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

I. 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.

II. 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 denominator by z. Thus the terms in z cancel out, leaving the value of [C.sup.obs] unchanged, as required by homogeneity of degree zero in the shadow prices.

Equation (EK12) is reproduced here as, (2) [Mathematical Expression Omitted]

Scaling all the shadow prices by z does not affect the values of the [Mathematical Expression Omitted], and the remaining terms in z cancel out of (2), leaving the values of the actual cost shares unchanged.

E-K state that they derive the system of equations they estimate by substituting the equations for total shadow cost and shadow cost shares into the logarithms of (1) and (2). Performing the indicated substitutions yields the equations (3) [Mathematical Expression Omitted] and (4) [Mathematical Expression Omitted]

Equations (3) and (4), which are equivalent to the corresponding equations in Atkinson and Halvorsen [1], are homogeneous of degree zero in the shadow prices as required.

However, instead of the correct equations (3) and (4), E-K write their final equations, (EK18) and (EK19), as (5) [Mathematical Expression Omitted] and (6) [Mathematical Expression Omitted]

Equations (5) and (6) are not homogeneous of degree zero in the shadow prices, contrary to the logic of the derivations Therefore, all parameter estimates obtained by E-K must be considered to be contaminated by misspecification of the estimated equations.

The misspecification of (5) might also have resulted in the use of incorrect formulas for observed marginal cost and output cost elasticities, but the corresponding equations. (EK27) and (EK28), appear to have been derived from the correct intermediate equation, (1), rather than the incorrect final equation, (5). However, the term [[Sigma].sub.i] [[Delta].sub.ik] ([w.sub.i]/ [Mathematical Expression Omitted]) should be divided by ([Mathematical Expression Omitted]) in both (EK27) and (EK28), and the first term on the right-hand side of (EK-27) should be [C.sup.obs] rather than [C.sup.sh].

Homogeneity of degree zero in shadow prices of the correctly specified equations for observed total cost and observed cost shares, (3) and (4), implies that only ralative shadow prices can be estimated. In order to identify the relative shadow prices, it is necessary that one of them be normalized. The choice of the shadow price to be normalized is purely arbitrary and made without loss of generality.

The homogeneity of degree zero of the observed cost and cost share equations, and the arbitrary character to the choice of normalization, both follow from the same basic conclusion of economic theory, namely that only relative prices matter in determining the cost minimizing quantities of inputs. This is noted by E-K in their equations (EK2) and (EK3), which show the cost minimizing and observed input choices being determined by the equality of the corresponding price ratios and marginal rates of technical substitution. As indicated by (EK2) and (EK3), the cost minimizing and observed marginal rates of technical substitution, and thus the cost minimizing and observed relative quantities of inputs, will be equal if all market and shadow price ratios are equal. This is all that is necessary for cost minimization.

Two important implications for the interpretation of shadow prices follow from the homogeneity properties and the economic theory underlying them. First, the conclusions concerning the efficiency of input use that can be drawn directly from the estimated shadow prices are limited to statements about the relative quantities of inputs. For example, if the estimation results indicate that (7) [Mathematical Expression Omitted] it can be concluded that less than the cost minimizing amount of input i is used relative to input j, but no conclusion can be drawn concerning the efficiency of the absolute of input i.(2)

Thus the statement by E-K that [Mathematical Expression Omitted] "results in underemployment of input [x.sub.i]" [2,584] is incorrect. Because only the relative values of shadow prices can be estimated, the relationship between the absolute values of the shadow and market prices of input i is arbitrarily determined by the choice of normalization and reveals nothing about the efficiency of the absolute amount of input i. Of course, if the normalization used is that [Mathematical Expression Omitted], then [Mathematical Expression Omitted] implies (7), and it V can be concluded that input i is underemployed relative to input j.

Second, the condition that [Mathematical Expression Omitted] for all inputs is sufficient, but not necessary, for [C.sup.obs] and [C.sup.min] to coincide and for cost minimization with respect to all inputs to be attained. The necessary and sufficient condition in both cases is that (8) [Mathematical Expression Omitted] where z is an arbitrary factor of proportionality. The equality of the observed and minimum cost functions given (8) can be demonstrated by substituting (8) in (3) and noting that the observed cost function reduces to the minimum cost function. This in turn implies that cost minimization with respect to all inputs is attained.

Accordingly, the statement, "[i]f any [Mathematical Expression Omitted], then there is systematic allocative inefficiency, and [C.sup.obs] > [C.sup.min]" [2,585] is not in general correct. If (8) is satisfied, efficiency is attained even though [Mathematical Expression Omitted] for all i if z [is not equal to] 1. However, using E-K's additive specification for [Mathematical Expression Omitted], (8) can be satisfied only if [[Theta].sub.i] is equal to 0 of all i, and therefore the value chosen for the normalized [[Theta].sub.i] has to be set equal to zero.(3)

If follows that the statement, "[t]here is conditional efficient employment of the ith input if, given the level of other inputs, [[Theta].sub.i] = 0" [2,586], as well as the related assertion that the assumption "that hospital labor is employed at the cost minimizing level [given the level] of capital and physicians" implies that the [Theta] for hospital labor should be constrained to equal zero [2,588], are not correct. One of the shadow prices has to be normalized to identify the relative values of the others, and given the use of E-K's specification of [Mathematical Expression Omitted], the normalized value has to be set equal to zero to allow for the possibility that efficiency is attained. But the choice of input to normalize is arbitrary, and equating its [Theta] to zero does not imply conditional efficiency of the normalized input. Scott E. Atkinson University of Georgia Athens, Georgia Robert Halvorsen University of Washington Seattle, Washington (1)If a function, f(p,q), is homogeneous of degree t in p, the function f'(p) = [Delta] f/[Delta] p is homogeneous of degree t - 1. (2)However, the effects of inefficiency on the quantity demanded of each input, holding output constant, can be calculated from the observed cost function, as discussed in Atkinson and Halvorsen [1]. (3)Because relative market prices vary across observations, but the [[Theta].sub.i] are constrained to be constant across observations, the only value for the additive shadow price divergence parameters that is consistent with conditions (8) is zero.

References

[1]Atkinson Scott E. and Robert Halvorsen, "Parametric Efficiency Tests, Economies of Scale, and Input Demand in U.S. Electric Power Generation." International Economic Review, October 1984, 647-62. [2]Eakin, B. Kelly and Thomas J. Kniesner, "Estimating a Non-minimum Cost Function for Hospitals." Southern Economic Journal, January 1988, 583-97.