Optimal Fiscal Policy. in a Business Cycle Model without Commitment

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1 Optimal Fical Policy in a Buine Cycle Model without Commitment Jeú Fernández-Villaverde Univerity of Pennylvania Aleh Tyvinki Univerity of Minneota Federal Reerve Bank of Minneapoli November 12, 2002 Abtract Thi paper tudie optimal taxation in the tochatic growh model when the goverment cannot commit. We ue recurive game theory to characterize the et of Sutainable Equilibria and to build trategie that upport equilibrium payoff. We calibrate our model to match U.S. data and compute both the et of utainable equilibria payoff, trategie that implement them and trigger. We alo look at the Bet Equilibrium under no commitment and compare it with the Markov Perfect Equilibrium and with the Ramey Equilibrium. Key word: Optimal Fical Policy, Buine Cycle, Recurive Game Theory, Computational Method. JEL claification: C73, E32, E62. Correponding Author: Jeú Fernández-Villaverde, Department of Economic, 160 McNeil Building, 3718 Locut Walk, Univerity of Pennylvania, Philadelphia, PA jeufv@econ.upenn.edu. Thi i a preliminary and incomplete draft. Future verion will be poted at Thank to V.V. Chari, Kenneth Judd, Narayana Kocherlakota, Chri Phelan, Joé-Victor Río-Rull, Sevin Yeltekin and participant at eminar at SED 2002, SITE 2002, Atlanta Fed, Rutger and Penn State for ueful comment. Beyond the uual diclaimer, we mut notice that any view expreed herein are thoe of the author and not necearily thoe of the Federal Reerve Bank of Minneapoli or of the Federal Reerve Sytem. 1

2 1. Introduction Thi paper addree the following quetion: how hould taxe be et over the buine cycle when there i no commitment for the government? The anwer to our quetion in the cae in which the government ha acce to a commitment technology wa developed in an important contribution by Chari, Chritiano and Kehoe (1994) who explored the quantitative implication of the theory. That contribution i complemented by Zhu (1992) that howed ome intereting theoretical propertie and by Stockman (2001) when the government mut follow a balanced-budget rule. However it i alo well undertood that, in general, the optimal policy precribed by the theory i time inconitent. Once capital i accumulated, taxing it i unditortionary and a benevolent government will deviate to a high capital tax to eae the (ditortionary) taxation of labor. Time inconitency i not a theoretical curioity. Caual obervation of contitutional arrangement in mot countrie ugget that, in fact, government are not uually bound in their choice of taxe for more than a relatively hort period of time and that change in the tax arrangement are frequent. Thi empirical obervation raie naturally the problem of how we can characterize the optimal fical policy over the cycle when no commitment device i available. Thi paper how how to find the et of utainable equilibria for a calibrated verion of the tochatic growth model uing the tool developed by Abreu, Pearce and Stacchetti (1990) and by Phelan and Stacchetti (2001). With thi et we can build the trategie aociated with any particular point in that et and explore the quantitative implication for aggregate quantitie and welfare. Sutainable Equilibrium, a definedbychariandkehoe(1990) ithe natural olution concept i thi context ince we want to aure equential rationality for the government and that we alway remain in a competitive equilibrium where conumer are mall enough to affect, each one of them by itelf, aggregate outcome. Our reult how a relatively mall et of utainable equilibria. The trategie computed for the bet equilibrium alo imply poitive taxation of capital and labor, departing from Ramey, but with a taxation on capital much lower than in a Markov equilibrium and labor tax rate auming mot of the effect of the tochatic hock We can alo ue for framework for anwer everal related quetion. We meaure the improvement from introducing a commitment device and we look at the optimal repone 2

3 function of the government to tochatic hock. Our paper relate to a growing literature on the tudy of optimal policy in the abence of commitment. We can highlight everal contribution. Klein and Río-Rull (2002) and Klein, Kruel and Río-Rull (2002) explore thee quetion when Markov Perfect equilibria are ued a the olution concept. Markov equilibria are natural and intuitively appealing. Alo the trategie that upport them are very imple to decribe. However, ince by contruction, Markov equilibria abtract from reputational mechanim it i difficult to ae how much i lot looking at thi ubet of equilibria. Alo ome time it i difficult to build thi type of equilibria. Benhabib and Ruticcini (1997) argue, in the context of a determinitic model, for the ue of optimal control when the problem of the government i appropriately augmented by an additional incentive compatibility contraint. Optimal control implifie the analyi but we do not have a contructive procedure to write down the value of the deviation needed for the incentive contraint. Thi limit it utility in practical application. Cloer to u we find Chari and Kehoe (1990), Sleet (1997) and Chang (1998). Chari and Kehoe (1990) propoe a method to check if a given tax policy i time-conitent. Sleet (1997) develop imilar tool than Phelan and Stacchetti (2001) and explore the et of equilibria in an overlapping generation economy. Chang (1998) ue recurive game theory to tudy optimal monetary policy in a very imilar pirit to our approach. We feel that our approach complement the exiting reult in the literature. With repect to the paper on Markov equilibria we add the poibility of exploring ytematically all the et of equilibria and ae whether reputational mechanim are of quantitative importance and, if they are, whether they are plauible or not. With repect to the literature in optimal control we offer a contructive procedure to find the value aociated with a deviation. Finally with repect to the theoretical reult, we offer quantitative aement in the framework of the tochatic neoclaical growth model. Itiimportanttonotethatweabtractfromeveralimportantfeatureofthedata. Firt, for computational reaon 1, we exclude the tudy of public debt. It i unknown to u how retrictive thi abtraction i. Luca and Stokey (1983) howed in a model without capital the role that public debt may have a a ubtitute for commitment. Whether any imilar intuition can be extended to the tochatic growth model i not intuitive. Alo in term of welfare, ince debt work a a hock aborber in the reult of Chari, Chritiano and 1 Including debt expand our tate pace by one additional variable. A we will how below that extenion caue a ubtantial computational burden we want to overcome in the hort future. 3

4 Kehoe (1994), we may be getting an inaccurate meaureofthevalueofcommitment. Second we abtract from any heterogeneity of agent. Different type of agent may raie intereting political-economic iue that deerve careful tudy. We feel however that we need to explore firt the implication of the theory in the implet cae of homogeneou agent. The ret of the paper i organized a follow. Section 2 preent the tochatic neoclaical growth model with government and taxation. Section 3 dicue how we can generate a recurive formulation of the problem that it i eaier to work with than the original formulation. Section 4 outline ome detail of the computation ued to find the et of utainable equilibria and the trategie that implement them and the aociated trigger. The economy i calibrated to match certain characteritic of the U.S. data in ection 5. Section 6 preent our finding and ection 7 conclude. An appendix provide further computational detail and ome proof omitted in the main text. 2. The Economy 2.1. The Environment We will conider a production economy populated by a meaure one of identical, infinitely lived conumer. In each period t =0, 1,..., the economy experience one poible realization of an tochatic proce S where the initial realization 0 i given. To implify the analyi we will work with finite et S. Notethatofarwedonotrequireanyparticular tructure in the tranition equation for t. The realization in period t of the proce i denoted by t and the hitory of t realization from 0 through t by t =( 0, 1,..., t ).Given the tructure of uncertainty we can ue a commodity pace in which good are indexed by hitorie. The probability of each of thee hitorie i given by π ( t ). With thi probability we can define conditional probabilitie of future event given a particular hitory t that we will call π ( t+j t ). For future convenience we alo define the binary relation  between two realization of the hock t and t+j, t+j  t,iftheevent t+j i compatible with hitory t, i.e. if π ( t+1 t ) > 0. Alowedefine t+j º t if either t+j  t or t+j = t. In each period a competitive-behaving firm ha acce to a technology to produce the final good, y ( t ), uing capital, k ( t 1 ) and labor, l ( t ), given by the neoclaical production function: F k t 1,l t, t (1) 4

5 Note that we index capital at period t by the hitory t 1 ince it i a predetermined variable at the beginning of the period. Alo the hitory t i an argument of the production function. Thi allow u to think about productivity hock a poible (but not necearily the only) event in S. In general we will concentrate in cae where only t i relevant to determine the current period technology. Competitive pricing enure that input price equate marginal productivitie: r t = F k k t 1,l t, t (2) w t = F l k t 1,l t, t (3) The final output can be ued for private conumption, c ( t ), government conumption, g ( t ) and invetment good i ( t ). The law of motion for capital k ( t ) i given by: k t = i t +(1 δ) k t 1 (4) where δ i the depreciation rate on capital and where we impoe i ( t ) 0. We will dicu below the role of thi nonnegativity contraint. The preference of each conumer over equence of conumption and labor are repreentable by a time-eparable utility function: X X β t π t u c t,l t (5) t=0 t where 0 < β < 1 i the dicount factor, 0 l ( t ) 1 i the labor upply and the period utility function u atifie uual aumption (trict concavity, differentiability and Inada condition). To ave on notation we pick normalized utility function (i.e. already multiplied by (1 β)). The conumer budget contraint aociated to thi objective function i: c t + i t = 1 τ l t w t l t + 1 τ k t r t k t 1 (6) k 1 > 0 given (7) where τ l ( t ) i a proportional tax on labor income, τ k ( t ) a proportional tax on the return of capital and k ( 1 ) i the initial endowment of capital of (almot all) conumer. To ave 5

6 in notation we omit in (6) the trading of Arrow-Debreu ecuritie by private agent. Since all our conumer are identical thee ecuritie will not be traded in equilibrium. It i important, however, to remember that the ecuritie exit and could be ued out of equilibrium. We will dicu the abence of public debt in more detail below. There i a benevolent government that maximize: v = X X β t π t [u c t,l t + G(g( t ))] t=0 t where G i a (alo normalized by 1 β) function of the government conumption g ( t ).We will aume that G( ) i increaing and concave. We could, equivalently, added that lat term in the objective function of the conumer. Separability and the mallne of each conumer that take g( t ) a given make both choice equivalent. We feel our choice i marginally clearer for the analyi below 2. In each period the government et g equal to the revenue raied by the two taxe g t = τ l t w t l t + τ k t r t k( t 1 ) Thi equation implie that we impoe a balanced-budget rule event by event. A explained in the introduction, eliminating public debt allow u to reduce the tate pace of the recurive problem we will develop later. Computational reaon trongly ugget thi choice. The balanced budget aumption i a deviation from the baic model of Chari, Chritiano and Kehoe (1994) that allow public debt with one-period maturity and a tate-contingent return. Stockman (2001) ha tudied the effect of introducing thi balanced-budget rule for the cae with commitment. We will contrain the freedom of the government to et taxe to thoe rate that belong to the interval T = {[τ, τ] 0 < τ < τ < 1}. We again depart from the baic framework of Chari, Chritiano and Kehoe (1994), pecially ince we do not allow taxe to be negative. Thi contraint maybe binding (Zhu (1992) how that the probability that the optimal tax rate on capital i le than zero in the Ramey problem i trictly poitive). However ince thegovernmentmutbalanceitbudgeteventbyeventand,awewilldecribebelow,taxe are announced before the conumer make their choice, a ubidy to an input (a negative tax) 2 Alo thi problem i lighlty different from the bauc Chari, Chritiano and Kehoe (1994) economy in which the government need to finance an exogenouly given tream of public conumption. In our formulation the government can ubtitute the public conumption intertemporally. 6

7 might not be financed for certain choice of the conumer. Even if thi would not happen in equilibrium, we would need to pend ome time fixing thi poibility without gaining further inight in the problem. Let u finih the decription of the economy mentioning the exitence of an exogenou, uniform [0, 1], erially uncorrelated random variable X t with a publicly oberved realization x t at the beginning of each period t and given initial value x 0. Thi random variable will be helpful later to convexify the et of equilibria Competitive Equilibria Evenifweareconcernedwiththecaewherethereinocommitment,itturnouttobeconvenient to think about competitive equilibria where the government can commit to a certain arbitrary tate contingent policy. Firt the government announce a contingent policy. Call that policy τ = {τ l ( t ), τ k ( t )} t=0. Given that policy we can define an allocation rule a a equence of function a (τ) that map policie into allocation {y ( t ),c( t ),l( t ),k( t ),g( t )} for each hitory t. Analogouly we can define price rule w (τ), r (τ) that map policie into price ytem for all hitorie. A competitive equilibrium i a tate contingent policy τ, an allocation rule a ( ) and price rule w ( ) and r ( ) uch that (1) given price, conumer olve their problem, (2) input price equate the marginal productivitie, (3) the government atifie it budget contraint period by period and (4) market clear Sutainable Equilibria Now we are ready to decribe the game aociated with our environment, Γ (k ( 1 ), 0 ).We will have two type of player, the government and the continuum of anonymou conumer 3. Anonymity implie that the government cannot oberve individual choice of conumer but only their ditribution. Similarly conumer oberve their own action, the ditribution of other conumer action and the government choice. The timing protocol i a follow. Firt, in each period t, afterx t and t are realized, the government pick the tax rate for the period, τ l ( t ), τ k ( t ) uch that the rate belong to T. The fact that the government only decide taxe for the current period embodie our concept 3 We alo have the repreentative firm but thi player only make trivial deciion equating marginal productivitie to price, to implify the expoition we omit further dicuion of it. 7

8 of lack of commitment 4. Alo, becaue of anonymity, the tax rate are common for all agent. Then the conumer pick c ( t ), l ( t ) and k ( t ) uch that their budget contraint i atified. Thee choice imply a value for g ( t ) that i conumed by the government after tax revenue ha been raied. It i natural to define a public hitory for the game a h t =(h 0,h 1,..., h t ) (not to be confued with the hitory of the tochatic proce t )whereh t =( t,x t, τ l ( t ), τ k ( t )). Thi public hitory keep track of the realization of the tochatic variable and of the choice of the government 5. A trategy for the government i a meaurable function σ G ( ) that map h t 1, t and x t into tax rate for each period t, (τ l ( t ), τ k ( t )) = σ G (t)(h t 1, t,x t ). A ymmetric public trategy for the conumer i a meaurable function σ C ( ) that map h t into c ( t ), l ( t ) and k ( t ), (c ( t ),l( t ),k( t )) = σ C (t)(h t ). The pair of trategie σ G ( ) and σ C ( ) i called a ymmetric public trategic profile σ. If Σ G i the et of trategie for the government and Σ C the et of ymmetric public trategie for the conumer, we can define the et of ymmetric public trategic profile a Σ = Σ G Σ C. Here we want to make two comment. Firt notice that we are retricting our attention to public trategie in which the conumer ignore their private information in chooing their action. Thi contraint avoid the need to account for private hitorie that will be irrelevant along the equilibrium path becaue of the convexity of the problem (ee Phelan and Stacchetti (2001) for detail). It till allow equilibria where a player contemplate deviating to a nonpublic trategy although the conumer prefer not to do o. Fudenberg, Levine and Makin (1994) provide further explanation. Second, we allow all the trategie to depend on the random variable x t. The role of thi public randomization i to aure, later on, that the et of equilibria that we characterize preent deirable convexity propertie. A a olution concept for the game Γ (k ( 1 ), 0 ) weadapttheconceptofutainable equilibrium introduced by Chari and Kehoe (1990) to deal with ymmetric public trategie and propoe the following definition. A ymmetric public trategy profile σ for the game Γ (k ( 1 ), 0 ) i a utainable equilibrium if for any hitory t (with correponding capital k ( t 1 )): 4 Of coure we could allow the government to announce future rate but, ince it can change that deciion in each future event, that announcement would be irrelevant. 5 We could alo account for the ditribution of variable a capital (remember that individual conumer action are not oberved and then they can not be included). However the convexity of the conumer problem imply that all of them will choe the ame action and that no nontrivial ditribution will be ever oberved (and conequently that thoe poible deviation do not have any effect on the equilibrium path). Then from h t we can alway recover all the value of the other variable. 8

9 1. The continuation equilibrium payoff for the government i higher than the payoff from any deviation to a different trategy. 2. Given k ( t 1 ) and the tax policy, the equence {y ( t ),c( t ),l( t ),k( t ),g( t )} j º t and {w ( t ),r( t )} j º t contitute a Competitive Equilibrium. The rational behind thi olution concept i imple. The government follow a trategy from which there i no profitable deviation after any hitory. Thi requirement derive directly from equential rationality. The conumer alway repond to the government trategy with deciion that imply a competitive equilibrium ince thi i the only ituation compatible with feaibility and individual optimization. Note that we are explicitly ruling out the poibility of conumer colluding to punih the government after a deviation with a behavior that implie a lower value of v than the wort competitive equilibrium. Since all the conumer are anonymou, each of them will have an incentive to deviate from uch a colluion agreement and maximize it own utility. Conequently any attempt in cooperation cannot be an equilibrium. Thi point i important for our quantitative analyi becaue it ubtantially reduce the wort available punihment after a deviation and thu the aociated et of utainable equilibria. It i alo intereting to compare thi definition with that of Ramey allocation. In both cae the government pick a policy to which the conumer repond with a competitive equilibrium, limiting the implementable allocation. The difference i that in Ramey the government earche over the pace of tate-contingent policie to findtheoptimaloneandit tick with it. In a utainable equilibrium, the requirement i trengthen to aure that the government, in cae of being able to retate it plan after ome particular hitory, cannot find any new deviation that increae it value. 3. Recurive Formulation of the Problem In thi ection we how how to rewrite the problem of the conumer in a recurive way. Working with the original problem i a challenging tak in particular becaue we need to keep track of the hitorie for the game h t intead of jut the tate, t and k ( t 1 ),ainthe tandard model with commitment. A recurive formulation allow a better undertanding oftheproblemandmoreimportantly,toapplythetoolofrecurivegametheoryawe will do in ection 4. To achieve our goal, firt we will preent the equence problem of the conumer that come directly from our decription of the environment. Then we will propoe 9

10 a recurive verion of the ame problem and we will finih howing the equivalence between the two formulation The Sequence Problem Remember that the conumer problem, for a given government policy τ l ( ) and τ k ( ), wa ubject to max X X β t π t u c t,l t (8) t=0 t c t + i t = 1 τ l t w t l t + 1 τ k t r t k t 1 (9) k 1 > 0 given (10) We will call thi problem the equence problem. Given our aumption about the utility and production function, a et of firt order condition for the conumer problem i given by 6 : u c t = β X t+1 t u c t τ k t+1 r t+1 δ (11) t+1 t π u l t = 1 τ l t w t u c t (12) where we ue the notation u x ( t ) to denote the marginal utility with repect to the variable x evaluated at the allocation in hitory t.thefirt equation, 11, i the uual intertemporal Euler condition where we um over all poible realization of the hock that can follow the current tate. The econd equation, 12, i the tatic optimality relation between labor marginal diutility and conumption marginal utility The Recurive Problem In order to build our recurive problem, we define firt a new variable: m t u c t 1+ 1 τ k t r t δ 6 For implicity in the expoition we alo ignore for the moment the retriction that i ( t ) 0. In our computation below we would include, though, thi contraint. 10

11 What i m? The marginal value of capital, the increae in the conumer utility if it had tarted with an additional unit of capital and pent all the additional income on conumption. Thi variable will be key in our recurive formulation below. The obervation that the marginal value of capital allow the writing of policy problem recurively come from Kydland and Precott (1980) and it ha been applied by Marcet and Marimón (1994) and Phelan and Stacchetti (2000). Another way to think about m i a the payoff to the conumer in parallel to v a the payoff to the government. Then we can conider the following recurive problem alo for the ame given government policy. For any t and k ( t 1 ) the conumer olve: uch that: max u c t,l t + β X π t+1 t m t+1 k t c( t ),k( t ) t+1 t c t + i t = 1 τ l t w t l t + 1 τ k t r t k t 1 k t = i t +(1 δ) k t 1 Given the aumption we made on the utility and production function thi problem ha an interior olution and the firt order condition of the conumer problem are given by: u c t = β X t+1 t m t+1 (13) t+1 t π u l t = 1 τ l t w t u c t (14) 3.3.TheEquivalenceoftheSequenceandRecuriveProblem To how the equivalence between the two formulation notice that, if we plug the definition of m ( t+1 ) in (13) we get: u c t = β X t+1 t u c t τ k t+1 r t+1 δ (15) t+1 t π u l t = 1 τ l t w t u c t (16) andthenwegettheameetoffirt order condition that olve the recurive problem than the one that olve the equence problem. 11

12 Then, if we can how that an appropriate tranverality condition i atified, the olution to both problem mut be the ame. In particular if lim t X X t=0 t β t π t m t+1 k t =0 (17) i.e. if the dicounted marginal of value capital goe to zero a time goe to infinity. Showing that thi tranverality condition hold in general cae i difficult. For our inherently quantitative purpoe we feel nothing of ubtance i lot if we retrain ourelve to the ituation where the intertemporal utility function ha the form (up to a contant): u c t,l à t c ( t ) 1 σ = (1 β) 1 σ + γ l! t if σ > 1 = (1 β) log c t + γ l t if σ =1 Thi pecification of preference i both compatible with the exitence of a balanced growth path and, from the perpective of public finance, it atifie a number of intereting propertie related with uniform commodity taxation (ee Chari and Kehoe (1999)). Lemma 1. If the utility function ha the form propoed above, then the tranverality condition (17) hold. Proof.Firtnotethatinceu c ( t )=c( t ) σ and r ( t )=F k (k ( t 1 ),l( t ), t ) we have m t+1 = c t+1 σ 1+ 1 τ k t F k k t,l t+1, t+1 δ and hence we need to how that lim t X X t=0 t β t π t c t+1 σ 1+ 1 τ k t F k k t,l t+1, t+1 δ k t =0 To do o... [to be completed] Nowthatwehavehowntheequivalenceoftherecuriveandtheequenceproblemand to ave on notation, when there will be no rik of ambiguity, we will ue a to denote the value ofavariableinthiperiodanda 0 to denote it value in the next. In that way and uing the equivalence reult, for example we can expre the value for the government v() given the 12

13 current tochatic hock a v() =u(c(),l()) + G(g()) + β X 0 Â π( 0 )v( 0 ) Alo with thi notation, and for a exogenouly given et of m( 0 ) for all 0 Â, k, and taxe τ k (), τ l (), we can think of the problem of the conumer max u (c (),l()) + β X π( 0 )m ( 0 ) k () c(),k() 0 Â uch that: c ()+i () = (1 τ l ()) w () l ()+(1 τ k ()) r () k k () = i ()+(1 δ) k, i () 0 plu the problem of the repreentative firm and the government policy, a an aociated tatic economy Ξ (, k, τ k (), τ l (),m( 0 ) for all 0 Â ) indexed preciely by the marginal value of capital, the initial amount of capital, the tochatic tate and the tax rate. It i eay to ee that c(),l(),k(),g(),w(),r() contitute a competitive equilibrium for Ξ (, k, τ k (), τ l (),m( 0 ) for all 0 Â ), and denoted by (c(),l(),k(),g(),w(),r()) CE (Ξ (, k, τ k (), τ l (),m( 0 ) for all 0 Â )), if and only if: u c () = β X π( 0 )m( 0 ) 0 Â (18) u l () = (1 τ l ())w()u c () (19) r () = F k (k, l (),) (20) w () = F l (k, l (),) (21) g() = τ l () w () l ()+τ k () r () k (22) k() = (1 τ l ()) w () l ()+(1+(1 τ k ()) r () δ) k c() (23) k() (1 δ) k (24) The interpretation of thee condition i imple. The firt two contraint (18) and (19) are the conumer firt order condition. Their preence aure that conumer optimize given price and their forecat about the future value of the variable. Remember here our previou 13

14 dicuion of how, not matter how the government behaved, in the whole dynamic game, anonymou agent can only anwer with a competitive equilibrium where they atify thee two condition. Analogouly (20) and (21) make inputpriceequaltomarginalproductivitie a implied by a competitive equilibrium. The lat piece of the definition of an equilibrium aregivenbythe definition of government conumption, (22), and by (23), the law of motion for capital (plu the nonnegativity of invetment 24), that provide market clearing. Alo appropriately linked (i.e. with the right law of motion for k, m and ) tatic competitive equilibria a the one decribed are a competitive equilibrium of the original equence economy. Thi linkage allow u to think of the original economy a one equivalent to the repetition of tatic economie with endogenouly changing tate variable and exogenou tochatic hock. 4. Self Generation Abreu, Pearce and Stacchetti (1990) howed a procedure to exploit a recurive formulation of a repeated game. In particular they noted that, in a repeated game, we do not need a complete pecification of the whole equence of future action. Intead a continuation value from the next period on can ummarize all relevant information about the future. In an equilibrium that continuation value i alo the equilibrium payoff for the repeated game beginning next period. In our context, a vector of payoff would be the one generated by a utainable plan if they are equal to the current period payoff pluthepayoff of a continuation utainable equilibrium. Phelan and Stacchetti (2000) have extended that contribution to problem of optimal fical policy. In particular they point out two iue related with the direct application of the Abreu, Pearce and Stacchetti framework. Firt, the optimal fical problem i a dynamic, not a repeated, game becaue of the preence of tate variable, a in our cae capital and the tochatic hock. Second ince we have a continuum of conumer we need an infinitely dimenional value correpondence. With repect to the firt iue, we only need to define a payoff correpondence in k and. Thi idea ha the important advantage of being rather eay to extend to more general tate pace (a thoe with public debt, nondegenerated income ditribution or other). The econd problem require further treatment. The main idea, already hinted before, i to ue the marginal value of capital, m, a the econd payoff (with the firt being the payment to 14

15 the government v). The intuition for the procedure i that m ummarize the marginal value of wealth (a good concept of payoff for the conumer) and that it carrie the information needed to make optimal intertemporal deciion. We define a correpondence that map all pair of poible value of the tate (k, ) into et of payoff (m, v) generated by a utainable plan. For a given et of tate (k, ) we will call thi et the equential equilibrium value correpondence V (k, ). Becaue of the preence of the publicly oberved random variable X t thi correpondence i convex for given value of k and (any point in the linear combination between the payoff to two utainable plan in pure trategie repreent the payoff to the utainable plan that appropriately mixe the two pure trategie). Alo in mot application of interet thi correpondence would not be trivial (with only one point) but it will include a large number of poible equilibrium payoff reflecting the omehow annoying fact that mot of thee dynamic game place relatively little tructure in the prediction regarding obervable. We alo define an arbitrary value correpondence a any mapping from all pair of poible value of the tate (k, ) into et of payoff (m, v). We can repreent one of thee value correpondence a W : < + S < 2. We will denote by A the pace of all value correpondence W (, ). Define an operator B : A A in the following way: ( ) (m, v) τ k (), τ l (),c(),l(),k(),g(),w(),r(), B(W )(k, ) =co and m( 0 ),v( 0 ) W (k(), 0 ), for all 0 Â where m() =u c ()(1+r()(1 τ k ()) δ) (25) (c(),l(),k(),g(),w(),r()) CE (Ξ (, k, τ k (), τ l (),m( 0 ) for all 0 Â )) (26) ( v() = u(c(),l()) + G(g()) + β X ) π( 0 )v( 0 ) π(k, ) (27) 0 Â The interpretation of the operator and the contraint i a follow. B( ) i the convex hull of the payoff m() and v() uch that there are aociated value of the taxe, conumption, labor upply, next period capital, government expenditure, input price and next period payoff that belong to the value correpondence for every poible realization of the hock compatible with the current tate and that atify certain condition. The firt contrain 15

16 i jut our definition of m() in recurive notation. The econd contrain aure that the value of all thee variable are compatible among themelve and with agent optimization, i.e. that the contitute a competitive equilibrium in the aociated tatic economy. Finally the incentive contrain (27) retrict the admiible payoff v() to thoe weakly bigger than the value of a certain tate-dependent function π(k, ) that, a it will clear hortly, we call the wort punihment function. Why do we impoe thi additional lat contraint and why do we call it the wort punihment function? Abreu (1986 and 1990) howed that any equilibrium in a repeated game can be utained through the ue of a trigger trategy that revert to a wort equilibrium after a deviation. Extending thi logic to our context, a payoff v can only be implied by a utainable equilibrium if it provide a higher utility than a deviation plu a reverion to the wort competitive equilibrium (wort undertood a given the lowet utility). Note that we require, a mentioned in ection 2, that the reverion i alway to a competitive equilibrium, the only ituation compatible with anonymity of the conumer. Thi particular type of reverion make the wort punihment not o bad and it reduce ubtantially the et of utainable equilibria 7. The tak i then to define thi function κ(, ) and to characterize it 8. The wort punihment i given by: κ(k, ) = max τ k (),τ l () ( min u(c(),l()) + G(g()) + β X ) π( 0 )v( 0 ) c(),l(),k(),v( 0 ),m( 0 ) 0 Â uch that {m( 0 ),v( 0 ) W (k(), 0 ) for all 0 } (28) (c(),l(),k(),g(),w(),r()) CE (Ξ (, k, τ k (), τ l (),m( 0 ) for all 0 Â )) (29) where the firt contraint, (28) limit the continuation payoff tobethoebelongingtothe value correpondence and the econd aure that all the choice of the conumer and the government are a competitive equilibrium. 7 If, for example, conumer could collude among themelve and, after a deviation, revert to an equilibrium with zero (or arbitrarily mall) conumption, the continuation utility for the government would be o low that, rather trivially, Ramey would be upported. 8 If we knew the function from the beginning we could follow Benhabib and Ruticcini (1997) and olve directly for the problem of the government augmented by a contraint that force the value for the government of an admiible policy to be greater or equal than thi wort punihment. 16

17 The intuition behind Abreu propoal i that we can manipulate the believe about future action in uch a way that the agent think that after a deviation from the government they will deviate to the wort equilibrium. To compute thi function, we fixed ome value for the tate and then minimize over the choice of the conumer and poible continuation payoff and maximize over taxe (the government trie to get the bet poible value out of the wort equilibrium) being ure, of coure, that we are alway inide a competitive equilibrium. With rather traightforward extenion of the exiting reult in Abreu, Pierce and Stacchetti (1990) and in Phelan and Stacchetti (2000). we can tate ome theorem that are quite ueful to characterize the equential equilibrium value correpondence Theorem 2. If W B(W ) then B(W ) V. Proof: [to be completed]. Thi reult, known a elf-generation, together with the fact that the equential equilibrium value correpondence i elf-generating V B(V ) imply that V i a fixed-point of the operator. Even more importantly, the next reult (factorization) tell u it i the larget of uch fixed point. Theorem 3. The equential equilibrium value correpondence V i the larget value correpondence W uch that W = B(W ). Proof: [to be completed]. Theorem 4. B( ) i monotone, i.e. for W, W 0 A uch that W W 0, B(W ) B(W 0 ) Proof: See appendix. Nearly a important a thee reult i that, if we define W n+1 = B(W n ) we can how, that for an initial W 0 V, Theorem 5. W = lim n B(W n ). Proof: [to be completed]. 17

18 5. Computation In thi ection we dicu how we implement in a computer the operator B( ) decribed above to find the equilibrium value correpondence and how to build trategie that upport particular point in that correpondence. We cover in certain detail the logic of the procedure and how it can be eaily extended to generate further reult than the one here reported. Reference for thi ection include Cronhaw (1997), Judd, Yeltekin and Conklin (2000) and Sleet and Yeltekin (2001) Finding the Equilibrium Value Correpondence The firt tak i to find the equilibrium value correpondence with all the poible payoff. Repreenting a multidimenional correpondence in a computer i a challenging endeavour. A imple way to implify thi objective i to dicretize the tate pace. Along the dimenion thi i not an iue ince we aumed that S i finite. For capital we define a grid k grid = [k 1,..., k n ] where k n i fixed to be ufficiently high that doe not affect the computation. After thi dicretization, intead of a correpondence W : < + S < 2 we have a new one W : k grid S < 2. It i equivalent to think about thi correpondence a n # {S} convex et W (k i, j ),wherek i k grid and j S, a much impler collection of object than the original W. But even after having dicretized the tate pace we till face the problem of how to repreent each of thee et and how to manipulate them to reproduce the operator B( ). Judd, Yeltekin and Conklin (2000) propoe two alternative procedure (an outer approximation and an inner approximation algorithm) to find the equilibrium value correpondence of a repeated game. We extend their reult for our cae where we have a dynamic game Outer Approximation The Outer Approximation approach build on the idea that we can ue a et of hyperplane H to define a convex polytope a the interection of the half-plane generated by H evaluated at ome ditance with repect to an origin of coordinate. A way to approximate an arbitrary convex et i to find the mallet convex polytope that include it generated by H and ome ditance. The great advantage of thi approach i that the only information required to keep track of the approximated et i preciely H and the correponding ditance, an information eaily torable. 18

19 Formally we will denote by W o (k i, j ) the outer approximation of W (k i, j ) defined by an outward et of hyperplane H R D,2 and a vector of ditance from the origin C W (ki, j ) (note that the hyperplane are contant acro the different tate value k i and j while the ditance may change). Then we ay that a point w() =[m(),v()] W 0 (k i, j ) if H 0 w() C W (ki, j ). However ince we are uing an outward approximation we need to remember that, generically, we will have point w 0 () uch that H 0 w 0 () C W (ki, j ) but w 0 () / W (k i, j ). Anoutlineofthealgorithmuedtocomputethiouterapproximationiafollow. 1. Fix H. 2. Start with an initial large C 0 W (k i, j ) that define W 0 (k i, j ) V (k i, j ). 3. For n =1,... update CW n (k i, j ) at each row (direction) z of H. Thi tep deliver the new et W n (k i, j )=B(W n 1 (k i, j )). 4. Iterate until convergence, V o (k i, j )=B(V o (k i, j )) = lim n W n (k i, j ),wherev o (k i, j ) i the D direction outer approximation of V (k i, j ). Several point deerve further dicuion. Firt, when we fix the et of hyperplane H, we need to decide the number D of direction to ue and how are we going to ditribute them acro the plane. With repect to the number of direction we face the tandard trade-off between accuracy and efficiency. McClure and Vitale (1975) prove convergence of the outer approximation for mooth convex et when D. With repect to the direction we want to accumulate more in the part of the plane where the et ha quicker change in hape. Step 2 i required to atify that the initial gue of the equilibrium correpondence W 0 (, ) i a trict uperet of the true correpondence and hence, by our reult in the previou ection, due to converge after repeated application of the operator B( ). Note that thi gue i facilitatedbythefactthatweknowthatnopayoff for the government higher than thoe provided by the Ramey equilibrium can ever be utained. Step 3 in the outline involve two tak: to compute the wort punihment conditional on the current gue of the value correpondence, κ n (, ), and to update the ditance CW n (k i, j ). To find κ n (, ) we olve κ n (k, ) = max τ k (),τ l () ( min u(c(),l()) + G(g()) + β X ) π( 0 )v( 0 ) c(),l(),k(),v( 0 ),m( 0 ) 0 Â 19

20 uch that m( 0 ),v( 0 ) W n 1 (k(), 0 ) for all 0ª (30) (c(),l(),k(),g(),w(),r()) CE (Ξ (, k, τ k (), τ l (),m( 0 ) for all 0 Â )) (31) thatitheameproblemthanthedefinition in ection 4 except that now the continuation payoff value mut belong to W n 1 (, ). Thi lat tep i one of the main difference with repect to the work of Benhabib and Ruticcini (1997) ince we have a contructive procedure to computationally generate κ n (, ) intead of relying of an exogenouly given function or on ome gue and verify method that can be difficult to implement in practice. Oneimportantiuewhenolvingforκ n (, ) i that k() may not be one of the point in k grid.tofix thi problem we could linearly interpolate between point in the k grid to find admiible value for m( 0 ),v( 0 ). Vitale (1979) how that if the correpondence i continuou and exactly known at the point in the grid, then a linear interpolation cheme converge to the true correpondence. Unfortunately there are not joint convergence reult when we do not know the exact value of the correpondence atthegridbutweapproximate themuing aetofhyperplane H. In fact example can be built where the computed correpondence i not an outer approximation outide the point in the grid (ee Sleet and Yeltekin (2001)). We will dicu alternative our alternative interpolation cheme in the appendix. Alo it i important to remember that through contraint (31) we alway impoe that invetment i nonnegative. Thi contraint i only going to be relatively important in the firt period of the game, where the government will tax capital heavily ince doing that i cloe to levying a lump-um tax. Nonnegativity of invetment reduce the incentive of the government to do o, making the problem of the government non-trivial in the firt period 9. After the firt period the contraint i unlikely to bind even if the government deviate to an equilibrium with higher tax rate. In thi cae the conumer know that the government will tax more heavily capital in the future and hence they want to reduce their capital holding but they prefer to do o progreively without net diinvetment. Otherwie they 9 Although ince the government cannot carry debt, the incentive to tax heavily in the firt period, invet in claim againt the private ector and ue the interet to reduce future taxe (and the ditortion aociated with them) i abent. Thi mechanim i, for intance, quite important in the characterization of optimal fical policy in Chari, Chritiano and Kehoe (1994) and it account for mot of the welfare gain of a witch from a ytem like that of the United State to Ramey taxation. 20

21 will ubtantially reduce their marginal utility of current conumption and they could not atify their Euler equation between conumption today and the next period 10. However the preence of the contraint greatly help to avoid ome numerical intabilitie of the algorithm. Thi pragmatic conideration jutifie our choice of including the contraint. The econd tak of the tep 3 i to update the ditance CW n (k,)( ). Baically we want to find, for each direction D, themaximumpayoff equilibrium value, given the retriction that we tay in a competitive equilibrium and that equential rationality i atified, for the current approximation of the equilibrium value correpondence. Since we can think of each direction a a particular weighting of the payoff, the update of the ditance come from olving the following auxiliary problem for each direction z: uch that CW n (k,)(z) =maxh(z, 1)m()+H(z, 2)v() (32) x x =(τ k (), τ l (),c(),l(),k(),m( 0 ),v( 0 ) for all poible 0 Â ) m() =u c ()(1+r()(1 τ k ()) δ) (c(),l(),k(),g(),w(),r()) CE (Ξ (, k, τ k (), τ l (),m( 0 ) for all 0 Â )) v() =u(c(),l()) + G(g()) + β X 0 Â π(0 )v( 0 ) κ n (k, ) C n 1 W (k(), 0 ) (z) H0 [m( 0 ),v( 0 )] The interpretation of the contraint i imilar to previou dicuion. The firt one i jut a definition of the vector x and the econd one the definition of the marginal value of capital. We again require that the different variable are part of a competitive equilibrium and that the government doe not have an incentive to deviate with the wort punihment function a computed. The only new contraint i the lat one, that ay that the future payoff earch a argument of the maximization indeed belong to the current gue of the equilibrium correpondence and hence they are admiible continuation utilitie. Finally tep 4 involve the update of the gue of the equilibrium correpondence and the iteration of the previou tep until convergence i achieved up to the approximation error involved in the outer nature of our procedure. 10 Notice that thi equation need to hold with equality ince there are enough reource to finance current conumption and ome capital can be tored for the future becaue τ k () < 1. The contraint may be binding, though, if the government deviate to higher tax rate for only one period. 21

22 Inner Approximation The econd approach to find the equential equilibrium value correpondence i to ue an inner approximation. Here the idea i to find q point (vertice) p in the frontier of the et and build their convex hull. Clearly the et contructed in a uch a way will be alway be a ubet of V and generically a trict ubet and hence we will have point w 0 () uch that w 0 () / co(p) but w 0 () V (k i, j ). Anoutlineofthealgorithmiafollow. 1. Fix a number q of point and H. 2. Start with an initial p 0 mall enough uch that p 0 V (k i, j ). Thenwehavethat co(p 0 )=W 0 (k i, j ) V (k i, j ). 3. For n =1,... update CW n (k i, j ) at each row (direction) z of H. The payoff m() and v() give u the new vertice p n. 4. Make co(p n )=W n (k i, j ). 5. Iterate until convergence, V I (k i, j )=B(V I (k i, j )) = lim n W n (k i, j ),wherev I (k i, j ) i the q point inner approximation of V (k i, j ). The algorithm preent a trong reemblance with the outer approximation and only need minor comment. In addition of picking D hyperplane now we need to elect q point balancing accuracy and peed and ome appropriate collocation criteria looking to aure that in fact all the choen point are point of V (k i, j ) 11. The role of the hyperplane in thi algorithm i only to provide the weight to be ued in the optimization problem in tep 3. Then we built the convex hull of thee point p 0 a the initial gue W 0 (k i, j ). The building of the convex hull of a et of point i a well undertood problem for which a wealth of efficient algorithm exit (ee the dicuion in de Berg et al. (2000)) that we do not dicu in detail. The main part of the algorithm i the tep 3, but ince thi tep i exactly equivalent to tep 3 in the outer approximation method (including olving for the wort punihment function) we omit detail and jut refer to our explanation above. The only important point to emphaize i that in the outer approximation we ue the value of the maximized function CW n (k,)(z) of the problem (32) to update the ditance while here we ue the argument m() and v() that olve the maximization. 11 In contrat with the initial gue in the outer approximation thi may not be an eay tak. 22

23 Mixing an Outer and an Inner Approximation A nice integration of both outer and inner approximation can be achieved a follow. We gue an initial large W 0 (k i, j ) and apply the outer approximation algorithm until we find V I (k i, j ). Then we can pick the equilibrium value generated in the tep 3 during the lat iteration and ue them a the original point p 0 of an inner approximation (we know that in fact thee point belong, up to the convergence criterion and floating point arithmetic) to V (k i, j ) and then co(p 0 )=W 0 (k i, j ) V (k i, j ). Thi make ure that the algorithm work properly. Alo thi initial value are very good guee and the inner approximation converge quickly. The main advantage of having both outer and inner approximation i that we have an upper and a lower bound on the true V (k i, j ) oweknowforurethatomepointare not equilibrium payoff and that ome are. We will have a region of ambiguity where we cannot determine the nature of a point but in general that region would be quite mall and, if trategie atify ome continuity criterion with repect to payoff (i.e. mall change in payoff imply mall change in trategie that upport them) the effect of the ambiguity would be minor Building Strategie Even if the et of utainable equilibria payoff provide valuable information, in general we are intereted in looking at particular trategie that implement one equilibrium. Only knowing the trategie can we make quantitative tatement about how the optimal fical policy look over the buine cycle, which i the optimal repone to ome particular hock or how doe the data compared with the model prediction. We will decribe here a procedure to compute a trategy that utain the bet equilibrium (one giving the highet value to the government). Why i it the bet equilibria intereting? Baically becaue of two reaon. Firt it i a focal point for the player. If the government can chooe an trategy and the conumer their belief, it i plauible to think that they may a well pick the one that implement the bet allocation. Second becaue we learn a lot about a et from looking at extreme and learning about the bet trategie convey a lot of information about all the other trategie. 12 For example, in our computation, the trategie that implement the bet equilibrium converge much more quickly than the equential equilibrium value correpondence. 23

24 An outline of the procedure we propoe i: Pick an initial point in the eat frontier of the et. Thi point i aociated with ome hyperplane h. Compute the next period variable implied by the optimal choice in the point uing the olution x from the update of ditance in tep 3 of either the outer or the inner approximation in the lat iteration of the approximation at the direction determined by the direction h. Find the new point in the value correpondence and the aociated hyperplane. In the cae that we fall into an interection point of two hyperplane in the outer approximation (or we fall in a vertice in the inner approximation) randomize with appropriate weight. Iterate. Aitcanbeeeninthioutlinefinding trategie that upport a vector of utainable equilibrium payoff i direct once we have olved the approximation decribed above. Alo it i important to notice that each point in an equilibrium value correpondence may be implemented by more than one trategy, o not only we have in general a large et of equilibria payoff but alo different way to achieve them. However in the cae of the tochatic growth model, the problem ha enough convexity that we have only one trategy (although it may involved randomization) to implement each particular point. Alo we can ue the olution aociated with finding the wort punihment function κ n (, ) to recurively built the trategie aociated with the wort equilibrium (the trigger trategie). Thi trategie are intereting, among other thing, to evaluate how plauible each equilibrium i (the hypothei being than le plauible equilibria require more involved trigger) and to compare reputational mechanim with ome more impler olution concept a a Markov Perfect Equilibria. 6. Calibration We will parametrize our model to match baic propertie of the U.S. economy under Ramey taxation. We take a our utility function, following Klein, Kruell, and Río-Rull (2001)) u(c, l) =(1 β) ((1 α p )α c ln c +(1 α p )(1 α c )ln(1 l)) 24