Vol. 36, No. 2

2000 Quarter 2

Understanding the Fiscal Theory of the Price Levelby Lawrence J. Christiano and Terry J. Fitzgerald

Understanding the Fiscal Theory of the Price Levelby Lawrence J. Christiano and Terry J. Fitzgerald

I. Introduction to the Fiscal Theory of the Price LevelWhat Distinguishes the FTPL?Assessing the FTPLOther Issues Addressed by the FPTLFrank Ramsey and the FTPL

II. Fiscal Theory in a One-Period EconomySargent and Wallace’s Unpleasant Monetarist ArithmeticThe Fiscal TheoryInterpreting Ricardian and Non-Ricardian Fiscal PolicyIs the FTPL Sensible? An Analogy to MicrosoftIs the Non-Ricardian Assumption Empirically Plausible?The Price Level in a World with No Government-Provided Money

III. Fiscal Theory in General EquilibriumIs the Price Level Overdetermined in the FTPL?Is the FTPL Fragile?FTPL with Stochastic Fiscal PolicyThe FTPL and the Control of Average Inflation

IV. Fiscal Theory and the Optimal Degree of Price InstabilityThe ModelThe Ramsey EquilibriumA Numerical ExampleSummary

V. Conclusion

VI. Appendix: The Logical Coherence of Fiscal TheoryThe Lucas–Stokey Cash/Credit-Good ModelConstant Money GrowthFixed Interest Rate Policies

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Understanding the FiscalTheory of the Price Levelby Lawrence J. Christiano and Terry J. Fitzgerald

Lawrence J. Christiano is a professor ofeconomics at Northwestern Universityand a visiting scholar at the FederalReserve Bank of Cleveland; Terry J.Fitzgerald is an economist at the Bank. The authors benefited fromdiscussions with Stefania Albanesi,Marco Bassetto, Charles Carlstrom,Martin Eichenbaum, and TimothyFuerst. They thank John Cochrane,Christopher Sims, and MichaelWoodford for their tireless efforts toeducate them on the fiscal theory.

I. Introduction to theFiscal Theory of thePrice Level

Price stability is an important goal of publicpolicy. To reach this goal, two key questionsmust be addressed: • How can price stability be achieved?

And, • How much price stability is desirable?

Standard monetarist doctrine offers a simpleanswer to the first question: Make sure the cen-tral bank has an unwavering commitment toprice stability. Recently, though, some econo-mists have begun to rethink the foundations ofthis doctrine, giving rise to an alternative viewin which a tough, independent central bank isnot sufficient to guarantee price stability. In thisview, price stability requires not only an appro-priate monetary policy, but also an appropriatefiscal policy.1 Because fiscal policy receives somuch attention in this new view of price-leveldetermination, Michael Woodford has called itthe fiscal theory of the price level.2 Throughoutthis review, we refer to it as the FTPL.

Monetarist doctrine also recognizes thatboth fiscal and monetary policy must be

selected appropriately if price stability is to beachieved. However, this doctrine holds that ifthe central bank is tough, the fiscal authoritywill be compelled to adopt an appropriate fiscalpolicy.3 The FTPL denies this. It says that unlesssteps are taken to ensure appropriate fiscalpolicies, the goal of price stability may remainelusive no matter how tough and independentthe central bank is.

The FTPL has significant implications for theway central banks conduct business. The con-ventional view prescribes that central bankersshould stay away from fiscal authorities toreduce the likelihood of being pressured into

� 1 Cochrane (2000) goes so far as to say that monetary policy maybe irrelevant to price determination. In his view, government-providedtransactions assets are a vanishing component of all financial assetstraded.

� 2 Benhabib, Schmitt-Grohe, and Uribe (2000), Cochrane (1998a,2000), Dupor (2000), Leeper (1991), Sims (1994, 1999), and Woodford(1994, 1995, 1996, 1998a,b,c, 1999) all advocate the FTPL, while Buiter(1999), Carlstrom and Fuerst (2000), Kocherlakota and Phelan (1999),and McCallum (1998) provide critical reviews.

� 3 This classic statement is from Sargent and Wallace (1981); seeespecially the last paragraph of their paper.

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making poor monetary policy decisions. TheFTPL implies that central bankers with a man-date to foster price stability must do more thansimply make sure their own house is in order;they also must convince the fiscal authority toadopt an appropriate fiscal policy.

The FTPL literature also draws attention tothe second question—how much price stabilityis desirable?—which is both important and dif-ficult. Sims (1999) and Woodford (1998a) pointout that allowing the price level to fluctuatewith unexpected shocks to the governmentbudget constraint produces public financebenefits.4 For example, a bad fiscal shock suchas a war or natural disaster drives up the pricelevel; this is equivalent to taxing the holders ofthe government’s nominal liabilities. This pro-motes efficiency to the extent that it allowsthe authorities to keep labor tax rates smooth.In practice, this benefit is likely to be miti-gated by whatever distortionary costs maybe associated with price instability.5 Cochrane(1998b) is mindful of these costs when, in hisanalysis, he simply takes for granted that com-plete price stability is a fundamental socialobjective.6 Sims (1999) claims that publicfinance benefits overwhelm the distortionarycosts associated with volatile prices, and so heconjectures that complete price stability is non-optimal. A convincing answer to the secondquestion awaits a quantitative study that care-fully balances benefits and costs.

This paper explains the FTPL and elaborateson its implications for the two questions posedabove, as well as for other issues. In theremainder of this introduction, we provide anoverview of our analysis. We first discuss thecrucial assumption that differentiates the FTPLfrom the conventional view. Next, we summarizesome of the key issues that any assessment ofthe FTPL must confront, then briefly describeother issues addressed in the FTPL literature.Finally, we emphasize the connection betweenthe FTPL and the traditional Ramsey literatureon optimal monetary and fiscal policy.

What Distinguishesthe FTPL?

The difference between the conventional viewand the FTPL does not lie in any error of logic.7

Instead, the two differ in their views of thegovernment’s intertemporal budget equation.That equation says the value of governmentdebt is equal to the present discounted value offuture government tax revenues net of expen-

ditures (that is, surpluses), where both debt andsurpluses are denominated in units of goods.This equation is expressed as

(1.1) B—P

= present value of future surpluses,

where B is the outstanding nominal debt of thegovernment and P is the price level. The con-ventional view holds that this equation is a con-straint on the government’s tax and expendi-ture policy;8 that is, policy must be set so theright side equals the left, whatever the value ofP. According to this view, when equation (1.1)is disturbed, the government must alter itsexpenditures or its taxes to restore equality.FTPL advocates, however, argue there is noinherent requirement that governments treatthis equation as a constraint on policy. In theirview, the intertemporal budget equation isinstead an equilibrium condition: When some-thing threatens to disturb the equation, themarket-clearing mechanism moves the pricelevel, P, to restore equality.

Michael Woodford has called this assumption—that government policy is not calibrated to satisfy the intertemporal budgetequation for all values of P—the non-Ricardianassumption. Another way of stating thisassumption is that if the real value of govern-ment debt were to grow explosively, no adjust-ments to fiscal and monetary policy would bemade to keep it in line.9

� 4 For previous discussions, see Chari, Christiano, and Kehoe(1991), Judd (1989), and Lucas and Stokey (1983).

� 5 See Woodford (1998a, pp. 59–60) for an elaboration of this point.

� 6 Cochrane (1998b) emphasizes the need for some type of govern-ment security whose payoff fluctuates with shocks to the governmentbudget constraint but does not generate the sort of distortionary costsassociated with a fluctuating price level.

� 7 Some authors are concerned with the possibility the FTPL may belogically incoherent (see Buiter [1999]). We address these concerns, inpart, by presenting a class of economic environments in which the FTPLis logically coherent.

� 8 Our notion of taxes includes seignorage revenues and taxes onthe return to government debt, that is, default.

� 9 Technically, we are exploiting the equivalence between theintertemporal budget equation and a certain transversality condition. Wediscuss this equivalence later in the text and in the appendix.

The FTPL does not anticipate exploding debt. Rather, as long as thereis absolutely no doubt about the government’s commitment to not adjust-ing fiscal policy in the face of exploding debt, then prices will respond insuch a way that the debt will not explode in the first place.

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Assessing the FTPL

To evaluate the FTPL, it is useful to focus onthe following positive and normative issues.Is the non-Ricardian assumption empiricallyplausible?10 Does the FTPL offer a compellingexplanation for episodes of high inflation? Andfinally, does the FTPL provide useful input intothe design of socially efficient policies?

Clearly, the non-Ricardian assumption is nota good characterization of policy in all timesand places. Often governments do seem readyto adjust fiscal policy when the debt gets toolarge. For instance, when the U.S. governmentdebt began to increase in the 1980s and 1990s,there was considerable pressure for some com-bination of a tax increase and expendituredecrease to bring the debt back in line.11 Like-wise, according to the Maastricht Treaty, mem-bers of the European Union formally recordtheir intention to adjust fiscal policy in theevent their debts grow too large. Anotherexample is provided by the InternationalMonetary Fund. That organization uses an arrayof sanctions and rewards to encourage membercountries to keep their debts in line by suitablyadjusting fiscal policy.

For the FTPL to be an interesting positivetheory, it need not hold in all situations. AsWoodford (1998b) emphasizes, it may providea useful characterization of actual policies insome contexts, even if it does not in others. Forexample, the government budget constraintwas essentially absent from standard macroeco-nomic models of the 1960s and 1970s, and itplayed little role in Keynesian policy analysis(Sargent [1987, p. 112]). As a result, it is perhapsreasonable to suppose the non-Ricardianassumption held for that period.12 Loyo (1999)argues that Brazilian policy in the late 1970sand early 1980s was non-Ricardian and that theFTPL provides a compelling explanation forBrazil’s high inflation during that time.13

Even if, in practice, policy has never beennon-Ricardian, the FTPL might still hold interestas a normative theory, for two reasons. First,optimal policies might themselves be non-Ricardian.14 Second, the FTPL could serve asuseful input into policy design, even if non-Ricardian policies are, in practice, bad. To seewhy they might be bad, consider legislatorsliving in a non-Ricardian regime. Understand-ing that tax cuts or increases in governmentspending do not necessarily have to be paid forwith higher taxes later, they might be temptedto embrace policies that imply too much spend-ing and too much debt. Restricting fiscal policy

by limiting government debt may be an effec-tive way to deal with this problem.15 By estab-lishing the logical possibility of non-Ricardianpolicy, the FTPL implies that such policiescould occur, in the absence of specific mea-sures to rule them out. As a result, the FTPLcan be used to articulate a rationale for thetype of debt limitations imposed by the IMFand by the Maastricht Treaty.

Other IssuesAddressed by theFTPL

Although we stress the implications of the FTPLfor the two questions posed above, they arenot the exclusive or even primary focus of theliterature. The FTPL has plenty to offer, evenfor those with no interest in our two questions.FTPL advocates emphasize the value of theirframework for understanding price-level deter-mination when traditional quantity-theoreticreasoning breaks down or does not apply. This

� 10 The empirical plausibility of the non-Ricardian assumptionposes a special challenge, because it cannot be assessed based on timeseries alone. For further discussion, see “Is the Non-Ricardian AssumptionEmpirically Plausible?” on page 8 of the present article.

� 11 Woodford (1998b) acknowledges that the political reaction togrowing debt in the 1980s and 1990s (as well as other considerations)indicates U.S. policy during the past two decades is probably not well-characterized as non-Ricardian (see also our section “The FTPL and theControl of Average Inflation” on page 18). He argues that earlier episodesin U.S. postwar history—for example, the 1965–79 period—might bebetter characterized in this way.

� 12 See Woodford (1998b) for an elaboration of the argument thatU.S. policy may have been non-Ricardian in the 1960s and 1970s.

� 13 One wonders whether it makes sense to assume that a theorylike the FTPL, which focuses on the long-run properties of fiscal policy,holds for some periods and not others.

� 14 See Sims (1999) and Woodford (1998a).

� 15 Chari and Kehoe (1999) describe a model in which countriesform a monetary union, and, absent debt constraints, the result is exces-sive debt. Woodford (1996) argues that a union without debt constraints islikely to end up with excessive price volatility. His reasoning uses the kindof logic surveyed in this paper. He notes that if policy is non-Ricardian,then fiscal shocks must show up as shocks to the price level, regardless ofmonetary policy. (We call this “Woodford’s really unpleasant arithmetic” in“The FTPL with Stochastic Fiscal Policy” on page 15.) He argues thatprice-level instability arising from this source is likely to be excessive in amonetary union that adopts a non-Ricardian policy. It is precisely becausehe believes non-Ricardian policy is a realistic possibility—a possibilitywhich, in this case, he thinks is bad—that he approves of the explicit debtrestrictions incorporated into the Maastricht Treaty. For another similardiscussion of the potential dangers of non-Ricardian policy, see Woodford(1998a, p. 60).

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could happen, for example, if the monetaryauthority adopted a policy of pegging the rateof interest, so that the money supply wouldrespond passively to demand. This casedeserves emphasis because interest rate tar-gets are thought to play an important role inmonetary policy in practice (see Taylor[1993]).16 Another scenario of interest occurswhen private transactions involve no use ofgovernment-provided money. This review pre-sents examples to illustrate the interest ratepegging and cashless economy scenarios.

Frank Ramsey andthe FTPL

Our cashless economy example highlightsthe parallels between the FTPL and the tradi-tional literature on optimal monetary and fiscalpolicy inspired by Frank Ramsey (1927) andreintroduced into macroeconomics by Lucasand Stokey (1983).17 In the Ramsey literature,government “policy” is a sequence of actions(tax rates, expenditures, etc.) indexed by thedate and (in models with uncertainty) by therealized value of shocks. Because these policiesare not functions of past prices, prices existsuch that the debt explodes and householdsrefuse to buy it. Such possibilities are of noconcern in the Ramsey literature because thegovernment is viewed as having selected itspolicy before prices are determined, and it istaken for granted that only equilibrium pricesoccur.18 In equilibrium, demand equals supplyin all markets, including those for governmentdebt.19 We think of non-Ricardian policies ascorresponding to the type contemplated in theRamsey literature.

In the Ramsey literature, there is a concernthat policies may not be time consistent, in thesense that they are not consistent with the gov-ernment’s incentives to implement them in realtime. We believe these concerns may also applyto the FTPL. Consider again the situation inwhich the price level rises when there is a badshock to the government budget constraint. Inthis situation, private agents may suspect thegovernment will resort to high prices as an easyway to renege on debt. In this case, the policybackfires, with agents refusing to accumulategovernment debt in the first place. To avoidthis outcome, it is necessary to convincepotential holders of government debt that theywill receive subsidies when good things hap-pen to the government constraint.20 However,there may be times and places in which the

institutional and other social structures neededto achieve the required degree of credibility donot exist.

The remainder of this review is organized asfollows: Part II makes most of our points in aone-period environment. By adopting such asimple setup we are able to get to the basicideas without technical complications. At thesame time, some issues simply cannot be dis-cussed in a one-period environment; we deferthese to part III. Part IV presents a simplemodel for thinking about the desirability ofprice fluctuations under the FTPL in an envi-ronment with no government-provided money.Part V provides concluding remarks.

� 16 For a recent analysis of the case in which the interest rate is notpegged but is allowed to move around with variations in the state of theeconomy, see Benhabib, Schmitt-Grohe, and Uribe (2000).

� 17 Sims (1999) and Woodford (1998a) emphasize these parallels.

� 18 Implicitly, we are taking a particular stand on what constitutesgovernment policy in a Ramsey–optimal policy setting. Strictly speaking,Ramsey theory tells us only what government actions are taken in the bestequilibrium. There may well be many government policy rules that resultin the same Ramsey equilibrium, where a policy rule specifies governmentactions as a function of the realization of exogenous and endogenousvariables. Different policy rules imply different policy actions out ofequilibrium. Woodford (1998a, 1999) clarifies the distinction between thegovernment’s policy rule and the Ramsey-equilibrium outcomes. Hecomputes Ramsey-equilibrium outcomes and then searches for Taylor-likeinterest rate rules which support those outcomes as an equilibrium. In thispaper, we take the position that the government’s policy actions in aRamsey equilibrium are the government’s policy rule.

� 19 In Ramsey theory and in the FTPL, market prices exist wheregovernment policy commits to infeasible actions. For example, there maybe prices where the government commits to paying for goods with moneyfinanced from new debt issues, which no one buys. By focusing onequilibrium prices only, standard practice ignores these possibilities.Bassetto (2000) argues this is a mistake and proposes alternativeequilibrium concepts to deal with the problem.

� 20 See Chari, Christiano, and Kehoe (1991) for a detailed analysis.

�

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II. Fiscal Theory in aOne-Period Economy

The FTPL is defined by its non-Ricardianassumption on fiscal policy. The best way to understand this assumption and to quicklyget to the heart of the FTPL is to examine aone-period model.21 That’s what we do here.

We begin with the conventional wisdom,the classic Sargent and Wallace (1981) analysis.We go on to explain how the FTPL differs fromthis conventional wisdom: Sargent and Wallaceadopt the Ricardian view, while the FTPL adoptsthe non-Ricardian view of policy. We exploreseveral interpretations of these assumptionsand conclude that the non-Ricardian assump-tion requires that the government is able tocommit to its policy actions in advance. Wethen ask, how can we assess the empiricalplausibility of the non-Ricardian assumption?Finally, we explain how the FTPL can be usedto study the price level in an economy with nogovernment-provided fiat money.

Sargent and Wallace’s Unpleasant Monetarist Arithmetic

Suppose that in the morning of the only day inthis model, private agents hold a given amountof government debt, b. Here and throughoutthis review, we assume government debt isnon-negative: Agents cannot borrow from thegovernment. In the Sargent and Wallace model,debt is fixed in real terms; it represents acommitment to pay a fixed real amount ofgoods—for instance, corn.

The government’s budget constraint is given by

b� + s f + sm =b.

The left and right sides of this equation sum-marize the sources and uses, respectively, ofcorn to the government. The first source offunds, b�, is corn the government receives fromhouseholds that purchase new debt in theevening. The second term, s f, denotes taxesnet of spending, and the third term, s m, isseignorage from government-supplied fiatcurrency. The right side of the budget con-straint, b, is the principal and interest on pastgovernment debt.

Optimizing households will obviously neverchoose b� > 0, and they are constrained fromsetting b�< 0 by assumption. Therefore, house-

hold optimization implies that b� must be zero.By imposing this result, we get the intertemporalgovernment budget equation,

(2.1) b = s f + sm.

Sargent and Wallace’s main conclusions canbe understood from this equation. Suppose a“loose” fiscal policy is adopted—that is, s f isreduced. Simple arithmetic dictates the mone-tary authority must increase s m. Under normalcircumstances, this translates into an increasein inflation.22 In a multiperiod model, there issome discretion over timing. The rise in inflationcan occur sooner or later, or it could be spreadout over time. Whatever the timing, though, ifthe fiscal authority reduces s f, the arithmeticnecessitates inflation must go up at some point.Hence the title of Sargent and Wallace’s famouspaper, “Some Unpleasant Monetarist Arithmetic.”

The same arithmetic suggests a solution tothe inflation problem: Design central banks sothey can credibly commit to not “caving in” toan irresponsible fiscal authority that sets s f toolow. Governments around the world havesought to implement this solution by makingcentral banks independent and directing themto assign a high priority to inflation. With themonetary authority completely committed to afixed value for sm, the arithmetic forces thefiscal authority to adopt a fiscal policy consistentwith that sm. This is the basis for the currentconventional view that to achieve a stable pricelevel, it is sufficient to have a tough, independentcentral bank that is focused on prices.

The Fiscal Theory

According to FTPL advocates, the Sargent andWallace framework may not be relevant foreconomies like the United States. In practice,government debt is a commitment to deliver a

� 21 We are grateful to Marco Bassetto for suggesting this to us.

� 22 By “normal,” we mean that the economy is on the “right” side ofthe Laffer curve. Seignorage is the nominal increase in the money stockdivided by the price level. A convenient formula becomes available if wetemporarily reinterpret our model as a multiperiod model in a steady state.Let the demand for real balances be m = exp(–�π), where � > 0, π is thegross inflation rate from this period to the next, and m is the money stockdivided by the price level. Seignorage, then, is just m (1–1/π ) =exp(–�π)(1–1/π ). For inflation rates below π* = (� + � 2 + 4� )/(2� ),seignorage is increasing in inflation, and for inflation rates above thispoint, seignorage is decreasing. The “right” side of the Laffer curve refersto inflation rates below π*.

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certain amount of domestic currency, not goods.This creates new possibilities.

Let’s revise the previous analysis, replacing bwith B, nominal debt. The government’s budgetconstraint becomes

(2.2) B�+P (s f + sm) = B.

As before, optimizing households will not buygovernment debt in the evening. With demandat zero, equilibrium can be reached only at B�=0. Equation (2.2), with B�= 0, is the govern-ment’s intertemporal budget equation,

(2.3) B =P (s f + sm).

Now, P is an endogenous variable. If the fiscalauthority makes s f small, there is no arithmeticto compel the monetary authority to raise sm.If the monetary authority holds fast to sm whilethe fiscal authority reduces s f, the equation canbe satisfied as long as P jumps. This is whatFTPL advocates expect would happen.

Interpreting Ricardian and Non-Ricardian FiscalPolicy

At this point, we must clarify two key concepts.Fiscal and monetary policy are said to be non-Ricardian if s � s f + sm is chosen in a way thatdoes not guarantee the intertemporal budgetequation (2.3) is satisfied for all possible prices.In contrast, s is a Ricardian fiscal policy if it ischosen so that the intertemporal budget equa-tion is satisfied no matter what P is realized. Inour single-period model, this can happen onlyif s is a particular function of the price level,s (P) =B/P. The assumption that fiscal andmonetary policy are non-Ricardian definesthe FTPL.

How are we to interpret non-Ricardianpolicy? In principle, two interpretations arepossible. The first may seem natural, initially;however, on further reflection it makes nosense. In this interpretation, the government isunconcerned with the intertemporal budgetequation when it chooses s : Either it is unawareof its existence, or it simply does not care. Ifthe government were completely unconcernedwith intertemporal budget balance, it wouldbe impossible to understand why we have

taxes. Absent concerns that stem from the exis-tence of the intertemporal budget equation,borrowing is always more appealing than raisingtaxes because the latter produces deadweightlosses. If governments didn’t raise taxes, how-ever, s would be negative and there would beno positive value of P to satisfy the inter-temporal budget equation. If we adopt thisinterpretation of non-Ricardian policy, theapparent existence of equilibrium is a puzzle.This interpretation deserves no furtherconsideration.

Can the government’s concern for intertem-poral budget balance be reconciled with thenotion that s is set exogenously, not as a func-tion of P ? Yes, if we imagine the governmentcommits to s in advance, before P is deter-mined. We can illustrate this in two ways.The first is based on the parable of the Walrasianauctioneer, who helps the economy find theequilibrium price level. Under the non-Ricardianassumption, the government announces sbefore the Walrasian auctioneer finds the market-clearing price level. When the governmentselects s, it fully understands that householdswill buy zero B � in equilibrium. However,because of its first-move advantage, the govern-ment knows it can force the auctioneer tochoose P so that P = B /s.

Our second illustration is drawn fromeveryday life. A pedestrian who wants traffic tostop at a crosswalk will sometimes step into thestreet, making a show of being unconcernedabout oncoming cars. Is such a person reallyunconcerned with the prospect of being struckand killed? Of course not. He expects theoncoming cars, seeing his commitment to crossregardless of the consequences, to stop ratherthan suffer the horror of an accident. Under anon-Ricardian fiscal policy, the government’sapproach is analogous to that of the pedestrian.The government’s “policy” is simply an action,s. In principle, a value of P could occur thatwould put the government in the fiscallyexplosive situation of offering debt that themarket refuses to absorb—that is, B�>0. How-ever, if the market is completely convinced ofthe government’s commitment to s, then, likethe car that stops for the pedestrian, the marketwill generate a value of P to guarantee debt isnot excessive (in this case, “excessive” simplymeans greater than zero). The non-Ricardiangovernment banks on the idea that the market

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abhors non-equilibrium P as much as driversabhor hitting pedestrians.23

Although the word “commitment” in thiscontext is consistent with the technical eco-nomics literature, it may nevertheless confusethe reader because it has so many meanings ineveryday language. By saying the governmenthas commitment, we mean only that it movesfirst, before prices are set. We do not mean toimply the government’s motives are laudable,or its ability to move first reflects strength ofcharacter on the part of policymakers. Forexample, a government that is perpetuallyin gridlock because legislators cannot reachagreement acts with commitment, in our usageof the term.24

Now consider Ricardian policies. For thepurpose of our analysis, we take no positionon the relationship between these policiesand the government’s ability to commit. Still, wesuspect that Ricardian policies are consistentwith any degree of commitment.

Although Sargent and Wallace don’t use thislanguage, it seems fair to say they adopt aRicardian specification of policy. If we think oftheir analysis as applying to a realistic moderneconomy, then we must think of real govern-ment debt in their model (that is, b in equation[2.1]) as B/P. For different values of P, the valueof b changes, leading to adjustments in sm + s f

under the Sargent and Wallace analysis. There-fore, we interpret Sargent and Wallace asadopting the Ricardian assumption.

Is the FTPL Sensible? An Analogy toMicrosoft

Under the FTPL, the price level is determinedby equation (1.1) or equation (2.3). A reasonablequestion at this point is, is there any sensibleinterpretation of the FTPL? At first glance,determining P in this way may seem likeaccounting gimmickry without substantiveinterest. But this is not the case. As Cochrane(2000) emphasizes, the price of Microsoft sharesis determined the same way! Under the FTPL,the government’s relationship to its bondholdersis somewhat like Microsoft’s relationship to itsequity holders.

Microsoft works to set aside real output forequity holders, though its motives for doingso are different from the government’s.Microsoft does not calibrate its dividend stream

to guarantee the present-value formula for itsstock price will hold for all possible stockprices. Instead, the mechanism operates in pre-cisely the opposite direction: Market tradersforecast what Microsoft will generate for them,then calculate the ratio of that amount to thenumber of shares outstanding, and that’s thestock price! FTPL advocates argue the price levelin an actual economy is determined in exactlythe same way. The government does notcalibrate s to ensure its present-value budgetequation (2.3) for all values of P. Instead,bondholders figure out how many goods (s)the government is setting aside for them andthen calculate the price level as the ratio ofB to s.

Is the Non-RicardianAssumption Empirically Plausible?

In assessing the FTPL as a positive theory fora particular time period, the plausibility of thenon-Ricardian assumption must be considered.A simple examination of time-series data willnot help. Under both the non-Ricardian and theRicardian assumptions, we expect to see s =B/P.The only direct way to distinguish the twoassumptions is to see how s responds when theeconomy is out of equilibrium. According tothe Ricardian assumption, s adjusts with P to

� 23 As the analogy suggests, there can be trouble if commitment isnot credible. If the oncoming traffic is not completely convinced of thepedestrian’s commitment (drivers believe the pedestrian is sneakingglances at the oncoming traffic, ready to make adjustments in case some-thing goes wrong), then miscalculations can lead to a tragic collision. Weargue (see especially “Summary” on page 24) that this is possible if privateagents are not completely convinced of the government’s commitment tonon-Ricardian fiscal policy. In this case, markets might produce the“wrong” prices, leading to excessive government debt in that the privatesector refuses to purchase it. Buiter (1999) seems to be concerned withthis kind of outcome: He refers to the “painful” fiscal adjustments that mustbe made when the “Ricardian reality dawns” and the private sector refusesto buy government debt. We do not mean to suggest that catastrophe willalways occur if there is uncertainty about government policy. As “The FTPLwith Stochastic Fiscal Policy” (page 15) shows, the non-Ricardianassumption is perfectly consistent with stochastic fiscal policy.

� 24 In private communication, Christopher Sims pointed out thatour notion of commitment encompasses “the commitment of two pedestri-ans who enter a crosswalk while engaged in a fist fight. Few doubt that theyare not watching traffic.”

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preserve s =B/P. According to the non-Ricardianassumption, s is like a utility-function parameter:Its value remains unchanged, so that s � B/P outof equilibrium. This sounds like an easy thingto check—just compare s and B/P out of equi-librium. The problem is that, according to thetheories considered here, only equilibrium val-ues of s are recorded in the data.25

This does not mean there is no way tochoose between the non-Ricardian and theRicardian assumptions. In fact, we think thereare two ways to go. One is to extrapolate whatis reasonable out-of-equilibrium behavior,based on what we see in equilibrium.26

Another way is to view the FTPL as a startingpoint for a natural set of auxiliary assumptionsthat restrict time-series data and then test thoseassumptions.27 If the non-Ricardian assumptionleads to a useful set of theories, this would tipthe balance in favor of that assumption. Wenow discuss these two approaches.

Extrapolating Out-of-Equilibrium Behaviorfrom Equilibrium

According to the non-Ricardian assumption,the government’s policy is a commitment to aparticular action, s. Under the Ricardianassumption, policy is a strategy for setting s asa function of real debt. If governments directlyrecorded their policies in writing, we couldbetter discriminate between the two assumptions.There are two situations where this seems tohave occurred, and, with one important caveat,the results appear to favor the Ricardian overthe non-Ricardian assumption. The MaastrichtTreaty requires members of the EuropeanUnion to adjust their fiscal variables when theirreal debt gets too large. The IMF works in thesame way, pressuring its members to adjust fis-cal variables if their real debt gets out of hand.We think it is fair to say that if a non-equilibrium P were somehow called out, thesearrangements would generate an adjustmentin s. Casual examination of the (admittedly,equilibrium) time-series data suggests the same.In practice, when the debt gets large, politicalpressures come into play to adjust the surplusto bring the debt back in line. This happenedin the United States in the late 1980s and 1990s,when the federal debt began to grow signifi-cantly, producing political support for raisingtaxes and/or reducing spending.

Now, for the caveat: These examples suggestthe non-Ricardian assumption may be animplausible characterization of current policy inEurope, the United States, and some emerging-market economies. However, as our introductionemphasized, these examples do not establish

the non-Ricardian assumption as implausiblefor all times and places.

Is the Non-Ricardian Assumption a Good Starting Point?

Another way of assessing the empirical value ofthe non-Ricardian assumption asks how gooda platform it is for developing useful, testablerestrictions. Space does not permit us to pursuethis idea here, beyond mentioning that interest-ing work is under way. In particular, Canzoneri,Cumby, and Diba (1998), Cochrane (1998a,b),Loyo (1999), and Woodford (1998b) havepursued the assumption of statistical exo-geneity of the government surplus. This is notan implication of the non-Ricardianassumption per se, though that assumptiondoes naturally lead one to it.

This approach can best be understood byan analogy attributed to Benjamin Friedman(see Cochrane [1998a] ). Consider the equa-tion of exchange, MV = PY, where M is money,V is velocity, and Y is output. As it stands, thisequation has no testable implications; withoutadditional assumptions, it simply defines V.Still, if incorporating simple, plausible assump-tions converts the equation into a theory thatallows us to understand the data better, thenthe equation of exchange is empirically use-ful.28 Similarly, the non-Ricardian assumptionmay be a good starting point for identifyingsimple auxiliary assumptions that convert theFTPL into a useful, testable theory. If so,this would help vindicate the non-Ricardianassumption as a useful empirical assumption.

� 25 This result holds even if there are multiple periods anduncertainty.

� 26 There are examples of models in which the equilibrium time-series data contain information about what happens out of equilibrium.For example, in Green and Porter (1984), limited information has theconsequence that events occur in equilibrium that are observationallyequivalent to agents’ having deviated from the equilibrium. Althoughagents don’t actually deviate in the equilibrium, they must, nevertheless,be punished as though they had as a credible signal of what wouldhappen if a deviation really did occur. In this sense, the events in equilib-rium provide evidence of what would happen out of equilibrium.

� 27 For a thorough discussion of this strategy, see Woodford (1998b).

� 28 An example of such an assumption is the specification that Vhas a simple functional relationship to the nominal rate of interest.

10

Although we are inclined to be skeptical ofthe non-Ricardian assumption, the FTPL is stillvery much in its infancy. It remains to be seenwhere the FTPL will take us and what observa-tions it will help us to explain. The initial resultsare promising, though not uncontroversial.Cochrane (1998a,b) argues that an FTPL thatassumes a statistically exogenous surplusprocess helps us understand the dynamics ofU.S. inflation in the 1970s, and Loyo (1999)argues that it is useful for understandingBrazil’s high inflation in the 1980s.

Another literature, begun by Calvo (1978),Kydland and Prescott (1977), and Barro andGordon (1983), posits that the absence of com-mitment in government policy can account forthe high-inflation episodes mentioned in theprevious paragraph. One way to assess the FTPLis to compare its ability to account for suchexperiences with that of the time-consistencyliterature. The outcome of this comparison isnot obvious. McCallum (1997), among others,argues that time inconsistency is not a usefulexplanation for high-inflation episodes. Ireland(1998) argues the other way, that absence ofcommitment is useful.29

The Price Level in a World with No Government-Provided Money

Some FTPL advocates claim that an importantvirtue of the theory is that it provides a way ofthinking about the price level that works evenin a world where supply and demand for gov-ernment fiat money are nonexistent. Cochrane(1998a, 2000) argues this is of interest because,to a first approximation, we have alreadyreached that point.30

The basic pieces of the argument are alreadyin place: Under the non-Ricardian assumptionthat s f + sm is exogenous, equation (2.3) deter-mines the price level. This conclusion can bereached without reference to money or whetherit is even present in the economy. That’s thetip-off for the result to come: The price levelcan be pinned down, even if there is nogovernment-provided money in the economy.To see this, imagine that trade in the economyis carried out by barter. Equivalently, onecould think of a scenario in which trades arefinanced with the exchange of financial claimson privately held assets. These trades could

even be denominated in “dollars,” even thoughgovernment-provided money (“dollars”) doesnot exist.

What is nominal, dollar-denominated gov-ernment debt in this world with no dollars?Clearly it is not a pledge to deliver government-provided money, because there is none! Instead,it is a pledge to deliver B dollars’ worth of goodsto the bearer of B. The formal obligation leavesopen exactly how many goods B dollars corre-sponds to, because the price level is unspeci-fied. In this sense, it is like real-world U.S. gov-ernment debt.31 The price level that is realizedis determined by the government’s fiscal deci-sions. Fiscal decisions result in real surplus, s,which is what the government actually hasavailable for paying off bondholders. With theamount of goods available to pay bondholdersequal to s and the nominal value of debt equalto B, the natural definition of the price level isP =B/s.

At this point, the price level in a worldwithout government-provided money mayseem a useless appendage. In part IV, we shallsee that the price level in such an economycan play an important role, helping to imple-ment an efficient fiscal policy.

� 29 For further discussion, see Albanesi, Chari, and Christiano(1999) and Christiano and Gust (2000a,b).

� 30 According to Woodford (1998a,b,c), the assumption thatmoney demand and supply are literally nonexistent is too extreme. He prefers to analyze a “cashless limit.” This is an economy in which thedemand for money is so small that seignorage is negligible and can, to a first approximation, be safely ignored in the government’s budgetconstraint (both the flow-budget constraint and the intertemporal budgetequation). However, the demand for money is sufficiently large that thecentral bank can still control the rate of interest.

� 31 The U.S. government also offers indexed debt, which correspondsto a commitment to deliver a specific amount of a basket of goods. Indexeddebt is a small portion of the government’s portfolio of liabilities.

1111

III. Fiscal Theory inGeneral Equilibrium

Here we address issues that could not beaddressed in the one-period example. The firstissue will likely concern any reader who hasmade it this far. Part II showed how an equa-tion that is not usually used in the context of price determination—the government’sintertemporal budget equation—can pin downthe price level. But don’t we already have atheory of the price level? If we adopt the non-Ricardian assumption on policy, won’t the pricelevel be overdetermined? It might be, depend-ing on how we flesh out the rest of the econ-omy. If the price level were overdetermined,there would be no equilibrium, except in thefortuitous case in which the government hap-pens to pick just the right value for s. If thiswere the situation for all reasonable ways ofmodeling the rest of the economy, the FTPLwould be in trouble: It would not be a logicallycoherent macroeconomic model. But this is notthe case. In the following sections, we flesh outthe economy in what appears to be a reason-able way, and we find the price level isuniquely determined. (This issue is addressedmore rigorously in the appendix.)

We then turn to an issue of greater concern.We present evidence suggesting that to use theFTPL, one must take the non-Ricardian assump-tion very seriously. Seemingly minute departuresfrom that assumption—in the direction ofallowing for some sensitivity in the surplus tothe real debt—collapses the FTPL’s ability topin down the price level.

The final section revisits the central bank’sability to control average inflation under theFTPL. It shows that conventional views abouthow to control average inflation could actuallyspark an explosive hyperinflation under theFTPL. So, although the central bank can fea-sibly control average inflation, its method ofdoing so must be designed with care.

Is the Price LevelOverdetermined inthe FTPL?

We begin this section by providing a generaldiscussion of the issues involved in determiningthe price level. We then turn to a specific exam-ple in which the price level is uniquely deter-mined by the FTPL.

General Discussion

It is easy to find examples of the FTPL in whichthe price level is overdetermined. Recall theequation of exchange, discussed previously.For convenience, we reproduce it here: MV = PY.In traditional, old-fashioned monetarism, V isassumed to be fixed by technology, Y is deter-mined exogenously, and monetary policy takesthe form of a choice of M. In this model, P isobviously determined by the equation ofexchange. If the rest of the economy werecharacterized by these assumptions, then alogically coherent FTPL would be impossible.32

Each of these assumptions, however, hasbeen rejected on empirical grounds. First, Vexhibits substantial fluctuations in the data; theassumption that V is fixed is replaced in modernmodels by the assumption that V is an increas-ing function of the nominal rate of interest.Additionally, expected inflation plays an impor-tant role in determining R. With these two fea-tures, a logically coherent FTPL is possible.These changes cause expected future valuesof P to enter the equation of exchangethrough V. This creates the possibility thatthere are many P processes that can satisfythe equation, leaving room for the non-Ricardianassumption to pin down one of them. This possi-bility is illustrated through an example in theappendix.33

Second, the assumption that Y is exogenoushas been questioned. There is general agree-ment that at least short-run movements in Y areinfluenced by movements in V, P, and M.When models are constructed to capture this,expected future values of P enter into thedetermination of Y. As in the example of the

� 32 Finite-horizon models in which the simple quantity equationholds in the last period, but in which velocity is an increasing function ofthe nominal interest rate in the previous periods also have the property thatthe price level is overdetermined (see Buiter [1999]). The reasoning is sim-ilar to that used in our text. In those models, the equilibrium price levelmust satisfy a first-order difference equation. All equilibrium prices arethen pinned down by the fact that the price level is pinned down in the lastperiod. The likelihood that this price level coincides with the one producedby the intertemporal budget equation of the government seems small. Themost likely outcome is that the price level will be overdetermined.

� 33 Brock (1975), Obstfeld and Rogoff (1983), and Matsuyama(1991) present similar examples. An example taken from Woodford (1994)can be found in the appendix.

1212

previous paragraph, there can be multiple Pprocesses that satisfy the equation of exchange.Again, this leaves room for the non-Ricardianassumption to pin down one of them.34

Third, there is a nearly universal consensusthat exogenous M poorly characterizes monetarypolicy. For example, Taylor (1993) has arguedthat, in practice, monetary policy is best thoughtof as a rule for setting the rate of interest. Inthis case, M becomes an endogenous variable.We can see in the equation of exchange that if Ris the exogenous policy variable (as opposed toM ), then V is pinned down. But now there aretwo endogenous variables, M and P, in thisequation. Generally, under these circumstances,P and M are not pinned down. There is, in asense, a missing equation. Again, there is roomfor the FTPL to fit in.

An Example

Next, we present a simple, multiperiod modeleconomy in which the price level is uniquelydetermined in the FTPL. There is no last period,and time is indexed by t = 0, 1, 2, .... Supposethat output, Y, is the same for each date, t.Money demand depends on the rate of interest,

(3.1) Mt = ARt– �, � > 0.

Pt

The parameter A captures other factors (likeincome) that affect money demand but areassumed to be constant here. Mt is the moneystock at the beginning of period t ; Pt is the pricelevel during period t ; and Rt is the nominalrate of interest on government bonds held fromthe beginning of period t to the beginning ofperiod t +1. The Fisher equation holds

(3.2) 1+r = (1+Rt )Pt .

Pt+1

The expression on the right (l+r) is the realrate of interest on bonds paying a nominal rateof return, Rt , and r > 0 is the rate at whichhouseholds discount future utility. This pinsdown the real rate of interest.

A reasonable specification of monetarypolicy is that the central bank targets the nomi-nal rate of interest. For purposes of exposition,we adopt an extreme version of this specifica-tion, in which the central bank pegs the rateof interest to a constant, R >0. The central bankaccomplishes this by supplying whateveramount of money the private economydemands at this rate of interest.

The interest rate peg pins down seignorage:

stm � Mt – Mt –1 = Mt – Pt–1 Mt–1 .

Pt Pt Pt Pt –1

Imposing the money-demand and Fisher equa-tions [(3.1) and (3.2)] and the policy rule Rt =R,we find35

(3.3) sm = AR –� R –r , t = 0,1,2, ....1+R

Consistent with the FTPL, we assume the primarybudget surplus, s

tf, is non-Ricardian. We adopt

the simplest such policy, one in which stf is

simply a constant, s f. Thus, net governmentrevenues from all sources, excluding interestpayments, are given by

(3.4) st = s = s f + sm > 0.

To complete the description of the government,we present the period-t budget constraint. Weassume government debt is composed of one-period discount bonds; that is, the amount ofborrowing in period t is Bt+1/(1+R), and theamount paid in period t+1 is Bt+1. The period-tgovernment budget constraint is

(3.5) Bt+1 +Pt s =Bt , t = 0, 1, 2, ....1+R

The terms on the left of the equality representthe government’s sources of funds, and theterms on the right denote the uses of funds topay off the debt.36 It is convenient to rewritethis expression in real terms, that is, in terms ofbt �Bt /Pt . Dividing equation (3.5) by Pt , takingthe Fisher equation (3.2) into account, and re-arranging the terms, we obtain

(3.6) bt +1= (1+r )(bt – s ).

� 34 A cash-in-advance model displayed in Christiano, Eichenbaum,and Evans’ (1998) illustrates this. Because this is a cash-in-advance model,velocity is fixed and the factors discussed in the previous paragraph areruled out. Chistiano, Eichenbaum, and Evans show that for different speci-fications of the monetary policy rule for selecting Mt , the model has a con-tinuum of equilibria. If the non-Ricardian assumption were adopted inthis model, the equilibrium would be pinned down. For other exampleslike this, see Carlstrom and Fuerst (1998).

� 35 For simplicity, we assume the interest rate peg was in place inperiod –1, too. A more rigorous treatment which does not make thisassumption can be found in the appendix.

� 36 An alternative representation, which has some theoreticaladvantages, expresses the government’s budget equation in terms ofits total nominal liabilities, Bt + Mt –1 . We work with this alternativerepresentation in the appendix.

1313

Finally, we develop the multiperiod analogof B�=0 in part II. Recall the logic we usedthere: First, B�>0 is not optimal, since house-holds could increase utility by raising consump-tion and financing it with a reduction of B �.A negative value of B � is also not optimal,since we have removed it from the feasibleset by assumption. We continue to assumethat holdings of government bonds must benon-negative; that is, households only lend tothe government, they do not borrow from it.

The analog of B�=0 in this setting is

(3.7) lim BT = 0.T →� (1+R)T

We establish that household optimizationimplies this condition by the same reasoningused to establish B�=0. The limit cannot bepositive, for otherwise households could increaseutility by reducing their holdings of governmentdebt. To see this, suppose the limit is positive.Eventually, government debt would grow at therate of interest, that is,

Bt =Bt* (1+R )t–t*, t � t * for some t *.

At this point, the government is engaged inwhat is called a Ponzi scheme with households.The principal and interest on debt coming dueare financed entirely and forever with newlyissued debt. Under these circumstances,households can do better by saying no to the

Ponzi game, consuming the principal andinterest on debt coming due in one period andthen never holding any more government debt.The household is better off, because the actionallows a one-time increase in consumptionwithout the need to reduce consumption atany other date. An optimizing householdwould not pass up an opportunity like this;therefore, household optimization implies thelimit cannot be positive. But the limit cannotbe negative either, because Bt<0 is not allowed.Equation (3.7) is called the transversalitycondition. It is convenient for us to express thiscondition in real terms, after substituting out forthe nominal rate of interest from the Fisherequation (3.2). Using that equation, we find37

(1+R)t = (1+r )t Pt , t =1, 2, ...,P0

so that BT /(1+R )T = P0bT /(1+r )T. The transver-sality condition can then be written as

(3.8) lim bT = 0, bT = BT .T→� (1+r )T PT

We have now stated the entire model. Thehousehold’s part is given by equations (3.1),(3.2), and (3.8) and by the condition Bt �0.The government is summarized by its policy,equation (3.4), and by its flow-budget constraint,equation (3.6). Does this economy uniquelydetermine the price level? To see that it does,first note that the money-demand equation andthe government’s policy of pegging the interestrate have the effect of pinning down real bal-ances, but not M or P separately. Double M andP, and those equations remain satisfied. Thesame is true of the Fisher equation: Double Pat all dates, and it continues to hold, too. So,the level of the money stock and the price levelare not pinned down. It turns out that the non-Ricardian specification of government policy,together with the household’s transversalitycondition, is sufficient to pin down the pricelevel uniquely.

To see that the price level is uniquely deter-mined, consider figure 1, which illustrates thegovernment budget equation, b� = (1+r)(b–s).The vertical axis measures b� and the horizontalaxis measures b. (The 45-degree line is includedin the figure for convenience.) The intercept forthe budget equation is negative, and it cuts the45-degree line from below. Its slope is steeperthan 45 degrees because we assume r > 0.

F I G U R E 1Debt Evolution Under Non-RicardianFiscal Policy

End-of-period real debt

Beginning-of-period real debt

b*

45°

� 37 For example, for t = 2,

(1+R )2 = �(1+R ) P1 � �(1+R ) P0� P2 = (1 + r ) 2 P2 .P2 P1 P0 P0

Here, we describe one concern about thefragility of the FTPL, based on Canzoneri, Cumby,and Diba (1998). We show that small, plausibleperturbations of non-Ricardian policy collapsethe FTPL’s ability to pin down the price level.40

Consider the following alternative to the canon-ical non-Ricardian policy of setting s to a con-stant. Suppose s =εb , where 0 < ε�1. With thispolicy, b �=(1+r)(1–ε)b, or

bt = (1–ε)t B0 , (1+r)t P0

so that the transversality condition is satisfiedfor all P0>0. Clearly, this is a Ricardian policy;the FTPL does not pin down the price level.Now, this policy may appear to be a significantperturbation of the policy st = s. Perhaps so, butit has close cousins in which the perturbationappears to be much smaller.

Consider the following alternative to thecanonical non-Ricardian policy:

(3.10) st = � – �

+1+r –�

bt bt > b–

,1+r 1+r

s bt � b–

where

0 �� <1,

1+r s <b–

<�

.r 1– �

In this case, as long as real debt remains belowsome upper bound, b

–, then the policy is the

constant-surplus policy that we have beenanalyzing. But, as soon as bt exceeds b

–, fiscal

policy adjusts to bring the debt back in line.

�

14

Figure 1 shows what happens to b over timefor any initial value of b.38 Denote the value ofb where the budget equation intersects the45-degree line by b*,

�(3.9) b*= 1+r s = � s .

r t=0 (1+r)t

As the last equality indicates, b* is the presentvalue of future surpluses.

The value of b in period 0, b0, is now anendogenous variable. Although the nominaldebt, B0, is predetermined at date 0, the pricelevel is not. Consider three possibilities:Suppose 0 � b0<b*. Figure 1 indicates that bquickly spirals into the negative zone, violatingthe non-negativity constraint on the household’sholdings of debt.39 Next consider b0>b*. In thiscase, figure 1 indicates the debt diverges to plusinfinity. To see how the debt’s growth rateevolves, divide equation (3.6) by bt :

bt+1 = (1+r)�1– s � . bt bt

As bt grows, s becomes relatively small, and thegrowth rate of bt eventually converges to 1+r .At this point, the debt becomes so large that sis, by comparison, insignificant. The govern-ment is now running a Ponzi scheme. For thereasons we have given above, it is not in thehousehold’s interest to participate in thisscheme (technically, the household’s transver-sality condition, equation [3.8], is violated).Since households will not hold this debt, weconclude that all b0 >b* do not correspond toequilibria.

This leaves only b0 =b* to consider. Since thelevel of real debt is fixed in this case, the trans-versality condition is now trivially satisfied.Thus, only P0 =B0 /b* is consistent with equilib-rium. We conclude this version of the FTPL isan internally consistent theory of the price level.

Is the FTPL Fragile?

The assumptions underlying economists’theories are, at best, only approximations.We don’t think of them as being exactly true.Therefore, we trust theories more if their centralimplications do not change when we alter theassumptions a little. But, if key implicationsevaporate with small changes—particularlychanges that are arguably in the direction ofgreater empirical plausibility—then there isreason for concern. In this case, we say a theoryis fragile.

� 38 To see this, specify an initial value for b on the horizontal axis.Proceed vertically to the budget line, move horizontally to the 45-degreeline, move vertically to the budget line, and so on.

� 39 Implicitly, we have ruled out the possibility that a negative bimplies a negative P. This cannot be, since P0 is positive for 0 � b 0 < b*.The Fisher equation (3.2) then pins down the price path for Pt and cannotproduce a negative Pt if P0 > 0.

� 40 Woodford (1998a) argues there may be a local sense in whichthe FTPL’s ability to pin down the price level survives the sort of perturba-tions we consider here. He also discusses versions of the model withlearning which may survive these perturbations.

15

Although the algebraic representation of thispolicy may seem forbidding, it is easy to analyzeit with the help of figure 2. For bt >b

–, real debt

evolves according to bt +1=�bt +�.If real debt followed this equation forever, it

would eventually converge to b= �/(1– �). Thisequation, as well as equation (3.6), is graphedin figure 2.

Figure 2 simulates the evolution of real debtunder the fiscal policy in equation (3.10). Thesimulation is initiated with the indicated valueof b0. Real debt initially follows the steep linewith slope 1+r >1 until it passes b

–, at which

point it follows the flatter line with slope � <1.All paths with b0�b* are consistent with thetransversality condition because they convergeto a finite value, either b* or b. With the givenchange in policy, the FTPL cannot pin downthe price level.

This perturbation of the non-Ricardian policyseems realistic. At low levels of debt, fiscalpolicy is exogenous, as it is in the canonicalnon-Ricardian policy. If the debt gets out ofline, then fiscal policy adjusts to bring it undercontrol. This rings true in light of the U.S. expe-rience in the 1980s and 1990s and the provisionsof the Maastricht Treaty, which limit the realdebts of European Union member countries.

The FTPL with Stochastic FiscalPolicy

Thus far, we have illustrated non-Ricardian fiscalpolicy with st = s, a constant. But the essenceof non-Ricardian fiscal policy is simply that stis not calibrated to satisfy the intertemporalbudget equation for all prices; it is compatiblewith a much larger class of specifications forst than st = s. Here, we study non-Ricardianpolicies in which surpluses, st , are random.We use this specification to make three points.

First, Barro’s (1979) famous policy ofabsorbing fiscal shocks by raising taxes in thefuture can be represented as non-Ricardianfiscal policy.41 This is an important example,partly because it clarifies the definition of non-Ricardian policy as it is used in the FTPL.Clarification is necessary because one mightmistakenly attach other meanings to the term“non-Ricardian,” based on economists’ everydayusage of the term “Ricardian.”

Second, unless policy takes the form advo-cated by Barro, fiscal shocks cause the inflationrate to fluctuate randomly about its average.The average value of inflation is determined bythe value of the monetary authority’s interestrate peg.

Third, we describe an important result fromWoodford (1996, 1998a). He shows that underthe FTPL, instability in fiscal policy must affectthe price level, no matter how committed themonetary authority is to price stability.42 We callthis striking result “Woodford’s really unpleas-ant arithmetic,” to contrast it with Sargent andWallace’s famous title. Woodford’s arithmetic iseven tougher than that of Sargent and Wallace,who argue that if the central bank is weak, thenthe fiscal authority can push it into producingprice instability. However, Sargent and Wallace’spessimistic (“unpleasant”) conclusion is balancedby their optimism that, if the central bank justhangs tough, the problem of price stability willbe solved. From the perspective of the FTPL,Woodford argues that no matter how toughthe central bank is, it still cannot stabilize theprice level.43

� 41 It seems obvious that the Barro policy can also be representedas Ricardian, so we do not discuss it here.

� 42 Implicitly, we have in mind non-Ricardian policies other thanthose advocated by Barro (1979).

� 43 Recall, however, our second point, that the central bank cancontrol the average inflation rate.

F I G U R E 2Debt Evolution Under PerturbedFiscal Policy

End-of-period real debt

Beginning-of-period real debt

b* bo b_

45°

b ~

16

Random Fiscal Policy

Suppose the surplus obeys the first-orderautoregressive representation,

(3.11) st+1 = (1–ρ)s +ρst +εt+1 .

In this equation, εt +1 is an independently andidentically distributed white-noise process independent of st–j , j � 0. A positive realizationof εt induces a change in the date-t governmentsurplus and in the expected value of futuregovernment surpluses. Let j denote this effectat date t+j for j � 0:

(3.12) j εt =Et st +j –Et –1st+j , j �0, 0�1.

Et denotes the expectation operator, conditionalon information available at date t (Et st =st ).When the surplus has the time-series represen-tation, equation (3.11), then j =ρ

j. Of course,equation (3.12) applies more generally, evenwhen st does not have the time-series represen-tation given in equation (3.11). The presentvalue of εt ’s impact on current and expectedfuture surpluses can be defined as

� 1 �εt � εt+1 εt +

2 εt+... .1+r 1+ r (1+ r )2

(Note here that (.) is a function.) In the caseof equation (3.11), this is

� 1 � =1+ r

.1+r 1+ r –ρ

When ρ=0, so that st is independently andidentically distributed, then the present-valueterm is just unity. In this case, the effect of aninnovation in the surplus is limited to the currentsurplus only. As ρ increases above zero, thenthe present-value term increases to take intoaccount the future effects of innovation. Nega-tive values of ρ cause the present-value termsto fall as innovations in the current surplus gen-erate expected reductions in the future surplus.

It is interesting to compare the fiscal policyconsidered in equation (3.11) with that advo-cated by Barro (1979). He argues that a neg-ative shock to government finances (due, forinstance, to war) should be met by a largeincrease in debt, coupled with a constantincrease in the labor tax rate that is sufficientto pay off the interest and principal on that debtover time. In particular, he advocates fiscalpolicies of the form

� 1 � = 0 .1+r

For example,44

0=1, 1= – (1+r), or,

0=1, i= – �1+r �i,i �1;

2that is,

st=s +εt –(1+r)εt–1, or,

�st=s +εt –� �1+r �

iεt–i .

i=1 2

These examples head off misunderstandingsabout the definition of “non-Ricardian” policy.In everyday discussion, the word “Ricardian”is used in a variety of senses. For instance,economists may refer to a policy as Ricardianwhen a current tax cut is financed by increasesin future taxes that are large enough in presentvalue to match the current cut.45 It is clear fromthe preceding discussion that this type of policycan be part of a non-Ricardian regime.

Under the fiscal policy just discussed, theprice level is insulated from fiscal shocks.Shocks to the real primary surplus arefinanced by appropriate movements in theopposite direction later. In the next section,we will show that when �0, surplus shocksare at least partially financed by movementsin the price level.

� 44 See, for example, Woodford (1998a), footnote 18.

� 45 Hayashi (1989) is one example.

17

Inflation with Random Fiscal Policy

We continue to assume that policy pegs Rt=R,so that the seignorage component of st is theconstant value given in equation (3.3). As aresult, the random nature of st in equation(3.11) reflects randomness in fiscal policy. TheFisher equation still holds, although it must beadjusted to take into account uncertainty,

1+r = (1+R)EtPt

,Pt +1

where Et is the conditional expectation, giveninformation available at time t. This expressionshows the central bank controls the expectedrate of deflation through its choice of R. Thistranslates into control over the average rate ofdeflation by the fact E [Et(Pt /Pt +1)]=E (Pt /Pt +1).Imposing the suitably adjusted version of thehousehold’s transversality condition, equation(3.8), on the government’s flow-budget equa-tion, the intertemporal budget equationbecomes46

�

(3.13)Bt=Et � st+j =s �1+r �� 1–ρ �+� 1+r � st .Pt j=0 (1+r) j r 1+r –ρ 1+r –ρ

We now have a completely specified theory of the price level and inflation. One way tounderstand it is to use the model to simulate asequence of prices for a given realization of pri-mary surpluses. Suppose we have a time series,s0, s1, ..., sT , from equation (3.11) and an initiallevel of nominal debt, B0.

47 P0 is computed byevaluating equation (3.13) at t =0. B1 is thencomputed from the government’s flow-budgetequation, Bt +1= (1+R )(Bt –Pt st ), for t =0. Asequence, P0, P1, ..., PT , is obtained by performingthese calculations in sequence for t =0,1,2, ..., T.

The interest rate peg guarantees that theexpected rate of inflation (actually, deflation) isconstant in these simulations. As a result, the rateof inflation itself will be approximately uncorre-lated over time, an artifact of the constant interestrate peg. If the interest rate rule were insteaddependent on past interest rates and/or past infla-tion, then persistence would presumably appearin the model’s inflation process.48

One can gain further insight into equation(3.13) by subtracting Et –1Bt/Pt :

(3.14) Bt –Et –1� Bt � = � 1+r � (st – Et –1st )Pt Pt 1+r –ρ

= � 1 �εt

1+r.

This says that a date-t shock in the primarysurplus induces a contemporaneous change inthe real value of the debt equal to the presentvalue of the shock.49 Since Bt is predeterminedat time t, the change is brought about entirelyby a change in the price level.50

� 46 To see how this is derived, consider first the expression tothe right of the first equality in equation (3.13). Note from the Fisherequation (3.2):

1 = Et +j � 1 � , all t , j �0.Pt +j (1+Rt+j ) (1+r )Pt +j +1)

Then, the government’s flow-budget constraint can be written

Bt + j = st + j + Bt +j+1 = st + j + 1 = Et + jBt +j+1 ,

Pt +j Pt +j (1+Rt +j ) 1+r Pt + j +1

or, after applying the law of iterated mathematical expectations,

EtBt + j = E t s t +j + 1 E t

Bt +j +1 (**) .Pt + j 1+r Pt + j +1

Substitute this, for j =1, into the period-t flow-budget constraint of thegovernment:

Bt = st + 1 Bt +1 ,Pt 1+ r Pt (1+R )

= st + 1 + EtBt +1 .

1+ r Pt +1

Applying (**) repeatedly to this expression, for j =1, 2, ... results in theexpression to the right of the first equality in equation (3.13), if we applythe transversality condition, lim E 0 bT / (1+r )T = 0.

T→�

To obtain the expression to the right of the second equality in (3.13),first solve equation (3.11) to find

EtSt + j = s � � 1

�j– �

ρ�

j

� + �ρ

�jst ,

(1+r ) j 1+r 1+r 1+r

for j = 0, 1, 2, ... Then substitute this into equation (3.13) and apply the geometric sum formula.

� 47 It should be obvious how this procedure could be adapted toaccommodate any other time-series representation for st.

� 48 Loyo (1999) emphasizes this in his discussion of the persistentrise in inflation observed in Brazil in the 1980s.

� 49 The analysis of price determination under the FTPL is similar tothe analysis of consumption in the permanent-income hypothesis. SeeChristiano (1987).

� 50 Divide both sides of equation (3.14) by Bt and take into accountthat Et–1Bt = Bt to set

1 – E t –1 � 1 � = 1 � � 1 �εt .Pt Pt Bt 1+r

18

Fiscal policies like equation (3.14) under-score the fact that movements in the price levelare an alternative to Barro’s way of financingshocks to the primary surplus. A jump in theprice level acts as a capital levy on holders ofgovernment bonds, which helps to financegovernment spending just as surely as the sortof taxes included in the primary surplus. Wedescribe an environment with this type ofefficient fiscal policy in part IV.

Woodford’s Really Unpleasant Arithmetic

Woodford’s argument—that instability in fiscalpolicy must affect the price level—is a simpleproof by contradiction. Suppose the monetaryauthority could perfectly stabilize inflation andthe price level. This implies Pt+1 = Pt , so that thenominal rate of interest is fixed and equal to thereal rate. This, in turn, implies that seignorage,sm, is zero and, as a result, st = st

f. Now, supposefiscal policy is stochastic, with [1/(1+r)] ≠ 0.According to equation (3.14), Pt responds toinnovations in st . But this contradicts ourassumption that Pt is constant. It follows thatwith shocks to fiscal policy, it may not be feasiblefor the monetary authority to insulate the pricelevel from those shocks.

Bear in mind that the monetary authoritycan control the expected rate of inflation inthe FTPL. For Woodford’s really unpleasantarithmetic to be truly unpleasant, shocks tothe realized price level must have socially in-efficient consequences. This is not the case inmany economic environments, where onlythe expected inflation rate matters (see, forexample, Chari, Christiano, and Kehoe [1991]).Shocks to the realized price level are costlyin environments with nominal rigidities and inenvironments with heterogeneous agents.51

The FTPL and theControl of AverageInflation

The previous section described how the mone-tary authority can control the average rate ofinflation by pegging the nominal interest rate toan appropriate value.52 In policy discussionsabout inflation, it is sometimes suggested thatinflation can be controlled more effectivelywith an interest rate rule that responds aggres-sively to inflation. In this section, we show howsuch a monetary policy could, in fact, lead todisaster if fiscal policy were non-Ricardian.

Suppose the monetary authority adjusts theinterest rate according to the rule

1+Rt =�0+�1πt, πt = Pt /Pt –1 .

The monetary authority implements this ruleby adjusting the money supply so that moneydemand is satisfied at the targeted rate of interest.An “aggressive” interest rate rule is one in which�1 is large. For example, Taylor (1993) hasargued that �1 should be around 1.5. Thismeans that if inflation rises 1 percentage point,then the central bank raises the nominal interestrate by 1.5 percentage points. According toconventional wisdom, an aggressive interestrate rule such as this is a good way to keepinflation under control. As we shall see, this isnot necessarily true if policy is non-Ricardian.

We suppose the rest of the economy corre-sponds to the example in “Is the Price LevelOverdetermined in the FTPL?” (page 11). As inthat model economy, we assume there is nouncertainty, since it is not essential to theanalysis here. Combining the interest rate rulewith the Fisher equation (3.2), we obtain thefollowing expression, which must hold inequilibrium:

πt +1 = α0 +

α1 πt . 1+r 1+r

Consider an aggressive interest rate rule withα1/(1+r) >1. The relationship between πt +1and π t is illustrated in figure 3. There is aparticular inflation rate, π *, such that if πt = π *,then πt +1 = π *. However, if the initial inflationrate is greater than π *, then πt grows without

� 51 See Woodford (1996) for an environment with endogenous pro-duction and sticky prices. With sticky prices, shocks to the aggregate pricelevel distort the allocation of resources across the production of differentgoods. They also distort aggregate output.

� 52 See “Inflation with Random Fiscal Policy” on page 17.

19

bound. This possibility is shown in figure 3,in which inflation starts at π0 in period 0 andthen explodes.

As in our previous model economy, theinitial price level is determined by fiscal policyaccording to the intertemporal budget equation(3.9). Technically, s is no longer constantbecause variations in inflation cause seignorageto vary over time, too. However, we assumethat seignorage revenues are small enough toignore, so that s comprises only s f. We continueto assume that s f is constant.53 The price levelin period 0 is determined by P0 = B0/b *, whereB0 is the initial nominal debt and b * is definedin equation (3.9).

With P0 determined by the intertemporalbudget equation and P–1 determined by history,π0 is uniquely pinned down. However, thereis no way to rule out the possibility that thisvalue of π0 lies to the right of π *, in which caseinflation explodes.54

One way to gain insight into the mechanicsof this exploding inflation is to focus on thegovernment’s budget constraint, equation (3.5).From that equation, we see that a higher nomi-nal interest rate leads to a more rapid increasein the nominal debt, Bt+1. Assuming the outlookfor the fiscal primary surplus does not change,the real value of the debt remains constant.With the nominal debt growing more quicklyand its real value constant, inflation must rise.

The central bank’s monetary policy respondsto the rise in inflation by driving the interestrate up even further, leading to an additionalincrease in inflation. This circular, self-reinforcingprocess produces the explosion in inflation.

The possibility just outlined, whereby anaggressive interest rate rule leads to explodinginflation, may seem peculiar. Loyo (1999) refersto it as a “tight money paradox.” According tothe model, if the central bank, instead of beingaggressive, adopts a more accommodatingstance by choosing a value of �1 substantiallyless than unity, then exploding inflation cannotoccur. In the previous section, with �1=0,inflation fluctuated around a constant value. Itis easy to confirm, using the logic of figure 3,that the same is true for 0<

�1 <1. Relative to asimple monetarist perspective, it is certainly aparadox that adopting an aggressive stanceagainst inflation by increasing �1 could convertstable inflation into an exploding inflation.55

However, we have just seen that it can occur ina coherent economic model. Moreover, Loyoargues the model captures the driving forcesbehind Brazil’s inflation take-off in the early1980s. Although we are skeptical that toughmonetary policy caused Brazil’s high inflation,the hypothesis certainly does seem intriguing.

Woodford (1998b, pp. 399–400) uses theexploding-inflation scenario to understand thenature of fiscal policy in the United States overthe past two decades. He observes that econo-metric estimates of the Federal Reserve’s policyrule in the 1980s and 1990s place �1 substan-tially above unity (see Clarida, Gali, and Gertler[1998]), and there is no evidence of instabilityin U.S. inflation. He concludes that policyin the United States during this time must nothave been non-Ricardian.

� 53 Here we are assuming the economy is in the “cashless limit”discussed by Woodford (1998 a,b,c) and defined in footnote 30 of the pre-sent paper.

� 54 In this situation, both fiscal policy and monetary policy areactive in the sense defined by Leeper (1991). Our analysis is consistentwith Leeper’s, which concludes that for almost all values of fiscal policy(s f ), there is no stationary equilibrium inflation rate.

� 55 Tight money paradoxes also exist in environments withRicardian fiscal policy. For example, Sargent and Wallace (1981) showedtight monetary policy may lead to an immediate rise in inflation in such anenvironment. See Kocherlakota and Phelan (1999) for a discussion of atight money paradox in the FTPL.

F I G U R E 3Evolution of Inflation under Aggressive Interest Rate Rule and Non-Ricardian Policy

Inflation in the current period

Inflation in the previous period

π* π o

45°

1+r

2020

IV. Fiscal Theory andthe Optimal Degreeof Price Instability

The FTPL literature has drawn attention to thepossibility that some price instability may bedesirable when unavoidable shocks to the gov-ernment budget constraint occur (Sims [1999],Woodford [1998a] ). When there is nominalgovernment debt, unanticipated shocks to theprice level act as capital levies on bondholders.The idea is that it is efficient to absorb unantici-pated shocks with capital levies rather than bychanging distortionary taxes.

We illustrate these observations in the simplestpossible model.56 Relative to the one-periodmodel of part II, this model incorporates twoessential complications. First, we must take intoaccount the distortionary effects on the bond-accumulation decision that may arise fromprice-level instability. For this reason, we adopta two-period model. The bond-accumulationdecision is taken in the first period, and thegovernment-spending shock and price-leveluncertainty occur in the second period. Second,the model must capture the notion that taxesare distortionary. Accordingly, we assume thelabor supply is endogenous and taxes are raisedusing a proportional tax on labor income.

The model is an example of the FTPLbecause government policy—the choice oflabor tax rates—is non-Ricardian. We illustratehow FTPL advocates study the optimal degreeof price stability by examining the “best” equi-librium of such a model (see, for example, Sims[1999] and Woodford [1998a]). The literature onoptimal fiscal and monetary policy (see, forexample, Lucas and Stokey [1983]) calls thisequilibrium the Ramsey equilibrium.

First, we describe the model. To simplifythe analysis, the model does not includemoney; as a consequence, the model againillustrates price determination in an economywith no government-provided money. Next, wecharacterize the best (that is, the Ramsey) equi-librium in this economy. We then present anumerical example to illustrate the role of priceinstability in bringing about efficient resourceallocation in the model. We assess the results ina summary section.

The Model

The economy comprises firms, households, anda government. Households and firms interactin competitive markets. The government mustfinance an exogenously given level of expen-ditures by levying a distortionary tax rate on laborand possibly by issuing debt. There is no un-certainty in the first period. However, there isuncertainty in the second period’s level of gov-ernment spending. Spending could be highor low, with probability 1/2 each, with theuncertainty being resolved at the beginningof the second period. Consistent with thenon-Ricardian assumption, the governmentcommits to its policies before the first period.Trade occurs by barter, and there is no moneyin the model.57

Firms have access to a linear productiontechnology,

y =n , y h =nh, y l =nl ,

where y and n denote output and labor,respectively, in the first period, and y i, ni

denote output and labor in the second period, i=h,l. The superscript h or l indicates the sec-ond period when government spending is highor low, respectively. The linearity in the pro-duction function guarantees the real wage isalways unity in equilibrium; henceforth, wesimply impose this result and do not refer tofirms any more.

Preferences of households over consumptionand labor during the two periods take the form

(4.1) U (x) =c –1

n2 +1β ��ch –

1(nh)2�

2 2 2

+ �cl –1

(nl )2� � , 0 � β � 1 ,2

where c denotes consumption in the first periodand β is the discount rate, with β=(1+r) –1.Similarly, ci denotes consumption in the sec-ond period, conditional on the realization ofgovernment spending, i =h, l. β is the discountrate of the household, and the fraction “1/2”

� 56 The example illustrates the results on the desirability of taxsmoothing and volatile prices reported in Chari, Christiano, and Kehoe(1991).

� 57 We could imagine there is “inside money” and trade isaccomplished through an efficient exchange of IOUs.

2121

preceding β corresponds to the probability ofthe h or l state of the world. Finally,

(4.2) x = (c,ch,cl,n,nh,nl ).

The linear-quadratic structure of preferences ischosen to ensure a simple analysis. The house-hold’s period-1 budget constraint is

(4.3) B� +Pc � B+P (1–τ)n,1+R

where P is the period-1 price level, B is thenominal bonds the household inherits from thepast, and R is the nominal rate of interest. Also,τ denotes the tax rate on labor and B� denotesbonds acquired from the government. Thehousehold’s budget constraint in the secondperiod, conditional on the realization of uncer-tainty, is

(4.4) P h ch �B�+P h (1–τh)nh,

P l cl �B�+P l (1–τ l )nl .

Again, superscripts indicate the realization ofthe exogenous government-spending shock.There is no government-supplied money in thiseconomy.

The household maximizes utility by itschoice of non-negative values for B�,c,ch,cl,n,nh, and nl. It must respect the budget con-straints just specified, and it takes prices andthe interest rate as given and beyond its control.The Euler equations associated with the house-hold’s optimal choice of labor and bonds are

(4.5) n=1–τ,nh =1–τh,nl =1– τ l ,

1 = 1 β � 1 + 1 � . (1+R )P 2 P h P l

The last of these equations tell us that theexpected gross real rate of return on bondsmust be 1/ β. That this is true, independent ofthe intertemporal pattern of consumption,reflects our assumption that utility is linear inconsumption.

The government’s budget constraints in thefirst and second periods are given by

(4.6) B�

+P τn �B +Pg1+R

P h τhnh �B�+P hg h

P lτ l n l �B�+ P lg l.

Here, g denotes government consumption inthe first period, and gi denotes period-2government consumption, i =h, l . In the equa-tions of our model, R appears everywhere as(1+R)/P h, (1+R)/P l, or B�/(1+R).58 Thus, wecannot pin down R, P h, P l, and B�separately.For this reason, we adopt the normalizationR =0 from here on. Government policy is avector of three numbers, π, where

π = (τ, τh, τ l ).

This is a non-Ricardian policy because thereis no set of values for π that will satisfy the gov-ernment’s intertemporal budget equation (seebelow) for all prices.

Combining the government and householdbudget equations, we obtain the resourceconstraints:

(4.7) c+g � n, ch+gh � nh, cl+g l � nl.

There are 10 variables to be determined inequilibrium: P, Ph, Pl, B�, c , ch, cl, n, nh, andnl. They are determined by the three house-hold budget constraints ([4.3] and [4.4]), evalu-ated with a strict equality; the four householdEuler equations ([4.5]); and the three resourceconstraints ([4.7]). These 10 equations, togetherwith the requirement P, Ph, Pl > 0, characterizethe equilibrium (if one exists!) associated with agiven government policy. The mapping from πto these variables is single valued. We denotethe function relating the last six variables to πby x (π), where x is defined in equation (4.2).

� 58 The statement is obviously true in the case of the householdEuler equation in (4.5). To see that it is also true of equations (4.3),(4.4),and (4.6), replace B� with

�B = B�/(1+R ) and divide the period-2 budget

constraints by 1+R .

22

� 59 See Bizer and Judd (1989), Chari, Christiano, and Kehoe(1990), Judd (1989), and Lucas and Stokey (1983), among others.

� 60 Here, κ = 2 � 1 – g + 1 β (1– g h – g l) � . 2 2

To see how we obtain this expression, note that c –(1/2)n 2 is y–g–(1/2) n 2 after using the resource constraint, y=c +g. Imposingn=1–τ , then, yields that c –(1/2) n 2 is (1/2)(1– τ 2)–g.

� 61 Chari, Christiano, and Kehoe (1991) confront the same problem,which they deal with by setting the initial debt to zero. In our context, thatcreates a problem because it leaves us with no ability to pin down P.

� 62 Our model differs from Sims’ (1999) model in two respects.First, ours has only two periods, while Sims’ has an infinite horizon. (It is trivial to extend our model to the infinite horizon.) Second, we modelagents at the level of preferences and technology, while Sims adopts areduced-form representation analogous to the one in Barro (1979). Ourreduced-form utility function coincides with Sims’, but our budget con-straint does not. Sims models taxes as lump-sum in the budget constraint,whereas we take into account the distortionary effects of taxation. Forexample, Sims would have τ in the budget constraint, rather than τ (1–τ),as we do. The conclusions of the analysis are not sensitive to thesedifferences.

22

The Ramsey Equilibrium

The Ramsey equilibrium is associated with thepolicies, π, that solve the problem

maxU [x (π)] ,π

subject to the requirement that prices be strictlypositive, B��0, and the elements in x be non-negative.59 The Ramsey equilibrium is easy tocompute in this model economy.

After substituting out for the endogenousvariables in terms of π in equations (4.7) and(4.5), the utility function is represented by

(4.8) U [x (π)] = – τ2 – 1β � �τh �2+�τ l �2�+κ ,2

where κ is a constant.60 To complete the state-ment of the Ramsey problem, we need a sim-ple representation of the restrictions placed onπ by the positive-price requirement. Before wedo this, we must confront a technical issue.

It is well known in the literature on Ramseyequilibria that it is efficient to renege on theinitial nominal debt, B, by selecting policiesthat produce an infinite first-period price level.Allowing this would plunge us into exoticmathematical issues, distracting us from thecentral focus of the example: the desirability ofletting prices in the second period react to therealization of government spending in thatperiod. With this in mind, we simply fix theperiod-1 price level at P =1. Since the nominaldebt, B, is given from the past, it follows thatwe have fixed the initial real debt. It is impor-tant to emphasize, however, that we do not fixthe second-period price levels.61

The restriction on π implied by P =1 is easyto determine by expressing the government’sfirst-period intertemporal budget equation interms of π. Combine the household’s intertem-poral Euler equation (4.5) with the govern-ment’s budget constraints (4.6),

(4.9) B �τ (1–τ) –g + 1 β �τh �1–τh� –gh

2

+τ l �1–τ l � –g l � ,where we have imposed P =1. The restrictionson second-period prices come from theintertemporal government budget equationsthat obtain

(4.10) τh �1–τh� –gh �0, τ l �1–τ l � –gl �0

in those periods. The Ramsey problem, modi-fied to incorporate the restriction P =1 is set upin Lagrangian form:

max – τ2 – 1 β � �τh �2+ �τ l �2 �τ , τh,τ l 2

+ λ �τ (1– τ ) –g+ 1 β �τh (1– τh) –gh

2

+τ l (1– τ l ) –gl � –B�

+�h �τh (1– τh) –gh � +�l �τ l (1– τ l ) –gl � ,

where λ, �h, and �l �0 are Lagrange multi-pliers.62 The necessary and sufficient conditionsassociated with the maximum are the inequalityconstraints on the multipliers, λ, �h, and �l �0;the inequality constraints in equations (4.9)and (4.10); the “complementary slackness”conditions,

(4.11) 0=λ �τ (1– τ ) –g +1β �τh (1– τh) –gh

2

+τ l (1– τ l ) –gl �–B� ,

0=µh �τh (1– τh) –gh � ,

0=µ l �τ l (1– τ l ) –gl � ;

23

and the three first-order conditions corres-ponding to τ , τh, τ l. After rearranging, these are

(4.12) λ = 2τ ,1–2τ

µh= β � τh– τ � ,

1– 2τh 1– 2τ

µ l= β � τ l – τ � .1– 2τ l 1– 2τ

We solve the (constrained) Ramsey problemby finding multipliers, λ, µh, µ l, and policies,τ , τh,τ l, that satisfy these conditions.

Once the Ramsey policies have been iden-tified, n, nh, and nl are obtained from equation(4.5) and c, ch, and cl from equation (4.7).Then, B�, Ph, and P l are obtained by solvingequation (4.6). Several qualitative features ofthe solution are immediately apparent. First,the weak inequality in equation (4.9) is sat-isfied as a strict equality.63 This is not sur-prising—otherwise, taxes would be higher thannecessary and, given the form of preferences,this would be counterproductive. Also, becausethe period-0 intertemporal budget equation issatisfied as a strict equality, it would have beenoptimal to inflate away the initial debt by settingP =�, had we not imposed the requirement thegovernment pay off B with P =1.64 Second,

ignoring the requirements of the non-negativityconstraints on prices in the second period, theoptimal outcome is τ=τh=τ l. To see this, notethat the first-order conditions in this case areequation (4.12) with µh � µl � 0. Inspecting thesecond two equations in (4.6), it is obvious thatP h >P l as long as gh>gl. Third, in practice, theconstant tax rate policy is not necessarily feasi-ble, since it may conflict with the positive-pricerequirement. In this case, however, the pricefluctuations across states of nature are evenmore extreme.

Suppose, for example, the constant tax ratepolicy is inconsistent with the first of the twoinequalities in equation (4.10). Then, µh > 0,τh > τ , and, by equation (4.11), τh (1–τh)–gh=0.The latter implies the government inflates awaythe debt completely in state h, with Ph= �. Toensure that households still have an incentiveto accumulate debt in the first period, equation(4.5) indicates P l must satisfy P l = (β/2)P(1+R)=β/2 in this case. That is, the real rate ofreturn on debt into state l must be high.

A Numerical Example

This section studies a numerical example toillustrate the properties of P h and P l in theRamsey equilibrium. A natural benchmarkto consider is the no-debt equilibrium:τ is selected so that B�= 0, and τh and τ l areselected so the constraints in equation (4.10)are satisfied as exact equalities. With this asa benchmark, we evaluate the Ramsey equilib-rium in which B�>0 and consider Ph and P l.

To see how taxes are determined in thebenchmark equilibrium, consider figure 4,which graphs τn =τ (1–τ) for τ ∈ (0,1). We havea single-peaked Laffer curve in our modeleconomy. The horizontal lines indicate the

� 63 Here is a proof by contradiction. Suppose the weak inequality inequation (4.9) were a strict inequality. Then, by the first expression inequation (4.11) and in equation (4.12), we have λ = 0 and τ = 0. The strictinequality in equation (4.9) implies that at least one of the weak inequalitiesin (4.10) is strict. That implies, by (4.11), the associated multiplier is zero.Equation (4.12) implies the associated tax rate is zero, but this contradictsthe non-negativity of the primary surplus in that period.

� 64 In that case, the constraint would have been equation (4.11)without the term –B.

F I G U R E 4

Revenues from Labor Tax

Revenues

Labor tax rate ( τ )0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

0.20

0.15

0.10

0.05

0.25

B + g

0.30

g h

g l

l hτ ττ

24

revenue requirements in the first and secondperiods. We assume the first-period revenuerequirement, B+g, is 0.20. The second-periodrevenue requirement is g h = 0.15 when gov-ernment spending is high and g l = 0.05 whengovernment spending is low. The benchmarkequilibrium requires that τ, τh, and τl be set asindicated on the horizontal axis. In particular,τ = 0.28, τh = 0.18, and τl = 0.05, after rounding.The value of equation (4.8) in this equilibriumis –0.0941, ignoring κ and setting β = 0.97. Taxrates are very uneven over time and over statesof nature.

Now consider the Ramsey tax rates. Weproceed under the conjecture (subsequentlyverified) that they are optimally chosen to be aconstant, τ *, across dates and states. We usethe fact, established in the previous section,that the constraint, equation (4.9), is binding.Two constant tax rates solve equation (4.9)

evaluated as a strict equality. Given prefer-ences, equation (4.8), we go with the lowerone, τ * = 0.19, after rounding. To verify thissolves the Ramsey problem, we must confirmthat equation (4.10) is satisfied. Indeed it is, withτ*(1–τ*)– gh = 0.0008 and τ*(1– τ*) – gl =0.10.

Solving the first expression in equation (4.6),we find that B�=0.05. In addition, we find fromthe second two expressions in equation (4.6)that Ph =64.67 and Pl =0.49. Essentially, thegovernment reneges on the debt in period hand pays an attractive 100 percent rate of returnin state l. Finally, the utility of this equilibriumis –0.0674. These results, plus the consumptionand labor allocations, are summarized in table 1.By issuing debt, it is possible to stabilize employ-ment and consumption across dates. By issuingthe debt in nominal terms and allowing theprice level to fluctuate, it is possible to makethe payoff on that debt state-contingent inreal terms.

Summary

We have described a model in which an efficientfiscal program issues nominal debt and thenallows the price level to fluctuate. Although wedemonstrated this finding in an economy withno government-provided money, this feature ofour model plays no fundamental role in theresult. The same result was obtained by Chari,Christiano, and Kehoe (1991) and by Woodford(1998a) using models with money.

In our model, the equilibrium is equivalentto one in which the government issues debtwhose payoff is denominated in real terms inthe first period, and where the payoff is explic-itly contingent on the realization of governmentspending in the second period.65 From thisperspective, the natural question is, why notuse the state-contingent-debt strategy, ratherthan going to the trouble of issuing nominaldebt and allowing the state contingency to arisebecause of fluctuations in the price level?

� 65 Lucas and Stokey (1983) emphasize the desirability of this type of debt.

Two Equilibria

T A B L E 1

Variable Benchmark, no-debt Ramseyequilibrium (B�=0) equilibrium

P 1 1

P h — 64.67

P l — 0.49

n 0.72 0.82

nh 0.82 0.82

nl 0.95 0.82

c 0.52 0.62

ch 0.67 0.67

cl 0.90 0.77

τ 0.28 0.19

τh 0.18 0.19

τ l 0.05 0.19

g +B 0.20 0.20

gh 0.15 0.15

gl 0.05 0.05

B� 0 0.05

Utility –0.0941 –0.0674

25

To address this question, we must invokeconsiderations that are not included in themodel. One advantage of the nominal-debtstrategy is that it is likely to have lower costs ofadministration and information acquisition,because the appropriate response of the realpayoff on the debt to shocks is achieved auto-matically as a by-product of price fluctuationsgenerated in the market-clearing process (Sims[1999] and Woodford [1998a]).

This may provide an overly optimistic viewof the nominal-debt strategy. For example, ifthere are sticky prices, then fluctuations in theprice level could distort resource allocations.In addition, price volatility may interfere withprivate contracts by inducing reallocations ofwealth among private agents. Presumably, aversion of the Ramsey problem that incorpo-rates those costs would still exhibit price fluctu-ations, though they would likely be smaller.66

Designing a fiscal system that properly balancesbenefits and costs would presumably be verydifficult, reducing the cost advantages of thenominal-debt strategy we allude to above.

There is another reason to question theadvantages of both the nominal and real state-contingent-debt strategies. Unless the govern-ment has substantial ability to commit to itspolicies, either strategy could backfire, a possi-bility that can be seen in the example. It isefficient in the first period for the governmentto inflate away the debt. But when time movesforward one period, the second period becomesthe first period. When that time arrives, it isagain in the government’s interest to inflateaway the debt! Households that understand thisin the first period may well choose not toaccumulate debt in the first place.67

Now, this case was excluded in our analysisbecause of the assumption that policy is non-Ricardian: The policy is just a sequence ofnumbers (tax rates) through time, and the pos-sibility of adjusting them ex post is ruled out.Is this a realistic assumption? Does it assumethat governments have more commitmentpower than they actually have? The literatureon Ramsey policy has generally concluded theanswer is yes, and has moved on to equilib-rium concepts that do not presume as muchcommitment power.68

In principle, one can make the case that thedegree of commitment needed for the policyto work is not implausibly large. This might beso if the required price fluctuations occurredautomatically, in a way that legislatures havedifficulty interfering with. For example, Judd(1989) suggests that price movements in theU.S. economy correspond roughly to therequirements of an efficient fiscal program.He notes that good shocks to the governmentbudget constraint, such as technology shocks,tend to produce a negative shock to the pricelevel, generating transfers to holders of govern-ment bonds. Similarly, bad shocks, like a jumpin government spending due to war or naturaldisaster, tend to drive the price level up, taxinggovernment bond holders.

Our point is not that the degree of commit-ment required for the volatile price strategy isnecessarily too great. Our point is only thatcommitment is a fundamental assumption ofthe volatile price strategy. In the absence ofcommitment, the strategy is likely to backfire.

� 66 It would be interesting to investigate this question inquantitative models. There is a possibility that the efficient degree ofvolatility in prices would be reduced to zero if price volatility introduceddistortions. Chari, Christiano, and Kehoe (1991) argue that, in principle,there are many ways to achieve state contingency in fiscal policy. If therewere costs to using the price level for this, then the efficient thing to dowould be to use another way. Only if there were costs associated with allways of achieving state contingency in fiscal policy would some volatilityin prices be desirable.

� 67 Sims (1999) considers a proposal that Mexico adopt theU.S. dollar as its national currency. He criticizes the proposal on thegrounds that, with the Mexican national debt denominated in a foreigncurrency, the Mexican government loses the fiscal benefits of the policydescribed in the text. That is, it would not be able to periodically renege onand subsidize holders of its debt through fluctuations in the Mexican pricelevel. Our point here is that giving up this option may not be very costly toMexico, if the Mexican government lacks credibility. Indeed, giving up theoption may be a good thing. In the absence of credibility, attempts to usethe option may lead to the disastrous situation in which everyone refusesto buy Mexican government debt.

� 68 Research on optimal policy that presumes a lack ofcommitment includes Chari and Kehoe (1990) and Stokey (1991). In contrast, Lucas (1990) argues forcefully in favor of implementingRamsey-optimal capital taxation, a theme continued in Atkeson,Chari, and Kehoe (1999).

26

Regarding the second question, Sims (1999)has stressed the potential benefits of pricevolatility.71 Variations in the price level inresponse to fiscal shocks have the effect of taxingand subsidizing holders of nominal governmentliabilities. Under certain circumstances, thiscan enhance the overall efficiency of gov-ernment fiscal and monetary policy. But thisresult also raises questions, because it isobtained in an enviro-ment with few of the frictions observed in actualeconomies that make price volatility costly.Whether the result would survive the introduc-tion of a realistic set of frictions—and a realisticset of alternative methods for dealing with fiscalshocks—is unclear at this time.

26

V. Conclusion

What insights does the FTPL provide into thetwo questions about price stability we posed atthe beginning of this review? That is, how canprice stability be achieved? And, how muchprice stability is desirable? Conventional wisdomholds that if there is no doubt about the centralbank’s commitment to low and stable inflation,then low and stable inflation is exactly whatwill happen.69 According to the FTPL, however,this overstates the central bank’s power. Still, itremains an open question just how severe thelimitations on central banks’ powers are. Theselimitations may not be very great for modern,developed economies. In the FTPL models thatwe have studied, the central bank can deter-mine the average rate of inflation. However, itcannot perfectly control the variance of infla-tion because it cannot eliminate the impact ofshocks to fiscal policy on the price level. But ina modern Western economy, the stock of out-standing nominal government liabilities is quitelarge—Judd’s (1989) estimate for the UnitedStates puts it at one year’s GDP. Therefore, arelatively small change in the price level canabsorb a fairly large fiscal policy shock.70 Inpractice, then, the conventional answer to thefirst question may be roughly the right one,even under the FTPL.

� 69 See Sargent and Wallace (1981), last paragraph.

� 70 Sims (1999) also stresses this point.

� 71 Woodford (1998a) has made a similar suggestion. The resulthas also been obtained in Chari, Christiano, and Kehoe (1991).

2727

VI. Appendix: TheLogical Coherence of Fiscal Theory

An important concern regarding the FTPLhas to do with its internal logical consistency.When the FTPL uses the intertemporal govern-ment budget equation to pin down the pricelevel, is that price level consistent with theone determined by the rest of the economy? Insome cases, the answer is no.72 Do these caseswarrant the conclusion that the FTPL is notlogically coherent? We think not, as enoughinteresting examples can be constructed inwhich the fiscal theory is logically coherent.One example is given in the body of thisreview. The point is also illustrated in severalarticles of a special issue of Economic Theoryin 1994. In this appendix, we present anotherexample.

The model we work with is the cash/credit-good model of Lucas and Stokey (1983). Weexamine a range of parameter values, includingthe empirically plausible ones, according toestimates reported in Chari, Christiano, andKehoe (1991). We skip detailed proofs in certainplaces, though never without providing theintuition for the argument. Readers who wishto see an extensive and rigorous treatment ofthe properties of the equilibria of this modelshould consult Woodford (1994). This appen-dix presents an extended example to illustratehis Propositions 2 and 10 at the level of anadvanced undergraduate or first-year graduateeconomics course.

We first consider the case in which monetarypolicy is characterized by a constant money-growth rate. We show the model has a uniqueequilibrium when the non-Ricardian assumptionis adopted. We then consider the case in whichthe monetary authority pegs the interest rate.Like the example in the text, the model has aunique equilibrium when the non-Ricardianassumption is adopted. When that assumptionisnot adopted, the model fails to exhibit a uniqueequilibrium. In this case, the model reproducesthe classic Sargent and Wallace (1975) result:The price level is indeterminate. From a techni-cal standpoint, the non-Ricardian assumption isa device that can eliminate the price-level inde-terminacy associated with interest rate peggingthat Sargent and Wallace analyze.73

The first section below describes the agentsof the model and defines equilibrium. The fol-lowing section addresses the case in whichmonetary policy is characterized by constantmoney growth. The final section addresses thecase of interest rate pegging.

The Lucas–StokeyCash/Credit-GoodModel

Households

THE HOUSEHOLD PROBLEM AND CONSTRAINTS

The model abstracts from differences amonghouseholds by assuming they are all identical.In addition, households are assumed to liveinfinitely long. This assumption can be inter-preted, following Barro (1974), as reflectingthat each household actually lives a finiteamount of time but cares in a particular way forits offspring.

The preferences of the representative house-hold are given by

�� βtu(ct), u(c)=log(c), 0<β <1,t=0

c = � �1–σ �c v1 +σc v

2 �1–v ,

where 0 <σ <1 and ct denotes consumptionservices. In our analysis, we restrict υ to therange 0 <υ <1. Chari, Christiano, and Kehoe(1991) argue this is the empirically relevantcase; based on postwar U.S. data, their pointestimates are σ =0.57 and υ =0.83.

Consumption services are generated by theacquisition of two market-produced goods, asindicated. The first, c1t , is called a “cash good”and the other, c2t , is a “credit good.” To pur-chase the cash good, households need to setaside cash in advance.

To make the notion of “in advance” precise,the model adopts a particular timing. Eachperiod is divided into two parts. In the first part(the “morning”), the household participates inan asset market, and in the second part (the“afternoon”) the household participates in agoods market. The cash that households needto purchase c1t in the afternoon of a given daymust be set aside at the end of asset-market

� 72 A simple example, in which the traditional quantity theoryholds, is presented in the body of this review. In the model of Obstfeldand Rogoff (1983), there is a countable set of equilibria. The Ricardianassumption does not sit comfortably with that model, because the like-lihood of the fiscal and monetary authorities choosing a fiscal policyconsistent with one of those equilibria seems slim. Buiter (1999) providesanother example.

� 73 This is how Kocherlakota and Phelan (1999) interpret the FTPL.

2828

trading the same morning. They hold thesebalances idle until the morning of the followingday, when actual payment is due. Credit goodswork differently. For these goods, the householdhas no need to accumulate cash in advance.The household simply pays for the goods withcash in the next morning’s asset market.

Do not be misled by the labels used toidentify these goods. It is not that one can bebought “on credit” in the traditional sense, andthe other cannot. Both goods are paid in cashthe morning after the purchase. No credit isoffered by the seller in either case. The differ-ence is simply that in the case of the cashgood, the household must forfeit interest: Tobuy the cash good, the household must carryidle cash in its pocket throughout the after-noon. From the point of view of the seller, thegoods are completely the same. The terms ofthe transaction are identical—cash only, to bedelivered the morning after the sale.

The distinction between cash and creditgoods may at first seem artificial. In fact, it isa clever device for capturing the idea thattransactions in some goods are more cash-intensive than in others. It will produce ademand for money, one that is a function ofthe interest rate.

We assume the marginal rate of transforma-tion in production is unity between the twogoods. Therefore, the price of the two goods,Pt, is identical in equilibrium. Moreover, in anyequilibrium it must be that Rt�0 and Pt>0. Mar-ket clearing is impossible if either of these twoconditions fails to be satisfied.

Let At denote the household’s financial assetsat the end of asset-market trading. In the firstperiod, t =0, this is simply a given number, A0 .The household can allocate At as follows:

(A1) Mdt +

Bdt +1 +Tt � At, t =0, 1, 2, ...,

1+Rt

where M dt denotes money balances; B d

t +1denotes government debt, which costs B d

t +1/(1+Rt ) today and pays off B dt +1 in the

next period’s asset market; and Tt denoteslump-sum taxes. The household does not setM d

t to zero because it must set aside cash inadvance, Pt c1t �M d

t , if it wishes to consumecash goods. Assets at the beginning of the nextperiod are

(A2) At +1=M dt +Pt (y –c1t –c2t ) +B d

t+1 .

Here, M dt is the cash balance carried into the

previous period’s goods market; Pt y denotesthe receipts from the sale of y in the previousperiod’s goods market; Pt (c1t +c2t ) representsthe bill of goods purchased in the previousperiod’s goods market; and B d

t +1 is the receiptsfrom government debt purchased in the previ-ous period’s asset market.

It is useful here to follow Woodford’s (1994)suggestion to write equations (A1) and (A2) ina slightly different form. “Spending” is defined as

St= Pt c1t + Pt c2,t + �1 – 1 � �Md

t –Pt c1t �.1+Rt 1+Rt

In this measure of spending, excess holdings ofmoney balances, above what are needed forthe cash-in-advance constraint, have a positiveprice if Rt >0. The relative “prices” of c1t and c2taccurately reflect that the former involves sacri-ficing interest earnings. “Income,” It , is definedas

It =Pt y

–Tt .1+Rt

Divide both sides of equation (A2) by (1+Rt )and substitute out for B d

t +1 / (1+Rt ), usingequation (A1) to obtain

(A3) At +1 � (1+Rt) (At+It –St ).

The accumulation of household assets obeys theusual simple equation one finds in a non-mon-etary, single-good model economy.

A lower-bound constraint must be placed onAt to ensure the household has a bounded con-sumption set. We impose the assumption thatthe current value of assets must eventually benon-negative,

(A4) lim qT AT �0,T→�

where

qT = 1

,q0 =1.(1+R 0) (1+R 1)... (1+R T–1)

2929

It is easy to verify that equations (A3) and (A4)are equivalent to the usual single present-valuebudget constraint for consumption, St , andincome, It .

74

We suppose that at each date the householdchooses c1t+j ,c2t+j �0,M d

t+j B dt +1+j �0, to maxi-

mize its utility, subject to the restrictions justdescribed, and takes At , Rt +j , Pt +j , j � 0, asgiven and beyond its control.75

NECESSARY AND SUFFICIENT CONDITIONS FOR

HOUSEHOLD OPTIMIZATION

The household first-order conditions are

(A5) u2,t =β

u1,t +1

Pt Pt +1

and

(A6) u1,t =1+Rt .u2,t

Here, ui,t denotes the partial derivative of utilitywith respect to cit , i =1, 2. To understand whythe first of these Euler equations is implied byhousehold optimization, consider the followingargument: Suppose the household reducesits purchases of credit goods in the period-tgoods market by one dollar and applies thatdollar to additional cash-good consumption inperiod t +1. Credit-good consumption todaydrops by 1/Pt , which translates into an immedi-ate decrease in utility of u2,t /Pt . This reductionof expenditures frees one dollar in the assetmarket in the next period, which can be appliedtoward the cash-in-advance constraint for pur-chasing 1/Pt +1 units of the cash good in nextperiod’s goods market. The utility benefit fromthe standpoint of period t is βu1, t+1/Pt +1. If thegain exceeded the cost, the household couldnot be optimizing, or we would have found achange in its plan that would improve utility.

Similarly, if the gain were less than the cost,the household could raise utility by increasingcredit-good consumption in period t and reduc-ing cash-good consumption in period t +1.Optimization requires that neither of thesestrategies raises utility, and this is why the firstEuler equation above (A5) is an implication ofhousehold optimization.

The second Euler equation (A6) is alsoimplied by household optimization, establishedby an argument similar to the one in the previ-ous paragraph. The argument exploits thetrade-off between cash and credit goods withinthe same period. The household can increasecurrent-period cash-good consumption byreducing its acquisition of government debt.

This reduces its cash receipts in the next period’sasset market, thus reducing the cash availablefor credit-good consumption today. This Eulerequation makes considerable sense: When Ris high, it implies that u1 is relatively high, sothat c1 is relatively low. This makes sensebecause high R raises the cost to householdsof purchasing c1.

There is also a condition associated with thecash-in-advance constraint, which we write as

(A7) Rt �Pt c1t –Mdt � =0.

As noted above, only the case Rt �0 must beconsidered. Since Pt c1t–Md

t cannot be negative,equation (A7) is a mathematically concise wayof stating that if Rt >0, it follows that Pt c1t =Md

t ,while if Rt =0, then all we know is Pt c1t �Md

t .From the point of view of the analysis below,the key is that when Rt >0, the cash-in-advanceconstraint holds as a strict equality. This makessense: When the interest rate is positive, it isinconsistent with optimization to carry cash inthe afternoon that is not absolutely necessary.

In addition to equations (A5)–(A7), the trans-versality condition is also implied by householdoptimization:

(A8) lim qT AT =0.T→�

The intuition for this condition is straightforward.To see that the limit cannot be positive, suppose,on the contrary, that it is. In this case, At growsfaster than the interest rate. It would then befeasible for households to increase spending

� 74 By recursive substitution, equation (A3) implies

T–1

qT AT �A0 +�qt (It –St ) .t = 0

Driving T→ � , we obtain

� ��qt St �A0 + �qt It .t =0 t =0

This shows that equations (A3) and (A4) imply the standard single-equation budget constraint. To establish the reverse, simply show that if{St , It}, t = 0, 1,... satisfy the budget constraint, then they also satisfy (A3)and (A4). That the present value of income is finite will be a feature ofequilibrium. Otherwise, demand would be unbounded and no equilibriumcould exist.

� 75 It is easy to verify that in any equilibrium, it must be that Rt � 0and Pt > 0. Market clearing is impossible if either of these conditions is notsatisfied.

30

in one date without reducing it in another.If this extra spending were financed by a loan,the power of compound interest would causethe resulting debt to spiral upward at a rateequal to the interest rate. However, with totalassets rising at an even greater rate, the house-hold’s net asset position would remain consistentwith equation (A4). The increase in consump-tion financed in this way raises utility becauseof nonsatiation, and so we have a contradiction.Thus, optimization implies the above expressioncannot be positive, but it also cannot be nega-tive because of the restriction of equation (A4).

For purposes of our analysis, it is convenientto write the transversality condition in a differ-ent form. Combining equations (A5) and (A6),we find u1,t =β (1+Rt)u1,t +1Pt /Pt +1. Substitutingthis into the expression for qt , we find

(A9) qt= � P0 � β tu1, t , t = 0, 1, 2, ....u1,0 Pt

After multiplying both sides of equation (A8) bythe positive constant, u1,0/P0, the transversalitycondition reduces to

(A10) lim βT u1,TAT

= 0.T→� PT

Equations (A5)–(A10) are not just necessary foroptimization, they are also sufficient. This iseasily established with a suitably adjusted ver-sion of the proof to Stokey, Lucas, andPrescott’s theorem 4.15 (1989).

Government

The government purchases no goods, it onlyparticipates in the asset market. Its sourcesof funds in the asset market are new debtissues, tax revenues, and newly created money,Ms

t –Mst–1 . It uses these funds to pay its out-

standing debt obligations, B st . Equating sources

and uses of funds gives us

Bst+1 +Tt +Ms

t –Mst –1=Bs

t .1+Rt

At time t, the government takes Mst–1 and B s

t asgiven, t = 0, 1,.... At date 0, M s

–1 +B s0 =A0.

Government policy is a sequence of B st +1,Tt ,

and M st that satisfies this flow-budget con-

straint, which can also be written as

Ast+1 +Tt +

Rt Mt = Ast .1+Rt 1+Rt

Here, Ast measures total nominal assets,

Ast = Bs

t +Mst–1, and As

0 =A0. Recursively substituting this expression

forward, we find that for each fixed T,

T–1(A11) qT As

T +� qt �Tt +Rt Mt � =A0.

t =0 1+Rt

The presence of Rt Mt /(1+Rt ) reflects the inter-est costs the government saves when it issuesmoney rather than bonds. The government’sintertemporal budget equation is representedby the above expression, with qT As

T absent andT–1 replaced by �:

�(A12) � qt �Tt +

Rt Mt � =A0.t =0 1+Rt

The only restriction we have placed ongovernment policy is that the flow-budgetconstraint is satisfied for all possible values ofprices, {qt , Pt , Rt , t � 0}. That is, we requirethat equation (A11) hold. But no assumptionhas been made that equation (A12) holds for allpossible prices. Government policy is said to beRicardian if (A12) holds for all possible prices,and it is non-Ricardian if (A12) holds only atequilibrium prices. (We shall see that, at equi-librium prices, [A12] must be satisfied regard-less of whether government policy is Ricardianor non-Ricardian. This follows from equation[A8] and the fact that, in equilibrium, As

t =At .)Equation (A11) converges to equation (A12)

if and only if

(A13) qT AsT → 0.

We can equivalently define a Ricardian policyas one that enforces equation (A13) at all possibleprices and a non-Ricardian policy as one thatdoes not.

Firms

Firms in this economy are simple. They buy yfrom households and transform it into cash andcredit goods. Given the assumed linearity of theproduction technology, the resource constrainthas the form

(A14) c1t + c2t = y.

31

Equilibrium

A general equilibrium for this economy is asequence of prices and interest rates, Pt and Rt ;a sequence of consumptions, c1t , c2t ; and asequence of money supplies and bonds, Mt +1and Bt +1 , such that households optimize, thegovernment flow-budget constraint is satisfied,and markets clear. Bond-market clearingrequires

B st +1=B d

t +1=0 ,

and money-market clearing requires

Mst =Md

t =Mt ,

say, for t � 0. These conditions imply that At +1 =M d

t +B dt +1=A s

t +1. Goods-market clearingcorresponds to the resource constraint,equation (A14).

A feature of equilibrium that will be useful inthe analysis is

(A15) 1+Rt =1–σ 1

�1, wt �c1t ,σ wt

1–v c2t

which we obtain from equation (A6) and ourparametric form for the utility function. WhenRt >1, we can rewrite this to obtain the model’s“money-demand” function. The binding cash-in-advance constraint, c1t =mt , and the resourceconstraint imply wt = mt /(y –mt ) where

mt = mt

Pt

Solving equation (A15) for mt yields

(A16) mt =y .

1+ � σ �1+Rt ��1–σ

Constant MoneyGrowth

Here, we consider the set of equilibria associ-ated with a fixed money growth rate policy. Weshow there is one equilibrium in which inflationis constant and equal to money growth. Thereis also a continuum of equilibria with explosiveinflation.

We suppose the government sets B st +1=0 for

all t �0 by paying off the entire stock of debtin the first period. In addition, it sets M s

t =µM st –1

for t = 0,1, ..., where µ �1. Money growth isaccomplished by means of lump-sum taxtransfers. In particular,

T0= B s0 – �µ –1 �M s

–1,

Tt = – �µ –1 �Mst –1, t �1.

It is straightforward to verify that, with thisspecification of policy, there are many pricesequences that satisfy equation (A10). Tech-nically, it does not fit into our formal definitionof a Ricardian policy, because (A10) is notsatisfied for all prices. Under this policy, Atcomprises only the money supply. Thus, equa-tion (A10) would be violated if the price levelfell sufficiently rapidly. Still, for practical pur-poses we will think of this as a Ricardian policy.

It is useful to rewrite the household’sdynamic Euler equation by multiplying bothsides of equation (A5) by Mt and using Mt +1 = µMt to obtain

(A17) u2,t mt =β

u1,t +1mt +1 .µ

A sequence of prices and quantities representsan equilibrium if and only if equations (A7)–(A14) and Pt > 0, Rt , c1t , c2t � 0 are satisfied.

A Characterization Result for Equilibria

We now simplify the equilibrium conditions toobtain a useful set of sufficient conditions forequilibria in which the cash-in-advance con-straint binds. In this case, equation (A14) allowsus to express equation (A17) as a differenceequation in mt and mt +1 alone. Because thecash-in-advance constraint binds, wt in equation(A15) can be written as

wt =mt .

y – mt

Using this notation, equation (A17) can beexpressed as a difference equation in wt andwt +1,

(A18) a (wt ) = b (wt +1),

where

(A19) a (w) =σ w ,

(1– σ)w v+σ

a�(w)=σ

[(1–σ)(1–v)wv+σ][(1–σ)w v+σ]2

and

(A20) b (w) =β (1–σ ) w v ,

µ (1– σ)w v+σ

b�(w)=β (1–σ ) wv –1 vσ .

µ � �1–σ �wv+σ � 2

11–v

32

Here, a� and b� represent the derivatives ofa and b, respectively, with respect to w. Thetransversality condition reduces, in the presentnotation, to

(A21) lim βTb (wT)=0.T→�

We are now in a position to state our charac-terization result.

Proposition A1: Suppose that wt �0, t =0,1,2, ..., satisfies equations (A18), (A21), and(A15). Then, wt corresponds to an equilibrium.

Proof: Write

mt= ywt , Pt=

Mt

1+wt mt

Rt=σ 1

, c2t= y

,c1t= c2t wt ,1–σ wt1–v 1+wt

and verify that all equilibrium conditions aresatisfied at these prices and quantities. QED.

MULTIPLE EQUILIBRIA WITH RICARDIAN CONSTANT

MONEY GROWTH

We use the characterization result to show thereis a continuum of equilibria in this economy

when the money growth rate, µ , is constantand greater than unity. From equation (A18),there is exactly one equilibrium with wt =w * for all t,

1—

(A22) w* = �1–σ β � .σ µ

It is easily verified that this satisfies the con-ditions of the characterization result. For example,substituting w * into equation (A15) yields apositive interest rate with 1+R =µ/β . This isgreater than unity by our assumptions on µ andβ. Because real balances are constant in thisequilibrium, the rate of inflation is equal to µ .

The intuition underlying equation (A22) isstraightforward: The relative quantity of cashgoods consumed in the equilibrium (that is, w*)is increasing in 1–σ, which is the relative weightin utility on these goods. It is decreasing in themoney growth rate, µ , because increases inµ raise the nominal rate of interest, in turn in-creasing the cost of the cash good. Finally,consider v →1. This is easiest to interpret whenσ = 1/2. In this case, the two consumptiongoods are perfect substitutes. Consequently, ifthe cash good is more expensive than the creditgood, as is the case when µ �1, zero cash goodswill be consumed, and w * = 0 as v →1.

There are other equilibria in which inflationexceeds µ . To show this, we first study theproperties of the functions a(w) and b(w).

According to equation (A19), a (0)=0 and a�(0)=1. Also, a�(w) > 0 for all w � 0. At the sametime, equation (A20) indicates that b (0) = 0, b�(w) → � as w → 0, and b�(w) > 0 for w > 0.These observations establish that a(w) andb(w) coincide at w =0, with b rising moresteeply than a for small values of w.

From the discussion leading up to equation(A22), we know there is a unique value of w>0—namely, w* in equation (A22)—where a(w) =b(w). Since the two functions are continuousfor 0 <w <w*, it follows that b(w) > a(w) for win this interval, and that a is steeper than b atw =w*. The latter observation can be confirmedby direct differentiation, which yields

1 v

a�(w*) = �1– σ �1-v

�1–v � �β �1-v

+ 1 > 1.b�(w* ) σ v µ v

The strict inequality reflects that the expressionimmediately after the equality is positive andthat 1/v >1 because 0< v <1.

Our results on the a and b functions aresummarized in figure A. Note how b risesabove a and then crosses once. Eventually, thetwo curves are parallel, since a�(w) and b�(w)

F I G U R E A

Equilibrium in the Cash/Credit-Good Model

a(w), b(w)

a(w)

b(w)

w2 w1 w0 w*

1–v

33

both converge to zero as w →�. We can usethis figure to study the set of equilibria forthe model.

Consider an arbitrarily selected w0<w*. Todetermine the value of w1 implied by equation(A18), draw a vertical line up to a (w0). Then,identify w1 such that b (w1) equals a(w0). Thiscan be found by following a horizontal line tothe left of a(w0) until it intersects b . The prop-erties of these curves guarantee that such anintersection will occur for a positive value of w.With w1 in hand, compute w2 in the same way,and so on.

It should be clear that the sequence of wtcomputed in this way converges to 0. Alongthis path, b (wt) � 0 and b (wt)→0 as t → �.Because b is bounded above along the path,equation (A21) is satisfied. Because wt declinesmonotonically, R >0 at w* and wt >0 for a givent, equation (A15) implies that Rt >0 for each t .This establishes that the sequence just computedconstitutes an equilibrium.76 The same argumentcan be applied for each 0<wt< w*; in each ofthese equilibria there is a hyperinflation as wt→0.77

UNIQUE EQUILIBRIUM WITH NON-RICARDIAN, CONSTANT MONEY GROWTH

The previous section showed that with a particular Ricardian policy, constant moneygrowth results in a continuum of equilibria.Here is a particular non-Ricardian policy:

Tt =Pts – Rt Mt ,

1+Rt

where s is a positive constant. It is easy to verifythat the set of equilibria under this policy is astrict subset of the set of equilibria analyzed inprevious section. Thus, we conclude that withconstant money growth, a non-Ricardian policydoes not lead to an overdetermined price level.

Fixed Interest RatePolicies

This section considers two representations ofpolicy in which the government pegs the nomi-nal rate of interest to a constant value, R >0. Inthe first representation, fiscal policy is Ricardianand there exists a continuum of equilibria. Inthe second, policy is non-Ricardian and theequilibrium, if it exists, is unique.

The fixed value of R pins down m (seeequation [A16]), c1, and c2 :

c1=m, c2=y –c1 .

As a consequence, the marginal utility of thecash good is constant, so that

Pt+1 =β (1+R )Pt

for all t. Consider two specifications of policy,

(A23) Tt = – R

mPt + εAt1+R

and

Tt = – R

mPt + dPt ,1+R

where d is a non-negative constant and 0<ε�1.As we will show, the first policy is Ricardian,while the second is not. These policies may ini-tially appear strange, but the motivation behindthem will soon become clear. To determinewhether a policy is Ricardian requires us todetermine whether equation (A13) holds for allpossible prices or only for equilibrium prices.

To investigate this further, it is convenient towrite the flow-budget constraint in real terms,

βat+1+τt + R

m=at.1+R

Here, at+1=At+1/Pt+1 and τt=Tt /Pt . Substitutingthe first specification of policy in equation (A23)into the flow-budget constraint gives us

(A24) at+1=1–ε

at .β

We seek to understand how �at �β tat evolves ast →�. By substituting from equation (A9), wehave

(A25) qT AT =P0 βT AT =P0

�aT .PT

Recall that a policy in which qT AT → 0 for all

� 76 Recall, in constructing equation (A18) we assumed the cash-in-advance constraint is binding. This assumption has been verifiedfor w0 <w *.

� 77 Our results would not be significantly affected if we allowedlabor to be endogenous. Introducing labor as a third argument in the utilityfunction has the effect of adding an extra Euler equation, –u3 /u2 = f�(l ),where f�(l ) denotes the marginal product of labor, l, and u3 denotes themarginal utility of labor. Feasibility restricts l to some subspace, l ∈D(for example D might be the unit interval). Also, y = f (l ) denotes theproduction function. Combining the new Euler equation with the resourceconstraint produces a function, l=F (w ), where F has a nice analyticalcharacterization with standard preferences and technology. To find anequilibrium, one would still start by looking for values of wt that solve thedifference equation, A (wt ) = B (wt +1). One would then have to verify F (w ) ∈D, in addition to the other conditions listed in the characterizationresult, to verify that the values of wt represent an equilibrium.

34

possible prices corresponds to a Ricardianpolicy, and one in which this occurs only forequilibrium prices is a non-Ricardian policy.Multiplying both sides of equation (A24) byβ t+1, we find

�at +1= (1–ε)�at=(1–ε)t �a0,

or, using equation (A25),

qT AT =(1– ε)TA0→0.

This establishes that the first policy in equa-tion (A23) is Ricardian. The government’s policyprevents the debt from exploding too fast,regardless of what happens. As a result, theintertemporal budget equation provides nouseful restriction for pinning down prices.

Now consider the second policy in equation(A23). For this policy, total real assets evolveaccording to

at +1=1

(at –d ).β

The policy makes the evolution of total assetsexogenous, while letting the private economydetermine the breakdown of real assets betweenmoney and bonds to be consistent with theinterest rate peg. Solve for at and thenmultiply by β t,

βtat =a0 –d

+d

βt,1–β 1–β

so that

βtat →A0 –

d ,

P0 1–β

where a0=A0/P0. This is a non-Ricardian policybecause β tat →0 for only one value of P0—theone that satisfies

A0= d .

P0 1–β

We can now summarize our results for theinterest rate peg. If it is accompanied by aRicardian policy, the price level is not pinneddown by the intertemporal budget equation,nor by the rest of the model. The model pinsdown only Mt /Pt and Pt +1/Pt , but not thenumerator and denominator terms. Under thenon-Ricardian policy, the intertemporal budgetequation supplies the extra equation needed.Once again, the price level is not overdeter-mined under the non-Ricardian policy.

3535

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Erratum

Figures were reported incorrectly in “PopulationAging and Fiscal Policy in Europe and theUnited States,” by Jagadeesh Gokhale and BerndRaffelhüschen, in Economic Review, vol. 35, no.4 (1999 Quarter 4). The first column on page 14should have read (corrections italicized):

By 2015, more than a third of the people living in these three countries will be 60 orolder.? In Italy, four out of every nine per-sons will be 60 or older by 2035! In Sweden,Austria, and Germany, two out of every fivepersons will be elderly by our criterion. Incomparison, the U.S. population will bemuch younger, with only one of every threepersons falling into the elderly category.