Dual allocative efficiency parameters

Rolf Fare • Daniel Primont

Published online: 3 September 2011

� Springer Science+Business Media, LLC 2011

Abstract Estimation of either price or quantity allocative

efficiency parameters provides an empirical method for

taking into account the presence of allocative inefficiency.

We show that the researcher can either (a) estimate a

system of input demand functions of the form, x ¼hðy; k�wÞ; obtain estimated values of the price allocative

efficiency parameters (the k0s), and then derive the quantity

allocative efficiency parameters with a simple calculation

or (b) estimate a system of shadow price functions of the

form, w ¼ gðy; d�xÞ; obtain estimated values of the quan-

tity allocative efficiency parameters (the d0s), and then

derive the price allocative efficiency parameters with a

simple calculation. These input-specific efficiency mea-

sures are then related to the standard measures of efficiency

that have appeared in the literature.

Keywords Allocative efficiency � Production theory

JEL Classification D2

1 Introduction

Estimation of either price or quantity allocative efficiency

parameters provides an empirical method for taking into

account the presence of allocative inefficiency. In this

paper, we explore the relationship between these two sets

of parameters in the framework of a competitive, cost-

minimizing firm.

Suppose we are given an observation consisting of

an output vector, y ¼ ðy1; . . .; yMÞ; and input vector, x0 ¼ðx0

1; . . .; x0NÞ; and an input price vector, w0 ¼ ðw0

1; . . .;w0NÞ:

The choice of x0 may not be allocatively efficient (and

hence, not cost-minimizing) given (y,w0). To account for

this allocative inefficiency one may introduce price allo-

cative efficiency parameters, k1; . . .; kN ; with the property

that x0 is allocatively efficient given ðy; k �w0Þ; where

k�w0 ¼ ðk1w01; . . .; kNw0

NÞ: In other words, k�w0 is a sha-

dow price vector for x0. This sort of empirical strategy was

used by Lau and Yotopoulos (1971), Atkinson and Hal-

vorsen (1980), and Lovell and Sickles (1983) in the context

of profit maximization and by Toda (1976), Atkinson and

Halvorsen (1984), and Atkinson and Cornwell (1994) in

the context of cost minimization.

Alternatively, one can introduce quantity allocative

efficiency parameters, d1; . . .; dN ; with the property that

d � x0 is allocatively efficient given (y, w0), where d � x0 ¼ðd1x0

1; . . .; dNx0NÞ: In other words, w0 is a shadow price

vector for d � x0: This approach was employed by Atkinson

and Primont (2002) and by Atkinson et al. (2003). These

two papers employ an input distance function.

In this paper, we answer the following research ques-

tion. What is the relationship between the price efficiency

parameters and the quantity efficiency parameters? It will

be shown that the researcher can either (a) estimate a

R. Fare

Department of Economics, Oregon State University,

Corvallis, OR 97331, USA

e-mail: rolf.fare@orst.edu

R. Fare

Department of Agricultural and Resource Economics,

Oregon State University, Corvallis, OR 97331, USA

D. Primont (&)

Department of Economics, Southern Illinois University

Carbondale, Carbondale, IL 62901, USA

e-mail: primo@siu.edu

123

J Prod Anal (2012) 37:233–238

DOI 10.1007/s11123-011-0240-4

system of input demand functions of the form, x ¼hðy; k �wÞ; obtain estimated values of the price efficiency

parameters and then derive the quantity efficiency param-

eters with a simple calculation or (b) estimate a system of

shadow price functions of the form, w ¼ gðy; d � xÞ; obtain

estimated values of the quantity efficiency parameters and

then derive the price efficiency parameters with a simple

calculation. These input-specific efficiency measures are

then related to the standard measures of efficiency that

have appeared in the literature.

For a simple example of an application of this paper

consider the Averch–Johnson model (Averch and John-

son 1962). Firms that are subject to rate-of-return regula-

tion are induced to over-invest in capital relative to other

inputs. To measure this effect one could estimate an input

demand model of the form x ¼ hðy; k �wÞ. If the value of

the k coefficient for capital is less than one then the firm

acts as if the price of capital is less than its actual price

and thus overuses capital—a result consistent with the

Averch–Johnson prediction. But this result does not tell us

how much overuse has occurred. Our simple calculation of

the d’s from the k’s will yield a d value for capital that

is less than one. Overusage is then easily calculated as

xn - dnxn where input n is capital usage.

2 Two basic lemmas

This section presents and proves two lemmas that will lead

to the required simple calculations. The technology is given

by input requirement sets that are defined by LðyÞ ¼x 2 RN

þ : x can produce y� �

for each y 2 RMþ : The input dis-

tance function is defined by Diðy; xÞ ¼ supk k : ðx=kÞ 2fLðyÞg. Fix the output vector at y and let the input price vector be

some arbitrary vector w. The cost function is derived from the

input distance function by the following cost minimization

problem (hereafter referred to as the CMP)

Cðy;wÞ ¼ minx

wx : Diðy; xÞ� 1f g ¼ w � hðy;wÞ ð1Þ

where the solution, h(y, w), is the N 9 1 vector of optimal

input demands with components:

hnðy;wÞ; n ¼ 1; . . .;N: ð2Þ

As is well known, hðy; �Þ is homogeneous of degree zero in

the input price vector. Moreover, it can be shown that

Cðy;wÞDiðy; hðy;wÞÞ ¼ w � hðy;wÞ: ð3Þ

The input distance function can be recovered from the

cost function by the following shadow pricing problem

(hereafter referred to as the SPP)

Diðy; xÞ ¼ minw

wx : Cðy;wÞ� 1f g ¼ gðy; xÞ � x: ð4Þ

where the solution, g(y, x), is the N 9 1 vector of shadow

prices with components, gnðy; xÞ; n ¼ 1; . . .;N:

Lemma 1 If x� solves the CMP (1) at prices, w0, i.e., if

x� ¼ h y;w0ð Þ then the normalized price vector w0/C(y,w0)

solves the SPP (4) at quantities, x�; i.e.,

w0n

Cðy;w0Þ ¼ gnðy; x�Þ; n ¼ 1; . . .;N: ð5Þ

Proof The normalized price vector w0/C(y,w0) is clearly

feasible since C y;w0=Cðy;w0Þð Þ ¼ 1: Next note that the

constraint in the CMP holds with equality, i.e.,

Di y; x�ð Þ ¼ 1. (For any x such that Di y; xð Þ[ 1 it is

feasible to reduce cost by scaling x down. Thus at the

optimum x� we must have Di y; x�ð Þ ¼ 1:) Therefore

w0x� ¼ Cðy;w0ÞDiðy; x�Þ

or

w0x�

Cðy;w0Þ ¼ Diðy; x�Þ;

and therefore w0/C(y,w0) also yields the optimal solution to

the SPP (4) at quantities, x�. h

Again fix the output vector at y and let the input vector

be some arbitrary vector x0. The input distance function is

derived from the cost function by the shadow pricing

problem (SPP)

Diðy; x0Þ ¼ minw

wx0 : Cðy;wÞ� 1� �

¼ w� � x0

¼ gðy; x0Þ � x0; ð6Þ

where the solution, w� ¼ gðy; x0Þ; is the N 9 1 vector of

shadow prices with components

w�n ¼ gnðy; x0Þ; n ¼ 1; . . .;N: ð7Þ

The shadow price function, gðy; �Þ; is homogeneous of

degree zero in the input quantity vector.

Note that the constraint in (6) will be binding and, thus,

Cðy;w�Þ ¼ 1. The cost function can be recovered from the

input distance function by the cost-minimization problem

(CMP)

Cðy;w�Þ ¼ minx

w�x : Diðy; xÞ� 1f g ¼ w� � hðy;w�Þ ð8Þ

where the solution, hðy;w�Þ; is the N 9 1 vector of optimal

input demands with components, hnðy;w�Þ; n ¼ 1; . . .;N:

Moreover,

Cðy;w�ÞDiðy; x0Þ ¼ w�x0 ð9Þ

since Cðy;w�Þ ¼ 1:

Lemma 2 If w� solves the SSP (6) at quantities, x0, i.e., if

w� ¼ g y; x0ð Þ then the normalized input vector, x0/

Di(y, x0), solves the CMP (8) at prices, w�; i.e.,

234 J Prod Anal (2012) 37:233–238

123

x0n

Diðy; x0Þ ¼ hnðy;w�Þ; n ¼ 1; . . .;N: ð10Þ

Proof The normalized input vector is feasible since

Di(y, x0/Di(y, x0)) = 1. Moreover, (9) implies that

w�x0

Diðy; x0Þ ¼ Cðy;w�Þ;

and therefore x0/Di(y,x0) also yields the optimal solution to

(8). h

3 Efficiency parameters

Suppose a data point is given by (w0, x0, y). Since the

observed, normalized input vector, x0/Di(y, x0), may differ

from the optimal input vector, x�; we introduce N quantity

efficiency parameters defined by

dn ¼x�n

x0n=Diðy; x0Þ ; n ¼ 1; . . .;N: ð11Þ

Then

x�n ¼dnx0

n

Diðy; x0Þ n ¼ 1; . . .;N: ð12Þ

or

x� ¼ d � x0

Diðy; x0Þ ¼d1x0

1

Diðy; x0Þ ; . . .;dNx0

N

Diðy; x0Þ

� �; ð13Þ

where d � x0 is the Hadamard product.

Alternatively, we may account for the difference

between the normalized input price vector, w0/

C(y, w0), and the optimal shadow price vector, w�; by

introducing N price efficiency parameters defined by

kn ¼w�n

w0n=Cðy;w0Þ ; n ¼ 1; . . .;N: ð14Þ

Then we have

w�n ¼knw0

n

Cðy;w0Þ ; n ¼ 1; . . .;N; ð15Þ

or

w� ¼ k �w0

Cðy;w0Þ ¼k1w0

1

Cðy;w0Þ ; . . .;kNw0

N

Cðy;w0Þ

� �; ð16Þ

where k �w0 is the Hadamard product.

Lemmas 1 and 2 and the definitions of the price and

quantity allocative efficiency parameters are illustrated in

the following four-quadrant diagram (Fig. 1).

An isoquant for the output vector y is given in the

quantity space and is defined by the equation, Di(y,x) = 1.

It is the set of input vectors that can just produce y. The

corresponding isoquant for output vector y in price space is

defined by the equation, C(y,w) = 1. It is the set of input

price vectors at which the firm can hire the least-cost

combination of inputs that can produce y for a cost of one

dollar.

Begin with x0 in quadrant II (quantity space). Deflation

by Di(y,x0) results in the point x0/Di(y,x0) whose shadow

price vector is w�: Hadamard multiplication by d results in

d � x0=Diðy; x0Þ whose shadow price vector is w0. In

quadrant III (I) draw a rectangular hyperbola given by the

equation w1x1 = C(y,w0) (w2x2 = C(y,w0)).

The endpoints of the isocost line through the point x� ¼d � x0=Diðy; x0Þ are given by C(y,w0)/w1

0 and C(y,w0)/w20. If

these two points are projected onto the corresponding rect-

angular hyperbola and then projected into price space these

two ratios are ‘‘flipped over’’ and we end up at the point w0/

C(y,w0). Thus, the input vector, x� solves the CMP at w0 and

w0/C(y,w0) solves the SSP at x�: This is just Lemma 1.

Similarly, the endpoints of the isocost line through the point

x0/Di(y,x0) are given by Cðy;w�Þ=w�1 and Cðy;w�Þ=w�2: If

these two points are projected onto the corresponding rect-

angular hyperbola and then projected into price space these

two ratios are ‘‘flipped over’’ and we end up at the point

w�=Cðy;w�Þ ¼ w� since Cðy;w�Þ ¼ 1: Thus, the input vec-

tor, x0/Di(y, x0), solves the CMP at w� and w� solves the SPP

at quantities x0/Di(y,x0). This is a result of Lemma 2 and the

zero-degree homogeneity of the shadow price function in the

input quantities.

In an analogous way, begin with w0 in quadrant IV (price

space) depicted in Fig. 2. Deflation by C(y,w0) results in the

point w0/C(y,w0) whose supporting hyperplane is defined by

x�: Hadamard multiplication by k results in k�w0=Cðy;w0Þwhose supporting hyperplane is given by x0. In quadrant III

(I) draw a rectangular hyperbola given by the equation

w1x1 = Di(y,x0) (w2x2 = Di(y,x0)).

The endpoints of the isocost line through the point w� ¼k�w0=Cðy;w0Þ are given by Di(y,x0)/x1

0 and Di(y,x0)/x20. If

these two points are projected onto the corresponding

rectangular hyperbola and then projected into quantity

space these two ratios are flipped over and we end up at the

point x0/Di(y,x0). Thus, the input price vector w� solves the

SSP at quantities x0 and the input vector x0/Di(y,x0) solves

the CMP at w�: This is just Lemma 2.

Similarly, the endpoints of the isocost line through the

point w0/C(y,w0) are given by Diðy; x�Þ=x�1 and Diðy; x�Þ=x�2:

If these two points are projected onto the corresponding

rectangular hyperbola and then projected into quantity space

they are flipped over and we end up at the point x�: Thus, the

input price vector w0/C(y,w0) solves the SSP at quantities x�

and x� solves the CMP at prices w0/C(y,w0). This is a result of

Lemma 1 and zero-degree homogeneity of the input demand

functions in the input prices.

J Prod Anal (2012) 37:233–238 235

123

In both Figs. 1 and 2, Hadamard multiplication by dinduces a movement along the isoquant in quantity space

and corresponds to a dual movement along the isoquant in

price space induced by Hadamard multiplication by k. This

graphical relationship is further characterized in our first

main result.

Theorem 1 The price and quantity allocative efficiency

parameters defined in (14) and (11), respectively, satisfy

the following set of equations.

kn ¼gnðy; x0Þ

gn y; d�x0ð Þ n ¼ 1; . . .;N: ð17Þ

and

dn ¼hnðy;w0Þ

hn y; k�w0ð Þ n ¼ 1; . . .;N: ð18Þ

Proof It is useful to summarize the two lemmas as

follows. They say that if

x�n ¼ hn y;w0� �

; n ¼ 1; . . .;N ð19Þ

and

w�n ¼ gn y; x0� �

n ¼ 1; . . .;N; ð20Þ

x x 0

D i y , x 0

x 0

D i y , x 0

w 1

w 2

x 2

w 0

w

x 0

w 0

w 0

C y , w 0

w k w 0

C y , w 0

D i y , x 1

C y , w 1w 1x1 C y, w 0

x1

w 2x2 C y, w 0

Quantity Space

Price Space

III

IIIIV

Fig. 1 Quantity space

x x 0

D i y, x 0

x 0

D i y, x 0

w 1

w 2

x 2

x 0

w 0

w 0

C y, w 0

w k w 0

C y, w 0

D i y, x 1

C y, w 1

x1

Quantity Space

Price Space

I II

IIIIV

x 0

x

w 1 x 1 D i y, x 0

w 2 x 2 D i y, x 0

Fig. 2 Price space

236 J Prod Anal (2012) 37:233–238

123

then

w0n

C y;w0ð Þ ¼ gn y; x�ð Þ n ¼ 1; . . .;N ð21Þ

and

x0n

Di y; x0ð Þ ¼ hn y;w�ð Þ n ¼ 1; . . .;N: ð22Þ

Now put (12) into (19) to get

dnx0n

Diðy; x0Þ ¼ hnðy;w0Þ; n ¼ 1; . . .;N; ð23Þ

and put (16) into (22) to get

x0n

Diðy; x0Þ ¼ hn y;k�w0

Cðy;w0Þ

� �; n ¼ 1; . . .;N;

or, since each hn is homogeneous of degree zero in prices,

x0n

Diðy; x0Þ ¼ hn y; k�w0� �

; n ¼ 1; . . .;N: ð24Þ

Divide (23) by (24) to get (18).

Next, put (15) into (20) to get

knw0n

Cðy;w0Þ ¼ gnðy; x0Þ; n ¼ 1; . . .;N; ð25Þ

and put (13) into (21) to get

w0n

Cðy;w0Þ ¼ gn y;d�x0

Diðy; x0Þ

� �; n ¼ 1; . . .;N; ð26Þ

or, since each gn is homogeneous of degree zero in inputs,

w0n

Cðy;w0Þ ¼ gn y; d�x0� �

; n ¼ 1; . . .;N: ð27Þ

Divide (25) by (27) to get (17). h

In general, the kn’s will be functions of the output

vector, the observed input vector, and the vector of the

dn’s; the dn’s will be functions of the output vector, the

observed input price vector, and the vector of the kn’s.

4 The standard measures of efficiency

The dn’s are input-quantity-specific measures of allocative

efficiency and the kn’s are input-price-specific measures of

allocative efficiency. In this section we relate these to the

standard measures of total and allocative efficiency as

introduced by Farrell (1957) and others.1

The usual (primal) definitions of input-oriented effi-

ciency are:

Technical Efficiency: TE ¼ 1

Diðy; x0Þ ð28Þ

Allocative Efficiency: AE ¼ C y;w0ð ÞDiðy; x0Þw0x0

ð29Þ

Overall Efficiency: OE ¼ C y;w0ð Þw0x0

ð30Þ

(See Farrell (1957)). As is well-known,

OE ¼ AE � TE: ð31Þ

The dn’s were defined by

dn ¼x�n

x0n=Diðy; x0Þ ; n ¼ 1; . . .;N; ð32Þ

where x� solves the cost minimization problem at the

observed prices w0, i.e.,

w0x� ¼ C y;w0� �

¼ minx

w0x : Diðy; xÞ� 1� �

: ð33Þ

Theorem 2 Allocative efficiency is

AE ¼P

dnw0nx0

nPw0

nx0n

: ð34Þ

Proof

Pdnw0

nx0nP

w0nx0

n

¼P x�n

x0n=Diðy;x0Þ

� w0

nx0n

Pw0

nx0n

using (32),

¼P

w0nx�nDiðy; x0ÞP

w0nx0

n

¼ w0x�Diðy; x0Þw0x0

¼ C y;w0ð ÞDiðy; x0Þw0x0

using (33),

¼ AE using ð29Þ:

h

Corollary

OE ¼P

dnw0nx0

nPw0

nx0n

� 1

Diðy; x0Þ : ð35Þ

Proof This follows directly from (28), (31) and (34).

h

Dual counterparts to TE, AE, and OE have been given

by Fare (1984). These dual definitions of input-oriented

efficiency are

Dual Technical Efficiency: DTE ¼ 1

Cðy;w0Þ ð36Þ

Dual Allocative Efficiency: DAE ¼ Diðy; x0ÞCðy;w0Þw0x0

ð37Þ1 We wish to thank an anonymous referee who suggested that we

explore this relationship.

J Prod Anal (2012) 37:233–238 237

123

Dual Overall Productive Efficiency: DOPE ¼ Diðy; x0Þw0x0

ð38Þ

It is obvious from these definitions that

DOPE ¼ DAE � DTE: ð39Þ

The kn’s were defined by

kn ¼w�n

w0n=Cðy;w0Þ ; n ¼ 1; . . .;N; ð40Þ

where w� solves the shadow-pricing problem at observed

quantities

w�x0 ¼ Diðy; x0Þ ¼ minw

wx0 : Cðy;wÞ� 1� �

: ð41Þ

Theorem 3 Dual allocative efficiency is

DAE ¼P

knw0nx0

nPw0

nx0n

: ð42Þ

Proof

Pknw0

nx0nP

w0nx0

n

¼P w�n

w0n=Cðy;w0Þw

0nx0

nPw0

nx0n

using (40),

¼P

w�nx0nCðy;w0Þ

Pw0

nx0n

¼ w�x0Cðy;w0Þw0x0

¼ Diðy; x0ÞCðy;w0Þw0x0

using (41);

¼ DAE using (37).

h

Corollary

DOPE ¼P

knw0nx0

nPw0

nx0n

1

Cðy;w0Þ ð43Þ

Proof This follows directly from (36), (39) and (42).

h

From Eqs. 29 and 37 we see that AE = DAE. Then

Theoerems 2 and 3 imply:

Theorem 4P

dnw0nx0

nPw0

nx0n

¼P

knw0nx0

nPw0

nx0n

:

5 Closing remarks

The answer to the question posed in the introduction is

contained in Eqs. 17 and 18. If one has estimated the input

demand system along with the price allocative efficiency

parameters then the nth quantity allocative efficiency

parameter is easily calculated as the nth input demand

evaluated at the observed input price vector divided by the

nth input demand evaluated at the shadow input price vector.

If one has, instead, estimated the inverse input demand

system along with the quantity allocative efficiency param-

eters then the nth price efficiency parameter is easily calcu-

lated as the nth inverse input demand evaluated at the

observed input vector divided by the nth inverse input

demand evaluated at the optimal input vector. Of course, one

can also estimate both systems. It might be interesting to

compare allocative efficiency parameters that are directly

estimated to those that are found by the above calculations.

But perhaps this should be left for further research.

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