Confidence Collapse in a Multi-Household, Self-Reflexive DSGE Model
Federico Guglielmo Morelli, Michael Benzaquen, Marco Tarzia, Jean-Philippe Bouchaud
CConfidence Collapse in a Multi-Household, Self-Reflexive DSGE Model
Federico Guglielmo Morelli a,b,c , Michael Benzaquen b,c,d,1 , Marco Tarzia a , and Jean-Philippe Bouchaud d,c,e a LPTMC, UMR CNRS 7600, Sorbonne Université, 75252 Paris Cedex 05, France; b Ladhyx, UMR CNRS 7646 & Department of Economics, Ecole Polytechnique, 91128Palaiseau Cedex, France; c Chair of Econophysics and Complex Systems, Ecole polytechnique, 91128 Palaiseau Cedex, France; d Capital Fund Management, 23-25, Rue del’Université 75007 Paris, France; e Académie des Sciences, Quai de Conti, 75006 Paris, FranceThis manuscript was compiled on July 18, 2019
We investigate a multi-household DSGE model in which past aggre-gate consumption impacts the confidence, and therefore consump-tion propensity, of individual households. We find that such a mini-mal setup is extremely rich, and leads to a variety of realistic outputdynamics: high output with no crises; high output with increasedvolatility and deep, short lived recessions; alternation of high andlow output states where relatively mild drop in economic conditionscan lead to a temporary confidence collapse and steep decline ineconomic activity. The crisis probability depends exponentially onthe parameters of the model, which means that markets cannot effi-ciently price the associated risk premium. We conclude by stressingthat within our framework, narratives become an important monetarypolicy tool, that can help steering the economy back on track.
Significance statement:
Despite their inability to cope with theGlobal Financial Crisis and various subsequent calls to “Rebuild Macroe-conomics”, Dynamics Stochastic General equilibrium (DSGE) modelsare still at the forefront of monetary policy around the world. Likemany standard economic models, DSGE models rely on the figment ofrepresentative agents, abolishing the possibility of genuine collectiveeffects (such as crises like the 2008 GFC) induced by heterogeneities andinteractions. By allowing feedback of past aggregate consumption on thesentiment of individual households, we pave the way for a class of morerealistic DSGE models that allow for large output swings induced byrelatively minor variations in economic conditions and amplified by in-teractions. Several important conceptual messages follow from our work,for example the de facto impossibility to price extreme risks and the im-portance of narratives, which may be an efficient depression-preventionpolicy tool whenever confidence collapse is looming.
1. Introduction
In spite of their poor performance during the Global Finan-cial Crisis (GFC), Dynamics Stochastic General Equilibrium(DSGE) models still constitute the workhorse of monetarypolicy models around the world (see e.g. (1) for an insight-ful introduction and references). Many ingredients that weremissing in previous versions of the model (such as the absenceof a financial sector) have been added in the recent years, inan attempt to assuage some of the scathing criticisms thatwere uttered post GFC (see for example (2–5) and (6–8) forrebuttals). However, the whole DSGE framework seems to be– partly for technical reasons – wedded to the RepresentativeAgent (or Firm) paradigm, and to a (log-)linear approxima-tion scheme that describes small perturbations away from afundamentally stable stationary state. ∗ In other words, crisesare difficult to accommodate within the scope of DSGE, asituation that led to J.C. Trichet’s infamous complaint:
Modelsfailed to predict the crisis and seemed incapable of explain- ∗ Quoting O. Blanchard in (4):
We in the field did think of the economy as roughlylinear, constantly subject to different shocks, constantly fluctuating, but naturallyreturning to equilibrium over time. [...]. The problem is that we came to believethat this was indeed the way the world worked. ing what was happening [...], in the face of the crisis we feltabandoned by conventional tools (3).Agent Based Models (ABMs) provide a promising alter-native framework to think about macroeconomic phenomena(9–13). In particular, ABMs easily allow for heterogeneitiesand interactions. These may generate non-linear effects andunstable self-reflexive loops that are most likely at the heartof the 2008 crisis, while being absent from benchmark DSGEmodels where only large technology shocks can lead to largeoutput swings. Unfortunately, ABMs are still in their infancyand struggle to gain traction in academic and institutionalquarters (with some major exceptions, such as the BoE (14)or the OECD). In order to bridge the gap between DSGEand ABMs and allow interesting non-linear phenomena, suchas trust collapse, to occur within DSGE, we replace the rep-resentative household by a collection of homogeneous butinteracting households. Interaction here is meant to describethe feedback of past aggregate consumption on the sentiment (or confidence) of individual households – i.e. their futureconsumption propensity. Low past aggregate consumptionbegets low future individual consumption. This opens thepossibility that a relatively mild drop in economic conditionsleads to a confidence collapse and a steep decline in economicactivity.We establish the “phase diagram” of our extended modeland identify regions where crises can occur. When the feedbackeffect is weak enough, standard DSGE results are recovered.As the strength of feedback increases, the economy can un-dergo rare, short-lived crises, where output and consumptionplummet but quickly recover. For even larger feedback, mildtechnology shocks can induce transitions between a high out-put state and a low output state in which the economy canlinger for a long time. In such conditions, output volatilitycan be much larger than the total factor productivity. Asportrayed by B. Bernanke (15), this is a “small shocks, largebusiness cycle” situation. We show that these endogenouscrises exist even when the amplitude of technology shocks isvanishingly small. But in this limit the probability for suchcrises is exponentially small and hence, we argue, unknowableand un-hedgeable. Our model thus provides an interesting ex-ample of unknown-knowns, where the crisis is a possible stateof the world, but its probability fundamentally un-computable.Our work relates to various strands of the literature thatemphasises the role of multiple equilibria and self-fulfillingprophecies, in particular the work of Brock & Durlauf on socialinteractions (16). Technically, our modelling strategy is akin tothe “habit formation” or “keeping up with the joneses” (KUJ)literature (17–19), although our story is more about keeping To whom correspondence should be addressed. E-mail: [email protected] | July 18, 2019 | a r X i v : . [ q -f i n . GN ] J u l ig. 1. Numerical simulation of the model for increasing values of the confidence threshold c and for fixed values of θ = 5 , σ = 0 . and η = 0 . . Top graphs: temporaltrajectories of the log output x t := log c t with a horizontal dot-dashed red line located at x and dashed black lines at x >,< ; Bottom graphs: (log-)probability distribution p ( x ) of the output, with the corresponding positions of x and x >,< . From left to right: e x = 0 . ( A phase, no crises, Gaussian distribution of output); e x = 0 . ( B + phase, short crises, increased volatility and skewed distribution of output); e x = 0 . ( C phase, long recessions, bi-modal distribution with most weight on x > ); e x = 1 . ( C phase, long recessions, bi-modal distribution with most weight on log x < ). Dashed lines: effective potential V ( x ) /σ , defined in Sec. 5. “down” with joneses (KDJ), as we are more concerned byself-reflexive confidence collapse than by consumption sprees(which also exist in our model, but in a regime that does notseem to be empirically relevant). We also specifically studythe interplay between the co-existence of two stable equilibriaand technology shocks, that can induce sudden transitionsbetween these equilibria (i.e. crises).Several other scenarii can lead to the coexistence of staticequilibria, corresponding to high/low confidence (20–22),high/low output (11, 23, 24), or high/low inflation expec-tations (23, 25), trending/mean-reverting markets (26–29),etc. with possible sudden shifts between the two. Multipleequilibria can be either a result of learning from past events,or from strong interactions between individual agents (director mediated by markets) – for a review, see e.g. (30). Another,distinct line of research explores the consequences of having anindeterminate equilibrium, i.e. a stationary solution aroundwhich small fluctuations can develop without being pinnedby initial conditions (for a recent review and references, see(31)). These fluctuations are not related to any real economicdriving force, but rather the result of self-fulfilling prophe-cies. In our present model, fluctuations are triggered by realtechnology shocks, but are then amplified by a self-reflexivemechanism. Nothing would prevent, however, the existence offurther indeterminacy around different stationary points.
2. A Multi-Household DSGE Model
We assume that each household i ∈ [[1 , M ]] is characterised bya utility function U i ( c it , n it ) that depends on its (unique good)consumption c it and amount of labour n it as: U i ( c it , n it ) = f it ( c it ) − ς − ς − γ i ( n it ) φ φ , [1]where γ i is a factor measuring the disutility of labour, and ς ∈ ]0 ,
1] and φ > i -independent parameters suchthat the utility function has the correct concavity. Standardchoices are ς = 1 (log-utility of consumption) and φ = 1.The quantity f it is a time-dependent factor measuring theconfidence of household i at time t , and hence their propensity to consume. This “belief function” (31) will be responsible forthe possible crises in our model (see below).Each infinitely lived household maximises its future ex-pected discounted utility with a discount factor β <
1, subjectto the budget constraint: p t c it + B it r t ≤ w t n it + B it − , [2]where p t is the price of the good, w t the wage (assumed tobe identical for all households) and B ti the amount of bondspaying 1 at time t + 1, purchased at time t at price (1 + r t ) − ,where r t is the one-time period interest rate (set by the centralbank). The maximisation is achieved using the standardLagrange multipliers method over the quantities c it , n it and B it . This gives the household’s state equations Eq. (3), Eq. (4)and the Euler equation Eq. (5): (cid:0) c it (cid:1) ς = f it λ i p t [3] (cid:0) n it (cid:1) φ = u t p t λ i γ i [4] f it (cid:0) c it (cid:1) − ς = β (1 + r t ) E t " f it +1 (cid:0) c it +1 (cid:1) − ς π t +1 , [5]where u t = w t /p t is the wage expressed in price units, π t := p t /p t − − λ i the Lagrange multiplier.The total consumption is P Mi =1 c it := C t and the total numberof work hours is P Mi =1 n it := N t .The unique firm has a technology such that its production Y t is given by: Y t = M α z t N − αt − α , [6]where z t is the total factor productivity and α another param-eter, often chosen to be 1 /
3. The scaling factor M α is thereto insure a correct limit when M → ∞ that allows marketclearing, i.e. total production and total consumption must beboth proportional to the number of households M . We willwrite z t := ¯ ze ξ t , where the log-productivity ξ t is assumed to | F. G. Morelli| F. G. Morelli
3. The scaling factor M α is thereto insure a correct limit when M → ∞ that allows marketclearing, i.e. total production and total consumption must beboth proportional to the number of households M . We willwrite z t := ¯ ze ξ t , where the log-productivity ξ t is assumed to | F. G. Morelli| F. G. Morelli et al. ollow an AR(1) process: ξ t = ηξ t − + p − η N (cid:0) , σ (cid:1) , [7]where η modulates the temporal correlations of the technologyshocks, and σ the amplitude of these shocks.Each time period the firm maximises its profit with theassumption that markets will clear, i.e. that Y t ≡ C t . Profitis given by P t := p t C t − w t N t . Maximisation of P t yields u t = z t ( M/N t ) α , i.e., the firm hires labour up to the point whereits marginal profit equals the real wage (1). Now, assuming forsimplicity that f it and γ i are all equal (homogeneous beliefsand preferences) leads to c it = c t = C t /M , n it = n t = N t /M , γ i = γ and f it = f t . We can use Eqs. (3,4) and Eq. (6) with Y t = C t to find c t , n t and u t as a function of f t and z t . Inthe following, we choose standard values φ = ς = 1, α = 1 / † c t = z t (cid:18) f t γ (cid:19) / . [8]
3. Animal Spirits and Self-Reflexivity
Now, the main innovation of the present work is to assumethat the sentiment of households at time t (which impacts theirconsumption propensity f t ) is a function of the past realisedconsumption of others , in an “animal spirits” type of way. Ifhousehold i observes that other households have reduced theirconsumption in the previous time step, it interprets it as asign that the economy may be degrading. This increases itsprecautionary savings and reduces its consumption propensity.Conversely, when other households have increased their con-sumption, confidence of household i increases, together withits consumption propensity. ‡ A general specification for thisanimal spirits feedback is f it −→ F M X j =1 ,j = i J ij c jt − ! , [9]where F ( . ) is a monotonic, increasing function and J ij weighsthe influence of the past consumption of household j on the con-fidence level of i . In this work, we will consider the case where J ij = J/M , i.e. only the aggregate consumption matters. § This corresponds to a mean field approximation in statisticalphysics, see e.g. (33). While it neglects local network effects,it captures the gist of the mechanism we want to illustrateand furthermore allows us to keep the household homogeneityassumption (different local neighbourhoods generally lead todifferent consumption propensities).Combining Eq. (8) and Eq. (9) yields: c t = e ξ t G ( c t − ) , with G ( x ) := ¯ z (cid:18) F ( x )4 γ (cid:19) / . [10] † Other values can of course be considered as well, but do not change the qualitative conclusions ofthe paper. ‡ In this work, we assign confidence collapse to households, i.e., to the demand side. However, onecould argue that confidence collapse in 2008 initially affected the supply side. One can of courseeasily generalise the present framework to account for such a “wait and see” effect as well. For another attempts to include behavioural effects in DSGE, see (32). § In the following, we will consider the large M limit such that J/M P Mj =1 ,j = i c jt − → Jc t − . Extensions to non-homogeneous situations will be considered in a future work, but manyfeatures reported here will also hold in these cases as well. Equation (10) is a discrete time evolution equation for theconsumption level. In order to exhibit how this dynamics cangenerate excess volatility and endogenous crises, we assumethat G ( x ) is a shifted logistic function (but it will be clearthat the following results will hold for a much larger family offunctions). To wit: G ( c ) = c min + c max − c min e θ ( c − c ) , [11]where c min , c max , c and θ are parameters with the followinginterpretation:• c min > ξ t =0).• c max > c min is the maximum level of goods that house-holds will ever consume when productivity is normal (i.e. ξ t = 0).• c is a “confidence threshold”, where the concavity of G ( c ) changes. Intuitively, c > c tends to favour a highconfidence state and c < c a low confidence state.• θ > θ → + ∞ , one has G ( c < c ) = c min and G ( c > c ) = c max . ¶ The standard DSGE model, where the animal spirit feedbackis absent, is recovered in the limit θc → −∞ , in which case G ( c ) ≡ c max = cst . The dynamics of our extended model fallsinto four possible phases, that we will call A , B + , C and B − (see Figs. 1, 2), and discuss their properties in turn. In thefollowing, we will use the notation ∆ := c max − c min .This phase corresponds to the DSGE phenomenology, wherethe only solution of e ξ G ( c ) = c is a high consumption solution c > c for all values of ξ . Even for large negative shocks ξ < σ (cid:28)
1, the consumption remains around the value c > solution of G ( c > ) = c > , and one can linearise the dynamicsaround that point: δ t +1 ≈ G ( c > ) δ t + ξ t , δ t := c t − c > c > . [12]This leads to the following expression for the consumptionvolatility: V [ δ ] = σ − G > ηG > − ηG > , G > := G ( c > ) . [13]In other words, the output volatility is proportional to theamplitude σ of the technology shocks – small shocks lead tosmall volatility (note that G > < A phase).However, the feedback mechanism leads to excess volatility ,since as soon as G > >
0, one has σ − G > ηG > − ηG > > σ . [14] ¶ In this work we shall fix θ and vary c . Note however that fixing c and varying the “temperature" θ would also be of interest to investigate the effect of population heterogeneity. As mentioned earlier,we leave the study of household heterogeneity for a future work. F. G. Morelli et al. | July 18, 2019 | Left: Phase diagram of the model, with analytically determined boundaries.Phase A : High Output, No Crises; Phase B + : High Output with Short-Lived Re-cessions; Phase C : Long-Lived Booms & Recessions; Phase B − : Phase B − :Low Output with Short-Lived Spikes. The colour level encodes the distance ratio ( c > − c ∗ ) / ( c ∗ − c < ) . This ratio is large in the yellow region, small in the blue regionand equal to one along the dot dashed black line. Right: Graphical representation ofthe iteration c t +1 = e ξt G ( c t ) in the different phases. The plain line corresponds to ξ t = 0 .
4. Phase Diagram
Phase A : High Output, No Crises The relative position of theboundary of the A phase depends on whether θ ∆ is larger orsmaller than 2. In the first case – corresponding to Fig. 2,the boundary is with the B + phase (to be described below).In the ( c , θ ) plane, the A phase is located on the left of thehyperbola defined by θc = 1 + 2 c min ∆ . [15]In the case θ ∆ <
1, the boundary is with the B − phase andthe A phase corresponds to c ≤ c min + ∆ / Phase B + : High Output with Short-Lived Recessions In thisphase B + there is still a unique equilibrium state when pro-ductivity is normal, i.e. a unique solution to G ( c > ) = c > with c > > c . However, downward fluctuations of productiv-ity can be strong enough to give birth to two more solutions c < < c ∗ < c , one unstable ( c ∗ ) and one stable ( c < ). Withsome exponentially small probability when σ → c > and crash into a low output state, in which it will remaintrapped for a time of the order of T η := 1 / | ln( η ) | , i.e. theauto-correlation time of ξ t . In other words, sufficiently largefluctuations of output are initially triggered by a relativelymild drop of productivity which is then amplified by the self-referential “panic” effect. But since the low output state isonly a transient fixed point, the recession is only short lived. Phase C : Long-Lived Booms & Recessions Phase C is suchthat equation G ( c ) = c has two stable solutions c < , c > andone unstable solution c ∗ . This phase is delimited, in the( c , θ ) plane, by a parabolic boundary (see Fig. 2) with c → ( c min + c max ) / θ ∆ → + and c → c min or c max when θ → ∞ . The lower boundary C → B + corresponds to c < → c ∗ before both disappear, leaving c > as the only solution, whereasthe upper boundary C → B − corresponds to c > → c ∗ beforeboth disappear, leaving now c < as the only solution.In the absence of fluctuations ( σ = 0), the economy inphase C settles either in a low output state or in a highoutput state. But any, however small, amount of productivityfluctuations is able to induce transitions between these two states. The time needed for such transitions to take place ishowever exponentially long when σ → T ( c >,< → c <,> ) = W ( c >,< → c <,> ) σ + O ( σ ); [16]where W ( c > → c < ) and W ( c < → c > ) are computable quanti-ties (see section 5 below and Fig. 3). This is clearly the mostinteresting regime: the economy can remain for a very longtime in a high output state c > , with relatively mild fluctua-tions (in fact still given by Eq. (13)), until a self-fulfilling panicmechanism throws the economy in a crisis state where outputis low ( c < ). This occurs with a Poisson rate 1 /T ( c > → c < ).Unless some explicit policy is put in place to restore confi-dence, the output will linger around c < for a Poisson time ∼ T ( c < → c > ) which is also very long when σ → ‖ Note that T ( c > → c < ) is the average time the system remains around c > before jumping to c < . The actual time needed to transitis itself short, and the resulting dynamics is made of jumpsbetween plateaus – see Fig. 1. A downward jump thereforelooks very much like a “crisis”.As we discuss below, recession durations are much shorterthan the time between successive crisis when c ∗ − c < < c > − c ∗ ,i.e. when the low output solution is close to the unstablesolution, which plays the role of an escape point (see below).As c grows larger, one will eventually be in a situation where c ∗ − c < > c > − c ∗ , in which case recession periods are muchlonger than boom periods, see Fig. 2. As σ grows larger, theoutput flip-flops between c < and c > at an increasingly fasterrate, see Fig. 3. While it becomes more and more difficultto distinguish crisis periods from normal periods, the outputvolatility is dramatically amplified by the confidence feedbackloop. Phase B − : Low Output with Short-Lived Spikes Phase B − isthe counterpart of phase B + when c is to the right of thephase boundary. In this case, the only solution to G ( c ) = c is c < : confidence is most of the time low, with occasionaloutput spikes when productivity fluctuates upwards. Theseoutput peaks are however short-lived, and again fixed by thecorrelation time T η . ∗∗ Phase Diagram: Conclusion
Although quite parcimonious, ourmodel is rich enough to generate a variety of realistic dy-namical behaviour, including short-lived downturns and moreprolonged recessions, see Fig. 1. We tend to believe that themost interesting region of the phase space is in the vicinityof the B + - C boundary, and that the 2008 GFC could corre-spond to a confidence collapse modelled by a sudden c > → c < transition. †† The behaviour of the economy in the B − phase,on the other hand, does not seem to correspond to a realisticsituation. One of our most important result is that the crisisprobability is exponentially sensitive to the parameters of themodel. We dwell further on this feature in the next section.
5. A Theory for Transition Rates
Discrete Maps
Let us now discuss in more detail one of themost important predictions of our model, namely the expo-nential sensitivity to σ of the crisis probability, Eq. (16). Such ‖ Note in particular that T ( c < → c > ) is much longer than T η : it is no longer the correlation timeof productivity fluctuations that sets the duration of recessions (at variance with the B + scenario). ∗∗ Note that there is no “ A − ” analogue of the A phase described above – this is due to the fact thatthe productivity factor z t can have unbounded upwards fluctuations but cannot become negative. †† The role of trust in the unravelling of the 2008 crisis is emphasised in the fascinating book of AdamTooze,
Crashed , Penguin Books (2018). | F. G. Morelli| F. G. Morelli
Crashed , Penguin Books (2018). | F. G. Morelli| F. G. Morelli et al. result can be obtained by adapting the formalism of (34) tothe present problem. In terms of x t := log c t , the map (10)reads: x t = H ( x t − ) + ξ t , [17]with H ( x ) := log G ( e x ). In the limit of white noise (i.e. η = 0in Eq. (7)), this is precisely the general problem studiedin (34) in the case where H ( x ) = x has two stable solutionsand an unstable one in-between. The authors show thatthe average time before jumping from one stable solution toanother is given, for small σ , by Eq. (16). They provide anexplicit scheme to compute (at least numerically) the quantity W , called the activation barrier in physics and chemistry.The idea is to find the most probable configuration of ξ t ’sthat allows the system to move from one stable position toanother. In a nutshell, this amounts in finding a heteroclinicconnection, in an enlarged space, between the starting pointand the intermediate, unstable fixed point x ∗ = log c ∗ (35).It is straightforward to generalise the approach of (34) andsee that the jump rate has the same exponential dependenceon σ when the correlation time T η is non-zero, as confirmedby Fig 3. However, finding the value of W is more complicated.Approximation methods can been devised in the continuoustime limit, that we describe now. Continuous Time Limit
Let us slightly change the dynamics byassuming that x t depends not on the previous value x t − butrather on an exponential moving average ¯ x t − of past valuesof x , defined recursively as¯ x t − = (1 − ε )¯ x t − + εx t − [18]Eq. (17) instead reads x t = H (¯ x t − ) + ξ t . Eliminating x t yields ¯ x t − ¯ x t − = ε ( H (¯ x t − ) − ¯ x t − + ξ t ) , [19] l og T ( c > ! c < )
Plot of log T ( c > → c < ) (left row) and log T ( c < → c > ) (right row) vs σ − for different values c , and η = 0 (top graphs) and η = 0 . (bottom row). Thevalue of c increases with the points tonality becoming darker. The linear dependenceconfirms the validity of Eq. (16) . The inset shows the corresponding barriers W asa function of c . For η = 0 , we plot the continuous time prediction Eq. (20) with ε = 1 (solid red), which overestimates the true barriers (dotted red) by a factor ≈ . In the limit ε →
0, this equation becomes a Langevin (or SDE)equation for ¯ x t , for which a host of results are available. It isuseful to introduce a potential function V ( x ) such that V ( x ) = x − H ( x ). The potential V ( x ) has two minima (“valleys”)corresponding to log c < and log c > and a maximum (“hill”)corresponding to log c ∗ (see lower panels of Fig. 1). With thisrepresentation, the dynamics of ¯ x t under Eq. (19) becomestransparent: for long stretches of time, ¯ x t fluctuates aroundeither x < = log c < or x > = log c > , until rare fluctuations of ξ t allow the system to cross the barrier between the two valleys.Calculating the rate Γ of these rare events is a classic problem,called the Kramers problem (for a comprehensive review, see(36)). In the limit T η = 0 where the noise is white, the finalexact expression is, for σ → x > → x < ) = p | H ( x > ) H ( x ∗ ) | π exp (cid:16) − Wεσ (cid:17) ,W := V ( x ∗ ) − V ( x > ) , [20]and mutatis mutandis for Γ( x < → x > ). Such a prediction iscompared with numerical simulations in Fig. 3; it overesti-mates the real barrier by a factor ≈
2. The most importantfeatures is the exponential dependence of this rate on theheight of the barrier W and on the inverse noise variance σ .The generalisation of Kramers’ result for so-called colourednoise (i.e. T η >
0) is also available, see (37). In this case,corrections to Eq. (20) can be systematically computed, butthe exponential dependence of Γ on σ − is preserved. Exponential dependence and “unknown knowns”
It is worth em-phasising the economic consequences of this exponential de-pendence of the probability of crises in our model. Clearly,any small uncertainty about the parameters of the model (i.e. c , c min , c max , θ ) or for that matter the precise specificationof the function G ( c ), or any other feature neglected in themodel, will no doubt affect the precise value of the barrier W . But in a regime where W/σ (cid:29) W is exponentially amplified. Take for example W/σ = 25; asmall relative error of 10% on W changes the crisis rate byone order of magnitude. Precisely as the famous butterflyeffect (i.e. the exponential sensitivity on initial conditions)forbids any deterministic description of chaotic systems, theexponential dependence of the crisis rate means that this rateis, for all practical purposes, unknowable. Since the probabil-ity of rare events cannot be determined empirically, it meansthat no market can provide a rational valuation of the corre-sponding risks. This is an interesting example of “unknownknowns”, where what may happen is known, but its probabil-ity impossible to quantify. The impossibility to price thesecatastrophic risks, argued to be at the heart of the excessequity risk premium (38), would then also be responsible forthe excess volatility in financial markets.
6. Inflation & Narrative-Based Monetary Policy
In the absence of frictions, the model is usually closed byassuming a Taylor rule for the interest rate, as r t = Φ π t − log β ,with Φ > π (1). Let us first assume that the crisisprobability is very small, so one can linearise the Euler equation F. G. Morelli et al. | July 18, 2019 | q. (5) with ς = 1. Solving forward in time leads to π t + κ > Φ ( δ t − δ t − ) = (cid:16) − κ > Φ (cid:17) ∞ X k =0 Φ − k − E t [ δ t + k +1 − δ t + k ] , [21]where δ t is the output gap defined in Eq. (12) and κ > :=3 G ( c > ) ≥
0. In the DSGE limit κ > → − κ > / Φ can become negative forsome range of parameters. Accounting for crises analyticallyis difficult in general, since the Euler equation cannot belinearised anymore. In order to make progress, we model thedynamics as follows: with probability p = T − ( c > → c < ) (cid:28) t and t + 1, and with probability1 − p it hovers normally around c > , with small fluctuations.We also assume that π t (cid:28)
1. Hence we approximate the righthand side of the Euler equation Eq. (5) as: F ( c t ) c > (1 + Φ π t ) (cid:16) (1 − p ) E >t [(1 − π t +1 − δ t +1 )] + p c > c < (cid:17) , [22]where E >t is an expectation conditional to remaining near thehigh output equilibrium. This eventually leads to an extraterm in Eq. (21) equal to δπ t = − p Φ − c > − c < c < . [23]As expected, anticipation of possible crises decreases inflation;provided c < (cid:28) c > this correction can be substantial evenwhen p (cid:28) C t = Y t : production and consumption will notmatch as confidence collapses. We leave these extensions forfuture research.In our opinion, however, the most important aspect ofour model is that it suggests alternative, behavioural toolsfor monetary policy, in particular in crisis time. Beyondadjusting interest rates and money supply, policy makers canuse narratives to restore trust, ‡‡ parameterised in our model bythe threshold c . If the economy lies in the neighbourhood ofthe C/B + phase boundary (see again Fig. 2), a mild decreaseof c , engineered by the Central Bank, can help putting backthe system on an even keel. ACKNOWLEDGMENTS.
We thank A. Amstrong, R. Farmer, X.Gabaix, S. Gualdi, A. Kirman, J. Scheinkman & F. Zamponi formany insightful discussions on these topics. This research wasconducted within the
Econophysics & Complex Systems
ResearchChair, under the aegis of the Fondation du Risque, the Fondationde l’Ecole polytechnique, the Ecole polytechnique and Capital FundManagement, and is part of the NIESR
Rebuilding Macroeconomics initiative. MT is a member of the
Institut Universitaire de France .
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