Consider a continuous time closed economy populated with infinitely lived iden- tical households. The size of the population is given by L(t) for all t 2 0. There is no technical change. The representative household has CRRA preferences with instantaneous utility cl -8 - 1 u(c) = 1 - 00 > 0 and 0 0. The discount factor is p > 0. The household supply labour and capital to firms; firms produce a final good used for investment and consumption with a constant-return to scale Cobb-Douglas production function with capital share o E (0, 1). Markets are competitive. It is assumed that 0 = a. 1. Set up the optimization problem of the household choosing consumption over time to maximize its lifetime utility, under the appropriate flow budget constraint and Non-Ponzi condition. Derive the optimality conditions of this problem and write down the transversality condition. (10 points) 2. Consider a competitive equilibrium. Derive the law of motion for con- sumption per capita and the capital stock per capital, c and k. Solve for the steady-state /balanced growth path of k = K/L and c = C/L. (15 points) 3. Find expressions for the growth rates of (a) the capital-output ratio, k(t)/y(t) and (b) the consumption-capital ratio, c(t)/k(t) in terms their levels and the parameters. (15 points) 4. Assume a constant consumption-capital ratio over time. Determine the time path of the capital-output ratio given its initial value and the path of output per capita given the initial value of k. (10 points)