Gambling to Preserve Price (and Fiscal) Stability

We study a model in which policy aims at aggregate price stability. A fiscal imbalance materializes that, if uncorrected, must cause inflation, but the imbalance may get corrected in the future with some probability. By maintaining price stability in the near term, monetary policy can buy time for a correction to take place. The policy gamble may succeed, with price stability preserved inde nitely, or fail, leading to a delayed, possibly large jump in the price level. The resulting dynamics resemble the models of a currency crisis following Krugman (1979) and Obstfeld (1986). Like in Obstfeld’s work, multiple equilibria arise naturally: whether or not price stability is preserved may depend on private agents’ expectations. The model can be reinterpreted as a model of partial default on public debt, in which case it is reminiscent of Calvo (1988).


Introduction
Sometimes, a very bad disturbance such as a pandemic or a war has a sharp economic impact, creating a need for socially costly adjustment.But there is a chance that the economy will bounce back quickly, which would reduce the required scale of adjustment, or at least diminish its cost.In such situations, macroeconomic policymakers face a choice that may be approximated as "adjustment now" versus "deeper adjustment in the future with some probability" in the case when the economy fails to improve sufficiently.Delaying adjustment may be the better policy decision, even though it amounts to taking a gamble on the economic recovery.In this paper, we study a model in which the outcome naturally depends not only on luck but also on expectations of private agents.
We are writing this paper for a conference in honor of Maurice Obstfeld at a challenging time for macroeconomic policy.First, the COVID-19 pandemic has had important economic repercussions, in particular because it led to a steep increase in government borrowing when public debt in some countries was high to begin with.Next, Russia's invasion of Ukraine raised further doubts about the prospects of a much needed global recovery.Both the COVID-19 pandemic and Russia's war fit the description of "a very bad disturbance" from the previous paragraph.The research of Maurice Obstfeld, with its emphasis on the role of private agents' expectations as a source of currency and price level stability or instability, can help guide the response of macroeconomic policy to the two shocks, and it serves as an inspiration for this paper.

Celebrating Obstfeld's contributions to the theory of currency and price level (in)stability
Maurice Obstfeld developed an interest in international economics when, as a math student in Cambridge, he lived through the disruptions associated with the sterling devaluation after the end of the Bretton Woods system.Why was the exchange rate moving so dramatically?How was the currency crisis related to economic policy?In the collapse of Bretton Woods, the world arguably lost a promising mathematician, but it certainly gained a great economist.
Much of Maury Obstfeld's research revolves around questions that emerged from the demise of the Bretton Woods system: How best to manage capital mobility and balanceof-payments imbalances?How to pursue macroeconomic and financial stability, nationally and globally, in a world in which different monetary and currency regimes coexist around a hegemon, the United States?To answer these questions, Maury developed an approach that, in spirit, is profoundly "Keynesian," as it recognizes that building a stable global order requires a clear-eyed analysis of the dynamics of economic, technological and social fundamentals; a healthy degree of skepticism about the equilibrating properties of markets; and resilient international institutions that foster multilateralism and effective cooperation between countries.
Maury's contributions to the theory of currency and price level (in)stability feature preeminently in his overall vast research output.His work on this topic includes very technical papers (for the few) on foundational, closed-economy issues such as "dynamic seigniorage" and "ruling out speculative hyperinflations" (Obstfeld, 1991b, Obstfeld and Rogoff, 1983, 2021) as well as less technical open-economy papers (for the many) on "the logic of currency crises" (Obstfeld 1986(Obstfeld , 1991a(Obstfeld , 1994(Obstfeld , 1996(Obstfeld , 1997)).1 From the outset, the vulnerability of a fixed exchange rate policy to belief-driven speculative attacks has been a leitmotif in Maury's research.His first paper with this theme, Obstfeld (1986), can be seen as a response to Krugman (1979).Krugman (1979) questioned the idea, popular at the time, that currency crises are caused by the vagaries of irrational speculation, proposing a model in which the timing of a speculative attack is unique because of the rational anticipation by market participants of the policymakers' ability or willingness to defend a fixed exchange rate.Reversing Krugman's conclusion, Obstfeld (1986) showed, in a very similar setting, that the outcome can be indeterminate, depending on beliefs that are both rational and self-fulfilling.In his subsequent papers, Maury further developed the idea that there is feedback from private agents' expectations to the state of the economy and from the state of the economy to policy, which opens the door to multiple equilibria.The general lesson from his work in this area is that currency and price level stability are fragile, because they may be vulnerable to a self-fulfilling shift in expectations.

Summary of the paper
We study a dynamic model with private agents and a government.The government aims at aggregate price stability.Initially, the economy is in a steady state in which the price level is constant.At some point, there is a one-time unanticipated budget deficit -one can think of the deficit as being caused by a recession.If nothing else happens, the fiscal imbalance will be growing over time, because after new debt was issued to finance the deficit, the new debt will be rolled over at a strictly positive real interest rate.Therefore, the policy of price stability will be unsustainable -at some point, inflation will have to "pay" for the deficit. 2 However, we assume that a fiscal correction may take place in a future period with some probability, in which case the policy of price stability will become sustainable again.One can think of a potential correction as reflecting a strong recovery from the recession that caused the initial deficit.Thus, when the deficit materializes, monetary policymakers face the choice between "inflation now" and "inflation in the future with some probability."The policy of maintaining price stability in the near term, to buy time for a correction, amounts to a gamble to preserve price stability indefinitely.The gamble may well be worth taking, if the probability of a correction is not too low.
We use the model to understand what may happen after policymakers decide to take the gamble.Since the fiscal imbalance is growing, the correction or budget surplus necessary to preserve price stability is increasing over time.We suppose that monetary policy will delay inflation as long as the required correction is not too large.We also assume that the probability of a correction falls with the necessary size of the correction.In equilibrium, if a correction occurs soon enough, price stability is maintained indefinitely.Otherwise, the economy experiences a delayed jump in the price level that is larger than if the policy of price stability had been abandoned right away.Moreover, the outcome of the policy gamble depends also on private agents' expectations -the model has multiple equilibria.For the same sequence of exogenous shocks, the policy of price stability may continue indefinitely if private agents expect it to continue, and it may fail if private agents expect it to fail.A shift 2 We also study a version of the model with default on public debt instead of inflation.
in inflation expectations can be self-fulfilling because it feeds into the state of the economy (government bond prices change, which affects debt issuance for a given deficit and thus the size of the required correction) and from there to policy behavior (the probability of a correction changes).The gamble to preserve price stability may well be worth taking, but it makes the economy vulnerable to a self-fulfilling shift in expectations -a conclusion very much in the spirit of Obstfeld (1986).
The next section sets up the basic model.We then solve the model under three different assumptions about what can happen after the budget deficit materializes.Section 3 assumes that the fiscal imbalance will not be corrected.Next, we introduce the possibility that the imbalance may be corrected in the future.In Section 4, we analyze the simplest case in which the probability of a correction over time is independent of the state of the economy.
In this case, the model has a unique equilibrium in the spirit of Krugman (1979).In Section 5, we study the main case in which the probability of a correction over time is decreasing in the necessary size of the correction.This analysis is in the spirit of Obstfeld (1986), as just described.In Section 6, we reinterpret our model of inflation (which may be avoided) as a model of partial default on public debt (which may also be avoided).The parallel to models of currency crises with multiple equilibria à la Obstfeld (1986) becomes the parallel to models of sovereign default with multiple equilibria à la Calvo (1988), Section I. 3 The concluding Section 7 comments on how one can think of the recent, significant increase in inflation in most developed economies from the perspective of the model.
One contribution of the paper may be to help readers appreciate how different strands of literature connect (the "first generation" literature on currency crises following Krugman (1979), the "second generation" papers after Obstfeld (1986), the models of sovereign debt default following Calvo (1988), and the fiscal theory of the price level, or the FTPL4 ).
We make some specific points about the literature at the end of each Section 3-6.Here let us make the following general observations.While in the early models of Krugman (1979) and Obstfeld (1986) inflation is fiscally beneficial to the extent that it produces seigniorage revenues from steady expansion of the monetary base (or more specifically, of non-interest-bearing currency), the subsequent literature on fiscally-driven currency crises has broadened the scope of the analysis (e.g., Burnside et al., 2001, Daniel, 2001, and Corsetti and Maćkowiak, 2006).Inflation is also, or perhaps mainly, fiscally beneficial because it reduces the real value of two quantities -nominal public debt and government expenditure commitments such as pensions that are imperfectly indexed to the price leveland this effect can be large in most modern economies.To focus on this effect of inflation, we adopt the modeling approach of the FTPL.The FTPL literature recognizes that while monetary policy may be unable to preserve price stability, the central bank can choose the path of inflation for given fiscal policy -Cochrane (2022) makes this point a recurrent theme of his book (e.g., "Monetary policy can (...) shift inflation over time as it pleases," p. 70).However, the literature typically focuses on situations when inflation responds to a shock smoothly with little delay,5 whereas we stress the possibility that the price level can be constant or approximately so for several quarters after a shock, followed by a delayed large jump in the price level.This may be an appealing way to think about the period after the outbreak of the COVID-19 pandemic.Furthermore, in our model monetary policy may be able, depending both on luck and private agents' expectations, to buy enough time for a fiscal correction so that inflation never happens.

Model
We study a dynamic model, in discrete time, with a government and a continuum of private agents distributed on the unit interval.
The government issues nominal long-term bonds.We include long-term debt because bond prices can then drop, producing a transfer from private agents to the government, in response to news that there may be inflation in the future even if there is no inflation at present.Let B t denote the stock of government bonds at the end of period t.Let Q t be the period t price of a bond.Following Woodford (2001), a bond issued in period t pays a nominal coupon ρ k k + 1 periods later, for each k ≥ 0, where ρ is a parameter satisfying 0 < ρ ≤ 1.With this formulation, the bond's yield-to-maturity simply equals 1/Q t + ρ.
The flow budget constraint of a representative private agent reads where Y t is endowment income in period t, C t is consumption, P t is the price level, and S t is a lump-sum tax (or transfer, if negative).The flow budget constraint of the government reads Since the model abstracts from money, one can think of the price level as the rate of exchange between bonds and the consumption good.
Optimizing behavior of private agents implies that the following relations hold in equilibrium: T →∞ The first equation is the bond-pricing equation, where for simplicity we assume that private agents discount future payoffs with the discount factor β ∈ (0, 1).The second equation is the transversality condition.In addition, it is convenient to define the period t (short-term) nominal interest rate R t according to the equation When we solve forward the government flow budget constraint (2), using equilibrium relations ( 3) and ( 4), we arrive at the equation stating that in equilibrium the real value of public debt must equal the expected present value of lump-sum taxes, or primary budget surpluses: (1 We suppose that initially the economy is in a steady state in which the government pursues the policy of price stability.Monetary policy picks a price level target P > 0, and fiscal policy sets a path for the primary surplus such that equation ( 6) holds in every period with, on the left-hand side, the price level equal to P and the bond price equal to 3) in the steady state with a constant price level).For simplicity, we assume that fiscal policy sets a time-invariant primary surplus S > 0 with this property.
Relative to this baseline, we will study the consequences of an unexpected fiscal imbalance which may or may not be corrected in the future.We consider three alternative specifications of the model: the imbalance is there to stay (Section 3); the imbalance may be corrected with a constant probability in each future period (Section 4); the probability of a correction is decreasing in the necessary size of the correction (Section 5).

Delaying inflation
In our first specification, once a fiscal imbalance materializes, it cannot be corrected.Thus, the imbalance makes the policy of price stability unsustainable.The remaining monetary policy options are "inflation now" and "inflation in the future."Let us describe the two alternatives.6

Analytics
In period t = −1, the economy is in the steady-state defined at the end of Section 2 with price stability and a positive primary surplus S > 0. In period t = 0, there is a one-time unanticipated shock: the primary surplus falls by ∆ > 0, that is, the primary surplus equals S − ∆ (we refer to the period 0 primary balance as a deficit, assuming that S − ∆ < 0).7 One can think of the period 0 deficit as being caused by an unanticipated recession, or by a recession that turns out to be deeper than anticipated.
Just before the arrival of the shock, equation ( 6) reads for some B −1 > 0 inherited from the past.Upon the arrival of the shock, equation ( 6) changes to which fails to hold with both P 0 = P and Q 0 = Q.Either the policy of price stability must be abandoned in period 0 (unanticipated inflation, P 0 > P ) or it must be abandoned in a future period t ≥ 1 (an unanticipated drop in the bond price, Q 0 < Q, and delayed inflation).Note that inflation could not be postponed if all debt was short-term (ρ = 0), since anticipated future price level movements are priced in when the government issues new debt. 8 Let T ≥ 0 denote the period in which the policy of price stability is abandoned.We suppose that from time T onwards monetary policy targets the bond price Q t , or equivalently the interest rate R t .For simplicity, we also assume that, whether T = 0 or T ≥ 1, at the time when price stability is abandoned there will be a one-time jump in the price level as opposed to persistent inflation, that is P t = P T , t ≥ T .Thus, monetary policy sets or equivalently R t = 1/β, for t ≥ T .One could modify the specification of monetary policy to obtain a persistent increase in the nominal interest rate and persistent inflation starting in period T . 9 If the policy of price stability is abandoned in period T = 0, equations ( 7)-( 8) imply a simple solution for the equilibrium inflation rate: In this case of "inflation now," equation ( 8) holds with Q 0 = Q.Alternatively, if the policy of price stability is abandoned in a future period T ≥ 1, equations ( 7)-( 8) imply that the period 0 bond price must satisfy 8 Even in an economy where all public debt is short-term, there may be government expenditure commitments such as pensions that are imperfectly indexed to the price level.Their real value can be reduced via inflation even when inflation is anticipated. 9See Corsetti and Maćkowiak (2006) for the case in which starting in period T monetary policy makes the nominal interest rate react less than one-for-one to the inflation rate, which in equilibrium produces both a price level jump at T and some subsequent growth in the price level.
In this case of "inflation in the future," equation ( 8) holds with P 0 = P .
Let us solve for the time T price level in the delayed inflation case.We use equations ( 3)-( 4) to derive the equilibrium relation between the bond price and subsequent price levels: If the policy of price stability is abandoned in period T ≥ 1, this equation implies that the period 0 bond price must satisfy Equating expressions ( 9) and ( 11) yields a solution for the time T price level, P T , or the inflation rate, P T /P .Two properties of P T are worth emphasizing.First, P T is increasing in T so that delaying inflation implies higher inflation in the future.The reason is that the period 0 bond price Q 0 is pinned down by the deficit ∆, independent of T (equation ( 9)) and in the bond-pricing equation future inflation is discounted (equation ( 10)).Second, for given T ≥ 1 and ∆, P T is decreasing in ρ which implies that longer debt maturity allows lower future inflation.The reason is that with longer debt maturity a given future jump in the price level imposes a larger capital loss on bondholders.
To solve the rest of the model in the delayed inflation case (T ≥ 1), from equation ( 10) we compute for the path of the bond price between periods 1 and T − 1, Q t , t = 1, . . ., T − 1 (we already solved for Q 0 in the case of T ≥ 1, and we know that Q t = Q, t ≥ T ).Given the paths of the bond price and the price level, we use the government flow budget constraint (2) to compute the path of government debt, B t , t ≥ 0. In the case of T ≥ 1, the interest rate in period T − 1 follows from equation ( 5), R T −1 = P T /βP (t = T − 1 is the only period in which the interest rate R t does not equal 1/β).

A numerical example
To gain insight, consider a numerical example.One period in the model equals one quarter.
We set β = 0.995, ρ = 0.95, and S = 0.01, which implies an annual real interest rate of 2 percent, a debt duration of about 20 quarters, and a ratio of debt to annualized GDP of 50 percent in period −1 (we think of aggregate income as time-invariant and normalize it to 1 per quarter).We pick P = 1 and compute the initial condition B −1 > 0 from equation ( 7).
We assume ∆ = 0.2, which corresponds to a primary deficit equal to 5 percent of annualized GDP.
Figure 1 plots P T , 1/Q T −1 + ρ, R T −1 , and normalized B T −1 as a function of T ≥ 0, the period in which the policy of price stability is abandoned.By "normalized B T −1 " we mean B T −1 multiplied by (1 + ρQ) /4P ; the value of (1 + ρQ) B T −1 /4P can be compared directly with the debt-to-GDP ratio in the initial steady-state, which equals 0.5 in this parameterization.In analogy to Flood and Garber (1984), one can refer to P T as the "shadow" price level -this is the price level that would be observed in equilibrium in period T , if the policy of price stability were abandoned at T .Figure 1 illustrates the property that P T is increasing in T -the longer the delay, the more inflation there will be.One can show that the shadow price level P T goes to infinity at a finite T . 10Similarly, 1/Q T −1 + ρ, R T −1 , and B T −1 can be referred to as the shadow bond yield, the shadow interest rate, 10 See Corsetti and Maćkowiak (2006), Section 3.3.The economy from Section 3 with T = 10 (T is the period in which price stability is abandoned).
and the shadow debt.Q T −1 and R T −1 are simple functions of P T , Q T −1 = Q (P/P T ) and R T −1 = P T /βP .The debt ratio (1 + ρQ) B T −1 /4P keeps growing as the abandonment of price stability is postponed (the lower right-hand panel in Figure 1).This is the case even though there was a one-time budget deficit.The reason is that after new debt is issued to finance the deficit, the new debt is being rolled over at a strictly positive net real interest rate.Thus, the fiscal imbalance is growing -as is the size of a hypothetical fiscal correction, or budget surplus, that would be necessary to preserve price stability.
Figure 2 plots the path of the economy assuming, as an example, that the policy of price stability is abandoned at T = 10.The equilibrium price level equals P through T − 1 = 9 and jumps at T = 10 (P t = P , t ≤ T − 1, P t = P T , t ≥ T ).The shadow price level jumps at t = 0 and rises gradually through T = 10.The equilibrium price level and the shadow price level coincide at T = 10.Analogously, the equilibrium interest rate and the shadow interest rate coincide at T − 1 = 9 (in equilibrium R t = 1/β in every period, except that R T −1 = P T /βP ).The equilibrium bond yield, 1/Q t + ρ, rises discretely at t = 0 and then increases gradually through period 9; it jumps down to 1/Q + ρ at T = 10.The equilibrium value of debt, (1 + ρQ t ) B t−1 /4P t , decreases in period 0 (it equals the present value of primary surpluses, which falls at t = 0), and returns to 0.5 at t = 1 as the deficit vanishes.
The difference between the equilibrium value of debt and the debt ratio (1 + ρQ) B t−1 /4P provides a measure of the fiscal imbalance.Once again, note that the imbalance is growing while inflation is being postponed.

Discussion
The delayed inflation scenario, just analyzed, closely resembles the currency crisis model of Krugman (1979).In principle, monetary policy wants to maintain price stability, or a fixed exchange rate, but a fiscal imbalance makes this policy unsustainable.The shadow price level (exchange rate) rises over time.The only question is when the policy of price stability, or the exchange rate peg, will be abandoned -in other words, when will the shadow price level coincide with the equilibrium price level?
Krugman famously showed that the timing of devaluation, T , was uniquely determined in his model.He assumed that there was no (privately-held) government debt, the central bank started out with a finite stock of international reserves, and it would abandon the exchange rate peg when the reserves fell to zero.There are several equivalent ways to to reproduce Krugman's uniqueness-of-T result in our framework.For instance, we could assume that there is an upper bound on net debt of the public sector (1 + ρQ) B t−1 /4P (a generalization of Krugman's lower bound on reserves) and that price stability is abandoned in the period directly after debt reaches the threshold.Alternatively, as in Corsetti and Maćkowiak (2006), we could assume that there is an upper bound on the interest rate R t (raising the interest rate "in defense" of price stability is socially costly, because it amounts to raising the real rate of interest in a world with sticky prices) and that price stability is abandoned in the period directly after the interest rate reaches the threshold.
Either assumption implies a unique timing of inflation T , with T − 1 as the period of a "speculative attack" in which the interest rate rises in anticipation of the jump in the price level at T .11 It is also instructive to discuss the relation to the FTPL literature.In the language of Leeper (1991), in the initial steady state monetary policy is active and fiscal policy is passive.This policy mix is consistent with price stability.In period 0, fiscal policy switches to active and produces a budget deficit.The assumptions of no default on public debt and a constant post-T bond price amount to passive monetary policy.The FTPL literature (see, e.g., Leeper and Leith (2016), Section 2, for a review) recognizes that, with long-term debt and active fiscal policy, passive monetary policy can choose the path of the price level for given fiscal policy.Cochrane (2022) stresses the role of monetary policy in shaping the path of inflation when fiscal policy is active.However, as noted in the introduction, the literature focuses on models in which inflation responds to a shock smoothly with little delay, whereas in this model a period of price stability is followed by a delayed jump in the price level.
So far, there is no rationale in the model for postponing inflation once the fiscal deficit materializes, just like in Krugman's model it is unclear why the central bank is losing reserves instead of abandoning the exchange rate peg sooner.The next section adds to the model a rationale for delay.
4 Gambling to preserve price stability: Krugman (1979) equilibrium We now introduce the possibility that the fiscal imbalance may be corrected in the future, in which case price stability can and will be preserved.One can think of a potential correction as reflecting a strong recovery from the recession that caused the initial deficit.In this setting, there is a reason to wait as the monetary policy choice becomes "inflation now" versus "higher inflation in the future with some probability."The policy of delaying inflation amounts to a gamble to avoid inflation altogether -a gamble that may well be worth taking, if the probability of a correction is not too low.
In the rest of the paper, we want to understand the dynamics if policymakers decide to take the gamble.In particular, once we think of delay as potentially justified in terms of buying time for a fiscal correction, it is interesting to ask if expectations of inflation, or with the Krugman model.
expectations of price stability, can become self-fulfilling.In this section, we give an example of an environment where the answer is no, in the spirit of Krugman (1979).In Section 5 we provide another example, which we think is more appealing, where the answer is yes, in the spirit of Obstfeld (1986).

Analytics
Compared with Section 3, we modify the model by assuming that the policy of price stability will be abandoned in period T ≥ 1 unless a fiscal correction has occurred.We model a possible correction as follows.In every period t = 1, . . ., T , nature draws a realization of the primary surplus: with some probability the surplus equals S + ω t > S, where the value ω t > 0 is such that the economy returns immediately to the steady state with the price level equal to P ; and with the complementary probability, the surplus equals S. Once the economy is back in the initial steady state or the price level jumps in period T , all uncertainty has been resolved.
How long will monetary policy wait for a correction?Recall from the previous section that as inflation is being delayed, the fiscal imbalance is growing.Therefore, the correction or budget surplus necessary to preserve price stability, S + ω t , is increasing over time.It seems natural to suppose that monetary policy will wait as long as the required correction is not too large -there is a maximum correction that can be achieved, for economic or political-economy reasons, and the policy of price stability will be abandoned when this threshold is reached (and a correction has not occurred thus far).Specifically, we assume that monetary policy follows the rule "T is the first period t ≥ 1 in which ω t ≥ ω and a fiscal correction has not occurred," where ω > 0 is a constant.In this section, we also suppose that the probability of a correction is independent of the state of the economy and equals a constant, ψ ∈ (0, 1), in every period t = 1, . . ., T .
We can solve the model for a given time when price stability is abandoned absent a correction, T ≥ 1, and verify that a unique T is consistent with the policy rule stated in the previous paragraph.To solve the model for a given T ≥ 1, we proceed backward from time T to 0. In period T , with probability ψ the price level equals P = 1 and equation ( 6) reads We guess B T −1 (we will verify this guess) and we solve this equation for ω T , which is the fiscal correction necessary for the policy of price stability to become sustainable in period T .With probability 1−ψ the price level jumps to P T pinned down by equation ( 6), (1 + ρQ) B T −1 /P T = S/ (1 − β).We move to period T − 1, when the bond price satisfies 3)).We compute and B T −2 given a guess about B T −1 .Period T − 2 is very similar to T − 1, except that when Every period before is exactly analogous, until we arrive at time 0 when we compute B −1 from a slightly changed equation ( 2): We have solved for If the solution for B −1 differs from the actual initial condition B −1 , we adjust the guess for B T −1 .We can simulate the path of the economy either assuming that a correction never takes place and the price level jumps at T or assuming that a correction takes place in some period τ = 1, . . ., T and the price level never jumps.

A numerical example
We use the same numerical example as in Section 3, except that now we allow for a random correction with probability ψ = 0.1, subject to the constraint on the size of the correction with ω = 0.3.Figure 3 plots P T , 1/Q T −1 − ρ, normalized B T −1 , and ω T as a function of T ≥ 0, the period in which the policy of price stability is abandoned.In this new figure, and (1 + ρQ) B T −1 /4P are similar to Figure 1.The difference to Figure 1 is that now the possibility of a correction keeps P T , 1/Q T −1 and B T −1 lower at any given T .12As already emphasized, the new variable in Figure 3, ω T , is increasing in T -the budgetary improvement required to eliminate the imbalance is increasing over time.Note that ω T starts out slightly greater than 0.2 (the value of ∆), and becomes greater than or equal to 0.3 (the value of ω) in period 12, indicating the time when price stability will be abandoned absent a correction.  in case 2, the fiscal correction happens to occur in period 12 (in period 12, the surplus drawn by nature equals S + ω 12 instead of S) and price stability is preserved.Case 2 is only an example of an equilibrium path with a correction (a correction can take place with probability ψ = 0.1, shown in Figure 4, in any period t = 1, . . ., T ).In case 1, the price level P t jumps in period 12, while in case 2 the price level remains constant.The interest rate R t and the bond yield 1/Q t + ρ each follows the same path in case 1 and in case 2. The value of debt, (1 + ρQ t ) B t−1 /4P t , is identical in both cases until period T , when in case 2 the correction raises the real value of debt.As in the previous section, the difference between the equilibrium value of debt and the debt ratio (1 + ρQ) B t−1 /4P provides a measure of the fiscal imbalance.In case 2, the difference falls to zero in period T when the correction eliminates the imbalance.
Let us summarize the insights from this specification of the model.There is a reason The economy from Section 4: in case 1, price stability is abandoned in period 12; in case 2, the fiscal correction happens to occur in period 12 and price stability is preserved.
to maintain price stability in the near term in the face of a fiscal imbalance -to buy time for a possible correction that, if it materializes, will make price stability sustainable indefinitely.Whether or not price stability is preserved depends on luck.If a correction occurs soon enough, the gamble for price stability succeeds.Otherwise, the gamble fails and the economy experiences delayed inflation.For a given sequence of exogenous shocks (nature's draws of the budget surplus), the model predicts a unique outcome.In particular, if inflation does occur in equilibrium, its timing is determinate.Furthermore, by the logic of the previous section, if the gamble fails, the economy experiences more inflation than if the policy of price stability had been abandoned in period 0.

Discussion: Krugman (1979) redux
This version of the model, with a possible future fiscal correction, remains in the spirit of Krugman (1979).Yet, going beyond his model, it provides a rationale for why delaying inflation may be an attractive policy or, in the context of a fixed exchange rate, why policymakers may want to postpone abandoning an exchange rate peg for some time in the face of a budget deficit.In further contrast to the Krugman model, price stability may end up being preserved in equilibrium.A key feature of Krugman (1979) remains intact, however -the timing of inflation, absent a fiscal correction, is unique.
Several papers in the FTPL literature (Davig andLeeper, 2007, Bianchi andIlut, 2017, among others) have used models in which the fiscal-monetary configuration changes randomly over time.In this model, fiscal policy becomes active in period 0 but may switch back to passive in the future with some probability.And if fiscal policy switches back to passive, the fiscal correction implemented at that time will be sufficient to pay for the period 0 deficit with all accumulated interest.By remaining active and maintaining price stability in period 0 and for some time thereafter, monetary policy buys time for fiscal policy to return to passivity -if that happens, price stability can be preserved indefinitely.
The model predicts that the bond yield (the long-term interest rate) rises in period 0 as soon as the fiscal imbalance appears; thereafter, the long rate keeps rising as time T approaches.In contrast, the short-term interest rate increases only if and when the economy arrives in period T − 1. See Figure 4.The prediction that long-term interest rates rise, or the yield curve steepens, soon after a fiscal imbalance materializes may be counterfactual.
In the next section, we propose a specification of the model in which this prediction changes.
5 Gambling to preserve price stability: Obstfeld (1986) equilibria We now show that the policy gamble to preserve price stability may expose the economy to multiple equilibria.To highlight the mechanism, we make one change in the model from the previous section.Instead of assuming that the probability of a fiscal correction is independent of the state of the economy, we suppose that the probability of a correction falls with the required size of the correction.It seems natural that, as the necessary correction increases over time, achieving it becomes more difficult, and hence less likely.

Analytics
Specifically, the model is as in Section 4 except that the probability of a correction is timevarying, ψ t ≥ 0, t = 1, . . ., T , and it is a decreasing function of the necessary correction, The solution method is analogous to Section 4: we solve the model for a given time when price stability is abandoned absent a correction, T ≥ 1, and we find values of T that are consistent with the policy rule "T is the first period t ≥ 1 in which ω t ≥ ω and a fiscal correction has not occurred."It turns out that, in general, there are multiple such values in this specification of the model.
The key feature of this economy is that, in contrast to Section 4, private agents' beliefs affect the probability of a fiscal correction.This feedback from expectations to the probability of a correction opens the door to multiple equilibria where expectations of inflation, or expectations of price stability, can be self-fulfilling.Consider the economy in period 0 when the fiscal imbalance appears.If private agents are "optimistic," they hold the belief that price stability will be maintained for a long time absent a correction -they think that T is far from 0 ("even if the economy is out of luck for many quarters, price stability will be maintained").A correction is likely to be achieved during that long period.Given this belief, the bond price is high, debt issuance is low for a given deficit, the necessary correction is moderate, and the probability of a correction is high.In equilibrium, the optimism of private agents is validated and T is much greater than 0 -delaying inflation buys a considerable amount of time for a correction.However, if private agents are "pessimistic," they hold the belief that price stability will be maintained for a short time absent a correction -they think that T is close to 0. Then the bond price is low, debt issuance is high for a given deficit, the necessary correction is large, and the probability of a correction is low.In equilibrium, the pessimism of private agents is also validated and T is close to 0 -delaying inflation buys little time for correcting the imbalance. 13 13 An important question is why a similar mechanism does not yield multiple equilibria in Section 4.
Consider the equilibrium from Section 4 and imagine that, for whatever reason, private agents' expectations were to shift to "more pessimism."Such a shift in the regime of expectations should put upward pressure on the bond price and thus on debt issuance.However, as long as the shift does not affect the probability of a correction, more pessimism among private agents would not be validated.The equilibrium path of the bond price and debt issuance would remain unaffected.The economy from Section 5: in case 1, price stability is abandoned in period 10; in case 2, the fiscal correction happens to occur in period 10 and price stability is preserved.

A numerical example
Let us return to the numerical example from the previous section, except that now we set ω = 0.25 (this number is slightly lower compared with ω = 0.3 in Section 4).We also suppose that ψ t is determined by the function where σ > 0 is a parameter, and we set σ = 0.018. 14We find three equilibrium values of T , T = 10, 4, and 2. We simulate the model for each of these equilibrium values of T .
Figure 5 plots the equilibrium path of the economy when in period 0 private agents form the belief that T = 10, in two cases: in case 1, price stability is abandoned in period 10; in case 2, the fiscal correction happens to occur in period 10 (in period 10, the surplus drawn by nature equals S + ω 10 instead of S) and price stability is preserved.Case 2 is only an example of an equilibrium path with a correction given the belief that T = 10 (a correction 14 The error function appearing in the formula is the same as the cumulative distribution function of a truncated normal random variable, so that ψt (ωt) = 1 for ωt = ∆ and ψt (ωt) decreases smoothly to 0 as ωt rises away from ∆.The economy from Section 5: in case 1, price stability is abandoned in period 2; in case 2, the fiscal correction happens to occur in period 2 and price stability is preserved.
can take place with probability ψ t , shown in Figure 5, in any period t = 1, . . ., T ).To understand Figure 5, note that private agents hold the optimistic belief that price stability will be maintained for 10 quarters absent a correction.The optimism produces a low bond yield 1/Q t + ρ, a moderate required correction ω t , and a probability of a correction ψ t close to 1 for several quarters -remarkably, the bond yield remains essentially unchanged at its initial steady-state level for as long as five quarters after the deficit appears (t = 0, . . ., 4).
Thereafter, as the required correction keeps rising steadily, the probability of a correction drops and the bond yield increases.In case 1 (no correction), the price level jumps in period 10.In case 2 (the correction in period 10), the gamble to preserve price stability succeeds at the last possible moment, with the correction eliminating the gap between the equilibrium value of debt and (1 + ρQ) B t−1 /4P (the lower right-hand panel in Figure 5).
Figure 6 plots the path of the same economy when in period 0 private agents form the belief that T = 2, instead of T = 10, in two cases analogous to Figure 5: in case 1, price stability is abandoned in period 2; in case 2, the correction happens to occur in period 2 and The economy from Section 5: in case 1, price stability is abandoned in period 4; in case 2, the fiscal correction happens to occur in period 4 and price stability is preserved.
price stability is preserved.Now private agents hold the pessimistic belief that price stability will be maintained only for 2 quarters absent a correction.The bond yield jumps sharply already in period zero, which quickly raises debt issuance and the required correction by large amounts.As a consequence, the probability of a correction ψ t falls below 0.1 already in period one.In case 1, the policy of price stability does not survive past period 2. In case 2, the economy gets lucky and the correction takes place in period 2.
Figure 7 illustrates the third equilibrium in which in period 0 private agents form the belief that T = 4, in the two familiar cases: in case 1, price stability is abandoned in period 4; in case 2, the correction happens to occur in period 4 and price stability is preserved.This is an intermediate situation between Figures 5 and 6, with the probability of a correction ψ t starting out in period 1 at about 0.5, instead of close to 1 (Figure 5) or close to 0 (Figure 6).15 In this specification of the model, whether or not price stability is preserved may depend on private agents' expectations.For the same sequence of exogenous shocks (nature's draws of the primary surplus), the gamble to preserve price stability may succeed if private agents expect it to succeed, and it may fail if they expect it to fail.Consider the economy from our numerical example, and suppose that the correction will occur in period t = 6 if the policy of price stability is still in place.If private agents are optimistic (T = 10), then the policy of price stability is still in place in period 6, the correction occurs, and price stability is preserved.If private agents are pessimistic (T = 2) or in the intermediate case (T = 4), price stability will be abandoned before the correction can take place.
The timing of inflation, in the absence of a correction, also depends on expectations: inflation occurs with a delay of 2 quarters in the equilibrium with T = 2 and with a delay of 4 quarters in the equilibrium with T = 4, for the same realization of the random state of the economy over time.
Finally, this version of the model predicts that, if private agents are optimistic, longterm interest rates may not change perceptively for several quarters, even though everyone knows that there may be inflation in the future.The reason is that, for some time, the probability of inflation is very small if private agents are optimistic.Both this empirical prediction about long rates and the assumption that the probability of a correction depends on the state of the economy seem more realistic compared with Section 4.
While expectations of price stability help win the policy gamble, luck continues to matter in this economy.Even if initially inflation expectations are anchored and long-term interest rates remain low for a considerable amount of time, the economy may experience an unfavorable sequence of shocks, and it may end up with higher inflation than if the policy of price stability had been abandoned without delay.

Discussion: Obstfeld (1986) redux
In the previous sections, we drew a parallel between our model and the currency crisis model of Krugman (1979).In this section, we examined our preferred specification in which private the answer is "yes" for the "optimistic" equilibrium with T = 10 and for the "pessimistic" equilibrium with T = 2, and "no" for the intermediate equilibrium with T = 4.
agents' beliefs affect the probability of a fiscal correction.This version closely resembles Obstfeld (1986) and his subsequent work on rational and self-fulfilling currency crises.In the words of Obstfeld (1996), p. 1039, "speculation against a currency creates objective economic conditions that make liability devaluation more likely" -in terms of this model, unanchored inflation expectations raise the required size of a fiscal correction, and thereby increase the chance that price stability will be abandoned.Furthermore, "even sustainable currency pegs may be attacked and even broken" (Obstfeld (1996), p. 1038) -here, the policy of price stability may fail if private agents expect it to fail, and it may continue if they expect it to continue.The gamble to preserve price stability is vulnerable to a shift in expectations.
In Obstfeld (1986), a speculative attack directly triggers a shift in policy from price (exchange rate) stability to an inflationary regime, whereas in the absence of an attack the fixed exchange rate policy continues.In later work, Obstfeld modeled explicitly the feedback from beliefs to the state of the economy and from the state of the economy to policy.For example, in Obstfeld (1996) he showed that multiple equilibria can be ranked by the degree of market skepticism in the policy of currency stability, and by the resulting worsening of the state of the economy conditional on that policy remaining in place.Furthermore, he demonstrated that different equilibria entail different probabilities of the policy's collapse.
The model in this section shares both features with Obstfeld (1996), as we just described.
From the perspective of the FTPL literature, the model provides an example in which a policy regime switch (here, from active back to passive fiscal policy) reflects the state of the economy and it may or may not occur in equilibrium depending on private agents' expectations.Maćkowiak (2007)  6 Gambling to avoid default The model can be reinterpreted as a model of default on public debt instead of inflation.
The gamble to preserve price stability with multiple equilibria à la Obstfeld (1986) then becomes the gamble to avoid default with multiple equilibria à la Calvo (1988), Section I.

A reinterpretation of the model
Suppose that the price level is exogenously fixed at P = 1.Then the analysis of the model goes through unchanged provided that we introduce a new variable, the "recovery rate" for each private agent on a bond in period t ≥ 0, 0 < θ t ≤ 1, and assume that the recovery rate satisfies θ t = 1/P t , where P t is the period t price level from the model in Sections 2-5.
It may be helpful to note that private agents' optimality condition (3) then becomes while the intertemporal government budget constraint, equation (6), reads Delayed inflation becomes delayed partial default in this version of the model, with the timing and size of the haircut the same as in the case of the price level jumps in Sections 3-5.By the logic of Sections 4-5, conditional on a fiscal imbalance in period 0, it may be worthwhile to delay default in order to buy time for a possible fiscal correction that would make default avoidable.The gamble to avoid default may succeed or fail depending on private agents' expectations, exactly like in Section 5.16 6.2 Discussion: Calvo (1988) redux Under the interpretation as a model of default, Section 5 closely resembles the model of sovereign default with multiple equilibria in Calvo (1988), Section I. Default may occur or may be avoided for the same sequence of exogenous shocks, depending on private agents' expectations -a conclusion very much in Calvo's spirit.His insight that multiple equilibria arise naturally in the market for defaultable sovereign debt has recently been refined by Corsetti and Dedola (2016), Ayres et al. (2018), Lorenzoni and Werning (2019), and Corsetti and Maeng (2020), among others. 17Our dynamic model with long-term debt is close to Lorenzoni and Werning (2019).Default can be "a slow moving crisis" here similar to their work (think of the equilibrium with T = 10 in the numerical example in Section 5 in the case when a fiscal correction fails to materialize and default occurs with a delay of 10 quarters).
Uribe ( 2006) studies a "fiscal theory of sovereign risk" related to the FTPL.He analyzes a scenario similar to the delayed adjustment case in Section 3 in which partial default is postponed, noting the relation between the time of default and the haircut.In his model, default is unavoidable (like in Section 3) and there are no multiple equilibria.
The version of the model with default instead of inflation may be especially relevant for countries with significant foreign-currency-denominated public debt.One can also envision a version of the model in which both inflation and default can occur in parallel, absent a fiscal correction -the self-fulfilling feedback between private agents' beliefs and policy would remain.This specification would be particularly appropriate to discuss the role of the central bank in providing a monetary backstop to government bonds -see Corsetti and Dedola (2016) for a model with optimizing fiscal and monetary authorities.The recent euro area sovereign debt crisis has spurred theoretical and applied work on this issue.
17 Calvo (1988), Section II, studies a model of inflation with multiple equilibria, and therefore one may argue that Section 5 of our model with inflation is in the spirit of Obstfeld (1986) and Calvo (1988), Section II.See Corsetti and Dedola (2016) on the role of the cost of inflation in that section of Calvo's paper.

Conclusions
We studied a model of adjustment to a fiscal imbalance.The imbalance may be corrected in the future, which gives policymakers a reason to gamble that no adjustment will ever be necessary.The policy gamble may well be worth taking, but it exposes the economy to a self-fulfilling shift in expectations that precipitates inflation or default.
We think that this basic insight is likely to hold up in richer models, with optimizing discretionary policymakers.While here we focused on positive analysis, further insight could be derived from normative analysis in a setting with socially costly inflation and taxes, and with a realistic probability distribution of future growth outcomes.It would also be important to understand better, over time and across countries, which conditions make a shift in inflation or default expectations more likely and why.
We are writing at the time when inflation rates have already increased significantly in most developed economies.How can one interpret this development from the perspective of the model?One interpretation is that during the COVID-19 pandemic in 2020-21 policymakers took the gamble to preserve price stability, and the gamble has been lost due to bad luck -we are past time T .The policy challenge now is to ensure that the adjustment will involve mostly a one-time jump in the price level, without very persistent inflation.
Another interpretation is that a part of the gamble may be still ongoing.The economy experienced multiple very bad shocks, and we can think of each shock as triggering a revision of the required adjustment and the terminal date T .While some inflation has already occurred, so far inflation expectations have remained fairly well anchored.In the future, however, we may still see more inflation and unanchoring of inflation expectations.The model, and the lessons from the work by Maury Obstfeld, remind us that such outcomes may be self-fulfilling.

Figure 3 :
Figure 3: Delaying inflation when fiscal correction is possible

Figure 4
Figure4plots the equilibrium path of the economy in two cases: in case 1, the policy of price stability is abandoned at T = 12, which is the equilibrium value of T in this economy;

Figure 5 :
Figure 5: Gambling to preserve price stability: An equilibrium with T = 10

Figure 6 :
Figure 6: Gambling to preserve price stability: An equilibrium with T = 2

Figure 7 :
Figure 7: Gambling to preserve price stability: An equilibrium with T = 4