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Estimation of production technology when the objective is to maximize return to the outlay
European Journal of Operational Research 208 (2011) 170–176
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Estimation of production technology when the objective is to maximize return to the outlay Subal C. Kumbhakar ⇑ Department of Economics, State University of New York – Binghamton, Binghamton, NY 13902, USA
a r t i c l e
i n f o
Article history: Received 16 March 2010 Accepted 10 September 2010 Available online 29 September 2010 Keywords: Distance function Transformation function Translog function Technical inefficiency
a b s t r a c t This paper deals with estimation of production technology where endogeneous choice of input and output variables is explicitly recognized. To address this endogeneity issue, we assume that producers maximize return to the outlay. We start from a flexible (translog) transformation function with a single output and multiple inputs and show how the first-order conditions of maximizing return to the outlay can be used to come up with an ‘estimating equation’ that does not suffer from the econometric endogeneity problem although the output and input variables are chosen endogenously. This is because the regressors in this estimating equation are in ratio forms which are uncorrelated with the error term under the assumption that producers maximize return to the outlay. The analysis is then extended to the multiple outputs and multiple inputs case with technical inefficiency. Although the estimating equations in both single and multiple output cases are neither production nor distance functions, they can be estimated in a straightforward manner using the standard stochastic frontier technique without worrying about endogeneity of the regressors. Thus, we provide a rationale for estimating the technology parameters consistently using an econometric model which requires data on only input and output quantities. 2010 Elsevier B.V. All rights reserved.
1. Introduction Given that resources are limited, economists emphasize on ‘optimal utilization’ of them. Optimality in standard microeconomics textbooks is mostly discussed in terms of cost minimization and profit maximization behaviors. Little or no attention is paid to analyzing increased cost associated with non-optimal allocation of resources (allocative inefficiency). In addition to being allocatively inefficient producers can be technically inefficient as well. That is, producers may not be producing the maximum possible output given inputs or they may be using more inputs than the minimum required to produce a given level of output. In the production efficiency literature the focus is on measuring potential output loss, increase in cost, decrease in profit, decrease in revenue, etc., resulting from either technical inefficiency or both technical and allocative inefficiency. For recent surveys on these issues see various chapters in Fried et al. (2008), Kumbhakar and Lovell (2000), Coelli et al. (2005), among others. Even if one is not interested in measuring efficiency per se, the idea of introducing behavioral assumption is important both economically and econometrically. Economic agents are inherently optimizers, and therefore the observed data are generated from the technology along with some specific behaviors on the part of the producers. These behaviors are not often recognized while estimating the model econometrically. Given the technology, levels of input and output variables are determined in such a way that the value of the objective function based on some economic behaviors is optimized. Thus, quantities of the choice variables are decided endogenously. For example, if producers choose inputs to minimize cost and outputs are exogenously given, we argue that inputs are endogenously determined by the producers. Some examples of these are: electricity, telephone, transportation, and other services, supply of which are determined by demand because these outputs cannot be stored. Consequently, the econometric models to estimate the production technology in the service industries treat outputs as exogenous and use the cost minimization objective to determine input quantities endogenously. Thus, economic behavior often tells us which variables are endogenous and which variables are exogenous. If one estimates the technology ignoring economic behavior the resulting estimates of the technology parameters are likely to be biased. As a result, when it comes to estimating the technology econometrically using a single equation method, we need to make sure that the economically endogenous variables do not cause econometric endogeneity problem. That is, the regressors in the estimating equation are not correlated with the error term no matter ⇑ Tel.: +1 607 777 4762; fax: +1 607 777 2681. E-mail address: [email protected] 0377-2217/$ - see front matter 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.09.015
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whether one wants to estimate inefficiency or not and whether one uses the simple ordinary least squares (OLS) technique or more sophisticated methods like maximum likelihood (ML), generalized method of moments, etc. Thus, in estimating a production function the OLS and ML parameters will be inconsistent unless the inputs are uncorrelated with the error term.1 This is true for estimating any econometric relationship in which the regressors in the estimating function are to be uncorrelated with the error term to obtain consistent parameter estimates. In this paper we argue that input and output variables are decision (economically endogenous) variables. Although cost minimization and profit maximization behaviors are widely discussed in microeconomics, other objectives such as maximizing value of the firm, maximizing return on investment, maximizing sales (revenue), etc., are also quite popular in finance. Here we assume that producers maximize return to the outlay to choose optimal quantities of inputs and output(s), given the technology. That is, the decision variables are inputs and outputs, and are therefore endogenous from economic point of view. The question is whether this causes econometric endogeneity problem (correlation of the regressors with the error term) in the estimating equation2 which can be represented in various forms. We first show that this is indeed the case and then we show that the econometric endogeneity problem can be solved if the ‘estimating equation’ is expressed in a particular form in which the regressors are ratios of input and output quantities. This ‘estimating equation’ is thus neither a production nor a distance function. We are not aware of any paper that discusses this issue in a formal econometric model. Färe et al. (2002), Zofío and Prieto (2006), Yu and Fan (2008) use maximization of return to the dollar to decompose technical and allocative inefficiency using the distance function formulation which is estimated by a deterministic linear programming approach3 in which endogeneity issue does not arise. This is because enodogeneity issue is typical in econometric models, especially when economic behaviors are believed to affect the regressors (quantities of inputs and/or outputs in our case).4 We start by specifying the technology parametrically in the form of a transformation function. In particular, we use a flexible (translog) functional form. First, we consider a single output without inefficiency, which is then extended to accommodate inefficiency. Finally, we consider the technology with multiple outputs and multiple inputs with inefficiency. In all these cases we recognize economic endogeneity of input and output variables and come up with an ‘estimating equation’ which is neither a production function nor an input (output) distance function. The distinguishing feature of our ‘estimating equation’ is that it does not suffer from econometric endogeneity problem, thereby meaning that the regressors in the ‘estimating equation’ can be treated as exogenous in the sense that these are not correlated with the error term. Thus this function, for both single and multiple output cases, can be easily estimated by OLS when there is no inefficiency; and by the ML method in the presence of inefficiency. The models with inefficiency are expressed in the form of a stochastic cost frontier model, estimation details of which can be found in Kumbhakar and Lovell (2000) as well as in Coelli et al. (2005). To sum up, the main contribution of the paper is that we provide a strategy to estimate both single and multiple outputs production technologies using data on the input and output variables in a single equation framework. In doing so we treat output and input variables as endogenous in the sense that quantities of these variables are decided by the producers (not exogenously given) whose objective is to maximize return to the outlay. We start from a transformation function representation of technology and show that it can be expressed in many equivalent forms. These are different econometrically and one cannot use the standard econometric techniques (OLS or ML) to estimate the parameters consistently because the regressors are endogenous (correlated with the error terms in the estimating equations). Thus, one cannot use the standard production and distance function formulations unless the econometric endogeneity problem associated with the regressors is resolved first (Guan et al., 2009). We solve this endogeneity problem first by deriving a particular form of the estimating equation in which the regressors are ratios of inputs and outputs. We then assume that producers maximize return to the outlay to show that these ratios are uncorrelated with the error term in the estimating model, thereby solving the econometric endogeneity problem that arises in the standard input (output) distance function. Thus, this equation can be estimated consistently using either the OLS (without inefficiency) or the ML method (with inefficiency). Consequently, all the canned stochastic frontier softwares can be used to estimate the models with inefficiency proposed in this paper.5 The rest of the paper is organized as follows. The production technology in various forms is presented in Section 2. Conditions for maximizing returns to the outlay for the single output technology with and without full efficiency is discussed in Section 3. Section 4 generalizes this approach for multiple output technologies with inefficiency. Section 5 concludes the paper. 2. Representations of the transformation function: a single output case Assume that a producer uses J inputs x to produce a single output y. The functional relationship between x and y is usually described by a production function f : RJþ ! Rþ where
y ¼ af ðxÞ;
ð1Þ
where a is the efficiency parameter (function). We write the functional relationship between x and y in a more general form, viz., Af(y, x) = 1, and call it a transformation function instead of a production function which becomes a special case. 1 Zellner et al. (1966) used expected profit maximization behavior to show that inputs are uncorrelated with the error term in the production function, although inputs are endogenously chosen by the producers. 2 Kumbhakar (2001) used profit maximizing behavior to address endogeneity of input and output variables and derived the profit function that includes both technical and allocative inefficiency. The econometric model is a profit system estimation which requires data on input and output prices together with profit and profit shares. Kumbhakar and Wang (2006) developed a primal system based on cost minimizing behavior. Data on input and output quantities as well as input prices are used to estimate such a system. The system approach used in these papers takes care of endogeneity of the decision variables. 3 There are many efficiency papers published in the operational research journals (for example, Bhattacharyya et al. (1997), Reinhard et al. (2000), Seiford and Zhu (2002), Brissimis et al. (2010), among many others). Since most of these papers use data envelopment analysis, which is a linear programming technique, there is no endogeneity issue in estimation even if some explicit economic behaviors are modeled. 4 For example, endogeneity issue is discussed in many papers under profit maximization behavior. Some recent papers are: Krusell et al. (2000), Coelli (2000), Kumbhakar (2001, 2009), among many others. Endogeneity in the context of a distance function using system approach under profit maximization behavior is discussed in Karagiannis (2006) and Coelli et al. (2007). 5 An alternative to this single equation approach is to develop a system approach similar to the system in Kumbhakar and Wang (2006) and Kumbhakar (2009). The first-order conditions of maximizing return to the outlay will be the other equations in this system. However, the system approach is econometrically challenging and as of now there are no canned software that can estimate such a system with inefficiency.
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Assume that lnf(y, x) can be represented by the translog (TL) function, i.e.,
X X 1 1XX ln f ðy; xÞ ¼ ay ln y þ ayy ln y2 þ bj ln xj þ bjk ln xj ln xk þ djy ln xj ln y; 2 2 j k j j which gives the following TL transformation function
1 2
a0 þ v þ ay ln y þ ayy ln y2 þ
X
bj ln xj þ
j
X 1XX b ln xj ln xk þ djy ln xj ln y ¼ 0; 2 j k jk j
ð2Þ
where bjk = bkj and lnA = a0 + v. The stochastic term v includes unobserved inputs such as management (managerial ability), random exogenous shocks, etc. Some of these are likely to be known to the producers while the others are not. Note that all the coefficients of the above transformation function cannot be identified, and some normalizing (identifying) restrictions on the parameters are necessary. If one uses the normalizations ay = 1, ayy = 0, djy = 0, "j = 1, . . . , J in (2) the standard translog production function is obtained, which is
Production function : ln y ¼ a0 þ
X
bj ln xj þ
j
1XX b ln xj ln xk þ v : 2 j k jk
ð3Þ
If we rewrite lnf(x, y) given above as
" # X X X 1 1XX 2 ln f ðy; xÞ ¼ ay ln y þ ayy ln y þ bj lnðxj =x1 Þ þ b lnðxj =x1 Þ lnðxk =x1 Þ þ djy lnðxj =x1 Þ ln y þ bj ln x1 2 2 j k jk j j j ( ) " # X X X þ bjk ln xj ln x1 þ djy ln y ln x1 ; j
j
k
P
P P and use the following normalizations j bj ¼ 1; j bjk ¼ 0; 8k; j djy ¼ 0, we get the familiar input distance function (IDF) representation of the transformation function in (2) (Baños-Pino et al. (2002), Coelli et al. (2005), among others), viz.,
IDF : ln x1 ¼ a0 þ
X
bj ln ^xj þ
j
X 1XX 1 bjk ln ^xj ln ^xk þ ay ln y þ ayy ln y2 þ djy ln ^xj ln y þ v ; 2 j k 2 j
ð4Þ
where ^xj ¼ xj =x1 ; j ¼ 2; . . . ; J. It is clear from the above that both the production and input distance functions are derived from the same transformation function using different normalizations. Thus, mathematically they are the same. This shows that the transformation function is more general and various representations of the technology can be obtained from it by using different normalizations. Furthermore, no behavioral assumptions are used in deriving the production and the input distance function representations. The critical question is: Does this mean that one can estimate the technology parameters of the transformation function consistently from any one of these specifications? This is where the econometric endogeneity issue comes in and the details are discussed in the next two sections. Since our primary focus is on endogeneity of inputs and outputs from economic point of view, we bring economic behavior explicitly into the model. This is because econometric argument for endogeneity depends on how the input and output variables appear in the regression function and what is the economic behavior of the producers. 3. Specification of the single output technology when returns to the outlay is maximized 3.1. Producers are fully efficient We assume that producers maximize return to the outlay (return to the dollar) with respect to y and x subject to the transformation function in (2).6 The Lagrangean of the problem is L = (py)/(w0 x) l(1 Af(y, x)) where l is the Lagrange multiplier. The first-order conditions (FOCs) of the problem with respect to xj and y are: wj(py/C2) + lfj(y, x)Ah = 0; and p/C + lAhfy(y, x) = 0, where fj(.) are partial derivatives of f(y, x) with respect to xj, fy(.) is partial derivative of f(y, x) with respect to y; and C = w0 x, w being the vector of input prices. These FOCs can be expressed as
ðpy=CÞ þ lf ðy; xÞA@ ln f ðy; xÞ=@ ln y ¼ 0;
ð5Þ
ðpy=CÞðwj xj =CÞ þ lAf ðy; xÞ@ ln f ðy; xÞ=@ ln xj ¼ 0; which can be rewritten as
@ ln f ðy; xÞ=@ ln xj @ ln f ðy; xÞ=@ ln y ¼ wj xj w0 x;
j ¼ 1; . . . ; J:
These FOCs along with the transformation function in (2) with some normalizations will give us (J + 1) equations which can be used to solve for input and output quantities. These solutions will naturally depend on v. Consequently, the regressors in the production function specification (3) will be correlated with the error term v, and therefore the OLS estimates from (3) will be inconsistent. The same logic applies for the IDF in (4). We explore this issue further in search for an ‘estimating equation’ that does not suffer from this econometric endogeneity problem. For this we use the FOCs in (5) and rewrite them as
6
The basic idea of this goes back to Georgescu-Roegen’s (1951) notion of ‘‘return to the dollar”.
S.C. Kumbhakar / European Journal of Operational Research 208 (2011) 170–176
@ ln f ðy; xÞ=@ ln y þ
X
@ ln f ðy; xÞ=@ ln xj ¼ 0;
j ¼ 1; . . . ; J;
173
ð6Þ
j
which is obtained after summing the last J equations in (5) over j and then adding it with the first equation. Using the FOCs in (6) the transformation function in (2) can be rewritten as
X j
( bj þ
X
bjk lnðxk =x1 Þ þ
X
k
) bjk ln x1
þ ay þ ayy ln y þ
X
djy lnðxj =x1 Þ þ
j
k
X
djy ln x1 ¼ 0;
ð7Þ
j
P P P P which after using the normalizations j bj ¼ 1; k bjk ¼ 0; 8j; j djy ¼ 0 implies ay 1 þ ayy ln y þ j djy lnðxj =x1 Þ ¼ 0. This restriction will hold for any lny and ln(xj/x1) if and only if ay = 1, ayy = 0 and djy = 0, "j. With these restrictions in place, the transformation function in (2) can be written as
lnðx1 =yÞ ¼ a0 þ
X
bj ln ^xj þ
j
1XX b ln ^xj ln ^xk þ v ; 2 j k jk
ð8Þ
where ^xj ¼ xj =x1 ; j ¼ 2; . . . ; J. This is our ‘estimating equation’. Now we need to show that the regressor (^ xj ) in (8) are exogenous (independent of v) so that (8) can be estimated using OLS. For this we use the FOCs in (6) and rewrite them as
P P P bj þ k bjk lnðxk =x1 Þ þ k bjk ln x1 b þ b ln ^xk wj xj wj ^xj ¼ j Pk jk P P ¼ ) : w1 x1 b1 þ k b1k lnðxk =x1 Þ þ k b1k ln x1 w1 b1 þ k b1k ln ^xk
ð9Þ
These equations can be implicitly solved for ^xj which will be independent of v. Thus the regressors in (8) can be treated as exogenous, and the model in (8) can be estimated using OLS. Note that (8) is not an IDF because lny does not appear as a regressor and the dependent variable is not lnx1. Since ln y does not appear as a regressor and ln ^ xj are independent of the error term there is no endogeneity problem in estimating (8) using OLS. On the contrary, if one starts from an IDF in (4) in which lny appears as a regressor the OLS estimates will certainly suffer from the endogeneity problem since lny is correlated with the error term. 3.2. Producers are technically inefficient So far we assumed that producers are fully efficient. The v term in the transformation (production) function was introduced to capture production uncertainty. If producers are inefficient we can write A as lnA = a0 u + v where u P 0 represents technical inefficiency. Alternatively, we can model inefficiency by rewriting the transformation function as Af(y k, x) = 1 where k P 1 is the efficiency factor, and therefore lnk = u P 0 is output-oriented technical inefficiency which shows the rate at which actual output could be increased, ceteris paribus. Viewed this way y* yk P y is potential output that could be produced using the same input vector. With this reformulation the Lagrangean of the problem is L = (py)/(w0 x) l(1 Af(yk, x)) where l is the Lagrange multiplier. The FOCs of the problem are: wj(py/C2) + lfj(yk, x)A = 0; and p/C + lkAfy(yk, x) = 0, where fj(.) are partial derivatives of f(yk, x) with respect to xj, fy(.) are partial derivatives of f(y, x) with respect to y and C = w0 x. These FOCs can be expressed as
ðpy=CÞ þ lf ðky; xÞA@ ln f ðky; xÞ=@ ln ky ¼ 0; ðpy=CÞðwj xj =CÞ þ lAf ðky; xÞ@ ln f ðky; xÞ=@ ln xj ¼ 0; X @ ln f ðky; xÞ=@ ln xj ¼ 0: ) @ ln f ðky; xÞ=@ lnðkyÞ þ
ð10Þ
j
The translog transformation function with inefficiency can be written as
1 2
a0 þ v þ ay ln y þ ayy ln y2 þ
X j
bj ln xj þ
X 1XX b ln xj ln xk þ djy ln xj ln y ¼ 0; 2 j k jk j
ð11Þ
where y* = yk. Using (11) along with the same normalizations as before, the FOCs in (10) can be expressed as ay 1 þ ayy ln y þ P j djy lnðxj =x1 Þ ¼ 0. This restriction will hold for any lny and ln (xj/x1) if ay = 1, ayy = 0 and djy = 0, "j. With these restrictions in place, the transformation function in (11) can be written as
lnðx1 =yÞ ¼ a0 þ
X
bj ln ^xj þ
j
1XX b ln ^xj ln ^xk þ u þ v ; 2 j k jk
ð12Þ
where u = lnk. This is our ‘estimating equation’ in the presence of technical inefficiency. To show that regressors in this equation are independent of u and v we rewrite the FOCs in (10) as
P b þ b ln ^xk wj ^xj ¼ j Pk jk ; w1 b1 þ k b1k ln ^xk
ð13Þ
which show that solutions of ^xj are independent of u and v. Thus, the regression function in (12) can be estimated using the standard stochastic frontier estimation technique. Note that (12) is not an IDF (the dependent variable is not lnx1), and there is no endogeneity problem associated with lny because it does not appear as a regressor in (12). The special representation of transformation function in (12) can be estimated using the stochastic frontier technique. In fact, any of the canned software that estimates stochastic frontier functions can be used to estimate (12). For details on estimation of inefficiency see Chapter 3 of Kumbhakar and Lovell (2000).
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4. Multi-output case with inefficiency Now we assume that a producer uses J inputs x to produce a vector of M outputs, y. The functional relationship between x and efficiency adjusted output y* = yk is expressed as Af(y*, x) = 1 which is the transformation function where k represents output-oriented technical inefficiency. We specify f(y*, x) as a TL function, i.e.,
ln f ðy ; xÞ ¼
X
am ln ym þ
m
X XX 1XX 1XX amn ln ym ln yn þ bj ln xj þ bjk ln xj ln xk þ dmj ln ym ln xj ; 2 m n 2 m j j j k
ð14Þ
where bjk = bkj and amn = anm. Thus, the multi-output TL transformation function can be written as
" # X X XX 1XX 1XX ^m a0 þ v þ am ln y^m þ amn ln y^m ln y^n þ bj ln ^xj þ bjk ln ^xj ln ^xk þ am ln y1 þ dmj ln ^xj ln y 2 2 m m n m m j j j k " # " # " # " # " # X X X X X X X XX þ amn ln y^m ln y1 þ dmj ln ^xj ln y1 þ bj ln x1 þ dmj ln ^xj ln y1 þ dmj ln y1 ln x1 X
m
n
j
m
" # " # 1 XX 1 XX 2 þ b ðln x1 Þ þ amn ln y1 2 ¼ 0; 2 j k jk 2 m n
j
j
k
j
m
ð15Þ
^m ¼ ym =y1 ¼ ym =y1 . where y The Lagrangean of the problem is L = (p0 y)/(w0 x) + l(1 Af(y*, x)) and the FOCs are: wj(R/C2) + lfj(y*, x)A = 0; and pm/C + lkAfm(y*, x) = 0, where fm(.) are partial derivatives of f(y*, x) with respect to ym; C = w0 x, and R = p0 y, p being the vector of output prices. These FOCs can be expressed as
ðpm ym =CÞ þ lf ðy ; xÞA@ ln f ðy ; xÞ=@ ln ym ¼ 0; ðR=CÞðwj xj =CÞ þ lf ðy ; xÞA@ ln f ðym ; xÞ=@ ln xj ¼ 0; X X @ ln f ðy ; xÞ=@ ln ym þ @ ln f ðy ; xÞ=@ ln xj ¼ 0: ) m
ð16Þ
j
P P P P The FOCs given by the last equation in (16) implies the following parametric restrictions: m am þ j bj ¼ 0, k bjk þ m dmj ¼ 0; 8j, and P P P P P a ¼ 0; 8m. These restrictions together with the following normalizing restrictions j bj ¼ 1; j dmj þ k bjk ¼ 0; 8j; j dmj ¼ 0; 8m, P n mn P imply m dmj ¼ 0; 8j and n amn ¼ 0; 8m. With these restrictions in place, the transformation function in (15) becomes
ln ~x1 ¼ a0 þ
X
am ln y^m þ
m
X 1XX 1XX amn ln y^m ln y^n þ bj ln ^xj þ b ln ^xj ln ^xk þ u þ v ; 2 m n 2 j k jk j
ð17Þ
where ~x1 ¼ x1 =y1 and u = lnk P 0. This is our ‘estimating equation’ which is neither an output nor an IDF. ^m and ^ Next we show that y xj are independent of u and v so that one can estimate (17) using the standard stochastic frontier technique. Using the transformation function in (15) along with the restrictions discussed above, the FOCs in (16) can be expressed as
P P am þ n amn ln y^n þ j dmj ln ^xj pm ^m ¼ P P ; y p1 a1 þ n a1n ln y^n þ j d1j ln ^xj P P ^m b þ b ln ^xk þ m dmj ln y wj ^xj ¼ j P k jk P : ^m w1 b1 þ k b1k ln ^xk þ m dm1 ln y
ð18Þ
^m ; x^j ; m ¼ 2; . . . ; M; j ¼ 2; . . . ; J solutions of which are independent of These M 1 + J 1 = M + J 2 equations can be implicitly solved for y u and v. That is, these variables can be treated as exogenous in the regression (17). The regression function in (17) is nothing but a cost frontier function which can be estimated using the canned frontier softwares. That is, the standard ML method that is routinely used to estimate a stochastic frontier function can be used here to estimate the parameters of the transformation function as well as technical inefficiency without worrying about endogeneity problem discussed in Guan et al. (2009). Note that it is possible to solve, at least implicitly, for optimal values of y and x using the FOCs in (16) and the transformation function in (15) together with the restrictions implied by the FOCs and normalizing constraints. Such solutions will depend on lnk and v. However, the ratios of inputs and outputs will be independent of them. This is confirmed from the FOCs in (18). So the conclusion is that although lnx and lny are endogenous both economically and econometrically, the input and output ratios are econometrically exogenous. This has implications for estimating the standard production, input and output distance functions, which we discuss next. To put this discussion into proper perspective, we start from the transformation function which is slightly more general than the one in (15). We do so to incorporate both input- and output-oriented technical inefficiency, which is necessary to discuss both input and output distance functions, in the transformation function. That is, we write the transformation function as Af(y*, x*) = 1, where y* = yk, x* = xh; k P 1 is output-oriented technical inefficiency (as before) and h 6 1 is input-oriented technical inefficiency. Finally, f(y*, x*) is assumed to be translog (TL), i.e., the TL transformation function is
a0 þ v þ
X m
am ln ym þ
X XX 1XX 1XX amn ln ym ln yn þ bj ln xj þ bjk ln xj ln xk þ dmj ln ym ln xj ¼ 0: 2 m n 2 j k m j j
The above transformation function is assumed to satisfy the following symmetry restrictions, viz., bjk = bkj and amn = anm.
ð19Þ
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One can use the following normalizations, viz., a1 = 1, a1n = 0, "n, d1j = 0, "j, h = 1, to the TL transformation in (19) to obtain a pseudo TL production function, viz.,
ln y1 ¼ a0 þ
X
bj ln xj þ
j
X XX 1XX 1XX b ln xj ln xk þ am ln ym þ amn ln ym ln yn þ dmj ln ym ln xj þ u þ v ; 2 j k jk 2 m¼2 m¼2 n¼2 m¼2 j
ð20Þ
P P P P P P P where u ¼ ln k 1 þ m¼2 am þ m¼2 n¼2 amn ln yn þ m¼2 j dmj ln xj þ 12 m¼2 n¼2 amn ðln kÞ2 .
We noted before that the output and input variables, which are the regressors in the above production function, are correlated with v and lnk. Thus, one cannot use the standard frontier technique to estimate the parameters in (20). Furthermore, u is a complicated function of output-oriented inefficiency, lnk, as well as inputs and outputs, and therefore it cannot be assumed to have a constant mean and variance. In other words, estimated parameters in (20) will be inconsistent, which in turn can generate meaningless estimates of inefficiency. If we rewrite (19) as
ln f ðy ; x Þ ¼
X XX 1XX 1XX amn lnðym =y1 Þ lnðyn =y1 Þ þ bj ln xj þ bjk ln xj ln xk þ dmj ln xj lnðym =y1 Þ 2 2 m m n m j j j k " # " # " # X X X X X þ am ln y1 þ amn ln ym ln y1 þ dmj ln xj ln y1 ;
X
am lnðym =y1 Þ þ
m
m
and use the following normalizations representation,7 viz.,
ln y1 ¼ a0 þ
X
bj ln xj þ
j
P
n
m
m
j
am ¼ 1;
P
n
amn ¼ 0; 8m;
P
m dmj
¼ 0; 8j; h ¼ 1, we obtain the TL output distance function (ODF)
X XX 1XX 1XX ^m þ u þ v ; b ln xj ln xk þ am ln y^m þ amn ln y^m ln y^n þ dmj ln xj ln y 2 j k jk 2 m m n m j
ð21Þ
^m ¼ ym =y1 ; m ¼ 2; . . . ; M. where u = lnk < 0, y Estimated parameters from the above ODF is likely to be inconsistent because as argued before the input variables will be correlated with the error term, v and u (via lnk), although the output variables appear in ratio form and uncorrelated with the error term. Based on this we conclude that ODF is likely to suffer from endogeneity problem coming from the input variables. In other words, estimated parameters of the ODF will be inconsistent which might produce meaningless estimates of inefficiency. If we rewrite (19) as
ln f ðy ; x Þ ¼
X XX 1XX 1XX amn ln ym ln yn þ bj lnðxj =x1 Þ þ bjk lnðxj =x1 Þ lnðxk =x1 Þ þ dmj lnðxj =x1 Þ 2 2 m m n m j j j k " # " # " # X X X X X ln ym þ bj ln x1 þ bjk ln xj ln x1 þ dmj ln ym ln x1 ;
X
am ln ym þ
j
and use the following normalizations
ln x1 ¼ a0 þ
X j
bj ln ^xj þ
j
P
j bj
¼ 1;
m
k
P
k bjk
¼ 0; 8j;
P
j dmj
ð22Þ
j
¼ 0; 8m; k ¼ 1, we get the TL IDF representation,8 viz.,
X XX 1XX 1XX b ln ^xj ln ^xk þ am ln ym þ amn ln ym ln yn þ dmj ln ^xj ln ym þ u þ v ; 2 j k jk 2 m m n m j
ð23Þ
where u ¼ ln h > 0; x^j ¼ xj =x1 ; j ¼ 2; . . . ; J. Similar to ODF, estimated parameters from the above IDF is likely to be inconsistent because the output variables will be correlated with both u and v, although the input variables appear in ratio form and uncorrelated with the error term. Based on this we conclude that IDF is likely to suffer from endogeneity problem coming from the output variables. Consequently, estimated inefficiency from IDF might not be trustworthy and meaningful. Now we view our estimating function in (17) in the light of the production, and input/output distance functions. The regressors in this equation are ratios of inputs and outputs which are exogenous when producers maximize return to the outlay. Thus it is neither an input nor an output distance function. Furthermore, it does not suffer from any of the problems in the production, input and output distance functions. We now summarize the main results from Section 4. These results also hold for the single output model. (i) If the producers are fully efficient technically the only change that one needs to make is to drop the u term in (17). The model can then be estimated by OLS. (ii) If producers are technically inefficient, one can use the stochastic frontier technique to estimate both the parameters of the transformation function and technical inefficiency. For details on estimation of technical inefficiency see Chapter 3 of Kumbhakar and Lovell (2000). (iii) The formulation in Section 4 also works if input-oriented inefficiency is used. With input-oriented inefficiency the transformation function is written as Af(y, h x) = 1, where h 6 1 is input efficiency factor. The translog version of it can be written exactly as (17) when u = lnh P 0 since lnh 6 0. That is, the estimating functions are identical irrespective of whether an input- or output-oriented measure of inefficiency is used.
7 Note that these normalizing constraints make the transformation function homogeneous of degree one in outputs. In the efficiency literature one starts from a distance function (which is the transformation function with inefficiency built in) and imposes linear homogeneity (in outputs) constraints to get the ODF. Here we get the same end-result without using the notion of a distance function to start with. 8 These normalizing constraints make the transformation function homogeneous of degree one in inputs. In the efficiency literature one defines the IDF as the distance (transformation) function which is homogeneous of degree one in inputs. Here we view the homogeneity property as identifying restrictions on the transformation function without using the notion of a distance function.
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P P (iv) The restrictions implied by the FOCs in (16), viz., m @ ln f ðy ; xÞ=@ ln ym þ j @ ln f ðy ; xÞ=@ ln xj ¼ 0, makes the underlying technology homogeneous returns to scale (RTS)). Note that RTS in terms of the transformation function is defined by nP of degree oneo(unitary P RTS ¼ (Panzar and Willig, 1977). j @ ln f ðy; xÞ=@ ln xj m @ ln f ðy; xÞ=@ ln ym (v) Instead of starting from the transformation function, if one starts from the IDF and imposes unitary RTS restrictions on it, the resulting estimating equation will be identical to (17). Under unitary RTS exogeneity of input ratios (in the IDF) can be justified from cost minimizing behavior and exogeneity of output ratios can be justified from revenue maximization behavior (Coelli, 2000). However, exogeneity of both input ratios and output ratios can be justified from maximizing return to the outlay. This exogeneity condition is important because the regressors in the estimating functions are all in ratio form. 5. Conclusions Based on the behavioral assumption of maximizing returns to the outlay, we derived the appropriate ‘estimating equation’ that can be used to estimate the parameters of the transformation function. The estimating equations are expressed in a way that the regressors are not correlated with the error term. This enables us to use OLS, especially when producers are fully efficient. If they are not fully efficient, the stochastic frontier technique can be used to estimate the parameters associated with the underlying technology and technical inefficiency for each producer. The stochastic frontier approach also requires regressors (input and output ratios in our case) to be uncorrelated with the error terms, which is guaranteed by the first-order conditions. It is worth noting that our estimating equations are neither an input nor an output distance function (popularly used in the literature). In both cases estimation of the models require data on only input and output quantities. Furthermore, the models are flexible enough to accommodate both single and multiple output flexible technologies (such as the translog). References Baños-Pino, J., Fernandez-Blanco, V., Rodriguez-Alvarez, A., 2002. The allocative efficiency measure by means of a distance function: The case of Spanish public railways. European Journal of Operational Research 137, 191–205. Bhattacharyya, A., Lovell, C.A.K., Sahay, P., 1997. The impact of liberalization on the productive efficiency of Indian commercial banks. European Journal of Operational Research 98, 332–345. Brissimis, S.N., Delis, M.D., Tsionas, E.G., 2010. Technical and allocative efficiency in European banking. European Journal of Operational Research 204, 153–163. Coelli, T.J., Rao, D.S.P., O’Donnell, C.J., Battese, G.E., 2005. An Introduction to Efficiency and Productivity Analysis, 2nd ed. Springer-Verlag, New York. Coelli, T.J., 2000. On the econometric estimation of the distance function representation of a production technology. Discussion Paper 2000/42. Center for Operations Research and Econometrics, Universite Catholique de Louvain. Coelli, T.J., Hajargasht, G., Lovell, C.A.K., 2007. Econometric estimation of an input distance function in a system of equations. working paper. Centre for Efficiency and Productivity Analysis, School of Economics, University of Queensland, Brisbane, Australia. Färe, R., Grosskopf, S., Zaman, O., 2002. Hyperbolic efficiency and return to the dollar. European Journal of Operational Research 136, 671–679. Fried, H., Lovell, C.A.K., Schmidt, S. (Eds.), 2008. The Measurement of Productive Efficiency and Productivity Growth. Oxford University Press, New York. Georgescu-Roegen, N., 1951. The aggregate linear production function and its applications to von Neumann’s economic model. In: Koopmans, T. (Ed.), Activity Analysis of Production and Allocation. Wiley, New York, pp. 98–115. Guan, Z., Kumbhakar, S.C., Myers, R., Oude-Lansink, A., 2009. Excess capital and its implications in econometric analysis of production. American Journal of Agricultural Economics 91, 765–776. Karagiannis, G., Tsionas, E.G., Kumbhakar, S.C., 2006. Estimation of input distance functions in the presence of technical and allocative inefficiency: A system approach. Working paper. State University of New York at Binghamton, New York. Krusell, P., Ohanian, L.E., Ríos-Rull, J.-V., Violante, G.L., 2000. Capital-skill complementarity and Inequality: A macroeconomic analysis. Econometrica 68, 1029–1053. Kumbhakar, S.C., Lovell, C.A.K., 2000. Stochastic Frontier Analysis. Cambridge University Press, Cambridge, New York. Kumbhakar, S.C., 2009. Estimation of production technology using primal approaches. Paper presented at the 4th annual New York Camp Econometrics, Lake Placid, NY. Kumbhakar, S.C., 2001. Estimation of profit function when profit is not maximum. American Journal of Agricultural Economics 83, 1–19. Kumbhakar, S.C., Wang, H.-J., 2006. Estimation of technical and allocative inefficiency: A primal system approach. Journal of Econometrics 134, 419–440. Panzar, J.C., Willig, R.D., 1977. Economies of scale in multi-output production. The Quarterly Journal of Economics 91, 481–493. Reinhard, S., Lovell, C.A.K., Thijssen, G.J., 2000. Environmental efficiency with multiple environmentally detrimental variables; estimated with SFA and DEA. European Journal of Operational Research 121, 287–303. Seiford, L.M., Zhu, J., 2002. Modelling undesirable factors in efficiency evaluation. European Journal of Operational Research 142, 16–20. Yu, M.-M., Fan, C.-K., 2008. The effects of privatization on return to the dollar: A case study on technical efficiency, and price distortions of Taiwan’s intercity bus services. Transportation Research Part A: Policy and Practice 42, 935–950. Zellner, A., Kmenta, J., Dreze, J., 1966. Specification and estimation of Cobb-Douglas production function models. Econometrica 34, 784–795. Zofío, J., Prieto, A., 2006. Return to dollar, generalized distance function and the Fisher productivity index. Spanish Economic Review 8, 113–138.
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Interfaces with Other Disciplines
Estimation of production technology when the objective is to maximize return to the outlay Subal C. Kumbhakar ⇑ Department of Economics, State University of New York – Binghamton, Binghamton, NY 13902, USA
a r t i c l e
i n f o
Article history: Received 16 March 2010 Accepted 10 September 2010 Available online 29 September 2010 Keywords: Distance function Transformation function Translog function Technical inefficiency
a b s t r a c t This paper deals with estimation of production technology where endogeneous choice of input and output variables is explicitly recognized. To address this endogeneity issue, we assume that producers maximize return to the outlay. We start from a flexible (translog) transformation function with a single output and multiple inputs and show how the first-order conditions of maximizing return to the outlay can be used to come up with an ‘estimating equation’ that does not suffer from the econometric endogeneity problem although the output and input variables are chosen endogenously. This is because the regressors in this estimating equation are in ratio forms which are uncorrelated with the error term under the assumption that producers maximize return to the outlay. The analysis is then extended to the multiple outputs and multiple inputs case with technical inefficiency. Although the estimating equations in both single and multiple output cases are neither production nor distance functions, they can be estimated in a straightforward manner using the standard stochastic frontier technique without worrying about endogeneity of the regressors. Thus, we provide a rationale for estimating the technology parameters consistently using an econometric model which requires data on only input and output quantities. 2010 Elsevier B.V. All rights reserved.
1. Introduction Given that resources are limited, economists emphasize on ‘optimal utilization’ of them. Optimality in standard microeconomics textbooks is mostly discussed in terms of cost minimization and profit maximization behaviors. Little or no attention is paid to analyzing increased cost associated with non-optimal allocation of resources (allocative inefficiency). In addition to being allocatively inefficient producers can be technically inefficient as well. That is, producers may not be producing the maximum possible output given inputs or they may be using more inputs than the minimum required to produce a given level of output. In the production efficiency literature the focus is on measuring potential output loss, increase in cost, decrease in profit, decrease in revenue, etc., resulting from either technical inefficiency or both technical and allocative inefficiency. For recent surveys on these issues see various chapters in Fried et al. (2008), Kumbhakar and Lovell (2000), Coelli et al. (2005), among others. Even if one is not interested in measuring efficiency per se, the idea of introducing behavioral assumption is important both economically and econometrically. Economic agents are inherently optimizers, and therefore the observed data are generated from the technology along with some specific behaviors on the part of the producers. These behaviors are not often recognized while estimating the model econometrically. Given the technology, levels of input and output variables are determined in such a way that the value of the objective function based on some economic behaviors is optimized. Thus, quantities of the choice variables are decided endogenously. For example, if producers choose inputs to minimize cost and outputs are exogenously given, we argue that inputs are endogenously determined by the producers. Some examples of these are: electricity, telephone, transportation, and other services, supply of which are determined by demand because these outputs cannot be stored. Consequently, the econometric models to estimate the production technology in the service industries treat outputs as exogenous and use the cost minimization objective to determine input quantities endogenously. Thus, economic behavior often tells us which variables are endogenous and which variables are exogenous. If one estimates the technology ignoring economic behavior the resulting estimates of the technology parameters are likely to be biased. As a result, when it comes to estimating the technology econometrically using a single equation method, we need to make sure that the economically endogenous variables do not cause econometric endogeneity problem. That is, the regressors in the estimating equation are not correlated with the error term no matter ⇑ Tel.: +1 607 777 4762; fax: +1 607 777 2681. E-mail address: [email protected] 0377-2217/$ - see front matter 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.09.015
S.C. Kumbhakar / European Journal of Operational Research 208 (2011) 170–176
171
whether one wants to estimate inefficiency or not and whether one uses the simple ordinary least squares (OLS) technique or more sophisticated methods like maximum likelihood (ML), generalized method of moments, etc. Thus, in estimating a production function the OLS and ML parameters will be inconsistent unless the inputs are uncorrelated with the error term.1 This is true for estimating any econometric relationship in which the regressors in the estimating function are to be uncorrelated with the error term to obtain consistent parameter estimates. In this paper we argue that input and output variables are decision (economically endogenous) variables. Although cost minimization and profit maximization behaviors are widely discussed in microeconomics, other objectives such as maximizing value of the firm, maximizing return on investment, maximizing sales (revenue), etc., are also quite popular in finance. Here we assume that producers maximize return to the outlay to choose optimal quantities of inputs and output(s), given the technology. That is, the decision variables are inputs and outputs, and are therefore endogenous from economic point of view. The question is whether this causes econometric endogeneity problem (correlation of the regressors with the error term) in the estimating equation2 which can be represented in various forms. We first show that this is indeed the case and then we show that the econometric endogeneity problem can be solved if the ‘estimating equation’ is expressed in a particular form in which the regressors are ratios of input and output quantities. This ‘estimating equation’ is thus neither a production nor a distance function. We are not aware of any paper that discusses this issue in a formal econometric model. Färe et al. (2002), Zofío and Prieto (2006), Yu and Fan (2008) use maximization of return to the dollar to decompose technical and allocative inefficiency using the distance function formulation which is estimated by a deterministic linear programming approach3 in which endogeneity issue does not arise. This is because enodogeneity issue is typical in econometric models, especially when economic behaviors are believed to affect the regressors (quantities of inputs and/or outputs in our case).4 We start by specifying the technology parametrically in the form of a transformation function. In particular, we use a flexible (translog) functional form. First, we consider a single output without inefficiency, which is then extended to accommodate inefficiency. Finally, we consider the technology with multiple outputs and multiple inputs with inefficiency. In all these cases we recognize economic endogeneity of input and output variables and come up with an ‘estimating equation’ which is neither a production function nor an input (output) distance function. The distinguishing feature of our ‘estimating equation’ is that it does not suffer from econometric endogeneity problem, thereby meaning that the regressors in the ‘estimating equation’ can be treated as exogenous in the sense that these are not correlated with the error term. Thus this function, for both single and multiple output cases, can be easily estimated by OLS when there is no inefficiency; and by the ML method in the presence of inefficiency. The models with inefficiency are expressed in the form of a stochastic cost frontier model, estimation details of which can be found in Kumbhakar and Lovell (2000) as well as in Coelli et al. (2005). To sum up, the main contribution of the paper is that we provide a strategy to estimate both single and multiple outputs production technologies using data on the input and output variables in a single equation framework. In doing so we treat output and input variables as endogenous in the sense that quantities of these variables are decided by the producers (not exogenously given) whose objective is to maximize return to the outlay. We start from a transformation function representation of technology and show that it can be expressed in many equivalent forms. These are different econometrically and one cannot use the standard econometric techniques (OLS or ML) to estimate the parameters consistently because the regressors are endogenous (correlated with the error terms in the estimating equations). Thus, one cannot use the standard production and distance function formulations unless the econometric endogeneity problem associated with the regressors is resolved first (Guan et al., 2009). We solve this endogeneity problem first by deriving a particular form of the estimating equation in which the regressors are ratios of inputs and outputs. We then assume that producers maximize return to the outlay to show that these ratios are uncorrelated with the error term in the estimating model, thereby solving the econometric endogeneity problem that arises in the standard input (output) distance function. Thus, this equation can be estimated consistently using either the OLS (without inefficiency) or the ML method (with inefficiency). Consequently, all the canned stochastic frontier softwares can be used to estimate the models with inefficiency proposed in this paper.5 The rest of the paper is organized as follows. The production technology in various forms is presented in Section 2. Conditions for maximizing returns to the outlay for the single output technology with and without full efficiency is discussed in Section 3. Section 4 generalizes this approach for multiple output technologies with inefficiency. Section 5 concludes the paper. 2. Representations of the transformation function: a single output case Assume that a producer uses J inputs x to produce a single output y. The functional relationship between x and y is usually described by a production function f : RJþ ! Rþ where
y ¼ af ðxÞ;
ð1Þ
where a is the efficiency parameter (function). We write the functional relationship between x and y in a more general form, viz., Af(y, x) = 1, and call it a transformation function instead of a production function which becomes a special case. 1 Zellner et al. (1966) used expected profit maximization behavior to show that inputs are uncorrelated with the error term in the production function, although inputs are endogenously chosen by the producers. 2 Kumbhakar (2001) used profit maximizing behavior to address endogeneity of input and output variables and derived the profit function that includes both technical and allocative inefficiency. The econometric model is a profit system estimation which requires data on input and output prices together with profit and profit shares. Kumbhakar and Wang (2006) developed a primal system based on cost minimizing behavior. Data on input and output quantities as well as input prices are used to estimate such a system. The system approach used in these papers takes care of endogeneity of the decision variables. 3 There are many efficiency papers published in the operational research journals (for example, Bhattacharyya et al. (1997), Reinhard et al. (2000), Seiford and Zhu (2002), Brissimis et al. (2010), among many others). Since most of these papers use data envelopment analysis, which is a linear programming technique, there is no endogeneity issue in estimation even if some explicit economic behaviors are modeled. 4 For example, endogeneity issue is discussed in many papers under profit maximization behavior. Some recent papers are: Krusell et al. (2000), Coelli (2000), Kumbhakar (2001, 2009), among many others. Endogeneity in the context of a distance function using system approach under profit maximization behavior is discussed in Karagiannis (2006) and Coelli et al. (2007). 5 An alternative to this single equation approach is to develop a system approach similar to the system in Kumbhakar and Wang (2006) and Kumbhakar (2009). The first-order conditions of maximizing return to the outlay will be the other equations in this system. However, the system approach is econometrically challenging and as of now there are no canned software that can estimate such a system with inefficiency.
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Assume that lnf(y, x) can be represented by the translog (TL) function, i.e.,
X X 1 1XX ln f ðy; xÞ ¼ ay ln y þ ayy ln y2 þ bj ln xj þ bjk ln xj ln xk þ djy ln xj ln y; 2 2 j k j j which gives the following TL transformation function
1 2
a0 þ v þ ay ln y þ ayy ln y2 þ
X
bj ln xj þ
j
X 1XX b ln xj ln xk þ djy ln xj ln y ¼ 0; 2 j k jk j
ð2Þ
where bjk = bkj and lnA = a0 + v. The stochastic term v includes unobserved inputs such as management (managerial ability), random exogenous shocks, etc. Some of these are likely to be known to the producers while the others are not. Note that all the coefficients of the above transformation function cannot be identified, and some normalizing (identifying) restrictions on the parameters are necessary. If one uses the normalizations ay = 1, ayy = 0, djy = 0, "j = 1, . . . , J in (2) the standard translog production function is obtained, which is
Production function : ln y ¼ a0 þ
X
bj ln xj þ
j
1XX b ln xj ln xk þ v : 2 j k jk
ð3Þ
If we rewrite lnf(x, y) given above as
" # X X X 1 1XX 2 ln f ðy; xÞ ¼ ay ln y þ ayy ln y þ bj lnðxj =x1 Þ þ b lnðxj =x1 Þ lnðxk =x1 Þ þ djy lnðxj =x1 Þ ln y þ bj ln x1 2 2 j k jk j j j ( ) " # X X X þ bjk ln xj ln x1 þ djy ln y ln x1 ; j
j
k
P
P P and use the following normalizations j bj ¼ 1; j bjk ¼ 0; 8k; j djy ¼ 0, we get the familiar input distance function (IDF) representation of the transformation function in (2) (Baños-Pino et al. (2002), Coelli et al. (2005), among others), viz.,
IDF : ln x1 ¼ a0 þ
X
bj ln ^xj þ
j
X 1XX 1 bjk ln ^xj ln ^xk þ ay ln y þ ayy ln y2 þ djy ln ^xj ln y þ v ; 2 j k 2 j
ð4Þ
where ^xj ¼ xj =x1 ; j ¼ 2; . . . ; J. It is clear from the above that both the production and input distance functions are derived from the same transformation function using different normalizations. Thus, mathematically they are the same. This shows that the transformation function is more general and various representations of the technology can be obtained from it by using different normalizations. Furthermore, no behavioral assumptions are used in deriving the production and the input distance function representations. The critical question is: Does this mean that one can estimate the technology parameters of the transformation function consistently from any one of these specifications? This is where the econometric endogeneity issue comes in and the details are discussed in the next two sections. Since our primary focus is on endogeneity of inputs and outputs from economic point of view, we bring economic behavior explicitly into the model. This is because econometric argument for endogeneity depends on how the input and output variables appear in the regression function and what is the economic behavior of the producers. 3. Specification of the single output technology when returns to the outlay is maximized 3.1. Producers are fully efficient We assume that producers maximize return to the outlay (return to the dollar) with respect to y and x subject to the transformation function in (2).6 The Lagrangean of the problem is L = (py)/(w0 x) l(1 Af(y, x)) where l is the Lagrange multiplier. The first-order conditions (FOCs) of the problem with respect to xj and y are: wj(py/C2) + lfj(y, x)Ah = 0; and p/C + lAhfy(y, x) = 0, where fj(.) are partial derivatives of f(y, x) with respect to xj, fy(.) is partial derivative of f(y, x) with respect to y; and C = w0 x, w being the vector of input prices. These FOCs can be expressed as
ðpy=CÞ þ lf ðy; xÞA@ ln f ðy; xÞ=@ ln y ¼ 0;
ð5Þ
ðpy=CÞðwj xj =CÞ þ lAf ðy; xÞ@ ln f ðy; xÞ=@ ln xj ¼ 0; which can be rewritten as
@ ln f ðy; xÞ=@ ln xj @ ln f ðy; xÞ=@ ln y ¼ wj xj w0 x;
j ¼ 1; . . . ; J:
These FOCs along with the transformation function in (2) with some normalizations will give us (J + 1) equations which can be used to solve for input and output quantities. These solutions will naturally depend on v. Consequently, the regressors in the production function specification (3) will be correlated with the error term v, and therefore the OLS estimates from (3) will be inconsistent. The same logic applies for the IDF in (4). We explore this issue further in search for an ‘estimating equation’ that does not suffer from this econometric endogeneity problem. For this we use the FOCs in (5) and rewrite them as
6
The basic idea of this goes back to Georgescu-Roegen’s (1951) notion of ‘‘return to the dollar”.
S.C. Kumbhakar / European Journal of Operational Research 208 (2011) 170–176
@ ln f ðy; xÞ=@ ln y þ
X
@ ln f ðy; xÞ=@ ln xj ¼ 0;
j ¼ 1; . . . ; J;
173
ð6Þ
j
which is obtained after summing the last J equations in (5) over j and then adding it with the first equation. Using the FOCs in (6) the transformation function in (2) can be rewritten as
X j
( bj þ
X
bjk lnðxk =x1 Þ þ
X
k
) bjk ln x1
þ ay þ ayy ln y þ
X
djy lnðxj =x1 Þ þ
j
k
X
djy ln x1 ¼ 0;
ð7Þ
j
P P P P which after using the normalizations j bj ¼ 1; k bjk ¼ 0; 8j; j djy ¼ 0 implies ay 1 þ ayy ln y þ j djy lnðxj =x1 Þ ¼ 0. This restriction will hold for any lny and ln(xj/x1) if and only if ay = 1, ayy = 0 and djy = 0, "j. With these restrictions in place, the transformation function in (2) can be written as
lnðx1 =yÞ ¼ a0 þ
X
bj ln ^xj þ
j
1XX b ln ^xj ln ^xk þ v ; 2 j k jk
ð8Þ
where ^xj ¼ xj =x1 ; j ¼ 2; . . . ; J. This is our ‘estimating equation’. Now we need to show that the regressor (^ xj ) in (8) are exogenous (independent of v) so that (8) can be estimated using OLS. For this we use the FOCs in (6) and rewrite them as
P P P bj þ k bjk lnðxk =x1 Þ þ k bjk ln x1 b þ b ln ^xk wj xj wj ^xj ¼ j Pk jk P P ¼ ) : w1 x1 b1 þ k b1k lnðxk =x1 Þ þ k b1k ln x1 w1 b1 þ k b1k ln ^xk
ð9Þ
These equations can be implicitly solved for ^xj which will be independent of v. Thus the regressors in (8) can be treated as exogenous, and the model in (8) can be estimated using OLS. Note that (8) is not an IDF because lny does not appear as a regressor and the dependent variable is not lnx1. Since ln y does not appear as a regressor and ln ^ xj are independent of the error term there is no endogeneity problem in estimating (8) using OLS. On the contrary, if one starts from an IDF in (4) in which lny appears as a regressor the OLS estimates will certainly suffer from the endogeneity problem since lny is correlated with the error term. 3.2. Producers are technically inefficient So far we assumed that producers are fully efficient. The v term in the transformation (production) function was introduced to capture production uncertainty. If producers are inefficient we can write A as lnA = a0 u + v where u P 0 represents technical inefficiency. Alternatively, we can model inefficiency by rewriting the transformation function as Af(y k, x) = 1 where k P 1 is the efficiency factor, and therefore lnk = u P 0 is output-oriented technical inefficiency which shows the rate at which actual output could be increased, ceteris paribus. Viewed this way y* yk P y is potential output that could be produced using the same input vector. With this reformulation the Lagrangean of the problem is L = (py)/(w0 x) l(1 Af(yk, x)) where l is the Lagrange multiplier. The FOCs of the problem are: wj(py/C2) + lfj(yk, x)A = 0; and p/C + lkAfy(yk, x) = 0, where fj(.) are partial derivatives of f(yk, x) with respect to xj, fy(.) are partial derivatives of f(y, x) with respect to y and C = w0 x. These FOCs can be expressed as
ðpy=CÞ þ lf ðky; xÞA@ ln f ðky; xÞ=@ ln ky ¼ 0; ðpy=CÞðwj xj =CÞ þ lAf ðky; xÞ@ ln f ðky; xÞ=@ ln xj ¼ 0; X @ ln f ðky; xÞ=@ ln xj ¼ 0: ) @ ln f ðky; xÞ=@ lnðkyÞ þ
ð10Þ
j
The translog transformation function with inefficiency can be written as
1 2
a0 þ v þ ay ln y þ ayy ln y2 þ
X j
bj ln xj þ
X 1XX b ln xj ln xk þ djy ln xj ln y ¼ 0; 2 j k jk j
ð11Þ
where y* = yk. Using (11) along with the same normalizations as before, the FOCs in (10) can be expressed as ay 1 þ ayy ln y þ P j djy lnðxj =x1 Þ ¼ 0. This restriction will hold for any lny and ln (xj/x1) if ay = 1, ayy = 0 and djy = 0, "j. With these restrictions in place, the transformation function in (11) can be written as
lnðx1 =yÞ ¼ a0 þ
X
bj ln ^xj þ
j
1XX b ln ^xj ln ^xk þ u þ v ; 2 j k jk
ð12Þ
where u = lnk. This is our ‘estimating equation’ in the presence of technical inefficiency. To show that regressors in this equation are independent of u and v we rewrite the FOCs in (10) as
P b þ b ln ^xk wj ^xj ¼ j Pk jk ; w1 b1 þ k b1k ln ^xk
ð13Þ
which show that solutions of ^xj are independent of u and v. Thus, the regression function in (12) can be estimated using the standard stochastic frontier estimation technique. Note that (12) is not an IDF (the dependent variable is not lnx1), and there is no endogeneity problem associated with lny because it does not appear as a regressor in (12). The special representation of transformation function in (12) can be estimated using the stochastic frontier technique. In fact, any of the canned software that estimates stochastic frontier functions can be used to estimate (12). For details on estimation of inefficiency see Chapter 3 of Kumbhakar and Lovell (2000).
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4. Multi-output case with inefficiency Now we assume that a producer uses J inputs x to produce a vector of M outputs, y. The functional relationship between x and efficiency adjusted output y* = yk is expressed as Af(y*, x) = 1 which is the transformation function where k represents output-oriented technical inefficiency. We specify f(y*, x) as a TL function, i.e.,
ln f ðy ; xÞ ¼
X
am ln ym þ
m
X XX 1XX 1XX amn ln ym ln yn þ bj ln xj þ bjk ln xj ln xk þ dmj ln ym ln xj ; 2 m n 2 m j j j k
ð14Þ
where bjk = bkj and amn = anm. Thus, the multi-output TL transformation function can be written as
" # X X XX 1XX 1XX ^m a0 þ v þ am ln y^m þ amn ln y^m ln y^n þ bj ln ^xj þ bjk ln ^xj ln ^xk þ am ln y1 þ dmj ln ^xj ln y 2 2 m m n m m j j j k " # " # " # " # " # X X X X X X X XX þ amn ln y^m ln y1 þ dmj ln ^xj ln y1 þ bj ln x1 þ dmj ln ^xj ln y1 þ dmj ln y1 ln x1 X
m
n
j
m
" # " # 1 XX 1 XX 2 þ b ðln x1 Þ þ amn ln y1 2 ¼ 0; 2 j k jk 2 m n
j
j
k
j
m
ð15Þ
^m ¼ ym =y1 ¼ ym =y1 . where y The Lagrangean of the problem is L = (p0 y)/(w0 x) + l(1 Af(y*, x)) and the FOCs are: wj(R/C2) + lfj(y*, x)A = 0; and pm/C + lkAfm(y*, x) = 0, where fm(.) are partial derivatives of f(y*, x) with respect to ym; C = w0 x, and R = p0 y, p being the vector of output prices. These FOCs can be expressed as
ðpm ym =CÞ þ lf ðy ; xÞA@ ln f ðy ; xÞ=@ ln ym ¼ 0; ðR=CÞðwj xj =CÞ þ lf ðy ; xÞA@ ln f ðym ; xÞ=@ ln xj ¼ 0; X X @ ln f ðy ; xÞ=@ ln ym þ @ ln f ðy ; xÞ=@ ln xj ¼ 0: ) m
ð16Þ
j
P P P P The FOCs given by the last equation in (16) implies the following parametric restrictions: m am þ j bj ¼ 0, k bjk þ m dmj ¼ 0; 8j, and P P P P P a ¼ 0; 8m. These restrictions together with the following normalizing restrictions j bj ¼ 1; j dmj þ k bjk ¼ 0; 8j; j dmj ¼ 0; 8m, P n mn P imply m dmj ¼ 0; 8j and n amn ¼ 0; 8m. With these restrictions in place, the transformation function in (15) becomes
ln ~x1 ¼ a0 þ
X
am ln y^m þ
m
X 1XX 1XX amn ln y^m ln y^n þ bj ln ^xj þ b ln ^xj ln ^xk þ u þ v ; 2 m n 2 j k jk j
ð17Þ
where ~x1 ¼ x1 =y1 and u = lnk P 0. This is our ‘estimating equation’ which is neither an output nor an IDF. ^m and ^ Next we show that y xj are independent of u and v so that one can estimate (17) using the standard stochastic frontier technique. Using the transformation function in (15) along with the restrictions discussed above, the FOCs in (16) can be expressed as
P P am þ n amn ln y^n þ j dmj ln ^xj pm ^m ¼ P P ; y p1 a1 þ n a1n ln y^n þ j d1j ln ^xj P P ^m b þ b ln ^xk þ m dmj ln y wj ^xj ¼ j P k jk P : ^m w1 b1 þ k b1k ln ^xk þ m dm1 ln y
ð18Þ
^m ; x^j ; m ¼ 2; . . . ; M; j ¼ 2; . . . ; J solutions of which are independent of These M 1 + J 1 = M + J 2 equations can be implicitly solved for y u and v. That is, these variables can be treated as exogenous in the regression (17). The regression function in (17) is nothing but a cost frontier function which can be estimated using the canned frontier softwares. That is, the standard ML method that is routinely used to estimate a stochastic frontier function can be used here to estimate the parameters of the transformation function as well as technical inefficiency without worrying about endogeneity problem discussed in Guan et al. (2009). Note that it is possible to solve, at least implicitly, for optimal values of y and x using the FOCs in (16) and the transformation function in (15) together with the restrictions implied by the FOCs and normalizing constraints. Such solutions will depend on lnk and v. However, the ratios of inputs and outputs will be independent of them. This is confirmed from the FOCs in (18). So the conclusion is that although lnx and lny are endogenous both economically and econometrically, the input and output ratios are econometrically exogenous. This has implications for estimating the standard production, input and output distance functions, which we discuss next. To put this discussion into proper perspective, we start from the transformation function which is slightly more general than the one in (15). We do so to incorporate both input- and output-oriented technical inefficiency, which is necessary to discuss both input and output distance functions, in the transformation function. That is, we write the transformation function as Af(y*, x*) = 1, where y* = yk, x* = xh; k P 1 is output-oriented technical inefficiency (as before) and h 6 1 is input-oriented technical inefficiency. Finally, f(y*, x*) is assumed to be translog (TL), i.e., the TL transformation function is
a0 þ v þ
X m
am ln ym þ
X XX 1XX 1XX amn ln ym ln yn þ bj ln xj þ bjk ln xj ln xk þ dmj ln ym ln xj ¼ 0: 2 m n 2 j k m j j
The above transformation function is assumed to satisfy the following symmetry restrictions, viz., bjk = bkj and amn = anm.
ð19Þ
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One can use the following normalizations, viz., a1 = 1, a1n = 0, "n, d1j = 0, "j, h = 1, to the TL transformation in (19) to obtain a pseudo TL production function, viz.,
ln y1 ¼ a0 þ
X
bj ln xj þ
j
X XX 1XX 1XX b ln xj ln xk þ am ln ym þ amn ln ym ln yn þ dmj ln ym ln xj þ u þ v ; 2 j k jk 2 m¼2 m¼2 n¼2 m¼2 j
ð20Þ
P P P P P P P where u ¼ ln k 1 þ m¼2 am þ m¼2 n¼2 amn ln yn þ m¼2 j dmj ln xj þ 12 m¼2 n¼2 amn ðln kÞ2 .
We noted before that the output and input variables, which are the regressors in the above production function, are correlated with v and lnk. Thus, one cannot use the standard frontier technique to estimate the parameters in (20). Furthermore, u is a complicated function of output-oriented inefficiency, lnk, as well as inputs and outputs, and therefore it cannot be assumed to have a constant mean and variance. In other words, estimated parameters in (20) will be inconsistent, which in turn can generate meaningless estimates of inefficiency. If we rewrite (19) as
ln f ðy ; x Þ ¼
X XX 1XX 1XX amn lnðym =y1 Þ lnðyn =y1 Þ þ bj ln xj þ bjk ln xj ln xk þ dmj ln xj lnðym =y1 Þ 2 2 m m n m j j j k " # " # " # X X X X X þ am ln y1 þ amn ln ym ln y1 þ dmj ln xj ln y1 ;
X
am lnðym =y1 Þ þ
m
m
and use the following normalizations representation,7 viz.,
ln y1 ¼ a0 þ
X
bj ln xj þ
j
P
n
m
m
j
am ¼ 1;
P
n
amn ¼ 0; 8m;
P
m dmj
¼ 0; 8j; h ¼ 1, we obtain the TL output distance function (ODF)
X XX 1XX 1XX ^m þ u þ v ; b ln xj ln xk þ am ln y^m þ amn ln y^m ln y^n þ dmj ln xj ln y 2 j k jk 2 m m n m j
ð21Þ
^m ¼ ym =y1 ; m ¼ 2; . . . ; M. where u = lnk < 0, y Estimated parameters from the above ODF is likely to be inconsistent because as argued before the input variables will be correlated with the error term, v and u (via lnk), although the output variables appear in ratio form and uncorrelated with the error term. Based on this we conclude that ODF is likely to suffer from endogeneity problem coming from the input variables. In other words, estimated parameters of the ODF will be inconsistent which might produce meaningless estimates of inefficiency. If we rewrite (19) as
ln f ðy ; x Þ ¼
X XX 1XX 1XX amn ln ym ln yn þ bj lnðxj =x1 Þ þ bjk lnðxj =x1 Þ lnðxk =x1 Þ þ dmj lnðxj =x1 Þ 2 2 m m n m j j j k " # " # " # X X X X X ln ym þ bj ln x1 þ bjk ln xj ln x1 þ dmj ln ym ln x1 ;
X
am ln ym þ
j
and use the following normalizations
ln x1 ¼ a0 þ
X j
bj ln ^xj þ
j
P
j bj
¼ 1;
m
k
P
k bjk
¼ 0; 8j;
P
j dmj
ð22Þ
j
¼ 0; 8m; k ¼ 1, we get the TL IDF representation,8 viz.,
X XX 1XX 1XX b ln ^xj ln ^xk þ am ln ym þ amn ln ym ln yn þ dmj ln ^xj ln ym þ u þ v ; 2 j k jk 2 m m n m j
ð23Þ
where u ¼ ln h > 0; x^j ¼ xj =x1 ; j ¼ 2; . . . ; J. Similar to ODF, estimated parameters from the above IDF is likely to be inconsistent because the output variables will be correlated with both u and v, although the input variables appear in ratio form and uncorrelated with the error term. Based on this we conclude that IDF is likely to suffer from endogeneity problem coming from the output variables. Consequently, estimated inefficiency from IDF might not be trustworthy and meaningful. Now we view our estimating function in (17) in the light of the production, and input/output distance functions. The regressors in this equation are ratios of inputs and outputs which are exogenous when producers maximize return to the outlay. Thus it is neither an input nor an output distance function. Furthermore, it does not suffer from any of the problems in the production, input and output distance functions. We now summarize the main results from Section 4. These results also hold for the single output model. (i) If the producers are fully efficient technically the only change that one needs to make is to drop the u term in (17). The model can then be estimated by OLS. (ii) If producers are technically inefficient, one can use the stochastic frontier technique to estimate both the parameters of the transformation function and technical inefficiency. For details on estimation of technical inefficiency see Chapter 3 of Kumbhakar and Lovell (2000). (iii) The formulation in Section 4 also works if input-oriented inefficiency is used. With input-oriented inefficiency the transformation function is written as Af(y, h x) = 1, where h 6 1 is input efficiency factor. The translog version of it can be written exactly as (17) when u = lnh P 0 since lnh 6 0. That is, the estimating functions are identical irrespective of whether an input- or output-oriented measure of inefficiency is used.
7 Note that these normalizing constraints make the transformation function homogeneous of degree one in outputs. In the efficiency literature one starts from a distance function (which is the transformation function with inefficiency built in) and imposes linear homogeneity (in outputs) constraints to get the ODF. Here we get the same end-result without using the notion of a distance function to start with. 8 These normalizing constraints make the transformation function homogeneous of degree one in inputs. In the efficiency literature one defines the IDF as the distance (transformation) function which is homogeneous of degree one in inputs. Here we view the homogeneity property as identifying restrictions on the transformation function without using the notion of a distance function.
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P P (iv) The restrictions implied by the FOCs in (16), viz., m @ ln f ðy ; xÞ=@ ln ym þ j @ ln f ðy ; xÞ=@ ln xj ¼ 0, makes the underlying technology homogeneous returns to scale (RTS)). Note that RTS in terms of the transformation function is defined by nP of degree oneo(unitary P RTS ¼ (Panzar and Willig, 1977). j @ ln f ðy; xÞ=@ ln xj m @ ln f ðy; xÞ=@ ln ym (v) Instead of starting from the transformation function, if one starts from the IDF and imposes unitary RTS restrictions on it, the resulting estimating equation will be identical to (17). Under unitary RTS exogeneity of input ratios (in the IDF) can be justified from cost minimizing behavior and exogeneity of output ratios can be justified from revenue maximization behavior (Coelli, 2000). However, exogeneity of both input ratios and output ratios can be justified from maximizing return to the outlay. This exogeneity condition is important because the regressors in the estimating functions are all in ratio form. 5. Conclusions Based on the behavioral assumption of maximizing returns to the outlay, we derived the appropriate ‘estimating equation’ that can be used to estimate the parameters of the transformation function. The estimating equations are expressed in a way that the regressors are not correlated with the error term. This enables us to use OLS, especially when producers are fully efficient. If they are not fully efficient, the stochastic frontier technique can be used to estimate the parameters associated with the underlying technology and technical inefficiency for each producer. The stochastic frontier approach also requires regressors (input and output ratios in our case) to be uncorrelated with the error terms, which is guaranteed by the first-order conditions. It is worth noting that our estimating equations are neither an input nor an output distance function (popularly used in the literature). 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