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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 r |
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