A note on commodity price aggregation bias without separability

Abstract

This study identifies sources of price aggregation bias when separability restrictions do not apply. It shows that even though the assumption of the generalized composite commodity theorem guarantees aggregate integrability, it does not guarantee consistent price aggregation except in the homothetic translog model.

Notes

¹ The GCT applies for a variety of flexible forms including all homothetic utility functions, the almost ideal demand system, and the translog demand system.

² Although not shown explicitly, the analytical implications derived in this study for share demand equations apply also for systems that use quantities rather than shares. Similarly, the analytical implications extend to cases where the explanatory variables are not necessarily defined in logarithms. Logarithms of explanatory variables are used to be consistent with Lewbel (1996). However, the conditions of independence derived below for absence of bias would be different in non-logarithmic cases.

³ The GCT applies for a variety of flexible forms including all homothetic utility functions, the almost ideal demand system, and the translog demand system.

⁴ As in Equation 2, all variables are defined in logarithms.

⁵ Of course, a weaker condition for unbiasedness is where the sum of correlations of R_it with ( p_{ij t}  − R_it )( p_hvt − R_ht ), R_ht ( p_ijt − R_it ), and R_it ( p_hvt  − R_ht ) in Equation 4 is zero. Under the assumption of the GCT, the sum the covariances of R_it with the stochastic processes of R_ht ( p_ijt  − R_it ) and R_it ( p_hvt  − R_ht ) is larger than the case in which individual stochastic processes are considered if E( p_ijt )>E(R_it ), E( p_hvt )>E(R_ht ), and Cov(R_it , R_ht )>0; or if E( p_ijt )>E(R_it ), E( p_hvt ) < E(R_ht ), and Cov(R_it , R_ht ) < 0. Correlations could partially cancel one another if E( p_ijt )<E(R_it ), E( p_hvt )>E(R_ht ), and Cov(R_it , R_ht )>0; or if E( p_ijt )>E(R_it ), E( p_hvt )<E(R_ht ), and Cov(R_it , R_ht )>0. However, no plausible underlying force is apparent that would cause these correlations to cancel one another except by chance. Thus, such possibilities are ignored for the analytical purposes of this study.

⁶ If p_ijt  −  R _it is a zero mean stationary process, then E( p_ijt  −  R _it ) = 0 for all t (see Hamilton, 1994, p. 53).

⁷ Equation 6 is obtained by multiplying Equation 4 by p_ijt , and then expressing p_ijt as a deviation from an aggregate price index, e.g., p_ijt  = ( p_ijt − R_it ) + R_it .

⁸ For i = h, the assumption that R_it is a degenerate stochastic process, Var(R_it ) = 0, clearly conflicts with the nature of most economic data. Additionally, R_it is useless for evaluating price elasticities without variability across time.

⁹ A linear form such as in Equation 7 with prices in logarithms is consistent with integrability only if preferences are homothetic (Silberberg, 1990, p. 411).