Department of Economics and Institute for Policy Analysis University of Toronto 150 St. George Street Toronto, Ontario M5S 3G7 Canada

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1 Deparmen of Economc and Inue for Polcy Analy Unvery of Torono 5. George ree Torono, Onaro M5 3G7 Canada January 26, 999 WORKING PAPER NUMBER UT-ECIPA-FAIG-99- THE OPTIMAL TRUCTURE OF LIQUIDITY PROVIDED BY A ELF-FINANCED CENTRAL BANK by Mquel Fag Copyrgh 999 by Mquel Fag Deparmen of Economc Inue for Polcy Analy IN Auhor' e-mal: fag@cha.uorono.ca On-lne veron: hp://

2 THE OPTIMAL TRUCTURE OF LIQUIDITY PROVIDED BY A ELF-FINANCED CENTRAL BANK Mquel Fag 5. George ree Unvery of Torono Torono, CANADA, M5 3G7 e-mal: fag@cha.uorono.ca phone: (46) ABTRACT Cenral bank have conenly dfferenaed he reurn on he ecure hey have ued (money and naonal deb). In conra, fr be effcency demand ha hee ecure earn he ame reurn: he reurn on capal. A elf-fnanced cenral bank, whou capal and axe, canno acheve h fr be. The reulng gap beween he reurn on capal and he reurn on publc ecure are mplc axe. Thee axe ncreae he opporuny co of he commode fnanced wh he lqudaon of hee ecure, o hey are ndrec axe on hee commode. Becaue axe on nvemen are le effcen han axe on conumpon, ecure nenve n fnancng nvemen hould be axed a a lower rae han ecure nenve n fnancng conumpon. Th feable f naonal deb nvemen nenve. Then, h ecury hould earn nere and be mpoed arfcal co on econd hand radng. In addon, becaue money pecalzed n provdng hor erm lqudy, rang he reurn on naonal deb delay expendure oward he fuure. Hence, he paymen of nere on naonal deb brng a wndfall of reource durng ranon acro balanced pah n addon o he long erm welfare gan of h polcy. mlar argumen apply o hor and long erm maure of he naonal deb. 2

3 . INTRODUCTION Governmen provde a dvere offer of publc lable wh a dvere menu of reurn. For example, hey pay nere on naonal deb, epecally o long erm maure, whle hey pay no nere on cah. In conra, for fr be effcency, all publc lable hould earn he ame reurn (adjued for rk), and h hould be he ocal reurn on capal. ( ee Fredman' [969] opmum quany of money.) To mplemen h fr be, governmen mu fund her lable wh capal or axe, and hey mu be able o generae ncome from hee fundng ource effcenly. Realcally, for a varey of reaon, ncludng he ame ranacon co and nformaon mperfecon ha generae a demand for money, hee fundng ource are coly o admner. A a reul, governmen have never funded her lable o he exend ha all her reurn approach he reurn on capal. Could h fundng lmaon explan he prevalen polcy of ung a dvere offer of publc lable? To anwer h queon, I analyze he opmal offer of publc lable ued by a elf-fnanced cenral bank, whou capal or axe, acng a he only branch of governmen. Th cenral bank mbedded n a model wh overlappng producon acve where boh money and naonal deb provde lqudy o he economy. The model of h paper develop he framework n Woodford (99) n he followng fahon. Producon projec ake me o maure, and he varou ype of projec n he economy overlap. o n every perod, ome projec yeld a harve of oupu, whle oher projec requre In Fag [998b], I elaborae on he ulmae mplcaon of h propoon when he reurn on capal ha doyncrac rk. 3

4 nvemen of real reource for fuure harve o come. Indvdual can parcpae a mo n one projec a a me, and cred among hem blocked becaue a n Luca (98) cred conrac canno be enforced. In h envronmen, ndvdual wh a harve have no problem fnancng her conumpon and nvemen expendure, whle ndvdual who are beween harve have a lqudy problem becaue hey have no earnng on hand. Indvdual olve h problem by ellng a poron of her harve o acqure money and naonal deb. Then, n perod whou a harve, ndvdual ue hee lqud ae o fnance her expendure. In h model, he cenral bank may chooe o keep he nomnal upply of all publc lable conan. If h baelne polcy purued, n a balanced pah all publc lable earn he ame real rae of reurn: he growh rae of he economy. In h nance, he naonal deb earn no (nomnal) nere, and n pracce all publc lable are alke. The cenral bank may chooe o depar from h baelne polcy and pay nere on he naonal deb. To pay h nere permanenly, he elf-fnanced cenral bank mu cro-ubdze he reurn on he naonal deb wh money creaon. Wh h polcy, he exence of a demand for money n equlbrum requre ha he cenral bank mpoe arfcal co o econd hand ranacon of he naonal deb. (Or alernavely, he cenral bank could mpoe oher nconvenence for ung he naonal deb for hor erm holdng horzon.) The end reul of h polcy a pecalzaon of lqud ae by holdng perod, money provdng lqudy a he hor end of he pecrum, and he varou maure of he naonal deb coverng he re. The man focu of h paper o denfy poble raonale for h dvere offer of publc ecure. 4

5 The fr raonale I provde for dverfyng he reurn on publc ecure baed on a dvere nvemen neny for he prvae expendure fnanced wh hem. A elf-fnanced cenral bank canno pay he reurn on capal o all publc ecure. The reulng gap beween 2 he reurn on capal and he reurn on publc ecure are mplc axe. Thee axe ncreae he opporuny co of he commode fnanced wh he help of he lqudy ervce he publc 3 ecure provde. Becaue axe on nvemen are more neffcen han axe on conumpon, ecure nenve n fnancng nvemen hould be axed a a lower rae han ecure nenve n fnancng conumpon. Th feable f naonal deb nvemen nenve. Then, h ecury hould earn nere and be mpoed arfcal co on econd hand radng. A mlar argumen apple o he varou maure of he naonal deb. For example, an upward-lopng erm rucure of he reurn on he naonal deb, whch wha we normally oberve, a econd be polcy for a elf-fnanced cenral bank f long erm maure are more nenve fnancng nvemen han hor erm maure. A econd raonale for payng nere on he naonal deb baed he fac ha he naonal deb pecalzed n relavely long holdng perod. Becaue of h pecalzaon, rang he reurn on he naonal deb delay expendure oward he fuure. Hence, he paymen of nere on he naonal deb, and epecally o long erm maure of he naonal deb, brng a wndfall of reource durng ranon acro balanced pah. Wh careful polcy by he cenral 2 Thee axe fnance he negave ne worh of he cenral bank (he cenral bank ue lable bu ha no ae). 3 Th forward hfng doe no depend on wha ecure are fnally ued o purchae good, bu wha ecure are held durng he perod precedng he purchae. 5

6 bank, hee reource can lead o a Pareo mprovemen veru he baelne polcy of payng zero nere on he naonal deb. The effcency of dfferenang he reurn of publc lable by payng nere on naonal deb wa prevouly queoned by Bryan and Wallace (979), Bryan (98a), and Romer (993). Depe major dfference n he modelng raege, my reul compare ealy wh hee earler conrbuon. My reul agree wh he neffcency of payng nere on naonal deb f publc ecure do no help o fnance fuure nvemen, and we lm welfare o comparon acro balanced pah. Th neffcency reed n he conrbuon by Bryan and Wallace. My reul agree wh Romer ha here are condon under whch payng nere on naonal deb opmal. Romer' reul are baed on how he lqudy ervce of money and naonal deb ener he uly funcon of a repreenave ndvdual. Inead, my reul follow from he pecalzaon of lqud ae by holdng perod, and he dvere nvemen neny of publc ecure ha h pecalzaon allow. everal earler conrbuon have provded emprcal uppor for a qua-moneary role for he naonal deb, epecally for hor-erm maure. For example, Fred and How (983) argue ha h role a good explanaon ha a low frequence he real nere rae earned on Treaury Bll declne a he rae of nflaon re. mlarly, Fred (995) argue ha h role explan he premum of one perod reurn of long-erm Treaury Bll over hor-erm Treaury Bll, and explan he correlaon of h premum wh oher macroeconomc varable. Fnally, Banal and Coleman (996 ) argue ha h role can alo explan he equy premum puzzle. 6

7 Alhough, he man empha of he preen paper normave, renforce h emprcal uppor n provdng a plauble normave raonale for he prevalen qualave rucure of he reurn on publc lable. Th paper exend my earler work on he overlappng producon acve model (Fag [998a]) by prolongng he producon me o more han wo perod and nroducng mulple lqud ae. In boh paper, I borrow heavly from earler conrbuon. The role of money and he naonal deb n compleng marke are mlar o he role hey play n Foley and Helwg (975) and Bewley (98), and he rapdly growng leraure ha ha followed hee emnal conrbuon. In parcular, he role of publcly uppled lqud ae n fnancng nvemen expendure mlar o Woodford (99), excep for he fac ha n here he reurn on money and naonal deb affec no only he ze of nvemen bu alo compoon. The demand for lqudy baed on a coly or low lqudaon of alernave ae found n clacal 4 conrbuon uch a Baumol (952), Tobn (956), and Damond and Dybvg (983). Fnally, he endogenou growh feaure of he model have much n common wh he 'Ak' model n Barro (99). In ummary, h paper advance a new perpecve on he normave analy of he reurn of oude publc lable uch a: money and naonal deb. Th new perpecve emphaze he lqudy role of hee ecure o fnance prvae expendure, he pecalzaon 4 For recen conrbuon ung h modelng of lqudy and dealng wh ue relaed o h paper ee Ayagar and Gerler (99) chref and mh (997), and Holmröm and Trole (998). 7

8 of hee ecure by holdng perod, and her relave nene o fnance dvere ype of prvae expendure. In parcular, f he expendure fnanced wh naonal deb are more nvemen nenve han hoe fnanced wh money, econd be for a elf-fnanced cenral bank o cro-ubdze he paymen of nere on naonal deb wh money creaon. mlarly, an upward lopng erm rucure of he reurn on naonal deb econd be f he nvemen neny of he naonal deb ncreae wh erm o maury. Thu, h paper provde a plauble normave raonale for he prevalen qualave feaure of he rucure of publc lable. The re of he paper organzed a follow. econ 2 decrbe he overlappng producon acve model ued n h paper. econ 3 analyze he behavor of ndvdual n he framework of h model. econ 4 conruc a balanced pah equlbrum. econ 5 compare he welfare of a repreenave ndvdual acro balanced pah. econ 6 analyze he ranonal effec acro balanced pah for a racable pecal cae. Fnally, econ 7 conclude wh a bref ummary of he fndng and ome drecon for fuure reearch. 2. THE MODEL Conder an economy populaed wh a large number of ndvdual. In h economy, ndvdual poe a non ranferable producon echnology whch he only vehcle o employ he capal hey own. The oupu of hee echnologe homogeneou, and can eher be conumed or nveed. Inveed capal he only npu hee echnologe requre. Producon ake me. To oban a harve of oupu, ndvdual have o nve n her echnologe for everal conecuve perod. Alo, he varou producon acve overlap. o n every perod, 8

9 ome ndvdual have a harve, whle oher are nveng for he harve o come. Th overlappng rucure of producon mple ha ndvdual mu rade, o hey can conume and nve n he perod whou a harve of her own. Bu radng lmed by he fac ha ndvdual can hde her deny and her capal f n her be nere o do o, precludng cred conrac n her mulple form (morgagng capal, fuure marke, jon frm,... ). (ee Bryan [98b] for a relaed dcuon of he moral hazard problem behnd he demand for money). Thee radng conran are relaxed wh he exence of wo publc lable: money and naonal deb. Indvdual ell a poron of her harve o acqure hee ae, whch are pen o buy oupu n perod when hey do no receve a harve. The mperfec funconng of capal marke crucal for publc ecure o be valued n h model. Followng Luca (98), I preclude he exence of cred. A he co of complcang he model, h exreme aumpon could be replaced wh weaker alernave. For example, could be replaced wh ranacon co n eablhng or enforcng cred conrac. (ee for example Ayagar and Gerler (99) for a relaed model wh hee co.) Alo, could be relaxed wh a lmaon of cred o a poron of capal eay o denfy and hu uable a collaeral. For mplcy, hee alernave aumpon are no purued here. Money and naonal deb are dfferenaed by her lqudy. Money perfecly lqud, whle naonal deb compoed of dcoun bond wh econd hand radng co. By makng hee co uffcenly hgh, he governmen can creae a demand for money for hor erm holdng horzon even f money yeld a lower reurn han naonal deb. Lkewe, dfferen 9

10 maure of naonal deb can coex even f her reurn ncreae wh her erm a long a he co on econd hand radng hgh enough o dcourage ung long erm deb for hor erm holdng horzon. The co o econd hand rade of he naonal deb a maer of polcy. I mpoon a devce o dfferenae he reurn on money and naonal deb of varou maure. The governmen can elmnae h co and equae he reurn on all lqud ae. Indvdual wh a harve allocae her wealh no conumpon, capal, and lqud ae. The flow budge conran for a repreenave ndvdual a h age, o be denoed age, he followng equaon: c + k + m x () ; where c conumpon a age n perod ; k capal a age n perod ; x real wealh a he begnnng of perod ; and m real demand for lqud ae (of erm) n perod. Lqud ae money. The oher lqud ae are he varou pure dcoun bond compong he naonal deb. The relaonhp beween he real quane of lqud ae and her nomnal face value m Mq p (2).

11 where M nomnal face value nveed n lqud ae n perod ; q dcoun prce of lqud ae n perod (q for all ) ; and p prce of oupu n perod. Indvdual whou a harve ue her lqud wealh o fnance her conumpon and nvemen expendure. In prncple, hey could alo ue her lqud wealh o purchae new lqud ae. However, n all he deermnc equlbra condered n h paper h pobly no carred ou becaue of a combnaon of hree fac: The reurn on phycal capal exceed ha of lqud ae. The ranacon co for econd hand radng of he naonal deb are hgh enough o preven ung long erm deb for hor erm holdng horzon. And he expeced one perod reurn on lqud ae non decreang wh her erm. Conequenly, ndvdual no havng a harve pend he value of her lqud ae maurng n ha perod: M( ) (3) c + k m ( ) r ; p where r real gro reurn of lqud ae ha maure n perod ; ha. (4) c, k, and m, for all and all. r q p ( ) In all perod, conumpon, capal, and he demand for lqud ae mu be nonnegave: p In he equlbra condered where he reurn on lqud ae domnaed by he reurn on capal, an ndvdual hold no lqud ae a he begnnng of a perod wh a harve. The harve elf depend on he capal nveed n he precedng + perod. Therefore, he wealh of ndvdual wh a harve n + +

12 (5) where k he vecor x + + F( k ) ( k, k,..., k ) ; and + ( + ) F gro producon funcon. For eae of noaon, he undeprecaed capal ncorporaed n he funcon F. The funcon F pove, wce connuouly dfferenable, ncreang, concave, and homogenou of degree one. Alo, afe andard Inada condon for an neror oluon. The ndvdual' horzon nfne. In recurve form, he uly of an ndvdual havng a harve n perod U β c σ ( + ) σ + β + (6) ; U + + where U uly n perod ; ß dcoun facor (ß > ); and nvere of he ner-emporal elacy of ubuon (pove, and f, he nananeou uly hould be underood a logarhmc). The only governmen agency a cenral bank ha provde and manage he upply of lqud ae n he economy. The cenral bank can chooe o change he reurn on lqud ae by changng her upply hrough open marke operaon, and revng f neceary he ranacon co ha hnder econd hand radng of naonal deb. Thee acve, hough, mu be elf- 2

13 fnanced. Thu, he curren revenue from ellng bond and ung money mu equal he curren value of maurng bond. Equvalenly, he ne egnorage meaured n a cah-flow ba mu be zero n all perod: (7) M q + M M. 2 ( ) 2 The equlbrum concep o be appled ha of a recurve lqudy domnaed compeve equlbrum (henceforh equlbrum) where he equence of prce and reurn, he allocaon of harve, and he quane of lqud ae afy he followng condon: () ndvdual have perfec foregh abou prce and reurn and ake hem a gven; () ndvdual maxmze uly ubjec o conran () o (5); () all lqud ae are valued, her one perod reurn non decreang wh erm o maury and domnaed by he reurn on capal; and (v) all marke clear: ( c + k ) x, for all (8), and (9) M M ( ) + M for all and all. Among he e of equlbra, pecal aenon gven o balanced pah equlbra where all real varable grow a he ame rae and he real rae of reurn on lqud ae are conan over me. 3

14 3. INDIVIDUAL' BEHAVIOR Th econ characerze an opmal plan for a repreenave ndvdual. In h opmal plan, nvemen mu be neremporally allocaed o maxmze he ze of he nex harve. Tha, f he ndvdual ge a harve n perod, hen he vecor of capal ock relevan o he nex harve, k, mu olve he followng program: max F( k ) ubjec o k( + ) r ( + ) κ x () ; k where? proporon of wealh o be ued n fnancng capal up o he nex harve. The fr order condon for an neror oluon o h problem are F F r for,... ; where F ( + ) ( + ) F k (). ( + ) Th condon ae ha a age, he margnal reurn of nveng a un of oupu mmedaely mu be equal o he margnal compounded reurn of purchang a lqud ae o fnance nvemen a a laer age of he projec. In he opmal plan, conumpon mu be allocaed o afy andard Euler equaon. In perod wh a harve, he margnal rae of ubuon beween mmedae conumpon and conumpon a a perod before he nex harve mu be equal o he gro real reurn on he lqud ae connecng hee wo perod: 4

15 σ c (2) σ r ( + ), for,...,. c β ( + ) Alo, n all perod he margnal rae of ubuon beween mmedae conumpon and conumpon a he nex harve mu be equal o he gro margnal produc of mmedaely nveed capal: σ c( + ) (3) F ( ) c,,... + σ + for β. + + To enure he exence of an opmal pah, we mu rerc he parameer decrbng preference and echnology. Thee rercon mu enure ha here a mo one feable pah ha gve nfne uly o he ndvdual. Alo, hey mu enure ha here a lea one feable pah ha gve uly hgher han mnu nfny. The followng aumpon acheve boh 5 objecve for he equlbrum pah o be analyzed here (ee Alvarez and okey [998]) : * AUMPTION : Le F (µ) be he oluon o problem () for r µ. Preference and echnology mu afy: * + - () If <, hen Gµ > uch ha F (Gµ) # Gµ and ß Gµ <. * + - () If $, hen µ > uch ha F (µ) $ µ and ß µ <. 5 The heorem n Alvarez and okey (997) requre ha he rao of conumpon o wealh bounded away from zero for $. Therefore, her applcaon requre fndng pove lower bound for c / x ha are never bndng. Th can be ealy acheved along balanced pah and he ranonal pah n econ 6 wh an explc oluon. 5

16 Inuvely, he fr par of he aumpon enure ha echnology no oo producve when < o he uly of feable pah doe no dverge o nfny. The econd par of he aumpon enure ha echnology uffcenly producve when $ o here a lea a feable pah wh a uly hgher han mnu nfny. 4. THE BALANCED PATH EQUILIBRIUM In h econ, I wll conruc a balanced pah equlbrum along whch boh he rae of reurn on ae and he rao among real varable are conan. Fr, I wll conruc he allocaon along a balanced pah equlbrum for gven rae of reurn on lqud ae. econd, I wll how how he rae of reurn on lqud ae are relaed o moneary polcy. And hrd, I wll characerze he conran on moneary polcy mpoed by he equlbrum concep and he budge conran of he cenral bank. Throughou h econ, I wll drop he me ubcrp when unneceary. ~ For a repreenave ndvdual, le be a perod wh a harve. Denoe k, c~, and m~ he relave demand k( + ) x, c( + ) x, and m x repecvely. Ung (), he relave allocaon of capal { ~ ~ k k } (4) F F r, for,...,. { ~ ~ k k } deermned by he followng equaon: Becaue of conan reurn o cale, he margnal produc of capal depend only on he e of rao. Ung (3), he gro rae of growh of conumpon hen 6

17 g β σ F σ + ( ) (5). Becaue preference are homohec and he rae of reurn are conan, wealh mu grow a he ame rae a conumpon. Hence, ung (5), he proporon of a harve mmedaely allocaed o capal deermned by he followng condon: ~ ~ k k F, ~,..., ~ k ~ + (6) g. k k Ung (2), he e { c~ c~ } deermned by he followng equaon: ~ c σ ~ ( β r ), for,... c (7). Fnally, o complee a decrpon of he allocaon along a balanced pah, he proporon of conumpon conumed a age follow hen from he neremporal budge conran ha reul combnng (), (3) and (5) wh (4) o (7): ~c β + σ σ (8). β σ F r σ I decrbe nex how he rae of reurn on lqud ae are relaed o moneary polcy. By defnon, he dcoun prce of ae mu afy he Fher' equaon: 7

18 p r (9) r q p q. Noe ha he gro real reurn on money, r, he nvere of he gro rae of nflaon. Alo, he nvere of he dcoun prce q he nomnal reurn on ecury for he perod h ecury ake o maure. The real quane of lqud ae are mpled by he conran (3): ~ ( ~ ~ (2) m c + k ) / r, for,.... For he rae of reurn o be conan, h equaon mple ha he upple of all lqud ae mu grow a a common conan rae? along a balanced pah equlbrum. Marke clearng n he money marke hen mple ha? mu afy he andard relaon: γ g (2) r. f For a e of reurn { } r o be an equlbrum all marke mu clear. The money clear M p m ~ x, o he andard neuraly reul apple. The marke for he oher lqud ae clear f her nal upple afy he followng equaon: M m~ m~ q M, for 2,..., (22). Fnally, by Walra' Law, he oupu marke clear f he ne egnorage colleced by he cenral bank zero. Ung (2) and (9), equaon (7) ranformed no: 8

19 R r g m ~ (23) ; where R rao of ne egnorage over gro oupu. A e of reurn { } r o be conen wh he equlbrum concep defned n econ 2 mu afy no only equaon (23) bu alo he followng wo ncenve conran. Fr, lqud reurn mu be weakly domnaed by he reurn on capal, ha F r +. econd, he cenral bank mu be able o make each lqud ae he be nrumen o ranfer wealh for a perod equal o maury. For example, gven ha ranacon co on he naonal deb canno be negave, h rule ou money havng a hgher reurn han naonal deb. If h were he cae, he demand for naonal deb would be zero and money would be he only publc lably. Thu, h would be equvalen o a polcy wh zero nere on he naonal deb. Thee wo ncenve conran on he e of reurn are afed wh he baelne polcy of keepng a conan upply of all lqud ae. In h cae, he reurn of lqud ae along a balanced pah afe r g for,...,. o all ae have he ame one perod rae of reurn. Th one perod rae of reurn domnaed by he rae of reurn on capal a he followng argumen demonrae. Mulplyng by k boh de of () and (+) + aggregang over all mple F( k ) F r κx. Gven ha F( k ) x g x and F? #, he domnance of lqud ae by capal follow: + g + g r r (24)

20 5. THE ILVERED RULE Th econ characerze he moneary polcy ha maxmze he uly of a repreenave ndvdual n a balanced pah. If he governmen could ue lump-um axe o ubdze moneary ae, hen, a well known, effcency requre o rae he reurn on all moneary ae o equae he ocal reurn on capal. Th fr be polcy, known a he opmum quany of money afer Fredman (969), precluded n h paper by he lack of fund of he cenral bank and he conran ha curren egnorage canno be negave. The econd be problem o be olved n h econ mpoe conran (23) on he e of reurn on lqud ae a well a he oher conran conen wh a balanced pah equlbrum. Wh hee conran, he problem o be olved con n fndng he e of reurn ha maxmze he uly of a repreenave ndvdual havng a harve f he nhered capal ock of he ndvdual no havng a harve were auomacally adjued o he balanced pah level. The oluon o h problem a econd be modfed golden rule, whch for hor I wll name he lvered Rule. The ranonal effec abraced n h rule and he drbuonal conequence hey enal are deal n he nex econ. Le V be he ndrec uly funcon of a repreenave ndvdual n a balanced pah equlbrum. Conder a margnal reform n whch he reurn on lqud ae ( > ) raed and h ncreae fnanced wh a reducon on he reurn from money. Ung Roy' deny, he oal effec on V from h reform he followng expreon: 2

21 dv d lnr V g F m m d r x ~ ~ ln + + d lnr (25). Applyng he Implc Funcon Theorem on conran (23), we oban d lnr R ln R d lnr lnr (26). lnr Dfferenang (23), R lnr mr ~ ln g g lnr m~ r g r m~ + for,..., g lnr (27). Therefore, combnng (25) o (27), he effec of he reform on V dv d lnr V m~ x + g F r g r g ln g lnr ln g lnr m~ r m g ~ + m~ r m~ g + r g r g ln m ~ m~ lnr m~ ln m ~ m~ lnr m~ (28). The lvered Rule mple he baelne polcy of zero nomnal nere on he naonal deb, or equvalenly r g for all, when expreon (28) non pove for all lqud ae. To ee wha h mple we can mplfy (28) conderably. Applyng he Envelop Theorem o problem 2

22 ~ (), he elacy of g wh repec o r proporonal o he dcouned expene k r : ~ ln g kr ln r σ ( + ) κ for,..., (29). Ung (29) and evaluang (28) a he pon where r g for all, expreon (28) mplfe o (3) dv d lnr V m~ x + g F dv Therefore, f and only f d lnr ~ k c~ ~ + k ~ k r g m~ c~ ~ + k σ ( + ) κ. ~ k r g m~ c~ ~ + k ( + ) σ κ ~ k c~ ~. Tha, he lvered Rule mple zero + k nomnal nere on naonal deb when money weakly pecalzed n fnancng nvemen expendure. Converely, he lvered Rule mple payng nere on naonal deb when, realcally, money pecalzed n fnancng conumpon expendure. An neror lvered Rule where he nere pad on naonal deb pove and ncreang wh maury can be ealy characerzed ung (28) and (29). If he lvered Rule neror, all margnal reform of he or condered n dervng (28) mu have a zero effec on V. Th mple ha he numeraor of he fracon nde he quare bracke n (28) mu be equal o a conan? for all. Conequenly, ~ r k r g m~ r m m + g c + k + g ~ ln ~ (3) λ ~ ~ σ ( ) κ m~ for all. lnr 22

23 Thu he proporonal premum of r over g ncreang wh he pecalzaon of ae on nvemen expene (mddle erm n RH) and decreang wh he egnorage co ha a change on r nduce hrough he demand for lqud ae (la erm n RH). Th la erm vanhe when no ae are cro-ubdzed. The conan? nverely relaed o he Lagrange mulpler of he cenral bank' budge conran (23), and value enure ha h conran afed. Th conran mple ha f r > g for ome, hen here mu be anoher ecury for whch r < g. Hence f money he publc lably wh he lowe reurn, he gro rae of growh of he money upply? g / r mu be larger han one. A mpled by (28), he lvered Rule dffer from a zero nomnal nere polcy only f he reurn on lqud ae affec growh. However, n general he lvered Rule doe no maxmze he growh rae. The effec on he growh rae of a margnal reform n whch r ( > ) raed a he expene r (32) d ln g d ln r σ ( + ) κ ~ ~ d lnr kr + kr d ln r Comparon of (32) wh (25) ung (3) mple ha a margnal reform wh a non negave effec ~ ~ k k on V rae he growh rae f ~ ~ > ~ ~. Thu, a he lvered Rule he margnal reform c + k c + k condered dervng (32) ncreae he growh rae f ae pecalzed n nvemen expene n relaon o money. 23

24 6. AN EXAMPLE WITH TRANITIONAL DYNAMIC The lvered Rule of econ 5 maxmze an ncomplee meaure of welfare becaue abrac from he ranonal effec of long-erm polcy change. Unforunaely, a common n mo growh model, ranonal dynamc have no explc oluon n general. For h reaon, h econ addree hee ranonal effec for an nereng pecal cae for whch ranonal dynamc have an explc oluon. Ou of he balanced pah, an equlbrum can be explcly olved wh un elace of ubuon n conumpon and gro producon. For eae of expoon, I wll furher aume ha harve come hree perod apar, o here are only wo lqud ae money and a wo perod bond. Conequenly, F( k, k, k ) Ak k k α α α A hor comng of h pecalzaon ha rule ou paral deprecaon of capal wh he andard aumpon ha undeprecaed capal addve n he gro producon funcon. However, h pecalzaon doe no rule ou a hgh value for a whch mmc he fac ha a large poron of gro oupu reman nveed for he nex harve. Wh h pecalzaon of funconal form, ndvdual allocae her harve n hare ndependen of he rae of reurn on all ae: ~ 3 ~ 3 c θ ( β ) k α β ~ 3 (33),, and m θ ( β ) + α β 3 for and 2; where 2 θ β ( + β + β ) for,, and 2. Hence, he demand funcon for lqud ae conue a generalzed yem of quany equaon wh conan velocy. In he perod when ndvdual do no have a harve, hey allocae her lqud wealh n proporon alo 24

25 ndependen of rae of he rae of reurn on all ae: 3 3 θ ( β ) M α β M (34) c 3 3 and k 3 3 for and 2. θ ( β ) + α β p θ ( β ) + α β p The marke clearng condon are M ( ) p m M 2( 2) m γ 2 + ( γ ) m p (35) and ; where? he growh facor of he money upply n perod. The yem of equaon (33) o (35) mple ha he allocaon of x depend on?. In parcular, (36) ~ ~ θ ( β ) γ, α β γ, ~ 3 ~ 3 c θ ( β ) ϕ, and k α β ϕ ; 3 c 3 k where ϕ + ( γ )( m ~ m ~ 2). Wh a conan?, he allocaon of oupu reman conan over me. Thu, ung (33) and (36), he law of moon of x 2 ( ) 3 α α x + 3 ζaβ α x+ ; where ζ γ ϕ 2 (37). α (The ubcrp on? and n have been dropped becaue hee varable are held conan). The reurn on lqud ae affec he allocaon of capal, and o affec oal facor producvy. In h pecal cae, oal facor producvy proporonal o he polcy facor?. Furhermore, becaue growh ncreang wh he reurn on capal, he growh facor n a balanced pah equlbrum alo ncreang wh?: 25

26 g 2 A ζ β α α 3 α 2α 3 2. In h pecal cae, condon (3) characerzng an neror lvered Rule drop he la erm becaue he demand for ae ndependen from rae of reurn. The mplfed equaon can hen be ealy olved ogeher wh (2) and (23) o oban explc formulae for he growh facor of money and he growh rae of reurn on he naonal deb along he lvered Rule pah: γ * m~ ( ~ + ~ + m m2 R) ~ 3 3m + ( β )( α θ α θ ) ~ 3 3m 2( β )( α 2θ α θ 2) (38) ; and 3 * ( m ~ + m ~ 2 R) 3m~ 2 + ( β )( α 2θ α θ 2) r2 3 g m~ 2 3( m ~ m ~ + 2) ( β )( α 2θ α θ 2) (39). 2 (Noe ~m ~ and m2 are ndependen of? n h example ). Thu, a long a R, he lvered Rule * call for? > f a? 2 < a 2?. Th condon equvalen o naonal deb beng nvemen nenve. The defnon of? mple? >?, o a? < a? f a # a To conclude h econ, I wll how ha, a lea n h example, ranonal effec renforce he cae for payng pove nere on he naonal deb. uppoe an economy where money no nvemen nenve: a? # a?. In h economy, he lvered Rule eher call for 2 2 pove nomnal nere rae (a? < a? ) or a an neror oluon wh zero nomnal nere

27 rae (a? a? ). The followng argumen how ha f h economy n a balanced pah wh a 2 2 zero nomnal nere rae, here a mple moneary reform leadng o pove nomnal nere rae ha accomplhe a Pareo mprovemen. Th reform con n a margnal ncreae n he upply of lqud ae wh he ncreae n he upply of naonal deb arng n perod and he ncreae n he upply of money arng n perod 2. Becaue he demand for lqud ae nenve o nere rae, he prce level and he allocaon of oupu n he perod and unchanged by he reform. The dcoun on publc bond hough become pove a her nomnal upply ncreae and her real demand unchanged. Hence, even f he harve n perod 2 unchanged becaue depend on prevou nvemen, allocaon change. Wh he reform, real reource hf from ndvdual a age of producon (pendng money) o ndvdual a age 2 of producon (pendng bond). Conequenly, he harve n perod 3 ncreaed becaue producon ha ued he ame capal a age and, bu more capal a age 2. All harve afer perod 3 are alo ncreaed becaue of he law of moon (37) and he fac ha a?, he polcy facor? locally ncreae wh?. (Th follow mmedaely from he Implcaon Funcon Theorem appled o he expreon defnng? and n n [36] and [37]). To how ha all ndvdual are beer off wh he new polcy, conder fr he ndvdual wh a harve a he momen he money upply ar growng n perod 2. Before perod 2, he reform ha no effec on hee ndvdual. Afer perod 2, he dcoun facor of fuure conumpon good are gven by ρ β x2 ρ x ρ x x γ ρ 2 ρ x x ϕ 2 (4),, and ; 27

28 where? he dcoun facor from perod o perod for ndvdual a age of producon. A perod 2, he margnal change n he ndrec uly funcon a? ncreae (4) dv d ln γ dv d ln ρ d ln ρ d ln γ d ln ϕ V ρ m + m + d ln γ x 3 ( 3+ ) 2( 3+ 2) ( 3 + ) ( 3 + ) 2 m r d ln x d ln γ 3+ > V ρ m + m x 3 ( 3 + ) 2( 3+ 2) m ( 3+ 2) m γ 2( 3+ 2) ϕ. In h expreon, he econd lne ue Roy' deny and (4). The hrd lne ue he Implc Funcon Theorem appled o (36) and he pove effec on oupu nduced by he reform. Fnally, he la equaly follow from n when?. Idencal argumen apple o he ndvdual wh a harve n perod changng ubcrp appropraely. The argumen alo applcable o he ndvdual wh a harve n perod from perod 3 onward. Thee ndvdual, hough, ge an exra bonu n perod 2 when hey receve a hgher reurn on he deb hey purchaed a whou havng o endure a lower reurn on money n perod. Conequenly, he polcy reform leadng o pove nere rae accomplhe a Pareo mprovemen. 28

29 7. CONCLUION The opmal rucure of oude publc lable when hee ecure provde lqudy ervce canno dregard her pecalzaon by holdng perod, and her relave nene o fnance nvemen and conumpon expendure. When hee feaure are aken no accoun, here are plauble and emprcally eable condon o raonalze a dvere offer of publc lable, wh he naonal deb earnng pove nere,and wh he reurn on he naonal deb ncreang wh erm o maury. Th fndng doe no conradc he opmum quany of money rule. If he cenral bank of my model could be effcenly funded wh capal or axe, hen would be effcen o equalze he reurn on all lqud ae o he ocal reurn on capal. In h paper, I do no dcu how effcenly he governmen can manage capal or rae axe o fnance he cenral bank operaon. Inead, n h paper I ake a a gven he elf-fnancng conran of he cenral bank, and hen I nqure abou he poble raonale for dverfyng he offer of publc lable by payng nere on he naonal deb. A fr raonale for h dverfcaon ha he naonal deb nvemen nenve compared o money. Becaue axe on nvemen are le effcen han axe on conumpon, ecure nenve n fnancng nvemen hould be mplcly axed a a lower rae han ecure nenve n fnancng conumpon. Hence he gap beween he rae of reurn of capal and he naonal deb hould be narrower han he gap beween he rae of reurn of capal and money. A econd raonale for payng nere on he naonal deb baed he fac ha he naonal deb pecalzed n relavely long holdng perod. Becaue of h pecalzaon, rang he reurn on he naonal deb delay expendure oward he fuure. Hence, he paymen of nere on he naonal deb, 29

30 and epecally o long erm maure of he naonal deb, brng a wndfall of reource durng ranon acro balanced pah. Wh careful polcy by he cenral bank, hee reource can lead o a Pareo mprovemen veru he baelne polcy of payng zero nere on he naonal deb. mlar raonale can jufy an upward-lopng erm rucure for he reurn on he naonal deb. The elf-fnancng conran on he cenral bank convenen bu no eenal o he analy of h paper. A long a publc lable are no fully funded, he cenral bank wll no be able o equalze he reurn on he lable ue o he ocal reurn on capal, and a mlar analy o he one offered here characerze he radeoff beween mplc axe on money and naonal deb. If he ne worh of he publc ecor change over me, perhap due o proraced governmen budge defc, he mplc axe on money and he naonal deb neceary o fnance hem mu change a well. Typcally, when a governmen rele heavly on egnorage o fnance he publc ecor, nflaon wll be hgh and he real nere rae on he naonal deb wll be low. 6 Therefore, f long-run change n nflaon are drven by varyng relance on egnorage, nflaon and real nere rae on Treaury Bll wll be negavely correlaed a low frequence. In conra, f long-run change n nflaon are drven by arbrary change n he rae of growh of he money upply wh a conan relance on egnorage, nflaon and real nere rae on Treaury Bll wll be povely correlaed. The negave correlaon beween nflaon and real nere rae on Treaury Bll a low frequence documened by ummer (983) wa nerpre 6 Th propoon ealy eablhed ung (38) and (39) and remarkng ha around he lvered pah ncreae n he rae of growh of he money upply are aocaed wh a faer rae of growh. In general, however, h propoon requre ha he nvemen neny of money and naonal deb no oo enve o he reurn on lqud ae. 3

31 by Fred and How (983) a an ndcaon of he naonal deb renderng lqudy ervce. Th paper brng h nerpreaon one ep furher. Th negave correlaon no only an ndcaon ha he naonal deb render lqudy ervce, alo an ndcaon ha polcy maker are enve o he neffcence caued by he low reurn earned on lqud ecure. Thu, when he egnorage o be fnanced by mplc axe on money and naonal deb re, boh of hee axe are mulaneouly raed o mnmze her overall welfare co. Fuure reearch mu addre how he reul of h paper are modfed when here uncerany on fuure expendure need. An exenon along hee lne wll have o deermne he exac co o econd hand radng. In he preen deermnc framework, he exac hape and ze of hee co rrelevan a hey are never ncurred n equlbrum. However, wh unceran expendure need ndvdual reve her plan over me, o here an ex-po welfare co of makng dffcul for hem o lqudae her naonal deb earler han ancpaed. In uch an exenon, he opmal polcy wll have o balance he radeoff beween makng he naonal deb dffcul o rade a a prereque o rae reurn, and he ex-po welfare co arng when ndvdual are prevened o reve her plan. Fuure reearch mu alo addre how he reul of h paper are modfed when oude money (cah and naonal deb) coex wh nde money (depo). The exence of nde money can ealy be ncorporaed o he preen model by aumng ha ome ype of capal are uable a collaeral. If hee ype are relavely carce, nde and oude money may coex. In h economy, he opmal rucure of he moneary ecor mu addre no only he effcency 3

32 ue raed n h paper, bu alo he effcency ue raed by he lkely dpary beween he equlbrum rae of reurn earned by he ype of capal uable a collaeral and he ype of capal unuable a collaeral. The oluon o h problem wll provde no only a normave heory of nflaon and nomnal nere rae, bu alo a normave heory of requred reerve and oher bankng regulaon. 32

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