# NU ES_APPM 411 - Final Exam (7 pages)

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## Final Exam

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- Es_Appm 411 - Diff Eq Math Phy

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Christiano 411 Winter 2005 FINAL EXAM Answer three of the following equally weighted four questions If a question seems ambiguous state why sharpen it up and answer the revised question You have 1 hour and 55 minutes Good luck 1 Consider the Krusell Rios Rull model in which there are 4 possible strategies available to each agent at the beginning of her 3 period life young middle age old 1 be unskilled in each period 2 be unskilled in the same vintage for the first 2 periods and then be skilled in that technology in the 3rd period 3 be unskilled in the first period a fast learner in some vintage in middle age and skilled in that vintage when old 4 be an innovator in the first 2 periods and then skilled in the new technology in the last period The availability of strategy 4 depends on whether innovation is or is not allowed Whether innovation is allowed is determined by majority rule There is a mass of 1 for each generation so that the total mass of agents at a given date t is 3 Agents discount the future at the rate 0 1 and maximize their discounted income The aggregate production function for technology is Cobb Douglas f s u A s 1 u where s and u are the number of skilled and unskilled workers respectively in vintage Productivity improves with vintage A A 1 Consider the following modification to the structure of the economy seen in class Assume that innovators receive a transfer t from the government in each of the 2 periods of their life The transfer is financed through labor taxes If t s u is the gross wage paid to the unskilled worker at time t in vintage then the net wage is 1 t s u where is an exogenous constant and 0 1 1 a For given s u derive the gross wage paid to the unskilled workers working in the best technology in a given period of time t Derive the net profits of a skilled worker in vintage at time t b Derive the discounted lifetime income for the 4 possible strategies c Conjecture a stationary equilibrium where in any period t a constant fraction of the young choose to innovate and develop new technologies and the remainder 1 choose to remain unskilled Derive the ratio of skilled to unskilled workers at each period of time t as a function of Derive the wage of unskilled workers and the profits of skilled workers as a function of this ratio Derive the budget constraint the government must satisfy in each period t d Derive the equilibrium value of as a function Discuss how changes as we increase Provide economic intuition e Derive conditions under which the conjectured equilibrium is actually an equilibrium i e make sure that no one would deviate and choose strategy 2 or 3 f Suppose the economy is in a no innovation equilibrium at time t and a vote is held on whether to allow innovation 50 of votes in favor of innovation are needed to make innovation possible If innovation is allowed innovators get a subsidy as above Discuss the incentives each di erent agent in the economy has to vote in favor or against innovation as a function of her current and future income In particular how is the subsidy to innovation going to a ect this decision Explain carefully 2 Suppose there are 2 types of agents In period t type 1 agent receives endowment et and type 2 agent receives endowment 1 et Suppose et can take on only two values 3 4 and 1 4 Let st denote the exogenous uncertainty in period t with st et Let st denote the history of exogenous uncertainty from the first period date 0 until period t st s0 s1 st The probability of a particular history st is t st 2 A planner chooses a sequence of consumption allocations across the 2 types by maximizing the following objective function X X h t t st t 0 st 1 u1 c1 st i 2 u2 c2 st subject to the resource constraint c1 st c2 st 1 for all st and the participation constraints X X j 0 st j st X X t t j t 0 st t t j st j st u1 c1 st j t j s t t j s u2 c2 s X X t t j t 0 st X X t t j t 0 st st j st u1 e st j st j st u2 1 e st j for each st The utility functions u1 and u2 are strictly concave and increasing a Let i st 0 be the Lagrange multiplier on the ith agent s participation constraint and st be the Lagrange multiplier on the resource constraint in history st Write down the Lagrangian version of the planner problem Define Mi st i i s0 i s1 i st i 1 2 Rewrite the planner s lagrangian problem in terms of Mi st Use this formulation to argue that Mi st can be interpreted as the weight the planner assigns to agent i b Derive the first order conditions to the planner problem Define M2 st z s M1 st i st i st i 1 2 Mi st t 3 V1 st V2 st Derive a law of motion for z st as a function of z st 1 1 st and 2 st Interpret these results and in particular discuss how the optimal allocations c1 st and c2 st change as z st changes What does z st measure Provide economic intuition on the relationship between i st and z st c Write down the recursive formulation of the problem and define carefully the state variables Carefully analyze the role of each of them From now on consider a particular low probability history with the property et 3 4 for all t 0 1 2 d Suppose that given the above realization of et from t 0 to t t the social planner has chosen allocations up to time t such that t t c1 s 3 4 and c2 s 1 4 Prove that the participation constraint of one of the two agents will not be binding at t given st What about the participation constraint of the other agent Will it bind Provide economic intuition e Given in b d and given the your results realization of et and t t t 1 and c2 st 1 be c1 s c2 s specified above would c1 s greater or smaller than respectively c1 st and c2 st Provide a proof and economic intuition for your answer Hint study how z st changes 3 Consider the following 2 period economy The representative household maximizes u c1 c2 l subject to the following two budget constraints c1 k c2 1 Rk 1 l Here is the wage rate which corresponds to the marginal product of labor R denotes the rental rate of capital its marginal product and ci denotes consumption in period i i 1 2 One …

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