Stochastic frontier estimation of a CES cost function: the case of higher education in Britain

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Abstract

Many estimates exist in the literature of multi-product cost functions estimated across a sample of institutions of higher education. Typically these have not employed the appropriate frontier estimation techniques. The cost functions usually estimated — such as the constant elasticity of substitution (CES) function — are highly non-linear, and the standard software packages do not allow stochastic frontier estimation in such cases. We derive and maximise the likelihood function associated with this problem, and hence construct measures of economies of scale and scope which obtain in British higher education. We use the half-normal residuals generated by the stochastic frontier estimation to construct measures of technical efficiency for each university in our sample.

Introduction

The economic analysis of multi-product organisations has burgeoned following advances made in industrial economics by Baumol, Panzar, and Willig (1982). One area which has received particular attention has been the estimation of multi-product cost equations. Largely on account of the ease of data availability, the higher education sector has been studied particularly extensively. Studies include analyses based on data for Australia (Lloyd, Morgan, & Williams, 1993), Britain (Glass, McKillop, & Hyndman, 1995a, Glass, McKillop, & Hyndman, 1995b; Johnes, 1996a, Johnes, 1997), Japan (Hashimoto & Cohn, 1997), Turkey (Lewis & Dundar, 1995), and the United States (Cohn, Rhine, & Santos, 1989; de Groot, McMahon, & Volkwein, 1991; Dundar & Lewis, 1995; Nelson & Hevert, 1992). Typically these studies have involved estimating a constant elasticity of substitution (CES), quadratic or hybrid translog cost function using best-fit methods — such as least squares or a standard application of maximum likelihood (ML).

The cost curves of theory are drawn under a condition of technical efficiency. But empirical cost functions are estimated on data which are contaminated by inefficiency; worse, indeed, such inefficiency is likely to vary from observation to observation. Best-fit methods of estimation are not therefore appropriate in this instance. This point was noted in early work by Aigner and Chu (1968), and subsequently developed by Aigner, Lovell, and Schmidt (1977) and Jondrow, Lovell, Materov, and Schmidt (1982). In these papers, methods of frontier analysis are proposed which impose on the ML estimation process a structure of residuals which is non-normal. Moreover, residuals no longer sum to zero, and hence the estimated function may not be regarded as a best-fit estimate. Outturns which lie away from the estimated frontier may do so, at least partly, as a consequence of technical inefficiency. Under certain assumptions, the technical inefficiency associated with each observation may be measured.3

Using stochastic frontier methods to evaluate the parameters of a multi-product cost function would be straightforward only in the case of a simple specification. Otherwise, the specification of the model is non-linear, so that solution using frontier methods requires the construction and maximisation of a dedicated likelihood function. This task has not been attempted in the received literature. We undertake it in the present paper, using data for 99 UK universities for 1994–95. To be specific, we estimate a CES multi-product cost function using the stochastic frontier method assuming error terms which may be decomposed into two parts — a normal component (which may be assumed due to measurement error) and a half-normal component (which is characteristically supposed to be due to inefficiency). The results are compared with those obtained from estimating the same cost function with two-sided normal error terms (referred to as non-linear maximum likelihood in this paper), and our estimates are used to produce measures of scale and scope economies, and to provide information about the technical efficiency of each institution in our sample.

The issue of technical efficiency in higher education is of considerable policy interest. In the United Kingdom universities are autonomous non-profit organisations. While they are not nationalised entities, they do receive a significant proportion of their income from government. Government therefore shows an interest in their efficiency as it seeks to demonstrate to the taxpayer that resources are being wisely spent. Over the last decade and a half, there has been a clamour for higher education performance indicators (Johnes & Taylor, 1990), and the various Research Assessment Exercises and Teaching Quality Assessments exemplify this desire to measure performance (Johnes, 1996b). While these provide much information which was not previously available, a satisfactory single measure of cost efficiency has remained elusive.

The paper proceeds as follows. The next section reviews the literature on costs in higher education. Section 3 concerns our estimating method and a discussion of our results. The final section draws together our conclusions and suggestions for future work.

Section snippets

Literature review

Models of multi-product organisations first came under serious scrutiny as recently as 15 years ago with the seminal contribution of Baumol et al. (1982). This work placed the analysis of industrial structure in a new context, one in which firms are typically able to produce a variety of products and can switch their mix of output types in response to market forces. Indeed the essence of contestable markets is that firms face a menu of choices when deciding what to produce with a given set of

Method of analysis and results

We have applied the method outlined in the last section to data for the year 1994–95 in order to estimate the parameters of a CES cost function characterised as:y=α+(β1xγ112xγ223xγ334xγ44)ρ+ε,where

    y

    total expenditure (£000 per annum),

    x1

    undergraduate student load in (broadly defined) arts subjects,

    x2

    undergraduate student load in (broadly defined) science subjects,

    x3

    postgraduate student load,

    x4

    value of research grants and contracts received (£000 per annum).

Further information on the

Conclusions

The notion that there remains inefficiency in the British higher education system may be anathema to many academics. Yet it is implausible to suppose that a multiplicity of institutions, with a variety of production technologies, missions and corporate cultures, have all stumbled upon the same, unsurpassable, level of technical efficiency. By the same token, it would be equally unlikely that the institutions have nothing to gain from benchmarking exercises — which of course imply that gains in

Acknowledgements

Without implication, the authors are indebted to two referees and seminar participants at Leeds and Sussex for useful comments.

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