Efficiency and Productivity Measurement: Multi-output Distance and Cost functions

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 Shepherd’s 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 ui 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 di(xi, qi) 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 O’Donnell and Coelli (2005). • Coelli and Perleman (1999) makes a comparison of parametric and non-parametric approaches to distance functions.