Overlapping Generations and Pension Systems

Transcription

1 Overlapping Generaions an Pension Ssems I. Inrouion In he previous haper we suie he onsumpion-savings eision in he simples of all frameworks- a wo perio moel. Alhough useful for gaining inuiion, his moel is raher limie in aressing ineresing maro quesions. One lear problem wih his onsru is ha he worl oes no en in wo perios. In applie nami eonomi heor, here are wo sanar was o analze he onsumpion-savings eision. The are. The infiniel live families wih parens aring abou heir hilren s uili. 2. The long bu finie-live people who leave heir hilren no bequess. For some issues, suh as assessing he quaniaive onsequenes of he Amerian paas-ou-go soial seuri ssem, i maers whih absraion is use. For ohers, suh as assessing he onsequenes of ehnolog shoks, publi finane shoks or oil shoks for business le fluuaions, for all praial purposes i oes no maer. The infiniel live onsru is b far he simpler o analze.

2 This haper uses he long bu finie live onsru. The nex haper uses he infiniellive famil onsru. The moel ha is suie in his haper is he Overlapping Generaions Moel. The moel ha is suie in he nex haper is he Neolassial Growh Moel. Effeivel, he Neolassial Growh Moel is jus he Solow Moel bu where savings is eermine b he uili maximizing hoies of househs. II. The Overlapping Generaions Moel The overlapping generaions moel was evelope b Paul Samuelson in 957. In his moel, people no onl live for a finie number of ears or perios, bu in an perio, here are people of ifferen ages or generaions alive. The simples of OLG moels assumes ha people live wo perios. Thus, a an poin in ime, here are some people an oung people alive. Le N be he number of people born in perio. Thus, hese people are oung in perio an in perio +. In he simples OLG moel, whih will be he basis of he analsis in his haper, here are N oung people an N - people alive in perio. In wha follows, we will assume ha N =N= for all =,2,3,4. Noe ha in perio =, he firs perio of our moel, ha Eah generaion enjos onsumpion when he are oung, an onsumpion when he are,. Evenuall, we will assume ha eah generaion values leisure when oung an. For now, however, we will eal wih he simpler ase where uili oes no epen on leisure, namel 2

3 U(, ) log log () Noie ha we have impliil se he isoun faor β =. This jus simplifies he algebra. None of he resuls woul hange if we were o allow for β <. III. ENDOWMENT ECONOMIES We begin wih a ver simple worl wih no prouion. Insea, eah person sars eah perio of heir life wih a erain quani of he onsumpion goo. We all his an enowmen eonom. For he presen purpose, le us assume ha eah onsumer is enowe wih e unis of he onsumpion goo when oung an e unis of he onsumpion goo when. We will firs solve he equilibrium for his eonom when here is no governmen an hene no governmen poliies. We will see ha if people are beer enowe when oung, he equilibrium will no be ver esirable for househs. The reason for his is raher sraighforwar. Househs wan o have relaivel similar onsumpion over heir lifeime. If he are beer enowe when oung, hen he esire for smooh onsumpion means ha he wan o save. The problem in his eonom is ha here are no borrowers. The onl oher people alive in he perio are. You as a oung agen woul never len o an person beause sal he ie a he en of he perio an so are no aroun o pa off he loans. Moel. No Governmen Poli- AUTARKY. 3

4 In his firs eonom, here is no governmen poli. Given he ispariies in enowmens beween heir wo life perios, eah oung person woul ieall like o len so as o smooh ou onsumpion. Bu an he o his? Here is he beau or rik of he Overlapping Generaions moel: Who are he going o len o? The onl possible agens are he people alive in he perio. Bu his woul be a supi iea o len o he. Wh? Beause he won be aroun in perio + o pa bak he loans. Hene, he onl equilibrium for his eonom is one where everone eas heir enowmen, namel (, 2 ) = (e,e ). Thus, here is no rae, an for his reason, he eonom is referre o as one being uner Auark. Formall, a ompeiive equilibrium mus solve he following se of oniions Uili maximizaion.,, an s mus maximize uili given b Equaion () subje o his buge onsrain when oung an his buge onsrain when, namel, s (2) e An e s( r) (3) Marke Clearing o N, N, Ne, N e, (4) o N 0 (5) s The buge onsrains of he househ in he Overlapping Generaions moel is no ifferen han wha we suie in he previous haper on savings. The househ when 4

5 oung eermines how muh of is enowmen when oung o save, an when eas is enire enowmen an saving an ineres. Noe ha in ever perio, he goos marke lears as he bon marke or savings-loans marke lears. Beause onl oung perio agens an borrow or len in his worl, he oal savings or borrowing of he oung mus equal zero in equilibrium. To formall, haraerize he equilibrium, we sar wih he bon marke learing oniion. I is obvious, ha s*=0 in equilibrium. Using his resul an he househ s buge onsrain when oung we arrive ha =e. Using again he equilibrium resul ha s*=0 ogeher wih he househ s seon perio buge onsrain iels he resul ha =e. This means ha here is onl one oher hing o solve for in he equilibrium, he real ineres rae, r. I is ver eas o eermine he equilibrium ineres rae for his eonom. We simpl use he uili opimizing oniion ha U / U / e e. (6) r Thus, he impliaion ha people ea heir enowmens in eah perio is ha he real ineres rae will be negaive whenever e >e. Wh is he real ineres rae in he equilibrium negaive? Reall, ha as people are beer enowe when oung versus, he reall woul like o save. However, here are no borrowers in he eonom. In an equilibrium, people have o be unwilling o hange heir alloaions a he given pries. The real ineres rae mus be negaive in equilibrium. 5

6 This is neessar beause in equilibrium here is no borrowing or lening. Given he srong esire o len b he oung, he ineres rae mus be suffiienl low o issuae hem from borrowing. In his ase, i mus be negaive so ha for eah $ save he woul have sa $.50 in heir bank aoun when he are. Properies of he Equilibrium Pareo Opimali. Suppose for he sake of illusraion ha e =7 an e =. Then his woul be wha househs ea in eah perio, an he real ineres rae woul be r=e /e -=-.85. The househ woul obain uili over is lifeime equal o ln7+ln =.94. Now suppose ha a planner ook 3 unis of he goo awa from eah oung an gave i o an person. The = =4 an he uili over he househ s lifeime is ln4+ln 4= Ever generaion woul be happier wih his alloaion inluing he people who are alive in he firs perio of he moel. The alloaion wih = =4is learl feasible, as he resoure onsrain is ha he sum of oung an onsumpion anno exee 8 unis of oupu in an perio. Eonomiss have an expression when he ompeiive equilibrium alloaion an be ominae in he above sense. Speifiall, he sa he Compeiive Equilibrium is Pareo Ineffiien when here exiss an alernaive alloaion whih is feasible whih () makes a leas one househ sril beer off an (2) makes no one worse off. In he ase ha no suh alloaion exiss, we sa he Compeiive Equilibrium is Pareo Opimal. 6

7 The auarki equilibrium in he above examples is Pareo Ineffiien. Given everone 4 o ea when oung an 4 when Pareo ominaes =7 an =. In general, whenever he real ineres rae is negaive, he ompeiive equilibrium will be ineffiien in he above sense. The ineffiien in he alloaion eermine b pries means ha governmen poli an be esirable. We now explore his iea b onsiering wo pes of pension ssems: a pa-as-ou-go ssem an a full fune ssem. Governmen Poli We nex onsier wo pe of governmen poliies ha go uner he heaing of pensions ssems. These are a Pa-as-ou-Go/Define Benefis ssem, where pension or soial seuri axes ollee in he perio are pai ou o he reirees in he perio, an where he paou o reirees is inepenen of wha axes he pai when oung. The seon is a Full Fune Ssem wih Define Conribuions. The ke feaure of his ssem is ha he amoun he governmen pas ou o a reiree is equal o his onribuion o a governmen reiremen aoun ha pas an ineres rae. The efiniion of a full fune ssem is where he funs ou onribue when ou are oung o he pension ssem are he ones ha are use o pa ou our benefis when. As ou shall see, his is a bi of a suble issue wih he poli we su. A Pa-as-ou-Go/Define Benefi Pension Ssem Auall, he negaive ineres rae implies he ompeiive equilibrium is no Pareo Opimal when here is no populaion growh. Wih populaion growh, he neessar an suffiien oniion for he Compeiive Equilibrium o be Pareo Opimal is ha he ineres rae has o be greaer or equal o he populaion growh rae. 7

8 We nex inroue a ransfer/ax ssem whereb he governmen imposes a lump-sum ax on he oung an gives hose reeips bak o he alive in he perio in he form of a lump-sum ransfer. The governmen lives forever. The governmen s buge onsrain in eah perio is hus, 0 N Tx N Tr (7) Where Tx is he lump-sum ax on he oung in perio an Tr are he ransfers on he alive in perio. The buge onsrains of a generaion onsumer in perio an perio + are s e Tx (8) e Tr ( r s (9) ) An equilibrium mus saisf he following se of oniions Uili maximizaion.,, an s mus maximize uili given b Equaion () subje o his buge onsrain when oung an his buge onsrain when, Equaions (8) an (9) Governmen Buge Consrain given b Equaion (7) Marke Clearing o N, N, Ne, N e, o N 0 s Noie ha he marke learing oniions are unhange b his pe of goverenme poli. In he equilibrium, here will no be an borrowing or lening for he same reason as before. Hene, in eah perio people ea heir afer ax enowmen. 8

9 Again for he purpose of illusraion, le us assume ha e =7 an e =. Furher, le us assume ha he governmen axes from eah oung agen 3 unis of oupu an gives hose 3 unis of oupu o he agen. To full solve ou he ompeiive equilibrium, we again sar wih he impliaion ha s*=0. Then going o he onsumer s buge onsrain when oung an using he amoun of he ax, Tx=3, we onlue ha =4. Nex, we use he resul ha s*=0, an Tr=3 ogeher wih he buge onsrain when o onlue ha = 4. Finall, we use he uili maximizing oniion given b (6) o onlue ha (+r)=. People are surel happier in his worl han in he previous worl where (, ) = (7,). In fa, we oul easil eermine he ifferene in welfare b solving for he faor we mus sale onsumpion up in boh perios. Namel, we wan o fin λ suh ha log[( )7] log[( )] log( 4) log( 4). Pension Ssem#2 Full Fune/Define Conribuions The governmen poli is one whih now gives oung househs he opion o save in he form of a reiremen aoun foun. In oing so, he governmen effeivel issues eb. In his wa he governmen fills a ruial voi in he eonom- beoming a borrower in a worl shor on borrows. A riial par of he governmen poli is wha how i uses he funs ha oung agens pu in heir reiremen aouns. The assumpion here is he governmen akes hese loans an uses hem o pa off he eb obligaions i has wih he agens alive in he perio, 9

10 namel, pas for he pensions of he. This implies he governmen s buge onsrain is B (0) ( r ) B This is auall he onsrain in all perios exep for he firs perio of he moel. In he firs perio, here are N 0 people alive, who have no been par of he full fune ssem an have no reiremen aouns. Thus, he governmen has no pension obligaions o hem. The quesion sill remains as o wha he governmen shoul o wih he savings i akes in from he oung born in perio. Wha we assume is ha he governmen makes a lump sum ransfer o he alive in perio equal o he amoun of is borrowings. Namel, in = B N () 0Tr0 Formall, a ompeiive equilibrium mus solve he following se of oniions Uili maximizaion.,, an s mus maximize uili given b Equaion () subje o his buge onsrain when oung an his buge onsrain when, namel, s (2) e An e s( r) (3) Governmen Buge Consrains given b Equaions (0) an () Marke Clearing o N, N, Ne, N e, (4) o N s B 0 (5) 0

11 The ke oniion o noe is he bon marke learing oniion, whih is now he sum of privae borrowing/lening an he governmen borrowing/lening. Again, o rsalize he resuls, we reurn o solving he ompeiive equilibrium for our numerial example where e =7 an e =. On he governmen sie, we assume ha he Full Fune pension ssem is assoiae wih governmen borrowing 3 unis from eah oung agen, i.e. B =-3. In he firs perio, eah agen ha is alive herefore reeives a ransfer of Tr=+3. We now an solve for he ompeiive equilibrium. Again we begin wih he bon marke. For he urren poli, as we now assume ha he governmen borrows 3 unis of he goo from he oung, equilibrium implies ha s*=3. Nex we use his resul wih he buge onsrain of he oung an onlue ha *= 7 s* = 4. Solving for is a bi rik in his ase, beause he righ han sie of he buge onsrain of he househ when epens no onl on he savings bu he real ineres rae. The real ineres rae mus saisf he uili maximizing oniions, ha r whih is bes rewrien as r. (6)

12 To proee, we make use of an earlier resul ha he househ s wo buge onsrains iel he following ineremporal buge onsrain is r e e r (7) Using (6) wih (7) implies Now using he resul ha =4, we have so ha r*=0. I follows from (6) ha = =4. e 2 e (8) r (9) r Summar of Finings Noie he imporane of governmen in his worl shor of borrowers. The governmen an make people beer off b implemening a pa as ou go pe pension ssem or he full fune ssem. Moreover, he resuls are he same in boh ases. This equivalene of equilibrium resuls is a onsequene of his being an enowmen eonom. Reall, here is no prouion, an people o no value leisure. As suh he governmen poliies o no have an isorionar effes. We nex ompare hese wo alernaive soial seuri ssems in a worl wih prouion an wih leisure. Before inrouion prouion in he eonom, noe a surprising resul in he ase of he full fune ssem. As we have speifie he poli, he governmen has a efii in he firs perio. (I pas ou a ransfer of 3 o eah person, an pas for i b borrowing.) In all fuure perios, he governmen rolls over is eb. Thus, we have a siuaion where 2

13 a governmen efii is no me wih a fuure surplus. This ma erainl seem a onraiion of he resul we esablishe in he previous haper whih inroue he governmen buge onsrain. The seemingl onraior resul is an arifa of he real ineres rae uner auark being negaive. In his ase, he governmen an roll over is eb eah perio wihou seeing he eb exploe. This srange resul was firs shown b Rober Barro in a paper he publishe in he 994 Journal of Poliial ile Are Governmen Bons Ne Wealh? III. PRODUCTION ECONOMIES We oninue o mainain he assumpion ha iniviuals live for wo perios wih an equal number born ever perio. Now insea of being enowe wih goos when oung an, we assume ha he are enowe wih ime, whih he an alloae beween marke prouion an non-marke aiviies. We shall assume ha he have uni of ime ha he an spli beween work an leisure. The uili of a person born in perio epens on heir onsumpion when oung, onsumpion when, as well as heir leisure when oung an leisure when. We speifiall assume ha lifeime uili is given b: U(, l,, l ) ln l [ln l ] (20) We will se β= as in he pure exhange moel so as o simplif he algebra. The α parameers eermine how muh ou value leisure when oung an. The uili funion given b (20) is slighl ifferen from our haper on work hour ifferenes 3

14 where we were ha leisure in he uili an aking a logarihmi form. The swih o his funional form wih uili being a linear funion of leisure simplifies he algebra an allows us o onsier he ase where he househ full reires when. The moel is onl ineresing if we assume ha people value leisure more when ompare o oung, i.e. α > α. One wa o hink of his is ha he flipsie of leisure, namel working, requires a lo more effor when. As we shall show, he assumpion ha here is greaer isuili of working when is imporan o obaining he resul ha he equilibrium uner auark is no Pareo Opimal on aoun ha he eonom is shor on borrowers. Inuiivel, if i is less painful o work when oung, people woul like o work a lo when oung an no muh when. Wih a big inome in heir firs perio, he woul like o save so he oul enjo a high level of onsumpion when while no working muh. This effeivel gives us he same siuaion in he pure exhange eonom of a worl shor on borrowers. To omplee he moel, we nee o esribe he prouion sie of he eonom. We will oninue wih he assumpion we mae in he haper on work hour ifferenes ha prouion is linear in oal hours. Speifiall, C AH (2) S Moel #: Auark Now ha we have ae leisure, i is bes o begin wih a su of he worl where here is no governmen. Jus as before, here will be no rae beween generaions. This is a prouion eonom, so solving for he equilibrium requires a lile bi more work. To o 4

15 his, we go o he uili maximizing problem of a oung onsumer. The onsumer of generaion will hoose (,l,h, s,,l,h ) o maximize uili subje o is oung perio an perio buge onsrains s w h w h ( r) s an he ime use onsrains l h l h We now have hree uili maximizing oniions, one for he MRSC =MRSE beween an l, one for he MRSC =MRSE beween an l, an he las for he MRSC =MRSE beween an. These are w w U / U /. r In aiion o uili maximizaion, an equilibrium mus saisf profi maximizaion. Given he assumpion of he linear prouion funion, he profi maximizing oniion is simpl w =A. For he prouion eonom, here is also a labor marke learing oniion in aiion o he goos marke an bon marke. An equilibrium mus herefore saisf he following se of oniions Uili maximizaion. Profi maximizaion Marke Clearing 5

16 o N, N, C s A H o N 0 s o N h N h H Now we an solve for he equilibrium. We sar wih he uili maximizing oniions relae o onsumpion an leisure when oung an when. The MRSC=MRSE oniions impl =A /α an =A + /α. Assuming ha TFP oes no grow, i.e. A =A, we have ha (+r)=α /α. Noie ha hese are he equilibrium soluion as he are expresse as funions of he parameers. Moreover, ha he oniion for a negaive real ineres rae in equilibrium is α <α. The nex sep is o solve for h an h. Here we use he resul ha s*=0 in equilibrium as here are no borrowers in his eonom. This resul wih he resuls, =A /α an w=a in he oung perio buge onsrain allows us o solve for h =/α. Appling he same logi o he perio buge onsrain allows us o solve h =/α. An imporan poin is ha even if he househ haes o work when, i.e. α /α, is.0 or more, he househ mus work when. He mus ea when, an here is no oher wa o o his bu o work, regarless of how unpleasan he ask is. Pension Ssems Now we will reinroue a governmen ino his eonom. In oing so, we shall assume ha α < α so ha he auarki equilibrium is no Pareo Opimal. Aiionall, we onsier onl hose governmen poliies whih guaranee equal onsumpion for he househ when he is oung an. Given he preferenes, his implies ha he 6

17 equilibrium ineres rae mus be zero. Moreover we will have he pension ssems guaranee ha he househ will reire when. The riial impliaion of his is he oniion ha MRSC=MRSE beween an l no longer hs. As he househ full reires, he is on a orner wih respe o his opimizing oniions an a a orner he slope of he buge onsrain nee no equal he slope of he inifferene urve. This is shown graphiall below Figure : Corner Soluion 00-h U 0 To onsoliae on spae an reunan, we will onsier a ver general speifiaion of he governmen poli. We will speif he househ s buge onsrains when oung an uner his general speifiaion of poli, as well as he governmen s buge onsrain. We will hen speialize he governmen poli so as o onsier boh he pa-as-ou go ssem an full fune ssem. 7

18 The governmen poliies will onsis of a ax rae on labor inome ha is he same for boh oung an, (τ) borrowing/lening (B), an lump-sum Transfers o (Tr). These ransfers represen he soial seuri benefis ha he governmen pas o in he eonom. The buge onsrains of a person born in perio are s ( ) Ah ( ) Ah s Tr Noie ha we have use he profi maximizing resul ha w =A =A. We an solve for s in he perio buge onsrain o arrive a he ineremporal buge onsrain, namel, (assuming ha r=0) ( ) h ( ) h Tr The governmen s buge onsrain in eah perio is B Ah Ah Tr B Again, here we are imposing he assumpion ha r=0. This will be vali b resriing governmen poliies o hose ha guaranee ha onsumpion is he same in boh perios of a person s life. An equilibrium mus saisf Uili maximizaion Profi Maximizaion Governmen Buge Consrain Marke Clearing 8

19 o N, N, C s A H o N s B 0 o N h N h H Having esribe he eonom an equilibrium oniions for a general governmen poli, we speialize he moel o onsier he pa-as-ou go ssem an he full fune ssem. The Pa as You Go/Define Benefis Ssem There is no governmen eb in his ssem, i.e. B + =B = 0. The governmen simpl axes he labor of he oung a rae τ > 0, an hans he reeips Tr=τ A h o he aliven he perio in he form of a ransfer. Reall, ha he ke is ha he ax rae an soial seuri benefis are suh ha he guaranee ha =. To full haraerize he equilibrium, we use he resul ha s*=0. Then from he oung perio buge onsrain we have ha =(-τ)ah. From he perio buge onsrain, = Tr=τ A h. Using hese wo resuls ogeher wih =, we an see ha he ax rae τ=/2. We onl nee o solve for h o omplee he soluion. Here we use ha he MRSC=MRSE oniion beween oung onsumpion an leisure is he same as in he auarki worl exep ha he prie of leisure is (-τ)w. I follows ha =(-τ)/α. Sine τ=/2, A. Given his ax rae an given ha =(-τ)ah, i follows ha 2 h. Full Fune/Define Conribuion Ssem 9

20 There are no axes on he oung an no ransfers o he (exep for an iniial ). Insea he governmen issues eb an roles i over ever perio. (The alernaive Again, he eb is exal he amoun ha ensures ha onsumpion is equal in boh perios of a person s life. Sine here are no axes on oung inome, he MRSC=MRSE oniion is he same as in auark, namel, ha A. This is also. Using his resul wih he ineremporal buge onsrain, we an solve for h. This is 2 Ah, whih implies ha h 2. Young onsumpion is hus exal half of oung inome so ha savings A s, whih is he amoun of governmen eb. Summar an Conlusion The following able summarizes he equilibrium pries an quaniies beween auark an he wo governmen pension ssems h h w r C s B B/C s Auark A/α A/α /α /α A -α /α A/α +A/α 0 0 Pa as ou go.5a/α.5a/α /α 0 A 0 A/α 0 0 Full fune A/α A/α 2/α 0 A 0 2A/α A/α /2 I is no iffiul o show ha he uili of eah househ uner he full fune ssem is greaer han uner auark. For he pa-as-ou go ssem, we nee some aiional resriions on he parameers. 20

21 The big ifferene beween he wo programs is on he amoun of work hours b he oung, being wie as large in he ase of he full fune ssem. The reason for his harkens bak o our su of he isorionar effes of axes on labor hoies. There we foun ha axes on labor inome reues he suppl of labor when he ax reeips are reurne o he househ as a lump-sum ax. This pure wealh effe is wha rives he lower oung work hours in he ase of he pa-as-ou go ssem ompare o he full fune ssem. The iming of his ransfer is no imporan, as onl ineremporal wealh maers. This ransfer b making a oung househ feel wealh, means ha he wan o enjo more leisure when he are oung. The las ommen on hese poliies relaes o he size of he governmen eb in he full fune ssem. As he able shows, governmen eb is 50 peren of GDP in ever perio. Tha is a large eb o GDP raio, an e here is no risk ha he eb will no be pai off in our moel. This suggess ha i migh be somewha inorre o all he high eb o GDP raios in some ounries a problem. Problems. Consier he following enowmen Overlapping Generaions moel where people live wo perios. Suppose people are enowe wih -ε unis of he goo when oung an ε unis when. Assume ε < /2 Eah generaion has he same number of people. Preferenes are ln( ) ln( ) where β< a. Solve ou he equilibrium where he governmen implemens a pa as ou go soial seuri ssem wih lump sum axes so ha agens onsume /2 when oung an /2 when. 2

22 b. Can he governmen implemen a full fune ssem ha gives he same onsumpion for oung an as in he pa as ou go? Show wha happens o governmen eb if he Governmen borrows from he firs oung generaion.. Wha shoul be he governmen s pa as ou go ssembe in orer for here o be a full fune ssem where he governmen oul roll over is eb ever perio wih no hange in borrowing ever perio. 2. Consier he Overlapping Generaions moel wih prouion suie in he haper. Namel, Eah generaion has he same number of people. Preferenes are ln( ) [ h ] ln( ) [ h ]. Eah agen is enowe wih one uni of ime s in boh he firs perio an seon perio of his life. Prouion is given b : Ah. a. Normalize A= an alibrae he value of α so ha in he equilibrium wih he pa as ou go ssem oung people work.25. (This is he seon row in he Summar an Conlusion Table). b. Suppose α =2 α. Verif ha people are beer off in he pa as ou go ssem ompare o auark?. B how muh o ou have o sale up an in he ase of no governmen poli so ha people have he same uili as in he pa as ou go equilibrium. 3. Use he same Overlapping Generaions moel wih prouion suie in he haper wih α > α. Consier a slighl ifferen pa as ou go ssem. Speifiall, suppose he governmen imposes a ax on oung onsumpion o fun he ransfers o he alive in he perio. Suppose he ax on onsumpion an ransfers o he are suh ha () people o no work an (2) people enjo he same amoun of onsumpion when oung an when. Solve for he equilibrium. Appenix: Overlapping Generaions Moel- Mone The overlapping generaions moel has been a workhorse in he su of monear eonomis. The reason is relae o he resuls we esablishe before, namel, ha auark is ineffiien when he eonom is shor of borrowers. Young agens wan o ransfer heir inome o he seon perio of heir life, bu anno o so if bons are he onl pe of asses people an h. Mone provies a seon pe of asse whih an be use o ransfer inome aross perios. I, however, has one big avanage over bonsnamel, ha i an be effeivel reeeme in he nex perio. In onras, a bon anno. If a oung person bough a bon from an person, he anno reeem if for goos. 22

23 The simples monear version of he OG moel oes no inlue an leisure in he uili wih onl oung agens enowe wih he goo so ha he enowmen paern is (e,0). Preferenes are jus U(, ) ln ln The bigges hange is he househ s buge onsrains. Le m be he mone bough b a generaion househ, an le q enoe he prie of mone. (Quesion: an ou ell wha he relaion is beween he prie of mone an he prie level?) When oung he househ s buge onsrain is s q m e When, he buge onsrain is s ( r) q m We an solve for he saving (whih is in erms of bons/loans) in he perio buge onsrain an subsiue i ino he firs perio buge onsrain arrive a he ineremporal buge onsrain e r m q q r The firs hing o esablish is ha he reurn o hing mone an he reurn o hing bons mus be equal when mone is hel b househs. The reurn o saving in he form of a bon is +r. Alernaivel, ou oul ake uni of oupu an bu /q unis of mone. Nex perio ou an sell he mone for goos a prie q +. Hene he reurn o mone is q + /q. Now if he reurn o bons is greaer han he reurn o mone, ou woul never h mone. If ou h mone, i mus have a leas as grea a reurn as bons. In fa, i mus have he same reurn. Oherwise ou oul borrow a uni of oupu, 23

24 use i o bu mone, earn he real reurn on mone, pa of our loan an make a profi. This is alle an arbirage opporuni, an b engaging in his bu low, sell high pe of behavior ou oul make infinie profis. Everone woul r o o his, an so we oul never have his happening in equilibrium. Thus, if people h mone hen q r. q Solving for he Equilibrium If here is no governmen, hen savings (speifiall bons) mus be zero. Thus, all savings one b he househ mus be in he form of mone. This means we an rop bons/savings from he househ buge onsrains. The buge onsrains are hus e q m an q m To solve for he equilibrium, we an subsiue he wo buge onsrains irel ino he uili funion o ge U(, ) ln ln ln( e qm ) ln( q m ) This is a simple unonsraine maximizaion problem over he hoie of erivae an seing his equal o zero implies e q q m m. m. Taking he Wih some algebra, we arrive a he mone eman equaion. m 2 e. Alernaivel, he prie of mone is q 24

25 q 2m e Now in equilibrium mone suppl equals mone eman. If he iniial sar b hing M unis of mone. Also, e is oal oupu in he eonom, whih we shall enoe b Y. Then he above equaion implies ha 2M q. Y Now, reurning o he quesion I pose above, wha is he relaion beween he prie of mone an he prie level? The orre answer is P. Thus he equaion we have q above is jus a simple version of he quani heor of mone. PM= VY where V is he veloi of mone. The equilibrium resul is jus he quani heor of mone. 25