Consider the production function

Y = AKαL β

where Y represents output, K and L represent the inputs of capital and labor, and A is total factor productivity. The coefficients α and β are positive.

a. Show that this production function exhibits constant returns to scale if (and only if) α + β = 1.

b. Consider an economy with perfectly competitive firms. Assume all the firms have the above production function with α+β = 1. Show that the share of labor income in output is constant (i.e. how much of output goes to labor). Also, define and calculate the share of capital in output (i.e. how much of output goes to capital).

c. Show that for the economy of Part b, the marginal product of labor (MPL) is proportional to the average product of labor, and the constant of proportionality is the share of labor. Also, show that MPL can be expressed as a function of the capital-labor ratio.