1. 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 In Ct and U* = Et _ B* In Ct, where: Et is the expectation operator, B E (0, 1) is the (constant) subjective discount factor, and C and C* denote consumption. Production in each country is determined by: Y't = AtK" and Y = AtKa, where: A and A* are stochastic technology processes; and a E (0, 1). Furthermore: [In At In At where: p, p* > 0; v, v* > 0; Et (Et) = Et (et) = 0; and the standard deviations of & and * are, respectively, of and *. In the preceding, all variables are normalized by the world population, which consists of a unit mass. Also, the evolution of capital in each country is given by: Ktti = It + (1 -8) Kt and Ki+1 = 17 + (1 -8* ) KY, where I and I* denote investment and o and 8* are capital depreciation rates. A benevolent world social planner solves the following problem: max Et _ Bt [y In C+ + (1 -20) In C+]. Ct,Ct, Kt+1,Ki+1,St+1,Si+1 t= 0 such that: Ct + + St+1 S Y+ + (1 + rt) St C+ + If + SHI SY* + ( 1 + re ) St, where S and S* denote countries' net foreign assets and r is the world real interest rate. So, the benevolent world social planner is weighting the utility of each country by their relative size in the world economy, and in this setup there are no net exports per se; instead, international financial transactions are only reflected accounting-wise through economies' net foreign asset positions. Because utility is increasing and concave, then each country's budget constraint will bind and from the planner's perspective the sole relevant budget constraint is: Ct + Kt+1 - (1 - 8) Kt + St+1 + C+ + Ki+1 - (1-8") K; + St+1 = AtKo + (1 + re) St + At Ka + (1 + rt) St. Moreover, the current value Lagrangian is: Et _ etty In C+ + (1 -1) In Ct 1 = 0 AK" + ( 1 + r.) S+ + AtKa + ( 1 + re) St the - Co-Knit (1 - 8) KI - SHI - CH - Kit (1- 8" ) K; - SH (a) (10 points.) Derive in mathematical detail the values of A and A* in steady state assuming that: p7 p * * 0* *. (b) (7 points.) State the benevolent social planner's first order conditions.(c) (12 points.) Is it possible to conclude from the planner's rst order conditions that out of steady state in any period t the solution to the planning problem implies that: aAth'l + (1 a) = om; {mp1 + (1 5*) = 1 + rt? Make sure to explain in detail. Note that, intuitively, this (impression means that in every period the marginal product of capital net of depreciation is equal to the real interest rate, which is equalized across countries