addition - unbounded rationality

8/8/2019 Addition - Unbounded Rationality

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Journal of Economic Behavior and Organization 21 (1993) 331-352. North-Holland

A model of decision making underbounded rationality*Kent D. WallNaoal Postgraduutr School, Monterey, CA, USA

Received November 1991. final version received June 1992

A model of decision making under bounded rationality is presented that combines satisficingbehavior with learning and adaptation through environmental feedback. The aspirations, orgoals, of the decision maker dynamically adjust in response to the observed sequence of pastdecisions and their corresponding effects on the decison makers objective function. A simplelinear response model is employed to represent the beliefs of the decison maker concerning thecausal connection between his/her decisions and the resulting objective function value. Thecombination of these simple elements yields a decision process model rich in dynamic behavior;it can exhibit optimizing behavior in the long-run and chaotic pseudo-random search in theshort-run. As such, the model bridges the gap between substantive rationality and proceduralrationality.

1. IntroductionThere is no doubt mathematical optimization as a characterization of

long-run economic equilibrium is of value. Certainly such an approach isviable in situations where changes in the economic environment occur slowlyrelative to the speed with which agents can adjust, and agents are able toacquire accurate information sufficiently fast and relatively inexpensively.Under these conditions the long-run dominates the short-run because theperiod of adjustment is so transient as to be inconsequential. For allpractical purposes we may consider agents to be forecver in equilibrium,moving instantaneously from one to another as the environment changes andinformation dictates.

If, however, speeds of adjustment are sufficiently constrained, or inform-ation is either incomplete or imperfect so that learning effects cannot beignored, then the long-run is no longer dominant. In these situations it is of

Correspondmcr TV: Kent D. Wall, DRMI, code 64Wa, U.S. Naval Postgraduate School,Monterey, CA 93943. USA.

*The author wishes to thank James D. Hamilton and Maxim Engers for their help in framingthe issues presented in the introduction, and in an earlier version of the proof of Theorem I.Thanks are also due the anonymous referees for many constructive suggestions.0167~2681/93,506.00 ,I# 1993-Elsevier Science Publishers B.V. All rights reserved


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332 K.D. Wall, A model of derision making

interest to study the dynamic response of eocnomic agents and it becomesnecessary to model precisely just how decisions are actually made.Such considerations have not been lost on many prominent economists.Simon (1955, 1957), March and Simon (1958), Cyert and March (1962),Baumol and Quandt (1964), Day (1967) Winter (1971), Nelson and Winter(1973), and Radner (1975) among others have given theoretical direction.Theoretical research in this area, however, has not experienced the rapidgrowth of other subjects like rational expectations, partly owing to thelimited framework available for empirical investigations. An example of theempirical work done in this respect is Crain et al. (1984). Their findingsencourage further theoretical developments, but their nonparametric method-ology does not support the tesing of more detailed models.The primary purpose of this paper is to lay out a model that addressesthis empirical gap. The model has two primary advantages: (1) its formadmits econometric estimation; and (2) its framework contains the optimizingparadigm as a special case, thus permitting testing of this paradigm againstan alternative. The model is predicated on Simons concept of boundedrationality and embodies explicit mechanisms for learning, adaptation, andgoal formation, all in a form suited for empirical studies. Other researchershave touched upon this theme and elements of the model presented here owemuch to their efforts. Most notable among these are Day, March, and, ofcourse, Simon. Adaptive aspirations is attributed to Simon (1955) and Marchand Simon (1958). Simple first order adjustment schemes for the adaptationof aspirations can be found in Levinthal and March (1981) and March(1988). Representation of search and learning in a form analogous tononlinear programming optimization is due to Day (1967) and Day andTinney (1968). This paper constitutes an elaboration and synthesis of theseearlier efforts. The exposition begins with a statement of the conceptualframework and then proceeds to its implementation in concrete operationalterms. Some properties of the resulting model then are demonstrated, and abrief discussion of the method necessary to employ the model in empiricalresearch is given.

2. Bounded rationalityThe descriptive model of decision making presented here owes its con-

ceptualization to Simons theory of bounded rationality, the essence of whichmay be captured in eight statements:[A] Decision making is dominated by the effects of complexity on the

limited abilities of humans to process large amounts of information.Thus, information processing tends to be parsimonious, and solutionsare simple-minded.


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K.D. Wall, A model ofdecisionmaking 333[B] New solutions are synthesized by modifying the currently implemented

one; so search is local.[C] Alternatives are considered one at a time, not simultaneously, so searchis sequential.

CD] The search for a new and better solution is undertaken only when it isdeemed necessary; when it is observed that goals are not being met.

[E] A satisficing mode is used in searching; the first solution that is goodenough is implemented.

[F] Goals are stated in terms of aspirations, and these are formed byadaptation and learning from experience.

[G] Search strategies are developed on the basis of learning and adaptationthrough experience.[H] The attention the decision maker pays to the environment is theproduct of learning and adaptation driven by experience.

The rendering of these tenets in mathematical terms depends directly uponthe type of problem under study. In economics the underlying decisionproblem is that faced by a single decision maker who must select the valuesof N decision variables at time t so as to maximize the value of someobjective function, ,$ The decision variables are represented by an N-vector,x(t), indexed by t, the time at which their values have been tixed. Theobjective function is then written as a function of x(t),f(t) =f(x(t)). Thechoice of x(t) is limited by constraints and so the decision maker must selectx(t) from some set of feasible alternatives denoted by X(t). Time is assumedto evolve discretely so f takes its values from the set integers, _ . Thedecision variables are assumed to be continuously valued so x(t) takes itsvalues in 9. The objective function .f is assumed to be a continuouslydifferentiable mapping from ;R to .&I. This basic decision problem, givenbounded rationality, is stated not as a maximization problem, but one inwhich the decision maker must determine x(t) subject to x(t) c :1(t) suchthat ,f(t) = ,f(x(t)) 2 II, where a denotes currently held aspirations.

Given this framework, the desiderata of a behaviorally based modelincorporating bounded rationality are implemented as follows. First, con-sideration of [A] through [C] suggests decision rules specifying a change indecision variables relative to previous values:

x(t+1)=x(t)+dx(t+l)=x(t)+cc(t+l)d(t+l), (I)where d(t) represents the local search direction and a(t) denotes the steplength to be taken along this direction. Furthermore, these two quantities arechosen on the basis of an information set Y(t) gathered from experience:

d(t + 1) = Yu,(.Y-(t)), (2)ci(t + 1) = Yv,(.F(t)).


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334 K.D. Wall, A model o f decision making

Realistic modeling would suggest that Y(t) be very simple and limited in itsscope. The decision maker should not be assumed in possession of inform-ation not directly observable or easily inferred from directly observable data.For example, in a model of monopolistic producer behavior we wouldexclude elasticities of demand from Y(t), but include the past prices askedfor the product produced and the corresponding consumption observed.

In consideration of [D] through [F] the decision maker is presumed tohold at the end of time t an aspiration level a(t+ 1) representing a value ofthe objective function f( .) which he or she believes is satisfactory and areasonable goal to attain in the next time period. Motivation for continuedsearch is provided by failure to achieve the currently defined goal; i.e.,

Iff(x(t)) < u(t + l), continue to pursuesearching for a better alternative. (4)

On the other hand, if satisfactory performance has been achieved thenterminate searching; i.e.,

Iff(x(t))Za(t+ I), set cx(t+ I)=0 andstop the search process. (5)

Finally, adaptation and learning require that a(t + 1) be adjusted accordingto experience as captured in Y(t):

u(t+ l)= Y,(Y(t)). (6)

3. The operational modelAny empirical implementation of the model described by (1) through (6)

requires concrete form for F(t) and the mappings Pi, where i= 1,2,3. Broadguidance in such a task is provided by three overriding consierations: First,research in cognitive psychology has found that humans employ simplelinear relationships to capture causal connections between variables and thatlinear models capture a great deal of this behavior [Dawes and Corrigan(1974)]. Second, there is evidence of persistence and inertia in the formationof beliefs and in the processing of information used to modify relationships[Edwards (1968) Einhorn (1980)]. Third, perception and judgment sufferfrom significant biases through which objective data are selectively attendedto and subjectively interpreted [Kahneman, Slavic and Teversky (1982) partsII-IV]. The simplest possible information processing mechanisms should beemployed with the focus primarily on linear relationships.

With these considerations in mind, the information set is assumed toconsist of nothing more than observations on past and current values off(t)and x(t). Thus


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K.D. Wall, A model of decision making 335

~(r)={f(r),x(z);r=t,t-l,t-2 )... }. (7)This is to be contrasted with the information set assumed in the neoclassicaloptimizing paradigm where complete information about f is assumed; i.e.,the functional form of f(t) and all parameters, derivatives, and so forth areknown. A further restriction can be placed on (7) by requiring a finitememory, and this is done below by inclusion of only the current and Nprevious observations as required.

Eqs. (2) and (3) are replaced by a set of operations consistent with [D]and [G] but involving only simple linear functions. To wit, the decisionmaker at the beginning of period t+ 1, holding aspirations a(t + l), choosesx(t + 1) to force the value of the objective function, f, at the end of theperiod to equal a(t + 1). The function f( .), however, is not included in Y(t),only past observed values of f and corresponding x are available for use indetermining x(t + 1). The exact form of f, its parameters, and how its value isdetermined by x must be learned. Therefore, to reflect this state of incompleteinformation, the decision maker is hypothesized to possess an estimate of thisfunction, denoted f, and to use this in place of f. Algebraically thisamounts to requiring x(t + 1) satisfy

a(t + 1) =f(f + 1) =f-yx(t + 1)).Furthermore, the decision maker uses a linear representation of reality inconstructing f:

f(t+l)=f(t)+c:dx(t+l). (8)The desired increment to x(t) can then be obtained by computing theminimum norm solution to this equality constraint; i.e.,

Mr+j)=C4r+ l)-f(mlllct~~2

=a(t+ l)d(t+ 1).The step length is seen to depend upon the difference between the achievedobjective and the desired value of the objective. Thus, if the decision maker isclose to his or her goal then a small step is taken, while large discrepanciesbetween the actual and desired objective value will lead to more bold search,in the form of large changes in the decision variables. If the decision makerhad achieved the goal then no step is taken and satisticing behavior obtains.The search direction is seen to be a unit vector which approximates thedirection of steepest ascent but uses only backward looking information.


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336 K.D. Wall, A model oidecision making

While this unit vector never changes in magnitude, the direction in which itpoints does change with experience and therefore embodies the learning andadaptation required by [G].

The N-vector c, represents the decision makers best estimate of thesensitivity of f(t) to changes in x(t). It is comprised of nothing more thansimple finite first differences derived from F(t). This formulation follows afterthe work of Cyert and March (1962) and Day (1967) who argue that humansemploy finite first difference approximations when dealing with the conceptsof derivative and gradient. In the one-dimensional case where x(t) is a scalarc, is simply

c,=c(t)=[f(t)-f(t-l)]/[x(t)-x(t-l)].In the multi-dimensional case c, is defined by a set of N equations:

Af (t) =ci[x(t)-x(t- l)]

Af(t-N+l)=c;[x(t-N+l)--(t-N)],where the prime denotes transposition. Therefore, c, is any solution to

A f(t) = X(t)c,,where X (t) is the N x N matrix obtained by columnwise concatenation ofthe vectors x(t-j)-x(t--j- l), O


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K.D. Wall, A model ofdecisionmaking 331but also the perceived rate of change of the objective function that influenceaspirations. Goal formation is argued to be significantly affected by themagnitude of change of the objective with a rapidly rising (falling) f(t)leading the decision maker to increase (decrease) the value of a(t+ 1) above(below) what would have been the case by keying only on the level of f(t).

The inclusion of a rate of change effect does not appear to have beenconsidered in the existing literature, but is crucial to the performance of thismodel. The motivation for this term is based on two considerations. First,humans attend to rapidly changing factors far more readily than they do tothose changing only gradually over time [Kahneman and Tversky, inKahneman, Slavic and Tversky (1992, ch. 4), and Hogarth (1987, ch. 2)]. Oneneeds only to reflect upon the public reaction to a change in gasoline pricesfrom, say $0.90/gal to $1.30/gal that occurs over a period of two or threeweeks. The same magnitude of change that occurs over a period of two yearsevokes almost no reaction at all. Second the formation of goals is moredependent upon long term perceptions than the formation of f, which isonly concerned with what can be expected over the next decision period.Therefore, aspirations are more likely to be a function of what is perceived tobe possible to achieve over more than one decision period. Using f(t+ 1) asa base and i(t+ 1) as measure of the average rate of change in f(t) over thenext k periods, one may expect that it is possible to achieve l(t+ l)+i(t+ 1)kover the next k periods. By setting y =+k we obtain the expression fora(t + 1).

The model of decision making under bounded rationality can now besummarized:Step 0. Initialization.

Set t = t,,x(t,) =.x0, f(t,) =fo, and

Step I. Update the information set at end of period t (by observing f(t)and remembering x(t))

Y(t)={f(t),x(t),s(t- l)}. (9)Step 2. Update aspirations

At+ l)=Cl -Pl.m+Bf(o~ (10)i(t+l)=[l-fi]i(t)+G[f(t)-f(t-l)], (11)u(t+l)=[l-~]a(t)+qqt+l)+yi(t+l). (12)


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338 K.D. Wall, A model qf decision making

Step 3. Update linear representation of the environmentc,=X(t)df(t).

Step 4. Determine new decision to implement

h(t+l)=a(t+l)d(t+l),x(t+l)=x(t)+dx(t+l).

(13)

(14)(15)(16)(17)

Step 5. Implement new decisionSet t = t + 1, implement decision and return to Step 1.

It is important to note here that the model addresses in a behaviorallyattractive, but implicit way, all the elements of the neoclassical search theorydeveloped by Stigler. The model can be interpreted as balancing marginalgains (to be derived from continued search) against the cost of continuedsearch. The aspiration, a(t + l), contains all the marginal benefit calculations.The gap u(t+ 1) -f(t) represents the value of continued search. If it is large,then search is very worthwhile and large steps in decision space will be taken(subject to any hard constraints, like the physical limitations on the rate ofchange of capital). The cost of changing is well worth the potential for gain.If, on the other hand, a(t+ 1)-f(t) .s small, then small steps will be takenbecause there is little gain to be had. When u(t + 1) =f(t) there is no benefitto continued search and it ceases. Since u(t+ 1) is a function ofT(t + 1) +h?(t+ 1) we find that when there is an expectation of gain, thensearch is undertaken because it is deemed worth it.

4. Dynamic properties of the modelThe model presented above is inherently dynamic, representing the evolu-

tion of decision making over time by a set of difference equations. In theterminology of dynamic systems, it is capable of describing transientbehavior or the short-run fluctuations of the decision variables as they areincrementally altered in the search for a satisfactory solution. For suchmodels two overriding issues arise: (1) the existence of steady-state, or long-run, solutions; and (2) the stability of these solutions.For use of the model in economics it seems reasonable to demand that the


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model demonstrate neoclassical optimizing behavior in the long-run undercircumstances where learning and adaptation would produce an accumu-lation of knowledge leading to complete information. In other words, itseems reasonable to demand that the neoclassical optimum be a solution tothe model and that the model be able to converge to this solution. Thefollowing theorem, the proof of which is found in the appendix, demonstratesthe ability of the model to do this under certain conditions.Theorem 1. Consider f: %?)+.%l, strictly concave and differentiable on anopen bounded set 9 c 99. Let x* l satisfy f(x*) > f(x) for all XEY, x#x*,and take a(t) = f * = f(x*) for all t. Define x(t + 1) to be the element of%(t)cYi nearest to x(t+l)=x(t)+a(t+l)d(t+l) for t-1,2,3,... and wherethe initial conditions have been assigned so that f (0) > f ( - 1) > ... > f ( - N).Then the sequence {x(t)} converges to x*.The model is thus capable of converging to the neoclassical optimizingsolution given feasible starting values and the omniscience to know themaximum attainable value of the objective function. The learning andadaptation inherent in the naive search scheme of (13)-(17) is sufficient toseek out the optimum solution when f * is given.

It is unreasonable, however, to assume that f * is initially available. Inrealistic situations this information must be extracted from experience. Itthus becomes critical to see if the model can learn f * as the search proceeds.More specifically, it is of vital interest to ascertain the interaction between(13)-( 17) and the learning and adaptation equations for aspirations, (lo)4 12).If only a reasonable initial value for a(0) is required, and not f *, then amuch more interesting model obtains - one with far more empirical andtheoretical potential.

At the present time a proof of Theorem 1 with a(0) = f * replaced bya(0) > f(0) is not available. All that can be offered here are the results ofsome simulation studies. They illustrate diverse dynamic behavior anddemonstrate the capability required to learn, adapt, and converge to theoptimizing solution. This convergence is by no means guaranteed, for itdepends on a complex interplay between the initial conditions and theparameters governing adaptation and learning. Similar findings are present inthe simulation studies of economic behavior carried out by Witt (1986) interms of market behavior and firm survival.

Simulation Example 1. Consider the quadratic function in one decisionvariable presented in Day (1967):

f(x(t))= -am +8x(t)- 1.This function has a unique global maximum at x* = 4.0 with f * = 15.0. The


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340 K.D. W all, A model of decision making

simulation studies presented below are initialized with using x( - l)=O,x(O)= 1.0, and a(0) = 12. The decision maker is constrainted to select onlynon-negative x(t) and can change x(t) by no more than 10% in any oneperiod. The first simulation depicted in figs. 1 and 2 below used 4 =0.4,/?=0.4, and 6 =0.3, and y = 3@(h = 3). The first figure shows how a(t), f(t),and f(t) +hi(t) behave over time. The model appears to converge to themaximizing decision within 25 time periods. Note how expectations andaspirations rise, forcing search to proceed long enough for the actualmaximum achievable objective value to be found. After this time, aspirationsadapt to the mounting evidence that no greater objective value is possible.Eventually aspirations fall and converge towards the optimum. The secondfigure presents the x(t) trajectory through the first 600 time periods.Convergence, for all practical purposes is achieved, however, more than 2,000iterations are required before x(t) converges to x* in the strict sense. Thisresult is representative of the performance under a range of values for 4, fiand 6. Learning and adaptation suffice to lead the decision maker to withinsome neighbourhood of the neo-classical optimum.

The model exhibits other behavior, however, that illustrates the possibilityof sub-optimization through what might be called defective informationprocessing. For example, consider the behavior captured in figs. 3 and 4.Here h has been reduced to 1, representing a decision maker with a shorthorizon. In this situation f(t)+hi(t) does not rise fast enough to force a(t) tostay above f(t) long enough to permit the decision maker to learn themaximum achievable value of f(t). In this case f(t) catches up with a(t).The decision maker becomes satisfied with the achieved objective value toosoon and search ceases too early. Compare this situation with that exhibitedin fig. 1.

The simulations of figs. 1 through 4 depict that what might happen in astable environment where learning and adaptation result in an accumulationof knowledge that ultimately leads the decision maker to the neo-classicaloptimum, or some small neighborhood of it. The real-world, however, isbetter characterized as a changing environment and it is of some interest toinvestigate how the model behaves in such a situation. Therefore the problemconsidered above was repeated with a changing coefficient in the objectivefunction. Figs. 5 and 6 present model behavior in the case where thecoefficient on x(t) is allowed to change according to the sequence{8,9,9.5,8,7,6.5,5,4.5,4,4.5}. For this simulation 4=0.2, p=O.5, 6=0.3,y =24(h=2), and a(0) = 8. Fig. 5 illustrates the ability of the model to learnabout the maximum achievable value of the objective function and track itup or down. Fig. 6 presents the behavior of the decision variable. The solidpiecewise constant line represents the optimal decision value and the smallopen circles indicating the decisions taken by the model. The decisions followclosely the optimal values but never converge precisely to the exact value


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K.D. W all, A m odel of decision making 341

J0 10 20 30 40 50

Fig. 1. Aspirations, expectations, and profit as functions of time for the quadratic functionone-dimension.42) XXf(C)+ h?(C) v---v

f(t)

in

0 100 200 300 400 500 600Time ,n*ex

Fig. 2. The decision variable as a function of time for the quadratic problem in one-dimension.


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342 K.D. W all, A model of decision making

0 10 20 30 40 50Time Index

Fig. 3. Aspirations, expectations, and profit as functions of time for the quadratic problem withinadequate learning leading to suboptimal decisions.a(t) XXj(t) + hi(t) v---vf ( t )

m

0 10 20 30 40 50Time Index

Fig. 4. The decision variable as a function of time with inadequate learning and adaptationleading to suboptimal satisficing decisions.


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K.D. W all, A m odel of decision mak ing 343

0 20 40 60 60 100 120 140 160 160 200Time Index

Fig. 5. Aspirations, expectations, and achieved profit for the quadratic problem with timevarying environment.

4) XXj(C)+ k(t) v---vf(t)

m

N

00 20 40 60 60 100 120 140 160 160 200

Time IndexFig. 6. The decision variable as a function of time for the quadratic problem with time varying

environment.JEB.0 C


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K.D. Wall, A model of decision making

601 04 0.7 1.0 1.3 1.6x( 1): Capal-Labor Ratio

Fig. 7. Profit function contours for the two dimensional example.

before another parameter shift occurs. Note that there is evidence ofsatisticing at times, particularly between t = 115 and 126, and again betweent= 176 and 190.

Simulation Example 2. A very good two dimensional example is found inDay and Tinney (1968). A firm seeks to maximize its profit through themanipulation of two decision variables: xl(t), its capital-to-labor ratio duringperiod t, and x2(t), its production level during period t. At the end of eachperiod the firm observes its profit f(x(t)) =f(xl(t),x,(t)) = f(t) defined as:

f&(t)) =(TOXl(t)l -IT1-i?oll(X*(t)~W2Xl(t))I~OIe1-~02C(XZ(t)w1Xl(t))/001825009

and adds this to its information set composed of past observations on f(t)and x(t). It then decides on a new value for x and implements this for periodt + 1. The firm is not assumed to have any information concerning the formor parameters of the demand function and is not assumed to know its supplyfunction.

The difficulty presented to our model of decision making can be gaugedfrom the contour plot of f(x) displayed in tig. 7. Profit falls off rapidly for


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K.D. W all, A model of decision mak ing 345

capital-to-labor ratios less than 0.15 and production rates less than 15,000units. The ridges encountered just prior to these two precipices oftenconfound many search algorithms by inducing zig-zagging behavior. Moreimportantly, the protit function is strictly concave only within a neighborhoodof x* = CO.3067; 47,274.41] (at which point f* = $183,662.7). Thus for largevalues of x1 and x2, strict concavity does not hold.

The first experiment employs an initial aspiration level of a(O)=$150,000and initializing decisions of

x(O) = [l.So; 89,000], x( - 1) = [1.55,94,000], x( -2) = C1.60, 104,000].The parameters for the adjustment eqs. (10)+12) are set at:

fiZO.20, 6=0.10, Cp=o.40, y=5f$.Two restrictions are placed on x(t) to capture some elements of realism.

First, the firm is assumed to be unable to reduce its capital-to-labor ratiobelow 0.1 and drop production below 14,000 units. Second, the firms speedsof adjustment are limited. In any one period it cannot change its capital-to-labor ratio and its production by more than 10%. Thus the firm has tochoose x(t) subject to:

and 1dx,(t)/x,(t- 1) / 50.1, (dx,(t)/x,(t- 1) ( 50.1).Fig. 8 displays the decision history in terms of x,(t) versus x,(t). The

model finds the neoclassical optimizing solution after approximately 150iterations and, for all practical purposes, converges. The value of the profitfunction remains within one decimal place of the optimum, x1 is within twodecimal places of its optimizing value, and x2 matches its optimizing value towithin 0.50/,.

As with the first simulation example, one might ask what the behavior ofthe model is if it is embedded in a changing environment. Would it ever beable to find a time-varying optimum? Would the decision variables displaybehavior that could be interpreted as a sample from some stochastic process?Some food for thought is provided by figs. 9 and 10. Here the modeloperates in a shifting environment much like that leading to figs. 5 and 6.Two shifts in the parameters of the profit function are introduced, corres-ponding to shifts in the demand curve and in the factor input supply curves.One shift occurs at t = 120 and the other at t =250. Over the period of thesimulation there are now three different maxima that must be sought:

x* = CO.31; 47,274], f* = $183,663,


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K.D. Wall, A model oJdecision making

001 0.4 0.7 10 1.3 1.6Capital-Labor Ratio

Fig. 8. Convergent behavior in decision space for the two-dimensional profit maximizationproblem.

x* = CO.86; 38,893], f* = $135,660,x* = CO.68; 80,423], f* = $254,212,

For this simulation experiment all model parameters are kept at theirprevious values except C$= 0.05.

The model locates the three maxima to within some small neighborhood.Fig, 9 presents the path taken in decision variable space, clearly illustratinghow search proceeds from one solution to the next as information is receiveddiscontirming prior conceptions of where the optimal decision is located. Fig.10 shows that the model requires approximately 100 time periods to learnwhere the optimal decision resides.

5. Estimation frameworkFor the model to be truly useful in empirical research it must permit

estimation and testing in conjunction with time series data. This requirescasting eqs. (9)-(17) in a form suitable for econometric work and oneframework immediately suggests itself. Eqs. (lo)-( 12) comprise a recursive


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01 0.4 0.7 10 13 16Capital-Labor Ratlo

Fig. 9. Behavior in decision space for the two-dimensional problem with time varyingenvironment.In0 * f x I

-0 50 100 150 200 250 300 350nme index

Fig. 10. Aspirations, expectations, and achieved prolit as functions of time for the two-dimensional profit maximization problem with time varying environment.a(t) XX

f(t) + hi(t) v---vf(t)

JE.90. D


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348 K.D. Wall, A model of decision making

system of linear difference equations that bring to mind a state-spaceapproach employing a Kalman filter for generating model residuals.To see this, define as the state a 3 x 1 vector, s(t), where

s(t) = C?(t), i(t), a(t)and define z(t) to be a vector of (N + 1) exogenous, or predetermined, inputs:

z(t)=[f(t),f(t-1) )...) f(t-N);x(t),x(t-l))...) x(t-N)].A a state space form representation ofresults:

the decision model immediately

s(t) =Fs(t- 1) +Gz(t), (18)y(t) = H(z(t))s(t) + h(z(t)), where (19)r(1-B) 0 1 ra 0 o...o1F= 01 (l-6) 0 ,G=1 1 6 -6 O...O ,4 Y (l-4) 0 0 o...o 1H(z(t))=CO:O:ctlllCt1121,

and y(t) =x(t). The state space model is linear in the state but time-varying inthe output equations because of the terms involving c,/ll c, 11. It should benoted, however, that this time variation in the coefficient matrix H is duesolely to exogenous variables and may be treated as predetermined. Estimat-ing the model can be accomplished by a number of techniques, for examplethat used in Burmeister, Wall and Hamilton (1986), Kalaba and Tesfatsion(1980, 1988), or Pagan (1980). If one is willing to state stochastic hypothesesregarding additive disturbances in (18)419), then a Gaussian likelihoodfunction can be postulated and the methods of the first and last referencesabove apply. If one does not wish to introduce additive random errors thena least-squares approach can be taken and the methods of Kalaba andTesfatsion applied. In either case, the methods are well developed, tried andtrue.

The researcher applying this estimation framework needs to obtain timeseries data for f(t) and x(t) over the period of interest, say, {to ZL~:~}, anduse them to form the z(t) vector. The c, are constructed using (13) and thefirst N observations of f(t) and x(t) as initial conditions. The requiredpseudo-inverses can be accomplished by singular value decomposition.Finally, H(z(t)) and h(z(t)) are formed. The need for initial conditions in


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K.D. Wall, A model cfdecision making 349computing the c, means estimation proceeds using the data over the intervalI o + N - 1 stst/ }. Four parameters are estimated: {#,/I, 6,~). In addition,.~n estimate of the trajectory of a(t) over the sample period can be obtainedby coupling the parameter estimation algorithm to a state space mode1 filter/smoother algorithm like that of De Jong (1989). This allows the allimportant goal formation behavior to be revealed and can be used to shedlight on how aspirations have responded to stimuli over the sample.

As an illustration of formulating an estimation problem, suppose onewishes to study the behavior of the U.S. auto industry over the past fourdecades, with particular interest in how the industry responded to growingforeign competition. In this case behavior might be interpreted in terms ofthe decisions taken to adjust production capacity, employment, and price.Thus x(t) is a 3 x 1 vector composed of time series observations on each ofthcsc three variables. Furthermore, it is hypothesized that decisions are basedupon attention to profits; thus S(t) is the observed industry profit in period t.If it is hypothesized that decision makers focus on market share, then f(t) isthe observed market share of domestic auto producers. Depending on thefrequency of observation, t is indexed by months, quarters, or years.l.\timation of this mode1 requires four time series, and then their use ingenerating the three time series comprising the elements of c,. The fourehtimated parameters, together with the estimated trajectory for a(t), wouldyield information on the rate with which the industry responded to theforeign challenge, how industry goals were affected by changing marketconditions, or whether profit or market share better reflects the concerns ofindustry management.

6. Discussion and conclusionsThe primary concern of this paper is the presentation of an operational

model of boundedly rational decision making that can be used in empiricalstudies of dynamic economics. The models main attributes are: (1) itsinherently dynamic nature, for it naturally explains the evolution of decisionsby framing them in terms of difference equations; (2) its ability to depict bothneoclassical optimizing behavior and non-neoclassical suboptimizing be-havior; (3) its very modest assumptions concerning the information set of thedecision maker; (4) its very modest assumptions concerning the com-putational and cognitive abilities of the decision maker; and (5) its explicitincorporation of learning and adaptation and the simplicity of the mecha-nisms by which this takes place. Learning and adaptation are the dominantcomponents in the model; they determine whether one obtains the optimizingsolution or a suboptimal result. The former may be characterized as a speciallimiting case where, given a stable environment and sufficient time, learningand adaptation is adequate to guide the search to the overall maximum.


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350 K.D. Wall, A model qf decision making

Suboptimization occurs when learning or adaptation allows aspirations toconverge too rapidly to the actual objective function value thus terminatingsearch prematurely.

This last point brings out an important feature of the model. It demon-strates the need to strike a balance between reacting rapidly to changes inthe environment and overreacting to ephemeral disturbances. If aspirationsadapt too slowly to information, then decision making will proceed toocautiously and search will be prolonged unnecessarily. On the other hand, ifaspirations adapt too rapidly, then it is possible to converge to a sub-optimum. For example, consider a situation like that depicted in fig. 10 att= 120. If aspirations adjust too rapidly to the adverse change, then thedecision maker will be inclined to satisfice and accept a profit (of approxi-mately Sl20,OOO) lower than that which is ultimately achievable ($135,660). Asimulation with 4=0.5 bears this out; convergence to the first optimum($183,663) occurs very fast, but between t= 120 and t=250 the model endssearch at a profit of $124,755.

Perhaps the most important characteristic of the model is its ability toportray long-run behavior different from that of the neoclassical paradigm.The model presented here can display suboptimal satisficing behavior, as wellas optimizing behavior. The sole determinants of this are the parametervalues: p, 6, 4, and y (or h).

The model is but one operationalization of the key mappings in eqs. (2)(3) and (6). Other operationalizations can be specified, but the ones presentedin this paper are believed to be the most simple. Even so, these simplespecifications, when coupled with learning and adaptation demonstrateremarkable abilities.

A number of questions are to be investigated in the near future. First thereis the proof of convergence to the neoclassical optimum when a(t) is notfixed at f*. The convergence of the algorithm has been demonstrated insimulation, but is seen to depend in a complicated way on a(O), theparameters p, y, 6 and C#J nd the specification of X(t). A proof appearspossible with the aid of a theorem on algorithmic convergence by Zangwill(1967, ch. 11).

Next there is the investigation of the effects of a changing environment.Both examples show the algorithm capable of finding new optima as theenvironment undergoes shifts. It seems important to establish if limits existto the ability of the model to learn enough to keep pace with rapidlychanging environments. For example, At what speed of change in theenvironment does the model begin to lose its ability to catch up with theoptimum?

Finally, there are a number of normative questions that the model can beused to answer. For example, if we assume a boundedly rational decisionmaker, as above, how can the information set be ammended to help the


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K.D. Wall, .4 model of decision making 351

decision maker converge more rapidly to the maximum. In a rapidlychanging environment, how can the information set be altered to aid intracking the optimum and, hopefully, to catch up with it? In other words,what can be done to increase learning and speed adaptation? The modelappears ideally suited to help in answering these and similar questions.

AppendixProof of T heorem 1. The argument is by induction. Let the 1 x N gradientvector of ,f at a point p be denoted Df (p). At time t=O we know from theprescribed initialization that Df(x( -j)) x [x(-j+ 1)-x(-j)] >0 for 1 0 and Df(C%jx( j))[x* -x(O)] >Df(x(O))[x*-x(O)]. Therefore cb[x*-x(O)] >O and d(1) is an ascent direc-tion with a component along x* -x(O) in a direction towards x*. If the stepalong d( 1) is not too long, the algorithm will produce an x( 1) with ,f(l)>f(0). In fact any step of length less than 11 *-x(O) (( will produce this result.

To show that the algorithm gives such a step length note thatx(~)-x(O)=C.~*--~(X(O))IC~IIICO12=Df(~*)C~*-x(0)lco/llco II*>

and11x(l)-40) //

whereP = 11J(g*)((//I f(tzz)/.But f om the strict concavity of f, p < 1 andkb ;;x:;ic; i;;(;;;i)l; so the step to x(l) is not too large so as to overx .(o))

Now assume that for some t=n, ,f(n)>J(n- l)>f(n-2)>f(n-3)~ ..>f(n- N). By exactly the same arguments as above, ck[x*-x(n)] >O, sod(n+ 1) is a direction with a component along x*-x(n) and towards x*.Furthermore, I/x(n+l)-x(n))If(n). Thus f(n)converges monotonically to some upper bound, F, and by continuity, x(n)converges to some point z such that F =f(z). To see that this upper bound isindeed f*, and that z=x*, consider (16) rewritten as

IIc,//C~(~+1)-x(~~l=C.f*-f(~)lc,lllc,//In the limit as PZ-+CG,c,-Df(d, the lefthand side +O, and c,/IIc,/(+l. Thisimplies ,f(x(n))-.f* and continuity of f implies x(n)-+z=x*. Q.E.D.


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