Question

Consider the production function below.

?(?, ?) = ?^1/2 + ?^1/2

a) Find the demand for labor and capital
b) Draw the demand curve for labor
c) Does the production function exhibit diminishing marginal returns of labor?
d) Is the production function exhibiting increasing, constant or decreasing returns to scale?

Answer 1

a).

Here the production function is, => Y = F(K, L) = K^0.5+L^0.5. So, the marginal productivity of both inputs are given below.

=> MPL = dY/dL = 0.5*L^(-0.5), => the demand for labor is given by, => W/P = MPL.

=> W/P = 0.5*L^(-0.5), => 2W/P = L^(-0.5), => P/2W = L^0.5, => L = (P/2W)^2.

=> MPK = dY/dK = 0.5*K^(-0.5), => the demand for capital is given by, => R/P = MPK.

=> R/P = 0.5*K^(-0.5), => 2R/P = K^(-0.5), => P/2R = K^0.5, => K = (P/2R)^2.

So, the above two equations are the demand for labor and capital respectively.

b).

The following fig shows the demand for labor.

Here we have measured “L” on the horizontal axis and “W/P” on the vertical axis. So, the downward sloping curve Ld shows the demand for labor.

c).

Here the MPL is given by, => MPL = 0.5*L^(-0.5) > 0.

=> dMPL/dL = 0.5*(-0.5)*L^(-0.5-1) = (-0.25)*L^(-1.5) < 0, => the production function exhibits diminishing MPL.

d).

Here the production function is given by, => Y = F(K, L) = K^0.5 + L^0.5. Now, if we increase both the input with proportion “t” then we have.

=> F(tK, tL) = (tK)^0.5 + (tL)^0.5 = t^0.5*[K^0.5 + L^0.5] = t^0.5*Y, => “Y” increases by less than “t”, => the production function exhibits DRS.