finance discipline group
TRANSCRIPT
FINANCE DISCIPLINE GROUP
UTS BUSINESS SCHOOL
WORKING PAPER NO.183 August 2015
Modelling the “Animal Spirits” of Bank's Lending Behaviour Carl Chiarella Corrado Di Guilmi Tianhao Zhi
ISSN: 1837-1221 http://www.business.uts.edu.au/finance/
Modelling the “Animal Spirits” of Bank’s
Lending Behaviour
Carl Chiarella, Corrado Di Guilmi, Tianhao Zhi∗
University of Technology, Sydney
July 30, 2015
Abstract
The idea of “animal spirits” has been widely treated in the literaturewith particular reference to investment in the productive sector. Thispaper takes a different view and analyses from a theoretical perspective therole of banks’ collective behaviour in the creation of credit that, ultimately,determines the credit cycle. In particular, we propose a dynamic modelto analyse how the transmission of waves of optimism and pessimism inthe supply side of the credit market interacts with the business cycle. Weadopt the Weidlich-Haag-Lux approach to model the opinion contagion ofbankers. We test different assumptions on banks’ behaviour and find thatopinion contagion and herding amongst banks play an important role inpropagating the credit cycle and destabilizing the real economy. The boomphases trigger banks’ optimism that collectively lead the banks to lendexcessively, thus reinforcing the credit bubble. Eventually the bubblescollapse due to an over-accumulation of debt, leading to a restrictive phasein the credit cycle.
keywords: animal spirits; contagion; pro-cyclical credit cycle; financial fragility.JEL classification: E12, E17, E32, G21.
∗Corresponding author: Tianhao Zhi. Finance Discipline Group - University of Technology,Sydney - E-mail: [email protected].
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1 Introduction
“Banks are much more social creature than most people think. Un-derstanding banks is not about brute facts. The values on a balancesheet are dependent upon confidence, and when an institution is introuble those values are quite different from the figures when theinstitution is thought to be doing well. So value is dependent onconfidence, which is a social fact rather than a material property.”
- Rethel and Sinclair (2012), The Problem with Banks
The recent financial crisis has revived the interest in studying the role offinancial markets and institutions in amplifying business cycles in the real sector.While models in the Keynesian tradition have always featured an integratedfinancial sector, mainstream analysis has only recently been applied to studythe role of the financial sector beyond its role of generating frictions for theadjustment process. Recent history has shown once again that the financial andthe real sector are interconnected, and the boom-bust cycle in the asset marketplays an important role in influencing aggregate demand and amplifying thebusiness cycle. It is therefore crucial to build macroeconomic models that takeinto account of the factors such as the balance sheet composition of economicagents, in particular, their credit, debt, and leverage position.
In the traditional banking literature, the commercial bank is often modelledas a passive intermediary that channels funds from the ultimate borrower tothe ultimate lender (Allen and Gale, 2000; Bernanke et al., 1999; Fama, 1980).In reality however, the role of banks goes well beyond the intermediation be-tween supply and demand of savings. A bank functions as an active creditcreator (Taylor, 2004; Ryan-Collins et al., 2012). As recently remarked byMcLeay et al. (2014), the creation of a loan simultaneously creates a deposit,which endogenously enlarges the money stock, since deposit is part of the broadmoney (M3). In other words, the bank’s behaviour is not a passive reflectionof the conditions of the economy, but is in itself an important factor that influ-ences the economy through the creation of credit. Minsky (1975) puts the roleof credit creation of banks at the centre of his framework: it is the relaxing ofcredit conditions that drives the expansionary phase of the cycle and it is thecontraction in credit that exacerbates the downturn.
Another important aspect, which is often overlooked in the traditional bank-ing literature, is the role of the bank’s lending attitude1. An optimistic atti-tude in the banking sector collectively lowers the lending standard and promptsbanks to lend excessively2, which potentially leads to the development of acredit bubble. Eventually the bubble bursts due to an unsustainable level ofdebt (Kindleberger, 1989). On the contrary, a collectively pessimistic banking
1Exceptions are Asanuma (2013) and Berger and Udell (2004).2The lending attitude also involves the composition of the credit portfolio and the decision
of lending in increasing proportion to a particular sector, as the latest crisis dramaticallyshowed. On the pros and cons of diversification see for example Allen and Gale (2001) andBattiston et al. (2007).
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system not only hinders economic growth but also renders expansionary mone-tary policy ineffective, as we have observed in the recent financial crisis. As itis shown in Figure 1, in the aftermath of the crisis, the money base in the UShas grown nearly threefold due to three rounds of Quantitative Easing (QE),however, it has virtually no effect on the growth of broad money due to thenegative outlook (Koo, 2011).
Figure 1: The effect of Quantitative Easing on Money Base and M2. Source:the Federal Reserve statistics release.
The pessimism or optimism, which can potentially influence behaviour ofeconomic agents such as bankers, is referred to as animal spirits by Keynes(1936) in chapter 12 of the General Theory, in which he claims that: mostprobably, of our decisions to do something positive, the full consequences ofwhich will be drawn out over many days to come, can only be taken as a re-sult of animal spirits: of a spontaneous urge to action rather than inaction,and not as the outcome of a weighted average of quantitative benefits multi-plied by quantitative probabilities. This fact brings two relevant corolloraries.First, expectations are self-fulfilling: an optimistic/pessimistic sentiment willbring forth a positive/negative outcome to the market, which further reinforcesthe optimistic/pessimistic sentiment. Second, market sentiment is contagious:sentiment spreads and it eventually leads to herding amongst agents. The herd-ing behaviour in financial market is well-documented in empirical literature(Sharma and Bikhchandani, 2000; Haiss, 2005). There is also notable amountof literature that finds empirical evidence of herding amongst banks, particu-larly in the US and Japan (Liu, 2012; Nakagawa and Uchida, 2011). In different
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papers, Dow relates the Minskyan credit cycle to the psychology of lenders andinvestors (see for example Dow, 2011).
A number of papers in the last decades model the animal spirits in finan-cial and macroeconomic as herding behaviour. This modelling philosophy isrooted in the approach initially proposed by Weidlich and Haag (1983). Thebasic idea is to model heterogeneous agents that choose and switch betweentwo attitudes in probabilistic terms. A reduced-form Master equation that cap-tures the “average opinion” is applied to simplify the analysis of the stochasticsystem. Lux (1995) proposes a seminal work that examines the relationship be-tween investors’ sentiment and asset price bubble/crash. Franke (2012) termsthis approach “Weidlich-Haag-Lux” and extends the Lux model to the contextof macroeconomic dynamics. He studies the interplay between firm’s sentiment,inflation and output gap, which establishes an alternative microfoundation formacroeconomics in the Keynesian tradition. This model is further developed byCharpe et al. (2012), which propose a “Dynamic Stochastic General Disequilib-rium (DSGD)” model. The DSGD model examines the real-financial interactionby incorporating a speculative financial market populated by heterogeneous in-vestors and it takes a “disequilibrium” approach that models the dynamicaladjustment process, instead of assuming immediate equilibrium adjustment.3
This paper follows this strand of literature and examines the role of “animalspirits” as the determinant of banks’ lending decisions. The aim is to assesshow the contagious waves of optimism and pessimism contributes to the boom-bust of the credit cycle, via a modification of banks’ balance sheet positions andhow it amplifies the business cycle in the real sector. From this perspective ouranalysis integrates related contributions in the Minskyan tradition4 by focusingon the role of banking sector rather than on borrowers.
We present an aggregative model in which banks follow heterogeneous lend-ing strategies and, given their cognitive limits, are assumed to follow a herdingbehaviour. The banks’ opinion formation dynamics is modelled in the spiritof the Weidlich-Haag-Lux approach. The joint evolution of banks’ behaviour,credit supply and aggregate output are analysed in a dynamical system.
In our take of this approach, herding among banks is characterized by twodifferent behaviours. The first is the switching of banks between the two cate-gories of optimistic and pessimistic, which is more intense the bigger is the sizeof the majority group. The second regards the herding behaviour of banks forwhat concerns their decision about their loans-to-reserve ratio, which is largerduring optimistic phases5. In this way we further extend the Weidlich-Haag-Lux approach by including a further dimension of herding. In particular westudy two different scenarios. In the first banks are assumed to have a constant
3In addition to this strand of literature that models the “animal spirits” in disequilibriumprocess, there are numerous papers that take a more standard equilibrium approach. Forexample, DeGrauwe (2011) develops a DSGE model that features waves of optimism andpessimism by incorporating agentss cognitive limitations. For a study of the banks lendingattitude in an agent-based model see Asanuma (2013).
4See Charpe et al. (2011), Chiarella and Di Guilmi (2011) and Delli Gatti et al. (1993).5For an analytical investigation of the separate effects of switching and herding on market
volatility see Di Guilmi et al. (2013).
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loan-to-reserve ratio (assumed to be larger for optimistic banks) and, as a con-sequence, the variations in the supply of credit depends only on the switchingof banks from one group to another. In the second scenario, the loan-to-reserveratio for each group of banks follows the market mood, expanding and con-tracting. Within this second scenario we study separately the special case of aconstant ratio for pessimistic banks.
The paper involves two main novelties. First, while several other contri-butions have already treated the effects of herding and bounded rationality offirms and households on the business cycle, to the best of our knowledge thispaper represents the first attempt to model banking behaviour as influenced byanimal spirits in a dynamic setting. The second original aspect concerns theintroduction of heterogeneity in the credit sector, which represents a noveltyin this stream of aggregative dynamical models. In contrast to the traditionalbanking literature, we stress the role of the mechanism of credit-creation bybanks as a potentially destabilising factor.
Anticipating some of the results, the numerical study of stability of thesystem reveals that contagion and heterogeneous expectations amongst bankersplay an important role in destabilizing the system and propagating boom-bustof credit cycle and business cycle in the real sector.
The remainder of the paper is organized as follows. In the next section, wepropose a baseline model, where banks are categorized in optimistic and pes-simistic. The two-dimensional dynamical system consists of the average opinionand the output dynamics. In section 3, the loan-to-reserve ratio of the opti-mistic banks follows the market mood while the pessimistic banks’ one is heldconstant. The three-dimensional dynamical system consists of the optimisticbanks’ loan-to-reserve ratio, the average opinion, and the output dynamics. Insection 4 we extend the model by endogenizeing the behavioural rule for pes-simistic banks and analyse the role of heterogeneous lending strategies in thecredit cycle and in destabilizing the real sector in a four dimensional dynamicalsystem. In section 5 we further extend the model by incorporating a speculativereal sector, adopting the framework of Charpe et al. (2012). Finally, section 6offers some concluding considerations.
2 The Baseline Model
In this first section we present the main behavioural hypotheses of the model.Banks are classified into the two categories of optimistic and pessimistic. Bothtypes of banks are assumed to keep constant their loan-to-reserve ratio. Thefluctuations in the supply of credit (and consequently in the real output) aretherefore an effect of the switching of banks between the optimistic and thepessimistic state.
Table 1 illustrates the structure of a typical balance sheet of a commercialbank. On the asset side it consists of bank reserves, loans, and other assets suchas treasury bonds; on the liability side there are deposits, bank borrowing, andbank equity. When a bank makes loans, it simultaneously creates deposits.
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Table 1: A Simplified Balance Sheet of Commercial Bank
2.1 The Basic Set-up
Following Taylor (2004), we focus on the loan-to-reserve ratio (λs), which isdefined as the ratio between loans and unborrowed reserves6. The variable λs
reflects not only the banks’ lending attitude, but also the debt accumulationdue to banks’ credit creation. Specifically, we have that
Ls = λsTc, (1)
where
• Ls is the level of aggregate credit supply;
• λs is the loan-to-reserve ratio of banks;
• Tc is the total amount of unborrowed reserves, which is assumed to beexogenous.
The total number of banks is 2N , while n+ is number of optimistic banks andn− is the number of pessimistic banks (2N = n+ + n−). The optimistic bankslend at a loan-to-reserve ratio λ+ assumed to be larger than the pessimisticbanks’ one λ−. We assume that each bank holds the same amount of reserves,and the two loan-to-reserve ratios are initially set as constant as λ+ and λ−.Hence equation (1) becomes
Ls = R(n+λ+ + n−λ−), (2)
Tc = 2NR, (3)
where R indicates the reserves. Following Lux (1995), the difference in the sizeof the two groups is quantified by the index x
x = (n+ − n−)/2N. (4)
As 2N = n+ + n−, it is easy to derive that
n+ = N(1 + x) (5)
n− = N(1− x) (6)
6The modelling of an inter-banks market is beyond the scope of the present paper andconsequently we assume it away. Reserves therefore can only be unborrowed.
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The index x describes the average opinion or, in the context of this paper, thegeneral lending attitude of banks. When x = 0, there are equal number ofoptimistic and pessimistic banks. When x = ±1, it implies that all the banksare either optimistic or pessimistic. Since Tc = 2NR, Equation (2) is thereforemodified in terms of x as
Ls =Tc
2[(1 + x)λ+ + (1− x)λ−]. (7)
We postulate that the availability of credit Ls determines the aggregatedemand in the real sector. Furthermore, we assume that output (y) is demand-driven and it follows Blanchard (1981) as a stylized dynamic multiplier process.That is,
yd = yd0 + kLs (8)
y = σ(yd − y) (9)
where yd0 is the autonomous component of the aggregate demand.Substituting (7) and (8) into (9) we get
y = σ{yd0 + kTc
2[(1 + x)λ+ + (1− x)λ−]− y}. (10)
2.2 Opinion dynamics
In this sub-section we introduce the law of motion for the average opinion x. Letp+− be the transition probability for a pessimistic bank to become optimistic,while p−+ is the probability of the opposite transition. Accordingly, the changein the level of x depends on the size of each group multiplied by their transitionprobability. Thus we have
x = (1− x)p+− − (1 + x)p−+. (11)
Two factors affect the probability of transition of banks from one group toanother: the bankers average opinion x, which captures the contagion effect; theoutput gap in the real sector yd−y and a general financial index d, representingthe propensity of banks to switch. We assume d to be constant as it is relatedto institutional factors. Accordingly, we compose a switching index s as a linearcombination of the three factors
s(x, y, d) = a1x+ a2(yd − y) + d. (12)
The parameter a1 quantifies the effect of herding and plays an important rolein our story. Assuming that the relative changes of p+− and p−+ in response tochanges in s are symmetric, the probabilities can be written as
p+− = v · exp(s), (13)
p−+ = v · exp(−s). (14)
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Hencex = v[(1− x) exp(s)− (1 + x) exp(−s)]. (15)
Equations (12) and (15) show that the emergence of a lending behaviour asthe most popular is self-strengthening: more and more banks are assumed tofollow the strategy adopted by the largest number of banks. An increase in sincreases the probability that the pessimistic banks will become optimistic ones.Equation (15) models the lending behaviour of banks in fashion similar to thefamous Keynes’ beauty contest metaphor7. In a situation of uncertainty andless than perfect information, an anchor to the expectations is provided by thebehaviour of other agents.
It is therefore possible to represent the dynamics by means of the two-dimensional system composed by equations (9) and (15).
y = σ(yd − y) (16)
x = v[(1− x) exp(s)− (1 + x) exp(−s)] (17)
where
yd = yd0 + kLs = yd0 + kTc
2[(1 + x)λ+ + (1− x)λ−] (18)
s = a1x+ a2(yd − y) + d. (19)
2.3 Analysis of the two-dimensional system
In order to study the properties of the system (16),(17) we first set LHS=0 onboth equations and derive the following isoclines
y = yd0 + kTc
2[(1 + x)λ+ + (1− x)λ−], (20)
y =a1a2
x− 1
2a2ln
1 + x
1− x+ yd + d. (21)
It is difficult to obtain the close-form solution of equation (20-21). Yetin a special case when we set d = 0, we can easily obtain a neutral opinionequilibrium (x⋆ = 0, y⋆ = yd). Furthermore, there is potentially an emergenceof other equilibria (x⋆
+, y⋆+) and (x⋆
−, y⋆−), depending on the value of contagion
parameter a1 - as shown in Figure 2, which plots the two isoclines8. We analysethe local stability of the neutral opinion equilibrium by deriving the Jacobianof the system (16-17)
J =
(−σ σk Tc
2 (λ+ − λ−)−2va2 2v[a1 + a2k
Tc
2 (λ+ − λ−)]
). (22)
7“It is not a case of choosing those [faces] that, to the best of ones judgement, are really theprettiest, nor even those that average opinion genuinely thinks the prettiest. We have reachedthe third degree where we devote our intelligences to anticipating what average opinion expectsthe average opinion to be. And there are some, I believe, who practice the fourth, fifth andhigher degrees.” (Keynes, 1936).
8The following parameters are set for the isoclines: a1 = 0.9 (left), a1 = 1.4 (right),a2 = 0.4, Tc = 1, k = 0.1, d = 0, v = 0.7, yd0 = 10, λ− = 5, λ+ = 20. σ = 0.8.
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The trace and determinant of the Jacobian at neutral opinion equilibrium (x =0) is calculated as
Tr(J) = 2v[a1 + a2kTc
2(λ+ − λ−)]− σ (23)
Det(J) = 2v{a2σkTc
2(λ+ − λ−)− σ[a1 + a2k
Tc
2(λ+ − λ−)− 1]} (24)
The necessary and sufficient condition for local stability for this equilibrium isthat Tr(J) < 0 and Det(J) > 0. Otherwise it would become locally unstablein the form of repelling cycle or saddle node. The contagion parameter a1plays an important role in determining the local stability: this neutral opinionequilibrium is more likely to be stable when a1 is relatively small. The resultsof a representative simulation, together with the bifurcation analysis for thecontagion parameter a1, are provided in Figure 4. The parameters are set asfollows: a1 = 0.7, a2 = 2.1, σ = 0.8, k = 0.1, Tc = 1, yd0 = 10, λ− = 5, λ+ = 20,v = 0.4, d = 0.5. Note that in the numerical simulation the parameter dtakes a non-zero value. As we can see, the neutral opinion equilibrium becomesunstable and a limit cycle emerges as the contagion parameter increases andpasses through a1 ≈ 0.3. As a1 further increases, the system becomes stableagain, yet it converges to another equilibrium with higher value of x.
−1 −0.5 0 0.5 19.5
10
10.5
11
11.5
12
12.5
13
x
y
−1 −0.5 0 0.5 110
10.5
11
11.5
12
12.5
x
y
Figure 2: The Isocline of baseline model with y = 0 (red) and x = 0 (blue)
3 Herding among the optimistic banks
In this setting pessimistic banks are supposed to be inactive, in the sense thatthey keep a constant loan to reserve ratio, while optimistic banks become activesince they adjust their ratio9.
9By keeping one group of banks inactive in terms of a constant loan-to-reserve ratio, weare able to analytically assess the properties of the dynamical system with the varying loan-to-reserve ratio at least in one group.
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3.1 The dynamics of λ+
The active banks constantly adjust their balance sheet positions according tothe average opinion while the inactive banks are insensitive toward change inaverage opinion, which is reflected in the constant loan-to-reserve ratio λ−. Wefocus the analysis on the endogenous variables x and λ+. Given the bank’srole as a credit creator, the bank’s lending behaviour is not constrained by theavailability of deposit, but determined by its lending attitude. For simplicity, weassume that the relationship between the rate of change of λ+ and x is linear.For hypothesis, when x > 0, the active banks expand their balance sheet at aconstant speed γ; when x < 0, they contract their balance sheet position atthe same speed; when x = 0, the balance sheet position of active banks remainsunchanged. We assume that λ+ adjusts also by the change of real output. Hencethe law of motion for λ+ takes the following form
λ+ = γ1x+ γ2y. (25)
From this perspective the index x is interpreted by the banks as a proxy ofthe state of the economy and in particular of the capacity of the productivesector to repay loans in the future. In equation (25), γ1 captures the speed ofadjustment regarding average opinion x and γ2 captures the speed of adjustmentregarding changes to y. The first term can be interpreted as the stimulative roleof bank’s credit creation driving by banker’s sentiment, whereas the second termrepresents a passive, accommodative role of credit creation driven by the growthin the real sector10.
3.2 The dynamical system
Given the above assumptions, the three-dimensional system with variable λ+
can be written as
λ+ = γ1x+ γ2y (26)
y = σ(yd − y) (27)
x = v[(1− x) exp(s)− (1 + x) exp(−s)] (28)
Figure 3 illustrates the feedback loop of the system (26), (27), (28). Sup-pose that the index x is initially positive: it leads to an increase of leverage ofoptimistic banks, thus increasing the credit supply. This is accompanied by anincrease of aggregate demand, thus raising output. When bankers’ sentimentbecomes more optimistic, and output gap increases, thus raising s. This formsa positive feedback loop. In addition, we have another positive feedback loopvia the herding channel, where the bankers’s decision making is affected by ma-jority opinion. However, this does not last forever. The increase of bankers’leverage leads to an accumulation of debt, which has a negative impact on the
10Consistently with the literature about financial fragility and business cycle, a positivecorrelation between aggregate output and debt is assumed here.
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Figure 3: The feedback loop of the 3D system.
opinion formation index s. At the beginning, the positive feedback outweighsthe negative feedback. Up to a certain point when debt accumulation becomessignificantly large, the system turns in the reverse direction.
We provide in the following an assessment of the properties of the system.Proposition 3.1: The steady state of the dynamical system (26), (27), (28) isunique and given by λ∗
+ = − da2, y∗ = yd0 + k Tc
2 (λ− − da2), x∗ = 0.
Proof: By setting equations (26), (27), (28) equal to zero, we have
0 = γ1x+ γ2y, (29)
0 = σ(yd − y), (30)
0 = v[(1− x) exp(s)− (1 + x) exp(−s)]. (31)
Since y = 0, we derive that y∗ = yd∗ and x∗ = 0 in equation (27) and(29). Substituting to equation (28) we obtain v[exp(s) − exp(−s)] = 0, orexp(2s) = 1. Hence s = 0. Since s = a1x + a2λ+ + a3(y
d − y) + d, we derivethat λ∗
+ = − da2. Finally, according to equation (20) and (23) we derive that
y∗ = yd∗ = yd0 + k Tc
2 (λ− − da2).
Next we analyse the local stability of the system (26), (27), (28). TheJacobian of this system at equilibrium is derived as
γ2σkTc/2 −γ2σ γ1 + γ2σk(Tc/2)(− da2
− λ−)
σk(Tc/2) −σ σk(Tc/2)(− da2
− λ−)
2v(a2 + a3k(Tc/2) −2va3 2v(a1 + a3k(Tc/2)(− da2
− λ−)− 1)
,
which has the sign structure + − ++ − +? − ?
.
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Proposition 3.2: The system (17)-(19) is more likely to be locally stable nearequilibrium if a1, a3, γ1, γ2 are sufficiently small, and |a2| is sufficiently large.Proof : The trace (Tr), determinant (Det), and the principle minors (Ji) arederived as follows:
Tr(J) = γ2σkTc/2− σ + 2v(a1 + a3k(Tc/2)(−d
a2− λ−)− 1), (32)
Det(J) = γ2σkTc/2
∣∣∣∣ −σ σk(Tc/2)(− da2
− λ−)
−2va3 2v(a1 + a3k(Tc/2)(− da2
− λ−)− 1)
∣∣∣∣+ γ2σ
∣∣∣∣ σk(Tc/2) σk(Tc/2)(− da2
− λ−)
2v(a2 + a3k(Tc/2) 2v(a1 + a3k(Tc/2)(− da2
− λ−)− 1)
∣∣∣∣+ [γ1 + γ2σk(Tc/2)(−
d
a2− λ−)]
∣∣∣∣ σk(Tc/2) −σ2v(a2 + a3k(Tc/2) −2va3
∣∣∣∣ ,(33)J1 =
∣∣∣∣ −σ σk(Tc/2)(− da2
− λ−)
−2va3 2v(a1 + a3k(Tc/2)(− da2
− λ−)− 1)
∣∣∣∣ , (34)
J2 =
∣∣∣∣ γ2σkTc/2 γ1 + γ2σk(Tc/2)(− da2
− λ−)
2v(a2 + a3k(Tc/2) 2v(a1 + a3k(Tc/2)(− da2
− λ−)− 1)
∣∣∣∣ , (35)
J3 =
∣∣∣∣ γ2σkTc/2 −γ2σσk(Tc/2) −σ
∣∣∣∣ . (36)
According to the Routh-Hurwitz theorem, the necessary condition for the stabil-ity of the 3D sub-dynamics is that tr(J) < 0, J1+J2+J3 > 0, and det(J) < 0.11
In order to satisfy these conditions we need to have sufficiently small a1, a3, γ1,γ2, as well as sufficiently large |a2|.The system (17-19) is more likely to be stable when contagion amongst banks isweak (herding and switching are small), bankers are less tolerant toward debt,and banks’ opinion is less sensitive toward changes in the real economy.
3.3 Numerical simulations
In order to investigate the global features of the nonlinear 3D dynamics, wereformulate the baseline model in discrete time by using a standard Eulermethod12. The simulations involve two different scenarios: a1 = 0.3 (stablescenario) and a1 = 1.5 (unstable scenario). The other parameters are set as fol-lows: a2 = −0.02, a3 = 1.3, σ = 0.8, k = 0.1, Tc = 1, yd0 = 10, d = 0.5, v = 0.4,γ1 = 0.5, γ2 = 2, λ− = 5. As is shown in Figure 5, we simulate the stable sce-nario where herding effect is relatively weak (a1 = 0.3) on the left-hand panel,
11The necessary and sufficient condition for local stability requires an addition of−tr(J)(J1 + J2 + J3) + det(J) > 0 (Chiarella and Flaschel, 2000).
12The Euler method has the advantage of preserving the economic content of the discretisedequations (Chiarella et al., 2005).
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as well as an unstable scenario where the herding effect is relatively strong onthe right-hand panel (a1 = 1.5). We observe that, when the herding effect isrelatively weak, the system tends toward a stable equilibrium (x = 0, yd = y,and λ+ = d/(−a2)). In the second case where the contagion parameter a1 isrelatively large, the equilibrium becomes unstable. It is possible to observe rel-atively regular sequences of fluctuations of the three variables. The upper rightpanel captures waves of optimism and pessimism amongst bankers, accompa-nied by expansion and contraction of the bank’s balance sheet. The mid-rightpanel describes the interplay between x, y, and λ+. Most interestingly, the lowerright panel shows a three-stage cycle in the real sector: a gradual expansion, asudden slump, and a prolonged period of recession. The unstable scenario is inline with anatomy of a typical credit cycle as described by Kindleberger (1989)and Minsky (1963): the optimism at the initial stage triggers banks to lendmore, which ultimately leads to the growth in the real sector. However, dueto an over-accumulation of debt that modifies lenders’ expectations, the cycleeventually reverses until it reaches the state where a majority of banks becomepessimistic. The representative simulation also shows the crucial role of coor-dination failure, reflected in herding behaviour amongst banks, in propagatingthe credit cycle and destabilizing the real sector.
We further examine the dynamical interplay between output and debt byanalysing the debt/GDP ratio, which is defined as
(Debt/GDP ) ratio = Ls/y. (37)
The different patterns of the Debt/GDP ratio with different values of thecontagion parameter a1 are shown by in Figure 6. The other parameters areunchanged with respect to the previous section. The simulation shows that ahigher value of a1 tends to destabilize the system, which is reflected in a largermagnitude of boom-bust cycle both in the credit sector and the real sector. Forthe highest value of the parameter, also the shape of the orbit appears to changetoward an 8-shape.
Both fluctuations and limit cycle are of larger amplitude in this scenariocompared to the setting with only switching. It is possible to conclude that theherding within the group of optimistic banks further destabilises the economy.
3.4 Sensitivity and bifurcation analysis of the behaviouralparameters
In this section we further analyse the effect of changes in the behavioural pa-rameters. We confine our attention to the three cognitive parameters in theopinion formation index (a1, a2, and a3), as well as the two action parameters(γ1 and γ2) in equation (26), since they are the most relevant in determiningbankers’ behaviour. The scope of the sensitivity and bifurcation analysis is two-fold: first, it serves as a robustness test; second, it provides important insighton how bankers’ behaviour stabilises or destabilises the system.
13
We first simulate the dynamics of output (y) showing its dynamics for valuesof contagion parameter (a1) and keeping fixed the other parameters. As isshown in Figure 7, when a1 = 0.3, output tends toward the stable equilibrium;When a1 = 0.7, a period of growth and tranquillity is observable, followed bya persistent period of instability. In this setting, “stability breeds instability”,as Minsky famously wrote, as this pattern seems to replicate the one in manyof the recent crises. When a1 = 1.5, the dynamics of output becomes a three-stage cycle, which is in line with the previous representative simulation. Wefurther draw the bifurcation diagram for a1 in Figure 8. It can be seen thatHopf bifurcation occurs when a1 increases and passes through a1 ≈ 0.65, ceterisparibus. The system transits from a stable equilibrium to persistent fluctuations.The destabilizing effect further increases when a1 increases and passes throughapproximately 1.2.
Figure 9 and 10 are the bifurcation diagrams for the other two cognitiveparameters a2 and a3. The Hopf bifurcation points turn out to be a2 ≈ −0.025and a3 ≈ 1.4. The two bifurcation diagrams show that banker’s degree of debt-tolerance and their opinion over the change of real sector also play importantroles in destabilizing the system. The system tends to be unstable if the bankersare more tolerant toward debt and hold more sensitive opinion toward a changein the real sector.
The strong relevance of the herding behaviour of banks over the cycle in thereal economy is confirmed by figures 11 and 12, which reports the bifurcationdiagrams for the two action parameters γ1 and γ2. Figure 11 indicates thatHopf bifurcation occurs and the system transits from stability to instabilitywhen γ1 increases and passes through the value γ1 ≈ 0.6, or when γ2 increasesand passes through the value γ2 ≈ 2.5, holding the other parameters same as inthe previous reference simulation.
4 The convergence and divergence of heteroge-neous lending strategies
In this section the model is further extended by incorporating heterogeneouslending strategies between optimistic and pessimistic banks. The dynamics ofloan-to-reserve ratio of pessimistic banks (λ−) is now modelled as an endogenousvariable and added to the new system. We assume that
λ+ = γ1(x+ g(.)) + γ2y + γ3(λ+ − λ+), (38)
λ− = γ1(x− g(.)) + γ2y + γ3(λ− − λ−). (39)
There are two additional terms in the extended dynamics of λ+ and λ−.First, following DeGrauwe (2011), we introduce a function g(.) that capturesthe reaction gap between optimists and pessimists over the average opinion,which is quantified by
g(.) = ξ0exp(−ξ1x2). (40)
14
The optimistic banks react to x + g(.) while the pessimistic banks react tox − g(.). The parameter ξ0 determines the magnitude of opinion gap and ξ1captures the sensitivity of g(.) relative to the change of average opinion (x).Equation (40) captures a scenario where there is a convergence or divergence oflending strategies during the exuberant/calm period, with agents behaving in amore coordinated manner during a period of increasing optimism or pessimism.On the contrary, the lending strategies diverge during a relative calm period.However, the gap g(.) narrows and the lending strategies converge when theoptimistic or pessimistic sentiment grows over time.
Second, we add two mean-reverting terms γ3(λ+ − λ+) and γ3(λ− − λ−) tocapture the long-run adjustments: we assume that optimistic banks tend towarda higher loan-to-reserve ratio while the pessimistic banks tend toward a lower one(λ+ > λ−). The mean reverting process serves the purpose to provide a ceiling(floor) to the growth (decrease) of the lending ratio of optimistic (pessimistic)banks, which is necessary with endogenous λ+ and λ−.
Recalling that Ls = R(n+λ+ + n−λ−), n+ = (1 + x)N , n− = (1− x)N andTc = 2NR, the quantities Ls and yd are given by
Ls = Tc/2((1 + x)λ+ + (1− x)λ−), (41)
yd = yd0 + k[Tc/2((1 + x)λ+ + (1− x)λ−)]. (42)
Hence the new system is written as
λ+ = γ1(x+ g(.)) + γ2y + γ3(λ+ − λ+) (43)
λ− = γ1(x− g(.)) + γ2y + γ3(λ− − λ−) (44)
y = σ(yd − y) (45)
x = v[(1− x) exp(s)− (1 + x) exp(−s)] (46)
where
yd = yd0 + kLs = yd0 + k(Tc/2)[(1 + x)λ+ + (1− x)λ−], (47)
g(.) = ξ0exp(−ξ1x2), (48)
s = a1x+ a2+λ+ + a2−λ− + a3(yd − y) + d. (49)
We first analyse the steady state and local stability of the above 4D system.We notice that there is no closed form solution for the steady state condition13.Therefore, we consider a special case where the average opinion is neutral atequilibrium (x⋆ = 0). In this case, the general financial condition index dbecomes d = −a2+λ
⋆+ − a2−λ
⋆−14. In this special case, it is easy to derive the
close form solution for λ⋆+, λ
⋆−, and y⋆.
13By setting the LHS = 0, we derive that a2+[ γ1γ3
(x + ξ0e−ξ1x2) + λ+] + a2−[ γ1
γ3(x −
ξ0e−ξ1x2) + λ−] = 1
2ln[ 1+x
1−xe−2(a1x+d)]. This equation has no closed form solution.
14Setting the LHS = 0 in equation (42), we find that s = 0 when x⋆ = 0, hence d =−a2+λ⋆
+ − a2−λ⋆−.
15
Proposition 4.1: In the special case where x⋆ = 0, the steady state of thesystem (36-39) is unique and given by
λ⋆+ = λ+ +
γ1γ3
ξ0, (50)
λ⋆− = λ− − γ1
γ3ξ0, (51)
y⋆ = yd⋆ = yd0 + k(Tc/2)[(λ⋆+ + λ⋆
−)], (52)
x⋆ = 0. (53)
Proof: By setting LHS = 0, we have
0 = γ1(x+ ξ0exp(−ξ1x2)) + γ2y + γ3(λ+ − λ+) (54)
0 = γ1(x− ξ0exp(−ξ1x2)) + γ2y + γ3(λ− − λ−) (55)
0 = σ(yd − y) (56)
0 = v[(1− x) exp(s)− (1 + x) exp(−s)] (57)
Since x⋆ = 0 and y = 0, from equation (50) and (51) we derive that λ⋆+ =
λ+ + γ1
γ3ξ0 and λ⋆
− = λ− − γ1
γ3ξ0. Substituting this result to equation 47 we have
y⋆ = yd⋆ = yd0 + k(Tc/2)[(λ⋆+ + λ⋆
−)].We then analyse the local stability of the system. To make the system ana-lytically tractable, we consider a special case without the real sector by settingγ2 = 0, σ = 0, and a3 = 0. The Jacobians of equation (36), (37), and (39)sub-dynamics without the real sector is derived as −γ3 0 γ1
0 −γ3 γ12va2+ 2va2− 2v(a1 − 1)
,
which has the sign structure − 0 +0 − +− − ?
.
Proposition 4.2: The sub-dynamical system (43), (44), (46) without the realsector is locally asymptotically stable if a1, a3, γ1, and γ2 are sufficiently small,and a2, γ3 are sufficiently large.Proof: The trace Tr(J), determinant Det(J), and the three principle minorsJ1, J2, and J3 are derived as follows:
Tr(J) = 2[v(a1 − 1)− γ3], (58)
Det(J) = 2v[γ23(a1 − 1)− γ1γ3(−a2+ − a2−)], (59)
J1 = −2v[γ3(a1 − 1) + γ1a2−], (60)
J2 = −2v[γ3(a1 − 1) + γ1a2+], (61)
J3 = γ23 . (62)
16
According to the already cited Routh-Hurwitz theorem, the necessary andsufficient condition for the stability of the 3D sub-dynamics is that tr(J) < 0,J1 + J2 + J3 > 0, det(J) < 0, and −tr(J)(J1 + J2 + J3) + det(J) > 0.The parameter γ3, which captures the long-run adjustment of λ+ and λ−, playsan important role in stabilizing/destabilizing the system. The system tends tobe stable if γ3 is relatively large, as the reversion toward an average loan-to-reserve ratio is more pronounced.
As for the investigation of the global features of the complete 4D system, weturn to numerical simulations. Figure 13 provides a representative simulationof the extended model with the following parameter set: a1 = 1.5, a2+ = −0.3,a2− = −0.5, a3 = 1.3, σ = 0.8, k = 0.1, Tc = 1, yd0 = 11, d = 10, v = 0.4,γ1 = 0.3, γ2 = 0.4, ξ0 = 0.2, ξ1 = 3. In the top panel we observe that the opti-mistic/pessimistic banks become increasingly optimistic/pessimistic over time,until they settled down to two distinct limit cycles with higher value of λ+ andlower value of λ−. The mid-right panel shows the cyclical dynamics of g(.),which indicates a constant convergence and divergence of lending strategies be-tween two groups of banks. In this scenario the convergence of the strategiessmooths down the cycle, as shown by the milder swings in x compared with theprevious settings. Also, the variable g displays a pattern with a “double-peak”.Possibly, the first downswing in g during the cycle is not strong enough to revertthe pattern of λ+ (and therefore of y).
Figures 14-16 provide the bifurcation diagrams of a1, γ1, and γ2 of theextended model. We observe that the topological structure of the bifurcation inthe extended model is similar to the baseline model. However, contrary to thebaseline model, the relationship between γ2 and the magnitude of cycles seemsto be negative in the extended model and there is no identifiable stability range.A variable and dependent on herding loan-to-reserve ratio for pessimistic banksappears to be a further factor of instability. Figure 17 draws the bifurcationdiagram for the new long-run adjustment parameter γ3. We observe that Hopfbifurcation occurs at γ3 ≈ 0.045. A higher value of γ3 tends to stabilize thesystem. Also, the stability region for a1 is larger in the the present case.
5 Extension: The Interaction between Banksand a Speculative Real Sector
The baseline model proposed in previous section focuses exclusively on the roleof banks in destabilizing the business cycles. The real sector was assumed inits bare minimal structure. However, in order to capture more insights of real-financial market interaction in Minskian terms, one has to consider also thedestabilizing role of speculative financial market in relation to a real sector.This section extends the baseline model to address this issue by adopting theframework proposed by Charpe et al. (2012).
Specifically, we now adopt the following law of motion for output (y), re-
17
placing equation (9)15:
y = βy[(ay − 1)y + ak(pk − pk0)K + A] (63)
Where A is autonomous expenditure; K is the total capital stock; ay ∈ (0, 1)is the propensity to spend; ak > 0 captures the sensitivity of investment andconsumption demand to deviations between the actual and the equilibrium levelof capital stock.
As is pointed out by Charpe et al. (2012), the market for K is imperfectdue to information asymmetries, adjustment costs, or institutional constraints,hence price do not move instantaneously to clear markets. We assume that theprice of K moves according to the expected rate of return on the capital stock,ρek. The law of motion for capital price (pk) and the expected rate of return(πe
k) is given by:
pk = θLs(ρek − ρk0) (64)
πek = βπe
k[1 + xp
2pk − πe
k] (65)
ρek =by
pkK+ πe
k (66)
Where b is the profit share; ρk0 denotes the equilibrium level of expected rate ofprofit. here we assume that the profits are completely distributed as dividends.Furthermore, the aggregate level of loan issued by banking sector (Ls) entersequation (64) in the form of adjustment parameter (θLs), since credit expansionwill accelerate the rise or fall of asset prices.
We further adopts the Weidlich-Haag-Lux approach to model the opinionformation dynamics of speculative investors. Following Charpe et al. (2012),we assume that there are 2M number of investors. Of these, Mc are chartistsand Mf are fundamentalists so that Mc + Mf = 2M. Let m =
Mc−Mf
2 andx = m
M . We focus on the difference in the size of the two groups (normalisedby M). Let pf→c be the transition probability that a fundamentalist becomesa chartist, and similarly for pc→f . The change in xp depends on the relativesize of each population multiplied by the relevant transition probability (exp(sp)and exp(−sp)). To be precise, we define
xp = vp[(1− xp) exp(sp)− (1 + xp) exp(−sp)] (67)
sp = sxpxp + sxx− spk(pk − pk0)2 − sπe
k(πe
k)2 (68)
15See Charpe et al. (2012) for detailed discussion of this real sector dynamics.
18
The full 7D dynamics with a speculative real sector thus becomes:
λ+ = γ1(x+ g(.)) + γ2y + γ3(λ+ − λ+) (69)
λ− = γ1(x− g(.)) + γ2y + γ3(λ− − λ−) (70)
y = βy[(ay − 1)y + ak(pk − pk0)K + A] (71)
x = v[(1− x) exp(s)− (1 + x) exp(−s)] (72)
pk = θLs(ρek − ρk0) (73)
πek = βπe
k[1 + xp
2pk − πe
k] (74)
xp = vp[(1− xp) exp(sp)− (1 + xp) exp(−sp)] (75)
We conduct numerical simulations to investigate the global properties ofsystem (69-75). The parameters are set as follows: a1 = 1, a2+ = −0.3,a2− = −0.5, a3 = 1, d = 10, σ = 0.8, k = 0.1, Tc = 1 v = 0.4, γ1 = 0.3,γ2 = 0.4, γ3 = 0.03, λ+ = 15, λ− = 5, ξ0 = 0.2, ξ1 = 3, sxp = 0.9, spk = 2,sπe
k= 2, b = 0.1, K = 1, βy = 0.4, ay = 0.3, ak = 0.5, pk0 = 0.8, A = 2,
θ = 0.1, ρk0 = 0.7, βpek= 0.2, vp = 0.3, sx = 0.2, axp = 0.4. Figure 19 pro-
vides the simulation results. Again we stress that this is just one particularexample, providing results that are interesting from the perspective of the 7Dfull dynamics of real-financial interaction. The result would be highly sensitivewith different parameter settings. Yet it generates some interesting results thatmay have potentially shed light on the nature of real-financial market inter-action. The top-left panel shows a turbulent, chaotic swings of asset price inthe financial sector generated from this deterministic framework. The top-rightpanel shows the dynamics of “animal spirits” in both the banking (x) and thespeculative financial market (xp), where xp presents a higher degree of volatilitythan x. The mid panel shows the irregular dynamics of real sector (mid-left),as well as the positive relationship between output and stock price (mid-right),which resembles that of Charpe et al. (2012). More interestingly, in the bottompanel it shows that the mood swing of bankers is positively associated with thevolatility in the speculative financial sector. When x approaches neutrality, thefinancial market tends to be stable. When the sentiment in banking sector be-comes increasingly optimistic as x increases, More speculative behaviours startto emerge in the stock market, in the form of a larger swings of both xp and pk.This result once again shows the banking sector plays a crucial role in propa-gating financial instability in the speculative financial market, which ultimatelytransmits into the macroeconomic fluctuations in the real sector.
We further conduct bifurcation analysis for these opinion formation param-eters in both banking sector and speculative real sector. As for the bankingsector, we observe that, compared to previous simulation of 4D dynamics, thedestabilizing effect of opinion contagion parameter a1 and sensitivity towardchanges in real sector a3 have disappeared, since the speculative real sector isintroduced. Furthermore, the limit cycles become more irregular in the new dy-namics. As for the speculative real sector, we observe that the two parameters θand sxp, which capture the sensitivity of change in expected return and sensitiv-
19
ity toward change in price volatility play a key role in stabilizing/destabilizingthe system. When θ increases and passes through 0.07, the system transitsfrom stability to instability. Furthermore, the system transits from stability toinstability when sxp increases and passes through approximately 0.7. However,the system remains insensitive toward changes in spk, sπe
k. We notice that the
destabilizing effect of a1 (contagion parameter of the banking sector) is ham-pered since the introduction of the real sector.
We also consider the efficacy of policy interventions in the form of Tobin tax(at the rate τ). Following Charpe et al. (2012), the law of motion for capitalgain expectations (πe
k) thus becomes
πek = βπe
k[1 + xp
2(1− τ)pk − πe
k] (76)
Figure 21 provides the simulation results augmented by Tobin tax param-eter. From this bifurcation plot we observe that the Tobin tax parameter τhas stabilizing effect over the system. When τ increases and passes throughτ ≈ 0.24, the system switches from a regime of instability to stability with thedisappearance of limit cycle. It is clear that Tobin tax is able to stabilize thesystem by altering the adjustment speed of capital gain expectations.
6 Conclusion
The inherent instability of the credit cycle lies at the centre of financial crises(Minsky, 1975; Kindleberger, 1989). In particular, the commercial banks’s coor-dination failure, driven by waves of optimism and pessimism, ultimately leads tosub-optimal credit provision and amplifies the credit cycle. Keynes (1936) pointsout that “A sound banker, alas, is not one who foresees danger and avoids it, butone who, when he is ruined, is ruined in a conventional and orthodox way withhis fellows, so that no one can really blame him”. More recently, Chuck Prince,the CEO of Citibank, vividly depicts the collective behaviour of banks with hisanalogy: “as long as the music is playing, you’ve got to get up and dance.”. Bet-ter policy actions in the aftermath of a credit crunch require a sound conceptualunderstanding of the dynamical behaviour of banks (Aikman D., 2011).
This paper presents a simple model to investigate this issue. The analysis ofthe model shows that the self-fulfilling and contagious waves of optimism andpessimism amongst bankers lead to a boom-bust pattern for the credit cycle.
Summarising the results, we find that the switching and the herding be-haviours of banks are destabilising for the economy. The study of the twoscenarios with a constant or variable loan-to-reserve ratio for banks highlightsthe different destabilizing effects of herding and switching. Switching of banksbetween the two clusters with different levels in the supply of loan is able byitself to generate fluctuations in the aggregate output. Introducing herding inthe banks’ decisions about their loan-to-reserve adds to the instability and am-plifies the fluctuations. Instability and amplitude of fluctuations grow with theintensity of herding. A larger value in the herding parameter also causes higher
20
values of the aggregate leverage ratio. The study of the stability and the nu-merical analysis demonstrate that the parameters that measure the interactionof banks prove to be more relevant than other behavioural parameters, such asthe targeted loans to reserve ratio. The introduction of a convergent behaviourof banks in the long run noticeably smooths the cycle.
The next extension of the present framework involves a properly modelledspeculative real sector, adopting from the DSGD16 framework proposed byCharpe et al. (2012). Besides providing a better assessment of the destabil-ising effects of the credit cycle on the real sector, the interaction between thebanking sector and a speculative real sector can open interesting perspectivefor policy analysis in the present setting. The complete model will be also beuseful to study the feedback effect on the credit cycle of the phenomena ofover-investment and over-indebtedness.
Aknowledgments
The authors would like to thank Frederique Bracoud, Tony He, Steve Keen,Keith Woodward, the participants to the WEHIA 2013, NED 2013 and SNDE2014 conferences and to the SydneyAgents seminar at the University of Technol-ogy, Sydney for valuable feedback and comments. The usual disclaimer applies.
16Dynamic Stochastic General Disequilibrium.
21
References
Aikman D., N. B., Haldane A. (2011): Curbing the credit cycle, Prepared forthe Columbia University Centre on Capital and Society Annual Conference,New York, November 2010.
Allen, F. and Gale, D. (2000): Financial Contagion, Journal of PoliticalEconomy, 108(1): pp. 1–33.
Allen, F. and Gale, D. (2001): Financial Contagion, Journal of PoliticalEconomy, 108(1): pp. 1–33.
Asanuma, D. (2013): Lending attitude as a financial accelerator in a creditnetwork economy, Journal of Economic Interaction and Coordination, 8(2):pp. 231–247.
Battiston, S., Delli Gatti, D., Gallegati, M., Greenwald, B. andStiglitz, J. E. (2007): Credit chains and bankruptcy propagation in pro-duction networks, Journal of Economic Dynamics and Control, 31(6): pp.2061–2084.
Berger, A. N. and Udell, G. F. (2004): The institutional memory hypoth-esis and the procyclicality of bank lending behavior, Journal of FinancialIntermediation, 13(4): pp. 458–495.
Bernanke, B. S., Gertler, M. and Gilchrist, S. (1999): The financial ac-celerator in a quantitative business cycle framework, in: Taylor, J. B. andWoodford, M. (eds.), Handbook of Macroeconomics, Handbook of Macroe-conomics, vol. 1, chap. 21, pp. 1341–1393, Elsevier.
Blanchard, O. J. (1981): Output, the Stock Market, and Interest Rates,American Economic Review, 71(1): pp. 132–43.
Charpe, M., Flaschel, P., Hartmann, F. and Proao, C. (2011): Stabiliz-ing an unstable economy: Fiscal and monetary policy, stocks, and the termstructure of interest rates, Economic Modelling, 28(5): pp. 2129–2136.
Charpe, M., Flaschel, P., Hartmann, F. and Veneziani, R. (2012): To-wards Keynesian DSGD (disequilibrium) Modelling: Real-Financial MarketInteractions with Heterogeneous Expectations Dynamics, IMK Working Pa-per 93-2012, IMK at the Hans Boeckler Foundation, Macroeconomic PolicyInstitute, URL http://ideas.repec.org/p/imk/wpaper/93-2012.html.
Chiarella, C. andDi Guilmi, C. (2011): The financial instability hypothesis:A stochastic microfoundation framework, Journal of Economic Dynamics andControl, 35(8): pp. 1151–1171.
Chiarella, C. and Flaschel, P. (2000): The dynamics of Keynesian mone-tary growth: macrofoundations, Cambridge University Press.
22
Chiarella, C., Flaschel, P. and Franke, R. (2005): Foundations for adisequilibrium theory of business cycle: qualitative analysis and quantitativeassessment, Cambridge University Press.
DeGrauwe, P. (2011): Animal Spirits and Monetary Policy, Economic Theory,47(2-3).
Delli Gatti, D., Gallegati, M. and Gardini, L. (1993): Investment confi-dence, corporate debt and income fluctuations, Journal of Economic Behavior& Organization, 22(2): pp. 161–187.
Di Guilmi, C., He, X.-Z. and Li, K. (2013): Herding, TrendChasing and Market Volatility, Research Paper Series 337, Quantita-tive Finance Research Centre, University of Technology, Sydney, URLhttp://ideas.repec.org/p/uts/rpaper/337.html.
Dow, S. C. (2011): Cognition, market sentiment and financial instability, Cam-bridge Journal of Economics, 35(2): pp. 233–249.
Fama, E. F. (1980): Banking in the theory of finance, Journal of MonetaryEconomics, 6(1): pp. 39–57.
Franke, R. (2012): Microfounded Animal Spirits in the New MacroeconomicConsensus, Studies in Nonlinear Dynamics & Econometrics, 16(4): pp. 1–41.
Haiss, P. R. (2005): Banks, Herding and Regulation: a Review and Synthe-sis, Tech. rep., Paper presented at the Workshop on Informational HerdingBehavior, Copenhagen, Denmark.
Keynes, J. M. (1936): The General Theory of Employment, Interest, andMoney, London: Macmillan.
Kindleberger, C. P. (1989): Manias, Panics, and Crashes: A History ofFinancial Crises, London: Macmillan.
Koo, R. (2011): The world in balance sheet recession: causes, cure, and politics,Real-World Economics Review, 58(1).
Liu, C. (2012): Herding Behavior in Bank Lending: Evidence from U.S. Com-mercial Banks, European finance association (EFA) working paper, URLhttp://northernfinance.org/2012/program/papers/282.pdf.
Lux, T. (1995): Herd Behaviour, Bubbles and Crashes, Economic Journal,105(431): pp. 881–96.
McLeay, M., Radia, A. and Thomas, R. (2014): Money creation in themodern economy, Bank of England Quarterly Bulletin, 54(1): pp. 14–27,URL http://ideas.repec.org/a/boe/qbullt/0128.html.
23
Minsky, H. P. (1963): Can “It” Happen Again?, in: Carson, D. (ed.), Bank-ing and Monetary Studies, Richard D Irwin, Homewood, reprinted in Minsky(1982).
Minsky, H. P. (1975): John Maynard Keynes, New York: Macmillan.
Minsky, H. P. (1982): Inflation, recession and economic policy, New York:ME Sharpe.
Nakagawa, R. and Uchida, H. (2011): Herd Behavior by Japanese BanksAfter Financial Deregulation in the 1980s, Economica, 78(312): pp. 618–636.
Rethel, L. and Sinclair, T. J. (2012): The problem with banks, London: ZedBooks.
Ryan-Collins, J., Greenham, T., Werner, R. and Jackson, A. (2012):Where does money come from? - a guide to the UK monetary and bankingsystem, The New Economics Foundation.
Sharma, S. and Bikhchandani, S. (2000): herd Behavior in Financial Mar-kets - A Review, IMF Working Papers 00/48, International Monetary Fund,URL http://ideas.repec.org/p/imf/imfwpa/00-48.html.
Taylor, L. (2004): Reconstructing Macroeconomics: structuralist proposalsand critiques of the mainstream, Cambridge, Massachusetts: Harvard Uni-versity Press.
Weidlich, W. and Haag, G. (1983): Concepts and Models of a QuantitativeSociology. The Dynamics of Interacting Populations, Berlin: Springer.
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A Simulation Results
Figure 4: Simulations of the baseline model
25
0 200 400 6000
0.2
0.4
0.6
0.8
x
Stable Scenario (a1=0.3)
0
0.5
10.511
11.50
20
40
xy
λ +
0 200 400 60010
10.5
11
11.5
outp
ut
0 200 400 6005
10
15
20
25
λ +0 200 400 600
−1
0
1
x
Unstable Scenario (a1=1.5)
0 200 400 6000
20
40
λ +
−10
1
1012
140
20
40
xy
λ +
0 200 400 60010
11
12
13
14
outp
ut
Figure 5: Herding amongst optimistic banks: representative simulation of stable(left) and unstable (right) cases.
26
10 10.5 11 11.5 12 12.5 13 13.5 140
0.5
1
1.5
2
2.5
3
output
debt
−to
−G
DP
rat
io
output vs debt−to−GDP ratio
Figure 6: Herding amongst optimistic banks: dynamics of the Debt/GDP ratio:a1 = 0.8 (blue), a1 = 1.2 (black), a1 = 1.7 (red).
27
0 0.5 1 1.5 2 2.5 3 3.5 4
x 104
10.5
11
11.5
12
12.5
13
13.5
time
y
Figure 7: Herding amongst optimistic banks: effect of varying a1 on output:a1 = 0.3 (blue), a1 = 0.7 (black), a1 = 1.5 (red).
Figure 8: Herding amongst optimistic banks: bifurcation diagram for a1.
28
Figure 9: Herding amongst optimistic banks: bifurcation plot for a2.
Figure 10: Herding amongst optimistic banks: bifurcation diagram for a3.
29
Figure 11: Herding amongst optimistic banks: bifurcation diagram for γ1.
Figure 12: Herding amongst optimistic banks: bifurcation diagram for γ2.
30
Figure 13: Extended 4D Model with λ− > 0: representative simulation. Topleft panel: simulation over 600 periods. Other panels: magnification over 50periods.
31
Figure 14: Extended 4D Model with λ− > 0: bifurcation diagram for a1.
Figure 15: Extended 4D Model with λ− > 0: bifurcation diagram for γ1.
32
Figure 16: Extended 4D Model with λ− > 0: bifurcation diagram for γ2.
Figure 17: Extended 4D Model with λ− > 0: bifurcation diagram for γ3.
33
Figure 18: Extended 4D Model with convergence: dynamics of output for dif-ferent ξ0 and ξ1. Values of the parameters 0 (blue), = 0.5 (black), 1 (red)
34
time200 300 400 500 600
pk
0.2
0.3
0.4
0.5
0.6
0.7p
k
200 300 400 500 600-0.5
0
0.5
1x (blue) and x
p (green)
x-0.5 0 0.5 1
x p
-1.5
-1
-0.5
0
0.5
1
x-0.5 0 0.5 1
pk
0.2
0.3
0.4
0.5
0.6
0.7
time200 300 400 500 600
y
2.52
2.54
2.56
2.58
2.6
y2.52 2.54 2.56 2.58 2.6
pk
0.2
0.3
0.4
0.5
0.6
0.7
-2
-1
0
1
Figure 19: Extended 7D Model with a speculative real sector
35
Figure 20: Bifurcation Diagram of the 7D dynamics
36
Figure 21: Bifurcation Diagram: the effect of Tobin tax
37