Temple University
Department of Economics
Econometrics I
Translog Answer Key
As a specific example of the origin of the cost frontier consider the following example
Production

Cost Constraint

First Order Conditions

Substitue the demands back into the production function

Now solve for Cost to get

This is the factor price cost forntier. It is a function of input prices and output. The unknowns could be estimated using non-linear least squares. You could now do a Taylor series expansion around the quintuplet y, pl, pk, pf, pm and see that the resulting form is approximated by what you started with in the homework exercise.  Had you not known the explicit functional form you would have started with an approximation like that in the homework. 

The following is the answer key for the homework assignment.
Given the following approximation to a factor price frontier

a. Homothetic function: Due to the symmetry in the cost forntier we only need to consider a pair of the first order conditions.  

If we were to construct the ratio of these two derivatives then there would be a term involving y in both the numerator and denominator.  If the ratio to depend only on prices then the necessary restrictions are 
gfy = gky = gLy = gmy = 0.
b. If gyy = 0 then the cost function is homgeneous of degree ay in output.
c. In addition to the above there would need to be the additional restriction that ay = 1.
d. Cost shares.  To do this we just need to differentiate logrithmically.  We already have two of them above, reproduced here for completeness and followed by the other two.

Labor's share

Material's share

e. Symmetry and equality
The equality restrictions with which we need to concern ourselves are

The symmetry restrictions with which we need to concern ourselves are the fact that across the k and f shares we must observe that g12 must be the same, across the k and L shares g13 and that across the k amd m shares we must see that g14 are the same. There are more of these as well: g23, g24, g34 can all be estimated in two ways. So there are six symmetry constraints.

The matter of equality in the first five constraints and symmetry in the next six is examined using t-tests.

e.i. Sum of the intercepts
Sum of the intercepts equals one:

This is a small t-stat, so we fail to reject the null.
e.ii.  Coefficients within an equation
Sum of the price coefficients within an equation is zero is tested using the following restriction and the coefficients covariance matrices.

This is quite a large t-stat, so we reject the null.
Labor 

Do not reject the null.  The withing equation restrictions do not fail for Labor.
Energy 

This is quite a large t-stat, reject the null hypothesis.

This is quite a large t-stat, so we reject the null.
e.iii Test the symmetry restrictions.  By construction we have imposed independence between the equations.  Therefore we need only the appropriate coefficients and the corresponding variances in order to construct the t tests.
 g12 

g13 

g14 

g23 

g24 

g34 

We reject the symmetry restriction in two of the materials coefficients.
f. Derive the rest of the unknowns.  We don't need to do this since the data set was changed and we could estimate the coefficients directly.
g. and h. Derive the Allen Elasticities - These are the elasticities of substitution along an isoquant.  Are these elasticities different from zero?

 From Hirofumi Uzawa,  Production Functions with Constant Elasticities of Substitution, Review of Economic Studies, Vol. 29, No. 4 (Oct., 1962), pp. 291-299 we know that we can write the Allen elasticities as 
and 

Although not all of the symmetry restrictions are rejected, we'll construct the Allen elasticities for the multiplicity of estimates evaluated at the 1971 shares. Since the question was not explicit you might have made a different choice about the shares that you used.

K-F

g. & h. A note: In the K-F I did above I treated the share as constructed as though it were a new random variable.  Another possibility would be to state the null and alternate hypotheses as 

Ho: gij = -SiSj
H1: gij <> -SiSj

In this way you can construct the observed t in the usual way and compare it to an appropriate critical value. 

Yet another approach would use all of the share observations.  To start construct a new random variable as the average of the elasticies. 

The varaince of this random variable would be

Now compute the t-stat as usual.  For the purposes of this answer key I'll stick to the simple method shown above.

K-L

K-M

E - L

E - M

L - M

i. Own and cross price elasticities
Cross

j. and k.  Calculate the elasticities at 1971.  Are any different from zero?
Cross price elasticities
K-F

K-L

K-M

E - L

E - M

L - M

Own Price Elasticities

K 

L 

E 

M