Essays on Secular Stagnation

Transcription

1 Essays on Secular Sagnaion Manuel Corrêa de Barros de Lancasre Nova School of Business and Economics Thesis presened as par of he requiremens for he Degree of Docor of Philosophy in Economics A Thesis carried ou on he Ph.D. Program in Economics, under he supervision of Prof. Pedro Brinca, and Prof. Francesco Franco Ocober 2016

2 Agradecimenos Agradecer a oporunidade de uma eapa nova e ão rica no meu percurso de vida é reconforane. Cronológicamene, começo pelo Francesco Franco a quem devo o encorajameno inicial, bem como o apoio perspicaz e exigene ao longo desa caminhada. Agradeço ao corpo docene e aos meus companheiros da Universidade Nova que ão nauralmene receberam um aluno e colega um pouco mais velho que o normal. Desaco o Iliyan Georgiev que me volou a fazer senir paixão pela maemáica, e a Susana Perala sempre disponivel para conversar. Agradeço ao Pedro Vicene o empurrão que me deu para ir para os Esados Unidos, e à Isabel Hora Correia que ão disponivelmene me apresenou às pessoas que deerminaram a minha vinda para Brown. Em paricular o Ricardo Nunes, que me pôs em conaco com o Gaui Eggersson, pela sua permanene disponibilidade. Muio obrigado ao Pedro Brinca que me em acompanhado durane ese úlimo ano, com pragmaismo, energia, e ambição relaivamene ao meu fuuro académico. Finalmene, o meu mais profundo agradecimeno e reconhecimeno ao Gaui Eggersson a quem devo o privilégio do seu convie, da sua orienação sempre disponivel, deses empos ão enriquecedores e produivos em Brown, e principalmene da consequene oporunidade desa experiência americana, ambém ão relevane para a familia que parilho com a Carolina, em paricular para os nossos cinco filhos, Teresa, Manel, Kiko, Mana e Caro. Manuel C. B. Lancasre Boson, Ouubro 2016 i

3 Acknowledgmens Thanking such an enriching period of my career and academic pah is comforing. Chronologically, I hank Francesco Franco for he moivaion o sar his advenure, as well as his demanding and insighful suppor. I hank all he faculy and fellow sudens a NOVA who received an older han average suden so naurally. I wan o highligh Iliyan Georgiev who rekindled my passion for mahemaics, and Susana Perala who was always available o cha. I also hank Pedro Vicene for encouraging me o go o he US, and also Isabel Hora Correia who generously inroduced me o people who would be crucial in my coming o New England. Specially Ricardo Nunes, who inroduced me o Gaui Eggersson, for his permanen suppor. Thank you o Pedro Brinca who has paienly accompanied me during his las year, wih energy, and ambiion regarding my academic fuure. Finally, my deep and hearfel hanks o Gaui Eggersson o whom I owe he privilege of his inviaion and of his guidance during his enriching and producive ime a Brown, as well as he resuling life experience ha my family - Carolina, and our children Teresa, Manuel, Francisco, Mariana and Carolina - have been enjoying so much on his side of he Alanic. Manuel C. B. Lancasre Boson, Ocober 2016 ii

4 Conens Agradecimenos Acknowledgmens Inroducion i ii vii 1 Disorionary Taxaion in Sable Recessions Inroducion Secular Sagnaion model wih disorionary axes Model wih labor and consumpion axes Increasing he naural rae of ineres Offseing a deleveraging shock Couneracing a sable recession wih disorionary axes Moneary and fiscal policy consisency Sicky wages, sable recessions, and labor ax on firms Reurn on Capial Income ax Final remarks Age Milesones and Low Ineres Raes, an Analyic approach Inroducion An Endowmen Economy wih age milesones Real ineres rae derivaives wih respec o age parameers Duraions of relevan lifeime economic periods Age milesones bounding he duraion of he model Age milesones bounding labor income duraion Inergeneraion ransfers Inergeneraional Alruism Social Securiy Quaniaive calibraion iii

5 CONTENTS iv OLG model wih endogenous oupu and capial Quanifying derivaives of r wih respec o age milesones Final remarks A Proposiion 1: Presen Value and Aggregae Consumpion B Proposiion 2: Excess Borrowing Seady Sae Properies C Proposiion 3: Real Ineres Rae Derivaives w.r. age parameers D OLG model wih endogenous oupu and capial: Inequaliy and Real Ineres Raes Inroducion Inequaliy, Marginal Borrowing/Saving raes, Real Ineres Raes Secular Sagnaion Endowmen Economy Model wih Bequess Decreasing real ineres raes, wih increasing income inequaliy Borrowing consrains, inequaliy, and real ineres raes Bequess, inequaliy, and real ineres raes Quaniaive calibraion of he model Final remarks A Endogenous Oupu and Capial B Inergeneraions Uiliy of Consumpion Bibliography 137

6 Lis of Tables 2.1 Iniial seady sae parameers for Derivaives wih same paramerizaion, β = 0.978: changing r Derivaives seing r = 4.40%, adjusing β for equilibrium Robusness Analysis Relaive Risk Aversion Simulaion Resuls Summary of model condiions for increasing inequaliy o decrease r Parameers and Simulaion Resuls Robusness analysis v

7 Lis of Figures 1.1 The Naural rae of Ineres and he Zero Lower Bound Allowing a Firs Bes Soluion wih Fiscal and Moneary Policy Secular Sagnaion Equilibrium One period aggregae supply conracion shock Life Expecancy and effecive Reiremen Age: EU and US Relevan Life-cycle Periods and Age Milesones US income shares wih capial gains : Top 0.1% and 1% Saving Raes and Income Real Ineres Raes and Income Inequaliy in US Equilibrium in he Loan Marke vi

8 Inroducion Ineres raes and inflaion have been persisenly low in relevan world economies. When he zero lower bound for he nominal ineres rae prevens he real ineres rae o reach an evenually negaive full employmen level - he Naural Rae of Ineres - an economy may ener ino a sable recession, according o recen work on Secular Sagnaion [34] and Liquidiy raps [31]. Moreover, when he zero lower bound for nominal ineres raes binds, shor erm moneary policy may lose effeciveness. Undersanding he facors ha may drag down he Naural Rae of Ineres in a permanen way is a fundamenal sep in order o design he mos adequae policies o overcome or o avoid he undesirable consequences of a prolonged economic slump. Some of hose facors have been explored in recen lieraure, where we can namely highligh diminishing borrowing limis during longsanding deleveraging shocks in Eggersson and Mehrora [21] recen work on Secular Sagnaion, or demographic changes inspeced for example by Carvalho e al. [12]. Inspecing he mechanisms dragging down he naural rae of ineres in a longsanding way may require alernaive modeling opions o he sandard single agen RBC and New Keynesian frameworks, where negaive real ineres rae levels are generally emporary by consrucion. Moreover, he policy prescripions o counerac a emporary recession may no necessarily be applicable when dealing wih a lasing economic slump, where he impac of a policy mus be permanen. For example, when a vii

9 INTRODUCTION viii higher level of inflaion is no a desirable policy oucome o allow a firs bes soluion consisen wih a negaive naural rae of ineres, Correia e al. [16] propose an emulaion of he role of higher consumer price inflaion hrough a emporary sable increase of consumpion axes in a single agen NK model. The ransiory naure of ha prescripion migh no be useful if he naural rae of ineres is permanenly low; in conras wih alernaive policies ha may permanenly level-up he naural rae of ineres o a level where moneary policy becomes effecive again. Our research aims a complemening recen work on couneracing persisen recessions, as well as analyically inspecing mechanisms wih persisen impac on he naural rae of ineres. In paricular, our conribuion in Chaper 1 is o inspec he role of disorionary axes in avoiding a permanen slump, complemening recen work on Secular Sagnaion, and on he role of disorionary axaion in emporary recessions. Our conribuion in Chapers 2 and 3 is o inspec analyically he impac of changing age milesones and inequaliy on he naural rae of ineres, by deriving explici algebraic relaions beween real ineres rae changes and changes on hose facors. In he firs Chaper we formalize he role of disorionary axaion in avoiding a sable recession characerized by he Secular Sagnaion framework proposed by Eggersson and Mehrora [21], based on a hree generaions OLG model wih borrowing consrains. We compare our resuls wih he ones obained by Correia e al. [16] ha propose a soluion based on he same se of disorionary axes in a sandard single agen New Keynesian model wihou borrowing consrains, o counerac a liquidiy rap ha is by consrucion emporary. We find reversed resuls. Our mechanism is based on a wealh re-disribuive policy using disorionary axes o increase he naural rae of ineres so ha i becomes achievable given he moneary policy arges, by increasing labor axes on he middle age employed and redisribuing he ax proceeds o he populaion in general by reducing consump-

10 INTRODUCTION ix ion ax. Insead, Correia e al. [16] emulae inflaion in consumer prices using an increasing pah of consumpion axes, so ha he ineremporal condiion allows an achievable negaive naural rae of ineres, and he liquidiy rap is neuralized. We use he same fiscal oolbox wih differen approaches. We increase he naural rae of ineres o a level consisen wih moneary policy effeciveness, insead of emporarily allowing a firs bes soluion compaible wih a negaive naural rae of ineres. As an alernaive o he sandard fiscal policy prescripions o counerac an economic downurn, based on Keynesian increases of public expendiures, more public deb and ax cus o simulae demand 1, and complemening he papers referenced above, he main purpose of our analysis is o show how fiscal policy based on disorionary axaion can be effecive in avoiding persisen recessions, even when increasing public expendiures and deb are no policy opions available. In he second chaper we formalize he relaion beween real ineres raes and relevan age milesones of an agen s life, using an overlapping muli-generaions model where one generaion correspond o one year. Alhough he impac of age srucure in relevan World economies has been a recurren opic covered in recen lieraure, in paricular o explain he persisen decline of ineres raes and economic sagnaion, here has no ye been an aemp, o he bes of our knowledge, o formally derive he analyic relaions of real ineres raes o each age milesone. The purpose of his chaper is o fill ou his gap, by deriving racable algebraic real ineres rae elasiciy expressions wih respec o each age parameer, in order o formalize he mechanisms by which real ineres rae changes occur. This allows, for example, o algebraically derive precisely by how much he naural rae of ineres may be permanenly dragged down evenually o negaive levels, by any combinaion of increasing life expecancy, posponing firs child birh, lowering he 1 Besides oher non-fiscal approaches, namely he one proposed by Eggersson and Woodford [23] where he cenral bank commis o keep ineres raes a a lower level even afer a recession resuling from a liquidiy rap is over.

11 INTRODUCTION x reiremen age, increasing adulhood age, or reducing he age of firs job. Moreover, we can use our framework o quanify explicily he impac on he reducion of real ineres rae level in recen years, of he evoluion of populaion age srucure. The main underlying mechanism relaing age milesones and real ineres raes in our model relies on he relaive place and duraion of labor income wih respec o life expecancy when agens smooh consumpion. For example, a longer reiremen period, resuling from a reducion of he reiremen age or an increase of life expecancy, makes households save more, hus expanding he supply of loans ha drags down he real ineres rae ha ensures equilibrium in he loan marke. We also inspec how exogenous parameers o our model, namely he elasiciy of ineremporal subsiuion, produciviy growh, income growh pah of households, and populaion growh, may amplify or miigae he impac of age milesones changes on he naural rae of ineres. In addiion we show how iner-generaion ransfers are affeced by age milesone changes. For example, why and how increasing life expecancy may reduce endogenous beques levels, decrease he propensiy o help children, or increase he willingness o help parens. Laerally o our main conribuion in his chaper, he analyic formulaion of ineres rae changes wih respec o age milesones, we also developed a racable algebraic oolbox o solve overlapping muli-generaions opimizaion problems. Finally in Chaper 3, we formalize he relaion beween increasing income inequaliy and low real ineres raes using an overlapping generaions model wih borrowing consrains and a beques moive. Our main conribuion in his chaper is o presen in a single simple framework an explici formalism linking low real ineres raes wih increasing income inequaliy, by gahering and building on some relevan opics in recen lieraure, namely increasing income inequaliy[33], bequess[4][7], he quesion of wheher higher-lifeime income levels lead o higher marginal propensiy o save[19] and lower marginal propensiy o borrow[1][32], and secular sagnaion[21][34]. The underlying mechanism in our model relaing real ineres raes

12 INTRODUCTION xi and inequaliy is based on empirical evidence in recen lieraure ha households marginal borrowing and saving raes are respecively negaive and posiive funcions of income, so ha he ne effec on aggregae borrowing and savings of a permanen increase of income inequaliy is respecively a ne conracion and a ne expansion, ha may lead o a persisen reducion of he naural rae of ineres. In paricular, he borrowing mechanism in our model is based on he concaviy of he marginal propensiy o borrow, and on binding borrowing consrains, boh consisen wih empirical observaions in recen lieraure[1][32]. In addiion, he saving mechanism is illusraed wih an endogenous propensiy of households o leave a beques when hey expec he nex generaion o be poorer. So ha wealhier households are more generous, making heir marginal savings rae higher han he poor ha in general leave much lower or no bequess a all, as also observed by Hendricks [28].

13 Chaper 1 Disorionary Taxaion in Sable Recessions Absrac Increasing public spending, generaing budge deficis, or raising he level of public deb, may no be opions available for an economy rying o avoid a recession. Complemening recen work on Secular Sagnaion and on fiscal policy during liquidiy raps, we use an overlapping generaions New Keynesian model wih borrowing consrains o explore how disorionary axes can be used o circumven a persisen economic slump, by raising he full-employmen equilibrium real ineres rae so ha i becomes achievable when he nominal ineres rae zero lower bound is binding. We propose a wealh redisribuive fiscal policy, by axing labor income and reducing consumpion or capial income axes, leading o a conracion of savings ha riggers an increase of he naural rae of ineres 1. We compare our resuls wih an alernaive approach in recen lieraure based on emulaing inflaion in consumer prices using an increasing pah of consumpion axes, so ha he ineremporal condiion allows an achievable negaive naural rae of ineres o neuralize a liquidiy rap. 1 Full-employmen equilibrium real ineres rae. 1

14 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS 1.1 Inroducion During a liquidiy rap, when inflaion is low and he zero lower bound for nominal ineres raes prevens he real ineres rae o reach is full employmen level, shor erm moneary policy may lose effeciveness. To circumven a liquidiy rap, sandard fiscal policy prescripions in he lieraure have been based on Keynesian increases of public expendiures and ax cus o simulae demand 2, and hus increase inflaion, allowing for a reducion of he real ineres rae owards he naural rae of ineres 3. More recenly, Correia e al. [16] proposed an alernaive approach based on he use of disorionary axes, wih no need o use inefficien policies such as waseful public spending or fuure commimens o low ineres raes. To allow a firs bes soluion compaible wih a negaive naural rae of ineres when he zero lower bound for nominal ineres raes is binding, heir recipe consiss of emulaing he role of higher consumer price inflaion, by a sable increase in consumpion axes. They use a sandard single agen New Keynesian model, where a slump is by consrucion emporary. In his paper we also inspec how fiscal policy based on disorionary axaion may circumven a liquidiy rap, bu in he conex of a permanen slump. Our approach is based on he recen Secular Sagnaion lieraure, in paricular on he work of Eggersson and Mehrora [21], o which we add a module of disorionary axes. We use heir proposed hree generaions OLG model, where he naural rae of ineres may be persisenly negaive. In conras wih he inflaion emulaion fiscal policy ype proposed by Correia e al. [16], ha increases gross consumpion prices hrough an increase of consumpion axes o allow a model firs-bes soluion, he disorionary axaion prescripion o avoid a liquidiy rap in a permanen slump 2 Besides oher non-fiscal approaches, namely he one proposed by Eggersson and Woodford [23] where he cenral bank commis o keep ineres raes a a lower level even afer a recession resuling from a liquidiy rap is over. 3 The naural rae of ineres has been defined in he lieraure as he equilibrium fullemploymen real ineres rae. 2

15 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS using he model proposed by Eggersson and Mehrora [21] is based on a wealh redisribuion mechanism from an endowed middle aged generaion, o young credi consrained households, and o he older generaion living a he same ime. An increase of income axes on he middle-age would reduce heir ne available funds o consume and o save, consequenly 4 riggering an increase of he naural rae of ineres o a higher sable level. If he proceeds from his ax increase are used o reduce he consumpion ax, hen he borrowing consrained younger households, as well as he old, would be able o consume more. Ineresingly, alhough his paper and he one of Correia e al. [16] boh use disorionary axaion o allow firs bes, full-employmen soluions o circumven he poenial damages of a liquidiy rap, he prescripions are reversed. Correia e al. [16] neuralize he effecs of he zero lower bound o achieve negaive real ineres rae levels, by inducing inflaion in consumer prices, he ones ha maer for ineremporal decisions, wih an increasing pah of consumpion axes over ime, simulaneously reducing labor axes such ha producer price inflaion is kep a zero. They use a single agen sandard New Keynesian model, wih no borrowing consrains, where a slump is by consrucion ransiory. In conras, in our model here are hree ypes of households, where he younger generaion is credi consrained. We use disorionary axaion o increase he naural rae of ineres o an achievable level, insead of neuralizing obsacles o achieve a negaive level. Insead of inducing inflaion in consumer prices o achieve negaive real ineres raes, our prescripion is wealh redisribuive in order o increase he naural rae of ineres. The fiscal policy we propose could be implemened wih lump-sum axes and ransfers, from middle age o young and old agens. Lump-sum axaion is hen effecive in our heerogeneous agens world wih borrowing consrains, bu no in a sandard single agen New Keynesian framework. Las bu no leas, we propose an 4 assuming a sandard conex where he derivaive of excess savings wih respec o he real ineres rae is posiive. Excess savings being defined as he difference beween loan supply and demand. 3

16 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS increase of labor axes o conrac savings and increase he naural rae of ineres, allowing also for a reducion of consumpion axes o he relaive benefi of younger and older agens; in conras wih a reversed prescripion by Correia e al. [16] based on increasing consumpion axes and reducing labor axaion. This chaper is hen organized as follows. In he nex secion we inspec how disorionary axes may affec he naural rae of ineres, by using a general version of he model inroduced by Eggersson and Mehrora [21] wih Consan Relaive Risk Aversion household preferences. We show ha emulaing inflaion wih a sable increasing pah of consumpion axes would no be able o increase he naural rae of ineres back o posiive ground on a sable basis, alhough i could be effecive in he shor erm, bu only when he elasiciy of ineremporal subsiuion is greaer han one. In addiion, we show ha he fiscal policy a he core of his chaper, based on axing income of middle age households, is effecive in he shor and long run, ensuring an increase of he naural rae of ineres o a sable level irrespecively of he elasiciy of iner-emporal subsiuion 5. Wihou loss of generaliy, we assume EIS 6 is equal o one hrough he res of he chaper because i allows closed-form expressions in paricular for he equilibrium real ineres rae. We hen use log-uiliy of consumpion preferences o inspec he role of each fiscal insrumens in avoiding a liquidiy rap by susaining he naural rae of ineres a posiive levels. As an example, we presen closed-form soluions for he changes required for each ax in order o offse he impac of deleveraging shocks on he naural rae of ineres. In he hird secion we inroduce nominal prices and endogenous rigid wages o model a Secular Sagnaion equilibrium as in Eggersson and Mehrora [21]. Capial and a ax on capial income are inroduced in he las secion. We inspec he role of disorionary axaion in avoiding a sable recession, driven in his model by a 5 Same assumpion as before abou he slope of excess savings wih respec o real ineres rae. 6 EIS is he acronym of Elasiciy of Ineremporal Subsiuion 4

17 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS deflaion mechanism ha prevens a imely reducion of nominal rigid wages, ha riggers an increase of real wages o a sable level away from he full-employmen flexible wage, hus dragging he economy ino a persisen recession. We remind[21] ha moneary policy can ensure he exisence of a full-employmen equilibrium, bu canno by iself avoid is coexisence wih a Secular Sagnaion one, o which he economy can ransi, namely during a deleveraging shock, if fiscal policy is no used in a sufficienly asserive way. Adequae fiscal policy measures based on disorionary axaion promoing he expansion of aggregae demand, possibly combined wih he conracion of aggregae supply, may effecively preven he economy o fall ino a slump, or ensure ha i ransis ou of one. Reducing consumpion and capial income axes expand aggregae demand, which combined wih an increase of labor axes, may raise he naural rae of ineres, evenually o achievable levels alogeher neuralizing a sable recession. Furhermore, and reminding he Paradox of Toil[20], a demand expansion can be combined wih a supply conracion ha in he shor erm can boos inflaion and force he ransiion from a sable recession wih deflaion, o a posiive inflaion full-employmen equilibrium. This can be achieved by increasing he labor ax on firms ha would lead firms o reduce nominal wages in order o susain profiabiliy. The nominal wage rigidiy prevens he nominal wage full adjusmen riggering a furher increase of real wages which conracs supply even furher, hus creaing a posiive pressure on inflaion ha may bring back he economy o a higher employmen level. The resuls presened in his chaper are relaed o a closed economy, alhough he combinaion alernaives of disorionary axes o avoid a permanen slump, for example by using a reducion of he ax on capial income insead of he consumpion ax, sugges ha his framework could be applied o oher conexs, namely o open economy frameworks. 5

18 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS 1.2 Secular Sagnaion model wih disorionary axes In his secion we briefly describe and derive a Secular Sagnaion overlapping hree generaions model wih real prices proposed by Eggersson and Mehrora [21], o which we add a module of disorionary axes. The aim is o find a general expression for he impac of ax changes on he naural rae of ineres, so ha i can be kep a a sufficienly high achievable level, for example during a deleveraging shock. We use he Implici Funcion Theorem o derive a general expression for he parial derivaives of he naural rae of ineres wih respec o policy insrumens, o explore he effeciveness of fiscal policy opions in avoiding a liquidiy rap, by manipulaing he naural rae of ineres level. In paricular we show ha inflaion emulaion fiscal approach based on an increasing pah of consumpion axes proposed by Correia e al. [16] is no effecive in couneracing a persisen recession in he conex of our model Model wih labor and consumpion axes We use a 3 periods overlapping generaions model[21] ha allows for seady sae equilibria wih persisen negaive real ineres raes, when populaion growh is low enough or when borrowing consrains of he younger generaion increase. Households go hrough hree sages of life: young, middle aged and old. The young generaion borrows B y from he middle-aged; he middle-aged save B m by lending o he young and o he governmen, and pay back heir loans o he previous generaion, he old. The old receive back wih ineres wha hey have len o he young and he Governmen when middle aged. Consumpion of he young C y, middle age C m, and old C o is axed by a consumpion ax τ c. I is assumed ha income is only earned by he middle aged hrough firm profis Z, and labor W L 6

19 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS which is axed wih τ l. Borrowing is consrained by a binding deb limi D faced by he young, exogenously deermined. The Governmen budge is balanced. The household objecive funcion is given by: max E {U(C y C y ) + βu(c+1) m + β 2 U(C+2)} o (1.1),Cm +1 Co +2 s.. (1 + τ c )C y = B y (1.2) (1 + τ c +1)C m +1 = Z +1 + W +1 L +1 (1 τ w +1) (1 + r )B y + B m +1 (1.3) (1 + τ c +2)C o +2 = (1 + r +1 )B m +1 (1.4) (1 + r )B y D, an exogenous borrowing limi. (1.5) Where U(C) is a consan elasiciy of iner-emporal subsiuion uiliy funcion expressed by U(c) = c1 σ. We assume ha he borrowing consrain of he young 1 σ generaion is binding, or (1 + r )B y = D. In his credi consrained environmen, alhough he amoun borrowed by a young agen does no direcly depend on any fiscal insrumen bu jus on he real ineres rae and he borrowing consrain, B y = D 1+r, he consumpion of he young will inversely direcly depend on he level of consumpion axaion, given by: (1 + τ c )C y = B y = D 1 + r (1.6) So unless an increase of consumpion axes were followed by a sufficien reducion of equilibrium real ineres raes, hey would resul in a consumpion conracion of younger agens. The same could be said abou he old, bu in his case wih no relaion wih curren equilibrium ineres rae, since heir consumpion depends on savings from previous period, ne of consumpion axes: (1 + τ c )C o = (1 + r 1 )B m 1 (1.7) 7

20 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS Consequenly an increase of consumpion axaion, as proposed by Correia e al. [16], in his environmen would lead o a conracion of consumpion of he younger and older generaions, wih an expansion of middle age consumpion, assuming sable governmen spending and full-employmen. A negaive welfare effec if we assume ha middle age are wealhier han he young and old. As he consumpion of he old is deermined by previous period equilibrium, unconsrained middle age consumpion is deermined by he model s Euler equaion given by: 1 + r = 1 β E U c (C m ) U c (C+1) o 1 + τ c τ c (1.8) This Euler equaion is similar o he one derived by Correia e al. [16], and would sugges ha an increase of consumpion axaion could have an increasing effec on he equilibrium real ineres rae as well. Nex we show why his is no necessarily he case in our borrowing consrained economy where persisen recessions are possible Increasing he naural rae of ineres The model is compleely deermined wih he budge consrains, he previous Euler equaion, and he loan marke equilibrium equaion given by: B y = B m (1.9) where B y = L d is he demand for loans by he young generaion, which in equilibrium mus be equal o he supply of loans from he middle aged B m = L s. Le Excess Savings be a coninuous and differenial funcion for r ] 1, + [ defined as S = L s L d, such ha a loan marke equilibrium S = 0. Using he he Implici Funcion Theorem we can explici he parial derivaive of he naural rae of ineres wih 8

21 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS respec o any ax parameer: L d = L s S = 0 r = τ S τ S (1.10) r We now inspec each componen of he previous expression. i) Loan Demand Toal borrowing in period is equal o he oal demand for loans, and is given by oal borrowing of young households and he governmen, from he middle age: B = N B y + N 1 B g (1.11) where B g is governmen borrowing per middle age household, N is he size of he young generaion a ime, and N 1 is he size of middle age generaion also a ime. Then he demand for loans per middle age agen is given by: L d = B N 1 = (1 + g )B y + B g = 1 + g 1 + r D + B g (1.12) where 1 + g = N /N 1, and g is populaion growh a ime. In ha case a reducion of he borrowing limi of he young would cause a conracion of loan demand, ha could be counerbalanced by an increase of public deb. Assuming ha governmen borrowing is exogenously deermined, loan demand would no depend on any ax insrumen in his credi consrained environmen, implying ha S τ = Ls τ. Moreover, loan demand is a negaive funcion of he real ineres rae, wih a sricly negaive parial derivaive given by: L d r = 1 + g (1 + r ) 2 D = Ld (1 B g /L d ) 1 + r (1.13) Because we assumed ha governmen borrowing B g is an exogenous parameer of his economy, we can already simplify he general expression for he parial derivaive 9

22 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS of he naural rae of ineres wih respec o ax insrumens o conclude ha heir relaive sign depends on he impac of he ax insrumen on loan supply. r = τ S τ S r L s τ = S (1.14) r As we show laer, we can realisically assume ha excess savings in he denominaor is an increasing funcion of he real ineres rae. In ha case any ax change having a conracion effec on loan supply, would raise he naural rae of ineres. We hen derive loan supply. ii) Loan Supply Firms in he model are in perfec compeiion and hire labor o maximize profis on a period by period basis. For now here are no disorionary axes on firms. The firm problem is given by: Z = max L Y W L s.. Y = F (L ) sricly concave (1.15) The soluions for W and Z are given by: W = F L (L ) and Z = F (L ) L F L (L ) (1.16) We hen ge a presen value expression for consumpion of middle age and old by combining consrains (1.3) and (1.4) wih he borrowing consrain (1.5), and replacing he soluions for W and Z : (1 + τ c )C m + (1 + τ c +1)C o r = F (L ) τ w F L (L )L D 1 (1.17) Loans supply is derived by replacing expressions for middle age and old consumpion 10

23 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS given by (1.8) and (1.7) in previous equaion: L s = B m = [ ] F (L ) τ l F L (L )L D 1 [ ] (1 + r β σ 1 ) 1+τ c 1 1 σ 1+τ+1 c (1.18) We see direcly from his expression ha an increase of labor axaion on middle age is enough o rigger a conracion of loan supply, independenly of he elasiciy of iner-emporal subsiuion EIS 1. By realisically assuming ha excess sav- σ ings increases wih he real ineres rae 7 a permanen increase of labor axaion on he middle age would hen rigger a permanen increase of he naural rae of ineres. Consumpion axes would have o be reduced in order o keep governmen budge balanced, in curren and following periods. Furhermore, if we assumed ha consumpion axes would remain consan in he presen and fuure, hen loan supply would no depend on consumpion axes, bu on labor axes alone which would become he only fiscal insrumen direcly deermining loan supply: L s = [ F (L ) τ l F L (L )L ] D (1 + r β σ 1 ) 1 1 σ (1.19) An inflaion emulaion policy wih a sable increase of consumpion axes can here be given by a commimen o increase consumpion axaion in he nex period. The impac on loan supply depends on EIS, as can also be observed direcly from expression (1.18). If EIS 1 σ < 1, hen commiing o increase consumpion axaion in he nex period expands loan supply in he curren period, and reduces he naural rae of ineres. Wih a low elasiciy of iner-emporal subsiuion, he income effec relaed o an expeced increase of consumpion axaion in he nex period would lead o an increase of savings in he curren period o susain fuure 7 As we show laer, a sufficien condiion for a posiive slope of excess savings S wih respec o r is ha he raio of governmen borrowing over oal borrowing in he economy Bg does no L g exceed he elasiciy of iner-emporal subsiuion 1 σ. This is a reasonable assumpion in developed economies, given he sandard esimaions of EIS usually above 0.5[27]. If EIS is close o 1, or for low level of governmen deb B g his becomes a rivial assumpion. 11

24 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS consumpion. This effec would here prevail over a subsiuion effec resuling from he fac ha consumpion in he fuure would become more cosly han curren consumpion. The opposie is rue if EIS > 1, which may no be he mos usual case aking he resuls of Havranek e al. [27]. Then a subsiuion effec would prevail wih a commimen o increase consumpion axes in he fuure, leading o a conracion of curren savings, and o a consequen increase of he naural rae of ineres. Bu in his case, for he budge o balance during he nex period labor axes would have o decrease, creaing a negaive pressure on he naural rae of ineres as seen before, ha would need a furher commimen o increase consumpion axaion in he fuure, and so on. This fiscal policy, besides being effecive only for EIS > 1, would force an unimplemenable recurren policy commimen from he governmen, wih no opporuniy for reversion in he conex of he curren model were a recession may be persisen. iii) Slope of Excess Savings wih respec o he real ineres rae: S r > 0? We firs derive an expression for he derivaive of loan supply wih respec o real ineres rae: ( ) L s 1 L s = r σ r ( 1 + β 1 σ 1 1+r 1+τ c +1 1+τ c ) σ 1 σ (1.20) Loan supply is posiive sloped for EIS > 1. Since Ss r EIS > 1 hen S r = Ls r Ld r, and Ld r < 0, if is posiive. If EIS < 1 hen loan supply is negaive sloped wih respec o r. A sufficien condiion for S r We assume ha inequaliy Bg L d < 1 σ > 0 is Bg L d < EIS 8. is valid hrough he res of he paper. This implies ha he sign of he parial derivaive of he naural rae of ineres wih respec o any ax insrumen is opposie o he sign of he parial derivaive of loans supply 8 S r > 0 L d > L s Ld (1 Bg /Ld ) 1+r > ( 1 1 σ 1 1 σ Bg L d < 1 σ. ) L s 1 1+r ( 1+β σ 1 1+τ 1 +1 c 1+r 1+τ c ) σ 1 σ 1 B g /L d > 12

25 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS wih respec o ha same insrumen. If we furher discard from our analysis policy measures suppored by increasing (or decreasing) pahs of consumpion axes, since we checked hem previously, we can assume wihou loss of generaliy, and for he purpose of algebraic racabiliy, ha EIS = 1, and U(C) = Log(C) Offseing a deleveraging shock In his secion we explore each fiscal insrumen in erms of is role in offseing any facor dragging down he naural rae of ineres. We sar wih labor axaion on households. i) Labor ax on households: τ l For σ = 1, a closed-form expression for he naural rae of ineres can be derived direcly from loan marke equilibrium given by equaion (1.12): 1 + r n (1 + g )D = [ ] β (1 ατ l 1+β )Y f D 1 B g (1.21) Where we assume ha oupu is now expressed by Y = L α, and in paricular, fullemploymen oupu is given by Y f = L α when he equilibrium real ineres rae is equal o he naural rae of ineres r n. We can observe direcly from expression (1.21) ha he naural rae of ineres decreases wih he conracion of aggregae borrowing, eiher because he Governmen needs o reduce public borrowing B g, or because of higher credi consrains imposed on young households hrough lower borrowing limis D. If he governmen is prevened o increase public deb, hen a sufficien increase of labor axaion τ l could offse he impac of a deleveraging shock, by riggering a conracion of aggregae savings o preven he naural rae of ineres from falling o non-achievable (negaive) levels. In paricular, o counerac he impac of a deleveraging shock, from a reducion of public borrowing, on he 13

26 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS naural rae of ineres, he required change of labor ax would be: τ l = 1 α ( 1 + β β ) B g Y (1.22) and o offse he impac on he naural rae of ineres of a deleveraging shock, due o a lower borrowing limi of young households, he required change of labor ax would be 9 : τ l = 1 α [ 1 + ( ) 1 + β 1 + g β 1 + r n ] D Y f (1.23) ii) Consumpion ax: τ c Wih no capial in he model, aggregae demand is he sum of aggregae consumpion C and governmen spending G. If he governmen is prevened o spend more han an exogenous upper limi G, hen consumpion C canno be lower han C f = Y f G in order o ensure full-employmen. By assuming ha public spending is exogenously deermined we nex explore how o susain consumpion a is full-employmen level, combining disorionary axes so ha he naural rae of ineres is also susained a viable levels. Toal consumpion in period is given by: C T oal = N 1 C = N C y + N 1 C m + N 2 C o Combining he previous expressions wih young and old budge consrains (1.6),(1.7), he euler equaion (1.8), and loan marke equilibrium (1.12), consumpion is also expressed by: C = τ c {[( 1 + β β ) ] [ 1 + g B g D + D r β r ]} 1 B g g 1 (1.24) Consumpion is direcly deermined by he borrowing consrains of he governmen 9 This is a seady sae expression. 14

27 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS and he young generaion. Because he borrowing consrains are binding, consumpion inversely depends on consumpion ax. Then, wih a conracion of he borrowing limi of younger households, he effecive fiscal insrumen ha susains he consumpion level from falling is τ c, if he real ineres rae is o be susained oo. Furher below, and considering he governmen budge consrain, we will show ha he change in labor income ax required o preven he real ineres rae from falling afer a deleveraging shock is consisen wih he required reducion of he consumpion ax ha prevens consumpion level from falling, given he same deleveraging shock. The consumpion ax changes required o susain he same level of consumpion, given a reducion of households borrowing limi, or of governmen deb, assuming he real ineres rae is susained oo, are respecively given by he following expressions in seady sae: [( ) ] 1 + β 1 + g D τ c = β 1 + r + 1 Y G [ 1 τ c = β r ] s B g 1 + g Y G (1.25) (1.26) Direcly from expressions above, o counerac a deleveraging shock, he fiscal adjusmen would need o be sronger for governmens ha spend more. iii) Governmen Budge consrain Unil now we used wo approaches o offse he impac of a deleveraging shock respecively on he naural rae of ineres, and on consumpion. The firs one is based on an increase of labor axes o susain he naural rae of ineres a an achievable level, and he second on reducing consumpion axes o susain consumpion a is full-employmen level. Boh approaches are equivalen given he budge consrain 15

28 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS of he Governmen, given by: B g = G T r g 1 B g 1 (1.27) Where oal axes per middle age household can be expressed as a funcion of oupu: T = τ c C + τ l w L = τ c (Y G ) + ατ l Y = (τ c + ατ l )Y τ c G (1.28) By replacing he above expression (1.28) in (1.27) we obain an alernaive expression for he budge consrain connecing he fiscal insrumens. We call his equaion he Governmen budge rule: G Y = ατ l + τ c 1 + τ c + 1 [ B g 1 + r ] 1 B g 1 Y 1 + g 1 (1.29) wih a seady sae expression given by: G Y = ατ l + τ c + Bg 1 + τ c Y [ ] g r 1 + g (1.30) Imagine here is a worsening of he credi consrains in his economy. If he Governmen is prevened o increase spending as well as public deb, hen, in order o susain he real ineres rae a he same level, savings would have o conrac hrough an increase of labor axaion on he middle age. The Governmen budge rule would hen imply a reducion of he consumpion ax, which would be he same required o susain consumpion a he same level, if he real ineres rae would remain unchanged. 16

29 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS 1.3 Couneracing a sable recession wih disorionary axes In his secion we inroduce he noion of persisen recession based on he Secular Sagnaion model proposed by Eggersson and Mehrora [21], and inspec he role of fiscal policy based on disorionary axaion o keep (or o bring back) he economy o a full-employmen equilibrium. In order o preven he economy o reach is firs bes full-employmen allocaion we inroduce nominal prices in he model, and he possibiliy of a binding zero lower bound for he nominal ineres rae ha can preven a negaive naural of ineres o be achieved, hus riggering a recession. We derive a simple sufficien condiion o mainain a firs bes equilibrium possible, by ensuring consisency beween moneary policy arges and disorionary axaion based fiscal policy. Then we inroduce nominal wage rigidiies ha allow he appearance of a second sable equilibrium in he model, in his case a persisen recession. We revisi he previous sufficien condiion o susain he economy a is firs bes allocaion, and also inspec how an adequae use of disorionary axes, in paricular a labor ax on firms, can help he economy move from a slump o a full-employmen equilibrium Moneary and fiscal policy consisency The maximizaion problem wih nominal prices is given by: max E {log(c y C y ) + β log(c+1) m + β 2 log(c+2)} o (1.31),Cm +1 Co +2 s.. P (1 + τ c )C y = P B y (1.32) P +1 (1 + τ c +1)C m +1 = Z +1 + W +1 L +1 (1 τ l +1) (1 + i )P B y + P +1 B m +1 (1.33) P +2 (1 + τ c +2)C o +2 = (1 + i +1 )P +1 B m +1 (1.34) 17

30 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS And an exogenous borrowing limi: (1+i )P B y P +1 D. Using he Fisher equaion 1 + i = (1 + r ) P +1 P, he Euler Equaion has he same expression as before, as well as he expressions for he real ineres rae, and for he Governmen budge rule. The Euler Equaion, in parcular, is given by: C m = 1 [ ] β E C+1 o (1 + τ+1) c P +1 1 = 1 [ ] (1 + τ c (1 + τ c ) P 1 + i β E +1 ) C+1 o (1 + τ c ) 1 + r (1.35) Assuming ha he nominal ineres rae follows a Taylor rule, 1 + i = max{1, (1 + i ) ( Π Π ) φπ 1 }, where φπ > 1, hen by using he Fisher equaion we ge he following expression for he real ineres rae: 1 + r = 1 + i Π { 1 = max, 1 + i Π Π ( Π Π ) φπ 1 } (1.36) Where Π and i are he moneary policy arges for gross inflaion and he nominal ineres rae. Noe ha when he zero lower bound is binding, he real ineres rae is jus a funcion of he curren inflaion level, and does no depend on any moneary policy insrumen given by inflaion and nominal ineres rae arges. From he previous expression, and given he nominal ineres rae zero lower bound, as well as his specific Taylor rule, a real ineres rae lower bound is given by: 1 + r (1 + i ) 1 φπ Π = 1 + r kink (1.37) If he naural rae of ineres r n is lower han he real ineres rae lower bound hen i would no be achievable, and a firs bes full-employmen allocaion would no be possible (see figure 1.1). r kink hen works like a lower bound for he naural rae of ineres o allow a full-employmen equilibrium in his economy. Anoher sufficien condiion o ensure ha he naural rae of ineres is achievable is o use he implici moneary policy arge real ineres rae level r as a lower bound for achievable levels of he naural rae of ineres r n. Le 1 + r = 1+i Π be he implici 18

31 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS Figure 1.1: The Naural rae of Ineres and he Zero Lower Bound moneary policy arge real ineres rae. Assuming ha φ π 1 hen r r kink : 1 + r kink = (1 + i ) 1 φπ Π = 1 + i Π (1 + i ) 1 φπ r, for i 0, φ π 1 (1.38) If he naural rae of ineres r n, deermined by and adequae fiscal policy selecion of disorionary ax levels, is greaer han he moneary policy implici real ineres rae arge r = 1+i, hen a full-employmen equilibrium is possible. In oher words, Π fiscal and moneary policy are complemens o fulfill full-employmen condiions. From an aggregae demand perspecive we can reach he same conclusion abou consisency beween moneary and fiscal policy o allow firs-bes full-employmen allocaions. Noe ha aggregae demand Y d (Π ) can be expressed by: [ Π > Π kink : Y d = 1 (1 + β)(1 + g )D Π φπ kink 1 + τ c β Π φπ 1 [ Π Π kink : Y d = τ c +D 1 ] + G (1.39) ] (1 + β)(1 + g )D Π +D 1 + G (1.40) β Where Π kink = Π (1+i ) 1 φπ Π. Also, Y kink = Y (Π kink ) is he upper bound of he se 19

32 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS of admissible aggregae demand levels. Then, if he naural rae of ineres has an achievable level, r n ({τ }) r kink (Π, i ), hen Y d ({τ }) = Y f Y kink (Π, i ), and a full employmen is allowed in his economy. Noe ha aggregae demand expands wih lower consumpion axes, an inuiive saemen in his credi consrained environmen. Noe in addiion ha if inflaion is above Π kink, hen aggregae demand Y d is a negaive funcion of inflaion, and depends on moneary policy arges. Aggregae demand is a negaive funcion of inflaion because i is a negaive funcion of he real ineres rae 10, and he real ineres rae in his case increases wih inflaion by means of he Taylor Rule as seen above. Oherwise, if inflaion is below Π kink hen aggregae demand becomes a posiive funcion of inflaion, since in his case he gross real ineres rae is he inverse of gross inflaion. More deflaion means higher real ineres raes and lower demand. Furhermore, he lower segmen of aggregae demand does no depend on any moneary policy insrumen. An example: susaining he economy a full-employmen Imagine ha for a given fiscal policy {τ h } and a binding borrowing limi D h he naural rae of ineres r n ({τ h }, D h ) is achievable, such ha r n ({τ h }, D h ) rh n r = r(π, i ), where r is an implici moneary policy real ineres rae arge. Le s furher assume ha rh n = r. If a deleveraging shock occurs and D falls from D h o D l, hen o mainain he economy on a full-employmen seady sae i is sufficien o find a fiscal policy {τ l } so ha he resuling naural rae of ineres is susained a he same level equal o he moneary arge real ineres rae. Then from expression (1.21) and (1.46) a full employmen seady sae equilibrium is susained if: rl n = rh n = r 1 + τ l c 1 + τh c = D l = 1 ατ l l D h 1 ατh l (1.41) 10 Given he same assumpions for he elasiciy of ineremporal subsiuion, excess savings is posiively sloped wih respec o he real ineres rae, and consequenly curren demand is negaively sloped wih respec o he same variable. 20

33 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS Which is equivalen o: r n = 0 τ l = D D h ( ) 1 α τ h l and τ c = D D h (1 + τ c h) (1.42) If τh c = τ h l = 20% and labor share α = 0.7, hen a 5% reducion of D would have o be offse by an increase of he labor ax from 20% o 26%, ogeher wih a reducion of consumpion ax from 20% o 14%. A combinaion of fiscal and moneary policy may aenuae he fiscal effor needed o offse a deleveraging shock in order o susain employmen (see figure 1.2). Bu only fiscal policy may preven he appearance of Figure 1.2: Allowing a Firs Bes Soluion wih Fiscal and Moneary Policy a secular sagnaion seady sae o where he economy can be dragged if fiscal and moneary policy agens are no asserive enough. We will see how nex, when wages are sicky Sicky wages, sable recessions, and labor ax on firms We now inroduce in he model sicky wages and a wage ax on firms τ w. he ax works similarly o he labor ax on households in he way i reduces heir ne income 21

34 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS and savings, having he same kind of impac on he naural rae of ineres. Bu τ w may also play a relevan role o avoid or o leave a secular sagnaion in he presence of wage rigidiies. The firm problem is now given by: Z = max L P Y W L d (1 + τ w ) s.. Y = A ( L d ) α (1.43) We coninue o assume for simpliciy ha A = A = 1. The soluions for w = W P and z = Z P are given by: w = α 1 + τ w Y L and z = (1 α)y (1.44) While he aggregae demand expression given by (1.39) and (1.40) remains unchanged wih he inroducucion of his ax, close-form expressions for he real ineres rae and he Governmen budge rule are given by: ( ) 1 + β (1 + g )D 1 + r = β (1 ατ lw )Y D 1 (1.45) lw G ατ = + τ c Y 1 + τ c (1.46) where τ lw = τ l +τ w 1+τ w which besides acing similarly o τ l is a labor ax index combining τ l and τ w in a single expression, in susaining he naural rae of ineres, can play a relevan role in leaving a Secular Sagnaion by inerfering direcly on he aggregae supply side of his economy, ha we describe nex. A wage rigidiy is inroduced in he model such ha households will no accep working for a wage lower han a nominal wage norm W given by: ( ) 1 γ W = W γ 1 P w flex (1.47) whereas he nominal wage will always be greaer or equal han he flexible labor 22

35 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS full-employmen nominal wage: W = max{ W, P w flex } (1.48) Wih an equivalen expression for aggregae supply given by: ( ) Y = min {Y f, Y f 1 + τ w γ α ( ) } 1 α γ 1 Y 1 Π 1 + τ w Y f (1.49) Noe ha, from he wage equilibrium equaion (1.44), an increase of he labor ax on firms riggers a downward pressure on he nominal wage o susain firms profis. Bu if he full-employmen nominal wage canno be fully achieved because of he nominal rigidiy, hen supply will adjus o a sub-employmen level. The corresponding seady sae expression is given by: Y = Y f min {1, Π γ 1 γ } α 1 α (1.50) which is an increasing funcion of inflaion for negaive inflaion levels, and does no depend on any fiscal insrumen, assuming ha a perpeual consan change of he labor ax on firms, alhough heoreically viable, is unrealisic. Noe ha he seady sae expression for aggregae supply does no depend on any moneary policy arge eiher. And neiher he lower segmen of aggregae demand given by expression (1.40). Figure 1.3 shows a graphical represenaion of a Secular Sagnaion equilibrium, deermined by he inersecion of he lower segmens of aggregae demand and supply. This equilibrium is sable, as described in deail by Eggersson and Mehrora [21], by assuming ha he wage rigidiy is sufficienly high in order o ensure ha he slope of he lower segmen of aggregae supply is smaller han he slope of lower aggregae demand. Figure 1.3 also shows ha moneary policy canno neuralize a sable recession, alhough i can provide he economy wih a full-employmen equi- 23

36 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS Figure 1.3: Secular Sagnaion Equilibrium librium by allowing an achievable negaive naural rae of ineres wih an adequae posiive inflaion arge. In fac, only fiscal policy is able o neuralize a sable recession in his economy. The fiscal mechanism o counerac a Secular Sagnaion equilibrium needs o boos inflaion, since any sable recession is characerized by an equilibrium in deflaion. This inflaion boos can be achieved by fiscally expanding demand, conracing supply, or a combinaion of boh. i) Aggregae Demand expansion o counerac a sable recession Aggregae supply in seady sae does no depend on any fiscal insrumen, hen only a sufficienly asserive permanen demand expansion can clear a recession from he se of available equilibria, hus forcing a permanen ransiion o a sable fullemploymen sae. Graphically i is easy o see ha a demand expansion mus be big enough so ha he lower segmen of aggregae demand does no inersec aggregae supply, which 24

37 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS is he same o say ha he lower segmen of aggregae demand mus achieve a fullemploymen level for an inflaion level smaller han zero. A sufficien condiion is o ensure ha he naural rae of ineres becomes posiive and achievable, corresponding o a negaive inflaion inersecion of he lower aggregae demand segmen wih full-employmen oupu. In addiion, a negaive naural rae of ineres in his economy implies he exisence of a Secular Sagnaion equilibrium 11. The sufficien aggregae demand expansion is achieved by a reducion of he consumpion ax given by 12 : τ c = (1 + τ c ss) [ (Yss ) ( G Π ss + Y f G 1 + β (1+β)(1+g) β (1+β)(1+g) ) 1 ] < 0 (1.51) The sufficiency of his resul presumes ha he resuling naural rae of ineres level r n is achievable from a moneary policy sandpoin, or r r n. ii) Aggregae Supply conracion o counerac a sable recession: Paradox of Toil A negaive naural rae of ineres implies he exisence of a Secular Sagnaion equilibrium in he model. Bu a full-employmen equilibrium may coexis if he naural rae of ineres is achievable by an adequae moneary policy such ha 1 + r < 1 + r n. From Figure(1.4) we can observe ha, in his model a necessary and sufficien condiion for he exisence of a secular sagnaion equilibrium is ha (i) he lower segmen of aggregae demand is seeper han he lower segmen of aggregae supply[21], and ha (ii) he inflaion level corresponding o shor erm aggregae supply deermined a full employmen, Π s,f, is lower han he inflaion level corresponding o aggregae demand deermined a full employmen 13, Π d,f = 1. Π s,f 1+r n is deermined by equaion (1.49), and he exising condiion for a secular 11 as he lower segmen of aggregae demand inersecs full employmen oupu a a posiive inflaion level, which by consrucion of he model ensures an inersecion wih he lower segmen of aggregae supply. 12 where he subscrip ss means Secular Sagnaion seady sae equilibrium, and f means fullemploymen seady sae. is by consrucion equal o he inverse of he gross naural rae of ineres. 13 Π d,f 25

38 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS Figure 1.4: One period aggregae supply conracion shock sagnaion is given by 14 : ( ) ( ) 1 α 1 + τ Π s,f w = Y f α 1 + τ 1 w Y 1 < r n = Π d,f (1.52) I is possible o force a ransiion from a sable recession o a sable full-employmen equilibrium, wih a ransiory conracion of aggregae supply, enough o clear a leas for one period he possibiliy of a Secular Sagnaion equilibrium (Figure 1.4). Using a similar mechanism as he previous one for aggregae demand, i is sufficien o conrac aggregae supply by increasing he labor ax on firms, such ha he lower segmen of aggregae supply, given by expression (1.49), inersecs full-employmen oupu a a gross inflaion level greaer han he inersecion of he lower aggregae demand segmen wih full-employmen oupu, given by a gross inflaion level equal 1 o he inverse of he gross naural rae of ineres. This condiion is direcly 1+r n 14 Noe ha if he labor ax on firms is unchanged from previous period, by consruion in a ( ) 1 α α α secular sagnaion equilibrium, > 1. ) 1 α Y f Y 1 < 1 1+r n 1 1+r n ( Y 1 Y f 26

39 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS derived from expression(1.52): 1 + τ w 1 + τ w r n ( Y 1 Y f ) 1 α α > 1 (1.53) Noe also ha he naural rae of ineres may remain consan if he increase of labor ax on firms 15 is compensaed from a budge perspecive by a reducion of he labor ax on households so ha he consumpion ax remains unchanged. Similarly o Correia e al. [16] a liquidiy rap is resolved wih an inflaion boos, bringing back he economy o a full-employmen firs bes equilibrium from a sable recession, bu wih a reverse usage of disorionary axes in his credi consrained closed economy environmen. iii) A fiscal rule o preven a secular sagnaion Noe ha he previous condiion (1.53) can be used as a fiscal rule o preven an economy o fall ino a persisen recession, when a full-employmen equilibrium and a secular sagnaion recessions are, boh, achievable seady sae equilibria. When moneary policy is ineffecive his fiscal rule would work like a Taylor rule, bu using fiscal, insead of moneary insrumens. In our economy, a secular sagnaion equilibrium given by Y < Y f, is driven by a sable deflaion seady sae where, from expression (1.49), oupu a ime is expressed by: α Y = Y f Π γ 1 α < Y f if Π < 1 (1.54) Combining he previous expression wih he condiion given by equaion (1.53), our 15 Noe also ha he exisence of a secular sagnaion equilibrium implies ha he lef-hand side of he equaion is greaer han one. 27

40 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS fiscal rule can hen be expressed by: Π 1 τ w τ w 1 (1.55) Π < 1 τ w Π γ = (1 + τ 1) w 1 + r n 1 > τ w 1 (1.56) This means ha he governmen commis o increase he labor ax on firms if inflaion becomes negaive, in order o ensure ha real wages do no increase above he flexible wage level, hus susaining full employmen, all in all assuming ha he naural rae of ineres is known and achievable. iv) Welfare implicaions The wo main mechanisms o move our economy ou of a slump may be welfare improving for all agens. An aggregae demand expansion via a consumpion ax reducion may clear he secular sagnaion equilibrium, alogeher no requiring an increase of labor axes from a Governmen budge equilibrium sandpoin. From he Governmen budge consrain given by equaion(1.27), a sufficien decrease in consumpion axes can increase aggregae demand o a full-employmen level wihou he need o reduce labor axes. If he naural rae of ineres in his economy is achievable, hen he hree ypes of agens would be beer off. The same resul may be obained using he fiscal rule, if he increase of he labor ax on firms is combined wih a reducion of he labor ax on households, in order o susain he labor ax index τ lw, as well as he naural rae of ineres which we assume achievable, a he same level Neverheless, if he naural rae of ineres mus be increased o be achievable (or susained a an achievable level during a credi shock), hen he combinaion of increasing labor axes wih decreasing consumpion axes is welfare improving for he old ha benefi from he consumpion ax reducion. The welfare implicaions for he young and middle age should be furher inspeced. 28

41 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS Reurn on Capial Income ax Inroducing capial and ax on capial in he model does no qualiaively change he fiscal policy opions o counerac a persisen recession we have derived unil ha poin. Bu by having an impac similar o he consumpion ax in expanding aggregae demand, reducing he ax on capial can be an effecive alernaive o a reducion of consumpion axes when furher promoing a consumpion expansion is no a desirable oucome. Alhough reducing he ax on capial has also an expanding impac on aggregae supply, he impac on demand prevails, so ha he ne effec is an increase of he naural rae of ineres, and of equilibrium inflaion. We use a version wih capial of he previous model[21] wih a ax on capial, ogeher wih all he oher axes already presened in his chaper. We inroduce disorionary axes including he ax on capial income in he budge consrains of he middle age and old, which are now given by: (1 + τ c +1)C m +1 = z +1 + w +1 L +1 (1 τ l +1) + K +1 [r k +1(1 τ k +1) 1] (1 + r )B y + Bm +1 (1.57) (1 + τ c +2)C o +2 = (1 + r +1 )B m +1 + K +1 (1 δ) (1.58) and he labor ax on firms is sill considered in he firm problem, now given by: Z = max L,K {P Y W L (1 + τ w ) P r k K } s.. Y = A L α K 1 α (1.59) where, w = W P = αlα 1 1+τ w = α Y 1+τ w L and r k = (1 α)a L α K α = (1 α) Y K. From he reurn on capial expression we can direcly observe ha a reducion of he ax on capial income reduces he cos of capial: r k = 1 1 τ k ( 1 1 δ ) 1 + r (1.60) Aggregae demand expands when if he ax on capial income decreases. This can be 29

42 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS direcly observed from he following expression of aggregae demand in real erms: Y d = 1 ατ lw 1 (1 α)b l [ ] 1 + β (1 + g )D D 1 β 1 + r (1.61) Where B l = 1 1+β 1 δ = (1 τ k r k β r +δ ) 1+β 1 δ. β r +δ Regarding aggregae supply, is expressions is equal o he one derived previously and given by expression (1.49). Bu now full employmen oupu is given by: Y f ( ) 1 α = A Lα K 1 α = LA 1 α 1 α α r k = [ ] 1 α 1 α (1 α)(1 τ α LA k α ) 1 1 δ 1+r (1.62) The difference lies on he aggregae supply expression for posiive inflaion levels which is no consan, and expands when τ k decreases, leading also o an aggregae supply expansion when inflaion is negaive. Alhough a reducion of he ax on capial income has an expanding impac on boh aggregae supply and demand, he impac on demand prevails. The resuling impac on inflaion, employmen and he naural rae of ineres are qualiaively similar o he ones derived for he consumpion ax, being an available adequae alernaive o his insrumen in couneracing a persisen recession. 1.4 Final remarks In his Chaper we formalized he role of disorionary axaion in avoiding a sable recession characerized by he Secular Sagnaion framework proposed by Eggersson and Mehrora [21], based on a hree generaions OLG model wih borrowing consrains. We compare our resuls wih he ones obained by Correia e al. [16] ha propose a soluion based on he same se of disorionary axes in a sandard single agen New Keynesian model wihou borrowing consrains, o counerac a liquidiy rap ha is by consrucion emporary. We find reversed resuls. Our 30

43 CHAPTER 1. DISTORTIONARY TAXATION IN STABLE RECESSIONS mechanism is based on a wealh re-disribuive policy using disorionary axes o increase he naural rae of ineres so ha i becomes achievable given he moneary policy arges, by increasing labor axes on he middle age employed and redisribuing he ax proceeds o he populaion in general by reducing consumpion ax. Insead, Correia e al. [16] emulae inflaion in consumer prices using an increasing pah of consumpion axes, so ha he ineremporal condiion allows an achievable negaive naural rae of ineres, and he liquidiy rap is neuralized. We use he same fiscal oolbox wih differen approaches. We increase he naural rae of ineres o a level consisen wih moneary policy effeciveness, insead of emporarily allowing a firs bes soluion compaible wih a negaive naural rae of ineres. As an alernaive o he sandard fiscal policy prescripions o counerac an economic downurn, based on Keynesian increases of public expendiures, more public deb and ax cus o simulae demand 17, and complemening he papers referenced above, he main purpose of our analysis is o show how fiscal policy based on disorionary axaion can be effecive in avoiding persisen recessions, in paricular when increasing public expendiures and deb are no policy opions available. 17 Besides oher non-fiscal approaches, namely he one proposed by Eggersson and Woodford [23] where he cenral bank commis o keep ineres raes a a lower level even afer a recession resuling from a liquidiy rap is over. 31

44 Chaper 2 Age Milesones and Low Ineres Raes, an Analyic approach Absrac Major age milesones like he age of firs job, reiremen age, or life expecancy, bounding relevan economic periods in a persons life, have been changing subsanially during he las decades. In parallel real ineres raes have been significanly declining in relevan world economies, reaching sable negaive levels in some cases. We propose an analyic approach o relae hose wo phenomena by using an overlapping muli-generaions model o find expressions for real ineres rae elasiciies o age parameers. The model formalizes he mechanisms supporing he relaion beween ineres raes and age, sheds ligh on he relaive imporance of each age milesone in explaining changes of real ineres raes, and how oher facors like elasiciy of iner-emporal subsiuion, populaion and produciviy growh, iner-generaional alruism, as well as a social securiy sysem, may miigae or amplify hose changes. 32

45 CHAPTER 2. AGE MILESTONES AND LOW INTEREST RATES 2.1 Inroducion During he las decades, he age srucure of he populaion in some of World s mos relevan economies has changed significanly. For example, alhough Life expecancy a birh increased by approximaely en years since he 70 s boh in US and EU, reiremen age has declined four and six years respecively, conribuing o raise he need o save in hose economies (Figure 3.1). Furhermore, he recen economic crisis ended o affec he average age of firs job as firms end o pospone hiring as a way o adjus down employmen level, which could lead o an increase of he borrowing needs of his populaion segmen. Changes in age milesones deermine many aspecs of relevan economic periods of a persons life, which hemselves may direcly impac real ineres raes hrough changes of borrowing and savings pahs. For example, for a higher effecive reiremen age, people need o save less for heir expeced reiremen period, leading o a conracion of savings and a consequen increase in equilibrium real ineres raes. In addiion, posponing he age of firs job increases he duraion of borrowing afer adulhood, pushing ineres raes upwards oo. Increasing boh parameers, age of reiremen and firs job, a he same ime and by he same amoun, alhough no changing he duraion of he working period, may impac he real ineres rae by affecing borrowing and saving pahs, and consequenly loan marke equilibrium and real ineres raes. Alhough he impac of age srucure in relevan World economies has been a recurren opic covered in recen lieraure, in paricular o explain he persisen decline of ineres raes, economic sagnaion and liquidiy raps, here has no ye been an aemp, o he bes of our knowledge, o formally derive he analyic relaions of real ineres raes wih respec age milesones. The general omission of changing demographic parameers in mos curren formal economic models ignores a poenially relevan facor influencing equilibrium condiions, and consequenly he ype 33

46 CHAPTER 2. AGE MILESTONES AND LOW INTEREST RATES Figure 2.1: Life Expecancy and effecive Reiremen Age: EU and US and even sign of soluions. For example, an increase of life expecancy can drag he full-employmen equilibrium real ineres rae from posiive o negaive. Since a negaive level may no be achievable when he nominal ineres rae zero lower bound is binding, a firs bes soluion may no more be available in such a model. The same can happen wih operaive beques moives, which may become inoperaive, for example if he reiremen age decreases, or life expecancy increases. The purpose of his paper is o fill ou his gap. By merging an age srucure framework wih an OLG model, we derive racable algebraic real ineres rae elasiciy expressions wih respec o each age parameer, o shed ligh on he demographic formal mechanisms ha influence real ineres raes, and inspec in paricular he examples menioned above. Moreover we provide a sraighforward alernaive o heavy compuaional quaniaive models, in order o illusrae he impac of demographic facors on general economic phenomena. Ikeda and Saio [29] sudy he effecs of demographic changes on he real ineres rae in Japan by capuring demographic dynamics by exogenous changes of he raio 34

47 CHAPTER 2. AGE MILESTONES AND LOW INTEREST RATES of workers o oal populaion. Bu mos of he lieraure covering he presen opic use perpeual youh ype models inspired by Blanchard and Fischer [7], using ransiion probabiliies beween age groups. This approach, ha faciliaes aggregaion of individual agens, hus ensuring analyically more racable life-cycle models, was adoped, for example, by Carvalho and Ferrero [11] o explain Japan s persisen deflaion, using ransiion probabiliies from worker o reired, and from reired o deah, and by Carvalho e al. [12] o inspec he mechanisms of how demographics affec real ineres raes. Similarly, Aksoy e al. [2] relae macroeconomic rends o demographic srucure wih a model o which hey add an addiional ransiion probabiliy from young o worker, afer conducing an empirical sudy where hey found evidence ha differences in generaion weigh across counries explain differences among main macro-economic variables. Neverheless, ransiion probabiliies in hose models end o be independen of age, and of ime since ransiion from previous age groups, which makes hem less appropriae o derive analyic relaions beween ineres rae and explici age milesones. This circumsance was recenly overcome by Eggersson and Robbins [22] who used a quaniaive overlapping muli-generaions model inspired by he work of Auerbach and Kolikoff [3] o invesigae he decline of real ineres raes in US. Similarly, we use an overlapping muli-generaions model, where mos relevan age milesones are exogenous parameers, allowing o analyically express he real ineres rae in erms of age srucure changes, surprisingly no affecing algebraic racabiliy, in order o shed ligh on relevan demographic mechanisms ha are dragging down real ineres raes. In wha follows, we begin by oulining an overlapping generaions deerminisic model in he conex of an endowmen economy, where agens are economically acive afer childhood unil heir age of life expecancy. The number of generaions of he model depends already on hose wo age milesones. A he age of adulhood 35

48 CHAPTER 2. AGE MILESTONES AND LOW INTEREST RATES agens sar borrowing o consume. From he age of firs job unil reiremen hey receive an income in he form of an endowmen, wih which hey pay back heir deb, consume, and save for reiremen 1. During ha working period, a a cerain momen in ime agens have payed back heir debs and sar saving for reiremen. Unil ha momen agens are borrowers, and afer hey become savers. The iniial savings age is an endogenous variable of he model. During reiremen hey use heir accumulaed savings o consume. We derive he equilibrium condiions and aggregae expressions for he main variables of he model, in paricular of excess borrowing, in erms of he real ineres rae and age milesones, which becomes zero for loan marke equilibrium. In he hird secion we use he excess borrowing expression a loan marke equilibrium in seady sae o formalize he analyic relaion beween he naural rae of ineres 2 and age srucure. We formalize he derivaives of real ineres rae wih respec o each age parameer, using he parial derivaives of excess borrowing wih respec o age milesones, and o he real ineres rae. We find ha excess borrowing decreases wih increasing ineres raes if he elasiciy of iner-emporal subsiuion is above a cerain accepable hreshold level ha depends on he relaive duraion of reiremen. We use his assumpion hroughou he paper, so ha he consisen negaive slope of excess borrowing wih respec o he real ineres rae allows he sign of age milesones elasiciies o be deermined by he signs of he parial derivaive of excess borrowing wih respec o each age parameer. In he fourh secion we inroduce inergeneraion ransfers in he form of bequess o children, of gifs o parens, and of a pay-as-you-go social securiy sysem, as hose conceps are closely relaed o agens age srucure, in paricular o he age when 1 In our model he age of adulhood and age of firs job may be differen. Afer adulhood and before he age of firs job agens have o borrow in order o consume. In he special case where hose wo age milesones are se he same, he algebraic expressions are simplified, and he calibraed oupus are no maerially differen. An alernaive no used in our model o dae, would be o consider endogenous ransfers from parens o children during ha specific phase of heir lives. 2 The naural rae of ineres is defined as he full-employmen equilibrium real ineres rae. 36

49 CHAPTER 2. AGE MILESTONES AND LOW INTEREST RATES heir children are born. We also analyze how inergeneraion ransfers parameers affec he elasiciies of real ineres raes wih respec o age milesones. Finally, in secion five we calibrae a model wih endogenous oupu and capial o quanify he analyic resuls of previous secions. We also es he impac of changing capial depreciaion on real ineres rae elasiciies wih respec o age milesones, as well as he impac of changes in age srucure on he capial-oupu raio. As we have already noed, his chaper focuses on he presenaion of a framework ha allows o derive formal algebraic relaions beween real ineres raes and age milesones, in order o inspec he influence of demographic facors in specific economic mechanisms. In paricular, we use our framework o explore how changing age srucure can swich an alruisic moive from helping children o supporing parens, or how an increase of life-expecancy, a reducion of reiremen age, and posponing of he age of firs job explain he decline of real ineres raes, furher quanifying hose phenomena. Alhough we keep our resuls focus on he demand side of OLG models, our framework can also be used, for example, o explore secular sagnaion mechanisms driven by demographic facors. In paricular, in curren work in progress, we exend our framework wih nominal prices, endogenous oupu, and nominal wage rigidiies, where when he naural rae of ineres becomes negaive, a second bes soluion wih a sub-opimal sable equilibrium oupu level is characerized by an endogenous persisen increase of he age of firs job. 2.2 An Endowmen Economy wih age milesones In his secion we describe and solve a muli-generaions OLG model where age milesones binding relevan economic periods of households, can exogenously change. We also derive some algebraic ools ha simplify he model soluion in closed-form expressions, and wih which he derivaives of he seady sae equilibrium real ineres rae wih respec o age milesones can be algebraically explicily derived. 37

50 CHAPTER 2. AGE MILESTONES AND LOW INTEREST RATES Consider an overlapping generaions model in he spiri of Eggersson and Mehrora [21] where new generaions sar every year. Imagine ha households live L d L years (where d L sands for duraion of life), bu are considered economically acive in he model only afer childhood, from he age of adulhood b l (b l sanding for lower borrowing age) unil he las year of heir lives a age d L. The number of overlapping generaions of he model T = d L b l + 1 is hen deermined by wo age milesones, bounding he period ha sars a he age of adulhood, and ending a he las year of heir lives. We sar by considering an endowmen economy where agens have no capial o inves in, bu where households can lend o one anoher. Afer childhood, a age b l households borrow from oher households o consume. During he middle-age period m l (m l sanding for lower middle age) hey receive an income in he form of endowmen y i=age which hey use o consume, o pay-back heir debs, and o save for reiremen by lending o oher households. In order o smooh heir life-ime consumpion pah, during he firs par of heir middle age period households are borrowers, becoming savers hereafer unil he end of heir lives. The iniial saving age s l [m l, o l [ is an endogenous parameer of he model. Households are reired from age o l o d L, having no endowmen and consuming wih he proceeds from heir savings during ha period. The model age srucure is illusraed in Figure 2.2. Age milesones in red are he boundaries of life economic periods wih duraions in green. We can look o an household from an income perspecive, saring his journey as a young borrower wihou income who needs o borrow from oher agens o be able o consume. The young borrower s period has a duraion in years of d b = m l b l. He hen eners ino middle age, wih a duraion in years of d m = o l ml, afer finding his firs job a age m l, and ges an income in he form of endowmen unil reiremen a age o l. Thereafer he will be reired for d o = T d m d b = L m h years. Alernaively we can look o an household from a borrowing/saving perspecive, which may faciliae he economic inuiion: in he beginning of heir journey hey are ne borrowers 38

51 CHAPTER 2. AGE MILESTONES AND LOW INTEREST RATES Figure 2.2: Relevan Life-cycle Periods and Age Milesones during d b years unil hey pay back heir loans, and become savers a he age s l for d s = d L s l + 1 years. I is he relaive weigh of borrowers and savers, or he ineracion beween loan demand and supply, ha will deermine loan marke equilibrium ineres rae level. Noe ha he iniial saving age s l, which deermines he relaive weigh of borrowers and savers, is an endogenous parameer of he model. s l iself depends on he relaive duraion of young borrowers, middle age and reiremen periods, wih duraions respecively given by b, m, and o respecively. In wha follows we will use duraions noaion b, m and model life span T = b + m + o o express mos of our findings, where b = m l b l = b h b l + 1, m = m h m l + 1, and o is reiremen duraion, here a dependen variable. Consider hen a represenaive household reaching adulhood a ime, wih he following uiliy funcion: max c bl +i +i E T 1 i=0 β i U(c bl +i +i ) (2.1) Where he U(c) is assumed o be a consan elasiciy of iner-emporal subsiuion 39