Endogenous crisis waves: a stochastic model with synchronized collective behavior

Stanislao Gualdi,1, 2 Jean-Philippe Bouchaud,3 Giulia Cencetti,4 Marco Tarzia,1 and Francesco Zamponi41Université Pierre et Marie Curie - Paris 6, Laboratoire de Physique Théorique de la Matière Condensée,

4, Place Jussieu, Tour 12, 75252 Paris Cedex 05, France2IRAMIS, CEA-Saclay, 91191 Gif sur Yvette Cedex, France

3CFM, 23 rue de l’Université, 75007 Paris, France,and Ecole Polytechnique, 91120 Palaiseau, France

4Laboratoire de Physique Théorique, École Normale Supérieure,UMR 8549 CNRS, 24 Rue Lhomond, 75231 Paris Cedex 05, France

We propose a simple framework to understand commonly observed crisis waves in macroeconomicAgent Based models, that is also relevant to a variety of other physical or biological situations wheresynchronization occurs. We compute exactly the phase diagram of the model and the location of thesynchronization transition in parameter space. Many modifications and extensions can be studied,confirming that the synchronization transition is extremely robust against various sources of noiseor imperfections.

Synchronisation is arguably among the most bafflingcooperative phenomenon in nature [1]. Examples aboundin physics and chemistry, but biological realizations areprobably the most relevant to us as they involve in partic-ular the synchronization of pacemaker cells in the heartor of neurons firing in the brain. Synchronization of clap-ping in concert halls or of flashing in assemblies of thou-sands of fireflies are also well known. The latter case isparticularly interesting: although the effect is so to sayvisible to the naked eye, its very existence is so counter-intuitive that for several decades no one could come upwith a plausible theory. As vividly recounted by Strogatz[1], scientists swayed between denial (assigning the flash-ing to a periodic quiver of the eyelids of the observer, orto the unavoidable presence of a “maestro”) and explana-tions that were “more remarkable than the phenomenonitself” [1]. The idea that interacting oscillators can gener-ically synchronize only slowly emerged at the end of thesixties [2], before becoming a well-established mathemat-ical truth with the work of Kuramoto [3], Strogatz &collaborators [4], and many others (for a review, see [5]).Still, the phenomenon is so unexpected (at least for

those only vaguely acquainted with these results) thatwhile working on a stylized agent based model (ABM)of the macroeconomy [6], we first disbelieved our results.We found a whole region of parameter space where thedynamics appeared to be (see Fig. 1) a nearly periodicsuccession of eras of prosperity interrupted by acute crisisof purely endogenous origin – indeed, no adverse exoge-nous shocks are present in our model; for details, see[6]. In view of the amount of heterogeneity and random-ness in our model (firms in our economy are all different,bankrupted firms are revived at Poisson random times,etc.), such a regular succession of crises waves is bothinteresting and surprising, and begs for a convincing the-oretical explanation. However, although highly stylized,our ABM is too complex to be amenable to an exactanalytical treatment, so further simplification is needed.This led us to the bare-bones model described below,

0 5 10 15

time / 1000

0

0.2

0.4

0.6

0.8

1

u

RUECFE

RU EC FE Θ

0 1.25 2.25

Figure 1: Typical trajectories of the unemployment rate u asa function of time for the macroeconomic agent based modeldescribed in [6], for different bankruptcy thresholds Θ. AsΘ is increased, the economy evolves from a phase of resid-ual unemployment (RU), endogenous crises (EC) and finallyfull employment (FE). In the EC phase, surprisingly, one ob-serves nearly periodic bursts of unemployment, correspondingto collective waves of bankruptcies.

which indeed exhibits – in fact, to our surprise – a syn-chronization transition. This model can be analyzed us-ing reasonably straightforward mathematical tools. Forinstance the location of the transition in parameter spacecan be computed exactly. Many modifications and exten-sions of the model can be studied, exactly or numerically.Their analysis confirms that the synchronization transi-tion is extremely robust, in particular against varioussources of noise and imperfections.

Our model is an alternative to the well-known Ku-ramoto model that has been studied inside-out [3–5], andwhich starts from the assumption that isolated individualelements are oscillators. Our setting is directly motivatedfrom the macroeconomic model we wanted to elucidate,and is in fact close in spirit to “integrate-and-fire” modelsof neurons [4]. It can also be rephrased as a mean-field

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model for epidemic dynamics [7], fiber bundles [8], de-pinning of elastic manifolds [9], or interbank default con-tagion [10]. In order to describe the model, we choose tokeep with the vocabulary of our macroeconomic model –transposition to other contexts is quite transparent andwill be discussed below. Each firm i = 1, . . . , N is charac-terized by its “financial fragility” measured by the ratioxi of its outstanding debt to total assets. We use theconvention that xi < 0 for indebted firms, and xi > 0 forcash/loan-rich firms. In the course of time, this ratio canincrease or decrease due to the success of its business,its needs for cash, etc. We posit [6] that when the fi-nancial fragility exceeds a certain threshold Θ (i.e. whenxi ≤ −Θ) banks are reluctant to restructure the debtof the firm, which then files for bankruptcy. (In a firstversion of the model, we assume this occurs with prob-ability 1 as soon as xi ≤ −Θ, but one can also considerthe case where bankruptcy is not certain and happenswith probability κ1xi≤−Θ per unit time, see below). Un-til firms hit the threshold, the evolution of xi is modeledas a biased random walk. The most important feature ofthe model is the feedback between bankruptcies and thedrift of these random walks. In the absence of money cre-ation (i.e. if the banking sector does not act as a buffer),the outstanding debt of the defaulted firm is spread be-tween remaining firms – thereby directly increasing theirfinancial fragility – and the household sector, leading toa decrease of its purchasing power and hence worsenedbusiness conditions for the surviving firms (see [6] for aconcrete implementation of this general idea). In bothcases, this gives a negative contribution to the drift ofxi’s. Finally, bankrupted firms are either revived or re-placed by new firms at a certain rate ϕ per unit time,and start with a zero initial fragility.Taking the limit of a large number of firms, the above

rules translate into the following Fokker-Planck equationfor the probability density P (x, t) of observing firms witha certain fragility between x and x+ dx at time t:

P (x, t) = DP ′′(x, t) + b(t)P ′(x, t) + J(t)δ(x−Θ), (1)

where we made a translation x → x + Θ, φ(t) =∫∞0 dxP (x, t) is the fraction of active firms at time t,J(t) is the flux of new-born firms at time t. The bound-ary condition P (0, t) = 0,∀t corresponds to case wherebankruptcy is immediate, as soon as the threshold isreached, and

b(t) = b+ βDP ′(0, t)Θ (2)

is the drift, that itself depends on the flux of bankrupt-cies DP ′(0, t) (i.e. the number of firms that cross x = 0at time t that spread an extra debt Θ) through a phe-nomenological, adimensional parameter β that measuresthe strength of the feedback. Note that with our signconvention, a positive b(t) corresponds to a drift towardsmore negative values of x. The reinjection current J(t) is

either (model I) a time-independent constant J(t) = ϕ,modeling the fact that new firms are created at a con-stant rate, independently of the number of existing firms,or (model II) given by J(t) = (1− φ(t))ϕ, meaning thatbankrupted firms have a probability ϕ per unit time tobe revived, in line with the choice made in the macroe-conomic model studied in [6].

In both cases the model contains 5 parameters:D, b, ϕ,Θ and the feedback parameter β, but two canbe fixed by the choice of units of time t and “length”x. This leaves us with three adimensional parameters:β, the Peclet number Pe = bΘ/D that compares driftto diffusion and z = b/(Θϕ) that compares the averagerevival time to the time needed to travel Θ under theaction of the drift b. Pe � 1 means that diffusion is asmall correction to drift in typical trajectories, whereasz � 1 means that revival is fast compared to the time tobankruptcy due to drift only. As implicitly assumed inthe above discussion, we will only consider the case b > 0(i.e. a negative drift on x), that allows the existence ofa stationary state in the absence of feedback (β = 0).This also describes the economy considered in [6]: fornon storable goods and in the absence of productivitygains or innovation, our myopic firms have a tendency toover-produce and lose money on average.

We therefore look for a stationary state P0(x), char-acterized by a constant bankruptcy flux J(t) = J0 andconstant φ(t) = φ0. Because DP ′0(x = 0) is the out-coming flux, and the fraction of active firms φ is timeindependent, it follows that DP ′0(x = 0) = J0 and thisleads to a constant drift b0 = b+βΘJ0. By setting P ≡ 0in Eq. (1), it is easy to show that:

P0(x) = J0

b0×

{(1− e−x) for x < Θ ,

(eΘ − 1)e−x for x > Θ ,(3)

where x = b0x/D and Θ = b0Θ/D are adimensional“lengths”. The stationary solution for P0(x) is thereforegiven in terms of the unknown parameters J0, φ0 and b0that are determined from:

φ0 ≡∫ ∞

0dxP0(x) = J0Θ

b0; b0 = b+ βJ0Θ. (4)

For model I, one has J0 = ϕ and φ0 and b0 are immedi-ately obtained. For model II, J0 = (1−φ0)ϕ gives a thirdequation. Using Eqs. (4), we can solve for b0 = b

1−βφ0

and J0 = φ0Θ

b1−βφ0

and obtain a self-consistent equation

φ0

Θb

1− βφ0= (1− φ0)ϕ , (5)

which leads to a second degree equation for φ0. If thetime needed for revival is much shorter than the lifetimeof the firms, z → 0, we expect that φ0 → 1, at least forsmall β’s. This allows one to choose the correct sign ofthe solution:

φ0 = 12β

[1 + β + z −

√z2 + 2z(1 + β) + (1− β)2

]. (6)

3

It can easily be checked that this solution is then alwayssuch that φ0 ∈ [0,min(1, 1/β)]. Therefore, the station-ary solution always exists, and b0 never diverges, whichwould be an obvious sign of an instability (see below fora case where this divergence actually happens). Still, thequestion is whether this stationary solution is dynami-cally stable. We therefore write:

P (x, t) = P0(x) + εP1(x, t), (7)

which induces corresponding O(ε) corrections to φ(t) =φ0 + εφ1(t) and similarly for J(t) and b(t). To order ε,the diffusion equation for P becomes, both for models Iand II:

[G−1P1](x, t) ≡ P1(x, t)−DP ′′1 (x, t)− b0P ′1(x, t)= b1(t)P ′0(x)− ϕφ1(t)δ(x−Θ) (8)

where G is the propagator of random walks with a wallat x = 0 and drift b0, given by the method of images [11]:

G(x, t|y, t− τ) = 1√4πDτ

{exp

[− (x− y + b0τ)2

4Dτ

]− eb0y/D exp

[− (x+ y + b0τ)2

4Dτ

]}(9)

Therefore, formally:

P1(x, t) =∫ ∞

0dτ

∫ ∞0

dyG(x, t|y, t− τ)

× [b1(t− τ)P ′0(y)− ϕφ1(t− τ)δ(y −Θ)] .(10)

This expression gives P1(x, t) in terms of b1(t) and φ1(t).Using the definition of b(t) and φ(t) then leads to self-consistent dynamical equations for b1(t) and φ1(t), whichare slightly different in models I and II. We focus onmodel II and we assume (and check self-consistently) thatφ1(t) = Φeαt and b1(t) = BΘeαt, where α is (a priori)a complex number with b20 + 4DRe (α) ≥ 0 to have con-vergent integrals. After simple algebra, one finally findsthat B and Φ have to obey the following equations:

Φ = −Bφ0

α

1− eKΘ

1−KD/b0− ϕΦ1− eKΘ

α;

B = −βϕΦeKΘ +Bβφ01− eKΘ

1−KD/b0, (11)

where K = b02D

(1−

√1 + 4αD

b20

). These have non-zero

solution only if α satisfies the equation [12]:

βφ0(α+ ϕ) = [α+ ϕ(1− eKΘ)]1−KD/b01− eKΘ . (12)

This equation has to be solved numerically in the com-plex plane, and one has to choose the solution with thelargest Re (α) that dominates the long-time evolution.

We find three different possibilities: Re (α) < 0, Im (α) =0: the stationary state P0 is linearly stable and relaxationtowards it is exponential; Re (α) < 0, Im (α) 6= 0: the sta-tionary state P0 is linearly stable and relaxation towardsit is exponential with oscillations; Re (α) > 0, Im (α) 6= 0:P0 is linearly unstable and the oscillations are the precur-sor of the synchronized state observed numerically. Anexample of the dependence of α as a function of ϕ forfixed β and z is given in Fig. 2-a. One sees that Re (α)crosses zero continuously for a critical value ϕc, for whichIm (α) 6= 0. The corresponding evolution of fraction ofinactive firms, 1 − φ(t), is given in Fig. 2-b, where weshow a numerical integration of the Fokker-Planck equa-tion, Eq. (1), but identical results are obtained from arandom walk simulation of 105 firms obeying the samedynamics. One clearly sees that the synchronized be-havior sets in exactly at the value ϕ = ϕc ≈ 35.15 (forβ = 1.3 and z = 0.002) predicted by the linear stabil-ity analysis. The transition is found to be continuous,with an amplitude of the oscillations that vanishes atthe transition, and a frequency given by Im (αc). Thephase diagrams in the (Pe, β) plane (for a fixed z) andin the (z, β) plane (for a fixed Pe) are given in Fig. 3.The conclusion of our analysis is that crises waves andsynchronization indeed appear in our skeleton macroeco-nomic model[15]. What is quite non trivial is that thistransition survives the fact that firms perform indepen-dent Brownian motion and are reinjected in the systemat random Poisson time. Similar conclusions of coursehold for the Kuramoto model as well.

The above stability analysis can be extended in differ-ent directions. First, the reinjection flux can be taken asa constant J0 = ϕ (Model I), leading to an even moreunstable system (see [12]). Second, the reinjection fluxdoes not need to be localized on x = Θ but it can bespread out over a certain region, i.e. one can replace theterm J(t)δ(x−Θ) in Eq. (1) by J(t)f(x−Θ), where f(x)is a function peaked in zero, normalized to one, and withfinite width w, with only quantitative changes. In fact,starting with a δ function in the synchronized phase, onecan induce the transition by increasing the width w be-yond some critical value wc. Third, one can replace thedeterministic bankruptcy condition by a stochastic one,by adding a term −κθ(−x) to the Fokker-Planck equa-tion and setting b(t) = b+βκ

∫ 0−∞ dx(Θ−x)P (x, t). The

above case corresponds to the limit κ = ∞. One findsthat the synchronization phenomenon survives at finiteκ. For example, when β = 1.3, z = 0.002, ϕ = 80, syn-chronization occurs for κ > κc ≈ 100. It is important(but again not very intuitive) that the synchronizationphenomenon does not sensitively depend on the presenceof a well defined threshold – one does not expect neu-rons or fireflies to be perfectly tuned to a precise firingthreshold. Finally, we have investigated the case wherethe bankruptcy feedback depends on the number of ac-

4

0 20 40 60 80ϕ

-2

-1

0

1

α /

ϕ

Re[α1]Im[α1]

Re[α2]Im[α2]

0 1 2t

0

0.2

0.4

0.6

0.8

1

1 -

φ

ϕ = 10

ϕ = 30

ϕ = 40

ϕ = 120

1 - φ0

Figure 2: (Top) Numerical solutions of Eq. (12): for smallvalues of ϕ there is only one linearly stable solution withoutoscillations (α1); for intermediate values of ϕ a second linearlystable solution appears with oscillations (α2); for high valuesof ϕ there is only a linearly unstable solution with oscillationscorresponding to the appearance of the synchronized behav-ior. (Bottom) Typical trajectories of the fraction of bankruptfirms 1− φ as a function of time for different values of ϕ. φ0is the theoretical value of stationary fraction of active firms,given by Eq. (6). For ϕ > ϕc the synchronized behaviorsettles in. The plot corresponds to z = 0.002 and β = 1.3.

tive firms, i.e. b(t) = b + βφ(t)DP

′(0, t)Θ, correspondingto the case where the debt of the failing firms is spreadamong the surviving firms only. In this case, we find thatthe above stationary state P0(x) ceases to exist as soonas β > 1, which becomes the threshold for synchronizedbehavior. In this case, the transition is found to be firstorder.Finally, we want to mention different potentially in-

teresting interpretations of our model. First, interbankdefault contagion, which has become a major theme sincethe 2008 crisis [10]. Here, the translation is almost im-mediate: since the assets of one bank is the liability ofanother, the default of one bank reduces the equity of itslenders, therefore pushing themselves closer to default.Treating the model in mean-field immediately leads to aFokker-Planck equation like Eq.(1). In a recent study ofa similar model, J. Bonart [13] has shown that the de-fault rate can diverge after a finite time, yet another sig-

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

β

-6

-4

-2

0

2

4

6

Log[P

e]

z = 0.02z = 0.002

0 0.02 0.04z

1

1.2

1.4

1.6

β

unstable

damped oscillations

unstable

stable

stable

damped oscillatio

ns

Figure 3: Phase diagram of the model in Eq. (1) in the (β, Pe)plane as given by the solutions of Eq. (12) with z = 0.02(black lines) and z = 0.002 (red lines). Dashed lines separatethe region where the maximal solution has Re (α) < 0 andIm (α) = 0 (stable) from the region where Re (α) < 0 andIm (α) 6= 0 (damped oscillations). Full lines separate the lat-ter region from the one where Re (α) > 0 (unstable). In theinset we plot the phase diagram in the (β, z) for Pe = 0.5.

nal of the collective synchronization effects studied here.Second, one can consider an epidemic model [7] wherexi gauges the level of infection of individual i. Whenxi > Θ, the illness declares itself and the disease be-comes strongly contagious. The flux J(t) would thenmodel non-infected new immigrants in the population.The model then predicts the possibility of sporadic out-breaks of the epidemics. Yet another interpretation of ourmodel is in terms of the fiber bundle model for fracture[8], allowing for “self-healing”, i.e. the possibility for bro-ken links to reform in time. Finally, one can interpret Eq.(1) as a mean-field description of the depinning transition[9], where each particle i is subject to an increasing forcefi = xi until fi reaches a local depinning threshold Θ.At this point, the particle advances and thereby relaxesthe force acting on it, and gets trapped again. However,if one assumes that particle i is elastically coupled to itsneighbours j, the forces fj will increase as particle i de-pins. In mean-field, we once again end up with Eq.(1),which predicts a transition from a smooth overall pro-gression when the external drive (here modeled by thedrift b) is large enough to a jerky stick-slip motion forsmall drive. An interesting generalization is to considerthat the feedback does not affect the drift b, as above,but the diffusion constant D, much as in [14]. This couldbe relevant for soft glassy matter or granular materials,where a localized yield event is often supposed to act asan effective temperature for the rest of the system. This,and other extensions, are left for future investigations.

Acknowledgements We thank J. Bonart, E. Bouchaudand J. Donier for very insightful conversations. This workwas partially financed by the EU “CRISIS” project (grantnumber: FP7-ICT-2011-7-288501-CRISIS)

5

[1] S. H. Strogatz, SYNC: the Emerging Science of Sponta-nenous Order, Hyperion, New York (2003).

[2] A. T. Winfree, J. Theor. Biol. 16, 15 (1967).[3] Y. Kuramoto, Chemical Oscillations, Waves and Turbu-

lence, Springer, New York (1984).[4] S. H. Strogatz, Physica D 143, 1 (2000), and refs. therein.[5] for a recent review, see: J. A. Acebrón, L. L. Bonilla,

C. J. Pérez Vicente, F. Ritort, R. Spigleri, Reviews ofModern Physics, 77, 139 (2005), and the extensive list ofreferences therein.

[6] S. Gualdi, M. Tarzia, F. Zamponi, J.-P. Bouchaud,Journal of Economic Dynamics and Control (2014),http://dx.doi.org/10.1016/j.jedc.2014.08.003i

[7] for a recent review, see: R. Pastor-Satorras,C. Castellano, P. Van Mieghem, A. Vespignani,arXiv:1408.2701v1.

[8] for a recent review, see: S. Pradhan, A. Hansen, B.Chakrabarti, Rev. Mod. Phys. 82, 499 (2010).

[9] D. S. Fisher, Physics Reports, 301, 113 (1998).[10] see e.g. J. Lorenz, S. Battiston, F. Schweitzer, Eur. Phys.

J. B 71, 441 (2009).[11] S. Redner, A guide to first passage time problems, Cam-

bridge University Press (2001).[12] For model I, the final equation is even simpler and reads:

βφ0 = (1−KD/b0)/(1− eKΘ).[13] J. Bonart, private communication and in preparation.[14] P. Hébraud and F. Lequeux, Phys. Rev. Lett. 81, 2934

(1998).[15] In order to understand the “reentrant” nature of the

phase diagram shown in Fig. 1 – i.e. the fact that thesystem is stable, then unstable and stable again as Θis increased, one should bear in mind that the values ofβ, Pe, z needed to describe the macroeconomic ABM of[6] all depend on Θ itself. One expects that in the pros-perous FE phase, the feedback parameter β is reduced,which indeed leads to a disappearance of the oscillations.