a two-dimensional non-equilibrium dynamic model

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Structural Change and Economic Dynamics 20 (2009) 221–238

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Structural Change and Economic Dynamics

journa l homepage: www.e lsev ier .com/ locate /sced

two-dimensional non-equilibrium dynamic model

rlando Gomes ∗

scola Superior de Comunicacão Social [Instituto Politécnico de Lisboa] and Unidade de Investigacão em Desenvolvimentompresarial – Economics Research Center [UNIDE/ISCTE - ERC], Campus de Benfica do IPL, 1549-014 Lisbon, Portugal

r t i c l e i n f o

rticle history:eceived February 2008eceived in revised form April 2009ccepted April 2009vailable online 9 May 2009

EL classification:321241

a b s t r a c t

This paper develops a non-equilibrium dynamic model (NEDyM) with Keynesian features(it allows for a disequilibrium between output and demand and it considers a constantmarginal propensity to consume), but where production is undertaken under plain neoclas-sical conditions (a constant returns to scale production function, with the stocks of capitaland labor fully employed, is assumed). The model involves only two endogenous/prognosticvariables: the stock of physical capital per unit of labor and a measure of market dise-quilibrium (MMD). The two-dimensional system allows for a careful analysis of local andglobal dynamics. Points of bifurcation and long-term cyclical motion are identified. The mainconclusion is that the disequilibrium hypothesis leads to persistent fluctuations generated

62

eywords:EDyMndogenous business cyclesonlinear growth

by intrinsic deterministic factors. These fluctuations may reflect some of the features fre-quently encountered in observed business cycles, once the model is conveniently adaptedto this purpose.

© 2009 Elsevier B.V. All rights reserved.

eynesian macroeconomicsyclical dynamics and chaos

. Introduction

The long lasting debate on macroeconomics about theources of business cycles has been built upon succes-ive disagreements and also some consensus (see Mankiw2006) for a survey). The Keynesian tradition, opposed tohe classical view of market clearing markets and exter-al shocks over fundamentals, stresses the presence ofisequilibria in the economic system. Firms and house-olds, instead of choosing optimally, often use rules ofhumb when deciding about price adjustments, how mucho invest, how to distribute consumption over time or how
o allocate time between work and leisure.
This paper analyzes a two-dimensional macroeconomicodel that combines classical and Keynesian features. Theodel is dynamic and purely deterministic. The main struc-

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954-349X/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.strueco.2009.04.001

ture of the model is based on Hallegatte et al. (2008)(hereafter HGDH), who present a problem designated asNEDyM (non-equilibrium dynamic model). As in HGDH, theobtained long-term outcome will depend on the particu-lar economic scenario that is furnished by a given array ofparameter values; we can have both a fixed-point balancedgrowth outcome (as in the neoclassical growth model) andendogenous fluctuations generated by the nonlinear natureof the relation between endogenous variables (as in a Key-nesian disequilibrium setup).

Our aim is to point out that, in opposition to what theReal Business Cycles theory claims, the presence of busi-ness cycles is not necessarily explained by random shockson the supply side (e.g., technological innovation). A mis-alignment between supply and demand that persists over
time may be the fundamental piece in explaining everlast-ing fluctuations that are not necessarily fed by exogenousperturbations. Although the endogenously generated fluc-tuations do not allow for a direct fit between the obtainedtime series and empirical evidence, after the character-
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Econom

propensity to consume, IS curve relation) become dom-inant. The main additional contribution that the presentpaper achieves is that it is able to obtain such a set of resultswithout departing from a simple two-equation model,

1 In Hallegatte et al. (2008), this variable is called ‘goods inventory’;we exclude this term because it can be equivocal in the sense that it isused. It is rather a measure of delivery lags or selling lags. We also avoid,at this level, the designation ‘utilization rate’. Changes in the utilizationrate of capital in time may generate this disequilibrium, but they are notthe disequilibrium itself. Capacity utilization generally emerges with themodelling of the production function; it calls the attention for the fact that

222 O. Gomes / Structural Change and

ization of the model a last section discusses how theobtained chaotic series may be adapted in order to rep-resent macroeconomic time series with the well knownproperties concerning volatility, co-movement and persis-tence that are found empirically.

According to HGDH, a NEDyM is a growth model builtupon a standard Solow (1956) model, but where multipleinefficiencies arise in the several markets that are consid-ered. In this analysis, agents do not have perfect foresightand markets do not clear, and the main reason pointed outfor such is the inertia that the economic system under-goes. Inertia implies a delay on the adjustment betweenproduction and demand, on one hand, and, on the otherhand, a suboptimal investment process. Investment deci-sions are linked with short-run profits and these may givesigns that differ from the reality attached to the long-termoptimal scenario. Furthermore, the labor market is sub-ject to relevant inefficiencies, which are translated into aPhillips curve that relates nominal wages with labor supply.Consumer decisions are not optimal, instead they dependon the available stock of real balances and on the Solow’sconstant rate of savings.

The HGDH model is, therefore, a large collection ofKeynesian relations built upon a minimal classical growthstructure; this consists just on a production function thatfully employs available inputs and on a conventional capi-tal accumulation difference equation. The authors are ableto find a route to chaotic motion and, thus, for differentparameter values, it is analytically possible to observe afixed-point stable equilibrium or cycles of any periodic-ity and completely a-periodic cycles. Such a co-existencecan be interpreted under the idea that, for certain arrays ofparameters, classical economics dominate, while for oth-ers the inertia factors become sufficiently relevant in orderto generate endogenous business cycles. Under this inter-pretation, we encounter a two-fold explanation for thepersistence of business cycles: in the scenario in whichclassical economics prevail, business cycles can only occuras the result of external shocks; when Keynesian eco-nomics dominate, the role of technology shocks (or others)will have a relatively smaller relevance since some marketinefficiency or inertia is able, by itself, of producing andperpetuating fluctuations.

By modelling simultaneously the dynamics of the goodsmarket, the labor market, the behavior of firms withinvestment as a function of profits and the behavior ofhouseholds as a function of real balances, the problemproposed by HGDH become an eight-dimensional systemwith eight endogenous variables (or prognostic variables,as the authors call them). Additionally, 11 other vari-ables (diagnostic variables) are modelled as functions ofthe endogenous state variables. With such a high dimen-sion, the problem cannot be analyzed in general terms;only through numerical particular examples one may inferabout the behavior of the economy. Thus, what the authorsgain in terms of completeness they evidently lose in what
concerns tractability.
Here, the main distinction relatively to the analysis ofHGDH, is that our model is more compact (it is just atwo-dimensional model), allowing for the general analy-sis of local dynamics, as well as for the investigation of

ic Dynamics 20 (2009) 221–238

the long-term global asymptotic behavior of the assumedendogenous variables.

The features we maintain in this version of the NEDyMare, on one hand, the neoclassical production function andthe capital accumulation process that is present in anygrowth optimization problem and, on the other hand, themost relevant Keynesian features; basically, we assume, asin the HGDH model, that an element of inertia is present inthe goods market: production and demand are not alwaysadjusted to one another, and thus a market disequilibriumpersists in time. This implies the need to assume a non-equilibrium variable, which plays a fundamental role in theobtained results. To this variable, we attribute the designa-tion of measure of market disequilibrium (MMD).1

Differently from the HGDH model, investment and con-sumption decisions are not explicitly modelled; instead,consumption is given just as a constant share of income(the good old constant marginal propensity to consumeis taken into account), while investment is the result ofa behavioral rule that takes into account the firms’ reac-tion to price changes and to variations on the value of theMMD. Demand is defined as consumption plus investment,and the dynamics of the system can be addressed oncedemand and output are connected through a short-runmacroeconomic relation. This relation is the HGDH mar-ket equilibrium adjustment equation. The analysis of thelabor market is neglected, by assuming that a fixed amountof labor is in every moment available to produce.

The framework that arises from the previous assump-tions is a two-dimensional deterministic system withphysical capital (per unit of labor) and the MMD (alsoper unit of labor) as endogenous variables. Relatively tothis model, one can address both local and global dynam-ics. Local analysis allows for perceiving that bifurcationpoints are eventually crossed, a necessary requirement toencounter long-term nonlinear motion. The global analy-sis, although less generic, confirms the generation of areasof endogenous cycles, that occur with a flip bifurcation.As in the HGDH problem, areas of fixed-point stabilitycan be interpreted as representing the balanced growthpath that is characteristic of classical growth models, whileregions where complex behavior is evidenced are the oneswhere the Keynesian features of the model (inertia, lackof alignment between production and demand, constant

in booms equipments and machinery are used more intensively than inperiods of recession. Although we will resort to a neoclassical productionfunction with full employment of inputs, in Section 6 we make a briefreference to how different utilization rates can be inserted in the proposedframework in order to help making the model useful to address volatilityproperties of business cycles.

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O. Gomes / Structural Change and

he dimension in which most of the classical models arexplored (e.g., the Ramsey capital-consumption model).

Other approaches to the generation of endogenous busi-ess cycles (EBC) have been extensively discussed in the

iterature. The origins of this literature can be traced backo Medio (1979), Stutzer (1980), Benhabib and Day (1981),ay (1982), Grandmont (1985), Boldrin and Montrucchio

1986) and Deneckere and Pelikan (1986). The main con-ept had to do with the idea of competitive chaos; that is,sing the benchmark optimization problem (intertempo-al or with overlapping generations) and introducing somelight changes to the conventional presentation (e.g., byhanging the shape of the production function), endoge-ous cycles were generated. These cycles corresponded,ost of the times, to chaotic time series, i.e., time series

xhibiting sensitive dependence on initial conditions.The notion of competitive chaos has been further devel-

ped by a group of mathematical economists, who claimhat nonlinearities can be found in conventional dynamiclassical models without the need of considering any kindf inefficiency. In Nishimura et al. (1994) and Nishimura andano (1994, 1995), among others, extreme conditions underhich the competitive growth scenario can generate long-

erm nonlinear motion are addressed (e.g., unrealisticallyigh intertemporal discount rates).

The literature on EBC has gained an important newreath with the model by Christiano and Harrison (1999),ho proved the existence of chaos in a standard deter-inistic Real Business Cycles model with production

xternalities. This line of research, where a utility maxi-ization control problem is taken into account (and where

onsumption and leisure are the arguments of the utilityunction), has been further developed by Schmitt-Grohé2000), Guo and Lansing (2002), Goenka and Poulsen2004), Coury and Wen (2005) and others. A similar strandf literature is the one that investigates the presence ofifurcation points and nonlinearities in overlapping gener-tions growth models, which are also subject to productionechnologies showing increasing returns. This work is asso-iated with the following references: Cazavillan et al.1998), Aloi et al. (2000), Cazavillan and Pintus (2004) andloyd-Braga et al. (2007), among others.

The models in the EBC literature are simple generalquilibrium models, which assume that the individualrm faces a positive external effect from production inociety; thus, the representative firm faces an increasingeturns technology. This simple idea is capable of produc-ng endogenous cycles, but it leaves unanswered many ofhe questions raised by Keynesian macroeconomics. Otheruthors search for additional features capable of introduc-ng nonlinear dynamics in simple optimization models; onelternative to the benchmark model consists in depart-ng from the idea of a representative agent. This is donen Goeree and Hommes (2000) and Onozaki et al. (2000,003), who develop macroheterogeneous agents models.ther hypothesis has to do with learning mechanisms;
ellarier (2006) replaces the optimal planner problem by aonstant gain learning mechanism that generates endoge-ous fluctuations.
Closer to a Keynesian setup is the analysis of Dosi etl. (2006). These authors develop a model where endoge-

ic Dynamics 20 (2009) 221–238 223

nous fluctuations are the result of the way firms behave. Inaccordance with what empirical evidence shows, invest-ment decisions are lumpy and constrained by the financialstructure of firms; moreover, firms are boundedly rationalwhen forming expectations about future events. Additionalingredients of Keynesian nature are added by Hallegatte etal. (2008), who introduce the term NEDyM, and mix classi-cal and Keynesian features in a way we explore further inthe next sections.

The remainder of the paper is organized as follows.Section 2 presents the basic structure of the NEDyM. Sec-tion 3 characterizes local dynamics. In Section 4, specificfunctional forms for the neoclassical production functionare proposed in order to obtain additional, more concrete,results. Section 5 explores global dynamics for a reasonablecalibration of parameter values. In Section 6, we discussthe relevance of the generated endogenous fluctuationsfrom an empirical point of view; despite the fact that theobtained cycles are not immediately compatible with thefluctuations in observed data, some additional assumptionscan be added to generate such compatibility. Finally, Sec-tion 7 concludes. Proofs of propositions are left to finalAppendix A.

2. The NEDyM: basic structure

Consider an economy populated by a large numberof households and firms. Households consume, in eachtime moment t = 0, 1,. . ., a constant share of the availableincome, ct = byt; variables ct ∈ IR + and yt ∈ IR + representreal per capita consumption and real per capita income,respectively. We assume that a constant amount of laboris available to produce (and, to simplify, that this coincideswith total population); normalizing this quantity to unity,there is a coincidence between per capita values, per laborunit values and level values. We will, interchangeably, useany of these terms. Parameter b ∈ (0,1) respects to a constantmarginal propensity of consumption.

Output or income is generated by a neoclassical pro-duction function of the type yt = f(kt), with kt ∈ IR + physicalcapital per unit of labor. In Section 4, particular results arederived for two specific functional forms of the productionfunction. For now, we just postulate that this is a neoclas-sical production function, by assuming that:

(a) f is a continuous and differentiable function (f ∈ C2) andit exhibits positive and diminishing marginal returns:f′ > 0, f′′ < 0.

(b) Inada conditions are satisfied: limk→0f ′ = +∞; lim

k→+∞f ′ = 0.

The accumulation of capital is driven by a process ofinvestment. Letting it ∈ IR+ represent investment per laborunit and ı> 0 a capital depreciation rate, the process ofcapital accumulation is given by Eq. (1).

k − k = i − ık k given. (1)
t+1 t t t 0
Essential to the characterization of the capital accumu-lation process is the rule that establishes the evolution ofinvestment over time. This will emerge from the assump-tion that the goods market does not clear, i.e., that a

Econom

order to eliminate the inflation rate from the analysis, oneverifies that, under the assumed conditions,


disequilibrium between output and demand persists overtime. Here, we follow closely HGDH, who explain the mis-alignment between yt and demand (dt = ct + it) through theintroduction of the MMD variable, ht ∈ IR (this variable, asall the others, is defined in per capita terms).

The dynamics of the MMD is determined by the dif-ference between production and demand and, therefore,it can assume both positive and negative values. In the caseof a positive MMD, ht > 0, there is a selling lag, i.e., tem-porary overproduction exists, which can be the result, forinstance, of the time needed to sell the goods. A negativeMMD, ht < 0, indicates the presence of underproduction ora delivery lag, and can be interpreted as the time requiredfor the consumer to get the goods she ordered.

Selling and delivery lags may be interpreted as a nor-mal fact of economic activity, but additionally they can bethought as the result of the presence of inertia that turnsdifficult to change the productive capacity that exists in agiven moment. An equilibrium situation will be the onein which ht = 0, a scenario that characterizes a competitivemarket. In the developed model, equilibrium will not nec-essarily exist in the long-term, i.e., the system may convergeto a steady state where despite the coincidence betweenoutput and demand, there is systematic under or overpro-duction.

As stated, changes in the MMD are the result of the dif-ference between output and demand; this is expressed inEq. (2),

ht+1 − ht = yt − dt, h0 given. (2)

Eq. (2) just states that when output is above demand,the value of the MMD rises; it will fall in the opposite cir-cumstance.

To complete the model, we need to characterize theimpact of market imbalances over the evolution of pricesand to understand how investment is determined by thefirms’ reaction to changes on prices and on the MMD. Theprice dynamics equation is similar to the one in HGDH, thatis, we assume that price changes are determined by theMMD per unit of demand:

pt+1 − pt = −� htdtpt, p0 given, � > 0. (3)

The interpretation of expression (3) is straightforward:for a positive MMD there is a delay in the selling of pro-duction, meaning that market power is on the side of theconsumers, who force prices (pt ∈ IR+) to decrease. If, oth-erwise, the MMD is a negative amount, then the lag is ondelivery, making sellers to concentrate market power, andtherefore producers can trigger a rise in prices. In practice,this equation just presents one of the most basic mech-anisms in economics: the mechanism of adjustment of
prices given the absence of equilibrium between supply anddemand. When supply exceeds demand (ht > 0) prices willfall as the market adjusts to equilibrium; when demandexceeds supply (ht < 0) prices will rise towards the marketequilibrium point.
ic Dynamics 20 (2009) 221–238

Defining�t ≡ pt+1−ptpt

,�t ∈ IR, as the inflation rate, Eq. (3)is rewritten as,

�t = −� htdt

(4)

Price evolution can be suppressed from the model’sanalysis once we introduce a behavioral equation concern-ing investment. We will adopt the following investmentfunction:

it = i∗exp(��t) −�ht, � > 0 (5)

Function (5) involves two components:

(i) Firms react to price changes. In the absence of infla-tion (and neglecting for now the second term in theright hand side of (5)), firms will invest an amountof resources corresponding to the potential invest-ment level: i* = (1 − b)y*; potential output, y*, will beassumed as the long-run value of output under opti-mality conditions (it will be interpreted, in Sections4 and 5, as the steady state value of income thatis derived from a standard neoclassical optimizationproblem with intertemporal consumption utility maxi-mization). If prices rise (�t > 0), firms will be motivatedto invest more than the potential level (because theycan sell the produced good at higher prices); if theinflation rate is negative, then the opposite occurs, i.e.,investment falls below the potential level.

(ii) There is an adjustment term related with the change onthe MMD (�ht = ht+1 − ht): if the selling lag becomeslarger or the delivery lag is shortened (a rise in ht), firmswill be willing to invest less (as inventories rise, firmswill adjust their behavior by reducing investment); inthe opposite case (a fall in ht), lower inventories orwider delivery lags will stimulate firms to increaseinvestment. This second component of the investmentfunction can be interpreted as involuntary investmentby businesses; thus, the first component will corre-spond to the part of investment that emerges from theconscientious and rational decisions of firms.

Recall that the definition of demand is dt = ct + it andconsumption was presented as ct = byt. Thus, it = dt − byt;combining the investment function (5) with the previousdefinition, one encounters the relation

xt = ��t (6)

with xt ∈ IR the output gap, i.e., the difference between thelogarithm of effective output and the logarithm of poten-tial output (or, similarly, the logarithm of the ratio betweeneffective and potential output).

By combining relation (6) with the inflation Eq. (4), in

xt = −��htdt

(7)

Positive levels of demand require one of the two follow-ing scenarios:

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(i) The MMD is positive and the effective level of output issmaller than the potential level;

ii) The MMD is negative and the effective level of outputis larger than the potential level.

Thus, we are stating that periods of recession (defineds the time periods in which the output gap is negative)re periods of temporary overproduction: producers wanto sell the generated goods, but demand is too low to coveruch requirement; in this case, excess supply is character-stic of periods of recession. Periods of expansion, in whichhe output gap is positive, are periods of underproductionr a negative MMD: people want to buy more, but the sell-ng capacity is constrained, what originates delivery lags or,n other words, an excess demand.

Eq. (7) requires a few more remarks. First, note the role ofarameters; because they can assume only positive values,oth a stronger reaction of firms to price changes in whategards investment decisions (a larger �) and a higher sen-itivity of prices to changes in the MMD (a larger �), implyhat demand will rise for a given MMD – output gap ratio.

A second relevant comment relates to the characteri-ation of market equilibrium. Market clearing exists whent = 0 and xt = 0; therefore, the demand equilibrium levelannot be withdrawn from Eq. (7). In the hypothetical mar-et clearing situation, because the MMD is constant andqual to zero, dynamic relation (2) states that demand isqual to the level of income (which coincides with out-ut). Therefore, Eq. (7) is a disequilibrium relation, whichllows relating aggregate demand, the output gap andhe MMD when effective output differs from potentialutput and a non-zero MMD value holds. Consequently,y resorting to the investment Eq. (5) to eliminate thenflation rate variable from the dynamic analysis, we areot stating an equilibrium condition; on the contrary, were presenting a relation that measures market disequi-ibria in relation with the difference between what isffectively produced and the potential capacity of the econ-my.

The simple case we are considering also implies thatrices do not change (�t = 0) in the Walrasian equilibrium,ince in this circumstance the MMD is zero and potentialnd effective output are identical.

A positive price change is found for a positive output gapnd deflation will exist in scenarios of negative output gap.hese remarks are just the interpretation of Eq. (6). A moreealistic approach would require adding a constant posi-ive level of inflation to the left hand side of expression (6).n this way, a zero output gap would not mean zero infla-ion, and we could have a recession (negative output gap)ithout having necessarily a scenario of deflation. We omit

his parameter, since it would introduce just an additionalnnocuous element to the structure of the model; thus, ouretting is one in which market equilibrium (yt = dt; ht = 0) islso a synonymous of xt = 0 and �t = 0.

A last remark concerning Eq. (7) relates to the discrete
ime nature of the model and to the behavior of variables inhe transitional dynamics phase. If the model’s equationsere presented in continuous time (as in HGDH) it woulde possible to consider eventual scenarios in which phasesf recession (xt < 0) and phases of expansion (xt > 0) co-exist
ic Dynamics 20 (2009) 221–238 225

in the same transitional process. In discrete time, accordingto Eq. (7), the transition from one state to the other is notfeasible because it would imply that in some moment onewould have xt = 0 and to this would correspond an infinitelevel of demand.

Therefore, given a set of initial values (k0,h0) such thath0 > 0 and x0 < 0 or, alternatively, h0 < 0 and x0 > 0, the con-vergence to the steady state will be such that the outputgap remains below zero (in the first case) or above zero(in the second case). Transitional dynamics does not con-template circumstances in which, starting from one type ofdisequilibrium, the equilibrium line is in some point in timecrossed such that in the long-run the other type of disequi-librium is achieved. This is particularly true if one considersa trivial dynamic process with a fixed-point stable steadystate (and, thus, with convergence from (k0,h0) to a unique(k, h)). As we will see below, less trivial dynamic pro-cesses arise when a period doubling bifurcation introducesbounded instability into the long-run outcome; neverthe-less, the basic idea is maintained: this is a framework ofpersistent disequilibria and the equilibrium case emergesas a particular case that cannot be contemplated by Eq. (7)and that is described by the scenario yt = dt, ht = 0, xt = 0,�t = 0.

In this discussion, one must recover the neoclassical –Keynesian debate as presented in Section 1. Business cyclesare commonly defined as the settings in which overpro-duction and underproduction alternate. According to thelogical arguments of the above two paragraphs, the tran-sitional dynamics and the fixed-point steady state thateventually generated do not correspond to a cycles’ setting;instead, this is the neoclassical outcome in which cycles donot exist under the assumed deterministic setup (they willoccur only as the result of exogenous disturbances that weignore). Obviously, when adding a constant term to Eq. (6)it is possible to have a transitional dynamics phase in whichthe steady state corresponds to a situation of recession(expansion) when the initial state was one of expansion(recession), because in this case the infinite demand resultwould not coincide with xt = 0 but with xt equal to suchconstant; however, this does not change the main argu-ment: business cycles do not occur if the outcome of themodel is of the neoclassical kind (endogenous fluctuationsare absent).

The endogenous business cycles outcome will occur ifinstead of a fixed-point result, fluctuations of some peri-odicity or even complete a-periodicity characterize thelong-term scenario. The Keynesian result, that emerges forsome combinations of parameters, will in fact be com-patible with the conventional interpretation of cycles asa sequence of alternate moments of overproduction andunderproduction, as the figures in Section 5 (particularly,Fig. 7) allow to confirm. In such figures, the moments ofoverproduction (ht > 0) dominate relatively to the time peri-ods in which underproduction (ht < 0) prevails. With theinclusion of the referred constant, it would be possible to
find circumstances in which overproduction and under-production situations would exist in more or less identicalnumber and following an alternate pattern.
We are now in conditions of stating the dynamic prob-lem,

Econom
Definition 1 (The NEDyM). The two-dimensional growthsystem, that combines Keynesian and classical features, iscomposed by Eqs. (1) and (2). In Eq. (2), output originateson a neoclassical production function and demand can beobtained through the investment Eq. (5) plus the simpleconstant marginal propensity consumption function. Priceevolution, as described by (4), has also a relevant role sinceit allows for establishing a relation between the output gapand demand that serves the purpose of closing the model.

Relatively to the problem in the definition, note twothings:

(i) The described dynamical system not only is a two equa-tions system, it also has only two endogenous variables:capital and the MMD;

(ii) As referred in the introduction, there is a clear co-existence between Keynesian and classical elements.The first relates to the shape of the consumptionfunction, the lack of equilibrium between output anddemand and the consideration of a behavioral invest-ment function; the second is present in the shape of theproduction function and on how capital accumulationis modelled.

3. Results on local stability

The low dimensionality of the model allows for obtain-ing some generic local dynamic results. In the presentationof these results, we simplify notations by taking = ��.

We begin by characterizing the steady state. This isdefined as follows,

Definition 2 (Steady state). A steady state or balancedgrowth path is a set {k, h, y, d, �, c, i} of constant val-ues, which can be determined by imposing conditions k ≡kt+1 = kt and h ≡ ht+1 = ht to Eqs. (1) and (2).

By applying Definition 2, it is straightforward to arriveto the following outcome,

Proposition 1. The steady state exists, it is unique andit is characterized by the group of relations that follows:

(i) y = d; (ii) f (k)k

= ı1−b ; (iii) � = 1

� ln(f (k)y∗

); (iv) h =

− 1

ln(f (k)y∗

)f (k); (v) c = bf (k); (vi) i = (1 − b)f (k).

Proof. See Appendix A.

The steady state relations deserve some comments: first,note that independently of the long-term value of the MMD,production and demand assume identical values; second,the average product of capital is constant in the steady stateand it is as much higher as the larger are the values of thedepreciation rate and of the marginal propensity to con-sume; third, prices rise in the long-run if a positive outputgap persists and decline otherwise; fourth, the MMD isnot only negative for a positive output gap, but it is also
as more negative as the larger is the value of the effec-tive output (a symmetric result can be established); fifth,because demand and income are identical in the long-run,investment can be expressed in the form of income times aconstant marginal propensity to save (i.e., in the long-run,
ic Dynamics 20 (2009) 221–238

households’ savings are integrally used by firms in theirinvestment projects); this motivates the presentation ofpotential investment in (5) as i* = (1 − b)y*.

To study local dynamics, one needs to linearize the sys-tem in the vicinity of {k, h}. The linearized system is[kt+1 − kht+1 − h

]= J

[kt − kht − h

],

with J =

⎡⎣ 1 − ı−

(b+ 1

x

)f ′(k) −

x(1 + 1

x

)f ′(k) 1 +

x

⎤⎦ (8)

with x = ln(y/y∗).An important result regarding local stability is presented

in Proposition 2.

Proposition 2. The existence of a negative output gap is anecessary condition for local asymptotic stability.


In the chosen terminology, the expression ‘local asymp-totic stability’ refers to any circumstance in which there isa coincidence between the stable eigenspace and the statespace of the system. In other words, the term is associatedto the case in which the two eigenvalues of the Jacobianmatrix in (8) lie inside the unit circle. This result is inde-pendent of how the convergence to the steady state takesplace: monotonically (if the two eigenvalues are real andpositive), through improper oscillations (if the two eigen-values are real and they are not both positive) or througha spiral movement with decreasing amplitude in time (ifthe eigenvalues are a pair of complex values). Proposition3 makes the distinction between node stability and focusstability.

Proposition 3. Assume that x < 0. If a stable fixed-pointexists, this corresponds to a stable node if the following con-dition is satisfied:

2

(2x)2−

[1− ı

2−1

2

(b−1x

)f ′(k)

]

x−1+ı+

(b+ 1

x

)f ′(k)

+[

1 − ı

2− 1

2

(b+ 1

x

)f ′(k)

]2

> 0

If the above inequality is of opposite sign, it becomes a nec-essary condition for the equilibrium point to be a stable focus(i.e., for the convergence to the stable equilibrium to occur inspiral).


Two remarks about Proposition 3: first, we reemphasizethat the presented condition is a necessary condition for thefixed-point to be a stable node (we have not yet imposedadditional conditions that ensure the presence of asymp-totic stability). Second, the expression in the propositionwas presented in such a way that it can be solved for the
combination of parameters . Relatively to this value, thenecessary stable node condition will be the area above thecorresponding parabola.
Sufficient conditions for local asymptotic stability arethe ones in Proposition 4.

Econom

Px

(

P

rafiPc

f

Crb

t

P

Pd

-

-

P

spiflauseNpb

I16

Pvs

P


roposition 4. Local asymptotic stability holds if, besides¯ < 0, the following inequalities are satisfied,

(i) [2 − ı+ (1 − b)f ′(k)] x + 4 − 2ı+ 2(b+ 1

x

)f ′(k)> 0;

ii) [1 − ı+ (1 − b)f ′(k)] x − ı−(b+ 1

x

)f ′(k)< 0.

roof. See Appendix A.

Compiling the results in Propositions 2–4, the stabilityesult is the following: the fixed-point is a stable node ifll the displayed conditions in Propositions 2–4 are satis-ed; the fixed-point is a stable focus if the inequality inroposition 3 is of opposite sign and the other referredonditions hold.

In what concerns the value of , Proposition 4 has theollowing corollary,

orollary of proposition 4. Consider again x < 0. Stabilityequires the combination of parameters to be bounded fromelow and from above:

∈(ı+ (b+ (1/x))f ′(k)1 − ı+ (1 − b) f ′(k)

x; −4−2ı+2(b+(1/x))f ′(k)2−ı+ (1−b) f ′(k)

x

).

Evidently, the lower bound will be zero if the first value ofhe set is negative.


Regarding the absence of stability,

roposition 5. In the case x < 0, two additional localynamic results are obtainable, besides asymptotic stability,

Saddle-path stability, under > − 4−2ı+2(b+(1/x))f ′(k)2−ı+(1−b)f ′(k)

x;

Instability, under < ı+(b+(1/x))f ′(k)1−ı+(1−b)f ′(k)

x.


The transition of regions of asymptotic stability toaddle-path stability or instability implies that bifurcationoints are crossed. The point in which asymptotic stabil-

ty gives place to saddle-path stability corresponds to aip bifurcation point. In this case, one of the eigenvaluesssumes the value −1, while the other remains inside thenit circle. A Neimark–Sacker bifurcation occurs in the tran-ition between the stability area and the area in which theigenvalues are complex with modulus larger than one.ote that, according to Proposition 5 (or the Corollary ofroposition 4), the unique required condition for any of theifurcations to occur is that the specified border values ofmust be larger than zero.Let us turn to the case in which the output gap is positive.

n this case, one of the eigenvalues of J is always larger thanand, therefore, asymptotic stability is absent. Propositionstates the possible local dynamic results.

roposition 6. Let x > 0. Saddle-path stability holds for aalue of inside the set presented in the Corollary of propo-ition 4. For values of outside the set, instability prevails.


ic Dynamics 20 (2009) 221–238 227

The instability result may correspond to two differenttime trajectories, depending on the stability condition thatis violated. If 1 + Tr(J) + Det(J)< 0, along with 1 − Tr(J) +Det(J)< 0, then one of the eigenvalues is higher than 1 andthe other lower than −1, and they are both real values. Inthis case, the trajectories will oscillate improperly as thesystem departs from the fixed-point. When the determi-nant of the Jacobian matrix is above unity, the divergenceprocess is determined by the existence of an unstable focusfixed-point.

A better understanding of the previous set of results isachieved through a graphical illustration of the stabilitypossibilities. Fig. 1 is a diagram that relates the values ofthe trace and the determinant.

In Fig. 1, we draw the three bifurcation lines; the areainside the inverted triangle formed by these three lines isthe area of stability. The two bold lines represent the twocases in Proposition 2: the one in which asymptotic stabil-ity is possible (to the left of the bifurcation line 1 − Tr(J) +Det(J) = 0) and the one in which asymptotic stability is notadmissible (to the right of this bifurcation line). In the firstcase, asymptotic stability can give place to a saddle-pathresult, if the flip bifurcation line is crossed; instability alsoarises for values of parameters such that the determinantof the Jacobian matrix becomes a value larger than 1. Whenthe condition 1 − Tr(J) + Det(J)> 0 is no longer verified,saddle-path stability holds as long as the other two stabilityconditions are satisfied. Otherwise, if any of such conditionsfails to hold, asymptotic instability will prevail according towhat was established in Proposition 6.

The stability case is straightforward to characterize froma dynamic analysis point of view. Independently of the ini-tial state of the system (k0,h0), if this is in the vicinity ofthe steady state, then both variables will converge to thelong-term steady state. Such result is coincident with theneoclassical growth outcome of a balanced growth path:given the decreasing returns to capital, the economy con-verges to a constant long-term value of capital and output(and, consequently, constant levels of consumption andinvestment). The main difference relatively to the neoclas-sical model is that this outcome is achieved for a level ofoutput below the optimal (this is an intuitive result if werecall that we have introduced a series of inefficiencies inour formulation, alongside with a low impact interest ratepolicy) and for a steady state MMD that is above zero (pro-duced output is sold after a delay, which is also a reflectionof our model’s inefficiencies).

From the point of view of local analysis, the situationof saddle-path stability delivers some interesting results.Thus, let us suppose that y > y∗ and that the condition inthe corollary of Proposition 4 holds.

Proposition 7. If the system is saddle-path stable, the saddletrajectory is

h − h = − (1 + (1/x))f ′(k)(k − k) (9)
t 1 − ε1 + ( /x) t
with ε1 the eigenvalue of J inside the unit circle.


228 O. Gomes / Structural Change and Economic Dynamics 20 (2009) 221–238

dynami
Fig. 1. Characterization of local
If x > 0, the stable trajectory in (9) is negatively sloped,meaning that if the convergence to the steady state is donethrough the saddle trajectory, then as the amount of capitalrises, the value of the MMD declines (or vice-versa).

The steady state may be disturbed by changes in any ofthe parameter values. For instance, if the impact of inflationover investment decisions increases (i.e., � rises and, thus, also rises), we know from Proposition 1 that the steadystate stock of capital remains unchanged, while the MMDbecomes a smaller value. From (9), the slope of the stabletrajectory decreases in absolute value, that is, the trajectorybecomes flatter. Therefore, when the reaction of invest-ment to changes in inflation increases, this will reduce thelong-term level of the MMD (that in the considered case ispositive) and the impact over the convergence to the steadystate is such that for a given change in the stock of capi-tal, the change in the MMD will be less pronounced. Fig. 2illustrates the case.

The set of propositions derived along this section tendsto associate stability predominantly with case ht > 0 and
xt < 0. From an economic intuition point of view, and takingin consideration market inefficiencies and the policy inter-vention that are characteristic of Keynesian economics, thisis a reasonable outcome: the economy tends to rest in a
Fig. 2. Saddle-path trajectory. The effect of an increase in �.

cs. Trace-determinant diagram.

state where production capabilities are not completely ful-filled and where some inventories always exist.

4. Specific production functions

To better understand the dynamics of the two-dimensional NEDyM model, we now adopt two explicitfunctional forms for the production function. We also con-sider a specific value for the potential output. Potentialoutput is defined as the steady state level of output thatcan be derived from an optimal control problem of utilitymaximization. We present this problem as

Max+∞∑t=0

ˇtU(ct) subject to kt+1 − kt = f (kt) − ct − ıkt,

k0 given (10)

with ˇ∈ (0,1) the discount factor.Taking a simple logarithmic utility function, U(ct) = ln ct,

the computation of first order conditions of this Ramseyproblem leads to the well known equation of motion forconsumption ct+1 = ˇ(1 − ı+ f ′(kt+1))ct . The evaluation ofthis equation in the steady state will give us the optimallong-term constant value of capital, which obeys to f ′(k∗) =1/ˇ − (1 − ı). Potential output is, then, defined as y* = f(k*).

4.1. Cobb–Douglas production function

The first case we consider takes a Cobb–Douglas produc-tion function: f (kt) = Ak˛t . Parameter A > 0 is a technologicalindex, and ˛∈ (0,1) is the output-capital elasticity. Withthis production function, the potential output is explicitlypresentable as y∗ = A1/˛(˛/(1/ˇ − (1 − ı)))˛/(1−˛).

Now, the steady state results can all be given as functionsof the assumed array of parameters. Recall from Proposition1 that, in the steady state, income and demand are identical,
what allows for presenting the steady state stock of capitalas k = ((1 − b)A/ı)1/(1−˛). The long-term capital stock riseswith the level of technology and with the output-capitalelasticity and it falls as the marginal propensity to consumeand the depreciation rate increase.

Economic Dynamics 20 (2009) 221–238 229

tiinTilftocltrii

hbs(rpeittipsif

l

Pb

P

otcaTrabn

smtoimrd

Table 1Calibration in the Cobb–Douglas case.

Parameter Value Source

A 1 Cellarier (2006)˛ 1/3 Hallegatte et al. (2008)

ces is straightforward:

ε1,2=0.7325+2.7924 ±0.5√

31.189 2−4.976 +0.2861(b = 0.7);


Consumption and investment are, respec-ively, given by c = bA1/(1−˛)((1 − b)/ı)˛/(1−˛) and= [(1 − b)A]1/(1−˛)ı−˛/(1−˛). Both consumption andnvestment steady state levels benefit from a better tech-ology level and from a lower rate of capital depreciation.he impact of the propensity to consume over steady statenvestment is also unequivocal (a higher b damages theong-term capacity to invest), but it is not so straight-orward in terms of long term consumption; computinghe derivative of the steady state consumption level inrder to b, one gets a positive value for b < 1 −˛; hence, weonclude that the marginal propensity to consume benefitsong-run consumption only if this constant is lower thanhe output-labor elasticity. A higher b means that too manyesources are withdrawn from the productive processn order to guarantee that consumption rises with anncreasing share of consumption.

In what concerns the MMD, the steady state becomes¯ = −(1/ )(˛/(1 − ˛))A1/(1−˛)((1 − b)/ı)˛/(1−˛) ln((1 −) · (1/ˇ − (1 − ı))/˛ı). Finally, we can look at the steadytate inflation rate: � = ˛/((1 − ˛)�) ln((1 − b) · (1/ˇ −1 − ı))/˛ı). As in the general case, the most meaningfulesult regarding this steady state value is the fact thatrices rise with a positive output gap and decline oth-rwise. Relatively to the last two steady state results, its not straightforward to perceive the impact of some ofhe parameters over those results. Note, as an illustration,he role of the discount factor: the more intensely futures discounted (lower ˇ) the higher is inflation (if this isositive) or the lower is deflation (for a negative price rise);imilarly, a higher discount rate lowers the MMD level (ift is positive, it becomes closer to zero; if it is negative, italls even more).

Concerning the sign of the output gap, we have the fol-owing result,

roposition 8. A positive steady state output gap requires< (1−ˇ)/ˇ+ı(1−˛)

(1−ˇ)/ˇ+ı .


According to the stability results in Section 3, the signf the output gap is of fundamental importance. Asymp-otic stability requires a negative output gap and, thus, foronstant values of ˇ, ı and ˛, stability is found for a rel-tively high value of the marginal propensity to consume.hus, the economy has no advantage in allocating too manyesources to investment, because although this allows forhigher steady state income level, it also raises the possi-ility of losing stability, implying that the steady state willo longer be accomplished.

To address local dynamics, we should note that for thepecific technology under appreciation, the steady statearginal product of capital is f ′(k) = ˛ı/(1 − b). Replacing

his, and the several steady state values, in the propositions
f Section 3, we would obtain conditions for the character-zation of local stability. Since this exercise does not add
uch information to the precedent generic results, we justemark that the relation in Fig. 1 between the trace and theeterminant of matrix J is, with the Cobb–Douglas technol-

ı 0.067 Guo and Lansing (2002)ˇ 0.962 Guo and Lansing (2002)b 0.7; 0.9 2

ogy,

Det(J) = Tr(J) − 1 − ı(1 − ˛)2

˛ ln((1 − b)(1/ˇ − (1 − ı))/˛ı) (11)

A negative output gap will allow (11) to cross the stabilityarea.

The exploration of a numerical example conductsto more tractable results. The calibration in Table 1 isconsidered.2

We let the combination of parameters be any positivevalue, that is, we elect this as the bifurcation parame-ter. With the above values, we compute steady states forthe various variables. First note that the potential outputis y* = 1.7691. The steady state level of capital comes k =9.4748 for b = 0.7 and k = 1.8234 for b = 0.9. To these capi-tal levels correspond the following output values: y = 2.116(b = 0.7) and y = 1.2217 (b = 0.9). We confirm that the lowerpropensity to consume implies a positive output gap (and,thus, the impossibility of asymptotic stability), while thelarger propensity to consume leads to a negative outputgap.

The other steady state values are: c = 1.4812 (b = 0.7),c = 1.0995 (b = 0.9) (observe that the second steady statelevel of consumption is lower than the first, despite thefact that in the second case the propensity to consume ishigher); i = 0.6348 (b = 0.7), i = 0.1222 (b = 0.9). The infla-tion rate comes � = 0.1791/� (b = 0.7), � = −0.3702/�(b = 0.9); as we should expect, inflation exists when theoutput gap is positive, and deflation arises for a nega-tive output gap. Finally, concerning the MMD, we get h =−0.3789(1/ ) (b = 0.7), h = 0.4523(1/ ) (b = 0.9).

To address local dynamics, it is possible to present theJacobian matrices of the system, considering each one ofthe propensities to consume. These matrices are:

J(b=0.7) =[

0.4651 −5.5847 0.4902 1 + 5.5847

];

J(b=0.9) =[

1.3352 2.701 −0.3799 1 − 2.701

]

The computation of the eigenvalues of the above matri-

2 b needs to be higher than 0.7907 to exist a region of stability(Proposition 8); therefore, we consider two values that produce two dif-ferent outcomes.

Econom

ε


1,2=1.1676−1.3505 ±0.5√

7.2954 2−2.2937 +0.1124(b = 0.9).

Asymptotic stability requires both eigenvalues to beinside the unit circle. For the first set of eigenvalues thisdoes not happen. One of the eigenvalues is always above1 for positive values of . The other eigenvalue lies aboveminus one for any positive value of , and below 1 for a below 1007.6. Since it does not make much sense to assumesuch a huge value for the combination of parameters (this would require an unreasonably high impact of infla-tion over investment decisions), we can guarantee that forb = 0.7 saddle-path stability is found (one of the eigenvalueslies inside and the other one outside the unit circle).

In the other case, b = 0.9, one of the eigenvalues is alwaysinside the unit circle, while the other eigenvalue lies insidethe unit circle as long as < 0.8843. When the combinationof parameters reaches this value, a flip bifurcation occurs(the eigenvalue assumes the value −1). When is above0.8843, then local dynamics are characterized by saddle-path stability. This result is confirmed with the globalanalysis of the following section. For the assumed parame-terization, the Neimark–Sacker bifurcation does not occurunder any positive value of . We can state an additionalresult by recovering Proposition 3. The fixed-point is a sta-ble node for values of obeying 7.2954 2 − 2.2937 +0.1124> 0, that is, for < 0.0674 ∧ 0.2537 < < 0.8843.Any other value of for which there is asymptotic stabilitycorresponds to a stable focus equilibrium.

Let us return to the case b = 0.7 in order to obtainthe expression of the stable trajectory in the saddle-path case. Recalling Eq. (8), that gives us the saddle-trajectory, one has in the present case: ht = −(0.3789/ ) +(4.6445/�) − (0.4902/�)kt , with � = 0.2675 + 2.7923 +0.5

√31.189 2 − 4.976 + 0.2861. Note that � < 0 would

mean that the slope of the stable trajectory is positive,being negative in the symmetric case. Since � > 0, ∀ > 0,under the imposed conditions and calibration, the stabletrajectory is negatively sloped; as the stock of capital risestowards equilibrium, the MMD value falls.

4.2. CES production function

In this section, we consider an alternative neoclassi-cal production function (as the Cobb–Douglas function, itexhibits positive and diminishing marginal returns and theInada conditions hold). This is a constant elasticity of sub-stitution (CES) production function, and we present it as inBarro and Sala-i-Martin (1995),

f (kt) = A[a(mkt)� + (1 − a)(1 −m)�]1/� (12)

In production function (12), A > 0 is again the technologyindex, and 0 < a < 1, 0 < m < 1, �∈ (−∞,1)\{0}. The elasticityof substitution between capital and labor is 1/(1 −�). TheCES function has, as limit cases, other shapes of produc-
tion functions. When �→ 0, the elasticity of substitutionapproaches 1, and the production function approaches aCobb–Douglas form. When �= 1, the production functionbecomes linear (the elasticity of substitution is infinite).Finally, when �→ −∞, we approach a Leontief production
ic Dynamics 20 (2009) 221–238

function with a fixed-proportions technology (the elasticityof substitution is zero).

The CES production function is more demandingto deal with analytically. In the Appendix A.10, wecompute the potential output as defined earlier. Theoutcome is y∗ = A((1 − a)(1 −m)�z/(z − am�)), with z ≡(1/ˇ − (1 − ı)/(Aam�))�/(1−�).

The proposed model implies, as a generic result,that in the steady state income and demand are equal,and therefore it is once again straightforward to obtainthe long-term stock of capital from the capital accu-mulation equation. This is given by k = ((1 − a)1/�(1 −m)/[ı/((1 − b)A)� − am�]1/�). Observe that, as we shouldexpect, the impact of parameters A, b and ı over thesteady state capital stock is qualitatively the same asin the Cobb–Douglas case. The steady state level of

output is y = A((1 − a)(1 −m)�ω/(ω − am�))1/�, with ω ≡(ı/((1 − b)A))�.

Steady state values of consumption and investment are,

respectively, c = bA((1 − a)(1 −m)�ω/(ω − am�))1/� and

i = (1 − b)A((1 − a)(1 −m)�ω/(ω − am�))1/�.The steady state MMD and inflation rate depend on the

output gap and, as discussed in the general case, we observethat a positive output gap implies a negative MMD level anda positive inflation rate.

Proposition 9. With a CES technology, the necessary con-dition for stability x < 0 implies the following constrainton b, b > (ϑ1/�A− ı/(ϑ1/�A)), with ϑ ≡ (am�z�/(z� −(1 − a)1−�(1 −m)�·(1−�)(z − am�)�)).


Proposition 9 shows that, similarly to the Cobb–Douglastechnology case, a lower bound is imposed on the marginalpropensity to consume in order to asymptotic stability tobe feasible.

An example illustrates the CES case. The values ofparameters A, ı and ˇ are the ones considered in theCobb–Douglas example, and we take a = 0.4 and m = 0.7. Theelasticity of substitution between capital and labor is 0.9 (avalue near the Cobb–Douglas case); this elasticity of sub-stitution means that �= −1/9. Once again, the value of isleft to be the bifurcation parameter. To choose a value for b,we first look at Proposition 9 under this particular example.For the selected array of parameters, z = 1.146, and y* = 1.077.It is also true that ϑ = 1.3942. Computation implies the fol-lowing necessary stability condition: b > 0.0074.

Thus, in the case in appreciation, the steady state out-put gap is negative for every value of the propensity toconsume, except extremely low values, which from anempirical plausibility point of view are negligible. This is asignificant departure from the Cobb–Douglas case. Despitethe chosen elasticity of substitution in the CES case beingclose to the one in the Cobb–Douglas scenario, the value of
the propensity to consume required to find stability can besignificantly different. Because in the present case any rea-sonable propensity to consume implies a long-term statewhere output is below potential, we select a reasonablevalue for b; this is b = 0.7.

Economic Dynamics 20 (2009) 221–238 231

ωisfl

c0

J

ε

stcor0f

ctaiklsap

ftoddrc

5

adtrou

bmcmf

between the two endogenous variables, for a value of under which chaotic motion exists ( = 1.05). Note that,although we have chosen to work with the case in whichthe output gap is negative and the MMD is positive, since

3


With the selected array of parameters, one computes= 1.1812 and y = 0.597. We confirm that the output gap

s negative, meaning that we should encounter an area oftability for a given interval of values of . Additionally, aip bifurcation will be identified.

The Jacobian matrix is, for the system under appre-iation (this is directly computed from (7), with f ′(k) =.0787),

=[

1.0113 1.6949 −0.0547 1 − 1.6949

]

The eigenvalues of the Jacobian matrix are:

1,2 = 1.0057 − 0.8474

± 0.5√

2.8727 2 − 0.3325 + 0.0001

One of the eigenvalues is inside the unit circle for< 21.005, that is, it lies inside the unit circle for any rea-

onable parameterization. The other eigenvalue lies insidehe unit circle for < 1.213. When = 1.213, the systemrosses a flip bifurcation and, consequently, the possibilityf endogenous fluctuations arises. The stable fixed-pointespects to a stable node equilibrium when 2.8727 2 −.3325 + 0.0001> 0, i.e., for 0.0003 < < 0.1154. A stableocus will mean that 0 < < 0.0003 or 0.1154 < < 1.213.

In the case of saddle-path stability ( > 1.213), one mayompute the expression of the stable trajectory. As an illus-ration, assume that = 1.5; for this value, the eigenvaluebove −1 is ε1 = 0.9557 (the other is ε2 = −1.4866). Recover-ng the stable arm in (8), this comes ht − h = −0.2782(kt −¯ ). In this example, assuming that the stable path is fol-owed, a one point increase in the stock of capital occursimultaneously with a 0.2782 points decrease in the MMD,s the convergence to the steady-state eventually takeslace.

In an overall evaluation, and despite the differenceound about the constraint bounding parameter b in ordero separate the cases of positive and negative steady stateutput gap, we find similar results when comparing theynamics of the model when to its structure underlie twoifferent production technologies. In both cases, stabilityequires to be lower than a bifurcation point that oncerossed leads to saddle-path stability.

. Comparing global dynamic results

In this section, we resort to the numerical examples ones presented earlier to make a graphical evaluation of globalynamics. We find that, for both types of production func-ions, the flip bifurcation gives place to a period doublingoute to chaos, such that one may identify the presencef endogenous cycles for certain arrays of parameter val-es.

The graphical analysis includes the presentation of a
ifurcation diagram, long-term attractors, time series of theost relevant variables and the computation of Lyapunov
haracteristic exponents (LCEs). LCEs are a well acceptedeasure of sensitive dependence on initial conditions, a

eature that constitutes one of the main properties of

Fig. 3. Bifurcation diagram [Cobb–Douglas technology] (kt , ).

chaotic systems.3 We begin by analyzing the Cobb–Douglascase, under the parameterization in Table 1.

Relevant global dynamic results only exist for b > 0.7907,the case in which the condition 1 − Tr(J) + Det(J)> 0 issatisfied. Thus, we work with b = 0.9. Recall that for thispropensity to consume, local dynamics has pointed to sta-bility under < 0.8843 and saddle-path stability otherwise.Fig. 3 displays the bifurcation diagram of variable kt as wechange the value of .4

The bifurcation diagram furnishes a visual confirma-tion of the existence of a stability area to the left of thebifurcation point (the steady state value of the capital vari-able that one has computed in Section 4, k = 1.8234, isobtained) and, once the bifurcation takes place, it is pos-sible to observe that cycles of growing periodicity arise asthe value of rises. Chaotic motion is found for values of slightly above 1. The presence of chaos is confirmed withthe presentation of LCEs in Fig. 4.

In a two-dimensional system, two LCEs can be com-puted. If one of them is a positive value, then there isexponential divergence of nearby orbits, that is, time seriesare sensitive to their initial values (a small difference inthe initial values means, for a chaotic system, completelydifferent trajectories over time). Thus, an LCE above zero issynonymous of the presence of chaotic motion. We observethat the contents of Fig. 4 confirm, in fact, the informa-tion furnished by Fig. 3. In particular, one of the LCEsassumes a positive value for most of the interval ∈ (1;1.06).

Fig. 5 presents the long-run attractor of the relation

See Alligood et al. (1997), Lorenz (1997) or Medio and Lines (2001)for detailed analysis of chaotic systems and respective applications toeconomics.

4 This figure, and all the following, are drawn using IDMC software(interactive dynamical model calculator). This is a free software programavailable at www.dss.uniud.it/nonlinear, and copyright of Marji Lines andAlfredo Medio.

http://www.dss.uniud.it/nonlinear

232 O. Gomes / Structural Change and Economic Dynamics 20 (2009) 221–238

Fig. 4. Lyapunov characteristic exponents [Cobb–Douglas technology](0.75 < < 1.06).

this is the case that allows for stability and for a bifurcationthat generates endogenous cycles, we observe in the figurethat the MMD can assume negative values, as variable ht

fluctuates in a region bounded above by 2.3 and below by−0.3 (approximately). Thus, although the MMD is, on aver-age, around 1.3, fluctuations will imply that the MMD canfall below zero, even in the circumstance one is consideringof a negative output gap. Another curious and relevant fea-ture in Fig. 5 is the negatively sloped shape of the attractor.This seems to make sense if one thinks that more capitaldirectly leads to increased output, and with more outputthe higher is also the value of the output gap (recall that thepotential output is modelled as a constant); therefore, theinformation in the figure is in accordance with the inverserelation one has established between the output gap andthe MMD.

Figs. 6 and 7 display the long-term time series (the first
10,000 observations are excluded) of the physical capitaland MMD variables for the same value of that allowed fordrawing the previous attractor. Now, one directly observesthe presence of endogenous fluctuations, that we haveinterpreted earlier as the result of a prevalence of the Key-
Fig. 5. Attracting set [Cobb–Douglas technology] (kt , ht); = 1.05.

Fig. 6. Time series of kt [Cobb–Douglas technology]; = 1.05.

nesian features of the model, relatively to the neoclassicalproperties, which in turn dominate in the balanced growthcase, found for lower values of .

One final figure is presented for the Cobb–Douglas case.This calls the attention for the need of selecting initial val-ues of the endogenous variables that allow for convergenceto the long-run state (being this a fixed-point, any periodicpoint or a chaotic attracting set). As we see in the basin ofattraction of Fig. 8, not all combinations of initial values arefeasible. If one starts from a point in the dark area (outsidethe basin of attraction), the system just diverges to infinity.

Relatively to the CES case, the qualitative results are notsignificantly different from the ones just obtained for thecase with a Cobb–Douglas production function. To save inspace, we just present the bifurcation diagram, similar tothe one in Fig. 3, and the attractor, which has also a sameshape as the one in Fig. 5.

To present the bifurcation diagram in Fig. 9, we takethe same set of parameters used in the local dynamics
example. In this, asymptotic stability was guaranteed under < 1.213. Then, a flip bifurcation occurs and, locally, saddle-path stability sets in. The figure confirms these results, andit reveals that also in this case, the flip bifurcation originates
Fig. 7. Time series of ht [Cobb–Douglas technology]; = 1.05.

O. Gomes / Structural Change and Economic Dynamics 20 (2009) 221–238 233

aw

etav

tltwnta

ftb

en

Fig. 8. Basin of attraction [Cobb–Douglas technology]; = 1.05.

process of cyclical motion with increasing periodicity andhere a region of chaos is observable.

Comparing Figs. 3 and 9, one realizes that differences areminently quantitative; for the selected parameter values,he steady state stock of capital is larger in the CES case,nd, also in this case, the flip bifurcation occurs for a higheralue of the combination of parameters .

Observing Fig. 9, we see that, for instance, for = 1.5here is chaotic motion. Fig. 10 presents, for this value, theong-term attractor (once again, the first 10,000 observa-ions are withdrawn). As one would expect, the similaritiesith the attractor in Fig. 5 exist. What one has said aboutegative values for the MMD and for the negative relationhat is established in the long-term, applies to the CES cases well.

It is possible to conclude that the type of the productionunction does not change the main dynamic properties ofhe model under a global analysis point of view, because
oth production functions are neoclassical in nature.
The economic interpretation for the emergence of thevidenced endogenous cycles might reside on the perma-ent tension between the real side and the monetary side

Fig. 9. Bifurcation diagram [CES technology] (kt , ).

Fig. 10. Attracting set [CES technology] (kt , ht); = 1.5.

of the economy. According with Eq. (4), prices change inorder to adjust for real inertia; however, by changing pricesthere is, according to Eq. (5), an automatic effect over thedecisions of private agents, because price changes will bereflected in investment decisions and these determine howmuch to produce relatively to the economy’s potential interms of generation of income.

In this way, we can identify a never ending process thatis likely to perpetuate a cyclical motion process: misalign-ments between supply and demand lead to price changes(e.g., temporary underproduction implies a rise in prices)which in turn triggers an investment change with directeffects over production. This process of mutual influenceguarantees, under some combinations of parameter val-ues, a convergence to the market equilibrium result but, asone as discussed, everlasting fluctuations around the steadystate value may subsist.

6. Business cycles stylized facts

The model developed in previous sections reveals thatthe dynamic relation between endogenous variables gen-erates a process of period doubling bifurcations leading toa region of periodic and a-periodic fluctuations that sug-gests the presence of endogenous business cycles undersome specific economic conditions (for some combinationsof parameter values). As Fig. 6 clearly evidences, the gener-ated cycles cannot reflect, in a straightforward manner, realeconomic conditions; this figure presents a time series forthe stock of capital and, obviously, such dramatic changesin the amount of accumulated capital as the ones displayed,are far from what one can effectively expect in terms of thesteady state behavior of this variable (a relatively smootherevolution would be expected).

The behavior of the capital stock time series is thenpassed on to output, consumption and investment. The out-put variable is given by the production function and this
is a straightforward relation between the stock of capitaland the income level. Consumption is obtained by multi-plying output by the marginal propensity to consume andinvestment will correspond to output times the marginalpropensity to save (in the steady state).

234 O. Gomes / Structural Change and Econom

the time series of output, consumption and investment;after eliminating the transient phase, one considers aset of 50 observations for each of the variables (seeAppendix A.12).6

Fig. 11. Time series of kt with three capital varieties [Cobb–Douglas tech-nology]; = 1.05.

A way to circumvent the unappealing features of thecapital stock evolution in time when the chaotic region isreached would be to consider that the proposed frameworkis applied to a specific economic sector rather than to theeconomy as a whole; then, the sum of all the capital stocksof the various sectors would lead to an aggregate stock ofcapital with smaller relative volatility in time. To illustratethis idea, consider that the economy is composed by threesectors with stocks of capital kt

I, ktII, kt

III; these sectorsare ruled by the same dynamic process that was consid-ered throughout the previous sections. Parameter valuesare the same for each one of the sectors and the only fea-ture that distinguishes them is the initial level k0. Underchaotic motion, sensitive dependence on initial conditionswill mean that the three time series will follow completelydistinct paths and, therefore, large variations in one timeseries can be offset by variations in the opposite directionon the other capital variables, making the changes in theaggregate capital stock smoother over time.5

Fig. 11 presents the time path of such an aggregate cap-ital variable (the contribution to production of each capitalvariety is assumed identical). The standard deviation ofthe series in Fig. 6 is approximately 0.805; in Fig. 11, thestandard deviation of the corresponding series is 0.694.As one adds more sectors, the standard deviation will fallprogressively. Thus, instead of considering a unique capitalvariable, the consideration of distinct capital stocks (thatare distinguished solely by their initial values) can justifya relatively smooth (with low volatility) aggregate capitaltime series. The process by which smoothness increases
relates to the process of compiling various chaotic timeseries generated by the same deterministic process.
Having referred above that, in our model, output, con-sumption and investment time series are directly related

5 The various sectors assumption is useful to address the volatility con-cern. However, one should keep in mind that in this setting demand iscreated by aggregate income, and therefore treating each sector separatelyand simply adding the corresponding output values will be just an over-simplification, assumed to illustrate how a smoother aggregate incometime path may eventually arise as a result of production heterogeneity.

ic Dynamics 20 (2009) 221–238

to the time path of the stock of capital in the long run,one can look at how and in what extent the various timeseries reflect business cycles main stylized facts. In Kingand Rebelo (1999), a detailed presentation of the most rel-evant stylized facts on business cycles is offered. Althoughbusiness cycles differ in duration and amplitude, there areobservable regularities in the nature of macroeconomicfluctuations. These regularities involve the following items:

(1) Co-movement. Macroeconomic series relating realaggregates are typically pro-cyclical, i.e., one shouldobserve a positive contemporaneous correlationbetween consumption, investment and output. Thisis not a relevant issue under the assumptions of ourmodel, because in the steady state consumption andinvestment are constant shares of output. Thus, thecontemporaneous correlation between aggregatesshould be equal to 1, i.e., the motion of variablesover time must be simply coincidental. When in thechaotic region there is, however, a change relativelyto this result: output and consumption effectivelydisplay contemporaneous correlation but the perfectcorrelation between output and investment ariseswith a one period lag.

(2) Volatility. Empirical evidence points to a relationbetween the evolution of key macroeconomic vari-ables during the different phases of the business cycles.Relatively to the variables involved in our model, thefollowing relations are significant:- consumption is less volatile than output;- investment is three times more volatile than output.

More specifically, King and Rebelo (1999) presentdata for the US economy indicating that the relativestandard deviation between consumption and outputis 0.74 and that the relative standard deviation betweeninvestment and output is 2.93. At a first look, theproposed model seems unable to capture these differ-ences in volatility, because, as referred, consumptionand investment emerge in the steady-state as shares ofoutput. However, the formation of endogenous cyclesgenerates a departure of the consumption and invest-ment time series volatility relatively to output. For thesame benchmark example of Sections 4 and 5 thatallowed to present the chaotic capital time series ofFig. 6 (i.e., the Cobb–Douglas case with the parametervalues in Table 1 and = 1.05), one can compute as well

6 Observe, relatively to the investment time series, that negative valuesalternate with positive ones. Although the presence of negative invest-ment values seem an unappealing feature of the model’s outcome, theyare in fact the direct consequence of having capital cyclical motion arounda constant trend (in Fig. 6 the capital stock increases and decreases sys-tematically, and the periods in which it falls are necessarily periods ofdisinvestment or negative investment); a way to circumvent this out-come, is to assume that all the variables we are considering are detrendedvariables, i.e., capital, output, investment and consumption may grow, inaverage, at a constant rate, and once removed the trend of growth we endup with the on average stationary variables we are considering.

Econom
By computing the standard deviations of consump-tion and investment relatively to output, resorting tothe information in Appendix A.12, one obtains, respec-tively, 0.80 and 7.29. These values are qualitatively inaccordance with real world data (consumption is lessvolatile than output and output is less volatile thaninvestment) and although the investment result is rela-tively far from being accurate [the Real Business Cycles(RBC) theory, according with the estimates in Kingand Rebelo (1999), point to a relative standard devia-tion of investment equal to 2.95], the result concerningconsumption is much closer to the observed relativestandard deviation than the one in RBC theory (0.44).The remarkable feature of the obtained volatility resultsis that they are found under a purely deterministic set-ting; Keynesian endogenous fluctuations can explainbusiness cycles even in the absence of random distur-bances that certainly influence the time movement ofmacroeconomic aggregates.

There are additional hypotheses one can take aboutconsumption and investment in this kind of frame-work. Below, these are briefly referred; however, theyare not pursued in detail since although they sophisti-cate the basic model, they will not change significantlythe obtained volatility results.

The most obvious sophistication consists in assum-ing, as in the RBC theory, instead of a constant savingsrate, an intertemporal consumption utility maximiza-tion setup. In Gomes (forthcoming), a model involving asimilar disequilibrium framework to the one adopted inthis paper takes such optimal control framework; there,it is evidenced that a same kind of bifurcation arisesleading eventually to a same type of region of endoge-nous cycles in the space of parameters. The propertiesof the consumption and investment time series in thepossibly achievable chaotic zone do not differ from theones discussed above.

Alternatively to the optimization setup, to reproducevolatility stylized facts one could take non-constantad-hoc consumption and investment rules capableof reflecting differences in volatility. For instance, byimposing a rule under which all the feasible investmentis undertaken when output exceeds the correspond-ing trend value but some investment capacity is notfulfilled when the output level is lower than thebenchmark trend, this would allow to manipulate thecharacteristics of the volatility of investment. Pro-cyclical changes in capacity utilization are known tobe a relevant ingredient on the explanation of businesscycles (see Burnside et al. (1995)).

Finally, one could, as well, model the savings rate as afunction of the interest rate (rt): 1 − bt = f(rt), with f′ > 0.Assuming that the interest rate is somehow linked toinflation (e.g., through a Taylor rule) and that inflationrelates to the MMD and to demand through condi-tion (4), it is possible to turn the marginal propensity
to consume into an endogenous variable. After test-ing each one of these possibilities numerically, onefinds no significant qualitative differences. Obviously,because the specifications of the models differ, bifurca-tions will take place at distinct points and the regions
ic Dynamics 20 (2009) 221–238 235

of cyclical motion will be found in different locationsof the parameters’ space; nevertheless, in every one ofthe cases the relative standard deviations found in thechaotic zone are similar to the ones presented for ourbenchmark model.

(3) Persistence. Macroeconomic variables display persis-tent behavior in the sense that there is a high first orderserial correlation for the referred variables. Apparently,looking at Figs. 6 and 11, one does not find a highdegree of autocorrelation. We recall that these figuresare drawn for variable kt, but the motion of yt does notdiffer significantly; in our particular numerical exam-ple, yt = kt

1/3 (in the Cobb-Douglas case). However, weshould remark that the values of parameters in Table 1,in particular the depreciation rate and the discountfactor concerning the problem from which potentialoutput is withdrawn, are annual values, while the men-tioned facts, specifically this fact about persistence,applies to quarterly data.

Consider the data in Fig. 11; finding the time series ofoutput directly from this and assuming that between each(annual) observation there are three more observationssuch that, within each year, income grows at a constantrate, one obtains a time series with quarterly data, relativelyto which it is straightforward to compute serial correla-tion. The obtained value is around 0.85. King and Rebelo(1999) explicitly state that the first order serial correlationfor most detrended quarterly variables is a value close to0.9. Our value is not only near to empirical evidence butalso to estimates obtained from RBC models. One shouldkeep in mind, however, that this result is obtained underthe strong assumption that within each year, relatively tothe information we do not have, the growth rate is con-stant.

In synthesis, the proposed theoretical structure basi-cally furnishes an essential ingredient to understandbusiness cycles: endogenous fluctuations in the stock ofcapital that, given the production technology, propagateto output and from this, given the propensity to consumeand the demand imbalance mechanism, to consumptionand investment. The main stylized facts on business cycles,concerning co-movement, volatility and persistence arepossible to replicate once the model is properly adapted.Relative volatility features are the ones that the model bestreplicates, without the need of further additional assump-tions. Relating absolute volatility and persistence, a betterfit with reality requires: (i) taking multiple capital vari-ables (that can be generated under the same technology)in order to decrease the strong volatility that a singlechaotic time series displays; (ii) assuming that there ispersistence of growth within each considered annual timeperiod.

7. Conclusions

Keynesian economics can be characterized as the anal-ysis of non-equilibrium situations in aggregate marketrelations. Following recent literature on the theme, wehave developed a NEDyM with only two dynamic equa-tions, one respecting to capital accumulation and the

Econom
other to the adjustment of output and demand over theMMD. Behind this reduced form there is a set of neoclas-sical (market clearing) and Keynesian (non-equilibrium)assumptions.

We were able to address patterns of growth and to real-ize that, by combining neoclassical growth features withKeynesian disequilibrium elements, a multitude of long-term results can be found, ranging from balanced growthstability to cycles of any periodicity and completely a-periodic cycles. While the classical components pull in thedirection of the stable outcome, the several inefficienciesthat were introduced led to the possibility of endogenousbusiness cycles. The main advantage of this approach rel-atively to other models in the area is that the used lowdimensionality allowed for finding some relevant genericresults, namely concerning local analysis.

A meaningful result concerns the idea that stability ispossible only for a negative output gap. This is intuitive ifone takes in consideration the set of inefficiencies that wereconsidered; the benchmark case is the neoclassical growthmodel (the potential output is the steady state level of out-put computed when assessing an optimal control utilitymaximization problem), thus, by introducing non equilib-rium components to the model, it seems obvious that thebalanced growth path that one can find must correspondto a long-run output level below the optimal one. Further-more, the assumptions of the model imply that along with anegative output gap, the MMD is positive, i.e., at each timemoment (and, in this case, at every time moment of thelong-run outcome) there are goods that are produced butnot sold. Thus, periods of recession (negative output gap)are periods of overproduction (demand is below the levelof available goods). This is also an intuitive result.

The most relevant conclusion is that the non-equilibrium features that are attached to the neoclassicalgrowth model are such that they introduce nonlinear rela-tions between variables, which are capable of generatingendogenous cycles for admissible parameter values. Thismay be used as an argument to justify the relevance ofKeynesian economics, under which no external shock isnecessary to trigger fluctuations.

Finally, we have asked if the generated endogenouscycles are capable of reproducing real world economicfluctuations. The answer is that the proposed frameworkprovides a rationale for fluctuations, but some additionalassumptions and considerations are needed if one issearching for a good fit with the available data and theknown stylized facts. Nevertheless, even without furtherassumptions and in the absence of any stochastic distur-bances, the model is capable of roughly replicating therelative volatility properties of output, consumption andinvestment.

Acknowledgments

Financial support from the Fundacão Ciência e Tecnolo-
gia, Lisbon, is gratefully acknowledged, under the contractNo POCTI/ECO/48628/2002, partially funded by the Euro-pean Regional Development Fund (ERDF). I also thank therelevant suggestions of two anonymous referees and of thejournal’s editor, F. Duchin, which helped to improve sub-
ic Dynamics 20 (2009) 221–238

stantially the contents of the paper; the usual disclaimerapplies.

Appendix A.

A.1. Proof of proposition 1

Just apply the conditions mentioned in Definition 2 toarrive to the group of relations in the proposition. Theuniqueness of the steady state is guaranteed by the con-cave shape of the neoclassical production function, whichmakes the average product of capital (which is a decreasingfunction in all of its domain) to intersect the constant valueı/(1 − b) in a single point.


The trace and the determinant of matrix J in (7) are,respectively,

Tr(J) = 2 − ı−(b+ 1

x

)f

′(k) +

x

Det(J) = 1 − ı−(b+ 1

x

)f ′(k) + (1 − ı)

x+ (1 − b)

xf ′(k)

From the above expressions, one withdraws a relationbetween trace and determinant,

Det(J) = Tr(J) − 1 + [(1 − b)f ′(k) − ı] x.

One of the necessary conditions for asymptotic stabilityis 1 − Tr(J) + Det(J)> 0. This condition will require expres-sion [(1 − b)f ′(k) − ı]( /x) to correspond to a positivevalue. Note that the expression may be presented as (1 −b)[f ′(k) − (f (k)/k)]( /x). This is positive if y > y∗ ∧ f ′(k)>(f (k)/k) or y < y∗ ∧ f ′(k)< (f (k)/k). This set of conditionscan be restricted by recalling the neoclassical nature ofthe production function. In this function, marginal returnsare positive but diminishing. This means that introduc-ing additional capital implies getting progressively smallerincrements on output. Therefore, the marginal product ofcapital will be lower than the average product of capitalfor any admissible value of this variable. Thus, by statingthat f ′(k)< (f (k)/k), we restrict the possibility of asymp-totic stability to the case in which the steady state outputlevel is below the corresponding potential level.


The parabola Det(J) = (Tr(J)/2)2 defines the case inwhich the two eigenvalues of J are identical and equal toTr(J)/2. Above this parabola [Det(J)> (Tr(J)/2)2] the eigen-values are complex, and below it [Det(J)< (Tr(J)/2)2] theyare two real values. Assuming that asymptotic stability pre-
vails, the last inequality defines the condition under whicha stable node exists. Applying this condition to the specificJacobian matrix in (7) and resorting to the trace and deter-minant expressions computed in the proof of proposition2, we get the expression in this proposition.

Econom

A

f1TPc

A

oetfu0aTbvwtoolfcs

A

i−br

A

121bhcttnmfi

A

aaw


.4. Proof of proposition 4

The two eigenvalues of J lie inside the unit circle if theollowing three conditions are simultaneously satisfied:− Tr(J) + Det(J)> 0; 1 + Tr(J) + Det(J)> 0; 1 − Det(J) > 0.he first condition was applied to arrive to the result inroposition 2. The other two correspond, respectively, toonditions (i) and (ii) in the proposition.

.5. Proof of the corollary of proposition 4

The expressions in Proposition 4 establish two boundsn the value of ; thus, we just have to rearrange thexpressions in the proposition to get the boundaries ofhe set in the corollary. The main issue resides in identi-ying which one is the lower bound and which one is thepper bound. To reach this result, observe that condition< f ′(k)< (f (k)/k) holds and that the steady state aver-

ge product of capital is the one derived in Proposition 1.he above condition implies that the terms that multiplyy in the two conditions of Proposition 4 are negativealues (keep in mind that the output gap is negative); thus,hen solving the inequalities in the proposition in order

o , the first one gives a value below some combinationf parameters, while the second gives a value above somether combination of parameters. If the first quantity isarger than the second, asymptotic stability is guaranteedor any value of inside the presented set. In the oppositease, asymptotic stability is absent from the possible steadytate results.


The conditions in the proposition are the ones thatmply that one of the eigenvalues becomes lower than1 (1 + Tr(J) + Det(J)< 0) and that the two eigenvaluesecome a pair of complex conjugate values (1 − Det(J)< 0),espectively.


In the case where a positive output gap exists, condition− Tr(J) + Det(J)> 0 is violated (see proof of proposition) and, therefore, one of the eigenvalues of J is larger than. Thus, at best we will have one stable dimension. This sta-le dimension exists if the other two stability conditionsold (1 + Tr(J) + Det(J)> 0 and 1 − Det(J)> 0). In our spe-ific system, these are the conditions that allow reachinghe interval in the corollary of proposition 4, according tohe proof of such proposition. If saddle-path stability doesot hold, no eigenvalue with modulus lower than 1 is deter-ined, implying instability or divergence relatively to the

xed-point, independently of initial conditions.


Assume that matrix J in (7) has, as eigenvalues, |ε1| < 1nd |ε2| < 1. In this case, a unique stable trajectory existsnd this is given by expression ht − h = (p2/p1)(kt − k),ith p1 and p2 the elements of an eigenvector asso-

ic Dynamics 20 (2009) 221–238 237

ciated with ε1. The eigenvector P =[p1 p2

]Tmay be

determined resorting to one of the lines of J. Taking thesecond line in consideration, the following relation applies:(1 + (1/x)f ′(k)p1 + (1 − ε1 + ( /x))p2 = 0. Choosing p1 = 1,

the eigenvector is P =[

1 − (1 + (1/x)f ′(k)1 − ε1 + ( /x)

]. From P, we

withdraw the elements necessary to present the slope ofthe stable arm, as displayed in the proposition.


The steady state output gap is x = ˛/(1 − ˛) ln(((1 −b)(1/ˇ − (1 − ı))/˛ı)). This is a positive value if the expres-sion inside the logarithm is larger than 1; by rearrangingthis condition, one arrives to the inequality in the proposi-tion.

A.10. Derivation of potential output in the CES case

The potential output was defined as the steady statevalue of output for an optimal growth problem with a log-arithmic utility function. Thus, after computing first-orderconditions, one arrives to the standard steady state relationf ′(k∗) = 1/ˇ − (1 − ı).

The marginal product of capital is, in the steady state,

f ′(k∗) = Aam�[am� + (1 − a)(1 −m)�k∗−�](1−�)/�. The rela-tion between the potential output and the steady statecapital stock is given by the production function: y∗ =A[a(mk∗)� + (1 − a)(1 −m)�]1/�, which can be rewritten

in order to k*, k∗ =[

(y∗/A)�−(1−a)(1−m)�

am�

]1/�. Replacing this

value of k* in the marginal product expression, the steadystate condition comes:

Aam�[am�+(1−a)(1−m)�

am�

(y∗/A)�−(1−a)(1−m)�

](1−�)/�

= 1/ˇ − (1 − ı).

Solving this last equation in order to the potential levelof output one obtains y∗ = A((1 − a)(1 −m)�z/(z − am�)),with z ≡ (1/ˇ − (1 − ı)/Aam�)�/(1−�).


This proof is just a matter of analytical calculation. Thesteady state negative output gap condition, y < y∗, writes

in the CES case as A((1 − a)(1 −m)�ω/(ω − a ·m�))1/�<

A((1 − a)(1 −m)�z/(z − am�)). Solving in order to ω, ω >am�z�/(z� − (1 − a)1−�(1 −m)�·(1−�)(z − am�)�). To sim-plify notation, denote the right hand side of the previousinequality by ϑ. Thus, given the definition of ω, it comesb > (ϑ1/�A− ı/(ϑ1/�A)).

A.12. Output, consumption and investment time series

In Table A.1, the time series of output, consumption andinvestment are presented for the Cobb–Douglas produc-tion function case, resorting to the benchmark parameter

238 O. Gomes / Structural Change and Econom

Table A.1Output, consumption and investment time series under chaos.

Obs. y c i

1 1.297 1.167 −0.4142 1.174 1.058 0.9083 1.34 1.208 −1.2864 0.985 0.888 1.6465 1.364 1.227 −1.7646 0.84 0.775 1.7517 1.32 1.188 −0.7788 1.107 0.999 1.319 1.372 1.235 −1.937

10 0.773 0.698 1.73211 1.292 1.164 −0.29912 1.198 1.077 0.78413 1.335 1.202 −1.07514 1.045 0.938 1.53115 1.376 1.238 −2.02316 0.375 0.662 1.70317 1.275 1.149 −0.02118 1.238 1.118 0.38119 1.292 1.161 −0.31820 1.193 1.072 0.79321 1.334 1.199 −1.08422 1.04 0.935 1.5423 1.371 1.235 −1.97524 0.758 0.685 1.71325 1.283 1.157 −0.20326 1.213 1.092 0.6327 1.318 1.185 −0.77828 1.107 0.996 1.30129 1.369 1.23 −1.89930 0.792 0.714 1.73231 1.3 1.17 −0.43332 1.174 1.056 0.94633 1.347 1.213 −1.35334 0.97 0.875 1.68435 1.362 1.227 −1.71736 0.861 0.776 1.76137 1.332 1.198 −0.96938 1.072 0.966 1.46439 1.379 1.241 −2.08140 0.708 0.637 1.68441 1.265 1.137 0.16142 1.267 1.14 0.09443 1.258 1.13 0.1744 1.268 1.141 0.02745 1.243 1.12 0.28546 1.277 1.151 −0.165
47 1.211 1.091 0.58248 1.307 1.177 −0.67349 1.127 1.012 1.17650 1.357 1.22 −1.669
values used in Sections 4 and 5. We consider a situation inwhich chaotic motion holds ( = 1.05). The first 1000 obser-vations are withdrawn and the time series are built with thefollowing 50 observations.

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a two-dimensional non-equilibrium dynamic model

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