Ramsey Monetary and Fiscal Policy: The Role of Consumption Taxation

Transcription

1 Economics Working Paper Series 23/4 Ramsey Moneary and Fiscal Policy: Te Role of Consumpion Taxaion Giorgio Moa and Raffaele Rossi Te Deparmen of Economics Lancaser Universiy Managemen Scool Lancaser LA 4YX UK Auors All rigs reserved. Sor secions of ex, no o exceed wo paragraps, may be quoed wiou explici permission, provided a full acknowledgemen is given. LUMS ome page: p://

2 R M F P: R C T Giorgio Moa Lancaser Universiy Raffaele Rossi Lancaser Universiy November 4, 23 Absrac We sudy Ramsey moneary and fiscal policy in a small scale New Keynesian model were governmen spending as inrinsic value, public deb is sae-nonconingen and e fiscal auoriy is consrained by using disorive axaion. We sow a Ramsey policy is remarkably alered wen consumpion axaion is considered as a source of governmen revenues alongside or as an alernaive o labour income axes. Firs, we sow a e opimal seady-sae size of e public spending is, ceeris paribus, greaer under consumpion axaion an under labour income ax. We furer sow a adoping consumpion axaion as enourmous long run welfare gains and a ese gains are increasing in e level of ousanding public deb. Tese welfare gains are no limied o e seady-sae, bu ey are also presen in e dynamic socasic equilibrium. Te reason is a e dynamic naure of consumpion axaion enables e policy-maker o affec e socasic discoun facor via modificaions of e marginal uiliy of consumpion. Tis exra wedge impacs on e pricing decisions of firms, and ence on inflaion sabilizaion, and grealy improves welfare in e socasic equilibrium. Keywords: Ramsey policy, Opimal public spending, Consumpion ax, Disorionary axaion. JEL: E3,E6, E62. We are paricularly graeful o Sarola Laczò and Emanuele Bracco for eir commens on a early version of e paper. Furermore, we benefi from discussions wi Micela Cella and Andrea Colciago, as well as all e paricipans a e deparmenal seminars a e Universiy of Milano-Bicocca, CEF 23 in Vancouver. Deparmen of Economics, Lancaser Universiy Managemen Scool, LA 4XY, Unied Kingdom. E- mail:g.moa@lancaser.ac.uk Deparmen of Economics, Lancaser Universiy Managemen Scool, LA 4XY, Unied Kingdom. r.rossi@lancaser.ac.uk

3 Inroducion Following e recen financial crisis, governmens around e globe implemened massive fiscal plans and a large number of fiscal reforms. Tis as conribued o increase e ineres of e economic discipline in fiscal policy issues. Along is line, is paper ries o address several policy quesions. Wa are e consequences for e seing of opimal moneary and fiscal policies of differen ax arrangemens? In paricular, wa are e welfare consequences of differen ax insrumens? Wa is opimal size of e public secor and ow is opimal size is affeced by differen fiscal arrangemens and differen levels of public deb? In order o address ese quesions, is paper sudies e opimal mix of moneary and fiscal policy in a New Keynesian model were public spending as inrinsic value and e fiscal auoriy can use consumpion and labour income axaion in order o finance public spending and o finance public deb. Te economic environmen considered in is paper feaures ree ineffi ciencies. Firs, firms ave marke power in e good markes wic allows em o carge a mark-up over marginal coss. Tis causes oupu o be below e effi cien level. Second, sicky prices in e good marke preven firms from fully adjusing eir prices in response o socks. Tird, fiscal policy as o use disorionary consumpion and labor income axes o finance public spending and ineres paymens on ousanding governmen deb. Public spending and governmen deb us ave addiional adverse effecs on economic aciviy. In order o address our policy quesions, we analyze wa is e bes way of joinly seing e differen policy insrumens, i.e. consumpion and labour income axaion, governmen spending, public deb and e sor erm ineres rae, in order o deal wi e disorions described in e previous paragrap. In urn, as we will discuss in deails laer, is model generaes several cannels for wic ere exis non rivial ineracions beween moneary and fiscal policy insrumens. Te policy analysis is divided ino wo main pars. In e firs one, we analyze Ramsey policy a seady sae. We sar by sowing a in order o replicae long run effi ciency moneary policy implemens a zero inflaion policy. Tis means a e Ramsey Planner does no find i opimal o use seady sae inflaion in order o decrease e real values of profis or o erode e real value of any ousanding deb/asse posiion. Furermore, effi ciency requires, independenly from e ax insrumen adoped, e Ramsey Planner o accumulae large asse posiions agains e privae secor. Te income from ese asses are en used o balance e governmen budge consrain and o correc e disorions generaed by monopolisic compeiion in e good marke. Tis resul implies a wen public deb is posiive, or a leas no oo negaive, e Ramsey Planner canno replicae e effi cien allocaion. In is second bes scenario wi posiive public deb, we find a opimal Ramsey policy calls for exremely ig consumpion axaion and labour subsidy, well above %, as e difference beween consumpion and wage income is usually small relaive o governmen spending. 2

4 However, is ax sceme may easily lead o wo ype of problems. Firs, a wage subsidy of e order of 5%, say, seems impracical, as i would lead o remendously ig coss associaed in verifying ours worked. Applying e same logic, a 6% consumpion ax rae would probably lead o a large amoun of unrepored barer. Terefore we leave e Ramsey Planner free o ax joinly labour income and consumpion wen large asse posiions can be accumulaed agains e public, wile in e second bes scenario were public deb is allowed o ake an arbirary and raer large value, we resric e policy maker o ax eier labour or consumpion. Under bo scenarios, we find a opimal policy implies zero seady sae inflaion. Tis resul does no dependen on e long run level of public deb, nor on e degree of monopolisic power in e good markes. Terefore, e Ramsey Planner never finds opimal o use inflaion in order o erode e real value of profis and public deb. Furermore we find a under bo fiscal scenarios, e Ramsey Planner acively uses governmen spending as a policy insrumen. In paricular, we sudy e opimal size of governmen consumpion relaive o oal oupu. By seing a cerain raio of governmen spending o GDP, e Ramsey Planner can influence privae secor beavior and erefore can reduce ineffi ciency and increase welfare. We sow a in e presen seing wic considers omoeic preferences over public and privae consumpion, e incenive for e Ramsey Planner o increase e governmen spending o GDP raio increases wi e ouseolds risk aversion. Te inuiion for is resul is e following. Consider, for insance, e case were only labour income axes are available o e policy-maker and assume ere is an increase in ineffi ciency, due, for example, o more monopolisic marke power or iger ousanding deb. Tis would pus economic aciviy away from e firs bes. Te Ramsey Planner by adjusing governmen spending o GDP raio and erefore e level of axaion, can influence ouseolds labour supply and privae consumpion and poenially reduce ineffi ciency. For example, by imposing a governmen o GDP raio lower an in e firs bes allocaion, e policy maker can susain a given level of ousanding deb wi lower axaion. Lower axaion increases labour supply. We sow a wen consumers risk aversion is low e increase in e labour supply a is policy brings abou booss privae consumpion and ence reduces ineffi ciency. On e oer and, wen e degree of risk aversion is ig, an increase in labour supply decreases privae consumpion. In is case, e Ramsey Planner as e incenive o se e governmen spending o GDP raio above is firs bes counerpar. Tis policy decreases labour supply and puses privae consumpion owards is effi cien level. A similar logic can be applied o e case wen only consumpion axaion is available. In is case, governmen size, and erefore axaion, direcly impacs on privae consumpion so a lower ax raes are associaed wi iger consumpion levels. Ceeris paribus, we sow a wen e degree of risk aversion is low, iger consumpion implies a iger labour supply. In is case, we find a e Ramsey Planner ses e governmen spending o GDP raio below is firs bes counerpar. Tis policy booss bo consumpion and labour supply and in urn 3

5 puses e economy closer o effi ciency. On e oer and, wen consumers risk aversion is ig, iger consumpion is associaed wi lower labour supply. In is case opimal policy ses e governmen spending o GDP raio above is firs bes counerpar. Wen we compare e seady sae opimal allocaion under e wo ax scemes, i.e. labour income and consumpion axaion, we find a under consumpion axaion e opimal sare of governmen spending is always iger en under labour income axaion. Furermore, we find a, for a given level of ousanding deb-o-gdp raio, ere is an enormous welfare gain in axing consumpion raer an labour income. In urn, we sow a ese gains are increasing in e level of seady sae public deb and in e risk aversion parameer. In e second par of e paper we analyze e dynamic beavior of e economy under ecnology socks. Firs we sudy e Ramsey dynamics under e assumpion a e governmen is allowed o accumulae large asses posiions agains e public so a e seady sae allocaion is effi cien. Wen e policy maker as access o bo labour income and consumpion axaion, e Ramsey soluion perfecly coincides wi e effi cien equilibrium along e ransiion pa. Hence, e Ramsey Planner can commi o a sae coningen policy a perfecly replicaes e firs bes allocaion. In doing so, e wo ax insrumens need o move in opposie direcions, i.e. if consumpion ax increases, e labour income ax rae mus decrease of e same amoun. Tis policy does no disor e marginal rae of subsiuion beween consumpion and leisure and e real wage is free o adjus according o e ecnology process. Furermore, e Ramsey Planner mus use consumpion axaion and e nominal ineres rae in order o replicae e sae coningen reurn in e bond marke via e real socasic discoun facor in e Euler equaion, and ereby eliminae any incenive for firms o cange prices. A e same ime governmen spending as o move so a e marginal uiliy of public and privae consumpion are e same. Ten we sudy a scenario were e Ramsey Planner is consrained o using one ax insrumen wile e oer ax rae is fixed a zero bu accumulaes asses suc a e seady sae is effi cien. By losing consumpion or labour income axaion, e Ramsey dynamics are no fully effi cien any longer, and, as a consequence, ouseolds suffer welfare losses in e socasic equilibrium. For example, e policy-maker canno offse anymore e disorive consequences of a ax cange on e marginal rae of subsiuion beween consumpion and leisure. As a resul, opimal policy allows for deviaions from full price sabiliy a in urn modify e reurn in e bond marke. Tis policy is mean o pus e dynamic equilibrium owards e effi cien allocaion. Wile is canno be compleely obained, under consumpion axaion is policy is muc more effecive. Te reason is a e dynamic naure of consumpion axaion enables e policy-maker o affec e real socasic discoun facor via modificaions of e marginal uiliy of consumpion. Tis exra wedge elps e Ramsey Planner o pus e dynamic allo- As i is well known, e.g. Benigno and Woodford 23 and Sci-Groé and Uribe 24, wen prices are sicky, opimal policy abandons e radiional ax smooing beaviour. 4

6 caion closer o e effi cien one. In urn, e welfare loss of being consrained of keeping labour income ax fixed a zero are rougly.6% as big as wen e policy-maker is consrained in keeping fixed a zero consumpion axaion. Furermore, i is ineresing o noe a ere is very lile use of governmen spending as a sabilizaion ool in e socasic equilibrium. Similarly, wen e policy maker is consrained o use only one ax insrumen and public deb is posiive, e welfare losses under consumpion axaion are muc smaller an under labour income ax. As in e previous case, e superioriy of consumpion axaion is due o e dynamic naure of is policy insrumen. Tis paper links o e exising lieraure in several ways. Firs of all, i is closely relaed o Benigno and Woodford 23, Scmi-Groé and Uribe 24, 27 and Adam 2. Tese works also analyze opimal Ramsey policy in a New Keynesian seing and sow, as ere, a nominal rigidiies preven e governmen from using price level canges as an imporan source of sae-coningen axaion in e presence of nominal governmen deb. As a resul, governmen deb opimally follows a near random walk, as in Barro 979 and Aiyagari e al. 22. However, none of ese works consider e possibiliy of consumpion axaion. An unforunae aspec of is resricion, oug, is a i rules ou a ax rae a is clearly used by many counries. As repored by Gordon and Li 29, e average consumpion ax rae in developed counries is around 6% and i reaces 25% in Hungary and Denmark. Secondly, is paper is relaed o e opimal policy lieraure on consumpion axaion. Coleman 2 sudies Ramsey policy under consumpion axaion in a neoclassical grow model. He finds a replacing income axes wi a consan consumpion ax leads o a welfare gain a is only sligly lower an a aained by a dynamic policy a axes consumpion and income. Correia 2 exends is resul in a seing wi eerogenous agens and also finds large welfare gains in adoping consumpion axaion. Correia, Nicolini and Teles 28 sow a in a New Keynesian model wi consumpion and labour axaion, full price sabiliy can be acieved via an appropriae mix of labour and consumpion axaion. Furermore, Correia, Fari, Nicolini and Teles 23 sow a wen e economy is consrained a e zero lower bound on e nominal ineres rae, consumpion axaion may replace e moneary policy insrumen as demand managemen ool. However, unlike ere, ey sudy a framework in wic governmen spending is exogenous and ence ey do no analyze ow differen axaions and public deb levels affec opimal governmen spending. We believe a analyzing opimal fiscal policy wi endogenous governmen spending may be imporan for wo reasons. Firs, governmen spending represens an imporan sare of GDP of all indusrialized counries and i is one of e main fiscal insrumens used by policymakers. I is erefore ard o undersand ow i can be reaed as exogenous from a normaive perspecive. Second, as Teles 23 poins ou, differen assumpions abou e endogeneiy of governmen spending may lead o sarp differences in opimal policy prescripions and welfare calculaions. 5

7 Te paper is organized as follows. Secion 2 ses up e model. Secion 3 presens e opimal policy exercises. Secion 4 concludes. 2 Te model We add a fiscal policy block o e sandard sicky-price casless DSGE framework, similar o e workorse model in e.g. Clarida e al. 999 or Woodford 23. Public consumpion as inrinsic value for e agens as in Galí and Monacelli 28, Adam and Billi 28 and Adam 2. Governmen spending mus be financed wi linear labour income and/or consumpion axes. Lump-sum axes or ransfer are ruled ou. We resric public nominal deb o be of one-period mauriy and o be sae-nonconingen as in Scmi-Groé and Uribe 24 and Correia, Nicolini and Teles 28. Besides presening e model ingrediens, is secion derives e implemenabiliy consrains caracerizing opimal privae secor beavior, i.e., i derives e opimaliy condiions deermining ouseolds consumpion and labor supply decisions, firms price seing decisions, as well as e governmen s budge consrain. 2. Privae Secor 2.. Houseolds Tere is a coninuum of infiniely-lived ouseolds wi preferences E β,, g, = were β is e discoun facor, c represens individual consumpion, denoes ours worked and g is governmen spending. E idenifies e raional expecaions operaor. We impose a uiliy is separable in is ree argumens and >, c <, u g >, u gg <, u < and u. Were u x defines e derivaive of e uiliy funcion wi respec o e generic variable x. Furermore, we assume omoeic preferences over public and privae consumpion, so a gugg u g = cucc. Tere is a coninuum of goods, indexed by i. Eac i good eners wi e same weig in e Dixi-Sigliz aggregaor. Tis can be wrien as c = c i, wile e aggregae consumpion price index is P = ] di ; i, ], >, 2 ] P i, di. 3 6

8 Hence, e demand for good i follows were, is e price elasiciy for differeniaed goods. Pi, c i, = c, 4 P Houseolds maximize subjec o e following period budge consrain P c + τ c + R B + + E Λ,+ Q + = B + Q + P d + P w τ. 5 In eac ime period, ouseolds can purcase any desired sae-coningen nominal paymen Q + in period + a e dollar cos E Λ,+ Q +. Te variable Λ,+ denoes e socasic discoun facor beween period and +. Here e only role of sae-coningen securiies is o define sae-coningen prices. We assume a sae-coningen claims are in zero ne supply. Real dividends are denoed by d, wile B is e quaniy of risk-less nominal bonds purcased in period a price R and paying one uni of e consumpion numeraire a period +. Taxes on consumpion and labour income are, respecively, τ c and τ, and w is e real wage. Te soluion for e opimizing ouseold problem is sandard and i can be wrien as:, = λ + τ c, 6 were λ sands for e Lagrangian muliplier associaed wi is programme. Labor supply is deermined by wile e Euler equaion is, + τ c u, τ = w, + τ c, 7 = βe,+ + τ c + R π + ], 8 were π = P P is e gross inflaion rae. Te socasic discoun facor is defined as u E Λ,+ = βe c,+ P +, and absence of arbirage profis in e asse markes implies a E Λ,+ = R Firms P +τ c, +τ c + A generic good i is produced in a monopolisically compeiive marke wi ecnology y i, = a i,, 9 7

9 were a is a common exogenous ecnology process. Firm i s real marginal coss are: mc = w a. We assume a firms, wen reseing eir prices, incur in quadraic adjusmen coss a lá Roemberg 983, i.e., ϕ 2 2 P Pi,+s, P i,+s were ϕ represens e degree of price sickiness. Te profi maximizing generic firm i s problem can be expressed as + max E β s,+s + τ c {P i, } u s= c, + τ c +s Pi,+s s.. y i,+s = y +s. P +s Pi+s P +s y i+s w +s y i+s a +s ϕ ] 2 Pi,+s 2 P i,+s We focus on e symmeric equilibrium in wic P i = P olds. Terefore, subsiuing for mc, e condiion for opimal pricing decision is { ] },+ + τ c a + w ] ϕ π π + ϕβe π, + τ c + π + = Aggregaion In e symmeric equilibrium i =. Hence e economy-wide producion funcion is 2 y = a. 3 Furermore, using e ouseold s budge consrain, we can obain e expression for aggregae profis as d = a w ϕ 2 π 2. 4 Combining e governmen budge consrain, e definiion of profis and e ouseolds budge consrain, one can obain e aggregae resource consrain as y = a = c + g + ϕ 2 π 2 5 8

10 2.3 Governmen secor Macroeconomic policies are implemened by wo auoriies. Tere is a Cenral Bank wic ses e nominal ineres raes on sor-erm nominal bonds. Furermore, ere is a governmen coosing e level of public expendiures, labor income ax, consumpion ax and public deb. Te governmen finances curren expendiure by raising linear labor income and consumpion axes and by issuing new one-period sae-nonconingen deb, i.e., Te fiscal auoriy credibly commis o repaying is deb. b π + g = b + R + w τ + c τ c. 6 governmen deb and ax policies are suc a e no-ponzi consrain and e ransversaliy consrain are bo saisfied. lim s + E +s i= In wa follows we assume a ] B +s =, 7 R i lim β +s uc,+s s + + τ c +s B+s P +s ] =, Raional Expecaion Equilibrium Definiion Raional Expecaions Equilibrium A Raional Expecaions Equilibrium REE is defined by a sequence of privae secor decisions {c, y,, w, π } = and economic policies { τ, τ c }, R, g, b + = a, given e iniial level of public deb b and e evoluion of ecnology, solves equaions 7, 8, 2, 3, 5, 6, 7 and 8. 3 Ramsey Policy 3. Firs Bes Allocaion Definiion 2 Social Planner s Program Te Social Planner s Program defines e firs bes allocaion and consiss in coosing {c,, g } = aking as given e ecnology process {a } =, in order o maximize e uiliy funcion of ouseolds as in subjec o e consrains imposed by e producion ecnology. Proposiion 3 Te Social Planner s allocaion is, = u, a = u g, 9 9

11 Proof. Please refer o e Appendix. In e firs bes equilibrium e marginal uiliies of privae and public consumpion mus equae o e marginal disuiliy of labour, were e laer is scaled by oal produciviy. Tis simple allocaion rule is opimal because i is equally cosly o produce public and privae consumpion goods. Definiion 4 Ramsey Problem: Te Ramsey Problem is o maximize over Raional Expecaion Equilibria. A Ramsey oucome is a Raional Expecaion Equilibrium a aains e maximum of. In is policy problem, e Lagrangian mulipliers associaed wi forward looking variables, i.e. E π +, E,+, E τ c + are addiional sae variables. Assuming ese saes iniial values equal o zero implies ransiory non-saionary componens in e soluion o e Ramsey problem, even in a non-socasic environmen. Tis is because in e iniial period e policy-maker may ave e empaion o emporary increase axes or generae inflaion so as o erode e real value of any ousanding governmen deb. 2 We do no analyze ese non-saionary deerminisic componens and concenrae insead on e ime-invarian deerminisic long-run oucome. Tis is wa we define as e Ramsey Seady Sae RSS encefor. Tis means a we are imposing an iniial commimen on e policy-maker no o generae surprise movemens in axes, governmen spending, or nominal ineres raes in period zero. Tis is sandard pracice in e opimal axaion lieraure, see e.g. Scmi-Groè and Uribe Analyical Resuls As i is well known, ere exiss a coninuum of RSS, eac of wic associaed wi an ousanding level of public deb. In oer words, wiou specifying e seady sae level of public deb, e model displays an indeerminacy of degree one. Proposiion 5 Opimal Inflaion: Despie is indeerminacy, all e RSS are caracerized by π = 2 and R = β 2 Proof. See Appendix. 2 However conrary o a flexible price economy, e Ramsey Planner does no find opimal o se P =. 3 Furermore, given e presence of sicky prices and sae-nonconingen govenrmen bond, our primal form of e Ramsey problem can no longer reduced o a unique ineremporal budge consrain in period and a feasibiliy consrain olding in every period. However we could sill wrie our policy problem in erms of a sequence of ineremporal implemenabiliy consrains. For a deailed discussion, see Aiyagari e. al. 22 and Scmi-Groé and Uribe 24.

12 I is erefore subopimal o use seady-sae inflaion o reduce e real value of any ousanding level of public deb or o erode e real value of profis. Tis is a common resul in e lieraure a considers is class of models, e.g. Adam and Billi 28, Adam 2. In order o idenify e indeerminacy problem, i suffi cies o recall e firs order condiion of e Ramsey problem wi respec o b +, i.e. βe γ 4,+ π + + γ 4, R =. 22 A seady-sae, e Euler equaion implies R = β. Terefore, a seady-sae, 22 is saisfied for any values of γ 4. Terefore, in order o pin down e RSS one as o fix e ousanding level of public deb suc a 22 becomes redundan, i.e. public deb becomes exogeneous a seady-sae. Tis means a in order o find e RSS, one can rewrie e 6 as x + g = w τ + c τ c 23 were x = β b represens e seady-sae level of real governmen liabiliies. As in Adam 2, is modified version of e Ramsey problem in leads o e same RSS as e general problem in Definiion 4 and is used o obain a number of analyical resuls. 4 Proposiion 6 Firs Bes Decenralized Equilibrium: e Ramsey Seady Sae and e Social Planner s allocaion a seady sae coincide under e following necessary condiions were x op = b op β Proof. See Appendix. π = 24 τ + τ c = wi τ, and τ c, 25 x op = c + τ c + τ wi x op < 26 Corollary 7 Te Ramsey Planner canno replicae e Social Planner Allocaion for a generic level of ousanding public deb wi τ, and τ c,. A few ings are wor sressing. Firs, from 25, e firs bes allocaion requires a leas one fiscal insrumen beween τ and τ c o be a subsidy, i.e. o be negaive. 5 Second, e firs bes allocaion requires negaive public deb. Pu i differenly, e Ramsey Planner canno 4 Formal proof of is can be found in e Appendix B, Subsecion In paricular, if τ >, en firs bes allocaion requires τ c <. However i is possible o find negaive values of τ for wic e firs bes requires τ c <. A e same ime, if τ c >, e effi cien allocasion requires τ <.

13 obain e firs bes allocaion wi a generic level of deb, as is would imply eier τ c < or τ >. 6 Moreover, using numerical soluions wic we describe in deails below, we find a e policy maker would always find a marginal welfare advanage in increasing consumpion axaion and subsidy labour wen e level of ousanding public deb is greaer an x op. However, wile is policy will never replicae e firs bes allocaion, i would creae exreme ax and subsidy posiions. 7 Hence, is policy would be exremely diffi cul o implemen due, for example, o e ig coss associaed in verifying ours worked. Similarly, a very ig consumpion ax rae would peraps lead o a significan amoun of unrepored barer. Terefore, we allow e policymaker o use bo ax insrumens wen x = x op, wile under a generic level of seadysae debasse-o-gdp, we sudy e cases were e Ramsey Planner is consrained in using only one ax insrumen, i.e. eier τ = or τ c =. 8 Proposiion 8 Under labour income axaion, e Ramsey Planner ses u g u 27 and u g g + x }{{}, 28 Proof. See Appendix. Tis proposiion sows a i is opimal o se public consumpion below e level suggesed by e Social Planner. Tis is because ere exiss a wedge beween e marginal uiliy of leisure and e marginal uiliy of consumpion. Tis wedge is composed of wo componens. Firs, e monopolisic power a firms old. o finance public spending and public deb. Second, e disorive naure of fiscal policy used By reducing governmen spending, e Ramsey Planner can lower e ax rae and ence srinks e wedge beween consumpion and leisure. Moreover, wile 28 does no give a precise analyical mapping beween u g and, we can nevereless provide some inuiions abou is relaionsip. Consider, for insance, any increase in ineffi ciency, due, for example, o more marke power or iger ousanding deb. Tis would pus down economic aciviy. Te Ramsey Planner by adjusing governmen size relaive o 6 Some readers may correcly wonder if is resul relies on e absence of profi axaion. As i is well known, e opimal profi axaion in is class of models is generally equal o %, i.e. e Ramsey planner finds i opimal o confiscae all e profis in e economy, see for insance, Correia, Nicolini and Teles 28 and Correia, Fari, Nicolini and Teles 23. We proof in e Appendix a even wi e inroducion of full profi axaion, i.e. τ d =, e effi cien level of public deb is negaive. 7 For example Coleman 2 finds a in a perfecly compeiive economy wi capial accumulaion, e Ramsey planner would se τ c = 692% and τ = 692%. Monopolisic disorion amplifies even furer ese fiscal posiions, i.e. wiin sandard paramerisaions consumpion axaion is iger an %. 8 Tis is common pracice in e lieraure wen opimal policy implies exreme ax posiions, e.g. Coleman 2, Correia 2 and Marin 2. 2

14 GDP and erefore e level of axaion, can influence ouseolds labour supply and privae consumpion and poenially reduce ineffi ciency. For example, by imposing u g >, i.e. a governmen o consumpion raio lower an e firs bes, e policy maker can susain a given level of ousanding deb wi lower axaion. Lower axaion would increase labour supply. 9 Te sign of c deermines weer a iger labour supply implies a iger or lower consumpion level. In e Appendix we sow a c c cc. 29 c x Te sign of 29 depends, iner alia, on e Arrow-Pra measure of risk aversion cucc, e degree of monopolisic compeiion and e level of ousanding deb x. Wen 29 is posiive, i.e. wen risk aversion is low, srinking e governmen size relaive o GDP below e firs bes, allows e Ramsey Planner o increases labour supply and privae consumpion, us reducing ineffi ciency. Te opposie is rue as 29 urns negaive, i.e. wen risk aversion is ig. In is case e policy-maker as a srong incenive o se e governmen spending-o- GDP raio greaer an e firs bes allocaion, i.e. u g <. Wi suc a policy e Ramsey Planner can induce an increase in consumpion and erefore reduce e wedge beween e marginal uiliy of privae consumpion and e marginal uiliy of leisure. Te desire for suc a policy, i.e. u g <, is decreasing bo in e degree of monopolisic compeiion and in e size of public deb. In e paricular case of perfec compeiion and no public deb, i.e. and x =, e RHS of 29 collapses o. In is case, if e uiliy is logarimic in privae and public consumpion, e Ramsey Planner finds i opimal o se e sare of governmen spending over oal oupu as in e firs bes. Nex we analyze e scenario were e Ramsey Planner as access only o consumpion axaion, i.e. τ =. Proposiion 9 Under consumpion axaion e Ramsey Planner ses u g > u. 3 Furermore, if cc > = > u g = = = u g 3 < = < u g and uc u g is decreasing increasing in x if cucc = for any ousanding level of public deb. u g > <. In e special case were cucc =, 9 We are implicily assuming a e subsiuion effec on labour supply always prevails, wic is consisen wi a Laffer curve in governmen revenues. In e case of log uiliy and posiive public deb and/or monopolisic compeiion, e Ramsey planner finds c i opimal o se e sare of public spending-o-gdp lower an e firs bes, i.e. >. 3

15 Proof. See Appendix. As under labour income axaion, e Ramsey Planner finds i opimal o se e level of governmen spending below e Social Planner level. Tis is due o e monopolisic feaures of e good markes and e disorive naure of fiscal policy. Furermore 3 clarifies weer e opimal allocaion requires e sare of governmen over oal oupu o be above or below e firs bes level. Ineresingly, is is now only funcion of e consumers risk aversion. If is is iger lower an, e opimal governmen spending-o-gdp raio is se above below e Social Planner level. By allocaing a ig sare of governmen spending, e Ramsey Planner is imposing, for a given level of ousanding deb, a iger ax rae. Assuming a consumpion is a normal good, iger axaion implies lower consumpion. In order o undersand e implicaions of lower consumpion on e labour supply scedule we need o sudy e sign of c = c u u + x ] c c ] 32 u u + u u x cucc In e Appendix we sow a e above expression is negaive wen cucc > and posiive oerwise. Tis means a wen cucc >, e lower consumpion generaed by iger axaion implies an increase in labour supply, wic in urn puses e economy closer o e firs bes. On e conrary, wen cucc <, e Ramsey Planner ses < u g. Te resuling lower axaion puses consumpion and erefore labour supply upward, i.e. c >. For e same reason, e opimal policy calls for an increase in e governmen spending-o- GDP raio wen public deb increases. Moreover, for e paricular case in wic risk aversion is, e Ramsey Planner, independenly of e ousanding level of public deb, ses e marginal uiliy of privae consumpion equal o e marginal uiliy of public spending, as prescribed by e Social Planner. A simple comparaive saic exercise can sow a for a given degree of risk aversion and a posiive or a leas no oo negaive ousanding deb, e opimal sare of governmen spending-o-gdp is greaer under consumpion axaion an under labour income ax. 3.3 Compuaional issues As discussed above, we assume a in period e economy is in e RSS. Wen an analyical expression is missing, we rely on non-linear numerical soluion algorims. 2 Firsly, we compue e exac non-linear RSS by using e OLS projecion approac proposed by Scmi-Groè and Uribe 24, 27, 22. Tis consiss in exploiing e insig a e Ramsey equilibrium condiions are linear in e vecor of Lagrange mulipliers, γ i. Ten, using e perurbaion meod proposed by Scmi-Groé and Uribe 24, we compue e accurae second-order Take for insance e case of logarimic preferences, i.e. cucc =. Under labour income axaion u g >, wile under consumpion axaion u g =. 2 Te differen Ramsey problems are described in deail in e Appendix. 4

16 approximaion of e Ramsey s FOCs around e non-socasic seady sae of ese condiions. We en use is soluion o simulae e Ramsey equilibrium in e face of a ecnology sock. 3 Te sock realizaions and all e oer srucural parameers used for e simulaion are kep consan roug e differen fiscal scenarios. Tis means a any differences beween fiscal arrangemens are aribuable enirely o e properies of e economic policies. Moreover, we measure welfare in erms of percenage of consumpion unis, i.e. ϖ required by a generic policy a o reac e same level of uiliy as under policy b, i.e. ] E β a + ϖ, a, g a ] = E β b, b, g b. 33 = 3.4 Paramerizaion We specify preferences o saisfy e condiions saed in Secion 2, i.e. =,, g = c σ σ ω +φ g σ + φ + ω g σ wi σ, φ >. Eac period represens a quarer wi e discoun facor, β, se o.993. Te elasiciy of demand is cosen in order ave a seady sae gross markup of.2 = 6, wic is in line wi e macro lieraure. Given e imporance of e CRRA parameer for our resuls, i.e. σ = cucc, we solve e model wi a se of values of risk aversion a are generally found in e lieraure, i.e. σ.8, 2. However, we se σ =, i.e. log uiliy, as a bencmark value. Te uiliy parameer ω is cosen so a e ouseolds supply beween one fif and one ird of eir ime o work in e decenralized equilibrium wen e seady-sae level of public deb is zero, i.e. ω = We furer fix ω g in order o ave in e firs bes allocaion, i.e. e Social Planner Equilibrium, a raio of governmen spending over oal oupu of 3%, i.e. ω g =.264. We se e inverse Frisc elasiciy of labour supply φ o, a value generally used as a bencmark in e macroeconomic lieraure, e.g. Adam and Billi 28. Te price sickiness parameer is seleced suc a e log-linearized version of e Pillips curve 2 is consisen wi e esimaes of Sbordone 22, ϕ = 7.4. Te quarerly sandard deviaion of e ecnology socks is.6% and i as a quarerly persisence equal o.7. Furermore, we sow e implicaions of varying e long run level of public deb. paramerizaion adoped. Table collecs e 3 In e case of effi cien seady-sae, i.e. wen e governmen is allowed o accumulae a large asse posiion, e second order approximaion approac gives e same welfare ranking of e corresponden linear-quadraic LQ represenaion of e problem, see Woodford 23. For is reason, par of our resuls are readily comparable wi e lieraure a adops e LQ framework. 5

17 3.5 Seady Sae Resuls Tis secion explores e quaniaive implicaions of differen fiscal scenarios for e RSS allocaions wi paricular focus on e Ramsey public spending and welfare. In paricular, ere we sow ow e various fiscal arrangemens inerac wi governmen spending and long-run level public deb in deermining e Ramsey allocaions. Figure presens e RSS allocaions wen public deb is allowed o vary from -% o 2% of GDP. Increasing e level of ousanding public deb implies an increase in e disorive ax rae and ence a loss of effi ciency. Higer axes imply a lower afer ax real wage. However, despie e consumpion ax being iger an e labour income ax, i as a lower impac on e real wage. Tis is because consumpion is generally more inelasic an leisure. Hence ouseolds respond less, ceeris paribus, o a variaion in consumpion ax an o a variaion in labour income ax. As a consequence, under consumpion axaion e increase in public deb implies a lower response of e gap variables. Figure 2 quanifies, in percenage of permanen seady sae consumpion unis, e loss suffered by ouseolds as seady sae public deb increases. Under bo fiscal scenarios, iger values of seady sae public deb imply an increase in ineffi ciency and ence a deerioraion of welfare. However ere is a surprisingly ig gain from using consumpion axaion over labour income axaion as a fiscal insrumen. Tis gain is increasing in e deb-o-gdp raio, passing from 24% wen governmen deb o GDP is % o 42% wen public deb is 2% of oal income. As public deb increases, e fiscal auoriy as o devoe more and more resources o pay is burden. Terefore, e lower disorive effecs of consumpion axaion generae a relaive gain as e ineffi ciency generaed by e burden of public deb increases. Figure 3 sows e difference beween e Ramsey governmen spending o GDP raio and e Social Planner allocaion wen public deb is a is bencmark value, i.e. 8% of GDP and we allow σ o vary in e inerval.8, 2] wile all e oer parameers are kep a eir bencmark values. Tis figure may be seen as a grapical presenaion of Proposiion 7 and 8. A negaive posiive number idenifies a scenario were e Ramsey Planner ses e raio beween public spending and oal oupu below above e firs bes, i.e. g g y < > y. Under bo fiscal scenarios, is difference depends criically on e parameer conrolling e risk aversion in e CRRA uiliy funcion. In paricular, as consumers become more risk adverse, opimal policy calls for reducing privae consumpion in favor of public spending. By direcly affecing e size of governmen spending and erefore e level of disorive axaion, e policy-maker can influence labour supply and privae consumpion and poenially reduce ineffi ciency. As discussed above, for a give level of risk aversion, opimal policy under consumpion axaion implies a iger sare of governmen spending over GDP an under labour income axaion. 4 Figure 4 presens e opimal public spending rule as public deb increases for differen 4 In paricular, under labour income ax g = g c for σ =.3868 = y y c x, i.e. c =. ε 6

18 degrees of risk aversion. Under labour income ax, e opimal sare of public spending is decreasing in public deb. Tis is because e srong disorionary effecs of iger ax raes prevail over e incenive of e Ramsey Planner o increase g/y as e degree of risk aversion increases. Tis is consisen wi 29, i.e. for a given level of risk aversion a iger level of seady sae public deb generaes an incenive o reduce e sare of public consumpion. Differenly, under consumpion axaion, e opimal spending rule depends crucially on weer risk aversion is below or above uniy, as presened in Proposiion 7. Terefore wen σ =.8, i.e. cucc <, e Ramsey Planner, by cuing e sare of governmen as public deb increases, can boos bo consumpion and labour supply. On e conrary, wen σ = 2, i.e. cucc >, e opimal policy increases e governmen spending-o-gdp raio in order o increase ours worked and erefore oal oupu. Finally, as presened in Proposiion 7, in e bencmark case of log uiliy in consumpion, i.e. cucc =, e opimal sare of governmen spending is always a e firs bes, independenly of e level of seady sae public deb. 3.6 Ramsey Dynamics 3.6. Firs Bes Seady Sae We now urn our aenion o e opimal policy in a socasic seing. To is end we perurb e model wi a ecnology sock as described in 3.4. Sudying Ramsey policy in e face of is ype of sock is sandard pracice in e lieraure e.g. Scmi-Groé and Uribe 24, Correia e al. 28, Adam 2, Lei and Wren Lewis 23 and allows us o beer disenangle our conribuion. We sar our analysis by considering a siuaion were e policy-maker implemens firs bes a seady-sae. Tis implies a e governmen is allowed o accumulae large asse posiions i.e. negaive deb and a one or bo axes can be negaive. Bo ese assumpions will be dropped laer. We refer e reader o Proposiion 5 in Secion 3.3 on ow a Ramsey Planner can acieve e firs bes in is environmen. In pracise, we fix e seady sae value of consumpion axaion o 6%. Tis value corresponds o e average level of consumpion axaion in e indusrialized counries found by Gordon and Li 29. Ten, e labour income ax rae and e level of e governmen asses are pinned down by 25 and 26 respecevely. Here we run ree policy exercises. In e firs one e policy maker can respond o socks by using bo axes. In e oer wo, we resric e fiscal auoriy o use only one ax insrumen in e face of socks. Te ree scenarios sare e same seady sae allocaion so a any difference can be aribued enirely o e dynamic properies of e ax insrumens adoped. Resuls are repored in Figure 5. Wen e policy maker as access o bo axes, firs bes is aainable in e socasic economy under consideraion. In oer words, e Ramsey Planner can commi o a saeconingen policy suc a in face of a ecnology sock e response in e decenralized economy coincides wi e Social Planner s allocaion. Absence economic policy, in a sickyprice environmen a negaive ecnology sock would imply an increase of inflaion and a posiive 7

19 oupu gap, i.e. price would increase bu no as muc as required in a flexible price world, us imply an ineffi cienly ig level of oupu. As i is well know from Galí 2, a welfare maximizing policy-maker would erefore igen aggregae demand as o pus oupu owards e effi cien level. Tis policy would also sabilize inflaion. Here e Ramsey Planner looses moneary policy on impac and promises o keep i above e seady sae level for some periods afer e sock. A e same ime, se increases e consumpion ax and promises o cu i omorrow. As we discuss in more deails laer, is coningen ax policy represens one of e main advanages of using consumpion axaion as a demand managemen ool. Indeed, e combined opimal use of moneary and fiscal insrumens allows e policy-maker o fully conrol e socasic discoun facor and ence offse firms desire o canging prices, us replicaing e flexible price equilibrium. Furermore, in order o mimic e Social Planner soluion, e Ramsey Planner as o avoid disoring ouseolds consumpion-leisure coice. In oer words, e policy-maker moves e consumpion and e labour income ax rae in e same measure and in opposie direcion so a ey do no creae a wedges beween e marginal uiliy of leisure and a of consumpion. Finally e Ramsey equaes e marginal uiliy of privae and public consumpion as required by e firs bes equilibrium by varying er asses posiions. In oer words, if e governmen is allowed o ake large asses posiion and can use bo consumpion and labour income axaion, se can offse bo e saic and e dynamic ineffi ciencies of e model. Table repors e model s campionary momens of is exercise. 5 Given e sape of e uiliy funcion σ =, dynamic effi ciency requires oupu, consumpion, governmen spending and real wages o be perfecly correlaed wi eac oer and wi e ecnology process, wile ours worked and inflaion sould remain a eir seady sae values. Tese resuls are closely relaed o Correia e al. 28. Tey sow a wi consumpion and labour income axaion, full profi axaion and exogenous governmen spending, e Ramsey Planner can mimic e flexible price equilibrium, so a any real allocaion is independen from e degree of nominal rigidiies. Here we sow a e firs bes allocaion can be implemened wiou profi axaion and wi endogenous governmen spending as long as e governmen is allowed o ake large asse posiions. Nex, we analyze fiscal scenarios in wic e governmen is consrained in using only one ax insrumen. We sar wi e case were only consumpion axaion is available. IRF s are presened in Figure 4. In is case, wile e policy-maker can obain long run effi ciency via asse accumulaion, e absence of labour income axaion makes i impossible o reac dynamic effi ciency. In paricular, e opimal ax policy required o balanced e governmen budge consrain disors e leisure-consumpion decision, us creaing an ineffi ciency wedge in e 5 Given e quasi-random walk properies of public deb, uncondiional momens are no available. Hence, like Scmi-Groé and Uribe 24, Arseneau and Cug 28 and Nieman and Picler 2, we calculae e campionary momens of e model. In paricular we simulae e model 5 imes for periods. For eac simulaion we calculae e saisics of ineres. Ten we repor e median value. 8

20 labour supply and erefore in e real wages. A e ime of e sock, oupu, consumpion and public spending all drop by rougly e same amoun as e case were e Ramsey Planner as access o bo axaion us resuling in very small movemens of gap variables. 6 policy, coupled wi an iniial cu in e nominal rae and an increase in consumpion axaion, is aimed o generae an increase on impac of inflaion. Tis elps o reduce e cu in e governmen asses necessary o balanced e governmen budge consrain. Tis As for e case wi wo ax insrumens, is policy is revered in e period afer e sock. Fiscal policy commis in reducing consumpion axaion wile moneary policy increases e nominal rae above is seady sae value. Tese combined policies affec e real socasic discoun facor and imply a one-period deflaion episode, wic in urn puses e price level near is seady sae value. Table repors e implied simulaed momens of is policy experimen. Compared o e previous scenario, e ineffi ciency generaed by e absence of labour income axaion generaes an incenive for e policy maker o reduce e volailiy of oupu, consumpion, governmen spending and real wages. However, wi e excepion of real wages, consumpion and governmen spending are perfecly correlaed wi oupu and almos perfecly correlaed wi ecnology. Te desire o inflae e sysem a e ime of e sock and deflae i in e period afer e sock implies a necessary increase in e volailiy of inflaion. Tis policy generaes also an increase in e volailiy of e nominal rae and an increase of is correlaion wi oupu and ecnology. Ineresingly, wiin is policy experimen, consumpion axaion displays a beavior a is very similar o e case were bo axes are available. Overall, e welfare cos of loosing labour income axaion is very small, around.5% of e effi cien seady sae consumpion unis. Finally we analyze e scenario were e Ramsey Planner is allowed o use only labour income axaion. A novely of is policy experimen wi respec o similar conribuions, e.g. Scmi-Groé and Uribe 24, Adam 2 and Lei and Wren-Lewis 23, is a we consider opimal labour income ax policy wi a firs bes seady sae obained via governmen asses accumulaion. As before, e IRF s analysis is displayed in Figure 6. As under consumpion axaion, under is scenario, e fiscal insrumen generaes an ineffi cien disorion in e marginal rae of subsiuion beween leisure and consumpion. However, now e policy-maker as less power o influence e ineremporal allocaion via e real socasic discoun facor. Tis as dramaic consequences on e opimal policy funcions as well as on e overall welfare. A e ime of e sock, consumpion, oupu and governmen spending all decrease. However, wile consumpion and oupu decrease less an in e effi cien allocaion, us resuling in posiive oupu and consumpion gaps, e governmen finds i opimal o cu governmen spending more an in e previous scenarios. Tis is refleced in lower sandard 6 As in Benigno and Woodford 23 and Adam 2, consumpion, oupu, governmen spending, axes and deb all follow a near ramdom walk paern. However, given e opimal consumpion ax policy, e long run values of ese variables are very close o eir seady sae counerpars. 9

21 deviaions of oupu and consumpion and a iger volailiy of governmen spending Table 2. Tis, coupled wi e negaive weal effec on labour supply, allows e policy-maker o cu axes and e nominal rae on impac us inflaing e sysem in e period of e sock. Tis is done in order o decrease e loss generaed by lower public asses. However, e lack of conrol on e ineremporal ouseolds decisions a consumpion ax brings abou, forces e Ramsey Planner o move inflaion away from seady sae only in e firs period and by a smaller amoun, ence generaing a lower volailiy of inflaion in equilibrium, and i excludes deflaionary episodes. Te welfare loss of using only labour income axaion amouns o.94% of seady sae effi cien consumpion, wic is rougly 6 imes larger an e scenario were e Ramsey Planner could use only consumpion axaion Generic Seady Sae In is secion we sudy e opimal policy wen e policy-maker faces a negaive ecnology sock and e economy is affeced by a posiive level of ousanding public deb. All e parameers are se a eir bencmark values and e public-deb o GDP raio is se o 8%. As discussed in deails above, wen public deb is posiive, even wi wo ax insrumens, e Ramsey Planner canno replicae e firs bes allocaion. Furermore, e second bes policy would imply exreme and unrealisic ax posiions. Terefore in is secion we analyze e scenarios were e policy-maker can use only one ax insrumen. Resuls of is policy exercise are repored in Figure 6 and 7. A e ime of e sock, under bo fiscal scenarios, oupu consumpion and governmen spending all decrease. Te dynamic beavior of ese variable are very similar under consumpion and labour income axaion. Noable differences across e wo scenarios emerge, owever, wen considering e opimal responses of inflaion, nominal ineres raes, axes, ours and governmen deb. Similar o e previous case, e opimal pa of consumpion axaion implies a sarp increase in e firs period and a decrease in e second period, wile under labour income axaion, axes increase only in e firs period and by a muc smaller amoun. As a consequence, e opimal level of public deb under consumpion axaion increases by less an under labour income ax, us reflecing a smaller incenive of e policy-maker o smoo axes across saes wi variaion in public deb posiions. Under bo scenarios, inflaion increases in e firs period. Tis elps o reduce e real value of mauring deb. However, e dynamic properies of consumpion axes imply a deflaionary episode afer e sock wic is compleely absen under labour income axaion. As i is well known from Scmi-Groé and Uribe 24 and Benigno and Woodford 24, i is subopimal in e presence of even small amouns of nominal rigidiies o bring abou large price level canges, i.e. o use nominal bonds as a sae coningen source of axaion and ence o fully smoo axes across saes. Tis resul sows up ere in e form of raer small movemens of inflaion. Table 3 repors e simulaed momens and e welfare calculaion of is policy exper- 2

22 imen. As in e case were public deb is a firs bes, e opimal volailiy of real as well nominal variables is closer o e effi cien response o sock, us confirming consumpion axaion as a beer policy insrumen even under a posiive level of public deb. Welfare calculaions presened in e Table 3 are formed by wo elemens, e seady sae loss of eac scenario and e loss a business cycle frequencies. Two ings are wor noicing. Firs of all, under bo scenario, posiive public deb ranslaes in a uge ineffi ciency in e opimal response o socks. Second, even wi posiive public deb, consumpion axaion remains a beer policy insrumen wen compared o labour income axaion. Te loss under e former amouns o.9% of permanen consumpion unis, wile under e laer fiscal scenario e sysem suffer a loss of.44%. 4 Conclusions Tis paper sudies e opimal mix of moneary and fiscal policy in a New Keynesian model were public spending as inrinsic value and e fiscal auoriy can use consumpion and labour income axaion in order o finance public spending and o repay e burden of public deb. We sow a e opimal policy mix is markably alered wen consumpion axaion is considered as a source of governmen revenues alongside or as an alernaive o labour income axaion. We find ineresing differences bo in e deerminisic and in e socasic allocaions. Firs, we sow a e opimal seady sae size of e public secor is, ceeris paribus, greaer under consumpion axaion an under labour income ax. We furer sow a adoping consumpion axaion as enormous long run welfare gains and a ese gains are increasing in e level of ousanding public deb. Ten we find a ese welfare gains are no limied o e seady sae, bu exend in e socasic equilibrium. Te reason is a e dynamic naure of consumpion axaion enables e policy-maker o affec e socasic discoun facor via modificaions of e marginal uiliy of consumpion. Tis exra wedge grealy improves e welfare in e socasic equilibrium weer e level of public deb is effi cien or no. References ] Adam, Klaus 2 Governmen Deb and Opimal Moneary and Fiscal Policy, E- E R, vol. 54, ] Adam, Klaus and Robero Billi 28 Moneary Conservaism and Fiscal Policy, J- M E, vol. 55, ] Aiyagari, Rao, Alber Marce, Tomas Sargen, and Jua Seppala 22 Opimal Taxaion Wiou Sae-Coningen Deb, J P E, vol.,