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Consider a set of T firms in an industry transforming the (n´ 1) input vector x into output y. The t-th firm is observed choosing inputs x_t and producing output y_t, t = 1, …, T. Using a nonparametric approach based on these data, the production frontier for the industry can be expressed as

F(x) º Max_l [S_t y_t l_t: S_t x_t l_t £ x, l_t ³ 0, S_t l_t = 1]. (1)

As shown by Afriat, within the range of the data, F(x) is a non-decreasing concave function of x which satisfies y_t £ F(x_t), t = 1, …, T. As such, it is the tightest non-decreasing concave function that provides an envelop to the data {(x_t, y_t): t = 1, …, T}. Hence, using F(x) is often called "data envelop analysis" or DEA.

Note that, by definition of the production frontier, technical feasibility implies that y £ F(x). This suggests that the nonparametric representation (1) can be alternatively expressed in terms the feasible set

S = {(x, y): y £ S_t y_t l_t: S_t x_t l_t £ x, l_t ³ 0, S_t l_t = 1}, (1')

for some feasible value of the l 's. This means that the input-output vector (x, y) is feasible if and only if (x, y) belongs to the feasible set S.

The production frontier F(x) in (1) (or the feasible set S in (1')) allow for variable return to scale (VRTS). This means that returns to scale can be increasing (IRTS), constant (CRTS) as well as decreasing (DRTS) for different input-output choices.

This suggests considering imposing restrictions on the nature of returns to scale. This can be done as follows

F₀(x) º Max_l [S_t y_t l_t: S_t x_t l_t £ x, l_t ³ 0] (2)

under constant return to scale (CRTS). Under CRTS, the magnitude of the l 's is unrestricted, meaning that F₀(x) allows for any proportional rescaling (either upward or downward) of inputs-output (as in the definition of global CRTS).

By definition of the production frontier, technical feasibility implies that y £ F₀(x) under CRTS. This suggests that, under CRTS, the nonparametric representation (2) can be alternatively expressed in terms of the feasible set

S₀ = {(x, y): y £ S_t y_t l_t: S_t x_t l_t £ x, l_t ³ 0}, (2')

for some feasible value of the l 's. This means that the input-output vector (x, y) is feasible under CRTS if and only if (x, y) belongs to the feasible set S₀.

 By comparing (1) and (2), it is clear that

F₀(x) ³ F(x) for all x.

This follows by noting that the two maximization problems (1) and (2) are very similar, except that (2) is less constrained in the sense that it does not impose a restriction on the sum of the l 's. Another way to express such a result is

S Í S₀,

meaning that the feasible set S is a subset of the feasible set S₀.

1/ Technical efficiency of firm #11

Given a production frontier f(x), consider the input-based technical efficiency index associated with the feasible input-output vector (x, y)

TE_I(x, y) = min_d [d : y £ f(d × x)] £ 1.

TE_I measures the proportional downward rescaling on all inputs that bring the production decisions (x, y) to the production frontier at the output level y. It follows (1 - TE_I) measures the proportional reduction in input (or in cost) that can be attained by moving to the production frontier while producing output y.

Under VRTS, using F(x) in (1) as a representation of the production frontier f(x) gives

TE_I(x, y) = min_d [d : y £ F(d × x)]

= min_d [d : (y, d × x) Î S]

= min_d,l [d : y £ S _t y_t l_t, S _t x_t l_t £ d × x, l_t ³ 0, S_t l_t = 1]. (3)

This is a simple linear programming problem that provides a basis for estimating the technical efficiency associated with the feasible input-output vector (x, y). Denote the solution of (3) by d* and l* = (l₁*, …, l_T*). The optimal d* is the smallest feasible value of the objective function in (3); it is also the technical efficiency index TE_I(x, y). And the non-zero optimal values of l * identifies the technically efficient firms that are "in the neighborhood" of (x, y).

Using GAMS, solving (3) for (d , l ) while setting (x, y) = (x₁₁, y₁₁) gives

d* = TE_I(x₁₁, y₁₁) = 0.9053

l₃* = 1.

This means that firm #11 is technically inefficient. It could reduce all its inputs (or its cost of production) by (100´ (1- 0.9053)) = 9.47 percent by becoming technically efficient while producing the same output. This identifies a change in inputs from x₁₁ = (29882, 35143, 8942) to (TE_I × x₁₁) = (27052, 31814, 8095) while producing output y₁₁ = 75033.

Finding l ₃* = 1 means that firm #3 (producing output y₃ = 76852 using inputs x₃ = (25445, 31814, 8079)) is the technically efficient firm that is "in the neighborhood" of firm #11. This means that firm #11 could learn from firm #3 to improve its access to the best available technology.

2/ Allocative efficiency of firm #11

Given a production frontier f(x), consider cost minimizing behavior under input prices r

C(x, y) = r' x^c(r, y) = min_x [r'x: y £ f(x)],

where x^c(r, y) are the cost minimizing input demand functions, and C(r, y) is the cost function measuring the smallest possible cost of producing output y.

Consider the allocative efficiency index

AE(r, y) = C(r, y)/[r' (TE_I(x, y) × x)]

= [C(r, y)/(r' x)]/[TE_I(x, y)] £ 1. (4)

AE measures the proportional downward rescaling of cost that can be achieved by moving from the technically efficient decisions (TE_I × x, y) to the cost minimizing decisions x^c(r, y). It follows (1 - AE) measures the proportional reduction in cost that can be attained by choosing inputs in a cost minimizing way.

Under VRTS, using F(x) in (1) as a representation of the production frontier f(x) gives

C(r, y) = r' x^c(r, y) = min_x [r'x: y £ F(x)],

= min_x [r'x: (x, y) Î S],

= min_x,l [r'x: y £ S_t y_t l_t, S_t x_t l_t £ x, l_t ³ 0, S_t l_t = 1]. (5)

This is a simple linear programming problem that provides a basis for estimating the cost function C(r, y) under VRTS. Denote the solution of (5) by x^c and l^c = (l₁^c, …, l_T^c). The optimal x^c are the cost minimizing input demands. C(r, y) in (5) is the smallest possible cost of producing output y. And the non-zero optimal values of l ^c identify the "more efficient" firms that are "in the neighborhood" of (TE_I × x, y).

Using GAMS, solving (5) for (x, l ) evaluated at r = r₁₁ and y = y₁₁ gives

x^c = (25445, 31814, 8079)

l₃^c = 1,

and

C(r₁₁, y₁₁) = r₁₁' x^c = 70003.59.

Substituting these results into equation (4) gives the following estimate of the allocative efficiency index AE

AE(r₁₁, y₁₁) = [70003.59/(r₁₁' x₁₁)]/TE_I(x₁₁, y₁₁)

= .8053/.9053

= .9751.

This means that firm #11 is allocatively inefficient. It could reduce its cost of production by (100´ (1- 0.9751)) = 2.49 percent by becoming allocatively efficient while producing the same output. This identifies a change in inputs from (TE_I × x₁₁) = (27052, 31814, 8095) to x^c = (25445, 31814, 8079) while producing output y₁₁ = 75033. This means that the technically efficient inputs (27052, 31814, 8095) use too much of input 1, the right amount of input 2, and just a little too much of input 3. The analysis would thus recommend that firm #11 reduces its relative use of input 1.

Finding l₃^c = 1 means that firm #3 (producing output y₃ = 76852 using inputs x₃ = (25445, 31814, 8079)) is the "more efficient" firm that is "in the neighborhood" of firm #11. This means that firm #11 could learn from firm #3 to improve its input decisions.

3/ Scale efficiency of firm #11

Consider the scale efficiency index SE

SE(r) = min_y{C(r,y)/y}/[C(r, y)/y] £ 1. (6)

SE measures the proportional downward rescaling of average cost that be achieved by moving from the allocatively efficient decisions (x^c(r, y), y) to the point that minimizes average cost (where CRTS holds at least locally and scale efficiency is attained). It follows (1 - SE) measures the proportional reduction in average cost that can be attained by choosing inputs in a scale efficient way.

Using (2) as the representation of the production frontier under CRTS, the cost function under CRTS is given by

C₀(r, y) = r' x₀^c(r, y) = min_x [r'x: y £ F₀(x)]

= min_x{r'x: (x, y) Î S₀},

= min_x,l [r'x: y £ S_t y_t l_t, S_t x_t l_t £ x, l_t ³ 0]. (7)

where x₀^c(r, y) are the cost minimizing input demand functions under CRTS, and C₀(r, y) is the cost function measuring the smallest possible cost of producing output y under CRTS.

By comparing (5) and (7), it is clear that

C₀(r, y) £ C(r, y) for any output y.

This follows by noting that the two minimization problems (5) and (7) are very similar, except that (7) is less constrained as it does not impose any restriction on the magnitude of the l 's. Thus, C₀(r, y) provides a lower bound on the cost function C(r, y). And, assuming a U-shape average cost function, the two functions become identical in the region exhibiting (local) CRTS.

Note that, under CRTS, average cost [C(r, y)/y] is a constant with respect to y. Since C₀(r, y) £ C(r, y), it follows that [C₀(r, y)/y] £ [C(r, y)/y] for any output y. If the average cost function has a U-shape, it follows that C₀(r, y)/y = min_y{C(r, y)/y}. Substituting this result into our definition of the scale efficiency index (6) yields the following expression

SE(r) = [C₀(r,y)/y]/[C(r, y)/y]

= C₀(r,y)/C(r, y) £ 1. (6')

The minimization problem (7) is a simple linear programming problem that provides a basis for estimating the cost function C₀(r, y) under CRTS. This estimate can be used in (6') to provide a measure of the scale efficiency index SE.

Denote the solution of (7) by x₀ and l₀ = (l₁⁰, …, l_T⁰). The optimal x₀ are the cost minimizing input demands under CRTS. C₀(r, y) in (7) is the smallest possible cost of producing output y under CRTS. And the non-zero optimal values of l ₀ identifies the scale efficient firms that are "in the neighborhood" of (x^c(r, y), y).

Using GAMS, solving (7) for (x, l ) evaluated at r = r₁₁ and y = y₁₁ gives

x₀ = (22188, 28000, 6350)

l₂⁰ = 0.782,

and

C₀(r₁₁, y₁₁) = 60554.41.

Substituting these results into equation (6') gives the following estimate of the scale efficiency index SE

SE(r₁₁) = 60554.41/70003.59

= .8650.

This means that firm #11 is scale inefficient. It could reduce its average cost of production by (100´ (1- 0.8650)) = 13.50 percent by becoming scale efficient.

Finding l₂⁰ = 0.782 means that firm #2 (producing output y₂ = 95921) is the "more efficient" firm that is "in the neighborhood" of firm #11. This indicates that firm #11 is too small: is output could expand from y₁₁ = 75033 to the scale efficient output 95921. This means that the firm #11 exhibits IRTS. It also means that firm #11 could learn from firm #2 to improve its output (or scale) decision.

4/ Evaluation

The three indexes of efficiency can be combined into a single overall index OE

OE = (TE_I ´ AE ´ SE)

= TE_I ´ [C(r, y)/r' (TE_I × x)] ´ [C₀(r, y)/C(r, y)]

= C₀(r, y)/[r' x]

= [C₀(r, y)/y]/[r'x/y] £ 1.

The overall index OE is the product of the three indexes TE_I, AE and SE. When applied to firm #11, we obtain OE = 0.9053 ´ 0.9751 ´ 0.8650 = 0.7636. It means that firm #11 is inefficient (since it has been found to be technically inefficient, allocatively inefficient, as well as scale inefficient). By becoming overall efficient (i.e., technically efficient, allocatively efficient as well as scale efficient), the firm could reduce its average cost by [100 ´ (1 - 0.7636] = 23.64 percent. This is a significant decrease in average that could contribute to lower price paid by consumers, which would greatly benefit consumer welfare. Thus, for firm #11, there are welfare gains that can be generated by improving technical efficiency (by using a better technology), allocative efficiency (by making a better choice of inputs) and scale efficiency (by making a better choice of output or scale of operation).