Redistribution and Optimal Monetary Policy: Results and Open Questions

Transcription

1 SAGGIO AD INVITO Redsrbuon and Opmal Moneary Polcy: Resuls and Open Quesons Sefana Albanes* Columba Unversy, NBER and CEPR Wha are he properes of opmal fscal and moneary polces wh heerogeneous agens? Ths s a pressng queson, gven he wealh of evdence on heerogeney n cash holdngs and labor ncome. Ye, unl recenly remaned largely unexplored. In hs paper, I show ha wh heerogeney he Fredman rule s opmal only f posve nomnal neres raes do no amelorae consrans on redsrbuon. Wh an emprcally plausble cross-seconal correlaon beween money holdngs and labor ncome, he Fredman rule s opmal f he governmen favors redsrbuon o he poor. I dscuss hese fndngs and propose several drecons for fuure research. [JEL Classfcaon: E5, E6, E63, H]. Qual sono le propreà delle polche moneare e fscal ome quando l eerogeneà nroduce un conflo ra redsrbuzone ed effcenza? S raa d una domanda pressane, dao l mone d evdenza sulla dsuguaglanza ne redd e nella rcchezza nella popolazone. Eppure, poch sud se ne sono snora occupa. Queso saggo dmosra che la Fredman rule, che prevede l azzerameno de ass d neresse nomnal, è oma solo quando un aumeno de ass d neresse non nfluenza vncol sulle polche redsrbuve. Con poes realsche sulla correlazone ra reddo ndvduale e domanda d monea, la Fredman rule è oma solo se l governo desdera redsrbure n favore de pover.. - Inroducon The opmaly of he Fredman rule s one of he man resuls n he leraure on opmal governmen polces wh commmen. Fredman (969) argumen s ha any posve value of he * <sefana.albanes@columba.edu>. 3

2 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 nomnal neres rae effecvely amouns o a dsorng ax on cash holdngs. Thus, equalzng he reurn on money o ha of oher asses mnmzes dsorons and mproves effcency. Ths logc presumes ha he governmen has access o oher less dsoronary sources of revenues, bu hs need no be rue n general. Phelps (973) argues ha n a publc fnance model n whch he governmen ses dsoronary axes on commodes and labor ncome, may well be opmal o ax cash holdngs f he neres elascy of money demand s lower han he prce elascy of oher commodes n absolue value. Char, Chrsano and Kehoe (996) adop Phelp s publc fnance approach and only allow for dsorng axes. They show ha Fredman rule s opmal for a class of preferences ha s broadly used n macroeconomc models. Essenally, he requremen s ha ncome elascy of money demand s equal or greaer han one, and ha preferences are weakly separable beween consumpon and labor. Ther argumen s an applcaon of he nverse elascy rule of commody axaon. If he ncome elascy on money demand s greaer han one, s opmal se he ax on cash holdngs lower han on oher commodes. Ths s mpossble gven he zero lower bound on nomnal neres raes, whch mples ha he Fredman rule holds for a large class of preferences. The opmaly of he Fredman rule s robus. For one, he requred resrcons on preferences are hose ha guaranee conssency wh balanced growh (Alvarez, Kehoe and Neumeyer, 004), and when hese resrcons are no sasfed, he opmal nomnal neres rae s very close o zero (Correa and Teles, 999). The Fredman rule fals o hold when he ax sysem s ncomplee. For example, a deparure from he Fredman rule mgh be opmal as a way o ax monopoly profs when a corporae ncome ax s no avalable for hs purpose (Schmd- Grohe and Urbe, 004a). Smlarly, he Fredman rule, whch requres ha nflaon s equal o he negave of he real neres rae, generaes relave prce dsorons n economes wh scky prces and hus may fal o hold (Schmd-Grohe and Urbe, 004b). However, f he relave prce dsorons can be removed 4

3 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. wh a full se of commody axes or subsdes, he Fredman rule s rensaed. Wha happens when redsrbuonal consderaons are nroduced? Ths s a pressng queson, gven he wealh of evdence on heerogeney n ransacon paerns, asse holdngs and labor ncome. Recen resuls sugges ha he rade-off beween redsrbuon and effcency genercally undermnes he opmaly of he Fredman rule. In hs paper, I sudy he opmal fscal and moneary polcy wh heerogenous agens and show ha he Fredman rule s opmal only f posve nomnal neres raes do no amelorae consrans on redsrbuon va labor axes. In parcular, for economes ha exhb emprcally plausble cross-seconal correlaons beween money holdngs and labor ncome, he Fredman rule s opmal only f he governmen wshes o redsrbue n favor of he poor. I begn by analyzng a smple one perod example ha clearly llusraes how resrcons on redsrbuon semmng from he srucure of labor axes or ncenve compably consrans generae deparures from he Fredman rule. I hen sudy an economy where heerogeney n ransacon paerns arses endogenously from dfferences n labor producvy. Low skll agens hold more cash as a fracon of oal purchases, conssen wh crossseconal evdence n he US (Mullgan and Sala--Marn, 000, Erosa and Venura, 00) and oher counres (Guso, Halassos and Jappell, 00). The governmen axes labor ncome a a proporonal rae and ses he nomnal neres rae. The model smplfes o he one consdered n Char, Chrsano and Kehoe (996) once heerogeney s removed. The Fredman rule urns ou o be opmal only f he Pareo wegh on low skll agens s hgh enough. Ths resul s based on a smple logc. The lneary of he labor ncome ax consrans he governmen s ably o redsrbue and nflaon hus assumes an auxlary role. The labor ncome ax weghs more heavly on hgh skll agens, whereas nflaon s a greaer burden for low sklled agens. Thus, f he governmen favors hgh skll agens, a deparure from he Fredman rule s opmal. If he governmen favors low skll agens, s opmal o subsdze cash holdngs. Snce he zero lower bound con- 5

4 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 sran on he nomnal neres rae prevens hs, he Fredman rule s he bes polcy f he Pareo wegh on low skll agens s hgh enough. da Cosa and Wernng (007) also sudy an economy where agens dffer n labor producves and he governmen ses labor ncome axes and he nomnal neres rae. They allow labor ncome axes o be non-lnear and assume ha ndvdual labor producves are prvae nformaon, followng Mrrlees (97). The governmen can use ndvdual specfc lump sum axes o acheve any amoun of redsrbuon. However, prvae nformaon on ndvdual ables nroduces ncenve compably consrans. In parcular, low skll agens labor supply mus be dsored o preven hgh skll agens from mmckng hose wh low sklls. The ncenve compably consrans lms redsrbuon and he nomnal neres rae could play an auxlary role f relaxes hese consrans. da Cosa and Wernng assume ha money and labor effor are gross complemens and show ha he Fredman rule s opmal. Under hs condon, he demand for money rses wh labor effor, for gven consumpon. Then, a reducon n he nomnal neres rae relaxes ncenve compably consrans, snce an ncrease n he demand for money also ncreases labor effor. Agan, gven he zero lower bound on he nomnal neres rae, he bes polcy o amelorae he ncenve problem o follow he Fredman rule. Whle hese resuls appear prma face nconssen, hey can be nerpreed as an applcaon of he unform commody axaon prncple. Aknson and Sglz (976) show ha, f he labor ncome ax schedule s suffcenly unconsraned or he ncome elascy does no vary across goods, all commodes should be axed a he same rae, rrespecve of agens wegh n he socal welfare funcon. A proporonal labor ax generaes a conflc beween effcency and redsrbuon ha nduces he governmen o abandon unform commody axaon f redsrbuonal goals are presen and ncome elasces vary across goods. Smlarly, wh non-lnear labor ncome axes and prvae nformaon on ndvdual labor producves, dfferenal commody axaon may be opmal as a screenng devce. In a 6

5 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. moneary economy, unform commody axaon ranslaes no opmaly of he Fredman rule. Moreover, he zero lower bound consran he nomnal neres rae mples ha he Fredman rule wll also hold n all hose cases ha would call for a lower ax rae on cash purchases relave o oher consumpon goods n a real economy. These resuls confrm he connecon beween opmaly of he Fredman rule n moneary economes and opmaly of unform commody axaon n real economes, esablshed by Charm, Chrsano and Kehoe (996) for represenave agen models. They also esablsh a general prncple for heerogeneous agen economes. For models ha dsplay an emprcally plausble correlaon beween cash holdngs and labor ncome, he Fredman rule s opmal only f he governmen wshes o dsrbue n favor of he poor. I prove hs resul under proporonal labor ncome axes n hs paper. The analyss n da Cosa and Wernng (007) suggess ha hs prncple also holds wh non-lnear labor axes. Under ularan socal welfare, hey prove ha he Fredman rule s opmal when cash holdngs and labor ncome are gross complemens. Ths resrcon on preferences mples a posve correlaon beween cash holdngs and labor ncome, whch s nconssen wh he emprcal evdence. If cash holdngs and labor ncome are gross subsues, a deparure from he Fredman rule would relax ncenve compably consrans and mprove effcency. However, he Fredman rule may sll be opmal f he governmen values dsrbuon owards low producvy agens enough. The paper proceeds as follows. Secon analyzes a smple one perod economy and llusraes he condons for he opmaly of he Fredman rule n relaon o he consrans on redsrbuon. Secon draws a connecon beween hese fndngs and he unform commody axaon prncple, as well as he opmaly of he Fredman rule n represenave agen economes. I also dscusses he emprcal evdence on ransacon paerns and asse holdngs and relaes o he heorecal fndngs. Secon 3 sudes opmal polces n a dynamc economy wh endogenous heerogeney n ransacon paerns. Secon 4 dscusses he me 7

6 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 conssency of opmal polces wh heerogeneous agens. Secon 5 dscusses he effec of aggregae and dosyncrac shocks, and Secon 6 concludes by ponng o numerous open quesons for fuure work.. - Redsrbuon and he Fredman Rule: A Smple Example To llusrae he forces shapng he opmal seng of moneary and fscal polces n economes wh heerogeneous agens, I begn by descrbng a smple one perod economy. A subse of commodes, cash goods, are purchased wh currency, whle he ohers, cred goods, are no subjec o hs requremen. The dsncon beween cash and cred goods s bul no preferences, as n Lucas and Sokey (983). The nomnal neres rae, R, smply corresponds o he relave prce of cash goods. Agens supply labor and dffer n labor producves. Leng T (l) denoe he ax on labor ncome, l, governmen polcy s gven by {R, T (l)}. In he frs example, T (l) s lnear n l. In he second example, labor axes are non-lnear and ndvdual producves are prvae nformaon. In boh examples, he opmal nomnal neres rae depends on he neracon beween he demand for cash goods and labor supply.. Lnear Income Taxaon The economy s populaed by a connuum of agens who dffer n labor producvy, ξ. Half of he agens have producvy, ξ, and he oher half, ξ, wh ξ > ξ > 0. Ther preferences are gven by: () U (c, c, l/ξ ) = u, (c ) + u (c ) l/ξ where c and c denoe cash and cred goods respecvely. Here u ( ) for =, are srcly ncreasng and srcly concave funcons, defned on he posve reals. 8

7 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. Agens choose {c, c, l } =, o maxmze () akng as gven he nomnal neres rae and he ax on labor ncome, {R, T (l)}. Ther budge consran s: () Rc + c l T(l ) Followng he radon of he Ramsey axaon leraure, I assume ha he labor ncome ax s affne, so ha T (l) = τ + τl. Here, τ > 0 corresponds o a lump sum ransfer and τ s he proporonal rae on labor ncome. The soluon o an agen s problem s hen characerzed by he followng hree equaons: (3) (4) (5) ξ u ( c ) = ( τ) Rc + c = l ( τ) + τ u ( c) = R u ( c ) (6) The resource consran s: 05. ( c + c l ) + g 0 =, where g > 0 s exogenous governmen consumpon. The governmen budge consran s: (7) g 05. T ( l ) Condons (3)-(7) for =, characerze an equlbrum for hs economy and represen a mappng beween governmen polcy, {R, τ, τ } wh R, and equlbrum allocaons. The consran on R corresponds o he zero lower bound consran on nomnal neres raes n moneary economes. The Ramsey equlbrum s smply he bes equlbrum from he sandpon of he socal welfare funcon. Le socal welfare be gven by: (8) ηuc (, c, l/ ξ ) =, =, 9

8 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 where η [0, ] wh η = correspond o he Pareo weghs for he wo groups of agens. Formally, he governmen chooses {R, τ, τ } o maxmze (8) subjec o (3)-(6) for =,. The mos effecve way o characerze he Ramsey equlbrum s o pose he governmen problem n he allocaon space. Ths requres expressng he se of equlbrum condons (3)-(6) n erms of allocaons only. Ths reformulaon of he problem s known as he prmal approach o Ramsey polces (see Char and Kehoe, 999). Ths approach s parcularly neresng for hs economy snce clarfes he consrans on redsrbuon ha are mplc n he polcy nsrumens avalable o he governmen. The frs consran can be derved by expressng (5) n erms of allocaons only, usng (3)-(4) o subsue for τ, τ and R, and combnng he resul wh (7). Ths gves rse o he followng consran: (9) u c c u c c l 05. ( ) + ( ) / =, ξ 0 Ths condon, known as mplemenably consran, does no capure all he consrans on Ramsey allocaons. There are wo wedges n hs economy, he cash-cred wedge, whch corresponds o (3), and he consumpon-labor wedge, capured by (4). Snce all agens face he same nomnal neres rae and he proporonal ax rae on labor ncome s no agen specfc, hese wedges mus be equalzed across agens. In addon, n any equlbrum R. The resulng consrans on he opmal allocaons are: (0) u ( c) u = ( c) u ( c ) u ( c ) () ξ u ( c ) = ξ u ( c ) The Ramsey allocaon problem corresponds o he choce of {c, c, l } =, o maxmze (8) subjec o (9)-() and (6). 0

9 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. Consrans (0)-() clearly denfy he lma o redsrbuon n hs economy. The lnear labor ncome ax nduces (), whch mples ha relave consumpon levels n he populaon are solely drven by relave producves and canno be nfluenced by he governmen. Sll, dfferences n labor producves may generae dfferences n consumpon paern. Snce value of R above effecvely correspond o a subsdy o cred good consumpon, hey could n par offse he resrcon mposed by (). Consrans (0)-() wll ypcally be bndng and hey shape he properes of Ramsey polces. To examne he effec of he consrans on redsrbuon mposed by he lneary of labor ncome axes and (0), s useful o frs solve a verson of he Ramsey allocaon problem where consrans (0)-() are dropped. Aachng mulplers µ 0 and λ 0 o (9) and (6), respecvely, he frs order necessary condons for hs relaxed problem are: () (3) η u j ( cj ) + µ u j ( cj ) + u j ( cj ) c j = λ ( η + µ ) + η = ξ 0 for, j =,, where µ = 0.5 µ and λ = 0.5λ. Equaon () mmedaely mples: u ( c) = u ( c ) for =,. Le assume for smplcy: u j (c) = c -σ / ( σ) for σ > 0. Combnng () for j = and (3) obans: ( ) = + σ ξ η + µ ( σ ) ( c ) η µ for =,. Ths condon clearly volaes () unless η = η. Thus, genercally, f he governmen has redsrbuonal goals, ha s η η, () wll be bndng.

10 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 Le s now mpose () on he Ramsey allocaon problem, whle sll gnorng (0). Denoe wh χ he Lagrangan mulpler aached o (). The frs order necessary condon for c ncorporang () s: (4) σ σ η + µ σ c + χ σ c ( ) ( ) ( ) ( ) = λ for =,, wh complemenary slackness condon: χ ξ u ( c ) ξ u ( c ) = 0 Combnng (4) and (3) obans: for =,. By (): σ χ( ) σ ξ ( c ) = ( η + µ ) η + µ ( σ) + c (5) η η χ = c ( η + µ ) ξ + ξ / σ Combnng (4) and () for =, : (6) σ ( c ) ( c ) = + ( ) σ χ + ( ) σ c η µ σ The nably of he governmen o redsrbue va labor ncome axes generaes a move for dsorng he cash-cred good wedge. Le s consder he case η > η, when by (5), χ > 0. The governmen favors redsrbuon o ype agens and can nduce such a redsrbuon, even wh a lnear labor ax rae, by subsdzng cash good consumpon for ype agens and axng for ype agens. Smlarly, f η > η and χ < 0, he governmen can redsrbue o ype agens by subsdzng her cash good

11 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. consumpon and axng for ype agens. However, he governmen canno se agen specfc subsdes o cash good consumpon and, n fac, canno subsdze cash good consumpon a all. By (0), he cash-cred wedge mus be equalzed across agens and mus be greaer han. Ths resrcon s clearly volaed n (6). Even f condon (6) does no hold when consran (0) s formally ncorporaed no he analyss, clearly denfes when wll be opmal o depar from he Fredman rule. If η > η, ha s when he governmen favors ype agens, a deparure from he Fredman rule relaxes he consran on redsrbuon mplc n he proporonal labor ncome ax. Insead, when he governmen favors redsrbuon owards ype agens, a subsdy o cash goods could play hs role. Snce such a subsdy volaes he zero lower bound on he ne nomnal neres rae, he Fredman rule wll be opmal for η > η.. Non-Lnear Income Taxes and Prvae Informaon Ths example makes wo changes relave o he prevous se up. Frs, ndvdual producves are assumed o be prvae nformaon. Second, he governmen selecs a labor ncome ax schedule T (l) ha s allowed o be arbrarly non-lnear. As n Mrrlees (97), prvae nformaon mples ha he opmal allocaon mus sasfy ncenve compably consrans o nduce agens o ruhfully reveal her ype. Ths requremen shapes he properes of he opmal ax schedule and may nfluence he opmal value of he nomnal neres rae. I wll rever o he general specfcaon of preferences, assumng ha U (c, c, l/ξ) s ncreasng n he frs wo argumens, decreasng n he hrd and srcly concave. In addon, he sngle crossng condon wll be mposed so ha he ndfference curves beween cred good consumpon and labor effor l/ξ become flaer as producvy ncreases. Under See SALANIÉ B. (000), for a dscusson. 3

12 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 he sngle crossng condon, he ncenve compably consran s bndng only for he hgh producvy ype and s gven by: (7) Uc (, c, l / ξ ) Uc (, c, l/ ξ ) The zero lower bound consran for he general uly specfcaon corresponds o: (8) U ( c, c, l / ξ ) U ( c, c, l / ξ ) U( c, c, l / ξ ) = U( c, c, l / ξ ) The axaon prncple holds n hs envronmen (see Guesnere, 998), so he governmen problem can be formulaed n he allocaon space drecly. The governmen chooses {c, c, l } =, o maxmze (8) subjec o (7), (8) and (6). Le s frs consder he governmen problem when (8) s dropped. Denong wh µ and λ he mulplers on (7) and on he resource consran, respecvely, he frs order necessary condons for hs problem are: (9) (0) () () ηu ( c, c, l / ξ ) µ U ( c, c, l / ξ ) = λ, for j =, j ηu ( c, c, l / ξ )/ ξ µ U ( c, c, l / ξ )/ ξ = λ 3 ( η + µ ) U ( c, c, l / ξ ) = λ, for j =, j j 3 3 ( η + µ ) U ( c, c, l / ξ )/ ξ = λ for λ = 0.5λ. The resulng expressons for he opmal comsumpon labor wedge are: (3) U( c, c, l / ξ ) = U ( c, c, l / ξ ) 3 4

13 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. (4) µ η 3 3 U( c, c, l / ξ ) 3 U ( c, c, l / ξ )/ ξ ξ ξ ξ ξ U ( c, c, l / )/ U ( c, c, l / )/ = + U( c, c, l / ξ ) U3( c, c, l / ξ )/ ξ + Condon (3) reproduces he cusomary no dsorons a he op. Snce ype agens do no have an ncenve o mmc ype agens, here s no need o dsor he allocaon for ype. Insead, ype 's wll be dsored as long as ype 's ncenve compably consran s bndng, as mpled by (4). Ths dsoron affecs he opmal allocaon for ype n a way ha makes undesrable for ype o msrepor her ype. In parcular, he dsoron corresponds o a posve margnal ax on ype 's labor ncome when: (5) U( c, c, l / ξ ) U ( c, c, l / ξ )/ ξ < 3 Snce by () and (9), c j < c j for j =, and l > l when he ncenve compably consran s bndng, (5) holds f he reducon n labor effor ncreases he demand for cred good consumpon, ha s when cred good consumpon and labor effor are gross subsues. By conras, f cred goods and labor effor are gross complemens, and ncrease n cred good consumpon wll ncrease labor effor, and he opmal margnal labor ncome ax on ype agens wll be negave. Le s now consder he mplcaons for he cash-cred wedge. By () and (9): (6) U( c, c, l / ξ ) = U ( c, c, l / ξ ) 5

14 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 (7) U( c, c, l / ξ ) = + U ( c, c, l / ξ ) µ U( c, c, l / ξ ) U( c, c, l / ξ ) η U( c, c, l / ξ ) U( c, c, l / ξ ) By (6), ype agens margnal rae of subsuon beween cash and cred goods s no dsored, whle here wll be a dsoron for ype agens f he ncenve compably consran s bndng and he expresson n square brackes n (7) s no zero. Thus, he raonale for dsorng he cash cred margn for ype agens s smlar han for he labor wedge. The consumpon paern s dsored o make less appealng for ype agens o repor o be ype. The sgn and he exen of he cash-cred dsoron for ype agens depends on preferences. Clearly, for U ( ) weakly separable n consumpon and labor he cash-cred wedge s undsored. More n general, here wll be an neracon beween he choce of labor and consumpon and wll be opmal o dsor he cash-cred wedge. Specfcally, he dsoron wll be posve when he margnal rae of subsuon beween cash and cred good consumpon: (8) U( c, c, l / ξ ) > U ( c, c, l / ξ ) Snce by () and (9), c j < c j for j =, and l > l when he ncenve compably consran s bndng, (8) corresponds o a uly specfcaon n whch a ype agen who cheas and reduces her labor effor wshes o ncrease her cash good consumpon more han her cred good consumpon. In oher words, wll be opmal o ncrease he relave prce of cash goods when cash good consumpon and labor effor are gross subsues. Thus, he margnal ax on cash consumpon wll reduce he margnal uly benef from reducng labor effor and relax he ncenve compably consran. If he oppose 6

15 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. nequaly holds, cash goods and and a reducon n labor effor are complemens, hus an ncrease n cash good consumpon wll generae a declne n he margnal uly value of a reducon n labor effor. In hs case, s opmal o subsdze cash good consumpon. Equaon (7) volaes (8) n boh cases bu pons o he condons under whch a deparure from he Fredman rule are opmal. If a declne n cash good consumpon relave o cred good consumpon ncreases he margnal uly value from reducng labor effor, R > relaxes he ncenve compably consran and mproves effcency. Insead, f a rse n cash good consumpon relave o cred good consumpon nduces a declne n he margnal uly value of reducng labor effor, a lower nomnal neres rae relaxes he ncenve compably consran. In he general governmen problem, he zero lower bound consran on he nomnal neres rae wll be bndng n hs case and he Fredman rule wll be opmal. Ths s exacly he case consdered by da Cosa and Wernng (007), who show ha when cash goods and labor effor are gross complemens, he Fredman rule s opmal Dscusson 3. Theory There s a srong lnk beween he opmaly of he Fredman rule and he unform commody axaon prncple n represenave agen economes. Char, Chrsano, and Kehoe (996) show ha when preferences are homohec n cash and cred goods and weakly separable n labor, he Fredman rule holds. Weak separably s he suffcen condon for opmaly of unform commody n a represenave agen economy (see Aknson and Sglz, 980). Homohecy also mples ha he ncome elascy of money demand s unary. If he homohecy assumpon s relaxed, he nverse elascy rule of commody axaon apples. Cash goods 7

16 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 should be subsdzed relave o cred goods whenever he ncome elascy of money demand s greaer han one, ha s when an ncrease n labor supply ncreases he demand for cash goods. Ths case corresponds o gross complemenary n preferences beween cash goods and labor effor. In a real nerpreaon of he economy, he opmal ax sysem would subsdze cash goods relave o cred goods. However, such a subsdy would volae he zero lower bound consran on nomnal neres raes n he moneary verson of he economy. Thus, he bndng zero lower bound on he nomnal neres rae mples ha he opmaly of he Fredman rule holds for a broader class of preferences han unform commody axaon. The unform commody axaon prncple and he bndng zero lower bound on he nomnal neres rae also play a role n he heerogeneous agen economy dscussed above. Aknson and Sglz (976) derve he condons for opmaly of unform commody axaon n heerogeneous agen economes. A key nsgh s ha when redsrbuon s consraned n some way, dfferenal commody axaon may be opmal o aan he desred dsrbuon of resources. The srucure of labor ncome axes and he assumpons on preferences are boh crcal for he properes of opmal commody axes. The case wh dfferences n producves and lnear ncome ax s one n whch dfferenal commody axaon s opmal n general wh redsrbuonal objecves. In parcular, commodes wh hgh ncome elascy should be axed more heavly, as hey wll be consumed more by hgh producvy agens. If nsead he labor ncome ax s nonlnear wh prvae nformaon on ndvdual producves, dfferenal commody axaon s opmal only f he paern of consumpon can serve as a screenng devce. If uly s homohec n consumpon and weakly separable n labor, hen he ncome elascy of demand s unary for all commodes, and unform commody axaon apples. Ths logc apples squarely o he examples consdered here. In he frs, he consran on redsrbuon semmng from he proporonal labor ncome ax movae deparures from he Fredman rule. Ths consran mples ha, n a real verson of 8

17 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. he economy, unform commody axaon would only hold wh equal Pareo weghs on boh ypes of agens. Snce n he moneary verson, he zero lower bound consran rules ou subsdes o cash goods, he Fredman rule holds for a range of Pareo weghs. The ncenve compably consran places a lm on redsrbuon wh non-lnear labor ncome axes. Deparures from he Fredman rule may be warraned f hey relax such consran. If cash goods and labor effor are gross subsues, ncreasng he relave prce of cash goods relaxes he ncenve compably consran, whle he oppose s rue f cash goods and labor effor are gross complemens. In a moneary verson of he economy, he zero lower bound consran on nomnal neres rae prevens he relave prce of cash goods from fallng below one. Thus, he Fredman rule s opmal for he class of preferences for whch a cash good subsdy would be opmal n a real verson of he economy. Ths explans he resul n da Cosa and Wernng (007) who show ha under weak gross complemenary beween cash goods and labor effor he Fredman rule s opmal. 3. Emprcal Evdence The prevous dscusson suggess ha he ncome elascy of moneary holdngs s key for he opmaly of he Fredman rule when redsrbuonal consderaons are presen. The emprcal evdence on he cross-seconal dsrbuon of currency can be used o dscplne assumpons on preferences and ransacons echnologes o derve realsc mplcaons for he ncome elascy of money demand and herefore opmal polces. The emprcal evdence srongly suggess a negave correlaon beween labor ncome (and wealh) and cash holdngs. Erosa and Venura (000) repor ha low ncome households use cash for a greaer fracon of her oal purchases relave o hgh ncome households n he US. Mullgan and Sala--Marn (000) esmae he probably of adopng fnancal echnologes ha hedge agans nflaon, based on US daa, and fnd ha s pos- 9

18 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 vely relaed o he level of household ncome and wealh, and o educaon. Aanaso, Guso and Jappell (00) fnd ha he probably of usng an neres bearng bank accoun ncreases wh educaonal aanmen, ncome and average consumpon, based on cross-seconal household daa for Ialy. Guso, Halassos and Jappell. (00) presen evdence from a broad se of counres ha confrms hs paern. The fac ha low ncome households hold more cash mples ha hey are more exposed o nflaon. Ths s conssen wh ndrec evdence on he dsrbuonal consequences of nflaon presened by Easerly and Fsher (000). Based on pollng daa for 38 counres, hey fnd ha he poor are more lkely han he rch o menon nflaon as a op naonal concern. Ths suggess ha low ncome household perceve nflaon as beng more cosly. Ths evdence srongly suppors models ha generae a negave cross-seconal correlaon beween labor ncome and moneary holdngs, gvng rse o less han unary ncome elascy of money demand. In he nex secon, I presen such a model and derve he mplcaons for opmal fscal and moneary polcy A Moneary Economy wh Heerogenous Agens The economy s populaed by agens, frms and a governmen. Agens consume, supply labor and rade n asses n each perod. They dffer n labor producvy bu have dencal preferences. Purchases are made wh currency or wh a cosly alernave paymen echnology. Perfecly compeve frms have access o a lnear echnology ha uses labor o produce consumpon goods. The governmen fnances an exogenous sream of spendng by axng labor ncome a a proporonal rae, ssung nomnal deb and prnng money. There s no aggregae or dosyncrac rsk. I now llusrae he model, a verson of he one analyzed n Albanes (005), n more deal. 0

19 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. 4. Frms There are wo ypes of compeve frms. All frms lve for one perod. Goods frms hre labor o produce a connuum of dfferenaed consumpon goods ndexed on he nerval [0, ]. The producon echnology s lnear, and dfferen consumpon goods are perfec subsues n producon. Perfec compeon mples: (9) P (j) = W for j [0, ], where P (j) s he prce of good j and W he nomnal wage per effcency un of labor a me. P = W wll denoe he prce of consumpon goods. Fnancal frms produce ransacon servces, enablng agens o purchase goods whou he use of cash. A fnancal frm s profs for provdng ransacon servces for he purchase of good j are: (30) π (j) W θ (j) where θ ( ) s measured n effcency uns of labor and sasfes θ' > 0 on he nerval [z, ], wh z 0. π s he dollar charge for arrangng purchases of consumpon good j whou currency. Prof maxmzaon mples: π (j) = W θ (j) for all and all j [0, ]. Ths specfcaon follows Presco (987). 4. Agens A un measure of agens s dvded no wo ypes, where 0 < v < s he fracon of ype agens, wh =, and Σ v =. All agens have dencal preferences defned over a consumpon aggregaor c and over hours of work l gven by: (3) = 0 β Uc (, l) ρ c = c ( j) dj 0 ρ

20 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 where ρ (0, ) for a agen of ype =,. I wll resrc aenon o preferences of he class: U (c, l ) = h (c ) + v (l ), where h s srcly ncreasng and srcly concave, whle v s srcly decreasng and concave. Agens of dfferen ypes dffer n labor producvy, denoed wh ξ, for =,. I wll assume ξ > ξ. In each perod, agens choose ransacon servces and consumpon levels, hey supply labor, accumulae currency and rade nomnal bonds. Gven (3) and he assumpon on ransacon coss, agens wll opmally choose z, he fracon of consumpon goods purchased whou he use of cash, and c,, c,, s he level of consumpon of goods purchased wh and whou currency, respecvely. Then: ρ ρ ρ c = ( z )( c ) + z ( c ) Gven M, an agen begnnng of perod cash holdngs, he cash n advance consran s: (3) P c, ( z ) M 0. The asse marke mees afer radng on goods and labor marke has closed. Durng he asse marke sesson agens receve labor ncome ne of axes, clear consumpon lables and rade bonds ssued by oher agens or by he governmen. Bonds purchased a me pay one un of currency n he + asse marke. The governmen and prvae agens are commed o deb repaymens, so ha agens are ndfferen beween holdng prvaely or governmen ssued bonds. The prce of a nomnal bond a me s Q. Ne new purchases of bonds by agen a me are denoed wh B + for =,.

21 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. The asse marke budge consran s: (33) M + + QB+ z M + B Pc, ( z ) Pc, z π () j dj + W ξ ( τ ) l z where τ s he ax rae on labor ncome and z π z () jdj he currency cos of arrangng purchases of consumpon goods wh cred. In addon, a no-ponz game condon: (34) + Q + 0 B + Qss, + Φ + s= 0, + M sw+ s s= + s Φ Φ ( τ ) ξ s also requred, wh Φ = Π - '=0, Q ', Φ 0 =. 4.3 Governmen The governmen fnances an exogenous sream of consumpon g and s subjec o he budge consran: (35) P g + M + B = Q B + + M + + W T where M, B are he supply of currency and bonds, respecvely, and: (36) T = v τ ξl 4.4 Compeve Equlbrum The mng of evens n each perod s as follows: 3

22 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007. Agens ener he perod wh holdngs of currency and deb gven by M and B for =,0,... and choose z.. Agens, frms and he governmen rade on he goods and labor markes. The agens purchases of cash goods are subjec o (3). Clearng on he goods marke requres: (37) v c z c z z, ( ) +, + θ( j) dj ξ l g = =, 3. Asse markes open. Agens purchase bonds and acqure currency o ake no he followng perod subjec o he consran (33). Clearng on he asse marke requres: (38) =, vb = B for s > 0 =, vm, + = M + DEFINITION. A compeve equlbrum s gven by a governmen polcy {g, τ, M +, B } 0, a prce sysem {P, W, Q, π (j)} 0, j [0,] and an allocaon {c,, c,, l, z, B } =,, 0 such ha:. gven he polcy and he prce sysem agens and frms opmze;. governmen polcy sasfes (35) and (36); 3. markes clear. The followng proposon characerzes he compeve equlbrum. PROPOSITION. An allocaon {c,, c,, l, z, B } =,, 0 and a prce sysem {P, W, Q, π (j)} 0, j [0,] consue a compeve equlbrum f and only f, for a gven governmen polcy {g, τ, M +, B } 0, (37), (35) and he followng condons are verfed: (39) Q P = β P +, +, 4

23 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. (40) (4) 0 < Q ul,, R + = ξ ( τ ) for 0 W = P, + = Q, + + +, ( R )( P c ( z ) M ) = 0 P c, ( z ) M (4) for ρ ρ θ 0 for z s = z ρ R ( zs) s = 0 for zs ( z, z) c, s 0 for zs = z 0, and: (43) P 0 c -0 ( z 0 ) M 0 (44) M 0 ˆ 0, + u0, P0 = β u c + u = 0 B P 0,,, c ˆ, + ul, l 0 for =,, wh z Cz ( ) = θ ( jdj ) z Here, cˆ uj, = U( c, l )/ cj,, ul, = U ( c, l ) and Cz ( ), = c + = u /( z ),ˆ u = u z j = / for,, z 5

24 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 Equaon (44) s he agens neremporal budge consran and ncorporaes he ransversaly condon. The proof of hs proposon s n Appendx A Ramsey Equlbrum The Ramsey equlbrum s he compeve equlbrum ha maxmzes socal welfare from he sandpon of me 0. The governmen selecs polces once and for all a me 0 for all fuure perods and s commed o followng hese plans. The socal welfare funcon s smply a weghed sum of he agens lfeme uly. The Pareo wegh on ype agens s η, wh η + η =. Pareo weghs are me-nvaran and he case η = v corresponds o a ularan governmen. As he one perod example, I solve for he Ramsey equlbrum by dervng he Ramsey allocaon problem, where he governmen chooses an allocaon a me 0 subjec o he consran ha consues a prvae secor equlbrum. Ths problem s choce varables are {c,, c,, l, z } =,, 0. PROPOSITION 3. An allocaon {c,, c,, l, z } =,, 0 and values of {R } 0 consue a Ramsey equlbrum f and only f hey solve he prmal problem: subjec o: max U { c,, c,, l, z } β η (, c l ) =0 =, =,, 0 (45),, = R, for =, (46) R (47) l,, u ξ l,, = u ξ (4) and (37) for all, as well as (44) and (43), for gven P 0. 6

25 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. The proof of Proposon 3 parallels he one for a represenave agens economy n Char, Chrsano and Kehoe (996). Consrans (45)-(47) are he analogue of (0) and () n he frs example n Secon. The level of P 0 should also be a choce varable, snce s no pnned down by he compeve equlbrum condons and compeve equlbra are ndexed by P 0. However, he qualave properes of Ramsey polces for > 0 do no depend on he value of P 0, so I ake as gven and rea as an nal condon for he purpose of hs analyss. 5. Opmal Polces The key properes of Ramsey polcy for > 0 are descrbed n he followng proposon. PROPOSITION 4. Assume: (A) U (c, l) = h (c) + v (l), wh h ( ) srcly ncreasng and srcly concave and v ( ) srcly decreasng and concave; (A) θ (j) s srcly ncreasng for j [z, ], wh z 0, and lm z z θ (z) = 0. Then, R = for n he Ramsey equlbrum, f and only f consran (47) s no bndng. Moreover, f (47) s non-bndng, η η wh 0 < η. The proof of hs proposon s an applcaon of Albanes (005) and s omed for brevy. I s smlar o he proof of he opmaly of he Fredman rule for a represenave agen economy n Chrsano, Char and Kehoe (996) and reles on he homohecy of he consumpon aggregaor and separably of uly n consumpon and lesure mposed n (A). I holds rrespecve of he funconal form of v ( ) or nal condons. Proposon 4 demonsraes ha he Fredman rule s no opmal when hgh producvy agens enjoy a relavely hgh Pareo wegh, ha s η, s low. The nuon s smlar o ha The proof of hs proposon mplcely also shows ha n a homogeneous agen verson of hs economy he Fredman rule would be opmal. In such an economy, he consrans (45) and (47) would no be par of he Ramsey problem. 7

26 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 for he frs example n Secon. The opmal labor wedge for each ype of agen s decreasng n η and n he value of he mulpler on he mplemenably consran, a measure of he cos of rasng dsoronary revenues. Lnear labor axaon forces hs wedge o be equalzed across agens n any equlbrum, whch resuls n consran (47). As n he one perod economy, hs consran wll be genercally be bndng. A deparure from he Fredman rule can relax hs consran wh heerogeney n ransacon paerns. The redsrbuonal effecs of moneary polcy and he opmal nomnal neres rae hnge crcally on he negave cross-seconal correlaon beween cash holdngs and labor producves, an mplcaon of he equlbrum ncome elascy of money demand n hs model. One can defne a shor run elascy, correspondng o he sensvy of money holdngs o consumpon for gven ransacon paern, ha s z. Ths depends on he properes of he uly funcon only and s equal o one n hs model, gven he homohecy n consumpon of U ( ) for gven z. The long run elascy ncorporaes he effec on z. Snce by (4) he average cos of ransacon servces s decreasng n he level of consumpon, he long run ncome elascy of money demand s smaller han one. Ths propery mples ha hgh producvy agens hold less cash as a fracon of consumpon. Consequenly, a deparure from he Fredman rule s opmal when he governmen wshes o redsrbue n her favor, ha s when he Pareo wegh on low producvy agens s low enough. Wha would be he predcons of hs model wh non-lnear axes and prvae nformaon on labor producves? da Cosa and Wernng (007) consder a general uly specfcaon of he form U (m, c, l) and show ha he Fredman rule s opmal f money and labor are gross complemens. For he model n hs paper, solvng he sub-opmzaon problem n whch for gven real balances, cred good consumpon and labor effor, agens choose z and cash good consumpon delvers an ndrec uly funcon of he form U (m, c, l), wh U separable n {m, c} and l. Ths separably mples ha he level of moneary holdngs does 8

27 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. no nfluence labor effor and hus he Fredman rule would be opmal wh non-lnear ncome axes. More n general, f uly s allowed o be non-separable n consumpon and labor, s sraghforward o show ha m and I are gross subsues for gven c n he mpled specfcaon of U, f z s ncreasng n c. Gross subsuably beween money holdngs and labor follows from he fxed cos of usng alernaves o cash o make paymens. In urn, hs feaure s essenal for generang an emprcally plausble cross-seconal dsrbuon of currency. Gross subsuably beween money holdngs and labor effor mples ha a rse n he nomnal neres rae relaxes he ncenve compably consran n da Cosa and Wernng s model, as llusraed n he smple example dscussed n Secon. da Cosa and Wernng assume a ularan governmen and do no consder redsrbuonal moves. Ths dscusson suggess ha wh assumpons on preferences ha generae an emprcally plausble cross-seconal dsrbuon of money, even wh non-lnear labor ncome axes, he Fredman rule s opmal only f he governmen wshes o redsrbue n favor of he low producvy agens. Oherwse, effcency consderaons would render posve nomnal neres raes opmal. Baacharya, Haslag and Marn (005) also analyze deparures from he Fredman rule n a varey of heerogeneous agen models wh ad hoc resrcons on fscal nsrumens and show ha he opmal moneary polcy s sensve o hose resrcons. 5. A Calbraed Example I s neresng o analyze he properes of opmal polces as a funcon of he dsrbuon of Pareo weghs for a calbraed example o quanavely evaluae he exen o whch deparures from he Fredman rule are opmal. I consder he followng specfcaon for uly and ransacons echnology: 9

28 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 (48) σ ( c ) Uc (, l) = + vl ( ), for =,, σ > 0 σ (49) vl ( ) = γ 0 ( l ) γ γ, γ, γ 0 > 0 (50) = θ 0 θ j z ( ) = j 0 for j z z j θ = for j for j ( z, z) z where 0 z < z. I se β = 0.97 and σ = 0.8. Oher parameers are chosen so ha n a seady sae wh τ = τ = 0.30 and R =.05 he model maches correspondng averages for he US economy. The fracon of low producvy agens n he populaon s se o 0.6 and her producvy s se o ξ =, whle ξ s se so ha he Gn coeffcen for consumpon n he model s equal o 5.5% 3. The properes of money demand depend on ρ, θ 0 and θ. I fx ρ = 0.5 and se θ 0 and θ o approxmae he neres elascy and he average velocy of ransacons accouns (currency plus checkable deposs, plus me and savngs deposs) as a fracon of personal consumpon expendures. These wo sascs are equal o 5.% and.37, respecvely, based on Flow of Funds daa for he pos-war perod. Inal real and nomnal deb holdngs are se o 0 and he dsrbuon of currency s symmerc. The parameers are summarzed n Table. Graph dsplays he resuls for he case wh lnear labor axaon. The opmal nomnal neres rae and labor ax rae are ploed as a funcon of η, for > 0. The Fredman rule s opmal for η v = 0.6. The ax rae on labor s ncreasng n η, even for η > v. Ths resul emerges snce he mulpler on he 3 More deals on he calbraon are avalable n ALBANESI S. (005). 30

29 Labor Tax R S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. TABLE PARAMETER VALUES Fxed β z z v ξ ρ Calbraed σ γ 0 γ ξ θ 0 θ mplemenably consran on ype falls (and he one for ype ncreases) as η rses. In oher words, he margnal value of ransferrng resources o ype falls, whch nduces a rse n he opmal labor ax rae. The ax rae on labor vares from 0.7 o 0.4, whle he ne nomnal neres rae from 6% o 0. The value of η need no be exremely small o movae emprcal plausble deparures from he Fredman rule. For example, for η = 0.5, he ne nomnal neres rae s equal o 7%. PROPORTIONAL LABOR TAX GRAPH η η 3

30 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO Tme Inconssency Wha are he mplcaons for me conssency of Ramsey polces? I s well known ha Ramsey polces are me nconssen n a represenave agen framework (Kydland and Presco, 977) 4. Wh dsoronary axes, a surprse revson of he neremporal pah of axes ha deprecaes he presen value of ousandng governmen lables can effecvely reproduce he mssng lump sum ax. Lucas and Sokey (983) show ha n a real economy hs ncenve can be elmnaed by appropraely resrucurng ousandng clams on he governmen. Ths opon s no avalable n a moneary economy. To ensure me conssency, boh real and nomnal governmen deb mus be non-zero for a gven pah of prces. Bu any posve level of nomnal deb generaes he empaon o nfle away wh a one me rse n he prce level. Lucas and Sokey conclude ha commmen o a pah for nomnal prces s necessary for me conssency n a moneary economy. Alvarez, Kehoe and Neumeyer (004) also examne hs ssue and prove ha opmaly of he Fredman rule s a necessary and suffcen condon for me conssency. The key sep n her argumen s ha, under he Fredman rule, a moneary economy s equvalen o a real economy. Then, seng he presen value of nomnal governmen lables o zero a all daes and saes removes he ncenve o change he pah of prces, whle an approprae choce of he maury srucure of real deb can remove he ncenve o change he pah of axes. Wh heerogeneous agens, snce agen specfc lump sum axes are ypcally no avalable, here could be wo moves for devang from prevously announced polces: ncreasng effcency and mprovng redsrbuon. Ths would seem o exacerbae he me nconssency problem. Albanes (005) analyzes me conssency of Ramsey polces n a verson of he model presened n Secon 4 and shows ha 4 In economes wh capal, here s an addonal ncenve o devae from prevously announced plans, snce he capal ax does no dsor pas nvesmen decsons. 3

31 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. Ramsey polces are me conssen. The proof follows he smple sraegy lad ou by Lucas and Sokey (983), based on he followng operaonal defnon of me conssency. For any 0, defne he Ramsey problem a perod analogously o he Ramsey problem for perod 0. Then, he Ramsey problem a perod s me conssen for perod +, f he soluon of he Ramsey problem a perod solves he Ramsey problem a +. The Ramsey equlbrum s me conssen f he Ramsey problem a me s me conssen for he Ramsey problem a + for 0. In pracce, snce he Ramsey equlbrum allocaon s a saonary funcon of he sae for, s suffcen o denfy a se of nal condons for he me problem exs ha would nduce he governmen a me o connue wh he allocaon ha solves he Ramsey problem a me 0. The me conssency of Ramsey polces wh heerogeneous agens s ha a one me change n he prce level has redsrbuonal effecs. The redsrbuonal coss can offse any effcency gans for an approprae dsrbuon of nomnal deb 5. Snce agens are ndfferen wh respec o her porfolo composon n equlbrum, s always possble o denfy a dsrbuon of deb ha guaranees he governmen wll sck o Ramsey polces n fuure perods. Wha happens wh non-lnear axes? In hs case, here are no ncenves o devae for he purpose of reducng he deadwegh burden assocaed wh governmen consumpon, snce lump sum axes are allowed. However, here wll be an ncenve o revse polces o amelorae he dsrbuon of resources. Wh a ularan socal welfare funcon, would be opmal o equalze consumpon across agens wh dfferen producves. However, hus polcy volaes ncenve compably consrans. More consumpon mus be promsed o hgh ably agens o nduce hem o reveal her ype, raher han mmc low ably agens. 5 Ths argumen was frs explored by ROGERS C.A. (986), who sudes opmal wage and capal axes n a wo-perod, mulple consumer economy. She fnds ha he ncenve o rase capal axes may be moderaed by redsrbuonal concerns. 33

32 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 Once ables have been revealed, however, he governmen has he ncenve o equalze consumpon, gvng rse o me nconssency. Ths problem was frs lad ou by Robers (984). How s he argumen affeced wh heerogeneous moneary holdngs? In he case consdered by da Cosa and Wernng (007), he complemenary of money holdngs and labor effor mples a posve correlaon beween moneary holdngs and ably. Then, a surprse ncrease n he prce level equalzes he dsrbuon of consumpon. By conras, when moneary holdngs and labor effor are subsues n preferences, hgh ably agens hold less money, as n he daa. The negave correlaon beween moneary holdngs and ably may gve rse o he ncenve o engneer a one me declne n he prce level. Thus, fscal polcy s me nconssen and so s moneary polcy Shocks The prevous analyss compleely absracs from he presence of aggregae and dosyncrac shocks. Sll, he resuls sugges a few nformed conjecured. Le s consder he case wh aggregae shocks frs. Wh non-lnear axes aggregae shocks can be smoohed by changng he level of he lump-sum componen of axes, so hey do no affec he analyss. Wh lnear axes, a fundamenal resul n represenave agen economes s ha nflaon exhbs very hgh volaly (Char, Chrsano and Kehoe, 99). Innovaons n nflaon are used o mpar he opmal amoun of sae conngency o he real value of nomnal bond reurns, hus allowng o manan labor ncome axes. Ths s opmal snce nnovaons n nflaon ac as a lump sum ax and dsorons assocaed wh he labor axaon are convex n he ax rae. The forces shapng he sochasc properes of opmal nflaon wh lnear axes are very smlar o hose ha lead o me nconssency. Ths suggess ha n a heerogeneous agen verson of he model opmal nflaon volaly may be subsanally reduced. Innovaons n nflaon redsrbue resources across 34

33 S. ALBANESI Redsrbuon and Opmal Moneary Polcy, ec. agens wh dfferen levels of nomnal asse holdngs. The hgh nflaon volaly ha would be requred o opmally smooh dsoronary axes n he face of aggregae shocks necessarly nvolves sysemac redsrbuon n alernave drecons. For example, f money s he only asse, snce low ably households hold more cash as a fracon of oal purchases, posve nnovaons n nflaon redsrbue agans hem and negave nnovaons redsrbue n her favor. Dependng on he srengh of redsrbuonal consderaon, he governmen may fnd opmal o lm nflaon volaly o preven he negave redsrbuonal effecs of nflaon nnovaons. The effec of dosyncrac shocks s more suble. Levne (99) s perhaps he frs o pon ou ha he Fredman rule may be subopmal wh dosyncrac shocks. Hs argumen s based on he fac ha mplemenng he Fredman rule requres a saonary rae of moneary conracon. Ths s subopmal when fnanced wh lump sum axes snce weghs more heavly on agens ha, due o adverse dosyncrac shocks, have low ncome. On he oher hand, Erosa and Venura (00) show ha an ncrease n he average nflaon rae can generae adverse dsrbuonal consequences n a verson of he model n Secon 4. In her economy, agens -hold money, nomnal bonds and physcal capal. Fxed coss of asse parcpaon mply ha low ncome agens hold a greaer share of money n her porfolo, whch n urn mples ha hey can reap lower rae of reurns wh posve nflaon. Snce dosyncrac shocks generae a precauonary move for asse accumulaon, he rae of reurn dfferenal assocaed wh posve nflaon mples ha low ably households have smaller opporunes for selfnsurance, leadng o a large ncrease n nequaly n wealh and welfare for revenue neural ncreases n average nflaon. These resuls sugges ha opmal moneary polcy wh dosyncrac shocks depends crucally on he se of avalable fscal nsrumens. Le s consder a verson of he model n Secon 4 where agens are ex ane dencal and are subjec o dosyncrac producvy shocks, and assume he governmen s ularan. Hgh labor 35

34 RIVISTA DI POLITICA ECONOMICA LUGLIO-AGOSTO 007 ncome axes provde nsurance n hs case. Moreover, lucky agens ha have enjoyed a long seres of good producvy shocks wll hold lle money, hgh levels of nomnal governmen bonds and have hgh consumpon relave o agens ha experenced a long seres of adverse shocks. Innovaons n nflaon wll amelorae he dsrbuon of resources f he dfferences n governmen deb holdngs are suffcenly greaer han hose n cash holdngs. On he oher hand, low nomnal neres raes on he margn wll redsrbue o unlucky agens. If he governmen can also provde (unform) lump sum ransfers, may be opmal o ncrease margnal axes o fnance posve lump sum ransfers, gven ha here wll be a posve socal value o ransferrng resources o hose agens ha receve adverse shocks n he curren perod. In dynamc models wh prvae nformaon and dosyncrac producvy shocks, moneary holdngs can play mulple roles. As n da Cosa and Wernng (007), hey can be seen as jus anoher commody wh a relave prce ha corresponds o he nomnal neres rae. Alernavely, money can play he role of an asse. Le s consder hese possbles n order. If money s jus anoher commody, opmal moneary polcy s ed once agan o he opmaly of unform commody axaon. Golosov, Kocherlakoa and Tsyvnsk (003) show ha wh weakly separable preferences n consumpon and labor, unform commody axaon s opmal n hs class of models, conssen wh he fndngs n Aknson and Sglz (976). If preferences are allowed o be non-separable, may be desrable o depar from he Fredman rule f hs relaxes ncenve compably consrans, as prevously dscussed. If money s an asse, ha s s held manly as a sore of value raher han for ransacon purposes, he pcure s que dfferen. Agens who experence a long hsory of good shocks wll end up accumulang hgh levels of money. Thus, money holdngs can serve as a sasc for an agen hsory. Moreover, agens can use money o self-nsure. Ths s desrable from an ndvdual sandpon, snce he opmal allocaon n hs class of models exhbs ncomplee nsurance due o he ncenve 36