CHAPTER 5 ANALYTICAL FOUNDATIONS AND MEASUREMENT OF FRONTIER
5.2 Conceptual framework of frontier efficiency
5.2.4 Economic efficiency
Economic efficiency is the measure of efficiency where a behavioural assumption such as cost minimisation, revenue maximisation or profit maximisation is imposed. Thus, depending on which behavioural objective is assumed, we can measure cost efficiency, revenue efficiency or profit efficiency. Cost minimisation is implied by profit maximisation.
(a) Cost minimisation and cost efficiency
Cost minimisation assumes that a firm seeks to incur the least cost of input combination to produce a given output bundle. In such situations, the production technology is adjusted to include input prices in addition to the input(s) and output(s). Inputs are the choice variables in the context of cost minimisation, hence an input-oriented measure of technical inefficiency, which focuses on input overuse, is assumed in the computation of cost efficiency. We note however that the attainment of technical efficiency is necessary, but not sufficient for the attainment of cost efficiency. This is because a technical efficient firm could use an input mix which, based on the input prices it faces, does not represent the least cost.
Accordingly, assessment of efficiency moves from the production frontier to a cost frontier. The cost frontier defines the minimum cost required to produce any level of output(s) based on input prices. The cost frontier serves as the standard against which to measure cost efficiency by comparing actual cost incurred by a firm to the minimum cost.
Given the objective of minimising cost subject to the production technology constraint, we can write the cost minimisation problem as:
∗(•, Ž) = Minimise W’X subject to F (Ž, •) = 0 (5.13)
whereC(•, Ž) is the minimum cost; • = vector of inputs; • = vector of input prices; Ž = output vector; and v(Ž, •) is the production function.
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The cost minimisation problem in equation (5.13) is solved to derive the input demand equations which can be substituted in the cost function to obtain the minimum cost, ∗(—, Ž). Hence ∗(•, Ž)is the cost frontier which gives the minimum cost based on input prices and observed output(s), and is the benchmark against which cost efficiency is measured.
Similar to the underlying properties of the production frontier and distance functions discussed in Sections 5.2.2 and 5.2.3, the cost frontier must satisfy the following properties (or regularity conditions) (Coelli et al., 2015):
l. Non-negativity: Costs can never be negative, that isC(W, Y) > 0.
2. Monotonicity in input prices: An increase in input price will not decrease costs. If ™ ≥
™ then (™ , X) ≥ (™ , X).
3. Monotonicity in output: An increase in output will not decrease costs. If X ≥ X then
(™, X ) ≥ (™, X ).
4. Homogeneity: A k-fold increase in input prices will cause a k-fold increase in costs. Formally, (u™, X) = u (™, X), 4lm u > 0.
5. Concavity: the cost function (™, X)is concave in input prices, that is, the input demand functions cannot slope upwards.
These regularity conditions must be satisfied in the empirical cost function estimation. Linear homogeneity is usually imposed prior to estimation of the cost function while monotonicity and concavity conditions can be tested post estimation.
Cost efficiency (CE) is therefore expressed as the ratio of minimum cost defined by the cost frontier to the actual cost of the firm as follows:
CE = ∗(™, X) W’Xš (5.14)
Cost efficiency therefore occurs at the point where a firm is both technically and allocatively efficient. If a firm is not on the cost frontier, then it is either technically inefficient or allocatively inefficient, or both.
The analysis of revenue efficiency and profit efficiency is similar to cost efficiency, except that the objective of cost minimisation is replaced by revenue and profit maximisation. We accordingly provide just a brief description of the main elements of revenue efficiency and profit efficiency below.
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(b) Revenue maximisation and revenue efficiency
Where revenue maximisation is the assumed behavioural objective, a revenue frontier is constructed as the benchmark to measure revenue efficiency. In revenue maximisation, outputs become the choice variables and so an output-oriented technical inefficiency is implicitly assumed, although we note that such technical efficiency is only necessary but not sufficient for achieving revenue efficiency.
Assume that a firm faces a positive vector of exogenous output prices ›and a given input vector•. If the firm seeks to maximise its total revenues, , then it will choose the output levels which will maximise revenue.
We express the revenue maximisation problem as follows:
∗(', Y) = Maximise ›′Ž such that v(R, U) = 0
(5.15)
Revenue-maximising output supply equations can be derived from the optimisation problem in equation (5.15) and substituted into the revenue function to obtain the revenue frontier in the form ∗(', Y). The revenue frontier provides a standard against which to measure revenue efficiency.
Revenue efficiency (RE) is therefore expressed as the ratio of a firm’s observed revenue to the maximum revenue:
k(', Y, X) =ž(R,Ÿ)›’Ž
(5.16)
Revenue efficiency can similarly be decomposed into (output-oriented) technical efficiency and allocative efficiency. Revenue inefficiencies can arise from either technical inefficiency, or allocative inefficiency (misallocation of outputs in the face of prevailing output prices), or both.
(c) Profit maximisation and profit efficiency
In cases where profit maximisation is assumed by producers, we can measure profit efficiency using the profit frontier. For profit maximisation however, both inputs and outputs become
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choice variables as producers must choose an appropriate input mix and also produce an appropriate output mix to maximise profit.
Firms face positive input prices (W) and output prices(P) and choose inputs (') and outputs
(X) in order to maximise profits (P&Y − W′X). The profit maximisation problem of the firm is specified as follows:
(Y, ™) = Maximise Y&X − ™&' sM.ℎ ¡ℎ ¡ v(', X) = 0 (5.17) The optimisation problem can also be solved to obtain the profit frontier which becomes the standard against which profit efficiency is measured. The measure of profit efficiency (PE) is given by the ratio of observed profit to maximum profit:54
Yk =
¢&‹G£1(¢,£)L• (5.18) Profit efficiency requires (either input-oriented or output-oriented) technical efficiency and both input allocative efficiency and output allocative efficiency.To sum up, the first part of this section has reviewed the conceptual framework of frontier efficiency using the structure of the production technology and its characterisation by production frontiers and distance functions to serve as standards for analysing technical efficiency. We then reviewed the concept of economic efficiency including cost, revenue and profit efficiency measures where behavioural objectives are imposed.
Most empirical applications have however focused on cost efficiency (see for instance Ferrier and Lovell, 1990; Berger and De Young, 1995; Fries and Taci, 2005; Zhao et al., 2010; Das and Drine, 2011; Mwega, 2011; and Molyneux and Williams, 2013, just to mention a few). Revenue efficiency is the least considered in empirical work based on the criticism that, by ignoring costs, it fails to effectively measure managerial capacity to optimally manage economic resources. Further, a revenue-efficient firm may derive such efficiency solely on the basis of high market concentration or market power which can enable it to exploit output prices through collusive behaviour. Profit efficiency, on the other hand, has received some attention in the empirical banking literature (see for instance, Maudos et al., 2002; Ariff and Can, 2008; Bonin et al., 2005; Kasman and Yildirim, 2006; Isshaq and Bokpin, 2011). It is
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The above specification of profit (Y, ™) is referred to as the standard profit measure as it assumes exogeneity of output prices in a perfectly competitive banking market. An alternative profit measure ( „(X, ™)) is used which assumes that due to market power of banks, output prices are not exogenous and therefore uses output levels instead of output prices (Berger and Mester, 1997; Maudos et al., 2002 )
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however also argued that profit efficiency could be attributable to the effects of market power or collusive behaviour by dominant banks or by the macroeconomic environment which result in high interest rate spreads, and so does not reflect true efficiency, which is particularly the case in most African countries, including Ghana. Further, it can also be argued that for some banks, especially state-owned banks, the objective may not be one of profit maximisation but to support the development of particular sectors or to foster financial inclusion in less developed areas. In all classes of banks, cost efficiency should remain a relevant objective. As noted by Koetter and Meesters (2013), cost minimisation is a necessary and sufficient condition not only for short term profitability, but also for long term survival of banks. It is in this context, that this study uses cost efficiency as the relevant economic efficiency measure in line most of the empirical literature.
The above analytical review forms the basis upon which models on the construction of efficiency frontiers and measurement of efficiency are developed, and to which we turn our attention to in the next section.