Measurement of Technical Efficiency in Public Education: A Stochastic and Nonstochastic Production Function Approach.

• Author: Chakraborty, Kalyan

Kalyan Chakraborty [*]

Basudeb Biswas [+]

W. Cris Lewis [++]

This paper uses both the stochastic and nonstochastic production function approach to measure technical efficiency in public education in Utah. The stochastic specification estimates technical efficiency assuming half normal and exponential distributions. The nonstochastic specification uses two-stage data envelopment analysis (DEA) to separate the effects of fixed inputs on the measure of technical efficiency. The empirical analysis shows substantial variation in efficiency among school districts. Although these measures are insensitive to the specific distributional assumptions about the one-sided component of the error term in the stochastic specification, they are sensitive to the treatment of fixed socioeconomic inputs in the two-stage DEA.

1. Introduction

   Efficiency in the public education system is a significant issue in the United States. Nationwide, real expenditure per student in public education increased more than 3% per year between 1960 and 1998, but output as generally measured by standardized test scores has not increased and in some cases (e.g., the verbal SAT score) has declined. [1] One explanation is that resources are not being utilized efficiently. There may be productive or technical inefficiency and/or allocative or price inefficiency (i.e., given the relative prices of inputs, the cost minimizing input combination is not used). This paper focuses on the former by evaluating technical inefficiency in public education using data from Utah school districts.

   The pioneering work by Farrell in 1957 provided the definition and conceptual framework for both technical and allocative efficiency. Although technical efficiency refers to failure to operate on the production frontier, allocative efficiency generally refers to the failure to meet the marginal conditions for profit maximization. Considerable effort has been made in refining the measurement of technical efficiency. The literature is broadly divided into deterministic and stochastic frontier methodologies. [2] The deterministic nonparametric approach that developed out of mathematical programming is commonly known as data envelopment analysis (DEA), and the parametric approach that estimates technical efficiency within a stochastic production, cost, or profit function model is called the stochastic frontier method.

   Both approaches have advantages and disadvantages, as discussed in Forsund, Lovell, and Schmidt (1980). DEA has been used extensively in measuring efficiency in the public sector, including education, where market prices for output generally are not available. For example, Levin (1974), Bessent and Bessent (1980), Bessent et al. (1982), and Fare, Grosskopf, and Weber (1989) used this method to estimate efficiency in public education. The stochastic frontier methodology was used by Barrow (1991) to estimate a stochastic cost frontier using data from schools in England. Wyckoff and Lavinge (1991) and Cooper and Cohn (1997) estimated technical efficiency using school district data from New York and South Carolina, respectively. Grosskopf et al. (1991) used the parametric approach to estimate allocative and technical efficiency in Texas school districts.

   The recent literature has seen a convergence of the two approaches and their complementarity is being recognized. [3] However, there is a lack of empirical evidence in the literature about the proximity of these two approaches in measuring technical efficiency. Policy formulations based on only one of these efficiency estimates may not be accurate because of the inherent limitations of each. Before any correctional measures are taken, the stability of the technical efficiency estimates obtained from a parametric method should be evaluated by comparing them against those found when using the nonparametric method.

   In this study, the technical efficiency estimates for each school district using the stochastic frontier method and Tobit residuals from the two-stage DEA model are compared. In the two-stage DEA model, technical efficiency scores obtained from DEA using controllable inputs are regressed on student socioeconomic status and other environmental factors. The residuals of this regression measure pure technical efficiency after accounting for fixed socioeconomic and environmental factors.

   The empirical analysis uses data from the 40 school districts in Utah for the academic year 1992-1993. The standardized test score for 11th-grade students is used as a measure of school output, and two classes of inputs are included. The first class is considered to be subject to control by school administrators and includes the student-teacher ratio, the percentage of teachers having an advanced degree, and the percentage of teachers with more than 15 years of experience. The second class includes such uncontrollable factors as socioeconomic status, education level of the local population, and net assessed real property value per student.

   This paper is organized as follows. First, the relevant literature is reviewed and then a definition of the educational production function is provided. Next, the stochastic and DEA specifications of technical inefficiency are reviewed. Finally, the data set is discussed and the empirical results are presented.

2. Background

   For a given technology and a set of input prices, the production frontier defines the maximum output forthcoming from a given combination of inputs. Similarly, the cost frontier defines the minimum cost for providing a specified output rate given input prices, and the profit frontier defines the maximum profit attainable given input and output prices. Inefficiency is measured by the extent that a firm lies below its production and profit frontier and above its cost frontier. Koopmans (1951) defines a technically efficient producer as one that cannot increase the production of any one output without decreasing the output of another product or without increasing some input. Debreu (1951) and Farrell (1957) offer a measure of technical efficiency as one minus the maximum equiproportionate reduction in all inputs that still allows continuous production of a given output rate (Lovell 1993).

   An early study that measured technical inefficiency in education production is that by Levin (1974, 1976). He used the Aigner and Chu (1968) parametric nonstochastic linear programming model to estimate the coefficients of the production frontier and found that parameter estimation by ordinary least squares (OLS) does not provide correct estimates of the relationship between inputs and output for technically efficient schools; that technique only determines an average relationship. Klitgaard and Hall (1975) used OLS techniques to conclude that the schools with smaller classes and better paid and more experienced teachers produce higher achievement scores. Their study also estimates an average relationship rather than an individual school-specific relationship between inputs and output.

   Among the studies on technical efficiency in public schools using the DEA method, one of the earliest was done by Charnes, Cooper, and Rhodes (1978), who evaluated the efficiency of individual schools relative to a production frontier. Bessent and Bessent (1980) and Bessent et al. (1982) made further refinements by incorporating a nonparametric form of the production function, introducing multiple outputs, and identifying sources of inefficiency for an individual school. Further extensions were made by Ray (1991) and McCarty and Yaisawarng (1993), who considered controllable inputs in the first stage of the DEA model to measure technical efficiency. Then the environmental (i.e., noncontrollable) inputs were used as regressors in the second stage using OLS or a Tobit model, and the residuals were analyzed to determine the performance of each school district.

   In these studies, it is postulated that all firms have an identical production frontier that is deterministic, and any deviation from that frontier is attributable to differences in efficiency. The concept of a deterministic frontier ignores the possibility that a firm's performance may be affected by factors both within and outside its control. That is, combining the effects of any measurement error with other sources of stochastic variation in the dependent variable in the single one-sided error term may lead to biased estimation of technical inefficiency. In response to this, the concept of a stochastic production frontier was developed and extended by Aigner, Lovell, and Schmidt (1977), Battese and Corra (1977), Meeusen and van den Broeck (1977), Lee and Tyler (1978), Pitt and Lee (1981), Jondrow et al. (1982), Kalirajan and Flinn (1983), Bagi and Huang (1983), Schmidt and Sickles (1984), Waldman (1984) and Battese and Coelli (1988). The basic idea behind the stochastic frontier model, as stated by Forsu nd, Lovell, and Schmidt (1980), is that the error term is composed of two parts: (i) the systematic component (i.e., a traditional random error) that captures the effect of measurement error, other statistical noise, and random shocks; and (ii) the one-sided component that captures the effects of inefficiency.

   Frontier production models have been analyzed either within the framework of the production function or by using duality in the form of a cost minimizing or profit maximizing framework. Barrow's (1991) study of schools in England tested various forms of the cost frontier and found that the level of efficiency was sensitive to the method of estimation. In their study of technical inefficiency in elementary schools in New York, Wyckoff and Lavinge (1991) estimated the production function directly and found that the index of technical inefficiency depends on the definition of educational output. For example, if output is measured by the level of cognitive skill of students rather than their college entrance test score, the index of...