Assume that you have data for a cross section of firms that produce the same output ( Y) using four inputs: land (L ), labour (M ), Energy ( E), and capital (K ). In addition to having data on the quantities of output produced and inputs used, you also have data on input prices, rL ,rM ,rE, and rK, respectively.
a. Formulate algebraically both a Cobb-Douglas cost function, and an Translog cost function that might be used to estimate conditional input demand functions and to test hypotheses about the elasticity of scale and the substitution between pairs of inputs.
b. Write down the estimating equations for each of the cost functions in part a). Be sure to discuss any appropriate restrictions on the parameters in each model. Please explain how many free parameters you have to estimate in the restricted models.
c. Suppose that you have an index of the quality of land (Lh ), for each observation in your data. Can you reformulate the Translog cost model in part a) so that the quality of land will affect the demand for the four inputs? If so, how?