Suppose a firm has the following production function: Q(L,K) = (L*K)1/4 where L

Question

Suppose a firm has the following production function: Q(L,K) = (L*K)1/4 where L is labor and K is capital. Suppose that W = 3 (price of one unit of labor) and R = 5 (price of one unit of capital). a) Please derive the expenditure-minimizing (L*, K*) if the firm wants to produce 10 units of the product. b) Please derive the expenditure-minimizing (L*, K*) if the firm wants to produce 20 units of the product. c) Please derive the expenditure-minimizing (L*, K*) if the firm wants to produce 30 units of the product. d) Based on your results in a), b), and c), please plot the firm’s expansion path. e) Based on your results in a), b), and c), please plot the firm’s cost function. Does this cost function display increasing, decreasing, or constant marginal cost? Please explain. f) Does this production function have increasing, decreasing, or constant returns to scale? Please explain. g) What is the relationship between your answers from parts e) and f)? Please explain.

Explanation / Answer

Production function: Q( L, K) = (L*K)1/4

W = 3, R= 5

Cost is minimized when MRTS = W/R ,( Assuming labor is on x-axis)

MRTS = (¼)*(K)1/4*(L)-3/4/(¼)*(L)1/4*(K)-3/4

MRTS = K/L

So at equilibrium

K/L = 3/5

K = (3/5)*L

Substitute K in the production function.

Q = (3/5)1/4*L1/2

L = (5/3)1/2*Q2

K = (3/5)1/2*Q2

For Q = 10. L* = 129.1, K* = 77.46

b) For Q = 20

L* = (5/3)1/2*(20)2 = 516.40

K* = (3/5)1/2*(20)2 = 309.84

c) For Q = 30 units

L* = (5/3)1/2*(30)2 = 1161.90

K* = (3/5)1/2*(30)2 = 697.14

*For solution to other parts please post as a different question. Can solve only these many sub-parts.

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