Problem 1 [5|] 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 nal goods producing rm Lifetime utility is given by 0;}10 lo' ': de= RZ_E[ in the Home and Foreign country, respectively, where: E is the expectation operator, ,3 E {[1,1] is the {constant} subjective discount factor [i_e., U s: ,3 s: 1, where E means \"belongs'1 and ,3 is the Greek letter "beta"], the parameter a" is strictly greater than zero {a is the Greek letter \"signia\"), and C" and C\" denote consumption in the Home and Foreign country, respectively (the price of consumption is normalized to 1, and all nonprice variables are normalized by the world population). Production in the Home and Foreign country is given hr 1?= dY*= thD respectively, where for the Home and Foreign countries, respectively, Y and 1\" denote out put, 3 and 3" denote exogenous productivity, K and Ii" denote the capital stock. Moreover, the parameter a E [i], 1} (is, D {i o: 4: 1, where E means "belongs\" and a: is the Greek letter \"alpha\"]- Furthermore: A ,. K: = \"$th and K: = [1 13} K}, where \"1'3 E {0,1} [1' is the Greek letter \"ganuna'i'j and If: is the total capital stock in the world economy in any period it. Therefore, E=m+m_ The preceding implies that capital is fully mobile across countries, meaning that in any period capital can be reallocated instantaneously across borders- All told, with is the Jfraction of world capital used in period t: to produce in the Home country, that is, K;, and {1 Tel R: is the fraction of world capital used in period t to produce in the Foreign country, that is, Kg. In addition, log: _ ,u. ln3_1 st ilnsi'lu\"l+l: :lllna1l+lei' where: gnu} 3 ll, p, p' :5- 1] and 1,1,1!\" :3- D[ Lu, is the Greek letter \"mu,\"1 p is the Greek letter \"rho:1 and v is the Greek letter "upsilon\" }_ The standard deviations of the error termsfshocksf innovations s and e' [s is the Greek letter \"epsilon\") are, respectively, 1.: E and (E. [q is the Greek letter \"zeta'- Moreover, iEt {st} = E; [5;] = 1} Each country produces capital, and the evolution of capital is given by KHIZI1+[1_IS}Kt and h=+UW {5 is the ISreek letter "delta"} in the Home and Foreign country, respectively, where I and I\" denote investment and d and 15* are the capital depreciation rate in the Home and Foreign country, respectively. A benevolent world social planner solves the following problem: maxEtZc {tilt CitE + {1 1,511} K}: 10' 10: i=0 where art E {0,1} (it: is the Greek letter "psi\") is the weight that the planner puts on the Home country in period t, subject to the following constraints: q+n+nngn and CHI: +Nx: a 1: where NX and NI\" denote net exports in the Home and Foreign country, respectively. In what follows, assume that, in addition to the usual variables, it: and y are choice variables as well {note that each of these variables has a time subscript in the setup of the problem]. Part 1.a [It] points}- State the benevolent social planner\"s current value Lagrangian us ing a single constraint and Jr. as the notation for the Lagrange multiplier. Please showfexplain in detail how you arrive at this constraint. Hints: 1. Some useful main references include Section 3 of the Lesson T lecture notes and prob lems ? and 3 from the Lesson 'T practice problems with solutions. Some useful sec ondary references include Section 2 of the Lesson 5 lecture notes, problem 5 of the Lesson 5 lecture notes, and problems 1 and 2 of the Lesson 5 homework. 2. The Lagrangian needs to be stated using a single {world} budget constraint- Remember that this constraint is equal to the sum of the Home and Foreign individual budget constraints having substituted out investment for each country using each country's respective capital equation of motion- 3. This problem is pretty much the same as the open economy REC models we've seen in class. The thing that's different is the presence of \"world capital-" \"That to do? Everywhere you have Home and Foreign capital, just go ahead and substitute these out by what they are in terms of world capital K per the info noted above in the problesz setup- In other words, you can start off by ignoring world capital, combine all the constraints as you would as usual, set up the current value Lagrangian, and once you're done with that substitute out Home and Foreign capital for what they are in terms of world capital. Part 1.13 {20 points}- State of the social planner's rst order conditions. Please show your work in detail. Hints: 1. Again, some useful main references include Section 3 of the Lesson 7 lecture notes and problems 7 and 8 from the Lesson 7 practice problems with solutions. Some useful secondary references include Section 2 of the Lesson 5 lecture notes, problem 5 of the Lesson 5 lecture notes, and problems 1 and 2 of the Lesson 5 homework. 2. Recall that capital is a state variable! 3. Recall as well that in this case v and y are choice variables, which means that, as a check of your work in part (a), your Lagrangian should be set up in a way that will clearly allow you to choose these variables via first order conditions. (a) All told, you should have first order conditions for C, C"*, v, y, A, and K (not for K and K* separately). Part 1.c (20 points). Now, assume that o = o*. Given your answers to part (b), solve explicitly for the planner's optimal choices of v and y. Note: your answers should be in terms of exogenous variables and/ or parameters, only. Please show all of your work in careful detail. Hints: 1. To arrive at the optimal value of v, use the first order condition for v and the first order conditions for consumption. 2. To arrive at the optimal value of y you just need to use the first order condition for y itself