Final answer:
a. The output of the economy is approximately 851.71 units. b. Capital's share of income α is approximately 0.333. c. Labor's share of income 1- α is approximately 0.667. d. The marginal product of capital MPK is approximately 2.703 units. e. The marginal product of labor MPL is approximately 0.865 units. f. If the economy increases the amount of capital, the capital share of income α will increase, the real rental price of capital will decrease, and the real wage will depend on the effect on labor productivity.
Step-by-step explanation:
a. To calculate the economy's output, we need to substitute the given values of capital and labor into the production function. Substituting 125 for capital and 64 for labor in the production function, we get:
Output = 100 * (125)^(1/3) * (64)^(2/3)
Calculating this using a calculator, we find that the output is approximately 851.71 units.
b. To calculate capital's share of income α, we need to find the ratio of capital income to total income. Capital income is the product of output and the share of capital in the production function. Labor income is the remainder. So, we have:
Capital's share of income α = (Capital income) / (Total income)
Capital income = Output * (Capital share) = 851.71 * (1/3) = 283.90 units
Total income = Output = 851.71 units
Capital's share of income α = 283.90 / 851.71 ≈ 0.333
Therefore, capital's share of income α is approximately 0.333.
c. Labor's share of income 1- α is the remainder of total income after deducting capital's share. So, we have:
Labor's share of income 1- α = 1 - α = 1 - 0.333 = 0.667
Therefore, labor's share of income is approximately 0.667.
d. The marginal product of capital MPK is the derivative of the production function with respect to capital. Taking the derivative of the production function, we get:
MPK = (∂Q/∂K) = (1/3) * 100 * (K^(-2/3)) * (L^(2/3))
Substituting the given values of capital and labor, we get:
MPK = (1/3) * 100 * (125^(-2/3)) * (64^(2/3)) ≈ 2.703 units
Therefore, the marginal product of capital MPK is approximately 2.703 units.
e. The marginal product of labor MPL is the derivative of the production function with respect to labor. Taking the derivative of the production function, we get:
MPL = (∂Q/∂L) = (2/3) * 100 * (K^(1/3)) * (L^(-1/3))
Substituting the given values of capital and labor, we get:
MPL = (2/3) * 100 * (125^(1/3)) * (64^(-1/3)) ≈ 0.865 units
Therefore, the marginal product of labor MPL is approximately 0.865 units.
f. If the economy increases the amount of capital:
i. The capital share of income α will increase, as capital becomes a larger factor in the production function.
ii. The real rental price of capital will decrease, as the increased supply of capital will lower its price.
iii. The real wage will depend on the effect of the increased capital on labor productivity. If the marginal product of labor increases, the real wage may increase; if it decreases, the real wage may decrease.