Problem 1 (50 points total). Consider two large open economies, Home and Foreign (Foreign variables that need not be equal to Home variables are denoted by an asterisk). Each economy is inhabited by a continuum of identical individuals grouped into an aggregate risk sharing household. In each country there is also a representative final goods producing firm. Lifetime utility is given by U =Et > Bt - and U* = Et 1 = 0 t = 0 1 - 0 in the Home and Foreign country, respectively, where: Et is the expectation operator, BE (0, 1) is the (constant) subjective discount factor (i.e., 0 0 and v, * > 0 ( is the Greek letter "mu," p is the Greek letter "rho," and v is the Greek letter "upsilon"). The standard deviations of the error terms/shocks/innovations & and * (e is the Greek letter "epsilon") are, respectively, Ce and Set (S is the Greek letter "zeta"). Moreover, Et (Et) = Et (et) = 0. Each country produces capital, and the evolution of capital is given by Kt+1 = It + (1 -8) Ktand Ki+ 1 = It + ( 1 - 8* ) Ky (8 is the Greek letter "delta") in the Home and Foreign country, respectively, where I and I* denote investment and o and * are the capital depreciation rate in the Home and Foreign country, respectively. A benevolent world social planner solves the following problem: M8 max Et But - + ( 1 - 2/ . ) ( CT ) 1 - 0 1 = 0 1 - 0 where vt E (0, 1) ( is the Greek letter "psi" ) is the weight that the planner puts on the Home country in period t, subject to the following constraints: Ct + + + NX+ SY+ and C+ + It + N X *