Ramsey model. Rationale. Basic setup. A g A exogenous as in Solow. n L each period.

Transcription

1 Ramsey mdel Rainale Prblem wih he Slw mdel: ad-hc assumpin f cnsan saving rae Will cnclusins f Slw mdel be alered if saving is endgenusly deermined by uiliy maximizain? Yes, bu we will learn a l abu cnsumpin/saving behavir and abu dynamic analysis by analyzing i. Basic seup f Ramsey mdel was described by Ramsey in 928. Dynamics were develped by Cass and Kpmans in a grwh cnex in 965. Basic seup Firms Maximize prfi Prduce Y; hire services f L and K frm husehlds wh wn hem Y FK, AL wih usual prperies A g A exgenus as in Slw usehlds Maximize uiliy Ren L and K firms inelasically Buy Y fr cnsumpin (C) and saving/invesmen Live frever dynasic inerpreain Size f each husehld grws by L n L each perid. All markes are assumed be perfecly cmpeiive wih perfec infrmain and perfec fresigh Only significan decisin in he mdel is husehlds decided when cnsume Saving/dissaving (r invesing) is he mechanism fr inerempral subsiuin usehlds maximize lifeime uiliy subjec lifeime budge cnsrain We will analyze his decisin prcess in several seps: Firs, w-perid discree-ime mdel (Diamnd mdel will use) Nex, exend many perids Then, exend cninuus ime and infinie lifeimes On he way, we will esablish sme impran implicains fr cnsumpin hery (frm Chaper 7 f Rmer, 6 f cursebk)

2 Inerempral budge cnsrain in discree ime Tw perids Le K be he amun f capial (he nly durable asse) ha a husehld wns a he end f perid (beginning f perid ) usehld can add K by saving: chsing C < incme W = wage incme in perid r = real ineres rae = reurn n capial per perid (annual cmpunding fr nw) K rk W C K r K W C r r K W C W C K r r K r W W K r C C W W2 C C2 K2 r r r r r r r r Iniial wealh + PV f labr incme = PV f cnsumpin + PV f erminal wealh If husehld leaves n beques, hen las erm is zer If, in addiin, rae f reurn is cnsan, hen r = r, and K W W C C r r r r Lef-hand side is exgenus PV f lifeime wealh (nn-human + human capial) Righ-hand side pses decisin fr husehld: hw much cnsume in perid ne vs. perid w? Graphing he w-perid budge cnsrain Le 2 Q2 r K r W W2 be ( + r) 2 imes he lef-hand side f budge cnsrain (equals lifeime wealh in presen value as f perid 2), hen C2 Q2 rc is he budge cnsrain relaing cnsumpin in he w perids. The budge cnsrain is a sraigh line wih slpe r and verical inercep Q 2 Exending n perids n n W C Kn K n r r r If we assume ha erminal wealh is zer, he las erm disappears 2

3 In infinie ime, we need assume ha he husehld s capial des n grw a a rae faser han he ineres rae, s he limi f he las erm is zer: W C Kn K lim n n r r r K W C r r Wih cnsan rae f reurn his becmes W C K r r Lifeime uiliy Insananeus uiliy and lifeime uiliy Tw perids: U uc uc 2 = marginal rae f ime preference (inernal discun rae), measures husehld s impaience = means husehld values cnsumpin nex perid as much as his perid >> means husehld is very impaien and discuns fuure uiliy heavily Exending n perids r infinie hrizn U U n u C u C Naure f he feliciy funcin (insananeus uiliy funcin) u: MU u C dmu u C dc (Psiive bu diminishing marginal uiliy f cnsumpin) Ne cnvex shape n graph f u vs. C Pssible funcinal frms ha have apprpriae derivaives: Linear desn wrk because u C Quadraic can frce u C bu des n have u C fr all C 3

4 Ms cnvenien frm urns u be Cnsan Rae f Risk Aversin frm: uc C,. Fr his funcin, u C C and u C C is rae f risk aversin ha gverns hw sharply he uiliy funcin bends = wuld be linear funcin wuld have kink = is special case in which frmula cnverges uc ln C. Uiliy funcin and cnsumpin smhing Suppse ha yu are cnsidering hw cnsume Q f wealh (ignre ineres and discuning) Shw ½Q in each perid and cmpare ½Q ± X in w perids. Average uiliy is lwer wih ½Q ± X in each perid han wih ½Q in each. Thus, husehlds wih cnvex uiliy funcins prefer smh planned cnsumpin ver lumpy cnsumpin wever, high (r lw) ineres rae migh emp hem cnsume mre in fuure (presen) Indifference curves fr he w-perid uiliy funcin Equain fr indifference curve: U uc uc 2 T ge slpe, differeniae equain ally wih du : du u C dc u C dc 2. Slve fr Because u 2 2 dc dc u C. u C du 2, he indifference curves are cncave frm abve as usual. Alng he 45-degree line frm he rigin, C C2, s uc uc and 2 u C, which means ha he slpe f he indifference curves alng ha u C2 ray is ( + ). Equilibrium in he w-perid mdel We can graph he indifference map alng wih he w-perid budge cnsrain and lcae he equilibrium a he angency. Suppse ha r = Slpe f budge cnsrain is ( + r) hrughu Slpe f indifference curve is ( + ) a C = C 2 4

5 Thus, if r =, hen he angency will ccur a C = C 2 and he husehld will chse a fla cnsumpin pah: equal cnsumpin in bh perids Suppse ha r > In his case, he budge cnsrain is seeper han he indifference curve a C = C 2 and he angency mus be abve C = C 2. The husehld chses higher cnsumpin in he fuure han in he presen The reward saving (r) exceeds he husehld s marginal disuiliy f pspning cnsumpin (), s i chses an upward-slping ime pah fr cnsumpin. Suppse ha r < Budge cnsrain is flaer han he indifference curve a C = C 2 Tangency mus be belw C = C 2 usehld chses higher cnsumpin nw and lwer in he fuure: a dwnward-slping cnsumpin ime pah The reward saving falls shr f he husehld s marginal disuiliy f pspning cnsumpin, s i cnsumes mre nw and less in he fuure. Effec f gverns he amun f curvaure in indifference curves igh sharp bend lile effec f r n cnsumpin pah Lw nearly linear IC srng effec Cnclusin: r C2 C r C2 C r C C 2 This is impran and quie general resul fr cnsumpin hery. Implicains f he w-perid mdel fr cnsumpin behavir Cnsumpin pah depends n w hings: Presen value f lifeime wealh (including fuure earnings) Deermines he heigh f he cnsumpin pah Ineres rae in relain marginal rae f ime preference Deermines wheher pah is upward r dwnward slping Effecs f emprary vs. permanen change in incme Temprary will have small effec n lifeime wealh Permanen will have large effec Permanen will have MPC near, emprary will have MPC near /T, where T is remaining years f life. Effecs f anicipaed vs. unanicipaed change in incme 5

6 If crrecly anicipaed, hen i is already in he perid planned cnsumpin pah and here will be n effec n he pah r n cnsumpin in he year f he change. If unanicipaed, hen enire pah will be revised when infrmain abu he change becmes available. If unanicipaed change is permanen, hen large change in cnsumpin pah If emprary, hen small change Only new infrmain a ime will cause cnsumpin a differ frm he level prjeced a ime. This is he basis f he all cnsumpin paper ha yu will read in a cuple f weeks. Cninuus-ime cnsumpin decisin in grwh mdel Budge cnsrain Recall infinie-hrizn budge cnsrain (wih limiing cndiin n erminal W C W C wealh): K r K r r r r if r is cnsan When we cnver cninuus ime, we change Nains frm C C() Summains inegrals (saring a = ) Frm annual cmpunding cninuus (insananeus) cmpunding r e r r d R e e r Infinie-hrizn, cninuus-ime budge cnsrain fr an individual persn K R R lking frward frm ime is e W d e Cd L, where W is he wage f ne wrker per perid and C is cnsumpin f ne K wrker per perid. is he amun f capial wned by ne wrker a L R ime. (Remember ha r if he reurn n capial is cnsan, which makes he discun facr mre familiar.) 6

7 Family size: There are husehlds in he ecnmy wih L individuals in each husehld a ime. Ne ha we assume ha ppulain grwh ccurs hrugh increases in husehld size (reprducin), n hrugh new husehlds enering (immigrain). This is impran because i means we can assume ha exising peple care abu heir children in a way ha hey prbably wuldn care abu unrelaed immigrans. The budge cnsrain a he husehld level is R R K L L e W d e C d, where we have used capial per husehld raher han capial per wrker and augmened he earnings and cnsumpin values in per-husehld measures by muliplying by he number f wrkers per husehld. Translaing in per-effecive-wrker unis. As in he Slw mdel, ur seady-sae equilibrium will have grwing levels f per-persn variables like W and C, bu sable levels f he crrespnding per-effecive-wrker variables: w c C A Ne ha W earnings per efficiency uni f labr and A cnsumpin per efficiency uni f labr. C = Tal cnsumpin / # f wrkers = CONS/L c = Tal cnsumpin / # f effecive wrkers = CONS/AL Using definiins abve, W w A and C ca R R K A L A L e w d e c d Accrding he equains f min f echnlgy and he labr frce A g g A Ae A L L If we define k n L L e K A L k. K n as in he Slw mdel, hen A L, s 7

8 Thus, R R A L A L A L k e w e e d e c e e d g n g n We can cancel u he AL/ erm ge ur final inerempral budge cnsrain: R gn R gn Ne ha he wage per efficiency uni is Y k e w e d e c e d f kk f k, s he budge cnsrain A L depends n he evluin f w variables ver ime: k and c. These will be he cenral variables f ur grwh-mdel analysis. Dynamic uiliy in cninuus ime Recall uiliy funcin frm discree-ime mdel: U In cninuus ime, he uiliy funcin changes mirrr hse f budge cnsrain Nains frm C C() Summains inegrals (saring a = ) u C Frm annual cmpunding cninuus (insananeus) cmpunding Cninuus-ime uiliy funcin a an individual level is U e u C d Fllwing he same sep as wih he budge cnsrain, we firs cnver he husehld level by muliplying uiliy by he number f peple per husehld: L U e uc d. Plugging in he CRRA uiliy funcin A c g c C uc A e, g c L n g c Le U e A e d e A e d n g A L c e c B e d d 8

9 Inerempral uiliy maximizain Frmal mahemaical maximizain prblem: c max B e, c d R gn R gn subjec k e w e d e c e d R ng and lim e e k. The las cnsrain is he n Pnzi scheme cnsrain ha prevens husehlds frm driving heir wealh infiniely negaive as ime passes. N ne wuld lend a husehld ha did his, s hey wuldn be able d i. This is a prblem in he calculus f variains. We fen call his kind f prblem dynamic cnrl hery in ecnmics. We wn explre he sluin mehd in deail. (Rmer skeches he sluin n page 54. The sluin fr he pimal cnsumpin pah cnsiss f w pars c r g The cnsumpin Euler equain, which describes he c grwh rae f c a each pin alng he cnsumpin pah. The Euler equain ells he slpe (in grwh rae erms) f he pah a each pin, bu desn deermine he level. The budge cnsrain 9 R gn R gn ha k e w e d e c e d deermines which f he infinie number f parallel cnsumpin pahs he husehld can affrd. (I chses he highes ne i can affrd.) Rmer shws in fne 9 n page 56 ha he iniial (ime ) value n he cnsumpin pah is R g n k e w e d c R n e d R g n k e f k k f k e d. R n e d Ne ha R() depends n r() a all pins in ime beween and, and ha r f k is he ne reurn n capial a ime. Thus, c() depends nly n he fuure ime pah f k (and he parameers f he mdel). Inuiin f he Euler equain

10 Ne ha c is cnsumpin per effecive wrker. T ge his back cnsumpin per wrker, we use C = ca. C c r g r This means g g. C c C means individuals are chsing a rising cnsumpin pah a mmen, s heir cnsumpin shrly afer is higher han a. Crrespndingly, C means ha an individual s cnsumpin is lwer a a mmen afer han a, and C means ha cnsumpin per persn is he same jus afer as a. Frm he equain: rc rc r C These cndiins crrespnd exacly hse f he w-perid mdel: r C2 C r C2 C r C C 2 The inuiin is he same: peple will chse a rising, fla, r falling cnsumpin pah per persn (a mmen ) depending n wheher he reward saving (reurn capial = ineres rae) exceeds, equals, r falls shr f heir marginal rae f ime preference. w much des a change in r affec he cnsumpin decisin? Change in r is change in slpe f budge cnsrain w far his changes he pimal cnsumpin pin depends n amun f curvaure in indifference curves Large ls f curvaure; implies sraigh line Large in denminar means given gap beween r and leads small change in cnsumpin pah Large means ha husehlds d n like subsiue cnsumpin ver ime: wan sick smh cnsumpin pah regardless f r / is he elasiciy f inerempral subsiuin Small nearly linear indifference curves Small in denminar means large change in cnsumpin resuling frm given gap beween r and Small means husehlds are OK wih subsiuing ver ime

11 Aggregae dynamics f Ramsey grwh mdel