) as /L
(B13) z[(,u - 1) -2kuOz] z-
1w
1
exp(2kuOz)+-exp(-2k.tOz).
When A < 1/0, the LHS of (B13) is null for
2
I- nt
E
exp[ilolx
xil]
which was derived for the logit model by Kenneth A. Small and Harvey S. Rosen (1981). Taking into account symmetry,the first- and second-orderconditionsare
z = 0 and negative for z > 0, while the RHS
of (B13) is null for z = 0 and positive for z > 0. In consequence(B13) admitsa unique solution z = 0. This implies that x = x = 1/2 correspondsto the global maximumof W( ). We now examine the case A > 1/0. The LHS of (B13) is a concave parabolaas shown in Figure B1, where LHS(1/2) 0. Since, in Figure B1, z = 0 is a solution to
(B15)
FOC=2/.(
- xl +ln(4x1 -1)
=
0
(B16) SOC 4- 2,u0(4x1-1) ? 0. At x1 = 1/2 we have FOC = 0 and SOC =
2- u0.
It follows that x1= 1/2 corre-
sponds to a local maximumof CS( ) for A < 2/0, and to a local minimumfor ,u> 2/0. As in our case, the maximumis also global. Thus the range over which
VOL. 84 NO. 3
DE PALMA ETAL.: IMPERFECTABILITY TO CHOOSE
Hotelling's principle of minimum differentiation holds in Anderson and de Palma (1988) is twice our range. The remaining conclusions are exactly analogous in both cases.
439
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