The Stochastic The Stochastic Nature of ProductionNature of Production

Lecture VIILecture VII

Stochastic Production Stochastic Production FunctionsFunctions

Just, Richard E. and Rulan D. Pope. Just, Richard E. and Rulan D. Pope. “Stochastic Specification of Production “Stochastic Specification of Production Functions and Economic Implications.” Functions and Economic Implications.” Journal of EconometricsJournal of Econometrics 7(1)(Feb 1978): 7(1)(Feb 1978): 67–8667–86

Our development of the random Our development of the random characteristics of the production function characteristics of the production function was largely one of convenience.was largely one of convenience.

We started with a production function that We started with a production function that we wanted to estimatewe wanted to estimate

In order to estimate each function, we In order to estimate each function, we multiplied or added a random term to multiplied or added a random term to each specificationeach specification

1 21 2 0 1 2

2 21 2 0 1 1 2 2 11 1 12 1 2 22 2

,

, 2

f x x x x

g x x a a x a x A x A x x A x

1 21 2 0 1 2 1 2 0 1 1 2 2

2 21 2 0 1 1 2 2 11 1 12 1 2 22 2

, ln , ln ln

, 2

uf x x x x e f x x x x u

g x x a a x a x A x A x x A x v

Just and Pope discuss three different Just and Pope discuss three different specifications of the stochastic specifications of the stochastic production functionproduction function

1

2

3

0

1

0

y F X f X e E

y F X f X E

y F X f X E

Each of these specifications has Each of these specifications has “problematic” implications. For example, “problematic” implications. For example, the Cobb-Douglas specification implies the Cobb-Douglas specification implies that all inputs increase the risk of that all inputs increase the risk of production:production:

1 2 1 2

22 1 2

1 2 0 1 2 0 1 21

,, 0

V f x xV f x x E x x e E x x e

x

Note that this expectation is complicated Note that this expectation is complicated by the fact the expectation of the by the fact the expectation of the exponential. Specifically, under log-exponential. Specifically, under log-normal distributionsnormal distributions

212E e e

Just and Pope propose 8 propositions Just and Pope propose 8 propositions that “seem reasonable and, perhaps, that “seem reasonable and, perhaps, necessary to reflect stochastic, technical necessary to reflect stochastic, technical input-output relationships.”input-output relationships.” Postulate 1: Positive production Postulate 1: Positive production

expectations expectations EE[[yy]>0]>0 Postulate 2: Positive marginal product Postulate 2: Positive marginal product

expectations expectations 0i

E yX

Postulate 3: Diminishing marginal product Postulate 3: Diminishing marginal product expectations expectations

Postulate 4: A change in variance for Postulate 4: A change in variance for random components in production should random components in production should not necessarily imply a change in expected not necessarily imply a change in expected output when all production factors are held output when all production factors are held constant constant

2

2 0i

E yX

0

E yV

Postulate 5: Increasing, decreasing, or Postulate 5: Increasing, decreasing, or constant marginal risk should all be constant marginal risk should all be possibilities possibilities

Postulate 6: A change in risk should not Postulate 6: A change in risk should not necessarily lead to a change in factor use for necessarily lead to a change in factor use for a risk-neutral (profit-maximizing) producer a risk-neutral (profit-maximizing) producer

0i

V yX

*

0iXV

Postulate 7: The change in the variance of Postulate 7: The change in the variance of marginal product with respect to a factor marginal product with respect to a factor change should not be constrained in sign change should not be constrained in sign a a priorprior without regard to the nature of the input without regard to the nature of the input

Postulate 8: Constant stochastic returns to Postulate 8: Constant stochastic returns to scale should be possible scale should be possible

0i

j

V y XX

F X F X

The Cobb-Douglas, transcendental, and The Cobb-Douglas, transcendental, and translog production functions are translog production functions are consistent with postulates 1, 2, 3, and 8. consistent with postulates 1, 2, 3, and 8. However, in the case of postulate 5However, in the case of postulate 5

2

2i ii i

E y f X E e V y f X V e

E y V yf E e f f V eX X

The marginal effect of input use on risk must The marginal effect of input use on risk must always be positive. Thus, no inputs can be always be positive. Thus, no inputs can be risk-reducing.risk-reducing.

For postulate 4, under normalityFor postulate 4, under normality

Thus, it is obvious that our standard Thus, it is obvious that our standard specification of stochastic production specification of stochastic production functions is inadequate.functions is inadequate.

2

10

2E y

f X eV

An alternative specificationAn alternative specification

4

20,

y F X f X h X

E V

Econometric SpecificationEconometric Specification

2

, ,

0, 1, 0

ln , ln

ln , ln

t t t t

t t t s

t t t

t t t

t t

y f Z h Z

E E E t s

f Z Z z

h Z Z z

Z Z X

Consistent estimationConsistent estimation Rewriting the error termRewriting the error term

So the production function can be rewritten So the production function can be rewritten asas

Where the disturbances are heteroscedastic.Where the disturbances are heteroscedastic.

,t t tu h Z

, 0t t t ty f Z u E u

b.b. Under appropriate assumptions, a Under appropriate assumptions, a nonlinear least-squares estimate of this nonlinear least-squares estimate of this expression yields consistent estimates of expression yields consistent estimates of αα. Thus, these estimates can be used to . Thus, these estimates can be used to derive consistent estimates of derive consistent estimates of uutt

ˆˆ ,t t tu y f Z

Consistent estimates of Consistent estimates of ββ are obtained in are obtained in the second stage by regressions on the second stage by regressions on uu. . Following the method suggested by Following the method suggested by Hildreth and HouckHildreth and Houck

2 2ˆ ,t tu h Z

Expanding the Specification Expanding the Specification to Panel Datato Panel Data

Going back to the simultaneity Going back to the simultaneity specificationspecification

This expression becomesThis expression becomes

01 2

uY Ax x e

1 2 1 1

1 1 2 2

2 2 3 3

ln ln ln ln

ln ln ln ln

ln ln ln ln

y x x A u

y x P W u

y x P W u

In order to discuss this specification, we In order to discuss this specification, we will begin with a brief survey of estimation will begin with a brief survey of estimation using panel data.using panel data. As a starting point of this model, we consider As a starting point of this model, we consider

a panel regressiona panel regression

1,2,

1,2,it it ity x i N

t T

This specification is implicitly pooled, the This specification is implicitly pooled, the value of the coefficients are the same for value of the coefficients are the same for each individual at every point in time.each individual at every point in time. As a starting point, we consider generalizing this As a starting point, we consider generalizing this

representation to include differences in constant representation to include differences in constant of the regression that are unique to each firmof the regression that are unique to each firm

it i it ity x

This specification can be expanded further to This specification can be expanded further to allow for differences in the slope coefficients allow for differences in the slope coefficients across firmsacross firms

it i i it ity x

Based on these alternative models, we Based on these alternative models, we conceptualize a set of nested tests. First we conceptualize a set of nested tests. First we test for overall pooling (i.e., the production test for overall pooling (i.e., the production function have the same constant and slope function have the same constant and slope parameters for every firm). If pooling is parameters for every firm). If pooling is rejected for both sets of parameters, we rejected for both sets of parameters, we hypothesize that the constants differ for hypothesize that the constants differ for each firm, while the slope coefficients are each firm, while the slope coefficients are the samethe same

Next, consider a random specification for the Next, consider a random specification for the individual constantsindividual constants

Hsiao, Cheng Hsiao, Cheng Analysis of Panel DataAnalysis of Panel Data New York: Cambridge University Press, New York: Cambridge University Press, 1986.1986.

it i t it ity x