Duality, Optimization, and Microeconomic Theory:...

Duality, Optimization, andMicroeconomic Theory:Pitfalls for the Applied Researcher

C. Robert Taylor

This article graphically illustrates the one-to-one duality mapping among theproduction function, the product supply equation, the derived factor demandequation, and the indirect profit function for the classical profit maximizationproblem. This pedagogical framework is then used to illustrate how empiricalapplication of conventional duality theory can lead to distorted empirical results if thetheory (e.g., Hotelling's lemma) does not apply because the firm is not a profitmaximizer or because envelope results from the wrong optimization model are used.Although the presentation is in terms of profit maximization, the basic concepts canbe extended to other maintained behavioral hypotheses such as cost minimization orutility maximization. Plausible reasons why a firm, even in a competitive market,may not behave according to the neoclassical maximization paradigm are brieflyreviewed.

Key words: duality theory, microeconomic theory applications, optimization.

Applications of duality theory to empiricalproblems are widespread. Some of the claimedadvantages of a dual approach are: (a) it opensup a richer class of operational functionalforms, especially for multiproduct, multifactorproduction; (b) it brings theoretical coherenceto the analysis, especially with respect to cross-commodity relationships, that is often lackingin nondual approaches; and (c) it is possibleto obtain factor demand and product supplyequations from an indirect profit function fit-ted to profit and price data without havingempirical observations of quantities demand-ed or supplied (Pope 1982b; Lau and Yoto-poulos 1971, 1972; Young et al.).

Most empirical studies have used dualitytheory associated with the conventional static,deterministic model of perfectly competitivefirm behavior (e.g., Binswanger; Lau and Yo-topoulos 1971, 1972; Trosper; Lopez 1984,1985; Weaver 1983; Collins and Taylor; Kako;Garcia, Sonka, and Yoo; Garcia and Sonka;Arif and Scott). Although the envelope theo-

The author is ALFA Professor of Agricultural and Public Policy,Auburn University.

Editorial comments by Bruce Beattie, Patricia Duffy, and anon-ymous reviewers are gratefully acknowledged.

rem always holds for optimizing behavior,conventional duality results obtained by ap-plying the envelope theorem to a particularmodel (i.e., Hotelling's lemma for the classicalprofit maximization model and Shephard'slemma for the classical cost minimizationmodel) do not necessarily hold in the case ofconstraints on profit maximization (e.g., Leeand Chambers), in the case of uncertainty (Pope1980, 1982a), or in the case of stochastic, dy-namic problems (Taylor). Furthermore, thereare plausible reasons for questioning the neo-classical maximization hypothesis, meaningthat we should question the validity of a dualapproach or a primal approach that uses first-order conditions for optimization.

This article graphically illustrates the one-to-one duality mapping among the single-in-put production function, the product supplyequation, the derived factor demand equation,and the indirect profit function for the classicalprofit maximization problem. This pedagogi-cal framework is then used to illustrate howempirical application of conventional dualitytheory may lead to distorted empirical resultsif an inappropriate duality mapping is exploit-ed. The graphical framework gives insight into

Western Journal of Agricultural Economics, 14(2): 200-212Copyright 1989 Western Agricultural Economics Association

Duality Pitfalls 201

distortions and how they can be minimized invarious cases. While pedagogical presentationin this article is in terms of the classical profitmaximization problem, the same basic con-cepts can be applied to the classical cost min-imization problem or to more complexoptimization such as expected utility maxi-mization.

For those who are not familiar with criti-cisms of the maximization hypothesis, the richbut partially obscure literature in this area isreviewed to establish that there are plausiblearguments for not taking the hypothesis as truea priori or as a tautology and thus to establishthat there are plausible reasons for questioningempirical application of duality theory. Prob-lems in empirically testing such a hypothesisare also briefly reviewed.

Graphical Exposition of Duality

Let us now turn to a graphical presentation ofduality. For a mathematical treatment of dual-ity for the classical profit maximization prob-lem, readers are referred to Henderson andQuandt; Varian (1984a); Beattie and Taylor;or Young et al.

To simplify graphical presentation of dual-ity, assume that a single product, y, is producedwith a single input, x, and that technology isgiven by the continuous, strictly concave pro-duction function, y = f(x). To further simplifypresentation, only product price, p, is variedwhile factor price, r, is held constant. Asterisksdenote profit maximizing levels of the vari-ables.

The Envelope Theorem

Since the envelope theorem as manifested byHotelling's lemma is the heart of duality the-ory, it is instructive to begin with a graphicalpresentation of this theorem and the relation-ship between the indirect profit function, lr* =7r*(p, r), and the direct profit function, r = py- rx. Figure 1 shows the indirect profit func-tion as related to product price. Since directprofit does not involve optimization, there isa family of direct profit equations that can bedrawn in figure 1; 7r0 and r,, which differ onlyby the fixed level ofx and thus y, illustrate twoequations in this family. The direct profitequations are linear because product price en-ters the direct profit equation linearly.

PO P1 PFigure 1. The envelope theorem applied to anindirect profit function

Under profit maximization, the highest prof-it is chosen for any (given) product price. Forproduct price level Po in figure 1, this profitmaximizing point is at A on the direct profitequation, 7r,. If product price changes to p,, adifferent direct profit equation is chosen be-cause output and factor quantities are adjustedin response to the new product price. The newmaximum profit level is at point B in figure 1.

The indirect profit function is the locus ofprofit maximizing points associated with allprices, thus forming an upper envelope of thefamily of direct profit equations.' Wordedanother way, for a given product price, say po,indirect profit will equal direct profit only ifdirect profit is evaluated at the profit maxi-mizing levels, x*(po, r) and y*(po, r). Thus, ifXo = x*(po, r) and Yo = y*(po, r), then 7ro infigure 1 will equal wr*(po, r) at point A. Thisrelationship will not hold, however, if anotherpoint on 7ro is selected or if another point ona different direct profit equation, say 7r-, is se-lected.

The envelope theorem follows from thisenvelope relationship. The partial derivative

'Note that in the multiple-input case, there exist direct profitequations that are everywhere below the indirect profit function.In the single-input case illustrated here, each direct profit functionwill be tangent to the indirect function at one point.

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Western Journal ofAgricultural Economics

Yo

xo x1 X2(a) Production function

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(b) Inverse factor demand (c) 45 degree line

Figure 2. Relationship of the production function, product supply, factor demand, and indirectprofit under profit maximization

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202 December 1989

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dr*l/p = y* = dro/dp = y,. In the case of theclassic profit maximization problem, this re-lationship is referred to as the product supplyproperty of Hotelling's lemma. Again, this en-velope relationship holds only for dr/Op eval-uated at x* and y*. For example, ify, # y*(po,r), then &dr*/Op # dar l/p = yi evaluated at po,since 7rl is not tangent to ir* at point A (ratherit is tangent at point B).

Duality Mappings

The five panels of figure 2 show the dualitymapping among the production function, theproduct supply equation, the derived factordemand equation related to output price, andthe indirect profit function. Panel (a) of thisfigure shows the production function, panel (b)is the inverse factor demand equation withrespect to output prices, panel (c) is a 45-degreeline to transfer product price from panel (b) topanels (d) and (e), panel (d) is the product sup-ply equation, and panel (e) is the indirect profitfunction. To graphically see how x*, y*, andr* are derived from the production function,

y = (x), consider an output price of Po. Thefirst-order condition for profit maximizationis dy/dx = MPP(x) = r/po, where MPP is mar-ginal physical productivity expressed as a func-tion of x. Graphically, the profit maximizinglevel of factor usage, x0, is associated with thepoint of tangency between the productionfunction and a line with slope equal to r/po.This level of factor usage is traced from theproduction function in panel (a) to a point onthe derived factor demand equation, x*(p, r),in panel (b). Panel (c) translates the profit max-imizing output level associated with x*, whichis yo, from the production function in panel (a)to a point on the product supply equation, y*(p,r), in panel (d). Indirect profit is given by eval-uating the direct profit function at the profitmaximizing input and output combination,which gives r* = py*(p, r) - rx*(p, r). Indirectprofit is traced from the production functionin panel (a) through panels (b), (c), and (d) topanel (e). Parameters and functions used forthe relationships in figure 2 are given in theappendix.

In an empirical setting, variation in p wouldgenerate points on the production function, thederived factor demand equation, the productsupply equation, and the indirect profit equa-tion. Given observations on all relevant vari-ables, empirical estimation of the relationships

in figure 2 could be carried out, in principle,with either a traditional approach or a dualapproach. With a traditional approach, obser-vations on x and y would be used, for example,to estimate the production function. Then, giv-en the production function, factor demand andproduct supply equations associated with prof-it maximization could be derived. The indirectprofit function could then be obtained by sub-stituting the functions x*(p, r), and y*(p, r), ob-tained from explicitly solving the maximiza-tion problem, into the direct profit functionfor x and y, respectively.

Pitfalls of the Dual Approach

Although there are many advantages of a dualapproach, there are potential pitfalls associ-ated with it (see e.g., Pope 1982b; Young; Lo-pez 1984; Chambers; Varian 1984a) if themaintained behavioral hypothesis is invalid,if constraint and information sets have not beencorrectly identified, or if the wrong dualitymodel has been specified. The graphical frame-work established in figure 2 gives insight intodistortions that can result from inappropriateuse of duality and also gives insight into howsuch distortions can be eliminated or reducedin certain applications.

There are three variations of the dual ap-proach to empirically estimating the equationsshown in figure 2. The approach commonlyused when there are observations on profit andprices, but no observations on the quantitiesx and y, is to estimate the indirect profit func-tion, then obtain x*(p, r) and y*(p, r) by Ho-telling's lemma (Binswanger; Lau and Yoto-poulos 1971; Lopez 1984, 1985; Shumway,Saez, and Gottret; Moschini). Because of theone-to-one mapping between the indirect prof-it function and the production function, it ispossible, at least in principle, to obtain theproduction function y = f(x) from r*(p, r),although in practice a closed form expressionof the production function can be obtainedonly for certain mathematical forms of the in-direct profit function. A second dual approachis to estimate the set of factor demand andproduct supply equations (Weaver 1983; Kako)then extract technical relationships using thedual mapping. The third dual approach is toestimate the system of factor demand andproduct supply equations jointly with the in-direct profit function (Trosper; Garcia, Sonka,

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Western Journal of Agricultural Economics

and Yoo; Lau and Yotopoulos 1972; Arif andScott). We now use the graphical frameworkin figure 2 to gain insight into distortions thatcan result from the three dual approaches men-tioned above if the maintained hypothesis ofprofit maximization does not hold or if thespecific manifestation of the envelope theo-rem, which in this case is Hotelling's lemma,breaks down for other reasons (see e.g., Pope1982a, b; Taylor; Lee and Chambers; Weaver1982).

Consider a case where the firm does not al-ways employ the profit maximizing level offactor x. A critical assumption for this illus-tration is that Hotelling's lemma does not hold.Whether the assumption does not hold be-cause (a) the firm is not a profit maximizer;(b) the firm has the wrong perception of tech-nology; (c) we have not identified appropriateconstraints on profit maximization; or (d) wehave used the wrong classical profit maximi-zation dual model rather than a constrainedmodel (Lee and Chambers), an uncertaintymodel (Pope 1980, 1982a), or a stochastic dy-namic model (Taylor) is not central to thegraphical analysis.

Figure 3, which is based on the same frame-work used in figure 2, illustrates a case wherethe firm applies the profit maximizing level ofx, xl, at a price of pi, but for a lower outputprice, say po; the firm responds by employingxg units of the factor, which is more than theoptimal amount, xO. Similarly at a price abovepi, say P2, the firm is assumed to employ xaunits of the factor, which is less than the op-timal amount, x2. Critical parameters associ-ated with the relationship in figure 3 are givenin the appendix.

A firm's response to variation in price wouldgenerate a time series of points along the in-verse factor demand curve, xa(p, r), shown inpanel (b); points on the product supply equa-tion, ya(p, r), shown in panel (d); and pointson the profit function, rTa(p, r), shown in panel(e) of figure 3. For comparative purposes, theindirect profit function, product supply, andderived factor demand equations associatedwith profit maximizing behavior are also shownin panel (e) as r*(p, r), in panel (d) as y*(p, r),and in panel (b) as x*(p, r), respectively.

For the case illustrated in figure 3, the em-pirical profit function, ra(p, r), is equal to theindirect profit function, 7r*(p, r), at a price ofpi, because it was assumed that the firm usedthe profit maximizing level of x at that price,

but not at other prices. If the firm does not usethe profit maximizing factor level at any price,the empirical profit function will always bebelow the indirect profit function.

Indirect Estimation of Demand andSupply Equations

Consider now a dual approach to estimatingthe profit function, ra(p, r) as an indirect wayof obtaining product demand and factor sup-ply equations. Note that in this case we areconsidering estimating only the profit equationwith demand and supply equations derived byapplication of Hotelling's lemma to the fittedprofit function, -ra(p, r).

A distortion resulting from the dual ap-proach to obtain demand and supply functionsfrom an empirically estimated profit functionis illustrated in figure 4, panels (a) and (b),which are expanded versions of panels (e) and(d) in figure 3. Subscripts and superscripts usedin figure 4 are consistent in definition withthose used in figure 3. The function ira(p, r) inpanel (a) of figure 4 is actual profit as relatedto price; the true indirect profit function is 7r*(p,r). Direct profit functions for two input-outputcombinations are also given in figure 4; 7rc isdefined by the profit maximizing pair (yo, xO)for a price of p, while rd is associated with thenonmaximizing pair (Yo, xo). Td is an implieddirect function tangent to the fitted profit func-tion at point D, while irc is the direct profitfunction tangent to the true indirect profitfunction, 7r*(p, r), at point C.

Consider a price of po in panel (a) of figure4. Application of the envelope theorem to thefitted profit function, ra(p, r), at the price Powould imply a tangency with the function ira(p,

r), at point D. The output level implied byHotelling's lemma would thus be y0, which isshown in panel (b). However, the true directprofit function for a price of Po is rc with as-sociated quantityyO is, in general, different thanyO. As graphically illustrated (see the appendixfor the mathematical functions used to con-struct the figures), the supply curve obtainedby applying Hotelling's lemma is y(p, r), whichis neither the product supply associated withprofit maximizing behavior, y*(p, r), or theactual product supply curve, ya(p, r). In gen-eral, yA # yg = yo. We do not obtain the profitmaximizing supply curve, y*(p, r), because ·r*is not everywhere equal to Tra, and we do notobtain the actual supply curve, ya(p, r), because

204 December 1989


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Figure 3. Relationship of the production function, product supply, factor demand, and profitwhen profit maximization does not hold

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Hotelling's lemma is not valid in this case.Elasticities of supply based on yd(p, r) willtherefore be distorted in this illustration.

Without profit maximization, we cannotmake any general statements about the cur-vature of the profit function even with a con-cave production function. However, an inter-esting feature of this particular example is thatthe actual profit function, ra(p, r), in panel (a)of figure 4 is not globally convex, which ismanifested in panel (b) by the negatively slopedsupply relationship (at low prices) obtained byapplication of Hotelling's lemma. In empiri-cally fitting a profit function in such a case,several errors could be made. First, we mightforce a convex function on the data set whichwould clearly lead to distorted estimates of theactual profit function. Second, data points couldspan the nonconvexity thereby hiding theproblem. Third, we could fit a flexible func-

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about specification and other biases rather thanleading us to consider the validity of the main-tained hypothesis.

Figure 5 illustrates a second departure frompromt maximization. (See tne appencdx tor nu-merical details.) In this case, only half the op-timal input level is used, as might happen ifthe firm had incorrect perceptions about tech-nology. In this case, the actual profit functionis convex for all prices, but the supply functionderived from the profit function by applicationof Hotelling's lemma results in a supply curvethat lies between the actual supply curve, ya(p,

r), and the supply curve associated with profitmaximization, y*(p, r). Data points generatedby the firm's response to varied price wouldgenerate points along ya(p, r) and not alongeither y*(p, r) or yd(p, r).

In the numerical example shown in figure5, it was assumed that the firm underappliedthe input resulting in a derived supply curve,yd(p, r), that was less elastic than the actualsupply curve. If the firm overapplies the input(not shown), then the actual supply curve willbe above the profit maximizing supply curve,Vlwhile the Clerive;xr llfnv curTllrveT wX7ill he hPlfw

*Po- P the actual and profit maximizing supply curves.(b) Product supply equations Also, the derived curve will be more elastic

than the actual supply curve in this case. ThisFigure 4. Application of the envelope theo-numerical example shows that without know-rem to obtain a factor demand equation in the ing ya(p, r), we cannot determine if applyingcase of nonmaximizing behavior Hotelling's lemma in the absence of profit

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206 December 1989

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maximization results in an upward or down-ward bias of supply elasticities.

The preceding discussion shows that fittinga profit function to observations on prices andassociated profit may give distorted estimatesof demand and supply functions if Hotelling'slemma does not hold. Although estimates ofthe actual profit function may be statisticallyunbiased (unless there is a functional form biasor inappropriate cross-price relationships suchas symmetry implicit in the dual approach),the graphical analysis suggests that supply anddemand equations derived from a fitted profitequation should be cautiously interpreted.

Without data on input and output quan-tities, we may have no choice but to derivefactor demand and supply equations on thebasis of a fitted profit function. However, it isimportant to recognize that the resulting de-mand and supply equations may not corre-spond to either actual behavior or to profitmaximizing behavior.

Direct Estimation of Demand andSupply Equations

Consider directly estimating product supplyand factor demand equations (but not the prof-it function) for the case depicted in figure 3.In this case a dual approach and a primal ap-proach that uses first-order conditions areequivalent, assuming consistent functionalforms. Assuming no functional form bias, thefitted demand equation would be xa(p, r) inpanel (b), and the fitted product supply equa-tion would be ya(p, r) shown in panel (d). Eventhough the fitted demand function and the fit-ted supply function differ from the functionsbased on profit maximizing behavior, they maynevertheless be valid supply and demand be-havioral relationships.

Without functional form bias, fitting eitherthe demand or supply curve will give an un-distorted estimate of that equation. However,fitting the set of equationswill give undistortedestimates only if the symmetry implicit in thedual or primal specification is appropriate.Symmetry of the behavioral relationshipswould imply that 0 ya(p, r)/Or = -xa(p, r)/dp.

Note that there are no logical reasons to ex-pect that, in general, symmetry holds withoutoptimization. For example, it is well knownthat in the classical expected utility (EU) mod-el of firm behavior, cross-price effects based

(a) Profit equations

- 2(b) Product supply equations

Figure 5. Application of the envelope theo-rem to obtain a factor demand equation in thecase of nonmaximizing behavior

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Western Journal ofAgricultural Economics

on ordinary demand functions are not sym-metric (because of the income effect). If thereis an expenditure constraint on profit maxi-mization (Lee and Chambers), it can be shownthat some cross-price relationships are notsymmetric because the cost constraint has thesame effect in this model as the budget con-straint in the EU model. Certain stochastic,dynamic characteristics of optimization prob-lems also introduce asymmetric cross-price ef-fects (Taylor). That is, the reciprocity condi-tions that are a byproduct of the envelopetheorem do not necessarily lead to symmetriccross-price effects for ordinary factor demandand product supply equations for some exten-sions of the classical profit maximization prob-lem.

In any empirical application, however, wecan statistically test for symmetry using stan-dard procedures and tests. But it is importantto note that if symmetry does hold, we cannotlogically conclude profit maximization be-cause certain kinds of symmetry hold for othermodels.

Simultaneous Estimation of All EconomicRelationships

A third variation of the dual approach is toestimate simultaneously a consistent set ofequations for all economic relationships. Withthis approach, a functional form for an indirectprofit function is specified, then the functionalforms for demand and supply equations arederived by application of Hotelling's lemma.The set of equations (supply, demand, and in-direct profit) are estimated as a system. Thisapproach is theoretically appealing because aconsistent set of equations that satisfy sym-metry and curvature properties is fitted. It couldbe argued that by using more equations, andthus all data, sharper estimates of parameterswill be obtained.

If Hotelling's lemma does not apply, how-ever, all of the equations will be distorted be-cause fitting the system will compromise allfitted equations. Supply and demand equa-tions will not fit actual data because this willdistort the profit equation away from datapoints; likewise, fitting the profit equation ex-actly will distort supply and demand equationsaway from the profit data points. Thus, fittedsupply and demand equations will lie some-where between the actual and maximizing re-lationships, while the fitted profit function will

lie between the actual function and the trueindirect function. The extent of the distortioncannot be determined a priori, or even ex postwithout knowing the actual relationships.

Derivation of the Production Function

Use of duality mappings to obtain the pro-duction function or related technical measuressuch as marginal physical productivity can alsogive distorted estimates with inappropriate useof Hotelling's lemma. This distortion can beseen in two different but equivalent ways. Oneway is to note that in the single-factor case, thefactor demand equation is the inverse of theMPP function; that is, solving the first-ordercondition for x gives x* = MPP-~(r/p) = x*(p,r). We can thus obtain MPP(x) by invertingthe function x*(p, r). However, if we use thefunction xa(p, r) or xd(p, r) rather than x*(p, r)to obtain estimates of MPP(x), it can be seenby comparing x*(p, r) to xa(p, r) or comparingx*(p, r) to xd(p, r) in panel (b) of figure 3 thatthese estimates are distorted.

A second way to view the distortion is thatthe dual approach infers that the slope of theproduction function is equal to the ratio of thefactor price to the product price. Thus, at xothe inferred slope of the production functionwould be r/po, which can be seen in panel (a)of figure 3 to be greater than the slope of thetrue production function at xg. Under the as-sumed nonprofit maximizing behavioral caseillustrated in figure 3, the inferred productionfunction will be more concave than the trueproduction function.

Validity of the Expected Utility Model

Belief in the expected utility maximization hy-pothesis by students of neoclassical micro-economics appears to be widespread. It couldbe euphemistically said that many economistsseem to belong to the Austrian school ofthought. As Caldwell notes, "Austrians ... in-sist that the [maximization] hypothesis is thefundamental axiom of human action which isknown to be true a priori but which nonethe-less has empirical content." The competitivemarket model, which is another element of theAustrian school, is often used as an argumentfor profit maximization; firms that do notmaximize profits are driven out of the marketby competitive forces.

208 December 1989


Due to the competitive nature of many ag-ricultural markets and the seemingly unques-tioned acceptance of the maximization hy-pothesis by some economists, it is appropriateto digress on why the hypothesis itself shouldbe questioned and thus establish why empir-ical use of certain envelope theorem resultsshould be questioned. For generality, the re-view is in terms of the EU model; expectedprofit maximization can, of course, be viewedas a special case of the EU model. The pur-poses of this review are simply to establish thatthere are plausible arguments for not takingthe maximization hypothesis as true a priorior as a tautology and to direct the interestedreader to relevant literature. Problems in em-pirically testing such a hypothesis, particularlyin the context of applications in agriculturaleconomics, are also briefly discussed. I beginwith consideration of the competitive marketargument then turn to the maximization hy-pothesis.

Competitive Market Argument

The classical model of perfect competition withits assumptions about perfect information andfree entry and exit leads to profit maximizationby all firms who remain in the market. Sincethere are no pure profits in a perfectly com-petitive market, firms who are not profit max-imizers will incur losses and exit from the mar-ket.

Extension of this textbook argument to com-petitive agricultural markets is not direct, how-ever, because the classical assumptions maynot be appropriate for the following reasons.First, profits earned by most agricultural firmsare affected by stochastic factors such as un-controllable crop yields, price instability, andthe recent instability of financial institutions.In a practical setting, the highly stochastic andlargely uncontrollable nature of returns coulddominate decisions that are not consistent withexpected profit maximization. For example,few, if any, economists would argue that thefinancial crisis in agriculture in the early 1980sweeded out only those farmers who were tra-ditionally viewed as "poor" managers. Sec-ond, wealth levels of firms can keep them vi-able for several years and perhaps forgenerations if the decision makers are espe-cially stubborn or reluctant to move out offarming. Third, if current firms and the poolof potential new firms do not have a profit

maximization objective, the market can bedominated by other kinds of behavior. Fourth,the pervasive influence of off-farm earningsmay dramatically alter agricultural decisions.Thus, there are plausible conceptual reasonsto find individual behavior in agriculturalmarkets that is not consistent with profit max-imization.

Even if the profit maximization hypothesisis valid, presence of imperfectly competitivemarket elements can obviously make dualitytheory break down. This can be seen in thelimiting case of monopoly. Since a monopolistcontrols price, output price is not even an ar-gument in the indirect profit function; rather,parameters of the demand function are argu-ments in the indirect profit function for a mo-nopolist.

Maximization Hypothesis

Consider now the validity of the neoclassicalmaximization model, the heart of which is theEU model. After an extensive review of thevariants, purposes, evidence, and limitationsof the EU model, Shoemaker concluded that"EU maximization is more the exception thanthe rule." From a descriptive perspective, heargued that the EU model failed on threecounts. First, people do not structure problemsas holistically and comprehensively as EU the-ory suggests. Second, they do not process in-formation, especially probabilities, accordingto the EU model. Burks notes that there arealso philosophical problems with the classicalnotion of probability. Third, EU theory poorlypredicts choice behavior in laboratory situa-tions. MacCrimmon and Larson note that"...many careful, intelligent decision makers doseem to violate some axioms of expected util-ity theory, even upon reflection of their choices

99

From a positivistic perspective, many schol-ars have noted that the EU model is useful forpredicting behavior, although accuracy of theprediction is often less than desirable. The samemight be said of empirical applications of dual-ity theory. Even though the EU model is ac-knowledged as having predictive value, un-derlying rationality assumptions have beenquestioned (Shoemaker). Even if the EU mod-el predicts well while its assumptions are wrong,the notion that only prediction matters is epis-temologically unappealing (Shoemaker; Sam-uelson).

Taylor


The cursory review given above providesplausible reasons for questioning the EU mod-el in general and the profit maximization mod-el in particular; therefore, in empirical appli-cation of duality theory we must alwaysquestion validity of the maintained hypothe-sis. Readers interested in additional readingon the neoclassical maximization hypothesisare referred to extensive references in Shoe-maker and in De Alessi.

Testing the Maximization Hypothesis

From a philosophical standpoint, there is noapparent agreement on whether the maximi-zation hypothesis is testable, as evidenced byan exchange between Boland (1981, 1983) andCaldwell. Some view the hypothesis as a tau-tology which is by definition untestable. Bo-land (1983) argues that the hypothesis is nota tautology but that no criticism of it will everbe successful. Caldwell agrees that the EUmodel is untestable. He states:

There are a number of problems associated with testingthe hypothesis; perhaps the most telling is that any directtest, including the revealed preference approach, re-quires that assumptions be made concerning the sta-bility of preferences of the choosing agent, as well as thestates of information confronting him. Since the contentof these assumptions ... are subject to change but arenot themselves directly testable, test results ... are notunambiguously interpretable. (pages 824-825)

Caldwell further argues that the EU model isuntestable because "utility" is an undefinedtheoretical term, but since "profit" is measur-able, profit maximization is logically testable.However, the above cited problems associatedwith assumptions about the states of infor-mation confronting the decision maker applyto testing absolute (as opposed to relative) prof-it maximization as well as testing the generalEU model.

Various parametric and nonparametric ap-proaches to testing for profit maximizationhave been investigated. The parametric ap-proach tests for departures from the first-orderconditions for profit maximization; that is, thetest is implicitly or explicitly based on a com-parison of price ratios to marginal physicalproductivities (e.g., Dillon and Anderson). Aweakness of the parametric approach is thatthe test is conditional on the functional formselected for the production function. Since

technical or biological theory rarely indicatesthe appropriate functional form to representtechnology, we are left with the difficult prob-lem of nonnested hypothesis testing. Non-nested model selection rules have a heuristicbase, and small sample properties of the rulesare virtually unknown (Judge et al.).

The nonparametric approach allows tests ofprofit maximization without any maintainedhypothesis of functional form for technology(Varian 1984b; Chavas and Cox; Hanoch andRothschild; Fawson and Shumway). However,the nonparametric tests are not ideal becausethey have a heuristic base and because theyonly determine whether observed behavior isconsistent or inconsistent with the null hy-pothesis.

Parametric and nonparametric tests of profitmaximization obviously complement eachother. Information gained from the tests, whilenot definitive, would complement empiricalapplication of duality theory and perhaps givea better intuitive understanding of whether themaintained hypothesis is valid. However, inan empirical setting seldom can we definitivelydiscriminate among alternative behavioral hy-potheses such as (a) unconstrained profit max-imization, (b) profit maximization subject toa cost constraint (Lee and Chambers), and (c)expected profit maximization in a dynamicsetting (Antle; Taylor). Therefore, which dual-ity model, if any, should be applied cannot bedefinitively established for any given empiricalproblem.

Summary and Concluding Remarks

Graphical analysis presented in this articlesuggests that potential distortions resultingfrom application of duality theory can be min-imized or even eliminated in certain cases. Onecase is where the aim of the empirical researchis to estimate factor demand and supply equa-tions directly as behavioral relationships (evenif profit maximization does not hold) and wherethe profit function is not of direct interest. Insuch a case, use of functional forms obtainedby applying Hotelling's lemma to a prespeci-fled indirect profit function may not result ina distortion as long as the profit function is notfitted along with the demand and supply equa-tions. Distortions can still result if symmetryof cross-price effects, which is implicit in the

210 December 1989


dual approach for the profit maximizationmodel, does not hold. However, symmetry canbe empirically tested using standard statisticalprocedures.

A second case is where the aim is to establishonly the profit function, perhaps for compar-ison to profit functions for other firms in themarket. In this case, distortions are minimizedby directly fitting only the profit function. Fi-nally, if the aim of the research is to establishsupply and demand equations and the profitfunction, then it must be recognized that all ofthe equations will be distorted if the main-tained hypothesis is not valid.

Major advantages of a dual approach to em-pirical problems are: (a) it opens up a richerclass of operational functional forms, espe-cially for multiproduct, multifactor produc-tion; (b) it brings coherence to the analysis,especially with respect to cross-commodity re-lationships, that is often lacking in nondualapproaches; and (c) it is possible to obtain fac-tor demand and product supply equations froman indirect profit function fitted to profit andprice data without having empirical observa-tions on the quantities demanded or supplied(Pope 1982b; Lau and Yotopoulos 1971, 1972;Young et al.). These advantages can indeed beempirically exploited if the maintained behav-ioral hypothesis is valid. Unfortunately, wecannot definitively establish validity of themaintained hypothesis in most if not all em-pirical studies.

It must be recognized that there are alsoweaknesses with some nondual approaches. Forexample, the nondual approach is not opera-tional without observations on quantities. Anondual approach also has pitfalls, especiallyif profit maximization (or other maintainedhypothesis) is valid, but a set of functionalforms for factor demand and product supplyequations that are inconsistent (with respect tocurvature or cross-price relationships) withprofit maximization is used; empirical resultswould therefore be distorted if in fact profitmaximization held. Additional appraisal ofadvantages and disadvantages of dual ap-proaches relative to other approaches is givenin Young et al. Selection among various dualapproaches and between dual and nondual ap-proaches for empirical application appears tobe more art than science. The graphical frame-work presented in this article allows analysisof distortions associated with different ways ofusing duality associated with classical profit

maximization. The framework can be extend-ed to other optimization models.

Duality is indeed useful, but empirical re-sults based on this theory should be cautiouslyinterpreted-much more cautiously than is ap-parent from recent literature.

[Received August 1988; final revisionreceived May 1989.]

References

Antle, J. M. "Incorporating Risk in Production Analy-sis." Amer. J. Agr. Econ. 65(1983):1099-106.

Arif, S. M., and J. T. Scott, Jr. Economic Efficiency inIntegrated Pest Management Decisions of Illinois CornFarmers. Staff Paper 86-E-356, Dep. Agr. Econ., Uni-versity of Illinois, June 6, 1986.

Beattie, B. R., and C. R. Taylor. The Economics of Pro-duction. New York: John Wiley and Sons, Inc., 1985.

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. "The Neoclassical Maximization Hypothesis:Reply." Amer. Econ. Rev. 73(1983):828-30.

Burks, A. W. Chance, Cause, Reason: An Inquiry into theNature of Scientific Evidence. Chicago: University ofChicago Press, 1977.

Caldwell, B. J. "The Neoclassical Maximization Hypoth-esis: Comment." Amer. Econ. Rev. 73(1983):824-27.

Chambers, R. G. "Relevance of Duality Theory to thePracticing Agricultural Economist: Discussion." West.J. Agr. Econ. 7(1982):373-78.

Chavas, J.-P., and T. L. Cox. "A Nonparametric Analysisof Agricultural Technology." Amer. J. Agr. Econ.70(1988):303-10.

Collins, G. S., and C. R. Taylor. "TECHSIM: A RegionalField Crop and National Livestock Econometric Sim-ulation Model." Agr. Econ. Res. 35(1983):1-18.

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Fawson, C., and C. R. Shumway. "A Nonparametric In-vestigation of Agricultural Production Behavior forU.S. Subregions." Amer. J. Agr. Econ. 70(1988):311-17.

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Judge, G. G., W. E. Griffiths, R. C. Hill, H. Lutkepohl,and T. C. Lee. The Theory and Practice of Econo-metrics, 2nd ed. New York: John Wiley and Sons,Inc., 1985.

Kako, T. "Decomposition Analysis of Derived Demandfor Factor Inputs: The Case of Rice Production inJapan." Amer. J. Agr. Econ. 60(1978):628-35.

Lau, L. J., and P. A. Yotopoulos. "A Test for RelativeEfficiency and Application to Indian Agriculture."Amer. Econ. Rev. 61(March 1971):94-109.

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Lee, H., and R. G. Chambers. "Expenditure Constraintsand Profit Maximization in U.S. Agriculture." Amer.J. Agr. Econ. 68(1986):857-65.

Lopez, R. E. "Estimating Substitution and Expansion Ef-fects Using a Profit Function Framework." Amer. J.Agr. Econ. 66(1984):358-67.

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APPENDIX

Relationships shown in figures 1-5 were generated fromthe following model:

(A.1)(A.2)

y = 2x 5 production function,rr = py - rx direct profit function,

where r = .2 and the graphs are for 0 < p - 1. The decisionfunction (which is also the factor demand equation) as-sumed for the suboptimization case shown in figures 3 and4 is

(A.3) xa = .25(x* - 6.25) + 6.25,

where x* is the profit maximizing level ofx. Note that forp = .5, xa = x*. The suboptimization case shown in figure5 is for

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(A.4) xa = .5X*.

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