UCLA Deparmen of Economics Spring 05 PhD. Qualifying Exam in Macroeconomic Theory Insrucions: This exam consiss of hree pars, and each par is worh 0 poins. Pars and have one quesion each, and Par 3 has wo quesions worh 5 poins each. All quesions are required. Answer each quesion in a separae bluebook. You should urn in (a leas) FOUR bluebooks, one (or more, if needed) bluebook(s) for each quesion.
Par Consider an economy populaed by a coninuum of idenical households wih he following preferences: lnc Aln l, 0, 0 where c is consumpion and l is leisure a dae. Households are endowed wih one uni of ime each period ha can be used for labor or leisure. In addiion, each household is endowed wih k 0 unis of capial in period 0 and can accumulae capial according o he law of moion k ( ) k i, 0, where i is invesmen a dae. The households sells labor o a compeiive firm and can work eiher a sraigh ime shif of lengh h, a sraigh ime plus overime shif of lengh h + h, or no a all (hus, labor is an indivisible commodiy). The echnology for combining capial wih sraigh ime and overime labor o produce oupu (y ) is given by z y e h k ( n n ) h k n, 0, where n is he number of households working only sraigh ime and n is he number of households working sraigh ime plus overime. Oupu can be used for curren consumpion or invesmen. The echnology shock, z, evolves according o z z, is an i.i.d. random variable wih mean 0. A. Carefully formulae he dynamic program ha would be solved by a social planner ha chooses capial, labor and consumpion sequences o maximize a social welfare funcion ha weighs all agen s uiliies equally. B. Prove ha in equilibrium he fracion of employed households ha work overime is a consan even when he economy is no in seady sae. (Hin: Do his by simply deriving his consan.) C. Suppose ha here are moving coss ha mus be incurred when he number of sraigh-ime workers d is changed, m n n ( ). The oupu available for consumpion and invesmen is, in his case, y m. Repea par A for his case and show ha saemen in par C no longer holds. D. Define a recursive compeiive equilibrium for he model of par (A) where agens rade employmen loeries. Be sure o compleely specify he problem solved by households and firms in your decenralized economy. E. Derive an expression for he sraigh ime hourly wage rae and he overime wage rae as a funcion of he prices deermined in par D. Nex, derive an expression for he overime wage premium, which is he raio of he hourly overime wage rae o he hourly sraigh-ime wage rae, in erms of he parameers of he model. Under wha condiions will he overime premium be greaer han one?
Par. Preferences. Time is discree {0,,,...} and here is no uncerainy. The economy is populaed by a uni measure of households who come in J differen ypes indexed by j {,..., J}. The measure of ype-j households is µ j and, since here is a uni measure of households, we le J j= µ j =. The ineremporal uiliy of an household of ype j is β [( α j ) log (c j, ) + α j log =0 where β (0, ) and α j (0, ). ( Mj,+ P )], Endowmens. Each household sars a ime = 0 wih an idenical endowmen of M0 unis of money, and b unis of mauring one-period real governmen bonds (i.e., he household is eniled o receive b unis of consumpion from he governmen a ime = 0). Every subsequen period, a household of ype j receives he endowmen y j. We denoe he aggregae endowmen by: y = J µ j y j. j= Governmen. There is a governmen who needs o finance g unis of consumpion. Every period he governmen mainains a consan supply of one-period real bonds, b, and levies consan lump-sum axes τ min{y i }. The governmen creaes or desroys money, M+ M, in order o mee is budge consrain. Noaions. In wha follows, we will denoe by P he nominal price of consumpion goods a ime, by r he real ineres rae beween and +, and by i he nominal ineres rae beween and +.. (3p) Definiions (a) (0.5p) Sae he governmen sequenial budge consrain. (b) (0.5p) Sae he ineremporal problem of a ype-j household. (c) (p) Define a feasible allocaion. (d) (p) Define a compeiive equilibrium.
. (4p) Consider now he economy wih one represenaive household, J =. (a) (0.5p) Derive he firs-order condiions of he household s ineremporal problem. (b) (0.5p) Show ha he real ineres rae is consan. (c) (0.5p) Guess ha he growh rae of money is consan and equal o γ, ha is: M + M = γ M, for all g 0, and where γ is o be deermined in equilibrium. Show ha he inflaion rae mus also be consan and equal o γ. (d) (p) Derive he aggregae money demand. (e) (.5p) Derive he equaion ha deermines he equilibrium inflaion rae, γ. How does i depend on he exogenous parameers g, τ, b, and α? Explain why. 3. (3p) Consider now he economy wih heerogenous households, J >. Guess ha here exiss an equilibrium wih he following feaures: money growh rae and inflaion are consan and equal o some γ, and he consumpion, c j,, and real money holding, M j,+ /P, of a household of ype j say consan, equal o c j and m j respecively. (a) (p) By combining he firs-order condiion of he household s problem, and he household s ineremporal budge consrain, find expressions for c j and m j as a funcion of α j, y j, τ, β/( + γ), and b + M 0 P 0. (b) (p) Le ᾱ J j= µ jα j, m = J j= µ jm j, and c = J j= µ jc j. Using he expression for c j and m j you found in he quesion above, calculae ( ( ᾱ) m β ) ᾱ c. + γ (c) (p) Derive he aggregae money demand. How does i differ from he aggregae money demand when here is a represenaive agen? How does i depend on he correlaion beween he level of money demand, α j, and he level of income, y j? Explain why.
Macro Comprehensive Exam Quesion 4 Summer 05 This quesion is worh 5 poins In his quesion, we examine he demand for money in a model wih heerogeneous agens and exogenous incomplee markes. Consider he following economy wih heerogeneous agens. Time is discree and denoed =,, 3,.... There is a coninuum of measure one of agens. Each period each agen experiences an idiosyncraic endowmen shock y drawn from he se Y = {y, y,..., y N }. Assume ha for each agen, hese endowmens follow a Markov process wih ransiion probabiliies φ ij = P rob(y + = y j y = y i ). Assume ha he iniial disribuion of endowmens across agens is given by he vecor η = (η, η,..., η N ) where η i denoes he fracion of agens wih iniial endowmen y 0 = y i. Assume ha η is a saionary disribuion of he Markov process described by ransiion probabiliies φ ij. Le π (h ) denoe he probabiliy as of dae 0 ha an agen experiences he hisory of endowmen realizaions h = (y 0, y,..., y ) and le c (h ) denoe he consumpion a ime of such agens. Assume ha agens choose sochasic processes for consumpion {c (h )} =0 o maximize heir discouned expeced uiliy wih period uiliy u(c) and ime discoun facor β <. Par : One poin Wrie an expression for agens expeced uiliy and wrie he feasibiliy consrains on he consumpion allocaion {c (h )} =0. When wriing expeced uiliy, noe ha agens know heir firs period endowmen y 0, so be careful o wrie heir expeced uiliy condiioning on ha iniial endowmen. Now assume ha he only asse available o agens is fia money. Assume ha a he sar of he firs period, each agen is endowed wih M pieces of fia money (nicely colored pieces of paper). Le he number of hese pieces of paper be fixed over ime. Le {P } =0 denoe he price a which agens exchange money for goods a each dae. We look o give a recursive definiion of a saionary equilibrium in which agens choose consumpion c and money holdings m each period o maximize he expeced discouned presen value of heir consumpion from he curren period on given endowmen y i and money holdings m a he sar of he period. In he saionary equilibrium, we assume ha he price level P is consan a P > 0. Par : One Poin In a saionary equilibrium, agens who sar he period wih endowmen y i and money holdings m choose consumpion c(y i, m) and money holdings o carry ino nex period m (y i, m) o maximize he expeced discouned presen value of heir consumpion subjec o he consrains c(y i, m) + P m (y i, m) = y i + P m
and m (y i, m) 0. Wrie a Bellman equaion describing he problem ha agens are solving o choose heir opimal plans for consumpion and money holdings c(y i, m) and m (y i, m). Par 3: One Poin Skech an argumen ha here exiss a value of money holdings m such ha he opimal choice of money holdings m (y i, m) m for all values of y i and m wih m m. In making ha argumen, you should make reference o he equilibrium real rae of reurn on money holdings in a saionary equilibrium. You should no ry o give a deailed argumen. Simply provide some inuiion for why here should be an upper bound on agens equilibrium money holdings. Le he join disribuion of agens holdings of money in real erm and endowmens a dae be given by F (m/p y i )η i where F (m/p y i ) is he cumulaive disribuion funcion of real money holdings for an agen condiional on ha agen having endowmen y i a dae. Recall ha η i represens a saionary disribuion of endowmens so ha his does no vary over ime. We define a saionary equilibrium as a value funcion and decision rules for agens V (y i, m), c(y i, m), m (y i, m) ogeher wih a saionary disribuion of real money holdings and endowmens across agens implied by hose decision rules F (m/p y i )η i (independen of ime) and a consan price level P such ha he goods marke and he money marke clear. Par 4: One Poin Wrie he marke clearing condiions for he goods and money markes in a saionary equilibrium. Also provide an argumen ha if you double he sock on money in he economy from M o M, hen he price level in he saionary equilibrium also doubles from P o P. Par 5: One Poin Now imagine ha we are comparing saionary equilibria across wo economies. In one of hese economies agens have no uncerainy over heir income. Tha is y = y 0 for all for all agens. In he oher, income is uncerain as described above. Presen an argumen ha, in he firs economy wih no uncerainy over endowmens, ha here is no saionary equilibrium wih finie, non-zero, and consan P (i.e. in which money is valued).
Neoclassical Growh and Heerogeneiy The economy is populaed by a large number of households, indexed by i. Preferences are given by: U i = =0 β c σi i σ i () Each household is endowed a dae 0 wih k i0 unis of capial, and he aggregae capial sock a dae is denoed as k, = 0,,,... Each household also is endowed wih one uni of labor. Le he aggregae supply of labor be normalized o. There are a large number of idenical and compeiive firms ha operae a echnology, denoed as f, which is homogeneous of degree in capial and labor, and ha produces oupu. The echnology is increasing in capial and labor, and is wice coninuously differeniable. Households receive income from supplying labor services and from rening capial. Assume ha capial markes and labor markes are perfec.. Define a compeiive equilibrium for his economy. Discuss he following saemen, and be as precise as you can: "Economis A says ha if he aggregae capial sock, k 0, is he seady sae capial sock of his economy, hen he iniial disribuion of wealh across households will be preserved, and he paern of consumpion inequaliy will mirror he paern of capial inequaliy, even hough preferences are differen. Economis B says ha for every k 0 > 0, each household s capial holdings will converge o he percapia seady sae capial sock because of diminishing marginal produciviy of capial and because each household has he same level of human capial and he same ownership in he firms."