Journal of Econometrics 34 (1987) 335-348. North-Holland

THE SPECIFICATION OF TECHNICAL AND ALLOCATIVE INEFFICIENCY IN STOCHASTIC PRODUCTION AND

PROFIT FRONTIERS*

Subal C. KUMBHAKAR

University of Texas at Austrn, TX 78712, USA

Received June 1985, final version received May 1986

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.

1. Introduction

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].

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-

*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.

0304-4076/87/$3.5001987, Elsevier Science Publishers B.V. (North-Holland)

336 XC. Kumbhakar, Stochastic production and pro$t frontiers

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.

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.

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.

2. Specification of technical and allocative inefficiency

We begin with the following Cobb-Douglas production function

y=A nxl?l exp(u), ( 1 i

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.

The production function (1) can easily be related to the frontier production

function by specifying A as

A = aOexp( T), 7 IO, (I’)

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

S.C. Kumbhakar, Stochastic production and profit frontiers 331

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).

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

wi =p . MP,;, (2)

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’

wi=p-MP,,-exp(ui). (2’)

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.

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.

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

‘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.

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.

% is assumed to be i.i.d. N(0, CT,‘).

338 XC. Kumbhakar, Stochastic production and profit frontiers

assumptions in place the first-order conditions for profit maximization in (2’) are4

k+i

where

a, = ln( ariaOp/wi),

Solving the (n + 1) equations (x,, x2,. . ., x,, _Y) gives

In xi = &lnbW) +

1 n +~,~l(aj+(1-r)6ij)'j+ j&”

i=1,2 n. ,-..,

in (3) and (1) for the (n + 1) unknowns

and

lna, lny=l_r+

r In p

- + j& ,$lajln(ajlwj) l-r

+

(4

(5)

where

Sij= 1 if i =j,

= 0 otherwise,

and r = cyZt=,aj which is less than one under the assumptions of perfect competition and profit maximization [Nerlove (1965, p. S)].

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

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.

XC. Kumbhakar, Stochastic production and profit frontiers 339

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).

For a profit-maximixm g firm the profit function [using (4) and (5)] is

~IT(P,W,O,~,U1,U2,...,~,)

=py - w’x

*“(p,w,v)=(p~o)“(‘-‘~ fi(~~/w~)~‘/‘~-~) {exp(u)-r}. (7) i i-1 1

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

Similarly, allocative inefficiency alone reduces (median) profit by

~,Jp,Vi,+,...,u,)

‘See also Lovell and Sickles (1983) for a similar conclusion.

!fro.

(8)

340

where

XC. Kumbhakar, Stochastic production and profit frontiers

Q = ( pa,)“(‘-‘) (9)

3. Estimation

In this section we initially consider the estimation issues in a single cross-section and then examine them in a panel data context.

3.1. Estimation using cross-section data

From the first-order conditions for profit maximization in (3) along with the production function in (1) and (1’) we get

In y,= a + Ccyiln xtf+ T,+ u,, i

(l---~~~)lnx~,=a~+ ~a,lnxk,+7,+Uj,, i=1,2 ,..., n, (10) k#i

where

a=lna, and a, = ln( aiaop/wi).

The subscript f indexes firms (f= 1,2,. . . , F). The system of equations in (10) is different from Zellner et al. (1966) since T

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.

3.2. Estimation using panel data

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)

6The log likelihood function is a special case (T= 1) of (16) - derived in the context of panel data in section 3.2.2.

‘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.

S. C. Kumbhakar, Stochastic production and profit frontiers 341

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.

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

(1 - cY,)lnx+= a;,,+ C cu,lnx,,,+ rr+ uift, k#i

01)

where

a = lna,, a if: = h(aOaiPfJwift),

and t indexes time periods (t = 1,2,. . . , r).

The distributional assumptions on the random terms in (11) are

’ ;) ur, - N(0, 0:); uft - N(0, 1) where uft = (Q, . . . , u,,,)‘;

(iii) r, is the non-positive value of a N(0, u,‘) variable; (iv) url, 7, and uft are independent of each other.

3.2.1. A single equation method

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).

Let

qfr=~Y,,-wp f=l,..., F,

I,, = In Xi/, - In xif’ t=l,..., T, i=l ,--., n,

342 S. C. Kumbhaknr, Stochastic production and profit frontiers

where

ln r/= Cm Y,JK

In xif= Clnxi,,/T. t

Rewriting (11) we get

(1 - “i)lifr - C &Jkj, + In wjft - In Wif- In pfr + In j$= U+ - Q,, k#i

(12)

where

ln ‘if= Cln wi/JT, 6,~ CvfJT, iii,= Cuift/T. t t t

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.

We can do the following to estimate u,‘, u,“, 2, u and 7. First, we obtain residuals from the first equation in (ll),

eft = In _Yft - Cgih Xift9 (13)

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/&,, we can form a con- sistent estimate of a as C = a^ + fi/&$, where c?~ is the estimate of a,. Finally, we can estimate E,( = r,+ u,~ by &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.

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 residuals. Both will give consistent estimates of 7 when both F and T-, 00 [see Schmidt and Sickles (1984)]. When 7,s are assumed random, consistency of the estimates of 7, require T + 00 as proved in Proposition 2 below.

XC. Kumbhakar, Stochastic production and proJit frontiers 343

3.2.2. The maximum likelihood method

In the following section we develop the maximum likelihood method. Given a cross-section of F firms observed over T periods, the composed error vector in (10) can be written as

Tf + uft

7f + Ulft Ll Tf+UZft .

Tf + Unft

(14)

In this section we drop the subscript f. We define zll, .z2* as zll = r + u, and zzl=Zr+u, where f,,,,,=(l,l,..., 1)’ to yield f(zlr,Z21,7)=f(uI,lr+ut,~)

[where f( .) indicates the joint pdfl since the Jacobian of the transformation is unity. With 2, = (zn,. . ., zlT)’ and Z, = (Zig, Zig,. . . , zZT)’ the joint pdf of Z,, Z, is

2 g(zl, z2) = (2m)(“T+T+1)/2auTa7,Z,T/2

+ C(h- h)‘X1(z2, - 17) + T’/u,’ dr t

2a,exp( -a */2) =

(2r) w+w24.g4T/2 @P( -k/e*>3

where

p, = u: cz,t/u,’ + CZ’Pzzt 3 1 t t 1

a* = CzQu,‘+ ~z3-‘z2, - a: CqJu, + CI’Z-‘Z2, 2, t t 1 t t 1

05)

and @( .) is the cumulative pdf of a standard normal variable.

344 S. C. Kumbhakar, Stochastic production and profit frontiers

Given a sample of F firms, the log likelihood function can be written as

L(% FT( n + 1)

~~,a,~,a,~,E)=Fln2- 2 ln(2a) - Tlnet

+ cln@( -p7/u*) + FT ln( 1 - ccx,), (16) f I

where zlft, zZi,, in a; and CL,, are to be replaced by their non-stochastic counterparts from eqs. (11) and (1 - &I,) is the Jacobian of the transforma- tion from (zu, zzt) to (In y,,ln x1,, . . . , In x,~).

Maximization of the above log likelihood function yields consistent and efficient estimates of the parameters CQ, a, u,“, u,‘, and 2. Given these estimated parameters we want to estimate 7 and u,. To do this we need to find the conditional pdf of 7 first. Once r is estimated U, can be estimated from eqs. (11).

Proposition 1. The conditionalpdf of r given Z,, Z2 is N(pL,, u:) truncated at zero.

Proof

h(W,, Z,) =f(Z&L ~)/g(-?J,)

2 (24 (T(n+1))/2,,"T,+,T/2

= (24 'T+'+"T'/2u,~u7,q T/2 2u,exp( -LZ*/~)@( -11.,/u*)

Xexp(-a*/2-(7-pT)2/2u:),

where

u: = + [l/u; + l/( Tu;) + Z’Z-‘11 -l,

/J, = u: CZ,Ju, + ~19-‘z,, . [ t t 1 Therefore,

h(TIZ,, Z,) = /@+*A - I%/u*)

exp(-(+r-k)2/2c*2)Y 07)

7 IO,

which is the pdf of a N(p,, u:) variable truncated at zero.

XC. Kumbhaknr, Stochastic production andprojt frontiers 345

The mean or mode of the conditional distribution of 7 can be used as a point estimator of 7. Thus,

where $( .) is the pdf of a standard normal variable, or

W%J2)=C1, if Cz,Ju,’ + Cl’E-lzzI IO, I I

= 0 otherwise, (19)

may be used as point estimator of 7. This is in the spirit of Jondrow et al. (1982) extended to the framework of panel data and the profit maximization hypothesis. For the more general case where 7 is the non-positive value of a N( b, u,‘) variable, the conditional pdf of r given Z,, Z, can similarly be derived (see appendix).

Proposition 2. The estimators defined in (17) and (18) are consistent (as T+ 00).

Proof

and

plim p, = plimu,2 Czi,/u,’ + Cl’z-’ Zzt t f 1

Therefore,

plim M( 7lZi, Z,) = 7,

if

7z1Ju:+ ~I’PZ,,IO, t

and plimE( TlZr, Z,) = 7.

5. summary

In this paper we have developed techniques for the estimation of technical and allocative inefficiency along with the production function parameters under the behavioral assumption of profit maximization. The methods of

346 XC. Kumbhakq Stochastic production andprojt frontiers

estimation include the production function and the first-order conditions for profit maximization. Since the production function is stochastic, so is profit. Consequently, the concept of ‘profit maximization’ has to interpreted differ- ently. We have considered maximization of the median value of profit as the behavioral objective of producers. The resulting input demand and output supply functions show that the presence of technical inefficiency reduces both input demands and output supply. This in contrast to the cost-minimization case of Schmidt and Love11 (1979), where input demand is increased due to technical inefficiency. One major advantage of the present approach is the assumption that the input demands are not independent of technical in- efficiency.

The maximum likelihood method of estimation is developed under (i) the original Aigner et al. (1977) framework, where technical inefficiency is the non-positive value of a normal (0, u,‘) variable, and (ii) a generalized case where technical inefficiency is the non-positive value of a normal (b, u,‘) variable [Stevenson (1980)].

Estimation methods are discussed in a single cross-section as well as in a panel data framework. Consistency of the estimates of technical inefficiency is proved in the context of panel data (as T + CO) both in (i) and (ii).

Appendix

Proposition A.1. If r is the non-positive value of a N(b, u,‘) variable truncated at zero, then the conditionalpdf of r given Z,, Z,, h(r[Z,, Z,) is N(p:, a**) truncated at zero.

Proof: Following the outline of the text we can find

a*exp(-b*/2)@(-p;/u*)

g(zl’ z2) = (2,)T(n+1)/2u~u~,B,T/2~( _b,u,) ’

where

cz,‘,/u,’ + ~I’Z:-1z21 + b*/uz - u**m**, t t I

(A-1)

(A-2)

m* = ~z,Ju, + ~I’E-1~2r i- b/u: , t t 1

SC. Kumbhakar, Stochastic production and profit frontiers 347

Now

@WV%) x a*exp(-b*/2)@(-pT/a*)@(-b/u,)

Xexp( -b*/2 - (T- ~:)~/20*‘)

1 = G-*g(_pt,u*)exp(-(~-~~)2/2u*2)r (A.3)

which is a N(pT, Use) variable truncated at zero.

The point estimators, similar to (18) and (19) of the text, are

E( 71Z1, Z,) = p; - u* &Wu*)

@(-IWu*) ’ (A.4

and

M(7jZ1,Z2)=~f if m*IO, (A.5)

= 0 otherwise.

Proposition A.2. The estimators defined in (A.4) and (A. 5) are consistent (as T-, 60).

Proof

plimu*2 = 0 (as in Proposition 2 of the text).

Therefore

F,/u,’ + ~K-t2JT+ b2/( 7’~;)

plirnpz = plim 1/u;‘+ l/(Tu;) + l’FIZ

+/uu’+ Lq

= (l/u;+ IL-‘I) =I-.

J.Econ-E

348 S.C. Kumbhokar, Stochastic production ondproJit frontiers

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