Problem 1 (50 points). Consider two large open economies, Home and Foreign. Each economy is inhabited by a continuum of infinitely lived identical individuals that are grouped into an aggregate risk sharing household. There are two goods in the world: X and Y. Good X originates from the Home country, only, and good Y originates from Foreign country, only. Moreover, the goods are tradable and each country gets utility from consuming both goods. In particular, Home country's instantaneous utility function is Ut = u (It , yt ) , where r is the amount of X consumed by Home, y is the amount of Y consumed by Home, u is increasing and concave in r, and u is increasing and concave in y. Similarly, instantaneous utility in Foreign is where r* is the amount of X consumed by Foreign and y* is the amount of Y consumed by Foreign. There is no uncertainty in this economy and, furthermore, this is an endowment economy: each period's total amount of X and Y "drops from the sky" and it is known with certainty how much X and Y will be available across periods. A benevolent social planner solves the economy's utility maximization problem, which is the following. Choose It, yt, ;, y; to maximize 1= 0 where A is the weight the planner puts on the utility of Home, A* is the weight that the planner puts on the utility of Foreign, and B e (0, 1) is the (constant) subjective discount factor, such that the following aggregate resource constraints hold: Xt = It+ It and Yt = yi + yi.Using I as the notation for the multiplier on the X constraint and ? as the notation for the multiplier on the Y constraint, the benevolent social planner's present value Lagrangian (for this problem we'll be considering the present value specification) is: Bt ( XU+ + X * UF ) + V. ( X 1 - 14 - I;) + 14 (14 - ye - 37)] Suppose that ( W ( 3 :2 ) # + ( 1 - W ) ( 3 ) 7 ) -1 Ut = 1 - 0 where w E (0, 1) is expenditure share on a (in very broad terms, though, you can just think of w as a reflection of the share of consumption of each good in the utility function), o is a strictly positive parameter that reflects risk aversion, and y is a strictly positive parameter that captures the elasticity of substitution across the two goods (so y is in essence a preference parameter, reflecting how much a country likes one good relative to the other). Similarly, instantaneous utility in Foreign is 1 1-0 (w (27) 27 + (1 - W) (32)37) Ut = 1 -0 where r* is the amount of X consumed by Foreign and y* is the amount of Y consumed by Foreign. Problem la (25 points). State the planner's first order conditions (please show your work in detail). Hint/tip: When dealing with utility, using the calculus chain rule will come in handy. So, for example, you can think of Home instantaneous utility as UL = 1 - where Ct = (D. )7 and De = w (It ) + + ( 1 - W) (30) 37. Hence, for example Out Out OCt OD: art act OD, OrtProblem 1b (25 points). Let 4, E (0, 1) denote the fraction of X, that Home consumes (and 1 - 4, is the fraction of X, that Foreign consumes) so that It = D X, and = = (1 -4,) X- 3 Similarly, let At E (0, 1) denote the fraction of Y that Home consumes (and 1 - ; is the fraction of Y, that Foreign consumes) so that Ut = MYt and of = (1 -A) Y. Derive mathematically what the values of &, and At are equal to. Please show your work in detail. Hint /tip: Here is a suggested path to solve the problem (this is literally a suggestion, only: you may find that other approaches work better for you). Start by going back to the first order conditions from problem la. Then, use the first order conditions to get at mathematical statements of how one country's consumption of a good is related to another country's consumption of a good. For example, where It is the proportion of Foreign's consumption of X, x*, consumed by Home (here, It is short-hand notation for a factor of proportionality that you can arrive at). Finally, use the fact that in each period nitr = Xt and Ut + yi = Yt. Also, for the purposes of the answer to this problem, you should find that &, and A, are functions of A and 1*