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} =
x^{2}**

**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**

**x ^{2} + y/4 =
100**

**Thus, equation of PPF
is**

**x ^{2} + y/4 =
100**

**slope of PPF = Rate of
product transformation(RPT) = - dy/dx = -
(-f _{x}/f_{y}) =
f_{x}/f_{y}**

**x ^{2} + 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 =
x ^{2} **

**50 =
x ^{2}**

**x = 50 ^{1/2} **

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

**option (b) is
correct.**

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