MEASURING PRODUCTION EFFICIENCY
Any analysis of production efficiency requires knowledge of the production technology. How can we obtain this knowledge?
Consider an industry facing a technology represented by the feasible set F. A firm in this industry produces outputs y Î R using inputs xÎ R. Given some reference bundle g Î R satisfying g ≠ 0, recall that technical feasibility (where (-x, y) Î F) implies that S(-x, y, g) £ 0, D(-x, y, g) ³ 0, DI(-x, y) ³ 1, DF(-x, y) £ 1, and DO(-x, y) £ 1, where
· S(-x, y, g) = minb {b: ((-x, y) - b g) Î F} is the shortage function,
· D(-x, y, g) = maxa {a: ((-x, y) + a g) Î F} is the directional distance function,
· DI(-x, y) = maxa {a: (-x/a, y) Î F} is Shephard input distance function,
· DF(-x, y) = minb {b: (-bx, y) Î F} is Farrell input distance function,
· DO(-x, y) = mina {a: (-x, y/a) Î F} is Shephard output distance function,
each function providing a measure of the distance between point z ≡ (-x, y) and the upper boundary of the feasible set F.
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) ³ 0} = {(-x, y): DI(-x, y) ³ 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. In other words, knowing any of these functions provide all the information necessary to evaluate the production technology.
1. Engineering Approach
The engineering approach consists in evaluating the technical feasibility of a production process where inputs x are used to produce outputs y based on the best available technology F. It relies on engineers who are asked to evaluate the feasibility of alternative combinations of inputs x and outputs y. The information obtained can be used to "trace out" the upper boundary of the feasible set F.
Also, when input prices r are known and inputs x are chosen to minimize cost, this can provide a simple way of evaluating economies of scale. In this case, for alternative configurations of outputs y, the engineering approach can provide an estimate of C(r, y). Then changing outputs y proportionally across configurations, traces out the ray-average cost C(r, qy)/q (or the average cost C(r, y)/y in the single output case when m = 1). The minimum of the ray-average cost corresponds to points where CRTS holds (at least locally), thus identifying efficient scales of operation.
The engineering approach is particularly useful when applied to a new technology that is not currently implemented by any firm and thus not directly observable.
2. Using firm data
When observations are available on firms in an industry, this information can be used to assess the underlying technology. In this context, consider a sample of T firms in the industry. Let (xt, yt) Î R denote the observation on inputs and outputs chosen of the t-th firm, t = 1, ..., T. Technical feasibility implies that S(-xt, yt, g) £ 0, D(-xt, yt, g) ³ 0, DI(-xt, yt) ³ 1, DF(-xt, yt) £ 1, and DO(-xt, yt) £ 1, t = 1, …, T. The industry sample data can then be used to estimate 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. (Note that this requires either that the technology exhibits free disposal, or that we restrict these functions to apply only in regions where there is no gap between the production function and the upper bound of the feasible set F (the gap being due to non-free disposal)).
2.1. Statistical Approach
We limit our discussion to the case where g = (0, …, 0, 1), and S(-x, y, g) = -D(-x, y, g) = ym - F(x, y1, …, ym-1), with F(x, y1, …, ym-1) = maxk {k: (-x, y1, …, ym-1, k) Î F} being the standard production function for the m-th output ym, conditional on inputs x and other outputs (y1, …, ym-1). Consider some parametric representation of the production function F(x, y1, …, ym-1) º f(X, a), where X = (x, y1, …, ym-1) and a is a parameter vector.
Technical feasibility implies S(-x, y, g) = -D(-x, y, g) = ym - f(X, a) ≤ 0. Define e º f(X, a) - ym. It follows that
ymt = f(Xt, a) - et, t = 1, …, T. (1)
Treating et as a random variable, equation (1) is a regression model where the regression line f(Xt, a) if the production function (or production frontier) for the m-th output. Assume that the random variable et has mean µ. But technical feasibility implies that et º f(Xt, a) - ymt ≥ 0. As long as the distribution of et is not degenerate (e.g., as long as it has a positive variance), it follows that et must have a positive mean: µ > 0. Since the error term et in (1) cannot have mean zero, expression (1) is thus not a standard regression model. It means that the parameters a in (1) cannot be consistently estimated by the least-squares method.
A common method to estimate a is to make some assumption about the probability distribution of the error term e ³ 0. Assuming that the et's are identically and independently distributed (i.i.d.), candidates for this distribution are the gamma distribution, the exponential distribution, and the half-normal distribution: they each satisfy the non-negativity constraint e ³ 0. Denote this probability function by h(e, b) where b is a parameter vector characterizing the probability function. The parameters (a, b) can be estimated by the maximum likelihood (ML) method, which chooses (a, b) so as to maximize the likelihood of the sample: Pt{h[f(Xt, a) - ymt, b]}. Under general conditions, the ML estimator of (a, b), (aML, bML), has desirable asymptotic properties: it provides a consistent and asymptotically efficient estimator of (a, b). Then, f(X, aML) is a consistent estimator of the production frontier. (Also, [yt/f(Xt, aML)] ≤ 1 provides a measure of technical efficiency for the t-th firm t, t = 1, 2, ..., T).
Note 1: What if the inputs and outputs are subject to measurement errors? This suggests the following decomposition of e:
e = e + v,
where e reflects measurement errors, and v ³ 0 reflects technical inefficiency. The random variable e is often assumed to be normally distributed with mean zero. In contrast, the random variable v is typically assumed to be distributed according to a gamma, exponential or half-normal distribution satisfying v ³ 0. Again, after assuming some probability distribution for e and v, the production frontier f(x, a) can be estimated by the ML method (as indicated above) provided that the sample information is rich enough to allow a "reliable estimation" of the parameters of the distribution functions of e and v...
Note 2: Given that et has mean E(et) = µ > 0, note that the statistical model (1) can be alternatively written as:
ymt = -m + f(Xt, a) + wt, t = 1, 2, ..., T, (2)
where wt = (µ - et) is a random variable with mean zero. As long as the x's are exogenous variables, model (2) satisfies the standard assumptions of the classical regression model. Thus, it can be consistently estimated by the least squares (LS) method. But the least squares estimates of a in models (1) and (2) differ only by their intercept. Thus, the least squares estimate of a in model (1) gives a consistent estimate of all the parameters except the intercept. This suggests that the estimation of the production frontier in (1) can be done by the corrected least squares approach:
1/ estimate model (1) by the least squares method, where aLS is the least squares estimate of a
2/ find the smallest value of the intercept shifter g which satisfies [ymt - {g + f(Xt, aLS)}] ³ 0 for all t. Denote this value by g0. A consistent estimate the production frontier is: [g0 + f(Xt, aLS)].
(Note that this procedure is no longer valid in the presence of heteroscedasticity where the variance of e is not constant and varies with x).
Also, note that, in the presence of measurement errors, the intercept shift should not be all the way up to the point where [ymt - {g + f(Xt, aLS)}] ³ 0 for all t. In this case, choosing the appropriate intercept shifter g requires a priori information on the distribution of the measurement errors...
Note 3: From duality, technical efficiency, allocative efficiency and scale efficiency can be estimated from a production function, a cost function or a profit function.
2.2. Nonparametric Approach
Assume that the technology is convex, i.e., that the feasible set F Ì R is convex (implying "diminishing marginal productivity"). By definition of convexity, for any netput z º (-x, y) Î F and z' º (-x', y') Î F, it must be that [l z + (1-l) z'] Î F for any l Î [0, 1], i.e. that any linear combination of z and z' is also in F. With z º (-x, y) and given the sample data {zt º (-xt, yt): t = 1, …, T}, consider the set
Fv = {z: z ≤ lt zt, lt = 1; lt ≥ 0, t = 1, …, T}. (3)
Expression (3) defines Fv as the set of netputs that are no greater than any linear combination of the sample netputs zt, t = 1, …, T.
Result 1: The set Fv in (3) is the smallest convex set representing technical feasibility under free disposal, and containing all sample data points {zt: t = 1, …, T}.
Proof: First, the set Fv given in (3) is convex (since any z Î Fv and z' Î Fv implies that [l z + (1-l) z'] Î Fv for any l Î [0, 1]. Also, the set Fv is closed (it includes its boundary). Next, assume that there exist a set F0 Ì Fv, F0 ¹ Fv, such that F0 and Fv are each convex, satisfy free disposal, and contain all T points. Consider the set Fv\F0, the complement of F0 in Fv. Fv being closed, if the set Fv\F0 is non-empty, then Fv\F0 Ì Fv must include some point z" Î Fv\F0 on the boundary of Fv. This can happen in one of three ways:1/ z" can be a "corner point" of Fv corresponding to one of the T observations; 2/ z" can be on a plane linking two or more "corners points"; or 3/ z" can be on a plane linking one or more corner points with points where some netputs are "freely disposed." But case 1/ cannot be because it would imply that F0 would not contain the observation defining the corner point. Case 2/ cannot be because it would imply that F0 is not convex. And case 3/ cannot be because it would imply that F0 does not satisfy free disposal. Thus, the set Fv\F0 must be empty. This implies that Fv is the smallest convex set satisfying free disposal and containing all points zt, t = 1, …, T.
Note that the set Fv in (3) involves the envelopment of a set of points. As a result, any analysis involving Fv in (3) is often called data envelopment analysis (DEA). It is nonparametric in the sense that it does not require the specification of a particular parametric form identifying the boundary of the feasible set F.
Note 1: What if the technology does not exhibit free disposal? Without free disposal, the smallest convex set containing all sample data points {zt, t = 1, …, T} is {z: z = lt zt, lt = 1; lt ≥ 0, t = 1, …, T}. And in the case where free disposal applies only to a subset of netputs, then the smallest convex set would be given by expression (3) after replacing the weak inequalities in [z ≤ lt zt] by an equality sign for the netputs that do not satisfy free disposal.
In general, Fv in (3) exhibits variable return to scale, VRTS (that is, it can exhibit increasing return to scale (IRTS), constant return to scale (CRTS) as well as decreasing return to scale (DRTS)). Alternatively, we can also consider the set Fc = {k z: z Î Fv, k ≥ 0}, where Fc exhibits constant return to scale (CRTS) and satisfies Fv Ì Fc. Using (3), the set Fc can be written as
Fc = {z: z ≤ lt zt; lt ≥ 0, t = 1, …, T}. (4)
Note 2: Fc in (4) is similar to Fv in (3), except that [lt] is now unrestricted: [lt] in (4) can take any value between 0 and ¥ (where lt > 1 means "rescaling up" while lt < 1 means "rescaling down").