## Abstract

In industry sectors where market prices for goods and services are unavailable, it is common to use estimated output and input distance functions to estimate rates of productivity change. It is also possible, but less common, to use estimated distance functions to estimate the normalised support (or efficient) prices of individual inputs and outputs. A problem that arises in the econometric estimation of these functions is that more than one variable in the estimating equation may be endogenous. In such cases, maximum likelihood estimation can lead to biased and inconsistent parameter estimates. To solve the problem, we use linear programming to construct a quantity index. The distance function is then written in the form of a conventional stochastic frontier model where the explanatory variables are unambiguously exogenous. We use this approach to estimate productivity indexes, measures of environmental change, levels of efficiency, and support prices for a sample of Australian public hospitals.

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## Notes

A small portion (<7 % in 2007–2008) of public hospital funding comes from insurance funds and other non-government sources.

In February 2011 the Commonwealth Government announced that it will fund a lower percentage of efficient prices.

If input prices are available then an estimated cost function will give the minimum cost of producing (i.e., the efficient price of) a vector of outputs. The first derivatives of the cost function with respect to individual outputs will give the efficient prices of those outputs under marginal cost pricing.

Let \(a=(a_1,\ldots,a_K)^{\prime}\). The notation

*a*≧ 0_{ K }means that*a*_{ k }≥ 0 for \(k=1,\ldots,K\). The notation*a*≥ 0_{ K }means that*a*_{ k }≥ 0 for \(k=1,\ldots,K\) and there exists at least one value \(k\in \{1,\ldots,K\}\) where*a*_{ k }> 0. The notation*a*> 0_{ K }means that*a*_{ k }> 0 for \(k=1,\ldots,K. \)The output and input distance functions are defined as \(D_O(x,q,z)=\inf\{\delta>0:(x,q/\delta)\in T(z)\}\) and \(D_I(x,q,z)=\sup \{\rho > 0:(x/\rho,q) \in T(z)\}\). If the production possibilities set is given by (1) then \(D_O(x,q,z)=\inf\{\delta>0:h(q/\delta)^r \le b(z)g(x)\}=\inf\{\delta>0:h(q)^r/\delta \le b(z)g(x)\}=\inf\{\delta>0:\delta \ge h(q)^r/[b(z)g(x)]\}\) =

*h*(*q*)^{r}/[*b*(*z*)*g*(*x*)] and \(D_I(x,q,z)=\sup \{\rho > 0:h(q)^r \le b(z)g(x/\rho)\}=\sup \{\rho > 0:h(q)^r \le b(z)\rho^{-r}g(x)\}=\sup\{\rho > 0:\rho \le [b(z)g(x)]^{1/r}/h(q)\}=[b(z)g(x)]^{1/r}/h(q)\).Coelli (2000, pp. 10–16) suggests that it is also fine in the case where firms minimise costs and there is no statistical noise (i.e., when the frontier is deterministic).

For a discussion of Lowe indexes and their properties see, for example, Hill (2008).

Most of these variants involve constraining the weights

*a*_{ it }in problem (22) to be strictly positive, something that can be done easily in Excel Solver. However, using Excel to solve these weight-restricted problems for every observation in the dataset can be time-consuming if the number of observations is large.These axioms include the linear homogeneity, identity, homogeneity of degree zero, commensurability, proportionality and transitivity axioms listed in O’Donnell (2012c), plus a weaker version of the monotonicity axiom listed in that paper. This list of basic axioms differs slightly from lists of axioms and tests found elsewhere in the literature [e.g., Balk (2008)] where quantity indexes are required to be functions of prices.

If there is no variation in the output vector then the minimisation problem (23) is equivalent to the standard data envelopment analysis (DEA) LP used to compute the reciprocal of the

*it*-th technical efficiency score under the assumption of constant returns to scale (CRS)—see, for example, O’Donnell (2010, p. 543, eq. 6.6). Thus, aggregate inputs can be computed quickly by (1) constructing an artificial dataset in which there is only one observation-invariant output, then (2) using this artificial dataset to solve the standard DEA technical efficiency problem under the assumption of CRS. Importantly, the ‘technical efficiency’ estimates reported by DEA software packages should not be interpreted as measures of efficiency—they are the reciprocals of aggregate inputs. In this paper we implemented this fast-computing procedure using the \(\hbox{DPIN}^{\rm TM}\) software.An alternative to the half-normal is the truncated normal distribution. The half-normal was chosen because (1) the truncated normal involves the estimation of an additional parameter, and (2) if the idiosyncratic error is normally distributed (as it is here), the truncated normal can make it harder to distinguish inefficiency from noise [Coelli et al. (2005, p. 252)].

The percentage increase in maximum aggregate output (and maximum revenue) per period, holding inputs and the output mix constant, is \( -\partial {\ln} {D_O}(x_{it}, q_{it}, z_{it})/\partial t = \partial \ln b(z_{it})/\partial t = \gamma_1 = \phi_1 r. \)

For information on the growing problem of antibiotic resistance, see Miller and Miller (2011). The failure of technological advances to keep pace with deteriorations in the physical environment is sometimes referred to as the Red Queen effect. This a reference to the Red Queen’s race in Lewis Carroll’s

*Through the Looking-Glass*. The Red Queen said “it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!”.If the technology is represented by (1) then the cost function is given by (4) \(\Rightarrow MC_{nit}\equiv {\partial C(w_{it},q_{it},z_{it})}/{\partial q_{nit}=C(w_{it},q_{it},z_{it})}\times {\partial \ln h(q_{it})/\partial q_{nit}} \Rightarrow p_{{nit}}^{*} = rMC_{{nit}} /D_{I} (x_{{it}} ,q_{{it}} ,z_{{it}} )^{r}.\) It follows that if

*D*_{ I }(*x*_{ it },*q*_{ it },*z*_{ it }) = 1 then \( p_{{nit}}^{*} = rMC_{{nit}} \).

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## Additional information

An earlier version of this paper entitled “An Econometric Approach to Estimating the Components of Productivity Change in Public Hospitals” was presented at the International Workshop on Efficiency and Productivity in Honour of Professor Knox Lovell, Elche, Spain, 4–5 October 2010.

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O’Donnell, C.J., Nguyen, K. An econometric approach to estimating support prices and measures of productivity change in public hospitals.
*J Prod Anal* **40, **323–335 (2013). https://doi.org/10.1007/s11123-012-0312-0

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DOI: https://doi.org/10.1007/s11123-012-0312-0

### Keywords

- Endogeneity
- Distance functions
- Shadow prices
- Efficient prices
- Total factor productivity
- Environmental change
- Technical efficiency
- Scale efficiency

### JEL Classification

- C43
- D24
- I12
- I18