The specification of technical and allocative inefficiency in stochastic production and profit frontiers

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<ul><li><p>Journal of Econometrics 34 (1987) 335-348. North-Holland </p><p>THE SPECIFICATION OF TECHNICAL AND ALLOCATIVE INEFFICIENCY IN STOCHASTIC PRODUCTION AND </p><p>PROFIT FRONTIERS* </p><p>Subal C. KUMBHAKAR University of Texas at Austrn, TX 78712, USA </p><p>Received June 1985, final version received May 1986 </p><p>In previous studies, frontier production models have been analysed either in a non-optimizing or cost-minimizing framework. In this paper we discuss estimation of technical and allocative inefficiency under the behavioral assumption of profit maximization. Specification and estimation issues are discussed in the context of the Cobb-Douglas production function. </p><p>1. Introduction </p><p>In recent years there has been a substantial change in the common approach to estimating production functions. An example of this is the stochastic frontier production model developed by Aigner et al. (1977) and Meeusen and van den Broeck (1977). The essential idea behind their approach is that the production function is viewed as a locus of maximum output levels from a given input set and thus the output of each firm is bounded above by a frontier. This frontier is assumed to be stochastic in order to capture exoge- nous shocks beyond the control of firms. Since all firms are not able to produce the frontier output, an additional (one-sided) error is introduced to represent technical inefficiency, something which is in the control of firms. Variants of these models now have been applied to a variety of industries [see Schmidt (1986) for a recent survey]. </p><p>Previous studies of frontier production models have been analysed either in a non-optimizing or in a cost-minimization frame work [Schmidt and Love11 (1979), Greene (1980), Bauer (1984), Melfi (1984), Kumbhakar (1986)]. Direct estimation of the production function (non-optimizing) gives consistent esti- </p><p>*This is a part of Chapter 4 of my Ph.D. dissertation submitted to the University of Southern California, 1986. I would like to thank Professors Dennis Aigner, Lee Lillard and Gerald Nickelsburg for their suggestions and encouragement. I am also thankful to the referees of the Journal for comments on an earlier version of this paper. They are, however, not responsible for remaining errors. </p><p>0304-4076/87/$3.5001987, Elsevier Science Publishers B.V. (North-Holland) </p></li><li><p>336 XC. Kumbhakar, Stochastic production and pro$t frontiers </p><p>mates of the production function parameters only when inputs can be treated as exogenous. The Zellner et al. (1966) argument of expected profit maximiza- tion can be used to treat inputs as exogenous only if technical inefficiency is completely unknown to the firm. Since technical inefficiency includes poor management, the composition and vintage of capital stock, local labor quality, etc., the assumption of unknown technical inefficiency may not be fully justified. If technical inefficiency is known to the firm the estimates of production function parameters obtained directly from the production func- tion will be inconsistent. This inconsistency can be avoided if a simultaneous equation approach is used. Moreover, such an approach permits us to estimate both technical and allocative inefficiency. </p><p>In this paper we discuss estimation techniques for production function parameters and both technical and allocative inefficiency under the behavioral assumption of profit m aximization. Such an exercise might seem trivial since the cost-minimization framework has been treated excellently by Schmidt and Love11 (1979). But there are some important differences which make the profit maximizing framework worthy of separate study. </p><p>The paper is organized as follows. In section 2 we specify the model with technical inefficiency and derive the input demand and output supply func- tions. Estimation issues are discussed in section 3. Section 4 summarizes the results. </p><p>2. Specification of technical and allocative inefficiency </p><p>We begin with the following Cobb-Douglas production function </p><p>y=A nxl?l exp(u), ( 1 i </p><p>where y is output, xi is input i (i = 1,2,. . . , n), u is general statistical noise that captures random exogenous shocks not in the control of the firm, and A is a technical efficiency parameter which is different for different firms. (Y~ (i= 1 ,-.*, n) are the parameters common to all firms. </p><p>The production function (1) can easily be related to the frontier production function by specifying A as </p><p>A = aOexp( T), 7 IO, (I) </p><p>where 7 represents technical inefficiency which differs from firm to firm, while cu,, is common to all firms. Technical inefficiency may be caused by poor management, low local labor quality, etc. Variation in r across tlrms leads to a different production function for each firm. Alternatively, T may be regarded as an input [e.g., management in Mundlak (1961) or care in gate checking in </p></li><li><p>S.C. Kumbhakar, Stochastic production and profit frontiers 331 </p><p>Forsund et al. (1980)] which varies across firms. One major difference between this and other inputs is that it is not observed by the investigator (since no data is available). </p><p>A profit-maximizing firm can be inefficient due to (i) technical inefficiency, (ii) allocative inefficiency and (iii) scale inefficiency [see Farsund et al. (1980)]. If a firm produces level of output y using input vector x0 available at fixed prices w, the production plan (y, x0) is allocatively efficient if f,(x)/f,(xO) = wi/wi where fi( x0) is the partial derivative of f(x) with respect to input i and f(x) is the production function. Similarly, the firm is scale efficient if P = WYO, WI/a y, where p is the output price and c(y, w) is the cost function. This two-step procedure of profit maximization (first choosing the optimal input proportions and then output supply) is equivalent to </p><p>wi =p . MP,;, (2) </p><p>where MP,, is the marginal product of input i. Since (2) is both necessary and sufficient for allocative and scale efficiency, an alternative way of representing technical inefficiency is </p><p>wi=p-MP,,-exp(ui). (2) </p><p>The interpretation of eq. (2) is straightforward if the production function is deterministic and technical inefficiency r is viewed as an input observed by the individual producers only. The random variable ui in (2) can be interpreted as allocative inefficiency that reflects the amount by which the first-order condition of profit maximization for the input i fails to hold. </p><p>If the production function is stochastic as in (1) with u representing random shocks not known to the firm, profit maximization has to be interpreted as expected (or median) profit maximization - the type introduced by Zellner et al. (1966). This is especially the case when the production process is not instantaneous since the effect of the random shock u on output cannot be known before the inputs are used in production. </p><p>In this paper we assume that (i) the firm maximizes the median value of profit, (ii) r is known only to the firm, (iii) u is an unknown random shock,3 and (iv) the input and output prices are known with certainty. With these </p><p>If one starts with the two-step procedure and the corresponding allocative and scale in- efficiency, they can be reparameterized in the form of (2) where the u, are well-defined functions of allocative and scale inefficiency. </p><p>2The choice of the expected or median value of profit depends on an assumption about the risk function of the decision process. Expected value is optimal if the risk function is quadratic and the median is optimal for the absolute value loss function. The problem with expected profit maximization is that expected profit becomes an increasing function of the variance of output. Because of this we prefer median profit maximization. </p><p>% is assumed to be i.i.d. N(0, CT,). </p></li><li><p>338 XC. Kumbhakar, Stochastic production and profit frontiers </p><p>assumptions in place the first-order conditions for profit maximization in (2) are4 </p><p>k+i </p><p>where </p><p>a, = ln( ariaOp/wi), </p><p>Solving the (n + 1) equations (x,, x2,. . ., x,, _Y) gives </p><p>In xi = &amp;lnbW) + </p><p>1 n +~,~l(aj+(1-r)6ij)'j+ j&amp; </p><p>i=1,2 n. ,-.., </p><p>in (3) and (1) for the (n + 1) unknowns </p><p>and </p><p>lna, lny=l_r+ </p><p>r In p - + j&amp; ,$lajln(ajlwj) l-r </p><p>+ </p><p>(4 </p><p>(5) </p><p>where </p><p>Sij= 1 if i =j, </p><p>= 0 otherwise, </p><p>and r = cyZt=,aj which is less than one under the assumptions of perfect competition and profit maximization [Nerlove (1965, p. S)]. </p><p>The input demand functions in (4) show that the presence of technical inefficiency reduces input demand (since r I 0) equiproportionately. This is opposite to the Schmidt and Lovell(1979, p. 345) result derived in the context </p><p>41f 7 is also unknown we get back the Zellner et al. (1966) result. This makes the single- equation method consistent and the within transformation in section 3.2.1 is unnecessary. On the other hand, if the firm knows v - especially when the production process is instantaneous - OLS after the within transformation will still be inconsistent. </p></li><li><p>XC. Kumbhakar, Stochastic production and profit frontiers 339 </p><p>of cost minimization. The reason for this apparent contradiction is that the demand functions in their work are conditional on exogenous output whereas the demand functions in (4) are unconditional. Since technical inefficiency implies underutilization of inputs, production of a given level of output requires the use of more inputs which is the Schmidt-Lovell result. On the contrary, the presence of technical inefficiency leads to a decrease in output supply [as shown in (5)] which in turn requires less inputs to be used. The net result depends on the magnitude of these two opposing forces. In the present case the net effect is to reduce input demand5 as shown in (4). </p><p>For a profit-maximixm g firm the profit function [using (4) and (5)] is </p><p>~IT(P,W,O,~,U1,U2,...,~,) </p><p>=py - wx </p><p>*(p,w,v)=(p~o)(-~ fi(~~/w~)~/~-~) {exp(u)-r}. (7) i i-1 1 </p><p>Since the cost and profit frontiers are stochastic we take their median values in order to derive an expression to calculate the cost of inefficiency. Thus, the reduction in (median) profit due to technical inefficiency alone is ~r*(l - exp( r/(1 - r))) where B * is the median value of the profit frontier </p><p>Similarly, allocative inefficiency alone reduces (median) profit by </p><p>~,Jp,Vi,+,...,u,) </p><p>See also Lovell and Sickles (1983) for a similar conclusion. </p><p>!fro. </p><p>(8) </p></li><li><p>340 </p><p>where </p><p>XC. Kumbhakar, Stochastic production and profit frontiers </p><p>Q = ( pa,)(-) (9) </p><p>3. Estimation </p><p>In this section we initially consider the estimation issues in a single cross-section and then examine them in a panel data context. </p><p>3.1. Estimation using cross-section data </p><p>From the first-order conditions for profit maximization in (3) along with the production function in (1) and (1) we get </p><p>In y,= a + Ccyiln xtf+ T,+ u,, i </p><p>(l---~~~)lnx~,=a~+ ~a,lnxk,+7,+Uj,, i=1,2 ,..., n, (10) k#i </p><p>where </p><p>a=lna, and a, = ln( aiaop/wi). </p><p>The subscript f indexes firms (f= 1,2,. . . , F). The system of equations in (10) is different from Zellner et al. (1966) since T </p><p>is transmitted to the input demand equations in (4). This makes a single-equa- tion method of estimation inconsistent. One can, however, use the maximum likelihood method. Estimates of the relevant parameters in (10) can be obtained by maximizing the log likelihood function.6 Finally, technical in- efficiency, T,, can be estimated from the mean or mode7 of the conditional distribution of T, given the residuals in (10) T,+ u, and T,+ uif, i = 1,2,. . . , n. </p><p>3.2. Estimation using panel data </p><p>We now consider methods of estimation in the context of panel data. Such methods are discussed by Pitt and Lee (1981) and Schmidt and Sickles (1984) </p><p>6The log likelihood function is a special case (T= 1) of (16) - derived in the context of panel data in section 3.2.2. </p><p>These expressions are derived in eqs. (18) and (19) in the context of panel data. The same results can be used here by putting T= 1. In estimating 7 we obtain information from the residuals of the production function as well as the first-order conditions for profit maximization. </p></li><li><p>S. C. Kumbhakar, Stochastic production and profit frontiers 341 </p><p>in a single-equation framework. Simultaneous equation models are considered by Schmidt (?984), Bauer (1984) and Melfi (1984) under a cost-minimizing hypothesis. Our approach differs from Pitt and Lee or Schmidt and Sickles because we consider inputs to be endogenous. It also goes beyond Schmidt (1984) who considered output to be exogenous. </p><p>With panel data an explicit assumption is to be made about the time behavior of technical inefficiency. Here we assume that the technical in- efficiency of each firm is invariant over time. With this qualification the system of equations in (10) in the context of panel data becomes </p><p>(1 - cY,)lnx+= a;,,+ C cu,lnx,,,+ rr+ uift, k#i </p><p>01) </p><p>where </p><p>a = lna,, a if: = h(aOaiPfJwift), </p><p>and t indexes time periods (t = 1,2,. . . , r). </p><p>The distributional assumptions on the random terms in (11) are </p><p> ;) ur, - N(0, 0:); uft - N(0, 1) where uft = (Q, . . . , u,,,); </p><p>(iii) r, is the non-positive value of a N(0, u,) variable; (iv) url, 7, and uft are independent of each other. </p><p>3.2.1. A single equation method </p><p>In stochastic frontier models the inputs xi and technical inefficiency r are assumed to be independent. In the present setup such an assumption is no longer valid since 7 affects input demand xi. This makes a single-equation method inconsistent. However, we can make the within transformation even though 7 is considered random [see Hsiao (1986)] in order to get rid of 7 in (11). </p><p>Let </p><p>qfr=~Y,,-wp f=l,..., F, </p><p>I,, = In Xi/, - In xif t=l,..., T, i=l ,--., n, </p></li><li><p>342 S. C. Kumbhaknr, Stochastic production and profit frontiers </p><p>where </p><p>ln r/= Cm Y,JK </p><p>In xif= Clnxi,,/T. t </p><p>Rewriting (11) we get </p><p>(1 - i)lifr - C &amp;Jkj, + In wjft - In Wif- In pfr + In j$= U+ - Q,, k#i </p><p>(12) </p><p>where </p><p>ln if= Cln wi/JT, 6,~ CvfJT, iii,= Cuift/T. t t t </p><p>Applying OLS to the first equation in (12) (which is the within transforma- tion of the production function) we can obtain a consistent estimator of a = (aI, a*, . . . , a,) since uft and url are independent by assumption. </p><p>We can do the following to estimate u,, u,, 2, u and 7. First, we obtain residuals from the first equation in (ll), </p><p>eft = In _Yft - Cgih Xift9 (13) </p><p>where ai is the OLS estimate of a. The method of moments can be used to estimate u, and u, in (13). 2 can then be estimated from the residuals of the last n equations in (11). By regressing er, on an intercept gives a^ - the OLS estimate of a. Since plim 6 = a + E(r) = a - a/&amp;,, we can form a con- sistent estimate of a as C = a^ + fi/&amp;$, where c?~ is the estimate of a,. Finally, we can estimate E,( = r,+ u,~ by &amp;ft = efl - a which can be used to estimate rf from the mean or mode of 7f conditional on ef, as develop in section 3.2.2 below. </p><p>When the 7/s are assumed to be fixed, one can estimate them together with other parameters only from the production function either by introducing firm dummies (suppressing the common intercept) or by making within transfor- mation and then recovering 7,s from the res...</p></li></ul>

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