Estimating a non-minimum cost function for hospitals: comment.

• Author: Atkinson, Scott E.
• Position: Comment on B. Kelly Eakin and Thomas J. Kniesner's article in the Southern Economic Journal, p.583, January 1988

1. Introduction

   In a recent issue of this Journal, Eakin and Kniesner [2] estimate a non-minimum cost function that is derived from a shadow cost function in which shadow and market prices for inputs are allowed to diverge. While the basic approach is sound, the authors make an error in the derivation that causes the equations for observed total cost and cost shares to violate necessary homogeneity conditions, and therefore their econometric model is seriously misspecified. Eakin and Kniesner (henceforth, E-K) also make several incorrect statements concerning the interpretation of the shadow prices.

2. Shadow and Observed Cost Functions

   E-K specify the shadow price of input [Mathematical Expression Omitted], to be equal to its market price [w.sub.i], plus a shadow price divergence factor, [[Theta].sub.i]. Total shadow cost, [C.sup.sh], is specified to be a translog function of the shadow prices, with linear homogeneity in shadow prices imposed.

   By Shephard's lemma, shadow-cost-minimizing input demand equations are derived by differentiating total shadow cost with respect to shadow prices, and therefore the observed quantities of inputs, [X.sub.i], are homogeneous of degree zero in shadow prices.(1) Similarly, logarithmic differentiation of the shadow cost function with respect to shadow prices yields the shadow cost shares, [Mathematical Expression Omitted], which are also homogeneous of degree zero in shadow prices.

   Homogeneity of degree zero of the observed input quantities implies that observed total cost, [C.sup.obs] = [[Sigma].sub.i] [w.sub.i] [x.sub.i], and the observed cost shares [Mathematical Expression Omitted], are also homogeneous of degree zero in shadow prices. The intermediate expressions derived by E-K for [C.sup.obs] and [Mathematical Expression Omitted] equations (EK11) and (EK12), respectively, are homogeneous of degree zero, but their final estimating equations, (EK18) and (EK19), do not satisfy this necessary condition, apparently due to an algebraic error in the derivation.

   Equation (EK11) is reproduced here for ease of reference. (1) [Mathematical Expression Omitted]

   Scaling all the shadow prices by an arbitrary constant, z, results in the first term of the righthand side of (1) [C.sup.sh], being multiplied by z (by linear homogeneity of the shadow cost function), leaves the numerator of the term in brackets unchanged (by homogeneity of degree zero of the shadow cost shares), and multiplies the shadow price in the...