Uncertainty and Factor ProportionsAuthor(s): Harry WhiteSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 19, No. 4(Nov., 1986), pp. 814-815Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/135330 .

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Uncertainty and factor proportions HARRY WHITE Northeast Louisiana University

Stewart (1978) examined the theory of the firm's factor proportion decision under input price uncertainty and concludes the uncertainty will induce the risk averse manager to hire more of the certain priced factor of production than his risk neutral counterpart. He then adds output price uncertainty to the model and concludes that it doesn't affect the direction of factor substitution for the risk averse firm.

The purpose of this paper is to show that the direction of factor substitution can be reversed when output price uncertainty is added to the model.

Stewart set up a model of the competitive firm producing a single output (q) using an input (X2), chosen prior to the production period, purchased at a known price (r2). In addition, the firm combines X2 with another input (XI), purchased in competitive markets at a price (rl) which is not known when the level of X2 is chosen. It is also assumed that r1 is subject to random fluctuations during the production period and has a subjective probability density 4>(rl), known to the firm's manager at the time X2 is chosen. Output and the price of X2 are known with certainty, so that the manager chooses X2 to maximize E u('7r) where u is a von Neumann-Morgenstem utility function with profit as it's argument and E is the expectations operator. In addition, the production constraint is solved for XI and substituted into the objective function so that

T= pq - rlg(X2,q) - r2X2, and a necessary condition for utility maximization is

E[u'()(-rI(agaX2) - r2)] = 0. (1) With the introduction of output price uncertainty, it is assumed that the

price of output (p) has a subjective probability density +(p) which is known to the firm's manager at the time X2 is chosen.

1 The sufficient conditions are satisfied for convex isoquants and a linear or concave utility function.

Canadian Journal of Economics Revue canadienne d'Economique, xIx, No. 4 November novembre 1986. Printed in Canada Imprime au Canada

0008-4085 / 86 / 814-815 $1.50 ? Canadian Economics Association

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Uncertainty and jfctor proportions 815

Stewart defines A = u'(v) and B = -rI(ag/aX2) - r2, so that equation (1) can be rewritten as

E(AB) = E(A)E(B) + cov (A,B) = 0. (2)

From (1), we see that E(B) = 0 for the risk neutral firm so that, - g/X2 =

r2/Erl. The risk averse firm will, therefore, use more X2 and less X1 than its risk neutral counterpart whenever E(B) < 0. From (2), E(B) < 0 when cov (A,B) > 0. The sign of cov (A,B) can be determined by examining the sign of cov ('r,r1).2

By definition, cov (v,rl) = E[ (v - Ev)(r1 - Er,) ], which, once we substitute for S and simplify, can be written as

cov (v",rl) = q cov (p,rl) X1 var (rl). (3)

For the risk averse firm with no output price uncertainty, cov (v,rl) < 0 which implies cov (A,B) > 0 so that E(B) < 0 and the manager hires factors to set -8g/8X2 < r2/Er, using more X2 and less X1 than the risk neutral counterpart.

With output price uncertainty, we see from (3) that if q cov (P,rl) < XI Var (rl), then cov (v,rl) < 0, and the factor substitution pattern for the risk averse firm is the same as with no output price uncertainty.

It is also the case from (3), that if q cov (p,rl) > X1 var (rl), then cov (V,rl) > 0, which implies cov (A,B) < 0. Then from (2), it must be the case that E(B) > 0, which implies the risk averse manager hires factors to set - ag/ 8X2 > r2/Erl, using more of the uncertain priced factor (X1) and less of the certain priced factor (X2). All that is necessary for this type of factor substitution pattern under uncertainty is that p and r1 be sufficiently positively correlated so that the first term on the right hand side of (3) dominates the second.

The risk averse firm is forgoing large random profits (p and r1 both rise) to protect against large random losses (p and r1 both fall). Since output pnces and input prices are typically highly positively correlated, one would expect to observe risk averse firms 'hedging' random revenues with the positively correlated random costs.

REFERENCES

Stewart, Marion (1978) 'Factor price uncertainty with variable proportions.' American Economic Review, 468-73

White, H. (1984) 'Exchange rate risk and the multinational firm.' Unpublished docto- ral dissertation, Texas A&M University, August

2 By definition cov (A,B) = cov [u'(Q), (-r1(dg/3X2) - r2) 3, which from the definition of a covariance can be simplified to Et [U'(fr) - Eu'('r) ][ (- dg/dX2)(r1 - Er,) ] ). This last term is equivalent to (-ag/dX2) cov (u'(w7),r1). Since ag/aX2 < 0, the sign of cov (A,B) is then equivalent to the sign of cov (u'(w7),r1). For the risk averse firm, if cov (u'(Q7),r1) < 0 then cov (7T,rl) > 0 and if cov (u'(Tw),r1) > 0, then cov (7T,rl) < 0.

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