Factor-price uncertainty with variable proportions: a note

Abstract

Marion Stewart recently examined the theory of the neoclassical firm under factorprice uncertainty in the same spirit as the well-known Leland-Sandmo theory of the firm under demand uncertainty. The main finding of his study is that, in most cases, risk-averse managers when faced with inputprice uncertainty would substitute fixed (riskless) factors of production for the factors. The principal methodology used consisted of a direct comparison of the input choices of the firm's manager under both risk neutrality and risk aversion. Stewart's results are important and have many obvious empirical implications, but they contain two serious deficiencies that may lead to errors and confusions. First, the formulation and solution of the multiple input case in the Appendix, as well as the uncertain price case in Section IV are flawed, insofar as the decision rules followed by the firm are suboptimal. Second, the first paragraph of Stewart's study implies that a riskindifferent manager operating in an uncertain environment is not going to be affected by the uncertainty with respect to his output and factor-proportions decisions. This implication is misleading. It is shown in this paper that input price uncertainty combined with risk indifference has a very definite effect upon the input choices of the firm as compared to the certainty case. This effect of uncertainty combined with risk neutrality must be distinguished from that of risk aversion proper because it has direct implications for empirical work. In empirical work at the firm or industry level, input prices are estimated by dividing the observed ex post cost of each input by the observed input use during the reference period. The result is an observed average price that, under certain assumptions, corresponds to an unbiased estimate of the expected ex ante input price. If (as it often happens) this observed price is used in the empirical work as if it were known ex ante with certainty, then in general a bias will be introduced even when the manager is risk indifferent. It is shown in this paper that in cases of practical interest this bias may or may not have the same sign as the effect of risk aversion, which means that the total effect of uncertainty upon input choices is very often unpredictable when compared to the predictions of the theory of the firm under certainty if the input and output prices are assumed equal to their mathematical expectations. This weakens somewhat the impact of Stewart's results. The other point made in this note is that in cases of more than one risky input, the method followed by Stewart (in his Appendix A) is incorrect, insofar as it leads to suboptimal decisions. Under variable proportions production the firm does not need to choose simultaneously the levels of all inputs. Hence, the inputs can be purchased after uncertainty has already been resolved, given the level of the riskless input. The selection of the latter must therefore incorporate the fact that inputs have been selected ex post under conditions of certainty. Similar remarks are also true when there is combined uncertainty in input and output prices. I shall consider a hypothetical firm that uses three' inputs, x1, x2, and X3, to produce the final product q, and let rl, r2, r3, and p denote their respective unit prices. Input X3 (plant and equipment) must be chosen prior to the other two, so that at the time X3 iS