Money in OLG Models. 1. Introduction. Econ604. Spring Lutz Hendricks

Transcription

1 Mne in OLG Mdels Ecn604. Spring Luz Hendricks. Inrducin One applicain f he mdels sudied in his curse ha will be pursued hrughu is mne. The purpse is w-fld: I prvides an inrducin he ke mdels f mne and i illusraes hw he mehds we develp can be used. Classical ecnmics psulaes ha mne is a veil, which is sa i des n have real effecs. N everne believes his, bu even hse wh d face a fundamenal quesin: Wh and when is mne valued in equilibrium? This is n bvius. Afer all, if anne handed u a piece f green paper wih a picure f a dead presiden, u wuld presumabl relucan hand ver anhing valuable in reurn. S wha is special abu hese bis f green paper? The challenge is paricularl severe, given ha mne des n pa ineres ( rae f reurn dminance ). Much f mnear ecnmics simpl assumes ha peple need mne fr cerain ransacins, bu ha apprach has prblems (Wallace 997). Frm he pin f view f OLG mdels here is an addiinal quesin: Can mne alleviae dnamic inefficienc? Cnsider a sandard endwmen ecnm wih w-perid husehlds. Wihu mne, here cann be an rade in equilibrium. Therefre, he allcain is auark: c = e and c = e 2. As we have seen, his ma n be Pare pimal. If he marginal rae f ransfrmain is high, u ( c ) / u ( c ) > + n, a ransfer frm curren ung curren ld wuld be a Pare imprvemen. Giving mne he ld allws hem rade wih he ung. Des his mean ha dnamic inefficienc ges awa?

2 2. An OLG Mdel f Mne Cnsider a sandard w perid OLG mdel wihu prducin r bnds. The ppulain grws a rae n. Mne is inrduced as fllws. In perid, he iniial ld are given M bis f green paper, each bearing a picure f a dead presiden and inscribed One U.S. dllar. In ever subsequen perid, he gvernmen prins addiinal paper and hands i he curren ld in prprin he quani f paper he chse when ung. Effecivel, mne pas (nminal) ineres if held frm ung ld age. This assumpin is f curse n realisic; he exensin he case where mne des n pa ineres is discussed in BF, p. 62. The mne grwh rae is cnsan: M / M = + θ. + I wuld seem ha here is n chance ha mne culd be valued in his ecnm. Suppse I were fllw he same plic f he gvernmen and hand u bis f green paper wuld u sell me ur huse fr i? (If es, please le me knw asap!). One f he ke insighs frm his mdel is: he nl reasn wh mne is valued is he belief ha i will sill be valued in he fuure when he husehld wans bain gds in reurn fr his mne hldings. In her wrds: mne is a bubble. Mne being valued is merel a self-fulfilling expecain. I s fragile. Shuld expecains ever change, he value f mne culd evaprae. There are f curse plen f hisrical examples f his (hperinflain episdes). 2. Husehld Excep fr he fac ha bnds are replaced b mne as he nl asse husehlds can hld, he husehld prblem is unchanged. Preferences are u c, c + ). The husehld receives endwmens ( e, e 2 f (perishable) gds. The price f gds in perid is P. The budge cnsrains are herefre c x, P + ( c+ e2 ) = x ( + θ). e = The ung budge cnsrain is wrien in real erms, he ld ne in nminal erms. The amun f nminal mne demanded b a ung husehld is P x. When ld, he husehld receives mne ransfers f θ P x. Tal wealh is hen used bu cnsumpin a price P +. The lifeime budge cnsrain is: e c = c+ e2 ( + θ) / + Ne ha mne acs exacl like a bnd ha pas grss ineres R + = ( + θ) / +. Se up a Lagrangean e Γ = + λ + 2 c u ( c, + c+ ) e c. R + 2

3 The FOCs are well-knwn: u ( ) = λ, u 2 ( ) = λ / R +. These can be cmbined in an Euler equain, ) = R u ( ). A sluin he husehld prblem is a riple c, c +, x ) which u( + 2 ( saisfies he Euler equain and he w budge cnsrains. Opimal behavir can be characerized b a savings funcin (which is nw a mne demand funcin) x ( + e2 = s R, e, ). ( A sluin he husehld prblem is a vecr c, c +, x ) ha saisf 2 budge cnsrains and he Euler equain. I is impran realize ha he husehld prblem is exacl he same as he ne wih ne perid bnds. This is ver general in he mdels we sud here. Since here is n uncerain, all asses are equivalen ne perid bnds. 3. Equilibrium The gvernmen is simpl described b a mne grwh rule: M / M = + θ. + Marke clearing: Mne suppl is exgenus (M ). Mne demand is x, per capia in real erms. If he ppulain size is N, hen M = N x r m = M /( N ) = s( R + ), where m denes real, mne balances per ung husehld. The gds marke clearing cndiin is 2 + n e + e /( + n) = c + c /( ), which is, f curse, he same as in he mdel wihu mne. T verif ha he mne marke clearing cndiin geher wih he husehld budge cnsrain implies gds marke clearing (Walras law), use he ung budge cnsrain f generain geher wih he ld ne f generain : c x and c e2 = x ( + θ) / e = mne marke clearing cndiin prvides he link beween x - and x : x = M /( N = ( + θ) M = ( + θ) x P ) /[ P N ( P ( + n)] / P ) /( + n). The Therefre, x ( + n) = ( e c )( + n) = c e2, which implies he gds marke clearing cndiin. ( Definiin: An equilibrium is a sequence c, c, x, P, M ) such ha. M bes he mne grwh equain. [ eqn] 2. Markes clear. [2 eqn, ne redundan] 3. The husehld chses cnsumpin and mne hldings pimall. [3 eqn] We have 5 (independen) equains in 5 unknwns (per perid). 3

4 4. Characerizing Equilibrium As in he case f he ecnm wih capial, we can characerize equilibrium b finding a difference equain fr he ecnm s sae variables. These are he mne sck and he price level. Hwever, nminal variables pla n rle in ur ecnm. Therefre, we nl need wrr abu real mne balances as he sae variable. The bvius place lk fr a difference equain in m is he mne marke clearing cndiin, m s( R ). = + T characerize he pah f m, we need subsiue u R +. Since R + = ( + θ) / +, we need an expressin fr inflain. Frm M + m N = M m N we have () R + = ( + θ) / + = ( + n) m + / m = s + n) m / m ) m (( + This is a difference equain ha cmpleel deermines he pah fr m (geher wih a bundar cndiin). Bu wha des i lk like? Nw cmes he rick: he ffer curve. Warning: This is unusual and smewha hard grasp (see BF fr an expsiin). The ke idea is use he husehld s inerempral cnsumpin allcain figure u hw mne evlves ver ime. Le s firs hink abu he husehld s cnsumpin chice, wihu an bvius mivain. The lifeime budge cnsrain ells us ha he husehld faces an inerempral budge cnsrain wih a slpe f R +. We nex pl hw he husehld s cnsumpin chices a bh daes var wih he ineres rae. This is dne in he lef panel f Figure. Each pin n he ffer curve is bained as he angenc f an indifference curve wih a budge line ha has slpe R + and ges hrugh he endwmen pin. Thinking abu he cnsumpin allcain allws us place resricins n he shape f he ffer curve. The ruble is ha here are incme and subsiuin effecs f changing R. Fr ver lw levels f R he husehld prbabl wans brrw. We dn need wrr abu ha case since he husehld cann d ha in his mdel. 4

5 If we sar wih he ineres rae where he husehld jus eas his endwmen and hen graduall raise i, he husehld will firs reduce c because subsiuin effecs dminae. In ha regin, s is increasing in R and s is bviusl R s. We herefre knw ha he ffer curve is undefined fr lw values f R and dwnward slping a leas iniiall fr higher levels f R. I ma evenuall bend backwards, bu i can inersec each line hrugh he endwmen pin nl exacl nce (he sluin he husehld prblem is unique). Nex we relae he ffer curve mne. This wrks because he ung budge cnsrain implies ha ung cnsumpin is relaed real mne demand in as m = s R + ) = e ( c. Smehing ver similar hlds fr he ld: ( + n) m + = R + s( R + ) = c+ e2. Wha happens is ver clear, if we absrac frm ppulain grwh fr a mmen. Yung savings cnsiue real mne demand in. Old dissaving a + becmes he mne suppl a +, which in equilibrium mus be purchased b he new ung. Hence, if he husehld wans pspne a large par f is cnsumpin, hen m + mus be large relaive m. We can herefre hink f he ffer curve as describing a relainship + n ) m + = F( m ). Here i is emping sa: F mus be linear because F(m ) = R + ( m. Bu ha is n useful i merel recvers he definiin f R. The righ panel f Figure mves he endwmen pin he rigin and shws a mirrr image f he firs graph. Since ever pin n he ffer curve is cnsruced ne he budge line fr a disinc R, he ffer curve inersecs ever ra hrugh he rigin exacl nce. The hriznal axis nw shws m and he verical ne shws (+n) m +. In equilibrium, s = m and herefre we can read ff R s = (+n) m + frm he ffer curve. Using a ra hrugh he rigin wih slpe (+n) allws us find m +, and s n. I fllws direcl ha here is ne mnear sead sae (inersecin f ffer curve and ra hrugh rigin) which is unsable. If he ecnm is perurbed awa frm he sead sae, he mne sck eiher cllapses zer r blws up infini. This happens hrugh inflain r deflain recall ha nminal M grws a a fixed rae. Wha are he prperies f he sead sae? Per capia real mne balances, m, are cnsan ver ime. Since he grss rae f reurn n mne is given b R = + θ) P / P = ( n) m / m, i + ( mus equal + n in sead sae. Hence, he ne reurn is negaive if n < 0. The sead sae inflain rae is given b P + / P = (+θ) / (+n), which ma als be negaive. 5

6 Offer curve Rs c 2 Offer curve R s(.) e 2 s(r) s (+n) c e m + m e s(r) Figure We can nw race u he evluin f he mne sck ver ime. Firs, hw is he iniial mne sck deermined? The answer is: i isn. Nhing guaranees ha mne is even valued in equilibrium. If mne is valued, i is nl because agens expec ha i will be valued in he fuure. Bu he equilibrium wih m 0 = 0 (which means P 0 = ) is a perfecl valid equilibrium. Bu i ges wrse: an m 0 in a cerain range ields an equilibrium here is a cninuum f equilibrium pahs. Wha range f m 0 values is accepable? Cnsider firs he case where m 0 is he righ f he sead sae. Such a pah wuld have deflain. Bu such a pah is acuall n feasible because evenuall real mne balances wuld exceed al upu. Cnsider nex he case where m 0 is less han he sead sae value. On his inflainar pah, R falls ver ime. Therefre, he inflain rae mus be acceleraing: he nl wa induce peple n save mre is lwer he rae f reurn. The bizarre implicain is ha mne isn valued asmpicall (P ). If he mne sck is exacl equal he sead sae level, peple save exacl he righ amun hld he mne sck cnsan ver ime. Tha is, he sead sae rae f reurn mus be n. Given he fixed nminal rae f mne grwh, his requires + n = ( + θ) /( + π) r apprximael n = θ π. Bu nw suppse m is a bi belw he sead sae. Then he equilibrium rae f reurn mus be belw n; herwise peple wuld demand much mne. Bu ha means he inflain rae mus be higher han he ne ha mainains a cnsan sck f mne. S nex perid m will be even smaller and he required π even larger. This psiive feedback ccurs frever, shrinking he mne sck a an ever increasing rae. 4. Dnamic Efficienc Hw mne relaes dnamic efficienc urns u depend n he slpe f he ffer curve a is rigin. There are w cases. In he Samuelsn case, he slpe f he ffer curve a he rigin is less 6

7 han (+n). The figure abve implicil assumes his case. In he Classical case, he slpe is greaer han (+n). In he Samuelsn case he nn-mnear ecnm is dnamicall inefficien. T see his, ne ha he ineres rae in he nn-mnear ecnm is deermined b he Euler equain β + e2 ( + r + ) u'( c ) = u'( c ). Bu in his ecnm, here is n rade. Therefre, + r = u'( e ) /[ βu'( )]. This means ha he ineres rae is given b he slpe f he indifference curve ha ges hrugh he endwmen pin. The ecnm is dnamicall inefficien, if he ineres rae is lwer han n, which exacl means ha he ffer curve is flaer han he (+n) line. In he Classical case here is n inersecin f he ffer curve wih he (+n) line (excep a zer, f curse) and a mnear equilibrium des n exis mne is never valued in equilibrium. A he same ime he nn-mnear ecnm is dnamicall efficien. The main resul is herefre: Mne is valued in equilibrium nl in an ecnm ha wuld be dnamicall inefficien wihu mne. Ne furher ha he mnear sead sae is dnamicall efficien. The ineres rae equals he ppulain grwh rae. I is herefre n pssible make husehlds beer ff b ransferring resurces frm ung ld r vice versa. Mrever, he mne grwh rae des n affec he sead sae allcain mne is super-neural. This wuld be quie differen, if mne did n pa ineres (see BF, chaper 4). T summarize: Saring frm a dnamicall inefficien nn-mnear ecnm, giving peple mne resuls in a dnamicall efficien sead sae (if mne is ever valued and if he sead sae is ever reached). Bu if he ecnm is dnamicall efficien wihu mne, mne will never be valued in equilibrium. 5. Is his a gd her f mne? The bis f green paper in his mdel are n reall mne. The have ne impran feaure f mne: he are inrinsicall wrhless. Bu he dn have he secnd feaure ha makes mne ineresing: rae f reurn dminance. This refers he fac ha he nminal reurn n mne in he real wrld is zer and is herefre dminaed b he reurns paid b her asses (such as -bills). The her des accmplish ne impran ask: i highlighs hw mne migh be valued jus because everbd expecs i be valued mrrw. Esseniall, mne can be a bubble: i has n inrinsic value, bu is acceped b everne purel because f hese expecains. The her als highlighs hw fragile he rle f mne in such an envirnmen is. 7

8 The her fails miserabl, hwever, when i cmes explaining wh anne wuld hld mne. If ne adds anher asse he ecnm, mne will have pa he same real rae f reurn if i is be valued. I is mre challenging (and sill an pen issue) answer wh mne wuld be valued in equilibrium, if here are her asses ha pa higher raes f reurn (Wallace 997). 6. References CM 4 has maerial n mne, bu beer references are: SLj ch. 8, BF ch. 4., 5.4; MW ch. 0 Wallace, Neil (997). A Dicum fr Mnear Ther. Federal Reserve Bank f Minneaplis Quarerl Review. An ineresing exensin (BF p. 62-3) is he case where mne des n pa ineres. The ld simpl receive a lump-sum ransfer f mne in each perid. In ha case, i can be shwn ha mne is n superneural. Tha is, he rae f mne grwh affecs he allcain. The mnear sead sae is als n lnger Pare-pimal. 8