Question

Suppose the production function in medieval Europe is Y 5 K 0.5L0.5, where K is the amount of land and L is the amount of labor. The economy begins with 100 units of land and 100 units of labor. Use a calculator and equations in the chapter to find a numerical answer to each of the following questions. a. How much output does the economy produce? b. What are the wage and the rental price of land? c. What share of output does labor receive?

Answer 1

Answer:

a) Y = 500

b) Wages: 2.5

   Rental price: 2.5

   

c) labor Share of output: 0.370511713 = 37.05%

Explanation:

$Y = 4K^{0.5} \times L^{0.5}$

if K = 100 and L = 100

$Y = 5(100)^{0.5} \times (100)^{0.5}$

$Y = 50 \times 10$

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)

$ax^b = bax^{b-1}$

$Y = 5K^{0.5} \times L^{0.5}$

$Y' = 5K^{0.5} \times 0.5 L^{-0.5}$

$Y' = 2.5K^{0.5} \times L^{-0.5}$

$Y' = 2.5(\frac{K}{L})^{0.5}$

With K = 100 and L = 100

$Y' = 2.5(\frac{(100)}{(100)})^{0.5}$

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)

$Y = 5K^{0.5} \times L^{0.5}$

$Y' = 2.5K^{-0.5} \times L^{0.5}$

$Y' = 2.5(\frac{L}{K})^{0.5}$

L = 100 K = 100

$Y' = 2.5(\frac{100}{100})^{0.5}$

Y' = 2.5

c) we use logarithmic properties:

$Y = 50 \times 10$

$log500 = log(50 \times 10)$

$log500 = log50 + log10$

50 was the land while 10 the labor

2.698970004 = 1.698970004 + 1

share of output to labor: 1/2.698970004  = 0.370511713