Question

A country has 100 units of labor (L) and production functions x = (Lx)^0.5 and y = 4Ly, where Lx and Ly describe the labor allocation. When the country divides its labor equally between producing x & y, what is the rate of product transformation (slope) for the PPF?

a. 49.3

b. 56.7

c. 61.1

d. 71.0

Answer 1

The production function of x is

x = L_x^0.5

x^1/0.5 = L_x  

L_x = x²

The production function of y is

y = 4L_y  

y/4 = L_y  

L_y = y/4

since country has 100 units of labor therefore

L_x + L_y = 100

x² + y/4 = 100

Thus, equation of PPF is

x² + y/4 = 100

slope of PPF = Rate of product transformation(RPT) = - dy/dx = - (-f_x/f_y) = f_x/f_y

x² + y/4 = 100

f_x = 2x

f_y = 1/4  

slope of PPF = RPT =    f_x/f_y = 2x/(1/4) = 8x

since country divides its labor equally between producing x and y therefore

L_x = 50

Ly = 50

but Lx = x²  

50 = x²

x = 50^1/2  

Hence slope of PPF = 8x = 8(50)^1/2   = 56.5688  

option (b) is correct.