Question

1). Consider the production function Q= L0.4 * K0.1 The price of labor is 8, the price of capital is 2, and the price of output is 160. The firm incurs $80 in fixed costs.

a.) Find the marginal product of labor.

b.) Find the marginal product of capital.

c.) Find MRTS.

d.) find the relationship between optimal level of capital and labor to be used.

e.) find the total cost function.

f) find the marginal cost.

g) what output will this firm produce?

Answer 1

Q= L^0.4 K^0.1

Price of labor is 8

Price of capital is 2

a)

Marginal product of labor= differentiation of Q wrt L= 0.4L^-0.6 K^0.1

b)

Marginal product of Capital= 0.1L^0.4 K^-0.9

c)

MRTS= marginal product of labor/marginal product of capital= 4K/L

d)

Optimal quantity of L and K arises where:

MRTS= Price of L/Price of K

4K/L = 8/2

8K= 8L

K= L Relationship between K and L at optimal situation

e)

Total cost function:

Total cost= L x Price of L+K x price of K+fixed cost

Total cost= 8L+2K+80

Use K=L in Q:

Q= L^0.4 L^0.1

Q= L^0.5

Squaring both sides:

L= Q²

K=Q² (From L=K)

Use L= Q² and K= Q² in total cost:

Total cost= 10Q² +80 Total cost function

f)

Marginal cost= differentiation of total cost wrt Q= 20Q

g)

Firm will produce output where:

Marginal cost = Marginal revenue (when price is constant it is equals to Marginal revenue)

20Q= 160

Q= 8 output firm will produce