Pitfalls in the estimation of a cost function that ignores allocative inefficiency: A Monte Carlo analysis

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<ul><li><p>Journal of Econometrics 134 (2006) 317340</p><p>explicitly in the cost function biases the estimates of: (i) the cost function parameters,</p><p>producers always operate efciently, in reality these producers are not always</p><p>ARTICLE IN PRESS</p><p>www.elsevier.com/locate/jeconom</p><p>Corresponding author. Tel.: +22782 2791x323; fax: +2 2785 3946.0304-4076/$ - see front matter r 2005 Elsevier B.V. All rights reserved.</p><p>doi:10.1016/j.jeconom.2005.06.025</p><p>E-mail addresses: kkar@binghamton.edu (S.C. Kumbhakar), hjwang@econ.sinica.edu.tw</p><p>(H.-J. Wang).(ii) returns to scale, (iii) input price elasticities, and (iv) cost-inefciency.</p><p>r 2005 Elsevier B.V. All rights reserved.</p><p>JEL classification: C15; C21</p><p>Keywords: Technical inefciency; Allocative inefciency; Stochastic frontier</p><p>1. Introduction</p><p>Although the mainstream neoclassical paradigm in economics assumes thatA Monte Carlo analysis</p><p>Subal C. Kumbhakara, Hung-Jen Wangb,</p><p>aDepartment of Economics, State University of New York Binghamton, Binghamton, NY 13902, USAbInstitute of Economics, Academia Sinica, Taipei 115, Taiwan</p><p>Available online 15 August 2005</p><p>Abstract</p><p>In the stochastic frontier literature, it is a widely held view that allocative inefciency can be</p><p>lumped together with technical inefciency in the estimation of cost frontiers. Therefore, a</p><p>one-sided error term in the cost function is believed to capture the cost of overall (technical</p><p>plus allocative) inefciency. In this paper we challenge that view through a detailed Monte</p><p>Carlo investigation. The results show that failure to include the cost of allocative inefciencyPitfalls in the estimation of a cost functionthat ignores allocative inefciency:</p></li><li><p>efcient. Consequently, two otherwise identical producers may never produce the</p><p>ARTICLE IN PRESS</p><p>S.C. Kumbhakar, H.-J. Wang / Journal of Econometrics 134 (2006) 317340318same output, and their costs and prot are not the same. This difference in output,cost, and prot can be explained in terms of technical and allocative inefciency, andsome unforeseen exogenous shocks. Given the input quantities, a producer is said tobe technically inefcient if it fails to produce the maximum possible output.Similarly, a cost or prot maximizing producer is allocatively inefcient if it fails toallocate the inputs optimally, given input and output prices. Both types ofinefciency are costly in the sense that cost (prot) is increased (decreased) due tothese inefciency.The econometric estimation of technical inefciency (popularly known as the</p><p>stochastic frontier technique) originated in Aigner et al. (1977) and Meeusen and vanden Broeck (1977). In this framework one species the production technology interms of the production function, i.e.,</p><p>y f x1; . . . ; xk : b expv u, (1)where y denotes a single output, x1; . . . ; xk are k inputs used to produce y, f : is thedeterministic production frontier, and b is a technology parameter vector to beestimated. Finally, v is a random noise component (an exogenous shock unknown tothe producer) and the one-sided term u (uX0) captures technical inefciency.1 This islabeled as output-oriented (output-augmenting) technical inefciency, becauseoutput can be increased by reducing inefciency, ceteris paribus.The cost frontier is often used as an alternative representation of the production</p><p>technology. In this formulation, one species the model as</p><p>C Cw1; . . . ; wk; y : b expv u, (2)where C is actual/observed cost and Cw1; . . . ; wk; y : b is the cost frontier. Sinceinefciency increases cost, uX0 appears with a positive sign and cost efciency isdened as expup1 (Coelli et al., 1998, p. 210; Kumbhakar and Lovell, 2000,p. 139). The question is: Does u represent the cost of technical inefciency (increasein cost due to technical inefciency) or is it a measure of overall cost inefciency (costof both technical and allocative)? In other words, can one lump the costs of bothtechnical and allocative inefciency together and assume it to be an i.i.d. randomvariable (as in Coelli et al., 1998, Chapter 9.4; Kumbhakar and Lovell, 2000,Chapter 4)?The purpose of this paper is to show that if there is allocative inefciency and one</p><p>estimates the cost function in (2) and thereby either ignores the presence of allocativeinefciency or erroneously assumes that allocative inefciency can be lumpedtogether with technical inefciency in the estimation, then the estimated technologyand estimates of inefciency are likely to be wrong. In particular, we examine howthe exclusion of allocative inefciency affects (i) the measure of cost inefciency,(ii) the parameters of the cost function, (iii) returns to scale, and (iv) price elasticitiesof input demand. All we can say analytically is that parameter estimates are biased</p><p>1Alternatively, technical efciency is expup1. Consequently, technical efciency equals 1 minus</p><p>technical inefciency, especially when technical inefciency is small.</p></li><li><p>and inconsistent. Therefore, measures based on the biased parameter estimates are</p><p>Shephards lemma can be used to derive the inefciency-adjusted input demand</p><p>ARTICLE IN PRESS</p><p>S.C. Kumbhakar, H.-J. Wang / Journal of Econometrics 134 (2006) 317340 319functions, viz.,</p><p>qCw; yqwj</p><p> xjw; y expl. (6)</p><p>2If the underlying production technology is homothetic, then u in (1) is a constant multiple of l.3likely to be biased. Empirically, the issue is not only about bias, but also the degreeof bias in the estimates, especially in the measures of interest such as inputelasticities, returns to scale, etc. The extent of such biases can only be evaluatednumerically. Therefore, we conduct a Monte Carlo analysis to assess these effects.</p><p>2. Preliminaries</p><p>The cost function framework is based on the assumption that producers minimizecost given the output and the technology. Thus, the optimization problem for aproducer is</p><p>minx</p><p>w0 x s.t. y f x expl, (3)</p><p>where x and w are J 1 vectors of actual input quantities and prices, respectively,and lX0 measures input-oriented (input-saving) technical inefciency.2 Since thevector x denotes the actual quantities of the inputs and explp1, one can labelxj expl as the technical inefciency-adjusted level of input j. Alternatively, whenl is multiplied by 100, it shows the percent by which all the inputs can be reducedwithout reducing output. If producers fail to allocate inputs efciently, then the rst-order conditions of the above are f j=f 1awj=w1, which can be rewritten as</p><p>f j</p><p>f 1 expxjwj</p><p>w1 w</p><p>j</p><p>w1; j 2; . . . ; J, (4)</p><p>where f j is the partial derivative of f with respect to the jth input and xja0j 2; . . . ; J represents allocative inefciency for the input pair (j; 1).3One can view the above optimization problem from a different angle. The rst-</p><p>order conditions in (4) show that producers use the shadow prices (w) to determinethe technical inefciency-adjusted level of inputs. Thus, the above optimizationproblem can be viewed as though producers are minimizing shadow cost given theproduction technology, viz.,</p><p>C w0 x expl s.t. y f x expl (5)instead of the actual cost in choosing inefciency-adjusted input quantities. Thesolution to this problem gives the technical inefciency-adjusted input quantities,which in turn can be used to dene the shadow cost function, C Cw; y, in (5).An advantage of the shadow cost function approach is that once it is specied,If the production function is not homogeneous then f j=f 1 will depend on l.</p></li><li><p>This approach is an alternative to the primal approach outlined in (3) and (4) in</p><p>ARTICLE IN PRESS</p><p>S.C. Kumbhakar, H.-J. Wang / Journal of Econometrics 134 (2006) 317340320which a production function is specied rst, and then the corresponding rst-orderconditions of cost minimization are derived (Schmidt and Lovell, 1979).Since C is unobserved we have to establish the link between actual cost (C) and</p><p>C.4 For this, rst we dene the shadow cost share function of input j, viz.,</p><p>Sj wj x</p><p>j</p><p>Cw; y q lnCw; yq lnwj</p><p> !; where xj xjw; y expl, (7)</p><p>which gives</p><p>xjw; y Cw; ySj</p><p>wjexpl. (8)</p><p>Therefore, the actual cost is</p><p>C XJj1</p><p>wjxjw; y XJj1</p><p>wjCw; ySj</p><p>wjexpl</p><p>" #9</p><p> explCw; yXJj1</p><p>wj</p><p>wjSj 10</p><p> explCw; yXJj1</p><p>Sj expxj 11</p><p> C0Cal expl, 12where</p><p>C0 Cjl0;n0 (13)is the minimum cost5 in the absence of both technical and allocative inefciency, and</p><p>Cal Cw; yX</p><p>j</p><p>Sj expxj( ),</p><p>C0 (14)</p><p>measures costs due to allocative inefciency. Thus, overall cost inefciency (CI) canbe dened as</p><p>CI CC0</p><p> CCjl0;na0</p><p> Cjl0;na0C0</p><p> Cal Ctech, (15)</p><p>where Cal and Ctech measure the cost of allocative and technical inefciency,respectively. Alternatively, the overall cost efciency (CE) is (Farrell, 1957)</p><p>CE C0</p><p>C 1</p><p>Cal 1</p><p>Ctech AE TE, (16)</p><p>4The following derivations are based on Kumbhakar (1997).5This is also labeled as the deterministic cost frontier or the neo-classical cost function.</p></li><li><p>ARTICLE IN PRESS</p><p>S.C. Kumbhakar, H.-J. Wang / Journal of Econometrics 134 (2006) 317340 321where allocative and technical efciencies (AE and TE) are the reciprocals of Cal andCtech. For a well behaved cost function CalX1 and CtechX1, which in turn imply thatAEp1, TEp1, and CEp1. It is clear from (14) that Cal depends on allocativeinefciency as well as data on y and w, while Ctech expl is independent of data.From the above equation, the actual cost can be written as (appending a two-sided</p><p>random noise term v):</p><p>lnC lnC0 lnCal l v lnC0 u v, (17)where u lnCal l is a function of n, y, and w. Both the mean and the variance of uwill depend on w, y and the parameters of the cost function.In empirical applications it is often assumed that u (in (17)) follows a half- or a</p><p>truncated-normal distribution. The cost function in (17) is then estimated with thepresumption that u captures either the cost of technical inefciency (therebyignoring/assuming away costs due to allocative inefciency, i.e., assuming u l), orthe overall cost inefciency (the cost of both allocative and technical inefciency).Neither of these presumptions, however, is a reasonable representation of the truemodel in many cases. It can be clearly seen from (17) that assuming u l andignoring lnCal result in a mis-specied model. Since the true Eu depends on w andy via lnCal, and that the same variables (w and y) appear in the cost frontierfunction, the estimated coefcients of the cost function will be biased andinconsistent even if one assumes that there is no technical inefciency (i.e., l 0)and subsequently applies least squares to (17). On the other hand, if u lnCal l isassumed, then a half- or truncated-normal distribution on u cannot be correct,because lnCal is a function of data, parameters and random variables n. Due to thenon-linearity of lnCal, it is not possible to analytically derive the distribution of u(which is required for a maximum-likelihood (ML) estimation) from any reasonabledistributions of n and l. Consequently, if one uses the half- or truncated-normaldistribution with constant parameter(s), then the ML estimates will be biased andinconsistent. Recently, Coelli et al. (2003, p. 3640) also point out that productivitymeasurement based upon single equation cost functions will be biased when there aresystematic deviations from allocative efciency, due to the biased coefcients.Thus, the main problem in estimating the mis-specied model (in (17)) using least</p><p>squares or ML is that the estimated parameters will be inconsistent. The MLestimators might be more problematic, because the half (truncated) normalassumption on u is not consistent with a model that allows allocative inefciency.This parameter inconsistency will be transmitted to everything that is derived fromthe estimated technology such as returns to scale, elasticities of substitution, inputdemand functions, technical and allocative inefciency, etc. Although the degree ofinconsistency can also be affected by other factors when the data generation processis unknown (which is the case if real data are used), the problem can be avoided in acontrolled environment (Monte Carlo studies), which is what we do here. Wetherefore, we use an extensive Monte Carlo experiment to investigate theconsequences of improper treatments of lnCal in the model. In particular,we examine how the exclusion of the allocative inefciency component (or the</p><p>assumption of lumping both allocative and technical inefciency in u) affects (i) the</p></li><li><p>a, is the mis-specied model, i.e., one that estimates</p><p>ARTICLE IN PRESS</p><p>S.C. Kumbhakar, H.-J. Wang / Journal of Econometrics 134 (2006) 317340322lnC lnC0 u v, (24)where u is a random variable with a half-normal distribution. As discussed in theprevious section, this model implicitly assumes that u captures either the cost oftechnical inefciency (thereby ignoring/assuming away the cost of allocativeinefciency, i.e., u l), or the overall (both allocative and technical) cost inefciencyof the rm. Since neither of the presumptions can be justied in the light of the truecost inefciency measure, (ii) the technology parameters (the cost function), (iii)returns to scale, and (iv) input price elasticities.To do this, we compare not only the means, standard errors, and root mean</p><p>square errors of the concerned statistics, but also the distributions of the statisticsacross competing models. The equality of the distributions across models is testedusing a non-parametric test of Li (1996).</p><p>3. Simulations</p><p>3.1. Design of the experiment</p><p>We consider a translog cost function for lnC with J 2. To impose the linearhomogeneity in prices, we use w1 as our numeraire and dene w w2=w1. Allocativeinefciency of the input pair (x2, x1) is represented by x.We assume a translog functional form for lnC and obtain the following</p><p>relationships based on (13) and (14):</p><p>lnC0 b0 by ln y bw lnw 12byyln y2 12bwwlnw2 bywln y lnw, 18lnCal lnS1 S2 expx bwx bwwx lnw 12bwwx2 bywx ln y, 19</p><p>where S1 and S2 are</p><p>S1 1 bw bwwlnw x byw ln y, 20S2 bw bwwlnw x byw ln y. 21</p><p>To generate data we use the following distributions on the non-stochastic (ln y,lnw) and random (x, v, l) variables.</p><p>ln yNmy; s2y; lnwNmw;s2w; xNmx; s2x; vN0;s2v, 22l jlj; where lN0; s2l. 23</p><p>The notation of (23) indicates that the technical inefciency l has a half-normaldistribution.Given the parameter values (discussed later), we draw N observations based on the</p><p>above distributional assumptions for each of the R replications. Two differentmodels are then estimated using the generated data. The rst one, labeled as Modelmodel, the estimated model is mis-specied.</p></li><li><p>The second model, labeled as Model b, is the correctly specied model and it</p><p>lnC lnCal lnC0 l v, (25)</p><p>ARTICLE IN PRESS</p><p>S.C. Kumbhakar, H.-J. Wang / Journal of Econometrics 134 (2006) 317340 323where lnCal is a well dened function of parameters and data. Consequently,differences in the results of models a and b can be attributed to the effects ofimproper treatment of allocative inefciency, and they are not due to the use ofdifferent estimation techniques.We make sure that regularity conditions of the cost function are satised in the</p><p>simulation model. The regularity conditions require a cost function to be (Varian,1992): (1) nondecreasing in w, (2) hom...</p></li></ul>

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