On the Theory of the Competitive Firm under Uncertainty When Initial Wealth Is Random

Abstract

Since Sandmo's seminal paper on the theory of the competitive firm under output price uncertainty, there has been considerable research on this topic [2; 5; 6; 10; 11; 14]. In all these studies, it is assumed that, in the absence of complete contingent markets, output price risk is the only source of uncertainty for the firm. However, this may not be a realistic assumption. In particular, as argued by Kihlstrom et al. [7], it appears reasonable to consider the case where some of the initial wealth of the firm is held in risky assets. In that case, initial wealth is uncertain and should be modelled as a random variable. The objective of this paper is to investigate the introduction of random initial wealth in Sandmo's theory of the firm under price uncertainty. In Sandmo's model, a number of important comparative static results (e.g., the conditions under which the supply function is upward sloping [14, 69], or under which increased price uncertainty reduces supply [5] depends critically on the sign of a particular expression' stated in Lemma 2 in Ishii [5]. While Sandmo [14, 69] shows this expression is non-negative under Arrow-Pratt decreasing absolute risk aversion, it can be verified that his proof no longer holds when initial wealth is random. Therefore, the comparative static analysis conducted by Sandmo [14], Batra and Ullah [2] and Ishii [5] is no longer valid and gives ambiguous results under initial wealth uncertainty. The reason is that the Arrow-Pratt measures of risk aversion [1; 12] are too weak for making comparisons in situations where there is more than one source of uncertainty [13]. Then, what can be deduced from the theory of the competitive firm in the (realistic) case where both initial wealth and output price are uncertain? We will show here that meaningful results on the behavior of the competitiv firm under uncertainty can be obtained provided that the Arrow-Pratt measures of risk aversion [1; 12] are replaced by the Ross measures of risk aversion [13]. In particular, a comparative static analysis of the production decisions using Ross characterization of risk behavior is presented. A number of new results are derived on the properties of risk responsive supply and input demand functions under both price and initial wealth uncertainty.

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