Q2: (28 marks) Your production function is given by Y = AK$^\alpha$L$^{1-\alpha}$, where Y is gross
domestic product, K is the stock of capital, L is labor, A represents the productivity of labor,
and \alpha is the elasticity of output with respect to capital.
(a) Assuming constant returns to both labor (L) and capital (K) jointly, derive an expression for
gross domestic product per capita (y). (4 marks)
(b) When the depreciation rate increases, what happens to steady state capital per capita and
GDP per capita? Explain using your answer both in words and using a graph. (4 marks)
(c) When the labor growth rate decreases, what happens to steady state capital per capita and
GDP per capita? What happens to growth of GDP per capita in long run? Explain using
your answer both in words and using a graph. (4 marks)
(d) Suppose the national saving rate is 18%, the labor growth rate is 4%, the depreciation rate
is 5% the productivity of labor (A) is 1, and the elasticity of output with respect to capital
(\alpha) is 0.5. What are the steady-state capital per capita, gross domestic product per capita? (4
marks)
(e) All the conditions are the same to (d). When the total labor is 1M, what is K$_0$? If the saving
rate increases to 27%, what are the steady-state capital per capita(k)? What should be K$_1$?
What is the total capital change? how much of capital is needed for capital widening? How
much of capital is needed for capital deepening? (12 marks)
