A note on Allen–Uzawa partial elasticities of substitution: the case of the translog cost function

Abstract

This note provides a useful property of the Allen–Uzawa partials for the translog cost function. It also suggests how the main results extend to any functional form with certain properties. The curvature of the Allen–Uzawa matrix is the same as the curvature of the Hessian matrix. Intuitively and empirically, the Allen–Uzawa partials allow for the verification of curvature properties.

Notes

¹For the original definition of the elasticity of substitution, see Hicks (1932), p. 177 and Allen (1938), p. 340–3, p. 503–9. For an early review of the concept and its uses, refer to Morrissett (1953) and the citations therein.

²In defining the cost function, we use (t − t*) instead of t. This facilitates the imposition of local concavity in what follows (Ryan and Wales, 2000). They also show that this has no effect on the likelihood function, in estimation, or the elasticities of interest. For regularity conditions on C(w, y, t) see Diewert and Wales (1987, p. 45).

³Because of duality, all derivations that follow can easily be shown to hold for the indirect translog profit function, see Diewert (1974), Hertel (1984), Diewert and Wales (1987). The derivation of the price elasticities (own and cross) used below is found therein as well.

⁴More formally, a diagonal matrix is defined as , where S_i varies with i = 1, … , n; and .

⁵For i = j, we have η _ii /S_i  = σ _ii  = −1/S _i  + 1 + b _ii /S _i S _i , and for i ≠ j η _ii /S_j  = σ _ij  =1 + b_ij /S_iS_j . See also the Appendix.

⁶Diewert and Wales (1987) provide some additional insight. They refer to the work of Diewert, McFadden and Barnett.

⁷Since S_i  (w, y, t) = w _i x _i (w, y, t)/C(w, y, t) = w_iC_i /C, and rearranging, in matrix form it gives the desired expression.

⁸This is the so-called Cholesky decomposition (Lau, 1978). The appropriate curvature is imposed by substituting the original parameter matrix with its Cholesky decomposition. See Featherstone and Moss (1994) for an application.

⁹For a discussion of the differences between their method and the Cholesky decomposition, see Ryan and Wales (1998, p. 332). They also discuss several alternative methods for imposing curvature in recent literature.

¹⁰This follows because, after substitution for estimation purposes, the number of elements above the main diagonal in D just replaces the (n ²− n)/2 elements of the parameter matrix B. See also Theorem 1 in Diewert and Wales (1987, p. 46).