# 1 Efficiency and Productivity Measurement: Multi-output Distance and Cost functions D.S. Prasada Rao School of Economics The University of Queensland Australia.

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1 Efficiency and Productivity Measurement: Multi-output Distance and Cost functions D.S. Prasada Rao School of Economics The University of Queensland Australia

2 Duality and Cost Functions So far we have been working with the production technology and production function – this is known as the primal approach. Instead we could recover all the information on the production function by observing the cost, revenue or profit maximising behaviour of the firms. This is known as the dual approach to the study of production technology. In our lectures, we have looked at the possibility of describing the technology using input and output distance functions. Here we will focus on cost functions – see the textbook for revenue and profit functions (Chapter 2).

3 Duality and Cost Functions The cost function is defined as:

4 Duality and Cost Functions When dealing with multi-output and mlti-input technologies, we can use the cost-function to derive the input demand functions. From Shepherds Lemma, we have: The input demand function satisfies all the expected properties: (i) non-negativity; (ii) non-increasing in w; (iii)non-decreasing in q; (iv) homogeneous of degree zero in input prices; and (v) Symmetry Estimation of cost function – either as a single equation or a system of equations including input-share equations.

5 Cost frontiers Advantages: –captures allocative efficiency –can accommodate multiple outputs –suits case where input prices exogenous and input quantities endogenous –suits case where input quantity data unavailable Disadvantages: –requires sample input price data that varies –biased if frontier firms are not cost minimisers

6 Estimation of cost frontier Before we examine the estimation of multi-output and multi-input distance function, we briefly consider the estimation of cost-frontier. If we assume that the cost function is modelled using Cobb-Douglas functional form (we use this since it is possible to decompose cost efficiency into technical and allocative efficiency. We use where u i is a non-negative random variable representing inefficiency.

7 Estimation of cost frontier Imposing linear homogeneity in input prices: and re-writing the model after imposing this constraint we have: which can be written in a standard frontier model as: This model can be estimated using the standard frontier methods and the Frontier program.

8 Cost efficiency and decomposition Cost efficiency is measured in a way similar to what we did for technical efficiency using and other formulae to find firm-specific efficiencies. Decomposition of Cost Efficiency: –If we have input quantities or cost shares, cost efficiency can be decomposed into technical and allocative efficiency components. –In this case it is possible to model a system of cost-share equations for different inputs –The cost frontier has both allocative and technical efficiency combined and share equations have information on allocative efficiency but the relationships between these is quite complex –A simpler approach is possible in the Cobb-Douglas since in this case both the production function and cost function have the same functional form.

9 Cost efficiency (CE) decomposition Translog is difficult - because the function is not self-dual. In this case the options are: –solve a non-linear optimisation problem for each observation to decompose CE –estimate a system of equations Important references: –Kumbhakar (1997) –Kumbhakar and Lovell (2000)

10 SFA model as a basis for estimating distance functions Suppose we want to estimate the input distance function d i (x i, q i ) and suppose we assume that the distance function is of log- linear form, then The main problem in estimating this model is that the distance function is unobserved. But we know that: the distance function is non-decreasing, linearly homogeneous and concave in inputs; and non-increasing and quasi-concave in outputs. Linear homogeneity in inputs gives us the condition Then the model can be rewritten as

11 SFA model as a basis for estimating distance functions There are several issues that need further consideration and resolution: It is possible that the explanatory variables may be correlated with the composite error term – this can lead to inconsistent estimators. It may be necessary to use an instrumental variable framework. Coelli (2000) argues that in the case of Cobb-Douglas and translog specifications, this is not an issue provided revenue maximisation or cost minimisation behaviour is assumed. The distance functions need to satisfy the concavity or quasi- concavity properties implied by economic theory. Otherwise, it may lead to strange results. This requires a Bayesian approach – see ODonnell and Coelli (2005). Coelli and Perleman (1999) makes a comparison of parametric and non-parametric approaches to distance functions.

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