Estimation of Input Distance Functions: A System Approach

Transcription

1 MPRA Munich Personal RePEc Archive Estiation of Input Distance Functions: A Syste Approach Efthyios Tsionas and Subal C. Kubhakar and Eir Malikov Athens University of Econoics and Business, State University of New York at Binghaton, St. Lawrence University 015 Online at MPRA Paper No. 639, posted 5. February :57 UTC

2 Estiation of Input Distance Functions: A Syste Approach Efthyios G. Tsionas 1 Subal C. Kubhakar Eir Malikov 3 1 Departent of Econoics, Athens University of Econoics and Business, Athens, Greece Departent of Econoics, State University of New York at Binghaton, Binghaton, NY 3 Departent of Econoics, St. Lawrence University, Canton, NY First Draft: Septeber 11, 01 This Draft: February 17, 015 Abstract This article offers a ethodology to address the endogeneity of inputs in the input distance function (IDF) forulation of the production processes. We propose to tackle endogenous input ratios appearing in the noralized IDF by considering a flexible (siultaneous) syste of the IDF and the first-order conditions fro the fir s cost iniization proble. Our odel can accoodate both technical and (input) allocative inefficiencies aongst firs. We also present the algorith for quantifying the cost of allocative inefficiency. We showcase our costsyste-based odel by applying it to study the production of Norwegian dairy fars during the period. Aong other things, we find both an econoically and statistically significant iproveent in the levels of technical efficiency aong dairy fars associated with the 1997 quota schee change, which a ore conventional single-equation stochastic frontier odel appears to be unable to detect. Keywords: Cost Miniization, Dairy Production, Dairy Quota, Endogeneity, Input Distance Function, Stochastic Frontier JEL Classification: C33, D4, Q1 We would like to thank the editor, Brian Roe, and two anonyous referees for any insightful coents and suggestions that helped iprove this article. Any reaining errors are our own. Eail: tsionas@aueb.gr (Tsionas), kkar@binghaton.edu (Kubhakar), ealikov@stlawu.edu (Malikov).

3 Introduction The identification and estiation 1 of production functions using data on inputs and outputs is aong the oldest epirical probles in econoics dating back at least as early as the 19th century (Chabers, 1997). While production function forulation appears to be a natural fraework of odeling single-output production processes, distance functions (Shephard, 1953, 1970) are ore popular aong researchers in the instances of ultiple-output ultiple-input production. The priary benefit of odeling the production process via distance functions is that it frees researchers fro having to ake a judgent call of deterining which output (input) is to be treated as a lefthand-side endogenous variable in the production function (input requireent function) since the results are not invariant to these choices. However, like the production functions (Marschak and Andrews, 1944), distance functions suffer fro an inherent endogeneity proble if inputs and/or outputs are endogenous to the fir s decision-aking. When estiating the input distance function (IDF) researchers explicitly or iplicitly assue that outputs are exogenous to the fir s input allocation decisions. Such an assuption is usually justified by either the preise of cost iniizing behavior by firs (e.g., Kubhakar et al., 008) or the structure of the industry (e.g., Das and Kubhakar, 01). However, the endogeneity 3 of the input ratios, which enter the IDF under the linear hoogeneity noralization, is often assued away with little or no justification (e.g., Labert and Wilson, 003; Karagiannis et al., 004; Atsbeha et al., 01). Since the assuption of exogenous outputs appears to be soewhat ore reasonable 4 than that of exogenous inputs required by the output distance function (ODF) [an alternative to the IDF], in this article we focus on the issues related to the estiation of the IDF only. In this article, we propose addressing the endogeneity of inputs in the IDF forulation of the production process fro the perspective of the econoic theory. More specifically, we suggest invoking the assuption of the fir s cost iniizing behavior not only to justify the treatent of outputs as exogenous but to also tackle the endogeneity of the input ratios in the IDF. We do so by augenting the IDF of a flexible translog for with the set of independent first-order conditions (FOCs) fro the cost iniization proble, which we then estiate as a syste of siultaneous equations via full-inforation axiu likelihood (FIML). Our identification strategy relies on copetitively deterined input prices as a source of exogenous variation. The odel we develop also accoodates both technical and (input) allocative inefficiencies aongst firs. We show that in the presence of allocative disturbances, the input ratios are functions of not only input prices and outputs but also allocative inefficiencies. Consequently, if allocative inefficiencies are correlated with technical inefficiency and/or a rando productivity shock in the IDF, the standard single-equation ethods applied to the IDF are likely to produce inconsistent estiates of both the production technology and stochastic inefficiency. Even if the input ratios are uncorrelated with the coposite error ter in the IDF, there are advantages to using a syste approach. First, since additional equations (the FOCs) do not contain any extra paraeters, the syste-based 1 Throughout this article, we focus on the estiation of stochastic production processes only. Our discussion of endogeneity and ethods of tackling it does not apply to the fitting of deterinistic production and/or distance functions by the definition of the latter. Färe and Priont (1995) deal with theoretical underpinnings of the ulti-output ulti-input distance function odels. However, since they provide little coverage of the estiation of distance functions, the endogeneity issue never arises in their discussion. 3 That is, the correlation with the error ter as a result of the siultaneity of the input allocation. 4 The cost iniization preise is in line with the econoic theory. Furtherore, it ay be relatively easier to justify exogeneity of outputs in heavily regulated industries, such as airlines or electric power distribution, since outputs are clearly not firs choice variables in these instances. Adittedly, the endogeneity in outputs ay still continue to persist even in the case of such regulated industries should there be an oitted variable that correlates with output quantities included in the regression equation. We thank an anonyous referee for pointing this out.

4 estiator is ore efficient. Second, technological etrics obtained via the cost syste approach are likely to be ore eaningful because the econoic behavior is ebedded into the syste through the FOCs. The endogeneity in the stochastic IDF has also been considered in a soewhat different fraework by Atkinson and Priont (00) and Atkinson et al. (003). The two papers propose addressing the endogeneity proble using Generalized Method of Moents (GMM). In the case of a single-equation IDF, Atkinson et al. (003) suggest using input (and possibly output) prices to instruent for inputs (and outputs). Siilar to our article, Atkinson and Priont (00) estiate a cost syste of the IDF where, consistent with the cost iniization assuption, input prices and output quantities are instead used as instruents. In practice, the GMM approach is applicable only if technical inefficiency is to be treated as a fixed-effect type inefficiency. That is, one needs to assue technical inefficiency to be either tie-invariant or paraeterized as a fir-specific function of tie in the spirit of Cornwell et al. (1990). In this article, we explore an alternative approach to odeling technical inefficiency, where the latter is treated as a tie-varying rando effect in the spirit of a stochastic frontier analysis. Hence, unlike Atkinson and Priont s (00) GMM forulation, we estiate the syste of the IDF and cost-iniizing FOCs via FIML. We note that the estiation of the IDF via a FIML syste approach has also been discussed in several related papers. For instance, Karagiannis et al. (006) consider a syste of the IDF along with share equations where allocative inefficiency is defined in ters of shadow input prices. Other studies whose priary interest lies in quantifying allocative errors (as opposed to resolving the endogeneity issue) via a syste approach using share equations have been found to still suffer fro the endogeneity proble (see Coelli et al., 008, for an excellent review). Our article differs fro the earlier work in at least two respects. As opposed to considering share equations, our prial syste contains the FOCs (in explicit for) where we use observed input prices. 5 Further, we odel allocative inefficiency in the spirit of Schidt and Lovell (1978), i.e., in ters of input quantities as opposed to input prices. 6 Our contribution to the literature is threefold. First, we offer a flexible syste approach to a consistent estiation of the IDF based on the (behavioral) cost iniization assuption consistent with the econoic theory, while also allowing for a rando-effect type technical inefficiency. The odel we develop also accoodates allocative inefficiency. Second, we present the ethodology for quantifying the cost of allocative inefficiency obtained fro the prial IDF syste. Lastly, we showcase our odel by applying it to study the production technology of Norwegian dairy fars during the period. Production Technology under Cost Miniization Consider a production process in which K inputs X R K + are transfored into M outputs Y R M + via the production technology described by Af(θX, λy) = 1, (1) where f( ) is the transforation function, and A captures observed and unobserved factors which affect the transforation function neutrally. We specify the production technology in ters of the transforation function f( ) because the latter is ore general than the production/distance/input requireent function (e.g., see Caves et al., 1981). Here, scalars θ 1 and λ 1 capture input 5 The unavailability of input prices is the priary reason why Karagiannis et al. (006) and others consider a syste of share equations, which are not based on the cost iniizing FOCs unlike our approach. 6 This also differentiates our cost syste fro that considered by Atkinson and Priont (00). 3

5 and output technical inefficiency, respectively. For instance, if θ = 0.9, inputs are said to be 90% efficient, i.e., the use of each input could be reduced by 10% without reducing outputs if inefficiency is eliinated. Siilarly, if λ is 1.1, each output could be increased by 10% without increasing the use of any input if inefficiency is eliinated. Since in general θ and λ are not jointly identified, researchers consider either of the three special cases (depending on the choice of noralization): (i) input-oriented technical inefficiency θ < 1 in the case of λ = 1; (ii) output-oriented technical inefficiency λ > 1 in the case of θ = 1; and (iii) hyperbolic technical inefficiency corresponding to the case of λθ = 1 which seeks a siultaneous expansion in outputs and a reduction in inputs at the sae rate. All three are related to one another (e.g., see Färe et al., 00; Kubhakar, 013). In this article, we focus on the first case, i.e., identifying and consistently estiating the input-oriented technical inefficiency. Specifically, defining Ŷ λy and X θx, we can rewrite the transforation function (1) in the logarithic for, i.e., ln A + ln f(ŷ, X) = 0. () We assue that ln f( ) takes the translog functional for, i.e., ln f(ŷ, X) = α ln Ŷ + 1 α n ln Ŷ ln Ŷn + β k ln X k + n k 1 β kl ln X k ln X l + γ k ln Ŷ ln X k, (3) k l which satisfies the following syetry restrictions: β kl = β lk and α n = α n. Note that we cannot identify, and hence estiate, all paraeters in (3) (even after iposing the syetry restrictions). Specifically, there are (M + K + ) unidentified paraeters. We therefore need to ipose identifying restrictions on the transforation function (3). We first rewrite (3) as ln f(ŷ, X) = α ln Ŷ + 1 α n ln Ŷ ln Ŷn + β k ln(x k /X 1 ) + n k= 1 β kl ln(x k /X 1 ) ln(x l /X 1 ) + γ k ln Ŷ ln(x k /X 1 ) + k= l= k= [ ] β k ln X 1 + [ ] β kl ln X k ln X 1 + [ ] γ k ln Ŷ ln X 1. (4) k k l Iposing the following (M + K + ) noralizations on (4): β k = 1; k k β kl = 0 k l γ k = 0 ; λ = 1 (5) k yields an identified IDF representation of the production technology (), i.e., ln X 1 = α 0 + β k ln X k + 1 β kl ln X k ln X l + α ln Y k= k= l= 1 α n ln Y ln Y n + γ k ln Y ln X k u + v 1, (6) n k= k 4

6 where X k X k /X 1 k =,..., K, ln A = α 0 + v 1, and u ln θ 0 easures technical inefficiency. Note that the noralizations in (5) except for λ = 1 are ost coonly referred to as the linear hoogeneity (in inputs) property of IDF. The reaining restriction λ = 1 provides an input-oriented interpretation of the inefficiency ter u. Cost Miniization with No Allocative Inefficiency Under the assuption of cost-iniizing behavior, according to which outputs and copetitively deterined input prices are exogenous to the fir s input allocation, the fir s objective is defined as in X W X : Af( X, Ŷ) = 1, (7) where W R K + is a vector of input prices. The above objective function yields (K 1) independent first-order conditions (FOCs): ) W k X ln f (Ŷ, X / ln X k k = ) k =,..., K. (8) W 1 X 1 ln f (Ŷ, X / ln X 1 Define, for convenience, Xk X k /X 1. Given the IDF representation of the production technology in (6), the FOCs (8) take the following for: W k W 1 Xk = β k + l= β kl ln X l + γ k ln Y 1 k= β k 1 k= l= β kl (ln X k + ln X ) l k= γ k ln Y k =,..., K, (9) which can be rewritten in a ore copact for using the identifying restrictions in (5), i.e., W k Xk = β k + l= β kl ln X l + γ k ln Y W 1 β 1 + l= β 1l ln X l + γ k =,..., K. (10) 1 ln Y We note that equation (10) by no eans iplies that paraeters β 1, β 1l and γ 1 are identified and can be estiated. Rather, they can be recovered using the (deterinistic) identifying restrictions in (5). The syste of the above FOCs along with the translog IDF in (6) can be used to solve for the input deand functions conditional on outputs. These solutions will be functions of input price ratios and outputs only. Since both W and Y are assued to be exogenous in a cost iniizing fraework, the input ratios X k k =,..., K are not affected by either the inefficiency u or stochastic productivity shock v 1 (which is absorbed in paraeter A). Hence, one can argue in support of treating the regressors in IDF (6) as exogenous (in the absence of allocative disturbances) thereby justifying the use of standard stochastic frontier ethods in order to estiate production technology and inefficiency via a single-equation IDF. 7 Cost Miniization with Allocative Inefficiency The cost iniization fraework can be relaxed to allow for the presence of allocative inefficiencies, i.e., optiization errors. Following Schidt and Lovell (1978), we augent the FOCs in (10) with 7 Coelli (000) reaches the sae conclusion. 5

7 allocative disturbances, i.e., W k Xk exp(v k ) = β k + l= β kl ln X l + γ k ln Y W 1 β 1 + l= β 1l ln X l + γ 1 ln Y k =,..., K, (11) where v k R k =,..., K represents the allocative inefficiency for the input pair (X k, X 1 ). Unlike in the case of full allocative efficiency (as considered in the previous subsection), the solution to the syste of the FOCs in (11) along with the IDF can be shown to yield conditional input deand functions that are functions of not only input price ratios and outputs but also allocative inefficiencies. Consequently, if the allocative disturbances v k k =,..., K are correlated with technical inefficiency u and/or the productivity shock v 1 in the IDF, the standard stochastic frontier ethods applied to a single-equation IDF (6) would produce inconsistent estiates of both the technology and inefficiency. Even if the input ratios X k k =,..., K are uncorrelated with both error ters in the IDF, thereby iplying that allocative inefficiencies are uncorrelated with both technical inefficiency and the productivity shock (stochastic noise in the IDF), there are advantages to using a syste approach. First, since additional equations (the FOCs) do not contain any extra paraeters, the syste-based paraeter estiates are likely to be ore precise. Second, technological etrics obtained based on the estiated IDF are likely to be ore eaningful because the econoic behavior is ebedded into the syste through the FOCs. That is, if one believes in the econoic behavior of the producers, the FOCs ought to be used in the estiation. However, firs ay pursue econoic objectives that differ fro the cost iniization. One such exaple is revenue axiization, under which outputs are endogenous to the firs decisionaking whereas inputs are exogenous. Under this scenario, for a syste approach to reain valid, one solely needs to replace inputs with outputs and vice versa. That is, the IDF needs to be replaced with an equivalent ODF based representation of the production process, and the FOCs are to be derived for outputs fro the revenue axiization proble. Mechanically, the latter iplies switching X and Y in our discussion above. If both inputs and outputs are endogenous to the fir s decisions, then the endogeneity proble reains no atter whether one estiates a single-equation IDF or the cost syste we propose in this article. Given that the joint endogeneity of inputs and outputs corresponds to profit-axiizing behavior by firs, a plausible identification strategy would be to augent the IDF with the FOCs obtained fro the profit axiization proble. We do not consider such an instance here, leaving the latter for a different paper. Econoetric Model We consider the econoetric odel consisting of the translog IDF (6) and the FOCs with allocative inefficiency in (11). Assuing the existence of panel data and denoting the log values of variables by their lower-case versions (except for the tie trend), e.g., x k ln X k, the odel (syste) takes the following for: x 1,it = α 0 + β k x k,it + 1 α y,it + k= 1 β kl x k,it x l,it + k= l= α n y,it y n,it + γ k y,it x k,it + n δ t t + 1 δ ttt + k= k= ϕ k x k,it t + ψ y,it t u it + v 1,it (1a) 6

8 ( x k,it + ln β k + β kl x l,it + l= ) ( γ k y,it + ϕ k t ln β 1 + β 1l x l,it + l= γ 1 y,it + ϕ 1 t ) = w k,it + v k,it k =,..., K, (1b) where, for convenience, we have denoted w k,it = ln (W k,it /W 1,it ); variable t is the tie trend to allow for technical change which captures a teporal shift of the production frontier; and subscripts i = 1,..., N and t = 1,..., T index firs and tie periods, respectively. Under the assuptions of copetitive factor arkets and the cost-iniizing behavior by firs, which are said to iniize cost subject to given outputs and input prices, the syste in (1) is exactly identified. Specifically, the nuber of endogenous variables in our case, an input x 1 and (K 1) input ratios x k k =,..., K equals the nuber of independent siultaneous equations, i.e., (K 1) FOCs and the IDF itself. Hence, the order condition is satisfied. In addition to exogenous variation generation by K copetitively deterined input prices W k k = 1,..., K, the distributional assuption about a K-variate vector of stochastic disturbances v it provides an additional source of identification. Our identification strategy can also be explained in a ore conventional instruental variable (IV) fraework. In particular, K endogenous inputs X are being instruented for by K exogenous input prices W, whereas exogenous outputs Y and a tie trend t instruent for theselves. Estiation We estiate the cost syste in (1) using the FIML ethod. To ipleent it, we assue v it = [v 1,it, v,it,..., v K,it ] [ v 1,it, v it ] i.i.d. N(0, Σ). (13) The first issue we need to deal with during the estiation is the non-unit Jacobian of the transforation when x it = [x 1,it, x,it,..., x K,it ] [x 1,it, x it ] are the endogenous variables. Then the Jacobian of the transforation fro v it to x it is equal to the Jacobian fro v it to x it since v 1,it x 1,it define [ = 1, and v k,it x 1,it = 0 k =,..., K. To express the Jacobian in a ore copact for, we β k + β kl x l,it + ] 1 γ k y,it + ϕ k t A k,it, k =,..., K; a it [A,it,..., A K,it ] l= B [β kl ; k, l =,..., K] ; b [β 1k ; k =,..., K] Q it a it ι K 1; R ι K 1 b, where ι K 1 is the row vector of ones with (K 1) eleents. Using the above notation we can write the Jacobian as J it (θ, x it, y it ) = B Q it A 1,it R I K 1, (14) where I K 1 is the identity atrix of a (K 1) diension. While the Jacobian is easy to copute in this for, it is still a function of all paraeters as well as a function of the data. The density function of endogenous variables x it conditional, aong other things, on technical inefficiency u it can be written as f (x it θ, Σ, u it, Ξ) = (π) K/ Σ 1/ exp { 1 } ξ itσ 1 ξ it J it (θ, x it, y it ), (15) 7

9 [ uit where ξ it v it 0 K 1 ], Ξ denotes the data (including input prices), and θ collectively denotes the paraeters α, α n, β k, β kl, γ k and ϕ k k, l =,..., K; = 1,..., M. Next, we write the syste in (1) in the following general notation: F it (θ) = v it u it j K ξ it, (16) where j K = [1, 0,..., 0] is a (K 1) vector, and the eleents of F it = [F 1,it, F,it,..., F K,it ] are given by F 1,it x 1,it α 0 β k x k,it 1 β kl x k,it x l,it α y,it k= k= l= 1 α n y,it y n,it γ k y,it x k,it n δ t t 1 δ ttt k= ( F k,it x k,it + ln ln ( β 1 + l= k= ϕ k x k,it t β k + l= β 1l x l,it + β kl x l,it + ψ y,it t γ 1 y,it + ϕ 1 t γ k y,it + ϕ k t ) ) (17a) w k,it k =,..., K. (17b) Foral integration with respect to u it in (15) produces a density that will be unconditional of technical inefficiency. For the sake of exposition, suppose u it i.i.d. N + ( 0, σ u ) independently of the data and the eleents of v it. 8 Then, it is straightforward to show that f (x it θ, Σ, Ξ) = (π) K/ Σ 1/ (σ u σ ) 1 Φ (û it /σ ) { exp 1 ( F itσ 1 F it σ ( F it Σ 1 ) ) } ι K J it (θ, x it, y it ), (18) where σ σu 1+σuι K Σ 1 ι K, û it σ F it Σ 1 ι K, and Φ( ) is the standard noral distribution function. Note that F it Σ 1 F it is a failiar su of squares fro a standard treatent of the nonlinear siultaneous equations odel or nonlinear seeingly unrelated regressions. The density in (18) is used to obtain the likelihood function which is axiized with respect to (θ, Σ). In (18), the ter F it Σ 1 F it σ ( F it Σ 1 ) ι K = F it Σ 1 MF it, where M I K σ J K Σ 1 and J K is a square atrix of diension K, all eleents of which are equal to unity. Although foral concentration with respect to Σ 1 is not feasible, we can use the Cholesky decoposition Σ 1 = C C and treat the non-zero eleents of C below the ain diagonal as unrestricted paraeters. Further, we can show that u it θ, Σ, Ξ N + (ûit, σ ) for which we extend the Jondrow et al. (198) forula to obtain observation-specific estiates of technical inefficiency E [u it θ, Σ, Ξ] which we can copute using properties of the truncated noral distribution. In our discussion of the syste approach to the (consistent) estiation of the IDF, we odel technical inefficiency u it as a rando-effect type inefficiency in the spirit of the stochastic frontier analysis (Kubhakar and Lovell, 000). However, given the IV interpretation of our identification strategy outlined in the previous subsection, one ay wonder if syste (1) can be alternatively ( ) 8 In our application, we consider a ore general case of u it i.i.d. N + µu,it, σu allowing the ean of inefficiency to vary with the dairy quota regie in Norway. For ore details, see application. 8

10 estiated via GMM as done in Atkinson and Priont (00). The latter is eaningful only if u it is to be treated as a fixed-effect type technical inefficiency. That is, u it is to be assued to be either tie-invariant ( the fixed effect) or paraeterized as a fir-specific function of tie in the spirit of Cornwell et al. (1990). For instance, Atkinson and Priont (00) pursue the latter route. The rando-effect treatent of (one-sided) technical inefficiency would however require a two-stage GMM procedure, in which u it is to be recovered via axiu likelihood (ML) under the distributional assuption about stochastic disturbances. The two stages can surely be cobined into a one-step ultiple-equation GMM fraework by augenting the IDF by the log-likelihood score functions fro the second stage. Regardless whether a one or two-step GMM procedure is eployed, it would still require a distributional assuption about u it and the rando productivity shocks. Given the asyptotic equivalence of ML and GMM under this scenario, eploying ML has the benefit of superior efficiency. Hence, in the stochastic frontier fraework (as in this article), the use of FIML which we opt for sees to be optial. Coputing Cost of Allocative Inefficiency Given syste (1) where we allow for allocative disturbances, we can also copute the cost of allocative inefficiency (CAI) which we define as the predicted difference between actual and frontier cost, coputed as a fraction of the predicted frontier cost. Both predicted actual and frontier costs are optial in the sense of being associated with the fir s optial input allocation. Predicted frontier cost is the cost of (predicted) optial input quantities in the absence of allocative inefficiencies, while predicted actual cost is the cost of optial input quantities in the presence of allocative inefficiencies. In this section, we show how to copute these optial values of input quantities with and without allocative inefficiency which we then use to evaluate CAI. Suppressing the (i, t) subscripts for notational convenience, we rewrite the set of the FOCs in (1b) as ( ( ln c k + l= β kl x l ) ln c 1 + l= β 1l x l ) = x k + w k + v k k =,..., K, (19) where c κ β κ + γ κy + ϕ κ t for κ = 1,..., K. The (sub)syste in (19) ay be written as or G k ( x) = x k + w k + v k k =,..., K (0) G ( x) = x + w + v, G : R K 1 R K 1, (1) where G( ) = [G ( ),..., G K ( )] the eleents of which are the left-hand sides of the FOCs in (19), and w = [w,..., w K ]. We next show the existence and uniqueness of an optial input-allocation solution. We first derive the Jacobian atrix for the syste in (1): J = [ G k ( x) / x l ; k, l =,..., K]. Fro (19), it follows that G k ( x) / x l = [ŝ 1 ŝ k ] 1 (β kl ŝ 1 β 1l ŝ k δ kl ŝ 1 ŝ k ), where ŝ κ = c κ + l= β κl x l for κ = 1,..., K, and δ kl is the Kronecker delta. Further, using the noralizing restrictions in (5) we can show that k= exp {G k ( x)} = 1. 9 Hence, we can su all FOCs in (19) in exponential for to obtain 1 ( c 1 + l= β ) 1l x l c 1 + l= β = exp { x l + w l + v l }. () 1l x l l= 9 The equality is evident fro equation (9). 9

11 Dividing (19), after exponentiating, by the above expression yields g k ( x) c k + l= β kl x l 1 ( c 1 + l= β ) = exp { x k + w k + v k } 1l x l l= exp { x l + w l + v l } k =,..., K. (3) We use (3) to show the existence and uniqueness of a solution to the cost iniization syste. c Specifically, since g k ( x) = k + l= β kl x l k=(c k +, we have that g l= β kl x l) k : R K 1 (0, 1). Showing that g k ( x) attains any value in (0, 1) would enable us to clai that there exists an optial inputallocation solution. Suppose τ = [τ,..., τ K ] (0, 1) K 1. Then, consider a syste g k ( x) = τ k for k =,..., K, which can be written in the for of D x = p, where D = [a kl ] and p = [b k ] with a kl = β kl + τ k β 1l, b k = τ k (1 c 1 ) c k for k, l =,..., K. Equations l= a kl x l = b k are equivalent to l= β kl x l = τ k (1 c 1 ) c k 1+τ k for k =,..., K after substituting the restrictions. This syste has a unique solution provided the (K 1) (K 1) atrix B = [β kl, k, l =,..., K] is invertible. In fact, this is so by the translog regularity conditions. This concludes a proof of the existence of a unique input-allocation solution. We next proceed to the discussion of how to copute it. Let x(v ) denote the solution to (19). We can approxiate x(v ) using a Taylor expansion as x(v ) x (0 K 1 ) + x (0 K 1 ) v, where 0 K 1 denotes the zero vector of diension (K 1), and x (0 K 1 ) is a solution to the syste (19) at v = 0 K 1 (i.e., in the absence of allocative inefficiencies) which exists and is unique as we have shown above. Denote the syste in (19) as H ( x) = v so that the solution satisfies the identity: H ( x(v )) = v. Differentiating the latter identity with respect to v we obtain: H ( x(v )) x(v ) = I K 1, fro which it follows that x(v ) = H ( x(v )) 1. It further follows that x(0 K 1 ) = H ( x(0 K 1 )) 1 assuing that the Jacobian of the syste J 0 = H ( x(0 K 1 )) is non-singular. Therefore, we have x(v ) x (0 K 1 ) + J 1 0 v. (4) Fro (4), it follows that the input distortions are ζ x(v ) x (0 K 1 ) J 1 0 v (5) to the first order of approxiation around the expected value of v, where ζ = [ζ,..., ζ K ]. Since the Jacobian is already available fro the FIML estiation, the additional coputation involved in (5) is inial. Also note that one does not need to obtain x(v ) separately for v 0 K 1 and v = 0 K 1, since we can obtain their difference ζ directly fro (5). We also obtain v = J 0 ζ fro (5), whereas fro (4) we can obtain x (v ) provided we have x (0 K 1 ). We can get the latter by solving (1) for x at v = 0 K 1, i.e., G ( x (0 K 1 ) ) w = x (0 K 1 ). (6) We solve syste (6) using a fixed-point iteration starting fro the observed value of x. We obtain x 1 (v ) and x 1 (0 K 1 ) fro (4) using the IDF in (1a) by a siple substitution assuing u = 0. This provides us with ζ 1 = x 1 (v ) x 1 (0 K 1 ), i.e., the allocative distortion of the nueraire input. We then copute the (observation-specific) cost of allocative inefficiency as CAI = k w k ζ k. (7) 10

12 Table 1: Suary statistics for Norwegian dairy production, Variable Units Mean Median SD.5% 97.5% Outputs Milk Liters 97, , , , , Other Outputs Real EUR 43, ,55.45, , , Inputs Land Hectares (ha) Feed Real EUR 15, , , , ,570.5 Labor Hours (hr) 3, , , ,9.70 6,37.7 Materials Real EUR 10, , , , , Capital Real EUR 6, , , , , Input Prices Price of Land Real EUR/ha Price of Feed Real EUR Price of Labor Real EUR/hr Price of Materials Real EUR Price of Capital Real EUR Data We focus on the analysis of dairy fars in Norway. The data are a large unbalanced panel with 7,866 observations on 1,104 fars observed during the period. Our far-level data coe fro the Norwegian Far Accountancy Survey adinistered by the Norwegian Agricultural Econoics Research Institute. The survey includes the inforation on far production and econoic data collected annually fro about 1,000 fars fro different regions, far size classes and types of fars. Participation in the survey is voluntary. There is no liit on the nuber of years a far ay be included in the survey. Approxiately 10% of the fars surveyed are replaced every year. The fars are classified according to their ain category of faring, defined in ters of the standard gross argins of the far. Hence, the dairy fars in our saple are the fars, the largest share of the total standard gross argin of which is attributed to dairy production. Dairy fars are often involved not only in the production of ilk but also in other far production activities such as the production of various types of eat, crop, etc. Following Sipiläinen et al. (014), we consider two outputs: Y 1 ilk, easured in liters sold, and Y a single easure of all other outputs, which includes cattle and crop products. Since Y includes several outputs we easure it in onetary value ters, i.e., revenue fro all these outputs. To covert the noinal value into the real ters, we first deflate Y to real 000 Norwegian Kroner (NOK) using a weighted price index for cattle and crops, which we then convert fro NOK to Euros (EUR) using the average exchange rate. Further, we specify the following five inputs: X 1 land, easured in hectares, X own and hired far labor, easured in hours, X 3 purchased feed, X 4 aterials, which include the cost of fertilizer, pesticides, preservatives, cost related to anial husbandry, etc., and X 5 physical capital, which includes far achinery. Siilar to Y, purchased feed, aterials and capital are easured in real 000 EUR. All are deflated using a respective price index. The inforation on input prices W k k = 1,..., 5 is taken fro either the far survey when available (i.e., the price of land follows the regional average rents, the price of labor is obtained fro the value of paid labor) or the agricultural sector of the national accounts. Table 1 reports suary statistics for the variables used in our analysis. Dairy faring in Norway is highly regulated (Jervell and Borgen, 000). Throughout the past 11

13 decades, various regulatory schees have been set up to align aggregate ilk production to doestic deand. A quota-based regulatory schee was set up in Fro 1991, quotas of dairy farers exiting the industry were used to reduce national ilk supply, and there was no redistribution of quotas. The individual quotas have been reduced on several occasions in order to adjust total supply to doestic deand. The exit rate fro the dairy sector was however slow for any years. Partly as a reaction to this outcoe, a liited quota-trading schee with quota prices defined by the governent as well as the adinistrative reallocation of quotas was introduced in The objective of the change was to introduce greater flexibility to the quota syste and to encourage structural changes such as higher efficiency and productivity. To aintain the regional distribution of production, the country was divided into ilk-trade regions, and quota transfers were restricted to a given region only. However, for any years, only a sall fraction of the ilk quota was reallocated due to the reduction in the total quota. Since 00, a farer interested in selling (a portion of) quota had to sell a portion to the governent to be reallocated. Epirical Results We start by exaining whether endogeneity is indeed a proble in our application. To do so, we consider the IDF in (1a) alone [without the FOCs in (1b)]. For the tie being, we also abstract away fro the technical inefficiency ter in the equation, provided the latter is distributed independently fro the data (right-hand-side covariates). A coon ethod to test for endogeneity of the covariates in the IDF is to eploy a Wu-Hausan test perfored on the difference between the OLS and two-stage least squares (SLS) [or IV] estiates of the IDF paraeters. To obtain the SLS estiates of the IDF, one would need to first obtain the fitted values of x k,it k =,..., K fro the respective reduced for equations (in ters of exogenous y,it = 1,..., M, w k,it k =,..., K and t), which would then be used to replace x k,it in (1a). However, since the IDF and the reducedfor equations are nonlinear it is unclear whether this is the best approach. In fact, Terza et al. (008) advocate for a ore robust alternative: a two-stage residual inclusion (SRI) estiation of Hausan (1978). 10 In accordance with SRI, the first-stage residuals fro the reduced-for equations for each of the potentially endogenous covariates are to be included as additional regressors while keeping all original covariates intact (i.e., no replaceent with first-stage fitted values). Intuitively, the first-stage residuals are the estiates of all the other factors that affect the endogenous variables in addition to the exogenous instruents that are explicitly accounted for. Those factors are also likely to appear in the rando error of the ain equation (here, the IDF). Hence, by including these residuals in the IDF, we are able to control for the correlation between endogenous regressors and the rando error thereby, essentially, avoiding the oitted variable proble. The test for endogeneity then boils down to a joint F-test on the first-stage residuals. In the case of linear odels, SRI and SLS are equivelent; in nonlinear odels (like ours) SRI is generically consistent whereas SLS is not. The four included residuals fro the reduced-for equations for x k k =,..., 5 are highly significant both individually and jointly. The joint exclusion F-statistic is Since the test is not exact, we use pair panel bootstrap to obtain critical values. Specifically, we use 10,000 rando draws (with replaceent) of cross-sections fro our data to reestiate both stages and recopute the F-statistic for the joint exclusion of the first-stage residuals fro the IDF. The 99% bootstrap critical value, which we copute as the 99th percentile of bootstrap distribution of the F-statistic, 10 The SRI procedure is siilar in its nature to Petrin and Train s (010) control function approach to handling endogeneity in consuer choice odels. 1

14 Table : Estiates of selected technological etrics Metric Mean Median SD.5% 97.5% Single-Equation IDF SE TC Tech. Ineff Cost Syste of IDF SE TC Tech. Ineff CAI is 3.1. As a robustness check, we repeat the test while also controlling for fixed effects in the IDF. The corresponding F-statistic is even larger and equals Thus, the data lend strong support in favor of endogeneity present in the IDF. 11 We next proceed to the discussion of the ain results. Instead of looking at the paraeter estiates, we rather focus on ore eaningful technological etrics such as scale econoies (SE), technical change (TC), technical inefficiency and cost of allocative inefficiency (CAI). Observationspecific estiates of technical inefficiency are obtained fro the estiated conditional ean of u in the spirit of Jondrow et al. (198), and the estiates of CAI are obtained as described earlier. Measures of scale econoies and technical change are coputed based on the estiated IDF, the available data and the duality results (Färe and Priont, 1995; Karagiannis et al., 004) as follows: SE ln C ln Y = ln f(ŷ, X, t) ln Y = [ α + n α n y n + k= γ k x k + ψ t ] TC ln C t = ln f(ŷ, X, t) t = δ t + δ tt t + k= ϕ k x k + ψ y. Scale econoies are defined as the su of output elasticities of cost thus capturing an increase in the cost (in %) resulting fro a proportional one-percent increase in all outputs. The instance of SE > 1 corresponds to decreasing returns to scale, while SE < 1 indicates increasing returns to scale. Lastly, technical change is defined as a secular decline in cost over tie. We estiate two odels: (i) the cost syste (1) via FIML and (ii) a single-equation IDF (1a) via ML. We estiate a single-equation odel, which suffers fro siultaneity, in order to exaine the degree to which results get distorted should the endogeneity of the input ratios reain unaddressed. We however acknowledge that this coparison is eaningful only if our identification strategy is valid. As discussed earlier, dairy faring in Norway is regulated via a quota-based schee, which underwent a (policy) regie change in the late 1990s. Specifically, a quota-trading syste was introduced in 1997 with the objective of encouraging structural changes that would lead to higher efficiency and productivity in this agricultural sector. To investigate whether the far-level technical inefficiency has responded to this regie change, 1 when estiating the syste in (1) or a singleequation IDF in (1a) we odel inefficiency u it in a way that perits its ean to vary with the 11 We note that we are unable to test for exogeneity of the input price ratios and/or outputs (our instruents) since our odel is exactly identified, and the test for over-identifying restrictions would require additional instruents. 1 We thank the editor and one of the referees for this insightful suggestion. 13

15 ( quota regie. Forally, we assue that u it N + µu,it, σu) with the ean µu,it paraeterized as µ u,it = η 0 + η 1 H it, where H it is an indicator which equals one for years fro 1997 onward and zero otherwise. 13 Table reports suary statistics of the technological etrics coputed fro both the cost syste of the IDF and a single-equation IDF. In order to not confine our analysis to ean estiates only (which are not that inforative), we also provide the distributions for each of the etrics in figures 1 and. The kernel densities are constructed using the second-order Epanechnikov kernel with the optial bandwidth selected via the data-driven least-squares cross-validation (Silveran, 1986). Coparing the scale econoies estiates fro the syste and single-equation IDF odel, we find that a single-equation approach coonly eployed in the literature (e.g., Labert and Wilson, 003; Karagiannis et al., 004; Atsbeha et al., 01), which takes endogeneity of the input ratios for granted, produces unreasonable estiates of SE. In particular, the ean single-equation-based estiate of SE is 0.58 with the corresponding.5% 97.5% interval ranging fro 0.03 to That is, dairy fars are predicted, on average, to enjoy increasing returns to scale of an astounding agnitude 1.7(=1/0.58). Furtherore, about % of the single-equation-based estiates of SE are negative (see figure 1) as a result of the violated IDF regularity conditions. In contrast, the systebased estiates of SE are uniforly positive and suggest the presence of increasing returns to scale (i.e., ostly SE < 1) of reasonable agnitudes. For instance, the edian syste-based estiate of SE is 0.85 with the standard deviation of 0.07, which is significantly larger than ore dispersedly distributed estiates fro the single-equation odel. To su, epirical evidence suggests that increasing the scale of fars ay considerably iprove fars perforance. In stark contrast to the results on scale econoies, the estiates of TC are virtually indistinguishable across the two odels. Both the syste and single-equation odels suggest that the Norwegian dairy sector has experienced technical change (progress) at an average rate of 1.3% per annu over the course of the saple period. The Li (1996) test confirs the equality of the two epirical distributions of the TC estiates. We next consider epirical results on the cost of allocative inefficiency. Figure plots the distribution of the CAI estiates obtained fro the cost syste (1) estiated via FIML. The distribution is highly skewed to the right. The CAI estiates average at 9.5% and range fro 0.4% to 30.% (see table ). These estiates indicate econoically significant isallocation of inputs which, if eliinated, ay potentially reduce fars costs by the edian of 6.9%. Also note that no CAI estiates are obtained fro the single-equation IDF odel since the latter does not allow the identification of allocative disturbances due to the oission of the cost iniizing FOCs. We conclude our analysis by exaining the results on technical inefficiency aong fars and studying the potential effect (if any) of the 1997 quota regie change on the forer. When pooling the technical inefficiency estiates over the entire saple period, we find that, on average, the syste FIML approach produces estiates of technical inefficiency of values twice larger than those fro a standard single-equation ML approach: 5.% vs..4% with the corresponding standard deviations of 4.3% and.1% (see table ). The syste-based estiates are considerably ore dispersed and skewed to the right, as can also be seen in figure 1. However, when differentiating between the technical inefficiency estiates corresponding to the pre-1997 period (prior to regie change) fro those corresponding to the period fro 1997 onward (post regie change), we see a substantially different picture. Figure 3 presents box-plots of the distributions of the technical inefficiency estiates for the two tie periods across the two odels we estiate. Based on our preferred cost syste FIML odel, we docuent both an econoically 13 The above specification of µ u,it essentially odels a structural break. 14

16 Figure 1: Scale econoies, technical change and technical inefficiency 15

17 Figure : Cost of allocative inefficiency and statistically significant iproveent in the levels of technical efficiency aong dairy fars associated with the 1997 quota schee change. The ean estiate of technical inefficiency has declined fro 8.5% in the pre-1997 period to 3.5% in the post-1997 period. This decline is nonnegligible as also confired by the statistical significance of the coefficient η 1 on H it in the equation for the ean of u it. Table 3 reports the estiates of the ean function paraeters with their corresponding standard errors. Fro table 3 and figure 3, it is evident that the single-equation ML odel however fails to detect this structural iproveent in the fars efficiency levels. Adittedly, these differences in the results ay not necessarily be due to the failure of a single-equation odel to account for the endogeneity of inputs. One ay alternatively argue that the post-1997 period is characterized by greater freedo for producers to choose ilk outputs, which can potentially lead to the endogeneity in outputs thus coproising the consistency of our syste-based estiator. It is possible that a single-equation ethod, which fails to account for endogeneity of either inputs or outputs and thus is always inconsistent, is ore insulated to such changes than our preferred syste approach, which becoes inconsistent only during the post-1997 period. The detected iproveents in technical efficiency ay therefore be argued to be spurious. 14 However, for any years after the regulatory regie change only a sall fraction of the ilk quota was reallocated due to the reduction in the total quota. Furtherore, since 00, a farer interested in selling (a portion of) quota had to sell it to the governent to be reallocated. In the light of the above, we believe it is soewhat unlikely that the structural iproveent in the fars efficiency levels that our preferred syste-based odel estiates is spurious. Conclusion In this article, we offer a ethodology to address the endogeneity of inputs in the IDF forulation of the ulti-output ulti-input production process fro the perspective of econoic theory. We 14 We would like to thank the editor for bringing this alternative explanation to our attention. 16

18 Figure 3: Technical inefficiency pre- and post-regie change Table 3: Mean of technical inefficiency (µ u ) Coeff. Estiate SE Single-Equation IDF η η Cost Syste of IDF η η

19 suggest invoking the assuption of the fir s cost iniizing behavior not only to justify the treatent of outputs as exogenous but to also tackle endogeneity of the input ratios in the IDF. We do so by considering a flexible (siultaneous) syste of the translog IDF along with the FOCs fro the fir s cost iniization proble. The odel we develop also accoodates both technical and (input) allocative inefficiencies aongst firs. There are advantages to using our syste approach over the standard single-equation odel even if the input ratios are uncorrelated with the rando productivity shocks in the IDF. Since additional equations (the FOCs) do not contain any extra paraeters, the syste-based estiator is ore efficient. Furtherore, technological etrics obtained via the cost syste approach are likely to be ore eaningful because the econoic behavior is ebedded into the syste through the FOCs. We showcase our odel by applying it to study the production of dairy fars in Norway during the period. Having failed to reject the presence of endogeneity in the data, we find that the single-equation IDF odel produces unreasonable estiates of scale econoies as well as fails to detect a change in the level of technical inefficiency associated with the quota schee change in In contrast, the technological estiates obtained fro the cost syste indicate the evidence of increasing returns to scale of reasonable agnitudes (the average of 1.18). Using the syste approach, we are also able to docuent both an econoically and statistically significant iproveent in the levels of technical efficiency aong dairy fars associated with the 1997 quota schee change. The ean estiate of technical inefficiency has declined fro 8.5% in the pre-1997 period to 3.5% in the post-1997 period. The edian cost of allocative inefficiency is estiated to be 6.9%. References Atkinson, S. E., Cornwell, C., and Honerkap, O. (003). Measuring and decoposing productivity change: Stochastic distance function estiation versus data envelopent analysis. Journal of Business and Econoics Statistics, 1(): Atkinson, S. E. and Priont, D. (00). Stochastic estiation of fir technology, inefficiency, and productivity growth using shadow cost and distance functions. Journal of Econoetrics, 108:03 5. Atsbeha, D. M., Kristofersson, D., and Rickertsen, K. (01). Anial breeding and productivity growth of dairy fars. Aerican Journal of Agricultural Econoics, 94: Caves, D. W., Christensen, L. R., and Swanson, J. A. (1981). Productivity growth, scale econoies, and capacity utilization in US railroads, Aerican Econoic Review, 71(5): Chabers, R. G. (1997). Applied Production Analysis: A Dual Approach. Cabridge University Press. Coelli, T. J. (000). On the econoetric estiation of the distance function representation of a production technology. Discussion Paper 000/4, Center for Operations Research and Econoetrics, University Catholique de Louvain. Coelli, T. J., Hajargasht, G., and Lovell, C. A. K. (008). Econoetric estiation of an input distance function in a syste of equations. Working Paper, University of Queensland. Cornwell, C., Schidt, P., and Sickles, R. C. (1990). Production frontiers with cross-sectional and tie-series variation in efficiency levels. Journal of Econoetrics, 46(1 ):