Question

Suppose the production function of a firm is given by q=L^1/4 K^1/4. The prices of labor and capital are given by and w=10 and r=20, respectively.

1. Write down the firm’s cost minimization problem.
2. What returns to scale does the production function exhibit? Explain.
3. What is the Marginal Rate of Technical Substitution  (MRTS) between capital and labor?
4. What is the optimal capital to labor ratio? Show your work.

Answer 1

Ans a.) As we know that the ,

Wage (w) = 10

Rent of capital (r) = 20

q = L^1/4K^1/4

Total cost of Production = wL + rK

Where ,

L = labor units

K = capital units

A firm need to minimize its cost of production for a given level of output which is as follows:

min.TC = wL + rK st. q = L^1/4K^1/4

M = 10L + 20K + (q = L^1/4K^1/4)....eq(1)

The eq(1) represents the cost minimization problem for a given level of output and input price.

Ans b.) To find the return to scale , we multiply the inputs by a certain proportion say 's'

q(sL, sK) = (sL)^1/4(sK)^1/4

q(sL, sK) = s^1/2L^1/4K^1/4

As the power of 's' is 1/2 which is less than 1. Therefore, the production function exhibits decreasing returns to scale.

Ans c.) Marginal rate of technical substitution is the change in the units of capital required when the labor changes by one unit keeping the level of output constant.

Mathematically,

MRTS = = MP_L / MP_K

Where,

MP_{L =} marginal product of labor

MP_K = marginal product of capital

MU_L = d(y)/d(L)

= 1/4(L)^{(1/4- 1)}K^1/4

MU_L = 1/4L^-3/4K^1/4

Similarly,

MU_K = d(y)/d(K)

= 1/4L^1/4K^-3/4

MRTS = (1/4L^-3/4K^1/4) ÷ (1/4L^1/4K^-3/4)

MRTS =  K/L

  

Ans d.) The optimal capital-labor ratio is hired where,

MRTS = ratio of input price = w/r

MU_L/MU_K = w/r

K/L = 10/20

K/L = 1/2

The optimal capital-labor ratio is 1/2.