Advanced and Contemporary Topics in Macroeconomics I

Transcription

1 Advaned and Conemporary Topis in Maroeonomis I Alemayehu Geda ag2526@gmail.om Web Page: Class Leure Noe 2 Neolassial Growh Theory wih Endogenous Saving Ramsey-Cass-Koopmans & OLG Models Based on he maerials: by Douglas Hibbs2004; Vahagn Jerbashian 204 and David Romer 2009/202 Addis Ababa Universiy Deparemen of Eonomis PhD Program, 204

2 A. The Ramsey-Cas-Koopman Model Inroduion Aim: Show dynami ineffiienly/over saving is ruled ou when maroeonomi rae of saving onsumpion is grounded by he opimal deision of HH using deeper parameers suh as uiliy/preferene/profi maximizaion/raionaliy e The sae of he world is he same as in he Bare Bones model exep ime is now oninuous and produion is governed by unspeified neolassial ehnology. As before, he eonomy is populaed by a large number of idenial firms and households so we jus wor on represenaive firm and HH.

3 Inroduion... Con d Firms aim o maximize profis and hey ren apial, hire labor and sell oupu a ompeiive pries in ompeiive mares. Households aim o maximize a Uiliy of ineremporal onsumpion funion..are dynasi generaions lined hrough ime by alruisi bequess They own he firms he apial so and so profis arue o hem

4 Households The Uiliy Objeive: Households are dynasi wih generaions lined hrough ime by alruisi bequess. They behave idenially and see o maximize he imeseparable ineremporal uiliy funion U 0 0 u ~ e n e d 0 u ~ e n d

5 Where household size, ; is N 0 given exogenously by C ; ~ denoes household onsumpion per adul, woring member; he disoun rae exeeds he growh rae of household size neessary o ahieve a onvergen inegral 0 n. N Insananeous uiliy "feliiy" has properies N N n e u u'[ ~ ] 0, u''[ ~ ] 0, u'[ ~ ], as 0 u'[ ~ ] 0, as he Inda ondiions in he ~ &u spae C ~ N

6 U 0 is he presen-value disouned inegral of noional sream of insananeous period uiliies feliiies of onsumpion per household member. u[ ~ ] e n u[ ~ ] N gives he household s gross uiliy of onsumpion ha is he household s aggregae uils of onsumpion per period. Beause households are idenial, he seup gives aggregae or soial uiliy whih maps naurally o aggregae onsumpion, saving, apial formaion and growh.

7 The Budge Consrain: Eah household member inelasially and suessfully supplies one uni of labor o mare per uni of ime, whih ommands a mare wage inome of w. Unlie he seup of he bare-bones model, we herefore absra from opimal labor supply deisions.

8 Households an borrow and lend a an ineres rae given by he mare, r: Household inome per adul member a eah period is herefore. w r a na where a is he so of ne asses or finanial wealh per member; i may be negaive or posiive in a given period. a0 is given bequess from earlier household generaions.

9 Everyhing is real and. expressed in unis of onsumables. The household s budge onsrain is: a NB: 2 is derived as: w r a na ~ w [ r n] a ~ Given a=a/n, A evolves as =A =ra+wn-c =A //N=ra+w- 2a - Given ha a =da=da/n=[da.n-dna]/n 2 = - da/n-[dn/n]a/n=a /N-N /NA/N=A /N-na - a +na=a /N hen replae A/N in 2a by his and solve for a 2

10 . If households ould borrow wihou limi a he mare rae of ineres, inenive would exis o pursue Ponzi sheme- hain leer sraegies of finaning in perpeuiy arbirarily high onsumpion levels unseured by presen and fuure inome flows. The budge onsrain would herefore no bind.

11 . We assume ha he redi mare rules ou suh sraegies by imposing a "no Ponzi sheme" resriion requiring ha in he limi he presen value of asses mus be non-negaive:** lim a exp [ r r0 n] d 0 3

12 . This resriion jus means ha deb per household member a < 0 anno in he long run grow faser han he per member rae of reurn o saving [r - n]; in oher words aggregae household deb, N a; anno grow faser han r. [See ahead for furher evaluaion of his expression.]

13 Soluion Seup for he Household s Program The Hamilonian: We shall solve he household s onsrained opimizaion problem by seing up a presen-value Hamilonian H 0 u[ ~ ] e n v [ a ] 4 u[ ~ ] e n v [ w [ r n] a ~ ]

14 where v are oninuous. ime Lagrangian mulipliers giving he presen-value shadow pries of wealh, ha is, he value of an inremen o asses a ime-, measured in unis of ime-zero uils. In he lingo of engineer s, is he "onrol deision or hoie variable; a wih a0 given is he "sae" variable, and v is he "o-sae" variable. NB. The presen value Hamilinion some imes is also se up wih urren value Hamilinion: o do ha jus muliply boh sides of H by n e where H n beomes H=H e and a new Lagrangian q=v ~ n e

15 . H an be hough of as he uiliy prospe of deisions abou ~ evaluaed a deision period = 0. For given shadow prie v, he Hamilonian represens he oal onribuion o ime-zero uiliy of a noional hoie of ~.The prospe is he sum of wo omponens. 2 The firs righ-side erm, u[ ~ ] e gives he ime-zero uiliy value he value oday of eah onsiuen of a noional onsumpion sream. I obviously rises falls wih a higher lower hoie of bu has an opporuniy os benefi given by he seond righ-side erm ie. v [ a ] n ~

16 . v [ a ] 3 The seond righ-side erm,, gives he ime- value alibraed in ime-zero uils of an inremen o ime- wealh whih affes subsequen onsumpion possibiliies ha is produed by a onsumpion hoie. ~ In oher words, he seond-erm is he value o he uiliy prospe of he hange in asses produed by a onsumpion hoie.

17 The Firs Order Condiions: Two FOCs and a erminal or ransversaliy" ondiion mus be saisfied. The maximum priniple Ponryagin e al. 962 requiers The firs FOC H ~ ha is 0, u' [ ~ ] e implying ha v all - ρn u' [ ~ ] e v - ρn

18 The firs FOC indeed helps find a maximum 2 beause H. by assumpion [ ] of he onaviy of uiliy. 2 u''[ ~ ] 0 The FOC implies ha opimal behavior requires ha he marginal uiliy of onsumpion a ime-, disouned ba o he presen ime-zero, be equal o he presen value he value o ime-zero marginal uiliy of onsumpion of he assoiaed hange o asses a ime- So he disouned inrease in marginal uiliy owing o a hoie of higher onsumpion lower saving mus equal he marginal uiliy os of he hange in wealh -v ha is aused by his onsumpion hoie.

19 The seond FOC is: whih is nown as he Ramsey rule of opimal saving. ] [ 8 0 n r v v a H v v a H

20 I implies ha he hange in he value of an. inremen o ime- asses o ime-zero marginal uiliy he "apial gain" in marginal uils mus be offse exaly by he ime-zero marginal uiliy value of he ime- reurn of an inremen o asses he "inome" in ime-zero marginal uils. Noe he resemblane o invesmen behavior, where agens should value equally ax onsideraions aside an inremen of apial gain and an inremen of inome from apial. In Invesmen leures ahead!

21 Finally we have a ransversaliy/ erminal ondiion lim v a 0 9 whih means ha he ime-zero uiliy value of asses held a he end of he world is zero; eiher he shadow prie, v; or asse so, a, mus go o zero asympoially. If his were no he ase here would be erminal "wase" in he family s opimal uiliy of onsumpion program

22 Inegraing he resul of he seond FOC, ύ = -v [r n] ; over ime gives** v v0exp 0 r n d 0 Subsiuing v ino he ransversaliy ondiion, lim v a 0,allows as o express lim lim v a 0 a exp 0 as r n d 0

23 . Noe ha if we evaluae he inegral erm in erms of he average ineres rae beween 0 and ; namely hen he presen value exponenial erm is d r 0 d r r 0 n r n r e e d n r 0 2, exp

24 So he ransversaliy ondiion implies ha lim a 0 3 r n e whih means ha wealh per household member, a, does no asympoially grow as fas as reurns o saving per member, [r - n], or equivalenly ha aggregae household wealh, n N, a e a does no grow as fas as aggregae r reurns o household saving,. e

25 Opimal Consumpion-Saving: We obain he Euler equaion for opimal hoie of onsumpion over ime by differeniaing he firs FOC,**[o ge v-do & lin wih he 2 nd FOC as boh are required o max H] ] '[ ~ ] ~ ''[ ~ v ] '[ ~ ] ~ ''[ ~ ] [ ] '[ ~ ] [ ~ ] [ ~ ]] '[ ~ [ ] [ ] '[ ~ ]] '[ ~ [ 4 ] '[ ~ [ : Wih respe oime,inorder ofind, ] '[ ~ u n u e e u n u e d e d u d d d u d e d e d u d u d e d e u d d dv v e u v n n n n n n n n n

26 Nex, subsiue his resul ino he seond FOC,. 7 ] '[ ~ ] ~ ''[ ~ ] ][ '[ ~ side: he exponenials on eah Geingrid of 6 ] '[ ~ ] ~ ''[ ~ ] '[ ~ from he firs FOC: ] '[ ~ Now subsiuein for 5 ] '[ ~ ] ~ ''[ ~ ] [ - u n u n r u u n u e n r e u e u v u n u e n r v n n n n

27 Nex, solve for ineres rae, r,. u''[ ~ ] ~ '[ ~ n u'[ ~ ] r n u ] u ''[ ~ '[ ~ ] ~ u ] ~ Muliplying he righ - side by ~,: [ ~ ~. ''[ ~ ] ~ '[ ~ ] r u ] u 8 8b 9

28 Alernaively, 9 ould be wrien as ~ ~ u' [r- ] ''[ ~ [ ~ ] ~ ] u The firs erm in he righ hand sided is he elasiiy of iner-emporal subsiuion [is also he inverse of he elasiiy of marginal uiliy]. I measures he sensiiviy of onsumpion growh o [r-ρ] Noe ha ċ/ > 0, if [r-ρ] >0 9b The sensiiviy of he growh of onsumpion o [r-ρ] is higher he lower he s erm in he righ hand u' side of he equaion 9/9b ie ''[ ~ [ ~ ] ~ ] u [See also Romer for same resul using anoher simple approah/9b is also alled he Euler equaion for his maximizaion problem of our households]

29 . The s righ-side erm in braes is he elasiiy of iner emporal subsiuion. This means his elasiiy is a measure of he proporional responsiveness of onsumpion o a proporional hange in marginal uiliy ie.; i measures he willingness o deviae form onsumpion smoohing. We an show ha by noing ha elasiiy of onsumpion o he marginal uiliy is given by d mu d u' u'.. dmu u'' u'' -The s erm in he righ hand side of 9b u''[ ~ ] ~ will be he same as he above definiion of elasiiy if we divide boh sides of eqn 9b by [r-ρ] and hen subsiue for /[r-ρ] in he lef hand side of he resul, he soluion ha is obainable by solving 9b for [r-ρ] -Noe in solving for [r-ρ] above ha he rae of reurn o saving [r-ρ] equals he erm in he righ hand side whih ould be aen as he rae of reurn o onsumpion. A he opimum, households are indifferen beween and herefore u' [ ~ ]

30 The Uiliy Funion: Consider he above resul for r. Noe ha if he rae of ime preferene equals he ineres rae he prie or ime value of money ; hen mus be zero; r r households will opimally hoose a onsumpion ime pah.as we shall see in subsequen leures, he fla onsumpion ime pah may equal he annuiy value of wealh; a resul forming he ore of Milon Friedman s famous Permanen Inome Hypohesis. For o be onsan for r held a some value ; he elasiiy of marginal uiliy, u''[ ~ ] mus be onsan. u'[ ]

31 Hene he populariy of he Consan Relaive Ris Aversion CRRA or Consan Ineremporal Elasiiy of Subsiuion CIES uiliy funion: u ~ under CIESmarginaluiliy is u' ~ ~ ~, 0, 20 2 and he elasiiyof ~ '' ~ ' ~ u u. ~ ~ marginaluiliy see ~ eqn 9/bis onsan,and is equal o 22 :

32 Under CIES uiliy, opimal onsumpion ondiion. 25 ]. [ ~ ~ raeis s opimalonsumpion growh he household' ly, Consequen 24 ~ ~ 23 ~ ~ hereforeurns ou o be, ~ ~ ] ''[ ~ ] '[ ~ ~ r r u u r

33 Noe ha: he opimal ime pah of onsumpion is herefore deermined by he gap beween he ineres rae and he rae of ime preferene, weighed by he ineremporal elasiiy of subsiuion he inverse of he negaive of he elasiiy of marginal uiliy. For a given gap beween r and ρ ; he bigger is he propensiy o subsiue onsumpion ineremporally he larger is, he bigger is he response of he opimal ime pah of onsumpion. I sems from he fa ha he bigger is σ, he bigger is he gain he smaller is he deline in marginal uiliy generaed by a proporional rise in onsumpion and, onsequenly, he more willing are households o deviae from a.a onsumpion ime pah. If r is onsan a r over some inerval from say = 0 o = ; hen inegraing he opimal soluion for ~ gives he onsumpion ime pah: ~ 0 e r 26

34 Firms Produion is neolassial wih he usual CRS properies Q F K, AN 27 and where as before g A A0 e,a0 28 Marginal Produiviies: As in earlier leures q and denoe inensive form variaes: q =Q /AN ; =K/AN. We already now ha he marginal produiviy of aggregae apial and inensive apial are equal F K = f ;

35 . and so ha apial s share in basi and inensive form are naurally idenial F ' K K f ' Q q and ha by CRS degree homogeneiy of produion we an wrie Q A Nf 29

36 So ha labor s marginal produ is **. ' ' ' 30 2 f f e f f A N A K Nf A f A N f N A f A N Q g

37 , The marginal produ of labor, and hene he wage, herefore grow wih he sae of ehnology, ha is wih he exogenous engine of eonomi growh. Noe he same is no rue of he reurn o apial I ould no be oherwise; wages have grown seularly under mare apialism whils he reurns o apial exhibi no suh seular inreases.. Q K f ' 3

38 Profi: The firm s profi per uni ime is Q r K wn 32 where he user os of apial is he mare ineres rae plus he rae of apial depreiaion per uni ime.

39 Subsiuing in We obain: Q ANf and w Q N A f-f' A Nf [ r ] K f [ r ] A N [ f f ' ] A N f ' [ r ] A [ f f ' ] N 0 We ge he sandard resul ha profis are zero in ompeiive equilibrium beause apial will be deployed up o he poin a whih is marginal produ, f ; equals is user os,. [ r ] 33

40 Equilibrium We now join he resuls for households and firms o analyze he ompeiive equilibrium. Remember everyhing is real and pried in unis of onsumables and ha households own he apial so. So we an equae asses per adul per worer in he household seor o apial per worer in he firm seor: a Asse N K N ~ A e g 34 Nb ~=K/N and =K/AN

41 The equilibrium is esablished by solving a pair of ordinary differenial equaions ODEs; one for inensive form apial, he oher for inensive form onsumpion. We now ha apial evolves as sf g n Sine sf f invesmen per effeive worer equals oupu per effeive worer no onsumed, he ODE for apial wrien in erms of onsumpion is f g n 35

42 From he soluion o he household s opimizaion problem, we earlier derived he famous resul ondiioned on he CIES uiliy funion: see Equaion 25 ~ ~ [ r ]. Given our parameerizaion of he evoluion of ehnial g progress, A e, we now ha** ~ ~ g 36 And from he opimal behavior of firms [see eqn 33] we now ha f ' [ r ] and herefore r f '

43 Hene he ODE for onsumpion in inensive per effeive worer form is: To find an equilibrium we also will need he ransversaliy ondiion, wrien earlier as [37] ' ' ~ ~ g f g f g 0 ] [ lim n r e a

44 Afer maing he subsiuions a and A e g -g e. r We an wrie he ransversaliy ondiion as lim e f ' [ f ' ng] 0 38

45 . The resul says ha asympoially he mare rae of ineres he marginal produ of apial less he depreiaion rae exeeds he growh rae of K ha is, exeeds n + g. Hene as, he denominaor goes o infiniy whereas he numeraor is finie, onverging o *: Pu anoher way, a seady-sae i mus be rue ha f ' n g 39 This ondiion beomes imporan jus below, when we evaluae he seady-sae of he sysem, and i disinguishes endogenous saving behavior from he "golden rule" saving rae implied by he Solow-Swan model

46 The ransiional dynamis of he model in, spae is haraerized by he Jaobian of he sysem evaluaed in he neighborhood of he seady sae see he phase diagram afer he nex slide, hus Transiional dynamis and Phase diagram of he model To haraerize he model a is sead sae ae eqn 35 and eqn 37 muliply 37 by This should give us: f g n g [ f ' ] 35b 37b

47 Thus, J f ' g f '' n A he seady sae ċ=0 {see eqn 35b}, hus he Jaobian he deerminan of his marix is: J ss f ' ss. f g '' ss ss n 0. f Sine de of J=μ*μ2 where hese are eigen values of he marix J we have μ, μ2 wih differen sign whih means saddle pah wih one sable arm orresponding o he ve & unsable arm orresponding o he +ve [Noe for 2x2 marix eigen values μ, μ2 ould be found from de M= μ*μ2 & rae of M= μ+μ2 {!apply ha in he above marix. Phse diagram of his follows: [ g f ' ] ''. ss ss 0

48 Phase Diagram of he Ramsey Model

49 Saring a any level of apial 0>0, he eonomy maes a disree jump o he saddle-pah hrough /=0 &/=0 & moves along ha pah o he seady sae ie. Given 0>0, onsumer sele 0 so ha he eonomy is on a saddle-pah; any oher 0 violaes eiher he Euler equaion or he TVC. Seady Saes: Remars: The equilibrium exiss and is unique. The seady-sae growh raes g*,g* & gq* are zero and he seady-sae values of *; * and herefore q* are onsan, jus as Solow-Swan exogenous saving model

50 . So he Solow-Swan resul did no hinge on he saving rae being given exogenously. Remember ha his means ha a seady-sae he orresponding per worer variables, g *, g, * and g * q grow a rae g he exogenous growh rae of ehnologial progress, and ha he aggregae level variables K,C and Q grow a rae n + g.

51 The seady-sae ondiion means ha seady sae onsumpion is And sine saving rae is: f. * g n * * f * g n s f * 0 40 ha he seady sae g n * s* f * and he seady saeou pu is of q* f* 4 ourse 42

52 These resuls mimi Solow-Swan and herefore deliver no value added. And, as in Solow-Swan, o say somehing more explii we need o speify f, he funional form of neolassial produion. However, value added is delivered by he fa ha 0 A he seady sae, ha is a.see equaion 37. Hene we arrive a a ondiion ha is someimes alled he "modified golden rule": f ' g [ f ' ] g 0 43

53 . I follows ha he saving rae and he assoiaed seady-sae apial so when onsumpionsaving hoies are made by opimizing households wih a posiive rae of ime preferene will be less han he "golden-rule" saving rae and apial so. Therefore we will never observe dynamially ineffiien over-saving in seady-sae as in he prooypial Salinis enral Planning' profile in whih he eonomy is direed so as o amass huge sos of apial a he expense of susainable urren onsumpion.

54 . Reall from he Solow-Swan seup and observe in he phase graph ha he goldenrule is defined by a saving rae and seadysae apial so saisfying f ' * gold g n By onras in he endogenous saving neolassial model see eqn 38/ransversaliy ondiion he MPK is g f ' * g n 44

55 whih holds by he ransversaliy ondiion in eq.38; in pariular in he denominaor of 38 we had f ' g n Hene, in he endogenous saving, marginal produ of equilibrium apial, f ; is bigger han ' * 0,as f ' *gold g n. and, herefore By he onaviy of neolassial produion, i mus follow ha he endogenous saving, seady-sae apial so is smaller han he exogenous saving, golden rule seady-sae apial so**. g as *

56 The upshos of f > f * gold in he presen endogenous saving se-up are:. *s* * gold s* gold 45 q** q*gold* gold 46 * s, q* * gold s * gold, q * gold 47 s* s*gold 48

57 . The resuls above sem from he fa ha opimizing households are "impaien" and disoun fuure onsumpion benefis a g effeive rae. Consequenly hey are unwilling o save enough o defer enough urren onsumpion o obain a higher level of susainable onsumpion, * gold, whih ould be reahed if onsumpion was wihou ime value.

58 Finally, noe ha i is diffiul o say muh abou. he saving rae during he ransiion o seadysae. Even if we ommi, say, o a Cobb-Douglas produion funion and a CIES uiliy funion, he ransiion behavior of saving and, here-fore, he onvergene behavior of he model will depend on he magniude of he uiliy parameer in relaion o ha of he produion parameer : see Barro andsala-i-marin for an exellen depiion END END of RCK model END

59 Par B: The Diamond Model/The Overlapping Generaion [OLG ] Model Par B: The Diamond Model or The Overlapping Generaion [OLG ] Model

60 Par B: The Diamond Model/The Overlapping Generaion [OLG ] Model OLG:- General Assumpions of he Model Turnover in populaion due o oninuous birhs and deahs Time is disree raher han oninuous as opposed in RCK People have wo period eonomi lives and do no are abou fuure generaions. Hene here are no bequess, no dynasi behavior, no alruism. In period agens wor and save he "young". In period 2, hey onsume and hen die he "old". A every period here is an overlapping generaion of one young and one old ohor of agens. A "period" is hen a generaion, and so should be hough of as around 30 or 35 years of annual ime. Everyhing is real and expressed in unis of onsumables.

61 Households and heir Uiliy: The Uiliy maximand is analogous o ha in he Ramsey-Cass-Koopmans leure, exep i has a finie, wo period horizon and ime is disree: U, u [, ] u [ 2, ] where, 0 is he onsumpion of generaion when young in period and 2, is he onsumpion of generaion when old in period +. Uiliy will be spei.ed as CIES, i.e., U, 2,, 0, 2

62 The Budge: OLG.Con d Agens inelasially and suessfully supply one uni of labor when young and reeive exogenously given wage inome w. Sine here are no bequess, he iniial asses of he young are zero. he onsumpion of he young is herefore jus wage inome less saving: w S 3,

63 OLG.Con d and he onsumpion of he old is given by he wealh aumulaed from savings when young: 2, r r w S, Subsiuing hese onsumpion values in he wo-period Uiliy program we obain: U w S [ r S ], 4 5

64 OLG.Con d The FOC: Wage inome w and he ineres rae r + are given, and so he hoie "onrol" variable deermining lifeime uiliy of onsumpion is savings ou of inome when woring, S. The FOC for lifeime Uiliy maximizaion is U S w S [ r S ] r 0 6

65 OLG.Con d Equaion 6 implies [ r S ] w S r 7 The lef-side erm wihin braes is 2;+, and he seond righ-side erm wihin parenheses is ; ; hene he FOC implies r 2,, 8

66 OLG.Con d Noe also ha Equaion 8 implies a relaion beween marginal uiliies a + and ha have seen before and will see again: u' [, ] u'[, r 2 9 Solving he FOC [eqn 8] for he growh of inergeneraional, lifeime onsumpion yields ] 2,, r 0

67 OLG.Con d Equaion 0 Implies Taing logs we obain 2,, r The above equaions should be now familiar see equaion 25 in RCK model. 2, ln r ln, -

68 OLG.Con d σ is he ine-ermporal elasiiy of subsiuion [he inverse of he negaive of he elasiiy of marginal uiliy, see eqn 25 in RCK model] Eqn implies he growh rae of onsumpion [2/] depends on he grwoh rae of he mare ineres rae relaive o he ime preferene of onsumpion [+r+/+ρ] weighed by σ. You an ge an idea of his elasiiy if you solve above for σ and ae aime differenial whih will give you [proporional hange in ]/[proporional hange in r relaive o ρ]

69 OLG Con d: The Saving-Inome Relaion: From he FOC we now equaion 7 [ r S ] r w S So we an find he relaion b/n saving and wage inome. Following, as usual, a lo of derivaion seps, he dependene of saving on inome is:

70 OLG.Con d. Where 2 ] [ r w S w r S w r r S w r r S S w r S r

71 OLG.Con d In he overlapping generaions model he dependene of aggregae saving on aggregae wage inome herefore is S w S w, 0 3 The resul ould hardly be oherwise. S w > violaes he budge onsrain. If S w = he young would sarve and never mae i o reiremen. More illuminaing is a boundary ase in whih we have log uiliy,, and he fuure is no disouned, 0 This senario implies see nex slide

72 OLG.Con d The above senario implies 0 0 r 2 4 implying ha he young would pu aside half heir woring life inome o finane onsumpion in old age and, in fa, enjoy higher onsumpion in reiremen beause he inome saved would grow by a faor of + r +.

73 OLG.Con d The Response of Saving o he Ineres Rae: r r w r r w r r w r w r S S r S S he r w S Given ] [ 5 is rae hange, savingresponseo an ineres and

74 OLG.Con d The resul has imporan inerpreaion. Remember ha σ is he ineremporal elasiiy of subsiuion. [See he Ramsey model leure noes & he disussion above] 7 S an be wrien, sine r 2 S r w w S Whih

75 OLG.Con d Hene: If σ >, S r > 0: A rise in he ineres rae indues opimal agens o redue urren onsumpion and inrease urren saving. So 2;+ beomes marginally more desirable han ; ; fuure onsumpion is subsiued for urren onsumpion. The "subsiuion effe" dominaes he "inome effe" of higher reurns o saving. If σ <, S r < 0: A rise in he ineres rae indues opimal agens o ae benefi of more urren onsumpion and redue urren saving. The inome effe dominaes he subsiuion effe

76 If σ =, S r = 0: OLG.Con d This is he log uiliy ase. The inome and subsiuion effes exaly offse eah oher.

77 Firms OLG.Con d We have he usual losed eonomy, neolassial environmen, bu everyhing is now in disree ime. Produion is Q q Tehnologyand labor foregrow a raes: A N r f A F[K,N N g n A ]

78 OLG.Con d Profi maximizaion means firms hire labor up o he poin where is marginal produ equals he given wage w Q N A [ f f ' ] see equaion 29&30 in RCK Model And firms deploy apial up o he poin where is user os equals mare ineres rae: r f ' see eqn 37 &38 in RCK Model 22 23

79 Equilibrium OLG.Con d The Dynamis of Capial: The apial aumulaion equaion is K K w N r K, 24 Now reall ha from he household seor we have ; = w - S and 2; = + r S -. So he dynamis of apial an be expressed N K K w N r K w S N - r S- N- 2, N 25

80 OLG.Con d and herefore he so of apial is K r K r K w N S N w We need an iniial ondiion for he apial so o ge hings sared a, say, period =. The following iniial ondiion senario is imposed a = see he nex slide S S N N - r S - N - 26

81 OLG.Con d We now from he budge onsrain ha he seond period onsumpion of people who are old a = equals he reurn o inome saved when hey were young, ha is, he reurn o heir = 0 saving: N r S 2, 0 0N0 27 Households own he apial so, and he = aggregae apial so, K =, is owned by he N =0 people who are old a =, and i is equal o heir aggregae savings during period = 0 when hey were young and woring: K S N

82 OLG.Con d The onsumpion of hose N =0 old people in period = herefore saisfies I follows from he apial aumulaion equaion [eqn 26] ha a period = 2 he apial so is: , K r N S r N 3 2, Hene N S K N S N S N S N S r N S N S K r K

83 OLG.Con d This resul implies ha apial so equals he aggregae savings of he young. I follows from generaional "selfishness": The old generaion owns he apial so, and he old maes no bequess o heir desendans. Consequenly, as hey exi he world o he grea beyond[mos liely hell, for no bequess], hey sell he apial so o he nex generaion of reirees. The apial so herefore mus be purhased wih he savings of he young. The dynami repliaes iself from generaion o generaion, yielding he equaion above.

84 OLG.Con d Capial per effeive worer in he overlapping generaions seing is whih follows from our assumpions abou he dynamis of labor fore growh and ehnologial hange: A + = + g A and N + = + n N. A g n S A g N N S N N A S N A K 32

85 OLG.Con d Reall ha given CIES uiliy and neolassial produion ' ] ' [ f r f f A N Q w r w S

86 OLG.Con d Hene apial per effeive worer an be wrien + is herefore a nonlinear differene equaion, and i an be solved in losed form only in speial ases of he produion and uiliy funions. ] ' [ ] ' [ ] ' [ ] ' [ 33 g n f f f A g n f f f A A g n w

87 OLG.Con d Cobb-Douglas Produion and Log Uiliy: We onsider he ase in whih produion is Cobb- Douglas and uiliy is logarihmi σ = : q u[ ] ln[ ] Maing hese subsiuions ino he above soluion for, we obain [ [2 ] ] n g 36 [2 ] n g

88 OLG.Con d The Seady-Sae: 38 ] [2 * is saeouou per effeiveworer,seady * Sine ] [2 37 ] [2 * * g n q q g n g n

89 OLG.Con d The orresponding apial per worer and oupu per worer ondiional on seady-sae are, ~ * *and q~ * q q* are ofourse A * and A q* Noe ha he growh raes n and g perain o raes over half a lifeime in he 2-period OLG model and, onsequenly, hey have orrespondingly larger magniudes han in oninuous ime models or disree ime modes wih onvenional periodiiy. see ahead

90 OLG Con d: Convergene Speed when is near *: In he Cobb Douglas-logarihmi uiliy speifiaion of he OLG model i is no hard o learn somehing abou he speed of onvergene. The differene equaion for + in eq.36 was Where f [2 ] n g Taing a firs-order Taylor approximaion of he funion for + a in he viiniy of * we have

91 OLG.Con d * * 40 * * ion abovean be wrien Hene he approxima. and, impliesha in eq.37 * Theseady- sae resul 39 * * * - * * *

92 OLG.Con d We also an easily derive resuls for he speed of onvergene from some iniial ondiion, jus as we did in he analogous exerise wih he Solow-Swan model. Eq.40 implies L * 4 Muliplying he equaion 4 hrough by +α L + α 2 L α L we obainsee he nex slide

93 OLG.Con d and herefore * 42 * * * 0

94 OLG.Con d From overlapping generaion o overlapping generaion he apial so approahes seadysae geomerially, wih onvergene rae ha rises as he share of apial inome,α ; ges smaller. Remember, however, ha a "period" is a generaion. See he nex slide

95 OLG Dynami Ineffiieny and he Golden Rule: Reall ha he exogenous saving of Solow-Swan admis ineffiien over-saving "dynami ineffiieny". By onras, over-saving an never our in he endogenous saving model of Ramsey-Cass-Koopmans wih a finie number of infiniely-lived, opimizing households. In he OLG ase, saving is also endogenous bu households have a finie "2 period" horizon, whereas he eonomy goes on forever. Consequenly, i is possible ha on he balaned growh pah he apial so may exeed he golden rule level a oordinaion problem ha a enral soial planner ould help reify, alhough he behavioral reord of enral invesmen planning does no, o say he leas, inspire onfidene.

96 OLG.Con d Reall ha he golden rule saving rae produing he highes susainable level of onsumpion generaes a apial so saisfying f ' * g n 44 whih in he Cobb-Douglas ase is α * g n 45

97 OLG.Con d So he golden rule Cobb-Douglas apial so is: * gold g n 46 The OLG seady-sae apial so eq. 37 is * OLG [2 ] n g

98 OLG.Con d We have dynami ineffiieny "over-saving" if f ' * OLG f ' * gold [owing o onaviy of f] implying ha dynami ineffiieny is presen when 47 * OLG * gold 48 i.e., [2 α ρ] n g δ α g n

99 OLG.Con d One way o sor ou he maer is o plug in some reasonable parameer values for ρ, n, g and δ, remembering ha we mus express parameers in magniudes suied o ime in generaions in order o ompare he OLG seady-sae apial so wih he golden rule seady-sae apial so. For Example A annual values of ρ = 0.02, n = 0.0, g = 0.02 and δ= 0.05 he proper values in erms of he Diamond model generaion would be approximaely 0.8, 0.35, 0.8 and 0.785, respeively, for generaions of 30 years **

100 Foo noe: Generaional onversion For omparing OLG wih RCM model we need o equae annualized value o generaional value assuming a generaion is 30 years/annualized form. Thus we nee o equae: x _ Annual x _ Annual 30 x _ Generaion x _ Generaion For he generaion-long rae of depreiaion for insane noe ha K K implies K K K _generaion I K K K 29 j0 30 j I 29 j 29 j0 j I 29 j

101 OLG.Con d Insering he above values in he inequaliy ondiion above implies ha If ha is, if * OLG * gold α

102 OLG.Con d Sine α is he share of apial in oal oupu, even under a narrow onepion of apial jus physial apial his ondiion is unliely o be saisfied. Hene, we may onlude ha ineffiien over-saving is no plausible in he OLG framewor wih Cobb-Douglas produion and CIES uiliy.

103 OLG.Con d One an also approah he issue empirially and in a more general onex, and he same onlusion appears o follow. The golden rule requires f ' * g n or [ f ' * ] g n where g + n is he growh rae of aggregae oupu a seadysae, and f ' * is he equilibrium real mare rae of ineres he rae a whih.firms an borrow in he apial mares and no he.ris free. rae a whih, say, he US governmen an borrow. Hene if [ f ' * ] r* g n he impliaion is ha saving is below he golden rule level and, herefore, ouside he dynamially ineffiien region.

104 OLG.Con d In maure eonomies, oupu growh raes over he yle average around 2-3 peren per annum, whih asual empiriism suggess is signifianly less han he real os of apial o he ypial firm. See Romer, haper 2, for addiional disussion. Finally, if he sandard OLG model is amended o allow for bequess "alruism" and he beques moive is reasonably srong, he model delivers resuls essenially he same as Ramsey-Cass- Koopmans. See Barro and Sala-i-Marin, haper 3.8, for demonsraion...end.nex Slide

105

106 End Of OLG See nex Slide for General Conlusion Than You

107 Summary: Growh models wih endogenous savings Savings are endogenously deermined by he onsumpion deisions of households and individuals, whih are derived from uiliy opimizaion. The eonomy onverges o a balaned growh pah, feauring he same properies as he Solow balaned growh pah, whih under erain assumpions abou he disoun rae is below he golden rule level. The speed of onvergene is generally faser in models wih endogenous savings han in he Solow model sine saving adjus o hanges in oher parameers in he models.

108 Summary: Growh models wih endogenous savings Savings are endogenously deermined by he onsumpion deisions of households and individuals, whih are derived from uiliy opimizaion. The eonomy onverges o a balaned growh pah, feauring he same properies as he Solow balaned growh pah, whih under erain assumpions abou he disoun rae is below he golden rule level. The speed of onvergene is generally faser in models wih endogenous savings han in he Solow model sine saving adjus o hanges in oher parameers in he models. END.END.END