Answer:
a) Y = 500
b) Wages: 2.5
Rental price: 2.5
c) labor Share of output: 0.370511713 = 37.05%
Explanation:
[tex]Y = 4K^{0.5} \times L^{0.5}[/tex]
if K = 100 and L = 100
[tex]Y = 5(100)^{0.5} \times (100)^{0.5}[/tex]
[tex]Y = 50 \times 10[/tex]
Y = 500
wages: marginal product of labor = value of an extra unit of labor
dY/dL (slope of the income function considering K constant while L variable)
[tex]ax^b = bax^{b-1}[/tex]
[tex]Y = 5K^{0.5} \times L^{0.5}[/tex]
[tex]Y' = 5K^{0.5} \times 0.5 L^{-0.5}[/tex]
[tex]Y' = 2.5K^{0.5} \times L^{-0.5}[/tex]
[tex]Y' = 2.5(\frac{K}{L})^{0.5}[/tex]
With K = 100 and L = 100
[tex]Y' = 2.5(\frac{(100)}{(100)})^{0.5}[/tex]
Y' = 2.5
rental: marginal product of land = value of an extra unit of land
dY/dK (slope of the income function considering K variable while L constant)
[tex]Y = 5K^{0.5} \times L^{0.5}[/tex]
[tex]Y' = 2.5K^{-0.5} \times L^{0.5}[/tex]
[tex]Y' = 2.5(\frac{L}{K})^{0.5}[/tex]
L = 100 K = 100
[tex]Y' = 2.5(\frac{100}{100})^{0.5}[/tex]
Y' = 2.5
c) we use logarithmic properties:
[tex]Y = 50 \times 10[/tex]
[tex]log500 = log(50 \times 10)[/tex]
[tex]log500 = log50 + log10[/tex]
50 was the land while 10 the labor
2.698970004 = 1.698970004 + 1
share of output to labor: 1/2.698970004 = 0.370511713
