PRODUCTION EFFICIENCY - CONCEPTS
- Technical Efficiency
Consider a technology represented by the feasible set F, where z ≡ (-x, y) F means that inputs x R can be used to produce outputs y R. Given some reference bundle g R 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 zis then located below the production functionF(x, y1, …, ym-1), meaning that it is possible to increase the 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 firmproducing 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 thatTEI(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 thatTEO(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 . When (-x, y) F, it follows that
TEI(x, y) TEO(x, y) under .
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 and output prices p = (p1, …, pm) R. The firm'sactual cost is: r x = ri xi. And its actual revenue is: p y = pjyj.
In the case where [-TEI(x, y) x, y] is technically efficient, define the firm's technically efficient cost as 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
= TEI(x, y).
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 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
= TEO(x, y).
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.
- Allocative Efficiency
2.1.Input-allocativeefficiency
Consider a competitive firm producing netputs (-x, y) F and facing input prices r = (r1, …, rn) R.
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) = ≤ 1.
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The firm is input-allocatively efficient if and only if AEI(r, x, y) = 1. Otherwise, it is input-allocatively inefficient: AEI(r, x, y) < 1.Then, [1 - AEI(r, x, y)] measures the proportional reduction in cost 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) =
= 1.
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-allocativeefficiency
Consider a competitive firm producing netputs (-x, y) F and facing output prices p = (p1, …, pm) R.
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
AEO(p, x, y) = ≤ 1.
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The firm is output-allocatively efficient if and only if AEO(r, x, y) = 1. Otherwise, it is output-allocatively 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:
TEO(x, y) AEO(p, x, y) =
= 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).
- 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.
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, ky) 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).
2/ Cc(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
SEI(r, y) = ≤ 1,
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 first-order necessary condition for an interior solution is
∂C(r, y)/∂ - C(r, y)/ = 0,
or
[∂C(r, y)/∂y] y = C(r, y),sinceC/ = [C/(y)] y, and C/y = [C/(y)] ,
or
[∂C(r, y)/∂yj] yj = C(r, y),(4a)
or, when C > 0 and y > 0,
∂lnC(r, y)/∂ln(yj) = 1. (4b)
Recall that the scale elasticity can be measured as [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 ∂lnC(r, y)/∂ln(yj) < 1 for scale inefficient firms that are "too small" (they exhibit IRTS); and that ∂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 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 py - 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 ∂lnC(r, y)/∂ln(yj) < 1. From (4a) and (4b), it follows that [∂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 ∂lnC(r, y)/∂ln(yj) > 1. From (4a) and (4b), it follows that [∂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 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)
= ,
= ≤ 1.
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 {py: (-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
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, {pY/: (-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
SEO(p, x) = ≤ 1,
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
[∂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).
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)
=
= ≤ 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.
- 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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