The Ramsey Model. Reading: Firms. Households. Behavior of Households and Firms. Romer, Chapter 2-A;

Transcription

1 Th Ramsy Modl Rading: Romr, Chaptr 2-A; Dvlopd by Ramsy (1928), latr dvlopd furthr by Cass (1965) and Koopmans (1965). Similar to th Solow modl: labor and knowldg grow at xognous rats. Important diffrnc: capital stock is dtrmind by optimization dcisions of housholds and firms. Firms Thr ar many idntical firms, ach with th sam production function Y FK,AL. Th production function displays th sam proprtis as bfor. Firms hir labor and capital in comptitiv markts. For simplicity, w assum thr is no dprciation of capital ( 0). ousholds idntical infinitly-livd housholds. Th siz of ach houshold grows at rat n. Each houshold mmbr supplis on unit of labor and rnts its capital to firms. oushold maximizs its lifitim utility: U t uct Lt t0 dt whr uct is th instantanous utility of ach mmbr of th houshold, of mmbrs of th houshold and is th discount rat. Assum constant rlativ risk avrsion (CRRA) utility function: uct Ct1 1 whr 0and n 1 g 0. Not: uc C 0 u C C 1 0 so that th cofficint of rlativ risk avrsion is : C u C uc Bhavior of ousholds and Firms C C1 C. Lt is th numbr Firms ar comptitiv and arn zro profits. Firms hir capital and labor and pay thm thir marginal products. Th ral intrst rat thn is:

2 Th ral wag is Wt FK,AL L rt fkt..sincy ALfk and k K/AL: Wt Afk Akfk Afkt ktfkt. Th wag pr unit of ffctiv labor thn is wt fkt ktfkt. ousholds Budgt Constraint ousholds tak r and w as givn. Th budgt constrain stipulats that th prsnt valu of consumption cannot xcd th sum of initial walth and prsnt valu of labor incom. Dfin t Rt rd 0 so that on unit of th good invstd at tim 0 is worth Rt units at tim t; Rt capturs th ffct of continuously compounding intrst ovr th priod 0,t. Similarly, on unit of output at som futur tim t is worth Rt units at tim 0. oushold budgt constraint thn is: Rt Ct Lt K0 dt t0 Rt Wt Lt t0 dt or K0 Rt Wt Ct Lt dt 0. t0 Th intgral can b rwrittn as a limit: K0 lim s s Rt Wt Lt Lt Ct dt 0. t0 ousholds capital stock at any tim s is Ks K0 Rs s RsRt Wt Ct Lt t0 dt, that is, houshold walth at any tim quals to th intrst-compoundd valu of its initial walth and its savings (positiv or ngativ). This can b rwrittn as Ks K0 Rs s Rt Wt Ct Lt t0 dt so that th houshold budgt constrain bcoms lim s Rs Ks 0. This implis that housholds cannot follow a path of consumption and invstmnt that would rsult in ngativ nt prsnt valu of walth (no-ponzi-gam condition).

3 ousholds Maximization Problm ousholds maximiz thir liftim utility subjct to th budgt constraint. All housholds ar idntical, thrfor all will choos th sam path of consumption and invstmnt. Dnot consumption pr unit of ffctiv labor ct so that Ct Atct (ach workr has A units of ffctiv labor) and Ct 1 1 oushold lif-tim utility function bcoms U t Ct 1 t0 1 t0 Atct1 1 A0gt 1 ct 1 1 A0 1 1gt ct 1 1. Lt dt t A0 1 1gt ct 1 1 A0 1 L0 B t0 t0 t ct 1 1 dt L0 nt dt t1gtnt ct 1 1 dt whr B A0 1 L0 and n 1 g. Not that w assumd to b positiv. Th budgt constraint, Rt Ct Lt K0 dt t0 Rt Wt Lt t0 dt can b rwrittn in trms of capital, consumption and wag pr ffctiv labor: Rt ct AtLt dt k0 A0L0 t0 Rt wt AtLt dt t0 Rt ct ngt A0L0 dt k0 A0L0 t0 Rt wt ngt A0L0 t0 t0 Rt ngt ctdt k0 Rt ngt wtdt. t0 Th limit vrsion of th budgt constraint can b also rwrittn lim s Rs Ks 0 lim s Rs ngs ks A0L0 0 lim s Rs ngs ks 0 dt

4 ousholds Bhavior ousholds choos th path of ct that maximizs thir liftim utility subjct to th budgt constraint. Bcaus additional consumption always incrass utility, u C 0, th budgt constraint will b mt as quality. Th Langrangan: B t0 t ct 1 1 dt k0 Rt wt ngt dt Rt ngt ctdt t0 t0 Th houshold chooss ct at any point in tim according to th following FOC: B t ct Rt ngt and according to th budgt constraint. Taking logs of th FOC: lnb t lnct ln Rt n gt Taking drivativs w.r.t. t or Substituting n 1 g lnb t lnct ln rd n gt. 0 ċt ct rt n g ċt ct rt n g. ċt ct rt g rt g Sinc Ct Atct, consumption pr workr grows at th rat of growth of ct plus th rat of growth of knowldg, g: Ċt Ct rt. nc, consumption pr workr grows if th intrst rat xcds th discount rat and falls othrwis. This quation is rfrrd to as th Eulr quation; it dscribs how ct volvs for any givn valu of c0. Th houshold chooss c0 so as to satisfy th budgt constraint: th prsnt valu of liftim consumption must qual th initial walth plus th prsnt valu of savings. Th Dynamics of th Economy Dynamics of c: Th Eulr quation can b rwrittn using rt fkt t

5 ċt fkt g. ct ċt 0 whn fkt g. Dnot th lvl of k for which this is th cas k. Consumption is incrsing for all k k and falling for k k. S Figur 2.1 in th book. Dynamics of k: Rcall that, as w drivd for th Solow modl, k t sfkt n gkt. r, assuming no dprciation and allowing savings to vary: k t fkt ct n gkt. For any k, k 0 whn c fk n gk. nc, k will rmain constant if consumption quals th diffrnc btwn output and brak-vn invstmnt. Th first trm is incrasing in k with diminishing rturns, th scond trm is linar in k. Thrfor, th lvl of consuption that kps k constant is hump-shapd in k and paks at such k for which fk n g (goldn-rul lvl of k). If consumption is lowr, k is incrasing, and vic vrsa. S Figur 2.2 in th book. Stady Stat: Th valu of k is givn by fk g. Th goldn-rul k is givn by fk GR n g. Byassumption n 1 g 0 or g n g. Thrfor, k k GR. Th lins charactrizing ċ 0andk 0 can b combind in a phas diagram. For vry k0 0, thr is a uniqu lvl of c that is consistnt with th houshold s intrtmporal optimization and will bring th conomy to th stady stat. Th st of all such combinations of c and k is rfrrd to as th saddl path. S Figurs in th book. Balancd Growth Path Solow and Ramsy modls display similar proprtis in quilibrium: (1) Capital, output and consumption pr unit of ffctiv labor ar constant. Th savings rat, yc y, is also constant (bcaus both y and c ar constant). (2) K, Y and total consumption growth at rat n g. (3) Capital pr workr, output pr workr and consumption pr workr grow at rat g. nc, th basic prdiction of th Solow modl ar rproducd also in th Ramsy modl: in th stady stat, th rat of growth of output pr workr is dtrmind ntirly by tchnological

6 progrss. Not, howvr, that th goldn-rul lvl of k will not b attaind in th Ramsy modl. In th Solow modl, th savings rat is xognous and thrfor any lvl of k, including th goldn-rul on, can constitut a stady stat. In th Ramsy modl, th quilibrium is such that k k GR. In th Ramsy modl, th savings rat is th outcom of housholds intrtmporal optimization rathr than bing xognous. Choosing k GR would lad to highr c, but sinc housholds discount futur consumption, this is not optimal. Fall in th Discount Rat Considr an conomy that is on th balancd growth path. Suppos th discount rat,, falls unxpctdly. Th discount rat only affcts th quation for consumption, ċt fk g. ct Th stady-stat capital is givn by fk g. If falls, this mans that that th nw k is highr than th original quilibrium. To bring th conomy on th saddl path, c must initially fall and thn ris gradually along th saddl path to th nw quilibrium. Onc th nw quilibrium is attaind, both k and c ar highr than in th original quilibrium. S Figur 2.6 in th book. Adjustmnt aftr changs in th discount rat is similar to changs in th savings rat in th Solow modl: growth acclrats during th transition to th nw quilibrium; onc th adjustmnt is compltd, all variabls grow at th sam rats as bfor. owvr, th savings rat is not constant during th transition in th Ramsy modl.

7 Govrnmnt Expnditur and Consumption Smoothing Considr impact of govrnmnt xpnditur and spcially of unanticipatd changs in govrnmnt xpnditur on th lif-tim path of consumption. Assum govrnmnt purchass and consums Gt pr unit of ffctiv labor. This xpnditur is financd by lump-sum tax also qual to Gt: th budgt is always balancd. Govrnmnt xpnditur dos not affct utility from privat consuption: govrnmnt spnding is pur wast, or it financs public goods that do not rplac privat consuption. Dynamics of capital: k t fkt ct Gt n gkt. Graphically, th introduction of govrnmnt spnding implis that th k 0 curv shifts down; th ċ 0 curv rmains unaffctd. Budgt constraint: Rt ngt ctdt k0 t0 Rt ngt wt Gtdt. t0 Th ffct of unanticipatd prmannt incras in Gt: Assum th conomy is in quilibrium. Gt incrass unxpctdly and th incras is prcivd to b prmannt. Th k 0 curv shifts down whil th ċ 0 curv rmains unaffctd. Consumption jumps immdiatly to th nw quilibrium. Th quilibrium k rmains th sam. Bcaus rt fkt th ral intrst rat dos not chang as a rsult of a prmannt incras in Gt. S Figur 2.8 in th book. Th ffct of unanticipatd tmporary incras in Gt: Th k 0 curv shifts down whil th ċ 0 curv rmains unaffctd, but th nw quilibrium is only sn as tmporary. Bcaus th instantanous utility is concav, housholds prfr to smooth consumption ovr tim. Thrfor, th fall in consumption will b only partial. Aftr th initial adjustmnt, consumption will gradually ris and k will fall, so as to bring th conomy onto th saddl path by th tim govrnmnt xpnditur falls again. Onc G rturns to th original lvl, th conomy convrgs along th saddl path back to th initial quilibrium. Th siz of th initial fall in c dpnds on th xpctd lngth of th incras in G. Aftr th incras in G, k first falls and thn incrass again. Thrfor, r first gradually incrass and thn gradually falls back to th original lvl. S Figur 2.9 in th book. nc, th intrst rat should only rspond to tmporary incrass in G but not to prmannt ons.

8 Barro (JME 1987) tsts this proposition using intrst rats and war-rlatd military xpnditur in th UK ovr th priod 1729 to 1918, and finds vidnc consistnt with th thortical prdiction.