Ca’ Foscari University of Venice

Nota di lavoro 2000.08 September 2000

Department of Economics S. Giobbe – Cannaregio 873 30121 Venezia (Italy)

A utility-frontier production function approach for the efficiency measurement of multi-output public and non-profit organisations With an application to the academic departments of the University of Venice Dino Rizzi Ca’ Foscari University of Venice - Department of Economics E-mail address: [email protected] Abstract In the presence of multiple outputs the use of duality theory allows the estimation of cost or profit functions if prices for inputs and outputs are known. If prices are not available, as in the case of public and non-profit organisations, duality theory cannot be applied. The solutions proposed so far by the literature are based upon representations of the technological connections among outputs. The present paper proposes a different solution based explicitly on the fact that the output mix is an autonomous choice made by public or non-profit organisations. An aggregation of outputs is obtained by considering that the quantities and the mix of outputs can be valued with the use of an objective function (a utility function), defined by the management or by a hierarchically superior organisation, such as a governmental agency. Once the aggregated output is obtained, the usual frontier production function estimation can be used. With respect to the usual frontier approach, the presence of a parametric utility function allows us to perform a wider range of exercises. As the utility function parameters can be either assigned exogenously or chosen to maximise the efficiency, the procedure can be seen as a bridge between the parametric and the non-parametric approaches. The paper ends with an empirical application to the academic departments of the University of Venice. Keywords: Multi-output; efficiency measurement; non profit organisation; public agency JEL-code:

1.

Introduction

The usual economic problem of a firm is to maximise the level of profits given a production function and the prices of outputs and inputs. With given quantities of inputs, the problem reduces a maximisation of profits or revenues, constrained to the production function and the prices of outputs. If outputs are not homogeneous and their market prices are not available, then the solution of the economic problem is just to maximise all outputs or an objective function based on them. In this framework, duality theory cannot be used and the solutions proposed so far by the literature are based upon representations of the technological connections among outputs. In the present paper, the solution proposed to obtain an aggregation of outputs is the explicit consideration that the mix of outputs is one of the choices that can be made by public and non-profit organisation (henceforth NPO). The aggregation of many outputs into a scalar measure can be found by considering that the quantities of outputs and their mix can be valued with the use of an objective function, a utility function, defined by the management or by a hierarchically superior organisation, such as a governmental agency, an authority, or public opinion. The article is organised as follows. The next section briefly describes a technological representation of a multi-output and multi-input organisation. Section 3 and 4 introduce the

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utility function, the concept of equally distributed equivalent output and the way in which they can be used to estimate a multi-output frontier production function and to derive the level of efficiency. Section 5 describes the possible uses of the combined utility and frontier production approach by presenting six problems: - the maximisation of the efficiency of each organisation, - the estimation of the efficiency when the utility function is known, - the maximisation of the average efficiency of the sample, - the maximisation of the goodness of fit of the frontier production function estimation, - the maximisation of the efficiency of each organisation with known frontier production function, - the computation of efficiency levels when both the utility and frontier production functions are known. Section 6 contains an empirical application of the new methodology (with the use of a deterministic frontier analysis) to the efficiency measurement of the University of Venice academic departments. 2.

Technology of multi-output organisations

It is assumed, as usual, that the observed organisations are sufficiently homogeneous, so that they use similar technologies in order to obtain similar outputs. The technology of the j-th NPO can be written as an implicit function: T (y j , x j , z j ) = 0

(1) where

y j = [ y1 j ,..., y kj ,..., y Kj ] is the K-vector of outputs ( k = 1,..., K ),

x j = [ x1 j ,..., xij ,..., x Ij ] is the I-vector of inputs ( i = 1,..., I ), z j = [ z1 j ,..., z hj ,..., z Hj ] is the H-vector of other exogenous variables ( h = 1,..., H ) that can affect the performance of the organisation. Technology (1) is a description of the way in which outputs can be obtained from inputs, given some exogenous variables. If the function (1) is separable in outputs, then: (2)

T ( y j , x j , z j ) = g( y j ) − f (x j , z j )

and the technology can be written in the standard form: (3)

g( y j ) = f (x j , z j )

in which g ( y j ) is a function of outputs and f ( x j , z j ) is a production function. If just one output is produced, the function of outputs g ( y j ) reduces to the output itself, so that it is possible to write the usual single-output production function y j = f ( x j , z j ) . In this case the methodology for the estimation of the frontier production function and the measurement of efficiency is available in the literature. In the presence of multiple outputs the use of duality theory allows the estimation of cost or profit functions if prices for inputs and outputs are known. If prices for some outputs or for some inputs are not available, as in the case of public and non-profit organisations, duality theory cannot be applied, so that other tools have to be

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found. Possible solutions are the use of the distance function1 or the use of the ray frontier production function (Löthgren, 1997a, 1997b). The aim of these methods is to reduce the K-vector of outputs into a scalar measure through a representation of the technological connections among outputs. After that, the usual parametric analysis can be applied. In the present paper, a different solution is proposed in order to obtain an aggregation of outputs, based explicitly on the fact that the management of the public or non-profit organisation is not indifferent about the output mix produced. On the contrary, the output mix is generally one of the most important choices of an autonomous management. Instead of looking for a technological representation, an aggregation of outputs into a scalar measure can be found considering that the quantities and the output mix can be valued with the use of a utility function defined by the management of the organisation itself or by some other external assessing body, such as a review committee or a governmental agency. 3.

The utility function and the equally distributed equivalent output Let the utility function of the j-th NPO be: U j = U ( y j, α j )

(4)

where α j = [α1 j ,..., α kj ,..., α Kj ] is a vector of weights assigned by the j-th NPO to the K outputs. In order to obtain the usual well-behaved indifference curves, it is convenient to assume positive and decreasing marginal utility for all outputs. The utility function is used by the NPO to choose the best mix of outputs among all possible points of the production possibility curve (see figure 1).

Figure 1 – Production possibility curve (PPC) and the best output mix y2

y =y 1

y2j

2

U(y ) j

PPC y1j

1

y1

See, for instance, Färe-Primont (1995) and Coelli-Prasada Rao-Battese (1998, ch. 3).

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The utility function can be written in any flexible form, but to simplify the exposition we use the CES (or isoelastic) utility function: 1−ρ j

∑k =1 α kj y kj K

Uj =

(5)

1− ρ j

∑k =1 α kj = 1 K

and ρ j ≥ 0 . The parameter ρ j represents the degree of aversion of the j-th NPO against unequal levels of its (weighted) outputs. A value ρ j = 0 with the constraint

gives a linear utility function with linear indifference curves, so that the NPO is indifferent about the output mix. With a value of ρ j = 1 the utility function becomes the standard CobbDouglas in log-linear form, with hyperbolic indifference curves: U j = ∑k =1 α kj ln y kj K

(6)

With higher values of ρ j the aversion against different levels of output increases, so that the NPO reaches higher utility levels by producing similar quantities of all weighted outputs. To simplify the definitions of meaningful weights α , it is useful to normalize all outputs dividing each value by the maximum reached in the sample2. After this normalization the value of each output in each NPO will range between 0 and 1: y kj ∈ [0,1] . We define now the K-vector Y jE = [ y Ej ,..., y Ej ,..., y Ej ] as the vector that gives the actual level of utility of the j-th NPO when all outputs are set at the same quantity y Ej , therefore: (7)

U j = U ( y j , α j ) = U (Y jE , α j ) Solving for y Ej , we obtain:

(8)

[ (

)]

y Ej = U −1 U j y j , α j = V ( y j, α j )

The scalar value y Ej can be defined as the equally distributed equivalent (EDE) output3. The EDE output is a monotonic transformation of the level of utility Uj, so that it can be seen as the level of utility reached by the j-th NPO measured with the metric of the output. In figure 2 the EDE output is represented in the case of two outputs. If the weights have the same value the indifference curve is symmetric with respect to the bisecting line. If the weights are set at different values the indifference curve is no longer symmetric and the EDE output changes because the outputs with higher weights have more influence on the level of utility. In the example of figure 2, the EDE output increases, with respect to the equal weights case, because the higher weight is associated with the larger output. With the CES utility function (5) the level of utility is: α ( y Ej ) ∑ k =1 kj = K

(9)

2

Uj

1−ρ j

1− ρ j

An average or an exogenously given value can also be used. Note the analogy with the equally distributed equivalent income introduced by Atkinson (1970) and used in the literature on the measurement of income inequality.

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and, given the constraint

∑k =1 α k = 1 , the related EDE output becomes: K

[

]

1 / 1−ρ j y Ej = (1 − ρ j )U j =

(10)

5

[∑

α y 1−ρ k kj kj

]

1 / 1−ρ j

In the example of the Cobb-Douglas utility function the EDE output is:

(

y Ej = exp(U j ) = exp ∑ k α kj ln y kj

(11)

)

With the sum of weights constrained to unity, the EDE output is a homogeneous function of degree one, so that if all outputs are doubled then the EDE output is doubled.

Figure 2 – Indifference curves and EDE output y1 = y2 y

2

y

2j

yE

α >α

j

2j

1j

α =α 2j

y1 j

4.

yE j

1j

y

The frontier production function and the level of efficiency The EDE output defined above is a scalar measure of output, so that if we define

y Ej = g ( y j ) then the production function (3) is: y Ej = f ( x j , z j )

(12)

It is now possible to choose an econometric method in order to estimate the efficiency of the NPOs. Both the deterministic and the stochastic frontier approaches can be used. Using the class of log-linear production functions, equation (12) becomes: ln y Ej = β 0 + β ln x j − u j

(13)

is the vector of (transformation of) inputs, β 0 is a constant term, β = [β1 ,..., β i ,..., β I ] is the I-vector of input coefficients, while the error term uj is defined

where

xj

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according to the estimation method used and can contain some exogenous variables zj that are supposed to be negatively correlated with the aggregate output in order to explain part of the inefficiency. The frontier production function can be written as: ln yˆ Fj = βˆ 0 + βˆ ln x j

(14)

where ^ denotes the estimated coefficients. The frontier output yˆ Fj represents the maximum potential value of the EDE output that the j-th NPO could reach, if the inefficiency due to the exogenous variables z and the residual inefficiency were not present. As usual, we define technical efficiency the ratio between the actual and the potential EDE output: y Ej

exp(βˆ 0 + βˆ ln x j − uˆ j ) = = exp( −uˆ j ) ej = exp(βˆ 0 + βˆ ln x j ) yˆ Fj

(15)

5.

Uses of the utility-frontier production function approach

With respect to the usual frontier production function approach, the presence of a parametric utility function allows us to perform a wider range of exercises. The ingredients of the combined approach are the utility function parameters (i.e. output weights and the degree of aversion to unequal output mix) and the frontier production function parameters. In both functions it is possible: • to allow for different values of the parameters for each individual NPO or, alternatively, to impose that the parameters must be common to all NPOs; • to estimate the parameters or to assign exogenous utility or frontier production functions. The combinations of choices lead to formulate several problems, that cannot be stated in the usual frontier analysis, and to approximate the flexibility of the non parametric approach. Some examples of possible problems are presented below, with the aim of showing that different procedures can be used according to different points of view. In fact the analysis can be performed, for instance, from the point of view of the managers of the NPOs, of an external assessing authority or of a neutral researcher. A summary of the problems that can be solved by means of this new utility-frontier production function approach (UFDF) is presented in table 1.

Table 1 – Summary of possible UFDF problems Frontier production function Estimated

Exogenous

Individual

Problem 1

Problem 5

Common

Problems 3-4

Problem 5a

Individual

Problem 2a

Problem 6a

Common

Problem 2

Problem 6

Utility function weights Free

Constrained

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Problem 1: Maximum individual efficiency The first problem is to choose the weights of the utility function α j = [α1 j ,..., α kj ,..., α Kj ] in order to maximise individual efficiency of the j-th NPO. The problem is similar to a DEA maximisation of the efficiency score for the j-th NPO, but in this case the efficiency is given by the ratio between the EDE output and the efficient frontier output of the same NPO, the latter estimated using all the N observations of the sample and assigning the weights of the j-th NPO to all the NPOs. The optimal weights α j are obtained by looking for the maximum efficiency: max e j =

(P1.1)

αj

y Ej yˆ Fj

in which the EDE output is obtained from the actual outputs and the utility function: y Ej = V ( y j , α j )

(P1.2) while the efficient frontier output is:

ln yˆ Fj = F (βˆ 0 , βˆ , x j , z j )

(P1.3)

in which the parameters come from the estimation of the frontier production function: (P1.4)

[

[

]

ln y E = β 0 + β ln x1 ,..., ln x j ,..., ln x I − u

]

E where y E = y1E ,..., y nE ,..., y N , y nE = V ( y n, α j ) for all n=1,…,N and u = [u1 ,..., u n ,..., u N ] . The solution can be found with a grid search procedure, in which the level of efficiency is tested for a number of values of the weights α j in the range [0,1]. The procedure is repeated

for each NPO. The results are: - the optimal values α j , -

the maximum efficiency level e j ,

- the associated frontier production function for each j-th NPO in the sample. Like in DEA, the solution is the best level of efficiency that can be obtained by the NPO and the estimated frontier production function can be different for each NPO.

Problem 2: Efficiency with a known utility function In the second problem we assume the presence of an external assessing authority that states its preferences about the vector of weights of the utility function α = [ α1 ,..., α k ,..., α K ] . In this case the vector of EDE output can be computed given the exogenous weights, so that the problem is solved simply through the estimation of the frontier production function. The level of efficiency is computed for the j-th NPO: (P2.1)

ej =

y Ej yˆ Fj

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with the constraints that the EDE output is obtained from the actual outputs and the assigned utility function: (P2.2)

y Ej = V ( y j, α )

and (P2.3) (P2.4)

ln yˆ Fj = F (βˆ 0 , βˆ , x j , z j ) ln y E = β 0 + β[ln x1 ... ln x i

... ln x I ] − u

in which y nE = V ( y n, α ) for all n=1,…,N. The solution can be found just by means of one regression. The results are: - the efficiency level e j for each j-th NPO, - the associated frontier production function. The solution gives the efficiency levels that the NPOs can obtain given the exogenous weights, as in an output-oriented DEA problem with output virtual weights fully constrained. A simple variant of problem 2 is the use of different exogenous weights defined for each NPO (problem 2a). Problem 3: Maximum average efficiency In the third problem involves the choice of the utility function and the frontier production function parameters in order to obtain the maximum average efficiency, with the constraint that the two functions are common to all the NPOs. So that the problem is to choose a vector of weight α = [α1 ,..., α k ,..., α K ] that maximises the average level of efficiency:

∑ j =1 y Ej max e = N ∑ j =1 yˆ Fj N

(P3.1)

with the constraints of problem 1, but with y nE = V ( y n, α) for all n=1,…,N. As in problem 1, the solution can be found with a grid search procedure testing the average level of efficiency for a number of values of the common weights α in the range [0,1]. The results are: - the optimal values of the common weights α , - the maximum level of average efficiency and the level of efficiency for each NPOs, - the associated frontier production function. Problem 4: Maximum goodness of fit of the frontier production function The researcher may be interested in finding the utility parameters and the associated frontier production function that maximise an indicator of the goodness of fit of the regression. The resulting utility weights are such to obtain the best estimation of the production function. The procedure is the same as in problem 3, with the difference that the value to maximise is a goodness of fit indicator, the R2, of the regression.

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Problem 5: Maximum individual efficiency with a known frontier production function The problem is similar to problem 1, but in this case frontier production function is known, for instance from a similar study based on other data or from the results of problems 3 or 4. In this case the solution is obtained by looking for the best individual weights with the grid search procedure described for problem 1, in which the efficient production level is simply computed from the assigned frontier production function. A similar problem (problem 5a) can be solved if one looks for a common vector of weights, in order to maximise the average efficiency as in problem 3. Problem 6: Efficiency with known utility and frontier production functions The levels of efficiency for all NPOs can be computed also if both a common utility function and the frontier production function are exogenously given. Obviously, a variant can be the possibility of different individual utility weights (problem 6a). 6.

UFDF with COLS: an application to the Ca’ Foscari University of Venice academic departments

This section shows some empirical results obtained by means of the UFDF approach procedure presented above. The data used concern the activity of the 17 departments of the University of Venice in the academic year 1997/984. The outputs considered are an indicator of teaching load (Tj), an indicator of students’ satisfaction (Sj) and an indicator of fund-raising for research activity (Rj), while the inputs are the budget assigned by the University (Bj) and an indicator of teaching, technical and administrative personnel resources (Pj). A low level of efficiency can be associated to the relative low number of students, so that the lecturers/students (Lj) ratio is used to explain some part of the technical inefficiency. For simplicity, the utility function of the departments is assumed to be a Cobb-Douglas in log form: U j = ∑ k =1 α k ln y kj K

(17) with the constraint

∑k =1 α k = 1 . K

In order to apply a robust method that gives a sure result, deterministic frontier analysis (DFA) and the method of corrected ordinary least squares (COLS)5 is used. The choice is due to the necessity of obtaining a regression for each iteration of the grid search. In this case a method like MLE would not assure the convergence and hence a value for the efficiency measure for any vector of weights. The COLS method used is extended in order to allow for the decomposition of inefficiency in two parts: one part explained by some exogenous variables and a residual part non explained. As the production function is assumed to be a Cobb-Douglas function, the EDE output is defined by: (18)

4

ln y Ej = β 0 + β ln x j + u j

Information about the data used can be found in Rizzi (1999). Richmond (1974), Försund-Lovell-Schmidt (1980). For a description of DFA and COLS see Lovell-Schmidt (1987) and Coelli-Prasada Rao-Battese (1998). 5

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in which β 0 is the constant term, β = [β1 ,..., β k ,..., β K ] is a I-vector of coefficients, while the error term uj depends upon some exogenous variables zj that explain a part of inefficiency6, and upon random variables v j ~ N (0, σ v2 ) assumed to be independently and identically distributed: u j = γ ln z j + v j

(19)

with coefficients γ = [ γ1 ,..., γ h ,..., γ H ] . The production function can be written as: ln y Ej = (β 0 + µ) + β ln x j − (µ − u j )

(20)

where the term µ is the correction to the constant needed to obtain an efficient frontier that envelops all the sample observations. The new error term (µ − u j ) is always positive and measures the inefficiency as the distance of the actual output from the production frontier. From the OLS estimation of the production function (18)-(19) it is straightforward to obtain the deterministic frontier production function: ~ (21) ln y Fj = β 0 + βˆ ln x j that represents the maximum potential value of the EDE output of department j, obtainable if ~ ~ is the definition of the constant the last term of (20) is not present. The term β = βˆ 0 + µ ~ is according to the COLS method7, in which the symbol ^ indicates OLS estimates and µ given by: (22)

~ = max[vˆ ] + H max[ γˆ ln z ] = v µ ∑ h =1 j h hj max + Z max j

j

The first part of (22) is the maximum positive residual obtained from the OLS estimation, while the second part is the sum of all the maximum positive influences on output coming from the exogenous variables (or minimum negative influences, according to the sign). The sum of maximum values is used instead of the maximum of the sum in order to avoid compensations of effects with opposite sign. Substituting definitions (19) and (22) into (20) we obtain: ~ (23) ln y Ej = β 0 + βˆ ln x j − [Z max − γˆ ln z ] − v max − vˆ j

[

]

in which the actual output is explained as the sum of a potential output, a loss of output due to the exogenous variables z and a loss of output not explained..

6

In order to explain the inefficiency, variables z should be negatively correlated with output, so that a negative sign is expected for γ . 7 Greene (1980) demonstrated the consistency of the constant of a production frontier obtained with COLS. In the present case, the constant incorporates also Zmax that contains the estimates of γ . These estimates are consistent as they derive from the OLS procedure.

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The technical efficiency indicator is then obtained by dividing the actual level of output by the maximum potential output: ~ y Ej exp β 0 + βˆ ln x j − ( Z max − γˆ ln z j ) − (v max − vˆ j ) (24) ej = = ~ exp( β 0 + βˆ ln x j ) y Fj

[

]

= exp( γˆ ln z j − Z max + vˆ j − v max )

Given definition (19), the maximum level of efficiency (100%) is obtained by the unit with the maximum value of vˆ j and, at the same time, with the minimum negative influence (or the maximum positive influence) of variables zj. If the effect of exogenous variables where not present, the level of output would be: ~ (25) ln y zj = β0 + βˆ ln x j − v max − vˆ j

[

]

and the corresponding level of efficiency would be: ~ y zj exp β 0 + βˆ ln x j − (v max − vˆ j ) z ej = = = exp v j − v max (26) ~ exp(β 0 + βˆ ln x j ) y Fj

[

]

(

)

The maximum level of efficiency is therefore decomposable into three parts: the technical efficiency reached (ej), the inefficiency explained by variables z: (27)

I zj = e zj − e j

and the residual inefficiency: (28)

z I res j = 1− e j .

It is important to notice that the efficiency computed is a relative value, as it is determined within a given sample of observations. The Appendix contains the results of the empirical application8. In the first exercise (Problem 1 - Maximum individual efficiency) each academic department chooses the best utility function weights (table 1.1), and the associated frontier production function (table 1.2), in order to maximise its level of efficiency. Table 1.3 presents the decomposition of inefficiency. Like in DEA results, output weights chosen are quite different among departments. In addition, each department proposes a different estimation of the frontier production function. The efficiency results are the best that each department can reach given the sample. The results for Problem 2 (Efficiency with a known utility function) are obtained by assigning fixed weights to the utility function. The weights assigned are: 25% to the teaching load indicator, 25% to the students’ satisfaction indicator and 50% to the fund-raising indicator. The budget and the staff coefficients COLS estimates are significant and very closed (0.494 and 0,500). The regression shows an adjusted R2 of 0.758 (table 2.2). Table 2.3 shows the efficiency results.

8

The results presented in this section and the Appendix are computed with UFPF, a DOS computer program written in Fortran. The computer program is available on line at the site: http://helios.unive.it/~rizzid/effic.htm.

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The third exercise (Problem 3: Maximum average efficiency) leads to a higher level of average efficiency (more than 84%) but to a very poor estimation of the COLS frontier production function (table 3.1-3.2-3.3). The opposite result can be found with problem 4 (Maximum goodness of fit of the frontier production function) as the adjusted R2 reaches the level of 0.824 (table 4.2) while the level of average efficiency is 42%. The results of Problem 5 (Maximum individual efficiency with known frontier production function) are found by assigning the coefficients of the frontier production function estimated in problem 4. Each department chooses the best weights that, like in Problem 1, are very different among departments. The last exercise (Problem 6 - Efficiency with common exogenous weights and exogenous frontier production function) uses the utility weights of Problem 2 and the frontier production function coefficients of problem 49. 7.

Conclusions

This paper presents the utility-frontier production function approach to the measurement of efficiency for multi-input multi-output decision making units. The approach is particularly useful when it is not possible to use duality theory because of the difficulty of defining output prices, as in the case of public and non-profit organisations. The usual frontier production function estimation is coupled with a manager utility function, based upon the output quantities, from which an aggregate measure of output is obtained. The approach seems promising as it combines a non-parametric part (the choice of the best utility function weights, like in a DEA problem) with the standard parametric frontier production function estimation. It is then possible, for instance, to state the problem in such a way that each organisation can choose its best weights to maximise its efficiency while, at the same time, a unique frontier production function is estimated, with the possibility to account for noise in the data and to conduct conventional tests of hypotheses. Obviously, the need of specifying a functional form still remains, even made worse by the presence of a utility function.

References Atkinson A.B. (1970) “On the Measurement of Inequality”, The Journal of Economic Theory, 2, pp. 244-263. Coelli T., D.S. Prasada Rao, G.E. Battese (1998) An Introduction to Efficiency and Productivity Analysis, Kluwer Academic Publishers, Boston. Färe R., D. Primont (1995) Multi-Output Production and Duality: Theory and Applications, Kluwer Academic Publishers, Boston. Försund F.R., C.A.K. Lovell, P. Schmidt (1980) “A Survey of Frontier Production Functions and Their Relationship to Efficiency Measurement”, Journal of Econometrics, 13, pp. 5-25. Green W.H. (1980) “Maximum Likelihood Estimation of Econometric Frontier Functions ”, Journal of Econometrics, 13, pp. 27-56. Lovell C.A.K., P. Schmidt (1987) “A Comparison of Alternative Approaches to the Measurement of Productive Efficiency”, in A. Dogramaci, R. Färe Applications of Modern Production Theory, Kluwer Academic Publishers, Boston. Löthgren M. (1997a) A Multiple Output Stochastic Ray Frontier Production Model, Stockholm School of Economics, W.P. Series in Economics and Finance, No. 158, February 9

In problem 4 the utility weights were different from those used in problem 6, so that the COLS correction should be computed according to definition (22).

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Löthgren M. (1997b) “Generalized Stochastic Frontier Production Model”, Economics Letters, 57, pp. 255-259. Richmond J. (1974) “Estimating the Efficiency of Production”, International Economic Review, 15, pp. 515-521. Rizzi D. (1999) The Efficiency of the University of Venice Departments via DEA and DFA, Department of Economics, Ca’ Foscari University of Venice, Working Paper 99.09. www.unive.it/dse/notelavoro/9909.htm.

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Appendix – Results of the empirical applications: the efficiency of the Ca’ Foscari University of Venice academic departments (academic year 1997/98)

Problem 1 - Maximum individual efficiency Table 1.1 Problem 1: Estimated frontier production function coefficients (t-test in italics) Departments 1

Chemistry

2

Physical Chemistry

3

Business Adm. and Managem.

4

Philosophy

5

Italian Studies and Rom. Ph.

6

Applied Mathematics

7

Environmental Sciences

8

Ancient and Classical Studies

9

Economics

10

Law

11

Statistics

12

Art History and Criticism

13

Anglo-american studies

14

Eurasian Studies

15

Oriental Studies

16

European linguistic studies

17

History

COLS Constant

Budget

Staff

-4.0513 -4.0513 -8.6752 -6.7670 -5.9140 -1.4786 -4.0513 -3.6541 -14.761 -1.6656 -8.8469 -1.4786 -3.9513 -1.4786 -1.4786 -1.4786 -1.4786 -

0.29781 2.2243 0.29781 2.2243 0.41941 2.0172 0.47870 3.2010 0.42153 2.9612 0.10120 0.74288 0.29781 2.2243 0.26878 2.0215 0.98507 3.7185 0.10967 0.81440 0.60998 3.5561 0.10120 0.74288 0.28753 2.1632 0.10120 0.74288 0.10120 0.74288 0.10120 0.74288 0.10120 0.74288

0.14860E-01 0.82201E-01 0.14860E-01 0.82201E-01 0.88680 3.1591 0.11638 0.57638 0.85062E-01 0.44258 0.47854E-02 0.26020E-01 0.14860E-01 0.82201E-01 0.62757E-02 0.34960E-01 0.52550 1.4693 0.26507E-01 0.14580 0.22245 0.96053 0.47854E-02 0.26020E-01 0.21429E-01 0.11941 0.47854E-02 0.26020E-01 0.47854E-02 0.26020E-01 0.47854E-02 0.26020E-01 0.47854E-02 0.26020E-01

Lecturers/ students 0.68002E-02 0.75077E-01 0.68002E-02 0.75077E-01 -0.91573 -6.5106 -0.13338E-02 -0.13184E-01 0.35692E-03 0.37063E-02 -0.98771E-01 -1.0718 0.68002E-02 0.75077E-01 -0.69495E-03 -0.77265E-02 -0.16668 -0.93008 -0.11851 -1.3010 -0.44201E-01 -0.38092 -0.98771E-01 -1.0718 -0.68189E-02 -0.75833E-01 -0.98771E-01 -1.0718 -0.98771E-01 -1.0718 -0.98771E-01 -1.0718 -0.98771E-01 -1.0718

D. Rizzi – Working paper 2000.xx

15

Table 1.2 Problem 1: Efficiency, goodness of fit and best output weights

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Departments

Efficiency %

R2 %

Chemistry Physical Chemistry Business Adm. and Managem. Philosophy Italian Studies and Rom. Ph. Applied Mathematics Environmental Sciences Ancient and Classical Studies Economics Law Statistics Art History and Criticism Anglo-american studies Eurasian Studies Oriental Studies European linguistic studies History

70.765 65.539 90.546 98.873 99.972 72.285 70.935 89.918 95.884 69.508 67.993 85.143 86.961 83.207 95.972 93.112 92.836

39.679 39.679 85.386 62.008 57.335 16.875 39.679 34.800 74.719 22.046 69.478 16.875 39.133 16.875 16.875 16.875 16.875

Teaching load α1 0.03 0.03 0.55 0.00 0.01 0.13 0.03 0.04 0.00 0.14 0.00 0.13 0.04 0.13 0.13 0.13 0.13

Students’ satisfaction α2 0.70 0.70 0.00 0.54 0.59 0.80 0.70 0.72 0.00 0.78 0.40 0.80 0.70 0.80 0.80 0.80 0.80

Fundraising α3 0.27 0.27 0.45 0.46 0.40 0.07 0.27 0.24 1.00 0.08 0.60 0.07 0.26 0.07 0.07 0.07 0.07

Explained inefficiency % 0.000 0.000 0.000 0.084 0.028 0.000 0.000 0.040 0.000 0.000 0.000 5.496 0.241 3.398 3.919 3.802 5.993

Residual inefficiency % 29.235 34.461 9.454 1.043 0.000 27.715 29.065 10.042 4.116 30.492 32.007 9.361 12.798 13.395 0.109 3.086 1.171

Table 1.3 Problem 1: Efficiency results

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Departments

Actual YEDE

Frontier YEDE

Efficiency %

Chemistry Physical Chemistry Business Adm. and Managem. Philosophy Italian Studies and Rom. Ph. Applied Mathematics Environmental Sciences Ancient and Classical Studies Economics Law Statistics Art History and Criticism Anglo-american studies Eurasian Studies Oriental Studies European linguistic studies History

0.52532 0.48918 0.68889 0.73388 0.75769 0.62435 0.58170 0.80441 0.70276 0.58148 0.52544 0.68398 0.73379 0.67986 0.77782 0.80061 0.78174

0.80042 0.75446 0.80441 0.56068 0.64371 0.86373 0.85223 1.00900 1.04290 0.82346 0.43536 0.80333 0.85506 0.81708 0.81046 0.85984 0.84206

70.765 65.539 90.546 98.873 99.972 72.285 70.935 89.918 95.884 69.508 67.993 85.143 86.961 83.207 95.972 93.112 92.836

16

D. Rizzi – Nota di lavoro 2000.08

Problem 2 – Efficiency with a known utility function Table 2.1 Problem 2: Assigned output weights Outputs indicators Teaching load Students’ satisfaction Fund-raising

Weight 0.250 0.250 0.500

Table 2.2 Problem 2: COLS estimation results Coefficient OLS constant Budget Staff Lecturers/students ratio Constant shift COLS constant Number of observations Log of Likelihood Function F-statistics (3,13)

β0 β1 β3 γ1 µ β0 + µ

17 6.126 17.673

Estimated coefficient -10.375 0.494 0.500 -0.431 2.075 -8.300

t-test -6.232 3.034 2.278 -3.913

R2 Adjusted R2

0.803 0.758

Table 2.3 Problem 2: Efficiency results Departments 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Chemistry Physical Chemistry Business Adm. and Managem. Philosophy Italian Studies and Rom. Ph. Applied Mathematics Environmental Sciences Ancient and Classical Studies Economics Law Statistics Art History and Criticism Anglo-american studies Eurasian Studies Oriental Studies European linguistic studies History Simple mean efficiency Weighted mean efficiency* * See definition P1.1

Actual YEDE

Frontier YEDE

Efficiency %

0.27252 0.20009 0.63557 0.34984 0.40326 0.52201 0.35200 0.63559 0.85426 0.33978 0.25676 0.22829 0.40522 0.25786 0.32946 0.43395 0.43329

0.71615 0.60105 0.83365 0.45954 0.54793 0.83493 0.90366 1.05090 0.96089 0.56521 0.39664 0.48137 0.61099 0.52255 0.46809 0.89113 0.76216

38.054 33.290 76.239 76.130 73.596 62.521 38.953 60.481 88.903 60.115 64.735 47.425 66.323 49.347 70.383 48.696 56.850 59.532 59.532

Explained inefficiency % 32.031 28.022 0.000 23.870 23.076 0.000 32.788 18.964 0.000 0.000 0.000 14.870 12.644 9.408 13.418 9.284 17.825

Residual inefficiency % 29.915 38.688 23.761 0.000 3.328 37.479 28.258 20.555 11.097 39.885 35.265 37.706 21.033 41.245 16.199 42.020 25.324

D. Rizzi – Working paper 2000.xx

17

Problem 3: Maximum average efficiency Table 3.1 Problem 3: Best output weights Outputs indicators Teaching load Students’ satisfaction Fund-raising

Weights 0.072 0.864 0.064

Table 3.2 Problem 3: COLS estimation results OLS constant Budget Staff Lecturers/students ratio Constant shift COLS constant

Coefficient

γ1 µ β0 + µ

Estimated coefficient -1.343 0.101 -0.818E-01 -0.184E-03 0.180 -1.163

Number of observations Log of Likelihood Function F-statistics (3,13)

17 8.492 0.196

R2 Adjusted R2

β0 β1 β3

t-test -0.927 0.712 -0.428 -0.192E-02

0.043 -0.177

Table 3.3 Problem 3: Efficiency results Departments 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Chemistry Physical Chemistry Business Adm. and Managem. Philosophy Italian Studies and Rom. Ph. Applied Mathematics Environmental Sciences Ancient and Classical Studies Economics Law Statistics Art History and Criticism Anglo-american studies Eurasian Studies Oriental Studies European linguistic studies History Simple mean efficiency Weighted mean efficiency* * See definition P1.1

Actual YEDE

Frontier YEDE

Efficiency %

0.60835 0.57790 0.67414 0.81321 0.83235 0.63805 0.64635 0.88083 0.69385 0.60113 0.56499 0.76652 0.82472 0.77410 0.83322 0.85960 0.84756

0.84934 0.84432 0.88117 0.83573 0.83261 0.87733 0.84743 0.93339 0.87736 0.88746 0.84523 0.84591 0.92108 0.86044 0.86449 0.85967 0.85033

71.626 68.445 76.505 97.306 99.968 72.727 76.271 94.369 79.084 67.736 66.844 90.615 89.539 89.965 96.383 99.993 99.674 84.532 84.528

Explained inefficiency % 0.019 0.018 0.000 0.011 0.012 0.000 0.020 0.011 0.000 0.000 0.000 0.011 0.007 0.007 0.007 0.007 0.012

Residual inefficiency % 28.355 31.537 23.495 2.683 0.020 27.273 23.709 5.620 20.916 32.264 33.156 9.375 10.454 10.029 3.610 0.000 0.314

18

D. Rizzi – Nota di lavoro 2000.08

Problem 4: Maximum goodness of fit of the frontier production function Table 4.1 Problem 4: Best output weights Outputs indicators Teaching load Students’ satisfaction Fund-raising

Weight 0.674 0.000 0.326

Table 4.2 Problem 4: COLS estimation results Coefficient OLS constant Budget Staff Lecturers/students ratio Constant shift COLS constant Number of observations Log of Likelihood Function F-statistics (3,13)

β0

β1 β3 γ1 µ β0 + µ

17 0.921 26.049

Estimated coefficient -12.012 0.292 0.968 -1.085 4.843 -7.169

t-test -5.312 1.321 3.246 -7.257

R2 Adjusted R2

0.857 0.824

Table 4.3 Problem 4: Efficiency results Departments 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Chemistry Physical Chemistry Business Adm. and Managem. Philosophy Italian Studies and Rom. Ph. Applied Mathematics Environmental Sciences Ancient and Classical Studies Economics Law Statistics Art History and Criticism Anglo-american studies Eurasian Studies Oriental Studies European linguistic studies History Simple mean efficiency Weighted mean efficiency* * See definition P1.1

Actual YEDE

Frontier YEDE

Efficiency %

0.11207 0.07880 0.79483 0.21764 0.25473 0.47767 0.18897 0.33566 0.85879 0.34024 0.20293 0.15558 0.20728 0.14391 0.28958 0.32896 0.29917

0.84679 0.68450 0.90715 0.49616 0.63646 0.92364 1.16540 1.00200 1.11450 0.52551 0.39097 0.50509 0.50914 0.52971 0.44942 1.08550 0.91644

13.234 11.512 87.618 43.866 40.023 51.717 16.215 33.499 77.053 64.745 51.904 30.803 40.713 27.167 64.435 30.306 32.645 42.203 41.667

Explained inefficiency % 48.389 42.092 0.000 43.324 39.529 0.000 59.289 33.085 0.000 0.000 0.000 30.422 22.472 14.995 35.565 16.727 32.241

Residual inefficiency % 38.377 46.396 12.382 12.811 20.448 48.283 24.496 33.417 22.947 35.255 48.096 38.775 36.815 57.838 0.000 52.967 35.114

D. Rizzi – Working paper 2000.xx

19

Problem 5: Maximum individual efficiency with known frontier production function Table 5.1 Problem 5: Exogenous production function coefficients (see Problem 4 – Table 4.2) Coefficient COLS Constant Budget Staff Lecturers/students ratio

β0 + µ β1 β3 γ1

Estimated coefficient -7.169 0.292 0.968 -1.085

Table 5.2 Problem 5: Efficiency and best output weights

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Departments

Efficiency %

Chemistry Physical Chemistry Business Adm. and Managem. Philosophy Italian Studies and Rom. Ph. Applied Mathematics Environmental Sciences Ancient and Classical Studies Economics Law Statistics Art History and Criticism Anglo-american studies Eurasian Studies Oriental Studies European linguistic studies History

21.476 21.476 87.656 50.311 46.795 56.601 21.476 50.311 86.648 65.091 54.975 46.256 55.570 42.448 64.435 33.070 36.960

Teaching load α1 0.00 0.15 0.55 0.35 0.47 0.55 0.35 0.21 0.54 0.55 0.55 0.26 0.34 0.24 0.61 0.55 0.54

Students’ satisfaction α2 0.30 0.52 0.00 0.30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.70 0.15 0.75 0.00 0.00 0.00

Fundraising α3 0.70 0.33 0.45 0.35 0.53 0.45 0.65 0.79 0.46 0.45 0.45 0.04 0.51 0.01 0.39 0.45 0.46

Table 5.3 Problem 5: Efficiency results

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Departments

Actual YEDE

Frontier YEDE

Efficiency %

Chemistry Physical Chemistry Business Adm. and Managem. Philosophy Italian Studies and Rom. Ph. Applied Mathematics Environmental Sciences Ancient and Classical Studies Economics Law Statistics Art History and Criticism Anglo-american studies Eurasian Studies Oriental Studies European linguistic studies History

0.42386 0.32901 0.72836 0.35905 0.28022 0.47886 0.26690 0.65423 0.88518 0.31332 0.19688 0.57951 0.33784 0.59915 0.27676 0.32880 0.31046

1.97360 1.53200 0.83093 0.71365 0.59882 0.84603 1.24280 1.30040 1.02160 0.48136 0.35812 1.25280 0.60796 1.41150 0.42952 0.99426 0.84000

21.476 21.476 87.656 50.311 46.795 56.601 21.476 50.311 86.648 65.091 54.975 46.256 55.570 42.448 64.435 33.070 36.960

Explained inefficiency % 78.524 78.524 0.000 49.689 46.216 0.000 78.524 49.689 0.000 0.000 0.000 45.685 30.672 23.429 35.565 18.253 36.503

Residual inefficiency % 0.000 0.000 12.344 0.000 6.989 43.399 0.000 0.000 13.352 34.909 45.025 8.059 13.759 34.123 0.000 48.677 26.538

20

D. Rizzi – Nota di lavoro 2000.08

Problem 6. Efficiency with common exogenous weights and exogenous frontier production function

Table 6.1 Problem 6: Assigned output weights Outputs indicators Teaching load Students’ satisfaction Fund-raising

Weight 0.250 0.250 0.500

Table 6.2 Problem 6: Assigned production function coefficients and COLS constant (see Problem 4 – Table 4.2) Coefficient Constant Budget Staff Lecturers/students ratio COLS constant

β0 β1 β3 γ1 ~ β0

Estimated coefficient -12.012 0.292 0.968 -1.085 -6.764

Table 6.3 Problem 6: Efficiency results Departments 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Chemistry Physical Chemistry Business Adm. and Managem. Philosophy Italian Studies and Rom. Ph. Applied Mathematics Environmental Sciences Ancient and Classical Studies Economics Law Statistics Art History and Criticism Anglo-american studies Eurasian Studies Oriental Studies European linguistic studies History Simple mean efficiency Weighted mean efficiency* * See definition P1.1

Actual YEDE

Frontier YEDE

Efficiency %

0.27252 0.20009 0.63557 0.34984 0.40326 0.52201 0.35200 0.63559 0.85426 0.33978 0.25676 0.22829 0.40522 0.25786 0.32946 0.43395 0.43329

1.26890 1.02580 1.35940 0.74351 0.95376 1.38410 1.74640 1.50160 1.67020 0.78750 0.58589 0.75689 0.76296 0.79379 0.67348 1.62660 1.37330

21.476 19.507 46.754 47.053 42.281 37.715 20.157 42.329 51.148 43.146 43.825 30.161 53.112 32.485 48.919 26.678 31.551 37.547 36.340

Explained inefficiency % 78.524 71.323 0.000 46.471 41.758 0.000 73.699 41.805 0.000 0.000 0.000 29.788 29.315 17.930 27.001 14.725 31.161

Residual inefficiency % 0.000 9.170 53.246 6.476 15.961 62.285 6.144 15.866 48.852 56.854 56.175 40.051 17.573 49.585 24.080 58.597 37.289