Testing expected utility maximization under price and quantity risk with a heterogeneous panel

Abstract

Refutable implications based on the curvature properties of the indirect utility function for the competitive firm operating under uncertainty are extended to the case of both price and quantity uncertainty. Using unit roots and cointegration tests for heterogeneous panels, a model of US agricultural production is developed based on the time-series properties of a panel of state-level data. Most refutable hypotheses under output price and output quantity risk are not rejected, but symmetry conditions implied by a twice-continuously-differentiable indirect utility function are rejected. The same test conclusions are obtained from a traditional model that presumes stationarity in all variables.

Acknowledgements

The authors are, respectively, a graduate research assistant and professor, School of Economic Sciences, Washington State University. The authors express appreciation to Eldon Ball and Wallace Huffman for access to the data used in this article and to Holly Wang, Douglas Young, Atanu Saha and an anonymous reviewer for constructive comments on earlier drafts of the manuscript.

Notes

¹It joins work by Chakrabarti (2003), Bandiera et al . (2000), Sarantis and Stewart (2001) and McCoskey and Kao (1999).

²Examples are Abdulkadri et al . (2006), Key et al . (2006), Ozanne (1998), Saha and Shumway (1998), Saha et al . (1994), Love and Buccola (1991), Chavas and Holt (1990) and Myers (1989).

³With a single source of risk, this assumption defines the distribution properties on the first-order moment of risk. For example, under output price risk, let denote random output price with mean and random variable . The random component of wealth is defined by , where Y is output quantity. Then, assumption (2) is equivalent to which is a commonly maintained hypothesis in the economics literature. However, under the scenario of both output price and output quantity risk, assumption (2) imposes a strict condition on the nature of the market and on empirical properties of the data when testing hypotheses derived from expected utility maximization. We will relax this assumption and demonstrate its importance for testing under both price and quantity risk in a later section.

⁴The following notation is used throughout this article: h_x denotes the partial derivative of h(·) with respect to x, h_xy represents the Hessian matrix whose ij-th element is ∂² h/∂x_i ∂y_j where h(·) is a real-value function of vectors x and y. Transpose notation for vectors and matrices is not used.

⁵This specification provides the convenient, but not necessary, result that the expected values of the stochastic terms are zero.

⁶Because they implicitly treated assumption (2) as a necessary condition, Saha and Shumway (1998) did not demonstrate that their propositions and corollary were applicable for testing the expected utility maximization hypothesis under all types of risk, market structure and aggregation level.

⁷In both cases, the null hypothesis is the same, i.e. that the variables are not cointegrated for each cross-sectional member. The alternative hypothesis is different for the two test categories. The alternative for the panel statistics category is that the stationary autoregressive parameters are homogeneous; the alternative for the group mean statistics allows them to be heterogeneous.

⁸The group-ρ statistic is slightly undersized and empirically the most conservative in small panels. The panel-v tends to have the best power relative to the others when the panel is fairly large. The other statistics lie between these two extremes and have little comparative advantage in terms of testing power, either in small or large panels. In our case the N dimension exceeds the T dimension, which may cause all the statistics to become overly conservative (Pedroni, 2004).

⁹The theory of expected utility maximization applies to the individual, in this case the individual firm. Although tests of utility maximization have not been reported for state-level data, Lim and Shumway (1992) failed to reject the hypothesis that each of the states acted as though they were profit-maximizing firms. They used nonparametric testing procedures on annual data for the period 1956–1982, which overlaps with the first 23 years of our data period.

¹⁰Significant (5% level) groupwise heteroskadasticity was still found in the scaled data.

¹¹This routine is available from the Estima website, http://www.estima.com.

¹²Although evidence was found that significant heteroskedasticity still remained across states, we were unable to transform the data to remove cross-sectional heteroskedasticity because we had more cross-sectional units than time periods.

¹³An additional dummy variable was included in each input demand equation in the time-series-based model for the production year 1983 to pick up the effects of the PIK program.

¹⁴See Lusk et al . (2002) for an example of the frailties of measurement error in causing violation of dual relationships implied by economic theory.

¹⁵The importance of sufficient price variability in recovering dual relationships has long been known. See, for example, Quiggin and Bui-Lan (1984).

¹⁶For example, Park and Antonovitz (1992a, b) found empirical support for the hypothesis of constant absolute risk aversion (CARA) for California feedlots. Ozanne (1998) found that the presumed CARA parameter was significant for aggregate US agriculture. Abdulkadri et al . (2003) found that the hypothesis of CARA was not rejected for Kansas irrigated corn farmers.

¹⁷For example, Dalal (1994) criticized Park and Antonovitz (1992b) conclusion supporting CARA and argued that their results contradicted rather than confirmed the existence of CARA. Chavas and Holt (1990, 1996) rejected the null hypothesis of CARA and supported DARA for US aggregate corn and soybean producers. Abdulkadri et al . (2003) rejected the hypothesis of CARA for Kansas dryland wheat and dairy farmers.