pgy + wgx . (5). The left hand side is the industry measure of profit efficiency which is equal to the right hand side which is the sum over firm profit efficiencies.
An Optimistic Note on Aggregation of Efficiency Indices Rolf F¨are Department of Economics and Department of Agricultural and Resource Economics Oregon State University Corvallis, OR and Shawna Grosskopf Department of Economics Oregon Sate University Corvallis, OR and Valentin Zelenyuk Department of Economics Oregon Sate University Corvallis, OR January 2001 JEL:
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Introduction
In a recent article in this journal, Blackorby and Russell (1999) reach a pessimistic conclusion concerning our ability to aggregate efficiency indices. From their summary: ‘Unfortunately, our results are discouraging, indicating that very strong restrictions on the technology and/or the efficiency index itself are required to enable consistent aggregation (or disaggregation).’ (p. 5) In this note we try to convince the reader that with respect to aggregation of efficiency indices, the glass is ‘half full’ rather than nearly empty. We begin with a theorem by T. C. Koopmans (1957) which states that the industry profit is the sum of firm profits. From this theorem we can derive industry and firm profit efficiency measures (which we refer to as Nerlovian profit efficiency). We show that the industry profit efficiency is the sum of the firm profit efficiencies. Furthermore, if all firms and the industry are allocatively efficient, we conclude that the sum of the firm indices of technical efficiency is equal to the industry technical efficiency. We note that our success stems in part from the fact that we adopt what we call directional distance functions as our measures of technical efficiency; in contrast, Blackorby and Russell use Shephard type distance functions as measures of technical efficiency. The directional distance functions are closely related to what Luenberger (1992) calls benefit functions in the consumer context. The advantage of these for aggregation was noted by Luenberger (1992) ‘The single normalization of the benefit function theory can be applied to all consumers, while the distance function approach requires that a given price vector be normalized differently for each consumer.’ (p. 480) In our context, the advantage of the directional distance function is that we can choose one direction (and therefore one associated normalization) for the evaluation of each firm’s efficiency; the Shephard type distance functions allow each firm to be evaluated in a different direction (namely that consistent with its input or output mix).
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The Details
Like Balckorby and Russell we define the industry technolgy T as the sume of the firm technologies, i.e., K 3
T =
T k,
(1)
k=1
where T k = {(xk , y k ) : input xk ∈
N +
can produce output y k }.
Koopmans (1957) proves that industry profit Π(p, w) = max{py − wx : (x, y) ∈ T }. x,y
(2)
is the sum over firm profits, i.e., K 3
Π(p, w) =
Πk (p, w),
(3)
k=1
where, given input and output prices (w, p), firm k : s profit is defined by Πk (p, w) = max{py k − wxk : (xk , y k ) ∈ T k }.
By subtracting observed profit (p
K
k=1
yk − w
K
k=1
(4)
xk ) from both sides of (3) and
normalizing with (pgy + wgx ) where (gx , gy ) is the direction in which technical efficiency is to be measured yields
k Π(p, w) − (p K k=1 y − w pgy + wgx
K
k=1
xk )
=
K 3
Πk (p, w) − (py k − wxk ) . pgy + wgx k=1
(5)
The left hand side is the industry measure of profit efficiency which is equal to the right hand side which is the sum over firm profit efficiencies. These efficiency measures, 2
introduced by Chambers, Chung and F¨are (1998) are called Nerlovian measures of profit efficiency. The Nerlovian profit efficiency index may be expressed as the sum of an allocative and a technical index. The technical index is defined for firm k as D k (xk , y k ; gx , gy ) = max{β : (xk − βgx , y k + βgy ) ∈ T k },
(6)
where (gx , gy ) is the directional vector. This index takes values greater than or equal to zero for feasible (x, y). As usual, the allocative efficiency index (AEk ) is defined as a residual, thus for firm k we have Πk (p, w) − (py k − wxk ) = AE k + D k (xk , y k ; gx , gy ). pgy + wgx
(7)
The industry decomposition is similar, with its directional distance function defined on the industry technology T .
k Π(p, w) − (p K k=1 y − w pgy + wgx
K
k=1
xk )
= AE + D(
K 3
k=1
xk ,
K 3
y k ; gx , gy ).
(8)
k=1
Now (5), (7) and (8) together with the assumption of allocative efficiency AE = k
AE = 0, k = 1, . . . , K yield D(
K 3
k=1
xk ,
K 3
y k ; gx , gy ) =
k=1
K 3
D k (xk , y k ; gx , gy ).
(9)
k=1
Thus (5) and (9) show that we can derive aggregate industry efficiency from firm efficiencies, a result which we hope that Blackorby and Russell find encouraging.
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Of course, our fairly optimistic result rests on the assumption of allocative efficiency, which may be inappropriate in some applications. If we relax that assumption, we can still derive the following relationship: D(
K 3
k=1
xk ,
K 3
k=1
y k ; gx , gy ) ≥
K 3
D k (xk , y k ; gx , gy ).
(10)
k=1
This result follows from the fact that K 3
k=1
(xk − D k gx , y k + D k gy ) = (
K 3
k=1
xk −
K 3
k=1
D k gx ,
K 3
k=1
yk +
K 3
k=1
D k gy ) ∈ T
(11)
and the definition of the industry distance function. Thus, even if one is not willing to assume allocative efficiency, we have shown that the sum of the firm technical efficiency measures based on directional distance functions will never be greater than the corresponding industry technical efficiency measure.
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References [1] Blackorby, C. and Russell, R. R. ‘Aggregation of Efficiency Indices.’ Journal of Productivity Analysis 12, 5-20 (1999). [2] Chambers, R.G., Chung, Y. and F¨are, R. ‘Profit, Directional Distance Functions and Nerlovian Efficiency’. Journal of Optimization Theory and Application, 98, 351-364 (1998). [3] Koopmans, T. C. Three Essays on the State of Economic Analysis. New York: McGraw-Hill (1957). [4] Luenberger, D.G. ‘Benefit Functions and Duality’.Journal of Mathematical Economics 21, 461-481 (1992).
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