UNIVERSI rv OF CALIFORNIA DAV'S MAY Agricultural Economics Library. JJCD ~partment. of Agricultural Economics ..._ WORKING PAPER SERIES

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1 UNIVERSI rv OF CALIFORNIA DAV'S MAY Agricultural Economics Library JJCD ~partment of Agricultural Economics...._ WORKING PAPER SERIES

2 University of California, Davis Department of Agricultural Economics Working papers are circulated by the author without formal review. They should not be quoted without permission. All inquiries should be addressed to the author, Department of Agricultural Economics, University of California, Davis, California TO DUAL OR NOT TO DUAL? by Rulon Pope Working Paper No. 78-5

3 To Dual or Not to Dual? Perhaps no single area of inquiry has had greater impact on production economics methodology than the duality results of Shepard (1970), McFadden (197lp and Uzawa (1964). That is, given convexity of production sets and competition, there is a correspondence between production, cost and profit sets. Recently, a number of flexible functional forms have been developed which enlarge the set of testable hypotheses, e.g., Christensen et al., (1973), Diewert (1973). However, as Burgess's (1974) study illustrates for functional forms which are not self dual (such as the translog), one must choose between the primal problem with production represented by the functional form or the indirect profit or cost function (dual) representation with the same functional form. Even for self dual forms, the decision between production and cost function approaches is substantive. Both methodologies are routinely employed with each possessing desirable properties. As Uzawa (1962) and Binswanger (1974) indicate, the partial elasticities of substitution are difficult to estimate with the primal problem, but relatively easy to estimate with the cost function; conversely, marginal products are most easily estimated from the primal problem. The cost or indirect profit function approaches appear to have widespread use because of the above mentioned properties and because prices are easily obtained and rationalized as exogenous. This simplifies estimational procedures to a constrained version of the Zellner-Parks iterative least squares procedure (see aboye mentioned references). The purpose of this note is to analyze which of the two procedures (primal or dual) is most advantageous for certain classes of risk problems. Since there is mounting evidence that risk aversion characterizes much behavior under uncertainty, it would be helpful if on a priori grounds,

4 2 one could choose an appropriate methodology, It will be shown that when output prices are risky and questions relating to constant output demand functions are of interest (e.g., separability is to be tested or production or factor demand elasticity estimates are desired), then the cost function should be used because it is robust to nonrisk neutral behavior, That is, under risk aversion, though the levels of outputs are affected by risk aversion, relative input changes in response to factor price changes are given by the duality results of certainty theory for the cost function. This is not true for the restricted profit function developed by McFadden (1971) and Lau (1969) and applied by Lau and Yotopoulos (1971) and others. That is, if risk neutrality is incorrectly assumed, inconsistent parameter estimates are obtained. An Ex Ante Model Assume that the firm maximizes expected utility of profits, E[U(n)] with the following notation defined Profit = n = Pq - y'x = Pq - m L yjxj j-=l q = F(X) denotes a production function where q = output and X is a vector of input levels P = price of output (which is random) y = vector of input prices. It will be assumed that P E(P) + = P + where E(E) 0, First order conditions for the above firm are: (1) E[U'(n)(PFj.- yj)] - Fj(P + cr) - rj c 0 j 1 m where Fj = df/oxj and cr a Cov [U'(n), P]/E[U'(n)]. Under risk aversion cr < 0 (Sandmo 1971) and expected value of the marginal product exceeds

5 3 1/ factor price at optimum.- It is clear from (1) that if risk neutrality is incorrectly assumed and equations are estimated on the basis of o 0 (that is, PFj yj), then inconsistent parameter estimates of F and Fj are obtained if factor demands are correlated with production errors as in the case of nanagement bias (Marschak and Andrews (1944), Hoch (1958)). Cost Minimization If (1) is solved for P +a, then (1) implies that that is, marginal rate of substitution equals the factor price ratio. Thus, as Batra and Ullah (1974) and Blair and Heggestad (1977) note, (2) implies that if q * is the risk averse optimal output, then the problems (3) Min r yjxj "" C -= Cost x j (4) Max E[U(7r)] x * s.t. q.. q are consistent. Thus, expected utility maximization is consistent with cost minimization and though the optimal output (q*) is lower under risk aversion than under risk neutrality, the risk averse firm still minimizes costs for the level of chosen output. The above results are established in an intuitive way as follows. Under certainty, it is well known that for the Lagrangean };./ Note that (1) does not imply that PFj Kyj where K is a fixed constant. In the context of equation (1), K [1 - (o/yj)fj]. Hence, tests for inefficiencies based upon the hypothesis K 1 are not appropriate since K is not fixed but depends on risk preferences, parameter estimates in Fj and X (such tests are found in Hoch (1958) and Lau and Yotopoulos).

6 4 L c Pq - y'x + A {q - F(X)] - n +A {q - F(X)] minimization of cost over X in the first stage (min y'x +A {q - F(X)]) and maximization of L over q in the second stage yields optimal q and X. In point of fact, the two stages will be congruous for any U(n) where U'(TI) > 0. That is, for any monotonically increasing transformation of profit n, cost minimization is appropriate. Finally, note that expected utility is a monotonically increasing transformation since probabilities are nonnegative. Hence, cost minimization must be consistent with expected utility maximization. It can easily be verified that cost minimization is appropriate for the multiproduct case as well. Further, suppose production consists of multiproduct stochastic production functions of the form where q 6 is the quantity of output s, F 6 (X 6 ) denotes the production function and ws is the random disturbance in the production function. It is easily shown that cost minimization is also appropriate in this case even though both prices and production are uncertain and where costs are minimized subject to a fixed level of expected output. Hence, factor demands (ex ante) are still recovered from the cost function. For expository convenience, the remainder of the paper will consider only the single product price uncertain case. Since cost minimization is appropriate under risk aversion, it is noted that in the single product price uncertainty case, the solu'tion to (3) may be written Xj Xj (q*, y). Hence the cost function is given by * (5) c - t yj xj (q, y). j ~--~ ,...,.,- 1...,..., - -- ~

7 --- ~ 5 Applying the envelope principle yields (Samuelson (1947), Silberberg (1974)): (6) ac - xj" * ayj This is the well-known duality result known often as Shepard's Lemma. Applying these results to the translog case indicates that the following sets of equations are appropriate under risk averse or risk neutral behavior m (7) Si ai + r bij ln yj ja:l i l m-1 where the ai and bij are constants to be estimated, lnc is the unit cost function and Si is the cost share. It can further be verified that the usual symmetry and homogeneity conditions hold on the bijand a 1 (i.e., bij bji' r bij O, r bij O, and E ai"" 1). i j i Hence, when one is unsure whether risk averse behavior is present, it seems as though the cost function approach is preferable to the primal problem if one is interested solely in information conveyed in the equations in (7). Conversely, if one is interested in absolute levels of inputs (or output supplies) as functions of all prices where revenue as well as substitution effects apply (nonconstant output demand function), then risk preferences are needed and the cost function (e.g., (7)) conveys little helpful information. Biased Duality Procedures Using Profits In this section, the application of McFadden's Lemma (indicating * - Xj) to the risk averse firm is examined. It will be informative to develop an expression indicating the bias when risk preferences are incorrectly assumed linear (neutral) ~-. 'T _ ~..,-~_,, : "'-~ -..--~ '"'--_, "":- ":; _ ~- -.-,._ :~

8 6 Blair and Lusky (1977) have shown that ae[u(tt)]/ayj -- Xj * E IU' (7T)] and ae[u - * 1 (7T)]/aP q E[U'(n)). These expressions follow directly from an application of Samuelson's envelope theorem. However, the expressions are not particularly useful in empirical work, nor do they aid in addressing the nature of the bias in duality procedures under risk aversion. Since profit, n, is observable, ex post, we will seek an expression for d1t */ayj under risk aversion. The indirect profit function is given by (8) 2/ Hence,- n * PF(X *(y, q))- y'x *(y, q). * Under certainty, PF 1 - y 1 0 and thus d7t /ayj - - Under uncertainty and risk neutrality, P is replaced by Pin (8) and (9) and we obtain ae(7t *)/ayj = - Xj. * PFi - Y = - F a i i or in terms of observed prices, PF - y - F 1 (cr- ). i i In general, risk aversion indicates (from (1)) that J/ However, it is interesting to note that for particular expected utility functions ae[u(7t)]/ayj = - Xj: * such are of the form T E[U(TT)] "" E(7T) + E a crt, t 2 t th where at are constants and crt is the t--=--'- central tooment of profit. Therefore, * ae[u(tt)] - - xj. ayj Hence, factor demands are given explicitly by differentiating the indirect utility function.

9 7 Hence, from (9) (10) (o-e:)e i -- Therefore, (10) indicates that on average -- * * er~ - xj.. ay j Given risk aversion o < O. If factors are normal, then oq /oyj < 0 (it should be noted that oq */oyj is unaabiguously negative for a large class of utility functions, Pope (1977)). that on average - Given these two conditions, it is clear * * an /ayj overstates factor demands, Xj. * The representation of (10) in terms of factor shares gives (11) ~ 1T * - s + j where Sj = yjxj/n * * are the factor shares. 2j_ 1T * (o-e:) Under risk neutrality, on average ' - olnn */olnyj s sj. Thus factor shares are inappropriate estimates of para- meters of the indirect profit function. Under risk aversion and oq */oyj < O, it is clear that on average the jth factor share understates - olnn*/olnyj. Hence, parameters of 1T * (and hence F_) * cannot be estimated without formally including risk preferences and higher order moments of price than P in the indirect profit function. Concluding Remarks There are many situations in applied economics where one suspects the possibility of risk averse behavior. In nearly every sector of an economy, one is likely to observe behavior which is ioore consistent with risk aversion

10 8 than risk neutrality (e.g., insurance, options markets, and some forms of diversification), yet contingent claim markets in production may not be adequate or exist at all. In such instances, if one is not fundamentally interested in the relationship between risk aversion and output supplied, then the cost function approach seems in many ways superior to other approaches when output price is random (or production has a random multiplicative disturbance), because it is robust in the absence of specific knowledge as to risk preferences. If one is fundamentally interested in output effects as they relate to risk aversion and factor demands, the primal problem appears to be superior (see Just and Pope (1978)) for an analysis of econometric estimation of the structural equations of the primal system). Finally, since many studies have employed the indirect profit function in econometric models of efficiency (or inefficiency), the results of this paper suggest that consistent estimates of parameters cannot be obtained following their procedures if risk aversion is present (see (1) and (11)). Hence, such tests of economic efficiency are invalid unless they specifically include the possibility of risk averse behavior. gg 5/16/78 _.,_..., ~. T-----~-~ ~~~ :_- w~f '-., ~

11 9 References Batra, R. and A. Ullah, 1974, "Competitive Firm and the Theory of Input Demand Under Price Uncertainty", Journal of Political Economics, pp Binswanger, H., 1974, "The Measurement of Technical Change Biases With Many Factors of Production", American Economic Review, 64, pp Blair, R. and A. Heggestad, 1977, "The Impact of Uncertainty Upon the Multiproduct Firm", Southern Economic Journal, 44, pp Burgess, D., 1975, "Duality Theory and Pitfalls in Technologies", Journal of Econometrics, 3, pp Christensen, L., D. Jorgenson and L. Lau, 1973, "Transcendental Logarithmic Production Frontiers", Review of Economic Statistics, 55, pp Diewert, W., 1973, "Separability and a Generalization of the Cobb-Douglas Cost, Production and Indirect Utility Fllllctions", Technical Report No. 86, Institute for Mathematical Studies in the Social Sciences, Stanford University. Hoch, I., 1958, "Simultaneous Equation Bias in the Context of the Cobb Douglas Production Function", Econometrica, 26, pp Just, R. and R. Pope, 1978, "On the Relationship Between Input Decisions and Risk", J. Rournasett and M. Boussard, eds., in Risk, Uncertainty and Agricultural Development, forthcoming, University of California Press. Lau, L., 1969, "Some Applications of Profit Functions", Memorandum No. 86A-B, Center for Research in Economic Growth, Stanford University Mimeograph. - _,._ ~-----~~ - - _.,,, - - ~ _,

12 10 Lau, L. and P. Yotopolous, 1971, "A Test for Relative Efficiency and Application to Indian Agriculture", American Economic Review, 61, pp Marschak, J. and W. Andrews, 1944, "Random Simultaneous Equations and the Theory of Production", Econometrica, 12, pp McFadden, D., 1974, Cost Revenue and Profit Functions: An Econometric Approach to Production Theory (North-Holland, Amersterdam), forthcoming. Pope, R., 1977, "The Generalized Envelope Theorem and Price Uncertainty", University of California at Davis, Working Paper, Department of Agricultural Economics. Samuelson, P., 1947, Foundations of Economic Analysis (Harvard University Press, Cambridge, Massachusetts). Sandmo, A., 1971, "On the Theory of the Competitive Firm Under Price Uncertainty", American Economic Review, 41, pp Shepard, R., 1953, Cost and Production Functions, Princeton University Press, Princeton, New Jersey. Silberberg, E., 1974, "A Revision of Comparative Statics Hethodology in Economics", Journal of Economic Theory, 7, pp Uzawa, H., 1962, '~roduction Functions with Constant Elasticities of Substitution", Review of Economic Studies, 29, pp ~ >## _,

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