\fl. lConsider an economy composed of heterogeneous agents who live for two periods. Agents derive utility from consumption :1: according to utility function EEG}, and discount utility of the second period at a discount rate of 3. .F'Lgents enter the economy with zero assets. In the rst period, they receive an endowment 101 = to; and in the second period.I they have a probability of ll.5 to receive endowment to: = [1+ lm. and a probability of {1.5 to receive endowment to; = {l :5}u.l._ where D {I .5 -'.T 1. agents can save but cannot borrow. Let n denote the amount ofassets carried to the second period1 and assets earn a gross return of 1+ 1". Further denote the first period consumption as {:1 and the second period consumption as 433. [all tVrite down the agents' problem. Your solution should contain the choice variables and the budget constraint for each period. b Find. the Euler e tion qua [cl Under the assumptions {1 + 1"]5 = l and EEC} = $116+ W361. where 7'31 I? l}. 11': -'.T I]. and I: 11' ll" what is the optimal level ofassets carried to the second period? {d} Under the assumptions [1 +T]I5 = l and ulc} = G1_\"_.-"[l I;T:|, where LT 12* I}. is the optimal level of asset carried to the second period equal to D or greater than I]? Provide a proof and an intuitive explanation for your answer. [ell Under the assumptions [1 + 1"}3 = l and My} 2 cl_al."{l LT], where IF 2" (I, suppose now agents face greater uncertainty.r about future income1 i.e. have a probability of [lull to receive endowment to; = [l + Ellis, and a probability of [1.5 to receive endowment to; : [l Jill's}, where {I -'.'. .5 of will s: 1. '53:?on agents save more. save less, or save the sarne amount cornpared to the ease in the previous question [part d}? Provide a proof and an intuitive explanation. \f3'. Consider an economy where there are two agents, indexed by t = 1,2, living for one period. Let 5' E {3.3} denote the state of the world for this period. Exante, each state has an equal probability {I15} to happen. Eatpost, when S = A, agent 1 receives endowment mi, and agent 2 receives endowment mi, where we use superscript to indicate the agent number; when S = .3, agent l. receives endowment tog, and agent 2 receives endowment 1613. Suppose the market is complete, and agents make tinie-tl [before knowing 3] trading of arrow securities. Denote the price of arrow security that pays one unit of goods for state 3 as 3}, and holdings of this security as 9:. Similarly, let c; denote the consumption for agent 1' in state S. The period utility function is Ink], the same for both agents. {a} Dene a competitive equilibrium, in particular, explicitly write market clearing condi tionlzs}. {b} \"trite the consumer problem for agent 1. Note that you need to specify the choice variables and the constraints that this agent faces. {c} Find the rst order conditions for agent 1. {d} Note that by replacing 'E' = l with i = 2 in the previous parts, we will have the rst order conditions for agent 3' = 2. lGiven these rst order condition, will the share of total 1 endowment consumed by agent i diEer across two states, i.e. whether fwj- equals .-1. .-. 1 #3\" {e} According to the rst welrc theorem, as the preference is locally nosatiated,the com petitive equilibriLm: is Pareto optimal. This moans, we can nd a set of Pareto weight, denoted by N, that makes the solution to the social planner problem to be the same as the solution to the competitive equilibrium To apply this theorem, rst dene the social planner problem. if} Solve the social planner problem and express consumption as a function of endowment and Pareto weight. {g} Use the information derived from previous steps to solve the prices as a function of endowments. 'What is the relationship between price p, and the total endowment of state .5. Provide an intuitive explanation for your nding. 4. Ono-sided search with learninghydoing Tim: Discrete1 innite horizon. Demographic A single worker who faces the possibility of death with probability .5 each period. 1|nil-"hen dead he gets ll utility forever. Preferences: The worker is risk neutral lie. BEE} = I] and discounts the future at the rate l\". Endowment s: The worker can be high or low' skilled. He is born unemployed and lowskilled. He becomes highskilled only by being employed as a Low-skilled worker. I \"men unelnployed [regardless of skill level} he receives the "benet", If, each period. I \"Then rst unemployed he has lD'H" skill level and with probability o he gets a lowskilled job that pays 10L each period. I Employed lowskilled workers become highskilled with probability "- each period. I High skilled employed workers receive war each period. I Once the worker becomes high-skilled he is high Skilled for life. a Employed workers [regardless ofskill level] loose their job with probability .'a. I Unemployed highskilled workers get highskilled jobs [paying mg] with probability 0.. N019 1: Events that ean happen. to an individual are mutually exclusive {i.e. not indepen dent}. Assume that 13+ 1- +115 of land I5+CL c: l. N019 I: The worker can be in one of 4 possible states: unemployed lowskilled, employed lowskilled__ unemployed highskilled and employed highskilled. a. 1|Write down the ow asset 1value [ow Bellman} equations for the worker. [Notatiul use I: for the 1value to being in skill level, I = L, H and employment status gr = 3,8.) -L E b. Obtain an expression for NEH Tl in terms of the model parameters. us. Use the result from part b. to obtain a restriction on the value of WL such that the lowskilled worker will not reject a job o'Eer. d. Now suppose there is mass one of such individuals and that anyone who dies is replaced by a newborn. {Newborns are unskilled and unemployed.) Draw a diagram to show the population ows between the 4 states. e. 1'Write down a system of equations that can be solved for the steady state populations [you don't need to solve them]
