Suppose that an economic production function cobb-douglas with parameter alpha = 0.3
a) what fraction of income do capital and labour receive
b)suppose that immigration increase the labour force by 10%. What happen to total output(in percent)?, The rental price of capital ? , The real wage?
Show your working clearly.
a) To determine the fraction of income that capital and labor receive in a Cobb-Douglas production function, we need to use the parameter alpha. In this case, alpha is given as 0.3.
The production function is represented as: Y = K^alpha * L^(1-alpha)
Where:
Y = total output
K = capital
L = labor
To find the fraction of income received by capital and labor, we can use the share equation:
Share of capital = (K^alpha * L^(1-alpha)) / Y
Share of labor = (L^(1-alpha) * K^alpha) / Y
substituting the given value of alpha as 0.3, we can calculate the share of income that capital and labor receive.
b) If immigration increases the labor force by 10%, we need to calculate the impact on total output, the rental price of capital, and the real wage.
1. Total Output:
To calculate the impact on total output, we can calculate the percentage change in total output using the formula:
% Change in Total Output = (New Total Output - Old Total Output) / Old Total Output * 100
2. Rental Price of Capital:
The rental price of capital can be determined using the marginal product of capital (MPK). MPK represents the additional output that an additional unit of capital produces. In a Cobb-Douglas production function, the MPK is given by the formula:
MPK = alpha * (Y / K)
To find the impact on the rental price of capital, we can calculate the percentage change in the MPK when the labor force increases by 10%.
% Change in Rental Price of Capital = (New MPK - Old MPK) / Old MPK * 100
3. Real Wage:
The real wage can be determined using the marginal product of labor (MPL). MPL represents the additional output that an additional unit of labor produces. In a Cobb-Douglas production function, the MPL is given by the formula:
MPL = (1 - alpha) * (Y / L)
To find the impact on the real wage, we can calculate the percentage change in the MPL when the labor force increases by 10%.
% Change in Real Wage = (New MPL - Old MPL) / Old MPL * 100
By substituting the appropriate values into the formulas above, we can calculate the impact on total output, the rental price of capital, and the real wage when the labor force increases by 10%.