Question

# A country has 100 units of labor (L) and production functions x = (Lx)^0.5 and y...

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

The production function of x is

x = Lx0.5

x1/0.5 = Lx

Lx = x2

The production function of y is

y = 4Ly

y/4 = Ly

Ly = y/4

since country has 100 units of labor therefore

Lx + Ly = 100

x2 + y/4 = 100

Thus, equation of PPF is

x2 + y/4 = 100

slope of PPF = Rate of product transformation(RPT) = - dy/dx = - (-fx/fy) = fx/fy

x2 + y/4 = 100

fx = 2x

fy = 1/4

slope of PPF = RPT =    fx/fy = 2x/(1/4) = 8x

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

Lx = 50

Ly = 50

but Lx = x2

50 = x2

x = 501/2

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

option (b) is correct.

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