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:
where Y = total cost and X = output. The additional explanatory variables Xf and Xf are derived from X.
a. Express the data in the deviation form and obtain (X'X), (X'y), and b. Estimate 2, 3, and 4.
c. Estimate the var-cov matrix of p.
d. Estimate fa. Interpret fa in the context of the problem.
e. Obtain R2 and R2.
f. A priori, what are the signs of fa, fa, and fa? 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 fa = fa = 0, what is the nature of the marginal cost function? How would you test the hypothesis that 3 = 4 = 0?
j. How would you derive the total variable cost and average variable cost functions from the given data?
C.10. In order to study the labor force participation of urban poor families
(families earning less than $3943 in 1969), the data in Table C.7 were obtained from the 1970 Census of Population.
a. Using the regression model Yi = fa + fa X2i + fa X3i + fa4X4i + u, obtain the estimates of the regression coefficients and interpret your results.
b. A priori, what are the expected signs of the regression coefficients in the preceding model and why?
APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL 953
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