Question 2 (20 points) Consider an economy that consists of two islands, i = (1, 2). Each island has a large population of infinitely-lived, identical agents, normalized to the unit. There is a unique consumption good, say, coconuts, which is not storable across periods. Although within each island agents have identical preferences over consumption, across islands there is a difference: Agents in island 2 are more patient. More precisely, the lifetime utility for the typical agent in island i is given by where 8, E (0, 1), for all i, and & > A. Due to weather conditions in this cconomy, island 1 has a production of e >0 units of coconuts in even periods and zero otherwise, and island 2 has a production of e units of coconuts in odd periods and vero otherwise. Agents cannot do anything to boost this production, but they can trade coconuts, so that the consumption of the typical agent in island i, in period f, is not necessarily equal to the production of coconuts on that island in that period (which may very well be xero). Assume that shipping coconuts across lulands is cout less. al Describe the Arrow-Debree equilibrium (ADE) allocations in this economy. You can use any method you like, but I strongly recommend that you exploit Negishi's method. by Describe the ADE prices in this economy. c Plot the equilibrium allocation for the typical agent in bland i, ie., tap f = [1, 2), against t. Is there any period & in which & - ed If yes, please provide a closed form solution for that value of t."1 Incomplete Markets and Asset Prices This problem investigates the effects of market incompleteness on asset pricing following Constantinides and Duffie (1996). There is a continuum of individuals with identical CRRA preferences: u (c) = cl-7/ (1 - y) and subjective discount factor is 3. (a) Write down the Euler equation for individual i and asset j. This equation should relate the gross return Ri+1 to the growth rate of consumption for individual i: 41/c. (b) Now assume that the growth rate of individual consumption satisfies, (1) Ct where c+1/G is the growth rate of aggregate consumption. Here i+1 represents the idiosyn cratic component of consumption growth. Conditional on off1, at1/c, and Ri41, assume i+1 is independent across individuals and log zit, is distributed N(-207+1,07+1). The cross sectional variance parameter, off1, is a random variable from the point of view of time t, it becomes known at time t + 1. Note that of , is not assumed to be independent of ct+1/c and Ry (i.e. the three variables may be correlated with each other). Use the individual Euler equation from (a) to show that: 1 = BE. + (071 ) CHI ) RU41 (2) where (c) Specialize the above by assuming that, Oil = A- Blog Show that, holds with, 1 = 7+57 ( 1 + 7 ) B 3 = Bexp 57 (1 + 7)4) (d) How can these results help explain the equity premium puzzle?2 Two-Sided Lack of Commitment: Stationary Alloca- tions Consider the environment of Alvarez and Jermann (2000), where the economy is populated by equal number of two types of agents with perfectly negatively correlated endowment, both with lack of committment. Focus on the two-state case. Based on the result that for the symmetric case with 2 agents and 2 shocks the optimal allocation eventually reached a "memory-less" (history independent) allocation, we now seek to characterize optimal stationary symmetric distributions. (a) Given (c', c?), show that V' (c', () = I- Btwo(c') + (1 -w) u(c?)} where w = 1 - Bp 1+ 3 -2p3 Moreover, show that " is decreasing in & and increasing in p . (b) We call a stationary symmetric allocation feasible if it satisfies the resource and partici- pation constraints: d'+d =e V' (d, 2) 2 Vi (y',y?) (3) V? ( @, ( 7) 2 V2 (y', y? ) (4) Notice that autarky is always feasible. Show that in any symmetric allocation (4) never binds. That is, show that whenever (3) 2 holds then (4) automatically holds with strict inequality. (c) Show that full risk sharing is attainable if and only if: u (e/2) 2 wu (y' ) + (1 - w) u(y?) (5) (d) Here we use the comparative static results for w found in part (a) and the result in part (c) to examine the parameters that affect the feasibility of risk sharing. How do 3 and p affect the likelihood of full-risk sharing being feasible? Show that for small enough spread between y' and y? (holding e constant) full risk sharing is not possible. Let utility take the form u (c) = el-s/ (1 -o) show that if o is sufficiently close to 0 full risk is not feasible. (e) If full risk sharing is not attainable we are interested in the best allocation that is feasible. Using your results from part (b) show that if (5) is not satisfied the best symmetric allocation satisfies d+d = e wu (c') - u(y')] + (1-w) [u(c) -1(3?)]=0 and y?