Question

The following table gives data on output and total cost of production of a commodity in the short run. (See Example 7.4.) Output Total cost, $ 1 193 2 226 3 240 4 244 5 257 6 260 7 274 8 297 9 350 10 420 To test whether the preceding data suggest the U-shaped average and marginal cost curves typically encountered in the short run, one can use the following model: Yi = β1 + β2Xi + β3X2 i + β4X3 i + ui where Y = total cost and X = output. The additional explanatory variables X2 i and X3 i are derived from X. a. Express the data in the deviation form and obtain (X
X), (X
y), and (X
X) −1 . b. Estimate β2, β3, and β4. c. Estimate the var-cov matrix of βˆ. d. Estimate β1. Interpret βˆ 1 in the context of the problem. e. Obtain R2 and R¯ 2 . f. A priori, what are the signs of β2, β3, and β4? Why? g. From the total cost function given previously obtain expressions for the marginal and average cost functions. h. Fit the average and marginal cost functions to the data and comment on the fit. i. If β3 = β4 = 0, what is the nature of the marginal cost function? How would you test the hypothesis that β3 = β4 = 0? j. How would you derive the total variable cost and average variable cost functions from the given data?