On the Estimation of Technical and Allocative Inefficiency Using Stochastic Frontier Functions: The Case of U.S. Class 1 Railroads

Economics Department of the University of PennsylvaniaInstitute of Social and Economic Research -- Osaka University

On the Estimation of Technical and Allocative Inefficiency Using Stochastic FrontierFunctions: The Case of U.S. Class 1 RailroadsAuthor(s): Subal C. KumbhakarSource: International Economic Review, Vol. 29, No. 4 (Nov., 1988), pp. 727-743Published by: Wiley for the Economics Department of the University of Pennsylvania and Instituteof Social and Economic Research -- Osaka UniversityStable URL: http://www.jstor.org/stable/2526830 .

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INTERNATIONAL ECONOMIC REVIEW Vol. 29, No. 4 November 1988

ON THE ESTIMATION OF TECHNICAL AND ALLOCATIVE INEFFICIENCY USING STOCHASTIC FRONTIER

FUNCTIONS: THE CASE OF U.S. CLASS 1 RAILROADS*

BY SUBAL C. KUMBHAKAR1

1. INTRODUCTION

Since the publication of the pathbreaking article of Farrell (1957), a great deal of attention has been focused on the estimation of technical and allocative inefficiency in production. Increasing availability of micro data in recent years has facilitated growing popularity of the measurement of these inefficiencies. In the stochastic frontier production model as introduced by Aigner, et al. (1977) and Meeusen and van de Broeck (1977), technical inefficiency is introduced by a one- sided error (non-positive) along with the general statistical noise. Schmidt and Lovell (1979, 1980) generalized this to incorporate allocative inefficiency under the cost-minimization framework. They modelled allocative inefficiency by allow- ing a rcandomii disturbance in the first-order conditions for cost minimization. An alternative approach of modelling allocative inefficiency is to introduce a non- unitary factor of proportionality in the first-order conditions (see Lau and Yoto- poulos 1971; Levy 1981; Schmidt 1984). One advantage of the latter approach is that the existence of allocative inefficiency can be tested without making the first order conditions deterministic (see Schmidt and Lin 1984). In other words, in the latter approach allocative inefficiency can be separated from random errors in the first-order conditions. This is, however, not possible in the Schmidt-Lovell model. Separation of random errors in optimization and allocative inefficiency is impor- tant because the former comes from, e.g., measurement errors, uncertainty in input prices, and quality of inputs, etc. which are not under the control of the firm, whereas the latter comes from, e.g., managerial errors out of inertia, ignorance, etc. which are under the control of the firm. Thus lumping the effect of exogenous shocks like uncertainty together with measurement errors and allocative inefficiency into a single error term and labelling it as "allocative" inefficiency is somewhat questionable.

The purpose of this paper is to introduce a flexible functional form of the production technology that permits elasticity of output to vary across firms and to introduce allocative inefficiency separate from random errors in optimization.2 Maximum likelihood method of estimation is also developed in a panel data framework. This permits us to estimate input- and firm-specific allocative inef-

* Maniuscript received September 1986; revised February 1987. l I would like to thank two anonymous referees for their helpful comments and suggestions.

2 A similar procedure is explored in Schmidt (1984) where both technical allocative inefficiency are represented by fixed firm effects.

727



728 SUBAL C. KUMBHAKAR

ficiency along with technical inefficiency for each firm. As an empirical illustra- tioin we uise panel data on 42 U.S. Class 1 railroads (1951-1975). Because of the regulated nature of the industry we consider cost minimization as the objective of the individual roads. The entire sample period is divided into five subperiods to take into account possible structural changes due to change in regulatory environment over the span of 25 years. This allows us to make efficiency comparisons of each r-oad over the subperiods.

The paper is organized as follows. The econometric model is presented in the following section. Estimation issues are discussed in Section 3. Section 4 contains the empirical results. We summarize the results of the present study in Section 5.

2. THE ECONOMETRIC MODEL

We consider the generalized production function (GPF) developed by Zellner and Revankar (1969)

(1) F(y) = yeo-1, = I H X e r(- l ,

i = 1, 2, ..., n

where v is output, Xi are inputs, and xo, xoc are the parameters to be estimated. v is the general statistical noise introduced to capture the exogenous shocks un- known to the producer. (<0) is a one-sided random variable that represents differences in technical efficiency of the firms arising out of variation in the vintages of capital stock, entrepreneurial ability, etc. Thus T = 0 in (1) defines the stochastic production frontier as developed by Aigner, et al. (1977) and Meeusen and van den Broeck (1977).

Direct estimates of the production function parameters in (1) may be obtained by the maximum likelihood method. Introducing specific probability distributions for i' anid , assuming that z and v are independent and that Xi are exogenotus, the asymiptotic properties of the maximum likelihood estimators can be proved in the usual way. On the other hand, if inputs are endogenous and firms do have some knowledge about technical inefficiency, the direct estimation of (1) will result in inconsistent parameter estimates under expected profit maxi- mization behavior.3 Again, if the objective of the firm is to minimize cost when output is treated as exogenous (e.g., the regulated industries)) and inputs endogenous, the production function cannot be estimated directly. The alternative is to estimate either the cost function or estimate a system of equations consisting of the prodtuction function together with the equations representing the first- order conditions for cost minimization (see Schmidt and Lovell 1979, 1980; Kumbhakar 1987b). In the latter approach simultaneity is required for the inter- nal consistency of the model and not . simply to increase the efficiency of the parameter estimates, by exploiting the cross-equation restrictions that implicitly arise because the production (or cost) function parameters appear in the first- order conditions" (Schlmidt 1986, p. 309).

' See KoLmbhakar (1987a) for the appropriate estimation method.



ESTIMATION OF INEFFICIENCY 729

In this paper we assume that the objective of the firm is to minimize cost the reason being that we will be dealing with railroads which are regulated. Since prices of passenger and freight services are fixed by the regulatory bodies, in the short run profit can be maximized by minimizing cost.

A firm is said to be allocatively efficient if it equates the marginal rate of substitution between each pair of inputs with the input price ratio. Departure from this optimality condition can be explained by (i) consistent under or over- utilization of inputs resulting from the failure to minimize cost exactly because of other some institutional, structural, or managerial problems which we call allocative inefficiency; and (ii) uncontrolled random exogenous shocks such as uncertainty in input and output prices, quality of inputs, etc. Allocative inefficiency is modelled as

(2) = kK )e i 2, ..., n

where factor of proportionality ki are firm and input specific, ui are random errors in cost minimization, MPX are marginal products of Xi, and wi are input prices. Thus exact cost minimization (except for the random error) is a special case when ki = 1 (i = 2, ..., n) and non-unitary ki represents allocative inefficiency in the input pair (1, i). Specification (2) goes beyond Lau and Yotopoulos (1971), Levy (1981), Lovell and Sickles (1983) since we separate allocative inefficiency from random errors. Equations (1) and (2) constitute the main body of our econometric model in this paper.

One way of avoiding the above simultaneous equation system and yet getting consistent estimators of the production function parameters is to estimate the cost function. This follows from the fact that there is a unique relationship between the cost function and the production function parameters. Solving Xi from equations (1) and (2) yields the following input demand functions.

,,

(3) ln Xi = ai + (j/r - ij) ln k i 2

1 )' n 1 + ln F(y) + E (cty/r) ln (wj/wi) + E (/r - i)uj - - (c + v)

1 ~~j1 j =2 r

where

a' = ln c, - (ln o0 + L gj ln oc r,

1=1 ~ ~~ = r f = , 2c, and

(lj= if i =j j 1, n

/0 otherwisei=1, 2 ..,n




The input demand functions in (3) show that the presence of technical inefficiency increases input demand by -z/r percent. But the impact of allocative inefficiency on input demand is indeterminant. The percentage change in the demand for input i is

E (a/r) In kj - In ki - 0 depending on kj and c, (j = 2, ..., n).

However, whatever the effect is, it varies across inputs. Since any kind of inefficiency leads to an increase in cost, it is worth investi-

gating the impact of each of these inefficiencies on the cost of production. In this vein we derive the cost function

(4) In C= - (In x + EcY In 0j( + - In F(y) + - E or In w.

+ In oc 1 + EY jeuJ)?e + - I (j uj - (? + v) + E

where

E = In La1 + E gjl(k/ + - Z gj In ki-nIn al + E ocie"

The stochastic cost frontier is given by putting -E = ln kj = 0 in (4). Thus the presence of technical inefficiency increases cost by - -/r percent. E = 0 if kj= 1 (j = 2, n) which is the case where firms operate on the least cost expansion path. Non-negative value of E can be viewed as the percentage increase in cost due to allocative inefficiency.

The distinguishing features of the present modelling strategy lies in accommo- dating the first-order conditions for cost minimization, where statistical noise is separated from allocative inefficiency; and at the same time estimating both firm- and input-specific allocative inefficiency. Functional form of the production technology is assumed to be scale flexible. Though otherwise inflexible, the major advantage is that we can derive analytically the expressions for both technical and allocative inefficiency and increase in cost associated with each of these.4 Once the relevanit parameters are estimated, each of these components can be estimated separately. The only problem is that the effect of random error in cost-minimization (l1) cannot be disentangled from the cost of allocative inefficiency.

3. ESTIMATION

There are two approaches to estimation. One is to use the production function along with the first-order conditions of whatever the firm optimizes. The other one is to use either the cost function only or the cost function together with the

4 With more flexible functional forms such as the translog, the present methodology cannot be applied. This limits its applicability to a particulalr class of production function.




share equations (see Greene 1980; Melfi 1984; Bauer 1984; and Kumbhakar 1987b). In a single equation framework, one can estimate the cost function (4) by the OLS if the interest is in getting consistent estimates of 0 and ai. Since the moments of the eriors in (4) are intractable, there is no way by which one can manipulate them to obtain consistent estimates of the other parameters of interest.

To avoid this problem we consider the system involving equations in (1) and (2). Introducing the firm and time subscript f and t (with f= 1, ..., F and t = 1,

T) respectively in (1) and (2) yields (after taking logarithms):

f ln YTt? + 0vf,t = In go + Z ci n Xift + TI + lift

(S) }i In Xlft -In X't = In (oci) + ln (wiJft/wjft) + ln (kjf) + t1jft

We assume Tf to be a one-sided randonm variable and time invariant. Similarly,

kf are firm- and input-specific but invariant over time. Now we need to specify the probability distributions of the error vector in (5).

We assume that

(i) f is the non-positive value of a N(O, o7) variable,

(ii) l ft is i.i.d. N(0, L) where lift = (t12f ttzft)"

(iii) A f t is i.i.d. N(0, K)

(iv) yf , ift , and Aftt are independent among themselves.

With the above distlibtutional assumptions, the log likelihood ftunction5 (for a sample of F firms observed over T years) can be written as

FTii F(T -l) 112 F 2 a2 (6) LiTFI II(2 )-II T 1n2 --In (K? + T)T )

__ ~~~~~~~~~~FTI - ~~2ZEgA8f +?E In P(D-a,fJ)+FT In 2--In I n -I E- EZt E ftZ

t f f

' ftf

wlhete A = IT- U 211/(oa + ToQ), X (1 ..., 1)', IT is a T x T identity matrix,

J~~~ ( I,

1 it

? T

( *fi ... I with

'/'t = Tf + I ft f a Z r E ?ft/ .( a + Tu2)"'2

()( is the cuLmLlative density of a standard normal variable, and finally

(7) In X1 ft --In X2 -In i,2ft

+ In Vll ft -In a 1 + In a2 - In k2

ln X1,ft-In X,IJ.-ln ivnft+ln v,1.ft-ln a+ln cx, -ln k,,1 J

5 See Pitt anid Lee (198 1) for a derivation of the likelihood fuLnctioni where inlpUts are exogenious.




The computational burden of obtaining maximum likelihood estimates can be reduced by concentrating the above log likelihood function with respect to kif and S. At the point of maximum of L, we have

ln kif = ln X -lnf if lnw Wif + l n xw3f-l n c o+lnci (8) ~ d 1ZZZyZ;

(8) X~and E = FTEE Zft1 ~a~ FT f t

where Z*t = Zft - Zf, with

Zf = Y Zft/T, ln Xif = ln Xift/T, and ln Wjf = ln WJft/T. I t t

Substituting these values back into the log likelihood function Ll, the con- centrated log likelihood function can be written as

F(T - 1) 2 F 1 02 7 (9) L2 = const. - 2 In o7 -- In (a2 + Ta2)- 2 ZE ' Aef 2 2 v ~ ~~~~2 2U2 f f

?Zln D(-af) + FT ln r- 2 In FTI Z*t Z* f ft

f

which can be maximized to obtain estimates of a *-, ? 0, a2, and a2. The log likelihood function L2 is much simplified because Z* does not depend on the parameters. A typical element (ith) of Z* is

Z*t = ln X* -ln X* ln wIf*-lnw*t

where the '*' over a variable means deviation from its mean (over time). Thus the last term in (9) is a constant. Once ac, , ..., a, UV, and o2 are estimated, maximum likelihood estimates of kif and E can be estimated from (8).

Our next concern is how to estimate technical inefficiency, -Cf. To do so we consider the following Lemma first.

LEMMA 1. The conditional pdf of Tf given ef, f(f I ef) is a N(Mf, a2) variable truncated it 7ero where

f= 5 E eft/( v + Tol) and r2 = o v/(o1 + Tl).

PROOF. The proof follows directly from the definition off(zf I ef) and the joint pdf of (rf, ?f) and the marginal pdf of -, .6

The above result is an extension of the Jondrow, et al. (1982) result under a cost minimization hypothesis in a panel data framework. Now a point estimator of If can be obtained either by the mean or the mode of Tf . Thus either

(10) E(zf1?f) = f- _ _

6 See Kumbhakar (1986) for details.




or

Y Eft < 0

( 11l ) M(rfIl ?f ) = otherwise.

can be used as a point estimator of If. Q.E.D.

4. EMPIRICAL RESULTS

In the present study we use panel data on 42 U.S. Class 1 railroads.7 The data set covers the period 1951-1975. Output is an aggregate of freight-ton-miles and passenger-miles. Inputs used are capital, labor, and fuel. Capital is a weighted index of materials and equipment. The quantity index of fuel is a weighted average of eight different types of fuel. Finally, the labor quantity index is com- puted as a weighted log-change index of straight-time and overtime hours of seven occupational groups. The data appendix of Caves, Christensen, and Swan- soIn (1979) describes in detail the sources and construction of the present data set.

Because of the regulated nature of the railroad industry we follow the tradition of treating output as exogenous. As mentioned before, if output is exogenous and inputs endogenous, direct estimation of the production function is not appropriate. Because of this we use the maximum likelihood method to estimate the system in (5).

In estimating the model, allowance has been made for incorporation of exogenous technical progress (Hicks neutral technical progress) by introducing time as a regressor, the coefficient of which gives thc estimate of such technical progress.

We choose the GPF over the Cobb-Douglas (CD) specification of the production technology. Since the former is a special case of the latter one can formally test whether the functional form is CD by hypothesizing 0 = 0. The other advantage is that it does not restrict elasticity of output to be constant for all the roads. The elasticity coefficient ('i) is given by the expression i1 = (OCK + 01L

+ 0F) (1 + 0y), where subscripts in ot refer to the inputs capital, labor, and fuel respectively. So far as estimation of inefficiency is concerned the main advantage is that one can go back and forth from the production side to the cost side keeping things analytically tractable thereby being able to estimate technical and allocative inefficiency as well as increase in cost due to them separately.

To allow the possibility of structural change mainly due to change in the regulatory environment over the period of 25 years, the sample period is divided into five subperiods each covering a period of five years. This helps to compare technical and allocative inefficiency for each road over these subperiods, especially to see whether there is any tendency for the roads to move close to the frontier. Furthermore, the assumption of unchanging technical inefficiency over

I would like to thank Robert Windle of Christensen Associated for providing me with these data. A list of these railroads is given in the Appendix.




the period of 25 years is too restrictive. For comparison purposes only we estimate the model over the full period (1951-1975) which we call Model I. Models 1I-VI cover periods 1951-1955, 1956-1960, 1961-1965, 1965-1970, and 1971- 1975 respectively.

The maximum likelihood estimates8 of the production function parameters are presented in Table 1. Asymptotic standard errors are given in parentheses. First, we look at the estimates of 0. In all the models (except for Model VI) a non-zero value of 0 is statistically significant at any reasonable level of significance. Thus the CD specification is rejected. A close look at the parameter estimates for the Models 1I-VI reveal that there have been substantial changes in the parameter estimates over the 25-year period. Thus estimating one relation over the full period as in Model I is quite inappropriate. Finally, estimates of a 2 are quite small compared to G2 but are all statistically significant at any reasonable level of significance thereby showing that the presence of technical inefficiency cannot be ignored.

Table 2 reports estimates of elasticity for each road (evaluated at the mean value of output). Estimated elasticities vary quite substantially between railroads and are greater than unity for most of the roads. In Model I the minimum value is .41 (PC) and the maximum 1.47 (DSS). For other models (II-VI) the minimum and maximum values of elasticity are .57 (PPR), 1.24 (DSS); .60 (PPR), 1.24 (DSS); .62 (PPR), 1.22 (NWP); .56 (PC), 1.09 (NWP); and .77 (PC), 1.19 (CLN), respectively. However, it is to be noted that for none of these roads estimates of elasticity have changed much over the subperiods.

Since the main motivation behind the study of production frontiers is estimation of inefficiency, we now present the estimates of technical allocative inef-

TABLE I ESTIMATES OF THE PRODUCTION FUNCTION PARAMETERS IN DIFFERENT MODELS*

Model In io 0 IK OXL O(F IT Ur a%,

I -22.251 .030 .352 .889 .247 .055 .009 .365 (.316) (.001) (.048) (.051) (.041) (.032) (.003) (.038)

11 -19.41 .020 .391 .794 .078 .031 .050 .302 (.956) (.006) (.153) (.193) (.010) (.019) (.006) (.080)

III -19.390 .020 .338 .883 .027 .048 .048 .145 (.662) (.003) (.085) (.127) (.009) (.010) (.008) (.026)

IV -19.205 .019 .504 .557 .167 .076 .028 .235 (1.223) (.007) (.179) (.201) (.176) (.015) (.019) (.113)

V - 16.096 .010 .215 .675 .207 .028 .070 .012 (.379) (.001) (.065) (.366) (.031) (.007) (.017) (.009)

VI -18.745 .006 .445 .665 .099 .018 .029 .305 (2.023) (.005) (.141) (.438) (.062) (.013) (.013) (.174)

* Output is naeasured in 109 freight-ton-miles. Asymptotic standard errors are in parentheses.

8 Since some of the roads are observed for less than the full 25 years we allow T to vary with roads. If a road is observed for less than 3 years in any subperiod we drop it from that group.




ficiency. We use (10) as the point estimator of -u and drop the negative sign on z for convenience. In the case of GPF, it is not quite straight forward to give an interpretation of technical inefficiency r as in the CD case where r represents percentage decrease in output. Since in the GPF r represents percentage reduction in F(y), the percentage reduction in y is r/(1 + Oy), which can be evaluated at the mean value of y for each road to calculate the percentage reduction in

TABLE 2 ESTIMATES OF ELASTICITY AND TECHNICAL INEFFICIENCY

Elasticity ('i) Technical Inefficiency (T)

Railroad I II III IV V VI I II III IV V VI

ACL 1.11 1.03 1.03 .99 -- .24 .20 .12 .13 ATS .67 .73 .72 .70 .74 .92 .20 .13 .07 .10 .04 .12 B&O .82 .79 .81 .82 .85 1.03 .31 .16 .10 .12 .12 .10 BM 1.37 1.17 1.17 1.17 1.06 1.19 .41 .36 .31 .22 .49 .13

BN .50 _- _ .83 .06 .07 C&E 1.38 1.20 1.19 1.17 1.06 - .11 .15 .07 .12 .07 C&O .76 .75 .75 .77 .81 1.03 .10 .08 .04 .07 .01 .12 CB .95 .91 .93 .91 .89 - .23 .14 .10 .12 .08

CLN 1.38 1.20 1.20 1.18 1.06 1.19 .02 .05 .02 .04 .00 .04 D&H 1.35 1.15 1.16 1.16 1.06 - .10 .12 .06 .08 .08 -

DLW 1.31 1.14 1.15 -- .17 .22 .20 DRG 1.24 1.12 1.12 1.11 1.01 1.15 .08 .09 .04 .07 .01 .05 DSS 1.47 1.24 1.24 - -- .04 .08 .04 EL 1.06 - .99 .95 1.12 .32 - .20 .32 .12

ERI 1.16 1.04 1.06 - - .18 .21 .17 - - FEC 1.43 1.22 1.21 1.20 1.08 - .19 .30 .23 .16 .08 -

GN .97 .90 .93 .93 .92 - .20 .12 .07 .10 .14 - L&N .91 .96 .92 .88 .85 .99 .20 .15 .06 .10 .02 .11

MIL 1.00 .93 .95 .96 .93 1.09 .37 .21 .12 .15 .25 .11 MON 1.44 1.22 1.22 1.21 1.08 .09 .13 .10 .13 .22 MP .91 .94 .89 .86 .87 .20 .16 .07 .10 .02 NCS 1.40 1.19 - .09 .14 -

NKP 1.15 1.03 1.03 1.06 - .20 .16 .10 .13 - NP 1.08 1.00 1.00 .99 .95 - .28 .20 .10 .13 .26 NW .74 .91 .87 .76 .72 .92 .09 .09 .03 .05 .01 .08 NWP 1.46 1.23 1.23 .122 .109 - .05 .13 .06 .09 .12 -

NYC .68 .66 .71 .73 .77 - .26 .15 .12 .15 .13 -

PC .41 .56 .77 .06 .01 .12 PPR .57 .57 .60 .62 .69 - .15 .10 .06 .09 .06 -

RDG 1.27 1.10 1.10 1.11 1.04 1.18 .41 .24 .19 .18 .48 .18

RF 1.43 1.22 1.22 1.20 1.08 - .10 .16 .10 .11 .21 -

ROC 1.00 .97 .96 .96 .92 1.08 .27 .15 .08 .13 .06 .07 SAL 1.12 1.04 1.03 1.00 - .20 .16 .11 .12 - - SCL .77 .83 1.00 .14 .05 .10

SOO 1.26 1.11 1.02 .10 - .11 .06 SP .60 .74 .75 .63 .67 .84 .14 .15 .07 .07 .01 .08 SPS 1.39 1.19 1.19 1.18 1.06 - .07 .09 .05 .09 .13 -

T&P 1.31 1.14 1.15 1.13 1.04 - .22 .20 .11 .14 .13 -

UP .67 .72 .74 .73 .75 .90 .16 .10 .05 .09 .01 .07 VGN 1.32 1.15 .05 .06 -- WAB 1.23 1.09 1.10 1.08 .19 .17 .13 .15 - - WP 1.32 1.15 1.16 1.14 1.04 1.17 .10 .09 .05 .08 .08 .09




output due to technical inefficiency. The estimates of T for each road is given in the second half of Table 2.' One interesting aspect of these results is that road CLN is technically most efficient in all the models and BM is the most inefficient in all the models except in Models I and VI. Technical inefficiency for CLN has reduced over time reaching the frontier during 1966-1970. This, however, is not the case with other roads. Declining tendency of technical inefficiency is observed for roads ACL and FEC. Except for the last subperiod (1971-1975) such down- ward tendency is also observed for ATS, CB, and SP. It is to be noted that technical inefficiency has declined for almost all the roads from 1951-1955 to 1956-1960. Unfortunately, no such regularity is observed for other subperiods.

Instead of - it is more interesting to report ESTT (=,/(l + Oy), which shows the percentage reduction in output due to technical inefficiency. This is reported in the first part of Table 3. In Model I the minimum value of ESTT is .017 (PC) and the maximum is .378 (BM) thereby showing that the presence of technical inefficiency reduces output of the most efficient road by 1.7% and the least efficient road by 37.80. The corresponding figures for Models II-VI are 4.83% (C&O), 33.1%'-, (BM); 2%Yo (CLN), 28.9% (BM); 3.1% (NW), 20.870//0 (BM); 00%c, (CLN), 60.9% (BM); and 4.50%0 (CLN), 18.9% (RDG),-respectively. Because of the unique correspondence between the production and cost functions as we have noted in Section 2 the percentage increase in cost due to technical inefficiency is c/r. Estimates of such costs are given in the second half of Table 3. It can be seen tlhat in Model I for the most efficient road (CLN) cost is increased by 1.10%/o and for the least efficient road (BM) it is 26.71%(/o. Similarly, for Models JI-VI such costs for the most efficient road (CLN) are 3.85%, 1.87"/%, 3.26%/., 0'(Y, and 4.27', respectively. For the most inefficient roads such costs are 28.20% (BM), 24.92% (BM), 18.21% (BM), 49.03% (BBM), and 18.43% (RDG) respectively. Increase in mean cost due to technical inefficiency for these subperiods are 12. 15")0, 7.79%, 9.380r, 10.56?/, and 8.230o respectively.

Finally, we look at the estimates of allocative inefficiency in Table 4. We use capital as numeraire. Therefore kL and kF estimate whether capital/labor and

capital/fuel ratios deviate from the optimal proportions. Values of these proportionality factors greater than unity indicate excessive use of capital relative to labor and fuel. Similarly, values less than unity imply that capital/labor and capital/fuel are used less than optimal proportions. In all the subperiods and for most of the roads we observe that allocative inefficiency due to labor and fuel is greater than unity thereby showing that capital is excessively used relative to labor and fuel. This problem of over-capitalization is aggravated by the Interstate Commerce Commission regulation that no road can abandon unprofitable lines without its permission. That the rail industry is overcapitalized and operates with

9 Statistical test of goodness of fit does show that the assumption of z being a N(O, U2) truncated at

zero is not qulite utnr-easonable. We split the values of z into 4 cells 0 < T < .10, .10 < z < .20, .20 < T < .30, anid .30 < z anid calcuLlate the expected cell couLnts for Models II-VI based on z - N(0,

o 2) truncated at zero (l < 0). The /2 statistics of goodness of fit are 11.95, 11.74, 8.46, 10.01, and 5.98, whereas the critical valuLe of /2(4) at 1 level of significance is 13.28. ThuLs these inefficienicies do not

look too different fromii drawings froim a N(0, 12) trLncated at zero.




substantial excess capacity is well-documented by Keller (1974). Under the above circumstances it is not surprising to have excess of capital relative to labor and fuel.

Wide variations in allocative inefficiencies are observed among the roads not all being over-capitalized for each of the subperiods. However, for the industry as a whole over-capitalization is observed for each of the subperiods. The average

TABLE 3 ESTIMATES OF ESTT AND COST OF TECHNICAL INEFFICIENCY

ESTT =z/(1 + 0y) Cost = z/r

Railroad I II III IV V VI I II III IV V VI

ACL .17 .17 .09 .10 .16 .16 .09 .10 ATS .09 .08 .04 .06 .02 .11 .13 .11 .05 .08 .03 .09 B&O .17 .10 .06 .08 .09 .10 .21 .13 .07 .10 .11 .08 BM .38 .33 .29 .21 .61 .13 .27 .28 .25 .18 .45 .11

BN .02 - - .07 .04 - .06 C&E .10 .14 .07 .11 .07 .07 .12 .05 .09 .07 C&O .05 .05 .02 .05 .01 .12 .07 .06 .03 .06 .01 .10 CB .15 .10 .07 .08 .07 .16 .11 .08 .09 .07

CLN .02 .05 .02 .04 .00 .04 .01 .04 .02 .03 .00 .03 D&H .09 .10 .05 .08 .07 .06 .09 .04 .07 .07

DLW .15 .20 .18 - - .12 .18 .16 - DRG .06 .07 .03 .06 .01 .05 .05 .07 .03 .05 .01 .04 DSS .04 .08 .04 - .03 .06 .03 EL .23 - .16 .28 .12 .21 - .16 .29 .10

ERI .14 .17 .14 - - .12 .17 .14 - _

FEC .18 .29 .22 .15 .08 .13 .24 .18 .12 .07 GN .13 .08 .05 .08 .11 .13 .09 .06 .08 .12 L&N .12 .11 .05 .07 .01 .11 .13 .12 .05 .08 .01 .09

MIL .24 .16 .09 .12 .21 .11 .24 .17 .10 .12 .23 .09 MON .08 .13 .09 .13 .28 .06 .10 .08 .11 .26 M P .12 .12 .05 .07 .01 .13 .12 .06 .08 .01 NCS .09 .13 - - _ - .06 .11 - __

NKP .15 .13 .07 .11 .13 .13 .08 .10 NP .20 .14 .08 .11 .22 .19 .15 .08 .11 .24 NW .04 .06 .02 .03 .00 .08 .06 .07 .02 .04 .01 .07 NWP .05 .13 .06 .09 .12 .03 .10 .05 .07 .11

NYC .12 .08 .07 .08 .90 .17 .12 .10 .12 .12 PC .02 .01 .12 .04 - .01 .10 PPR .06 .05 .03 .05 .04 .10 .09 .05 .08 .05 -

RIDG .35 .21 .17 .16 .46 .18 .28 .19 .15 .15 .44 .15

RE .10 .15 .10 .11 .20 .07 .13 .08 .09 .19 ROC .18 .12 .06 .10 .05 .08 .18 .12 .06 .10 .05 .07 SAL .15 .13 .07 .10 .13 .12 .08 .10 SCL .07 - -- - .04 .10 .09 - - - .05 .08

SOO .08 - .10 .06 .07 .09 .06 S P .05 .09 .04 .04 .00 .08 .09 .12 .06 .06 .01 .07 SPS .06 .09 .05 .08 .12 .05 .07 .04 .07 .11 T&P .19 .16 .10 .13 .12 .15 .15 .09 .12 .11

U P .07 .06 .03 .05 .01 .07 .10 .09 .04 .07 .01 .06 VGN .05 .05 - .04 .05 -

WAB .16 .15 .11 .13 .13 .14 .10 .12 W P .08 .08 .04 .08 .08 .09 .06 .07 .04 .07 .08 .07




values of kL for Models II-VI are 1.04, 1.47, 1.02, 1.01, and 1.50. Similarly, for kF

these values for 1.56, 1.77, 1.59, 1.65, and 1.26 respectively. Thus on the average, the degree of allocative inefficiency in labor is less than that of fuel.

These allocative inefficiencies are reflected also in the cost function. The problem in calculating the cost of allocative inefficiency is that random errors ui cannot be disentangled, and it can act both favorably and unfavorably depending

TABLE 4 ESTIMATES OF ALLOCATIVE INEFFICIENCY IN LABOR AND FUEL

Allocative Inefficiency in Labor (kL) Allocative Inefficiency in Fuel (kF)

Railroad I 11 III IV V VI I II III IV V VI

ACL 1.56 1.11 1.14 1.05 - 1.43 1.58 1.21 1.32 ATS 1.43 .95 1.28 .85 .81 1.25 1.19 1.21 1.32 1.11 1.24 .51 B&O 1.95 1.20 1.51 1.26 1.14 1.75 1.46 .87 1.36 2.00 1.82 1.43 BM 1.76 .97 1.84 1.37 1.02 1.38 2.27 1.37 2.19 2.52 2.45 1.48

BN 1.22 - .88 1.84 - 1.14 C&E 1.86 1.17 1.65 1.16 1.21 2.08 3.50 2.53 2.16 2.08 C&O 1.51 .98 1.38 .93 .84 1.27 1.64 .95 1.98 1.88 2.07 .86 CB 1.32 .92 1.19 .78 .82 - 1.19 .98 1.36 1.19 1.54

CLN 1.54 .89 .97 .98 1.16 1.73 1.25 1.03 1.02 1.03 1.31 .99 D&H 1.25 .86 1.13 .89 .79 1.42 1.45 1.25 1.73 1.35 1.64 .73

DLW 1.06 .83 1.12 - 2.36 1.83 1.94 DRG 1.35 .88 1.16 .97 .99 1.07 1.35 1.21 .90 .83 .89 .29 DSS 1.40 1.18 1.38 2.57 2.37 1.78 EL 1.55 1.27 .87 .97 1.19 1.59 1.02 1.36 1.88 1.19

ERI 1.37 1.05 1.49 1.74 2.40 1.93 FEC 1.38 .93 1.34 1.57 1.11 - 1.59 1.42 1.79 1.87 1.85 - GN 1.29 .89 1.23 .88 .89 1.10 .87 1.26 1.20 1.35 L&N 1.49 .97 1.14 1.03 1.00 1.63 1.28 1.16 1.44 1.62 1.77 .85

MIL 1.66 1.10 1.45 1.02 .97 1.29 1.82 1.01 2.25 1.85 1.86 1.16 MON 1.79 1.58 1.83 1.24 1.19 1.31 2.58 2.48 2.28 2.14 -- MP 1.80 1.04 1.61 1.19 1.05 1.81 1.26 .87 1.29 1.49 1.83 .81 NCS .99 .86 .89 2.22 2.61 2.08

NKP 1.26 .97 1.26 .88 1.33 1.22 1.32 1.72 NP 1.12 .97 1.10 .70 .96 1.29 1.12 1.08 1.12 1.21 NW 1.48 1.03 1.13 1.01 1.07 1.28 1.77 1.10 1.26 1.42 1.96 1.70 NWP 1.64 1.07 1.45 1.24 .99 2.12 1.67 2.06 2.11 2.54

NYC 1.52 .99 1.43 1.06 .98 1.18 1.46 1.47 1.78 1.99 PC 1.61 - - - .91 1.25 1.74 - - 2.03 1.49 PPR 1.53 1.04 1.48 1.00 .88 1.97 1.09 2.21 2.75 2.13 RDG 1.37 .89 1.25 .88 .89 1.09 2.25 1.69 3.01 3.31 3.51 2.65

RF 1.34 1.45 1.02 .87 .95 1.62 2.76 1.50 1.28 1.29 ROC 1.85 1.11 1.65 1.00 1.16 1.65 1.46 1.27 1.38 1.19 1.40 1.04 SAL 1.41 .95 1.38 .94 1.61 1.92 1.54 1.27 SCL 1.66 .97 1.21 1.29 1.29 1.39

SOO 1.44 .97 .98 1.12 1.38 1.63 SP 1.59 .98 1.47 .94 .91 1.32 1.14 1.47 1.20 1.17 1.34 .98 SPS 1.34 1.10 1.46 .87 1.05 1.40 1.39 1.79 1.23 1.20 T&P 1.87 1.25 1.59 1.13 1.15 1.41 1.93 1.39 1.08 1.49

UP 1.58 1.05 1.44 .96 .89 1.28 .92 1.34 .94 .83 1.08 1.04 VGN 1.03 .89 .97 - l - 1.09 1.35 .173 - WAB 2.05 1.48 1.83 1.37 1.06 2.02 2.12 2.22 WP 1.62 .97 1.42 .99 1.05 1.26 1.57 1.75 .149 1.27 1.57 .99




on the values of ui which is not under the control of any road. However, to get some idea about the increase in cost due to allocative inefficiency we evaluate E defined in equation (4) at the mean value of u;. These estimates are reported in Table 5. For some of the roads cost of allocative inefficiency is greater than that of technical inefficiency. The roads having lower cost of technical inefficiency do not necessarily have lower cost due to allocative inefficiency. Individual roads vary substantially in terms of the cost of allocative inefficiency. For example, during 1951-1975 (Model II) NP performed best and C&E worst their cost of

TABLE 5 ESTIMATES OF COST OF ALLOCATIVE INEFFICIENCY

Railroad I II III IV V VI

ACL .23 .14 .15 .15 ATS .18 .10 .13 .10 .11 .15 B&O .30 .13 .17 .25 .22 .30 BM .37 .08 .24 .34 .31 .22

BN .15 .10 C&E .33 .26 .20 .25 .22 C&O .24 .10 .15 .18 .23 .16 CB .17 .09 .11 .09 .15

CLN .21 .07 .08 .10 .15 .28 D&H .18 .07 .10 .11 .16 .19

DLW .27 .11 .10 DRG .12 .07 .11 .06 .07 .11 DSS .25 .19 .15 _

EL .24 - - .11 .20 .16

ERI .27 .17 .17 FEC .25 .11 .14 .18 .22 GIN .15 .07 .12 .10 .12 L&N .20 .06 .11 .17 .20 .25

MIL .29 .12 .16 .19 .21 .18 MON .35 .30 .24 .28 .28 MP .25 .10 .19 .19 .21 NCS .20 .15 _ _

NKP .12 .08 .12 .16 NP .10 .05 .10 .08 .10 NW .16 .05 .10 .15 .24 .21 NWP .33 .12 .16 .26 .32

NYC .20 .08 .16 .19 .23 PC .31 .26 .19 PPR .29 .11 .17 .29 .27 RDG .35 .10 .13 .34 .37 .23

RF .21 .17 .09 .10 .10 ROC .28 .13 .20 .13 .17 .26 SAL .22 .13 .15 .13 - -

SCL .23 .12 .17

SOO .18 -_- .12 .17 SP .20 .08 .16 .12 .13 .18 SPS .19 .13 .16 .09 .09 T&P .25 .19 .19 .15 .18

UP .18 .10 .16 .10 .10 .17 VGN .08 .06 -_ _

WAB .34 .24 .30 .30 __

WP .23 .13 .15 .13 .18 .17




allocative inefficiency being 5.02% and 25.71 % respectively. For most of the roads costs of allocative inefficiency tend to increase over time. For the industry such costs are 12.03'S), 15.2%, 16.6%, 19.1Oo, and 20.4%0 for Models II-VI. We would like to mention that the estimates of allocative inefficiency and cost increase therefrom may not be definite, especially in the following situation. If the roads minimize cost in a dynamic framework and are allocatively efficient, the static measure that we have adopted here will give rise to allocative inefficiency.

Though the present analysis does not consider institutional factors that might affect efficiency, there are several things that indirectly validate the empirical results. First, we consider financial viability of the roads. We define a road to be financially viable if its capital is earning an opportunity cost (Keeler 1983, p. 4). One would expect efficiency to be directly related to the rate of returns on investment. For most of the railroads in our study estimates of such returns can be found in Keeler (1983, pp. 9-10). The roads which are rated high in terms of returns (CLN, DRG, NW, UP) are also rated high in terms of efficiency (have lower cost of inefficiency). The most efficient road in our study, CLN, enjoyed the highest rate of return in the post-sample period (1976-1979). Similarly, some of the roads rated 'in states of bankruptcy' are MIL and BM, which are also rated very poor in terms of efficiency. Costs of technical and allocative inefficiency for MIL over the whole sample period are 24.2% and 29.1% while for BM these figures are 28.90/o and 37/0. Moreover, some of the roads which were bankrupt during the 1970s, for example RDG and EL, were highly inefficient before 1970. Thus the estimates of inefficiency perform well as an indicator of financial viability. This can also be verified from the correlation coefficient between rates of return on investment and costs of technical inefficiency. These correlation coef- ficients for Models I-VI are -.607, -.656, -.562, -.762, -.585, and -.415 based on samples of 17, 17, 17, 18, 18, and 13, respectively. The figures are high enough to suggest a strong inverse relationship between inefficiency and rate of return oll investment.

Secondly, one would expect to see better performance from merger by "cost savings and marketing opportunities that may accrue to the combined system" (Adams 1978, p. 91). Thus it might be worthwhile to consider performance of the roads having major acquisitions/sales during the years 1951-1975. For example, the L&N acquired the Nashville, Chattanooga, and St. Louis in 1957, and in 1971 it also acquired the Chicago, Indianapolis, and Louisville. Compared to 1951-1955, its performance during the next three subperiods were very en- couraging. After 1971 its cost went up slightly, but on the average it was doing better that what it did during 1951-1955. Similar is the case with NW which has major acquisitions in 1959 and 1964. Its cost of technical inefficiency reduced substantially after 1960. During 1976-1979 its return on investment was above opportunity cost of capital (Keeler 1983, p. 9). The same is true for MP which had a major acquisition in 1956.

Finally, there is a dramatic reduction in technical inefficiency over time for FEC compared to other roads in this study. It fought against union work rules, wages, and labor practices which increased its cost considerably. It, finally, man-




APPENDIX

List of Railroads

Abbeviation Railroad Used

Atchison, Topeka. and Santa Fe ATS Atlatntic Coast Litne ACL Baltimore and Ohio B&O Boston and Maine BM BuLrlington Northern BN Chesapeke and Ohio C&O Chicago, BuLrlington, and Quincy CB Clhicago and Easterni Illinois C&E Chicago, MilwauLkee, and St. Paul MIL Chicago, Rock Island., and Pacific ROC Clinchfield CLN

aged to introduce work rules and wages different from those set by the unions. The pay-off is reflected in reduced technical inefficiency in the subsequent years. However, there are obvious limitations of suchv inferences from a single road, since there could be other factors for some of these differences.

5. CONCLUSION

In this paper we have considered specification and estimation of technical and allocative inefficiency in a cost-minimizing framework using panel data. Some distinguishing features of the model are: (i) the functional form used is flexible enouglh to estimate elasticity for each firm; (ii) allocative inefficiency arising from, e.g., managerial errors out of inertia, ignorance, etc. are separated from random errors representing measurement errors, uncertainty in input and output prices, quality of inputs, etc.; (iii) the panel nature of the data made it possible to estimate input- and firm-specific allocative inefficiency together with technical inefficiency; and finally (iv) division of the full sample period (1951-1975) into five stibperiods allowed to make efficiency comparison of the individual roads over subperiods and to check the stability of the parameter estimates.

The empirical results using 42 U.S. Class 1 railroads can be summarized as follows. (i) Estimate of elasticity of output varies substantially between roads, and the CD specification is rejected for all the subperiods except the last (1970-1975). (ii) The mean cost of technical inefficiency over the subperiods are 12.15%, 7.79/0, 9.38%', 10.56(%,, and 8.23%, respectively. (iii) In all the subperiods and for most of the roads we observe over-capitalization relative to labor and fuel. For the industry the degree of allocative inefficiency in labor is less than that of fuel. (iv) Individual roads vary substantially in terms of the cost of allocative inefficiency evaluated at the man values of random errors in cost-minimization (ui). For the industry such costs for the subperiods (Models 1I-VI) are 12.03%, 15.2%, 16.6(Y(,, 19.1 ̀ X?, and 20.4%0 respectively.

Unfit,ersity of Texas at Austini, U.S.A.




APPENDIX (Continued)

Abbeviation Railroad Used

Delaware and Hudson D&H Delaware, Lackawana, and Western DLW Denver, Rio Grande, and Western DRG DuLlutll South Shore DSS Erie ERI Erie-Lackawana EL Florida East Coast FEC Great Northern GN Loulisville and Nashville L&N Missouri-Pacific MP Monon (Chicago, Indianapolis, MON

and Loulisville) Nashville, Chattanooga, and St. Louis NCS New York Central NYC New York, Chicago, and St. Louis NKP Nor-folk and Western NW Northern Pacific NP Nor-thwesternl Pacific NWP Penn-Central PC Pennsylvania PPR Reading RDG Richmond, Fredricksburg, RF

and Potomac Seaboard Airline SAL Seaboard Coast Line SCL SOO Line SOO SoLIthern Pacific SP Spokane, Portlaid. and Seattle SPS Texas and Pacific T&P Union Pacific UP Virginia VGN Wabash WAB Western Pacific WP

REFERENCES

ADAMS, BROCK, Prospectuts for Chalntge in the Freiglht Rtailroad Industry, U.S. Department of Trans- portation (1979).

AIGNER, D. J., C. A. K. LOVELL, AND P. SCHMIDT, "Formulation and Estimation of Stochastic Fronitier Production Fuinction Models," Journal of Economnetrics 6 (1977), 21-37.

BAUER, P. W., "An Analysis of Multi-Product Firm Technology and Efficiency Using Panel Data," mimeo., Department of Economics, University of North Carolina at Chapel Hill (1984).

CAVES, D. W., L. R. CHRISTENSEN, AND J. A. SWANSON, "Productivity in U.S. Railroads, 1951-1974," Discussion Paper 7909, Social System Research Institute, University of Wisconsin at Madison (1979).

FARRELL, M. J., Tlhe Measurement of Productive Efficiency, Journial of the Royal Statistical Society Series A, 120 (1957), 253-290.

GREENE, W., "On the Estimation of a Flexible Frontier Production Model," Journal of Econometrics 13(1980), 101-116.

JONDROW, J., C. A. K. LOVELL, I. S. MATEROV, AND P. SCHMIDT, "On the Estimation of Technical Inefficiency in the Stochastic Frontier Production Function Model," Jour-nial of Econiometrics 19 (1982), 233-238.

KEELER, T. E., Railroads, Freigiht, ansd Puiblic' Policy, The Brookings Institution (1983). , "Railroad Costs, Returns to Scale, and Excess Capacity," Review of Econiomtiics and Statistics

LVI (1974), 201-208.




KUN1BHAKAR, S. C., "The Specification of Technical and Allocative Inefficiency in Stochastic Pro- duction and Profit Frontiers," Joturna til of Econo)netrics 34 (1987a), 335-348.

"Production Frontiers and Panel Data: An Application to U.S. Class 1 Railroads," Journal of Business andti Econiomtiic Statistics 5 (1987b), 249-255.

"Estimatioin of Inefficieincy in Production: Methodological and Empirical Issues," unpub- lislhed Ph.D. dissertation, University of Southern California (1986).

LAU, L. J. AND P. A. YOTOPOULOS, "A Test for Relative Efficiency and Application to Indian Agricul- ture," Aiiiericani Econiomiic Reviewv 61 (1971), 94-109.

LEVY, V., "On Estimating Efficiency Differential between the Public and Private Sectors in a Devel- opinlg Ecoinomy - Iraq, Journal of Comnparative Econiomnics 5 (1981), 235-250.

LOVELL, C. A. K. AND R. SICKLES, "Testing Efficiency Hypothesis in Joint Production: A Parametric Approach," Review of Econoinics tanid Statistics 65 (1983), 51-58.

MEEUSEN, W. AND J. VAN DEN BROECK, "Efficiency Estimation from Cobb-Douglas Production Func- tions with Composed Error," Initernlationial Economiic Review 18 (1977), 435-444.

MELFI, C. A., "Technical aind Allocative Efficiency in Electric Utilities," Working Paper, Department of Economics, Indiana University (1984).

PITT, M. AND LUNG-FEI LEE, "The Measurement and Sources of Technical Inefficiency in the Indo- nesian Weaving Industry," Joirnual of Developmnent Economlics 9 (1981), 43-64.

SCHMIDT, P., "Estimation of a Fixed-Effect Cobb-Douglas System Using Panel Data," unpublished paper, Michigani State University (1984).

,"Frontier Pr-oductioin Fulnctions," Econometric Review 4 (1986), 289-328. AND C. A. K. LOVELL, "Estimating Technical and Allocative Inefficiency Relative to Sto-

chastic Production and Cost Frontiers," Jour nail of Economnetrics 9 (1979), 343-366. 'Estimating Stochastic Production and Cost Frontiers When Technical and Allocative Inef-

ficiency are Correlated," Journabll of Economnetrics 13 (1980), 83-100. AND T.-F. LIN, "Simple Tests for Alternative Specifications in Stoclhastic Frontier Models,"

Journal of Econ7om0ietrics 24 (1984), 349-361. ZELLNER, A. AND N. S. REVANKAR, 'Generalized Production Functions," Review of Economnic Studies

36 (1969), 241-250.



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On the Estimation of Technical and Allocative Inefficiency Using Stochastic Frontier Functions: The Case of U.S. Class 1 Railroads