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