A Game-Theoretic View of the Fiscal Theory of the Price Level Marco Bassetto  July 12, 2000 Abstract The goal of this paper is to probe the validity of the scal theory of the price level by modeling explicitly the market structure in which households and the governments make their decisions. I describe the economy as a game, and I am thus able to state precisely the consequences of actions that are out of the equilibrium path. I show that there exist government strategies that lead to a version of the scal theory, in which the price level is determined by scal variables alone. However, these strategies are more complex than the simple budgetary rules usually associated with the scal theory, and the government budget constraint cannot be merely viewed as an equilibrium condition. 1 Introduction This paper stems from a recent heated debate on the relationship between the price level and scal policy. This relationship has a long tradition in macroeconomics. Milton Friedman stressed extensively that in ation is chie y a monetary phenomenon and that price stability can be achieved by stabilizing the money supply.1 Sargent and Wallace 16 showed that monetary and scal policy are intertwined through the government budget constraint; the objective of a stable money supply is inconsistent with a persistent scal de cit. Sargent 15 studied several in ationary episodes and argued that scal de cits were primarily responsible for the ultimate recourse of policymakers to the printing press. A recent string of papers2 have pushed the link between the price level and scal policy further, developing a scal theory of the price level". These papers observe that in low and  Preliminary and incomplete. Comments welcome. I am indebted to Lawrence Christiano and Larry Jones for helpful comments and discussions. Address: Department of Economics, Northwestern University, 2003 Sheridan Road, Evanston, IL 60208; phone 847 491-8233; email m-bassetto@nwu.edu 1 See e.g. Friedman and Schwartz 7 . 2 To my knowledge, Leeper 11 started this line of research, and Sims 19 and Woodford 21 are the seminal contributions. Woodford has developed the idea further in 22, 23, 25, 24 . Cochrane 4 has extended the analysis to long-term debt, and Dupor 6 to the exchange-rate determination in an open-economy framework. Loyo 12 has applied the theory to study in ation episodes in Brazil. 1 moderate-in ation countries governments borrow mainly by issuing nominal bonds. The presence of nominal bonds introduces an additional link between scal and monetary policy, and the revenues that the government can achieve by implicitly defaulting on its debt through in ation are much larger than the seigniorage revenues emphasized by Sargent and Wallace. However, the key distinction between the traditional" view of in ation and the scal theory of the price level is much deeper than the mere presence of nominal debt. According to the scal theory, money is completely secondary in determining the price level, which is instead driven by the sequence of primary surpluses and de cits. The price level is simply the instrument through which the real value of debt stays in line with the present value of future government surpluses. The key di erence between the scal theory and the traditional view lies in the interpretation of the government budget constraint, which links the real value of debt to the present value of primary surpluses the government will run in the future. The advocates of the theory view this link as an equilibrium condition: an imbalance between the real value of debt and the surpluses would trigger changes in the price level that would lead back towards an equilibrium, either by reducing or by increasing the value of the nominal debt. The traditional view interprets the link as a constraint on policy, which forces government action, either through a scal adjustment or through a default on debt or through money-induced in ation, whenever the real value of debt and the present value of primary surpluses tend not to be equal. It is this di erence that has spurred the major controversy.3 The goal of this paper is to reach a clearer and less controversial understanding of the con- straints imposed on monetary and scal policy by their interdependence. I describe the entire economy as a game, and I provide a market microstructure that shows how prices arise from the actions of the players in the economy. Speci cally, prices are formed by the bidding process of households and the government on specialized trading posts where goods and assets are traded pairwise. While the market structure I describe is highly stylized, it is able to clearly set apart constraints on the set of actions that the government can take from relations that hold only in equilibrium, thereby shedding light on the key source of controversy. In a companion paper 1 , I show that the standard de nition of a competitive equilibrium and of a commitment equilibrium fail to describe out-of-equilibrium paths, and I provide a more complete de nition of an equilibrium for an economy with a large player the government and many atomistic players. In this paper, I apply the de nition to a speci c game which is well suited to address the validity of the scal theory of the price level. I show that, in the environment I describe, there exist government strategies that lead to a version of the scal theory, in which the price level is determined by scal variables alone. However, these strategies are more complex than the simple budgetary rules usually associated with the scal theory, and the government budget constraint cannot be merely viewed as an equilibrium condition. Section 2 illustrates the scal theory of the price level and the theoretical criticism against it. Section 3 describes the market structure I assume. Section 4 contains the main the results of 3 Among the authors that have attacked the view that the government budget constraint is purely an equilib- rium condition are Buiter 2 . Other papers that express similar views are by McCallum 13 and Kocherlakota and Phelan 10 . 2 the paper, section 5 talks about extensions that are in progress and section 6 concludes. 2 Ricardian and non-Ricardian Policy Rules In this version of the paper, I study a cashless economy, in which money is purely a unit of account. This speci cation is often pursued by the papers that adopt the scal theory of the price level, consistently with their idea that money as a medium of exchange is secondary in determining the price level. I choose a cashless speci cation because it is simpler and still captures the main insights of the debate. In section 5, I discuss how I plan to introduce a transaction role for money in a future extension and what aspects of the debate on the scal theory can only be addressed by this introduction. Let us consider an economy with a continuum of identical households that live for two periods 1 and 2 and a government. Households receive a constant exogenous endowment of a single homogeneous good in each period, which we normalize to 1. A nonconstant endowment and production could be easily introduced without altering the results, but they would make the notation more cumbersome and would introduce many more markets to keep track of in the game-theoretic version. Each household starts the rst period with B1 units of government bonds. A government bond is a claim to 1 dollar", which is just a unit of account. All debt is assumed to mature in one period; once again, this is not an important assumption, but saves on notation considerably. The government has access to lump-sum taxes in both periods; with the tax revenues T1 and T2 , it nances some exogenous government spending in either period G1 and G2 , as well as repayment of its original debt. We assume no uncertainty. Households have preferences given by uc1  + uc2  1 where u is a strictly increasing and concave function and c is consumption in period j . We j use lower-case letters for variables that refer to a single household, and upper-case letters for the corresponding aggregates. We only look for symmetric equilibria, in which each household is taking the same action; therefore, lower-case and upper-case variables will always coincide in equilibrium. Government spending does not enter in the households' utility; as usual, it could be added in a strongly separable way without a ecting the results. The household's ow budget constraints are b2 P1 c1  P1 1 , T1  + B1 , d R1 2 P2 c2  P2 1 , T2  + b2 d Pj is the price level, i.e., the inverse of the value of a dollar; R1 is the nominal interest rate in the economy and b2 is the amount of newly-issued government bonds with period-2 maturity that d the household demands in period 1. 3 The government budget constraint for this economy is4 B2 P1 G1 = P1 T1 + R1 , B1 3 P2 G2 = P2 T2 , B2 where B2 is the supply of bonds in period 2. A competitive equilibrium is an allocation C1; C2; B2 , a price system P1; P2; R1 and D a government policy T1 ; T2; B2 such that: i given the price system and the government policy, the allocation maximizes the households' utility subject to the budget constraint 2; ii the government budget constraint 3 is satis ed; iii Markets clear, i.e. B2 = B2 . D The de nition of a competitive equilibrium describes the actions taken by the households and the government at the equilibrium; it does not specify what would happen if the government took a di erent policy, or if the price system were di erent from the equilibrium one. We de ne a scal policy rule as a mapping from the price P1 into T1, and from the vector P1; P2 to T2 . While this economy does not have money, we still de ne a monetary policy rule as a mapping from the price P1 to an interest rate R1 . The rationale behind this de nition is the perception that the cashless economy is only a limiting concept and that the central bank retains the ability to peg the nominal interest rate as we drive the economy to the cashless limit. In the game we describe below, the ability of the government to peg the interest rate will explicitly come out of the model. The de nition of a scal and monetary policy rule here is more limited than the one in Woodford 21, 22 or in Kocherlakota and Phelan 10 , as I specify which variables the government is targeting in its rule. This is only done for simplicity of exposition. We de ne a policy rule to be the combination of a scal and monetary policy rule. The literature distinguishes two types of rules, which I will call Ricardian and non-Ricardian, following Woodford 22 . A policy rule is Ricardian if it satis es the government budget con- straint for any price vector; it is non-Ricardian otherwise. This de nition is justi ed by the fact that, in any Ricardian rule, the present value of taxes payments from the households to the government less the value of debt present value of payments from the government to the households is identically equal to the present value of government spending, a constant that does not depend on the price levels P1 ; P2. With a Ricardian rule, an increase in P1 that decreases the value of nominal government debt held by the households is matched by a reduction in the present value of taxes, and does not a ect the households' choices, provided the real interest rate remains constant. While the previous argument justi es the name Ricardian", the key distinction from our perspective is that a non-Ricardian policy rule allows the government to violate its budget 4In what follows, I do not allow the government to waste any resources other than spending itself.... The analysis would be similar if the government had access to free disposal; in that case, violations of 3 would only be a problem when taxes are too small. 4 constraint out of equilibrium, whereas a Ricardian rule meets the budget constraint both in and out of an equilibrium. If government spending were allowed to vary, instead of being exogenous, the name Ricardian vs. non-Ricardian would no longer capture the key di erence. Proponents of the scal theory of the price level assume that the government can commit to non-Ricardian policies. While their arguments are not cast in a model that properly speci es out of equilibrium behavior, their reasoning is a variation of the following. For any price P1 , tax T1 and interest rate R2 0, it is possible to nd a supply of government debt B2 such that the ow budget constraint is satis ed in period 1. If the policy rule is non-Ricardian, then there are some price vectors P1; P2 for which the budget constraint in period 2 is not satis ed; at this price vector, the government would o er" bonds B3 that mature after the end of the economy to meet its ow budget constraint. Since nobody is willing to buy these bonds, there is excess supply and prices will have to adjust. The opponents of the scal theory5 insist that any rule that is non-Ricardian is simply a misspeci cation: no matter what the prices are, the government should always choose a policy that satis es its intertemporal budget constraint, which includes the transversality condition B3 = 0. In order to deem non-Ricardian rules admissible, it is necessary to interpret the intertem- poral budget constraints di erently: the households' budget constraints are viewed as binding both in and out of equilibrium, whereas the government budget constraint is interpreted as a government valuation equation" that only holds at the equilibrium price see e.g. Cochrane 5 . Woodford 25 justi es this asymmetry with two arguments: i if the households were not subject to budget constraints, they would demand an in nite amount of goods, so there would be no equilibrium; the same is not true for the government, which for exogenous reasons has an interior satiation point; ii households are price takers, whereas the government is a big player capable of moving prices. Neither of these arguments is compelling. The possibility or impossibility of violating the budget constraint out of equilibrium should not have anything to do with preferences. Having the ability to a ect prices is not the same as having the ability of violating a budget constraint for any given price vector. The admissibility of non-Ricardian rules has dramatic implications on the determinacy of the price level, which we now turn to. Proposition 1 If the government adopts a Ricardian policy rule, P1 is indeterminate; more precisely, given any strictly positive value, there exists a competitive equilibrium in which P1 attains that value. Proof. Under a given policy rule, a symmetric competitive equilibrium is characterized by the following equations: 5 See e.g. Buiter 2 , Kocherlakota and Phelan 10 . 5 i rst-order conditions for the household's problem P1 R1 0 u0 C1  = u C2  4 P2 ii household budget constraints at equality B2 P1 C1 = P1 1 , T1  + B1 , d R1 5 P2 C2 = P2 1 , T2  + B2 d iii government budget constraints at equality 3 iv market clearing conditions C1 = 1 , G1 C2 = 1 , G2 6 B2 = B2 d v policy rule speci cation: T1 = T1P1 , R1 = R1 P1 and T2 = T2 P1; P2.  Let P1 be a strictly positive value. We show that, if the policy rule is Ricardian, there exists a   unique competitive equilibrium in which P1 = P1. Given P1 , the policy rule speci es a unique value for T1 and R1. We can substitute these values to obtain a unique value for the supply of bonds B2 from the government budget constraint. Consumption and the demand for bonds can be uniquely determined by the market clearing conditions; these choices satisfy the household budget constraint in period 1 by Walras' law, as can be veri ed by substitution. The price level in the second period is determined by 4: even though the government cannot set the initial price level, it controls in ation through the choice of the nominal interest rate. If the policy rule is Ricardian, T2P1; P2 is consistent with the period-2 budget constraint of the government; nally, the household budget constraint in period 2 is redundant because of Walras' law. QED. Proposition 1 is the cashless counterpart to the well-known result that, in many monetary models, nominal interest-rate targeting leads to price indeterminacy. While in a Ricardian regime the scal policy cannot help in determining the initial price level, the result obviously changes when we no longer require T2 P1; P2 to be such that the government budget constraint is met at all prices. The scal theory of the price level is most often derived by assuming that the government sets the real value of taxes T1 and T2 and the nominal interest rate R1 independently of the prices. Proposition 2 Assume that the policy rule speci es unconditional values for T1 , T2 and R1 and that B1 0. There exists at most one competitive equilibrium that is consistent with such a rule; the equilibrium exists provided T1 or T2 are su ciently large. 6 Proof. A competitive equilibrium must satisfy the same equations we listed in proposition 1. As before, we can uniquely determine consumption from the market clearing conditions. We can solve 3 and 4 as a system of 3 equations in P1 , P2 and B2 , which yields the following unique result: B1 P1 = T1 , G1  + T2 , G2 00  2  u u C 1 C B1 R1 P2 = 0 1 7 T1 , G1  0 2  + T2 , G2 u u C C B1 R1 T2 , G2  B2 = T1 , G1  00 2  + T2 , G2 u u 1 C C This system yields positive prices P1 and P2 if T1 or T2 are large enough. Finally, market clearing implies that B2 = B2 , and the household's budget constraints are satis ed by Walras' law. QED. d The policy rule described in proposition 2 is consistent with a competitive equilibrium only if the initial real value of debt takes a particular value. This is the source of the scal theory of the price level: if taxes do not respond to meet the government budget constraint, then the price level must do so to guarantee that the real value of debt acts as the residual variable. Taxes must not be too low, for otherwise they would require a negative real value of debt, which is ruled out assuming B1 0 as prices must be positive. The scal theory of the price level follows from the assumption that the policy rule in propo- sition 2 or variants of it, as in Loyo 12 , where the interest rate reacts to in ation is a good description of the actual policy rule followed in many countries. Accordingly, the papers that advocate the scal theory view the price level as being primarily determined by the dynamics of government de cits surpluses and debt. Both the papers that advocate the scal theory and those who deny its possibility or plausibil- ity contain discussions of policy rules and often vague descriptions of out-of-equilibrium dynamics and adjustment to the equilibrium. However, all of these papers de ne an equilibrium as a com- petitive equilibrium, which is not a good concept to address the consequences of deviations from the equilibrium path. To my knowledge, no paper has attempted to cast the problem in an environment in which it is possible to explicitly discuss the household and government behavior out of the equilibrium path. By writing the economy as a game, I am able to answer explicitly the following questions: is it possible for the government to commit to non-Ricardian policy rules? Can price determinacy be achieved through the scal policy when the monetary policy is characterized by interest-rate targeting? What actions lead to out-of-equilibrium prices, and what is the evolution of the economy out of equilibrium? 3 A Game-Theoretic Version of the Economy In order to model the economy we described above as a game, we need to be explicit about the way prices are formed from the actions by the households and the government. In what follows, 7 I model the market structure as a version of trading posts that is similar to Shubik 18 .6 While I make a number of assumptions on the details of how trading takes place, it is straightforward to show that these details could be changed without a ecting the results. What can potentially make a di erence is the main assumption that trading takes place simultaneously and through trading posts.7 The players of the game are households and the government. Every time a player wishes to trade, it has to submit a bid to a specialized trading post, which I will equivalently call a market". Each market deals with pairs of goods or assets, and there is a market for any exchange that the government and the households may wish to entertain. Accordingly, in period 1 there are 3 trading posts: in the rst, goods are exchanged for maturing bonds; in the second, goods are exchanged for newly issued bonds that mature in period 2; in the third, maturing bonds can be exchanged for newly issued bonds that mature in period 2. In period 2, the only trading post is one where goods are exchanged for maturing bonds. As in Shubik 18 , each household that wants to trade must submit an unconditional bid for the amount it wishes to sell on a given market. The bid must represent a quantity of the good or bond sold, rather than bought, because only in this way households can meet their binding obligation at any price. In equilibrium, households have perfect foresight about the relative price in each market, and a single household cannot alter any price through its actions. For this reason, households would be strictly indi erent between using unconditional bids or more-sophisticated bid schemes. In some of the markets, the government has more degrees of freedom in submitting bids than the household do: as a seller of future bonds, the government is not constrained by a limited endowment, as it can freely print as many bonds as it wishes. For this reason, in such markets the government can either submit a sale bid for a speci c quantity, or set a price at which it is ready to meet any demand. I assume that the government submits unconditional sale bids in all markets except the one where maturing bonds are exchanged for newly-issued bonds, in which the government sets the price. This assumption retains the analogy with the previous section in which the government targeted interest rates. The results I establish are independent of this assumption. Being a large player, the government could potentially have an interest in submitting more- complex bids than just setting a price or a quantity o ered. As an example, it could submit complicated bids, in which rationing is sometimes involved. However, I show that the government can attain price determinacy even by using the simple bidding scheme proposed above, so nothing would be gained if the government were to resort to more-complicated mechanisms. Each trading post except the one that determines the nominal interest rate clears simply by setting the price equal to the ratio of the supply of the two objects to be exchanged; at that 6 I assume enough symmetry that these trading rules yield the Walrasian outcome. As Shubik 18 points out, this is far from guaranteed in general. A more-complicated version with multilateral trading posts could overcome this problem. 7 An alternative model of the microstructure of the determination of prices in a competitive equilibrium is provided by the search-theoretic approach developed by Rubinstein and Wolinsky 14 and Gale 8, 9 . However, this approach is considerably more cumbersome to deal with, and introducing a government in their environment would require signi cant adaptations that are currently beyond the scope of this project. 8 price, market clearing is achieved as an identity, independently of the bids, and exchange takes place. As in the previous section, lower-case variables refer to single households and upper-case variables refer to aggregates. The timing of the economy is as follows. i Households start with 1 unit of the period-1 good and B1 units of government debt maturing in period 1. The government levies a rst installment of period-1 taxes, T11 2 0; 1 and sets a price P 1 2 at which it stands ready to exchange maturing bonds for new bonds. B B From here on, I index prices by the objects that are being exchanged at each trading post. The government submits a sale bid for C1 1 units of goods in the market for maturing B bonds, subject to C1 1  T11. It also submits a sale bid for B2 1 units of new bonds in B C exchange for goods. While we assume here that the government submits its bids rst, nothing would change if we assumed that the bids are submitted jointly by the government and the households; this is true because we only look at commitment equilibria in which the government speci es its strategy ex ante. ii Trading opens. There are bilateral trading posts for each possible exchange; in our case 3 exchanges are possible: goods for maturing government bonds, goods for new bonds issued by the government and maturing bonds for new bonds. Each household may submit a sale bid for b1 1 units of bonds in the market for goods, and another sale bid for b1 2 units C B of bonds in the market for new bonds maturing next period, subject to the constraint that b1 1 + b1 2  b1 B1 , i.e., the sale bids cannot exceed the total amount of bonds the C B household starts with. I use superscripts to indicate the object the player wishes to buy in each market: e.g., C1 represents period-1 goods, B2 represents bonds maturing in period 2. There is no point in distinguishing between lower- and upper-case on the superscript, as it only refers to the type of good, not the quantity; for this reason, I always use upper-case letters. Each household may also submit a sale bid of c1 2 units of goods in exchange for B new bonds, subject to the constraint that c1 2  1 , T11. B iii For the markets in which the price is not set by the government, the ratio of the quantities of the unconditional bids sets the price and exchange takes place. The government meets the demand of new bonds in the market in which it sets the price. We thus have B 1 C P 1 1 = 11 C1 C B B B 1 C 8 P 1 2 = 22 C1 C B B B2B1 = B 2P B 1 1 2 B B The relative price of goods and maturing bonds P 1 1 determines the value of the unit of C B account the dollar" for the cashless economy. For this reason, I interpret P 1 1 as the C B general level of prices; it thus corresponds to P1 as de ned in section 2. I explain below that 9 this may be di erent in a model in which there is money and I explain how the analysis will be generalized. P 1 2 is the relative price of the unit of account in the two periods, B B i.e., it is the nominal interest rate in the economy, which we called R1 in the previous section. Here and throughout the rest of the paper, prices are not de ned on markets in which either side contains no bids; any positive bid on a market where no bids are posted on the other side is wasted. iv The government levies a second installment of taxes or transfers T12 2 ,T11 + C1 1 , B B2 1 P 1 2 ; 1 , T11 + C1 1 , C1 2 . The bounds ensure that the government has enough C C B B B resources to carry out the transfer or the households have enough resources in the aggregate to meet the tax obligation. If an individual household bids more than the others, it might not have enough resources to meet the tax obligation at this stage. We assume that the government can in ict an arbitrarily negative punishment to any household that is unable to meet its tax obligations, so it is always optimal for a household to plan to have enough resources left to pay for taxes.8 Any unmet tax obligation is distributed evenly across remaining households.9 v Consumption and government spending take place. Each household consumes b1 1 C c1 = maxf0; 1 , T1 , c1 2 + B g 9 P 1 1 C B where T1 = T11 + T12 and starts period 2 with b2 = b1 2 P 1 2 + c1 2 P 1 2 units of nominal B B B B C B bonds. The government spends G1 = T1 + B2 1 P 1 2 , C1 1 C C B B 10 units in the rst period. vi Households start with 1 unit of the period-2 good. The government levies a lump-sum tax T2 2 0; 1 . In the second period, we do not distinguish between a rst and a second install- ment in taxes, although we could do so. In the last period, the government cannot raise any resources by borrowing and hence cannot face an unexpected shortfall in its resources; as a consequence, distinguishing between a rst and second installment is super uous. The only market open in period 2 is the one where maturing bonds are traded for goods. The government submits a bid C2 2  T2 , G2. B vii Each household submits a bid b2 2  b2 . C viii The price is determined as before by the ratio of bids, i.e. 2 = B2 2 C P 2 2 11 C2 C B B 8 B B If limc!0 uc = ,1 and T12 1 , T11 + C1 1 , C1 2 , a su cient punishment is for the government to tax away any residual endowment the household has in that period. 9 The bounds on T 2 guarantee that there will be enough resources to be raised even if they are not evenly 1 spread in the population, so that the government strategy is feasible even out of equilibrium. 10 ix Each household consumes b 2 C c2 = 1 , T2 + 2 12 P 2 2 C B The government spends G2 = T2 , C2 2 B 13 The household's preferences over the outcomes are described by 1. As for the government, the papers that address the scal theory of the price level do not model its preferences explicitly. In line with the exogenous policy they take, I look for strategies that let the government achieve  an exogenous target" level of taxes, which I normalize to T in both periods.10 De nition 1 A competitive equilibrium is an allocation C1; C2; T11; T12; T2 ; B2; B1 1 ; B1 2 ; C1 2 ; B2 2 ; C1 1 ; B2 1 ; B2 1 ; C2 2  C B B C B B C B and a price system P 1 1 ; P 1 2 ; P 1 2 ; P 2 2  C B B B C B C B such that: i Given the price system and taxes T11; T12; T2 , C1; C2; B2 ; B1 1 ; B1 2 ; C1 2 ; B2 2  solves the C B B C household maximization problem: max C1 B2 B2 C2 7 uc1  + uc2  s.t. 1 2 2 1 c ;c ;b ;b ;b 1 1 ;c ;b 2 2R+ b1 1 C c1 = 1 , T11 , T12 + , c1 1 B P 1 1 C B b2 2 C c2 = 1 , T2 + 14 P 2 2 C B b1 1 + b1 2 C B  b1 b2 = b1 2 P 1 2 + c1 2 P 1 2 B B B B C B b2 2  b2 C c1 1  1 , T11 B ii The government's actions satisfy the feasibility requirements T11 2 0; 1 C1 1B 2 0; T11 T12 2 ,T11 + C1 1 , B2 1 P B C 1 2; 1 C B , T11 + C1 1 , C1 2 B B 10 The assumption of a constant target can be easily relaxed without a ecting any of the results. 11 iii Markets clear and the government budget constraints hold, i.e. equations 8, 11, 10 and 13 are satis ed. As usual, the de nition of a competitive equilibrium only involves only the outcome of the game. The information a competitive equilibrium gives us is that each household would optimally choose the prescribed allocation if it expects everybody else to choose the same allocation, the government to follow the speci ed policy and the price system to be the one included in the de nition. A competitive equilibrium does not convey any information on how the households or the government would react if people behaved di erently. Compared with the de nition of a competitive equilibrium in section 2, the only di erence is that we need here to specify the trade volume and the relative price in each market. The set of consumption levels C1; C2, prices P1 = P 1 1 ; P2 = P 2 2 ; R1 = P 1 2 , government taxes T1 ; T2 and period-2 bond holdings C B C B B B B2 = B2 compatible with a competitive equilibrium is the same under both de nitions; the d latter de nition only speci es more details of how trading actually takes place within the market structure assumed here. A household strategy is the following: 1. bids b1 1 ; b1 2 ; c1 2  as functions of the actions taken by the government up to that node, C B B i.e. T11; P 1 2 ; C1 1 ; B2 1 ; B B B C 2. a bid b2 2 as a function of the government choices T11; T12; T2; P 1 2 ; C1 1 ; B2 1 ; C2 2 , of C B B B C B the aggregate bids by the households in period 1 B1 1 ; B1 2 ; C1 2  and of its previous bids C B B b1 1 ; b1 2 ; c1 2 . C B B Consumption was not included, as it can be deducted mechanically from 9 and 12. A government strategy is the following. 1. A tax T11, bids C1 1 , B2 1 and a price P 1 2 . B C B B 2. A tax T2 as a function of the previous actions taken by the government T11; P 1 2 ; C1 1 ; B2 1  B B B C and by households B1 1 ; B1 2 ; C1 2 . The actions taken by each individual household are C B B unobservable except to the household itself; only their aggregates are common knowledge. I dropped T12 and C2 2 from the de nition of a government strategy: they are determined as a B residual by 10 and 13. I assume that the government can commit to a strategy before the game begins; time in- consistency is not an issue I am interested in, since government preferences are not explicitly modeled. In this paper, I am only studying whether there exists a strategy in the game that corresponds to the scal theory of the price level. Establishing whether such a strategy is part of a plausible equilibrium would require to model more completely the government preferences and is beyond the scope of this work. In this setup, commitment means that there is an additional stage at the beginning of the game in which the government picks commits to the strategy it will follow throughout the game I described. After this initial stage, the government's actions are entirely determined by it, so that in the subgame that ensues only the households are players. This de nition corresponds 12 to the one in Schelling 17 . In a companion paper Bassetto 1 , I discuss more in detail some issues relating to the existence of a subgame perfect equilibrium in the game with commitment, and I contrast the de nition of a commitment equilibrium given here with that contained in Chari and Kehoe 3 and Stokey 20 . 4 Ricardian and non-Ricardian Strategies in the Game It is interesting to study two di erent cases. In the rst case, government spending is identically zero; in this case, the target level of taxes always exceeds spending and there is never a need for the government to raise additional resources through borrowing.11 Government debt exists in this case only as an initial condition, and is repaid using the revenues in excess of spending. In the second case, we maintain the assumption that G2 = 0, but we assume that G1 T : in  the rst period, the target level of taxes in insu cient to nance government spending, and the government needs to raise additional resources by borrowing. We do not consider the case in  which G2 T : this would only be possible if the government started with negative debt B2 , which we rule out. I am interested in knowing when and whether the government can adhere to its target level of taxes both in and out of the equilibrium, and what are the minimal" deviations that are needed if it is impossible to keep faith to the target. 4.1 No Government Spending Proposition 3 If G1 = G2 = 0 and B1 0, there exist government strategies in which taxes are  T both in and out of equilibrium. If the government adopts any such strategy, there is a unique sequential equilibrium outcome.12 Furthermore, any such strategy achieves the same initial price level P 1 1 , whereas in ation and hence the price level P 2 2 depends on the particular strategy. C B C B The complete proof is in the appendix; I describe here the outline and the intuition. The  government strategy sets T11 = T , and the nominal interest rate P 1 2 at any strictly positive  B B level. The government bids the entire amount C1 1 = T in exchange for maturing bonds while B it does not submit any bid on the market between goods and new bonds. In period 2, the   government levies a tax T2 = T and uses the revenues to bid C2 2 = T in exchange for bonds B maturing in period 2. It can be immediately veri ed from the description of the game that these actions can be taken independently of the choices by the households, and that they deliver the target level of taxes independently of the household actions and hence both in and out of equilibrium. With the given government strategy, there is a unique equilibrium, in which the unit of account the dollar" has a well-de ned value. As in Cochrane 5 , government debt in this example is essentially an entitlement to a future payo and a dollar" simply represents a share This analysis could easily be extended to cases in which government spending is positive but below the target 11 level of taxes in both periods. 12 While I adopt sequential equilibrium as the equilibrium concept here, I never specify beliefs. In the two-period game, beliefs are irrelevant, as the optimal choice for each household is to bid all of its debt holdings at time 2. 13 of the debt; in equilibrium, households will submit bids such that these shares are correctly priced as if they were any other asset. We want next to establish whether the suggested government strategy is Ricardian. If we write the government budget constraint adapted from 3, we obtain   TP 2 2 B1 = T P 1 1 + C B 15 C B P 1 2 B B which only holds at the equilibrium price level. For prices that are out of equilibrium, 15 is violated, so the strategy is non-Ricardian according to the de nition in section 2. However, prices only deviate from the equilibrium values when households fail to make their equilibrium bids. There are two types of deviations: in the rst type, households fail to redeem part of the debt. As an example, they bid less than B2 in the second period, in which case P 2 2 C B decreases and the present value of taxes seems to exceed the value of debt. This excess is only apparent, for it is the result of many households failing to claim their parts of repayments: if we only count debt that is presented for redemption, the government budget constraint holds. In the second type of deviation, households do not waste any of their debt, but they misallocate B1 across the two markets, redeeming too many bonds and rolling over too few or vice versa. Substituting 8, it can be easily veri ed that 15 always holds for prices that follow this type of deviation; the strategy is Ricardian" with respect to this type of deviations. By studying the market structure behind a competitive equilibrium, we are able to see that the government is subject to budget constraints that must hold in and out of an equilibrium: equations 10 and 13. Equation 15 is instead not a true government budget constraint, because it assumes that all of the debt will be redeemed: this is a correct assumption on the equilibrium path, but may be violated out of equilibrium. In section 2, we argued that a policy rule that satis es 3 in and out of equilibrium is called Ricardian because the present value of taxes net of debt repayment is independent of the price level, which did not happen for non-Ricardian rules. However, in this example the validity of 15 out of equilibrium is not connected to the present value of taxes net of debt repayment; in fact, this present value is independent of the price level for the government strategy we analyze, but 15 may be violated because it assumes all debt has to be repaid. 4.2 Variable Government Spending In the case discussed above, all of the debt is inherited from the past, and the government is only  setting terms to repay it. We now look at the case in which G1 T . In this case, the government would like to run a primary de cit in the rst period. In the previous example, the government participated in the markets only by buying government debt, which would have otherwise been worthless to the households; in this example, the government needs to buy goods in the rst period, and must thus persuade the households to trade resources that are intrinsically valuable to them. For the sake of simplicity, we retain the assumption that G2 = 0. While the government was able to meet its target in and out of equilibrium when spending was less than taxes in both periods, it is trivial to see that this is not possible when target spending 14 exceeds the target level of taxes. No matter what the government strategy is, households have the option of not participating in the markets where goods are traded for future bonds. If households do not participate in this market, equation 10 implies G1  T1 . In this case, there  is thus no government strategy that includes T1 = T independently of the history of play. In the environment we study, any rule that unconditionally requires the government to set spending above taxes in any given period is meaningless. The previous observation seems to defeat the scal theory of the price level. In all of the papers that I am aware of, an unconditional path for taxes and spending is assumed. Nonetheless, the following proposition rescues the scal theory by showing that the government can adopt a strategy that leads to a unique equilibrium in the game; in such an equilibrium, taxes are at the target level and the price level is uniquely determined by spending and taxes. Proposition 4 Assume that there exists a competitive equilibrium in which T1 = T2 = T and  that B1 0. Then the government can commit to a strategy such that the unique outcome of a sequential equilibrium in the subgame following the commitment coincides with such a competitive equilibrium. The complete proof is contained in the appendix. I present here the outline and the intuition behind the result. Let ~ ~ ~ ~  ~ ~ ~ ~ ~ ~ ~ ~ ~ C1 ; C2; T11; T12; T ; B2; B1 1 ; B1 2 ; C1 2 ; B2 2 ; C1 1 ; B2 1 ; B2 1 ; C2 2  C B B C B B C B 16 be the competitive equilibrium allocation and let the associated price system be ~ ~ C B ~ B B ~ P 1 1 ; P 1 2 ; P 1 2 ; P 2 2  C B C B 17 A government strategy that achieves the desired result is the following. In period 1, the ~ ~ government sets T11 = T11. It bids C1 1 units of goods in exchange for maturing bonds and B2 1 B ~ C ~ units of new bonds in exchange for goods, and sets the nominal interest rate at P 1 2 . The B B second installment of taxes T1 is set so that 10 holds; this installment depends thus on the 2 household bid C1 2 . Independently of what happened in period 1, the government sets taxes at B  ~  T and bids C2 2 = T in exchange for bonds maturing in period 2; it follows that G2 0. B The intuition behind this strategy is simple. The government cannot guarantee that borrow- ing will raise enough resources to cover the target level of spending. In order to obtain a unique ~ equilibrium outcome in which the appropriate amount C1 2 is raised through borrowing at period B 1, the government needs to ensure that the rate of return on debt looks very attractive when households lend less new resources than expected and vice versa. The government is o ering a xed amount of goods in period 2, which would naturally deliver this result if new lenders were the only claimants to that surplus; however, the period-2 surplus must be shared between new lenders in period 1 and households that roll over their initial debt. O ering a xed amount of bonds to new lenders independently of the resources they lend is su cient to ensure to guarantee them a share of the future surplus that is not too diluted if they fail to lend enough or that does not increase too much if they lend in excess of what is expected: the rate of return will thus move in the opposite direction of the amount lent, delivering uniqueness of the equilibrium outcome. 15 However, the government strategy o ers new lenders a xed amount of bonds maturing in period 2. Through this strategy, when households lend less more than the desired amount to the government, the rate of return on the debt becomes automatically very unattractive, which rules out a second equilibrium with lower higher lending. In obtaining this result it is crucial that the government is able to partially separate the resources promised to new lenders from those reserved to previous lenders that roll over their debt. In period 1, the government is selling claims to its future surplus on two markets. By choosing P 1 2 and B2 1 , it is controlling the share of that surplus that goes to either group of creditors. B B C Ceteris paribus, a higher P 1 2 and or a lower B2 1 implies a smaller share for new lenders and B B C a larger share for previous debt holders, which leads people to lend fewer new resources to the government. In order to raise exactly the target level of revenues, the government must attain the appropriate mix of debt on the two markets. The initial price level is determined by the households bids in redeeming debt for goods in period 1. These bids in turn depend on the amount of goods the government is o ering in period 1 and on the share of the future surplus that is o ered to them through new bonds. Once again, Cochrane's 5 analogy between the price of government debt and the price of stock is very well suited for the microstructure I am introducing. However, this does not imply that the government budget constraint can be viewed simply as a government valuation equation": out of equilibrium, the government is forced to raise taxes above its target level and it is the ability of adjusting its use of resources in a very speci c way that leads to uniqueness of the equilibrium. Cochrane is correct in claiming that no budget constraint forces Microsoft or Amazon.com! to adjust future earnings to match current valuations " emphasis added, but overlooks the fact that Amazon.com would indeed have to adjust their earnings if their valuation and the households' willingness to subscribe their capital changed.13 To the extent that the adjustment in their earnings would not match any alternative valuation, the uniqueness of the equilibrium valuation holds. 5 Extensions 5.1 Many periods The extension of the results derived above to a multiperiod economy is straightforward. It is particularly interesting to extend the analysis to in nite-horizon economies. The ow budget constraint of the government in an in nite-horizon economy becomes PG t t = P T + B +1 , B ; t = 1; 2; ::: t t R t t 18 t 13Microsoft may be able to promise the same earnings independently of its valuation because it is quite possible that all of its periods of negative cash- ow pertain to the past. In other words, Microsoft may just be similar to our rst example, in which the government does not spend and is only repaying old debt. The budget constraint hits the company and the government when it needs to raise fresh resources, not while it is able to nance internally any investment. 16 Unlike in the nite-horizon case, the sequence of ow budget constraints does not imply that the intertemporal budget constraint is satis ed; for this to happen, the sequence of taxes and debt that is o ered must also satisfy the transversality condition ,1 Y 1 t lim B !1 =0 19 t t s =1 R s Given a sequence of taxes, spending and prices, it is now always possible to nd a sequence of government debt that satis es the ow budget constraint in any period; it is formally no longer necessary for the government to o er bonds that mature after the end of the economy. Nonetheless, a generic sequence of taxes, spending and prices will imply a sequence of debt that violates the transversality condition, which is exactly the analogous of the condition B3 = 0 in our two-period economy. In an in nite-horizon economy, a policy rule is thus called Ricardian if it satis es the transversality condition independently of the sequence of prices, and non-Ricardian otherwise. We assume now that the household preferences are described by14 1 X t uc  t 20 t=0 with 1. The appendix shows that the results we obtained for a two-period economy extend to the in nite-horizon economy under the additional assumption that government spending is bounded   away from the endowment, i.e., 9G : G  G 1 8t. t In particular, the following holds.  Proposition 5 Assume that there exists a competitive equilibrium in which T = T 8t. Then t the government can commit to a strategy such that the outcome of the unique equilibrium in the subgame following the commitment coincides with such a competitive equilibrium. Moreover, if   G  T 8t, one such strategy commits the government to raise exactly T in every period both in  t and out of equilibrium. If there is some period t0 for which G 0 T , there is no government  8t both in and out of equilibrium. t strategy that implies T = Tt Proposition 5 con rms the following two key results that we obtained the preceding sections. i The government can play a strategy in which the price level is uniquely determined by spending and the target level of taxes; the initial price level satis es   P1 ~ s u01 , G1  B1 + =1 B +1 Qs C ~ 1 PB 1 X ,1 0 s j =1 j Bj +1 ~ u 1 , G C s : s = s B 21 P 1 1C B s =1 s s 14The introduction of a discount factor is important; to obtain our results, it is necessary that the present value of the current and future endowment be nite at the equilibrium prices. 17 The numerator on the left-hand side is the nominal value of all bonds outstanding and all of the bonds that the government will issue in exchange for goods fresh borrowing, discounted at the nominal interest rate; the right-hand side is the real value of all repay- ments to bondholders that the government will make. In a standard government budget constraint, such as 3, only net debt ows appear. Equation 21, which is based on the actual trading strategy the government adopts on the markets in which it participates, keeps the gross ows into and out of the debt stock explicit, emphasizing what the gov- ernment can control directly new issues of bonds and the amount of goods it repays to bondholders. In this modi ed version, equation 21 is an equilibrium condition and not a constraint upon the government behavior, as emphasized by the scal theory of the price level. ii Unless taxes always exceed spending, the government cannot set a xed and exogenous level of surplus de cit in each period and maintain it both in and out of equilibrium, as it is assumed by all of the papers on the scal theory of the price level. Out of equilibrium, an unexpected shortfall in revenues from borrowing must be covered through additional taxes. By making the timing of moves explicit, the game-theoretic description of the economy con- vincingly shows that there is no di erence between a nite- and in nite-horizon economy and that the transversality condition plays no special role in our analysis. Both in the nite- and in nite-horizon economy, the crucial issue the government is facing is whether households will be willing to lend the right" amount of resources in exchange for debt. This is a problem that the government faces in any period and that requires an immediate reaction, independently of whether the economy will last a nite or in nite number of periods. The notion that the govern- ment could solve the shortfall by issuing additional unbacked debt at out-of-equilibrium prices is simply awed. The transversality condition only plays a role in determining the households' will- ingness to purchase the debt in equilibrium, just as the two-period-horizon counterpart B3 = 0 does. 5.2 Money The introduction of money is very important to compare the economy I present here to a standard monetarist model in which the price level is essentially determined by the quantity of nominal balances in the economy. However, the scal theory of the price level stems precisely from the failure of such models to deliver price determinacy in many instances. In particular, I have assumed throughout this paper that the monetary authority" follows an interest rate peg. Such a policy typically leads to indeterminacy in both the nominal money supply and the price level. Money plays an important role also in Buiter's 2 criticism of the scal theory. In his framework, a non-Ricardian policy is interpreted as a policy that defaults on part of the debt; as a consequence, debt trades at a discount over its nominal value. In our cashless economy, it is impossible for the debt to trade at a discount, as we de ned the value of a dollar precisely in terms of debt; in order for this to be a possibility, it is necessary to introduce a second nominal asset money whose price relative to debt may not be xed. 18 Money can be introduced in the game described above through a cash-in-advance" tech- nology that prevents some barter trading posts from opening. Households are divided into n symmetric groups, with n even, that lie on a circle. Each group i produces a good that cannot be bartered with the good at the opposite extreme", i.e. i  n=2, so these trades require money. Because each group is still formed by a continuum of households, each household behaves as price taker. I now assume that each household likes to consume all n types of goods. Trading posts are open for all pairwise combinations of goods except for the opposite extreme goods, for all goods vs. money, for goods vs. bonds and money vs. bonds. In this case, a dollar" is the price of money, not national debt. While this is work in progress, I conjecture that the price of a dollar of money and a dollar of debt will coincide identically only if the government explicitly pursues a policy that pegs the relative price; such a policy implies a commitment to monetization of the debt should households wish to get rid of it by selling it on the market rather than rolling it over. The scal theory of the price level is unlikely to survive if accompanied by a money-supply rule, which is inconsistent with any monetization. However, an interest-rate peg is consistent with a peg of the relative price of money and debt, as the government can freely adjust the supply both of money and bonds; this suggests that a strategy similar to that described in section 4 may achieve price determinacy through appropriate management of debt. 6 Conclusion While this research is unlikely to lay to rest the dispute on the validity of the scal theory of the price level, it shows how the question can at least be cast in a more complete model in which the de nition of an equilibrium is not controversial. In this paper, I show that the usual version of the government budget constraint is not ade- quate to describe the restrictions on the government policy out of equilibrium. Nonetheless, the government does face budget constraints on its actions even out of equilibrium; the policy rules postulated by proponents of the scal theory violate these constraints and are thus misspeci ed. I rescue the scal theory by displaying a strategy in which the scal side of the economy determines the price level in an environment in which the traditional monetarist analysis would imply indeterminacy. This strategy is very much in the spirit of the scal theory of the price level: the government guarantees a stream of real payments to the current holders of debt independently of the current or future price level. A Proof of proposition 3 I solve the household's problem backwards. When submitting its bid in period 2, each household inherits as a given its previous con- sumption c1 and its level of nominal bonds b2 . At this stage, the household can only choose how much of b2 to bid in exchange for additional period-2 goods; the price it expects on that market is given by 11, which is a strictly positive number and is independent of its bid assuming 19 C2 B2 0. The household will thus bid all of its b2 bonds and consume c2 = 1 , T2 + C22B2 . P b In period 1, the household has to submit 3 bids. Given that the government does not o er new bonds in exchange for goods, the household expects a price P 1 2 = 0 any bid on that C B market to be wasted, so it will choose c1 2 = 0. The household is thus left with the problem to B allocate the initial amount of bonds b1 between the bid for new bonds and that for goods. From the perspective of an individual household, each unit bid for goods yields 1=P 1 1 units of theC B consumption good, and each unit bid for new bonds yields P 1 2 units of new bonds. While B B P 1 1 is not known to the household ex ante, in equilibrium the household has perfect foresight C B about it.15 The household also knows that each unit of new bonds will fetch 1=P 2 2 units of C B period-2 goods. Its problem becomes thus exactly 14. The mechanism I designed corresponds to a Walrasian economy from the perspective of each household: each household is simply taking prices as given and maximizing by allocating its resources.16 While mathematically the problem is identical, conceptually a household faces a more-complex problem in the economy I consider: it has to form beliefs not only about future prices, as in a dynamic Walrasian equilibrium, but also about current prices, which are determined only after the bid has been submitted. The rst-order condition for household bids at an interior yields:17 P P u0c2  u0 c  = 1 2 1 1 1 B B C B 22 P 2 2 C B which is the standard Euler equation, together with B1 1 + B1 2 = B1. C B An equilibrium in the subgame in which the government strategy is speci ed, as above, by      T1 = T , C1 1 = T , B2 1 = 0, B 2 1 = B , T2 T , C2 2 T is characterized as follows. From B C B B C2  the government strategy, C2 B2 = T after any history. From the government strategy, 11 and B 2P 12 we obtain C2 = 1 independently of the household bids. Notice that this is a result on C2 , which is average consumption; in principle, each household could consume more or less than 1. Similarly, the government strategy, 8 and 9 imply C1 = 1 independently of the history. Using C1 = C2 = 1, we see from 22 that in ation is equal to the nominal interest rate chosen by the government. This is because consumption is constant and there is no discount factor, so the real interest rate must be 0. We can solve for the bids and the initial price using 8, 11, B1 1 + B1 2 = B1 and B2 = C B B1 2 P 1 2 , from which we obtain B1 1 = B1 2 = 1=2. The initial equilibrium price is P 1 1 = B B B C B C B B 1 : it is uniquely determined and is independent of the nominal interest rate chosen by the 2 government. QED. T There is no uncertainty because the government is not playing mixed strategies, and the households' choices 15 are uncorrelated and, in equilibrium, the households play pure strategies too. 16 There is no market for private debt, which makes households borrowing constrained; this is irrelevant in my setup with identical households. 17 In equilibrium, households must be choosing an interior point when allocating maturing bonds to the 2 markets. If this were not the case, there would be one period in which goods are o ered in exchange for maturing bonds, but no bonds are redeemed; it would then be enough to bid an arbitrarily small amount to obtain the goods essentially for free. 20 B Proof of proposition 4 Let ~ ~ ~ ~  ~ ~ ~ ~ ~ ~ ~ ~ ~ C1 ; C2; T11; T12; T ; B2; B1 1 ; B1 2 ; C1 2 ; B2 2 ; C1 1 ; B2 1 ; B2 1 ; C2 2  C B B C B B C B be the competitive equilibrium allocation and let the associated price system be ~ ~ ~ ~ P 1 1 ; P 1 2 ; P 1 2 ; P 2 2  C B B B C B C B We prove the proposition for the case in which the government participates in all markets: ~ ~ C1 1 ; B2 1  0. The proof of the other cases is analogous, except that prices are not de ned in B C the markets in which the government does not participate; in those markets, household correctly expect any bid they submit to be wasted, and hence in equilibrium they would not submit bids. ~ Consider the following government strategy. In period 1, the government sets T11 = T11. It bids CB~1 1 units of goods in exchange for maturing bonds and B2 1 units of new bonds in exchange ~ C ~ for goods, and sets the nominal interest rate at P 1 2 . The second installment of taxes T12 is set B B so that 10 holds; this installment depends thus on the household bid C1 2 . Independently of B  ~  what happened in period 1, the government sets taxes at T and bids C2 2 = T in exchange for B bonds maturing in period 2; it follows that G2 0. We now look at the household response if the government commits to the strategy above. In period 2, households will bid all of their maturing bonds against goods, independently of the previous history, so for each household b2 2 = b2 and in the aggregate B2 2 = B2 , independently C C of the previous history. In a competitive equilibrium in which G1 T1 it is necessarily the case ~ ~ that B2 0 and T2 0, so we know B2 0 and hence P 2 2 2 0; +1. C B Each household has beliefs about the bids that will be submitted by the others, and uses 8 and 11 to get a belief about the prices that will arise in each trading post. Given its beliefs about prices, the household solves 14. In a symmetric equilibrium, the solution to 14 must coincide with the belief that the household has about the behavior of other households. In a symmetric equilibrium, the bids submitted by the households can be derived from the following requirements. i First-order conditions for 14: u0 C1  =u  2 2 2 P 1 2 + ; 0 1,T +B P B B P 1 1C B P 2 2 C B C B 23   0 if B1 1 0;   0 if B 1 C 1 B1 C P u0 C1  = u0  1 , T + B2 P 2 2 P C1 B2 + ; C B C2 B2 24   0 if C1 2 B 1, T11 ;   0 if C1 2 B 0 21 B 1 C C1 = 1 , T1 + 1 , C1 2 B P 1 1 B1C1 +B 2 =B B C B 25 1 1 B2 = B1 2 P 1 2 + C1 2 P 1 2 B B B B C B where  and  are Kuhn-Tucker multipliers; ii Equations 8 and 11, which describe the price formation at the trading posts. iii The decisions to which the government is committed: ~ T11 = T11 B B ~ P 1 2=P 1 2 B B ~ B2 1 = B2 1 C C 26 B ~ C1 2 = C1 2 B ~ C2 2 = C2 2 = T B B iv The government budget constraint T1 = T11 + T12 = G1 , C1 1 + C1 2B B 27 The allocation and price system in 16 and 17 form a competitive equilibrium, which implies that equations 23, 24, 25, 8, 11, 26 and 27 must hold. The competitive equilibrium we are considering is thus an equilibrium outcome of the subgame in which the government committed to the strategy above. The household strategy in this equilibrium calls ~ ~ B C B ~ for bidding B1 2 ; B1 1 and C1 2 in the rst period, and bidding all of the period 2 bonds in the second period independently of the previous history. We next need to prove that this is the unique symmetric equilibrium. Notice that, in an equilibrium, we must have   0 and   0. If  were greater than 0, equation 23 implies that households would not be bidding maturing bonds in exchange for goods. In this case, a single household could capture the entire government bid of goods by submitting an arbitrarily small bid on the market shunned by all others: it would face an arbitrarily favorable price on that market, which would contradict the optimality of not submitting a bid. Similarly, we have   0: since the government is o ering new bonds in exchange for goods, households must be submitting strictly positive bids on that market. There are thus four cases, depending on whether either constraint is binding. In all four cases, repeated substitution shows that there exists a unique solution to the system of equations 23, 24, 25, 8, 11, 26 and 27, which yields the desired result. QED. It is worth noticing that, in the more natural case in which  = 0 and  = 0, 23 and 24 imply P 1 2 B B =P 1 2 =PC1 B1 C B 28 22 This relationship stems from the fact that, from the perspective of a single household, this economy has redundant markets. The same consumption vector can be achieved either by rolling some debt over or by redeeming it for goods while at the same time purchasing new bonds with goods. In equilibrium, a household must be indi erent between the two strategies in order to participate in all markets, and this links the prices on the 3 markets that are open in period 1.18 C The In nite-Horizon Economy and Proof of Proposi- tion 5 As in the two-period economy, we keep the assumption of a unit endowment of the consumption good in each period. Each household starts the rst period with B1 units of government bonds; we continue to study an economy with only one-period debt. The government must nance an exogenous sequence of spending fG g1 that is bounded =1 t t away from 1 so that consumption is bounded away from 0 in equilibrium. Lump-sum taxes are denoted by T . t Households have preferences given by 20, where u is a function that satis es the same assumptions I introduced for the 2-period case. We now describe the sequence of actions within each period t = 1; 2; : : : . Since there is no longer a last period, the same number of markets three is now open in each period. i Each household starts with 1 unit of the period-t good and b units of government debt t maturing in period t. While in equilibrium all households will have the same amount of goods, in principle b may vary from household to household. The government levies a rst installment of period-t taxes, T 1 2 0; 1 and sets a price P t t+1 at which it stands ready t t B B to exchange maturing bonds for new bonds. The government submits a sale bid for C t B units of goods in the market for maturing bonds, subject to C t  T 1. It also submits a t B t t sale bid for B +1 units of new bonds in exchange for goods. C t t ii Trading opens. Each household may submit a sale bid for b t units of bonds in the market C t for goods, and another sale bid for b t+1 units of bonds in the market for new bonds B maturing next period, subject to the constraint that b t + b t+1  b . Each household may t C B t t t also submit a sale bid of c t+1 units of goods in exchange for new bonds, subject to the B constraint that c t+1  1 , T 1. t B t t iii For the markets in which the price is not set by the government, the ratio of the quantities of the unconditional bids sets the price and exchange takes place. The government meets Equation 28 is analogous to a no-arbitrage condition, but arbitrage is precluded in this environment because 18 households cannot sell goods or assets short. 23 the demand of new bonds in the market in which it sets the price. We thus have t =B C P t t t C t C B B t B t C 29 P t t+1 = +1 t C t+1 C B B t B +1 = B t+1 P t t+1 t B t B t B B As before, P t t is the price level of this economy and P t t+1 C B is the nominal interest rate. B B iv The government levies a second installment of taxes or transfers T 2 2 ,T 1 + C t , B B +1 P t t+1 ; 1 , T 1 + C t , C t+1 .19 t t t Ct B B t C B t t t v Consumption and government spending take place. Each household consumes t + b C c = maxf0; 1 , T , c B t+1 t g 30 t t t P t t C B where T = T 1 + T 2 and starts period t + 1 with b +1 t t t t = b t+1 P t t+1 + c t+1 P t t+1 B t B B B t C B units of nominal bonds. The government spends G = T + B +1 P t t+1 , C t t t C t C B B t t 31 vi Period t ends and the economy starts from the rst step in period t + 1. De nition 2 A symmetric competitive equilibrium is an allocation fC ; T 1; T 2; B +1 ; B t ; B t+1 ; C t+1 ; C t ; B +1 ; B +1g1 t t t t C t t B t t =1 B t t B B t C t t and a price system fP t t ; P t t+1 ; P t t+1 g1 C B =1 B B C B t such that: i Given the price system and taxes, fC ; B +1; B t ; B t+1 ; C t+1 g1 solves the household =1t t C t B t B t t maximization problem: 1 X max Bt+1 Bt+1 t uc  s.t. fct ;bt+1 ;bCt ;bt t t ;c t g1 0 t=1 t=1 t = 1 , T1 , T2 + b C c t t t P t t t ,c B t t 32 C B bt tC + bt t+1 b1 B  bt+1 = bBt+1 PBt Bt+1 t +c B t t+1 P t t+1 C B c Bt t 1 T1  , t 19The same assumptions that I made in section 3 apply if a household does not have enough resources to meet tax obligations. 24 ii The government's actions satisfy the feasibility requirements T 1 2 0; 1 t C t B t 2 0; T 1 t T2 2 t ,T 1 + C t , B +1 P t t t B C t C B t t+1 ; 1 , T1 + C t , C t B t t Bt+1 iii Markets clear and the government budget constraints hold, i.e. equations 29 and 31 are satis ed for any t. In order to de ne strategies, we need a notation that keeps track of the nodes and information sets of the game. We de ne the histories of our game as follows. =; h1 g h= h ; T 1; P t t+1 ; C t ; B +1 ; t = 1; : : : h t g t t t B B B t C t h +1 = h ; b t ; b t+1 ; c t+1  t = 1; : : : g t h t C t B t B t subject to the following restrictions: T 1 2 0; 1 ; t = 1; : : : t P t t+1 2 R + ; t = 1; : : : B B C t 2 0; T 1 ; t = 1; : : : t B t B +1 2 R + ; t = 1; : : : t t C b t : 0; 1 ! R + ; t = 1; : : : C t b t+1 : 0; 1 ! R + ; t = 1; : : : B t b t i + b t+1 i  b i; i 2 0; 1 ; t = 1; : : : C t B t t 33 b1 i B1 given; i 2 0; 1 t,1 c ,1 i ; i 2 0; 1 ; t = 2; : : : t B b i b ,1 t,1 t + B t t iP B t t B B t C t C ,1B Z 1 t C ,t1 t B c ,t1 idi; i 2 0; 1 ; t = 2; : : : B t 0 c t+1 i 2 0; 1 , T 1 ; i 2 0; 1 ; t = 1; : : : B t t Households are indexed by i 2 0; 1 , and boldface letters describe the behavior of each house- hold.20 20For obvious technical reasons, the game only looks at histories in which all of the boldface functions are measurable. As a matter of fact, we will later restrict our attention to a much smaller set of histories, namely those after which all household have taken the same action except at most a set of measure 0. 25 The government is called to move after histories of the type h . However, the government only g t observes the actions by the households up to sets of measure 0: for this reason, a deviation by a single household will not be detected. On the other hand, deviations by numerous households more precisely, by positive-measure sets are detected. The households move after histories h ; h t each household i observes its own actions and the actions by all other households in the economy up to measure 0 sets. Each household is thus capable of remembering its own deviations, but cannot notice deviations by other individual households. Let H be the set of histories after which the government moves and H the set of histories g h after which households move. A government strategy is a mapping from H to a vector of g g actions T 1; P t t+1 ; C t ; B +1  such that: t B B t B t C t i is measurable with respect to the information available to the government; g ii T 1; P t t+1 ; C t ; B +1 satis es 33. t B B t B t C t A household strategy is a mapping from H to a vector of actions b t i; b i h C t B t t+1 i; cB t t+1 i such that: i is measurable with respect to the information available to household i; i ii b t i; b t+1 i; c t+1 i satis es 33. C t B t B t We call a strategy pro le  ; ji 2 0; 1  symmetric if g i 8h 2 H ; 8i; j 2 0; 1 n h s h o b t i; b t+1 i; c t+1 i = b t j ; b C t B t B t C t B t t+1 j ; cB t t+1 j  8s  t  h  = h i h t j h t In words, a strategy pro le is symmetric if, at each node of the game, it prescribes the same actions to households that made the same choices in the past. Because information in the game is not perfect, the appropriate equilibrium concept we will use is that of a symmetric sequential equilibrium. However, notice that the behavior of measure 0 sets of households has no impact on the aggregate economy, and hence a deviation by such a set does not in uence the payo of either the government or any other player. For this reason, beliefs about what each individual household did are irrelevant. As before, the government strategy is taken as exogenous; we look for equilibria in the game in which the government has committed to a given strategy. Within this game, we now prove proposition 5. Proof of proposition 5. Let ~ ~ ~ ~ ~ t ~ ~ ~ t ~ t =1 fC ; T 1; B +1 ; B t ; B t+1 ; C t+1 ; C t ; B +1 ; B +1g1 t t t C B t B t B t B t C t t and ~ fP tC B ~ ~ t ; PBt Bt+1 ; PCt Bt+1  g1 =1 t 26 ~ ~ t ~ be a competitive equilibrium. We will assume C t 0; B +1 0; B t+1 0 8t. The proof can B t C t B t be easily repeated for all other cases, except of course the fact that the equilibrium price will be unde ned in markets in which no exchange takes place. A competitive equilibrium satis es the following conditions at each time t: u0 C  u0 C +1 P t t+1 = t + ; t B B P t t C B P t+1 t+1 C 34 B t   0;  = 0 if B t t B t C t t u0C +1 P t t+1 u0 C  = + t C B P t+1 t+1 35 t t C B   0  = 0 if C t+1 1 , T 1 t t B t t lim u0C  B = 0 !1 t t 36 t t P t t C B T2 = G t t , T1 + C t , C t B t B t t+1 37 B t + B t+1 C t B t =B t 38 B +1 = B t+1 P t t+1 + C t+1 P t t+1 t B t B B B t C B 39 t =B C P t t t 40 C t C B B t t B +1 C P t t+1 = t 41 C t+1 C B B t =1,G C t 42 t In what follows, we assume that the Lagrange multiplier  is zero for the competitive equi- t librium that we are considering. If this is not the case, then we can construct another price system that is identical to the former except for P t t+1 , which is set such that 34 holds with B B  = 0. It is trivial to check that this new price system forms a competitive equilibrium with the t same allocation as before.21 21 Intuitively, if 34 holds with inequality, the government is o ering a very unattractive rate of return on rolling over debt, so each household strictly prefers redeeming all of its maturing debt at time t. In this case, nothing changes if the government raises the rate of return up to the point at which households still redeem all of their debt at time t, but are exactly indi erent at the margin between redeeming it or rolling it over. 27 We consider the following government strategy. At each time t, independently of the past ~ ~ ~ ~ t history of play, the government chooses the vector T 1; P t t+1 ; C t ; B +1 . With this strategy, B B B C ~ t t t t if B +1 0, total tax revenues in period t are G + C t , C C t t t+1 and depend thus on the actions B t B t private households take at time t. Intuitively, whenever the government expects to raise revenues through fresh borrowing, taxes must be adjusted if these revenues fall short of or exceed the target. We now show that, if the government adopts this strategy, then there exists a unique sequential equilibrium in the game that ensues among the private households. Because there is a continuum of households and the actions of each of them are not observ- able individually, each household perceives that the future actions by all other players will be una ected by whatever sequence of actions it takes. As a consequence, each household takes as given the actions of the government and of other households when choosing its moves. In particular, this implies that each household expects a sequence of prices and taxes that follows from everybody else's strategies but is independent of its own actions: therefore, in equilibrium, each household behaves as in a competitive equilibrium and solves 32. For this reason, any outcome of a sequential equilibrium must be a competitive equilibrium. In order to prove the proposition, we thus need to prove the following: i An equilibrium exists.22 This means that, even on information sets out of equilibrium, the households' strategy prescribes a best reply to what they expect the government and other households to do. ii There is a unique allocation and price system that satis es equations 34-42 at all times t together with t B B B t t ~ ~ C t ~ ~ t T 1 ; P t t+1 ; C t ; B +1  = T 1; P t t+1 ; C t ; B +1  t B B 43 B t t C i To be completed. ii Using repeated substitution in the system of equations 34-42 and 43, the entire al- location, price system and sequence of taxes can be derived uniquely as a function of the initial price level P t t . From this system it also follows that C B " ,1  ! ,1 ,1 u0 1 , G  B u0 1 , G1  X 1 X ,1 0 t t t t = B1 + ~ s B +1 Q C , ~ u 1 , G C s s B t P t t C B P 1 1 C B =1 s s =1 ~ P j j+1 =1 s j B B s s s 44 We can now use the transversality condition 36 to obtain a unique solution for the initial price level:   P1 ~ s u0 1 , G1  B1 + =1 B +1 Qs ~B B 1 C j =1 j j +1 s s = 21 P P 1 1 P1 ,1 0 ~ s u 1 , G C C B s B s =1 s s 22As I argue in Bassetto 1 , the government could commit to strategies such that there exists no sequential equilibrium in the subgame following the commitment. We want to check that this is not the case here. 28 Notice that the second in nite sum in 21 is always convergent provided G is bounded t away from 1; the rst in nite sum must be convergent in order for equation 21 to have a ~ solution. By assumption P 1 1 satis es equation 21, given that it is part of a competitive ~ C B equilibrium. 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