Question

3. Consider the production function Q = K²L , where L is labor and K is capital.

a.[4] What is the Marginal Product of Capital for this production function? Is it increasing, decreasing, or constant? Briefly explain or show how you arrived at your answer.

b.[4] Does this production function exhibit increasing, constant or decreasing returns to scale? Briefly explain or show how you arrived at your answer.

c.[5] If the firm has capital fixed at 15 units in the short run and the firm must produce 8,000 units of the good, find the cost-minimizing quantity of Labor. If labor is paid $200 and capital is rented at $400, what is the Total Cost at this short run equilibrium?

d.[10] Find the long-run cost minimizing quantities of Labor and Capital when labor is paid $200, Capital is rented at $400, and the firm must produce 8,000 units of the good. What is Total Cost at the long run equilibrium?

4. Consider the production function for a blueprint, B that can be produced using either 1/2 hour of computer time (C) or three hours of a manual draftsman’s time, D.

a.[4] Which of the following equations represents the production function for blueprints? Choose either B = ½ C + 3 D, or B = 2 C + D.

b.[4] Does this production function exhibit increasing, constant or decreasing returns to scale? Briefly explain or show how you arrived at your answer.

c.[3] On the axes given, draw the isoquant that produces exactly 20 Blueprints.

d.[3] On the same graph, draw at least one isocost line (including the cost minimizing one) if the price of computer time is $10 per hour and the price of a draftsman’s time is $5 per hour.

e.[5] What is the cost minimizing bundle of computer and draftsman time? What is the total cost at that equilibrium?

5. Consider a firm with total cost: TC = 10Q³ – 100 Q² + 300Q, where Q is output.

a.[4] Find the equations for Marginal Cost and Average Total Cost.

b.[3] What is the efficient scale of production for this firm?

c.[3] At which quantities does this firm exhibit economies of scale?

answer as many as possible please. Thank you

Answer 1

(3)

Production function: Q = K²L

(a)

Marginal product of capital (MPK) = Q/K = 2KL

With increase in K, the value of (KL) increases, hence MPK increases. The MPK function is increasing in K.

(b)

When both inputs are increased N times, new production function is:

Q* = (NK)²(NL) = N² x K² x N x L = N³ x K²L = N³ x Q

Q*/Q = N³ > N

Since increasing both inputs by N times increases output by more than N times, there is increasing returns to scale.

(c)

Plugging in K = 15 in short-run production function with Q = 8,000:

(15)² x L = 8,000

225L = 8,000

L = 35.56

Short run Total cost ($) = wL + rK = 200 x 35.56 + 400 x 15 = 7,112 + 6,000 = 13,112

(d)

Long run cost minimization condition is: MPL/MPK = w/r = 200/400 = 1/2

MPL = Q/L = K²

MPK = 2KL (from part a)

MPL/MPK = (K²) / (2KL) = K/2L = 1/2

2L = 2K

L = K

Plugging in production function with Q = 8,000:

(L)²L = 8,000

L³ = 8,000

L = 20

K = L = 20

Long run Total cost ($) = 200 x 20 + 400 x 20 = 4,000 + 8,000 = 12,000

NOTE: As per Answering Policy, 1st question is answered.