04 Jun Question 1. For each of the following production functions, state and show whether it is Constant/Increasing/or Decreasing Retur
Question
1. For each of the following production functions, state and show whether it is Constant/Increasing/or Decreasing Returns to Scale: [Hint: to show Constant/Increasing/or Decreasing Returns to Scale, see how production changes when all inputs increase by a factor of ? where ? > 1. ]
(a) F(K,L) = K^1/3 L^2/3 + K + L has___________returns to scale.
(b) F(K,L) = 5(K^1/2?*3?L^1/2? has__________ returns to scale.
(c) F(K,L) = K^1/3 L^2/3 + 2 has _________returns to scale.
(d) F(K,L) = (K^1/3 + L^1/3)^2 has________returns to scale.
2. Using the same production function from question 1d, and given wages w = 3 and rental rate of capital r = 6:
(a) solve for the cost function in terms of q when the firm is free to choose both labor and capital. (You will need to solve the cost minimization problem and find the optimal K,L in terms of q in order to find cost in terms of q).
(b) Solve for the marginal cost function using your result from part a).
(c) Suppose p = 30 and that the firm is a price taker. What is the optimal amount of output the firm should produce?
3. Suppose there is a fixed amount of capital K¯ = 10. Find the short run cost function CFK(q) when the wage is 6 and the rental rate of capital is 3 for a firm whose production function is F(K,L) = K^3/5L^2/5.
What does this say about the marginal return to labor as you increase the amount of labor used?
4. Using the production function: F(K,L) = K^1/3L^2/3:
(a) Solve for the firm’s maximum profit in terms of w, r and p.
(b) Suppose w=2, r=4 and p=10. How much labor and capital will the firm use? How much output will the firm produce? What will the firm’s profit be?
• L =
• K =
• q =
• profit =
(c) Suppose w=5, r=5 and p=8. How much labor and capital will the firm use? How much output will the firm produce? What will the firm’s profit be?
• L =
• K =
• q =
• profit =
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