American Economic Association

Recurrent Hyperinflations and Learning Author(s): Albert Marcet and Juan P. Nicolini Source: The American Economic Review, Vol. 93, No. 5 (Dec., 2003), pp. 1476-1498 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/3132138 Accessed: 27/07/2009 06:29

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http://www.jstor.org RecurrentHyperinflations and Learning

By ALBERTMARCET AND JUAN P. NICOLINI*

We use a model of boundedly rational learning to account for the observations of recurrent hyperinflations in the 1980's. In a standard monetary model we replace the assumption of full rational expectations by a formal definition of quasi-rational learning. The model under learning matches some crucial stylized facts observed during the recurrent hyperinflations experienced by several countries in the 1980's remarkably well. We argue that, despite being a small departure from rational expectations, quasi-rational learning does not preclude falsifiability of the model, it does not violate reasonable rationality requirements, and it can be used for policy evaluation. (JEL D83, E17, E31)

The goal of this paper is to develop a model prompting, with a concomitant sensitivity of that accounts for the main features of the hy- outcomes to details of adaptive algorithms."A perinflations of the 1980's and to study the side contributionof the paperis to show with an policy recommendationsthat arise from it. The example that,contrary to Sargent's statement,if model is standard,except for the assumptionof certain rationality requirements are imposed, quasi-rationallearning. Moder macroeconom- learning models can be useful to understand ics has been reluctantto use boundedlyrational real-time transitiondynamics. expectations models to match empirical obser- The long-run relationship between money vations. It is commonly believed that such mod- and prices is a well-understood phenomenon. els are not falsifiable and expectations are not The price level and the nominal quantity of consistent with the model. This view is stated money over real output hold an almost propor- clearly in the following quotationfrom Thomas tional relationship so that the inflation rate is J. Sargent (1993): "... the literature on adaptive essentially equal to the growth rate of money decision processes seems to me to fall far short supply minus the growthrate of output.There is of providing a secure foundation for a good widespread consensus in the profession that theory of real-time transitiondynamics. There successfully stopping inflation involves sub- are problems of arbitrarinessand the need for stantial reductions in money growth rates. On the other hand, long periods of high money growth rates are associated with large seignor- * Marcet:Department of Economics and Business, Uni- collection to finance Trias age required government versitat Pompeu Fabra, C/Ramon Fargas, 23-25, deficits. A about is CREI, and CEPR (e-mail: simple story hyperinflations 08005, Barcelona, Spain, is unable to [email protected]);Nicolini: UniversidadTorcuato Di often told: when the government Tella, C/Miiiones 2177, C1428ATG Buenos Aires, Argen- either reduce its fiscal deficit or finance it tina (e-mail: [email protected]).We thank Tony Braun,Jim through the capital market, high seignorage is Bullard, George Evans, Seppo Honkapohja,Rodi Manuelli, and inflation rates are unavoid- Tom Stacia Harald required high Ramon Marimon, Sargent, Sowerby, able. This is the behind the advice of the Uhlig, Neil Wallace, and Carlos Zarazagafor helpful con- logic versations and Marcelo Delajara and Ignacio Ponce InternationalMonetary Fund (IMF) to countries Ocampo for research assistance. Any errors are our own. experiencinghigh inflationrates. Cross-country Partof this work was done when both authorswere visiting evidence very strongly supportsthis story. Hy- the FederalReserve Bank of Minneapolis.Most of the work have occurred in countries with Marcet was done when CEMFI, Madrid. Re- perinflations by visiting and countries that suc- search supportfrom DGICYT (Spanish Ministryof Science high seignorage, many and Education),CIRIT (Conselleria d'Ensenyament Generali- cessfully stopped inflation did so by eliminat- tat), CREI, the Barcelona Economics Program of CREA, ing the fiscal imbalance that required high ANCT (Argentina, PICT 98-02-03543), and HCM (Euro- of the can seignorage. pean Union) is greatly appreciated.Many proofs this fails when we be found in AppendixC, availablefrom the authorsor in the However, simple story working paper version of this article. closely look at time series of inflation and sei- 1476 VOL.93 NO. 5 MARCETAND NICOLINI:RECURRENT HYPERINFLATIONS AND LEARNING 1477 gnorage for very high-inflationcountries. Coun- facts with models of boundedly rational learn- tries that undergo very rapid price increases ing. Some exceptionsare Evans and Honkapohja typically exhibit periods of relatively high but (1993), Allan Timmermann(1993, 1996), and stable inflation rates, followed by a sudden ex- Jasmina Arifovic et al. (1997). Some papers plosion in the rate of inflation; this often hap- looking at policy implications are Hee-Taik pens without any importantchange in the level Chung (1990) and Evans et al. (2001). How- of seignorage. We observe inflation rates mul- ever, with the partial exception of Evans and tiplying by 8 or 10 in a couple of months while Honkapohja(see our discussion following Def- seignorage remains roughly the same or even inition 3), none of them formally addressedthe decreases. This could challenge the validity of critique to boundedly rational models that is the IMF advice to hyperinflationarycountries to commonplace in today's macro literature and decrease their seignorage. that is clearly stated in the above quote from In this paper we develop a model that ac- Sargent.This critique says that using models of counts for this and other crucial observations boundedly rational learning would entail prob- that occurred during the hyperinflationsof the lems similar to those found in models of adap- 1980's. These episodes involve very high infla- tive expectationsof the prerational-expectations tion rates (for instance, inflationin Argentinain era, namely: (i) too many degrees of freedom June 1989 peaked at 200 percent a month) and are available to the economist, so the model is all we know about the welfare effects of infla- not falsifiable; (ii) agents' expectations are in- tion suggest that they are very costly. consistent with the model, so rational agents Sargent and Neil Wallace (1987) explained would eventually abandon their ad hoc expec- these hyperinflationsas bubble equilibria.Their tations;and (iii) the model does not predicthow model generates a standardLaffer curve with expectation formation will change if there is a two stationaryrational expectations equilibria; change in policy. hyperinflationscould occur as speculative equi- We addressthese criticisms by restrictingthe libria converging to the high-inflation steady learningmechanisms to produce good forecasts state. Their paper explains how inflation can within the model. We only consider learning grow even though seignorage is stable; but it mechanismsthat producesmall departuresfrom fails to explain other facts observed in the hy- rationality within the model, in a way that is perinflationaryepisodes. Our work builds upon precisely defined in the paper.We show that the Sargent and Wallace's by introducinglearning; model has empirical content and that expecta- we show that, with this modification,the model tions are endogenous to policy.2 matches observationsmuch better.Our model is Quite a few papers have presented models consistent with the very high hyperinflations, that explain some of the facts we consider, their recurrence, the fact that exchange rate among others, Zvi Eckstein and Leonardo rules temporarily stop hyperinflations, the Leiderman (1992) and Benjamin Bental and cross-country correlation of inflation and sei- Eckstein (1997) explain the very large inflation gnorage, and the lack of serial correlation of rates in Israel with an ever-increasing Laffer seignorage and inflation in hyperinflationary curve, and Carlos Zarazaga (1993) develops a countries. model of endogenous seignorage. These papers The last decade has witnessed a renewed account for some, but not all, the facts we interest in learning models in macroeconomics, describe in the paper. ,Their stories could be mostly focusing on issues of convergence to combined with the story of the currentpaper. rational expectations.1This literaturehas made The paper is organized as follows. Section I enormous progress, and convergence of learn- presentsthe stylized facts and provides support- ing models to rationalexpectations can now be ing evidence. Section II presents the model studied in very general setups. But few attempts and characterizesrational expectations equilib- have been made to explain observed economic ria. Section III discusses the lower bounds in

2 Recent literatureimposing consistency requirementsin See Sargent (1993), Ramon Marimon (1997), and learningmodels are Evans and Honkapohja(1993); Morde- George Evans and Seppo Honkapohja (1999, 2001) for cai Kurz (1994); Drew Fudenberg and David K. Levine reviews. (1995); and Cars Hommes and GerhardSorger (1998). 1478 THEAMERICAN ECONOMIC REVIEW DECEMBER2003 rationality in a general setup. Section IV dis- Argentina cusses the behavior of the model under the Moanlfy I1Nflao Rate (Tnlas) lower bounds on rationality. The paper ends 0.284 86 88 2 with some concluding remarks. 84 86 88 90 92

Bolivia I. Evidenceon RecurrentHyperinflations Monthly Illatan Rate (in logs) 0.4

A numberof countries, including Argentina, 01 ....- Bolivia, Brazil, and Peru experiencedduring the 81 82 83 84 85 86 87 1980's the highest average inflation rates of Brazil their history. While the durationand severity of MonthlyIljafn Rate n Lbs) the hyperinflationsand the policy experiments 0.2 differ there are several 0.1 substantially, stylized 0 facts that are common to those experiences 85 86 87 88 89 90 91 92 (and, to some extent, to those of some European countries after the first world war, and those of Peru Monthly Rate East European countries after the end of the Ilktion (in logs) 0.6 - cold war). These stylized facts are: 0.4 0.2 ;...... 1. Recurrence of hyperinflationaryepisodes. 85 86 87 88 89 90 91 Time series show relatively long periods of FIGURE 1. MONTHLY INFLATION RATE (IN LOGS) moderate and steady inflation, and a few short periods of extremely high inflation rates. 2. Exchange rate rules (ERR) stop hyperinfla- Our summaryof stylized facts should be un- tions. But often an EER only lowers inflation controverted.3Facts 1 and 2 are clearly shown temporarily,and new hyperinflationseven- in Figure 1, which presents data on the infla- tually occur. tionary experiences of Argentina,Bolivia, Bra- 3. During a hyperinflation,seignorage and in- zil, and Peru in the 1980's.4 Periods when an flation are not highly correlated. explicit fixed ERR was in place are indicatedby 4. Average inflation and seignorage are shaded areas. The end of the shading indicates strongly correlatedacross countries. Hyper- the date in which the ERR was explicitly inflations only occur in countries where sei- abandoned. gnorage is high on average. Fact 3 has been well documented in the lit- erature and it has motivated quite a bit of re- Facts 2 and 4 can be combined to state the search including, for example, that of Sargent following observationon monetarypolicy: sta- and Wallace (1987). Hyperinflations did not bilization plans based on ERR-"heterodox" happen together with high peaks in seignorage; policy-that do not permanentlyreduce average very differentlevels of seignorage were present seignorage, may be successful in substantially duringdifferent hyperinflations in a given coun- reducing the inflation rate only in the short try and, in some instances, seignorage even run. Some stabilization plans not only relied decreased while a hyperinflation was taking on the fixing of the exchange rate but also place.5 permanently reduced the deficit-"orthodox" policy-and the need for seignorage. It is now relatively well accepted that this combination of both orthodox and heterodox ingredients has been successful at stopping hyperinfla- 3 See Michael Bruno et al. (1988) and (1991). 4 InternationalFi- tions To our ours is Inflation rates were computed from permanently. knowledge, nancial Statistics consumer price indices. that 5 the first economic model satisfactorily See a plot of inflation and seignorage for Argentinain explains the above facts and is consistent with the 1980's in the working paper version of this paper this policy recommendation. (Marcet and Nicolini, 1998). VOL. 93 NO. 5 MARCETAND NICOLINI:RECURRENT HYPERINFLATIONS AND LEARNING 1479

II. The Model pegs the nominal exchange rate by buying or selling foreign reserves at an exchange rate e, A. Money Demand and Money Supply satisfying

The in this subsection are stan- assumptions - dard.The model consists of a pf =P portfolio equation -1 et-1 for the demand of real money balances, a gov- ernment budget constraintrelating money cre- where 3 is the targeted inflationrate, and Pf is ation and changes in reserves, and a rule for the price level abroad. Assuming full mobility establishing fixed exchange rates.6 of goods, purchasingpower parity implies The demand for real balances is given by P pe pe (3) -=3 Pt--1I - 3 if - (1)(1) P,Pt Pt yp>0 =0 otherwise and the targetedinflation rate is achieved. In the case that targeted inflation (3 is the same as where y, 4 > 0 are parameters,Pt, Md are price foreign inflation, the government announces a level and nominal demand of money; Pt+1 is fixed exchange rate. Otherwise, a crawling peg the price level that agents expect for next is followed. period. Under ERR, equilibriumprice level is deter- Money supply is driven by the need to fi- mined by (3). Given this price level and an nance seignorage. On the other hand, govern- expectations hypothesis, (1) determines money ment's concern about currentlevels of inflation demand.In general,this money demandwill not prompts the adoption of ERR when inflation match money supply as determinedby (2). As it gets out of hand or to restore equilibrium. is standardin fixed exchange rate models, in- If no ERR is in place at t, the government ternationalreserves (Rt) adjustso the right level budget constraintis given by of money balances is achieved. Thus, instead of (2) money supply is now given by: (2) Mt = M,- + dtP,. (4) Mt = M,t + ddtP + et(Rt- Rt 1). Seignorage is given by an exogenous indepen- dently and identically distributed (i.i.d.) sto- Finally, we impose the rule that government chastic process {dt}to and it is the only source acts to satisfy of uncertaintyin the model.7 Equations(1) and (2) plus a hypothesis of expectations forma- Pt tion, determine the equilibrium values for (5) {Mr, P}t =o in periods of floating exchange Pt-1 rates. If an ERR is in place at t, the government where ,u is the maximum inflation tolerated. ERR is only imposed in periods when inflation would otherwise violate this bound or in periods 6 where no level Appendix A shows that the following equationscan be positive price clears the market rationalized as the equilibrium conditions of an overlap- if Rt = R_ 1. ping generations (OLG) monetary model of a small open Our model makes the implicit assumption economy. that ERR can be enforced. In 7 The i.i.d. is made for always fact, gov- assumption simplicity. For exam- ernments run out of ple, if d, were a Markov process, Pte+ would have to may foreign reserves, and depend on dt for the learning scheme to satisfy the lower they may be unable to enforce ERR for a suf- bounds on rationality,and agents would have to learn about ficiently long period. Hence, we are making the at least two parameters.It would be interestingto generalize implicit that the con- the model to this assumption nonnegativity case, especially since seignorage is, in- straint on reserves is never deed, serially correlatedin the data. We that the foreign binding. conjecture Since we will choose the main results of the paper would go through with serially target inflation rate correlatedseignorage, but some analytical results would be ,3 to be the lower stationaryrational expecta- harderto prove. tions equilibriumsteady-state inflation, the loss 1480 THEAMERICAN ECONOMIC REVIEW DECEMBER2003 of reserves is small in our simulations and it is real governments would be the existence of likely to be small for most parametervalues. institutions that can implement this measure Modeling reserve accumulationformally is un- quickly, while lowering government expendi- likely to change our main results, but it opens tures or increasingtaxes often takes a long time. up a host of interestingissues. For example, the An importantpolicy decision is how long to government may run out of reserves during a maintain the ERR. Obviously, the longer the hyperinflation, so that "orthodox" measures ERR is maintained,the closer expected inflation cannot be avoided, a feature that is consistent will be to (3. In our simulations, we hold the with our model. Alternatively,by increasingthe ERR until expected inflation is close to 3 in a length of the ERR after a hyperinflationthe sense to be made precise below. monetary authority could accumulate reserves In summary, the government in our model since the real value of the money stock is in- sets money supply to finance exogenous sei- creased after the stabilization.8 gnorage;if inflationis too high, the government We have modeled policy in this way because establishes ERR. The parametersdetermining it mimics the broadfeatures of policies followed government policy are /3, 3U, and the distri- by South-Americancountries during the 1980's. bution of d,. The issue of why these countries followed this kind of policy is not addressedformally in this B. Rational Expectations paper, but we can advance three possible justi- fications for using this rule within our model. If we assume that agents form expectations First, the fact that ERR has been established rationally, the model is very similar to that of only after some periods of high inflation is Sargent and Wallace (1987) (henceforth,SW). justified because then the value of foreign re- As long as seignorageis not too high, the model serves is high, and a large part of the domestic has two stationaryequilibria with constant ex- money can be backed with existing reserves.9 pected inflation levels (called low- and high- Second, in principle, any reduction in the gov- inflationequilibria), and a continuumof bubble ernment deficit of et(R, _ Rt_1) units would equilibria that converge to the high-inflation also fix the inflation to (3 in periods of ERR. equilibrium.10 In fact, the reduction in seignorage that is The main motivationbehind the work of SW needed to achieve an inflationequal to 3 is often was to explain fact 3 in Section I as rational quite moderate, which raises the issue of why bubble equilibria.1l Their original model does governmentshave used ERR instead of lower- not allow for recurrenceof hyperinflations(fact ing the fiscal deficit (and seignorage) suffi- 1), but the work by Funke et al. (1994) shows ciently. One possible answer is that the exact that recurrencecan be explained by introducing value of et(Rt - R_-), can only be inferred a sunspot that turnsrational bubbles on and off. from knowledge of the true model and all the Even if one accepts rational sunspots as an parameter values, including those that deter- explanation, fact 1 is not matched quantita- mine the (boundedly rational) expectations tively: for reasonable parameter values, the Ptl, and all the shocks. By contrast, an ERR magnitude of the hyperinflationsthat can be can be implementedonly with knowledge of the generatedwith this model is very small.12Fact foreign price level and the policy parameters(8, 4 is contradicted:the long-run inflation rate in (U3). A third advantageof establishingERR for any rational bubble equilibriumis lower when seignorage is higher, so the model under ratio-

8 For instance,Central Bank reserves grew, in Argentina, 'o our model in from 1991 (year in which the Convertibility Plan was We reproducesome of these results for launched)to 1994 from 500 million dollars to more than 12 Appendix 2, available from the authors or in the working billion. paper version of the paper. existence of 9 This interpretationwould suggest that the burst in in- 1 There has been some work testing the of the flation at the beginning of 1991 in Argentinawas crucial for rationalbubbles in the Germanhyperinflation 1920's. the success of the ConvertibilityPlan launched in April of A summary of the literature and a test of bubble versus can be found in the same year, because it substantiallyreduced the value of stationary equilibria in the SW model the money stock to a point where, at a one dollar = one peso SelahattinImrohoroglu (1993). 12 discussion of 3 in exchange rate, the governmentcould back the whole money This is documented in our Figure stock. Section IV, subsection E. VOL. 93 NO. 5 MARCETAND NICOLINI:RECURRENT HYPERINFLATIONS AND LEARNING 1481 nal expectations (RE) predicts that hyperinfla- icism is hyperbolized by the sentence: "Any tions are less severe in countries with high economic model can match any observationby seignorage. choosing expectations appropriately";the sec- The papers of Maurice Obstfeld and Kenneth ond criticism is typified by the sentence "Eco- Rogoff (1983) and Nicolini (1996) maintain nomic agents do not make systematicmistakes." rational expectations and introduce ERR that RE becamethe way to overcomethese criticisms. goes into effect if inflation goes beyond a cer- In this paper we use a boundedly rational tain level and, therefore, these papers can be learning model to explain stylized facts, so a used to address fact 2. Their results show that naturalquestion is: are we slipping into a use of just the threat of convertibility eliminates bub- learningmodels that is as objectionableas, say, ble equilibria altogether and that the ERR, un- adaptive expectations? der rationalexpectations equilibria,never takes The term boundedlyrational learning (which, place. Thus, once ERR is a credible threat the in this paper, we use as synonymous with the rationalexpectations equilibrium is inconsistent term learning) is used to denote learningmech- with the existence of hyperinflations.But this anisms that place upper bounds on rationality. was certainly a credible threat in these coun- For example, agents are assumed not to know tries, since ERR did take place. the exact economic model or to have bounded Marcet and Sargent (1989) studied stability memory. But this admits too many models of of rational expectations equilibria in the SW learning.Indeed, once we rule out RE, anything model underleast-squares learning. They found can be a boundedly rational learning scheme that the low-inflation equilibriumis locally sta- and we could be falling back into old mistakes ble and the high-inflationequilibrium is always and the "wildernessof irrationality."14 unstable. Taken literally, these results would Our approachis to allow for only small de- say that bubble equilibriacannot be learned by viations from rationality,both along the transi- agents. Therefore, none of the above facts is tion and asymptotically. Given an economic appropriatelymatched if we restrict our atten- model we only admit learning mechanisms tion to rational expectations equilibria that are that satisfy certain lower bounds on rational- stable under learning.13 ity within this model. In Section IV we will In the next section we propose several criteria show how this small departurefrom rational- to assess models with quasi-rational learning ity generates equilibria in the model of Sec- and to addressthe criticisms of learningmodels tion II that are quite different from RE, commonly found in the literature. precisely in the direction of improving the match of empirical observations. Il. Learningand Lower Boundson Rationality We now set up a general framework and define lower bounds that we place on rational- Before the "rationalexpectations (RE) revo- ity. Assume that an economic model satisfies lution," economic agents' expectations were specified in macroeconomics according to ad (6) x, = g(x,_,,x+ ,, t, ,q) hoc assumptions; one popular alternative was "adaptive expectations." We explained in the where g is a function determined by market introductionthat this was criticized because it equilibriumand agents' behavior,xt contains all leads to: (i) too many degrees of freedom, (ii) the variables in the economy, xe+I is agents' irrational expectations, and (iii) expectations expectationof the futurevalue of x, g, is an ex- that are exogenous to the model. The first crit- ogenous shock, and r is a vector of parameters,

13 Marcet and Sargent (1989) is a special case of the 14 It might seem that Bayesian learning is a way out of when present paper uncertaintyis eliminated, fu is arbi- this dilemma, but the literaturehas described several para- and forecast trarilyhigh, agents Pi by regressingit on Pi i. doxes and shortcomingsof this approach.See, for example, These authors noted that if inflation goes beyond the high MargaretM. Bray and David M. Kreps (1987), Marimon state it steady may enter an unstable region where inflation (1997), and David Easley and Aldo Rustichini (1999). tends to grow withoutbound. This featureof the model with '5 Marimon (1997) and Easley and Rustichini (1999) constitutes the core of the learning dynamics in the current also argue that learningcan be used for more than a stability paper. criterion. 1482 THE AMERICANECONOMIC REVIEW DECEMBER2003 including the parametersof governmentpolicy where Et is the true conditional expectation and the parametersthat govern the distribution under the learning model and P is probability. of t. For example, in our model, xt is inflation The first lower bound on rationalitywe pro- and real balances, t is seignorage, the function pose is: g is given by the demand for money (1), the governmentbudget constraint(2), and the ERR Definition 1 AsymptoticRationality (AR): The rule, while the vector of parametersr includes expectationsgiven by (z, f, It) satisfy AR in the > y (,,3U,9 and the parametersof the distribu- model (g, r) if, for all e 0, tion of seignorage. Assume that agents' expectationsare given by rT --- 1 as T ---> o.

(7) x + 1= z(l, (f), x,) This requires the perceived forecast to be asymptotically at least as good as the forecast where P8t(gi)is a vectorof statisticsinferred from from the conditional expectation in terms of past data and z is the forecastfunction. The sta- sample mean square prediction error. In this tistics p8are generatedby a learningmechanismf case, agents would not have any incentive to and learningparameters ,t accordingto change their learning scheme after they have been using it for a sufficiently long time. (8) jlt(/-)f()ft- = (10),xt, It). AR can be viewed as a minimal requirement in the sense that it only rules out behavior that The learning mechanism f dictates how new is inconsistentforever. It rules out, for example, informationon xt is incorporatedinto the statis- learning mechanisms where a relevant state tics p. The learning parameters JLgovern, for variable is excluded from the forecasting rule z example, the weight that is given to recent in- (this feature would exclude adaptive expecta- formation.For now, (z,f, Ji) are unrelatedto the tions, for example, if dt were serially corre- true model (g, 17),but later in this section we lated). It is satisfied by least-squares learning will define bounds on rationalitythat amountto mechanisms in models where this mechanism imposing restrictions on the space of (z, f, ,j) converges to RE and certaincontinuity assump- given a model (g, r). tions are satisfied.16Similar concepts of consis- In the context of our model in Section II, the tency can be found in the literature.'7 function z will be defined as However, AR admits learning mechanisms that generate very bad forecasts along the tran- (9) pe+l = tPt, sition for very long periods. For example, ordi- nary least squares (OLS) in a model with where 3, is expected inflation, estimated some- recurrenthyperinflations would generate very how from past data. bad forecasts every time a hyperinflationstarts, Equations(6), (7), and (8) determinethe equi- because least-squares learning gives less and libriumsequence for given learningparameters j,. less importance to recent events as time goes Obviously,the process for xt dependson the pa- by, so it would take longer and longer for agents rametersjp. This dependencewill be left implicit to realize that a hyperinflationis starting.Even in most of the paper,and we will writext only if worse, under OLS the agents' expectations we want to make the dependenceexplicit. would adjust more slowly for each subsequent Let ,TrT be the probability that the perceived outburstin inflation. errorsin a sampleof T periodswill be within e > 16 "rational 0 of the conditionalexpectation error. Formally: Perhapssurprisingly, AR excludes many equi- libria"in the terminologyof Kurz (1994), which allows for T agents to make systematicmistakes forever, as long as these I mistakesare not contemplatedin the priordistribution. - 17 This was in the liter- (10) TE,T-_ : Ixt+I xte+ l1 requirement implicitly imposed t=1 ature on stability of RE under learning, where the use of least squareswas often justified because of its optimality in the limit. Also, AR is related to the (e - 8) consistency of Fudenbergand Levine (1995), where agents in a game are < E(xti+l)l2i - + E requiredto only accept small deviations from best response t=l asymptotically. VOL 93 NO. S MARCETAND NICOUNI: RECURRENTHYPERINFLATIONS AND LEARNING 1483

To restrict learning mechanisms so that they Thus, if IC is satisfied, agents are doing al- generate good forecasts along the transitionwe most as well as possible within the learning impose the next two lower bounds. mechanism specified after T periods, so that they are likely to stay with /. Definition 2 Epsilon-Delta Rationality (EDR): IC is, in general, more restrictive than AR, The expectations given by (z,f, ,u)satisfy EDR since it requires that good forecasts are gener- for (E, 8, T) in the model (g, r]) if: ated along the transition,not only at the limit. As in the case of EDR, it only makes sense to > 1 - ',TET 8. study IC in the context of "moderatelyhigh" T. The first two bounds compare the perfor- If EDR is satisfied for small E, 8 > 0, agents mance of the learning mechanism used by are unlikely to switch to another learning agents relative to an external agent who knows scheme after period T, even if they were told the best prediction that can be computed from "the whole truth."'8 knowledge of (f, p,, z, g, rq). The bound IC, It is only interestingto study EDR for "mod- instead, compares the learningmechanism with erately high" values of T. If T is too low the forecasts that use the same family of mecha- sample means have no chance to settle down. If nisms f but are allowed to pick alternativepa- T is large enough and AR is satisfied, EDR is rameter values ,I. This last bound contains also satisfied. The precise empirical application some of the intuitionof rationalexpectations, in that the researcherhas in mind should suggest the sense of looking for an approximatefixed an interesting value for T. For example, in our point in which agents' expectations minimize application below, we choose T = 10 years, the errors within the mechanismf. Notice that which is the length of the hyperinflationarype- this restrictionwill, in general,imply that agents riod in many of the countries studied. under different policy environmentsuse differ- AR is unambiguouslysatisfied (there is a yes ent learning parameters jt, so that the learning or no answer), but EDR can only be satisfied in parameterthat satisfies IC is endogenous to the a quantitativeway, for certain e and 8. model and to governmentpolicy. For example, The next bound on rationalityrequires agents in our model, agents in high seignorage coun- to use learning parameters ,u that are nearly tries (say, Argentina in the 1980's) will use a optimal within the learning mechanismf. De- differentlearning parameter from agents in low note by At(3,, /j') the forecast producedby the seignorage countries (say, Switzerland). These learningparameter ,u' when all agents are using Definitions can be readily generalized to more the parameter,u. Formally, complicated models or to objective functions other than the average prediction error. At(IL,' )= Af(t-l (A, W'), xt, W) Imposing these lower bounds on rationalityis our way of relaxing rationalexpectations while Definition 3 Internal Consistency (IC): Given maintainingthe requirementthat agents do not (g, 7R),the expectationsgiven by (z,f, ,l) satisfy make mistakes forever. Agents have a certain IC for (E, T) if amount of forward-looking capabilities under Definitions 2 and 3 but far less than under T \ rationalexpectations. (11) E TE |X+- t(P,), XP)l|2 Rational expectations can be interpretedas i t=1 / imposing extreme versions of the second and third bounds. Obviously, RE satisfies AR. It would appearthat requiringEDR for all e, 8 > -< - 1 0, and all T is the same as rational min E T it=Ix - z(P3,(, .'), X)2 + E. imposing but a careful A' Tt=l / expectations, proof should be worked out. Also, if the RE equilibrium(REE) is recursive,if the appropriatestate variablesare 18 Bray and Nathan E. Savin (1986) study whether the learning model rejects the hypothesis of serially uncorre- lated errors prediction by assuming that agents run a Durbin '9 Evans and Honkapohja(1993) developed a very sim- and Watson test. That exercise carries the flavor of EDR. ilar criterion in a different context. 1484 THEAMERICAN ECONOMIC REVIEW DECEMBER2003 includedin z, if z is a dense class of functions(for t-i example,polynomials or splines),imposing IC for -I any e, T is the same as rationalexpectations. i=Oi=n aj Pt-i,_

IV. Learning Equilibrium so that past informationis now a weighted av- erage of past inflations, where the past is dis- In this section, we propose a learning mech- counted at a geometric rate.22 anism f that combines least-squares learning Least squares gives equal weight to all past with trackingand we show that, in the model of observations,while trackinggives more impor- Section II, it satisfies the three lower bounds on tance to recent events. Trackingproduces better rationalitydefined in the previous section. forecasts when there is a sudden change in the environmentbut it does not converge. OLS is A. The Learning Mechanism known to be a consistent estimatorin stationary setups but it reacts slowly to sudden changes. In the model of Section II with expectations Both alternativesare likely to fail the lower given by (9), we assume that the learningmech- bounds on rationalityof Section III in a model anism is given by the stochastic approximation that replicates fact 1, where periods of stability algorithm are followed by hyperinflations.Tracking per- forms poorly in periods of stability because perceived inflation is affected by small shocks = +I (12) ,- - even though, in truth, the shocks are i.i.d. and (12) P, P,-, (Pt,-PPt-- -- ) at Pt-2 they should not affect today's expected infla- tion: formally, tracking does not converge to for given 30. That is, perceived inflation 3t is RE and it does not even satisfy AR, while updated by a term that depends on the last OLS has a chance of converging and satisfy- prediction error20 weighted by the gain se- ing AR. quence l/at. Equation (12) together with the On the other hand, least squares does not evolution of the gains l/ca determinesthe learn- generate "good" forecasts along a hyperinfla- ing mechanismf in (8). tion, because it will be extremely slow in adapt- One commonassumption for the gain sequence ing to the rapidly changing inflation level. is at = at 1 + 1, for ao = 1. In this case, a = During hyperinflations"tracking" performs bet- t, and simple algebrashows that (with go = 0) ter. Least squares does not satisfy EDR or IC and its performanceis likely to worsen as there are more successive hyperinflations. 1 P We will a mechanism that + = t specify learning Pi= 1 I mixes both alternatives:it will use OLS in sta- ble periods and it will switch to "tracking"when So, in this case, perceived inflation is just the some instability is detected. This amounts to sample mean of past inflationsor, equivalently, assuming that agents use an endogenous gain it is the OLS estimatorof the mean of inflation. sequence such that, as long as agents don't Anothercommon assumptionfor the gain se- make large predictionerrors, at follows a least- where a quence is at = & > 1. These have been termed squares rule, but in periods large pre- fixed "tracking"or "constantgain" algorithms.21 In this diction error is detected, at becomes a case, perceivedinflation satisfies (with Po = 0)

20 22 to As usual in models of learning, we make the conve- In this simple model "tracking"is equivalent adap- a In a more model nient assumptionthat the last observationused to formulate tive expectations with delay. general from and it expectationsis dated at t - 1. Includingtoday's inflationin trackingis different adaptive expectations gen- 3, would make it even easier for the learning scheme to erates betterforecasts. For example, if seignorage is autore- inflationwould have to satisfy the lower bounds and to match the stylized facts, and gressive of order 1, expected depend to of the lower it would not change the dynamics of the model. on current seignorage in order satisfy any that would be fun- 21 Chung (1990), Evans and Honkapohja (1993), and bounds on rationality.In case, tracking from Sargent (1993) also discuss trackingalgorithms. damentallydifferent adaptive expectations. VOL. 93 NO. 5 MARCETAND NICOLINI:RECURRENT HYPERINFLATIONS AND LEARNING 1485 positive value -a > 1 as in "tracking."23For- Pt - mally, the gain sequence follows

Pt- I

= (13) a, at-1 + if< Pt-2

= a otherwise.

The learningmechanism is the same whetheror not ERR is enforced in a given period. The conventional wisdom that the importanceof an ERR is the effect it has on expectations is con- sistent with the model, since the exchange rate rule has an impact on expectations by its effect on the currentprice level and by setting the gain factor to its base value a. The learning mechanismf is fully described by equations (12) and (13). The learningparam- = Pt eters t - (v, a) and the statistics B, (j3, at). Pt-i

B. Learning and Stylized Facts

The variableswe need to solve for are {P,, t,, a,}. Simple algebra implies

Pt (14) = H(3,, 1 -,, d,) Pt- 1 where24

(15) H(13,, t _,, d,) - 1I- yl,_ 'Yt- high E(dt) r = u 1- yft,- dt,/ ifO< 1-y f,d,/- 4 < and 1 - > 0 FIGURE2. ACTUALINFLATION AS A FUNCTIONOF -y,- PERCEIVED INFLATION =p otherwise.

23 and define a of Evans and also the Equations(12), (13), (14) system Garey Ramey (1998) analyze prop- second-order difference erties of a learningmechanism that responds endogenously stochastic, equations. to the performanceof the predictionswithin the model and Characterizingthe solution analytically is un- within the realization. feasible since the is nonlinear. 24 system highly Notice that the second part of this equation, where We now provide some intuition on the be- ERR will prevail, applies if one of the following (mutually cases occurs: havior of inflation. Let h(3, d) = H(3, 3, d). exclusive) - - Case (i): 1 - y,t,- 1 < 0, which implies M,_ = 0, so Notice that if t, ,t_-, then P,/P,_ 1 h(3t, the budget constraint of the government is incompatible d), so that the graph of h in Figure 2 provides with the demand for real balances unless reserves adjust. an approximationto the actual inflation as a Case 1 - - < - (ii): yl, d/o 0 and 1 y3p,- > 0 so function of inflation and it can only a level clears the and perceived be negative price market, used to describe the 1 - y/3,- approximatedynamics of Case (iii): None of the above and > the model. 1- y,dt,- - 4/ p, such that the market generates a level of inflation The first graph correspondsto a low d,. The to the if reserves do not unacceptable government adjust. low rational expectations equilibrium 1"3E is 1486 THE AMERICANECONOMIC REVIEW DECEMBER2003 locally stable under least squares learning.25 a hyperinflationto occur even if inflation has The horizontal axis can be split into the inter- been stable for a while. Thus, a country with a vals S, U, and ERR. high average seignorage tends to have hyperin- If /, E S, actual inflationis on averagecloser flationary episodes more often, and fact 4 is to /,E than perceived inflation and the learning consistent with the model. mechanism pushes perceived inflation towards /RE. Roughly speaking, S is the stability set of C. AsymptoticRationality (AR) perceived inflation. On the other hand, if per- ceived inflation is in U, actual inflation is on To prove convergence to RE and that AR is averagehigher than P,, perceived inflationtends satisfied we need: to increase and a hyperinflation is likely to occur. Then, when the set ERR is reached, a ASSUMPTION 1: The support of d, is the fixed exchange rule is established and inflation interval [K-, K+], where K- > 0, K+ < k. is sent back to S. The economy may end up in the unstable set U due to a numberof reasons: ASSUMPTION 2: d, has a continuous density - > 0. a few high shocks to seignorage when 1/a, is fd, and fd(K+) not yet close to zero, initially high perceived inflation, the second-order dynamics adding Letting S(/3) - E(h(3, d,)), Appendix C momentumto increasing inflation, etc. [available from the authors,and in the working If a shock to inflationoccurs agents are likely paperversion of the paper(Marcet and Nicolini, to switch to tracking and set 1/a, = l/a, per- 1998)] proves that ceived inflationwill then be more heavily influ- enced by actual inflationand it is more likely to * S is increasing and convex; /u = - end up in U than under pure OLS. Hyperinfla- * as -> oo,S has an asymptote at 3 (1 tions promptagents to switch to trackingand to K+l)lIy; pay more attention to recent observations;this * S has at most two fixed points 3RE, /E; in turn makes hyperinflations more likely to * /iE, 82E are the stationary rational expecta- occur and predictionswith trackingbetter, thus tions equilibria; reinforcingthe switch to tracking in periods of * two fixed points exist iffd is close enough to instability.Only if 1/ao is very small relative to zero; the variance of inflation and if initial inflation * no fixed point exists iffd is large enough; startsout in S (and v is large enough), hyperin- * as we consider larger distributionsfor d, (as closer flations are impossible. f, shifts to the right) 3R, 32E get This intuition suggests that the model is con- together.Therefore, for largerd, the stable set sistent with stylized fact 1, since a number of S shrinksand /IE} is closerto the unstableset U; hyperinflationsmay occur in the economy be- * if two fixed points exist, least-squareslearn- fore it settles down. Also, it is clear that an ERR ing converges to /3E a.s. will end each hyperinflationtemporarily, so that fact 2 is foundin this model.Also, once /t is in the PROPOSITION1: In addition to Assumptions set U, inflationis likely to grow even if seignorage 1-2, assume that average seignorage and its does not, which is consistentwith fact 3. variance are low enough for two stationary To analyze fact 4, consider the second graph REE to exist, that ,3 E S (targeted inflation of Figure 2, for a high d,. Now, the unstable set belongs to the stabilityset) and that a and v are U is much larger. Furthermore,U is "danger- large enough.26Then 1, --> 1E a.s. and Asymp- ously" close to the rational expectations equi- totic Rationalityobtains. librium P3E where the economy tends to live, and it is likely for the model to end up in U and

26 & The assumptionon can be interpretedas saying that 25 This discussion assumes that S(j3) is close to h(3, d,), convergence occurs if the importancegiven to recent news is needed in order to which is approximatelycorrect if d, is close to its expecta- is never too high. This assumption of inflation in the first of the tion. The proofs of stability of least-squareslearning and of obtain a lower bound part on inflation can also be obtained for the propertiesof S are in an Appendix available from the proof. A lower bound in reasonable authorsand in the working paper version of the paper. unrestricteda by changing the model ways. VOL. 93 NO. 5 MARCETAND NICOLNI: RECURRENTHYPERINFLATIONS AND LEARNING 1487

PROOF: (12), and the second equality follows from the The theorem holds for fact that an ERR was establishedat t - 1 so that P,-l/P-_2 = p. Now, using t,_ < '-1, a, 1 - tyj a, d, KK and (16) we have (16) x > K - Y -_ _ d,d,e y K-K > O (3U yi (> a Y-1) + and v> - 1 1K-1 which, together with (17), that min I, implies P/ P,_ > 1. In case (iii), the condition on Pt_ 1 and simple In order to show that the learning mechanism algebra imply stays in the OLS form in all periods t > a, we first show that inflation is bounded below. For P, di-/d K-/1 each t and each realization, three cases are only P, 1- 't ,-d,/l4 1 - K-/ possible: Case (i): an ERR is activated at t, (ii): an ERR is not activated at t and 1 - y,t_ 1 - d,_ 1/+ 0, and (iii): an ERR is not activated at Therefore we find the lower bound P/P,_ I - = t and 1 - 3,-_ - d,_ l/ > 0. min[l, (K-l/~(1 K-~l), (] pL > 0. Since Notice that in cases (ii) and (iii) the first ,t is an average of past inflations we also have branchof (15) applies and we have 3,_ , < y- . ,t > pL for t > a large enough such that the Note also that 3, is a weighted average of past effect of the initial condition has disappeared. - inflations and pU is an upper bound of infla- For any v > IpU/3L 11we clearly have tion so that Pt < pU. We now find a pL > 0 such |(P,_ /P,_2 - f,t - )/3t_ < v with probability that p3 > pL for all t. one for all t > a, then c, = a,_- + 1 for all t In case (i), inflation is equal to 3. large enough and the learning mechanism stays In case (ii) in the OLS form. Now, let Cs {o E f : 3t(o) E S i.o.}, P E (17) CU - {( C : Pt(o) E U i.o.}, and CER Pt- I1 {w E fl : 3t(o) E ERR i.o.} we want to ar- gue that P(Cs) = 1. Clearly, any realization 1 - y7/,- o belongs to either CS, CU, or C . Consider a o E CERR;for any t such that E ERR, - P3,(o) I) dtl, an ERR is then enforced for the 1-'_ ' at Pt- 2- ' sufficientlylong beliefs to go back to the stable set, so that 3,t+j(o) = (3 E S for some j; therefore, it is clear that also w E Cs. Therefore CERRC CS. Now consider CoE CU, the theorem of Lennart a (P - Pt-i) + d,l/ Ljung (1977) implies that the differentialequa- 1 - = 1 - yo3, , 1 tion / S(p) - p in Appendix C (available from the authorsor in the workingpaper version where the first equality follows from (15) and of this article) governs the dynamics of 3,, therefore for t large enough inflation tends to grow and eventually goes into ERR, which would also C C CS. Therefore co E Cs For example, assuming that the government has the objec- imply tive of avoiding deflation and it achieves this by activating for all co, and 3, E S i.o. with probabilityone. an ERR and insures that P/P,_ i - 1 at the same time that In Appendix C we apply the o.d.e. approachfor reserves increase. A lower bound in v can be interpretedas convergence of schemes in that do not learning dynamic saying agents easily switch to tracking;a lower models to show that this that con- bound is necessary because, if v is too small, even if 3, implies t, to (3E almost is very close to O3E, it will eventually happen that verges surely. 1 The rest of the proof simply shows that, if the P-, > v, then a, = a, perceived inflationwill -2 1 learning scheme converges to 3 E, then the havehave positive variance, and convergence will never occur. sample mean squareerrors converge to the best 1488 THEAMERICAN ECONOMIC REVIEW DECEMBER2003 forecasts and AR obtains. Notice that 3, -> 13E forecasting errors whenever a hyperinflation a.s. and the fact that H is continuous at 13RE happened, since OLS does not weigh recent imply events more heavily.

D. Internal Consistency (IC) (RE - Pt-pP I1 1 =|H(P , ,,-2, d) In Section IV, subsection B, we explained intuitively why hyperinflationsare more likely -0 - H(3RE, 13RE,d,)| a.s. to occur with high 1/a. Also, a high value of I/a is likely to generate better forecasts during a as t -> oo, where we used the fact that, by hyperinflation.Therefore, there is potential for definition, PR/p -1 = H( RE, PRE, d). Now IC to be satisfied precisely for the 1/a's that generate hyperinflations. IC is the criterionwe use to define equilibria E, _1 P) IRE in the paper. The variables we have to deter- mine are the sequences of inflation, expected inflation, and nominal balances, together with = ) the a. Notice since a is deter- E. 1,( ) ( .E , ., --0 a.s. parameter that, mined as part of the equilibrium, the a that satisfies IC will vary as the process for d, = where 1RE E(PRE/PtRE1)is by definition, and changes so that the learning mechanism is en- convergence follows by Lebesgue-dominated dogenous to governmentpolicy. convergence and boundedness of inflation. Therefore,both (t and Et_ (P/P_ l) converge Definition 4: A stochastic process {P,, 3,, M,} to PRE. This, together with boundedness of together with a is an IC equilibrium for (e, T) if: prices and 3 implies that 1. Given a, {P,, 3,, M,} satisfy (7), (12), (13), -2 (14) for all t. 1 PT~ 2. a satisfies IC for (e, ).27 T1T P,P 1 t-- 1 Since the dynamics are highly nonlinear, T the a's T1 P P, characterizinganalytically equilibrium - - We solve the model simula- I - Et-P-->) 0 a.s. is impossible. by tion and search numericallyfor a that satisfy IC in a way to be described below. This will show as T -> oo, so that that IC does impose restrictionson the space of learningparameters, and that the resultingequi- P libria match the stylized facts of the hyperinfla- - well. t - tionary experiences remarkably " E. Characterizationof the Solution by 1 P, Simulation -E ,_ (P, ---+ la.s.> T T Pt-P,_ I=I Pt-7.I To generate simulations we must assign val- ues to the parametersof the money demand as T -> oo for any e > 0. equation (y, )) and the distributionof dt. We choose values (y = 0.4 and 4 = 0.37) in order Notice that AR imposes very few restrictions to replicate some patterns of the Argentinean on the learning scheme. In particularAR holds a's. Even if AR is for many satisfied, agents 27The careful readerwill note that we did not IC for ex- impose could be making systematic mistakes; on the learning parameter v in this definition or in the ample, in periods where a, is updatedaccording simulations we describe below. This was done only for to OLS, agents could be making very large simplicity. VOL.93 NO. 5 MARCETAND NICOLINI:RECURRENT HYPERINFLATIONS AND LEARNING 1489

032

02

0.16

0.1-

0.0 -

I--Low 8und RE -High8o SHndw REoInflo I

FIGURE3. SIMULATIONOF THE(LOG) INFLATION experience during the 1980's; for details see making it difficult for the model to departfrom Appendix B. We assume that seignorage is nor- RE, we have chosen values of the average sei- mally distributed, truncated to have positive gnorage for which a REE exists. In order to values of seignorage, with mean that varies quantify the relevance of average seignorage across experiments we perform and o'd = (fact 4), we performedour calculations for four 0.01.28 different values: E(dt) = 0.049, 0.047, 0.045, The parameterv was set equal to 10 percent. and 0.043. We also assumed that the government estab- First of all, we describe the typical behavior lished ERR whenever expectations were such of the model. A particularrealization is pre- that inflation rates would be above 5,000 per- sented in Figure 3. That realization was ob- cent, so that we set 3u = 50. The ERR is tained with E(dt) = 0.049 and 1/a = 0.2. We enforced until expected inflation is inside the will show below that this value of the learning stable set S. parameter satisfies IC. This graph shows the Since our purpose is to show that a small potential of the model to generate enormous deviation from rationalexpectations can gener- inflationrates. In the same graph, we also plot- ate dynamics quite different from and closer to ted two horizontal lines, one at each of the the data than RE, we choose as initial beliefs stationary deterministic rational expectation 30 = f3RE so that our simulations are biased in equilibria,to show how the model under learn- favor of looking like RE.29 For the specified ing can generate much higher inflation rates parameters,the maximum level of average sei- than the rationalexpectations version. gnorage in the deterministicmodel for which a This graphdisplays some of the stylized facts REE exists is E(d,) = 0.05. In the spirit of in the learning model.30In the first periods, the

28The results for 30 lower values of oa2 were similar. Of The behavior of the REE in this economy is clear: for were then less course, hyperinflations frequent. the stationary REE, inflation would be i.i.d., fluctuating 29 For it would be example, trivial to generate at least around the horizontal line of IE. For bubble equilibria, one > hyperinflationby choosing 10 3RE. inflation would grow towards the horizontal line of BE. 1490 THE AMERICANECONOMIC REVIEW DECEMBER2003 inflationrate is close to the low stationaryequi- 1/a' for each 1/a on the grid. Figure 4 shows librium.When a relatively large shock occurs, it the result of these calculations:in the horizontal drives perceived inflation into the unstable re- axis we plot 1/a, while the vertical axis plots gion U and a hyperinflation episode starts. I/&'. The intervalof alternativelearning param- Eventually, ERR is established and the econ- eters that generate a mean square error within omy is broughtback into the stable region. If no E = 0.01 of the minimum in each column is large shocks occur for a long while, 3, would be markedwith a dark area. An IC equilibriumfor revised accordingto the OLS rule a, = a,_ 1 + (E, T) = (0.01, 120) is found when the darkarea 1, and the model would converge to the rational cuts the 45 degree line. expectations equilibrium;however, since aver- Table 1 reportsthe probabilitiesof having n age seignorageis high for this simulation, 3RE is hyperinflationsin ten years for different values close to the unstable set (see Figure 2) and it is of average seignorage and for those values of likely that a new large shock will put the econ- l/a that satisfy the IC criterion. omy back into the unstable region and a new As Figure 4 shows, for a low value E(d,) = burst in inflation will occur. Clearly, we have 0.043, only l/a = 0 and 0.1 satisfy the IC recurrent hyperinflations, stopped by ERR requirement. It turns out that for those two (facts 1 and 2). Since seignorage is i.i.d., and values the probabilityof a hyperinflationin 120 since the graphshows some periodsof sustained periods is zero. Therefore,if IC is imposed, this increasesin inflation,it is clear thatthere is little value of average seignorage rules out hyperin- correlationof inflation and seignorage (fact 3). flations. Since hyperinflationsdo not occur, giv- In order to reduce (or eventually eliminate) the ing too much importanceto recent observations chances of having a new burst, the government does not generategood forecasts, so a low 1/a is must reduce the amountof seignorage collected a good choice within the model. If seignorage is (i.e., an "orthodox"stabilization plan) in order increased to 0.045, the criterion is satisfied for to increase the size of the stable set. This would all values of alpha between 0.5 and zero. As separatethe two horizontallines, it would place indicated by Table 1, for this average seignor- 3RE far from the unstable set and it would age there are equilibriain which the probability stabilize the economy permanentlyaround the of experiencing recurrent hyperinflations is low stationary equilibrium. Establishing ERR high, so that higher alternative a's generate just before a reduction in average seignorage good forecasts, and the hyperinflationarybehav- would help stabilize the expectations of agents ior is reinforced.Table 1 and Figure4 show that more quickly, so there is room for a positive as the mean of seignorage increases, quasi- effect of a "heterodox"intervention as well. rational learning is consistent with hyperinfla- To find the learning equilibriumwe look for tions. Furthermore,hyperinflations are more values of cathat satisfy the lower bound crite- likely when seignorage is high. This documents rion IC for (E, T) = (0.01, 120). This value of T how fact 4 is present in our model. is chosen to represent ten years, roughly the This exercise formalizes the sense in which length of the hyperinflationaryepisodes we are the equilibriawith a given learning mechanism studying. The value of E is just chosen to be reinforces the use of the mechanism. For in- "small";it will be clear below how the results stance, when seignorage is 0.49 and l/a = 0.2, may change if this parameterchanges. an agent using an alternative alpha equal to To find numerically those values of a that zero, which is the collective behavior that rep- satisfy IC we proceed as follows: we define a licates the REE, will make largerMSE than the = grid of I/a E [0, 1.2] separatedby intervals of agent using 1/a 0.2. The reason is that in length 0.1. The same grid is used both for 1/a equilibriumthere are many hyperinflations,and and the alternative learning parameters 1/a' the agent that expects the REE will make bad considered. We compute the mean squareder- forecasts. rors in the right side of (11) by Monte Carlo integration,31and we find the minimum over

realization, we compute the sample mean square error for 31 we over all More specifically, we draw 1,000 realizations of each alternative 1/a' in the grid, and average {d, ..., d120}, find the equilibriuminflation rates for each realizations. .

VOL.93 NO. 5 MARCETAND NICOLINI:RECURRENT HYPERINFLATIONS AND LEARNING 1491

Average S . .. .. 0.9 0,S 0.7 0.6 I: nAvI-)../ 0_4 ---~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i. 0.2 0.1 0.0. ns n 07 -1 AR I v

Av ;=4.5 percent ,i 0,9 0.8 0.7 0-. 0o5 04 0.3-- __-2__~ 0.1 , 0.0 na J 'tsn7 I VJ^' YV - I % 0.1V -41'"r iA ? 0.3w i 0.4 ~05.,, , I 0.6

= 4.7 percent 0.9 _.1 0.8 1:. . . 0.7 0.6 05 0.4 0.3 0...2 0.1 l r--'--- I m s I 0,0 I -M 9 - I nn I n n1 I ni nA n(; n.(1 n(7 I n( no I V V-. U.uu I / VxT N!-? t, . V- . Y V * x,,'t .

Average Seignorage= 4.9 percent 0,9 0.8 0.7

0.4 - 0.3 0.1 0.0), 0.0 0.1 0.2 0.3 0,4 0.5 0.6 0.7 0.8 0

FIGURE4. INTERNALCONSISTENCY Notes: Columns representpossible values for 1/a actually used by agents. Rows depict alternativevalues for 1/&'.Light gray cells indicate the 45-degree line. Dark gray cells indicate that the value for l/a is efficient. Black cells indicate fixed points on 1/a. 1492 THEAMERICAN ECONOMIC REVIEW DECEMBER2003

TABLE 1-PROBABILITIESOF HYPERINFLATIONSOCCURRING IN TEN YEARSFOR DIFFERENT DEFICIT MEANS AND LEARNINGPARAMETERS

Deficit mean = 4.5 percent

Probabilityof no Probabilityof one Probabilityof two Probabilityof three Probabilityof more than Alpha hyperinflations hyperinflation hyperinflations hyperinflations three hyperinflations 0.5 0.16 0.34 0.28 0.16 0.06 0.4 0.55 0.34 0.09 0.01 0 0.3 0.90 0.10 0 0 0 0.2 0.99 0.01 0 0 0 0.1 1 0 0 0 0 0 1 0 0 0 0

Deficit mean = 4.7 percent

Probabilityof no Probabilityof one Probabilityof two Probabilityof three Probabilityof more than Alpha hyperinflations hyperinflation hyperinflations hyperinflations three hyperinflations 0.4 0.09 0.26 0.30 0.22 0.13 0.3 0.45 0.37 0.15 0.03 0 0.2 0.82 0.14 0.04 0 0 0.1 1 0 0 0 0 0 1 0 0 0 0

Deficit mean = 4.9 percent

Probabilityof no Probabilityof one Probabilityof two Probabilityof three Probabilityof more than Alpha hyperinflations hyperinflation hyperinflations hyperinflations three hyperinflations 0.2 0.23 0.40 0.27 0.09 0.02 0.1 0.73 0.26 0.01 0 0 0 1 0 0 0 0

Wheneverequilibria with hyperinflationsexist, F. Epsilon-Delta Rationality (EDR) there is multiplicityof equilibria(several 1/a's satisfy IC). The behavior of inflation does not In this subsection we show that in the equi- change much for differentequilibrium 1/a's.32 libria with hyperinflationsdiscussed above, the The numerical solutions show that the criterionEDR is satisfiedif the highestadmissible chances of a hyperinflationduring the transition inflation 3U is large enough, for values of 8 that to the rationalexpectations equilibriumdepend are closely relatedto the probabilityof experienc- on the size of the deficit. The lower the deficit, ing a hyperinflation.This is because, when a hy- the lower the chances of experiencing a hyper- perinflationoccurs, the conditional expectation inflation. Notice how the equilibriumlearning can be arbitrarilyhigh due to the existence of an parameters depend on the size of average asymptotein the mappingfrom perceivedto ac- seignorage: higher seignorage corresponds to tual inflation(see Figure2) but, in fact, the actual higher equilibriuma' s, which are more likely to value of inflationis unlikelyto be ever so high in generate a hyperinflation.33 a given realization.Thus, for every realization when a hyperinflationoccurs, the learningforecast can do betterthan the conditionalexpectation with very high probabilityin finite samples. 32 Since we chose 30 = I3E when we set 1/&= 0 we have the REE. When initial beliefs are far apart from the REE, then 1/a = 0 will no longer satisfy IC. PROPOSITION2: Consider the model of Sec- 33We have simulatedthe model undermany othervalues 1-2 are then about tion II. If Assumptions satisfied, for for the parameters.The main results of this subsection the model and the behaviorof inflationare observedfor a wide range of the given parameter values of any such that parameters. (E, T), there is a 3U large enough VOL.93 NO. 5 MARCETAND NICOLINI:RECURRENT HYPERINFLATIONS AND LEARNING 1493

T> rr.e ?P(ERRat some t-T) This proves that E(P7+ IP,P)() = co, there- fore where ?P(ERRat some t < T) is the probability that the government implements ERR at some 1 1 -p2e t = 1,..., T. P- + I () P+ i period (19) 1 P, PROOF: t= l P,(w) Fix-E,T. We first consider the case that 3u = oo. Consider a realization where an ERR is + o -P,+l(6) I = -Et-Ep)()Pt ( Aw) + E,, established at some t O,..., T. Letting t + 1 P, (o) pt be thefirst period where this occurs, it has to be the case that 1 - y,+ l(o) - -dT+ ()/ < 0 and 1 - y(3(o) > 0. Clearly, we can only have because the right-handside is, in fact, infinite. the first inequality if d 4- (1 - y3i+l())) < So, (19) holds for all realizationswhere there is K+. Therefore,since inflation is given by equa- one hyperinflationand ir.ET > P [ERRat some tion (14) and (3t+ , ,3) are known with infor- t T71. mation available at t, we have that The case of Pf finite but arbitrarilylarge follows from observing that, with arbitrarily high probability, the sequences of the case (C) (18) -(P+ 3 = ooare below a certain bound 1; also, for arbitrarilyhigh Pu the conditional expectation e'K+ is arbitrarilyclose to the one with pU infinite, so that all the inequalities are maintainedwith ar- bitrarilyhigh probability. K- - Since hyperinflationsoccur with high proba- dt1- y/ (,) bility if average seignorage is high, this propo- - 1 y +7I+(C)-)- d dl ( sition shows that EDR is satisfied with high 8 i K- when seignorage is high. For example, Table 1 shows that the of at least + > probability having P[+ d]3. one hyperinflationis 0.84, 0.91, and 0.97 for average seignorage 0.045, 0.047, and 0.049, The integralin (18) correspondsto the values of respectively, so this proposition implies that d+ 1 for which thereis a positive price level that EDR is satisfied for 8 > 0.84, 0.91, and 0.97. clearsthe marketwithout ERR andthe firstbranch of (15) holds, while the second term accountsfor V. Conclusion those values of next period shock for which an exchangerate rule needs to be enforced. There is some agreement by now that the Now we show that the integral in (18) is hyperinflationsof the 1980's were caused by the unbounded.Using argumentssimilar to the ones high levels of seignoragein those countries,and used in Appendix C (available from the authors that the cure for those hyperinflationswas fiscal or in the workingpaper version of this article)to discipline and abstinence from seignorage. The show that S has an asymptote we have IMF is currentlyimposing tight fiscal controls on the previously hyperinflationarycountries 1 -t() a dF ) that are consistent with this view. Nevertheless, 1 -- WYo+1(') -- / dFd-(d) to our knowledge, no currentlyavailable model K- justified this view and was consistent with some basic facts of hyperinflations.In particular,the 1 fact that seignoragehas gone down duringsome -> (1 - y/-(&))Q(&) dx = oo x hyperinflationsmakes it difficult for the IMF to '~ o argue in favor of fiscal discipline. Our model is consistent with the main styl- for some finite constant Q(iO)and small -1. ized facts of recurrenthyperinflations and with 1494 THEAMERICAN ECONOMIC REVIEW DECEMBER2003 the policy recommendationsmentioned above: On the practical side, this paper shows that an exchange rate rule (ERR) may temporarily hyperinflationscan be stopped with a combina- stop a hyperinflation,but average seignorage tion of heterodox and orthodox policies. We must be lowered to eliminate hyperinflations have been working on this paper in the second permanently. half of the 1990's; at that time it might have The economic fundamentalsof the model are seemed that hyperinflationswere a purely aca- completely standard except for the use of a demic issue: South American countries seemed boundedly rational learning rule instead of ra- to have solid fiscal stances and hyperinflations tional expectations.We show thatif the learning were a thing of the past. Unfortunately, the rule is restricted to be quasi-rational in the recent events in Argentinaand the experienceof sense that it must perform fairly well within some Eastern European countries have lent the model at hand, the model is falsifiable, some immediate interest to the policy conclu- and the learning rule driving expectation for- sions of this paper. It still seems importantto mation is endogenous to government policy. have a solid model that can help judging the This deviation from rational expectations is reasonabilityof the IMF recommendations.The attractive because it avoids the strong require- methodological contributionof the paper is to ments on rationality placed by rational expec- show that, with adequate equipment for orien- tations, and because the fit of the model tation and survival, an expedition into the "wil- improves dramatically even if the deviation is derness of irrationality"can be quite safe and small. productive.

APPENDIXA

Households: To provide some microfoundations for the model in Section II, subsection A, we solve a deterministicsmall open economy version of a standardoverlapping generations model.34 Each cohort has a continuumof agents living two periods. There is one type of consumptiongood in the world. Preferencesare given by (ln cY+ A In c?+ ) where c; is consumptionof young agents at time t and c?+ is consumptionof old agents at time t + 1. Agents are endowed with (oy, co) units of consumptionwhen young and old respectively, where oy > (o > 0. Asset markets: There are two assets in the economy: domestic and foreign currency. In our hyperinflationary equilibria,domestic currencywill be return-dominatedby foreign currency.To ensure that money demandis positive we will impose a cash-in-advanceconstraint for local currencyon net purchases of consumption.

Mt > Pt+ (c+ I - o?) for t ' -1. This condition makes foreign currencyvalueless for households.Therefore, we can write the constraintsfor the household as

P,tc = Ptcy + M,

Mt= Pt + I(ct I - oo)

M ? 0.

Household's optimizationimplies

34This Appendix extends the closed economy results of Sargent and Wallace (1987). VOL. 93 NO. 5 MARCETAND NICOLINI:RECURRENT HYPERINFLATIONS AND LEARNING 1495

+ = P,tY Pt+l? M, oYA Pt+ o? WoY Pt + clt 9 if (Al) (1 +A)P, ' P, (1 +A) P, (1 +A) w P, M, Cy= Y, - =0 otherwise. Pt oWA (0? This a microfoundationto with = + A and = gives equation (1), ~ ITA y ~co Ak. Foreign sector: The world is inhabited by wholesale firms that can buy and sell goods in any country without transactioncosts and are not subject to cash-in-advance constraints. If we let X, (which can be negative) be the net numberof units of the consumptiongood bought domestically and sold abroad by firmj, profits are given by

7J = XP{e, - XP, where et is the nominal exchange rate and Pf the price of the consumptiongood abroad.Free entry into the business implies that profits must be zero, therefore

(A2) P{e, = P,.

If we let TB, be the trade balance in units of consumption,market clearing implies that

oY + o? = c, + c? + d, + TB, > where d, 0 is exogenously given governmentconsumption at time t. The government budget constraint: We assume the government does not tax agents,35it only generates income by seignorage and, occasionally, by changing its stock of foreign currencyR,. The budget constraintof the government is therefore given by - M, M,-_ e, (A3) = d, + (R,- R,_)-.

Equilibriumin all markets implies (R, - R,_ )e, = TB,P,. Government policy: Government policy must set money supply and reserves to satisfy (A3). Reserves can be set according to two regimes:

A Floating Regime.-In this regime the government does not change its position on foreign currency. Then, all the government expenditure is financed by means of money creation, so that TBt = 0 and

M, =d,P, + M,_ which together with the money demand (Al) solve for the equilibriumsequences of M, and P,. The nominal exchange rate is given by equation (A2).

35 Taxes and governmentdebt are easy to introduceby reinterpretingd, and the endowmentsw: all equationsare consistent with co denoting endowments net of age-dependent,constant, lump-sum taxes, and with d, being the primarydeficit of the government. Debt can be introduced, for example, if we assume that government debt is constant (perhaps because the government is debt constrained)and d, representsinterest payments on debt plus primarydeficit. 1496 THEAMERICAN ECONOMIC REVIEW DECEMBER2003

A Fixed ERR Regime.-In this regime the governmentbuys or sells foreign currencyat a given exchange rate. Given Pf, Pf,, e,_ and a desired level of inflation 13,the exchange rate is

-P{,1 e,- 3e,_l pf

Equation (A2) implies that with this policy the government achieves 3 = P/P,_ . The money demand(Al) determinesthe level of nominal money demandconsistent with the nominal exchange rate target. Given this level of money supply and d,, foreign reserves and, consequently, the trade balance adjust so as to satisfy (A3). Of course, ERR is only feasible if the constraints on the governmentasset position is never binding. We assume that the first regime is used if inflationachieves an acceptablelevel less than PU; the ERR regime is followed otherwise. The equilibriumis therefore given by equations (Al), (A2), and (A3) which are deterministic versions of equations(1) to (4) in the paper.The analogy between this deterministicversion and the stochastic one in the paper is only exact up to a linear approximation,a usual simplification in macroeconomicmodels under learning.

APPENDIXB

In this Appendix we explain the choice of parametervalues for the demandfor money used in the numericalsolution of Section IV. The money demandequation (1) is linear with respect to expected inflation. It is well known, though, that the linear functional form does not perform very well empirically.However, departingfrom linearity would make the analysis of the model impossible to deal with. While we do maintainlinearity, we want to use parametervalues that are not clearly at odds with the observations.Since we are interestedin the public finance aspect of inflation, we use observations from empirical Laffer curves to calibrate the two parameters.In particular,as one empirical implication of our model is that "high" average deficits increase the probability of a hyperinflation,we need to have a benchmarkto discuss what high means. Thus, a naturalrestriction to impose on our numbersis that the implied maximum deficit is close to what casual observation of the data suggests. We also restrictthe inflationrate that maximizes seignorage in our model to be consistent with the observations. We use quarterlydata on inflationrates and seignorageas a shareof GNP for Argentinafrom 1980 to 1990 from Hidelgort Ahumadaet al. (1993) to fit an empiricalLaffer curve. While there is a lot of dispersion, the maximum feasible seignorage is around5 percent of GNP, and the inflation rate that maximizes seignorage is close to 60 percent. These figures are roughly consistent with the findings in Miguel Kiguel and Pablo A. Neumeyer (1995) and other studies. The parametersof the money demand y and 4, are uniquely determinedby the two numbersabove. Note that the money demand function (1) implies a stationaryLaffer curve equal to

+ (B1) + m1M = + (1 - y( 7r)) where m is the real quantity of money and ir is the inflation rate. Thus, the inflation rate that maximizes seignorage is

1r*= 1-- 1

which, setting Tr*= 60 percent,implies y = 0.4. Using this figurein (B 1), and makingthe maximum revenue equal to 0.05, we obtain 4 = 0.37. VOL.93 NO. 5 MARCETAND NICOLINI:RECURRENT HYPERINFLATIONS AND LEARNING 1497

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