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.
To determine the fraction of income that capital and labor receive in a Cobb-Douglas production function with parameter alpha = 0.3, we need to use the formula:
Y = A * (K^alpha) * (L^(1-alpha))
where:
- Y represents total output
- A represents total factor productivity
- K represents capital input
- L represents labor input
- alpha represents the capital's share of income (0 < alpha < 1)
a) Fraction of income received by capital:
Since alpha represents the capital's share of income, the fraction of income received by capital is equal to alpha. In this case, alpha = 0.3, so capital receives 30% of the total income.
b) Impact of immigration on total output, rental price of capital, and real wage:
To analyze the impact of a 10% increase in the labor force due to immigration, we need to adjust the labor input (L). Let's assume the original labor force size is L0, and the increased labor force size is L1 = L0 + 0.1 * L0 = 1.1 * L0.
Total output:
To calculate the impact on total output, we compare the original and increased labor inputs while keeping other factors constant. Using the Cobb-Douglas production function formula, we can calculate the new total output (Y1) with the increased labor input (L1) and compare it to the original total output (Y0) with the original labor input (L0).
The percentage change in total output is given by:
[(Y1 - Y0) / Y0] * 100%
Rental price of capital:
The rental price of capital can be determined by taking the marginal product of capital (MPK) and multiplying it by the total output (Y). The formula is:
Rental price of capital = MPK * Y
The percentage change in the rental price of capital can be computed using the formula:
[(Rental price of capital1 - Rental price of capital0) / Rental price of capital0] * 100%
Real wage:
The real wage can be obtained by dividing the total output (Y) by the labor input (L). The formula is:
Real wage = Y / L
The percentage change in the real wage can be calculated using:
[(Real wage1 - Real wage0) / Real wage0] * 100%
By using these formulas and plugging in the appropriate values for Y0, L0, L1, and the Cobb-Douglas production function, we can evaluate the impacts of immigration on total output, rental price of capital, and real wage.