Macroeconomic Theory Ph.D. Qualifying Examinaion Fall 2005 Comprehensive Examinaion UCLA Dep. of Economics You have 4 hours o complee he exam. There are hree pars o he exam. Answer all pars. Each par has equal weigh. ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3 Consider an economy wih a coninuum of infiniely lived individuals each of who work one of wo work shifs or no a all. The shifs correspond o working only a sraigh ime shif ( h1 ) or sraigh ime plus overime ( h1+ h 2 ). Le n 1 be he fracion of individuals ha work only sraigh ime and be he fracion ha work sraigh ime plus overime. Each individual has preferences given by { 0,, } h h h + h. 1 1 2 n 2 E β c A = 0 [ log + log(1 h) ], where Oupu, which can be used for consumpion or invesmen, is produced using a Cobb- Douglas echnology ha is a funcion of capial, labor and a echnology shock. Tha z 1 1 is, θ y e K ( H θ θ = + H ), where K is he sock of capial, H is oal sraigh ime 1 2 H 2 hours worked and is oal overime hours worked. Oupu can be used as i consumpion or invesmen ( ). Invesmen in period become producive capial in period +1, and he sock of capial depreciaes a he rae δ. Finally, he echnology shock, z, evolves over ime according o a firs order auoregression, z+ 1 = ρ z + ε+ 1, 2 where ε N(0, σ ). ε A. Wrie down he social planner s problem for his economy as a dynamic program. The planner should give equal weigh o he uiliy of all individuals in is objecive funcion. B. Derive a se of firs order necessary condiions ha characerize a soluion o his problem. In paricular, find equaions ha deermine he following variables: 1 1
y (oupu), c, n1, n2, H, i and K + 1. Here, H is oal hours worked. Also, characerize he seady sae for a nonsochasic version of his problem. C. Suppose ha you are given he following saisics compued from U. S. daa: (1) average labor s share; (2) he average capial o oupu raio (annual); (3) he average invesmen o oupu raio; and (4) he average fracion of ime ha individuals spend working in he marke secor. Suppose ha a period is one quarer of a year. Your job is o calibrae he economy so ha he seady sae maches he U.S. averages in hese four respecs. Show how hese facs can be used o find values for A, β, θ, and δ. Tha is, give a se of equaions ha can be solved o obain values for hese parameers given values for and h. h1 2 D. For he se of equaions you obained in par B, wrie wo of hese equaions in erms of log deviaions from seady sae. Derive a linear approximaion of his se of wo equaions. E. Suppose you combined he equaions obained in par B so ha he linearized Euler equaion is a funcion of only k, k + 1, k + 2, z, and z + 1. The resuling equaion will be of he form 0 = E[ k + ak + a k + a z + a z ]. Show how his Euler equaion + 2 1 + 1 2 3 + 1 4 can be solved for he opimal decision rule, k = bz + b. In paricular, find he k + 1 1 2 parameers b1 and b2 as funcions of a1,, a4. Does your soluion saisfy he ransversaliy condiion? Explain. F. Define a recursive compeiive equilibrium for his economy. 2
represens PART TWO: ANSWER IN BOOK 2 WEIGHT 1/3 Answer all pars 1. Consider he following model in which here exis a represenaive family ha maximizes he uiliy funcion: (1) C U =, ρ 1 ρ 1 β = 1 1 where ρ is a posiive parameer, β is he discoun facor and C is consumpion. Oupu is produced from he echnology: α M (2) Y = A, P where α is a posiive parameer beween zero and one, A is a posiive parameer, M is he nominal quaniy of money held beween periods and +1 and P is he price of commodiies in erms of money. All oupu is consumed. The family holds zero bonds and M unis of money a he beginning of period 1. The money supply is increased each period according o he rule: (3) M = 1 μm, μ > + 1, M = M 0 and all new money is disribued o he represenaive family as a lump sum ransfer denoed T measured in unis of money. The family may choose o hold is wealh in he form of money, or governmen bonds, BB, where B B bonds held beween periods and +1. Bonds are in zero ne supply. The nominal ineres rae on a bond ha is held beween periods and +1 is denoed i. Le he presen value of a period s dollar in period be denoed by Q. s A. Wha is mean by a presen value price? The erm is defined o be idenically equal o 1. Wrie an expression ha defines ineres raes { i } s j j= for s >. Q s Q as a funcion of he sequence of B. Assume ha he family owns a represenaive firm and ha each period i sells oupu, buys consumpion goods and accumulaes money and bonds in a sequence of markes. Wrie down he budge consrain faced by he family/firm in each of hese markes. 3
C. Wrie down a single consrain for he family which consrains is expendiures and consumpion plans for periods 1 hrough T. [HINT: his expression should involve M erms in P,,, i C T, Q1 and Y, for = 1,... T ]. D. Wha is mean by a Ponzi scheme? Wrie down a consrain ha prevens he family from running a Ponzi scheme. E. Wrie down a single infinie horizon budge consrain for he family. [HINT: his M expression should involve erms in P,,, i C T, Q1 and Y, for = 1,... ]. F. Using equaion (2) and your answer o par (B) rewrie he he uiliy funcion in M erms of real balances P and real bonds B. [Assume ha he period budge P consrain holds wih equaliy.] Find a firs order condiion for he choice of real balances when he family firm maximizes he uiliy funcion you have derived wrien in erms of money and bonds. G. Find a difference equaion in real balances ha mus hold in a compeiive equilibrium. [Hin: Use he equilibrium condiions, C =Y and eliminae P from he problem using he money supply rule]. H. Find an expression for he unique seady sae equilibrium value of real balances as a funcion of he parameers of he problem. Does an increase in he money growh rae raise or lower real balances? Provide inuiion as o why his occurs. I. Wha is mean by indeerminacy of equilibrium? Explain in words wha condiion is necessary for his seady sae equilibrium o be indeerminae. 4
PART THREE: ANSWER IN BOOK 3 WEIGHT 1/3 ANSWER ONE of TWO EITHER Par III 1. Consider a version of our saionarized cash-in-advance economy in which he moneary injecion is received a he beginning of he period before he goods marke opens. Afer he goods marke closes in he period, hen he asse marke opens in he second half of he period. In his case he household's problem is given by ( ) ( ) ( ) ( ) ( ) ( ) ( ) V m, b, τ = max u c + v 1 l + β V m'/ 1 + τ, b'/ 1 + τ, τ ' h τ ' τ dτ ' subjec o ( ) pc + m' + q s b ' m + b + pl + τ, where h denoes he pdf on nex period's money injecion. The firm's problem is given by max l The marke clearing condiions are given by c= l, pl wl m' = l+ τ, b' = 0. A) Characerize he equilibrium of his economy. Try o explain how he curren money injecion will affec he real variables in he economy. Please ry and explain wha is he impac of beliefs abou τ ' on curren real variables. B) Consider a perfec foresigh version of his model. Wha moneary policies can lead o efficien equilibrium oucomes? Be sure o sae wha your noion of an efficien oucome is and why. 5
OR 2. Consider a version of he Ramsey problem of opimal axaion in which he governmen only has access o capial axes, bu no o labor axes. Assume moreover, ha he firs period axes on capial, τ is given exogenously. Assume ha he model is k 0 oherwise sandard. In paricular, assume ha he economy has he following feaures: Governmen's budge consrain: Represenaive Household's problem ( ) + 1 τ, g + 1 + r b = b + k qk. ( ) max E β u c, l subjec o ( 1 τ ) ( 1 ) ( 1 δ) k, + 1 + 1 0 and k, b, r given. + 1 0 0 0 wl + qk + + r b = c + k k + b k Represenaive Firm's Problem: Resource Consrain: ( ) max f k, l qk wl ( δ ) ( ) c + k+ 1 1 k + g = f k, l (Noe ha for simpliciy we have assumed ha depreciaion isn' deducible.) A) Consruc he se of consrains ha any primal approach o he governmen's c, l, k + ha saisfies hese problem mus saisfy. Be sure o show ha any allocaion { } 1 consrains can be mapped ino a policy for he governmen and associaed compeiive equilibrium. B) Se up he Lagrangian ha characerizes he soluion o he Ramsey problem, and derive he associaed firs-order condiions. Wha can you say abou axes in he long-run if his economy converges o a seady sae in which governmen expendiures was consan a g? 6