Question

Suppose a firm’s production function is given by Q = L^1/2*K^1/2. The Marginal Product of Labor and the Marginal Product of Capital are given by:

MP_L= K^1/2 / 2L^1/2 & MP_K = L^1/2 / 2K ^1/2

a) If the price of labor is w = 48, and the price of capital is r = 12, how much labor and capital should the firm hire in order to minimize the cost of production if the firm wants to produce output Q = 18?

b) What is the firm’s Total Cost function TC(Q)?

c) What is the firm’s marginal cost of production?

Please in detail show you work step by step.

Answer 1

As per the question the production function of a firm Q= L^1/2K^1/2

Marginal Product (MP) of Labour = K^1/2/ 2L^1/2

Marginal Product (MP) of Capital = L^1/2/ 2K^1/2

Marginal rate of technical substitution (MRTS) = MP of Labour / MP of Capital = (K^1/2/ 2L^1/2)/( L^1/2/ 2K^1/2) = 2K/2L = K/L

(A) if price of capital (r) = 12 and price of labour (w) =48, output (Q) =18

At cost minimising level of output or equilibrium level of output the MRTS = Price of labour (w) / Price of capital (r)

K/L=48/12

K/L=4

K=4L    and L=K/4

As per production function Q= L^1/2K^1/2

For producing 18 units of output the production function will be

18 = L^1/2K^1/2   (replacing the value of K =4L)

18 = L^1/2(4L)^1/2

2L=18

L=9

For producing 18 units of output the production function will be

18 = L^1/2K^1/2   (replacing the value of L=K/4)

18 = (K/4)^1/2K^1/2

K/2=18

K=36

Equilibrium units of labour (L) =9

Equilibrium units of Capital (K) =36

(B) As per the question the production function Q= L^1/2K^1/2

At cost minimising level of output or equilibrium level of output where, MRTS = Price of labour (w) / Price of capital (r)  

So we got

K=4L    and L=K/4

For producing Q level of output

Q= L^1/2K^1/2           (replacing the value of K =4L)

Q= L^1/2(4L)^1/2

Q= 2L   

L=Q/2

For producing Q units of output

Q = L^1/2K^1/2   (replacing the value of L=K/4)

Q = (K/4)^1/2K^1/2

K/2=Q   

K= 2Q

Total cost(TC) = price of labour (w) x units of labour(L) + price of capital (r) x units of capital(K)

TC for producing Q level of output

TC(Q) = 48(L) + 12(K)

TC(Q) = 48(Q/2) + 12(2Q)   (replacing the value of L=Q/2 and K=2Q)

TC(Q) = 24Q+ 24Q

TC(Q)=48Q

The firms total cost of production TC(Q)=48Q

(C)Marginal cost (MC) = Change in TC / Change in output = dTC/dQ

TC(Q)=48Q

MC= dTC/dQ = 48

The firms marginal cost (MC) of production is 48