1. Consider a production technology that takes three inputs: capital (K), labor (N ), and materials (M ). The production function is given by Y = AKαNβ Mγ , where α, β, γ > 0 and α + β + γ = 1.  For this production function,

(a) show that it exhibits constant returns to scale;

(b) show that it satisfies the Inada conditions for materials;

(c) derive a corresponding function in per capita terms (per unit of labor);

(d) derive a corresponding growth accounting equation;

(e) assume α = 2β = 2γ. Suppose you observe that over a period of a year output grew by 5%, capital increased by 4%, labor declined by 2%, and materials grew by 5%. What fraction of output growth came from the technological change and what was the growth rate of A?

2. Given the law of motion for the aggregate capital stock in the Solow growth model with technological progress, K efficient worker (k˜).

= sKY - δK, derive a law of motion for capital per

3. Explain why a saving rate above the golden rule saving rate creates dynamic ineffi- ciency.

4. Explain what the notion of "conditional convergence" is supposed to capture.

5. Consider an inflow of immigrants (a temporary jump in the population growth rate) in the Solow growth model without technological change.

(a) What are the effects on the steady state capital per worker, output per worker, consumption per worker? Use a diagram in your answer.

(b) How are these variables affected in the short run? Draw and explain the transition paths for

i. the growth rate of capital per worker;

ii. output per worker

iii. aggregate savings;

iv. the growth rate of aggregate output.