PRODUCTION EFFICIENCY - CONCEPTS
1. Technical Efficiency
Consider a technology represented by the feasible set F, where z ≡ (-x, y)  F means that inputs x
n
m
 R  can be used to produce outputs y  R m . Given some reference bundle g  R n
satisfying g ≠ 0,

we have defined the following functions:
 the shortage function: S(-x, y, g) = min {: ((-x, y) -  g)  F},
 the directional distance function: D(-x, y, g) = max {: ((-x, y) +  g)  F},
 Shephard input distance function: DI(-x, y) = max {: (-x/, y)  F},
 Farrell input distance function: DF(-x, y) = min {: (-x, y)  F},
 Shephard output distance function: DO(-x, y) = min {: (-x, y/)  F}.
Each function provides a measure of the distance between point z ≡ (-x, y) and the upper boundary
of the feasible set F. We have shown that these functions are related as follows: S(-x, y, g) = -D(-x, y, g),
D(-x, y, (x, 0)) = 1 - 1/DI(-x, y), DF(-x, y) = 1/DI(-x, y), and D(-x, y, (0, y)) = [1/DO(-x, y)] - 1. Also, we
know that technical feasibility (where (-x, y)  F) implies that S(-x, y, g)  0, D(-x, y, g)  0, DI(-x, y)  1,
DF(-x, y)  1, and DO(-x, y)  1. It means that, in general, finding S(-x, y, g) = 0, D(-x, y, g) = 0, DI(-x, y) =
1, DF(-x, y) = 1, or DO(-x, y) = 1 implies that point z ≡ (-x, y) is on the upper bound of the feasible set. In
addition, under free disposal, the production technology can be written as F = {(-x, y): S(-x, y, g)  0} =
{(-x, y): D(-x, y, g)  0} = {(-x, y): DI(-x, y)  1} = {(-x, y): DF(-x, y)  1} = {(-x, y): DO(-x, y)  1}. Then,
the upper bound of the feasible set can be represented by any of the implicit production functions: S(-x, y,
g) = 0, D(-x, y, g) = 0, DI(-x, y) = 1, DF(-x, y) = 1, or DO(-x, y) = 1.
Definition: Consider a firm in an industry facing a technology represented by the feasible set F. Let z ≡ (-x,
y)  F denote an observation on the inputs x and outputs y of the firm.
 The firm is said to be technically efficient if there is no z' ≡ (-x', y')  F, z' ≠ z, with z'  z.
 The firm is said to be technically inefficient if it is not technically efficient.
Result 1: A necessary (but not sufficient) condition for the technical efficiency of netputs z ≡ (-x, y) is that
z is located on the upper bound of the feasible set F.
Proof: Assume that point z that is below the upper bound of F. Then, there is a feasible move toward the
upper bound that increases outputs y and/or decreases inputs x, making point z technically
inefficient. However, note that there may be points on the upper bound of F that are not technically
efficient (e.g., points where free disposal does not hold).
This has the following implications:
1/ If point z ≡ (-x, y) is technically efficient, then S(-x, y, g) = 0, D(-x, y, g) = 0, DI(-x, y) = 1, DF(-x, y)
= 1, and DO(-x, y) = 1.
2/ If S(-x, y, g) < 0, D(-x, y, g) > 0, DI(-x, y) > 1, DF(-x, y) < 1, or DO(-x, y) < 1, then the point z ≡ (-x,
y)  F is necessarily technically inefficient.
Example: Consider the case where g = (0, …, 0, 1), D(-x, y, g) = [F(x, y1, …, ym-1) - ym], and F(x, y1, …,
ym-1) = maxk{k: (-x, y1, …, ym-1, k)  F} is the standard production function (or production
frontier) for the m-th output. Then, from 1/, technical efficiency of point z ≡ (-x, y) implies that ym
= F(x, y1, …, ym-1). Alternatively, from 2/, finding that D(-x, y, g) > 0 (or equivalently ym < F(x, y1,
…, ym-1)) implies that point z ≡ (-x, y) is technically inefficient. Indeed, point z is then located
below the production function F(x, y1, …, ym-1), meaning that it is possible to increase the
1
production of the m-th output by [F(x, y1, …, ym-1) - ym] > 0 holding other netputs (x, y1, …, ym-1)
constant.
1.1. Measuring technical efficiency
Each of the functions S(-x, y, g), D(-x, y, g), DI(-x, y), DF(-x, y) or DO(-x, y) can provide a
measure of the technical efficiency (or inefficiency) of a firm producing netputs (-x, y)  F:
 -S(-x, y, g) = D(-x, y, g) measures the largest number of units of the reference bundle g that can be
produced getting from point (-x, y) to the boundary of the feasible set F,
 1 - DF(-x, y) = 1 - [1/DI(-x, y)] measures the largest proportional reduction in inputs x that can be
obtained by moving from point (-x, y) to the boundary of the feasible set F,
 [1 - DO(-x, y)] measures the largest proportional increase in output y that can be obtained by
moving from point (-x, y) to the boundary of the feasible set F.
Considering a firm producing netputs (-x, y)  F, we limit the discussion below to measures of
efficiency based on DF(-x, y) and DO(-x, y).
1.1.1. Input-based index of technical efficiency
Define the input-based Farrell index of technical efficiency as
TEI(x, y) = DF(-x, y)
= min {: (-x, y)  F}.
The technical efficiency index TEI(x, y) has the following properties:
 In general, TEI(x, y)  1 if (-x, y)  F.
 Technical efficiency of netputs (-x, y)  F implies that TEI(x, y) = 1.
 TEI(x, y) < 1 means that netputs (x, y) is technically inefficient as [1 - TEI(x, y)] measures the
largest proportional reduction in inputs x that the firm can obtain while producing outputs y.
1.1.2. Output-based index of technical efficiency
Define the output-based index of technical efficiency as
TEO(x, y) = DO(-x, y)
= min {: (-x, y/)  F}.
= 1/maxk (k: (-x, ky)  F}.
The technical efficiency index TEO(x, y) has the following properties:
 In general, TEO(x, y)  1 if (-x, y)  F.
 Technical efficiency of netputs (-x, y)  F implies that TEO(x, y) = 1.
 TEO(x, y) < 1 means that netputs (x, y) is technically inefficient as [1 - TEO(x, y)] measures the
largest proportional increase in outputs y that the firm can produce while using inputs x.
 
 
Note: When z ≡ (-x, y)  F, we have shown that DF(z)   DO(z) under
 
 
 IRTS 


CRTS  . When (-x, y)  F, it
DRTS 


follows that
2
 
 
TEI(x, y)   TEO(x, y) under
 
 
 IRTS 


CRTS  .
DRTS 


Thus, it is always true that TEI(x, y) = TEO(x, y) under constant return to scale (CRTS), where a
proportional rescaling of inputs is equivalent to a proportional rescaling of output. However,
TEI(x, y) and TEO(x, y) in general differ when the technology that does not exhibit constant return
to scale.
1.2. Alternative interpretations
Consider a competitive firm producing netputs (-x, y)  F and facing input prices r = (r1, …, rn) 
R n  and output prices p = (p1, …, pm)  R m  . The firm's actual cost is: r x =
revenue is: p y =
as


m
j1

n
i 1
ri xi. And its actual
pj yj.
In the case where [-TEI(x, y) x, y] is technically efficient, define the firm's technically efficient cost
n
i 1
ri [TEI(x, y) xi]. It is the cost faced when the firm produces outputs y and inputs have been
rescaled down to the boundary of the feasible set. It follows that
technicall y efficient cost i1 ri [TE I (x, y) xi] TE I (x, y) [i1 ri xi]
= TEI(x, y).
=
=
n
n
actual cost
r
x
r
x
i
i
i
i
i1
i1
n
n
It means that [(1 - TEI(x, y)] can be interpreted as a proportional reduction in cost that the firm can achieve
while producing outputs y.
Similarly, in the case where [-x, y/TEO(x, y)] is technically efficient, define the firm's technically
efficient revenue as

m
j1
pj [yj/TEO(x, y)]. It is the revenue faced when the firm uses inputs x and outputs
have been rescaled up to the boundary of the feasible set. It follows that
TE O (x, y) [ j1 p j y j]
 j1 p j y j
actual revenue
= TEO(x, y).
= m
=
m
technicall y efficient revenue  p j y j /TE O (x, y)
p
y
 j1 j j
j1
m
m
Using netputs (-x, y), it means that [1 - TEO(x, y)] can be interpreted as the proportional increase in
revenue that the firm can achieve by becoming technically efficient.
2. Allocative Efficiency
2.1. Input-allocative efficiency
Consider a competitive firm producing netputs (-x, y)  F and facing input prices r = (r1, …, rn)  R n  .
Definition: The firm is input-allocatively efficient if it chooses its inputs x in a cost minimizing way, i.e. if
x  argminx {r x: (-x, y)  F).
An index of input-allocative efficiency is
AEI(r, x, y) =
allocative ly efficient cost Min x {r x : (x, y)  F}
≤ 1.
=
n
technicall y efficient cost
[
(x,
y)
]
r
TE
x
i
 i I
i1
The firm is input-allocatively efficient if and only if AEI(r, x, y) = 1. Otherwise, it is inputallocatively inefficient: AEI(r, x, y) < 1. Then, [1 - AEI(r, x, y)] measures the proportional reduction in cost
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that the firm can achieve at point [-TEI(x, y) x, y] by becoming input-allocatively efficient (i.e., by
choosing inputs x so as to minimize cost).
Note: TEI(x, y) and AEI(r, x, y) can be combined. Define [TEI(x, y) AEI(r, x, y)] as an index of economic
efficiency given outputs y. When (-x, y)  F, we have:
TEI(x, y) AEI(r, x, y) =
=

n
i1 i
r [TE I (x, y) xi] Min x {rx : (x, y)  F}
n
n
i1 ri xi
i1 ri [TE I (x, y) xi]
Min x {rx : (x, y)  F}  1.
n
i1 ri xi
In the case where (-TEI(x, y) x, y) is technically efficient, it follows that [1 - [TEI(x, y) AEI(r, x, y)]] can be
interpreted as measuring the proportional reduction in cost that the firm can achieve by becoming both
technically and input-allocatively efficient while producing outputs y.
2.2. Output-allocative efficiency
Consider a competitive firm producing netputs (-x, y)  F and facing output prices p = (p1, …, pm)  R m  .
Definition: The firm is output-allocatively efficient if it chooses its outputs y in a revenue maximizing way,
i.e. if y  argmaxy {p y: (-x, y)  F).
An index of output-allocative efficiency is
 j1 p j y j /TE O (x, y) ≤ 1.
technicall y efficient revenue
=
AEO(p, x, y) =
allocative ly efficient revenue Max y {py : ( x, y)  F}
m
The firm is output-allocatively efficient if and only if AEO(r, x, y) = 1. Otherwise, it is outputallocatively inefficient: AEO(r, x, y) < 1. Then, [1 – AEO(x, y)] measures the proportional increase in
revenue that the firm can achieve at point [-x, y/TEO(x, y)] by becoming output-allocatively efficient (i.e.,
by choosing outputs y so as to maximize revenue).
Note: TEO(x, y) and AEO(p, x, y) can be combined. Define [TEO(x, y) AEO(p, x, y)] as an index of
economic efficiency given inputs x. When (-x, y)  F, we have:

m
TEO(x, y) AEO(p, x, y) =
=

m
j1

m
j1
j1
pj yj

m
j1
p j y j /TE O (x, y)
p j y j /TE O (x, y) Max y {py : ( x, y)  F}
pj yj
Max y {p y : ( x, y)  F}
 1.
It follows that [1 - [TEO(x, y) AEO(p, x, y)]] can be interpreted as measuring the proportional increase in
revenue that the firm can achieve by becoming both technically and output-allocatively efficient while
using netputs (-x, y).
3. Scale Efficiency
Definition: A firm producing netputs (-x, y)  F is scale efficient if (-x, y) is in a region of the feasible set
F that exhibits constant returns to scale.
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3.1. Technology-based assessment of scale efficiency
Start with the technology represented by the feasible set F. In general, F can exhibit variable return
to scale (VRTS). Consider the associated technology that exhibits constant return to scale (CRTS): Fc = {-k
x, k y): (-x, y)  F, k ≥ 0}. It satisfies F  Fc.
Define TEIc(x, y) = min {: (-x, y)  Fc}, and TEOc(x, y) = min {: (-x, y/)  Fc} = 1/maxk
(k: (-x, k y)  Fc}. With F  Fc, it follows that TEIc(x, y)  TEI(x, y), and TEOc(x, y)  TEO(x, y). Thus,
TEIc(x, y) = TEI(x, y), or TEOc(x, y) = TEO(x, y) implies that the firm is scale efficient, as netputs (-x, y) are
in the region of the feasible set F exhibiting constant return to scale (CRTS). Alternatively, finding that
TEIc(x, y) < TEI(x, y), or TEOc(x, y) < TEO(x, y) implies that the firm is scale inefficient, as netputs (-x, y)
are in a region of the feasible set F that does not exhibit CRTS. (One way to know whether (-x, y) is in a
region exhibiting increasing return to scale (IRTS) or decreasing return to scale (DRTS) is to evaluate the
scale elasticity…).
3.2. Cost-based assessment of scale efficiency
Start with the cost function C(r, y) = minx {r x: (-x, y)  F} associated with the feasible set F
(which in general can exhibit variable return to scale (VRTS)). Alternatively, consider the cost function
associated with the feasible set Fc = {(-k x, k y): (-x, y)  F, k ≥ 0} which exhibits constant return to scale
(CRTS) and satisfies F  Fc. Assuming that a minimum exists, the associated cost function is
Cc(r, y) = minx {r x: (-x, y)  Fc}.
Since F  Fc, it follows that
Cc(r, y) ≤ C(r, y).
(1)
In addition, note that
Cc(r, y) = minx {r x: (-x, y)  Fc},
= minx, {r x: (-x, y)  F,  ≥ 0}, where  = 1/k,
= minX, {r X/: (-X, y)  F,  ≥ 0}, where X = x,
= min {minX {r X: (-X, y)  F}/,  ≥ 0},
= min {C(r, y)/:  ≥ 0}.
(2)
For  > 0, define the ray-average cost function
RAC(r, y, )  C(r, y)/.
It follows from (1) and (2) that
Cc(r, y)  min {C(r, y)/:  ≥ 0}  min {RAC(r, y, ):  ≥ 0} ≤ C(r, y).
(3)
This implies the following results:
1/ If the technology F exhibits CRTS, then
Cc(r, y) = C(r, y)
and
min {RAC(r, y, ):  ≥ 0} = C(r, y).
c
2/ C (r, y) < C(r, y), or equivalently min {RAC(r, y, ):  ≥ 0} < C(r, y), implies that the technology F
departs from CRTS (i.e., it must exhibit at least locally either increasing return to scale (IRTS) or
decreasing return to scale (DRTS)).
This suggests the following input-based index of scale efficiency
5
SEI(r, y) =
min θ {RAC(r, y, θ) : θ  0}
≤ 1,
C(r, y)
where SEI(r, y) = 1 means that the firm is scale efficient, while SEI(r, y) < 1 implies that the firm is not
scale efficient. This implies a simple way to identify a scale efficient firm: it is a firm that produces outputs
y at a point where the ray-average cost is minimized. In addition, under scale inefficiency (where SEI(r, y)
< 1), [(1 - SEI(r,y)] measures the proportional reduction in ray-average cost that the firm can achieved by
becoming scale efficient.
Note 1: Consider the case where the cost function C(r, y) is differentiable in y. Expressions (2) and (3)
involve the minimization of the ray-average cost: min {C(r, y)/:  ≥ 0}.The associated firstorder necessary condition for an interior solution is
∂C(r, y)/∂ - C(r, y)/ = 0,
or
[∂C(r, y)/∂y] y = C(r, y), since C/ = [C/(y)] y, and C/y = [C/(y)] ,
or

m
j1
[∂C(r, y)/∂yj] yj = C(r, y),
(4a)
or, when C > 0 and y > 0,

m
j1
∂lnC(r, y)/∂ln(yj) = 1.
Recall that the scale elasticity can be measured as [1/[
(4b)

m
j1
(∂ln(C)/∂ln(yj)]]. As expected, under
an interior solution, it follows from (4b) that the ray-average cost is minimized at a point where the
scale elasticity is 1, i.e. at a point where CRTS holds (at least locally).
Note 2: In the case where the function RAC( , , ) has a U- shape, it follows from (4b) that

m
j1
∂lnC(r,
y)/∂ln(yj) < 1 for scale inefficient firms that are "too small" (they exhibit IRTS); and that

m
j1
∂lnC(r, y)/∂ln(yj) > 1 for scale inefficient firms that are "too large" (they exhibit DRTS).
Note 3: In the single output case (m = 1) with y > 0, the minimization problem in (2) or (3) can be written
as min{C(r,  y)/:   0} = y MinY{C(r, Y)/Y: Y  0}, where Y =  y. This implies the
minimization of average cost [C(r, y)/y] with respect to output y. The associated first-order
condition is: ∂C(r, y)/∂y = C(r, y)/y, or marginal cost = average cost. Thus, the minimum of
average cost takes place at a point where marginal cost equals average cost, and CRTS holds (at
least locally).
When the average cost [C( ,y)/y] has a U-shape, this implies that scale efficient firms are the ones
that operate at a scale y that minimizes the average cost function [C(r, y)/y] where CRTS holds (at
least locally). Scale inefficient firms can exhibit either IRTS or DRTS. They exhibit IRTS when
their scale of operation y is in the region where the average cost function [C(r, y)/y] is declining in
y (i.e., they are "too small"). And they exhibit DRTS when their scale of operation y is in the
region where the average cost function [C(r, y)/y] is increasing in y (i.e., they are "too large").
To motivate the economic significance of scale efficiency, consider an industry facing free entry
and exit, where firms can enter and exit the industry without any cost. (A market exhibiting free entry and
6
exit is called a contestable market.) Also, define a long-run equilibrium for an industry as a situation
where there is no incentive for any change.
At the profit maximizing optimum for a competitive firm and assuming an interior solution,
outputs y > 0 are chosen according to the marginal cost pricing rule: p = C/y. This result combined with
the first-order condition associated with scale efficiency (4a) implies
p y = C(r, y),
or
profit  p y - C(r, y) = 0.
With zero profit for the incumbent firms, there is no incentive for entry no exit. Without entry or
exit, this corresponds to a long-run industry equilibrium. It follows that scale efficiency is consistent with
long-run industry equilibrium under free entry and exit.
Alternatively, consider the case of an industry where some firms are scale inefficient. They exhibit
either IRTS or DRTS. If they exhibit IRTS, then their scale elasticity is greater than 1, and
y)/∂ln(yj) < 1. From (4a) and (4b), it follows that

m
j1

m
j1
∂lnC(r,
[∂C(r, y)/∂yj] yj < C(r, y). Under marginal cost
pricing (with ∂C/∂y = p), it follows that p y < C. This means that profit is negative and that the firms have
incentives to exit. This cannot be a situation of long-run equilibrium. Alternatively, if the scale inefficient
firms exhibit DRTS, then their scale elasticity is less than 1, and
and (4b), it follows that

m

m
j1
∂lnC(r, y)/∂ln(yj) > 1. From (4a)
[∂C(r, y)/∂yj] yj > C(r, y). Under marginal cost pricing (with ∂C/∂y = p),
j1
it follows that p y > C. This means that profit is positive and that new firms have the incentive to enter the
industry. Again, this cannot be a situation of long-run equilibrium. Thus, under free entry and exit, only
scale efficient firms producing under CRTS (at least locally) can be expected to be present in a long-run
industry equilibrium. This also means that, with zero firm profits in the long-run, the welfare effects of any
economic changes would be entirely passed on to consumers.
Note 4: Define OEI(r, x, y) = [TEI(x, y) AEI(r, x, y) SEI(r, y)] as an input-based index of overall economic
efficiency. We have
OEI(r, x, y) = TEI(x, y) AEI(r, x, y) SEI(r, y)

n
=
r TE I (x, y) x i
i 1 i

n
r xi
i 1 i
=
min θ {RAC(r, y, θ) : θ  0}
,
C(r, y)
i1 ri TE I (x, y) x i
C(r, y)
n
min θ {RAC(r, y, θ) : θ  0}

n
≤ 1.
rx
i 1 i i
It follows that [(1 - OEI(r, x, y)] can be interpreted as the proportional reduction in ray-average cost
that can be achieved by a firm becoming technically efficient, allocatively efficient as well as scale
efficient.
3.3. Revenue-based assessment of scale efficiency
Start with the revenue function R(p, x) = maxy {p y: (-x, y)  F} associated with the feasible set F
(which in general can exhibit variable return to scale (VRTS)). Alternatively, consider the revenue
function associated with the feasible set Fc = {(-k x, k y): (-x, y)  F for any k ≥ 0} which exhibits constant
return to scale (CRTS) and satisfies F  Fc. Assuming that a maximum exists, the associated revenue
function is
7
Rc(p, x) = maxy {p y: (-x, y)  Fc},
Since F  Fc, it follows that
Rc(p, x) ≥ R(p, x).
(5)
In addition, note that
Rc(p, x) = maxy {p y: (-x, y)  Fc},
= many, {p y: (-x, y)  F,  ≥ 0}, where  = 1/k,
= maxY, {p Y/: (-x, Y)  F,  ≥ 0}, where Y = y,
= max {maxY {p Y: (-x, Y)  F}/,  ≥ 0},
= max {R(p, x)/:  ≥ 0}.
(6)
For  > 0, define the ray-average revenue function
RAR(p, x, )  R(p, x)/.
It follows from (5) and (6) that
Rc(p, x)  max {R(p, x)/:  ≥ 0}  max {RAR(p, x, ):  ≥ 0}  R(p, x).
(7)
This implies the following results:
1/ If the technology F exhibits CRTS, then
Rc(p, x) = R(p, x)
and
max {RAR(p, x, ):  ≥ 0} = R(p, x).
2/ Rc(p, x) > R(p, x), or equivalently max {RAR(p, x, ):  ≥ 0} > R(p, x), implies that the technology
F departs from CRTS (i.e., it must exhibit at least locally either increasing return to scale (IRTS) or
decreasing return to scale (DRTS)).
This suggests the following output-based index of scale efficiency
R(p, x)
≤ 1,
max θ {RAR(p, x, θ) : θ  0}
SEO(p, x) =
where SEO(p, x) = 1 means that the firm is scale efficient, while SEO(p, x) < 1 implies that the firm is not
scale efficient. This implies a simple way to identify a scale efficient firm: it is a firm that produces inputs
x at a point where the ray-average revenue is maximized. In addition, under scale inefficiency (where
SEO(p, x) < 1), [(1 - SEO(p,x)] measures the proportional increase in ray-average revenue that the firm can
achieved by becoming scale efficient.
Note 1: Consider the case where the revenue function R(p, x) is differentiable in x. Expressions (6) and (7)
involve the maximization of the ray-average revenue: max {R(p, x)/:  ≥ 0}.The associated
first-order necessary condition for an interior solution is
∂R(p, x)/∂ - R(p, x)/ = 0,
or
[∂R(p, x)/∂x] x = R(p, x),
or

n
i 1
[∂R(p, x)/∂xi] xi = R(p, x).
(8)
This shows that, at the maximum of the ray-average revenue, (8) corresponds to CRTS (at least
locally).
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Note 2: Define OEO(p, x, y) = [TEO(x, y) AEO(p, x, y) SEO(p, x)] as an output-based index of overall
economic efficiency. We have
OEO(p, x, y) = TEO(x, y) AEO(p, x, y) SEO(p, x)

m
=

m
j1
j1
pj yj
m
j1
p j y j /TE O (x, y)
p j y j /TE O (x, y)

m
=

j1
pj yj
max θ {RAR(p, x, θ) : θ  0}
R(p, x)
R(p, x)
max θ {RAR(p, x, θ) : θ  0}
≤ 1.
It follows that [1 - OEO(p, x, y)] can be interpreted as the proportional increase in ray-average
revenue that can be achieved by a firm becoming technically efficient, allocatively efficient as well
as scale efficient.
4. Profit-based measures of firm efficiency
Given output prices p > 0 and input prices r > 0, profit maximization gives
(p, r) = maxx,y {py – r x: (-x, y)  F},
where (p, r) is the indirect profit function. This can be alternatively written as
(p, r) = maxx,y {p y - r x: D(-x, y, g)  0},
= maxx,y {p y - r x + D(-x, y, g) [p gy + r gx]},
where D(-x, y, g) is the directional distance function, and g = (gx, gy).
For (-x, y)  F, this implies (p, r)  p y - r x + D(-x, y, g) [p gy + r gx], or 0  D(-x, y, g)  [(p, r) – [p y r x]]/[p gy + r gx]. This suggests the following profit-based measures of firm efficiency:
OE = (p, r) – [p y - r x]/[p gy + r gx]  0,
as a measure of overall inefficiency,
TE = D(-x, y, g)  0,
as a measure of technical inefficiency, and
AE = [(p, r) – [p y - r x]]/[p gy + r gx] - D(-x, y, g)  0,
as a measure of allocative inefficiency, where
OE = AE + TE.
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