A Dynamic Theory of Public Spending, Taxation and Debt

Transcription

1 This revisio: March 26 A Dyamic Theory of Public Spedig, Taxatio ad Debt Abstract This paper presets a dyamic political ecoomy theory of public spedig, taxatio ad debt. Policy choices are made by a legislature cosistig of represetatives elected by geographically-defied districts. The legislature ca raise reveues via a distortioary icome tax ad by borrowig. These reveues cabeusedtofiace a atioal public good ad district-specific trasfers (iterpreted as pork-barrel spedig). The value of the public good is stochastic, reflectig shocks such as wars or atural disasters. I equilibrium, policy-makig cycles betwee two distict regimes: busiess-as-usual i which legislators bargai over the allocatio of pork, ad resposible-policy-makig i which policies maximize the collective good. Trasitios betwee the two regimes are brought about by shocks i the value of the public good. I the log ru, equilibrium tax rates are too high ad too volatile, public good provisio is too low ad debt levels are too high. I some eviromets, a balaced budget requiremet ca improve citize welfare. Marco Battaglii Departmet of Ecoomics Priceto Uiversity Priceto NJ 8544 mbattagl@priceto.edu Stephe Coate Departmet of Ecoomics Corell Uiversity Ithaca NY 4853 sc63@corell.edu We thak Rolad Beabou, Faruk Gul, Per Krusell ad semiar participats at Berkeley, Cal Tech, Corell, Northwester ad Priceto for helpful commets. We also thak Jo Eguia for very detailed commets o a previous draft. Marco Battaglii gratefully ackowledges fiacial support from a NSF CAREER Award.

2 Itroductio This paper presets a dyamic political ecoomy theory of public spedig, taxatio ad debt. The theory is desiged to shed light o fiscal policy i political systems i which legislators have primary loyalty to the districts they represet as opposed to a atioal political party. The theory yields positive predictios cocerig the dyamic evolutio of public debt, taxatio, ad the allocatio of public reveues betwee atioal public goods ad pork-barrel spedig. It also provides predictios cocerig the size of the coalitios that pass legislatio. Furthermore, the theory delivers isights ito the ormative performace of political decisio-makig ad the case for fiscal restraits i the form of balaced budget requiremets. The theory cosiders a political jurisdictio i which policy choices are made by a legislature comprised of represetatives elected by sigle-member, geographically-defied districts. The legislature ca raise reveues i two ways: via a proportioal tax o labor icome ad by borrowig i the capital market. Borrowig takes the form of issuig oe period bods. The legislature ca also purchase bods ad use the iterest earigs to help fiace future public spedig if it so chooses. Public reveues are used to fiace the provisio of a public good that beefits all citizes ad to provide targeted district-specific trasfers, which are iterpreted as pork-barrel spedig. The value of the public good to citizes is stochastic, reflectigshockssuchaswarsor atural disasters. The legislature makes policy decisios by majority (or super-majority) rule ad legislative policy-makig i each period is modelled usig the legislative bargaiig approach of Baro ad Ferejoh (989). The level of public debt acts as a state variable, creatig a dyamic likage across policy-makig periods. There exists a uique political equilibrium ad the equilibrium distributio of public debt coverges to a uique ivariat distributio. There are two regimes of govermet policy-makig: busiess-as-usual (BAU) i which legislators bargai over the allocatio of pork ad resposiblepolicy-makig (RPM) i which legislators choose to forsake their parochial iterests for the atioal good. I the BAU regime, the level of public debt ad the tax rate are state idepedet. Public good spedig is resposive to chages i the value of the public good, but these spedig chages are fiaced etirely by adjustmets i pork-barrel spedig. Legislatio is passed by miimum wiig coalitios. I the RPM regime, legislators allocate all reveues to providig the public good ad servicig the debt. No pork is provided ad legislatio is passed uaimously. Chages

3 i the value of the public good lead to chages i taxes ad debt as well as public good spedig. The prevailig regime is determied by both the curret stock of public debt ad the value of the public good. Specifically, there is a cut-off value of the public good that is decreasig i the stock of debt. Below this cut-off the legislature is i the BAU regime, while above it RPM prevails. The structure of the equilibrium reflects the fact that reveues are costly to raise sice they must ultimately be fiaced by distortioary icome taxatio. Whe the value of the public good ad/or the stock of debt to be repaid is high, the opportuity cost of allocatig reveues to pork-barrel spedig is large ad hece legislators refrai from such spedig. Trasitios betwee the two regimes are brought about by shocks i the value of the public good. Periods of BAU are brought to a ed by a high realizatio of the value of the public good. This triggers a icrease i public debt ad taxes to fiace higher public good spedig as well as a cessatio of pork-barrel spedig. Oce i the RPM regime, further high realizatios of the value of the public good result i additioal icreases i debt ad taxes. The ecoomy returs to BAU oly after a suitable sequece of low realizatios of the value of the public good. The larger the amout of public debt that has bee built up, the greater the expected time before returig to BAU. I this way, the ecoomy cycles through periods of BAU ad periods of RPM. Both policy-makig regimes are persistet i the sese that the probability of remaiig i them is greater tha the probability of trasitioig from them. Whe the level of public debt chose i the BAU regime is positive, the ecoomy is i perpetual deficit, with the extet of the deficit spikig up after a sequece of high values of the public good. However, legislators do ot ecessarily borrow i the BAU regime. I some eviromets, they purchase bods with the aim of fiacig future public good spedig with the iterest earigs. I such eviromets, the govermet will ru budget surpluses i the BAU regime ad deficits will arise oly after a suitably log sequece of high public good values. The key feature of the eviromet determiig whether the legislature borrows or saves i the BAU regime is the size of the tax base relative to the ecoomy s desired public good spedig. Paradoxically, it is ecoomies with relatively large tax bases that experiece perpetual deficits. With respect to citize welfare, the equilibrium policy choices geerate a strictly lower level of utility tha those that would be made by a beevolet plaer. The plaig solutio ivolves the govermet gradually accumulatig a stock of bod holdigs sufficiet to allow it to fiace first best public good provisio i all states, without icome taxatio (Aiyagari et al (22)). By 2

4 cotrast, i equilibrium, the level of public debt ever coverges to a determiistic steady state ad is bouded below by the level of debt that legislators choose i the BAU regime. Eve whe this is egative, so that legislators acquire bods i the BAU regime, these bod holdigs are isufficiet to fiace first best public good provisio i all states. Thus, i equilibrium, taxes are too high ad public good provisio too low i the log ru. Moreover, taxes are too volatile. The theory also has implicatios for the desirability of balaced budget requiremets. We study a fiscal restrait that requires the legislature to esure that tax reveues equal public spedig i every period. We suppose the govermet iitially has o debt, so that uder the restrait spedig is just o public goods ad trasfers. We ask whe will citizes welfare be ehaced by the costrait that public spedig be fiaced solely by tax reveues? The key determiat of the desirability of a balaced budget requiremet is agai the size of the tax base relative to the ecoomy s desired public good spedig. Whe the tax base is relatively large, a balaced budget requiremet will ehace citize welfare, but whe it is relatively small, the opposite coclusio applies. The orgaizatio of the remaider of the paper is as follows. I the ext sectio we discuss related literature. Sectio three presets the model. Sectio four characterizes the political equilibrium ad develops the positive predictios of the theory. Sectio five studies the efficiecy of political equilibrium ad sectio six studies the desirability of a balaced budget requiremet. Sectio seve offers a brief coclusio. The Appedix cotais the proofs of the propositios. 2 Related literature Our theory builds o the well-kow tax smoothig theory of fiscal policy stemmig from Barro (979). Accordig to this view, the govermet should use budget surpluses ad deficits as a buffer to prevet tax rates from chagig too sharply. Thus, the govermet should ru a deficit i times of high govermet spedig eeds ad a surplus whe eeds are low. Uderlyig this theory are the assumptios that govermet spedig eeds fluctuateovertimeadthatthe deadweight costs of icome taxes are a covex fuctio of the tax rate. I a importat paper, Aiyagari et al (22) poit out that the tax smoothig logic does ot ecessarily imply the couter-cyclical theory of deficits ad surpluses that it had bee presumed to. I the absece of ad hoc limits o govermet bod holdigs, they prove that i some 3

5 eviromets the optimal policy is for the govermet to gradually acquire sufficiet bod holdigs so as to evetually be able to fiace ay level of spedig with the iterest earigs from these holdigs. This permits the fiacig of govermet spedig without distortioary taxatio. Iterest earigs i excess of spedig eeds are rebated back to citizes via lump-sum trasfers. The ecoomic eviromet uderlyig our theory is similar to that i Aiyagari et al (22). The oly differeces are (i) that we specify a stochastic value of public goods as opposed to a stochastic govermet spedig level ad (ii) that we iclude district-specific trasfers i the policy space. Our mai departure from the tax smoothig literature is that policy decisios are made by a legislature of elected represetatives rather tha a beevolet plaer. This iovatio produces a theory of fiscal policy cosistet with the origial ituitios from the literature, but without ad hoc limits o govermet bod holdigs. Thus, while the optimal policy i our eviromet ivolves the govermet gradually acquirig sufficiet bod holdigs to fiace all spedig eeds with iterest earigs, i political equilibrium the level of public debt fluctuates i accordace with the value of the public good ad serves to smooth icome taxes. Our theory also relates to the political ecoomy of deficits literature. 2 A key theme of this literature is that deficits ca arise because of redistributive ucertaity (Alesia ad Tabellii (99), Lizzeri (999)). 3 Such ucertaity arises whe citizes do ot kow whether they will beefit from redistributive trasfers i the future. Whe faced with such ucertaity, citizes will favor the trasfer of resources from the future to the preset if they are certai that these resources will be used to their beefit. This ca result i deficits. I our model, legislators face ucertaity as to whether i the ext period they will be i the BAU or RPM regimes. I additio, coditioal o beig i the BAU regime, they face ucertaity as to whether they will be icluded i the miimum wiig coalitio of districts that receive pork. This redistributive ucertaity meas that if those legislators who are curretly i the miimum wiig coalitio could simply Followig Lucas ad Stokey (983), Aiyagari et al (22) cosider a ifiite-horizo geeral equilibrium model with o capital, a liear tax o labor icome, ad stochastic govermet expeditures. Their model departs from Lucas ad Stokey i assumig that the govermet caot issue state-cotiget debt. 2 For more o this literature see Alesia (2), Alesia ad Perotti (995), Persso ad Svesso (989), ad Persso ad Tabellii (2). 3 More geerally, such redistributive ucertaity ca explai may iefficiet decisios i dyamic political ecoomy models. For example, public ivestmets that are potetially Pareto improvig may ot be udertake because those curretly holdig political power are ucertai as to whether those holdig political power i the future will share the fruits of the ivestmet (Besley ad Coate (998)). For further discussios of political failure i dyamic models see Acemoglu (23), Besley ad Coate (998), Coate ad Morris (999) ad Hassler et al (23). 4

6 trasfer a dollar costlessly from the future to the preset period, they would wat to do so. This i tur explais why it is the case that the equilibrium level of public debt (eve whe it is egative) is always above the efficiet level. The cotributio of our paper relative to the political ecoomy of deficits literature is that we imbed a sophisticated model of political decisio-makig ito a dyamic geeral equilibrium model that icorporates the key assumptios of the tax smoothig literature. Thus, our uderlyig ecoomic model icorporates a ifiite horizo, stochastic public good prefereces, distortioary icome taxatio, district-specific trasfers, ad public debt. This allows us to itegrate the political ecoomy ad tax smoothig literatures by developig a theory of fiscal policy with a rich set of ecoomic ad political predictios. Moreover, the theory permits a welfare aalysis of both the efficiecy properties of equilibrium ad the case for fiscal restraits. 4 Fially, our paper relates to the literature o the efficiecy of legislative policy-makig i political systems i which legislators have geographically-defied costituecies. I a well-kow paper, Weigast, Shepsle ad Johse (98) argue that pork-barrel spedig will lead to a govermet that is too large. They do ot model the process of passig legislatio, assumig istead that legislative policy-makig is govered by a orm of uiversalism. Uder this orm, each legislator uilaterally decides o the level of spedig he would like o projects i his ow district ad the aggregate level of taxatio is determied by the eed to balace the budget. Policy-makig the becomes a pure commo pool problem. A umber of authors have argued that this commo pool logic may also explai budget deficits - see, for example, Ima (99) ad vo Hage ad Harde (995). Velasco (2) formally models the accumulatio of public debt as a dyamic commo pool problem. While there is o social role for debt i his model, he demostrates the existece of a equilibrium i which deficits ad debt accumulatio cotiue uabated util the govermet s debt ceilig is reached. A umber of papers study the efficiecy of legislative policy-makig usig the legislative bar- 4 Lizzeri (999) cosiders a two period model with oe good ad Dowsia political competitio i each period. Social welfare is uaffected by the allocatio of resources across the periods. Alesia ad Tabellii (99) study the steady states of a ifiite horizo model i which i each period two political parties hold office with exogeous probability. There are two goods that may be publicly-provided, but each party s costituecy values oly oe. Accordigly, the goods ca be thought of as trasfers targeted to the two parties costituecies. Taxes are distortioary. I each period, the wiig party chooses taxes, debt ad how much to sped o the publicly provided good that its costituecy cares about. Our model geeralizes this set-up i two key ways. First, we have political decisio makers with distict costituecies who must collectively choose policy i each period via majority rule. Secod, we have a atioal public good with stochastic value as well as targeted trasfers. The latter assumptio creates a tax smoothig role for debt. 5

7 gaiig approach employed i this paper. Baro (99) shows that legislators may propose projects whose aggregate beefits are less tha their costs, whe these beefits ca be targeted to particular districts. Related models are elaborated by Persso ad Tabellii (2) ad Auste-Smith ad Baks (25). LeBlac, Syder ad Tripathi (2) argue that legislatures will uder-ivest i public goods. They make their argumet i the cotext of a fiitehorizomodeliwhichieach period a legislature allocates a fixed amout of reveue betwee targeted trasfers ad a public ivestmet that serves to icrease the amout of reveue available i the ext period. I a paper that lays some of the aalytical groud work for the theory preseted i this paper, Battaglii ad Coate (25) develop a ifiite horizo model of legislative policy-makig i which the legislature ca raise reveues via a distortioary icome tax ad these reveues ca be used to fiace ivestmet i a atioal public good ad pork-barrel spedig. They explore the dyamics of legislative policy choices, focusig o the efficiecy of the steady state level of taxatio ad the allocatio of tax reveues betwee pork ad ivestmet. They obtai coditios uder which the equilibrium size of govermet is too large ad the level of public goods too low. However, they also show that there are coditios uder which legislative decisios are efficiet ad/or govermet is too small. I cotrast to the preset paper, there is o public debt, ad it is ivestmet i the public good that creates the dyamic likage across policy-makig periods. Moreover, the value of the public good is determiistic. 3 The model A cotiuum of ifiitely-lived citizes live i idetical districts idexed by i =,...,. The size of the populatio i each district is ormalized to be oe. There are three goods - a public good g; cosumptio z; ad labor l. The cosumptio good is produced from labor accordig to the techology z = wl ad the public good ca be produced from the cosumptio good accordig to the techology g = z/p. Each citize s per period utility fuctio is z + Ag α l(+/ε) ε+,where α (, ) ad ε >. The parameter A measurestherelativeimportaceofthepublicgoodto the citizes. Citizes discout future per period utilities at rate δ. The assumptios o techology imply that the competitive equilibrium price of the public good is p ad the wage rate is w. Moreover, the quasi-liear utility specificatio implies that the iterest rate is ρ =/δ. At this iterest rate, citizes will be idifferet as to their allocatio of cosumptio across time ad hece their welfare will equal that which they would obtai if 6

8 they simply cosumed their et earigs each period. At wage rate w, each citize will work a amout l (w) =(εw) ε i each period, so that ε is the elasticity of labor supply. The associated per period idirect utility fuctio is give by u(w, g; A) = εε w ε+ ε + + Agα. () The value of the public good varies across periods i a radom way, reflectigshockstothe society such as wars ad atural disasters. Specifically,i each period,a is the realizatio of a radom variable with rage [A, A] (where < A < A) ad cumulative distributio fuctio G(A). The fuctio G is cotiuously differetiable ad its associated desity is bouded uiformly below by some positive costat ξ >, so that for ay pair of realizatios such that A<A,the differece G(A ) G(A) isatleastasbigasξ(a A). Thus, G assigs positive probability to all odegeerate sub-itervals of [A, A]. Public decisios are made by a legislature cosistig of represetatives from each of the districts. Oe citize from each district is selected to be that district s represetative. Sice all citizes are the same, the idetity of the represetative is immaterial ad hece the selectio process ca be igored. The legislature meets at the begiig of each period. These meetigs take oly a isigificat amout of time, ad represetatives udertake private sector work i the rest of the period just like everybody else. The affirmative votes of q represetatives are required to eact ay legislatio. The legislature ca raise reveues i two ways: via a proportioal tax o labor icome ad via borrowig i the capital market. Borrowig takes the form of issuig oe period bods with iterest rate ρ. 5 Thus, if the govermet borrows a amout b i period t, it must repay b( + ρ) iperiodt +. Publicreveuescabeusedtofiace the provisio of public goods but ca also be diverted to fiace targeted district-specific trasfers, which are iterpreted as (o-distortioary) pork-barrel spedig. 6 so that b ca be egative. The legislature ca also hold bods if it so chooses, To describe how legislative decisio-makig works, suppose the legislature is meetig at the begiig of a period i which the curret level of public debt is b ad the value of the public 5 Thus we do ot cosider state-cotiget debt as i Lucas ad Stokey (983). We feel that this is the appropriate assumptio for a positive aalysis. 6 The district-specific trasfers could be either direct grats to particular localities or earmarks for specific public projects that the districts would udertake ayway. I the latter case, the earmarks would be o-distortioary ad equivalet to a direct trasfer. 7

9 good is A. Oe of the legislators is radomly selected to make the first policy proposal, with each represetative havig a equal chace of beig recogized. A proposal is described by a +3-tuple {r, g, x, s,..., s },wherer istheicometaxrate;g is the amout of the public good provided; x is the proposed ew level of public debt; ad s i is the proposed trasfer to district i s residets. The reveues raised uder the proposal are x + R(r) where R(r) =rwl (w( r)) = rw(εw( r)) ε, (2) deotes the tax reveue fuctio. The proposal must satisfy the budget costrait that reveues must be sufficiet to cover expeditures. Lettig B(r, g, x; b) =x + R(r) pg ( + ρ)b (3) deote the differece betwee reveues ad spedig o public goods ad debt repaymet, this requires that B(r, g, x; b) X s i. The set of costraits is completed by the o-egativity i costraits that s i for each district i (which rules out fiacig public spedig via districtspecific lump sum taxes). If the first proposer s pla is accepted by q legislators, the it is implemeted ad the legislature adjours util the begiig of the ext period. At that time, the legislature meets agai with the differece beig that the iitial level of public debt is x ad there is a ew realizatio of the value of public goods. If, o the other had, the first proposal is ot accepted, aother legislator is chose to make a proposal. There are T 2 such proposal rouds, each of which takes a egligible amout of time. If the process cotiues util proposal roud T,adtheproposalmadeatthat stage is rejected, the a legislator is appoited to choose a default policy. The key restrictio o thechoiceofadefaultpolicyisthatitmustivolveauiformdistrict-specific trasfer. There are limits o both the amout the govermet ca borrow ad the amout of bods it ca hold. Thus, x [x, x] wherex is the maximum amout that the govermet ca borrow ad x is the maximum amout of bods that it ca hold. The limit o borrowig is determied by the uwilligess of borrowers to hold govermet bods that they kow will ot be repaid. If the govermet were borrowig a amout x such that the iterest paymets exceeded the maximum possible tax reveues; i.e., ρx >max r R(r),theitwouldbeuabletorepaythedebt eve if it provided o public goods. Thus, the maximum level of debt is certaily less tha this level, implyig that x max r R(r)/ρ. I fact, we will assume that x is slightly smaller tha 8

10 max r R(r)/ρ. Thisisbecauseifx equals max r R(r)/ρ the if govermet debt ever reached x it would stay there forever, because the legislature could ever pay it off. For our dyamic results, it is coveiet to assume away this (relatively uiterestig) possibility. The limit o the amout of bods that the govermet ca hold is determied costitutioally. The govermet is allowed to hold o more tha the amout of bods that would allow it to fiace the Samuelso level of the public good from iterest earigs. Thus, x = pg S (A)/ρ, where g S (A) is the level of the public good that satisfies the Samuelso Rule whe the value of the public good is A. 7 The Samuelso Rule is that the sum of margial beefits equal the margial cost, which meas that g S (A) satisfies the first order coditio that αag α = p. 4 Political equilibrium We look for a symmetric statioary equilibrium i which ay represetative selected to propose at roud τ {,..., T } of the meetig at some time t makes the same proposal ad this depeds oly o the curret level of public debt (b) ad the value of the public good (A). Such a equilibrium is characterized by a collectio of fuctios: {r τ (b, A), g τ (b, A), x τ (b, A), s τ (b, A)} T τ=. Herer τ (b, A) is the icome tax rate that is proposed at roud τ whe the state is (b, A); g τ (b, A) isthelevelof the public good; ad x τ (b, A) is the ew level of public debt. The proposer also offers a trasfer of s τ (b, A) to the districts of q radomly selected represetatives where (recall) q is the size of a miimum wiig coalitio. 8 Ay remaiig surplus reveues are used to fiace a trasfer for the proposer s ow district. We focus, without loss of geerality, o equilibria i which at each roud τ, proposals are immediately accepted by at least q legislators, so that o the equilibrium path, o meetig lasts more tha oe proposal roud. Accordigly, the policies that are actually implemeted i equilibrium are described by {r (b, A), g (b, A), x (b, A), s (b, A)}. To be more precise, {r τ (b, A), g τ (b, A), x τ (b, A), s τ (b, A)} T τ= is a equilibrium if at each proposal roud τ ad all states (b, A), the equilibrium proposal maximizes the proposer s payoff subject to the icetive costrait of gettig the required umber of affirmative votes ad the appropriate feasibility costraits. To state this more formally, let v (b, A) deote the legislators 7 The substative coclusios of the paper would be uaffected by assumig that the govermet could hold more bods tha this or eve that there was o upper limit o govermet bod holdigs. It is, however, importat for our coclusios cocerig the ature of the plaer s solutio that the govermet ca hold at least this level of bods (see Aiyagari et al (22)). 8 It should be clear that there is o loss of geerality i assumig that the proposer oly offers trasfers to q represetatives. 9

11 roud oe value fuctio which describes the expected future payoff of a legislator at the begiig ofaperiodiwhichthestateis(b, A). I additio, let v τ+ (b, A) deote the expected future payoff of a legislator i the out-of-equilibrium evet that the proposal at roud τ is rejected. The, for each proposal roud τ ad all states (b, A), (r τ (b, A), g τ (b, A), x τ (b, A), s τ (b, A)) must solve the problem max u(w( r),g; A)+B(r, g, x; b) (q ) s + δev (x, A ) (r,g,x,s) s.t. u(w( r),g; A)+s + δev (x, A ) v τ+ (b, A), (4) B(r, g, x; b) (q )s, s &x [x, x]. The first costrait is the icetive costrait ad the remaider are feasibility costraits. The legislators roud oe value fuctio is defied recursively by v (b, A) =u(w( r (b, A)),g (b, A); A)+ B(r (b, A),g (b, A),x (b, A); b) + δev (x (b, A),A ). To uderstad this recall that a legislator is chose to propose i roud oe with probability /. If chose to propose, he obtais a payoff i that period of (5) u(w( r (b, A)),g (b, A); A)+B(r (b, A),g (b, A),x (b, A); b) (q )s (b, A). (6) If he is ot chose to propose, but is icluded i the miimum wiig coalitio, he obtais u(w( r (b, A)),g (b, A); A)+s (b, A) ad if he is ot icluded he obtais just u(w( r (b, A)),g (b, A); A). The probability that he will be icluded i the miimum wiig coalitio, coditioal o ot beig chose to propose, is (q )/( ). Takig expectatios, the pork barrel trasfers s (b, A) cacel ad the period payoff isasdescribedi(5). For all proposal rouds τ =,.., T the expected future payoff of a legislator if the roud τ proposal is rejected is v τ+ (b, A) =u (w( r τ+ (b, A)),g τ+ (b, A); A)+ B(r τ+(b,a),g τ+ (b,a),x τ+ (b,a);b) +δev (x τ+ (b, A),A ). (7) This reflects the assumptio that the roud τ + proposal will be accepted. Recall that if the roud T proposal is rejected, the assumptio is that a legislator is appoited to choose a default

12 tax rate, public goods level, level of debt ad a uiform trasfer. Thus, ½ ¾ B(r, g, x; b) v T + (b, A) = max u(w( r),g; A)+ + δev (x, A ):B(r, g, x; b) &x [x, x]. (r,g,x) (8) Give a equilibrium {r τ (b, A), g τ (b, A), x τ (b, A), s τ (b, A)} T τ=, we call the iterval of debt levels [if (b,a) x (b, A), x] thepolicy domai. Levels of debt outside this rage will ever be observed except whe exogeously assumed at date zero. A equilibrium is said to be well-behaved if the associated roud oe legislators value fuctio satisfies the followig three properties: (i) v is cotiuous o the state space; (ii) for all A, v (,A) is cocave o [x, x] ad strictly cocave o the policy domai; ad (iii) for all b, v (,A)isdifferetiable at b for almost all A. We will restrict attetio to well-behaved equilibria i what follows, showig that there exists a uique such equilibrium. Heceforth whe we refer to a equilibrium it should be uderstood to be wellbehaved. Fially, ote that ecoomy-wide aggregate utility i a equilibrium at the begiig of some period i which the state is (b, A) isgivebyv (b, A). This follows from the fact that each district has a populatio of size ad represetatives obtai the same payoffs astheircostituets. 4. The equilibrium policy proposals The basic structure of the equilibrium policy proposals is easily uderstood. To get support for his proposal, the proposer must obtai the votes of q other represetatives. Accordigly, give that utility is trasferable, he is effectively makig decisios to maximize the utility of q legislators. The optimal policy will deped upo the state (b, A). If the level of public debt (b) ad/or the value of the public good (A) are sufficietly high, the eve though the proposer is oly takig ito accout the well-beig of q legislators, he will still ot wat to divert resources to pork. Pork requires reducig public good spedig or icreasig taxatio i the preset or the future (if fiaced by issuig additioal debt). Whe b ad/or A are sufficietly high, the margial beefit of spedig o the public good ad the margial cost of icreasig taxatio are both too high to make pork attractive. The proposer will therefore choose a policy package that does ot ivolve pork ad the outcome will be as if he is maximizig the utility of the legislature as a whole. If b ad/or A are lower, the the opportuity cost of pork is lesseed ad the collective utility of the q legislators will be maximized by divertig some resources to pork. Accordigly, the proposer will propose pork for the districts associated with his miimum wiig coalitio. I equilibrium, therefore, there will exist a cut-off value of the public good, iversely related

13 to the level of public debt, that divides the state space ito two rages. Above the cut-off, the legislature will be i the resposible-policy-makig regime (RPM) ad, i every proposal roud, the proposer will propose a o-pork policy package that maximizes aggregate legislator (ad also citize) utility. These proposals will be supported by the etire legislature. Below the cut-off, the legislature will be i the busiess-as-usual regime (BAU) ad, i every proposal roud the proposer chooses a policy package that provides pork for his ow district ad those of a miimum wiig coalitio of represetatives. The tax rate-public good-public debt triple maximizes the aggregate utility of q legislators, give that they appropriate all the surplus reveues. The trasfer paid out to coalitio members is just sufficiet to make them favor acceptig the proposal. Thus, oly those legislators whose districts receive pork vote for these proposals. To develop this more precisely, cosider the problem of choosig the tax rate-public goodpublic debt triple that maximizes the collective utility of q represetatives uder the assumptio that they divide ay surplus reveues amog their districts ad that the costrait that these reveues be o-egative is o-bidig. Formally, the problem is: max (r,g,x) u(w( r),g; A)+ B(r,g,x;b) q + δev (x, A ) s.t. x [x, x]. (9) Usig the first-order coditios for this problem together with equatios () ad (2), it ca easily be verified that the solutio is (r,g (A),x ) where the tax rate r satisfies the coditio that q = [ r r ], () ( + ε) the public good level g (A) satisfies the coditio that αag (A) α = p q, () ad the public debt level x satisfies q δe[ v (x,a ) ] (= if x < x). (2) x To iterpret these coditios, ote that ( r)/( r(+ ε)) measures the margial cost of taxatio - the social cost of raisig a additioal uit of reveue via a tax icrease. It exceeds uity wheever the tax rate (r) is positive, because taxatio is distortioary. For a give tax rate, the margial cost of taxatio is higher the more elastic is labor supply; that is, the higher 2

14 is ε. Coditio () therefore says that the beefit of raisig taxes i terms of icreasig the per-legislator trasfer (/q) must equal the per-capita cost of the icrease i the tax rate ( r)/( r( + ε)). Coditio () says that the per-capita beefit of icreasig the public good must equal the per-legislator reductio i trasfers that providig the additioal uit ecessitates. Coditio (2) says that the beefit of icreasig debt i terms of icreasig the per-legislator trasfer must equal the per-capita cost of a icrease i the debt level. This cost is that there is a higher iitial level of debt ext period. This coditio ca hold as a iequality, if the debt level is at its ceilig. Now defie the fuctio A (b, x) as follows: max{a : B(r,g (A),x; b) } if B(r,,x; b) A (b, x) = if B(r,,x; b) <. (3) Ituitively, A (b, x) is the largest value of A cosistet with the triple (r,g (A),x) satisfyig the costrait that B(r,g (A),x; b). It follows that if A A (b, x ), the proposer proposes the triple (r,g (A),x )togetherwithatrasferjustsufficiet to iduce members of the coalitio to accept the proposal ad the legislature is i the BAU regime. If A>A (b, x ), the the costrait that B(r, g, x; b) must bid ad the solutio equals that which maximizes aggregate legislator utility. This follows from the observatio that if the solutio to the problem of maximizig the utility of q represetatives does ot ivolve trasfers, the this solutio must also solve the problem of maximizig the utility of represetatives. The legislature is therefore i the RPM regime. Thus, we have: Propositio : Let {r τ (b, A), g τ (b, A), x τ (b, A), s τ (b, A)} T τ= be a equilibrium with associated value fuctio v (b, A). The, there exists some debt level x such that for ay proposal roud τ if A>A (b, x ) (r τ (b, A),g τ (b, A),x τ (b, A)) = arg max ad s τ (b, A) =,while if A A (b, x ) u(w( r),g; A)+ B(r,g,x;b) + δev (x; A ) B(r, g, x; b) &x [x, x] (r τ (b, A),g τ (b, A),x τ (b, A)) = (r,g (A),x ) 3

15 ad B(r,g (A),x ;b) if τ =,..., T s τ (b, A) = v T + (b, A) u(w( r,g (A); A) δev (x,a ) if τ = T. I the RPM regime (i.e., whe A>A (b, x )), Propositio implies that the equilibrium tax rate-public good-public debt triple is implicitly defied by the followig three coditios: αag α r = p[ ], (4) r( + ε) ad r [ r( + ε) ] δe[ v (x, A ) ] (= if x<x), x (5) B(r, g, x; b) =. (6) Coditio (4) says that the level of the public good should be such that the social margial beefit equals the price p times the margial cost of taxatio. The social beefit of raisig additioal reveue by issuig more public debt is that it saves raisig that reveue by taxes. Thus, coditio (5) says that the level of public debt should be such that the margial social beefit ofraisig public debt equals the expected margial social cost. I the Appedix (sectio 8.2), we show that i the RPM regime, the tax rate, public debt level, ad the level of the public good all deped positively o the value of the public good (A). I additio, the tax rate ad level of public debt deped positively o the curret level of debt (b), while the level of the public good depeds egatively o b. It is importat to ote that at A = A (b, x )thetriple(r,g (A),x ) maximizes aggregate legislator utility. To see this, ote that B(r,g (A),x ; b) equalszeroata = A (b, x ). Furthermore, usig the defiitio of r i () we may write the first order coditios () ad (2) as αag (A) α r = p[ r ], (7) ( + ε) ad r [ r ( + ε) ] δe[ v (x,a ) ] (= if x < x). (8) x Thus, the equilibrium policy proposal is a cotiuous fuctio of the state (b, A). Moreover, give the mootoicity properties of the solutio i the RPM regime, it follows that whe A>A (b, x ), 4

16 the equilibrium policy proposal ivolves a tax rate higher tha r, the provisio of a public good level below g (A), ad a level of debt that exceeds x. The level of debt x therefore forms a lower boud o the govermet s debt holdigs. Further progress ca be made by characterizig the debt level x. Propositio tells us that, i equilibrium, v (x, A) = max {r,g,z} u(w( r),g; A)+ B(r,g,z;x) + δev (z,a) B(r, g, z; x) & z [x, x] if A>A (x, x ). u(w( r ),g (A); A)+ B(r,g (A),x ;x) + δev (x,a ) if A A (x, x ) Thus, by the Evelope Theorem: v (x, A) ( = x r (x,a) r (x,a)(+ε) )( +ρ ) if A>A (x, x ) ( +ρ ) if A A (x, x ) (9). (2) The discotiuity that arises i the derivative of the value fuctio reflects the fact that a higher future level of debt reduces pork if the legislature is i the BAU regime ad icreases taxes if the legislature is i the RPM regime. Icreasig taxes is more costly tha reducig pork because of the margial cost of public fuds. Usig (2), we have that the expected margial social cost of debt is δe[ v Z (x, A) A ]=G(A (x, x r (x, A) )) + ( )dg(a) (2) x A (x,x ) r (x, A)( + ε) Combiig this with equatios () ad (8), the debt level chose i the BAU rage x must satisfy Z A q G(A (x,x r (x,a) )) + ( A (x,x ) r (x,a)( + ε) )dg(a) (= if x < x). (22) Our assumptio cocerig the maximum debt level x implies that A (x, x) <A. Thus, sice taxes exceed r i the RPM regime, the expected margial social cost of debt must exceed /q whe x = x. It follows that x < x ad coditio (22) holds as a equality. Coditio (22) provides importat isights ito the determiats of the debt level x.whe q<, it implies that A (x,x ) must lie strictly betwee A ad A. Ituitively, this meas that the debt level x must be such that the legislature will trasitio out of BAU with positive probability 5

17 ad stay i it with positive probability. Recall that A (x,x ) is implicitly defied by the equatio R(r ) ρx = pg (A). Thus, if R(r ) >pg (A), iterest paymets must be positive to soak up the excess tax reveues ad hece x is positive. O the other had, if R(r ) <pg (A), the iterest earigs are required to supplemet scarce tax reveues ad x must be egative. The key determiat of the magitude of x is therefore the size of the tax base as measured by R(r ) relative to the ecoomy s desired public good spedig as measured by pg (A). The greater the relative size of the tax base, the larger is the debt level chose i the BAU regime. 4.2 Existece ad uiqueess of equilibrium The foregoig aalysis of equilibrium policy proposals presumes that a equilibrium exists. The key to validatig this presumptio is to demostrate the existece of a roud oe value fuctio v (b, A) with the desired properties. I geeral, establishig the existece of a value fuctio i dyamic games is much more difficult tha establishig the existece of a value fuctio for a plaer s problem, because the equilibrium policy proposals do ot ecessarily maximize the players value fuctio. However, we ca exploit the structure of the equilibrium uveiled i the previous sectio to make the problem tractable. To prove the existece of a equilibrium we start by defiig F to be the set of all real valued fuctios v defied over the state space that are cotiuous ad cocave i x for all A. The, for all z [x, x], we defie a operator T z o F as follows: u(w( r),g; A)+ B(r,g,x;b) + δev(x, A ) T z (v)(b, A) = max (r,g,x) B(r, g, x; b), g g (A), r r,&x [z,x] Thus, give that future payoffs are described by δev(x, A ), the problem is to maximize average legislator utility, but subject to the costrait that the tax rate must be at least r, the public debt level must be at least z ad the public good level ca be o more tha g (A). By stadard argumets (see Stokey, Lucas ad Prescott (989)), T z is a cotractio ad T z (v) belogs to F. Thus, for all z, thereexistsauiquefixed poit v z which is cotiuous ad cocave i x for all A. From Propositio ad the subsequet discussio, it should be clear that if v is a equilibrium roud oe value fuctio ad x is the level of public debt that is chose i the BAU regime, the v = v x. Moreover, it must be the case that x maximizes x/q + δev x (x, A). The ext step, (23) 6

18 therefore, is to defie the correspodece M(z) = arg max{ x q + δev z(x, A)}, (24) ad to demostrate that there exists z such that z belogs to M(z ). We the show that the policy fuctios associated with the value fuctio v z are uique ad defie a equilibrium. Moreover, for all A, v z (,A) is strictly cocave o the policy domai [z, x] adforallx the fuctio v z (,A)isdifferetiable at x for almost all A. I this way, we establish: 9 Propositio 2: There exists a equilibrium. Importatly, we ca also prove that there ca be at most oe equilibrium. proceeds via cotradictio. The argumet Suppose that there were two equilibria with associated roud oe value fuctios v ad v. Let x ad x be the correspodig debt levels chose i the BAU regimes associated with each equilibrium ad suppose that x <x. The, we demostrate that it must be the case that for ay ρ (, ρ) adayx [x, x] δe[ v (x, A) ] δe[ v (x x x +ρ,a) ]. (25) Thus, the expected margial social cost of borrowig with a iitial debt level x i the high debt equilibrium exceeds that i the low debt equilibrium with a iitial debt level x (x x )/(+ρ ). From equatio (22), we kow that the expected margial social costs of borrowig at x ad x respectively, must equal /q; that is, δe[ v (x,a) ]= δe[ v (x,a) ]= q. (26) Combiig these two equatios, yields the coclusio that x x (x x )/( + ρ )-whichis a cotradictio. I this way, we obtai: Propositio 3: Thereexistsatmostoeequilibrium. 4.3 Dyamics Havig uderstood the structure of equilibrium policy proposals ad established the existece of a uique equilibrium, we are ow ready to explore the dyamic evolutio of fiscal policy. We will show that, irrespective of the ecoomy s iitial debt level, the same distributio of debt emerges i 9 Our strategy for provig existece suggests a simple two-step algorithm to computig a equilibrium. First, fid the value fuctio v z associated with each z. The, solve for a fixed poit of the correspodece M(z) i[x, x]. 7

19 the log ru. Moreover, this distributio of debt is o-degeerate: eve i the log-ru, shocks i the value of the public good iduce persistet cycles betwee the two policy-makig regimes. This is i sharp cotrast to the plaer s solutio for the ecoomy i which, as we will show i the ext sectio, the level of debt coverges to a uique degeerate value. Political decisio-makig therefore fudametally alters the dyamic patter of fiscal policy. Let {r (b, A), g (b, A), x (b, A)} be the equilibrium roud oe policy fuctios ad let x be the level of public debt chose i the BAU regime. The equilibrium policies determie a distributio of public debt levels i each period. Let ψ t (x) deote the distributio fuctio of the curret level of debt at the begiig of period t. The distributio fuctio ψ (x) is exogeous ad is determied by the ecoomy s iitial level of debt b. To describe the distributio of debt i periods t 2, we must first describe the trasitio fuctio implied by the equilibrium. First, defie the fuctio A b :[x, x] (x, x] [A, A] as follows: A if x<x (b, A) ba(b, x) = mi{a [A, A] :x (b, A) =x} if x [x (b, A),x (b, A)]. (27) A if x>x (b, A) Ituitively, b A(b, x) is the smallest value of public goods uder which the equilibrium debt level would be x give a iitial level of debt b. The, the trasitio fuctio is give by G( A(b, b x)) if x (x, x] H(b, x) =. (28) G(A (b, x )) if x = x Ituitively, H(b, x) is the probability that i the ext period the iitial level of debt will be less tha or equal to x [x, x] ifthecurretlevelofdebtisb. Usig this otatio, the distributio of debt at the begiig of ay period t 2isdefied iductively by Z ψ t (x) = H(b, x)dψ t (b). (29) b Our mai iterest is to uderstad how the equilibrium debt distributio evolves over time. I particular, does it coverge to some limit distributio? We say that the sequece of distributios hψ t (x)i coverges to the distributio ψ(x) if for all x [x, x], we have that lim t ψ t (x) =ψ(x). 8

20 Moreover, ψ (x) isaivariat distributio if Z ψ (x) = H(b, x)dψ (b). (3) We ca ow establish: b Propositio 4: Let {r (b, A),s (b, A),g (b, A),x (b, A)} be the roud oe equilibrium policy fuctios. The, the implied sequece of debt distributios hψ t (x)i coverges to a uique ivariat distributio ψ (x). Thus, o matter what the ecoomy s iitial debt level, the same distributio of debt emerges i the log ru. The lower boud of the support of this distributio is x - the level of public debt chose i the BAU regime. There is a mass poit at this debt level, sice the probability of remaiig at x havig reached it is G(A (x,x )) - which is positive. However, the distributio of debt is o-degeerate because there is a positive probability of leavig the BAU regime (sice G(A (x,x )) < ). If x is positive, the ecoomy is i perpetual deficit, with the extet of the deficit spikig up after a sequece of high values of the public good. Whe x is egative, the govermet will ru budget surpluses i good times (i.e., whe A is low) ad deficits oly after a suitable sequece of high realizatios of the value of the public good. To get a ituitive feel for the log ru dyamics of the system, suppose that the legislature is i the BAU regime i period t, implyig that the level of debt is x at the begiig of period t. If A t is less tha A (x,x ), the the legislature remais i BAU i period t. The tax rate will be r ad the amout of public good provided will be g (A t ). Govermet debt will be just rolled over ad expeditures o pork will be R(r ) pg (A t ) ρx. O the other had, if A t exceeds A (x,x ), the the legislature will trasitio to RPM. To meet the costs of the public good pg (x,a t ), the legislature will raise taxes ad borrowig; that is, the tax rate r (x,a t )will exceed r ad the level of debt x (x,a t ) will exceed x. Moreover, it will cease all pork barrel spedig. Assumig that A t exceeds A (x,x ), the legislature will remai i RPM i period t + if A t+ exceeds the threshold A (x (x,a t ),x ). The probability of remaiig i RPM exceeds the probability of trasitioig to it i period t sice A (x,x ) exceeds A (x (x,a t ),x ). If A t+ is such that the legislature returs to BAU, the tax rate will be reduced to r, the debt level I the preset eviromet, this defiitio is equivalet to the requiremet that the sequece of probability measures associated with hψ t (x)i coverges weakly to the probability measure associated with ψ(x) (seestokey, Lucas ad Prescott (989) Theorem 2.8). 9

21 r r * x x * g g * ( A H ) g * ( A L ) t t L t H time Figure : The dyamics of the political equilibrium. reduced to x ad the amout of public good provided will be g (A t+ ). The retiremet of the additioal debt is fiaced solely by a reductio i pork (as opposed to a cut back i public good spedig or a icrease i taxes). O the other had, if A t+ is such that the legislature remais i RPM, the higher curret level of debt will make the legislature less iclied to sped, i the sese that g (x (x,a t ),A)islessthag (x,a) for all A. Moreover, both taxes ad borrowig will be higher for ay give value of the public good (i.e., r (x (x,a t ),A) exceeds r (x,a)ad x (x (x,a t ),A) exceeds x (x,a) for all A). Thus, if the value of the public good remais as i period t, citizes will experiece a decrease i public good spedig ad further icreases i taxes ad debt. For a graphical aalysis of the dyamics of the system, let A L be less tha A (x,x )ad 2

22 A H larger tha A (x,x ). Agai, suppose that the legislature is i BAU i period t sothat the level of debt is x at the begiig of period t. Further suppose that i periods t through t L the value of the public good is A L ;iperiodst L +throught H the value of the public good is A H ;adiperiodst H + the value of the public good returs to A L. The, the dyamic patter of public debt, tax rates ad public good provisio is as represeted i Figure. At date t L + debt, taxes ad public good levels jump up i respose to the icrease i A. Durig periods t L + through t H, debt ad taxes cotiue to rise, while public good provisio falls. I period t H +, public good provisio drops i respose to the fall i A, overshootig its atural level g (A L ). After period t H + debt ad taxes start to fall ad public good provisio icreases. Evetually, the legislature returs to BAU. To summarize: i the log-ru, legislative policy-makig oscillates betwee BAU ad RPM. Periods of BAU are brought to a ed by a high realizatio of the value of public goods. This triggers a icrease i public debt ad taxes to fiace higher public good spedig ad a cessatio of pork-barrel spedig. Oce i the RPM regime, further high realizatios of the value of the public good trigger further icreases i debt ad higher taxes. Policy-makig returs to BAU oly after a suitable sequece of low realizatios of the value of the public good. The larger the amout of public debt that has bee built up, the greater the expected time before returig to BAU. Both policy-makig regimes are persistet i the sese that the probability of remaiig i them is greater tha the probability of trasitioig from them. 5 The efficiecy of political equilibrium To uderstad the theory s implicatios cocerig efficiecy, we focus o a compariso of the political equilibrium ad the policies that would be chose by a social plaer whose objective is to maximize aggregate utility. We refer to the plaer s solutio as the efficiet solutio. This is motivated by the fact that the plaer s solutio is the uique Pareto efficiet policy sequece i the set of policy sequeces that provide all citizes with the same expected payoff. Sice all citizes have the same expected payoff i political equilibrium, divergecies betwee the equilibrium ad the plaer s policy sequeces represet Pareto iefficiecies ad thereby costitute political failures i the sese defied by Besley ad Coate (998). 2

23 5. The efficiet solutio While the efficiet solutio could be derived from first priciples, it is istructive to derive it as a special case of our equilibrium model. The efficiet solutio is exactly that which would emerge i equilibrium if the legislature operated uder a rule of uamiity; that is, if q =. As oted earlier, represetatives obtai the same payoffs as their costituets ad whe q =, the equilibrium policy sequece maximizes aggregate legislator utility. It follows that the efficiet solutio has the same form as the equilibrium, except that the equilibrium variables are those associated with q =. Thus, the tax rate i the BAU regime (r ) equals ad the level of public goodsprovidedisthesamuelsolevel(g (A) =g S (A)) (see equatios () ad ()). Because q =, thelevelofdebtx that is chose i this regime satisfies the first order coditio (see equatio (22)) Z A =G(A (x,x r (x,a) )) + ( A (x,x ) r (x )dg(a). (3),A)( + ε) This equatio implies that at debt level x, the future tax rate must be with probability oe. Sice r (x,a) exceeds for all A>A (x,x ), this requires that A (x,x )=A. This i tur implies that x = x. At this debt level, the govermet s iterest earigs o its bod holdigs are always sufficiet to fiace the Samuelso level of public goods, implyig that o taxatio is ecessary. We coclude that the efficiet solutio has the followig form. Whe the state is such that A A (b, x), the tax rate is zero, the public good level is the Samuelso level ad the debt level is x. Surplus reveues (which will be positive assumig A<A (b, x)) are redistributed to citizes via (uiform) district-specific trasfers. Whe the state is such that A>A (b, x), the optimal policy ivolves positive levels of taxatio, the provisio of a public good level below the Samuelso level ad a level of debt that exceeds x. There are o surplus reveues ad hece o district-specific trasfers. It is clear from this discussio that the distributio that puts poit mass o the debt level x is a ivariat distributio. For oce the govermet has accumulated this level of bods, it ca provide Samuelso levels of the public good without distortioary taxatio. Ay surplus iterest paymets ca be redistributed as a uiform district-specific trasfer. These trasfers are lump-sum ad create o distortio. The legislature has o icetive to ru dow accumulatios by redistributig reveues via additioal trasfers because this would ecessitate the use of distortioary taxatio 22