UCLA Department of Economics Fall PhD. Qualifying Exam in Macroeconomic Theory

Transcription

1 UCLA Deparmen of Economics Fall 2016 PhD. Qualifying Exam in Macroeconomic Theory Insrucions: This exam consiss of hree pars, and you are o complee each par. Answer each par in a separae bluebook. All hree pars will receive equal weigh in your grade.

2 Par I Consider a real business cycles model wih a represenaive household ha lives forever and maximizes he following uiliy funcion: 0 E log c Alog L, 0 1 and A0. Here, c is consumpion and L is a convex combinaion of leisure in periods and 1. Each period, households are assumed o have one uni of ime ha can be allocaed beween marke work, h, and leisure. In paricular, le L a(1 h) (1 a)(1 h 1) where 0a 1. Oupu, which can be used for consumpion, invesmen ( i ) or governmen purchases, is z 1 produced according o a consan reurns o scale echnology, y e k h, where y is oupu and k is he sock of capial. The variable z is a echnology shock observed a he beginning of period ha evolves hrough ime according o a firs order auoregressive process wih mean zero innovaions. The sock of capial is assumed o depreciae a he rae each period. Invesmen in period becomes producive capial one period laer, k 1 (1 ) k i. Governmen spending is an exogenous random variable ha, like he echnology shock, follows a firs order auoregressive process, in his case wih an uncondiional mean of g and 2 uncondiional variance of g. Innovaions o his process are assumed o be independen of innovaions o he echnology shock process and he value of g is observed a he beginning of period. In addiion, governmen purchases are financed wih lump sum axes. Noe ha governmen purchases do no direcly affec preferences or he echnology; hey are simply hrown ino he sea. (a) Are he equilibrium allocaions for his economy he soluion o a social planner s problem? Explain. If so, wrie he social planners problem for his economy as a dynamic programming problem. Be specific abou he sochasic process (law of moion) for z and g. (b) Derive as se of equaions ha characerize a sequence c, h, L, k 1, y ha solves his 0 problem. Be sure ha you have he same number of equaions as unknowns. Explain he role of he ransversaliy condiion in deermining his opimal sequence. (c) Define a recursive compeiive equilibrium for his economy. (d) Assuming ha for a given variable x, x log x log x, where x is he non-sochasic seady sae value of x. Derive a linear expression for h as a funcion of k, c, z, and h 1. (e) As in par (d), derive a linear equaion expressing c as a funcion of k, z, g, h, and k 1.

3 (f) In a sandard real business cycle model, a 1. Using he equaion derived in par (d) and/or (e), explain how seing a 1 migh change he cyclical properies of he model economy. In paricular, focus on he size of flucuaions in hours worked relaive o z. Provide inuiion in your explanaion. (g) In a sandard real business cycle model, g is no included as a sochasic shock. Discuss how adding his feaure migh change he cyclical properies of he model economy. In paricular, focus on he correlaion beween hours worked and z. Again, provide inuiion.

4 Par Consider firs he McCall Model we sudied in class. There is an infiniely-lived risk neural worker wih discoun facor β (0, 1). The worker can be in eiher one of wo saes, unemployed ( U ) or employed ( E ). Every period when she is unemployed, she receives he unemploymen benefi b > 0 and draws exacly one wage offer from he CDF F (w). If she acceps he offer, she sars working nex period a ha wage. A he end of every period of employmen, he worker looses her job wih probabiliy δ (0, 1) and becomes unemployed. Le V E (w) denoe he maximum aainable uiliy of an employed worker, and V U he maximum aainable uiliy of an unemployed worker. (a) Wrie he Bellman equaions for an unemployed worker, V U, and for an employed worker a wage w, V E (w) (1p). (b) Show ha he opimal policy of he unemployed worker is o accep any offer above some reservaion wage (1p). (c) Find an expression for he reservaion wage in erms of V U (1p). (d) Consider a worker who has received an offer jus equal o he reservaion wage. Show ha, for his worker, i would be weakly opimal o accep he offer and qui afer jus one period of employmen. (1p) 2. Now assume ha job offers are heerogenous no only in erms of wage, bu also in erms of sabiliy. Tha is, when receiving an offer, a worker draws boh a wage w and a job desrucion rae δ. A higher δ hus corresponds o a less sable job. Assume for simpliciy ha w and δ are independenly disribued wih respecive CDF F (w) and G(δ). (a) Wrie he Bellman equaions for an unemployed worker, V U, and for an employed worker a wage w facing job desrucion rae δ, V E (w, δ). (1p) (b) Show ha he reservaion wage is he same regardless of he job desrucion rae δ. Explain why. Hin: use he insigh of quesion 1.d. (1p) (c) Wha is he impac of an increase in δ on V E (w, δ)? Show ha here are wo effecs going in opposie direcions. Explain hese wo effecs. Explain when each 1

5 of he effec dominae and why. (1p) 3. Now assume ha workers accumulae human capial on he job. Precisely, consider an employed worker wih human capial level h. If she keeps her job nex period, hen her level of human capial is h (1 + γ), for some small posiive γ. Assume as well ha wage offers, w, are per uni of human capial : ha is a worker wih human capial h and wage w receives he pay w h. Likewise, he unemploymen benefi b is per-uni of human capial: a worker wih benefi b and human capial h receives b h. (a) Wrie he Bellman equaion for an unemployed worker wih human capial h, V U (h) and for an employed worker wih human capial h, wage w, and facing job-desrucion rae δ, V E (h, w, δ). (1p) (b) Argue ha V U (h) = h v U and V E (h, w, δ) = h v E (w, δ). Wrie he Bellman equaion for v U and v E (w, δ). (c) Does he insigh of quesion 1.d coninue o hold? Why? (1p) (d) How does he reservaion wage depend on he job-desrucion rae, δ? (1p) Why? (e) Wha is he impac of an increase in γ on he value of an unemployed worker, v U? (1p) (f) Shows ha an increase in γ has wo impacs on he reservaion wage going in opposie direcions. Explain why. (1p) (g) Argue ha one effec always dominae for δ 1. Argue ha he oher effec can dominae for δ 0. (1p) 2

6 Par 3 - Taxaion and Economic Aciviy in an Opimal Growh Model Preferences for he represenaive household, which has one uni of ime available per period, are given by: max β {ln(c ) φh } (1) The aggregae producion echnology is given by: AK θ (X H ) 1 θ (2) The law of moion for he aggregae capial sock is given by: K +1 = (1 δ)k + I (3) The echnology process X is given by: X +1 = (1 + γ)x (4) A. Explain why you can - or canno - solve for he compeiive equilibrium allocaions by solving a social planning problem. (2 poins) B. Derive equaions ha can be used o solve for he planner s allocaions in a saionary version of his economy. Show all of your work (5 poins) C. Show he equaions ha characerize he seady sae of he planner s problem. (3 poins) Suppose ha he economy is in seady sae a dae j. Suppose ha from dae j onwards ha he governmen will ake g unis of resources every period, 1

7 and ransfer hose resources back o he household as a lump sum ransfer. Suppose ha he governmen has access o he following axes o obain hese resources: consumpion axes, labor income axes, or i can ax he capial sock. D. Show ha for a suffi cienly small per-period g ha here exiss a ax sysem such ha his economy remains in is original seady sae for all fuure periods, and ha welfare is unaffeced by he governmen s ax-ransfer policy. Noe: a ax sysem is defined as infinie sequences of ax raes on consumpion, labor income, and he capial sock. These ax raes can be negaive, zero, or posiive. Show a formula ha deermines he maximum size of g ha can be financed wih his welfare-preserving ax sysem, and call his g. (7 poins). E. Is your ax sysem equivalen o a lump sum ax sysem? Why or why no? (3 poins) 2