Appendix 1. The DEA Approach
The key constructs of a DEA model are the envelopment surface and the efficient projection path to the envelopment surface (see Charnes and others 1995). The projection path to the envelope surface is determined by whether the model is output-oriented (maximizing the level of output given levels of the inputs) or input-oriented (minimizing the use of inputs to produce a given level of output). The selection of the path depends on the production process characterizing the DMU. The input orientation is better suited to cross-country analysis aimed at expenditure rationalization. With input-oriented DEA, the linear programming model is configured so as to determine how much the input use of a firm could contract if used efficiently in order to achieve the same output level.
Different from parametric techniques, DEA calculates the frontier directly from the data without imposing specific functional restrictions and considers all deviations from the frontier explained by inefficiency. It assumes that different combinations of the observed input-output bundles are feasible. Thus, DEA constructs an envelope around the observed combinations by connecting all the efficient DMUs.
Figure 1 shows the DEA production possibility frontier, where X is the input and Y is the output. While the free disposal hull (FDH) approach builds the frontier with vertical steps-up and assesses efficiency of DMU A only against the peers B and C, DEA evaluates efficiency also against a virtual DMU D, which employs a weighted combination of A and D inputs to yield a virtual output. Therefore, while FDH would have considered A as efficient, DEA puts it behind the efficiency frontier defined by EBCF. The input-oriented technical efficiency of A is defined by the ratio YD/YA.
The envelopment surface will differ depending on the scale assumptions that underpin the model. These could be constant returns to scale (CRS) or variable returns to scale (VRS). The latter includes both increasing and decreasing returns to scale.11 The frontier EBCF exhibits VRS. In particular, while the segment EB is characterized by increasing returns to scale, the segments BC and CF reflect decreasing returns to scale. The CRS frontier can be visualized by a ray extending from the origin through DMU B, which would be the only efficient one.
More formally, Charnes and others (1978) defined a multi-factor productivity analysis model assuming that there are n homogeneous DMUs which efficiency has to be assessed. Each uses different quantities of different m inputs to produce different s outputs. Thus, in the presence of multiple input and output factors the relative efficiency of DMU p is measured with a ratio of a weighted sum of outputs to a weighted sum of inputs:
where k = 1,2,…s, j = 1,2, …m,ykp is the amount of output k produced by DMU p, xjp is the amount of input j used by DMU p, vk is the weight given to output k, and uj is the weight given to input j. However, without other constraints, (1) would be unbounded. Under the restriction that the efficiencies of all DMUs are less than or equal to one, and that all weights are non-negative, the optimal weights are defined by solving the programming problem:
The fractional program in (2) is equivalent to following linear one:
The problem in (3) will need to be solved n times to calculate the efficiency scores of all the DMUs. A score equal to one implies that the maximum output has been achieved with the available inputs. A score lower than one suggests some inefficiency. When we want the model to be input-oriented, the dual problem in (4) should be solved:
where λs represent the dual variables and Θ is the input-oriented technical efficiency score, measuring the extent to which every DMU could reduce inputs to obtain the same output.
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The views expressed herein are those of the authors and should not be attributed to the IMF, its Executive Board, or its management. I would like to thank Benedict Clements, Marek Jakoby, Gil Mehrez, Daria V. Zakharova and colleagues in the Expenditure Policy division for their insightful comments. I am also grateful for the suggestions made by participants in a seminar held at the Slovak Ministry of Finance in May 2012.
Differences in spending levels can also reflect differences in needs, rather than differences in priorities.
A clear example of an indirect cost of public sector provision inefficiency is the increase in the burden of taxation to collect those resources that are inefficiently spent (Afonso and Gaspar, 2007).
The composite indicators include a set of seven sub-indicators of public performance. On the one hand they look at administrative, education, health, and public infrastructure outcomes, and on the other they consider the “Musgravian” tasks for government, such as income distribution, economic stability, and economic performance.
The Programme for International Student Assessment (PISA) is a triennial OECD international survey of the knowledge and skills of 15-year-old pupils, an age at which students in most countries are near the end of their compulsory time in school. PISA ranks countries according to their performance in reading, mathematics, and science by their mean score in each area.
The rapidly changing age structure and the more general acceleration of the population aging process in the country is a well-documented issue. For further information see http://esa.un.org/unpd/wpp/index.htm.
Consequently, a larger number of observations contribute to better define the frontier.
The expenditure data used in this and the following subsections follow the Classification of the Functions of Government (COFOG) developed by the OECD. It classifies government expenditure data from the System of National Accounts by the purpose for which the funds are used.
Third-party payers pay physicians/ hospitals according to the cases treated rather than per service or per bed days.