Stochastic estimation of firm technology, inefficiency, and productivity growth using shadow cost and distance functions

Journal of Econometrics 108 (2002) 203–225www.elsevier.com/locate/econbase

Stochastic estimation of rm technology,ine$ciency, and productivity growth using

shadow cost and distance functionsScott E. Atkinsona ;∗, Daniel Primontb

aDepartment of Economics, University of Georgia, Athens, GA 30602, USAbDepartment of Economics, Southern Illinois University, Carbondale, IL 62901, USA

Received 13 February 1998; received in revised form 12 September 2001; accepted 13 November 2001

Abstract

It is well-known that a rm’s technology, allocative e$ciency, technical e$ciency, and pro-ductivity growth can be estimated using a shadow cost system, comprised of a shadow costfunction and its share equations, whose arguments are outputs and shadow input prices (pricesinternal to the rm). We provide a dual characterization to estimate these measures using ashadow distance system. This system is comprised of a shadow distance function, expressed interms of shadow input quantities and output quantities, plus the rst-order conditions from thecost-minimization problem. An advantage of the distance system over the cost system is that weobtain direct estimates of input ine$ciency with the former, but indirect estimates with the latter.We also show how to express cost function derivatives in terms of distance function derivatives,which allows calculation of returns to scale and price elasticities of demand from the estimateddistance system. Using panel data on US electric utilities, we estimate both systems and nd astrong similarity between their associated measures. c© 2002 Elsevier Science B.V. All rightsreserved.

JEL classi&cation: C13; C33

Keywords: Distance and cost frontiers; Allocative ine$ciency; Technical ine$ciency; Productivity change;Technical change

1. Introduction

Stochastic estimation of a shadow cost function, expressed in terms of shadow inputprices and outputs, has been addressed by Atkinson and Cornwell (1994) and others.

∗ Corresponding author. Tel.: +1-706-542-1311; fax: +1-706-542-3376.E-mail address: [email protected] (S.E. Atkinson).

0304-4076/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0304 -4076(01)00133 -6

204 S.E. Atkinson, D. Primont / Journal of Econometrics 108 (2002) 203–225

Shadow prices (internal to the rm) may diCer from market (actual) prices. Using ashadow cost system, where actual costs and shares are expressed in terms of shadowcosts, one can fully characterize the rm’s behavior by estimating the allocative e$-ciency (AE) and technical e$ciency (TE) of the rm, productivity change (PC), returnsto scale, and price elasticities of demand for inputs. One can further decompose PCinto e$ciency change (EC), which measures improvement relative to the frontier rm,and technical change (TC), which measures the shift in the frontier itself.A number of papers have also estimated the dual input or output distance function.

Among them are Grosskopf et al. (1997), Coelli and Perelman (1999), and Reinhardand Thijssen (1997). However, these studies do not establish the set of dual relation-ships between input distance and cost functions that would allow full characterizationof the rm’s behavior.Therefore, in this paper, we derive these fundamental dual relationships between a

shadow cost function and a shadow distance function. Using these dualities, we canfully characterize the rm’s behavior using a cost system or a shadow distance system.The shadow distance system is comprised of a shadow distance function, expressed interms of shadow input quantities and output quantities, plus the rst-order conditionsfrom the cost-minimization problem. One advantage of the shadow distance systemis that it allows direct estimation of the impact of allocative ine$ciency on relativeinput utilization. This is typically of greater interest than the direct estimation of relativeshadow prices using a shadow cost system. After appending error terms to the equationsin our cost and distance systems, we utilize panel data on US electric utilities toestimate each system. Since at least one of the variables in these regression modelsmust be endogenous, we use the Generalized Method of Moments (GMM) to testthe validity of our overidentifying restrictions. We nd a strong degree of similaritybetween the AE, TE, EC, TC, PC, returns to scale, and price elasticities of demandcomputed from each system.The remainder of this paper is organized as follows. In Section 2, we derive the

shadow cost and distance systems. In Section 3, we obtain the necessary equations toobtain scale economies and elasticities of substitution from the distance system. Forboth systems, in Section 4 we examine issues in the econometric estimation of AE,TE, and PC. An application to a panel of US utilities is presented in Section 5, andconclusions follow in Section 6.

2. Modelling allocative ine�ciency

In this section we show that shadow cost and shadow distance systems can bespeci ed to measure AE. First, consider the shadow cost system.

2.1. A shadow cost system

The (N × 1) input vector is denoted by x = (x1; : : : ; xN )∈RN+ and the output vector

is denoted by y = (y1; : : : ; yM )∈RM+ : The input requirement set is given by

L(y) = {x : x can produce y}: (2.1)

S.E. Atkinson, D. Primont / Journal of Econometrics 108 (2002) 203–225 205

Assuming shadow cost minimization, we obtain the shadow cost function as

C(y; p∗) = minx

[p∗x : x∈L(y)

]; (2.2)

where p∗=[p∗1 ; : : : ; p

∗N ]= [k1p1; : : : ; kNpN ] is a (1×N ) vector of shadow prices ∈RN

+.The kn parameters, n= 1; : : : ; N , measure divergence of actual from shadow prices forthe rm. Let h(y; p∗) be the cost-minimizing input vector that solves the problem in(2.2). Then p∗ is the price that equates h(y; p∗) to x, the actual input vector. Withpanel data, the kn parameters can be made time and rm-speci c.Consider the Lagrangian problem

L= p∗x− [D(y; x)− 1]; (2.3)

(where D is de ned in (2.9)) and use Shephard’s lemma to obtain

@C(y; p∗)@pn

= hn(y; p∗); n= 1; : : : ; N; (2.4)

where the notation @C(y; p∗)=@pn indicates the partial derivative of C(y; p) with respectto pn, evaluated at p∗

n .We use (2.4) to de ne actual costs, CA, as

CA =N∑

n=1

pnxn =N∑

n=1

pnhn(y; p∗) =N∑

n=1

pn@C(y; p∗)

@pn: (2.5)

Since C(y; p∗) is linearly homogeneous in p∗, via Euler’s theorem, the unobservedshadow cost function is

C(y; p∗) =N∑

n=1

p∗n@C(y; p∗)

@pn: (2.6)

Subtracting (2.6) from (2.5), we obtain actual costs in terms of shadow prices as

CA = C(y; p∗) +N∑

n=1

(1− kn)pn@C(y; p∗)

@pn: (2.7)

Given a Nexible functional form approximation to C(y; p∗), one can estimate alloca-tive ine$ciency using the N input demand equations in (2.4) jointly with the log of(2.7), after appending error terms to each equation. We term this set of equations thestochastic shadow cost system.Formulas for other statistics of interest, including elasticities of substitution and price

elasticities of demand for inputs in terms of shadow prices, are provided in Atkinsonand Halvorsen (1984). For the case of multiple outputs, returns to scale are computedfor the shadow cost (SC) system as

RTSSC =1∑

m(@ lnCS=@ym)ym

: (2.8)


2.2. A shadow distance system

The input distance function is de ned as

D(y; x) = sup�{�: (x=�)∈L(y)}: (2.9)

One might obtain non-parametric estimates of L(y) using linear programming DataEnvelopment Analysis techniques discussed in FOare et al. (1994) or Coelli et al. (1998).Here, however, our approach is parametric.We now reverse the roles of shadow prices and input quantities employed to derive

the shadow cost function. Assuming cost minimization, we alter (2.2) to obtain

C(y; p) = minx{px : D(y; x)¿ 1}; (2.10)

and let x∗ = [k1x1; : : : ; kN xN ] be the (N × 1) vector of shadow input quantities thatsolves the minimization problem in (2.10). In contrast to (2.2), we assume that actualinput prices equal shadow input prices, so that the rm’s problem is now characterizedin terms of “shadow input quantities” rather than shadow prices. In the shadow costmodel, actual and shadow costs typically diverge. However, with the shadow distancemodel, costs to society and shadow costs to the rm are equivalent. All costs are borneby the rm, whether or not it achieves AE. Hence, we refer to “cost minimization”rather than “shadow cost minimization” with this model.The rst-order condition corresponding to (2.10) is

pn = @D(y; x∗)

@xn; n= 1; : : : ; N; (2.11)

where is the Lagrangian multiplier and @D(y; x∗)=@xn indicates the partial deriva-tive of D(y; x) with respect to xn, evaluated at x∗n. Multiplying both sides by x∗n andsumming we obtain∑

l

plx∗l = ∑l

@D(y; x∗)@xl

x∗l : (2.12)

Since D(y; x∗) is linearly homogeneous in x∗, via Euler’s theorem,∑l

@D(y; x∗)@xl

x∗l = D(y; x∗): (2.13)

Since by assumption

1 = D(y; x∗); (2.14)

we can write (2.12) as

=∑l

plx∗l = C(y; p): (2.15)

This allows us to express the nth equation in (2.11) as

pn =

(∑l

plx∗l

)@D(y; x∗)

@xn; n= 1; : : : ; N: (2.16)


The rm might not use cost-minimizing amounts of inputs for a variety of reasons,including: (1) satis cing behavior; (2) actions by labor unions to limit labor input;(3) shortages of other inputs; (4) production delays; (5) regulated production; and(6) production quotas or target levels. We can estimate AE using the N equations in(2.16) plus (2.14), after appending error terms to each. We term this set of equationsthe stochastic input shadow distance system.

3. Elasticities of scale and substitution

Blackorby and Diewert (1979) establish the duality between expenditure, distance,and indirect utility functions as second-order approximations to each other. FollowingPrimont (1996), we extend these results to distance and cost functions by derivingelasticities of scale and substitution, de ned as partial derivatives of the cost function,in terms of derivatives of the shadow distance function. This allows us to estimatethese elasticities using estimates of only the shadow distance function.Our analysis requires a more compact notation. Thus, we reexpress (2.14) and the

rst-order conditions for the cost-minimization problem in (2.11) more succinctly as

D(y; x∗) = 1; (3.1)

∇xD(y; x∗) = p: (3.2)

It was rst shown by Shephard (1953), and again in (2.15), that the optimal value ofthe Lagrangian multiplier, (y; p), is equal to the cost function, C(y; p). In addition,Shephard’s Lemma states that x∗(y; p) = ∇pC(y; p). Thus, (3.1) and (3.2) can berewritten as

D(y;∇pC(y; p)) ≡ 1; (3.3)

C(y; p)∇xD(y;∇pC(y; p)) ≡ p: (3.4)

Clearly, (3.3) and (3.4) are identities in (y; p) and so we may diCerentiate them, rst with respect to prices and then with respect to outputs. Letting Rx∗ = ∇pC(y; p)and diCerentiating with respect to p we get

∇xD(y; Rx∗)∇ppC(y; p) = 0N ; (3.5)

∇xD(y; Rx∗)′∇pC(y; p) + C(y; p)∇xx D(y; Rx∗)∇ppC(y; p) = I; (3.6)

where 0N is a row vector of N zeroes, I is the N × N identity matrix, the Hessianmatrices are N × N , and all other vectors are row vectors. Eqs. (3.5) and (3.6) maywritten as[

0 ∇xD(y; Rx∗)∇xD(y; Rx∗)′ C(y; p)∇xxD(y; Rx∗)

][∇pC(y; p)

∇ppC(y; p)

]=

[0NI

]: (3.7)

Computing C(y; p)=∑

n pnxnkn, the bordered Hessian matrix in (3.7) may be invertedto yield the rst- and second-order derivatives of the cost function with respect to input


prices. Using the estimated distance function, we can compute own and cross-priceelasticities of demand as (@xn(y; p)=@pl)(pl=xn(y; p)), using the result that the typicalelement of ∇ppC(y; p) is @xn(y; p)=@pl; n; l= 1; : : : ; N .Next we diCerentiate the rst-order conditions with respect to the output vector. This

yields

∇yD(y; Rx∗) +∇xD(y; Rx∗)∇pyC(y; p) = 0M (3.8)

∇xD(y; Rx∗)′∇yC(y; p) + C(y; p)

[∇xyD(y; Rx∗) +∇xxD(y; Rx∗)∇pyC(y; p)] = 0N×M : (3.9)

After some rearrangement this becomes[0 ∇xD(y; Rx∗)

∇xD(y; Rx∗)′ C(y; p)∇xxD(y; Rx∗)

][ ∇yC(y; p)

∇pyC(y; p)

]=

[ −∇yD(y; Rx∗)

−C(y; p)∇xyD(y; Rx∗)

]:

(3.10)

If the same bordered Hessian is inverted we recover the rst derivatives of the costfunction with respect to outputs and we get the second-order cross partial deriva-tives with respect to input prices and outputs. Using (3.10), we can calculate returnsto scale from the estimated distance function by computing C(y; p)=(∇yC(y; p)y). Asa simpler alternative we compute returns to scale from the shadow distance (SD)function as

RTSSD =− D(y; x∗)∇yD(y; x∗)y

=− 1∇yD(y; x∗)y

: (3.11)

We now consider the dual problem to (2.10) given by

D(y; x∗) = minp

{px∗: C(y; p)¿ 1}: (3.12)

The rst-order conditions for (3.12) are

C(y; p) = 1; (3.13)

�∇pC(y; p) = x∗: (3.14)

It is easy to show that the optimal value of the Lagrangian multiplier, �(y; x∗), isequal to the input distance function, D(y; x∗). Multiply both sides of (3.14) by p.Then D(y; x∗) = px∗ (by de nition) =�∇pC(y; p)p= �C(y; p) (by homogeneity) =�(by (3.13)). In addition, the dual to Shephard’s Lemma states that p(y; x∗)=∇xD(y; x∗).Thus we can write the rst-order conditions as

C(y;∇xD(y; x∗)) = 1; (3.15)

D(y; x∗)∇pC(y;∇xD(y; x∗)) = x∗: (3.16)


Let Rp=∇xD(y; x∗) and diCerentiate (3.15) and (3.16) with respect to input quantitiesto get

∇pC(y; Rp)∇xxD(y; x∗) = 0N ;

∇pC(y; Rp)′∇xD(y; x∗) + D(y; x∗)∇ppC(y; Rp)∇xxD(y; x∗) = I (3.17)

or in matrix notation[0 ∇pC(y; Rp)

∇pC(y; Rp)′ D(y; x∗)∇ppC(y; Rp)

][∇xD(y; x∗)∇xxD(y; x∗)

]=

[0NI

]: (3.18)

There is a nice symmetry between (3.7) and (3.18). Of course, a cost-minimizing rm will always choose a production plan (y; x∗) such that D(y; x∗) = 1. Thus (3.18)becomes[

0 ∇pC(y; Rp)

∇pC(y; Rp)′ ∇ppC(y; Rp)

][∇xD(y; x∗)∇xxD(y; x∗)

]=

[0NI

]: (3.19)

If the bordered Hessian of the cost function can be inverted then we can solve for the rst- and second-order partial derivatives of the input distance function with respect toinputs.We can also diCerentiate (3.15) and (3.16) with respect to outputs. We get

∇yC(y; Rp) +∇pC(y; Rp)∇xyD(y; x∗) = 0M

∇pC(y; Rp)′∇yD(y; x∗) + D(y; x∗)

[∇pyC(y; Rp) +∇ppC(y; Rp)∇xyD(y; x∗)] = 0N×M : (3.20)

Again setting D(y; x∗) = 1, the above may be rearranged as follows:[0 ∇pC(y; Rp)

∇pC(y; Rp)′ ∇ppC(y; Rp)

][∇yD(y; x∗)∇xyD(y; x∗)

]=

[−∇yC(y; Rp)

−∇pyC(y; Rp)

]: (3.21)

Inverting the bordered Hessian in (3.21) will yield the partial derivatives of the in-put distance function with respect to the outputs and the second-order cross-partialderivatives of the input distance function with respect to inputs and outputs.

4. Econometric estimation

4.1. A stochastic translog shadow cost system

Introducing t as a time trend, t=1; : : : ; T , and letting f represent an individual rm,f = 1; : : : ; F , the stochastic translog shadow cost function can be written as:

ln [C(yft ; p∗ft ; t)h(�ft)]

= lnC(yft ; p∗ft ; t) + ln h(�ft)


=�o +∑m

�m ln ymft + (1=2)∑m

∑w

�mw ln (ymft) ln (ywft)

+∑m

∑n

�mn ln ymft lnp∗nft +

∑n

�n lnp∗nft

+(1=2)∑n

∑l

�nl lnp∗nft lnp

∗lft +

∑m

�mt ln ymftt

+∑n

�nt lnp∗nftt + �t1t + 0:5 �t2t2 + ln h(�ft); (4.1)

where

h(�ft) = exp(vft + uft): (4.2)

The composite error ln h(�ft) is an additive error with a one-sided component, uft ,and a standard noise component, vft , with zero mean. While the uft can be treatedas xed or random, neither approach dominates the other. With the xed eCects ap-proach, identi cation may be di$cult, since the number of parameters increases withF . However, with the random eCects speci cation, one must impose strong distribu-tional assumptions on both the vft and uft , as well as the unlikely assumption that theuft are uncorrelated with the explanatory variables.Since T is of roughly the same magnitude as F in our sample, we adopt the xed

eCects approach for time-varying ine$ciency proposed by Cornwell et al. (1990):

uft = "0 + "f0df + "f1 df t + "f2 df t2; (4.3)

where df is a dummy variable for rm f and the "f0; "f1, and "f2 are parameters tobe estimated for this rm. The "f0 capture time-invariant, rm-speci c diCerences inthe technology, whereas the "f1 and "f2 capture time-varying, rm-speci c diCerencesin technology. This approach avoids the distributional and exogeneity assumptions thatwould be required in a random eCects approach.To allow for the eCect of time in (4.1) in a Nexible manner, we include continuous

time interacted with logs of shadow prices and output quantities as well as rst- andsecond-order terms in time. For an alternative approach using time dummy variablesto achieve even greater Nexibility, see Baltagi and Gri$n (1988).Since shadow costs in (4.1) are unobservable, we must replace the shadow cost

function in (2.7) with its stochastic version from (4.1) to obtain the stochastic actualcost function. Taking logs we obtain

lnCAft = ln

[C(y; p∗; t) +

N∑n=1

(1− kn)pn@C(y; p∗; t)

@pn

]+ vft + uft : (4.4)

Because we wish to control for time-invariant, rm-speci c eCects in our estimatedmodel, we reexpress (4.4) as

lnCAft = ln

[C(y; p∗; t)+

N∑n=1

(1−kn)pn@C(y; p∗; t)

@pn

]+"f0df+vft+u∗ft ; (4.5)

where u∗ft = uft − "f0df.


A number of additional restrictions are required before we can estimate this model.First, we impose symmetry by setting

�mw = �wm; ∀m;w; m =w;

�nl = �ln; ∀n; l; n = l: (4.6)

Since C(y; p∗; t) is linearly homogeneous in p∗, the following restrictions are alsoimposed on the parameters in (4.5):∑

n

�n = 1;

∑n

�nl =∑l

�nl =∑n

∑l

�nl = 0;

∑n

�nt = 0;

∑n

�mn = 0; ∀m: (4.7)

Since we are estimating a cost function, we can measure only relative price e$ciency.This implies that for one n we must restrict knt to some constant ∀t, where we drop thetilde for notational simplicity. We arbitrarily restrict knt for input N . For the remaininginputs, we specify

knt = kn + kn1t + kn2t2 + kn3t3; ∀n; n= 1; : : : ; N − 1: (4.8)

The choice of the numeraire input has no impact on the log likelihood. For details seeAtkinson and Cornwell (1994).We substitute (4.6)–(4.8) into (4.5) and N input quantity equations derived using

(2.4). Without these input quantity equations, we would be unable to identify theunrestricted knt . After appending an error term with zero mean to each of the latterequations, this set of N + 1 equations comprises the stochastic translog shadow costsystem, which is estimated using GMM.

4.2. A stochastic translog shadow distance system

We can write the stochastic input distance function as

1 = D(yft ; x∗ft ; t)h(�ft): (4.9)

As a Nexible approximation to the underlying true distance function, the stochastictranslog shadow input distance function is written as

0 = ln[D(yft ; x∗ft ; t)h(�ft)]

= lnD(yft ; x∗ft ; t) + ln h(�ft)

= �o +∑m

�m ln ymft + (1=2)∑m

∑w

�mw ln(ymft) ln(ywft)


+∑m

∑n

�mn ln ymft ln x∗nft +∑n

�n ln x∗nft

+(1=2)∑n

∑l

�nl ln x∗nft ln x∗lft +

∑m

�mt ln ymftt

+∑n

�nt ln x∗nftt + �t1t + 0:5�t2t2 + ln h(�ft); (4.10)

where

h(�ft) = exp(vft − uft); (4.11)

and uft is de ned in (4.3).Analogous to (4.5), our stochastic shadow distance function can be written as

0 = lnD(yft ; x∗ft ; t)− "f0df + vft − u∗ft ; (4.12)

where u∗ft = uft − "f0df:As with the cost system, additional restrictions on (4.12) are necessary before esti-

mation. We impose symmetry from (4.6) and, since D(y; x∗; t) is linearly homogeneousin x∗, the parametric restrictions for linear homogeneity from (4.7). We also substitute(4.8) into (4.12), where as with the cost system, for one n, knt must be restrictedto some constant ∀t. We also substitute (4.6)–(4.8) into N input price equations de-rived using (2.16). After appending an error term with zero mean to each of the latterequations, this set of N + 1 equations comprises the stochastic translog shadow inputdistance system, which is estimated using GMM.There is an equivalent method of imposing linear homogeneity on (4.12). One nor-

malizes the left-hand side of (4.9) and all input quantities on its right-hand side bysome arbitrarily chosen input. See, for example, Coelli and Perelman (1999). Althoughthe estimated coe$cients and standard errors are identical using either method, theestimation of (4.12) subject to the parametric restrictions in (4.7) may be more conve-nient. For example, the calculation of returns to scale in Eq. (3.11) and TC, as shownbelow, is directly based on the antilog of the tted version of (4.12). One potentialdrawback to estimation of (4.12) is that it does not produce an R2. However, with thenormalized model, since the choice of the normalizing input is arbitrary, there are asmany R2 values as there are inputs.An alternative approach has been taken by Grosskopf et al. (1997) in estimating an

output distance function. Applying their approach to an input distance function, onewould rst divide the left-hand side of (4.9) by the Euclidean norm of the inputs andthen impose the restrictions in (4.7) to guarantee linear homogeneity in prices. How-ever, this normalization serves no valuable function and may introduce computationalerror by altering the scale of the data. 1

1 Other Nexible functional forms such as the Generalized Leontief and Generalized Symmetric McFaddencan also be employed to estimate the cost and input distance systems and all the techniques developed inthis paper are applicable to them. By construction, these two functional forms are linearly homogeneous ininput prices (for the cost system) and input quantities (for the distance system). All candidate functionalforms must be linearly homogeneous in the same arguments.


4.3. Orthogonality conditions

Consistent estimation of (4.5) and (4.12) using GMM requires that the model satisfythe moment conditions E(vft | zft)=0, where z is a vector of instruments. As explainedbelow in greater detail, the Hansen (1982) test of overidentifying restrictions is usedto determine the validity of the instrument set. We are unaware of any previous studyestimating distance functions that addresses the possibility of the endogeneity of atleast one variable on the right-hand side of (4.12).

4.4. Measuring allocative ine7ciency

For the shadow cost system, the derivatives of (2.3) with respect to xn yield thefollowing conditions for AE:

p∗nf

p∗Nf

=@D(yft ; xft ; t)=@xn

@D(yft ; xft ; t)=@xN; n= 1; : : : ; N − 1: (4.13)

With a shadow cost system, relative ine$ciencies for each input must be obtainedindirectly by rst estimating shadow prices from the cost system and then solving the tted demand equations for input quantities. For input n, one can then compute ratiosof ine$cient demands, obtained using the estimated values of knt , to e$cient demands,calculated by setting knt = 1.With the distance system, the conditions for AE are given by (4.13), where shadow

prices are replaced by actual prices and actual quantities are replaced by shadow quan-tities. For rm f at time t, we can directly estimate relative over- and under-utilizationof any pair of inputs, xnft and xlft , in comparison to the cost-minimizing ratio,(kntxnft)=(kltxlft), by computing knt=klt . This is the fundamental advantage of theshadow distance system over the shadow cost system. The researcher typically is in-terested in shadow quantities rather than shadow prices. He would generally use thelatter only to indirectly compute the relative over- or under-utilization of input quanti-ties, which is directly measured once the shadow distance system has been estimated.Examples include the eCects of import quotas or job hiring quotas on input quantities,ine$cient input usage due to rate of return regulation, and the extent of misallocationof public services by local municipalities.

4.5. Measuring TE, EC, TC, and PC

In order to compute TEft , ECft , TCft , and PCft , we require consistent estimatorsof the uft . These are obtained by rst computing the residuals from the estimation of(4.5) as "f0df + vft + u∗ft = vft + u ft . These residuals, which are consistent estimatorsof vft + uft as T → ∞, are then regressed on the right-hand side of (4.3), where forone f, we must impose a restriction on "f0; "f1; and "f2. The tted values of thisregression are consistent estimators of uft .Next we impose non-negativity on the residual used to compute technical ine$ciency.

We accomplish this by de ning u t=minf (u ft) as the estimated frontier intercept. Then


by adding and subtracting u t from the estimated version of (4.1) and using (4.2) weobtain

ln[C(yft ; p∗ft ; t) exp(�ft)] = ln C(yft ; p∗ft ; t) + vft + u ft + u t − u t ;

= ln C(yft ; p∗ft ; t) + vft + u t + uFft ;

= ln CF(yft ; p∗ft ; t) + vft + uFft ; (4.14)

where the tted frontier shadow cost function in period t, ln CF(yft ; p∗ft ; t), is de ned

as ln C(yft ; p∗ft ; t) + u t and uFft = u ft − u t¿ 0. We then estimate TEft as

TEft = exp(−uFft); (4.15)

where our normalization of uFft guarantees that 06TEft6 1, so that it represents thepercent e$ciency of a given rm’s input vector relative to the isoquant of its inputrequirements set. Given TEft in (4.15), we estimate ECft using the relation

ECft =TTEft = TEf; t+1 − TEf; t ; (4.16)

where ECft is simply the change in TEft from t to t + 1.

We estimate TCft as the diCerence between ln CF(y; x; t + 1) and ln C

F(y; x; t),

holding input and output quantities constant:

TCft = ln C(y; x; t + 1) + u t+1 − [ln C(y; x; t) + u t]

=∑m

�mt ln ymft +∑n

�nt ln xnft + �t1

+0:5�t2[(t + 1)2 − t2] + (u t+1 − u t): (4.17)

Thus, the time change in the frontier intercept, u t , aCects TCft as well as ECft . Finally,given ECft and TCft , we obtain PCft following Atkinson et al. (2000) who decomposePCft as

PCft = TCft + ECft : (4.18)

After estimation of the distance system, we again require consistent estimators ofthe u ft to compute TEft , ECft , TCft , and PCft . We proceed by rst calculating thenegative of the residuals from (4.12) as "f0df − vft + u∗ft = u ft − vft , which areconsistent estimators of uft −vft . These estimators are then regressed on the right-handside of (4.3), where for one f, we must impose a restriction on "f0; "f1; and "f2. The tted values of this regression are consistent estimators of the uft .Following Atkinson et al. (2000), we impose non-negativity on the one-sided error

used to compute technical ine$ciency by letting u t =minf (u ft) de ne the estimatedfrontier intercept in each period. We then add and subtract u t from our tted versionof (4.12) to obtain

0 = ln D(yft ; x∗ft ; t) + vft − u ft + u t − u t ;

= ln D(yft ; x∗ft ; t)− u t + vft − uFft ;

= ln DF(yft ; x∗

ftt) + vft − uFft ; (4.19)


where the tted frontier shadow distance function in period t, ln DF(yft ; x∗

ft ; t), is

de ned as ln D(yft ; x∗ft ; t)− u t and uFft = u ft − u t¿ 0. Thus, we can estimate TEft as

TEft = exp(−uFft); (4.20)

where our normalization of uFft guarantees that TEft has the same interpretation as withthe cost system. We estimate ECft as in (4.16). After eliminating our residuals, vft and

uFft , we estimate TCft as the diCerence between ln DF(y; x; t + 1) and ln D

F(y; x; t),

holding input and output quantities constant:

TCft = ln D(y; x; t + 1)− u t+1 − [ln D(y; x; t)− u t]

=∑m

�mt ln ymft +∑n

�nt ln xnft + �t1

+0:5�t2[(t + 1)2 − t2]− (u t+1 − u t): (4.21)

Thus, the time change in the frontier intercept, u t , aCects TCft as well as ECft . Finally,given ECft and TCft , we obtain PCft using (4.18).

5. Data and results

The sample consists of 43 privately-owned electric utilities operating in the USover the 37 year period 1961–1997, for a total of 1591 observations. Since tech-nologies for nuclear, hydroelectric, and internal combustion diCer from that of fossilfuel-based steam generation and because steam generation dominates total productionby investor-owned utilities during the time period under investigation, we limit ouranalysis to this component of steam electric generation.We thank Randy Nelson for making his data available to us. The de nition of

the variables is consistent with, but not identical to, that of variables in the data setdescribed in Nelson (1984). The input variables are fuel (E), labor (L), and capital(K). The price of fuel is computed as a weighted average of the cost per million BTUs.The price of labor is the wage rate, de ned as the sum of salaries and wages chargedto electric operation and maintenance, divided by the number of full time plus one halfthe number of part time employees. As a modi cation to Nelson, the price of capitalis the yield of the rm’s latest issue of long term debt adjusted for appreciation anddepreciation of the capital good using the Christensen and Jorgenson (1970) cost ofcapital formula. Also as a modi cation of Nelson, we distinguish between residential(R) and industrial–commercial (I) output. We use the ratio of sales to each categoryto total sales in order to decompose total steam output into residential and industrial–commercial components. Total output data was taken from data compiled by DanielMcFadden and Thomas Cowing and updated and complemented, if necessary, with datafrom Statistics of Privately Owned Electric Utilities in the US. In addition to thesemodi cations, we extended his data set to include the 1984–1997 time period.The primary sources for Nelson’s data are the US Federal Power Commission Stati-

stics of Privately Owned Electric Utilities in the US, US Federal Power Commission


Steam Electric Plant Construction Cost and Annual Production Expenses, and USFederal Power Commission Performance Pro&les—Private Electric Utilities in theUnited States: 1963–1970. Additional data were taken from Moody’s Public UtilityManual. Whenever necessary we accounted for missing data points by either using thevalue of the previous period or the average of the previous and the subsequent perioddepending on how related variables changed. After calculating total costs as the sumof total expenditure on inputs, but before estimating our cost and distance systems, wenormalize all price and quantity data by their means. Table 1 lists the 43 utilities inour sample, which constitutes a random sampling of all major utilities in the US.We employ GMM estimation, allowing for heteroskedasticity and autocorrelation of

unknown form by employing the Newey and West (1987) covariance matrix estimatorwith a lag of eight periods. The estimated shadow cost system is (4.5) together withthe N input quantity equations derived from (2.4), while the estimated shadow distancesystem is (4.12) together with the N input price equations derived from (2.16). Therestrictions for symmetry from (4.6) and linear homogeneity in input prices and inputquantities from (4.7) were imposed on their corresponding system. Eq. (4.8) de ningthe knt was substituted into the cost and distance systems, respectively, where werestricted kLt = 1 ∀t with both systems.We address the validity of the overidentifying restrictions using the Hansen (1982)

J test. A variety of instrument sets are examined, including various subsets of logsof prices and quantities for outputs and inputs. We fail to reject the null hypothesisthat the moment conditions are satis ed when we employ the following instruments:an intercept, df, dft, dft2, dft3 (f = 2; : : : ; F), time period dummies correspondingto the time periods 1960–1972, 1973–1981, 1982–1990, t2, t3, lnpnt , n = K; L; E andtheir squares and interactions, ln ymt , m= R; I and their squares and interactions, lnpnt

t; n = K; L; E, and ln ymt t; m = R; I . We exclude output prices and input quantities,which is consistent with the assumptions made by many other empirical researchers.The three time-period dummies represent the pre-oil-embargo period through 1972, thesubsequent period including a second oil price shock through 1981, and the followingperiod until the second major revision of the Clean Air Act in 1990.The value of the J statistic for the cost and distance systems was small enough

that we failed to reject the null hypothesis that the instruments are valid even atthe 0.99 level, with a chi-squared value of 211.38 for the shadow cost system and211.05 for the shadow distance system. The degrees of freedom for both tests is705. Other sets of instruments which included either output prices, input quantities,or both produced signi cant values for the J statistic and, hence, were rejected aslegitimate.Both shadow cost and shadow distance models t the data well. The computed R2

values for the tted actual cost equation and tted quantity equations for capital, laborand energy are 0.97, 0.65, 0.71, and 0.93, respectively. To obtain an estimated R2

for the distance system, we must estimate the normalized form of (4.9) along withthe input price equations for capital, labor, and energy from (2.16). The computed R2

values for the tted normalized distance equation ranges from 0.980 to 0.984, whilethe values for the tted price equations for capital, labor, and energy are 0.80, 0.85,and 0.86, respectively.


Table 1Utilities in the sample

Firm number Utility

1 Alabama PC2 Arizona PSC3 Arkansas PLC4 Paci c GEC5 SanDiego GEC6 PSC Colorado7 UIC Connecticut8 Delmarva PLC9 Potomac EPC10 Tampa EC11 Georgia PC12 C Illinois PSC13 PSC Indiana14 PC Iowa15 Kansas GEC16 Kentucky UC17 Louisville GEC18 C Louisiana EC19 C Maine PC20 Baltimore GEC21 Boston EC22 Detroit EC23 Mississippi PLC24 Kansas City PLC25 PSC New Hampshire26 Atlantic City EC27 PSEGC New Jersey28 PSC New Mexico29 Central Hudson GEC30 CEC New York31 Rochester GEC32 Carolina PLC33 Duke PC34 Cleveland EIC35 Ohio EC36 Oklahoma GEC37 DLC Pennsylvania38 Philadelphia PC39 West Penn PC40 S Carolina EGC41 Virginia EPC42 Appalachian PC43 Wisconsin EPC

We next examine the monotonicity and curvature properties of both shadow cost anddistance systems. The shadow cost function corresponds to a well-behaved productionor distance function only if it is monotonically increasing in shadow input prices andoutput quantities and concave in shadow input prices. Monotonicity and concavity


(examined by looking at estimated eigenvalues) are satis ed at the mean of the datafor all inputs.For the duality between input prices and quantities to be valid, the input shadow dis-

tance function must be monotonically increasing in inputs, monotonically decreasing inoutputs, and concave in inputs. Our estimated model satis es the required monotonicityand the curvature properties (again seen from examining estimated eigenvalues) at themean of the data.In Table 2 we report estimated coe$cients and asymptotic standard errors for the

structural parameters of the shadow cost and distance systems. All parameters aresigni cant at the 0.05 level, using a two-tailed test.Table 3 presents the estimated coe$cients and asymptotic standard errors for the

AE parameters in Eq. (4.8). All the coe$cients are signi cant at the 0.01 level witha two-tailed test.Table 4 presents the estimated values for the knt (n = K; E; t = 1; : : : ; T ), where

kLt =1 ∀t. For the shadow cost system, the estimated shadow prices make (4.13) holdsubject to actual quantities. Since we x kLt = 1, an estimate of kKt ¡ 1 means thatthe ratio of the shadow price of capital to that of labor is considerably lower than thecorresponding ratio of actual prices. This indicates over-utilization of capital relativeto labor throughout the entire sample period. There is variation in the degree of overutilization, but no clear-cut trend. The average ratio of the estimated kKt to kLt is0.41. Further, energy is under-utilized relative to labor throughout the entire sampleperiod, with an average ratio of the estimated kEt to kLt equal to 2:81. Finally, capitalis over-utilized relative to energy. The results for the cost system are not inconsistentwith those presented in Atkinson and Halvorsen (1984) for a cross-section of electricutilities estimated with a shadow cost system.The estimated allocative ine$ciencies from the shadow cost system are highly con-

sistent with those from the shadow distance system. From the cost system, capital isover-utilized relative to labor. With the distance system, when facing actual prices, rms wish to reduce the amount of capital relative to labor, since the average ratioof the estimated kKt to kLt is 0.46. Further, from the cost system, rms under utilizeenergy relative to labor. With the distance system, rms wish to increase energy usagerelative to labor, since the average ratio of the estimated kEt to kLt is 2.37. Finally,from the cost system, capital is over-utilized relative to energy, and with the distancesystem, rms wish to reduce the ratio of capital to energy.Technical e$ciency scores, computed using (4.15) and (4.20), are based on u ft

obtained from the tted version of Eq. (4.3). Before estimating this equation, we imposethe restrictions that "f0="f1="f2=0 for f=1. The values of the estimated parametersare available from the authors. The R2 values for the tted version of (4.3) are 0.84and 0.72 for the shadow cost and shadow distance systems, respectively.Weighted-average rm TEft scores are presented in Table 5 (where the weights

are the rm’s share of total output in each time period). Average e$ciency scoresranged from 0.48 to 0.96 for the cost function, with an average of 0.72, and from0.49 to 0.94 for the distance function, with an average of 0.67. Over allobservations, the e$ciency scores are highly correlated, with a simple correlationcoe$cient of 0.70. These levels of TEft for both systems are consistent with the


Table 2Estimated structural coe$cients

Cost Fn. Dist. Fn.

�0 0.6888 −0:4014(0.0026)∗ (0.0025)∗

�K 0.1118 0.1692(0.0014)∗ (0.0005)∗

�L 0.4272 0.1472(0.0016)∗ (0.0007)∗

�E 0.4610 0.6836(0.0019)∗ (0.0008)∗

�KK −0:0405 0.0420(0.0007)∗ (0.0002)∗

�KL 0.0683 −0:0250(0.0007)∗ (0.0001)∗

�KE −0:0278 −0:0170(0.0003)∗ (0.0002)∗

�LL 0.0395 0.1078(0.0007)∗ (0.0003)∗

�LE −0:1078 −0:0827(0.0004)∗ (0.0003)∗

�EE 0.1356 0.0997(0.0005)∗ (0.0005)∗

�R 0.2760 −0:2813(0.0036)∗ (0.0036)∗

�I 0.5907 −0:5566(0.0034)∗ (0.0034)∗

�RR 0.3894 −0:2324(0.0093)∗ (0.0093)∗

�II 0.2854 −0:1479(0.0102)∗ (0.0102)∗

�RI −0:3238 0.1780(0.0095)∗ (0.0096)∗

�KR 0.0051 −0:0049(0.0006)∗ (0.0003)∗

�LR −0:1557 −0:0503(0.0010)∗ (0.0005)∗

�ER 0.1506 0.0553(0.0009)∗ (0.0006)∗

�KI −0:0229 0.0068(0.0006)∗ (0.0003)∗

�LI 0.1485 0.0649(0.0009)∗ (0.0005)∗

�EI −0:1256 −0:0717(0.0009)∗ (0.0006)∗

�T1 −0:0159 0.0147(0.0002)∗ (0.0004)∗

�T2 −0:0001 −0:0014(0.0000)∗ (0.0000)∗

�KT −0:0014 −0:0006(0.0000)∗ (0.0000)∗

�LT −0:0014 0.0056(0.0001)∗ (0.0000)∗


Table 2 (Continued)

�ET 0.0028 −0:0050(0.0000)∗ (0.0000)∗

�IT −0:0089 0.0031(0.0001)∗ (0.0001)∗

�RT 0.0084 −0:0042(0.0001)∗ (0.0001)∗

Note: Asymptotic standard errors in parentheses.∗Denotes signi cance at the 0.05 level using a two-tailed test.

Table 3Estimated allocative e$ciency coe$cients

Cost Fn. Dist. Fn.

kK 0.5838 0.0748(0.0052)∗ (0.0042)∗

kK1 −0:0518 0.1001(0.0007)∗ (0.0010)∗

kK2 0.0033 −0:0060(0.0000)∗ (0.0001)∗

kK3 −0:0001 0.0001(0.0000)∗ (0.0000)∗

kE 6.0418 2.4886(0.0768)∗ (0.0045)∗

kE1 −0:8820 −0:1079(0.0120)∗ (0.0009)∗

kE2 0.0559 0.0032(0.0007)∗ (0.0001)∗

kE3 −0:0010 0.00003(0.0000)∗ (0.0000)∗

Note: Asymptotic standard errors in parentheses.∗Denotes signi cance at the 0.05 level using a two-tailed test.

local service monopoly that has protected technically ine$cient electric utilities fromcompetition.In Table 6 we provide weighted-average annual estimates of PCft , TCft , and ECft

(where the weights again are the rm’s output share in each time period), computedassuming that knt =1 ∀n, so that our estimates are based on actual rather than e$cientinput levels. 2 The estimates from the cost and distance systems are in reasonablyclose agreement. Both systems nd that ECft decreases over time and is negativeat the end of the sample period. Both weighted-average estimates of ECft are smalland negative: −0:42 percent for the cost system and −0:49 percent for the distancesystem. For both systems, TCft begins in 1961 and ends in 1996 with highly similarvalues. Over all years, average TCft is similar for both systems: 0:68 percent for the

2 Results conditional on estimated values for knt , ∀k, k �= L are similar and are available from the authors.


Table 4Allocative ine$ciencies over time

Year kKt kEt

Cost system Dist. system Cost system Dist. system

1961 0.5353 0.1690 5.2147 2.38401962 0.4929 0.2517 4.4937 2.28591963 0.4564 0.3235 3.8728 2.19451964 0.4255 0.3850 3.3462 2.11001965 0.3997 0.4369 2.9081 2.03261966 0.3789 0.4797 2.5525 1.96251967 0.3625 0.5141 2.2737 1.89991968 0.3504 0.5407 2.0657 1.84501969 0.3421 0.5600 1.9227 1.79791970 0.3373 0.5727 1.8389 1.75891971 0.3358 0.5794 1.8084 1.72821972 0.3371 0.5807 1.8252 1.70591973 0.3410 0.5773 1.8836 1.69231974 0.3471 0.5696 1.9778 1.68751975 0.3550 0.5583 2.1017 1.69171976 0.3645 0.5441 2.2497 1.70511977 0.3752 0.5276 2.4157 1.72801978 0.3867 0.5092 2.5940 1.76041979 0.3988 0.4898 2.7788 1.80271980 0.4111 0.4698 2.9640 1.85491981 0.4232 0.4498 3.1439 1.91731982 0.4349 0.4306 3.3127 1.99001983 0.4458 0.4126 3.4644 2.07331984 0.4556 0.3966 3.5932 2.16741985 0.4639 0.3830 3.6933 2.27241986 0.4704 0.3726 3.7587 2.38851987 0.4748 0.3659 3.7836 2.51601988 0.4767 0.3635 3.7622 2.65501989 0.4758 0.3660 3.6886 2.80571990 0.4718 0.3741 3.5569 2.96831991 0.4643 0.3884 3.3613 3.14301992 0.4531 0.4094 3.0960 3.32991993 0.4377 0.4378 2.7549 3.52941994 0.4178 0.4742 2.3324 3.74151995 0.3932 0.5191 1.8225 3.96651996 0.3634 0.5733 1.2194 4.20451997 0.3282 0.6372 0.5172 4.4558

Avg. 0.4104 0.4593 2.8094 2.3717

cost system and 1.16 percent for the distance system. Finally, average PCft over allperiods is reasonably similar, with 0.27 percent for the cost system and 0.67 percentfor the distance system. These results are not inconsistent those of Callan (1991), whoestimated a standard cost function (which assumed AE) for the 1965–1984 time periodfor a similar set of US electric utilities.


Table 5Average rm technical e$ciencies

Firm Cost Fn. Dist Fn.

1 0.8021 0.85062 0.5909 0.53893 0.6654 0.69824 0.5725 0.61705 0.7000 0.73656 0.7021 0.59137 0.7892 0.72528 0.7947 0.77109 0.5944 0.601210 0.7070 0.613911 0.5582 0.572912 0.7483 0.636013 0.7481 0.667214 0.8129 0.764515 0.9554 0.935516 0.7577 0.678317 0.7300 0.628118 0.8552 0.791019 0.7989 0.785020 0.7183 0.628021 0.6790 0.641622 0.5958 0.536323 0.7871 0.759024 0.7632 0.579325 0.7406 0.669126 0.8570 0.755627 0.5930 0.564628 0.7574 0.669829 0.8777 0.825630 0.4813 0.488731 0.8262 0.659032 0.6821 0.647933 0.6319 0.606534 0.6436 0.657635 0.6260 0.597836 0.7598 0.702337 0.5959 0.562338 0.6164 0.568939 0.7880 0.724340 0.7447 0.706041 0.6884 0.615542 0.7519 0.738243 0.6744 0.5982

Avg. 0.7154 0.6675

Weighted-average estimates of returns to scale (where weights are again based onoutput shares) are computed using (2.8) for the shadow cost function and (3.11)for the shadow distance function. The estimates from the two models are highly


Table 6Time varying PC, TC, and EC

Year Cost Fn. Dist. Fn.

PC TC EC PC TC EC

1961 −0:0443 −0:0530 0.0087 −0:0396 −0:0598 0.02031962 −0:0437 −0:0525 0.0088 −0:0368 −0:0574 0.02051963 −0:0420 −0:0511 0.0091 −0:0186 −0:0205 0.00191964 −0:0399 −0:0493 0.0093 −0:0152 −0:0144 −0:00081965 −0:0371 −0:0469 0.0098 −0:0145 −0:0138 −0:00071966 −0:0337 −0:0437 0.0100 −0:0142 −0:0138 −0:00041967 −0:0310 −0:0414 0.0103 −0:0132 −0:0130 −0:00021968 −0:0284 −0:0390 0.0106 −0:0127 −0:0128 0.00001969 −0:0242 −0:0350 0.0109 −0:0125 −0:0126 0.00011970 −0:0240 −0:0352 0.0112 −0:0112 −0:0115 0.00031971 −0:0179 −0:0134 −0:0045 −0:0102 −0:0107 0.00041972 −0:0103 0.0177 −0:0281 −0:0088 −0:0095 0.00061973 −0:0028 0.0226 −0:0254 −0:0086 −0:0094 0.00091974 −0:0044 0.0185 −0:0229 −0:0069 −0:0080 0.00101975 −0:0013 0.0194 −0:0206 −0:0078 −0:0091 0.00131976 0.0008 0.0194 −0:0186 −0:0052 −0:0068 0.00161977 0.0035 0.0201 −0:0167 −0:0044 −0:0062 0.00181978 0.0054 0.0203 −0:0149 −0:0030 −0:0051 0.00211979 0.0071 0.0203 −0:0132 −0:0019 −0:0043 0.00231980 0.0089 0.0205 −0:0117 −0:0005 −0:0032 0.00271981 0.0124 0.0225 −0:0100 0.0001 −0:0029 0.00291982 0.0146 0.0232 −0:0086 0.0018 −0:0013 0.00311983 0.0176 0.0247 −0:0071 0.0030 −0:0006 0.00371984 0.0202 0.0259 −0:0057 0.0041 0.0001 0.00401985 0.0208 0.0252 −0:0044 0.0086 0.0093 −0:00071986 0.0228 0.0259 −0:0030 0.0257 0.0477 −0:02191987 0.0241 0.0258 −0:0017 0.0292 0.0515 −0:02231988 0.0244 0.0248 −0:0004 0.0323 0.0550 −0:02271989 0.0244 0.0236 0.0008 0.0360 0.0589 −0:02291990 0.0277 0.0257 0.0020 0.0393 0.0620 −0:02281991 0.0298 0.0265 0.0033 0.0425 0.0652 −0:02271992 0.0318 0.0273 0.0045 0.0460 0.0686 −0:02271993 0.0310 0.0250 0.0060 0.0500 0.0722 −0:02221994 0.0425 0.0482 −0:0057 0.0533 0.0752 −0:02181995 0.0533 0.0740 −0:0206 0.0568 0.0782 −0:02141996 0.0585 0.0794 −0:0210 0.0600 0.0809 −0:0208

Avg. 0.0027 0.0068 −0:0042 0.0067 0.0116 −0:0049

similar and indicate modestly increasing returns to scale. Weighted-average returnsare 1.13 for the shadow cost equation and 1.15 for the shadow distance equation.The general consensus in the literature appears to be that modest, increasing re-turns prevails in this industry. See for example, the non-shadow cost system resultsof Callan (1991) using panel data and the shadow cost system results of Atkinsonand Halvorsen (1984) using cross-sectional data. Finally, we compute input price


elasticities of demand evaluated at the mean of the data for the cost and distancesystems. 3

6. Summary and conclusions

We have formulated shadow distance and shadow cost systems as dual approaches toestimating rm technology, allocative e$ciency, technical e$ciency, and productivitygrowth. The shadow input quantities from a shadow distance system and shadow inputprices from a shadow cost system provide dual measures of allocative ine$ciency.While estimation of relative input misallocation is performed directly using the shadowdistance system, it is performed indirectly using the shadow cost system. Using dualitytheory, we show how to recover cost function derivatives in terms of distance functionderivatives, which allows estimation of returns to scale and price elasticities of demandusing only the estimated distance function parameters. For both systems, technicaline$ciency is decomposed from noise, using a xed eCects frontier speci cation.Using panel data for 43 US utilities over 37 years, we estimate both shadow cost

and shadow distance systems. We nd general agreement between the two modelswith regard to the over-use of capital relative to labor and to energy and the under-useof energy relative to labor. The technical ine$ciency estimated by both models isconsistent with an industry characterized by local monopoly power subject to rateof return regulation. For both models, e$ciency change is small but negative, whiletechnical change and productivity change are small but positive. Finally, both models nd the existence of moderately increasing returns to scale and substitutability amonginputs.

Acknowledgements

We wish to thank Chris Cornwell for helpful comments on an earlier draft.

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Documents

Stochastic estimation of firm technology, inefficiency, and productivity growth using shadow cost and distance functions