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

I. Introduction

There were typographical errors in equations (18), (19), (27), and (28) in the theoretical section of our original manuscript [5]. The corrected equations are (18) [Mathematical Expression Omitted] (19) [Mathematical Expression Omitted] (27) [Mathematical Expression Omitted] (28) [Mathematical Expression Omitted] where T [Mathematical Expression Omitted] There were no programming errors and all estimates we reported in our original paper [5] are unchanged.

In what follows, we clarify three issues--that

1. If the shadow price of the ith input exceeds its market price [Mathematical Expression Omitted

then there is relative underemploymentof the ith input [X.sub.i];

2. If any [Mathematical Expression Omitted then there is empirical allocative inefficiency; and

3. if [Mathematical Expression Omitted] = [[Theta].sub.i] = 0 then there is conditional efficient

employment of input [X.sub.i] Our first two points should rectify possible confusion over the roles of absolute versus relative input prices in the non-minimum cost function. Our third point refines the concept of conditional efficient employment of an input.

II. Relative Underemployment of an Input

Empirical inefficiency first appeared in the Cobb-Douglas profit function of Lau and Yotopolus [6].Toda [10] modeled non-minimization of quadratic average cost. Eight years later Atkinson and Halvorsen [1] adopted Toda's approach in using a translog shadow cost function to investigate possible Averch-Johnson [2] overcapitalization in regulated electric power generation. The model use in [5] resembles the Atkinson and Halvorsen model with two differences. First, we choose an additive ([W.sub.i] + [[Theta].sub.i]) rather than a multiplicative [[Theta].sub.i] [W.sub.i]) parameterization of shadow input prices and second, we model a multiproduct firm and consequently choose a hybrid-translog multiproduct shadow-cost function that permits some outputs to be zero.

We emphasize that when using an estimated cost function to discuss economic efficiency over- or underemployment of an input is both conditional on the levels of the other inputs and is relative to the numeraire input. In our earlier paper [5] the estimated shadow price of capital ([kappa]) was below the observed price, [[Theta] [caret].sub.k] < 0, and the estimated shadow price of physicians' services (d) was above the observed price [[Theta] [caret].sub.d] > 0, while equality between the shadow and the observed prices of non-physician labor (l) and equality between the shadow and observed prices of materials (m), [[Theta].sub.l] = [[Theta].sub. m] = 0, was imposed ex ante. Consequently, [Mathematical Expression Omitted] and capital is overemployed relative to physicians, materials, and non-physician labor and physicians is underemployed relative to capital, materials, and non-physician labor.

We contend that a discussion of empirically absolute versus empirically relative input efficiency is meaningless and confusing. Cost-minimizing resource allocation depends only on relative input prices. Atkinson and Halvorsen [1,653] define absolute efficiency as occurring "if the value of the marginal product of each input is equated to the input's market price." This is the profit-maximization condition, which is neither necessary for cost minimization nor testable in the non-minimum cost function model. A similar confusion over the effects of price distortions occurs in the rate of return regulation literature where overemployment of capital is sometimes misinterpreted as either more capital or a greater capital-labor ratio than the unregulated firm would choose [2;3]. The correct interpretation of overemployment of capital, which we gave to our results [5], is that the ratios of capital to other inputs are greater than the cost-minimizing ratios for the given level of output.

III. Empirical Allocative Inefficiency

Not all absolute shadow input prices are retrievable empirically. Even in a profit-maximization context, including m output prices as well as n input prices, only (any) n + m - 1 independent relative prices affect resource allocation. Thus, there is no less of generality in our discussion [5, 585] of empirical allocative inefficiency even if the true model is a multiplication parameterization of the divergence observed input prices and shadow prices ([Mathematical Expression Omitted]).

IV. Conditional Efficient Input Employment

Conditional efficiency is closely related to the concept of the second best. In our judgement conditional efficiency is the cost-minimizing amount of input [X.sub.i] accepting distortions that exist in other input markets. Consider a three-input production function. If [Mathematical Expression Omitted] then [Mathematical Expression Omitted] results in conditional and complete efficiency where complete efficiency is defined as the minimization of the observed cost of producing a given vector of outputs [5, 586]. Further, if [Mathematical Expression Omitted], then [Mathematical Expression Omitted] is also conditionally and completely efficient. A first-best solution to distorted input markets is similar to Robinson's "world of monopolies" [8] and might be called a "world of monopsonies." Of course, one cannot empirically detect absolute shadow prices via a non-minimum cost function so there is no meaningful difference between completely efficient outcomes. Now, if [Mathematical Expression Omitted], and [Mathematical Expression Omitted] what value of [Mathematical Expression Omitted] gives conditional efficiency? In general, [Mathematical Expression Omitted] does not give conditional efficiency. That is, the second best outcome given the unequal distortions in markets 1 and 2 is not, in general, an undistorted market 3. The theory of the second best compels us to revise our conditional efficiency condition [5, 586] to recognize the possibility of partially offsetting distortions in input markets. The theory of second best also makes us reject the concept of "relative price efficiency" in [1, 653].

V. Homogeneity Properties of the Non-Minimum Cost Function

We close with some clarifying remarks about the homogeneity properties of the non-minimum cost function and the structure of shadow input prices. Economic theory requires that the shadow cost function be homogeneous to degree 1 in shadow input prices and, consequently, that factor demands and shadow cost shares be homogeneous to degree 0 in shadow input prices. By construction, if observed input prices do not change as shadow input prices change, observed cost is homogeneous to degree 0 in shadow input prices. However, observed input prices are a component of shadow input prices. That is, shadow input prices are theoretically equal to (1) Mathematical Expression Omitted] where [[Mu].sub.i] and endogenous Lagrangian multipliers.(1) Changing all shadow input prices proportionately while holding observed input prices constant is impossible. Given that observed input prices do change as shadow input prices change, then a proportional change in all shadow input prices may indeed cause observed cost to change. Furthermore, economic theory does not require that either the shadow or observed cost function be homogeneous in observed prices. With a multiplicative parameterization of shadow prices [Mathematical Expression Omitted] both the shadow and observed cost functions are homogeneous to degree 1 in observed prices. Neither the shadow nor the observed cost function is homogeneous in observed input prices if divergence between shadow and observed input prices is parameterized additively [Mathematical Expression Omitted].

Finally, the general theoretical form of shadow price [Mathematical Expression Omitted] emphasis that the divergences between shadow and observed input prices [Mathematical Expression Omitted] are endogenous. However, the empirical non-minimum cost function literature treats [Mathematical Expression Omitted] parametrically. The next generation of empirical non-minimum cost functions should attempt to incorporate the endogeneity of shadow price divergences, [[Theta].sub.i].(2) B. Kelly Eakin University of Oregon Eugene, Oregon Thomas J. Kniesner Indiana University Bloomington, Indiana (1)Shadow input prices given by equation (1) are first-order conditons to the Lagrangian optimization problem of the utility-maximizing entrepreneur facing imperfect input markets. Shadow prices have also been called virtual prices For details on the theoretical derivation of shadow prices see Neary and Roberts [7] and Thornton and Eakin [9]. (2)A excellent reference on the econometrics of endogenous firm-specific random heterogeneity is Breusch, Mizon, and Schmidt [4].

References

[1]Atkinson, Scott and Robert Halvorsen, "Parametric Efficiency Tests, Economics of Scale and Input Demand in U.S. Power Generation." International Economic Review, October 1984, 647-61. [2]Averch, Harvey and Leland L. Johnson, "Behavior of the Firm under Regulatory Constraint." American Economic Review, December 1962, 1053-69. [3]Baumol, William and Alvin Klevorick, "Input Choices and Rate-of-Return Regulation: An Overview of the Discussion." Bell Journal of Economics and Management Science. Autumn 1970, 162-90. [4]Breusch, Trevor S., Graham E. Mizon and Peter Schmidt, "Efficient Estimation Using Panel Data." Econometrica, May 1989, 695-700. [5]Eakin, B. Kelly and Thomas J. Kniesner, "Estimating a Non-Minimum Cost Function for Hospitals." Southern Economic Journal, January 1988, 583-97. [6]Lau, Lawrence and Plan Yotopolus, "A Test for Relative Efficiency and an Application to Indian Agriculture." American Economic Review, March 1971, 94-109. [7]Neary, J, Peter and K. W. S. Roberts, "The Theory of Household Behavior under Rationing." European Economic Review, 13, 1980, 25-42. [8]Robinson, Joan. The Economics of Imperfect Competition. London: MacMillan, 1934. [9]Thornton, James and B. Kelly Eakin, "Virtual Prices and a General Theory of the Owner-Operated Firm." Southern Economic Journal, April 1992. [10]Toda, Yasushi, "Estimation of A Cost Function When Cost Is not Minimum: The Case of Soviet Manufacturing Industries, 1958-71." Review of Economics and Statistics, August 1976, 259-68.

There were typographical errors in equations (18), (19), (27), and (28) in the theoretical section of our original manuscript [5]. The corrected equations are (18) [Mathematical Expression Omitted] (19) [Mathematical Expression Omitted] (27) [Mathematical Expression Omitted] (28) [Mathematical Expression Omitted] where T [Mathematical Expression Omitted] There were no programming errors and all estimates we reported in our original paper [5] are unchanged.

In what follows, we clarify three issues--that

1. If the shadow price of the ith input exceeds its market price [Mathematical Expression Omitted

then there is relative underemploymentof the ith input [X.sub.i];

2. If any [Mathematical Expression Omitted then there is empirical allocative inefficiency; and

3. if [Mathematical Expression Omitted] = [[Theta].sub.i] = 0 then there is conditional efficient

employment of input [X.sub.i] Our first two points should rectify possible confusion over the roles of absolute versus relative input prices in the non-minimum cost function. Our third point refines the concept of conditional efficient employment of an input.

II. Relative Underemployment of an Input

Empirical inefficiency first appeared in the Cobb-Douglas profit function of Lau and Yotopolus [6].Toda [10] modeled non-minimization of quadratic average cost. Eight years later Atkinson and Halvorsen [1] adopted Toda's approach in using a translog shadow cost function to investigate possible Averch-Johnson [2] overcapitalization in regulated electric power generation. The model use in [5] resembles the Atkinson and Halvorsen model with two differences. First, we choose an additive ([W.sub.i] + [[Theta].sub.i]) rather than a multiplicative [[Theta].sub.i] [W.sub.i]) parameterization of shadow input prices and second, we model a multiproduct firm and consequently choose a hybrid-translog multiproduct shadow-cost function that permits some outputs to be zero.

We emphasize that when using an estimated cost function to discuss economic efficiency over- or underemployment of an input is both conditional on the levels of the other inputs and is relative to the numeraire input. In our earlier paper [5] the estimated shadow price of capital ([kappa]) was below the observed price, [[Theta] [caret].sub.k] < 0, and the estimated shadow price of physicians' services (d) was above the observed price [[Theta] [caret].sub.d] > 0, while equality between the shadow and the observed prices of non-physician labor (l) and equality between the shadow and observed prices of materials (m), [[Theta].sub.l] = [[Theta].sub. m] = 0, was imposed ex ante. Consequently, [Mathematical Expression Omitted] and capital is overemployed relative to physicians, materials, and non-physician labor and physicians is underemployed relative to capital, materials, and non-physician labor.

We contend that a discussion of empirically absolute versus empirically relative input efficiency is meaningless and confusing. Cost-minimizing resource allocation depends only on relative input prices. Atkinson and Halvorsen [1,653] define absolute efficiency as occurring "if the value of the marginal product of each input is equated to the input's market price." This is the profit-maximization condition, which is neither necessary for cost minimization nor testable in the non-minimum cost function model. A similar confusion over the effects of price distortions occurs in the rate of return regulation literature where overemployment of capital is sometimes misinterpreted as either more capital or a greater capital-labor ratio than the unregulated firm would choose [2;3]. The correct interpretation of overemployment of capital, which we gave to our results [5], is that the ratios of capital to other inputs are greater than the cost-minimizing ratios for the given level of output.

III. Empirical Allocative Inefficiency

Not all absolute shadow input prices are retrievable empirically. Even in a profit-maximization context, including m output prices as well as n input prices, only (any) n + m - 1 independent relative prices affect resource allocation. Thus, there is no less of generality in our discussion [5, 585] of empirical allocative inefficiency even if the true model is a multiplication parameterization of the divergence observed input prices and shadow prices ([Mathematical Expression Omitted]).

IV. Conditional Efficient Input Employment

Conditional efficiency is closely related to the concept of the second best. In our judgement conditional efficiency is the cost-minimizing amount of input [X.sub.i] accepting distortions that exist in other input markets. Consider a three-input production function. If [Mathematical Expression Omitted] then [Mathematical Expression Omitted] results in conditional and complete efficiency where complete efficiency is defined as the minimization of the observed cost of producing a given vector of outputs [5, 586]. Further, if [Mathematical Expression Omitted], then [Mathematical Expression Omitted] is also conditionally and completely efficient. A first-best solution to distorted input markets is similar to Robinson's "world of monopolies" [8] and might be called a "world of monopsonies." Of course, one cannot empirically detect absolute shadow prices via a non-minimum cost function so there is no meaningful difference between completely efficient outcomes. Now, if [Mathematical Expression Omitted], and [Mathematical Expression Omitted] what value of [Mathematical Expression Omitted] gives conditional efficiency? In general, [Mathematical Expression Omitted] does not give conditional efficiency. That is, the second best outcome given the unequal distortions in markets 1 and 2 is not, in general, an undistorted market 3. The theory of the second best compels us to revise our conditional efficiency condition [5, 586] to recognize the possibility of partially offsetting distortions in input markets. The theory of second best also makes us reject the concept of "relative price efficiency" in [1, 653].

V. Homogeneity Properties of the Non-Minimum Cost Function

We close with some clarifying remarks about the homogeneity properties of the non-minimum cost function and the structure of shadow input prices. Economic theory requires that the shadow cost function be homogeneous to degree 1 in shadow input prices and, consequently, that factor demands and shadow cost shares be homogeneous to degree 0 in shadow input prices. By construction, if observed input prices do not change as shadow input prices change, observed cost is homogeneous to degree 0 in shadow input prices. However, observed input prices are a component of shadow input prices. That is, shadow input prices are theoretically equal to (1) Mathematical Expression Omitted] where [[Mu].sub.i] and endogenous Lagrangian multipliers.(1) Changing all shadow input prices proportionately while holding observed input prices constant is impossible. Given that observed input prices do change as shadow input prices change, then a proportional change in all shadow input prices may indeed cause observed cost to change. Furthermore, economic theory does not require that either the shadow or observed cost function be homogeneous in observed prices. With a multiplicative parameterization of shadow prices [Mathematical Expression Omitted] both the shadow and observed cost functions are homogeneous to degree 1 in observed prices. Neither the shadow nor the observed cost function is homogeneous in observed input prices if divergence between shadow and observed input prices is parameterized additively [Mathematical Expression Omitted].

Finally, the general theoretical form of shadow price [Mathematical Expression Omitted] emphasis that the divergences between shadow and observed input prices [Mathematical Expression Omitted] are endogenous. However, the empirical non-minimum cost function literature treats [Mathematical Expression Omitted] parametrically. The next generation of empirical non-minimum cost functions should attempt to incorporate the endogeneity of shadow price divergences, [[Theta].sub.i].(2) B. Kelly Eakin University of Oregon Eugene, Oregon Thomas J. Kniesner Indiana University Bloomington, Indiana (1)Shadow input prices given by equation (1) are first-order conditons to the Lagrangian optimization problem of the utility-maximizing entrepreneur facing imperfect input markets. Shadow prices have also been called virtual prices For details on the theoretical derivation of shadow prices see Neary and Roberts [7] and Thornton and Eakin [9]. (2)A excellent reference on the econometrics of endogenous firm-specific random heterogeneity is Breusch, Mizon, and Schmidt [4].

References

[1]Atkinson, Scott and Robert Halvorsen, "Parametric Efficiency Tests, Economics of Scale and Input Demand in U.S. Power Generation." International Economic Review, October 1984, 647-61. [2]Averch, Harvey and Leland L. Johnson, "Behavior of the Firm under Regulatory Constraint." American Economic Review, December 1962, 1053-69. [3]Baumol, William and Alvin Klevorick, "Input Choices and Rate-of-Return Regulation: An Overview of the Discussion." Bell Journal of Economics and Management Science. Autumn 1970, 162-90. [4]Breusch, Trevor S., Graham E. Mizon and Peter Schmidt, "Efficient Estimation Using Panel Data." Econometrica, May 1989, 695-700. [5]Eakin, B. Kelly and Thomas J. Kniesner, "Estimating a Non-Minimum Cost Function for Hospitals." Southern Economic Journal, January 1988, 583-97. [6]Lau, Lawrence and Plan Yotopolus, "A Test for Relative Efficiency and an Application to Indian Agriculture." American Economic Review, March 1971, 94-109. [7]Neary, J, Peter and K. W. S. Roberts, "The Theory of Household Behavior under Rationing." European Economic Review, 13, 1980, 25-42. [8]Robinson, Joan. The Economics of Imperfect Competition. London: MacMillan, 1934. [9]Thornton, James and B. Kelly Eakin, "Virtual Prices and a General Theory of the Owner-Operated Firm." Southern Economic Journal, April 1992. [10]Toda, Yasushi, "Estimation of A Cost Function When Cost Is not Minimum: The Case of Soviet Manufacturing Industries, 1958-71." Review of Economics and Statistics, August 1976, 259-68.

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Title Annotation: | response to Scott E. Atkinson and Robert Halvorsen, Southern Economic Journal, p. 1114, April 1992 |
---|---|

Author: | Kniesner, Thomas J. |

Publication: | Southern Economic Journal |

Date: | Apr 1, 1992 |

Words: | 1484 |

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