1. Consider an infinite horizon model where time is indexed by t = 0, 1, 2, ..., o. Each period, Lo two- period-lived consumers are born, and each is endowed with one unit of labor in the first period of life, and zero unites in the second period. In period 0, there are some old consumers alive who life for one period and collectively endowed with Ko units of capital. Preferences for a consumer born in period t are given by u (c, c+1) = log c + Blog +1. Young agents supply their labor to firms and old agents rent capital to firms in perfectly competitive markets. The representative firms operate via Cobb-Douglas production function Y = KL- where > 0 and a (0, 1). The first-order conditions for a profit maximization are FK (K, L) = rt and FL (K, L) = wt. Since F(,) is homogeneous of degree one, we can rewrite these conditions as f'(k) = r f(k) - kf'(k)= w. Capital will depreciate with = 1. (a) Define and solve a life-time utility maximization problem of a young agent born in period t. 1. Consider an infinite horizon model where time is indexed by t = 0, 1, 2, ..., o. Each period, Lo two- period-lived consumers are born, and each is endowed with one unit of labor in the first period of life, and zero unites in the second period. In period 0, there are some old consumers alive who life for one period and collectively endowed with Ko units of capital. Preferences for a consumer born in period t are given by u (c, c+1) = log c + Blog +1. Young agents supply their labor to firms and old agents rent capital to firms in perfectly competitive markets. The representative firms operate via Cobb-Douglas production function Y = KL- where > 0 and a (0, 1). The first-order conditions for a profit maximization are FK (K, L) = rt and FL (K, L) = wt. Since F(,) is homogeneous of degree one, we can rewrite these conditions as f'(k) = r f(k) - kf'(k)= w. Capital will depreciate with = 1. (a) Define and solve a life-time utility maximization problem of a young agent born in period t