inflation stabilisation and the consumption of durable goods

INFLATION STABILISATION AND THECONSUMPTION OF DURABLE GOODS �

JoseÂ De Gregorio, Pablo E. Guidotti and Carlos A. VeÂgh

Exchange rate-based stabilisations in chronic-in¯ation countries have often been characterisedby an initial consumption boom (which is most evident in the behaviour of durable goods)followed by a later contraction. This paper provides an explanation for such a boom-recessioncycle based on the timing of purchases of durable goods. The initial fall in in¯ation results in awealth effect which induces many consumers to bring forward their purchases of durablegoods, thus generating an aggregate consumption boom. Since most consumers replenishtheir stock of durable goods at the beginning of the programme, a later slowdown follows.

According to conventional wisdom, in¯ation can only be brought down at thecost of a recession. The only source of controversy lies in the magnitude of thecontractionary effect. Estimates for the United States of the `sacri®ce ratio'(de®ned as the cumulative percent output loss per percentage point reductionin in¯ation) lie anywhere from 3 to 18 (Sachs, 1985). Furthermore, thecontraction is expected to occur irrespective of whether the money supply orthe exchange rate is used as the nominal anchor (Fischer, 1986). The conven-tional wisdom, however, has not gone unchallenged. The most famous dis-senter is probably Sargent (1982) who argued, based on the Europeanhyperin¯ations of the 1920s, that if a stabilisation is accompanied by a crediblechange in regime, in¯ation should come down with only minor costs. Even ifthe notion that hyperin¯ations have been stopped at virtually no cost isaccepted (which some would not), hyperin¯ations are usually regarded asextreme episodes whose relevance for more mundane in¯ations is unclear.

A relatively less well-knownÐbut potentially more relevantÐchallenge toconventional wisdom has emerged from the experiences of disin¯ationaryprogrammes in chronic-in¯ation countries. In the late 1970s, exchange rate-based programmes in Argentina, Chile, and UruguayÐthe so-called `tabli-tas'Ðwere accompanied by an initial consumption boom (Fig. 1). The contrac-tionary effects usually associated with in¯ation stabilisation appeared only laterin the programmes. A similar phenomenon was observed in the exchange rate-based programmes of the mid-1980s in Israel and Mexico. More recently, theArgentine Convertibility plan of April 1991 also generated an initial boom.Indeed, Kiguel and Liviatan (1992) and VeÂgh (1992) have argued that a boom-recession cycle in private consumption has characterised most exchange rate-

The Economic Journal, 108 ( January), 105±131. # Royal Economic Society 1998. Published by BlackwellPublishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.

[ 105 ]

� The authors are grateful to Eduardo Engel, John Hey, Juan Pablo Nicolini, Roberto Perotti, CarlosZarazaga, two anonymous referees, and seminar participants at Johns Hopkins University, Texas A&M,UCLA, the University of Pennsylvania, the University of Chicago, the International Monetary Fund,CEMA (Argentina), and the Universidad de San AndreÂs (Argentina) for helpful comments anddiscussions. We thank Gil Bufman, Luis Cespedes, Julio Santaella, and Ernesto Talvi for providing data.De Gregorio thanks FONDECYT (Chile), Grant 1960009, for ®nancial support.

based programmes in chronic in¯ation countries, regardless of whether theprogrammes have been successful or not.

The boom in consumption has been particularly evident in the behaviour ofdurable goods, as suggested by Dornbusch (1986) and Drazen (1990). InChile, for instance, consumption of durable goods more than doubled fromthe beginning of the programme to the year in which consumption peaked,

Fig. 1. GDP, private consumption, and consumption of durable goods(in real per capita terms, logarithmic scale)

106 [ J A N U A R YT H E E C O N O M I C J O U R N A L

# Royal Economic Society 1998

while total consumption increased by `only' 26% (Fig. 1). In Israel, consump-tion of durable goods rose by 70% in the analogous period while total privateconsumption increased by 25%. In the recent Argentine Convertibility plan ofApril 1991, car sales (a proxy for consumption of durable goods) rose by astaggering 400% in four years, while private consumption rose by `only' 30%.Not surprisingly, the contraction in durable goods consumption is usuallymore severe than that in total consumption, as is particularly clear for theSouthern-Cone `tablitas' and the Israeli stabilisation.1

Ever since the Southern-Cone `tablitas' of the late 1970s, the boom-recessioncycle associated with exchange rate-based stabilisation has attracted the atten-tion of many researchers. An early paper by Rodriguez (1982), inspired by theArgentine `tablita', highlighted the role of low real interest rates in generatingthe initial boom. Dornbusch (1982) focused on the late recession that mayresult from a sustained real exchange rate appreciation. Later on, Calvo(1986), noting the short-lived character of many Latin American stabilisationexperiences, argued that lack of credibility may lie at the heart of the businesscycle associated with exchange rate-based stabilisation. In Calvo's (1986)model, lack of credibility is identi®ed with temporary policy. The publicexpects the programme to be discontinued in the future, and therefore actson the belief that the policy is temporary. A reduction in the rate of devalua-tion leads to a fall in the nominal interest rate and, thus, in the effective priceof consumption, because the opportunity cost of holding money is part of thecost of consuming (through a cash-in-advance constraint). Since the fall in theeffective price of consumption is perceived as temporary, intertemporal substi-tution leads to a temporary increase in consumption.

While some analysts (see, for instance, Kiguel and Liviatan, 1992) haveheralded the temporariness hypothesis as the most convincing explanation forthe boom-recession cycle associated with exchange rate-based stabilisation,others remain skeptical. In particular, since intertemporal elasticities of substi-tution are generally believed to be low, many doubt the empirical relevance ofthe temporariness hypothesis. In effect, Reinhart and VeÂgh (1995a) calibrate asimple model of temporariness, and ®nd that large declines in nominalinterest rates are required for the temporariness hypothesis to be able toaccount for actual consumption booms. In the case of the Southern-Cone`tablitas', for instance, the fall in nominal interest rates, which was in the orderof 5% per quarter, fails to account for any signi®cant fraction of the actualincrease in private consumption. Also, in view of the recent successful stabilisa-tions of Israel and Argentina, the temporariness hypothesis may not besatisfactory as it would imply that expectations are biased toward pessimismregardless of the success of the program. Bruno (1993), for instance, arguesthat there is evidence indicating that the Israeli plan was credible, andattributes the initial boom to a wealth effect caused by the increased value (asperceived by the public) of government bonds. In Bruno's (1993) view, the

1 Section 1 provides econometric evidence based on panel data which supports the existence of aboom-recession cycle in consumption, which is most evident in the behaviour of durable goods.

1998] 107T H E C O N S U M P T I O N O F D U R A B L E G O O D S


normal recession associated with in¯ation stabilisation was simply delayed bythis initial wealth effect. In the cases of the Chilean and Uruguayan `tablitas', ithas been argued that there is no reason why agents should have been pes-simistic about the future sustainability of the plans, given that the ®scal de®cithad been eliminated by the time the programmes were implemented.

Some observers have thus attributed the initial boom to a sense of monetarystability and higher perceived wealth brought about by the initial fall inin¯ation, especially when the stabilisation package is accompanied by impor-tant structural reforms, as in the Convertibility plan and the Chilean `tablita'.Analytically, wealth effects may result from lack of Ricardian equivalence(Helpman and Razin, 1987), future reductions in government spending(Drazen and Helpman, 1988), or supply-side effects (Lahiri, 1995; Roldos,1995 and Uribe, 1995). Wealth effects, however, cannot explain the laterecession. Hence, the late recession must be attributed to other factors, suchas the cumulative effects of the real appreciation of the domestic currency.Moreover, even the introduction of supply-side effects is not enough to gener-ate quantitative predictions which accord well with the actual increases inconsumption (Rebelo and VeÂgh, 1995). It has also been argued that exogen-ous shocks (i.e., foreign interest rate shocks in the `tablitas', and the Intifadain Israel) may have been responsible for the late recession.

The purpose of this paper is to provide an alternative explanation for theboom-recession cycle observed in exchange rate-based stabilisations, whichdoes not rely on the assumption of policy temporariness. Quite to the contrary,the boom-recession cycle turns out to be an unavoidable consequence of thevery success of the programme. The focus of the analysis is on the behaviour ofconsumption of durable goods which, as suggested by Fig. 1 and formallyshown in Section 1, has exhibited a more pronounced cycle than totalconsumption.2 As in previous analysis, our explanation also relies on an initialwealth effect following the implementation of the plan. Our model, however,exploits the difference between individual and aggregate patterns of consump-tion of durable goods. By so doing, the model is capable of accounting for theentire boom-recession cycle without having to introduce additional shocks toexplain the late recession. In fact, the initial boom and the late recession aresimply the two sides of the same coin.

The key feature of our analysis is that the presence of transaction costsimplies that individuals purchase durable goods only at discrete intervals oftime. At an aggregate level, however, the ¯ow of sales (and thus production) ofdurable goods is continuous since consumers buy durable goods at differentpoints in time. In this context, suppose that a stabilisation plan is implementedwhich, through an immediate fall in in¯ation, generates a wealth effect.3 The

2 While some models have also focused on durable goods, the key driving force has still beentemporariness (see, for example, Calvo (1988) and Drazen (1990)).

3 The channel through which the fall in in¯ation generates a wealth effect is not essential for ourstory. In the model, we assume that money reduces time-consuming transactions costs. The fall inin¯ation results in an increase in real money balances, which leaves consumers with more time to work.This increases output and, hence, households' income (since households own the ®rms).



wealth effect induces many consumersÐwhich would otherwise have waitedÐto bring forward the purchase of durables. In addition, consumers spend morethan they had planned to. In other words, next year's new Honda becomestoday's new Mercedes.4 The resulting `bunching' in the timing of purchasesgenerates an initial aggregate consumption boom. Interestingly, the initialconsumption boom sows the seeds of the future recession. Since so manyconsumers bring forward their purchases of durable goodsÐand it is only aftersome time that these new durable goods will need to be replacedÐaninevitable slowdown follows after the initial boom simply because consumersneed not buy new durable goods (i.e., no Mercedes will be sold next year).

Formally, it is assumed that consumers follow (S, s) rules for the purchase ofdurable goods. Each new purchase consists of D units of the durable good,which depreciates at a positive rate. The next purchase is done when d units ofthe durable good (which has no resale value) remain. In this framework,aggregate consumption is `smooth' in spite of the fact that individual con-sumption is `lumpy'. This point has been made, in a different context, byCaplin and Spulber (1987). They show, in a sticky-prices menu-cost model,that `lumpy' price adjustment may be consistent with monetary neutrality. Akey assumption underlying this conclusion, however, is that the economy is ina steady-state and shocks are such that they do not move the economy awayfrom this steady-state (i.e., the distribution of individuals in the state spaceremains unchanged). Caballero and Engel (1991) have dealt with this problemand provided a framework to analyse the aggregate dynamics of an economywith microeconomic `lumpiness' out of the steady state.5

Our framework, in turn, simpli®es signi®cantly the stochastic structure ofrelated models with the purpose of analysing the effects of an abrupt changein policy. We assume that the economy is initially in a steady-state with constantaggregate consumption, and then consider the effects of a permanent andunanticipated reduction in the rate of devaluation. We thus abstract fromindividuals' shocks and any other perturbations. This simple set-up allows us tostudy (S, s) rules and aggregation in the context of an intertemporal optimis-ing model, in which consumers optimally choose the trigger points (D and d).Furthermore, we also obtain explicit solutions for the trigger points and theevolution of aggregate consumption, which enables us to perform compara-tive-statics exercises.

The model thus generates a boom-recession cycle in consumption which isunrelated to the public's pessimistic expectations about future policy. Quite tothe contrary, it is precisely the fact that the stabilisation plan is successfulÐandthus generates a wealth effectÐwhich leads to the initial boom and laterslowdown. In our model, the boom-recession cycle hinges on the existence ofnon-convex adjustment costs for the purchase of durable goods. In contrast, if

4 As noted by Bar-Ilan and Blinder (1992), expenditure in durable goods during a given period oftime is the product of the average of individuals' purchases and the number of individuals that buydurable goods during that period. Fluctuations in the latter, as a response to income changes, couldaccount for most of the variability of aggregate durable expenditures. The earliest discussion of thisissue, in the context of inventory behaviour, is in Blinder (1981).

5 See also Bertola and Caballero (1990) and Caplin and Lehay (1991).



we were to consider a partial adjustment model, derived from convex adjust-ment costs, we could generate an initial boom, but the path of consumptionwould gradually converge to the steady-state without going through a re-cession.6 In the context of this model, the recessionary period would lookparticularly puzzling for observers (as was the case in Israel) since it isunrelated to any contemporaneous government policies or external factors. Asa result, policymakers might be forced to take some `corrective' actions when,in reality, the slowdown is the unavoidable consequence of their success inreducing in¯ation.

The paper is organised as follows. Before turning to the model, Section 1provides some formal econometric evidence on the existence and quantitativeimportance of the boom-recession cycle in durable goods consumption. Thefollowing three sections deal with the theoretical model: Section 2 analyses theconsumer's problem, Section 3 deals with general equilibrium considerations,and Section 4 examines the effects of an exchange rate-based stabilisationplan. Section 5 concludes.

1. Empirical Evidence

The main motivation behind our theoretical model is the behaviour of durablegoods consumption during exchange rate-based stabilisations. As illustrated inFig. 1, casual evidence suggests that the boom in private consumption isparticularly pronounced for durable goods. This section attempts to go beyondcasual empiricism and provide some formal econometric evidence on thebehaviour of durable goods.7 More speci®cally, the purpose of the econo-metric analysis is to show that (i) the boom-recession cycle associated withexchange rate-based stabilisation is mainly driven by durable goods consump-tion, and (ii) the cycle occurs irrespective of whether the stabilisation plan issuccessful or not. We take the ®rst ®nding as a formal motivation for ouremphasis on the role of durable goods, and the second ®nding as providingeconometric support for the main prediction of our model.

Any empirical analysis of the behaviour of durable goods consumption indeveloping countries faces a formidable challenge in terms of data availability.Few countries report data on consumption of durable goods and, when avail-able, the data often cover a short period of time. We gathered data for ®vechronic in¯ation countriesÐArgentina, Chile, Israel, Mexico, and UruguayÐfrom the late 1970s to the early 1990s. This data set covers seven major exchange

6 As argued by Caballero and Engel (1992, p. 364): `the partial-adjustment does a good job attracking the dynamics during normal times. However, it fails during periods of large ¯uctuations, likedeep recessions and brisk expansions.' In our case, it could also be argued that it fails under suddenand drastic changes in policy, such as that implied by a stabilisation from high in¯ation.

7 For all the theoretical literature on the subject, there is surprisingly little formal econometric workon the empirical regularities associated with in¯ation stabilisation in chronic in¯ation countries. Anexception is Reinhart and VeÂgh (1995b) who analyse empirically the real effects of exchange rate-basedstabilisation by using panel regressions and de®ning dummy variables for the early and late stages ofeach programme. They ®nd a boom-recession cycle in both real GDP and private consumption. Herewe extend this type of analysis to account for the behaviour of durable goods.



rate-based stabilisations in the last 20 years (the six illustrated in Fig. 1 and theUruguayan 1990 programme).8 In addition to annual real GDP and real privateconsumption for all ®ve countries, the data set includes real consumption ofdurables goods for Chile, Israel, and Mexico. For Argentina and Uruguay, wewill use car sales as a proxy for consumption of durable goods.9,10

As a ®rst (and less formal) pass on the data, we computed rates of growth ofreal GDP, real private consumption, and real durable goods consumption (allexpressed in per capita terms) before and after seven major exchange rate-based stabilisations (Table 1). We divided the period after the stabilisation intotwo stages (èarly' and `late') to account for the initial boom (which should showup as a higher rate of growth in the èarly' stage relative to `before') and the laterecession (which should show up as lower, possibly negative, rate of growth inthe `late' stage relative to the èarly' stage). More speci®cally, to capture theunderlying trend before the stabilisation plan was put into effect, the `before'

8 We will ignore the 1985 Austral Plan in Argentina, which was too short to be meaningful for ourpurposes.

9 The data were provided by the Central Banks of Argentina, Israel, Mexico, and Uruguay and theChilean Ministry of Finance. All data were transformed into per capita ®gures using the populationseries from International Finance Statistics (IMF).

10 Car sales appear to be a good indicator of consumption of durables. In Chile, for example, whilethe correlation of real car sales with GDP and consumption in the period 1985.1-1995.4 is 0.75 and0.79, respectively, the correlation with the consumption of durable goods is 0.97.

Table 1.The Boom-recession Cycle

(annual average rates of growth in per capita terms)

Real GDP Consumption Durable goods cons.Programme

Country (date) Before Early Late Before Early Late Before Early Late

Argentina Tablita(Dec. 1978)

ÿ0.7 2.5 ÿ5.9 ÿ3.6 8.8 ÿ6.0 ÿ7.2 31.9 ÿ37.8

Convertibility(April 1991)

ÿ4.1 6.6 0.0 ÿ4.4 8.4 ÿ1.6 ÿ21.8 61.8 ÿ12.9

Chile Tablita(Feb.1978)

ÿ2.7 5.6 ÿ5.5 ÿ5.4 5.0 ÿ3.0 NA 20.2 ÿ33.4

Israel 1985programme(July 1985)

2.2 3.3 0.4 ÿ1.0 10.1 0.4 ÿ11.4 27.9 ÿ5.5

Mexico 1987programme(Dec. 1987)

ÿ1.9 0.3 2.2 ÿ1.5 2.9 3.6 ÿ4.9 5.3 6.8

Uruguay Tablita(Oct. 1978)

2.9 5.5 ÿ4.6 ÿ0.3 6.4 ÿ4.5 NA 45.6 ÿ22.2

1990programme(Dec. 1990)

0.2 4.8 2.6 ÿ2.2 9.2 NA ÿ1.0 31.7 NA

Note: For Argentina and Uruguay, durable goods consumption is proxied by car sales. See text for thede®nitions of before, early, and late.



®gure is computed as the average rate of growth during the three years prior tothe implementation of the plan. The ®gure for the èarly' stage is the averagerate of growth during the year in which the stabilisation plan was implemented(if this happened in the ®rst six months of the calendar year) and in thefollowing two years. The ®gure for the `late' stage is the average rate of growthduring the two calendar years following the èarly' stage. It is important to noticethat the `late' stage is de®ned independently of whether the stabilisation planfailed or not (and, if it did, of the actual date in which the plan ended).

By and large, the ®gures in Table 1 are quite revealing. In the case of Israel,for instance, Table 1 indicates that the average annual rate of growth ofdurable goods consumption in the three years before the plan was ÿ11.4%.The growth rate jumped to 27.9% in the èarly' stages of the plan, only to fallprecipitously to ÿ5.5% during the late stage. With the exception of MexicoÐwhere the late slowdown does not show up in Table 1Ðthe data clearly suggestthe presence of an early boom in real GDP, consumption, and durable goodsconsumption followed by a late contraction.11 Furthermore, the cycle seems tobe much more pronounced for durable goods consumption.

We then conducted a more formal econometric exercise, in the spirit ofReinhart and VeÂgh (1995b). We constructed a balanced panel with fourcountriesÐArgentina, Chile, Israel, and UruguayÐfor the period 1978-93.12

We then regressed the per capita rate of growth of real GDP, real privateconsumption, and real durable goods consumption on country dummies(®xed effects), an èarly' dummy and a `late' dummy. This enables us to testfor the existence and magnitude of the boom-recession cycle relative to alinear trend. We allowed disturbances to vary across countries (groupwiseheteroscedasticity) and, to capture the impact of common shocks, for cross-country correlation of disturbances (cross-group heteroscedasticity).

In the spirit of the de®nition used for Table 1, the èarly' dummy takes avalue of one in the year in which the stabilisation plan was implemented (ifthis happened in the ®rst semester) and in the following two calendar years.For the case of the `late' dummy, we chose two alternative de®nitions (`latefail'and `lateall') to illustrate some important points. The dummy variable `latefail'takes a value of 1 in the year in which the stabilisation plan failed and in thefollowing year.13 Hence, `latefail' does not test for the late recession takingplace in successful programmes. However, since our model will predict arecession even in successful programmes, we wanted to test for this hypothesisas well. To that effect, we de®ned `lateall' as taking a value of 1 in the twocalendar years following the èarly' dummy de®ned above for all programmes.As an example, consider the Chilean `tablita' implemented in February 1978.The èarly' dummy takes a value of 1 in 1978, 1979, and 1980, and zerootherwise; the `lateall' dummy takes a value of 1 in 1981 and 1982, and zerootherwise; and the `latefai' dummy takes a value of 1 in 1982 (the year theprogramme collapsed) and 1983, and zero otherwise.

11 A slowdown in Mexico is noticeable only in 1992 (see Fig. 1).12 Mexico was excluded because data were not available for all the relevant period.13 Of the six stabilisation plans included in the panel, three failed (the Southern-Cone `tablitas').



Table 2 reports the panel regression estimates. The estimates with `latefail'appear in the top section of the table, while those for `lateall' are reported inthe bottom section. The coef®cients are signi®cant at standard levels in allregressions reported. The results thus support the hypothesis of a boom-recession cycle not only in failed programmes but, more importantly for ourpurposes, in all programmes. For example, the third column in the bottomsection indicates that the rate of growth of per-capita consumption of durablegoods is 29.06% higher (relative to trend) in the early stages of the programmeand 21.46% lower in the late stages. The corresponding coef®cients for GDPand consumption are considerably smaller, with the cycle in consumptionbeing more pronounced than the GDP cycle. The evidence thus suggests thatthe cycle in durable goods consumption is large both in absolute terms andrelative to that of GDP and consumption. Finally, it is worth noticing that themagnitude of the late recession is higher when `latefai' is used. This suggeststhat the failure of the programme induces a contractionary effect which is overand above that of the `normal' cycle (i.e, the cycle which takes place in bothfailed and successful programmes).

2. The Consumer's Problem

Consider a small open economy perfectly integrated with the rest of the world,which is inhabited by consumers blessed with perfect foresight. We ®rst discussthe consumer's preferences and constraints, and then proceed to solve theconsumer's optimisation problem.

Table 2.Panel estimates

Dependent variable

Growth in real GDPGrowth in real private

consumption

Growth in real privateconsumption of

durables

A. Recession in failedprogrammesEarly 2.44 (0.49) 6.02 (1.34) 21.10 (6.27)Latefail ÿ11.70 (1.44) ÿ10.44 (2.12) ÿ71.50 (9.28)Number of observations 64 64 64R-squared 0.59 0.46 0.51B. Recession in allprogrammesEarly 2.33 (0.66) 7.35 (1.43) 29.06 (7.46)Lateall ÿ3.10 (0.70) ÿ3.67 (1.77) ÿ21.46 (9.18)Number of observations 64 64 64R-squared 0.34 0.28 0.24

Note: Dependent variables are expressed in per capita terms. Standard errors in parenthesis. Method ofestimation was a two-step GLS procedure which allows for groupwise and cross-group heteroscedasticityand groupwise autocorrelation. The regressions include ®xed country effects (not reported). Thereported R-squared is for the ®xed-effects OLS regression.



Preferences

A typical consumer derives utility from the ¯ow of services of a durable goodwhich, for simplicity, is assumed to be the only good. The ¯ow of services isassumed to be proportional to the stock of durables, denoted by D. Thus, theintertemporal utility function is given by:

U t ��1

tlog Ds eÿr(sÿ t)ds, (1)

where r is the (positive and constant) rate of time preference.14 To simplifythe analysis, the instantaneous utility function is taken to be logarithmic.

The stock of durable goods is assumed to depreciate at an exponential rateä. Moreover, each time a new durable is bought, a transaction cost is incurred.Speci®cally, the old durable good becomes totally obsolete and has no resalevalue. The presence of transactions costs implies that consumers will not buygoods continuously (see, for instance, Bar-Ilan and Blinder, 1987, and Gross-man and Laroque, 1990).

The consumer is assumed to follow an (S , s)-type rule for the purchase ofdurables; namely, the individual buys a durable good, D, and keeps it until itreaches a lower value of d. Since there is no uncertainty, this is equivalent toassuming that, in equilibrium, the consumer buys D after a period of time oflength T has elapsed from the previous purchase, where T � log (D=d)=ä.Therefore, we will be able to characterise the utility derived from the consump-tion of durable goods as a function of only D and T.

Consider now the consumer's decision problem under the assumption thatconsumption of durables follows the above-mentioned (S , s)-type rule. Allexogenous variables will be assumed to be constant over time. Initially (i.e., attime t), the individual owns a stock of durables Dt , and must decide when tostart following the rule which involves buying D every time-interval of length T.Let ô denote the time at which the ®rst durable good is bought. In otherwords, the ®rst purchase is done after a time period of length ôÿ t has elapsedsince the beginning of the planning period. Of course, as it will be shownbelow, an individual starting with Dt � D will buy the new good whenôÿ t � T . The timing of the problem and the path of individual consumptionare illustrated in Fig. 2.

Taking into account the (S , s)-type rule followed by the consumer, theintertemporal utility function, given by (1), can be written as:

U (D, T , ô) ��ô

t[log Dt ÿ ä(s ÿ t)]eÿr(sÿ t)ds

� P1j�1

�ô� j T

ô�( jÿ1)T[log D ÿ ä(s ÿ ôÿ ( j ÿ 1)T )]eÿr(sÿ t)ds:

(2)

14 To eliminate inessential dynamics, the rate of time preference will be assumed to be equal to theworld real interest rate.



Equation (2) can be further simpli®ed to:

U (D, T , ô) � 1ÿ eÿr(ôÿ t)

rlog Dt � eÿr(ôÿ t)

rlog D

ÿ ä

r2� ä(ôÿ t)

reÿr(ôÿ t) � äT

reÿr(T�ôÿ t)

1ÿ eÿrT,

(3)

which provides the objective function for the consumer's problem.

Constraints

The consumer is endowed with one unit of time per period.15 In the spirit oftransactions costs models (see, for instance, Dornbusch and Frenkel, 1973), weassume that the consumer must devote time to the purchase of durablegoods.16 The time spent transacting depends negatively on the level of realmoney balances. The rest of the time is devoted to work, l. Thus, theconsumer's time constraint is given by:

15 To keep the analysis simple, we have assumed that the consumer does not value leisure.Incorporating leisure into the utility function would complicate the analysis without adding additionalinsights.

16 For simplicity, we assume that transactions costs take place continuously, even though the durablegood is only purchased at discrete intervals of time. Hence, transactions costs can be interpreted as timespent searching for the desired model or speci®cation and the best price. Alternatively, if a non-traded,non-durable good were introduced into the modelÐas discussed in the concluding sectionÐthen wecould assume that money is used to reduce the transactions costs associated with buying this secondgood.

Fig. 2. Timing and purchases of durables



1 � l s � v(ms) (4)

where v(m) satis®es v(m) . 0, v9(m) , 0, and v 0(m) . 0.Consider next the consumer's intertemporal budget constraint. The indivi-

dual can borrow and lend freely at an interest rate equal to r. In addition, theconsumer receives wage earnings, wl (where w denotes the real wage rate), realpro®ts from ®rms, Ð, real government transfers, g, and pays the in¯ation tax,im (where i denotes the nominal interest rate). Hence, his or her net realincome (excluding interest on net assets), y, is given by y � wl �Ð� g ÿ im.Since the individual purchases D at time ô (. t), and then repeats the samepurchase after a period of length T has elapsed, the present discounted valueas of time t of expenditure in durable goods is given by:

Deÿr(ôÿ t) � Deÿr(T�ôÿ t) � . . . � Deÿr(ôÿ t)

1ÿ eÿrT: (5)

Finally, since the individual bought his or her last durable good at timet ÿ x, income received since then has been saved at a yield r.17 Hence, thetotal value of assets accumulated as of time t, a t , equals:18

a t �� t

tÿxy0eÿr(sÿ t)ds � y0

r(e rx ÿ 1), (6)

where y0 denotes net real income before time t. The length of time x inequation (6) is related to the amount of durables bought in the last purchasebefore time t, D0, and to the stock of durables at time t, Dt , by the followingequation:

x � 1

älog

D0

Dt

� �: (7)

Taking into account equations (5) and (6), the consumer's intertemporalbudget constraint is given by:

yo

r(e rx ÿ 1)�

�1t

(ws l s �Ðs � g s ÿ i s ms)eÿr(sÿ t)ds � Deÿr(ôÿ t)

1ÿ eÿrT: (8)

Optimisation Problem

The consumer's problem consists in maximising utility function (3) subject toconstraints (4) and (8). This optimisation problem may be solved in two stages.In the ®rst stage, the optimal level of real money balances is chosen which, by(4), determines the labour effort supplied by the consumer. In the second

17 Note that we are implicitly assuming that the consumer was left with a zero net stock of assetsimmediately after the last purchase. This is equivalent to choosing an initial level of net assets equal tozero in any standard intertemporal model. Naturally, none of the results depend on this assumption.Similarly, we assume that the government's initial net stock of foreign assets is zero.

18 For simplicity, and with no loss of generality, we will assume that it is the government which ownsthe stock of real money balances. Consumers simply pay a rental fee of im per unit of time.



stage, the optimal (S, s)-rule for the consumption of durable goods is solvedfor.

The ®rst-order condition for real money balances follows from maximisingthe left-hand side of intertemporal budget constraint (8), subject to (4):

ÿwsv9(ms) � i s, (9)

which implicitly de®nes m � ~m(w, i). Substituting the latter into time con-straint (4) yields the labour supply function:

l s � ~l(ws, i s),@~l

@w. 0,

@~l

@ i, 0, (10)

which implies that labour supply is a decreasing function of the nominalinterest rate. It follows from (9) and (10) that if the real wage and the nominalinterest rate are constant over time, so are m and l. Thus, assuming for thetime being that Ð and g are constant over time, y (� w~l(w, i)�Ð �g ÿ i ~m(w, i)) is also constant over time. The intertemporal budget constraint(8) can then be rewritten as (with W denoting real wealth):

W t � a t � y

r� Deÿr(ôÿ t)

1ÿ eÿrT: (11)

Substituting D from (11) into (3), the optimal choice of T and ô followsfrom maximising:

log Dt

rÿ ä

r2� eÿr(ôÿ t)

r3

log W t � (r� ä)(ôÿ t)ÿ log Dt � log (1ÿ eÿrT )� äT eÿrT

1ÿ eÿrT

� �: (12)

Given the optimal values of T and ô, the optimal value of D is found fromequation (11). It is clear from equation (12) that the optimal value of Tfollows from the maximisation of the last two terms inside the square brackets.Therefore, the optimal T depends only on r and ä, and is thus independent ofwealth. The ®rst-order condition which yields the optimal value of T impliesthat:

r� ä

r� äT

1ÿ eÿrT: (13)

The following proposition characterises the optimal T:

PROPOSITION 1. The optimal T is positive and decreasing in r and ä.

Proof. See Appendix. i

Intuitively, if the rate of depreciation increases, then the consumer will holdthe durable good for less time. By (11), this implies that the increase in thefrequency of purchases is offset by a reduction in the size of the purchase.Similarly, higher impatience leads to more frequent purchases.



The ®rst-order condition associated with the choice of ô, together with (13),imply:

log W t ÿ log Dt � (r� ä)(ôÿ t)� log (1ÿ eÿrT )ÿ äT � 0: (14)

Equations (11), (13), and (14) determine the optimal values of T, ô, and D.In the initial steady-state equilibrium in which y � y0 and D � D0, (6), (7),

(11), and (14) imply that:

DeÿrT

1ÿ eÿrT� y

r, (15)

ôÿ t � T ÿ 1

älog

D

Dt

� �� T ÿ x: (16)

If Dt � D, it follows from (16) that ôÿ t � T . This is to be expected, sincethe individual ®nds it optimal to continue with the same (S, s)-rule. Theintuition behind (16) is illustrated in Fig. 2. Equation (16) says that thenext purchase will occur once the current durable good, Dt , reaches a valueequal to DeÿäT , which is the value at which the durable good is replaced.Since the lowest value for Dt is the value at which the durable is replaced,DeÿäT , the expression (1=ä) log (D=Dt) is always less than T, and henceôÿ t > 0. If one allows for an unanticipated change in income at time t(i.e., if y is different from y0), then ôÿ t may be negative, as discussed inSection 4. In such a case, there will exist `bunching' in the purchase ofdurable goods.

3. Firms, Government, and General Equilibrium

This section considers the behaviour of ®rms and the government, derivesaggregate constraints, and examines the equilibrium response of output tolower in¯ation.

Firms

It is assumed that ®rms produce durable goods using labour as the only input,according to a strictly-increasing, strictly-concave, twice-continuously differenti-able production technology, f (l). Firms' pro®ts are given by:

Ðs � f (l ds )ÿ ws l d

s : (17)

Pro®t maximisation requires that

f 9(l ds ) � ws , (18)

which implicitly de®nes the demand for labour as a decreasing function of thereal wage.



Assets and Goods Market Equilibrium

The economy is small and completely integrated in international capitalmarkets. Therefore,

i s � r� Es , (19)

where Es is the rate of devaluation of the nominal exchange rate (hereafterassumed constant over time) and r is the world nominal interest rate (i.e.,there is no world in¯ation). The price of durables is given by purchasing-powerparity. Hence, E is also the domestic rate of in¯ation.

Labour Market Equilibrium

In equilibrium, l � l d . Hence, for a given nominal interest rate, (10) and (18)determine the equilibrium real wage and labour supply. The latter, togetherwith the production function, determine equilibrium output as a function ofthe nominal interest rate. Moreover, using (10) and (18), it can be easilyshown that output, f (l), is a decreasing function of the nominal interest rate,and thus, by (19), of the rate of devaluation (and in¯ation), E. Intuitively, a fallin the nominal interest rate reduces time-consuming transactions costs, andtherefore frees time for productive activities. The real wage falls to accommo-date the increased labour supply. Hence, a reduction in the rate of devaluationleads to lower real wages, and higher employment and output.

Private Sector's Budget Constraint

We assume that there is a continuum of individuals in the interval (0, N ]. Inthe initial steady-state, individuals are distributed uniformly over time, witheach one buying a new durable good every time-interval of length T. Hence, ateach instant in time, there is a mass of N =T individuals purchasing a newdurable good.19 Aggregating budget constraint (8) over all individuals (andassuming that the economy is in a steady-state in which y0 � y and D0 � D)yields (after some manipulation):

y � D

T: (20)

Equation (20) simply indicates that, in per capita terms, income equalsexpenditure. Since each consumer spends D every time-interval of length T ,(per capita) expenditure per unit of time equals D=T .

Government

The government is assumed to rebate to consumers all the proceeds fromrevenues from money creation. (Recall that the government's initial net stock

19 In the following section, we will assume as a convenient normalisation that N � T , so that there isa unitary mass at each instant in time. Hence, in equilibrium, aggregate consumption per unit of timeequals D.



of foreign assets is zero.) The government's intertemporal budget constraint isthus

N

�1t

g s eÿr(sÿ t)ds � N

�1t

( _ms � Es ms)eÿr(sÿ t)ds: (21)

Integrating (21), imposing the standard transversality condition, and assumingthat the economy is in a steady-state equilibrium yields:

g � im: (22)

Aggregate Resource Constraint

From the ®rms' problem, it follows that wl �Ð � f (l). Combining the latterequation with equations (20) and (22), and recalling that y � wl �Ð� g ÿ im, the resource constraint obtains:

f (l) � D

T: (23)

In the absence of net foreign assets, (per capita) output equals (per capita)expenditure on durable goods. Although each individual follows an (S , s)-rulewhereby he or she purchases an amount D of durable goods every time-intervalof length T, the presence of a continuum of individuals ensures constantaggregate consumption of durable goods (in per capita terms) equal to D/T.Thus, the lumping of consumption which occurs at the household levelvanishes in the aggregate.

Finally, note that, in equilibrium, consumers' income equals output; that is,y � f (l). Therefore, since output is a decreasing function of the in¯ation rate,so is consumers' income; that is, y � ~y(E), and ~y9(E) , 0. The positive incomeeffect that results from a lower rate of devaluation will play a crucial role in theanalysis below.

4. Exchange Rate-based Stabilisation

Suppose that the economy is initially in the steady-state equilibrium describedin the previous section. In such an equilibrium, the rate of devaluation (andin¯ation) is expected to be E0 for the inde®nite future. Hence, each individualis spending D0 on durable goods every T periods, and receiving an income ofy0. Suppose that, at time t, there is a permanent and unanticipated reductionof the devaluation (and in¯ation) rate from E0 to E, where E0 . E. As discussedin the previous section, the reduction in in¯ation increases consumers'income from y0 to y (where y0 , y). Proposition 1 in Section 2 indicates thatthe optimal T is independent of the level of income. Therefore, in the newsteady-state equilibrium, the optimal T will remain unchanged in response tothe increase in y. Consumption of durable goods, however, will be different(i.e., D will be different from D0). Moreover, individuals must decide at time twhen to purchase the new durable and, hence, when to start with the new



(S , s)-rule.20 This implies that individuals will choose an optimal ô in responseto the change in income.

From (6), (7), and (14), it follows that

ôÿ t � T ÿ äx

r� äÿ 1

r� älog

y

y0� e rx ÿ 1

� �, (24)

where T is given by (13), and x is given by (7). At time t, individuals areindexed by x; that is, by how depreciated their durable good is at the time ofthe change in income. Equation (24) is the central piece of the analysis. Forvalues of the R.H.S. of the equation which imply that ô ÿt > 0, (24) shows thatoptimal ôÿ t is a decreasing function of both x and the ratio y=y0. Moreover,since x < T and y=y0 . 1, we have that ôÿ t , T ÿ x.21 This implies that everyindividual will anticipate the purchase of their new durable, D, in response tothe increase in income. (Recall that in the initial steady-state equilibrium,ôÿ t � T ÿ x, as indicated by (16).) Intuitively, since the increase in incomeprompts the consumer to move to a higher consumption equilibrium, he orshe also anticipates the purchase of the new steady-state level of durables.Indeed, the higher is the increase in income, the lower is ôÿ t. The negativerelation between ôÿ t and x is also intuitive. A higher x means that the olddurable is more depreciated, which implies that the transaction cost is lowerbecause the transaction cost equals the value of the durable good. Formally, itcan be shown that:

PROPOSITION 2. There is an x� 2 [0, T ) such that

T ÿ äx

r� äÿ 1

r� älog

y

y0� e rx ÿ 1

� �, 0, (25)

for all x > x�.Proof. See Appendix. i

This proposition shows that there is consumption `bunching'; that is,individuals with a high value of x will prefer to anticipate their purchase ofdurable goods to the moment when the stabilisation is launched. Moreover,the higher is the ratio y=y0, the higher is the fraction of individuals for which(25) holds. Since it must be true that ôÿ t > 0, those individuals for which(25) holds have a corner solution at ôÿ t � 0. These individuals (i.e., thosewho decide to purchase the new durable at time t) generate `bunching' in theconsumption of durable goods. This `bunching' effect is at the centre of theboom in aggregate consumption.

The individuals who bunch at time t are those who suffer the smallest loss inswitching from the old (S , s)-rule to the new one. Among these, those with

20 It should be noted that, since the rate of time preference equals the world real interest rate andthe policy change is permanent, the adjustment to the new (S , s)-rule will be instantaneous (i.e., thereare no transitional dynamics).

21 Note that, since in the initial steady-state equilibrium individuals were renewing their durableevery T periods of time, then x < T . Otherwise, optimality would be contradicted.



higher x are the richest, since they have the most assets accumulated as of timet. (Of course, the counterpart of these assets is that they will throw away an `olddurable'.) Therefore, the new steady-state consumption level for those indivi-duals who bunch at time t (which will be denoted by DB) is an increasingfunction of x.

Formally, the value of x which divides consumers who bunch from thosewho do not, denoted by x�, is that which makes the R.H.S. of (25) equal tozero. By (25), x� solves:

T � äx�r� ä

� 1

r� älog

y

y0� e rx� ÿ 1

� �, (26)

where T is given by (13). For x 2 [x�, T ], the new consumption level followsfrom (6) and (11):

DB � DB(x) � y

r� y0

r(e rx ÿ 1)

" #(1ÿ eÿrT ), (27)

where T is given by (13). From (27), it is clear that D9B(x) . 0.For those individuals who do not bunch (i.e., those for which x 2 (0, x�)),

the new consumption level (denoted by DN ) and corresponding ô are given by(6), (11) and (24), whence it follows that

DN � DN (x) � DB(x)y

y0� e rx ÿ 1

� �ÿr=(r�ä)

e rTÿärx=(r�ä): (28)

Moreover,

D9N (x) � ÿ(1ÿ eÿrT )e r(ôÿ t)ä( y ÿ y0) , 0: (29)

Notice that, in principle, there are two effects at work in determining the signof D9N (x). On the one hand, a higher x implies higher wealth at time t, whichwould tend to imply a higher DN . On the other hand, a higher x implies thatthe consumer anticipates more the purchase of the new stock of durable goods(i.e., a higher x implies a lower ôÿ t), which tends to imply a lower DN .Equation (29) indicates that the latter effect dominates.

According to the previous discussion, the new consumption level of durablesper individual, D, is given by:

D � D(x) � DN (x), x , x�DB(x), x > x�:

�(30)

Note that DB(x�) � DN (x�), as follows from (26) and (28). Hence, D(x) iscontinuous, though not differentiable, at x � x�. More importantly, however,D(x) exhibits the following property:

PROPOSITION 3. In the new steady-state, all individuals will purchase a durablegood which is greater than the one they were purchasing before; that is, D(x) . D0,for all x 2 [0, T ].




This proposition says that all individuals will buy a durable good greaterthan D0, even though they all anticipate the next purchase. Intuitively, thereare two effects which go in opposite directions. If individuals did not anticipatetheir purchase, they would buy a larger durable good (because they earn moreincome). However, the anticipation of the next purchase forces them (by thebudget constraint) to buy a smaller durable good. In principle, one couldimagine a situation in which an individual that brings forward the nextpurchase to time t (a `buncher') would be willing to buy a smaller durablegood. The last proposition, however, shows that the ®rst effect dominates.

We now turn to calculate the effect of in¯ation stabilisation on the aggregatelevel of consumption, denoted by DA.22 In the initial steady-state equilibrium,there exists a continuum of individuals of length T, with mass equal to one.Hence, aggregate consumption before in¯ation stabilisation was constant overtime and equal to D0. Formally, we index individuals by x, and assume N � T .(De®ne the population in any interval [0, x], by the function F (x), where:

F (x) ��x

0dx � x, (31)

F 9(x) � 1, and F (T ) � T � N , the size of the population.The new steady-state aggregate consumption is de®ned by a mapping from

calendar time in [t, t � T ] to the real line. Using (30) and the fact that (a) forx , x�, there is a unique value of s at which a new purchase will occurÐgivenby optimal ô in (24)Ðand (b) for x > x�, the new purchase of durables occursat time s � t, we can de®ne D in (30) as a function of s. Formally,DB(s � t) � DB(x) for x > x�; and DN (s) � DN [x(s)] for s 2 (t, t � t M ],where x(s) is obtained by inverting (24) after substituting ô by s; DN � 0 fors 2 (t M , t � T ]. Thus, aggregate consumption in the time interval [s, s � ds]is given by the function DA(s): [t, t � T ]! R�, such that:

DA � DA(s) �

�T

x�DB(x)dx, s � t

DN (s)h(s)ds, s 2 (t, t M ]0, s 2 (t M , t � T ],

8>>><>>>: (32)

where h(s) represents the population mass at time s in the aftermath ofstabilisation, and t M corresponds to the individual with x � 0 (that is, theindividual that just bought a durable and, hence, will be the last to renew it),and is given by:

t M � t � T ÿ 1

r� älog

y

y0

� �, t � T : (33)

Hence, in response to a lowering of in¯ation at time t, there is an initial22 Henceforth, we will assume, as a convenient normalisation, that N � T .



`bunching' of consumption at time t. Technically, there is a spike in consump-tion at time t, which is an order of magnitude larger than what happens therest of the time. In the data, this spike will show as a consumption boom in thequarter following the stabilisation. Furthermore, in reality there are otherfrictions, such as uncertainty about the persistence of the wealth shock, thatwill tend to spread out the consumption boom over time.

Individuals that do not bunch will anyway anticipate the purchase of thenext durable, as shown in (24). Before the stabilisation, we have normalisedthe mass of individuals per unit of time to one. After the stabilisation, we canshow that the mass of individuals that do not bunch per unit of time increases.This is summarised in the following proposition:

PROPOSITION 4. The proportion of individuals at each instant of time in(t, t M ] increases after the stabilisation is implemented; that is, h(s) . 1.


Thus, after the initial spike there is still a mass of individuals purchasingmore than D0 per unit of time, and later on there is no population mass and,hence, no consumption in the period s 2 (t M , t � T ]. This implies thatin¯ation stabilisation results in an initial consumption boom, as individualsnot only increase but also anticipate their consumption of durable goods.However, the anticipation of consumption and the resulting synchronisationof consumption patterns by a portion of the population results in a recessionlater on. Interestingly, the recession in aggregate consumption follows fromthe presence of (S , s) rules in durables consumption, rather than fromcontemporaneous government policies. Hence, from a policy point of view,the late recession should be no cause for concern since it re¯ects the naturaladjustment of the private sector to a credible stabilisation plan (see thediscussion below).

A hypothetical evolution of quarterly consumption, and considering a spikethat spreads out over time, is depicted in Fig. 3. The economy starts in a steadystate consisting of a ¯at consumption path. There is an initial boom explainedby the bunching of purchases at the onset of the stabilisation. This boom slowsdown, but still per capita expenditure in durables as well as the mass ofindividuals purchasing per unit of time is greater than in the initial cycleÐD0

per individual and 1 individual per unit of time. Finally, after t M has elapsed,no individual will make a purchase until the next cycle, which starts at t � T .Therefore, the economy enters into a recession. Since output increasespermanently, the cyclical behaviour in consumption caused by the stabilisationplan generates an initial current account de®cit followed by a current accountsurplus during the recession period. Furthermore, as it now stands, the modelwould predict a repetition ad in®nitum of the cycle in consumption (and,hence, in the current account). This feature is due to some simplifyingassumptions that were made to facilitate the analysis. In particular, given ourassumption of a log-utility function, the optimal T is not affected by changes in



wealth. However, if a more general utility function was chosen (say, a constant-relative-risk-aversion utility function), then the optimal T would not beindependent of wealth. Hence, the wealth effect that accompanies the stabili-sation plan would affect the optimal T of all consumers differently (since thesize of the wealth effect differs across individuals). This would imply that therewould be de-bunching over time, as those consumers who experienced thelarger wealth effect would buy the durable good more frequently. Theconsumption cycle would then disappear gradually over time. Alternatively, astochastic version of this economy with idiosyncratic shocks would generate de-bunching of consumption and, hence, a smoothing of the consumption cycle(see Caballero and Engel, 1991).

5. Final Remarks

Exchange rate-based stabilisations in chronic-in¯ation countries are oftenaccompanied by an initial consumption boom followed by a later recession.This boom-recession cycle is particularly evident in the behaviour of durablegoods. This paper has suggested an explanation for this phenomenon basedon the timing of purchase of durable goods. The initial fall in in¯ationgenerates a wealth effect which induces many consumers to bring forwardpurchases of durable goods, thus leading to an aggregate consumption boom.This initial bunching in purchases of durable goods sets the stage for a laterslowdown, as most consumers will not need to replenish their stock of durablegoods for a while.

Unlike the temporariness hypothesisÐwhich relies on expectations of future

Fig. 3. Evolution of consumption of durables



abandonment of the stabilisation plan to generate the boom-recession cycleÐthe explanation offered in this paper does not rely on the assumption ofperceived policy changes. The policy interpretations and implications aretherefore radically different. Under the temporariness hypothesis, the boom-recession cycle is a clear indication that policymakers have not done enough toconvince the public that the programme will be sustained over time. In sharpcontrast, under the explanation proposed in this paper, the boom-recessioncycle is a direct consequence of the ability of policymakers to implement acredible and successful plan. Hence, while the boom-recession cycle should because for much concern under the temporariness hypothesis, it should beviewed as the natural adjustment process to lower in¯ation under this alter-native interpretation.23

Three aspects of the model are worth discussing. The ®rst refers to thedistinction between money-based and exchange rate-based stabilisation. Itshould be emphasised that, in our model, ®xed exchange rates (and, thus, theendogeneity of the nominal money supply) are critical for the initial boom totake place. In effect, (4) shows that real money balances must rise toaccommodate the higher employment and, thus, higher output. In contrast toexchange rate-based stabilisations, both theory and evidence suggest thatmoney-based programmes are contractionary (see, for instance, Calvo andVeÂgh (1994)). Such a feature could be easily captured in our model byassuming that prices are sticky. In that case, and contrary to what wouldhappen in an exchange rate-based stabilisation, real money balances cannotrise initially in a money-based stabilisation, which would prevent the initialexpansion from taking place. Hence, our model could be made consistent withthe evidence on both exchange rate- and money based-stabilisation.

A second issue is the wealth effect that puts into motion the consumptioncycle. Clearly, the consumption-cycle (which follows from the (S , s) rule) isindependent of how this wealth effect comes about. In our model, the fall inin¯ation increases real money balances and reduces time-consuming transac-tions costs, thus freeing time for productive activities. Such a mechanism isanalytically convenient, and serves to illustrate how any wealth effect wouldwork. Although wealth effects are quite dif®cult to measure, casual evidencesuggests that they may be important. A boom in real estate markets is a well-known stylised fact of exchange rate-based stabilisations. In the ®rst two yearsfollowing the implementation of the Uruguayan 1978 stabilisation plan, forexample, the prices of housing and land (in dollar terms) rose by 181% and212%, respectively (Roldos, 1991). During the Chilean `tablita', real housingprices increased by 135% in the ®rst three years of the programme (Morandeand Soto, 1992). On the other hand, stock markets price indices rose by afactor of seven in the initial stages of the Argentine `tablita' and by three and ahalf in the Chilean `tablita' and the Argentina Convertibility plan. It should be

23 Of course, in many actual experiences the decline in consumption may have been the conse-quence of some other problems, such as loss of competitiveness, but our model suggests that theconsumption decline by itself may not necessarily re¯ect underlying problems.



pointed out, however, that for the purposes of our model the wealth effectneed not be necessarily large since there is a `multiplicative' effect broughtabout by the bunching in the purchase of durable goods. Hence, it isconceivable that a relatively small wealth effect may have a large impact ondurable goods purchases.

We also feel that our results are more general in that they should hold underother scenarios. Speci®cally, we believe that, in practice, liquidity constraintsare bound to play an important role. For instance, when in¯ation is falling,backward-looking indexation would result in higher real wages which, ifconsumers are liquidity-constrained, would lead to higher disposable incomeand thus generate the boom-recession cycle in durable goods. In a similarspirit, one could assume that consumers face a ®nancial liquidity constraintwhereby cash is needed to pay interests on consumer debt. Then a fall innominal interest rates, by reducing interest payments, would provide theconsumer with more liquidity and allow him or her to bring forward purchasesof durables. A related channel is the reappearance of credit in the aftermathof stabilisation programmes which, as argued by De Gregorio and Sturzeneg-ger (1995), could result from an increase in the supply of loans due to areduction in informational frictions caused by in¯ation. This effect couldprovide an additional channel which could trigger the consumption cycle aftera stabilisation plan is implemented.

A third issue is the behaviour of the real exchange rate in exchange rate-based stabilisations. It is by now a well-documented stylised fact that the realexchange rate appreciates considerably in such programmes (see Kiguel andLiviatan (1992) and VeÂgh (1992)). A simple extension of our model whichwould allow for a second, non-traded, good would capture such empiricalregularity. Suppose, for the sake of argument, that the supply of the homegood is constant over time. Then, a reduction in the devaluation rate whichincreases the consumption of the durable (i.e., traded) good would require anincrease in the relative price of home goods (i.e., a real exchange rateappreciation) to accommodate the higher consumption of traded relative tonontraded goods.

Universidad de Chile

Ministry of Finance, Argentina

UCLA

Date of receipt of ®rst submission: February 1995Date of receipt of ®nal typescript: August 1997

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Appendix: Proof of Propositions

Proof of Proposition 1: Characterisation of the optimal T.The ®rst order condition for T from the consumer's problem can be written as:

1� ä

rÿ äT

1ÿ eÿrT� 0: (A:1)

Hence, the second order conditions require that the following expressions be negative:

SOC � d

dT1� ä

rÿ äT

1ÿ eÿrT

� �: (A:2)

After some manipulations it can be shown that:

SOC � ÿä 1

1ÿ eÿrTÿ rT eÿrT

(1ÿ eÿrT )2

� �, 0: (A:3)

SOC is negative because zeÿz=(1ÿ eÿz) is decreasing for all z . 0 and is equal to onewhen z approaches zero, which implies that the expression in square brackets is alwayspositive. Hence, the optimal T exists and is unique.

We now show that T is a decreasing function of ä. Differentiating ®rst ordercondition (A.1) with respect to ä, we obtain:

1

rÿ T

1ÿ eÿrT� SOC

@T

@ä� 0: (A:4)

Since SOC is negative and the ®rst two terms are equal to ÿ1=ä by the ®rst-orderconditions, we have that

@T

@ä� 1

äSOC, 0: (A:5)

We now show that T is a decreasing function of r. Differentiate ®rst order condition(A.1) with respect to r to obtain:

ÿ ä

r2ÿ äT 2eÿrT

(1ÿ eÿrT )2� SOC

@T

@r� 0, (A:6)

which can be rewritten as:

@T

@r� ä

SOCr21ÿ (rT )2eÿrT

(1ÿ eÿrT )2

" #: (A:7)

It is easy to check that the term in square brackets is greater than 0 and less than 1,because the function z2eÿz=(1ÿ eÿz)2 is decreasing in the interval [0, 1] for all zgreater or equal than zero. Therefore, we have that:

@T

@r, 0: (A:8)

Proof of Proposition 2. Existence of bunching (x�, T ).Equation (24) implies that ôÿ t � 0 for x � x�, which is given by (26). Note that for



y0 � y, x� � T . To show that x�, T for y0 . y, ®rst note that the expressionlog (y=y0 � e rx ÿ 1)=(r� ä) is greater than xr=(ä� r), and, therefore, greater than Tat x � T . To recover the equality at ôÿ t � 0, the RHS must decrease. This is achievedwith a reduction in x since the RHS of (24) is decreasing in x. Hence, x�, T .

Proof of Proposition 3. Characterisation of D(x).Since DB(x) is increasing in x for x . x�, DN (x) is decreasing in x for x , x�, and

DN (x�) � DB(x�), it is enough to show that DB(x�) . D0, because the minimum ofthe function D(x) occurs at x � x�.

Using (15), for D � D0, and (27), we obtain:

DB(x�) � y

r� y0

r(e rx� ÿ 1)

" #(1ÿ eÿrT )

� y

y0� e rx� ÿ 1

� �D0eÿrT :

(A:9)

Substituting for T from (26), it follows that:

DB(x�) � D0 y

y0� e rx� ÿ 1

� �ÿr=(r�ä)

eÿräx�=(r�ä)

� D0 y

y0� e rx� ÿ 1

� �eÿrx�

� �ä=(ä�r)

� D0 1� y

y0ÿ 1

� �eÿrx�

� �ä=(ä�r)

. D0,

(A:10)

which completes the proof.

Proof of Proposition 4. Characterisation of h(s).The variable s, which represents the time at which a new purchase will occur for an

individual who is buying the good after t, is given by:

s � T ÿ äx

r� äÿ 1

r� älog

y

y0� e rx ÿ 1

� �, (A:11)

where x is uniformly distributed in the interval [0, x�). De®ne Ø in the interval[0, t M ], where t M is given by (33). According to (A.11), s is a monotonic transforma-tion of x. Both variables can be treated as random variables. Denoting the inverse of(A.11) as x � G(s), one can ®nd the distribution of s starting from the distribution of x(uniform):

F s(ó) � Prob (s < ó)

� Prob T ÿ äx

r� äÿ 1

r� älog

y

y0� e rx ÿ 1

� �< ó

" #

� Prob [x > G(ó)]

� 1ÿ G(ó)

x� :

(A:12)



Therefore, the density of the random variable s at a value ó is equal to ÿG9(ó)=x�.Since there are x� individuals who do not bunch, the mass of individuals per unit oftime will be given by:

h(s) � ÿG9(s): (A:13)

Note that h(:) is not constant, as was the case for the pre-stabilisation distribution.Before stabilisation, there was a unitary mass of people per unit of time. Now we willshow that the mass of individuals is greater than 1(h(s) . 1). The function G(s) isimplicitly de®ned by:

s � T ÿ äG(s)

r� äÿ 1

r� älog

y

y0� e rG(s) ÿ 1

� �, (A:14)

which after implicit differentiation yields:

ÿG9(s) � (ä� r) ä� r e rG(s)

e rG(s) � y

y0ÿ 1

24 35ÿ1

: (A:15)

Clearly, the term in square brackets is less than ä� r, hence, ÿG9(s) is greater thanone, which completes the proof that h(s) is greater than 1.



inflation stabilisation and the consumption of durable goods

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