In this chapter, we present the methodology to be used to estimate cost efficiency, technology gap, scale economies, and scope economies. As discussed by Berger and Mester (1997), the adoption of the economic efficiency concepts will
provide further insights into the problem of the economic optimization.3 The cost efficiency is undoubtedly an appropriate approach since the European financial markets have been more competitive and highly integrated. The main idea comes from Battese et al. (2004) while generalized to a cost frontier setting.
3.1 Stochastic Meta-frontier Cost Function
Cost efficiency is gauged by the extent to which a bank’s actual cost deviates from the efficient cost frontier. We first introduce the stochastic cost frontiers of the banking industry for each country. Suppose that there are R different countries under consideration, and that each country k has N banks that face input prices and k seek to minimize the cost which they incur in producing the outputs. The stochastic cost frontier model for each bank w of country k at time t can be given as are identically and independently distributed random variables. The former is assumed to be distributed as N( 0 , σ2 ν (k) ), capturing the statistical noise, and the latter is assumed to be a truncated normal distribution, a positive disturbance capturing technical inefficiency, to be specified shortly. For expository convenience, equation (1) is further formulated as
) The model, as proposed by Battese et al. (2004), assumes that there is only one
3 There are three economic concepts: cost, revenue and profit efficiencies.
data-generation process for the banks operating under a given technology for each country. The data is individually generated from the frontier models in the different countries. In general, the meta-frontier is assumed to have the same functional form as the stochastic frontiers in the different countries. Thus, the meta-frontier cost function for all banks is given by is the corresponding parameter vector associated with the meta-frontier cost function such that
)
* wt (k
wt X
X ϕ ≤ ϕ (4) The meta-frontier is defined as a deterministic parametric function such that its values must be less than or equal to the deterministic components of the stochastic cost frontier of the different countries involved. The inequality constraint of equation (4) is held for all countries and time periods. The meta-frontier is considered to be an envelope of the individual stochastic frontiers of the different countries. Figure 1 provides an illustration of how the meta-frontier envelopes the stochastic frontiers of the different countries. We will estimate the stochastic cost frontiers for each country, denoted by frontier1, frontier2 and frontier3 in the figure. Then, a meta-frontier is estimated as an envelope curve which surrounds the three stochastic frontiers from below using the pooled data over all countries.
3.2 Technology Gap and Efficiency Levels
Cost efficiency is determined by how close a bank’s cost lie to the overall cost frontier, namely the meta-frontier. Therefore, the measure of cost efficiency (CE*) for bank w in year t is formulated by the ratio of the minimum cost to observed cost, adjusted by the corresponding random error,
)
Substituting (2) into (5), we obtain
) technical efficiency (CE) relative to the stochastic frontier of country k ,
) It must lie between zero and one, because U is a nonnegative random variable by wt
construction. The second term on the right-hand side of equation (6) is the technology currently available technology adopted by its banks lags behind the technology available for all countries. We measure the TGR using the ratio of the potential cost that is defined by the meta-frontier function to the cost for the frontier function for country k given the observed outputs and input prices. It has a value between zero and one because of equation (4).
The cost efficiency measure of equation (5) can be expressed as
) CE* also lies between zero and one because CE and TGR are both between zero and one.
3.3 Formula of the Scale and Scope Economies
In the context of multiple outputs, a formal measure of scale economies is referred to as ray scale economies (RSE), developed by Baumol et al. (1982) and applied to banking by Berger et al. (1987). It is defined as
∑
∂∂ economies, constant returns to scale, or scale diseconomies.Economies of scope exists when total cost of a firm simultaneously producing more than one output are lower than the sum of the costs of firms producing each output separately. In the case of a bank producing two outputs, as suggested by
Mester (1996), the estimate of scope economies is defined as respectively, scope economies or scope diseconomies.
3.4 Estimation Procedure
Now that we have introduced the meta-frontier model, the next step is to estimate the technology parameters of the cost function. The estimation procedure is divided into three steps:
1. Obtain the maximum likelihood estimates,ϕˆ(k), of ϕ(k) in the stochastic cost frontier for country k . The stochastic frontier model proposed by Battese and Coelli (1992), which allows for time-varying technical efficiency, will be adopted.
2. Obtain the estimate of ϕ* in the meta-frontier. Battese et al. (2004) pointed out that there are two approaches to find out the best envelop curve. Detailed see below.
3. According to equations (6)-(11), calculate the cost efficiency, the technology gap, scale economies ,and scope economies, using ϕˆ(k) and ϕˆ* obtained by Step 1
and 2.
We now return to the estimation procedure on the meta-frontier. There are two alternative approaches can be applied to identify the best meta-frontier. One is based on the sum of absolute deviations of the meta-frontier values from those of the group
frontiers, and the other is based on the sum of squares of the same deviations.
I. Minimum sum of absolute deviations ˆ*
ϕ is estimated by solving the optimization problem:
∑∑
represents the reciprocal of the radial distance between the meta-frontier and the frontier of country k . The weights of the deviations for all banks in the sample are the same. One may notice that all the deviations are positive because of equation (13). Therefore, all the absolute deviations are exactly equal to the differences. Using equations (2) and (3), we can simplify the above optimization problem to the linear programming (LP) problem:
∑∑ ( )
II. Minimum sum of squares of deviations:The other approach minimizes the sum of squares of the deviations between the meta-frontier and the frontier of the individual countries. ϕˆ* is estimated by solving a quadratic programming (QP) problem:
∑ ∑ ( )
gap ratio of the bank is, the higher weight to the deviation is.Standard errors of the estimators for the two meta-frontier can be obtained by either simulation or bootstrapping methods. Bootstrapping method will be used in this paper.