Dynamics in a nonlinear Keynesian good market model.

Dynamics in a nonlinear Keynesian good market model Ahmad Naimzada and Marina Pireddu Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 24, 013142 (2014); doi: 10.1063/1.4870015 View online: http://dx.doi.org/10.1063/1.4870015 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Unbounded dynamics in dissipative flows: Rössler model Chaos 24, 024407 (2014); 10.1063/1.4871712 Unraveling chaotic attractors by complex networks and measurements of stock market complexity Chaos 24, 013134 (2014); 10.1063/1.4868258 Price game and chaos control among three oligarchs with different rationalities in property insurance market Chaos 22, 043120 (2012); 10.1063/1.4757225 Hierarchy of bifurcations in the truncated and forced nonlinear Schrödinger model Chaos 15, 013107 (2005); 10.1063/1.1831591 Nonlinear dynamics of a pulse-coupled neural oscillator model of orientation tuning in the visual cortex AIP Conf. Proc. 502, 80 (2000); 10.1063/1.1302369

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CHAOS 24, 013142 (2014)

Dynamics in a nonlinear Keynesian good market model Ahmad Naimzada1,a) and Marina Pireddu2,b) 1

Department of Economics, Quantitative Methods and Management, University of Milano-Bicocca, U7 Building, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy 2 Department of Mathematics and Applications, University of Milano-Bicocca, U5 Building, Via Cozzi 55, 20125 Milano, Italy

(Received 10 December 2013; accepted 19 March 2014; published online 31 March 2014) In this paper, we show how a rich variety of dynamical behaviors can emerge in the standard Keynesian income-expenditure model when a nonlinearity is introduced, both in the cases with and without endogenous government spending. A specific sigmoidal functional form is used for the adjustment mechanism of income with respect to the excess demand, in order to bound the income variation. With the aid of analytical and numerical tools, we investigate the stability conditions, bifurcations, as well as periodic and chaotic dynamics. Globally, we study multistability C 2014 AIP Publishing LLC. phenomena, i.e., the coexistence of different kinds of attractors. V [http://dx.doi.org/10.1063/1.4870015] In the economic and financial literature, instabilities and irregularities in the behavior of markets are usually attributed to two main causes: random shocks or nonlinearities in deterministic frameworks. Adopting the latter point of view, in the present paper we show how complex dynamics may arise in the basic Keynesian income-expenditure model, when a simple nonlinearity is introduced, for instance, in the adjustment mechanism of income with respect to the excess demand. More precisely, we find that complexity emerges through bifurcations, leading to chaotic behaviors and multistability phenomena. In order to get a more complete model, we then extend such setting by endogenizing the government expenditure. For both frameworks, we are able to prove both analytically and numerically the presence of chaotic dynamics and to obtain some comparative statics results that shed light on the stabilizing or destabilizing role of the various parameters in the model.

I. INTRODUCTION

Instabilities are known, both empirically and theoretically, to be features of all markets: the product markets, the labor market, and the financial markets. Persistent irregularities exhibited by macroeconomic variables are usually attributed to random shocks (see, for instance, Refs. 8 and 9). An alternative explanation is based on phenomena caused by nonlinearities in the deterministic frameworks (see, e.g., Refs. 5, 7, 12, 16, and 20). For instance, we recall that the deterministic standard Keynesian business cycle model is not able to produce rich dynamical behaviors given the linearity of the economic relations and the linearity of the income adjustment mechanism. Nonetheless, when introducing nonlinearities in that model, complex dynamics have a)

E-mail: [email protected]. Tel.: þ39 0264485813. Fax: þ39 0264483085. b) Author to whom correspondence should be addressed. Electronic mail: [email protected]. Tel.: þ39 0264485767. Fax: þ39 0264485705. 1054-1500/2014/24(1)/013142/11/$30.00

been obtained in Refs. 17 and 19, considering heterogeneous agents with extrapolative and regressive expectation formation rules; in Refs. 21 and 22, within a framework of interactive formation of individual consumption propensities; in Ref. 18, taking into account the consumer sentiment of the representative agent, dependent on the business cycle. In all the above quoted papers on the Keynesian business cycle model, it is however assumed that there is an instantaneous adjustment of production with respect to demand, so that the market may always be thought in equilibrium. Our aim consists in showing that, even discarding such equilibrium hypothesis, it is still possible to obtain interesting complex dynamics in the income-expenditure Keynesian setting, when introducing a suitable simple nonlinearity. More precisely, we consider a specific sigmoidal functional form for the adjustment mechanism of income with respect to the excess demand, in order to bind the income variation. In particular, in our model we introduce parameters representing the reactivity and the bounds of the income adjustment mechanism. Such setting is then extended by endogenizing the government expenditure via the introduction of parameters representing the reactivity and the income target of the fiscal policy authority. With the aid of analytical and numerical tools, we show for the model without endogenous public expenditure that increasing the reactivity in the adjustment mechanism destabilizes the system, while decreasing the bounds in the income adjustment mechanism has a stabilizing effect. Such stabilization occurs via a sequence of periodhalving bifurcations. For the extended model with public expenditure, we find that increasing the reactivity with respect to the difference between the income target and the current target has a destabilizing effect and that the steady state value increases if the reactivity increases, if and only if the income target is larger than the income steady state without public expenditure. From an economic point of view, the latter result means that, independently of the nature of the attractor, the weight of the endogenous public expenditure may increase the value of income, if the target of the fiscal policy authority is sufficiently large. We also find that the income target has no

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influence on the stability properties of the system, while it has a direct influence on the steady state position. For both cases, with and without public expenditure, we show that, differently from the linear case in which local instability implies diverging trajectories, with the introduction of the sigmoidal adjustment mechanism in the instability regime we have the emergence of an absorbing interval, i.e., an invariant interval which eventually captures all forward trajectories. Moreover, we show the existence of chaotic dynamics in the sense of Li and Yorke10 for both the scenarios considered. Globally, we find that endogenizing the public expenditure may imply multistability, i.e., the coexistence of different kinds of attractors. This feature may be considered, especially in a macroeconomic model, as a source of richness for the framework under analysis because, other parameters being equal, i.e., under the same institutional and economic conditions, it allows to explain different trajectories and evolutionary paths. The initial conditions, leading to the various attractors, represent indeed a summary of the past history, which in the presence of multistability phenomena does matter in determining the evolution of the system. Such property is also known as “path-dependence.”1 We stress that, since we consider a sigmoidal functional form for the adjustment mechanism of income with respect to the excess demand, we are led to work with a bimodal map. In the literature on one-dimensional dynamical systems, there are several papers on bimodal functions (see, for instance, Refs. 3, 11, and 13–15). Nonetheless, we chose to present a complete analytical and numerical analysis of our model, because the framework and/or the kind of study performed in the above quoted papers differ from the specific setting and investigation performed in the present work. The remainder of the paper is organized as follows. In Sec. II, we introduce the model with exogenous government spending; in Sec. III, we present analytical and numerical local results for this setting, and we show the existence of an absorbing interval; in Sec. IV, we investigate analytically the first flip bifurcation and the existence of Li-Yorke chaos; in Sec. V, we introduce the model with endogenous government spending and show how the results in Secs. III and IV can be adapted to that new framework; in Sec. VI, we present a global scenario with multistability phenomena; finally, in Sec. VII, we draw some conclusions and discuss our results.

Ytþ1 ¼ Yt þ cgðEt Þ;

We consider a model with a Keynesian good market of a closed economy with public intervention. It is assumed that investment and government expenditures are exogenously given and that consumption has an autonomous component, as well as a term depending on income. The basic static model is as follows: Y ¼ C þ I þ G; with G ¼ G;

C ¼ C þ cY;

(1)

(2)

where c > 0 is the real market speed of adjustment between demand and supply,23 g is an increasing function with g(0) ¼ 0, and Et ¼ Zt – Yt is the excess demand, with Zt the aggregate demand in a closed economy, defined as Zt ¼ Ct þ It þ Gt :

(3)

We stress that, due to the properties of the map g, the dynamical system generated by (2) has a unique steady state, corresponding to Et ¼ 0, whose expression is given by Y ¼

A ; 1c

(4)

where A ¼ C þ I þ G is aggregate autonomous expenditure 1 and 1c is the Keynesian multiplier (see Ref. 4). Imposing a linear adjustment mechanism, the corresponding dynamical system is described by Eq. (2) with g equal to the identity map, which can be rewritten as follows: Dtþ1 :¼ Ytþ1 Yt ¼ cEt ;

(5)

where Dtþ1 represents the income variation. We depict such a framework in Figure 1. Making all terms explicit, in the linear case Eq. (2) can be also rewritten as follows: Ytþ1 ¼ Yt þ cðZt Yt Þ ¼ Yt þ cðC þ I þ G ð1 cÞYt Þ ¼ Yt þ cðA ð1 cÞYt Þ:

II. THE MODEL

I ¼ I;

where Y is aggregate income, C is aggregate consumption, I is aggregate investment, and G is government expenditure. Investment and government expenditures are exogenous and respectively. In the consumption function, equal to I and G, C is autonomous consumption and c 僆 (0, 1) is the marginal propensity to consume. Equation (1) is the market equilibrium condition. In a dynamic framework, we assume a dependence of consumption at time t on the income in the same time period, i.e., Ct ¼ C þ cYt . The dynamic behavior in the real economy is represented by an adjustment mechanism depending on the excess demand. If aggregate excess demand is positive (negative), production increases (decreases), that is, income Ytþ1 in period t þ 1 is defined in the following way:

(6)

In order to have a converging behavior towards Y*, we need 1 < 1 cð1 cÞ < 1; 2 . For a larger value of c, we which is satisfied for c < 1c 2 we have have instead a diverging behavior, while for c ¼ 1c infinitely many period-two cycles. Looking again at Figure 1, we observe that Dtþ1 may grow unboundedly and, in particular, when Et limits to 61, the same does Dtþ1. However, this is an unrealistic assumption because of the material constraints in the production side of an economy. Indeed, when excess demand increases, capacity constraints will surely lead to lower increases in


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Inserting Zt from (3) into (2) and recalling the definition of Dtþ1 in (5), we obtain the dynamic equation of the real market in the nonlinear framework a1 þ a2 Dtþ1 ¼ ca2 1 a1 eEt þ a2 a1 þ a2 ¼ ca2 1 (8) a1 eðAYt ð1cÞÞ þ a2 that we depict in Figure 2 and that generates the dynamical system we are going to analyze in Secs. III and IV. III. LOCAL ANALYSIS AND THE ABSORBING INTERVAL

FIG. 1. The graph of Dtþ1 as a function of Et in the linear case in (5) for c ¼ 0.5 in blue and c ¼ 1.5 in yellow, respectively.

income, due to the limited expansion from time to time of capital and labor stock; when excess demand decreases, capital cannot be destroyed proportionally to excess demand as the only factors that may reduce productivity are attrition of machines from wear, time, and innovations. Assuming instead that the adjustment mechanism is Sshaped, we specify the function g as a1 þ a2 1 ; (7) gðEt Þ ¼ a2 a1 eEt þ a2 with a1, a2 positive parameters. With such choice, g is increasing and g(0) ¼ 0. Moreover, it is bounded from below by a2 and from above by a1. Thus, the income variations are gradual and this prevents the real market from diverging and it may create a real oscillator. We stress that an S-shaped function similar to the one in (7) has already been considered in the Kaldorian business cycle model (see, for instance, Ref. 2). Notice however that its use is different in the two contexts both from an economic and a formal viewpoint. As regards the economic point of view, in the Kaldorian model the S-shaped function is used as investment function, relating investment demand and income; in our paper, the adjustment mechanism described by the sigmoidal function relates instead the income variation to the excess demand. In regard to the formal viewpoint, in the Kaldorian model the S-shaped function enters in an additive way in the equation describing the income adjustment mechanism, the other components in the sum being linear in income; in the present context, we have instead a composition of mappings, among which the sigmoidal function is the most “external” one and whose arguments are consumption (linear in income), investment, and government spending (constant and exogenously given). Hence, although the presence of an S-shaped function both in the Kaldorian model and in our framework, the two resulting dynamical systems are qualitatively different and thus a new complete analysis is required.

In this section, we analyze the stability of the steady state in the nonlinear case. Moreover, we show the presence of an absorbing interval eventually attracting all forward trajectories. In view of the subsequent analysis, it is expedient to introduce the map F : Rþ ! R defined as a1 þ a2 1 ; (9) FðYÞ ¼ Y þ ca2 a1 eðAYð1cÞÞ þ a2 associated to the dynamic equation in (8). Proposition 3.1 The unique steady state Y* in (4) is locally asymptotically stable if 2ða1 þ a2 Þ 1 1 : ¼2 þ cð1 cÞ < a1 a2 a1 a2 Proof. The result immediately follows by solving the chain of inequalities 1 < F0 ðY Þ < 1 with respect to c(1 c). In ð1cÞ and fact, it is immediate to find that F0 ðY Þ ¼ 1 ca1aa12þa 2 thus, in order to get the stability of the steady state, we only need to impose F0 ðY Þ > 1. At first, we stress that recognizing in s ¼ 1 c the propensity to save, it is possible to rewrite the stability condition 2Þ in Proposition 3.1 as cs < 2ðaa11þa a2 : Hence it is evident that an increase in c or s has a destabilizing effect. Considering instead the stability condition in terms of c 1 þa2 Þ . Comparing then the thresholds for only, we get c < a2ða 1 a2 ð1cÞ 2 c in the linear and nonlinear cases, i.e., c ¼ ð1cÞ and 1 þa2 Þ c ¼ a2ða , respectively, we find that c < c if and only if 1 a2 ð1cÞ 2Þ 1 1 a1 þ a2 > a1a2. In other terms, since ðaa11þa a2 ¼ a1 þ a2 , we have that c < c when a1 and a2 are small enough. This is the

FIG. 2. The graph of Dtþ1 as a function of Et in the nonlinear case in (8) for c ¼ 0.5 in blue and c ¼ 1.5 in yellow, respectively.


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case in Figure 3, where we represent the stability regions in the (c, c)-plane for the linear and nonlinear formulations. An intuition for the fact that small values for a1 and a2 allow to enlarge the stability region can be obtained by observing that, when reducing a1 and a2, we decrease the current variation of output. In fact, a1 and a2 appear in the expression of the asymptotes, whose equations in the (x, y)-plane are y ¼ x – ca2 for x ! þ 1 and y ¼ x þ ca1 for x ! 1, and, as it is immediate to verify, such straight lines get closer when a1 and a2 decrease. Looking again at the shape of the stability region in Figure 3 for the linear (nonlinear) case, we observe that 2Þ when c < 2 ðc < 2ðaa11þa a2 Þ the steady state is stable for any c and that, when c increases, we need values for c closer and closer to 1 in order to stabilize Y*. Hence increasing c has a destabilizing effect, both in the linear and nonlinear cases. In Figure 4, we depict the bifurcation diagram with respect to c in the nonlinear framework, which confirms the destabilizing role of c, leading to the emergence of chaos. In Figure 5, we depict the bifurcation diagram with respect to c in the nonlinear framework, for the same parameter values in Figure 4, except for a larger value of a1. This change makes the loss of stability of the steady state happen for values of c smaller with respect to Figure 4, confirming what said above about the destabilizing role of increasing a1 (and a2). In fact, by the stability condition in Proposition 3.1 it follows that, choosing a1 and a2 large enough, it is possible to obtain the destabilization of the steady state in correspondence to any positive value of c, even for c 僆 [0, 1], like in Figure 5, where the destabilization occurs for c around 0.55. The same observation about the possibility of reducing c when suitably increasing a1 and a2 applies to the conditions in Propositions 4.1 and 5.2, as well.

FIG. 3. The stability regions in the (c, c)-plane for the linear and nonlinear cases (in dark yellow and pale yellow, respectively) with a1 ¼ 1 and a2 ¼ 1.2.

Chaos 24, 013142 (2014)

FIG. 4. The bifurcation diagram with respect to c 僆 [0, 2.6] for the map F with A ¼ 12, a1 ¼ 5, a2 ¼ 11, and c ¼ 0.6.

Since, as observed above, the steady states for the linear and nonlinear scenarios coincide with Y*, we depict in Figure 6 the graph of the maps corresponding to those two cases in a neighborhood of Y*, i.e., f : Rþ ! R;

f ðYÞ ¼ Y þ cðA ð1 cÞYÞ

and F in (9), respectively, when a1 þ a2 ¼ a1a2, that is, when also the stability thresholds for the linear and nonlinear scenarios coincide. Starting from this framework in which a1 þ a2 ¼ a1a2, we depict in Figure 7 the bifurcation diagram for the map F with respect to a1 ¼ a2 僆 [0, 2], which shows that decreasing a1 ¼ a2 from 2 (value for which the condition a1 þ a2 ¼ a1a2 is satisfied and both systems are unstable) to 0 has a stabilizing effect. Similarly, in Figure 8 we present the bifurcation diagram for the map F with respect to a1 僆 [0, 4], when a2 ¼ 43. That picture shows that, if we start with a1 ¼ 4 and a2 ¼ 43 (so that a1 þ a2 ¼ a1a2) and we then decrease a1 from 4 to 0, we obtain again a stabilizing effect, also in the asymmetric framework in which a1 and a2 may differ. We now describe some further dynamical features for the map F in Proposition 3.2 below. More precisely, we show the existence of an absorbing interval and thus,

FIG. 5. The bifurcation diagram with respect to c 僆 [0, 1] for the map F with A ¼ 12, a1 ¼ 50, a2 ¼ 11, and c ¼ 0.6.


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FIG. 6. The maps describing the dynamics for the linear and nonlinear cases (in red and cyan, respectively), together with the 45-degree line, depicted in a neighborhood of Y* when a1 ¼ 3 and a2 ¼ 1.5, so that a1 þ a2 ¼ a1a2.

differently from the linear case, in which local instability implies diverging trajectories (see Ref. 4), in our framework local instability may imply periodic and chaotic orbits. In view of the next statement, we observe that for the map F in (9) only two scenarios may occur: if it is not globally increasing, then there exists a unique positive local maximum point, followed by a unique local minimum point. In fact a1 þ a2 a1 þ a2 1 > ca2 1 ¼0 Fð0Þ ¼ ca2 a1 eA þ a2 a1 þ a2 and F0 ð0Þ ¼ 1

ca1 a2 ða1 þ a2 ÞeA ð1 cÞ ða1 eA þ a2 Þ2

;

which is positive for any A large enough. In particular, in the pictures in the present paper, we have A 1. In the cases considered, F is positive and locally increasing in a right neighborhood of 0. If F is not globally increasing, it has a local maximum point, followed by the steady state and then by a local minimum point, and after that F grows monotonically to infinity. We are now ready to state the following result.

FIG. 7. The bifurcation diagram with respect to a1 ¼ a2 僆 [0, 2] for the map F with A ¼ 12, c ¼ 8, and c ¼ 0.6.

FIG. 8. The bifurcation diagram with respect to a1 僆 [0, 4] for the map F with A ¼ 12, a2 ¼ 43, c ¼ 8, and c ¼ 0.6.

Proposition 3.2 If the map F in (9) is increasing, then the generated dynamical system is globally stable. Else, call m and M the unique positive local minimum point and local maximum point of the map F, respectively, and set m0 :¼ FðmÞ and M0 :¼ FðMÞ. Then the compact interval I ¼ ½m0 ; M0 is “globally absorbing,” i.e., for all x 2 Rþ there exists n 2 N such that Fn ð x Þ 2 I and for any x 僆 I, Fn(x) 僆 I, for all n 2 N. Proof. Let us assume at first that F is increasing and show that, given a generic starting point x in Rþ , its forward trajectory will tend to Y*. Since F(0) > 0 and Y* is the unique fixed point of F, then by continuity, F(x) > x, for every x < Y*, and F(x) < x, for every x > Y*. Hence, if 0 x x Þ will tend increasingly towards Y* as n < Y ; then Fn ð x Þ will tend decreasingly ! 1, while if x > Y , then Fn ð towards Y* as n ! 1. If the map F is not increasing, let us consider a generic starting point x in Rþ and show that its trajectory will eventually remain in I. If x 2 I, then by construction its forward orbit will be trapped inside I, as well. Let us now proceed with the two remaining cases, i.e., x < m0 and x > M0 . Since Y*僆 I and by continuity F(x) > x, for every x < Y*, and F(x) < x, for every x > Y*, if x < m0 , then its iterates will approach I in a strictly increasing way, while if x > M0 , then its iterates will approach I in a strictly decreasing way. Once that a forward iterate of x lies in I, then by construction all its subsequent iterates will be trapped inside I, as well. This concludes the proof. ⵧ As we shall see in what follows, in the absorbing interval many different dynamic behaviors may arise. We depict in Figure 9 the absorbing interval in the case the map F is not increasing, and indeed in that graph it holds F0 ðY Þ < 1. In fact, for the parameter configuration used in Figure 9, the bifurcation diagram in Figure 5 shows we are in the chaotic regime. Notice that, in Proposition 3.2, the first scenario (i.e., F strictly increasing) may be seen as a limit case of the second framework, in which the absorbing set collapses into a unique point, that is, the steady state. We conclude the present section by illustrating in some pictures the possible behaviors of the map F. In Figure 10(a), we show the case in which F is strictly increasing and thus


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FIG. 9. The interval highlighted on the x-axis is the absorbing interval I for the map F when A ¼ 12, a1 ¼ 50, a2 ¼ 11, c ¼ 0.6, and c ¼ 0.99. In this case F0 ðY Þ < 1 and the steady state is locally unstable.

the convergence to Y* is monotone. In Figure 10(b), we consider the framework in which, having increased c, the map F is no more globally increasing and the convergence to Y* is oscillatory in nature. Finally, in Figure 10(c) we deal with the case in which c is still larger and the forward orbit of a generic starting point gets trapped in the absorbing interval. IV. THE FLIP BIFURCATION AND CHAOTIC DYNAMICS

In Sec. III, we have investigated the existence of an absorbing interval, the stability conditions for the steady state, and what happens after its destabilization. Now we will describe how such destabilization originates, i.e., through a period-doubling bifurcation when the reactivity parameter is large enough, and we will show how complex dynamics do emerge when that parameter is even larger. We start by verifying that the map F undergoes a flipbifurcation at the unique steady state Y*. Proposition 4.1 For the map F in (9), a flip bifurcation 1 þa2 Þ 2 ¼ 1c ða11 þ a12 Þ. occurs at Y ¼ Y* when c ¼ a2ða 1 a2 ð1cÞ Proof. According to the proof of Proposition 3.1, the steady state Y ¼ Y* is stable when F0 ðY Þ > 1. Then, the map F satisfies the canonical conditions required for a flip bifurcation (see Ref. 6) and the desired conclusion follows. 1 þa2 Þ , then Y* is a Indeed, when F0 ðY Þ ¼ 1, i.e., for c ¼ a2ða 1 a2 ð1cÞ 1 þa2 Þ it is attracting non-hyperbolic fixed point; when c < a2ða 1 a2 ð1cÞ 1 þa2 Þ and finally, when c > a2ða , it is repelling. ⵧ 1 a2 ð1cÞ In the proof of Theorem 4.2, we will show the existence of chaos in the sense of Li-Yorke, as described in Conditions (T1) and (T2) in [10, Theorem 1] (from now on, Th1 LY). We recall that result in particular applies to self-maps of an interval J, which are continuous and have a periodic point with period 3. Actually, as observed in Ref. 10, Th1 LY can be generalized to the case in which F : J ! R is a continuous function that does not map the interval J onto itself, as long as it has a period-three point. In the next statement, given an interval I, we denote by int(I) its interior and by @(I) its boundary. Proposition 4.2 Let F be the map in (9). Fix A ¼ 12, a1 ¼ 50, a2 ¼ 11, c ¼ 0.6, c ¼ 0.99, and set J ¼ [42.6, 46.3].

FIG. 10. The first iterate for the map F with A ¼ 12, a1 ¼ 50, a2 ¼ 11, c ¼ 0.6, and c ¼ 0.15 in (a), c ¼ 0.55 in (b) and c ¼ 0.99 in (c), respectively.

Then for any point x 僆 J, it holds that y ¼ F(x), z ¼ F2(x), and w ¼ F3(x) satisfy w x > y > z and thus Conditions (T1) and (T2) in Th1 LY do hold true. In particular, for any x 僆 int(J), it holds that F3(x) > x, while for any x 僆 @(J) it holds that F3(x) ¼ x, that is, the extreme points of J have period three.


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Proof of Proposition 4.2: We show that the chain of inequalities w x > y > z is satisfied on J by plotting in Figure 11, the graphs of the identity map in blue, of F in red, of F2 in green, and of F3 in cyan (the higher the iterate, the lighter the color). A direct inspection of the picture shows that it is possible to apply Th1 LY on J and thus we immediately get the desired conclusions. ⵧ Notice that both in the statement of Proposition 4.2 and in Figure 11, we considered the same parameter values like in the bifurcation diagram in Figure 5 (except for c there varying on [0, 1]), in order to show that when c ¼ 0.99 both the computer simulations and Proposition 4.2 confirm the presence of chaos. We stress that it was possible to show the presence of Li-Yorke chaos also for smaller values of c, but we chose a parameter configuration for which the corresponding illustration was clear enough. In fact, Proposition 4.2 is robust, as the same conclusions hold for several different sets of parameter values, as well. Moreover, once that a result analogous to that proposition is proven for a certain parameter configuration, by continuity, the same conclusions still hold, suitably modifying the interval J, also for small variations in those parameters. Hence, Proposition 4.2 actually allows to infer the existence of Li-Yorke chaos for the map F when c lies in a neighborhood of 0.99 and for some suitable corresponding values of the other parameters. V. THE EXTENDED MODEL: ENDOGENOUS GOVERNMENT SPENDING

In the present section, we introduce a suitable modification of the model so far analyzed. In fact, instead of an exog in analogy to what enous government expenditure G ¼ G, done with consumption, we assume Gt ¼ G þ rðY F Yt Þ; where YF and r are positive constants denoting, respectively, the income target of the fiscal authority and the reactivity of the fiscal policy with respect to the difference between the income target and the current income. With this new formulation, the expression for the excess demand becomes

Chaos 24, 013142 (2014)

Et ¼ Zt Yt ¼ A þ cYt þ rðY F Yt Þ Yt ; and the corresponding dynamic equation is given by a1 þ a2 Ytþ1 ¼ Yt þ ca2 1 : a1 eðAþrY F Yt ð1cþrÞÞ þ a2

(10)

(11)

Similarly to what is done in Sec. III, we introduce the map Fr : Rþ ! R defined as a1 þ a2 1 ; (12) Fr ðYÞ ¼ Y þ ca2 a1 eðAþrY F Yð1cþrÞÞ þ a2 associated to the dynamic equation in (11). We observe that such dynamical system can be obtained by (9) via the simple parameter transformation n : Rþ ð0; 1Þ ! Rþ R; ðA; cÞ7!ðAr ; cr Þ ¼ ðA þ rY F ; c rÞ: We stress however that, in this way, cr does not necessarily belong to (0, 1) anymore, and thus it cannot be interpreted as the marginal propensity to consume. Moreover, the introduction of the new parameters r and YF allows us to obtain some interesting results on comparative statics. Hence, we will go through the main steps in Secs. III and IV, reformulating the results obtained therein according to the new framework with endogenous government spending, omitting the unnecessary details and adding the new results we are able to obtain. We start by noticing that in equilibrium the excess demand in (10) vanishes and thus we find the expression for the steady state in the next result. Proposition 5.1 The dynamical system generated by the map Fr in (12) has the unique steady state Yr ¼

A þ rY F : 1cþr

(13)

It holds that @Yr ð1 cÞY F A ¼ ; @r ð1 c þ rÞ2 A which is positive if and only if Y F > 1c ¼ Y in (4), and

@Yr r ; ¼ @Y F 1 c þ r which is always positive. We now analyze the stability and the first bifurcation of the steady state in Proposition 5.2. Some comments on the sign of the derivatives in Proposition 5.1 can be found in the subsequent discussion. Proposition 5.2 The steady state Yr in (13) is locally asymptotically stable if FIG. 11. The identity map in blue and the first three iterates of the map F, in red, green, and cyan, respectively (the higher the iterate, the lighter the color) for A ¼ 12, a1 ¼ 50, a2 ¼ 11, c ¼ 0.6, and c ¼ 0.99. The interval highlighted on the x-axis is J from Proposition 4.2.

2ða1 þ a2 Þ 1 1 : cð1 c þ rÞ < ¼2 þ a1 a2 a1 a2


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For the map Fr in (12), a flip bifurcation occurs at Y ¼ 1 þa2 Þ Yr when c ¼ a12ða a2 ð1cþrÞ. Proof. The first part of the result immediately follows by 0 solving the chain of inequalities 1 < Fr ðYr Þ < 1 with respect to c(1 c þ r). In fact, it is immediate to find that 0 Fr ðYr Þ ¼ 1 ca1 aa21ð1cþrÞ and thus, in order to get the stability þa2 0

of the steady state, we only need to impose Fr ðYr Þ > 1: ⵧ The occurrence of the flip bifurcation can then be proven via the same argument used in Proposition 4.1. ⵧ Hence, Proposition 5.2 allows to infer that the steady state is stable when both c and r are small enough. In particular, the stability condition in Proposition 5.2 may 1 þa2 Þ be rewritten in terms of c as follows: c < a12ða a2 ð1cþrÞ. Setting c ¼ 2ða1 þa2 Þ and comparing it with the threshold a1 a2 ð1cþrÞ

1 þa2 Þ c ¼ a2ða encountered in Sec. III, we notice that the pres1 a2 ð1cÞ ence of r makes the stability region smaller with respect to the case of exogenous government spending. We illustrate the destabilizing role of r in Figures 12(a) and 12(b) below, in which we represent the bifurcation diagrams with respect to r for the first iterate of Fr for A ¼ 1 and A ¼ 4, respectively, while keeping the other parameters fixed. In both cases, when r increases, the system gets

@Y

destabilized: notice however that @rr is positive in the first scenario and negative in the second one. In fact, as stated in @Y Proposition 5.1, it holds that @rr > 0 if and only if A ¼ Y , that is, the steady state value increases Y F > 1c (decreases) when r increases if and only if the target value YF is larger than the steady state value in absence of endogenous government spending. In regard to the dependence of the steady state on YF, @Y we saw in Proposition 5.1 that instead @Y rF is always positive. F Hence we stress that, although Y does not influence the stability of the steady state by Proposition 5.2 that parameter does influence its position. In particular, increasing YF moves Yr to the right. Through some pictures (see Figures 13(a)–13(c) below), we show that YF also influences the position of the critical points that simultaneously emerge when the map F is not globally increasing. Indeed, similarly to what argued for the map F in (9), it is possible to show that a1 þ a2 a1 þ a2 1 > ca2 1 Fr ð0Þ ¼ ca2 a1 þ a2 a1 eðAþrY F Þ þ a2 ¼0 and F

0

Fr ð0Þ ¼ 1

FIG. 12. The bifurcation diagrams with respect to r 僆 [0, 7] for the map Fr with a1 ¼ 3, a2 ¼ 1, c ¼ 0.6, c ¼ 0.95, Y F ¼ 5, and A ¼ 1 in (a) and A ¼ 4 in (b), respectively.

ca1 a2 ða1 þ a2 ÞeðAþrY Þ ð1 c þ rÞ ða1 eðAþrY F Þ þ a2 Þ2

;

which is positive for any A large enough. In the cases considered in the present paper, Fr is positive and locally increasing in a right neighborhood of 0. When Fr is not globally increasing, it has a local maximum point, followed by the steady state and then by a local minimum point, and after that Fr grows monotonically to infinity. Then, increasing YF moves, not only the steady state, but also the local minimum and maximum points to the right on the x-axis, and consequently the absorbing interval gets moved to the right, too, together with the attractor possibly contained in it. This means that, no matter what kind of attractor the system has, the income value levels corresponding to that attractor increase when the income target increases. In order to illustrate such phenomena, we represent in Figures 13(a)–13(c) the first iterate of Fr for YF ¼ 1, YF ¼ 5 and YF ¼ 10, respectively, while keeping the other parameters fixed. Also in this context with endogenous government spending, it would be possible to prove a result analogous to Proposition 3.2, about the existence of a globally absorbing set when the map Fr is not everywhere increasing. If Fr is instead increasing, the generated dynamical system is globally stable. Due to the similarity with the framework already analyzed in Sec. III, we omit such a result and we only depict in Figure 14 the absorbing interval when the map Fr is not increasing. In regard to the Li-Yorke chaos for the scenario with endogenous government spending, we just observe that it is


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Chaos 24, 013142 (2014)

FIG. 14. The interval highlighted on the x-axis is the absorbing interval I for the map Fr when A ¼ 1, a1 ¼ 3, a2 ¼ 1, c ¼ 0.6, c ¼ 0.95, r ¼ 5.4, and YF ¼ 5.

identity map in blue, of Fr in red, of F2r in green, and of F3r in cyan (the higher the iterate, the lighter the color). The interval J is the one highlighted on the x-axis. Similarly to what observed in relation to Proposition 4.2, by continuity, the latter finding actually allows to infer the existence of Li-Yorke chaos for the map Fr when r lies in a neighborhood of 5.4 and for some suitable corresponding values of the other parameters. Notice that in Figure 15 we consider the same parameter values like in the bifurcation diagram in Figure 12(a) (except for r there varying on [0, 7]), in order to show that when r ¼ 5.4 both the computer simulations and our findings confirm the presence of chaos. In this regard, we stress that, for instance, we could have analogously inferred the presence of Li-Yorke chaos for Fr on the interval J ¼ [6.39, 6.82] when A ¼ 4, a1 ¼ 3, a2 ¼ 1, c ¼ 0.6, c ¼ 0.95, r ¼ 5.4, and Y F ¼ 5, in agreement with the chaotic behavior detected in Figure 12(b) for those parameter values. VI. GLOBAL ANALYSIS

In Secs. III–V, we presented some theoretical results about the dynamics of our systems, both with exogenous and endogenous government spending. We saw that in both

FIG. 13. The graph of the first iterate of the map Fr with A ¼ 6, a1 ¼ 4, a2 ¼ 2.5, c ¼ 0.85, c ¼ 0.9, r ¼ 1, and YF ¼ 1 in (a), YF ¼ 5 in (b), and YF ¼ 10 in (c), respectively.

possible to prove its existence via the same argument used in the proof of Proposition 4.2, for instance, on the interval J ¼ [5.875, 6.303] when A ¼ 1, a1 ¼ 3, a2 ¼ 1, c ¼ 0.6, c ¼ 0.95, r ¼ 5.4, and YF ¼ 5. This conclusion can be immediately drawn by looking at the graphs in Figure 15 of the

FIG. 15. The identity map in blue and the first three iterates of the map Fr, in red, green, and cyan, respectively (the higher the iterate, the lighter the color) for A ¼ 1, a1 ¼ 3, a2 ¼ 1, c ¼ 0.6, c ¼ 0.95, r ¼ 5.4, and YF ¼ 5.


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FIG. 16. The bifurcation diagram with respect to r 僆 [0, 5] for the map Fr with A ¼ 3, a1 ¼ 3, a2 ¼ 4.6935, c ¼ 0.8, c ¼ 0.8, and YF ¼ 5, which highlights a multistability phenomenon characterized by the coexistence of a stable three-orbit with a chaotic attractor in four pieces. The green dots refer to the initial condition Y (0) ¼ 0.5, while the blue dots refer to the initial condition Y (0) ¼ 0.8.

frameworks different kinds of dynamics can occur, such as the existence of absorbing intervals, stable steady states, periodic cycles, and chaotic behaviors. In this section we investigate, using bifurcation and time series diagrams, the global behavior of the system as the parameter r increases. In particular, in Figure 16 we illustrate a numerical example in which, from a stable steady state to a chaotic attractor, a multistability scenario occurs. Indeed we show that there exists a sufficiently large value of r for which we have the coexistence of a stable three-orbit with a chaotic attractor in four pieces. In Figure 17, we present a magnification of Figure 16, in which we better illustrate that coexistence phenomenon. Moreover, in Figures 18(a) and 18(b) we show the time series corresponding to the period-three cycle with initial condition Y (0) ¼ 1 and to the chaotic attractor in four pieces with initial condition Y (0) ¼ 3, respectively. VII. CONCLUSIONS

In this paper, we showed how a rich variety of dynamical behaviors can emerge in the standard Keynesian income-

FIG. 17. A magnification of Figure 16 for r 僆 [3.7, 4.2], in order to better show the coexistence of a stable three-orbit with a chaotic attractor in four pieces.

FIG. 18. The time series corresponding to the period-three cycle in Figures 16 and 17 in (a) and the chaotic attractor in four pieces in (b), with initial conditions Y (0) ¼ 1 and Y (0) ¼ 3, respectively, for r ¼ 3.835.

expenditure model when a nonlinearity is introduced, even discarding the market equilibrium hypothesis in Refs. 17–19, 21, and 22, both in the cases with and without endogenous government spending. A specific sigmoidal functional form has been used for the adjustment mechanism of income with respect to the excess demand, in order to bind the income variation. With the aid of analytical and numerical tools, we investigated the stability conditions, bifurcations, as well as periodic and chaotic dynamics. Globally, we studied multistability phenomena, i.e., the coexistence of different kinds of attractors. Here, we proposed a simple framework, in order to improve our knowledge about the role of nonlinearities in the emergence of complex behaviors in disequilibrium models, where there is no instantaneous adjustment between income and the excess of demand. In this context, the convergence to the steady state is the exception, the rule being given by periodic and complex dynamics. Our simple model shows indeed how an economy can easily reach cyclical or chaotic orbits. We stress that irregularities and fluctuations are caused by the nonlinearity of the system, without the need of introducing external shocks, whose interpretation is connected with non-economic factors, like in the DSGE (Dynamic Stochastic General Equilibrium) models.24,25 Of course, our model may be extended in various directions, for instance, introducing further nonlinearities in the definition of the relevant economic variables. Moreover, subsequent work should focus on extensions of the present model in view of considering money supply, interest rate and the general price level, too, aspects that are taken into


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account in more sophisticated classes of models, such as the IS-LM and AD-AS frameworks. ACKNOWLEDGMENTS

The authors thank the anonymous Reviewers for their helpful and valuable comments. 1

W. B. Arthur, Increasing Returns and Path Dependence in the Economy (University of Michigan Press, 1994). 2 G. I. Bischi, R. Dieci, G. Rodano, and E. Saltari, “Multiple attractors and global bifurcations in a Kaldor-type business cycle model,” J. Evol. Econ. 11, 527–554 (2001). 3 B. Branner, “Iteration by odd functions with two extrema,” J. Math. Anal. Appl. 105, 276–297 (1985). 4 B. S. Ferguson and G. C. Lim, Dynamic Economic Models in Discrete Time: Theory and Empirical Applications (Routledge, London, 2003). 5 J.-M. Grandmont, Nonlinear Economic Dynamics (Academic Press, New York, 1988). 6 J. K. Hale and H. Koc¸ak, Dynamics and Bifurcations, Texts in Applied Mathematics, Vol. 3 (Springer-Verlag, New York, 1991). 7 C. Hommes, Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems (Cambridge University Press, Cambridge, 2013). 8 F. E. Kydland and E. C. Prescott, “Time to build and aggregate fluctuations,” Econometrica 50, 1345–1370 (1982). 9 F. E. Kydland and E. C. Prescott, “The econometrics of the general equilibrium approach to business cycles,” Scand. J. Econ. 93, 161–178 (1991). 10 T.-Y. Li and J. A. Yorke, “Period three implies chaos,” Am. Math. Mon. 82, 985–992 (1975). 11 R. S. Mackay and C. Tresser, “Some flesh on the skeleton: The bifurcation structure of bimodal maps,” Physica D 27, 412–422 (1987).

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A. Medio, Chaotic Dynamics. Theory and Applications to Economics (Cambridge University Press, Cambridge, 1992). J. Milnor, “Remarks on iterated cubic maps,” Exp. Math. 1, 5–24 (1992). 14 J. Ringland, N. Issa, and M. Schell, “From U sequence to Farey sequence: A unification of one-parameter scenarios,” Phys. Rev. A 41, 4223–4235 (1990). 15 J. Ringland and M. Schell, “The Farey tree embodied—in bimodal maps of the interval,” Phys. Lett. A 136, 379–386 (1989). 16 M. Wegener and F. Westerhoff, “Evolutionary competition between prediction rules and the emergence of business cycles within Metzler’s inventory model,” J. Evol. Econ. 22, 251–273 (2012). 17 F. Westerhoff, “Business cycles, heuristic expectation formation, and contracyclical policies,” J. Public Econ. Theor. 8, 821–838 (2006). 18 F. Westerhoff, “Consumer sentiment and business cycles: A NeimarkSacker bifurcation scenario,” Appl. Econ. Lett. 15, 1201–1205 (2008). 19 F. Westerhoff, “Heuristic expectation formation and business cycles: A simple linear model,” Metroeconomica 59, 47–56 (2008). 20 F. Westerhoff, “Interactions between the real economy and the stock market: A simple agent-based approach,” Discrete Dyn. Nat. Soc. 2012, 504840 (2012). 21 F. Westerhoff and M. Hohnisch, “A note on interactions-driven business cycles,” J. Econ. Interact. Coord. 2, 85–91 (2007). 22 F. Westerhoff and M. Hohnisch, “Consumer sentiment and countercyclical fiscal policies,” Int. Rev. Appl. Econ. 24, 609–618 (2010). 23 In the existing literature, it is often assumed that the adjustment coefficients belong to [0, 1]. Notice that we could make the same assumption also in the present context as all the results in the paper could be obtained for c 1 as well, suitably increasing a1 and a2 (see Secs. III–V): however, for the sake of generality, we chose to impose no upper bounds on the value of c. 24 J. Galı, Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework (Princeton University Press, 2008). 25 M. Woodford, Interest and Prices: Foundations of a Theory of Monetary Policy (Princeton University Press, 2003). 13


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