Consider the aggregate production function, Y = K^1/3 L^2/3. Assume the macroeconomy is in competitive equilibrium where capital and labor are paid their respective marginal products.
a. Show mathematically that aggregate production function has constant returns to scale.
b. Derive the intensive form of the production function: y = f(k) = k^1/3, y = Y/L & k = K/L.
c. Show that the aggregate wage bill = (2/3)Y and the capital share of national output is 1/3.
d. Suppose in a given year population growth is 3% and the capital stock grows 6%. Show that output grows 4% and output per worker grows 1%.
e. Show that in the steady state output grows at the same rate as population.