On output price uncertainty and the comparative statics of industry equilibrium.

• Author: Horbulyk, Theodore M.

1. Introduction

   The analysis of firm response to output price variability has, until recently, been incomplete due to a failure to consider explicitly the effect of firm response on the attainment of competitive equilibrium at the industry level. A recent paper by Appelbaum and Katz [1] has addressed the issue of industry equilibrium under output price uncertainty. They show how the optimal number of firms might change through the process of exit and entry, and they characterize the collective influence of firms' decisions on expected market price. However, their analysis focuses on firms with a single-stage production process which allows no production flexibility (in the form of adjustment of output levels) after output price is revealed.

   An earlier paper by Hartman [3] has shown that, at the firm level, the optimal response to output price uncertainty will depend on whether production flexibility exists. Specifically, if firms employ a two-stage production process which is characterized by the presence of a quasi-fixed factor, then firm-level output may increase or decrease in response to changes in output price variability even if firms are risk neutral. The assumption that production flexibility exists changes those results initially developed by Sandmo [7], and applied more recently by Appelbaum and Katz [1], which are based on a single-stage production process.

   This paper provides an extension of the Appelbaum and Katz analysis to admit production flexibility. Equivalently, it extends the analysis of Hartman to consider explicitly the effect of firm response on the attainment of competitive equilibrium at the industry level.

   Appelbaum and Katz show that for an industry of identical risk-averse firms with a single-stage production technology, the effect of a mean-preserving increase in price uncertainty is to decrease industry output and to raise market price. A more recent comment by Ishii [5] shows that output per firm will also fall, yet none of these authors shows unambiguously whether the number of firms increases or decreases. Although not explicitly stated, the analysis in Appelbaum and Katz suggests there would be no firm or industry response to a mean-preserving change in price spread if firms were risk-neutral (expected-profit maximizers).

   The model employed in this paper is analogous to that of Appelbaum and Katz with two principal exceptions: (i) here an industry is composed of firms all of which are either risk neutral or risk averse,(1) and (ii) firms employ a two-stage production process such as the one analyzed by Hartman. There are two parts to the first-stage decision: whether to produce at all, and if so, how much of a quasi-fixed (or capital) input to commit before output price is revealed. Once output price is revealed, the second-stage decision determines the level of variable input use, and thus the output level. The effects of price uncertainty on industry equilibrium are shown to vary according to the assumptions which are made about the firms' preferences and technology.

   In particular, it is shown for an industry of identical risk-averse firms which employ a two-stage production technology that, in long-run equilibrium, industry output might rise or fall in response to a mean-preserving increase in the dispersion of output price. This is in contrast to the result of Appelbaum and Katz who show that when risk-averse firms employ a single-stage production technology, the industry output will decrease unambiguously. in the special case of risk-neutral firms, conditions are given here under which industry output could increase or decrease in response to a mean-preserving increase in output-price uncertainty.

   It is implicit in this result, as in the work of Epstein [2], Hartman [3] Turnovsky [8]and Wright [9], that under a two-stage production technology firms may have an affinity for price variability. Such an affinity may be present even where firms are averse to variability of profits.

   For example, suppose the technology at the firm level is such that, under price uncertainty, increased use of a quasi-fixed factor ex ante increases a firm's short-run elasticity of supply ex post. An expected-profit-maximizing firm would choose to increase its use of the quasi-fixed factor in response to greater output price uncertainty. Provided that the factors used in each stage of production are technical complements, this will result in increased output as well.(2) For firms which are risk averse, there will be an offsetting tendency, described by Sandmo [7], to produce less output when price uncertainty increases. The optimal (short-run) firm response has thus been described in the literature in terms of the firm's preferences and technology. This paper extends that analysis by characterizing the long-run industry equilibrium. The direction of firm response will depend on the offsetting forces described above, as well as on adjustments in expected price which result from changes in output per firm and the number of firms.

   Section II characterizes industry demand and the behavior of individual firms. The following two sections describe two industry equilibria and provide comparative-static analyses of industry responses to a mean-preserving change in the dispersion of output prices. Section III considers an industry composed of identical firms which are risk-averse whereas section IV considers an industry of fisk-neutral firms.

   2. Industry Demand and Individual Firm Behavior

      Expected price will be inversely related to industry output and thus is influenced by entry and exit. Industry demand is stochastic, as given by

      [Mathematical Expression Omitted] (1)

      where [micro] is expected price and Q is the industry output of a homogeneous good such that [micro](Q)

      Consider an industry composed of representative (identical) firms which produce a homogeneous good, the demand for which is variable in successive periods. Production follows a two-stage process which requires some commitment of resources by a firm before its uncertainty about output prices is resolved. The firm employs two factors which may be denoted capital, K, and labor, L, and which have the following properties. Capital decisions must be made before a random output price is revealed, and, once made, represent sunk costs to the firm. Labor decisions are not binding until after the output price has been revealed, but before production is in fact completed. Production is a single-period process, with capital decisions made before the start of a period and labor inputs chosen at the start of the period.

      With such a two-stage production process, there are two parts to the first-stage decision: whether to produce at all, and if so, how much of the capital or quasi-fixed input to commit before output price is revealed. Once output price is revealed, the second-stage decision determines the level of variable input use, and thus the output level.

      Let the number of identical firms, n, be continuous, so that the output per firm, q, is given by Q/n where q = f(K, L) and where f is strictly concave. A firm which chooses not to produce in some period incurs fixed cost T but can avoid costs, c and w, associated with K and L, respectively. Thus, a firm's profit per period will be [pi] = pq - wL - cK - T = ([micro](Q) + [gamma][epsilon])f(K, L) - wL - cK - T.

      For firms which are risk averse, let U([pi]) be the von Neumann-Morgenstem utility function of each, such that U'([pi]) > 0 and U"([pi]) 0 and U"([pi]) = 0. Assuming U(0) = 0, the benchmark expected utility level for risk-neutral firms will be b = 0, which implies the expectation of (positive) economic profit shall signal entry of firms.

   3. Industry Equilibrium with Risk-Averse Firms

      Following Appelbaum and Katz, risk-averse firms will choose to enter the industry (or to stay in the industry) and to produce some output provided the expected utility gained from so doing is no worse than in the next best alternative activity. Entry and exit are not instantaneous, so the industry need not always be in equilibrium. However, in industry equilibrium, this condition must hold with equality. If firms choose to produce, they will commit capital to that level where the expected marginal utility from its product, net...