Bounds on price-setting

Kocherlakota, Narayana Rao Article Bounds on price-setting Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Kocherlakota, Narayana Rao (2021) : Bounds on price-setting, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 16, Iss. 3, pp. 979-1015, https://doi.org/10.3982/TE4367 This Version is available at: https://hdl.handle.net/10419/253535 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 16 (2021), 979–1015 1555-7561/20210979 Bounds on price-setting Narayana R. Kocherlakota Department of Economics, University of Rochester I study a class of macroeconomic models in which all firms can costlessly choose any price at each date from an interval (indexed to last period’s price level) that includes a positive lower bound. I prove three results that are valid for any such halfclosed interval (regardless of how near zero the left endpoint is). First, given any output sequence that is uniformly bounded from above by the moneyless equilibrium output level, that bounded output sequence is an equilibrium outcome for a (possibly time-dependent) specification of monetary and fiscal policy. Second, given any specification of monetary and fiscal policy in which the former is time-invariant and the latter is Ricardian (in the sense of Woodford 1995), there is a sequence of equilibria in which consumption converges to zero on a date-by- date basis. These first two results suggest that standard macroeconomic models without pricing bounds may provide a false degree of confidence in macroeconomic stability and undue faith in the long-run irrelevance of monetary policy. This paper’s final result constructs a non-Ricardian nominal framework (in which the long-run growth rate of nominal government liabilities is sufficiently high) that pins down a unique stable real outcome as an equilibrium. Keywords. Pricing bounds, monetary policy, fiscal policy. JEL classification. E52, E61, E62. 1. Introduction Most macroeconomic models assume that price-setting firms are able to choose any element of the positive reals. This paper instead considers a class of otherwise standard macroeconomic models in which all firms can choose their prices from a common positive interval that includes its lower bound. I show that regardless of how near to zero that lower bound is, these models with bounds on price-setting give rise to dramatically different answers to key macroeconomic questions about the long-run relevance of monetary policy and long-run macroeconomic stability. Narayana R. Kocherlakota: [email protected] This is a much revised version of a paper that previously circulated under the titles “Monetary Economies with Bounded Competition” and “The L-Shaped Phillips Curve: Theoretical Justification and Empirical Implications” (NBER WP 24086). I owe special thanks to three anonymous referees for their valuable feedback. I also thank Fernando Alvarez, Marco Bassetto, Mariacristina De Nardi, Martin Eichenbaum, Greg Kaplan, Chen Kan, Matteo Maggiori, Stephen Morris, Ivan Werning, and participants in seminars at Chicago-Booth, CREI, Cornell University, the European Central Bank, the Federal Reserve Bank of San Francisco, the London School of Economics, Northwestern University, University College-London, Yale University, and the University of Zurich for helpful comments. ©2021 The Author. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE4367 980 Narayana R. Kocherlakota Theoretical Economics 16 (2021) The specifics are as follows. I study the implications of a class of simple infinitehorizon macroeconomic models in which monopolistically competitive firms can all costlessly choose prices at each date from a positive interval (indexed to the prior period’s price level) that includes its lowest value. The numeraire is money, which is an asset issued (and withdrawn via lump-sum taxation) by the government. The models feature time- and state-invariant specifications of technology and preferences, so that equilibrium quantities would be time-invariant in a moneyless economy (with labor, for example, being the numeraire). I obtain three main results. They are all valid regardless of how close the lower bound is to zero. The first result is that, given any output sequence that is uniformly bounded from above by the time-invariant moneyless equilibrium output level, that bounded sequence is an equilibrium outcome for some (possibly time-dependent) nominal framework (time paths of interest rate rules and nominal liabilities). This result is straightforward to prove: given a consumption sequence, we need only pick a path of nominal interest rates that is consistent with that sequence and its associated inflation rate path. But it has important implications. Scientifically, the result implies that it is impossible to model the real economy accurately without having some minimal information about monetary policy. From a policy point of view, the result implies that even if prices are highly flexible, economies cannot obtain desirable real outcomes if monetary policy interventions are poorly designed. One response to this “extreme relevance of monetary policy” result is that its proof relies on the possibility that the central bank could follow “crazy” policies (specifically, time-dependent interest rate pegs). The second main result of the paper restricts attention to “sensible” nominal frameworks in which the central bank follows a timeinvariant monetary policy rule and in which fiscal policy is Ricardian1in the sense of Woodford (1995) (so that the intertemporal government budget constraint is satisfied for all time paths of inflation rates). I show that for any such nominal framework, there is a sequence of equilibria in which consumption (which equals output and labor) converges datewise to zero.2The corresponding sequence of household utilities, as evaluated at the initial date, converges to zero (which is the households’ utility level from a time path that delivers zero consumption and zero labor at all dates). This result implies that even if governments use what might appear to be sensible nominal frameworks, the dynamic complementarities in monetary economies can give rise to deviations from macroeconomic stability that are large in terms of both quantities and welfare.3 The third and final result describes how to design a nominal framework that uniquely implements the constant moneyless equilibrium real outcome. Again, the unique implementation is valid for any positive interval of firm pricing choices that 1See also Leeper’s (1991) highly related discussion of active fiscal policy. 2A sequence of consumption paths {(Ck t)∞ t=1}∞ k=1converges datewise to zero if limk→∞ Ck t=0for all t. 3This result (about the arbitrarily bad equilibria that are possible when fiscal policy is Ricardian) applies for any interest rate rule. Consequently, it has nothing to do with the existence of a lower bound—zero or otherwise—on the nominal interest rate, and so is not related to the results of Benhabib et al. (2001). Indeed, the range of indeterminacy exhibited in this paper is much broader than is established in those authors’ work. Theoretical Economics 16 (2021) Bounds on price-setting 981 includes its lower bound. We know from the second result (described in the prior paragraph) that the framework must be non-Ricardian. As in prior work on non-Ricardian fiscal policies by Benhabib et al. (2002), I use a fiscal policy that targets the (long-run) growth rate of nominal liabilities.4Specifically, I consider any nominal framework in which the following conditions are fulfilled: (i) The monetary policy rule is active (the implied real interest rate is a strictly increasing function of the inflation rate) when the inflation rate is above, at, or slightly below target. (ii) The growth rate of the government’s nominal liabilities converges over time from below to the target nominal interest rate (expected inflation is at target and the real interest rate equals the rate of time preference). (I do not impose any restrictions on monetary policy when the inflation rate is more than slightly below target to allow for the possibility of a lower bound on the nominal interest rate.) Given such a nominal framework, I establish that, in any equilibrium, consumption is constant at the benchmark moneyless equilibrium level. Note that the last enumerated requirement means that fiscal policy is non-Ricardian, because the government’s intertemporal budget constraint is not satisfied when the nominal interest rate is lower than the target nominal interest rate. This policy is being used to eliminate equilibria in which consumption is ever below its moneyless level. Within the class of models studied in the paper, money has no transaction role.5 Accordingly, the Friedman rule is always satisfied: the risk-adjusted real rate of return on money is the same as that on any other asset. The point of this paper is that even though the Friedman rule is always satisfied and prices are (arbitrarily) close to fully flexible, money can still be highly distortionary in this economy. The distortion arises because money’s endogenous desirability as an asset is tied to household expectations about future inflation and output. The policy conclusions in this paper are quite different from those reached in the New Keynesian literature (as elucidated in Gali 2015, for example). That literature takes as given that under any specification of monetary policy and Ricardian fiscal policy, aggregate outcomes remain close to a long-run zero inflation steady state. Its main conclusion is that, given this presumption, there are no equilibrium deviations from steady state under active monetary policy rules. In this paper, it is shown that all monetary policy rules, when combined with Ricardian fiscal policies, admit arbitrarily poor equilibrium outcomes. This finding about real outcomes echoes Cochrane’s (2011) argument 4I restrict fiscal policies to date-contingent lump-sum nominal transfers/taxes and assume that the only government liability is (interest-bearing) money. Given this restriction, whether a fiscal policy is Ricardian has to do with the present value of the long-run nominal liabilities of the government, when calculated using the lowest possible time path of nominal interest rates. If that present value is zero, then the government’s infinite-horizon intertemporal budget constraint is satisfied for any sequence of inflation rates, and becomes irrelevant for price level determination. Note that if the lowest time path of nominal interest rates is highly negative, then the long-run growth rate of nominal liabilities must also be highly negative (so that nominal liabilities converge to zero extremely rapidly) in a Ricardian fiscal policy. 5I relax this assumption in Appendix C without affecting the validity of the main results. 982 Narayana R. Kocherlakota Theoretical Economics 16 (2021) that if fiscal policy is Ricardian, then inflation is indeterminate in equilibrium for all monetary policy rules (active or not). So as to ensure macroeconomic stability, governments must follow non-Ricardian fiscal policy regimes (assuming that such regimes are possible). In the next two sections, I set up the baseline model and describe the key results. In Section 4, I discuss implications of the results (with supporting technical details provided in Appendix B), connections to related literature, and sketch how to include noninterest-bearing currency in the model (with supporting technical details provided in Appendix C). 2. Models with a price-setting lower bound In this section, I describe an infinite horizon monetary model in which all firms can choose prices from a common positive interval that includes its lower bound. The lower bound is defined relative to the prior period’s price level. Hence, it ends up serving as a constraint on the time path of inflation rates. I define and characterize equilibria in this economy. 2.1 Setup Consider an economy with a unit measure of households who live forever. Time is discrete and the households maximize the expected value of ∞  t=1 βt−1u(Ct)−v(Nt)0<β<1 where Ctis the consumption of a composite good in period tand Ntis labor in period t. Here, I assume that u(0)=v(0)=0and that u−uvv >0 lim c→0u(c) =∞ lim c→∞ u(c) =0 The composite good consists of a unit measure of consumption goods, indexed by j, andisdefinedas Ct=1 0 c(j)1−1/η djη η−1  η>1 Each household’s consumption of each good jis bounded from below by zero. Each consumption good jis produced by a monopolistically competitive firm. A typical firm jhas a technology at each date that converts xunits of labor into xunits of consumption good jfor any x≥0. The households own equal shares of all firms. There is no entry or exit. Theoretical Economics 16 (2021) Bounds on price-setting 983 Labor markets are competitive and so, at each date, firms all hire workers at the same wage Wt(denominated in terms of dollars). Given that wage, firms simultaneously set prices for their consumption goods in terms of dollars.6The firms’ problems are identical, and so they each choose the same price Ptin equilibrium; that price is also the aggregate price level. At date t, each firm jis constrained to choose its price subject to a lower bound: pt(j) ≥πLB tP∗ t−1 Here the bounds πLB =(πLB t)∞ t=1form an exogenously specified sequence. The firm treats them and last period’s (endogenously determined) price level P∗ t−1parametrically. I define the gross inflation rate πtas Pt/Pt−1and (without loss of generality) set P∗ 0=1. Monetary policy works as follows. Each household is initially endowed with ¯ M0dol- lars. Like reserves at many central banks, money is interest-bearing. Specifically, at the beginning of period (t +1), a household that has Mtdollars is paid (Rt(πt)−1)Mtdollars. Here the interest rate rule R=(Rt)∞ t=1is a sequence of exogenous (possibly timedependent) weakly increasing continuous functions that map period tinflation into a period tgross nominal interest rate.7The range of Rtfor any date tis restricted to be nonnegative (which translates into a zero lower bound on the gross nominal interest rate). Finally, fiscal policy works as follows. The government’s only liability is interestbearing money. Let {¯ Mt}∞ t=1be an arbitrary sequence of positive real numbers. At each date (t +1), the government levies a lump-sum tax, in dollars, equal to τt(πt)=Rt(πt)−1¯ Mt+(¯ Mt−¯ Mt+1) This tax ensures that the per-household level of nominal government liabilities at the end of period (t +1)is equal to ¯ Mt+1. 2.2 Equilibrium In this subsection, I define an equilibrium in this economy. To simplify the analysis, I restrict attention to nonstochastic but possibly time-dependent equilibria. I refer to an interest rate rule and fiscal policy (R ¯ M)collectively as a nominal framework. Given its specification, an equilibrium in this economy is a vector sequence (C∗N∗M∗P∗W∗),where(C∗N∗M∗)represent per-household consumption, labor, and money-holdings, and (P∗W∗)represent price levels and wages. Given this vector sequence, it is useful to define the implied inflation, taxes and profits as π∗ t=P∗ t/P∗ t−1 6Throughout, I treat money as the numeraire. If consumption or labor is the numeraire, then there is always an equilibrium in which the price of money in terms of that real numeraire is zero in all periods. 7As noted in the Introduction, there is no special transaction role for money. Hence, what I term “money” could also be seen as a perpetual bond, with a coupon payment in each period that is determined with reference to the inflation rate. 984 Narayana R. Kocherlakota Theoretical Economics 16 (2021) τ∗ t=¯ MtRtπ∗ t−1+¯ Mt−¯ Mt+1 ∗ t=P∗ t−W∗ tN∗ t The vector sequence satisfies the usual equilibrium conditions. First, (C∗N∗M∗) solve the household’s optimization problem, given prices, wages, taxes, and profits that it treats as exogenous: C∗N∗M∗=argmax (CNM) ∞  t=1 βt−1u(Ct)−v(Nt) s.t. P∗ tCt+Mt=Mt−1Rt−1π∗ t−1+W∗ tNt−τ∗ t−1+∗ t∀t≥1w.p. 1 CtMtNt≥0 Second, in any date, P∗ tsolves firm j’s pricing period tproblem, given W∗ tand last period’s price index (which shapes the lower bound): P∗ t=argmax PtP1−η t−W∗ tP−η t s.t. Pt≥πLB tP∗ t−1 Finally, markets must clear in all dates: C∗ t=N∗ t M∗ t=¯ Mt 2.3 A simple characterization of equilibrium In this economy, there are three decisions that are made each period: consumption– savings, consumption–labor, and price-setting. The first decision gives rise to the familiar Euler equation that leaves households marginally indifferent between consumption and money: uC∗ t=βRtπ∗ tuC∗ t+1 π∗ t+1  If we exploit goods–market clearing, the consumption–labor decision gives rise to a standard intratemporal first order condition uC∗ tw∗ t=vC∗ t Here w∗ trepresents the period treal wage w∗ t≡W∗ t/P∗ t Theoretical Economics 16 (2021) Bounds on price-setting 985 The household saving decision also gives rise to a transversality condition8that leaves households marginally indifferent to permanent increases/reductions in their moneyholdings: lim t→∞ βtuC∗ t¯ Mt/P∗ t=0 Finally, the price-setting decision on the part of the firm gives rise to the condition P∗ t=maxπLB tP∗ t−1(1−1/η)−1W∗ t In words, the firm follows the usual markup formula unless doing so violates the lower bound on prices. If we divide through by P∗ t−1, we can rewrite this price-setting condition as π∗ t=maxπLB t(1−1/η)−1w∗ tπ∗ t By combining these conditions, we can present the following conclusion. Proposition 1. Given a nominal framework (R ¯ M), a consumption–inflation–realwage sequence (C∗π∗w∗)is part of an equilibrium if and only if it satisfies the restrictions uC∗ t=βRtπ∗ tuC∗ t+1/π∗ t+1 w∗ t=vC∗ t uC∗ t π∗ t=maxπLB t(1−1/η)−1w∗ tπ∗ t in all dates and the households’ transversality condition is satisfied: lim t→∞ βtuC∗ t¯ Mt t  s=1 π∗ s =0 All proofs not provided in the text are given in Appendix A. Proposition 1 shows that the model does not imply a tight short-run or medium-run connection between money growth and inflation. (The household’s transversality condition does imply that, asymptotically, money growth cannot exceed the nominal interest rate.) This disconnect has nothing to do with the lower bound on inflation; rather, it is a consequence of assuming that money has no transaction role.9Empirically, it is consistent with recent data from many advanced economies, as inflation has remained low even though the monetary base has grown rapidly over the past 10 or more years. 8In writing the transversality condition in this way, I am implicitly restricting attention to equilibria in which the limit exists. See Kocherlakota (1992) for the relevant generalization. 9In Appendix C, I augment the model by adding a transaction role for currency (non-interest-bearing money). In that model, there is an intratemporal equilibrium restriction between the quantity of currency and the price level. 986 Narayana R. Kocherlakota Theoretical Economics 16 (2021) 2.4 Ricardian versus non-Ricardian nominal frameworks In what follows, it will be important to distinguish between nominal frameworks (R ¯ M) that are Ricardian and those that are non-Ricardian. 2.4.1 Definitions As in Woodford (1995, p. 26), a nominal framework is said to be Ricardian if the limiting present value of the government’s nominal liabilities is guaranteed to be zero for any possible sequence of inflation rates. Intuitively, this restriction means that, like a household in the standard definition of competitive equilibrium, the government’s (intertemporal) budget constraint is satisfied for all possible price level sequences. Since the nominal interest rule consists of a sequence of weakly increasing functions, the condition lim t→∞ ¯ Mt t  s=1 RsπLB s =0(1) is both necessary and sufficient to ensure that the nominal framework is Ricardian. Note that, for Ricardian nominal frameworks, the household’s transversality condition is implied by the other equilibrium conditions in Proposition 1, because 1 RtπLB t≥1 Rtπ∗ t=βuC∗ t+1 uC∗ tπ∗ t+1  This ensures that fiscal policy (that is, the specification of the path ¯ Mof nominal liabilities) plays no role in the determination of equilibrium. A non-Ricardian nominal framework is one in which the asymptotic growth rate of nominal liabilities is sufficiently high that the limit in (1) is positive. Under a non- Ricardian fiscal policy, it is impossible for the inflation rate to equal its minimal value for all dates in an equilibrium because such a sequence fails to satisfy the household’s transversality condition. Intuitively, if the nominal liabilities are growing so rapidly while paying such a low nominal return, households would find it optimal to lower their money-holdings permanently. 2.4.2 Examples Here are three examples that are intended to illustrate what “Ricardian” means.10 The first example demonstrates that apparently passive fiscal policies may not be Ricardian. Example 1. Suppose Rt(π) =1for all t≥1and all π, so that money pays no interest. Suppose too that ¯ Mt=¯ M1for all t≥1. Under this fiscal policy, taxes are equal to zero forever. Nonetheless, this (apparently passive) fiscal policy is not Ricardian. Indeed, the households’ transversality condition is not satisfied by any price level sequence, and so there is no equilibrium.11 Intuitively, money is an intrinsically useless object in this 10I thank an anonymous referee for suggesting these examples. 11Note that this same nonexistence result applies for any sequence {¯ Mt}∞ t=1such that limt→∞ ¯ Mt>0.If consumption were the numeraire, then there would always be an equilibrium in which the price of money is constant at zero. Theoretical Economics 16 (2021) Bounds on price-setting 993 Proposition 6 tells us that any such class must be restricted to non-Ricardian nominal frameworks. As noted earlier, many economists are uncomfortable with non- Ricardian fiscal policies. For those economists, Proposition 6 is really the end of the story of what happens once firms’ pricing choices are restricted by a positive lower bound. Others (such as Cochrane 2011) have argued that non-Ricardian fiscal policy is essential for price level determinacy; the following proposition is congruent with this thinking.14 Proposition 7. Consider a nominal framework (R ¯ M) with a time-invariant interest rate rule ˆ Rthat targets πTAR and such that for some >0, the gross real interest rate ˆ R(π)/π is strictly increasing for π∈[πTAR − πTAR ]. Suppose that fiscal policy ¯ Mtakes the form ¯ Mt=M0 t ˆ RπTARt so that the rate of growth of nominal liabilities asymptotes from below to (ˆ R(πTAR )−1). Then, in any equilibrium, C∗ t=Yreal for all t≥1. The proposition shows that is possible to eliminate the bad equilibria with a twopronged approach. First, fiscal policy is non-Ricardian: nominal liabilities grow so rapidly over time that households can improve their welfare by permanently reducing their money-holdings unless the long-run nominal return on money is at least ˆ R(πTAR )=β−1πTAR . Second, monetary policy is active when inflation is slightly below target. This ensures that if C∗ t<Yreal, inflation converges to some rate that is strictly below πTAR , so that the long-run nominal return on money has to be less than β−1πTAR . 4. Discussion In this section, I discuss three aspects of the above analysis: the broader lessons of the results (with supporting technical details in Appendix B), the relationship of the analysis to other work on adverse equilibria in macroeconomic models, and the robustness of the results to adding currency (with supporting technical details provided in Appendix C).15 4.1 Two lessons In this subsection, I describe two kinds of conclusions that can be drawn from the above analysis. 14Proposition 7 can be readily extended to eliminate real outcomes other than Yreal in stochastic (sunspot) equilibria. 15It is possible to generalize the results in Section 3 to a model with variable demand elasticity of the kind used by Arkolakis et al. (2019). That generalization is available on request. 994 Narayana R. Kocherlakota Theoretical Economics 16 (2021) 4.1.1 Lesson 1: Change in policy implications The most direct conclusion from the results in Section 3 is that the normative and positive implications of a standard class of macroeconomic models are highly sensitive to the inclusion of a lower bound (regardless of how small) into the constraint set of price-setting firms. Is this kind of lower bound a feature of actual economies? There is no clear evidence with which to address this question—one way or another—about whether boundary points are included in firm action sets. But that lack of evidence surely suggests that we should be willing to put some weight on the possibility that firm price-setting choice sets do contain a positive lower bound. Once we do so, the models’ implications for appropriate monetary/fiscal policy change radically. More explicitly, suppose a policymaker is choosing between any Ricardian nominal framework and a specific non-Ricardian nominal framework of the kind constructed in Proposition 7. If there is no lower bound included in the price-setting firms’ constraint sets, the policymaker can choose either of these nominal frameworks, because (per Proposition 2) they both lead to the same unique (stable) real outcome. But if it is even possible that the firms’ constraint sets include a lower bound, then the former (Ricardian) nominal framework could give rise to highly adverse macroeconomic outcomes. The policymaker would then find the non-Ricardian framework to be strictly optimal. 4.1.2 Lesson 2: Revelation of otherwise hidden strategic forces Proposition 2 proves that, regardless of the choice of nominal framework, there is a unique equilibrium allocation once we drop the lower bound on prices. In this subsection, I argue that this result is a (highly) misleading description of the possible strategic interactions among the firms in the model. Even if we remove the bounds, the model retains the powerful dynamic complementarities that play such a key role in Proposition 6.Thoseforces continue to push firms and households to coordinate on low-output–low-inflation outcomes, even though they are not formally in the equilibrium set. The following game is a simple illustration of this line of reasoning. Suppose two players simultaneously choose actions from the set (−∞∞), and player ireceives a payoff 4aiaj−a2 i,wherej= i. There is a unique (pure strategy Nash) equilibrium in this game, in which ai=aj=0. But this apparent uniqueness result is misleading. The best response function to this game is that player ichooses 2aj,wherej= i. If we iterate this function ktimes, starting at an initial vector () of actions, we arrive at the action vector (2k 2k).In this sense, the players find it desirable to coordinate on very large outcomes in absolute value, even though those outcomes are not in the equilibrium set. It is not surprising that if we add an upper bound to the action sets, so that they become [0amax],thereis another equilibrium {amaxamax}and it is the robust one. In Appendix B, I show how this same logic applies in (a numerical parameterization of) the model in Sections 2and 3when we drop the lower bound on firm pricing decisions. I focus on the case in which the interest rate rule is time-invariant and active (so that there is no lower bound on the net nominal interest rate). In this dynamic environment, an iterated best response involves simultaneous choices at all dates by all Theoretical Economics 16 (2021) Bounds on price-setting 995 households and firms. I show that iterating on the firms’ best response function results in a limit in which consumption is zero at all dates. Note that this is the same extreme limit that we found in Proposition 6. In summary, (standard) macro models feature strong downward pricing complementarities, as households’ perceptions about the real return to money depend on their expectations about inflation and their shadow real interest rates depend on their expectations about future consumption. These forces serve to create the possibility of a macroeconomic “death spiral” of sorts. Appendix B demonstrates that these forces are present even when the firms do not face pricing lower bounds. But Propositions 2and 6 together show that the full effect of these powerful complementarities becomes manifest in the equilibrium set only if firms’ pricing decisions are bounded from below. This is not the first paper to note the strong power of the intertemporal complementarities in representative agent macroeconomic models. They are, for example, the source of the so-called forward guidance puzzle (Del Negro et al. 2015 and McKay et al. 2016).16 4.2 Other related literature In this subsection, I discussed the connections between this paper and other recent work on the potential for highly adverse outcomes in macroeconomic models. Werning (2011) studies a continuous time version of a linearized Calvo model with an initial shock to the discount factor that persists over a finite horizon. As in Eggertsson and Woodford (2003), he considers an equilibrium in which output and inflation return to target after the shock ends. He shows that, conditional on this equilibrium, the initial response of output and inflation to a given shock becomes increasingly large as the fraction of price-changers grows closer to 1. (This is a sharp analytical characterization of what Eggertsson and Krugman 2012 call the paradox of flexibility.) Cochrane (2017) reconsiders the impact of the zero lower bound on nominal interest rates in linearized continuous-time Calvo models. Like Werning (2011), he points out that in response to a negative shock to the natural real interest rate, there are multiple Pareto-ranked equilibria for a given interest rate path. Unlike Werning, Cochrane argues in favor of a selection based on bounding the size of the endogenous response to an initial adverse shock. Applying Cochrane’s desideratum in this model would imply restricting attention to equilibria of the kind described in Proposition 4 (in which output equals Yreal and inflation is always equal to target). The current paper can be seen as an extension of Werning (2011)andCochrane (2017) in the following ways: *ThemainProposition 6 applies for any interest rate rule, regardless of whether it has a zero lower bound. 16Farhi and Werning (2019) and Gabaix (2020) analyze bounded rationality modifications of the standard model in which these intertemporal complementarities are dampened. There is an ongoing debate about how the magnitude of these intertemporal complementarities is affected by the introduction of incomplete financial markets; see, among others, Werning (2015) and Kaplan et al. (2018). 996 Narayana R. Kocherlakota Theoretical Economics 16 (2021) * As discussed in Section 4.1.2, this paper shows that the potential for highly adverse low-output outcomes is not tied to Calvo pricing per se. Rather, those outcomes are a natural consequence of the basic strategic complementarities associated with firms’ pricing decisions. * Cochrane suggests that non-Ricardian fiscal policy can be used to support his “muted response” equilibrium selection criterion. In contrast, I prove in Propositions 6and 7that unique implementation of desirable equilibrium outcomes requires the use of non-Ricardian fiscal policy. Like this paper, Bassetto and Phelan (2015) show how adding plausible constraints to a macroeconomic model can give rise to additional undesirable equilibria. However, their focus is different: they show how central bank limits on household borrowing can give rise to hyperinflationary outcomes. As in this paper, the additional undesirable equilibria can be eliminated through the appropriate use of a non-Ricardian fiscal policy. 4.3 Money and currency In the models described in Section 2, money has no liquidity role. Money is held only to pay lump-sum taxes levied by the government and pays the same real return as all other assets in the economy. It is nonetheless potentially distorting because households can contemplate off-equilibrium trades of consumption for money. In reality, households do hold non-interest-bearing currency, and banks can always trade their interest-bearing reserves with the government for that currency. How would adding currency to the model affect the results obtained in Section 3? Suppose, in particular, that households get momentary utility from the real value of their currency holdings Xaccording to a function um(X/P) This function is strictly increasing for x≤¯ xand satisfies um(x) =um(¯ x) for all x≥¯ x(so that ¯ xcan be viewed as a satiation level of real currency-holdings). In Appendix C, I extend the model of Sections 2and 3by adding non-interest- bearing currency in this fashion. I show that Propositions 3and 6generalize to this extended model. I also prove a version of Proposition 7, in which the targeted gross inflation rate is above, but arbitrarily close to, the discount factor β(the Friedmanian rate of deflation). This last restriction ensures that real balances are approximately optimal in equilibrium. 5. Conclusion Most macroeconomic models treat price-setting firms as being able to make their choices from the set of positive reals. In this paper, I instead consider a class of macroeconomic models in which firms choose prices from an interval that includes its positive Theoretical Economics 16 (2021) Bounds on price-setting 997 lower bound. I find that the normative and positive implications of the models are highly sensitive to this perturbation, regardless of how close to zero the lower bound is. If the models have no pricing lower bound, there is a strong (albeit well known) irrelevance result: the equilibrium level of output is completely independent of the specification of monetary and fiscal policy. In contrast, in models with some positive lower bound on price-setting, the nature of the nominal framework matters greatly for the set of equilibrium real allocations. In particular, governments can then ensure macroeconomic stability only if they follow (appropriate) non-Ricardian fiscal policies. Some readers may question the empirical relevance of the assumption that firms’ pricing sets include positive lower bounds. But the analysis in this paper contains a more general message: even without the bounds, there are strong strategic forces in this (standard macro) model that are inducing firms to coordinate on arbitrarily low-output outcomes (see the discussion in Section 4 and Appendix B). Basically, if we omit boundary points from firm action sets, we risk “hiding” what would otherwise be completely natural equilibria (in the sense of being generic limit points of iterations of the best response function). The model’s implications for appropriate policy should take these “missing” equilibria into account. Accordingly, the results in this paper place new emphasis on an old question: can governments follow arbitrary non-Ricardian fiscal policies? If they have this capability, do they exploit it in reality? Addressing these questions in a compelling fashion will likely require a deeper modeling and understanding of fiscal policy than is incorporated into current macroeconomic theory.17 Appendix A In this appendix, I gather the proofs of Propositions 1,6,and7. Proof of Proposition 1 The necessity of the first order conditions is straightforward. The necessity of the transversality condition follows from a standard argument. Suppose lim t→∞ βtuC∗ t¯ Mt/P∗ t=L>0 I claim that it is possible to find a budget-feasible perturbation that makes the household better off. Thus, given εin (0L),thereexistsTsuch that βtuC∗ t¯ Mt/P∗ t>ε for all t≥T. Consider a perturbation whereby the household increases consumption at date tby β−tε/u(C∗ t),lowersMtby β−tεu(C∗ t)−1/P∗ t,andlowersMt+s,s≥1,by εβ−tuC∗ t−1/P∗ t s  τ=1 R∗ t+τ =εβ−tuC∗ t−1/P∗ tβ−suC∗ t/uC∗ t+sP∗ t/P∗ t+s 17Bassetto (2002) represents an early effort along these lines. 998 Narayana R. Kocherlakota Theoretical Economics 16 (2021) =εβ−t−s P∗ t+suC∗ t+s <¯ Mt+s This perturbation is budget-feasible (because the household’s money-holdings remain positive in all future periods). The sufficiency of the price-setting first order condition as a solution to the firm’s problem is obvious. The sufficiency of the other conditions for household optimality is by contradiction. Suppose (CNM)is budget-feasible and dominates (C∗C∗¯ M), so that 0<lim T→∞ T  t=1 βt−1uC t−vN t−uC∗ t−vC∗ t We can apply the subgradient inequality for concave functions, 0<lim inf T→∞ T  t=1 βt−1uC∗ tC t−C∗ t−vC∗ tN t−C∗ t =lim inf T→∞ T  t=1 βt−1uC∗ tC t−C∗ t−W∗ tN t−N∗ t/P∗ t =lim inf T→∞ T  t=1 βt−1uC∗ tM t−1−¯ Mt−1R∗ t/P∗ t−M t−¯ Mt/P∗ t =lim inf T→∞ βT−1uC∗ T ¯ MT−M T/P∗ T ≤lim inf T→∞ βT−1uC∗ T¯ MT/P∗ T =0 where the penultimate step comes from the nonnegativity of M. This contradiction proves the proposition. Proof of Proposition 6 There are two distinct cases based on the magnitude of the parameter γ, defined as γ=β−1πmin/ˆ R(πmin) (3) Suppose first that γ≥1(so that the average real return to money is no higher than 1/β when inflation is at its lowest level). Define (λk)∞ k=1to be any strictly increasing sequence that converges to infinity with initial λ1>1. Define (Ck∗ t)∞ t=1via the Euler equation: uCk∗ t+1=λkγt−1uYrealt=12 Theoretical Economics 16 (2021) Bounds on price-setting 999 We can readily verify that for all (k t), uCk∗ t+1>u Yreal Define πk∗ t=πmin and wk∗ t=v(Ck∗ t)/u(Ck∗ t)for all (k t). Then we can verify, using the conditions in Proposition 1, that (Ck∗wk∗πk∗)is part of an equilibrium. Note that for any t, lim k→∞ uCk∗ t≥γt−1uYreallim k→∞ λk=∞ and so limk→∞ Ck∗ t=0=limk→∞ Uk∗. The second case is that, as defined in (3), γ<1(intuitively, the average long-run real return to money is higher than 1/β when inflation equals πmin). In that case, let ˆπsatisfy πmin <ˆπ<π TAR βˆ R(πmin)> ˆπ (There is such a value for ˆπbecause πTAR =βˆ R(πTAR)≥βˆ R(πmin)>π min.) Pick any horizon k>1.Givenk, define an inflation sequence πk∗recursively as πk∗ t+1=βˆ Rπk∗ tt≥k πk∗ k=ˆπ πk∗ t=πmint<k and define a consumption sequence Ck∗so that Ck∗ t=Yrealt≥k uCk∗ t=γt−k−1βˆ R(πmin)uYreal πk∗ k 1≤t<k Since γ<1,u(Ck∗ t)>u (Yreal)for t<k. We know that πk∗ k+1=βˆ R( ˆπ) ≥βˆ R(πmin)> ˆπ=πk∗ k We know too that since πTAR >π k∗ k,πTAR ≥πk∗ k+1. Since ˆ Ris weakly increasing, induction implies that πTAR ≥πk∗ t+1≥πk∗ t≥πmin for all t≥kand that the sequence (πk∗ t)t≥kconverges (as tconverges to infinity) to the smallest fixed point of βˆ Rthat is larger than ˆπ. We can then verify using Proposition 1 that (Ck∗πk∗)is part of an equilibrium. It is clear that limk→∞ πk∗ t=πmin for all t, and since γ<1,limk→∞ u(Ck∗ t)=∞for all t. It follows too that limk→∞ Uk∗=0. 1000 Narayana R. Kocherlakota Theoretical Economics 16 (2021) Proof of Proposition 7 There are two cases. Case 1: βˆ R(πmin)≤πmin.The proof for this case is by contradiction. Suppose C∗is part of an equilibrium and C∗ t<Yreal.Then uC∗ t+1=β−1uC∗ t ˆ R(πmin)π∗ t+1 =β−1uC∗ t ˆ R(πmin)/πmin π∗ t+1 πmin ≥uC∗ t Hence, by induction, π∗ t+s=πmin for all s≥0. The households’ transversality condition requires that 0=uC∗ tlim s→∞ βsuC∗ t+s¯ Mt+s uC∗ t(πmin)sP∗ t =uC∗ tlim s→∞ M0(t +s)−1ˆ RπTARs ˆ R(πmin)s =∞ which is a contradiction. Case 2: βˆ R(πmin)/πmin >1. Define ˆπ∈(πminπTAR −) so that it satisfies βˆ R( ˆπ) =ˆπ βˆ R(π) > π for all πin [πminˆπ). We know such a ˆπexists because βˆ R(πTAR −) < (πTAR −) and βˆ R(πmin)>π min. Suppose C∗ t<Yreal at some date t. I show first, by contradiction, that there is some s≥0such that C∗ t+s+1=Yreal. Suppose not. Then for all s≥0, uC∗ t+s+1=β−1ˆ R(πmin)−1πmins+1uC∗ t But this implies that u(C∗ t+s+1)is lower than u(Y real)for ssufficiently large, which is the desired contradiction. Hence, there is some s≥0such that C∗ t+s+1=Yreal and C∗ t+s<Yreal. It follows that π∗ t+s+1=βˆ R(πmin)uC∗ t+s+1/uC∗ t+s <βˆ R(πmin)uC∗ t+s+1/uYreal ≤βˆ R( ˆπ) =ˆπ Theoretical Economics 16 (2021) Bounds on price-setting 1001 Since βˆ R(π∗ t+s+1)>π ∗ t+s+1, we can conclude that uC∗ t+s+2 π∗ t+s+2 =β ˆ Rπ∗ t+s+1 π∗ t+s+1−1uYreal π∗ t+s+1 <u Yreal/π∗ t+s+1 and so C∗ t+s+2=Yreal.Hence, π∗ t+s+2=βˆ Rπ∗ t+s+1 <βˆ R( ˆπ) =ˆπ By induction, we can conclude that for all r≥1, C∗ t+s+r=Yreal π∗ t+s+r+1=βˆ Rπ∗ t+s+r where π∗ t+s+1is specified as above. The sequence (π∗ t+s+r)∞ r=1is strictly increasing and is bounded from above by ˆπ. Hence, it converges to ˆπ(the smallest fixed point of βˆ Rthat is greater than πmin). To be an equilibrium, the households’ transversality condition must be satisfied: 0=lim T→∞ ¯ Mt+s+T T  r=1 ˆ Rπ∗ t+s+r ≥lim T→∞ ¯ Mt+s+T/ˆ R( ˆπ)T =lim T→∞ ¯ M0ˆ RπTARt+s+T (t +s+T)ˆ R( ˆπ)T =∞ But this is a contradiction: the nominal liabilities are growing too fast to be consistent with an equilibrium in which inflation is bounded from above by ˆπ. It follows that there cannot be any C∗ t<Yreal. Appendix B In this appendix, I consider the limits of iterated best responses in a numerical example of the model in Sections 2and 3without a lower bound on firm prices. The utility functions (u v) are defined as u(C) =2(1−1/η)−1C1/2 v(N) =N2 1002 Narayana R. Kocherlakota Theoretical Economics 16 (2021) Under this parameterization, Yreal =1. There is a time-invariant interest rate rule ˆ R defined by ˆ R(π) =β−1πTAR πα+1 πTARα+1α>0 for some πTAR . It follows that ˆ Robeys the Taylor principle and that ˆ R(πTAR )/πTAR =1/β. Within this numerical example, I explore the limits of iterated best responses (where, in this dynamic setting, each iteration specifies choices for all firms and households at all dates). In the first iteration, I suppose that all firms at date tset their prices equal to (¯π1)t,where ¯π1is time-invariant and less than πTAR. (Recall that the initial price P∗ 0=1.) Households then best respond to the resulting low real interest rates through falling consumption sequences, under the presumption that fiscal policy is Ricardian. Through the households’ labor supply condition, those falling consumption choices give rise to implied real wages in each period, which become a sequence of nominal wages when multiplied by the price level sequence (( ¯π1)t)∞ t=1. All firms then choose their best response (¯π2)tby maximizing their profits in response to this nominal wage sequence. Suppose inductively that in a hypothetical kth iteration, all firms at date t≥1choose their prices equal to (¯πk)t,where ¯πk<π TAR. Suppose too that households react by demanding a falling consumption sequence that is everywhere below Yreal: Ck t=Ck 0¯πk πTAR 2αt t≥1 Ck 0≤Yreal It is readily verified that this shrinking consumption sequence satisfies the Euler equation for any specification of Ck 0. (The sequence is budget-feasible for any Ck 0because fiscal policy is Ricardian.) The resulting implied real-wage sequence wkis wk t=vCk t/uCk t=Ck 03/2¯πk πTAR 3αt (1−1/η)−1t≥1 This real-wage sequence, combined with the firms’ pricing choices in the kth iteration, implies that the nominal wage sequence in the kth iteration is Wk t=Ck 03/2¯πk πTAR 3αt ¯πkt (1−1/η)−1t≥1 The update is that a firm’s pricing best response in the (k +1)st iteration to this nominal wage sequence is given by Pk+1 t=Ck 03/2¯πk ¯πTAR 3αt ¯πktt≥1 Theoretical Economics 16 (2021) Bounds on price-setting 1009 lim k→∞ Ck∗ t=0 lim k→∞ Uk∗=0 lim k→∞ πk∗ t=πmin where Uk∗≡∞ t=1βt−1[u(Ck∗ t)−v(Ck∗ t)]. In these sequences of equilibria, real currencyholdings xk∗converge datewise to satiation, so that for all t, lim k→∞ xk∗ t=lim k→∞ Xk∗ t/Pk∗ t=¯ x Proof. There are two distinct cases based on the magnitude of the parameter γ,defined as γ=β−1πmin(4) Suppose first that γ≥1(so that the average real return to money is no higher than 1/β when inflation is at its lowest level). Define (λk)∞ k=1to be any strictly increasing sequence that converges to infinity with initial λ1>1. Define (Ck∗ t)∞ t=1via the Euler equation uCk∗ t+1=λkγt−1uYrealt=12 We can readily verify that for all (k t), uCk∗ t+1>u Yreal Define πk∗ t=πmin,wk∗ t=v(Ck∗ t)/u(Ck∗ t),and xk∗ t=¯ x for all (k t). Given this definition, it is readily verified that (Ck∗Xk∗Pk∗)is part of an equilibrium given (ˆ R ¯ M). Note that for any t, lim k→∞ uCk∗ t≥γt−1uYreallim k→∞ λk=∞ and so limk→∞ Ck∗ t=0=limk→∞ Uk∗. The second case is that, as defined in (4), γ<1(intuitively, the average long-run real return to money is higher than 1/β when inflation equals πmin). In that case, let ˆπsatisfy πmin <ˆπ<β Pick any horizon k>1.Givenk, define an inflation sequence πk∗recursively as πk∗ t+1=βˆ Rπk∗ tt≥k πk∗ k=ˆπ πk∗ t=πmint<k 1010 Narayana R. Kocherlakota Theoretical Economics 16 (2021) and define a consumption sequence Ck∗so that Ck∗ t=Yrealt≥k uCk∗ t=γt−k−1βuYreal πk∗ k 1≤t<k Since γ<1,u(Ck∗ t)>u (Yreal)for t<k. We know that πk∗ k+1=βˆ R( ˆπ) ≥β> ˆπ=πk∗ k We know too that since πTAR >π k∗ k,πTAR ≥πk∗ k+1. Since ˆ Ris weakly increasing, induction implies that πTAR ≥πk∗ t+1≥πk∗ t≥πmin for all t≥kand that the sequence (πk∗ t)t≥kconverges, with respect to t, to the smallest fixed point of βˆ Rthat is larger than ˆπ. We can then readily verify that (Ck∗πk∗)is part of an equilibrium. Define the equilibrium price level path to be Pk∗ t=(πmin)tt<k =(πmin)k−1ˆπ t =k =(πmin)k−1ˆπ t  s=k+1 πk∗ st>k Then we can define currency-holdings to be Xk∗ t=¯ xPk∗ tif ˆ Rπk∗ t=1 =Pk∗ tu−1 muCk∗ t−βuCk∗ t+1 πk∗ t+1if ˆ Rπk∗ t>1 We can readily verify that (Ck∗Xk∗Pk∗)is an equilibrium given (ˆ R ¯ M). It is clear that limk→∞ πk∗ t=πmin for all t, and since γ<1,limk→∞ u(Ck∗ t)=∞for all t. It follows too that limk→∞ Uk∗=0. In contrast, for all t,limk→∞ Xk∗ t/Pk∗ t=¯ x, so that real currencyholdings converge to satiation at each date. In this proposition, in the limit, the government is following the Friedman rule (the gross nominal interest rate equals 1) and households are satiated with real balances. But consumption and labor are zero in the limit. Thus, the economy may have extremely poor outcomes even though the central bank (approximately) follows the Friedman rule. C.7 Generalization of Proposition 7 In this subsection, I provide a generalization of Proposition 7 for the model with noninterest-bearing currency. As in that proposition, unique implementation requires the Theoretical Economics 16 (2021) Bounds on price-setting 1011 interest rate rule to be active for inflation rates slightly below target. But this rules out the possibility that households are satiated with currency when inflation is at target. The following proposition deals with this issue by showing that, given any δclose to zero but positive, it is possible to find an interest rate rule that both implements Yreal as a unique consumption–labor equilibrium allocation and uniquely implements a constant gross nominal interest rate (1+δ) in any equilibrium in which inflation is bounded from above. Proposition 11. Suppose δ>0such that (1+δ)β < 1, and suppose β>π min.Consider a nominal framework (R ¯ M) with a time-invariant interest rate rule ˆ Rδsuch that it targets πTAR δ=β(1+δ): ˆ RδπTAR δ=(1+δ) πTAR δ=β(1+δ) Suppose also that interest rate rule is active for inflation rates above or slightly below target, so that the gross pseudo-real interest rate βˆ Rδ(π)/π is strictly increasing for all π∈[β(1+δ/2) ∞). Finally, suppose that fiscal policy ¯ Mtakes the form ¯ Mt=M0 t(1+δ)t Then, in any equilibrium, C∗ t=Yreal for all t≥1, and in any equilibrium with bounded inflation, real currency-holdings satisfy X∗ t/P∗ t=u−1 muYrealδ 1+δ for all t≥1. Proof. We first prove that in any equilibrium, C∗ t=Yreal.Notethatβ>π min and ˆ Rδ(πmin)=1.Hence, βˆ Rδ(πmin)>π min βˆ Rδβ(1+δ/2)<β(1+δ/2) This implies that we can define ˆπδ∈(πminβ(1+δ/2)) so that it satisfies βˆ Rδ(ˆπδ)=ˆπδ βˆ Rδ(π) > π for all πin [πminˆπδ). 1012 Narayana R. Kocherlakota Theoretical Economics 16 (2021) Suppose C∗ t<Yreal at some date t. I show first, by contradiction, that there is some s≥0such that C∗ t+s+1=Yreal. Suppose not. Then for all s≥0, uC∗ t+s+1=β−1ˆ Rδ(πmin)−1πmins+1uC∗ t But this implies that u(C∗ t+s+1)is lower than u(Y real)for ssufficiently large, which is the desired contradiction. Hence, there is some s≥0such that C∗ t+s+1=Yreal and C∗ t+s<Yreal. It follows that π∗ t+s+1=βˆ Rδ(πmin)uC∗ t+s+1/uC∗ t+s <βˆ Rδ(πmin)uYreal/uYreal ≤βˆ Rδ(ˆπδ) =ˆπδ Since βˆ Rδ(π∗ t+s+1)>π ∗ t+s+1, we can conclude that uC∗ t+s+2 π∗ t+s+2 =β ˆ Rδπ∗ t+s+1 π∗ t+s+1−1uYreal π∗ t+s+1 <u Yreal/π∗ t+s+1 and so C∗ t+s+2=Yreal.Hence, π∗ t+s+2=βˆ Rδπ∗ t+s+1 <βˆ Rδ(ˆπδ) =ˆπδ By induction, we can conclude that for all r≥1, C∗ t+s+r=Yreal π∗ t+s+r+1=βˆ Rδπ∗ t+s+r where π∗ t+s+1=βˆ Rδ(πmin). The sequence (π∗ t+s+r)∞ r=1is strictly increasing and converges to ˆπδ. To be an equilibrium, the households’ transversality condition must be satisfied: 0=lim T→∞ ¯ Mt+s+T T  r=1 ˆ Rδπ∗ t+s+r ≥lim T→∞ ¯ Mt+s+T/ˆ Rδ(ˆπδ)T =¯ Mtlim T→∞t/(t +T)(1+δ) ˆ Rδ(ˆπδ)T =∞ Theoretical Economics 16 (2021) Bounds on price-setting 1013 But this is a contradiction: the nominal liabilities are growing too fast to be consistent with an equilibrium in which inflation is bounded from above by ˆπδ. It follows that in any equilibrium, C∗ t=Yreal for all t. I next show that in any equilibrium in which inflation is bounded from above, π1= πTAR δ=β(1+δ). Suppose not. Then there are three cases, depending on the initial level of inflation. In the first case, suppose βˆ Rδ(π1)≤π1<β(1+δ). Then the sequence defined by πt+1=βˆ Rδ(πt) is weakly decreasing and converges to a limit that is smaller than β(1+δ). But this is a violation of the transversality condition. In the second case, suppose π1<βˆ Rδ(π1)and π1<β(1+δ). Then the sequence defined by πt+1=βˆ Rδ(πt) is strictly increasing. It converges to a limit that is necessarily smaller than β(1+δ/2), and again this is a violation of the transversality condition. Finally, suppose π1>β(1+δ).Thenπ1>βˆ Rδ(π1)and the sequence defined by πt+1=βˆ Rδ(πt) is strictly increasing. If it were bounded, then the sequence would converge to a fixed point larger than πTAR δ, but there is no such fixed point (given that βR(π)/π is strictly increasing). We can conclude that inflation πtequals β(1+δ) for all t≥1in any equilibrium with bounded inflation. It follows that in any equilibrium with bounded inflation, the nominal interest rate is constant at ˆ R(β(1+δ)) =(1+δ). Accordingly, in any such equilibrium, real currency-holdings are constant at X∗ t/P∗ t=u−1 muYrealδ 1+δ The restriction to equilibria with bounded inflation in the above proposition is ad hoc. 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