Parameter-estimation uncertainty, risk aversion, and 'sticky' output decisions.

• Author: Horowitz, Ira

1. Introduction

   A thirty-year onslaught from a variety of launching pads has led to the well-known result that a risk-averse, profit-seeking, quantity-setting, single-product seller faced with uncertain demand will opt for a lower output rate than that which is optimal under risk neutrality.[1] What is neither well known nor even strongly suspected is the extent to which the two optima will differ. That is, from the practical standpoint of a real-world decision maker, is the issue worth worrying about in the first place?

   To attempt to answer that question with something beyond an all-purpose "It depends," this paper focuses on a particularly pervasive form of real-world uncertainty, notably, the parameter-estimation uncertainty that accompanies regression-based estimates of the demand curve. In this case, as more data come in over time and previous estimates are revised, uncertainty as to the "true" demand curve and its parameters is reduced and thus, one might suspect a priori, so too would be any discrepancy between the risk-neutral and risk-averse optima. The suspicion, however, turns out to be invalid. Rather, as will be shown, whether the discrepancy tends to decrease or increase with experience and additional information depends directly upon whether the risk-averse sellers' initial output sales are relatively low or high, because this will determine whether their subsequent output rates ultimately approach a long-term equilibrium level from either below or above. The risk-neutral seller follows no such can impact on output rates that are established through regression-based estimates of demand. This is done by first erecting a parsimonious theoretical structure that yields some interesting insights that are subsequently illustrated through some experimental results. A discussion of these results and the conclusions follow.

2. The Seller's Problem

   Consider a profit-seeking firm whose management has specified the von Neumann-Morgenstern risk-preference function V = v(W), W = [Pi] + w, where [Pi] is profit and w is wealth. Management is assumed to be non-risk preferring, so that dV/dW > O and [d.sup.2] V/[dW.sup.2] [is less than or equal to] 0.

   The firm's profit will be derived from sales of a new product whose demand is unknown. Management's intention is to operate as a quantity setter for whom the market determines price via the firm's unknown demand P = d (Q, [Epsilon]), where P is price, Q is output, and [Epsilon] is an all-encompassing random-error term whose variance is [[Sigma].sup.2] and expectation is [Epsilon] [bar].

   Output is obtained at a total cost of C = c(Q), so that [Pi] = PQ - c(Q). Once management specifies d(Q, [Epsilon]) and the probabilistic process that generates [Epsilon], it can determine the optimal Q by solving max (E[V]), where E is the expectations operator. It will be assumed that c(Q) is convex and d(Q, [Epsilon]) is concave in Q, so that the risk-neutral optimum occurs at the [Mathematical Expression Omitted] that satisfies [Delta] E [[Pi]]/[Delta] Q = 0.

   For present expositional purposes it will suffice for management to (a) take a second-order Taylor expansion of v(W) around E[[Pi]] + w, and (b) assume that [Mathematical Expression Omitted] are small (where R is remainder term in the Taylor expansion) in order to (c0 determine that the optimum Q = Q* must satisfy (1) [Mathematical Expression Omitted] where [Pi] [bar] = E [Pi], r is Arrow-Hart measure of absolute risk aversion evaluated at [Pi] [bar] + w, and [Mathematical Expression Omitted] is the profit variance.(2) For the risk-neutral firm, r = 0.

   Management believes the "true" demand curve to be linear and described by P = [B.sub.0] + [[Beta].sub.1] Q + [Epsilon], where [[Beta].sub.0] and [[Beta].sub.1 are parameters to which management has assigned an "information-less" normal prior density with variance [[Sigma].sup.2]/[n.sub.0] and [n.sub.0], which reflects the extent of management's prior information about the parameters, is equal to zero. To ease our computational burdens it will be assumed that because of its experience with other products, management is confident that it knows, with certainty, [[Sigma].sup.2], the variance of the normally-distributed random-error term; and, [Epsilon] [bar] = E[[Epsilon]]. The assumption that [[Sigma].sup.2] is known, as discussed in footnote 4 below, is nondistortive and easily relaxed. It will also be assumed, solely for computational convenience, that dc (Q)/dQ = c; or, marginal cost = average variable cost = c.

   Management's very vague ideas about demand are only partially crystallized by pre-production consumer surveys that have encouraged it to produce initial outputs of [Q.sub.1], . . . ,[Q.sub.k] through the first k production periods. The market-clearing prices realized during these periods are [P.sub.1], . . . ,[P.sub.k]. Then, in Bayesian fashion, management's informationless prior over [Beta] = ([[Beta].sub.0] > 0, [[Beta].sub.1]

   To determine [Mathematical Expression Omitted] management must now solve equation (1) with [Mathematical Expression Omitted], and [Mathematical Expression Omitted], which requires solving a cubic equation--given that management specifies a value for r, as we shall assume it does. Specifically, [Mathematical Expression Omitted] must satisfy (2) [Mathematical Expression Omitted]

   As successive outputs and random errors result in newly-observed prices, [b.sub.0] and [b.sub.1] are revised along with the historical mean, [Mathematical Expression Omitted]. As the total number of observations, n increases, n replaces k in equation (2), [Mathematical Expression Omitted] and (3) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the long-term equilibrium output rate. For the risk-neutral firm, [Mathematical Expression Omitted]

3. Some Initial Insights

   1. Equation (1) is a typical mean-variance formulation in which management is trading off "risk" for "return." The maximum expected profit occurs at...