questions

Question 5 a) Write down the Social Planner's problem, letting a e (0, 1) denote the Pareto weight that the planner assigns on the typical agent of country 1. The planner wishes to maximize the expression [ lastin(c!) + (1 - a) Byin(@)]. subject to c + c = e for all t, by choosing sequences of consumptions for the typical agent of each country. The Langrangian function for this problem is L = [asjin(c!) + (1 - a) Bin(c)] + Em(e-q -). 1=0 1=0 where , is the Langrangian multiplier for period t (there will be infinitely many such multipliers). The first order conditions withe respect to c and c, respectively, are given by: (1 - a)35 = and combining these we obtain the following relationship between consumption in the two countries: (9) But we also know that the consumptions must obey the feasibility constraint: c +c = e, for all t. Combining these facts, we can obtain the following result: For all t, the consumption of the typical agent in the two countries is given by:3. (20) Consider a discrete time variation of the Sidrauski monetary model with a constant population. Specifically, assume that the representative agent's lifetime utility is given by: Is [u(a)+ v (#) ] where U () and V () are concave, twice-differentiable functions, c denotes consumption and Me is money chosen in period t. Each period, agents use beginning of period nominal balances, the revenue from sales of output and a lump-sum monetary transfer to purchase consumption, investment and new money, In contrast to the Sidrauski model, both capital and money are used as inputs into the production process. Letting y denote output, the production function is given by: W = (1 -=( #))( K) where ?' (.) 0, = (0) = 1, lim =(M/P) = 0. The function f (k,) has standard properties. The money supply in this economy is growing at the constant rate # > 0 and capital depreciates at the constant rate of 6