Question

Hello, I was wondering if someone could show me the process as to how to answer this question. The answers were posted by the teacher assistant however I do not actually know how to solve the question: The production function of a firm is given as ? =K 1/2, L1/2 , capital is 200. The rental rate for both labor and capital is 5 and 20. There are total 20 firms that produce the same good. A. Write down the short run production function.

B. Write down the labor output relationship equation.

C. Write down the total cost function.

D. Compute the marginal cost function, that is to say, MC, which is a function of q.

E. Derive the supply function.

F. compute the market supply function.

G. Given market demand function, Q=20- 2p. Calculate the equilibrium price and quantity.

H. Calculate the profit for a firm.

I. find out the equilibrium employment.

Answer 1

A)

q = K^1/2L^1/2

K = 200

q = (200)^1/2L^1/2  

So short run production function is q = (200)^1/2L^1/2  

B)

q = (200)^1/2L^1/2  

q/(200)^1/2 = L^1/2  

squaring on both sides

q²/200 = L

So labor output relationship is

L = q²/200  

C)

Total cost is

C = wL + rK

C = 5L + 20K

C = 5(q²/200 ) + 20(200)

C = q²/40 + 4000

D)

C =  q²/40 + 4000

dC/dq = MC = 2q/40 + 0

MC = q/20

E)

supply curve of a firm is given by

p = MC  

p = q/20

20p = q

So supply function is q = 20p  

F)

Since there are 20 firms so market supply

Qs = 20q

Qs = 20(20p)

Qs = 400p

G)

Qd = 20 - 2p

Qd = Qs (in equilibrium)

20 - 2p = 400p

20 = 400p + 2p

20 = 402p

10 = 201p

p = 10/201

Q = 400p

Q = 400(10/201)

Q = 4000/201

Equilibrium price and quantity are p = 10/210 and Q = 4000/201

H)

we have calculated

q = 20p

q = 20(10/201)

q = 200/201

so quantity produced by a firm is q = 200/201

Profit of a firm

= pq - TC  

= pq - q²/40 - 4000

= (10/201)(200/201) - 1/40(200/201)² - 4000

= 2000/40401 - 1000/40401 - 4000

= 1000/40401 - 4000

= 0.024 - 4000

= - 3999.976

I)

L = q²/200 from part (B)

q = 200/201

L = (1/200)(200/201)²  

L = 200/40401

equilibrium employment is L = 200/40401