Econ 510a (second half)Yale UniversityFall 2005Prof. Tony SmithFINAL EXAMINATIONThis is a closed-book and closed-notes exam. You have three (3) hours to complete the exam.There are four (4) questions on the exam for a total of 100 points. The points allocated toeach part of each question are indicated below. Please put the answer to each question ina different blue book. To receive full credit, you must provide convincing explanations tosupport your answers. Please write as neatly as possible.1. Consider an exchange economy with two types of consumers, each of which comprisesone-half of the economy’s population. The economy lasts for two time periods (la-belled 0 and 1) and there is one nonstorable consumption good in each period. Eachtype-1 consumer is endowed with one (1) unit of the consumption good in each timeperiod. Each type-2 consumer is endowed with one (1) unit of the consumption goodin the first period. In the second period, however, each type-2 consumer has a stochas-tic endowment: with probability one-half he is endowed with two (2) units of theconsumption, and with probability one-half he is endowed with zero (0) units of theconsumption good. Each consumer has the same preferences over consumption goods:log(c0) + E[log(c1)], where ctis consumption in period t.(a) (8 points) Assume that markets are complete. Carefully define a sequentialcompetitive equilibrium for this economy (i.e., one in which consumers tradeArrow securities in period 0).(b) (8 points) Compute the competitive equilibrium allocations and prices (i.e., theprices of the Arrow securities). Interpret your answers.(c) (4 points) Use your answer from part (c) to find the price (in terms of period-0consumption goods) of a riskfree bond that delivers one unit of the consumptiongood in all states of the world in period 1.(d) (8 points) Suppose now that markets are not complete: consumers are allowedto trade only a riskfree bond in period 0. Define a competitive equilibrium andfind an equation that determines the price of the riskfree bond (you do not haveto solve explicitly for this price).12. Consider a (deterministic) neoclassical growth model in which the government taxescapital income at a proportional rate τ > 0 and returns the proceeds as a lump-sumtransfer to consumers. Consumers have identical preferencesP∞t=0βtu(ct), where ctisperiod-t consumption. Each consumer is endowed with k0units of capital in period 0and with one unit of time in each period. A typical consumer’s budget constraint inperiod t reads:(1 − τ)rtkt+ wt+ St= ct+ kt+1− (1 − δ)kt,where ktis period-t holdings of capital, Stis the lump-sum transfer, and rtand wtare the rental price of capital and the wage rate, respectively. Profit-maximizing firmsoperate a Cobb-Douglas production technology F (k, n) = kαn1−α, where n is laborsupply. The government balances its budget in every period.(a) (9 points) Carefully define a sequential competitive equilibrium for this economy.(b) (9 points) Find an algebraic expression for the steady-state capital stock incompetitive equilibrium.(c) (9 points) Is the steady-state capital stock in part (b) higher, lower, or the sameas the one that would be chosen by a social planner? E xplain fully.3. Consider a (deterministic) neo classic al growth model in which the representative con-sumer’s preferences are given by:∞Xt=0βtu(ct, `t, `t−1).That is, current-period utility depends on the current amount of consumption ct, thecurrent amount of leisure `t, and the lagged amount of leisure `t−1. Profit-maximizingfirms operate a Cobb-Douglas production technology: y = kαn1−α, where y is output,k is capital, and n = 1 − ` is labor supply. Capital accumulates according to: k0=(1 − δ)k + i, where i is investment.(a) (9 points) Carefully define a recursive competitive equilibrium for this economy,assuming that consumers own the factors of production.(b) (9 points) Display the Bellman equation for the social planning problem in thiseconomy.(c) (9 points) Find a pair of difference equations that determine, together with thetransversality condition, the competitive equilibrium behavior of capital and laborin this economy. (You do not need to display the transversality condition.)24. Consider a version of the Lucas “tree” model in which trees not only yield fruit (ordividends) but also enter the utility function directly. In particular, the representativeconsumer’s preferences are given by:E0∞Xt=0βt[log(ct) + A log(st)] ,where ctis period-t consumption, stis the number of trees held in period t, and A ispositive. Thus owning more trees leads to higher utility: trees are considered “beauti-ful” and consumers value beauty. The tree yields a stochastic dividend stream {dt}∞t=0.Assume that dtis independent and identically distributed (so that its realization todayis statistically independent of its past realizations) and assume that E(dit) = miforall nonzero integers i. Dividends are the only source of consumption goo ds in thiseconomy and they are not storable. Each consumer is endowed initially with one tree.Consumers can buy and sell trees in a competitive market.(a) (9 points) Derive the Euler equation of a typical consumer.(b) (9 points) Use your answer from part (a) to find the equilibrium price of atree as a function of the current dividend. (Hint: Guess that the price is equalto a constant B times the current dividend, and then solve for B in terms