In Problem we introduced the Cobb-Douglas production function of the form q = Ka Lb. The cost function that can be derived from this production function is: TC = Bq1/(a+b)va/(a+b)wb/(a+b), where B is a constant, and v and w are the costs of K and L, respectively.
a. To understand this function, suppose a = b = 0.5. What is the cost function now? Does these function exhibit constant returns to scale? How ‘‘important’’ are each of the input prices in this function?
b. Now return to the Cobb-Douglas cost function in its more general form. Discuss the role of the exponent of q? How does the value of this exponent relate to the returns to scale exhibited by its underlying production function? How do the returns to scale in the production function affect the shape of the firm’s total cost curve?
c. Discuss how the relative sizes of a and b affect this cost function. Explain how the sizes of these exponents affect the extent to which the total cost function is shifted by changes in each of the input prices.
d. Taking logarithms of the Cobb-Douglas cost function yields ln TC = ln B +[1/(a + b)] ln q+[a/(a + b)] ln v + [b/(a + b)] ln w. Why might this form of the function be especially useful? What do the coefficients of the log terms in the function tell you?
e. The cost function in part d can be generalized by adding more terms. This new function is called the ‘‘Tran slog Cost Function,’’ and it is used in much empirical research. A nice introduction to the function is provided by the Christenson and Greene paper on electric power generation references in Table 1 of Application 7.3. The paper also contains an estimate of the Cobb-Douglas cost function that is of the general form given in part d.
Can you find this in the paper?