Question

Suppose that the production function for your firm is given by: F(L,K)=L1/2K1/2 w=$1 and r=$1. In the long-run, how many workers and capital should you hire in order to produce Q units of output? Select one: a. L=2Q; K=Q b. L=Q; K=Q2 c. L=0.5Q; K=0.5Q d. L=Q; K=Q e. None of the above

Answer 1

As per the information given in the question

Production function F(L,K)=L^1/2K^1/2

Price of labour or Wage (w) = $1

Cost of capital (r) = $1

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

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

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

Marginal rate of technical substitution (MRTS_LK) = MP of Labour / MP of Capital

MRTS_LK = (1/2K^1/2/L^1/2) / (1/2L^1/2/K^1/2)=K/L

At equilibrium or profit maximising level of output, the MRTS_LK = Price of labour (w) / Price of capital (r)

MRTS_LK = Price of labour (w) / Price of capital (r)

K/L=$1/$1

K/L=1

K=L

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

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

Q=L      

Then L=Q

To produce Q unit of output labour required is L=Q

Similarly

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

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

Q=K      

Then K=Q

To produce Q unit of output capital required is K=Q

In the longrun to produce Q level of output the worker required is L=Q and capital required is K=Q

Answer Option-D. L=Q;K=Q