Continuing the analysis of Section 14.3.2, we find that a trans log cost function for one output and three factor inputs that does not impose constant returns to scale is

ln C = α + β1 ln p1 + β2 ln p2 + β3 ln p3 + δ11½ ln2 p1 + δ12 ln p1 ln p2 + δ13 ln p1 ln p3 + δ22 ½ ln2 p2 + δ23 ln p2 ln p3 + δ33 ½ ln2 p3 +γy1 ln Y ln p1 + γy2 ln Y ln p2 + γy3 ln Y ln p3 + βy ln Y + βyy½ ln2 Y + εc. The factor share equations are

S1 = β1 + δ11 ln p1 + δ12 ln p2 + δ13 ln p3 + γy1 ln Y + ε1,

S2 = β2 + δ12 ln p1 + δ22 ln p2 + δ23 ln p3 + γy2 ln Y + ε2,

S3 = β3 + δ13 ln p1 + δ23 ln p2 + δ33 ln p3 + γy3 ln Y + ε3.

a. The three factor shares must add identically to 1. What restrictions does this requirement place on the model parameters?

b. Show that the adding-up condition in (14-39) can be imposed directly on the model by specifying the Trans log model in (C/p3), (p1/p3), and (p2/p3) and dropping the third share equation. Notice that this reduces the number of free parameters in the model to 10.

c. Continuing Part b, the model as specified with the symmetry and equality restrictions has 15 parameters. By imposing the constraints, you reduce this number to 10 in the estimating equations. How would you obtain estimates of the parameters not estimated directly? The remaining parts of this exercise will require specialized software. The E-Views, TSP, Stata or LIMDEP, programs noted in the preface are four that could be used. All estimation is to be done using the data used in Section 14.3.1.

d. Estimate each of the three equations you obtained in Part b by ordinary least squares. Do the estimates appear to satisfy the cross-equation equality and symmetry restrictions implied by the theory?

e. Using the data in Section 14.3.1, estimate the full system of three equations (cost and the two independent shares), imposing the symmetry and cross-equation equality constraints.

f. Using your parameter estimates, compute the estimates of the elasticities in (14-40) at the means of the variables.

g. Use a likelihood ratio statistic to test the joint hypothesis that γyi = 0, i = 1, 2, 3.