# Exams

Please answer all questions. Each short question is worth 10% of the grade and each long question is worth 30%. Good luck.

## Short Questions

1. Suppose that

$$Y(t) = \text{exp}(g_At)F(\text{exp}(g_Kt)K(t), \text{exp}(g_Lt)L(t))$$,

where $$F$$ exhibits constant returns to scale. Suppose that $$\dot{L}(t)/L(t) = n$$ and $$\dot{K}(t) = sY(t)$$. Suppose also that $$F$$ is not Cobb-Douglas (more specifically, suppose the share of labor is not constant as the effective capital-labor ratio $$\text{exp}(g_K(t))K(t) / \text{exp}(g_L(t))L(t)$$ changes). Show that balanced growth, where output grows at a constant rate, is only possible if $$g_K = g_A = 0$$.

2. Consider the following overlapping generations model with competitive markets. There are $$N$$ generations, each of which lives for two periods. Agents from generation $$i$$ supply labor at time $$t = i$$ and live off