Empirical Production Analysis

*** Empirical Production Analysis

Production analysis can be done investigating a production function, a cost function, or a profit function. By duality, each approach is in principle equivalent to the others. Using econometric tools, the standard approach involves specifying parametric functional forms for a function, and finding a way of estimating the associated parameters using real world data. For convenience, it is often assumed that all functions are differentiable.

Definition: An empirical analysis is said to be flexible if the assumed functional form does not impose a priori restrictions on the Allen elasticities of substitution.

Note: For simplicity, we focus our attention on the single output case (m = 1). It should be kept in mind that all the results obtained can be extended to the multi-output case (m > 1).

Note: While we focus on parametric analysis, an alternative would be to use non-parametric representations of the associated functions. For example, consider a sample of T observations on inputs and output (xt, yt), t = 1, …, T, in a given industry. Then, a non-parametric representation of the production frontier under variable returns to scale is given by the solution to the linear programming problem

F(x) = maxy,{y: y t yt, x t xt, t =1, t 0, t = 1, …, T}.

Here the production frontier F(x) can be shown to be non-decreasing and concave in x, and satisfy yt F(xt), for all t = 1, …, T.

Alternatively, a non-parametric representation of the production frontier under constant returns to scale is given by the solution to the linear programming problem

F0(x) = maxy,{y: y t yt, x t xt, t 0, t = 1, …, T}.

Here the production frontier F0(x) can be shown to be non-decreasing, linear homogenous and concave in x, and satisfy yt F0(xt), for all t = 1, …, T. In addition, we have F(x)  F0(x) for all x  0. The representations F(x) and F0(x) provide the “tightest bounds” among all concave and non-decreasing functions for the data under variable and constant returns to scale, respectively. For that reason, they are called “data envelopment analysis” (DEA). Note that, by construction, F(x) and F0(x) are continuous, piece-wise linear functions. While continuous, they are not differentiable everywhere.

** Production function approach y = f(x)

Let y = f(x) denote the production function, where y = output, and x = (n1) vector of inputs.

* Cobb-Douglas production function

The Cobb-Douglas (CD) production function is

y = ,

where bi > 0, for all i = 0, 1, …, n. It can be alternatively written as

ln(y) = a0 + b1 ln(x1) + b2 ln(x2) + … + bn ln(xn),

where a0 = ln(b0). This form is particularly convenient for empirical analysis since it is linear in the parameters.

The CD production function implies that

ln(y)/ln(xi) = bi, i = 1, …, n.(1)

Since ln(y)/ln(xi) = (y/xi)(xi/y), it implies that

y/xi = bi y/xi, i = 1, …, n,

MRSij = (y/xi)/(y/xj) = (bi xj)/(bj xi), for all i, j = 1, …, n.

Using the first-order conditions (FOC) to cost minimization ((y/xi)/(y/xj) = ri/rj), this gives

(bi xj)/(bj xi) = ri/rj,

xi/xj = (rj bi)(ri bj),

ln(xi/xj) = ln(bi/bj) - ln(ri/rj),

implying that

ln(xi/xj)/ln(ri/rj) = -1, for all i  j = 1, …, n.

Since the Allen elasticity of substitution (AES) is ij = -ln(xi/xj)/ln(ri/rj), it follows that

ij = 1, for all i  j = 1, …, n.

Thus, the Cobb-Douglas (CD) production function implies unitary AES. As a result, CD is not a flexible functional form.

Also, from (1), the scale elasticity SE is

SE = ln(y)/ln(xi) = bi.

This shows that the Cobb-Douglas production function is a homogenous function of degree bi, with (globally constant) scale elasticity SE = bi.

Note: The cost function associated with a Cobb-Douglas production function takes the form

C(r, y) = A yk,

where ai = 1 (since C is linear homogenous in prices r), and k = 1/(bi) (since the Cobb-Douglas technology is homogenous of degree (bi)). Check that this cost function indeed implies unitary Allen elasticities of substitution (ij = 1 for i  j).

Note that this functional form for the cost function is similar to the functional form for the Cobb-Douglas production function. For that reason, the Cobb-Douglas production function is sometimes called “self-dual”.

*CES production function

The “constant elasticity of substitution” (CES) production function is

y =  [ K- + (1-)L-]-v/,

whereK = capital

L = labor

 = “efficiency parameter” ( > 0)

 = “distribution parameter” (0 <  < 1)

 = “substitution parameter” (-1 ).

It can be written as

ln(y) = ln() – (v/) ln[ K- + (1-) L-].

It follows that

ln(y)/ln(K) = (y/K)(K/y) = -(v/) [],

and

ln(y)/ln(L) = (y/L)(L/y) = -(v/) [].

This gives

MRS= (y/K)/(y/L) = .

Let r1 = price of capital, and r2 = wage rate. Using the first-order conditions (FOC) to cost minimization ((y/K)/(y/L) = r1/r2) yields

= r1/r2,

K/L = ,

ln(K/L) = [1/(+1)] ln(/(1-)) - [1/(+1)] ln(r1/r2),

implying that

ln(K/L)/ln(r1/r2) = -1/(+1).

Since the Allen elasticity of substitution (AES) between capital and labor is  = -ln(K/L)/ln(r1/r2), it follows that

 = 1/(+1).

Thus, the CES production function implies a constant but unrestricted AES between capital and labor, with  = 1/(+1). As a result, the CES is a flexible functional form when there are only two inputs.

Note: We have

ln(r1K/r2L) = ln(K/L) + ln(r1/r2)

=  ln(/(1-)) -  ln(r1/r2) + ln(r1/r2)

=  ln(/(1-)) + (1- ln(r1/r2).

This implies that ln[r1 K/(r2 L)]/ln(r1/r2) = 1-, meaning that a decrease in r1/r2 generates a decrease (increase) in (r1 K/(r2 L)) if < 1 (> 1). Historically, there has been a decrease in the capital price-to-wage ratio (r1/r2) and a decrease in relative value of capital compared to labor (r1 K/(r2 L)). This has been interpreted as evidence that the elasticity of substitution between capital and labor is less than 1. This has also been interpreted as evidence against the Cobb-Douglas specification.

For the CES production function, the scale elasticity SE is

SE = ln(y)/ln(K) + ln(y)/ln(L) = v.

This shows that the CES production function is a homogenous function of degree v, with (globally constant) scale elasticity SE = v.

Note that, if v = 1,

 = -1 implies  = , giving the linear production function y =  [ K + (1-)L],

 = 0 implies  = 1, giving the Cobb-Douglas production function with b1 + b2 = 1,

 =  implies that  = 0, giving the Leontief production function y = min{K, L}.

Thus, in the two-input case, the CES is a flexible functional form that generalizes the Cobb-Douglas production function, and includes as special cases both the linear production function and the Leontief production function.

Note: The cost function associated with a CES production function takes the form

C(r, y) = y1/v [A r11- + B r21-]1/(1-).

(Check that this indeed satisfies  = C (2C/r1r2)/[(C/r1)(C/r2)].)

One issue with the CES is that it does not easily generalize to more than 2 inputs while maintaining its “flexibility”.

* Other forms

Examples of “flexible functional forms” for the production function y = f(x) include:

- the quadratic form: y = b0 + bi xi + bij xi xj,

- the square-root form:y = c0 + ci (xi)1/2 + cij (xi)1/2 (xj)1/2,

- the translog form:ln(y) = d0 + di ln(xi) + dij ln(xi) ln(xj),

etc.

** Cost function approach C(r, y).

* Cobb-Douglas cost function

As we have seen above, the cost function C(r, y) associated with a Cobb-Douglas production function takes the form

C(r, y) = A yk,

where ai > 0,ai = 1. It can be equivalently written as

ln(C) = ln(A) + k ln(y) + ai ln(ri).

It is associated with a technology that is homogenous of degree (1/k). And it exhibits unitary Allen elasticities of substitution (ij = 1 for i  j). Thus, it is not a flexible functional form. From Shephard’s lemma, the associated cost minimizing input demand functions are

xic = A yk ai/ri, i = 1, …, n,

ln(xic) = ln(A) + k ln(y) + (ai-1) ln(ri) + aj ln(rj).

* Translog cost function

The translog cost function C(r, y) takes the form

ln(C) = ln(a0) + ay ln(y) + ai ln(ri) + ½ aij ln(ri) ln(rj) + ayi ln(y) ln(ri)

+ ayy [ln(y)]2.

It is associated with a general technology that imposes no a priori restriction on the Allen elasticity of substitution. Thus, it is a flexible functional form. It includes the Cobb-Douglas cost function as a special case when ayi = aij = ayy = 0 for all i and j = 1, …, n.

1/ Theoretical restrictions

symmetry: 2ln(C)/ln(r)2 = symmetric, implying that

aij = aji for all i  j,

generating (n2 – n)/2 restrictions.

homogeneity: C(r, ) is linear homogenous in prices r, implying that

ln(C)/ln(ri) = 1 (Euler equation),

In the translog case, we have

ln(C)/ln(ri) = ai + ayi ln(y) + aij ln(rj), i = 1, …, n,

implying that

[ai + ayi ln(y) + aij ln(rj)] = 1, for all y and r.

This gives the following homogeneity restrictions

ai = 1

ayi = 0

aij (= aji) = 0, for all j = 1, …, n.

2/ Special cases

- homothetic technology: obtained when ayi = 0

- homogenous technology: obtained when ayy = 0, and ayi =0, for all i, where 1/ay measures the degree of homogeneity (e.g., ay = 1 corresponding to CRST).

- Cobb-Douglas technology: obtained when ayy = 0, ayi = 0 (since the Cobb-Douglas technology is homogenous), and aij = 0 for all i and j (since the Cobb-Douglas technology implies unitary Allen elasticities of substitution; see below).

3/ Estimation

From Shephard’s lemma, we have

ln(C)/ln(ri) = (C/ri)(ri/C) = ri xic/C.

Let wic = ri xic/C denote the i-th cost share. It follows that, under cost minimizing behavior,

wi = ln(C)/ln(ri)

always.

In the translog case, this gives the following cost share specification

wi = ai + ayi ln(y) + aij ln(rj), i = 1, …, n.

In conducting econometric analysis, an error term is typically added to the above model, yielding the following econometric model

wi = ai + ayi ln(y) + aij ln(rj) + ei, i = 1, …, n,

where ei is a random variable with mean zero and finite variance. This is a system of n equations that can be estimated using standard estimation methods.

Note that, by definition, we have wi = 1. This implies that the dependent variables are linearly dependent, meaning that associated variance of the e’s is singular. This is often handled by simply dropping an equation before estimation, and thus estimating a system of (n-1) equations. The parameters of the equation dropped can then be recovered from the homogeneity restrictions mentioned above. In addition, the parameter estimates are invariant to the equation dropped when the model is estimated by the maximum likelihood estimation method.

4/Elasticities

For i  j, we have

ln(xic)/ln(rj) = (xic/rj)(rj/xi)

= (2C/rirj)(rj/xi) (from Shephard’s lemma)

= [2ln(C)/ln(riln(rj)][C/(rixi)] + [ln(C)/ln(ri)][ln(C)/ln(rj)][C/(rixi)]

= aij/wi + wi wj/wi, where wi = ln(C)/ln(ri).

This gives the following cross-price elasticities of the cost minimizing input demand functions xic

ln(xic)/ln(rj) = aij/wi + wj, for all i  j.

Being homogeneous of degree zero in prices, xic must satisfy

ln(xic)/ln(ri) = -ln(xic)/ln(rj) (from Euler equation)

= -[aij/wi + wj].

But-aij = aii from the linear homogeneity of the cost function in r, andwj = 1 - wi. This gives the following own-price elasticities of the cost minimizing input demand functions xic

ln(xic)/ln(ri) = aii/wi + wi – 1, for all i = 1, …, n.

Note: We have seen that the Allen elasticities of substitution (AES) between inputs i and j are ij = [ln(xic)/ln(rj)]/wj. For i j, this gives

ij = [aij/wi + wj]/wj

ij = aij/(wi wj) + 1, for all i  j.

Note: In the Cobb-Douglas case where ij = 1 for all i  j, this implies aij = 0 for all i  j. In addition, we have seen that the linear homogeneity of the cost function in r gives the restrictionaij = 0, or aii = -aij. In the Cobb-Douglas case, this implies that aii = 0. In other words, unitary Allen elasticities of substitution (ij = 1 for all i  j) implies that aij = 0 for all i, j = 1, …, n (e.g., under a Cobb-Douglas technology).

* Generalized Leontief cost function

The generalized Leontief cost function C(r, y) takes the form

C = h(y) ij (ri)1/2 (rj)1/2 + g(y) i ri.

It is associated with a general technology that imposes no a priori restriction on the Allen elasticity of substitution. Thus, it is a flexible functional form. It includes the Leontief technology as a special case (see below).

1/ Theoretical restrictions

symmetry: 2C/r2 = symmetric, implying that

ij = ji for all i  j,

generating (n2 – n)/2 restrictions.

homogeneity: C(r, ) is already linear homogenous in prices r. Thus there is no additional homogeneity restriction to impose.

2/ Special cases

- homothetic technology: obtained when i = 0 for all i.

- Leontief technology: obtained when ij 0 for all i  j. This gives

C = g(y) i ri.

It implies that Cr2 = 0, yielding ij = 0. This is the Leontief technology, with fixed proportions, and zero possibilities of substitution among inputs.

3/ Estimation

From Shephard’s lemma, we have

C/ri = xic.

In the generalized Leontief case, this gives the following cost minimizing input demand functions

xic = h(y) ij (rj/ri)1/2 + i g(y), i = 1, …, n.

In conducting econometric analysis, an error term is typically added to the above model, yielding the following econometric model

xic = h(y) ij (rj/ri)1/2 + i g(y) + ei, i = 1, …, n,

where ei is a random variable with mean zero and finite variance. This is a system of n equations that can be estimated using standard estimation methods. (Note that, in this case, there is no problem with the singularity of the variance of the e’s, and thus no need to drop an equation).

4/Elasticities

For i  j, we have

ln(xic)/ln(rj) = (xic/rj)(rj/xi).

This gives the following cross-price elasticities of the cost minimizing input demand functions xic

ln(xic)/ln(rj) = ½ h(y) ij (ri rj)-1/2 rj/xi, for all i  j.

Being homogeneous of degree zero in prices, xic must satisfy

ln(xic)/ln(ri) = -ln(xic)/ln(rj) (from Euler equation)

= -½ h(y) [ij (ri rj)-1/2 rj/xi].

This gives the following own-price elasticities of the cost minimizing input demand functions xic

ln(xic)/ln(ri) = -½ h(y) [ij (rj/ri)1/2 /xi], for all i = 1, …, n.

**Efficiency analysis

Efficiency analysis consists in comparing actual decisions with “efficient decisions”. Denote by (xa, ya) the actual decisions made by a given firm.

Assume that the firm faces the production fonction y = f(x, A), where A is a technology index satisfying f/A > 0. Denote by A* the technology index associated with “best production practices” in the industry, and let A be the index of technology actually used by the firm. Allowing the firm to use possibly “inferior technology”, we allow A  A*.

Definition: The production decision (xa, ya) is said to be technically efficient if A = A*. And it is technically inefficient if A < A*.

An index of technical efficiency is given by IT = A/A*, where 0 < IT 1. Then, (1 - IT)  0 is a measure of the distance between (xa, ya) and the production frontier y = f(x, A*).

Efficient decision rules include cost minimizing choices xc(r, y, A*), and profit maximizing choices x*(r, p, A*) and y*(r, p, A*). To allow for possible departures from production efficiency, consider that actual decisions may fail to respond to actual market prices. Instead, assume that they may respond to “effective prices”. Let (k rk) denote the “effective price” for the k-th input, where k > 0 is a parameter, k = 1, …, n. This allows the effective price to differ from the actual price whenever k 1 for some k. Then, under profit maximization, actual decisions can be represented by

xka = xk*[(r)/p, A)], k = 1, …, n,

and

ya = y*[(r)/p, A)],

where ( r) = (1 r1, …, n rn), and A  A*.

Alternatively, under cost minimization, actual decisions can be represented by

xka = xkc(r, y, A), k = 1, …, n,

where ( r) = (1 r1, …, n rn), and A  A*.

Definition: The production decision xka, k = 1, …, n, is said to be allocatively efficient if k = 1 for all k = 1, …, n. And it is allocatively inefficient if k 1 for some k: it uses “too little” xk if k > 1, and “too much” xk if k < 1.

An index of allocative efficiency is given by IA = C(r, y, A*)/[r’xc( r, y, A*)] , where 0 < IA 1. Then, (1 – IA)  0 measures the percentage reduction in cost of production that the firm can be achieved by becoming “allocatively efficient” (i.e., by minimizing cost).

Definition: The production decision ya is said to be scale efficient if the scale elasticity (evaluated at xc(r, y, A*) and y) is SE = 1. And it is scale inefficient if SE  1: it is “too small” if SE > 1, and “too large” if SE < 1.

An index of scale efficiency is given by IS = {miny[C(r, y, A*)/y]}/[C(r, y, A*)/y], where 0 < IS 1. Then, (1 – IS)  0 measures the percentage reduction in average cost of production that the firm can be achieved by becoming “scale efficient” (i.e., by minimizing average cost).

** Profit function approach

We have

(p, r, A) = maxx,y{p y – r’x: y = f(x, A)}

where A is a technology index satisfying f/A > 0. The associated profit maximizing decision rules are xk*(r/p, A), k = 1, …, n, and y*(r/p, A). The use of price ratio (r/p) reflects the fact that supply-demand functions are homogenous of degree zero in prices.

* Cobb-Douglas profit function

Consider the Cobb Douglas production function

y = A ,

with n variable inputs (x1, …, xn) and m fixed factors (z1, …, zm), the parameters A, ’s and ’s being all positive and satisfying i =  < 1 (i.e., with decreasing returns to scale). The profit function associated with this Cobb-Douglas technology is

(p, r, A, z) = maxx{p A - rixi}.

The associated first-order condition (FOC) with respect to xk gives

p y/xk = rk,

p ln(y)/ln(xk) = rk xk/y,

k = rk xk/(p y), k = 1, …, n.

This states that, under profit maximization and CD technology, the production parameter k equals the (observable) share rk xk/(p y), k = 1, …, n.

More specifically, under CD technology, the (FOC) with respect to xk are

p k A /xk = rk, k = 1, …, n.

Solving these n equations for x* gives the profit maximizing input demand functions (see homework. Hint: Note that (FOC) implies xi = xk rki/(rik); substitute this in the above expression and solve for xk).

xk* = , k = 1, …, n,

where  = i < 1.

Substituting this into the production function gives the profit maximizing output supply function

y* = .

Finally substituting these optimal decision rules into the objective function yields the indirect profit function

(p, r, A, z) = p y* - r’ x* = p (1-).

And the associated profit shares are

rk xk*/ = k/(1-), k = 1, …, n.

1/Efficiency analysis under a CD profit function

Let the actual decision rules be

xka = xk*[( r)/p, A, z]

= , k = 1, …, n,

and

ya = y*[( r)/p, A, z]

= ,

where ( r) = [( r1), …(n rn)] are “effective input prices”, k > 0 being the allocative efficiency parameter for the k-th input, k = 1, …, n, and = i < 1.

Noting that [p p/(1-)] = p1/(1-), the associated actual profit function is

a( r/p, A, z) = p ya - r’ xa

= .

And the associated actual profit shares are

rk xka/a = , k = 1, …, n,

rk xka/a = k(), k = 1, …, n,(B1)

where

k() = , k = 1, …, n.

It follows that

ln(a) = ln(1- (k/k)) + (1-)-1 ln(A) +(1-)-1 ln(p) + i(1-)-1 ln(i/i)

- i(1-)-1 ln(ri) + j(1-)-1 ln(zj),

ln(a) = K(, A) + (1-)-1 ln(p) + i* ln(ri) + j* ln(zj),(B2)

where

K(, A) = ln(1- (k/k)) + (1-)-1 ln(A) + i(1-)-1 ln(i/i),

i* = -i(1-)-1, i = 1, …, n,

andj* = j(1-)-1, j = 1, …, m.

Equations (B1) and (B2) consist of a system of (n+1) behavioral equations that can be used to conduct an econometric investigation of production efficiency.

Consider a set of observations on two groups of firms, say groups S and L, in a given industry. Assume that each group face the same parameters ’s and ’s, but possibly different A’s (AS and AL) and different ’s (S and L). How can we test hypotheses about the efficiency of these two groups?

1- H0: AS = AL, S = L: equal relative economic efficiency. This implies KS = KL.

2- H0: S = L: equal relative allocative efficiency. This implies Sk = Lk , k = 1, …, n.

3- H0: AS = AL, S = L: equal relative technical and allocative efficiency. This implies KS = KL, and SkLk, k = 1, …, n.

4- H0: S = 1: absolute allocative efficiency for group S. This implies Skk*, k = 1, …, n.

5- H0: L = 1: absolute allocative efficiency for group L. This implies Lkk*, k = 1, …, n.

6- H0: SE = i + j = 1: scale efficiency (i.e., CRTS). This implies j* = 1.

* Translog profit function

The translog profit function (p, r) is

ln() = b0 + bp ln(p) +bri ln(ri) + ½ brij ln(ri) ln(rj) + ½ bpp [ln(p)]2

+ bpri ln(p) ln(ri)

It is associated with a general technology that imposes no a priori restriction on the Allen elasticity of substitution. Thus, it is a flexible functional form. It includes the Cobb-Douglas profit function as a special case when brij = bpp = bpri = 0 for all i and j = 1, …, n.

1/ Theoretical restrictions

symmetry: 2ln(ln(r)2 = symmetric, implying that

brij = brji for all i  j,

generating (n2 – n)/2 restrictions.

homogeneity: (p, r) is linear homogenous in prices (p and r), implying that

ln()/ln(ri) + ln()/ln(p) = 1 (Euler equation),

In the translog case, we have

ln()/ln(ri) = bri + brij ln(rj) + bpri ln(p), i = 1, …, n,

and

ln()/ln(p) = bp + bprj ln(rj) + bpp ln(p),

implying that

[bri + brij ln(rj) + bpri ln(p)] + bp + bprj ln(rj) + bpp ln(p) = 1,

for all p and r. This gives the following homogeneity restrictions

bri + bp = 1

brij + bprj (=brji + bprj) = 0, j = 1, …, n.

bpri + bpp = 0.

2/ Special cases

- Cobb-Douglas technology: obtained when brij = bpp = bpri= 0 for all i and j = 1, …, n.

3/ Estimation

From Hotelling’s lemma, we have

ln()/ln(ri) = (/ri)(ri/) = -ri xi*/, i = 1, …, n,

and

ln()/ln(p) = (/)(p/) = p y*/

Let wi* = ri xi*/ denote the i-th input profit share, and wp* = p y*/ denote the output profit share. It follows that, under profit maximizing behavior,

wi* = -ln()/ln(ri), i = 1, …, n,

and

wp* = ln()/ln(p),

always.

In the translog case, this gives the following cost share specification

-wi* = bri + brij ln(rj) + bpri ln(p), i = 1, …, n,

and

wp* = bp + bprj ln(rj) + bpp ln(p).

In conducting econometric analysis, an error term is typically added to the above model, yielding the following econometric model

- wi* = bri + brij ln(rj) + bpri ln(p) + ei, i = 1, …, n,

and

wp* = bp + bprj ln(rj) + bpp ln(p) + ep.

where the e’s are random variables with mean zero and finite variance. This is a system of n+1 equations that can be estimated using standard estimation methods.

Note that, by definition, we have wp* - wi* = 1. This implies that the dependent variables are linearly dependent, meaning that associated variance of the e’s is singular. This is often handled by simply dropping an equation before estimation, and thus estimating a system of n equations. The parameters of the equation dropped can then be recovered from the homogeneity restrictions mentioned above. In addition, the parameter estimates are invariant to the equation dropped when the model is estimated by the maximum likelihood estimation method.

* Generalized quadratic profit function

Consider the normalized profit function

[(p, r)/p] = y* - (r/p)’ x*,

where y* and x* are the profit maximizing supply-demand functions. The generalized quadratic profit function is given by

/p = 0 + i (ri/p) + ½ ij (ri/p)(rj/p).