Country Performance Evaluation: The DEA Model Approach

Abstract

This study employs four data envelopment analysis (DEA) models to evaluate the performance efficiency of 21 OECD countries and assess whether the undesirable outputs are over-produced relative to desirable outputs. In evaluating the performance of OECD countries via super-efficiency models, this study focuses on two aspects. First, employing the concept of the Sharpe ratio, we propose another method to deal with undesirable outputs (the unemployment rate, inflation, and air pollution) in DEA. This approach can reveal the relative importance of desirable outputs and undesirable outputs, detect whether undesirable outputs are over-produced, and obtain more accurate efficiency scores. Second, we examine whether knowledge capital can improve a country’s efficiency scores. Our empirical results support the above arguments. In addition, research and development (R&D) expenditures, the proxy variable for knowledge capital, can indeed improve countries’ efficiency scores, implying that the endogenous growth theory is supported in OECD countries. Evidently, whether the undesirable outputs are included in the DEA models and are properly treated is crucial in the evaluation of efficiency values.

1 Introduction

In assessing a country’s competitiveness, the World Economic Forum (WEF) and the International Institute for Management Development (IMD) both take the management effectiveness of government into account. To improve their competitiveness, developed countries, such as the G7 countries, have been actively assessing their operating or executive efficiency. This trend is expected to spread to developing countries. Thus, the evaluation of a country’s operating efficiency is a topic worthy of study.

One way to evaluate country performance is to analyse whether a country utilises its resources in an economically efficient manner, i.e., whether a country uses minimum inputs (e.g., labour, physical capital, and land) to obtain maximum outputs (e.g., real per capita income and employment). Data envelopment analysis (DEA) is a well-known linear programming approach to evaluating the efficiency of a given set of similar decision-making units (DMUs) and is regarded as a powerful methodology for identifying the best practice frontiers and inefficiency sources and providing suggestions for performance improvement. This approach does not require that a functional form for the relationship between multiple inputs and multiple outputs be identified and can assess technology efficiency, pure technology efficiency and scale efficiency, which makes it possible to carry out a comprehensive evaluation of achievements. Thus, DEA has been extensively applied in various fields, including government organisation, manufacturing, finance, and hospital management (Banker and Morey 1986; Andersen and Petersen 1993; De Borger and Kerstens 1996; Grossman et al. 1999; Worthington and Dollery 2000; Steinmann and Zweifel 2003; Zelenyuk and Zheka 2006; Borge et al. 2007).

The CCR model (Charnes et al. 1978) and the BCC model (Banker et al. 1984) are two conventional DEA models. In the original CCR model, the relationship between inputs and outputs is a constant return to scale, and the efficiency of a DMU can be expressed as the maximum ratio of weighted outputs to weighted inputs, subject to the condition that the same ratio for all DMUs must be less than or equal to one. That is, the CCR model calculates an overall efficiency for each DMU, where both pure technical efficiency and scale efficiency are aggregated into a single value. The BCC model yields a measure of pure technical efficiency that ignores the impact of the scale size by only comparing a DMU to a unit of similar scale. That is, the BCC model extends the CCR model to account for technologies that exhibit variable returns to scale. Thus, the efficiency score obtained using the BCC model is greater than or equal to the score obtained using the CCR model.

In evaluating the operating efficiency of a group of countries, two problems are encountered using the conventional DEA models. First, they ignore the existence of undesirable outputs (e.g., unemployment, inflation, air pollution, and garbage release) or improperly deal with the co-existence of desirable outputs and undesirable outputs. There are some situations in which undesirable outputs are produced together with desirable outputs (i.e., the outputs generating positive utility), such as income, consumption, and profit. Färe et al. (1989) argue that the performance evaluation of DMUs is very sensitive to whether specific undesirable outputs are included. Thus, in evaluating the operating efficiency of DMUs, researchers need to consider desirable outputs and undesirable outputs simultaneously; otherwise, the evaluation results may be misleading.¹ Second, they neglect the role of knowledge capital in the operating performance of a country. Since the endogenous growth theory or new growth theory was proposed in the 1980s, it has attracted vast attention from economists (Romer 1986; Lucas 1988; Rebelo 1991). The new growth theory emphasises that economic growth is generated from within a system as a direct result of internal processes. Hulten (2000) argues that the endogenous growth theory relies on the new assumption that the marginal product of capital is constant rather than diminishing, as in the neoclassical growth theory. Capital in new growth models includes investments in knowledge, research and development of products, and human capital. That is, knowledge capital is an important production factor and plays a key role in a country’s economic growth or operating performance. Thus, the main objective of this study is to evaluate the operating performance of the selected countries by simultaneously resolving these two problems.

The previous literature has proposed some methods for measuring efficiency values in DEA models with undesirable outputs (Färe et al. 1989; Seiford and Zhu 2002; Silva Portela et al. 2004; Jahanshahloo et al. 2005; Amirteimoori 2006). However, these methods may encounter some constraints in empirical application. First, most of them involve complicated mathematical calculations and strict constraint conditions, which make their application inconvenient. For example, directional distance functions with a fixed directional vector will result in different efficiency scores when the dataset’s units of measurement are changed. Even if the change of a fixed directional vector is permitted, there are still rigorous sufficient and necessary conditions that need to be satisfied (Salnykov 2008). Second, methods that treat undesirable outputs as inputs may neglect the relative importance of desirable and undesirable outputs. (Ali and Seiford 1990; Tyteca 1996, 1997; Scheel 2001) That is, such methods cannot gauge whether undesirable outputs are over-produced relative to desirable outputs. This is especially important for a country that is trying to reduce air pollution (i.e., an undesirable environment output) for purposes of sustainable development.

Since the Sharpe ratio was initiated in 1966 by William Sharpe, it has been one of the most widely referenced financial performance measures. The ratio is defined as the excess return per unit of risk in an investment asset or portfolio. The higher the Sharpe ratio is, the better the asset or portfolio’s risk-adjusted performance should be. For the DMUs of production or management, the excess return is similar to the desirable output, and the risk corresponds to the undesirable output. For example, a country’s operating performance typically includes GDP and unemployment rate indices. The former represents a desirable output, and the latter represents an undesirable output. In this situation, one can apply the concept of the Sharpe ratio to integrate GDP and unemployment rate into a new modified desirable output, namely, the ratio of GDP to unemployment rate, to evaluate a country’s operating efficiency. Obviously, the higher the ratio of GDP to unemployment rate is, the more efficiency the country achieves. More importantly, combining the measured efficiency scores and the modified output slacks of DEA models, researchers can easily investigate whether each undesirable output is over-produced relative to a specific desirable output, given a specific set of inputs. Thus, it is appropriate for analysts to employ the concept of the Sharpe ratio to express a desirable output–undesirable output pair as a modified desirable output to evaluate the efficiency scores of the DEA model with undesirable outputs and avoid the constraints associated with the methods adopted in previous studies (Ali and Seiford 1990; Tyteca 1997; Scheel 2001; Salnykov 2008).

In addition to the improvement in the treatment of undesirable outputs in DEA models, this study also offers a new perspective on the inputs to DEA models. This study treats knowledge capital as a new input factor in DEA models to investigate how knowledge capital influences a country’s performance. This study explores how the input of knowledge capital and undesirable outputs (including the unemployment rate, inflation, and air pollution) influence efficiency scores in DEA models.² To deal with the coexistence problem of desirable and undesirable outputs and assess whether the undesirable outputs are over-produced relative to desirable outputs, the paper employs the concept of the Sharpe ratio. Moreover, this study utilises a super-efficiency model to perform the efficiency rankings of DMUs with efficiency scores of 1 in the conventional DEA model. Twenty-one OECD countries are selected as sample objects because these countries have similar levels of economic development, which satisfies the requirement for applying DEA models to the evaluation of efficiency scores.

The results show that the model employed, which considers a new input (knowledge capital) and two undesirable outputs (unemployment rate and air pollution) along with the Sharpe ratio to address the co-existence of desirable and undesirable outputs, is the best model to use to evaluate countries’ operating performance. In addition, the consideration of undesirable outputs has a significant influence on the estimation of OECD countries’ performances and demonstrates that undesirable outputs are over-produced relative to desirable outputs in some of the sample countries. Lastly, the inclusion of knowledge capital makes all OECD countries more efficient, regardless of their technical efficiency, pure technical efficiency or scale efficiency, implying that the endogenous growth theory is supported in OECD countries. These results cannot be obtained using conventional DEA models.

The remainder of this study is organised as follows. Section 2 briefly introduces various empirical models, including conventional DEA models (the CCR model and the BCC model) and the super-efficiency model. This study applies these models and various combinations of inputs, desirable outputs and undesirable outputs to assess the influences of knowledge capital and undesirable outputs on efficiency scores. Section 3 describes the selection of DMUs, inputs and outputs, and data sources. Section 4 presents important empirical results. The conclusions are summarised in the last section.

2 The Models

This section briefly introduces the DEA models used to evaluate countries’ operating performance.

2.1 CCR Model

The CCR model, based on the assumption of constant return to scale, is used to evaluate the technical efficiency (TE) of DMUs. This study uses x _ik , i = 1,…,m and y _rk , r = 1,…,s to denote the i-th input and r-th output of DMU _k (k = 1,…,n), respectively. The efficiency score of each DMU _k in an output-oriented CCR model can be derived from the following model.

$$\begin{array}{*{20}c} {Max} \hfill & {E_{k} = \theta + \varepsilon \left( {\sum\limits_{i = 1}^{m} {S_{ik}^{ - } + } \sum\limits_{r = 1}^{s} {S_{rk}^{ + } } } \right)} \hfill \\ {s.t.} \hfill & {\sum\limits_{k = 1}^{n} {\lambda_{k} x_{ik} + S_{ik}^{ - } = x_{ik} ,\,\;i = 1, \ldots ,m} } \hfill \\ {} \hfill & {\sum\limits_{k = 1}^{n} {\lambda_{k} y_{rk} - S_{rk}^{ + } = \theta \, y_{rk} ,\,\;r = 1, \ldots ,s} } \hfill \\ {} \hfill & {\lambda_{k} ,S_{ik}^{ - } ,S_{rk}^{ + } \ge 0,\;k = 1, \ldots ,n} \hfill \\ \end{array}$$

(1)

where E _k is the relative efficiency score of DMU _k ; $S_{ik}^{ - }$ and $S_{rk}^{ + }$ represent the i-th input slack and the r-th output slack of the DMU _k , respectively; $\lambda_{k}$ denotes the intensity variable of DMU _k , which is used to construct the best practice frontier and ε is an infinitesimal constant (usually 10⁻⁸). The evaluated unit DMU _k is efficient if the optimal objective value E _k  = 1, i.e., θ = 1, and inefficient if E _k  < 1.³

Another version of the conventional DEA model frequently used is the Banker et al. (1984) model, BCC.⁴ The BCC model is more flexible and allows variable returns to scale; consequently, it measures only pure technical efficiency (PTE) for each DMU. That is, for a DMU to be considered as CCR efficient, it must have both scale efficiency and pure technical efficiency. Moreover, the scale efficiency (SE) index can be derived by estimating the ratio of CCR efficiency to BCC efficiency.

In the evaluation of a country’s operating performance, this study allows for the coexistence of desirable and undesirable outputs, not just desirable outputs as in the specification of the conventional CCR and BCC models. Moreover, to highlight the importance of knowledge capital in influencing efficiency scores, the paper uses R&D expenditures, a proxy variable for knowledge capital, in the DEA model as a new input term.

2.2 Super-Efficiency Model

To rank the efficiency of the DMUs with efficiency scores of 1 according to conventional DEA models, this study applies the super-efficiency model proposed by Andersen and Petersen (1993).⁵ Super-efficiency indicates the extent to which the efficient products exceed the efficient frontier formed by other efficient units. The super-efficiency indices of an output-oriented DEA model with constant return to scale are derived from the linear programming model (2).

$$\begin{array}{*{20}c} {Max} \hfill & \theta \hfill & {} \hfill \\ {s. \, t.} \hfill & {\sum\limits_{k = 1}^{n} {\lambda_{k} y_{rk} \ge \theta y_{rl} } } \hfill & {r = 1, \ldots ,s} \hfill \\ {} \hfill & {\sum\limits_{k = 1}^{n} {\lambda_{k} x_{ik} \le x_{il} } } \hfill & {i = 1, \ldots ,m} \hfill \\ {} \hfill & {\theta \ge 0} \hfill & {} \hfill \\ {} \hfill & {\lambda_{ l} = 0} \hfill & {} \hfill \\ {} \hfill & {\lambda_{ k} \ge 0 \, } \hfill & {k = 1, \ldots ,n;\,k \ne l} \hfill \\ \end{array}$$

(2)

The derived efficiency score θ in model (2) indicates the super-efficiency of DMU k. In the output-oriented DEA model, the value of θ lies in the interval (0,1] for the identified efficiency DMUs, with smaller values indicating increasing efficiency, and lies in the interval (1, ∞) for the identified inefficiency DMUs, with larger values indicating decreasing efficiency.

If the aim of the research is to identify the production frontier, then a conventional technical efficiency measure in the interval [1, ∞) is appropriate, and efficient DMUs are compared only to themselves. In such a model, any efficient DMU that increases its outputs or reduces its inputs can increase its efficiency; however, the efficiency score of 1 remains unchanged, in spite of its improved performance. In contrast, in the super-efficiency model (2), when the inefficient or efficient DMUs change their performance, the efficiency scores also change. Moreover, super-efficiency scores always benchmark the target DMU on its efficient peers, regardless of its own efficiency level. Thus, this study applies model (2) to further rank the performance of the efficient DMUs in conventional DEA models, the CCR and BCC models.

3 Selection of DMUs and Variables

This section identifies the DMUs, inputs and outputs used in the efficiency evaluation in DEA models.

3.1 Decision-Making Units (DMUs)

OECD is an international organisation of developed countries that accept the principles of representative democracy and a free-market economy. Thus, this study assumes that the OECD member countries have a certain degree of homogeneity and are appropriate DMUs to use in an evaluation of countries’ operating performance. Due to incomplete data for some inputs and/or outputs, 21 of the 30 OECD member countries were selected as the DMUs: Austria, Belgium, Canada, the Czech Republic, Denmark, Finland, France, Germany, Hungary, Italy, Japan, Luxembourg, the Netherlands, Norway, Poland, Portugal, Spain, Sweden, Turkey, the UK and the USA. The DEA models are estimated using the latest available data from the year 2005.⁶

3.2 Input Variables

In deciding which inputs to use in the DEA models, the production function is a referenced instrument. Generally speaking, the main production factors are labour, physical capital, land, and entrepreneurship. Physical capital refers to factors of production, such as machinery, buildings, and computers. For a country, executive ability is an appropriate proxy variable for entrepreneurship and can be embedded in its operating performance. The executive area of a country, a proxy variable for land, is less variable and is thus exogenous for the country. Thus, physical capital and labour are appropriate inputs for gauging a country’s performance.

Moreover, to investigate how knowledge capital influences a country’s operating achievement, this study considers knowledge capital as another input.⁷ While a variety of methods can be used to measure knowledge capital, based on the considerations of simplifying the model and the available data, this study adopts the most commonly used item, R&D expenditures, as a proxy variable.

However, to concentrate the analysis on the role of undesirable outputs and knowledge capital in efficiency evaluation and to exclude the disturbance of inflation and the scale effect arising from different populations or employment levels, this study first expresses physical capital and knowledge capital in real terms, i.e., real physical capital and real knowledge capital, respectively, and then divides the two real terms by the number of workers.⁸ The real physical capital per worker and the real knowledge capital per worker are selected as the inputs. More importantly, the two constructed inputs can also be used to assess a country’s physical capital intensity and knowledge capital intensity.

3.3 Output Variables

Income is regarded as a measure of the welfare level of a country; therefore, it is also the best indicator of a country’s total operating performance. To focus the efficiency analysis on the issues of the existence of undesirable outputs and knowledge capital, the paper chooses only income as a desirable output of the DEA models.

As mentioned above, there are many circumstances in which undesirable outputs are jointly produced with desirable outputs. Thus, this study also considers the probable undesirable outputs produced by a country’s operating activities. Among the likely undesirable outputs, this study selects two important variables, namely, the unemployment rate and air pollutants (air pollution and greenhouse gases). The former is the major component of the misery index, while the latter can cause harm to humans and the environment. For the inhabitants of a country, these two generate disutility and are regarded as undesirable outputs. To assess the relative importance between desirable output and undesirable outputs or assess whether each undesirable output is over-produced relative to a specific desirable output, this study combines desirable output–undesirable output pairs into new modified desirable outputs. The procedures for constructing new modified desirable outputs are described below.

First, nominal income is replaced with real income per worker to exclude the influences of price level and employment on the nominal income. Second, employing the concept of the Sharpe ratio, real income per worker (i.e., the desirable outputs) is divided by the two undesirable outputs to construct two new modified desirable outputs, i.e., real income per worker over the unemployment rate and real income per worker over air pollutants, named output 1 and output 2, respectively.⁹

The definitions and measurement of the inputs and outputs and the data sources are listed in Table 1. The efficiency scores are evaluated using the DEAP programme (provided by Coelli 1996) and the EMS programme (provided by Scheel 2000).

Table 1 Data source and measurement

Using various combinations of inputs and outputs and various DEA models, the following four DEA models were used to investigate the roles of the unemployment rate, air pollution, and knowledge capital in influencing efficiency scores (see Table 2). In the benchmark (or standard) model, the input variable is the real physical capital per worker and the output variable is the real GDP per worker. In the model, it is convenient to utilise the input to measure the effect of the physical capital intensity or the physical capital equipment rate of employment on the efficiency score.

Table 2 The input(s) and output(s) in each DEA model considered

In Model I, the benchmark model is extended by adding a variable for real knowledge capital divided by the number of workers (or the real knowledge capital per worker) to evaluate its role in a country’s efficiency score. Model II is a modification of Model I in which the real GDP per worker is replaced with two modified desirable outputs, i.e., real GDP per worker divided by the unemployment rate and real GDP per worker divided by a measure of the air pollutants. This approach can properly assess the role of undesirable outputs in influencing efficiency scores. Lastly, the super-efficiency model has the same inputs and outputs as Model II.

4 Empirical Results

Four DEA models were employed to evaluate the performance efficiency of 21 OECD countries and assess whether undesirable outputs are over-produced relative to desirable outputs. Therefore, it is more appropriate to adopt an output-oriented DEA model to perform the empirical study than to adopt an input-oriented model. To further investigate the inefficiency sources of the DMUs, this study decomposes the technical efficiency (TE) into pure technical inefficiency (PTE) and scale efficiency (SE). The TE scores are evaluated using a CCR model, and the PTE scores are evaluated using a BCC model and the same data. That is, the TE efficiency score is divided by the PTE efficiency score to obtain the scale efficiency score. The evaluation results are described below.

4.1 Evaluation Results of Efficiency Scores

The evaluation results of efficiency indices are displayed in Tables 3, 4 and 5. There are some remarkable findings. First, comparing the evaluation results for the benchmark model and Model I yields insights into the role of knowledge capital in efficiency scores. Knowledge capital can indeed increase efficiency values on average, by increasing technical efficiency or pure technical efficiency and scale efficiency. Even for an individual country, the increase in efficiency indices that result from considering the knowledge capital input are universal. In particular, the number of countries with technical efficiency, pure technical efficiency, and scale efficiency significantly increases. This conclusion may partly support the argument of the endogenous growth theory that the accumulation of knowledge or human capital can create increasing return to scale.

Table 3 Technical efficiency evaluation in 2005

Table 4 Pure technical efficiency evaluation in 2005

Table 5 Scale efficiency evaluation in 2005

Second, for Model I and Model II, the average technical efficiency indices are 0.776 and 0.849, respectively, those of pure technical efficiency are 0.888 and 0.925, respectively, and those of scale efficiency are 0.868 and 0.919, respectively. Evidently, considering the appearance of undesirable outputs and our treatment of the coexistence of desirable and undesirable outputs, the technical efficiency, pure technical efficiency, and scale efficiency indices all increase on average. That is, neglecting undesirable outputs results in underestimating, on average, the technical efficiency, pure technical efficiency, and scale efficiency.

The erroneous efficiency evaluation phenomenon also occurs for each individual country. Among the sampled countries, technical efficiency, pure technical efficiency, and scale efficiency are underestimated for 57.14 % (=12/21), 38.10 % (=8/21), and 61.91 % (=13/21), respectively. In addition, 80.95 % (=17/21) of the countries are technical inefficient, according to Model I and Model II, and the decomposition indicates that technical inefficiency arises equally from pure technical inefficiency and scale efficiency.

Third, using Model I and Model II, some countries become technical efficiency units after consideration of undesirable outputs, i.e., the unemployment rate and air pollutants. These countries are Denmark (0.646 to 1), France (0.727 to 1), Turkey (0.566 to 1), and the USA (0.955 to 1). Moreover, the countries moving from pure technical inefficiency to pure technical efficiency are Denmark (0.779 to 1), Norway (0.878 to 1), Sweden (0.843 to 1), and Turkey (0.698 to 1). The countries moving from scale inefficiency to scale efficiency are Denmark (0.829 to 1), France (0.727 to 1), Turkey (0.811 to 1), and the USA (0.955 to 1).

Lastly, employing a super-efficiency model, one can straightforwardly rank the efficient units evaluated from conventional DEA models (benchmark model and Model I) and the constructed DEA model (Model II). Taking the countries with technical efficiency as an example (Table 3), there are four countries (Denmark, France, Turkey, and USA) with technical efficiency scores of 1 in Model II, i.e., all those located on the efficient frontier. Therefore, one can determine their ranking in terms of efficiency. The ranking of the countries from high to low by technical efficiency, according to the super-efficiency model, are France, Turkey, the USA, and Denmark. Evidently, after considering undesirable outputs (i.e., the unemployment rate and air pollution) and employing the Sharpe ratio to deal with the co-existence of desirable and undesirable outputs, France, Turkey, the USA, and Denmark are the top four countries in terms of operating performance. In addition, according to the technical efficiency scores produced by the super-efficiency model, one can classify the countries evaluated in the following five groups: Group I—France, Turkey, the USA, and Denmark, with scores in the interval of (0.7, 1.00); Group II—Canada, Sweden, Luxembourg, and the UK, with scores in the interval of (1.01, 1.10), Group III—Italy, Finland, Japan, and Germany, with scores in the interval of (1.11, 1.20); Group IV—Norway, Belgium, the Netherlands, Austria, and Portugal, with the scores in the interval of (1.21, 1.30), and Group V—Spain, Poland, Hungary, and the Czech Republic, with scores in the interval of (1.31, 2.40). The larger the technical efficiency score is, the worst the operating performance of a country is. Moreover, the countries with lower efficiency scores are those with relatively low real per capita income in the sample OECD countries (i.e., the countries in Group V). The rankings of pure technical efficiency scores in Table 4 are similar to the results in Table 3.

Overall, the knowledge capital input can indeed increase the efficiency scores of DMUs. Whether the undesirable outputs are included in the DEA models and are properly treated is crucial to the evaluation results of efficiency scores (Table 5).¹⁰

4.2 The role of Undesirable Outputs

Regarding the sources of inefficiency, it is detected that the output 1 and output 2 need to increase averagely. According to the definitions of the output 1 and output 2, this means that given a specific real disposal income per worker, the undesirable outputs in unemployment rate and air pollutions are over-produced. The phenomenon is especially obvious in Hungary, Austria, Portugal, Italy, and Belgium for output 1; Netherlands and Japan for output 2.

One can further integrate output slack, original desirable and undesirable outputs to assess the extent of over-produced in undesirable outputs. Take Hungary as an example, the slack in output 1 is 4.72 %, and the real GDP per employment and unemployment rate are 16951.67 thousand USD per year and 7.32 % (see Table 6), respectively; therefore, relative to the desirable output-real GDP per employment, the optimal unemployment rate should be reduced to 6.784 %. Employing similar method, the over-production of the other undesirable output can be traced out. The slack in output 2 for Netherlands is 2.24 %, and the real GDP per employment and air pollutions are 35110.66 thousands USD per year and 213316 thousand tons, respectively; therefore, relative to the desirable output-real GDP per employment, the optimal air pollutions should be abated to 208537.72 thousand tons.

Table 6 Slack outputs and actual outputs in 2005

It appears that employing the concept of the Sharpe ratio to combine desirable and undesirable outputs and construct new modified output terms could easily affect the assessment of the relative importance of desirable and undesirable outputs. This approach is almost unmentioned in previous studies.

5 Conclusions

The evaluation of a country’s performance is crucial to the country’s efforts to improve its international competitiveness. In evaluating the performance of countries using DEA models, most previous studies have assumed that inputs and outputs are all desirable. However, desirable outputs are frequently accompanied by undesirable outputs of the operating and production processes. That is, to measure a country’s operating performance correctly, the consideration of undesirable outputs and the proper treatment of undesirable outputs are necessary. Moreover, in the knowledge economy era, knowledge capital is an important input to the evaluation of the operating processes of a country. Therefore, the role that knowledge capital plays in influencing government operating efficiency and whether neglecting knowledge capital will lead to a biased efficiency evaluation deserves our attention.

The main purpose of this study is not to identify all of the inputs and outputs that influence a country’s performance but rather to provide a method for deal with the coexistence of desirable outputs and undesirable outputs and to investigate whether knowledge capital can improve efficiency in DEA models. To this end, the paper constructs four models to evaluate the operating performance of 21 selected OECD countries. The model that incorporates knowledge capital as a new input and deals with two undesirable outputs—the unemployment rate and air pollution—using the Sharpe ratio to combine desirable and undesirable outputs was found to be the best model for evaluating the operating performance of countries. The reason for this is clear. This model considers more appropriate inputs and outputs and applies a reasonable tool, the Sharpe ratio, to deal with the co-existence of desirable and undesirable outputs. Additionally, considering undesirable outputs (i.e., the unemployment rate and air pollutants, in the present study) does indeed exert a significant influence on the estimates of OECD countries’ performance and indicates that the undesirable outputs are over-produced relative to the desirable output. This phenomenon is especially obvious in Hungary, Austria, Portugal, Italy, and Belgium for output 1 and in the Netherlands and Japan for output 2. Moreover, after considering undesirable outputs and employing the Sharpe ratio to deal with the co-existence of desirable and undesirable outputs, France, Turkey, the USA, and Denmark are found to be the top four countries in terms of operating performance, and the countries with less efficiency are those with relative low real per capita income in the sample OECD countries. Lastly, the inclusion of R&D expenditures makes all the countries in OECD more efficient, regardless of their specific technical efficiency, pure technical efficiency or scale efficiency scores. This implies that the endogenous growth theory is supported in the OECD countries.

This study presents the following policy propositions: (1) increasing the accumulation of knowledge capital helps a country to increase its operating performance; (2) in evaluating a country’s operating performance, it is important to consider the effects of undesirable outputs and desirable outputs on efficiency simultaneously; (3) the Sharpe ratio can be used by researchers to measure the efficiencies of countries because employing the ratio to combine desirable and undesirable outputs and construct new modified output terms facilitates the assessment of the relative importance of desirable and undesirable outputs; and (4) based on the estimated slacks and the actual inputs, the managers of a country can adopt policies for reducing air pollution and increasing operating performance.

The method proposed in this study can be easily applied to the following research topics: (1) the performance evaluation of public security and/or social welfare (with some undesirable outputs) for local governments; (2) the linkage between evaluated environmental efficiency and governmental subsidy policy; and (3) the dynamic analysis of performance evaluation (i.e., the persistence of efficiency scores), which can confirm whether two-stage DEA estimation is required.