Cooperative Ph.D. Program in School of Economic Sciences and Finance QUALIFYING EXAMINATION IN MACROECONOMICS. August 8, :45 a.m. to 1:00 p.m.

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1 Cooperaive Ph.D. Program in School of Economic Sciences and Finance QUALIFYING EXAMINATION IN MACROECONOMICS Augus 8, 213 8:45 a.m. o 1: p.m. THERE ARE FIVE QUESTIONS ANSWER ANY FOUR OUT OF FIVE PROBLEMS. If you answer all problems, hen he firs four problems will be graded RESPOND TO A TOTAL OF FOUR QUESTIONS. You mus complee he examinaion wihin four hours. You will have 15 minues o read over he quesions before saring (8:45-9:). This exam is closed book. Calculaors and paper will be provided. Read he quesion carefully. Allocae your ime carefully. Pars wihin quesions will ofen vary in difficuly and weigh. Be sure o do all pars of each quesion chosen. If necessary, i is permissible o make clarifying assumpions, bu be sure o label hem explicily. (Grades will no ake unsaed assumpions for graned.) Also, label graphs and define noaion. Number your answer shees consecuively. Begin your answer o each quesion on a new page and idenify he quesions number. Leave ½-1 spacing around he edges of your paper. Wrie your exam ID number on EACH page of he exam.

2 Quesion 1: a) Individuals live wo periods. They are young in period 1 and old in period 2. When young, hey supply 1 uni of labor inelasically. When old, hey do no work. There is no populaion growh or ime discouning. Hence, individuals maximize uiliy from consumpion: U = log(c 1 ()) + log(c 2 ( + 1)) where c 1 is consumpion in he firs period and c 2 is consumpion in he second period. Aggregae oupu, Y (), is produced according o Y () = K() α L() 1 α where K() and L() sand for capial sock and labor, respecively, and α (, 1). There is no echnology growh. Capial depreciaes a rae δ. Show he individual s lifeime budge consrain, solve for he individual s opimal savings in period 1, show k( + 1) as a funcion of k(), and solve for he seady sae capial k. b) Now, individuals face a probabiliy of deah a he end of period 1. Therefore, hey have only probabiliy π of living in period 2. Hence, individuals maximize heir expeced uiliy from consumpion: E(U) = log(c 1 ()) + πlog(c 2 ( + 1)) When he individuals die, he governmen akes heir savings and wases hem. Oher assumpions are he same as in Par a). Show he individual s lifeime budge consrain, solve for he individual s opimal savings in period 1, show k( + 1) as a funcion of k(), and solve for he seady sae capial k. c) Under he scenario in Par b), how much does he governmen receive? d) Under he scenario in Par b), a new disease srikes and he probabiliy of dying increases. Show wha will happen o he seady sae capial k.

3 Quesion 2: a) Households maximize lifeime uiliy: Max e ρ c() 1 θ 1 θ d subjec o: ȧ() = r()a() + w() c() A() = A and lim e r(s)ds a() where c() is consumpion per capia, ρ is he discoun rae, a() are asses per capia, r() is he ineres rae, and w() is he wage. There is no populaion growh. Firms produce oupu according o: Y i () = A()K i () α [K()L i ()] 1 α where i sands for an individual firm, so Y i, K i and L i are oupu, capial sock and labor of he individual firm, respecively. A is echnology, K is aggregae capial sock, and < α < 1. There is no exogenous echnology growh. Capial depreciaes a rae δ. Derive he growh rae for a decenralized economy. Then, derive he growh rae under a social planner. Does he decenralized economy achieve a social opimum? Please explain why. b) Now, le s assume ha he producion funcion is: Y i () = A()K i () α L i () 1 α [ K() L() ]γ(1 α) where γ >. Oher assumpions are he same as in Par a). Derive he growh rae for a decenralized economy. Then, derive he growh rae under a social planner. Does he decenralized economy achieve a social opimum? Please explain why. c) Consider he following saemen: Large counries enjoy larger learning-by-doing effecs and, herefore, higher growh raes han small counries. Please commen on his saemen using models from Par a) as well as Par b).

4 Quesion 3 Consider an overlapping generaions economy in which each generaion lives for hree periods. There is a single nonsorable consumpion good in each period. The consumer born in period, =,1, 2,..., has goods endowmens ( ω, ω+ 1, ω+ 2) and sandard uiliy funcion uc (, c+ 1, c+ 2). 1 1 There is an iniial middle-aged consumer wih goods endowmens ( ω, ω1 ) and sandard uiliy funcion v ( c, c1 ) and an iniial old consumer wih goods endowmen ω and sandard uiliy funcion v ( c ). The iniial middle-aged consumer is endowed wih amoun m of fia money and he iniial old consumer is endowed wih amoun of fia money. a. Define an Arrow-Debreu equilibrium. b. Define a sequenial markes equilibrium. c. Characerize he soluions o he consumers problems in an Arrow-Debreu equilibrium and in a sequenial markes equilibrium. 2 m Now suppose ha 1 2 ( ω, ω, ω ) ( ωy, ωm, ωo) = for all. d. Lis he equaions ha a seady sae mus saisfy. (Hin: You should have 7 equaions in 6 unknowns.) 1 e. How many seady saes are here? Briefly explain. For wha values of m 2, m, and p is he sequenial markes equilibrium iniially in a seady sae? (You may assume ha you have already calculaed he unknowns in par (d).)

5 Quesion 4 Consider a sandard growh model wih axes and governmen spending. There is a represenaive consumer who is endowed wih unis of labor in each period and k unis of iniial capial. The consumer s uiliy funcion is β uc (, ), = where < β < 1 and u(,) is a sandard period uiliy funcion. There is a single good ha can be used for privae consumpion, privae invesmen, and governmen consumpion, so he resource consrain is c + k+ 1 (1 δ ) k + g = f( k, ), where < δ 1 and f (,) is a sandard producion funcion. There is a ax on he consumer s i e labor income of τ and a ax on he firm s labor expendiure of τ. The governmen purchases g unis of he good in period, where g is exogenous. The governmen can borrow from and lend o he consumer. a. Define an Arrow-Debreu equilibrium. b. Define a sequenial markes equilibrium. c. Characerize he soluion o he consumer s problem in an Arrow-Debreu equilibrium and a sequenial markes equilibrium. d. Show ha here exiss a sequence of axes { τ e } such ha he equilibrium allocaion wih ( i, e ) (, e i e i τ τ = τ ) is he same as he equilibrium allocaion wih ( τ, ) ( ˆ τ = τ,). Specify he value of τ as a funcion of ˆi τ. e e. Carefully specify how o calculae he percenage change in welfare, in erms of lifeime consumpion equivalens, in swiching from one ax sysem o anoher.

6 Quesion 5 Consider a simple model of he credi marke. Time is discree and he economy lass forever. A ime a measure of uni one of borrowers is born and a measure of one uni of lenders is born. Each lives for wo periods, young and old. The lender is endowed according o (K, ) in he wo periods of life, while he borrower is endowed (, E). They have he same preferences,. Lenders can make loans (L) and hold real money balances ( where L is loan volume, M is nominal money, and p is he price level. Assume he raes of reurn are equal, 11, where r is he reurn on loans and r m is he reurn on money. Borrowers can borrow a rae 1+r. The hree quesions below receive equal weighing of 1/3. A. Find he savings funcion of he lender. Le 1/(1+r). Wha is? Wha is?, where R = B. Find he borrowing funcion, B. Wha is db/dr? C. Wha happens o he ineres rae if here is a massive decline in bank capial, K? Discuss wha he cenral bank should do in ha case.