1. Introduction
Sandmo´s (1971) model of the competitive firm under output price uncertainty has been the object of ongoing interest by researchers since its appearance in 1971 [e.g., Lippman and McCall (1981), Honda (1985), Simpson and Sproule (2000), Moschini and Hennessy (2001), and Paulsson and Sproule (2002)]. 1
One measure of this interest is the fact that three distinct approaches to signing the marginal effect of risk have been identified [Simpson et al. (1995, pp. 313314)]. The first is to impose restrictions both on the von NeumannMorgenstern utility function, and on the functional form of meanpreserving changes in the probability density function [e.g., Ishii (1977)]. A second approach is to impose restrictions on the von NeumannMorgenstern utility function, and on the cost functions [e.g., Lippman and McCall (1981), and Honda (1985)]. A third approach is to impose restrictions on the functional form of meanpreserving changes in the probability density function alone [e.g., Eeckhoudt and Hansen (1980), and Meyer and Ormiston (1983 and 1985)].
The last approach has contributed to the development of new research into what is termed the "twomoment model" [e.g., Meyer (1987 and 1989), Sinn (1989 and 1990), BarShira and Finkelshtain (1999 and 2002), Ormiston and Schlee (2001), Gelles and Mitchell (2002), Wagener (2002 and 2003), and Eichner (2004)]. The essence of this enterprise is the idea that: "If a decision problem satisfies Meyer´s locationscale condition, then any utility representable preferences are also representable by a meanstandard deviation utility function" [BarShira and Finkelshtain (1999, p. 237)].
The present paper offers a new bridge between the Sandmo model of the competitive firm and twomoment models. In particular, the present paper outlines how the Sandmo model might be recast in the context of a twostate world. In doing so, we offer a generalization of an obscure model by Sakai (1977).
This paper is organized as follows. Section 2 outlines the preliminary results needed to define Sandmo´s model of the competitive firm in a twostate world. Section 3 presents both the marginal and global effects associated the conditions outlined in Section 2. Concluding remarks are offered in Section 4.
2. Preliminary Remarks
Suppose that the firm´s fixed cost function is b, and its variable cost function is C(y,w), where y denotes output and w denotes the unitcost of the firm´s only variable factor. In accordance with cost theory, we assume that In summary, the total cost function is defined as T(y,w,b) = C(y,w) + b.
Next suppose that output price, p, is distributed as a onetrial binomial variate, and suppose its expected value is E(p) = . In particular, the two states of the world are: (a) a low price, , that occurs with a probability of that occurs with a probability of 1  ?.
Given the mean and variance of output price, the mean and the variance of profits are:
2.1. The Optimization Problem: Suppose the decision maker (DM) has a von NeumannMorgenstern utility function, . Assume that is at least twice differentiable, and that > 0 and < 0 for all that is, the DM is risk averse. Moreover, assume that the DM exhibits decreasing absolute risk aversion (DARA) [viz., denotes the ArrowPratt measure of absolute risk aversion].
The DM´s goal is the maximization of the expected utility of profits with respect to output, viz.,2
Finally, provided that the FOC and SOC are met, there exists a unique value of y that solves Equation (2), that is:
Remark 3: The second and the third terms on the lefthand side of the last line in the proof of Lemma 4 are called: (a) the "physical part of marginal expected cost," and (b) the "marginal cost of risk bearing" [Sakai (1977 and 1978)]. Note that the "marginal cost of risk bearing" is the product of three terms: (a) the probability, 0 < < 1, (b) the scaling factor in the outcome space, , and (c) the "omega" function. We turn next to a discussion of the properties of this function.
2.2. The Omega Function: The properties of the omega function are outlined in the following six lemmas:
3. The Determinants Of The Optimal Level Of Output
This section outlines the determinants of the optimal level of output. Of special interest are the marginal and global effects of risk.
3.1 The ComparativeStatic Results: A perturbation from equilibrium may be captured by taking the total derivative of Equation (3), viz.,
4. Conclusion The present paper outlines how the Sandmo model might be recast in the context of a twostate world.
In doing so, we offer a generalization of an obscure model by Sakai (1977). In particular, by allowing for a symmetrical and asymmetrical binomial variate, this paper offers a new bridge between the Sandmo model of the competitive firm and the growing literature on twomoment models.
Footnotes
1 For textbook discussions of Sandmo (1971), see Takayama (1993), Levy (1998), Wolfstetter (1999), and Silberberg and Suen (2001).
2 The present model is a generalization of Sakai´s (1977) to the extent that Sakai assumes that and therefore that
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About the Author
Author: Robert Sproule Direction: Bishop´s University, Lenoxvile. Québec (Canada) Email: ra_sproule@hotmail.com
