Suppose that an economic production function cobb-douglas with parameter alpha = 0.3
a) what fraction of income do capital and labour receive. Show your working clearly
b) What is the marginal product of capital?
a) The fraction of income that capital and labour receive can be calculated using the Cobb-Douglas production function. The Cobb-Douglas production function is given by:
Y = K^α * L^(1-α)
Where Y is total output, K is capital, L is labour and α is the parameter.
In this case, α = 0.3.
Therefore, the fraction of income that capital receives is 0.3 and the fraction of income that labour receives is 0.7.
b) The marginal product of capital (MPK) is the change in output (Y) resulting from a one unit increase in capital (K).
MPK = (∂Y/∂K) * (1/K)
Using the Cobb-Douglas production function, we can calculate the marginal product of capital as follows:
MPK = (α * K^(α-1) * L^(1-α)) * (1/K)
MPK = α * K^(α-1) * L^(1-α)
In this case, α = 0.3.
Therefore, the marginal product of capital is 0.3 * K^(0.3-1) * L^(1-0.3) = 0.3 * K^(-0.7) * L^0.7