Consider a production function of Cobb-Douglas form: F(L,K)= LOK, for some a, (0, 1). (a) Plot...

  

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Consider a production function of Cobb-Douglas form: F(L,K)= LOK, for some a, (0, 1). (a) Plot the isoquant of F. (b) Derive that technical rate of substitution of F. Does Fexhibit diminishing technical rate of substitution? (e) Does F exhibit diminishing marginal productivity of labor? What about marginal productivity of capital? (d) Find out the conditions for a and 3 such that F is increasing return to scale, decreasing return to scale and constant return to scale, respectively. (e) Suppose that F does not exhibit constant return to scale, solve the firm's profit maximization problem with prices p, w, r: max pF (L,K)-wL-rK LK20 and express the optimal amount of labor and capital in terms of p, w, r. How does the optimal L change as w increases and as r increases. Give an economic interpretation of your answer. Now suppose that F exhibits constant return to scale and that the market price for the output is 1. (f) Use your answer in (d), rewrite 3 in terms of a. (g) Write down the first order condition for the firm's profit maximization problem. max F(L,K)-wL-TK. L.K20 (h) Can you solve the first-order conditions in (g)? Why or why not? If not always, derive a condition on the relationship between w and r such that the first-order condition in (g) does have solution(s). (i) For a firm with constant return to scale technology, the only possibility for it to have a profit maximizing solution is that the optimal profit is zero. Verify this observation with your answers above (Hint: Use the first order condition to rewrite the objective function). When does this happen? Consider a production function of Cobb-Douglas form: F(L,K)= LOK, for some a, (0, 1). (a) Plot the isoquant of F. (b) Derive that technical rate of substitution of F. Does Fexhibit diminishing technical rate of substitution? (e) Does F exhibit diminishing marginal productivity of labor? What about marginal productivity of capital? (d) Find out the conditions for a and 3 such that F is increasing return to scale, decreasing return to scale and constant return to scale, respectively. (e) Suppose that F does not exhibit constant return to scale, solve the firm's profit maximization problem with prices p, w, r: max pF (L,K)-wL-rK LK20 and express the optimal amount of labor and capital in terms of p, w, r. How does the optimal L change as w increases and as r increases. Give an economic interpretation of your answer. Now suppose that F exhibits constant return to scale and that the market price for the output is 1. (f) Use your answer in (d), rewrite 3 in terms of a. (g) Write down the first order condition for the firm's profit maximization problem. max F(L,K)-wL-TK. L.K20 (h) Can you solve the first-order conditions in (g)? Why or why not? If not always, derive a condition on the relationship between w and r such that the first-order condition in (g) does have solution(s). (i) For a firm with constant return to scale technology, the only possibility for it to have a profit maximizing solution is that the optimal profit is zero. Verify this observation with your answers above (Hint: Use the first order condition to rewrite the objective function). When does this happen?

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a Plotting the isoquant of F An isoquant represents all the combinations of labor L and capital K that yield the same level of output F Since the production function is of CobbDouglas form the isoquan... View the full answer