Problem 1 Consider a small open economy that lives for two periods, only, (0 and 1) and is inhab- ited by a continuum of identical individuals grouped into an aggregate risk-sharing house- hold. Each period, aggregate output Y, is produced via the function Y, = ZtL, where Z, is technology/productivity (which is exogenously determined), and L, is labor. There is no uncertainty about the level of technology/productivity in any one period, but it need not be the case that Z] = Z1. If output is not consumed or loaned out in any one period, then it spoils and cannot be carried over to the following period. This is a small open economy, so it can borrow and lend freely at the constant-across-periods international interest rate r. The household's lifetime utility is given by 1 5 1+_6 U=Zt(ct_71+Lts )3 i=0 where: {3 6 1(9; 1) is the household's (constant) subjective discount factor; C is consumption; 0, yL, '3 is the instantaneous utility function, whose form is known as "quasi-linear;" and 'y and e are parameters whose value is strictly greater than 0. Moreover, the household "owns" the production function, so its thinking with regards to its maximization problem is akin to that of a benevolent social planner. 1.1. What is the slope of an indifference curve in (00,01) space? (15 points total) 1.2. Consider the following version of the household's intertemporal utility maximization problem, Where A denotes the state variable "assets." The household chooses consumption, labor, and assets to maximize: l 5 H1: 2:? (Ct71+5L) t=0 such that at + At+1 5 (1+ 5") At + ZtLt Let A: denote the time t Lagrange multiplier. Based on the setup of this problem state the household's current value Lagrangian. (10 points total) 1.3. State the relevant first-order conditions (that is, the first-order conditions for C, L, and A) associated with the Lagrangian in your solution to (1.2). (15 points total)1.4. State the relevant transversality condition. What is the intuition behind this condition? (15 points total) 1.5. What is the optimal level of labor in any period, regardless of Whether the economy is open or closed, and what are optimal consumption Co and 01 equal to if the economy is in autarky? Show your mathematical derivations in detail. (15 points total) 1.6. Use your solution to (1.5) to state the open economy's lifetime budget constraint and derive the slope of the open economy's budget line in (Co, 01) space (remember that Ct and Lt are endogenous variables!). (15 points total) 1.7. Is the following claim true, false, or uncertain? When the economy is open, the solution to the household's intertemporal consumption problem is indeterminate from the point of View that it is optimal for the household to consume bundle (00,01) that lies on the open economy's budget line. Explain your answer in detail, including writing down any mathematical statements and/or derivations needed to support your conclusions. (15 points total)