Question

4. Suppose that the production function for haircuts is of the form ?(?,?) = (50?^1/2)(?^1/ 2), where ? is capital and ? is labour. The marginal products are ??? = 50 (1/2)(?^1/2)(?^−1/ 2), and ??? = (50(1/ 2)?^−1/2)(?^1/2).

a. What is the equation of an isoquant if labour is on the horizontal axis?

b. Calculate the technical rate of substitution of labour for capital.

c. If your current technical rate of substitution is -2 but you want to cut your use of capital by 4 units, how much more labour will you need to produce the same quantity of output?

Answer 1

a. Let Q = y where y is the output produced by a given isoquant.

So, (50?^1/2)(?^1/ 2) = y
So, (?^1/2)(?^1/ 2) = y/50
So, (KL)^1/2 = y/50
So, (K/L) = (y/50)^2
So, K/L = y²/2500
So, K = Ly²/2500 is the equation of isoquant if labor is on the horizontal axis.

b. Technical rate of substitution, RTS = MPL/MPK = [50 (1/2)(?^1/2)(?^−1/ 2)]/[(50(1/ 2)?^−1/2)(?^1/2)] = K^(1/2)+(1/2)/L^(1/2)+(1/2) = K/L
So, RTS = K/L

c. RTS = -2
This means 1 unit of capital is given up to gain 2 units of labor so that output remains same. Thus, if 4 units of capital are given up then increase in amount of labor = (2)*(4) = 8 units. So, 8 units more of labour will be needed to produce the same quantity of output.