Robert Sproule
Bishop´s University, Lenoxvile. Québec (Canada)

Reference: Received 8th April 2006; Published 26th June 2006.
ISSN 1579-1475

Este Working Paper se encuentra recogido en DOAJ - Directory of Open Access Journals http://www.doaj.org/

Resumen El modelo de Sandmo (1971) para la empresa competitiva bajo incertidumbre en el precio ha sido objeto de investigación desde su aparición en 1971, en parte por el hecho de que se hayan identificado tres enfoques diferentes para señalar el efecto marginal de riesgo. Uno de estos enfoques ha generado nuevas investigaciones en lo que denominamos modelo en dos fases. El presente artículo pone en relación el modelo de Sandmo para la empresa competitiva con el área de los modelos en dos fases. Concretamente, aquí examinamos cómo el modelo de Sandmo puede adaptarse al contexto de un modelo en dos fases. De este modo, ofrecemos una generalización de un desconocido modelo de Sakai (1997). Abstract Sandmo´s (1971) model of the competitive firm under output price uncertainty has been the object of on going research interest since its appearance in 1971. One measure of this interest is the fact that three distinct approaches to signing the marginal effect of risk have been identified. One of these has spawned new research into what is termed the two-moment model. The present paper offers a new bridge between the Sandmo model of the competitive firm and the domain of two-moment models. In particular, the present paper outlines how the Sandmo model might be recast in the context of a two-state world. In doing so, we offer a generalization of an obscure model by Sakai (1977).

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. 313-314)]. The first is to impose restrictions both on the von Neumann-Morgenstern utility function, and on the functional form of mean-preserving changes in the probability density function [e.g., Ishii (1977)]. A second approach is to impose restrictions on the von Neumann-Morgenstern 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 mean-preserving 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 "two-moment model" [e.g., Meyer (1987 and 1989), Sinn (1989 and 1990), Bar-Shira 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 location-scale condition, then any utility representable preferences are also representable by a mean-standard deviation utility function" [Bar-Shira and Finkelshtain (1999, p. 237)].    The present paper offers a new bridge between the Sandmo model of the competitive firm and two-moment models. In particular, the present paper outlines how the Sandmo model might be recast in the context of a two-state 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 two-state 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 unit-cost 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 one-trial 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 Neumann-Morgenstern 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 Arrow-Pratt 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 left-hand 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 Comparative-Static 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 two-state 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 two-moment 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 References Bar-Shira, Z., and I. 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(1977), "On the theory of the competitive firm under price uncertainty: Note," American Economic Review 67, 768-69. Levy, H. (1998), Stochastic Dominance: Investment Decision Making Under Uncertainty (Amsterdam: Kluwer Academic). Lippman, S., and J. McCall (1981), "Competitive production and increases in risk," American Economic Review 71, 207-11. Meyer, J. (1987), "Two-moment decision models and expected utility maximization," American Economic Review 87, 421-30. Meyer, J. (1989), "Two-moment decision models and expected utility maximization: Reply," American Economic Review 89, 603. Meyer, J., and M. Ormiston (1983), "The comparative statics of cumulative distribution function for a class of risk averse agents," Journal of Economic Theory 31, 153-169. Meyer, J., and M. Ormiston (1985), "Strong increases in risk and their comparative statics," International Economic Review 26, 425-437. Moschini, G., and D. Hennessy (2001), "Uncertainty, risk aversion and risk management for agricultural producers," in B. Gardner and G. Rausser. (Eds.), Handbook of Agricultural Economics, Volume 1 (Amsterdam: Elsevier Science Publishers). Ormiston, M.B., and E.E. Schlee (2001), "Mean-variance preferences and investor behaviour," Economic Journal 111, 849-861. Paulsson, T., and R. Sproule (2002), "Stochastically-dominating shifts and the competitive firm," European Journal of Operational Research 141 (1), 100-105. Sakai, Y. (1977), "Price uncertainty and the competitive firm: An elementary analysis," Seikei Ronso 26 (6), 49-76. Sakai, Y. (1978), "A simple general equilibrium model of production: Comparative-statics with price uncertainty," Journal of Economic Theory 19, 287-306. Sandmo, A. (1971), "On the theory of the competitive firm under price uncertainty," American Economic Review 61, 65-73. Silberberg, E., and W. Suen (2001), The Structure of Economics: A Mathematical Analysis, 3rd Edition (New York: McGraw-Hill). Simpson, W., and R. Sproule (2000), "The production responses of the competitive firm to three conventional distributional shifts: A unified perspective," Metroeconomica 51 (2), 168-81. Simpson, W., R. Sproule, and D. Hum (1995), "Can the sufficient conditions used to sign the global effect of risk be used to sign the marginal effect of risk?" Bulletin of Economic Research 47, 305-319. Sinn, H.-W. (1989), "Two-moment decision models and expected utility maximization: Comment," American Economic Review 89, 601-02. Sinn, H.-W. (1990), "Expected utility, - preferences and linear distribution classes: A further result," Journal of Risk And Uncertainty 3, 277-281. Takayama, A. (1993), Analytical Methods in Economics (Ann Arbor, MI: University of Michigan Press). Wagener, A. (2002), "Prudence and risk vulnerability in two-moment decision models," Economics Letters 74 (2), 229-35. Wagener, A. (2003), "Comparative statics under uncertainty: The case of mean-variance preferences," European Journal of Operational Research 151, 224-232 Wolfstetter, E. (1999), Topics in Microeconomics: Industrial Organization, Auctions, and Incentives, (Cambridge: Cambridge University Press).   About the Author Author: Robert Sproule Direction: Bishop´s University, Lenoxvile. Québec (Canada) E-mail: ra_sproule@hotmail.com

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