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Estimation of Technical and Allocative Inefficiency: A Primal System
Approach
Subal C. Kumbhakar
Department of Economics
State University of New York - Binghamton
Binghamton, NY 13902
Hung-Jen Wang
Institute of Economics
Academia Sinica
Taipei 115, Taiwan
February 8, 2004
Abstract
The estimation of technical and allocative inefficiency using a flexible (translog) cost system
is found to be quite difficult (not yet solved satisfactorily), especially when both inefficiencies are
random. In this paper we use the alternative primal system consisting of the production function
(translog) and the first-order conditions of cost minimization. The estimation of the primal system
is more straightforward and it enables us to estimate observation-specific technical and allocative
inefficiency. The impact of technical and allocative inefficiency on input demand and cost are also
computed. We use panel data on steam-electric generating plants from the U.S. to estimate the
model using both Cobb-Douglas and translog production functions.
Keywords: Cost and Production Functions; Cost and Production Systems; Technical Change;
Returns to Scale.
JEL Classification No.: C31, D21
1 Introduction
A quarter of a century ago Peter Schmidt and Knox Lovell (1979) (SL hereafter), in a seminal paper,
proposed estimating technical and allocative inefficiency jointly using a Cobb-Douglas production
function and a cost minimizing behavioral assumption. The system of equations they used consisted
of the production function and the first-order conditions of cost minimization. The estimation of such
a primal system can produce observation-specific estimates of technical and allocative inefficiency.
For a Cobb-Douglas production function (which is self-dual), the cost of technical and allocative
inefficiency (observation-specific) can also be derived analytically.
An alternative to this primal approach is to use a cost system consisting of a cost function and
the cost share equations, and also rely on duality results, especially when a flexible functional form
is used. Although widely used in productivity and efficiency studies, the cost system approach has
several limitations, especially when both technical and allocative inefficiency are introduced into the
model. Since both technical and allocative inefficiency increase cost, it is necessary to estimate the
increased cost associated with each type of inefficiency. To do so we need additional information
coming from either the cost share equations or the input demand functions. From a modeling point
of view, the main problem is: How to link allocative inefficiency (usually the errors in the cost
share equations) with cost of allocative inefficiency (in the cost function) in a theoretically consistent
manner?1 Establishing the linkage is necessary in order to formulate the econometric model based
on the cost system. The second problem is: What are the justifications and interpretations of the
noise/error terms (that are routinely appended before estimation) in the cost function and cost
share equations?2 Finally, is it possible to estimate the cost system (using the maximum likelihood
technique with cross-sectional data) assuming that both technical and allocative inefficiency are
random?
Some of the above-mentioned problems (outlined in details in section 2) are too difficult to address
in a cost system. Because of these difficulties, either strong assumptions are made regarding the
link between allocative inefficiency and increase in cost therefrom (Bauer (1985), Melfi (1984)), or
1Joint estimation of technical and allocative inefficiency in a translog cost function presents a difficult problem (Greene
(1980)). This problem is later labeled as the Greene (1980) problem by Bauer (1990). Recently Kumbhakar (1997) proposed
a solution for the Greene problem using a translog cost system, but empirical estimation of this model has been restricted
to panel data models in which technical and allocative inefficiency are either assumed to be fixed parameters or parametric
functions of the data and unknown parameters. Kumbhakar and Tsionas (2003), who used a Bayesian approach, is an
exception.2Note that the error/noise term in the production function does not always get transmitted to the cost function (in
log) in a linear fashion. It depends on the functional form used to represent the production technology.
1
producers are assumed to be allocatively efficient. We avoid these difficulties by using the primal
approach of SL (1979). We extend their approach and use a flexible (translog) production function.
In this approach (observation-specific) estimates of technical and allocative inefficiency can be easily
obtained after the model is estimated. Since the cost function associated with the translog production
function cannot be analytically derived, algebraic expressions for the cost of technical and allocative
inefficiency cannot be derived. To compute the impact of technical and allocative inefficiency on input
demand, we solve the primal production system numerically for input quantities (with and without
inefficiency). These results are then used to compute the effect of technical and allocative inefficiency
on cost. Thus, we obtain observation-specific estimates of technical and allocative inefficiency as well
as the increased cost associated with each of these inefficiency components.
The rest of the paper is organized as follows. In section 2 we describe the cost system approach that
accommodates both technical and allocative inefficiency and point out the difficulties in estimating
such a system. The primal production system with both technical and allocative inefficiency is
developed in section 3. Data is described in section 4. Section 5 discusses the results and Section 6
contains our conclusions.
2 Cost system approach with technical and allocative ineffi-
ciency
The production function for a typical producer (i) with output-oriented technical inefficiency (Aigner
et al. (1977), and Meeusen and van den Broeck (1977)) can be represented as
yi = f(xi) · e(vi−ui), i = 1, 2, ..., n, (1)
where f(·) is the production frontier, x is the vector of inputs, v is production uncertainty (noise),
and u ≥ 0 is output-oriented technical inefficiency, which can be interpreted as the percent loss of
output, ceteris paribus, due to technical inefficiency. The cost minimization problem for a typical
producer is (omitting the subscript i)
min w · x s.t. y = f(x)ev−u (2)
where w is the vector of input prices. The first-order conditions of the above problem can be expressed
implicitly as
fj
f1=
wj
w1eξj =
wsj
w1, j = 2, · · · , J, (3)
where wsj = wje
ξj and ξj �= 0 is allocative inefficiency for the input pair (j, 1). This definition of
allocative inefficiency, used by SL (1979), is appropriate, because a producer is allocatively inefficient
2
when it fails to allocate inputs in a way that equates the marginal rate of technical substitution
(MRTS) with the ratio of respective input prices.
The above minimization problem can be written as a standard neoclassical cost minimization
problem, viz.,
min ws · x s.t. yeu−v = f(x), (4)
in which the first-order conditions (given y, v, u and ξj) are exactly the same as above. The advantage
of positing the problem in the above format is that stochastic noise, as well as technical and allocative
inefficiency are already built-in to the above optimization problem, and there is no need to append
any extra terms in an ad hoc fashion at the estimation stage. Moreover, all the standard duality
results go through, although the model we consider here allows for the presence of statistical noise,
and technical and allocative inefficiency. For example, the solution of xj (the conditional input
demand functions) can be expressed as xj = xj(ws, yeu−v), j = 1, · · · , J . Furthermore, the cost
function, defined as cs(·) = ws · x(·), can be expressed as
cs(·) = c(ws, yeu−v). (5)
Note that cs(·) is neither the actual cost nor the minimum cost function. The former is the cost
of inputs used at the observed prices, while the latter is the cost of producing a given level of output
without any inefficiency. The cost function cs(·) is an artificial construct that is useful for exploiting
the duality results. Since cs(·) is derived from the neoclassical optimization problem in which the
relevant input prices are ws and output is yev−u, it can be viewed as the cost function when prices
are ws and output is yev−u. Thus, if one starts from the cs(·) function, Shephard’s lemma can be
used to obtain the conditional input demand functions (i.e., ∂cs(·)∂ws
j= xj(·)). Equivalently,
∂ ln cs(·)∂ ln ws
j
=ws
j xj(·)cs(·) ⇒ xj(·) =
cs(·)ws
j
ssj(·),
where ssj(·) =
∂ ln cs(·)∂ ln ws
j
.
(6)
The actual cost ca is now expressed as
ca =∑
j
wjxj(·) =∑
j
wsj xj(·)
(wj/ws
j
)= cs(·)
∑j
ssj(·)
(wj/ws
j
),
⇒ ln ca = ln cs(·) + ln
ss
1 +J∑
j=2
ssj(·)e−ξj
.
(7)
The above equation relates actual/observed cost with the unobserved cost function cs(·), and the
formulation is complete once a functional form on cs(·) is chosen. For example, if cs(·) is assumed to
3
be Cobb-Douglas, then the above relationship becomes
ln ca = α0 +∑
j
αj lnwsj + αy ln
(yeu−v
)+ ln
α1 +
J∑j=2
αje−ξj
= α0 +J∑
j=1
αj ln wj + αy ln y + αy (u − v) +J∑
j=2
αjξj + ln
α1 +
J∑j=2
αje−ξj
.
(8)
Furthermore, the actual cost share equations are
saj =
wj xj
ca=
cs(·)eξj
αj
ca= αje
−ξj
{α1 +
J∑k=2
αke−ξk
}−1
= αj +
{αj
[e−ξj
α1 +∑J
k=2 αk e−ξk
− 1
]},
⇒ saj ≡ αj + ηj(ξ), j = 2, . . . , J,
(9)
where the cost function errors (ηj) are functions of allocative errors (ξ), and they are therefore related
to the cost of allocative inefficiency defined below. The above relationship can also be directly derived
from the Cobb-Douglas production function along with the first-order conditions of cost minimization.
The cost function in (8) can be written as
ln ca = ln c0(·) + ln cu + ln cv + ln cξ, (10)
where ln c0 = α0+ln[∑J
j=1 αj
]+∑J
j=1 αj ln wj +αy ln y is the minimum (neo-classical) cost function
(frontier). The percentage increase in cost due to technical inefficiency is ln cu×100 = αy u×100 ≥ 0.
Similarly, the percentage increase in cost due to allocative inefficiency ln cξ (when multiplied by 100)
isJ∑
j=2
αjξj + ln
[α1 +
J∑j=2
αje−ξj
]− ln
[∑Jj=1 αj
]. Production uncertainty can either increase or
decrease cost since ln cv = −αy v > (<)0 , depending on v < 0(> 0).
It is clear from equations (8) and (9) that the error components in the above system are quite
complex. The allocative inefficiency term (ξj) appears in a highly non-linear fashion in both the cost
function and cost share equations. Consequently, an estimation of the model (the cost function and
the cost share equations specified above) based on distributional assumptions on ξj , u, and v is quite
difficult. The main problem in deriving the likelihood function is that elements of ξ appear in the
above system in a highly non-linear fashion. Alternatively, if one makes a distributional assumption
on the cost share errors ηj (which are also non-linear functions of the elements of ξ), then to derive
the likelihood function one has to express ξj in terms of η , because ξjs appear in the cost function.
Thus, even if one starts from a Cobb-Douglas production function, the estimation of the cost system
is difficult (a closed-form expression of the likelihood function is not possible).
4
Similar results are obtained if one uses a translog function for ln cs(·). The actual cost function is
ln ca = ln cs(·) +
ss
1 +∑
j
ssj(·) e−ξj
, (11)
where
ln cs(·) = α0 +∑
αj ln wsj + αy ln
(yeu−v
)+
12
∑j
∑k
αjk ln wsj ln ws
k +12αyy
{ln(yeu−v
)}2
+∑
j
αjy ln wsj ln
(yeu−v
),
(12)
and
ssj(·) = αj +
∑k
αjk ln wsk + αjy ln
(yeu−v
). (13)
The cost function in (11) can be written as
ln ca = ln c0 +∑
αjξj + αy (u − v) +∑
j
∑k
αjk ln wjξk +12
∑j
∑k
αjk ξjξk +∑
j
αjy ξj ln y
+∑
j
αjy ξj(u − v) + αyy ln y(u − v) +12αyy(u − v)2 + {ss
1 +∑
j
ssj(·)e−ξj},
(14)
where the frontier (minimum) cost function (c0(·)) is given by
ln c0(·) = α0 +∑
αj ln wj + αy ln y +12
∑j
∑k
αjk ln wj lnwk +12αyy ln y2
+∑
j
αjy ln wj ln y.
(15)
Finally, the actual cost share equations are
saj =
wjxj(·)ca
=cs(·)eξj
ssj(·)ca
=ss
j(·)e−ξj
ss1 +
J∑k=2
ssk(·)e−ξk
, j = 2, · · · , J (16)
≡ s0j (·) +
ss
j(·)e−ξj
ss1 +
J∑k=2
ssk(·)e−ξk
− s0j (·)
= s0
j (·) + ηj (ξ, u, v, and data) , (17)
where ssj(·) is written as
ssj(·) = s0
j +∑
k
αjkξk + αjy(u − v), (18)
and
s0j (·) = αj +
∑k
αjk ln wk + αjy ln y. (19)
5
It is clear from the above algebraic expressions that the translog cost function cannot be expressed
as
ln ca = ln c0(·) + ln cu + ln cv + ln cξ, (20)
which decomposes actual cost (in log) into the minimum (frontier) cost (ln c0(·)), an increase in
cost (log) due to technical (ln cu) and allocative inefficiency (ln cξ), and an increase/decrease in cost
(log) due to uncertainty in the production process (v). The terms u, v, and ξj term interact among
themselves in the cost function, which makes it impossible to decompose the log cost into the above
four components. In other words, an increase in cost due to technical (allocative) inefficiency will, in
general, depend on allocative (technical) inefficiency and the noise components.
The translog cost system is hence more complex than the Cobb-Douglas cost system, because
technical inefficiency (u) and the noise term (v) enter non-linearly in the cost system, and they also
interact with the allocative inefficiency components (ξj) as well as output and input prices. In other
words, it is impossible to estimate the translog cost system starting from distributional assumptions
on u, v and ξj . Note that none of the error terms in the present model is ad hoc.
If the underlying production function is homogenous, then it satisfies the following parametric
restrictions, viz., αyy = 0, αjy = 0 ∀ j. Consequently, the cost function is linear in u and v (as in
the Cobb-Douglas case), but the non-linearity in ξ is not eliminated. Therefore, the observed cost
can be expressed as ln ca = ln c0(·) + ln cu + ln cv + ln cξ. That is, the cost function (in log) can
be expressed as the sum of frontier cost plus a percentage change in costs (when multiplied by 100)
due to technical and allocative inefficiency and production uncertainty. An estimation of the model
is still difficult, if not impossible, because of the non-linearities in ξ. Unless the parameters of the
cost system are estimated consistently, the costs of technical and allocative inefficiency cannot be
computed (although algebraic expressions for an increase in cost due to technical and allocative
inefficiency are known). In other words, in the case of a homogeneous production function, the cost
system formulation has the advantage of obtaining analytical solutions of ln cu, ln cv, ln cξ, but an
econometric estimation of the model is still difficult (if not impossible).
We now consider the alternative modeling strategy, viz., the primal approach. Compared to
the cost system approach, the primal system is easier to estimate, although analytical expressions
for ln cu, ln cv, and ln cξ are not available. We compute them numerically by solving a system of
non-linear equations for each observation.
6
3 The primal system
It is possible to avoid the estimation problem discussed in the preceding section if one starts from
the production function and uses a system consisting of the production function and the first-order
conditions of cost minimization. Note that this system is algebraically equivalent to the cost system
for self-dual production functions. The only difference is that one starts from a parametric production
function, instead of a cost function. We start from the production function (either a Cobb-Douglas
or a translog)
ln y = ln f(x) + v − u, (21)
and write down the first-order conditions of cost minimization, viz.,
fj
f1=
wj
w1eξj ⇒ ∂ ln f
∂ ln xj÷ ∂ ln f
∂ ln x1≡ sj
s1=
wjxj
w1x1eξj ,
⇒ ln sj − ln s1 − ln(wjxj) + ln(w1x1) = ξj ; j = 2, ..J.
(22)
We interpret ξj(>=< 0) as allocative inefficiency for the input pair (j, 1). Thus, for example, if
ξ2 < 0 ⇒ w2eξ2 < w2, then input x2, relative to input x1, is over-used. The following figure shows it
graphically for two inputs.
1
2
ww
2
1
2
w
w eξ
A
B
x1
0y
x2
Figure 1: Allocative inefficiency without technical inefficiency
Assume that there is no technical inefficiency. The input quantities given by point A are used
to produce output level y0. However, the optimal input quantities are given by point B which is
7
the tangency point between the isoquant and the isocost line, i.e., MRTS = f1f2
= w1w2
. At point
A the equality f1f2
= w1w2
is not satisfied. The dotted isocost line is tangent to the observed input
combination (point A). That is, the observed input quantities are optimal with respect to the input
price ratio w1w2eξ2
. Departure from the optimality condition (MRTS = w1w2
) is shown by the difference
in slopes of the dotted and the solid isocost line. Failure on the part of the producer to allocate
inputs optimally by equating MRTS with the respective price ratios (given by point B) is viewed as
allocative inefficiency.3
Now assume that the producer is technically inefficient, i.e., y = f(x)e−u (ignoring the presence
of the stochastic noise component, v). Point A in Figure 2 shows an observed input combination
where the output produced is y0. We write the production function as yeu = f(x), which shows a
neutral shift of the isoquant from y0 to y0eu > y0.
B'
A'
A
B
x1
x2
O
1
2
ww
y0
y0eu
Figure 2: Allocative inefficiency with technical inefficiency
Allocative inefficiency can then be defined in terms of the new isoquant (y0eu). Given the prices,
the allocatively efficient input combination is shown by point B, whereas, the actual input combi-
nation is given by point A (which is made technically efficient by shifting the isoquant). Allocative
inefficiency is shown by the difference in the slope of the (y0eu) isoquant between points A and B.
Since technical inefficiency shifts the production function neutrally, the slope of the isoquant will
be unchanged. Thus, one can define allocative inefficiency with respect to the y0 isoquant by dropping
3It is worth mentioning that such failures may not be a mistake, especially if producers face other constraints in input
allocations (e.g., regulatory constraints).
8
down radially from points A to A′ and B to B′. Allocative inefficiency can then be represented by
the difference in the slopes of the y0 isoquant between points A′ and B′.4
The error structure of the primal system described in (21) and (22) is simple enough to derive the
joint probability density function of the error vector. The joint pdf is required to estimate the above
system using the maximum likelihood (ML) method, and it requires distributional assumptions to be
made on the error components. Here, we make the following assumptions (some of which are relaxed
later):
v ∼ N(0, σ2v), (23)
u ∼ N+(0, σ2), (truncated at zero from below) (24)
ξ ∼ MV N(0,Σ), (25)
ξj are independent of v and u. (26)
The assumptions on v and u are standard in the efficiency literature, although other distributions
such as exponential and truncated normal can be used for u (see, e.g., Kumbhakar and Lovell 2000).
Since ξj can take both positive and negative values (meaning that inputs can be over- or under-
used), it is natural to assume normal distributions on allocative errors.5 The final assumption of
independence is for simplicity.6
With the above distributional assumptions, the joint probability distribution of v − u and ξ is
f(v − u, ξ) = g(v − u) · h(ξ), (27)
where
g(v − u) =2σ
φ
{(v − u)
σ
}Φ{−(v − u)σu
σvσ
}, (28)
and φ(·) and Φ(·) are the pdf and cdf of a standard normal variable, respectively, and σ =√
σ2u + σ2
v .
The multi-variable normal pdf for ξ is given by h(ξ).
The likelihood function for the system in (21) and (22) is
L = g(v − u) · h(ξ) · |J |, (29)
where |J | is the determinant of the Jacobian matrix, viz.,
|J | =∣∣∣∣ ∂(v − u, ξ2, ξ3, ..., ξJ )∂(ln x1, ln x2, ..., ln xJ)
∣∣∣∣ . (30)
4Note that slopes of the isoquants at A and A′ (B and B′) are the same.5Note that the cost shares appear in logs in the first-order conditions and are in difference form. Thus, although the
shares are limited between 0 and 1, ln sj − ln s1 will not be constrained to take only positive or negative values.6Schmidt and Lovell (1980) allowed a positive correlation between u and |ξj |, j = 2, · · · , J .
9
The log-likelihood function, for a single observation, is
ln Li = constant − 12
ln σ2 + lnφ
((vi − ui)
σ
)+ ln Φ
(− (vi − ui)σu
σvσ
)
− 12
ln |Σ| − 12ξ′
iΣ−1ξi + ln |J |.
(31)
The likelihood function can be concentrated with respect to Σ. The elements of Σ, σjk, can be
obtained from
σjk =1N
∑i
ξjiξki, j, k = 2, · · · , J,
⇒ Σ =1N
∑i
ξiξ′i. (32)
Substituting (32) into the above log-likelihood function gives the concentrated log-likelihood function.
The observation sum of this concentrated log-likelihood function can be maximized to obtain the ML
estimate of the parameters.
After estimating the parameters, one would like to obtain (observation-specific) estimates of
technical and allocative inefficiency. Technical inefficiency u (for each observation) can be estimated
using the Jondrow et al. (1982) formula, viz., E{u|(v − u)}, which for the present model is
E{u|(v − u)} = µ∗ + σ∗ φ(µ∗/σ∗)Φ(µ∗/σ∗)
, (33)
where µ∗ = − (v−u) σ2u
σ2 and σ∗ = σuσv
σ . The estimates of u show the percent by which output falls
short of the frontier (maximum possible) output due to technical inefficiency.
Allocative inefficiency ξj for the input pair (j, 1) can be obtained from the residuals of the first
order conditions. The sign of ξj shows whether input j is over- or under-used relative to input 1. If
ξj > 0 , then input j is under-used relative to input 1. Unfortunately, the extent of over-use (under-
use) of inputs cannot be inferred from the values of ξj alone. For this we need to derive the input
demand function, which is not possible (analytically) for the translog production function. However,
one can compute the extent of over-use (under-use) of inputs numerically (discussed later).
While the estimates of technical and allocative inefficiency are useful economically, one might be
interested in computing the effect of technical and allocative inefficiency on cost. This is because
what matters most to cost-minimizing producers is by how much is cost increased due to inefficiency.
To address this issue, first we compute (numerically) the impact of u and ξ on lnxj . These results
are then used to compute the impact of u and ξ on cost.
10
4 Data
We use data on fossil fuel fired steam electric power generation plants (investor-owned utilities) in the
United States.7 A panel data (1986-1996) on 72 electric utilities are used in this study. The sources
of the data are: Energy Information Administration, the Federal Energy Regulatory Commission,
and the Bureau of Labor Statistics.
The output is net steam electric power generation in megawatt-hours which is defined as the
amount of power produced using fossil-fuel fired boilers to produce steam for turbine generators in a
given period of time. The variable inputs are: labor and maintenance (L), fuel (F), and capital (K).
The price of labor and maintenance (wL) is a cost-share weighted price for labor and maintenance.
Quantities of labor and maintenance are obtained from the cost of labor and maintenance divided
by its price. The price of fuel (wF ) is the average price of fuels (coal, oil and gas) in dollars per
BTUs. The fuel quantity (F ) is obtained from dividing the fuel cost by the fuel price. The price
of capital (wK) is calculated using the Christensen and Jorgenson (1970) cost of capital formula.
Finally, capital stock is measured using the estimates of capital cost discussed in Considine (2000).
We also use the time trend (t) to capture technical change.
5 Empirical results
5.1 Econometric model and results
Appending the time trend variable (t) as a regressor and introducing the firm subscript i, the translog
production function can be expressed as
ln yit = α0 +∑
j
αj ln xjit + αt t +12
∑
j
∑k
αjk ln xjit ln xkit + αtt t2
+
∑j
αjt ln xjit t
+ vit − uit, j = labor, fuel and capital.
(34)
The corresponding first-order conditions (using labor as the numeraire) are
ln sjit − ln s1it − ln(wjitxjit) + ln(w1itx1it) = ξjit, j = fuel and capital. (35)
where
sjit = αj +∑
k
ajk ln xkit + αjt t, j = fuel and capital.
7We thank Spiro Stefanou for providing the data to us. Details on the construction of the data set can be found in
Rungsuriyawiboon and Stefanou (2003).
11
The likelihood function of the model, which consists of the production function in (34) and the
first-order conditions in (35), is obtained from (31). We concentrated the likelihood function to get
rid of the parameters associated with Σ. The remaining parameters of the model are estimated by
maximizing the concentrated log-likelihood function. We estimate both the Cobb-Douglas and the
translog production functions. The parameter estimates are presented in Table 1. All the parameters
(except 6 out of 15 in the translog case) are statistically significant. The model with a Cobb-Douglas
production function can be obtained by imposing parameter restrictions (e.g., αjk = 0, αjt = 0,
and αtt = 0) on the model of (34) and (35). Such restrictions are overwhelmingly rejected by the
likelihood ratio test. Thus, the translog model (although it has some insignificant parameters) is
supported by the data.
Based on the estimated parameters we compute several statistics that are of interest to economists.
First, we estimate the returns to scale (RTS) from
RTS =∑
j
∂ ln y
∂ ln xj≡∑
j
sj(·), (36)
which is a constant (∑
j αj) for the Cobb-Douglas case, but is observation-specific for the translog
model. The Cobb-Douglas model predicts constant (unitary) returns to scale, while a slightly de-
creasing returns to scale (although not statistically different from unity) is observed at the mean
of the data for the translog model. Estimates of RTS in the translog model range from 0.89 to
1.12. These results suggest that most of the electric utilities in our sample are operating at their
efficient scale (minimum point of the average cost curve), and therefore are not likely to benefit from
increasing their size.
We also compute and report (in Table 2) technical change (TC) from
TC =∂ ln y
∂t≡ αt +
∑j
αjt ln xj + αtt t, (37)
which is a constant (∑
j αt) for the Cobb-Douglas case but is observation-specific for the translog
model. Both the Cobb-Douglas and the translog model predict technical progress at the rate of
about 2% per year. Large variations in TC are observed in the translog model (ranging from 0.86%
to 3.42% with a standard deviation of .57%). A 2% technical progress per year means that output
can be increased on average by 2% per year, holding everything constant.
To obtain an observation-specific estimate of technical inefficiency (u), we use the Jondrow et
al. (1982) result; that is, estimate u from u = E(u|v−u) in which (v−u) is replaced by the residuals
of the production function. To save space we report the mean values of u in Table 2 for both the
Cobb-Douglas and the translog functions. Both the Cobb-Douglas and translog functions show that,
on average, the electric utilities are producing 30% less than their maximum potential output due to
12
technical inefficiency. Given that electric utilities mostly operate with excess capacity (to meet peak
demand), the presence of a somewhat large technical inefficiency might not be surprising.
We next examine allocative inefficiency ξ for fuel and capital (relative to labor). Since the mean
values of ξF and ξK are negative (-0.01 and -0.013 for the Cobb-Douglas, and -0.006 and -0.016 for
the translog), this means that exp(ξF ) < 1 and exp(ξK) < 1, and labor/fuel and labor/capital ratios
are on average lower than the cost minimizing ratios. This result shows that capital is over-used
relative to both labor and fuel (the Averch-Johnson hypothesis). An over-use of capital (relative to
fuel and labor) follows from the fact that wK
wF
exp(ξK)exp(ξF ) < wK
wFand wK exp(ξK)
wL< wK
wLevaluated at the
mean values of ξF and ξF .
As mentioned before, the estimates of ξ for each pair of inputs tell us whether an input is over- or
under-used (relative to another input). However, the degree of over- (under-) use cannot be inferred
from the estimates of ξj . To do so, we need to derive the input demand functions, which for the
Cobb-Douglas model, are:
ln xj = aj +1r
J∑k=1
αk ln wk − ln wj +1r
ln y +1r
J∑k=2
αkξk − ξj − 1r
(v − u) , j = 2, ..., J
ln x1 = a1 +1r
J∑k=1
αk ln wk − ln w1 +1r
ln y +1r
J∑k=2
αkξk − 1r
(v − u)
where r =J∑
k=1
αk, aj = lnαj − 1r
[α0 +
J∑k=1
αk ln αk
], j = 1, · · · , J.
(38)
Thus, for example, due to allocative inefficiency demand, xj is increased or decreased by [ln xj |ξ =
ξ]− [lnxj |ξ = 0] = 1r
J∑k=2
αk ξk− ξj percent for j = 2, · · · , J , while for input x1 it is 1r
J∑k=2
αk ξk percent.
Since ξj can be positive or negative, the presence of allocative inefficiency can either increase or
decrease demand for an input. Note that ξj are residuals of the first-order conditions in (35).
The effect of technical inefficiency on input demand is obtained from [ln xj |u] − [lnxj |u = 0] =
(1/r)u ≥ 0 for j = 1, · · · , J . Thus, demand for each input is increased by (1/r)u 100 percent due to
technical inefficiency.
For the translog model an analytical solution of lnxj is not possible. We can, however, compute
the effects of technical and allocative inefficiency on input demand numerically. To do so, first we
solve for ln xj from the production function and the first-order conditions in (21) and (22) using the
estimated parameters and setting u = 0 and ξj = 0 ∀ j. We label the solution as lnxoj . We then
solve the system again setting u = u and ξj = 0 ∀ j, and label the solution as ln xj . Finally, we solve
the system setting u = 0, and ξj = ξj , and label the solution as ln xj . Using these solutions of lnx,
the impact of allocative inefficiency on xj is computed from ln xj − ln x0j for j = 1, · · · J . Similarly,
the effect of technical inefficiency on the demand for xj can be computed from ln xj − ln x0j .
13
5.2 Computing the cost of technical inefficiency and allocative inefficiency
Since both technical and allocative inefficiencies increase cost, it is desirable to compute the increase
in cost due to each of them. Such a cost difference can be obtained from the cost function with
and without inefficiency. Since the cost function has an analytical solution for the Cobb-Douglas
model, it is possible to get analytical solutions for the cost of technical inefficiency (ln cu) and cost
of allocative inefficiency ln cξ. For this we write the cost function as
ln ca = a0 +1r
ln y +1r
J∑i=1
αi ln wi − 1r
(v − u) + E − ln r
where a0 = ln r − α0
r− 1
r
(∑i
αi ln αi
), and
E =1r
J∑j=2
αjξj + ln
α1 +
J∑j=2
αje−ξj
− ln r.
(39)
Thus ln cu ≡ ln ca− [ln ca|u = 0] = u/r ≥ 0. Similarly, ln cξ = ln ca− [ln ca|ξj = 0 ∀ j] = E− ln r ≥ 0.
For a translog production function the corresponding cost function cannot be derived analytically.
Consequently, it is not possible to get analytical expressions for ln cu and ln cξ. We use numerical
solutions of x with and without inefficiency to compute ln cu and ln cξ. That is, the percentage
increase in cost due to technical inefficiency, ctech, is
ctech = (w′x)/(w′xo) − 1. (40)
Similarly, the percentage increase in cost due to allocative inefficiency, callo, is
callo = (w′x)/(w′xo) − 1. (41)
5.3 Technical and systematic allocative inefficiency
Given that the electric utilities are subject to rate of return regulation which leads to a system-
atic over-utilization of capital relative to any other input (the Averch-Johnson hypothesis), we now
allow non-zero means for allocative inefficiency (ξj). The basic model is the same, except that
ξ ∼ MV N(ρ,Σ). Thus, the only change in the likelihood function will be in h(ξ). Now the likeli-
hood function can be concentrated with respect to both ρ and Σ, i.e.,
ρj = ξj =1N
∑i
ξji, j = 2, · · · , J, (42)
and
σjk =1N
∑i
(ξji − ξj)(ξki − ξk), j, k = 2, · · · , J, (43)
14
where the subscript i = 1, · · · , N indicates observation. Substituting these expressions for ρ and Σ
into the log-likelihood function gives the concentrated log-likelihood function that is maximized to
obtain the remaining parameters of the model.
Results from this model (reported in Tables 3 a nd 4) are very similar to those of the previous
model. Mean allocative inefficiency is found to be not different from zero at the 5% level of signifi-
cance. This result is not surprising given that the means of ξF and ξK of the previous model were
close to zero. Returns to scale in the Cobb-Douglas model are found to be 1.005 (not different from
unity at any reasonable level of significance). Technical change is found to have taken place at the
rate of 2.3% per year. Mean technical inefficiency is found to be 31.3%. On average, an increase in
cost due to technical inefficiency is 39.6%, while allocative inefficiency increased cost (on average)
by 3.5%. The results for the translog model (at the mean of the data) come very close to the Cobb-
Douglas model (RTS = 0.998, TC = 2.3%, mean u = 31.6%). Again these are very similar to those
from the previous model.
6 Conclusion
In this paper we demonstrate how one can use a flexible functional form to estimate technical and
allocative inefficiency in a cost minimization framework. Instead of using the cost function and the
associated cost share functions (which turn out to be difficult to estimate), we follow the primal ap-
proach used by Schmidt and Lovell (1979) for the Cobb-Douglas production system. The production
system consists of the production function and the first-order conditions of cost minimization. We
consider both systematic and non-systematic allocative inefficiency.
In formulating the model we have taken into account production uncertainty as well as technical
and allocative inefficiency in a consistent and coherent manner. None of the error terms in the model
is ad hoc. This is in contrast to the cost system approach in which (at least for flexible functional
form) the noise term (production uncertainty) is added at the very end (for both the cost function
and cost share equations). This approach is quite ad hoc (lacks economic meaning) and is criticized
by McElroy (1987) and others. The main advantage of the production approach is that every error
term has a clear meaning and nothing is added at the end either to simplify the derivation of some
results or to make the estimation simpler.
Using the production system, we derive observation-specific estimates of technical and allocative
inefficiency (for each pair of inputs). These estimates (together with the estimated parameters of
the production system) are then used to obtain estimates of a cost increase (for each observation)
due to technical and allocative inefficiency. We used a panel of U.S. stem-electric generating plants
15
to estimate both the Cobb-Douglas and translog production systems. Our results (based on both
functional forms) show that, potential output on average is reduced by about 30% due to technical
inefficiency alone. This pushes actual cost up on average by about 39%. On the other hand, allocative
inefficiency results in an over-use of capital (relative to both labor and fuel), which increases actual
cost on average by 3.5% (for the Cobb-Douglas model) to 5.0% (for the translog model).
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16
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17
Table 1: Estimated Production Function Parameters
Cobb-Douglas Translog
var. coef. (std.err.) var. coef. (std.err.) var. coef. (std.err.)
cons 4.899*** (0.112) cons 1.786** (0.863)
l 0.167*** (0.003) l 0.272*** (0.049) lf 0.022** (0.011)
f 0.569*** (0.007) f 0.441*** (0.105) lk 0.004 (0.010)
k 0.269*** (0.004) k 0.794*** (0.089) lt 0.001 (0.001)
t 0.021*** (0.003) t 0.049* (0.029) fk 0.098*** (0.008)
ll -0.055*** (0.009) ft -0.002 (0.002)
ff -0.119*** (0.015) kt -0.0001 (0.001)
σ2u 0.154*** (0.012) kk -0.126*** (0.013) σ2
u 0.147*** (0.011)
σ2v 0.012*** (0.002) tt -0.002 (0.002) σ2
v 0.012*** (0.002)
log likelihood -933.985 log likelihood -840.597
Note: l = ln L, lf = ln L ln F , etc.; Significance: ***: 1% level, **: 5% level; *: 10% level.
Table 2: Summary statistics of inefficiency, RTS and TC
Cobb-Douglas Translog
mean (std.dev.) mean (std.dev.)
RTS 1.005 RTS 0.988
TC 0.021 TC 0.020
E(u) 0.307 (0.225) E(u) 0.301 (0.219)
ξF -0.010 (0.504) ξF -0.006 (0.661)
ξK -0.013 (0.509) ξK -0.016 (0.687)
Ctech Calloc Ctech Calloc
mean 0.396 0.035 mean 0.389 0.050
25% 0.146 0.006 25% 0.155 0.005
50% 0.263 0.016 50% 0.266 0.022
75% 0.486 0.035 75% 0.488 0.050
Note: Ctech = (w′x)/(w′xo) − 1, Callo = (w′x)/(w′xo) − 1.
18
Table 3: Estimated Production Function Parameters with Systematic Errors in Allocation
CD TL
var. coef. (std. err.) var. coef. (std. err.) var. coef. (std. err.)
cons 5.319*** (0.121) cons 5.249*** (0.863)
l 0.252*** (0.016) l 0.164** (0.075) lf 0.027** (0.013)
f 0.569*** (0.025) f 0.543*** (0.120) lk 0.027** (0.011)
k 0.188*** (0.024) k 0.319** (0.130) lt 0.004*** (0.002)
t 0.023*** (0.002) t -0.004 (0.027) fk 0.013** (0.006)
ll -0.077*** (0.013) ft -0.001 (0.002)
ff -0.020 (0.013) kt 0.001* (0.0001)
σ2u 0.160*** (0.012) kk -0.047** (0.020) σ2
u 0.162*** (0.012)
σ2v 0.009*** (0.002) tt -0.001 (0.002) σ2
v 0.010*** (0.002)
log likelihood -916.344 log likelihood -819.023
Note: Significance: ***: 1% level, **: 5% level; *: 10% level.
Table 4: Model Statistics: with Systematic Errors in Al-
location
CD TL
mean (std. dev.) mean (std. dev.)
RTS 1.009 RTS 0.998
TC 0.023 TC 0.023
E(u) 0.313 (0.235) E(u) 0.316 (0.233)
ξF -0.419 (0.504) ξF -0.257 (0.615)
ξK -0.777 (0.508) ξK -2.210 (0.939)
Ctech Calloc Ctech Calloc
mean 0.406 0.070 mean 0.410 0.184
25% 0.152 0.029 25% 0.150 0.087
50% 0.273 0.050 50% 0.283 0.151
75% 0.502 0.090 75% 0.530 0.233
Note: Ctech = (w′x)/(w′xo) − 1, Callo. = (w′x)/(w′xo) − 1.
19
