Koskela, Erkki

Puhakka, Mikko

Helsingin yliopiston digitaalinen arkisto

öMmföäflsäafaäsflassflassflasffffffffffffffffffffffffffffffffffffDiscussion PapersCycles and Indeterminacy in OverlappingGenerations Economies with Stone-GearyPreferencesErkki KoskelaUniversity of Helsinki, RUESG and HECERandMikko PuhakkaUniversity of OuluDiscussion Paper No. 105May 2006ISSN 1795-0562HECER – Helsinki Center of Economic Research, P.O. Box 17 (Arkadiankatu 7),FI-00014 University of Helsinki, FINLAND, Tel +358-9-191-28780, Fax +358-9-191-28781, E-mail info-hecer@helsinki.fi, Internet www.hecer.fiHECERDiscussion Paper No. 105Cycles and Indeterminacy in OverlappingGenerations Economies with Stone-GearyPreferences*AbstractWe explore dynamics in an overlapping generations economy with Stone-Gearypreferences. We show that a steady state exists, and furthermore and importantly,that there can be a multitude of two cycles even though intertemporal elasticity ofsubstitution in consumption exceeds unity.JEL Classification: E21, E32Keywords: Stone-Geary preferences, overlapping generations economy, cycles.Erkki Koskela                                       Mikko PuhakkaDepartment of Economics                   Department of EconomicsUniversity of Helsinki                           University of OuluP.O. Box 17                                         P.O. Box 4600FI-00014 University of Helsinki            FI-90014 OuluFINLAND                                             FINLANDe-mail: erkki.koskela@helsinki.fi         e-mail: mikko.puhakka@oulu.fi* We thank the Research Unit of Economic Structures and Growth (RUESG) at theUniversity of Helsinki and the Yrjö Jahnsson Foundation as well as the Academyof Finland  (grant No.100089 to Koskela) for financial support.11. IntroductionNecessities of consumption characterize a large set of goods. They are a key part of thelinear expenditure system, which can be derived from the Stone-Geary preferencesoriginally developed by Geary (1950-51). Stone (1954) utilized them in empirical work.These preferences have been used also in theoretical work (see e.g. Auerbach andHines, 2002, and Azariadis, 1996).Consumer with the Stone-Geary preferences gets utility from that part ofconsumption, which exceeds the subsistence level. These preferences are closely relatedto preferences with habit persistence. Lahiri and Puhakka (1998) have studied theimplications of these preferences in an overlapping generations (OG) model with pureexchange. Among other things they showed that the possibility for cycles, withrelatively weaker concavity of the utility function than in conventional specifications.Koskela and Puhakka (2006) introduced logarithmic Stone-Geary preferencesinto a standard overlapping generations economy with pure exchange, and studiedstability, indeterminacy and cycles. They showed that dynamics depend on the relativenecessities of current and future consumption. In particular, a stable, and at the sametime indeterminate, nontrivial steady state exists for parameter values, for which there isno such equilibrium in the model with purely logarithmic utility function. Moreover, thefeatures of saving behavior lead to the possibility of period-doubling bifurcation (a two-cycle).In this paper we generalize the model in Koskela and Puhakka (2006). Theirresults also hold with more general Stone-Geary preferences with an importantexception that there are a multitude of two cycles although intertemporal elasticity of2substitution exceeds unity. As shown first by Grandmont (1985) (see also Gale, 1973)an overlapping generations economy with pure exchange can exhibit a stable monetarysteady states and cycles, if intertemporal elasticity of substitution is smaller than unity.Section 2 specifies the more general Stone-Geary utility function and presents itsimplications for saving behavior. Section 3 explores the dynamic properties ofequilibrium. There is also a brief concluding section.2. General Stone-Geary utility function and saving behaviorWe analyze the dynamic implications of a perfect foresight overlapping generationseconomy under pure exchange, where consumers live for two periods. The generalStone-Geary preferences of the consumer are(1)?????????????1)(1)(),(12211121acacccvtttt ,where ?/1  is the intertemporal elasticity of substitution calculated for consumption overthe subsistence level, ,ac ? 1 ? is the discount factor, and ia  is the exogenous level ofsubsistence. The measure of concavity of the periodic utility function is)./('/'' accucu ??? ? It is decreasing in consumption and increasing in subsistence level.A representative consumer maximizes (1) subject to the periodic budgetconstraints(2i) 11 ysc tt ??(2ii) 212 ysRc ttt ?? ? ,1 This formulation of preferences was originally not due to Geary and Stone. Literature calls thesemore general preferences Stone-Geary preferences, since they also lead to linear expendituresystem (see e.g. Barro and Sala-i-Martin, 2004, p. 178).3where 1y ( 2y ) is the endowment in the first (second) period, ts  the saving, 1?tR  theinterest factor from period t  to period 1?t , and tc1 (tc2 ) the first (second) periodconsumption, respectively. For the problem to make sense the following inequality mustbe satisfied for the equilibrium interest factor(3)RyyRaa 2121 ??? .This means that the point ( 21,aa ) should lie inside the budget line. For (3) to hold it isnot necessary that 11 ya ?  and 22 ya ?  at the same time. If 11 ya ?  and 22 ya ? , we havethe decision problem with well-known properties.The decision problem becomes different when either 11 ya ?  or 22 ya ? . Figure1 describes the budget set, when 011 ?? ay  and 022 ?? ay , so that saving is positive.In Figure 1 we need to have RR ?  for the decision problem to make sense. InspectingFigure 1 reveals that )/()( 1122 ayyaR ??? .Figure 1.1c2c2a1a 1y2yR?slope=4The first-order condition for consumer’s optimum, ttt Racac ??? ?? ? )(/)( 1122 ,leads to the saving function(4)?????????1122111111)();(RRayayRRRRs??????? ,with the following property(5) )(11)()11(222111111121111ayRRRayRRRRs ?????????????????????????????????????If 11 ya ? and 22 ya ? we have the same result as with ,021 ?? aa i.e. with typicalCRRA preferences. In this case it is necessary to have 1??  for saving to be adecreasing function of the interest factor. But if 11 ya ?  or 22 ya ? , then saving can be adecreasing function of the interest factor even though 1?? .3. Dynamic EquilibriumWe introduce an outside asset into the economy by assuming that government borrows(lends) from (to) the public. Government debt (or assets) at the beginning of the period isdenoted by tb , and the primary deficit by td  so that government’s budget constraint is(6) tttt bRdb 11 ?? ?? .Since we want to concentrate on the fundamental dynamic implications of the modelwith Stone-Geary preferences, we assume the primary deficit to be zero, i.e. 0?td  forall t, and study the case with nonnegative government debt, i.e. 0?tb . If we had5positive deficits, there would be two stationary equilibria in this economy.2 Thus in theasset market equilibrium tt sb ? , which leads to the following difference equation(7) tttttt sRRsRRss 11121 )()( ????? ??? .We study the case presented in Figure 1, when 011 ?? ay  and 022 ?? ay , so thatsaving is positive. This is a more interesting case than the other one (i.e. 22 ya ? ) sincethe restriction ( 22 ya ? ) seems to be quite plausible in economies with retirementsystems, where the subsistence consumption exceeds the second period endowment.Because saving is always positive, there is no interest factor such that saving is zero.Hence in (7) there is only one steady state such that 1?tR  for all t, which is possiblewhen 1)/()( 1122 ???? ayyaR , i.e. under 022 ?? ya , and when the first periodendowment is sufficiently higher than the subsistence consumption.We analyze equation (7) by the geometric techniques of the reflectedgenerational offer curves developed by Cass, Okuno and Zilcha (1979). Inverting thesaving function and substituting for 1?tR  in (7) we obtain the reflected generationaloffer curve.To derive the equibrium dynamics we take into account the periodic budgetconstraints and the fact that under zero primary deficit ttt ssR /11 ?? ? . We can thenrewrite the first-order condition as an equilibrium condition(8) )()( 111112211tttt saysayss ????? ?? ??? ? ,which implicitly defines the reflected generational offer curve. The steady state saving is2 The appearance of more than one steady state in models with positive deficits is basically knownsince Bailey (1956), see pp. 102-105 and especially his Figure 2.6(9)????1111221)()(*????? ayyas .Using equation (8) and totally differentiating gives(10)11111111122111111)(1)(????????????????????????ttttttttssayssayssdsds1221111)(111??????????ttttsyasssay?? .Because the maximum amount consumer can save is 11 ay ? , the numerator of (10) isnegative. Since 022 ?? ya , we can see that (10) is negative for 1?? . If 2a  is largeenough, (10) can be negative for 1?? , which is not possible when 021 ?? aa .Evaluating (10) at the steady state defined in (9) gives the following stabilitycondition(11) )()211(21)()211(212211112yaay ??????????????????????? ??? ????? .3Suppose now that 1?? , 02 ?y  and the discount factor, ? , is very small. Then the left-hand side gets very small, but the right-hand side approaches the value of 2)2/1( a . Inparticular, as 0?? , the steady state is stable, and at the same time indeterminate.4 As1??  the stability condition becomes 2211 yaay ???  and is thus independent of thecoefficient, ? .Moreover, the stability of the economy is more likely, the smaller is 11 ay ?  andthe higher is 22 ya ? . As previously noted the condition, 022 ?? ya , is plausible in3 For stability the value of the slope must be greater than minus unity. To derive (11) we haveassumed that the offer curve is downward sloping, and initially 1?? .4 Guesnerie and Woodford (1992) discuss thoroughly the concept of indeterminacy in OG models, seechapter 5.7economies with retirement systems. The small difference, 011 ?? ay , describes anaspect of poor economies. Furthermore, it can be seen from equation (11) that highersubsistence requirements 1a  and 2a make the economy more stable, and decrease theintertemporal elasticity of substitution ( cac ?/)( ? ).We present an example by specifying 02 ?y , aaa ?? 21 , and where 1ya ?? .Then we can rewrite the stability condition as(12)??????????????????1)211(21)211(21)( 112LHS .The left-hand side is an increasing function of the discount factor, if 1?? , and1)1( ?LHS , 0)0( ?LHS , 1)(0 ?? ?LHS  for 10 ?? ? . Then the condition in (12)implies an upper bound for the discount factor. For that upper bound to be less thanunity, it must be the case that 2/1??  (see Figure 2). For discount factors '?? ?)1/()( ??? ??LHS , and the stability condition thus holds.)1/( ?? ?)(?LHSFigure 2.?1)(?LHS'?8 Cycles can emerge in equilibrium under preferences without subsistenceconsumption, if intertemporal elasticity of substitution is smaller than unity (see Gale1973, and in particular Grandmont 1985). We explore the existence of a period-doubling bifurcation (a two-cycle) for the case when intertemporal elasticity ofsubstitution is higher than unity, 1?? , by resorting to a numerical example.We fix the parameters such that the slope of the reflected generational offercurve is slightly less than minus unity at the steady state. Then we find a pair ofnumbers, x  and z  (different from *s ) fulfilling (8) for 1??  such that the followingequations hold(13)???111122 ???????????xzxayayz  and???111122 ???????????zxzayayx  .The slope at the steady state is(14)**)(1**)(1111111*1ssayssaydsdsststt?????????????????? .We choose the parameters as follows: 2/1?? , 95.??  and 11 ?y . We know from (14)that the slope is sensitive to the values of necessities. We set the slope to be –1.001, andchoose the value of 1a  such that the slope indeed equals –1.001. Such a value for 1a(and indeed we assume the same value for 2a ) is .332782. R  (in this case )/( 112 aya ? )is .4987. Using these values the steady state, *s , will be .441566. Solving equations in(13) for these parameter values we get 441564.?x  and 441568.?z . The respectiveequilibrium interest factors are .99999 and 1.000009 so that the computed two-cycle is9feasible in the case of general Stone-Geary preferences as well. Of course the steadystate is one solution for (13).4. ConclusionWe have used an overlapping generations model with pure exchange and Stone-Gearypreferences to study the dynamic properties of equilibrium in economies with positivelevels of saving and public debt. We have shown that a stable nontrivial steady stateexists. Furthermore, and importantly there can be a multitude of period-doublingbifurcations (two-cycles) in equilibrium even though the intertemporal elasticity ofsubstitution is greater than one.References:Auerbach, A.J. and J.R. Hines Jr., 2002, Taxation and Economic Efficiency, inAuerbach, A. and M. Feldstein, eds., Handbook of Public Economics, Vol.3(North-Holland, Amsterdam) 1347-1421.Azariadis, C., 1996, The Economics of Poverty Traps. Part One: Complete Markets,Journal of Economic Growth 1, 449-486.Bailey, M.J., 1956, The Welfare Cost of Inflationary Finance, Journal of PoliticalEconomy 64, 93-110.Barro, R.J. and X. Sala-i-Martin, 2004, Economic Growth, 2nd edition (MIT Press,Cambridge, MA).Cass, D., M. Okuno and I. Zilcha, 1979, The Role of Money in Supporting the ParetoOptimality of Competitive Equilibrium in Consumption-Loan Type Models,Journal of Economic Theory 60, 277-305.Gale, D., 1973, Pure Exchange Equilibrium of Dynamic Economic Models, Journal ofEconomic Theory 6, 12-36.Geary, R.C., 1950-1951, A Note on “A Constant-Utility Index of the Cost of Living”,Review of Economic Studies 18, 65-66.Grandmont, J-M., 1985, On Endogenous Competitive Business Cycles, Econometrica53, 995-1045.Guesnerie, R. and M. Woodford, 1992, Endogenous Fluctuations, in Laffont, J.-J., ed.,Advances in Economic Theory. Sixth World Congress. Vol. II (CambridgeUniversity Press, Cambridge) 289-412.Koskela, E. and M. Puhakka, 2006, Stone-Geary Preferences in OverlappingGenerations Economies under Pure Exchange: A Note, forthcoming in Journal of10Macroeconomics 29, (downloadable at:http://www.valt.helsinki.fi/raka/koskela/erkki.htm).Lahiri, A. and M. Puhakka, 1998, Habit Persistence in Overlapping GenerationsEconomies under Pure Exchange, Journal of Economic Theory 78, 176-186.Stone, R., 1954, Linear Expenditure Systems and Demand Analysis: An Application tothe Pattern of British Demand, Economic Journal 64, 511-527.

Cycles and Indeterminacy in Overlapping Generations Economies with Stone-Geary Preferences

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