ESTIMATION OF ALLOCATIVE INEFFICIENCY
AND PRODUCTIVITY GROWTH
WITH DYNAMIC ADJUSTMENT COSTS∗
Scott E. Atkinson
email: [email protected]
and
Christopher Cornwell
email: [email protected]
Department of Economics
University of Georgia
Athens GA 30602
Phone: 706-542-3670
FAX: 706-542-3376
December 28, 2008
Abstract
A substantial literature has been generated on the estimation of allocative and technical inefficiency using static production, cost, profit, and distance functions. We develop a
dynamic shadow distance system that integrates dynamic adjustment costs into a long-run
shadow cost-minimization problem, which allows us to distinguish static allocative distortions from short-run inefficiencies that arise due to period-to-period adjustment costs. The
set of estimating equations is comprised of the first-order conditions from the short-run
shadow cost-minimization problem for the variable shadow input quantities, a set of Euler
equations derived from subsequent shadow cost minimization with respect to the quasifixed inputs, and the input distance function, expressed in terms of shadow quantities.
This system nests within it the static model with zero adjustment costs. Using panel data
on U.S. electric utilities, we contrast the results of static and dynamic shadow distance
systems. First, the zero-adjustment-cost restriction is strongly rejected. Second, we find
that adjustment costs represent about .42 percent of total cost and about 1.26 percent
of capital costs. Third, while both models reveal that labor is not utilized efficiently, the
dynamic model indicates a longer period of over-use and less variance over time in the
degree of inefficiency. With the dynamic model, productivity growth is larger but more
stable.
JEL Categories: C13, C33
Key Words: Dynamic Estimation, Euler Equations, Allocative Inefficiency, Technical Inefficiency, Productivity Change, Technical Change.
1. Introduction
This paper draws together two related literatures—one involving the estimation of
allocative inefficiency, and the other, the estimation of dynamic cost minimization models.
A standard approach in the former is to assume the firm is a “shadow cost-minimizer”
that focuses on shadow (or internal) prices which may differ from actual (market) prices.
Atkinson and Cornwell (1994) show how to estimate a translog shadow cost system with a
parametric specification of both allocative and technical distortions. This framework can
be extended to accommodate time-varying specifications of inefficiency, so that productivity change (PC) can also be estimated and decomposed into technical change (TC) and
efficiency change (EC).
The literature on dynamic cost minimization is concerned not with allocative inefficiency per se, but incorporation of dynamic adjustment costs into the cost-minimization
calculus. Approaches to the specification and estimation of econometric models of dynamic
adjustment costs using cost functions are reviewed in Good, Nadiri, and Sickles (1999). An
early example is Pindyck and Rotemberg (1983) and a more recent example is MorrisonPaul and Siegel (1999). Both formulate a dynamic optimization problem which involves
minimizing the discounted value of the costs of adjusting levels of quasi-fixed factors plus
the standard (not shadow) costs of variable and quasi-fixed factors. Thus the system that
is estimated includes a set of Euler equations for the quasi-fixed factors along with the variable cost function and the static input demand equations for the variable factors. Cooper
(2001) provides more general guidance on dynamic structural modelling beyond the cost
minimization setup.
Bernstein et al. (2004) extend the standard dynamic setup to account for technical inefficiency. Their approach is to specify the production technology in terms of input changes
1
that are scaled by input-specific parameters. They assert that these parameters capture
net efficiencies from changes in input usage and adjustment costs. However, such a specification will generally conflate technical inefficiency, allocative inefficiency, and adjustment
costs associated with changes in input levels. In addition, although adjustment costs and
hence implicitly quasi-fixed inputs are assumed with the primal production function, a
variable cost function (without quasi-fixed inputs) is derived for the dual.
In this paper we argue that technical efficiency, allocative efficiency, and adjustment
costs for the quasi-fixed inputs are three separable effects that should be estimated separately. We posit a restricted cost function with quasi-fixed inputs and derive the corresponding dual distance function with the same quasi-fixed inputs as well as parameters
that measure allocative efficiency and the costs of adjustment for the quasi-fixed inputs.
We recover estimates of technical efficiency residually.
We first develop a dynamic shadow distance system that integrates dynamic adjustment costs into a long-run shadow cost-minimization problem. However, our empirical
implementation utilizes the dual shadow distance function. This means that the dynamic system of equations we estimate is comprised of the first-order conditions from the
short-run shadow cost-minimization problem for the variable inputs (expressed in terms
of shadow quantities), a set of Euler equations corresponding to subsequent shadow cost
minimization with respect to the quasi-fixed inputs, and the input shadow distance function. All equations are expressed in terms of the shadow distance function and its partial
derivatives rather than the shadow cost function and its derivatives. As a restricted case
of this dynamic system, we derive and estimate a static distance system, comprised of the
shadow distance system and its associated first-order conditions for short-run shadow cost
minimization, ignoring dynamic adjustment costs for the quasi-fixed inputs.
2
Our work complements Rungsuriyawiboon and Stefanou (2007), who formulate and
estimate a shadow-cost system that incorporates adjustment costs. However, they take
a different approach to estimation and are not concerned with productivity change and
its decomposition. We prefer the shadow distance system implementation to the shadow
cost formulation, because the former permits direct estimation of the effect of allocative
inefficiency on relative input usage and directly facilitates the calculation of TC, EC, and
PC.
We estimate both shadow distance systems using a panel of 78 U.S. electric utilities covering the 1988–97 period. Following Atkinson, Cornwell and Honerkamp (2003),
the model is estimated by the generalized method of moments (GMM). Then, from the
estimated parameters, we evaluate the role of adjustment costs and their effect on estimates of allocative and technical inefficiency as well as productivity change. We calculate
that adjustment costs represent a little more than .4 percent of total cost and 1.26 percent of capital costs, assuming that firms are using energy at optimal levels. However,
controlling for adjustment costs is important because estimates of allocative inefficiency
and productivity growth differ substantially between the static and dynamic models. In
particular, compared with the static model, the dynamic model indicates that the overutilization of capital and labor is less severe and average annual PC is more stable and
about 1.8 percentage point higher on average. Finally, we reject the static model, with its
zero-adjustment-cost restriction, in favor of the dynamic model.
3
2. Modelling Allocative Distortions and Adjustment Costs
a. Static Shadow Distance Systems
As noted at the outset, static models of cost minimization have been widely utilized
to estimate allocative inefficiency. Here we follow Atkinson and Primont (2002) to derive
a set of estimating equations for this purpose. We begin with an input distance function,
D(yt , xt ) = sup{λ : (xt /λ) ∈ L(yt )},
(2.1)
λ
where yt is a M -vector of outputs and xt is a N -vector of inputs. Then, we assume the
typical firm solves the following short-run cost minimization problem:
½
¾
C(yt , pt1 ; xt2 ) = min pt1 xt1 : D(yt , xt1 ; xt2 ) ≥ 1 ,
xt1
(2.2)
where xt1 and xt2 are N1 × 1 and (N − N1 ) × 1 vectors of variable and of quasi-fixed
“shadow” inputs, respectively, and pt1 is a 1 × N1 vector of variable-input prices. We
denote the shadow input quantities that solve (2.2) as x∗t1 = [k1t x1t , . . . , kN1 t xN1 t ], where
the knt (n = 1, . . . , N1 ) are parameters to be estimated. The shadow inputs will differ
from actual inputs to the extent the firm fails to equate marginal rates of substitution
to input-price ratios. This sort of failure may persistently occur in regulated industries
like electric-power generation where the empirical evidence suggests the nature of the
regulation induces over-capitalization. However, inefficient utilization of inputs may result
from contractual obligations or shortages as well.
The first-order conditions corresponding to (2.2) are
pn = µ
∂D(yt , x∗t1 ; xt2 )
,
∂xn
4
n = 1, . . . , N1 ,
(2.3)
where µ is the Lagrangian multiplier and
∂D(yt ,x∗
t1 ; xt2 )
∂xn
is the partial derivative of
D(yt , xt1 ; xt2) with respect to xn evaluated at the shadow values of the inputs, x∗ti . From
(2.3) we obtain shadow variable costs as
pt1 x∗t1
N1
X
∂D(yt , x∗t1 ; xt2 ) ∗
=µ
xn .
∂xn
n=1
(2.4)
Because D(yt , x∗t1 ; xt2 ) is linearly homogeneous in the variable inputs, we can apply Euler’s
theorem to obtain
N1
X
∂D(yt , x∗t1 ; xt2 ) ∗
xn = D(yt , x∗t1 ; xt2 ).
∂x
n
n=1
(2.5)
Further, since D(yt , x∗t1 ; xt2 ) = 1 by assumption,
µ = pt1 x∗t1 = C(yt , pt1 ; xt2 ).
(2.6)
Thus, the first-order conditions in (2.3) lead to the following price equations for the variable
inputs:
pn = (pt1 x∗t1 )
∂D(yt , x∗t1 ; xt2 )
,
∂xn
n = 1, . . . , N1 ,
(2.7)
evaluated at the shadow values of the inputs.
The firm’s long-run cost-minimization problem provides price equations for the quasifixed inputs. Conditional on the solutions to (2.2), we can formulate this problem as
£
¤
C(yt , pt1, pt2 ) = min C(yt , pt1 ; xt2 ) + pt2 xt2 ,
xt2
(2.8)
where C(yt , pt1 ; xt2 ) = pt1 x∗t1 (yt , pt1 ; xt2 ). The first-order conditions corresponding to
(2.8) are
pn = −
∂C(yt , pt1 ; x∗t2 )
,
∂xn
5
n = N1 + 1, . . . , N,
(2.9)
evaluated at the shadow quantities of the quasi-fixed inputs and x∗t2 is the value of xt2 that
minimizes (2.9). To state our problem strictly in terms of distance functions, we employ
Mahler’s Inequality, which says
pt1 xt1 ≥ C(yt , pt1 ; xt2 ) D(yt , xt1 ; xt2 ).
(2.10)
Because (2.10) holds with strict equality when evaluated at the shadow quantities, we can
write
C(yt , pt1 ; x∗t2 ) =
pt1 x∗t1
.
D(yt , x∗t1 ; x∗t2 )
(2.11)
It follows that
¤−2
∂C(yt , pt1 ; x∗t2 )
∂D(yt , x∗t1 ; x∗t2 ) £
= −(pt1 x∗t1 )
D(yt , x∗t1 ; x∗t2 ) ,
∂x2
∂x2
(2.12)
which, with (2.9) and D(yt , x∗t1 ; x∗t2 ) set to 1, leads to a set of price equations for the
quasi-fixed inputs given by
pn =
∗
∗
∗ ∂D(yt , xt1 ; xt2 )
(pt1 xt1 )
,
∂xn
n = N1 + 1, . . . , N.
(2.13)
The distance function in (2.1) combined with the price equations given in (2.7) for all
N inputs defines a set of estimating equations which we term our static shadow distance
system. We propose a GMM estimation strategy along the lines described in Atkinson,
Cornwell, and Honerkamp (2003) and Atkinson and Primont (2002).
b. Adding Adjustment Costs
We now incorporate adjustment costs into the firm’s decision calculus, replacing the
second stage of the cost-minimization problem described in (2.8) with the following dynamic optimization problem:
min E
xt2
∞
X
·
¸
dt C(yt , pt1 ; xt2 ) + pt2 xt2 + h(∆xt2 ) ,
t=1
6
(2.14)
where dt is the long-run discount rate and pt2 is a 1×(N −N1 ) price vector associated with
the quasi-fixed inputs and h(∆xt2 ) reflects adjustment costs. We denote the quasi-fixed
shadow input quantities that solve (2.14) as x∗t2 = [kN1 +1,t xN1 +1,t , . . . , kN t xN t ], where
the knt (n = N1 + 1, . . . , N ) are additional parameters to be estimated for the quasi-fixed
inputs. The first-order conditions for this problem are:
·
¸
∂h(∆x∗t+1,2 )
∂C(yt , pt1 ; x∗t2 )
∂h(∆x∗t2 )
+ pt2 +
− E dt+1
= 0.
∂xt2
∂xt2
∂xt2
(2.15)
Solving for pt2 , we obtain
·
pt2
¸
∂h(∆x∗t+1,2 )
∂h(∆x∗t2 ) ∂C(yt , pt1 ; x∗t2 )
= E dt+1
−
−
.
∂xt2
∂xt2
∂xt2
(2.16)
Then, repeating the argument from Mahler’s Inequality made in (2.10) and (2.11), we can
write (2.16) as
pt2
·
¸
∂h(∆x∗t+1,2 )
∂h(∆x∗t2 )
∂D(yt , x∗t1 ; x∗t2 )
= E dt+1
−
+ (pt1 x∗t1 )
.
∂xt2
∂xt2
∂xt2
(2.17)
By allowing for adjustment costs, the system of estimating equations is now comprised
of the price equations for the variable inputs in (2.7), the Euler equations given in (2.17),
and the distance function, (2.1). We term this set of equations our dynamic shadow
distance system. Note that the static model is a special case of the dynamic model when
adjustment costs are zero. As in the static case, the model is estimated by GMM.
3. Empirical Model and Estimation
a. Translog Distance Function
Expressed as an econometric model, the shadow input distance function for firm f
and time t can be written as
1 = D(yf t , x∗f t , t)g(²f t ),
7
(3.1)
where ²f t is a random error. Adopting the translog functional form for (3.1) leads to the
following empirical model:
0 = ln D(yf t , x∗f t , t) + ln g(²f t )
= γo +
X
XX
γm ln ymf t + .5
m
+
XX
m
+ .5
X
γn ln x∗nf t
n
n
X
γmw ln ymf t ln ywf t
w
γmn ln ymf t ln x∗nf t +
XX
n
+
m
γnl ln x∗nf t ln x∗lf t +
X
γmt ln ymf t · t
m
l
γnt ln x∗nf t · t + γt1 t + .5 γt2 t2 + ln g(²f t ),
(3.2)
g(²f t ) = exp(vf t − uf t ),
(3.3)
n
where n = 1, . . . , N and
such that vf t is a two-sided disturbance and uf t ≥ 0 captures technical inefficiency. Since
we employ panel data, we also include a set of firm dummies in the distance function
equation, (3.2), to control for unobserved time-invariant heterogeneity.
Using (3.2) we restate the static price equations from (2.7) in terms of the parameters
of the translog distance function as
(·
¸
X
X
∗
∗
pn = (pt xt ) γn +
γmn ln ymf t +
γnl ln xlf t + γnt t
m
·
× exp γo +
+
XX
m
X
l
γm ln ymf t + .5
m
XX
m
γmn ln ymf t ln x∗nf t
γmw ln ymf t ln ywf t
w
X
+
γn ln x∗nf t
n
n
XX
X
+ .5
γnl ln x∗nf t ln x∗lf t +
γmt ln ymf t · t
n
+
X
m
l
γnt ln x∗nf t
· t + γt1 t + .5 γt2 t
¸
2
n
·
× γo +
X
m
γm ln ymf t + .5
XX
m
w
8
γmw ln ymf t ln ywf t
+
XX
m
γmn ln ymf t ln x∗nf t +
X
γn ln x∗nf t
n
n
XX
X
+ .5
γnl ln x∗nf t ln x∗lf t +
γmt ln ymf t · t
n
+
X
m
l
¸−1
γnt ln x∗nf t · t + γt1 t + .5 γt2 t2
n
·
×
1
x∗nf t
¸)
, n = 1, . . . , N.
(3.4)
As proposed by Cornwell, Schmidt, and Sickles (1990), we specify the uf t in terms of
firm-specific linear and quadratic trends:
uf t = β0 + βf 0 df + βf 1 df t + βf 2 df t2 ,
(3.5)
where the df are firm dummies and the βf q , q = 0, 1, 2, are parameters to be estimated. We
also model the allocative inefficiency parameters in terms of linear and quadratic trends,
but we assume each is constant across firms:
knt = kn + kn1 t + kn2 t2 ,
n = 1, . . . , N.
(3.6)
Finally, we specify the adjustment-cost function, h(∆x∗n ), as
h(∆x∗n )
=
N
X
δn (x∗nt − x∗n,t−1 )2 /2,
(3.7)
n=N +1
where δn is a shadow price associated with changes in quasi-fixed input n. We interpret
adjustment costs as the cost of reduced output induced by changes in the levels of quasifixed inputs. For example, new capital equipment may go through a phase of testing,
calibration, and debugging before it is fully operational. Similarly, newly hired workers may
enter an initial training period. In contrast, costs associated with allocative inefficiency
arise when the firm hires too many or too few workers to operate and maintain the new
capital equipment. Because the static model is nested within the dynamic model, the
validity of the former can be assessed by testing the null hypothesis δn = 0, n = N1 +
1, . . . , N.
9
Using (3.2) and (3.7) we restate the Euler price equations, (2.17), for the quasi-fixed
inputs in terms of the parameters of the translog distance function as
¸−1 (
·
£
¤
E dt+1 pt+1,2 δn (x∗nt − x∗n,t−1 )
pn = 1 + δn (x∗nt − x∗n,t−1 )
½
+
(pt1 x∗t1 )
X
X
£
γn +
γmn ln ymf t +
γnl ln x∗lf t + γnt t]
m
l
X
XX
£
× exp γo +
γm ln ymf t + .5
γmw ln ymf t ln ywf t
+
XX
m
m
m
w
X
γmn ln ymf t ln x∗nf t +
γn ln x∗nf t
n
n
XX
X
γmt ln ymf t · t
+ .5
γnl ln x∗nf t ln x∗lf t +
n
+
X
m
l
γnt ln x∗nf t
· t + γt1 t + .5 γt2 t2
¤
n
X
XX
£
γmw ln ymf t ln ywf t
× γo +
γm ln ymf t + .5
m
+
XX
m
m
γmn ln ymf t ln x∗nf t
n
w
X
+
γn ln x∗nf t
n
XX
X
+ .5
γnl ln x∗nf t ln x∗lf t +
γmt ln ymf t · t
n
+
X
m
l
γnt ln x∗nf t · t + γt1 t + .5 γt2 t
¤
2 −1
¾)
¤
× [1/x∗nf t
, n = N1 + 1, . . . , N. (3.8)
n
For the dynamic system, price equations for the variable inputs, restated in terms of the
translog distance function parameters, are the same as in (3.4).
Identification requires a restriction on both the βf q and the knt . First, for one firm
we set βf q = 0, ∀q. Second, because we can only measure the over or under-utilization
of one input relative to another, for some input we must assign knt to a constant ∀ t. As
discussed in Atkinson and Cornwell (1994), the choice of numeraire is inconsequential.
Prior to estimation, other parametric restrictions are imposed on the shadow distance
system. Symmetry requires that
γmw = γwm ,
∀ m, w, m 6= w
10
γnl = γln ,
∀ n, l, n 6= l.
(3.9)
In addition, the linear homogeneity of the distance function must be imposed with respect
to the variable shadow input quantities. This implies
N1
X
γn = 1
n=1
N1
X
n=1
N1
X
γnl =
N1
X
γnl =
N1 X
N1
X
γnl = 0
n=1 l=1
l=1
γnt = 0
n=1
N1
X
γmn = 0,
∀ m.
(3.10)
n=1
Finally, to obtain identification of the firm dummies, one coefficient is arbitrarily normalized to zero.
We substitute the restrictions in (3.9) and (3.10) as well as (3.3) and (3.6) into the
stochastic translog shadow distance system (3.2), so that all derived first-order conditions
are also subject to these restrictions. The full stochastic specification appends random
errors to all first-order conditions. The econometric issues surrounding the estimation of
distance functions are detailed in Atkinson, Cornwell, and Honerkamp (2003). Their main
point is that treating both inputs and output as exogenous, as most previous work has
done, makes little sense either theoretically or empirically. Instead, they argue for a GMM
estimation strategy, which we follow here, that appeals to overidentification tests for model
validation.
Since we have introduced firm dummies into (3.2), we have subtracted them from ²f t
before estimation. After estimation we add the fitted firm dummies back to ²̂f t and then
regress the negative of this on the right-hand-side of (3.5) to obtain an unbiased estimator
11
of uf t , ûf t , which is used to compute technical inefficiency, PC, TC, and EC. Refer to
Atkinson, Cornwell, and Honerkamp (2003) for details.
b. Allocative and Technical Inefficiency
In most applications concerned with measuring allocative inefficiency, interest typically lies in the distortion of the input mix. Examples include the effects of import
restrictions, job hiring quotas, and government regulation on the misallocation of inputs.
Shadow cost systems yield shadow price ratios, which can be translated into measures of
relative over- or under-utilization of inputs. In contrast, with the shadow distance system,
ratios of the fitted equations in (2.7) for all input pairs satisfy the conditions for allocative
efficiency. So, for any firm in any time period we can directly estimate relative over- and
under-utilization of any pair of inputs, xnt and xlt , compared with the cost-minimizing
ratio, (knt xnt )/(klt xlt ), by computing k̂nt /k̂lt . The ability to directly measure relative
input misallocation is the fundamental advantage of the shadow distance system.
We estimate firm f ’s level of technical efficiency (TE) in period t as
TEf t = exp(−û∗f t ),
(3.11)
where û∗f t = ûf t − ût ≥ 0 and ût = minf (ûf t ) is the frontier intercept in period t.
Because we do not impose one-sidedness (non-negativity) on the uf t in estimation, we
do so afterwards by adding and subtracting ût from the fitted model. Let ln D̂(yt , x∗t , t)
represent the estimated translog portion of (3.2) (i.e., those terms other than h(²f t )).
Then, adding and subtracting ût yields
0 = ln D̂(yt , x∗t , t) + v̂f t − ûf t + ût − ût = ln D̂∗ (yt , x∗t , t) + v̂f t − û∗f t ,
(3.12)
where ln D̂∗ (yt , x∗t , t) = ln D̂(yt , x∗t , t) − ût is the estimated frontier distance function in
period t. This normalization guarantees that 0 ≤ TEf t ≤ 1.
12
c. Productivity Change
Following the estimation of (3.2), we compute PC as the sum of its technical and
efficiency change components:
PCf t = TCf t + ECf t .
(3.13)
Given the estimates of TEf t obtained from (3.11), changes in technical efficiency are calculated as
ECf t = ∆TEf t = TEf t − TEf,t−1 .
(3.14)
We calculate TC as a discrete approximation which involves computing the difference
between the estimated frontier distance function in periods t and t − 1 holding output and
input quantities constant:
TCf t = ln D̂∗ (yt , x∗t , t) − ln D̂∗ (yt , x∗t , t − 1)
=
X
γ̂mt ln ymf t (dt − dt−1 ) +
m
X
γ̂nt ln x∗nf t (dt − dt−1 )
n
+ γ̂t − γ̂t−1 + (ût−1 − ût ),
(3.15)
where dt is a time period dummy. Thus, the change in the frontier intercept, ût , affects
TC as well as EC.
4. Data and Results
The sample consists of 78 privately owned electric utilities operating in the U.S. over
the 10-year period 1988-97, for a total of 780 observations. Since technologies for nuclear,
hydroelectric, and internal combustion differ from that of fossil fuel-based steam generation and because steam generation dominates total production by investor-owned utilities
13
during the time period under investigation, we limit our analysis to this component of
steam electric generation.
The definitions of the variables are consistent with, but not identical to, those given
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, defined as the sum of salaries and wages charged to electric operation
and maintenance, divided by the number of full time plus one half the number of part
time employees. As a modification to Nelson, the price of capital is the yield of the firm’s
latest issue of long term debt adjusted for appreciation and depreciation of the capital
good using the Christensen-Jorgenson (1970) cost of capital formula. Output (y) is total
megawatts generated as compiled by Daniel McFadden and Thomas Cowing, updated and
complemented if necessary with data from Statistics of Privately Owned Electric Utilities
in the U.S. In addition to these modifications, we extended his data set to include the
1988-97 time period.
The primary sources for our data are the Federal Power Commission’s Statistics of
Privately Owned Electric Utilities in the U.S., Steam Electric Plant Construction Cost and
Annual Production Expenses, and Performance Profiles – Private Electric Utilities in the
United States. Additional data were taken from Moody’s Public Utility Manual . Whenever
necessary we accounted for missing data points by either using the value of the previous
period or the average of the previous and the subsequent period depending on how related
variables changed. After calculating total costs as the sum of total expenditure on inputs,
but before estimating our cost and distance systems, we normalize all price and quantity
data by their means. Table 1 lists the 78 utilities in our sample, which constitutes all
major privately owned utilities in the U.S.
14
The estimated static shadow distance system is (3.2) together with the price equations
in (3.4) for all inputs, while the estimated dynamic shadow distance system is (3.2) together
with (3.4) for the variable inputs–energy and labor–and (3.8) for the quasi-fixed input,
capital. In both we assume neutral technical change.
Three utilities—firms 11, 28, and 40—are owned by conglomerate parent companies.
They exhibit extremely low labor/output ratios compared to the other firms in the sample,
because they do not directly hire labor, but rather “rent” it from their parent companies.
Therefore, we control for ownership status, allowing for separate conglomerate-ownership
effects in the first-order input quantity and allocative efficiency parameters.
Estimation proceeds after substituting (3.3), (3.5), (3.6), (3.9), and (3.10) into the
estimating equations. As indicated above we must impose the following additional restrictions to achieve identification: kEt = 1 and β1q = 0, ∀q. Both systems are estimated by
GMM, allowing for heteroskedasticity and autocorrelation of unknown form by employing
the Newey and West (1987) covariance matrix estimator with a lag of two periods.
To identify the models, we specify an instrument set that includes first and secondorder terms in input prices, firm dummies, time dummies, firm and time dummies interacted, and interactions of prices and time dummies. Thus, output is treated as endogenous.
The only difference between the static and dynamic models is that all terms involving capital adjustment costs are eliminated. In both cases, the p-value of Hansen’s (1982) J
statistic is greater than .9, so we fail to reject the null that the overidentifying restrictions
have support in the data.
For the duality between input prices and quantities to be valid, the input shadow
distance function must be monotonically increasing in inputs and monotonically decreasing
15
in outputs. Our estimated model satisfies the required monotonicity properties for all
observations with both models.
Overall, both models are fit well. For the dynamic model, all but one of the technology
parameters (γtt ) are statistically significant at the .05 level, using a two-tailed test; for the
static model they all are statistically significant. Further, the firm and input-specific
parameters are generally very precisely estimated as well. The only exceptions in either
case are a handful of technical efficiency parameters and kL2 . Detailed results for both
distance systems are available on request.
First, we assess the role of adjustment costs in the electricity generation. The adjustment cost parameter, δK , is estimated to be .231 with a standard error of .007. Table
2 translates this estimate into total adjustment costs (TACk ) for the industry, which is
reported as a percentage of total capital costs (TCk ) and total variable costs (TVC), assuming the quantity of energy used is optimal for the firm. As shown in the second column
of Table 2, adjustment costs amounted to almost 1 percent of TVC early in the period, but
after the years 1989 and 1990, generally accounted for less than .4 percent. Averaging over
all the years of the sample, we find adjustment costs were .42 percent of TVC. Adjustment
costs were a considerably higher percentage of TCk , falling from 2.84 percent initially to
.08 percent in the final year. On average adjustment costs were 1.26 percent of TCk .
Second, we examine the effects of adjustment costs on the estimation of allocative and
technical efficiency. Table 3 presents the estimates of the allocative efficiency parameters
shown in (3.6), plus the estimates of the knd , which apply only to the conglomerate firms,
reflecting the differences between these and the other firms in terms of allocative efficiency.
With the exception of k̂Lt in the dynamic case, each is statistically significant at the .05
level with a two-tailed test. While the k̂K s produced by the two models are fairly similar,
16
there are substantial differences in magnitude between each k̂L . The k̂L from the dynamic
model is 1.058, which is more than twice as large as the static model estimate. Thus, at
least in terms of the time-invariant term, the dynamic model indicates labor usage is much
less distorted than suggested in the static case.
Given the positive and significant estimated coefficients for the conglomerate allocative efficiency dummy variables, the three conglomerate firms exhibit quite different levels
of allocative efficiency compared to the non-conglomerates. Under both models, the conglomerate firms should employ considerably more capital and labor relative to energy to
achieve allocative efficiency than is required by the non-conglomerates. Whether this reflects structural differences or mere accounting irregularities, we cannot say.
To see the implications of the differences in the estimated allocative efficiency parameters for the entire sample, for each model we compute (3.6) for each time period. Table
4 provides the results of this exercise. Since we fix kEt = 1, an estimate of kLt < 1, for
example, means that labor is over-utilized relative to energy. The pattern for static model
is k̂Lt and k̂Kt falling below 1 early in the sample period, while exceeding 1 by the middle.
The same pattern hold for k̂Kt in the dynamic case. Thus, the evidence indicates that
both capital and labor were under-utilized relative to energy from the early 1990s forward.
However, the picture painted by the dynamic model is that the misallocation with respect
to energy was less severe until the final period. Furthermore, the estimates from the dynamic model show less variability, particularly those associated with labor. The range of
the k̂Lt s fit from the dynamic model is much narrower, from 1.02 to 1.20, while k̂Lt s from
the static model vary from .68 to 1.26. Compared with labor, capital and energy are used
in much closer to the correct mix in every year. The estimated kKt s from both models fall
in a range of .98 to 1.03.
17
Table 5 reports the average technical efficiency for each utility in the sample using
(3.11), where we obtain ûf t using (3.5). First, for both the static and dynamic systems we
estimate (3.5) (with β1q = 0 ∀ q), using the residual from (3.2) as the dependent variable.
Then, we retrieve ûf t as the fitted value from this regression. Perhaps as we should expect,
controlling for adjustment costs has little effect on estimates of TE. The overall outputweighted average for the static and dynamic models is .47 and .46, respectively, and the
differences between the estimates are small at the firm level. Also, the rank correlation
between the two sets of estimates efficiencies is quite high.
To provide some context for our findings, we appeal to Rungsuriyawiboon and Stefanou (RS) (2007), who estimate their dynamic shadow-cost system using a similar set of
electric utilities covering roughly the same time period (1986-99). They find that capital
is over-utilized relative to labor, which is consistent with our results because k̂kt < k̂lt ,
but that energy is under-utilized relative to labor, which is not. This qualitative difference
may be explained by their choice to allow technical efficiency to vary across inputs. RS
implement the idea by normalizing the net investment efficiency parameter to one and
estimating the technical efficiency of variable input use relative to net investment. RS
report an average estimated relative technical efficiency of .767 and firm-specific values as
low as .24.
Directly comparing our estimated technical efficiencies with those produced by RS
is difficult, because technical efficiency is input-specific in their model. As an additional
assessment of our estimated efficiency levels, we examined the effects of vintage on technical
efficiency, controlling for capacity utilization as well as heating and cooling degree days.
Mean vintage is about 24 years and a histogram of utilities by vintage. We find that
vintage is strongly negatively correlated with estimated technical efficiency; utilities with
18
older plants are less technically efficient. This substantial plant age for each utility lends
credence to our argument that our single measure of technical efficiency for all inputs
should be lower than that for only the variable inputs, as reported by RS. It is reasonable
to infer that a utility operating at 50 percent efficiency would be tempted to tear down and
rebuild. However, new plants are subject to New Source Performance Standards, which
will invariably require scrubbers for SO2 and N Ox as well as a design and permitting
process that may be dragged out in court for many years. This makes the continued use
of older, technically inefficient plants a cost-effective option. Over time these plants will
be gradually retro-fitted and sometimes replaced with new ones.
Finally, we gauge the effects of adjustment costs on estimates of PC and its components. Table 6 presents output-weighted-average annual estimates of PCf t , TCf t , and
ECf t from the static and dynamic models. Both sets of results indicate the industry consistently exhibited positive PC after 1990, accounted for by improvements in TC. Often
EC is negative, indicating that less-efficient firms are unable to keep up with the outward
expanding production frontier. The primary difference between the static and dynamic
results is that the latter (again) show less variance and both PC and TC from the dynamic
model are smaller in absolute value in 5 of 9 cases. However, the dynamic model produces
a larger average annual rate of PC (4.3 vs. 2.6 percent) and TC (5.6 vs. 3.8 percent). The
estimates of EC from the two models, like their estimates of TE, are very similar.
19
5. Summary and Conclusions
Many studies have considered the estimation of allocative and technical inefficiency
using static production, cost, profit, and distance functions. Often the goal is to simultaneously measure firm or industry-level productivity growth. In this paper we employ a
dynamic optimization model of firm behavior, which allows us to distinguish static allocative distortions from short-run inefficiencies that arise from period-to-period adjustment
costs (shocks). Our dynamic model includes the Euler equations for the quasi-fixed inputs
and static input demand equations for the variable inputs, derived from the firms’s costminimization problem subject to a shadow-distance function constraint. Our static model,
which is nested within the dynamic model, is comprised of the shadow distance function
and the first-order conditions for the variable and quasi-fixed inputs.
Following Atkinson, Cornwell and Honerkamp (2003), we estimate both shadow distance systems by GMM using a recently constructed panel of 78 U.S. electric utilities covering the 1988–97 time period. Then, from the estimated parameters we construct measures
of adjustment costs, technical and allocative inefficiency, and productivity growth. First,
the zero-adjustment-cost restriction is strongly rejected. Second, we find that adjustment
costs represent about .42 percent of total variable cost and 1.26 percent of total capital
costs. Third, we show that controlling for adjustment costs has a substantial impact on the
estimated allocative inefficiency that would otherwise obtain. While both models suggest
that labor is not utilized efficiently, the dynamic model indicates less under-use of labor
and capital relative to energy and less variance over time in the degree of inefficiency. Finally, the dynamic model also indicates substantially greater but more stable productivity
growth than does the static model.
20
References
Atkinson, S. E., C. Cornwell, and O. Honerkamp, 2003, Measuring and decomposing productivity change: stochastic distance function estimation vs. dea, Journal of
Business and Economic Statistics 21, 284-294.
Atkinson, S. E. and C. Cornwell, 1994, Parametric measurement of technical and
allocative inefficiency with panel data, International Economic Review 35, 231-244.
Atkinson, S. E. and D. Primont, 2002, Measuring productivity growth, technical efficiency, allocative efficiency, and returns to scale using distance functions, Journal of
Econometrics 108: 203-225.
Bernstein, J. I., T. P. Mamuneas, and P. Pashardes, 2004, Technical efficiency and U.S.
manufacturing productivity growth, Review of Economics and Statistics 86, 402-412.
Christensen, L.R. and D. W. Jorgenson, 1970, U.S. real product and real factor input,
1928–1967, Review of Income and Wealth 16, 19-50.
Cooper, R.J., 2001, General Structural Dynamic Economics Modeling, Macroeconomic
Dynamics 5, 647-672.
Cornwell, C., P. Schmidt, and R.C. Sickles, 1990, Production frontiers with time series
variation in efficiency levels, Journal of Econometrics 46, 185–200.
Good, D., Nadiri, M., and R. Sickles, 1999, Index number and factor demand approaches to the estimation of productivity, in Handbook of Applied Econometrics,
Vol. II: Microeconomics, M. Pesaran and P. Schmidt, eds., 14-80.
Hansen, L. P., 1982, Large sample properties of generalized method of moments estimation, Econometrica 50, 1029–1054.
Moody’s Public Utility Manual, Moody’s Investors Service, New York, various years.
Morrison-Paul, C, and D. Siegel, 1999, Scale economies and industry agglomeration
externalities: a dynamic cost function approach, American Economic Review 89, 272–
290.
Nelson, R. A., 1984, Regulation, capital vintage, and technical change in the electric
utility industry, Review of Economics and Statistics 66, 59–69.
Newey, W. K., and K. D. West, 1987, A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica 55, 703–706.
Pindyck, R. S. and J. J. Rotemberg (1983), Dynamic factor demands and the effects
of energy price shocks, American Economic Review 73, 1066-1079.
Rungsuriyawiboon S. and S.E. Stefanou, 2007, Dynamic efficiency estimation: an
application to U.S. Electric Utilities, Journal of Business and Economic Statistics 25,
226-238.
U.S. Federal Power Commission, Steam Electric Plant Construction Cost and Annual
Production Expenses, Washington, D.C., annually.
U.S. Federal Power Commission, Statistics of Privately Owned Electric Utilities in the
U.S.–Classes A and B, Washington D.C., annually.
U.S. Federal Power Commission, Performance Profiles – Private Electric Utilities in
the United States: 1963–70 , Washington, D.C., annually.
21
Table 1: Utilities in the Sample
Firm Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
Utility
Alabama Power Co.
AmerenCIPS
AmerenUE
Appalachian Power Co.
Arizona Public Service Co.
Atlantic City Electric Co.
Baltimore Gas & Electric Co.
Boston Edison Co.
Carolina Power & Light Co.
Central Hudson Gas & Electric Corp.
CLECO Corp.
Central Maine Power Co.
Central Power & Light Co.
Cincinnati Gas & Electric Co.
Cleveland Electric Illuminating Co.
Columbus Southern Power Co.
Commonwealth Edison Co.
Consolidated Edison Co. of New York, Inc.
Dayton Power & Light Co.
Delmarva Power & Light Co.
Detroit Edison Co.
Duke Energy Corp.
Duquesne Light Co.
Entergy Arkansas, Inc.
Entergy Gulf States, Inc.
Entergy Louisiana, Inc.
Entergy Mississippi, Inc.
Entergy New Orleans, Inc.
Florida Power & Light Co.
Florida Power Corp.
Georgia Power Co.
Gulf Power Co.
Houston Lighting & Power Co.
Illinois Power Co.
Indiana Michigan Power Co.
Indianapolis Power & Light Co.
Interstate Power Co.
Kansas City Power & Light Co.
Kentucky Utilities Co.
22
Table 1: Utilities in the Sample (Cont.)
Firm Number
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
Utility
KGE, A Western Resources Company
Long Island Lighting Co.
Louisville Gas & Electric Co.
Minnesota Power & Light Co.
Mississippi Power Co.
Montana Dakota Utilities Co.
Montana Power Co.
New England Power Co.
New York State Electric & Gas Corp.
Niagara Mohawk Power Corp.
Northern Indiana Public Service Co.
Northern States Power Co.
Ohio Edison Co.
Ohio Power Co.
Oklahoma Gas & Electric Co.
Pacific Gas & Electric Co.
PacifiCorp
PECO Energy Co.
PP&L, Inc.
Potomac Edison Co.
Potomac Electric Power Co.
PSC of Colorado
PSC of New Hampshire
PSC of New Mexico
PSI Energy, Inc.
Public Service Electric & Gas Co.
Rochester Gas & Electric Corp.
San Diego Gas & Electric Co.
South Carolina Electric & Gas Co.
Southern California Edison Co.
Southwestern Electric Power Co.
Southwestern Public Service Co.
Tampa Electric Co.
Texas Utilities Electric Co.
United Illuminating Co.
Virginia Electric & Power Co. (Virginia Power)
West Penn Power Co.
Wisconsin Electric Power Co.
Wisconsin Public Service Corp.
23
Table 2: Adjustment Costs as a Percentage of Costs
Year
TACk /TCk
TACk /TVC
1989.
0.0253
0.0083
1990.
0.0246
0.0078
1991.
0.0043
0.0021
1992.
0.0012
0.0005
1993.
0.0174
0.0060
1994.
0.0104
0.0037
1995.
0.0067
0.0024
1996.
0.0113
0.0037
1997.
0.0008
0.0003
Avg.
0.0113
0.0038
24
Table 3: Estimated Allocative Efficiency Parameters
kK
kKd
kKt
kKt2
kL
kLd
kLt
kLt2
Static System
Dynamic System
0.9789
( 0.0098)∗
0.3927
( 0.0350)∗
0.0109
( 0.0042)∗
-0.0007
( 0.0004)∗
0.0001
( 0.0723)∗
1.2950
( 0.1680)∗
0.3817
( 0.0319)∗
-0.0290
( 0.0029)∗
0.9941
( 0.0000)∗
0.3413
( 0.0000)∗
0.0110
( 0.0000)∗
-0.0011
( 0.0000)∗
0.6988
( 0.0000)∗
1.1486
( 0.0000)∗
0.0946
( 0.0000)∗
-0.0052
( 0.0000)∗
Note: Asymptotic standard errors in parentheses. An asterisk denotes significance at the .05 level using a two-tailed
test.
25
Table 4: Average Time-Varying Allocative Inefficiency
Static
Dynamic
Year
kk
kl
kk
kl
1988.
1989.
1990.
1991.
1992.
1993.
1994.
1995.
1996.
1997.
0.9920
1.0003
1.0077
1.0134
1.0173
1.0206
1.0219
1.0217
1.0199
1.0169
0.3627
0.6563
0.8940
1.0729
1.1927
1.2575
1.2626
1.2099
1.0986
0.9300
1.0065
1.0139
1.0196
1.0228
1.0235
1.0228
1.0193
1.0137
1.0057
0.9957
0.7969
0.8748
0.9439
1.0019
1.0484
1.0870
1.1136
1.1298
1.1350
1.1302
Avg.
1.0131
0.9937
1.0144
1.0261
26
Table 5: Average Firm Technical Efficiencies
Firm
Static System
Dynamic System
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
0.5313
0.2606
0.3460
0.4961
0.1858
0.4078
0.2307
0.2142
0.3229
0.5110
0.5425
0.5560
0.3545
0.4535
0.2805
0.3166
0.2025
0.2272
0.2809
0.3303
0.3440
0.3311
0.2891
0.4596
0.4277
0.5745
0.4555
0.6455
0.3389
0.2779
0.5710
0.3765
0.4689
0.3322
0.3191
0.2943
0.5726
0.3006
0.4209
0.5174
0.2551
0.3354
0.4903
0.1788
0.4002
0.2219
0.2010
0.3126
0.5061
0.5891
0.5447
0.3473
0.4496
0.2791
0.3070
0.1939
0.2163
0.2723
0.3243
0.3329
0.3175
0.2818
0.4420
0.4207
0.5511
0.4443
0.6794
0.3300
0.2689
0.5587
0.3706
0.4612
0.3248
0.3113
0.2875
0.5692
0.2918
0.4145
27
Table 5: Average Firm Technical Efficiencies (Cont.)
Firm
Static System
Dynamic System
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
0.6674
0.2466
0.2751
0.3096
0.3961
0.4706
0.3786
0.4459
0.2800
0.1813
0.2254
0.3059
0.3365
0.7721
0.4536
0.2621
0.5298
0.2110
0.4621
0.2819
0.3563
0.3242
0.3492
0.2594
0.5137
0.1816
0.4863
0.4848
0.3219
0.3543
0.4694
0.5860
0.3013
1.0000
0.4886
0.3228
0.4785
0.2853
0.2907
0.6788
0.2432
0.2658
0.3040
0.3849
0.4732
0.3743
0.4382
0.2721
0.1686
0.2173
0.3000
0.3300
0.7707
0.4490
0.2522
0.5246
0.2009
0.4571
0.2722
0.3466
0.3151
0.3381
0.2524
0.5070
0.1684
0.4810
0.4859
0.3063
0.3464
0.4631
0.5793
0.2955
1.0000
0.4803
0.3124
0.4755
0.2798
0.2817
Avg.
0.3871
0.3807
28
Table 6: Time Varying PC, TC, and EC
(FirmWeightedAverages)
Static System
Dynamic System
Year
PC
TC
EC
PC
TC
EC
1988.
1989.
1990.
1991.
1992.
1993.
1994.
1995.
1996.
-0.1652
-0.1229
-0.0814
-0.0410
-0.0013
0.0384
0.0776
0.1181
0.1599
-0.1648
-0.1247
-0.0846
-0.0445
-0.0044
0.0356
0.0757
0.1158
0.1559
-0.0005
0.0018
0.0032
0.0035
0.0032
0.0028
0.0019
0.0023
0.0040
-0.0513
-0.0330
-0.0156
0.0010
0.0171
0.0334
0.0491
0.0661
0.0840
-0.0531
-0.0363
-0.0195
-0.0028
0.0140
0.0308
0.0476
0.0644
0.0812
0.0018
0.0033
0.0039
0.0038
0.0031
0.0025
0.0015
0.0017
0.0029
Avg.
-0.0020
-0.0044
0.0025
0.0168
0.0140
0.0027
29