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A new & simple model of currency crisis: Bifurcations and the emergence of a bad equilibrium
Journal Pre-proof A new & simple model of currency crisis: Bifurcations and the emergence of a bad equilibrium Partha Gangopadhyay
PII: DOI: Reference:
S0378-4371(19)31625-5 https://doi.org/10.1016/j.physa.2019.122860 PHYSA 122860
To appear in:
Physica A
Received date : 11 July 2018 Revised date : 12 July 2019 Please cite this article as: P. Gangopadhyay, A new & simple model of currency crisis: Bifurcations and the emergence of a bad equilibrium, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.122860. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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*Highlights (for review)
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The Highlights: This paper examines currency crisis in a dynamic setting with agents who don’t hold rational expectations. In a simple and new setting the paper highlights the fragility of the foreign exchange market: it demonstrates the possibility of a collapse of the currency due to a progressive loss of reserves. It is important to note that a collapse is not triggered by the two known sources - a bad policy, or bad luck, but due to a regime-shift from a stable to an unstable and unique steady state – bad equilibrium. The paper hence proffers a new explanation of currency crisis: a crisis that erupts even when there is no evidence of bad policy, or of multiple equilibria (bad luck). The model is further extended to a case of heterogeneous agents.
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*Revised Manuscript with Annotation Click here to view linked References
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A New & Simple Model of Currency Crisis: Bifurcations and the Emergence of a Bad Equilibrium
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Partha Gangopadhyay, University of Western Sydney, NSW, Australia.
Abstract: This paper considers the question of currency crisis in a dynamic setting in which agents don’t hold rational expectations. Under a sufficient condition the paper shows the possibility of a collapse of the currency due to a progressive loss of reserves. It
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is important to note that a collapse is not triggered by a bad policy, or bad luck, but due to a regime-shift from a stable to an unstable and unique steady state – bad equilibrium. In this sense the paper offers a new explanation of currency crisis: a crisis that erupts even when there is no evidence of bad policy, or of multiple equilibria (bad luck). The model is further extended to a case of heterogeneous agents.
Keyword: Bifurcations, Currency crisis, Dynamics of fundamentals, Rational
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irrationality, Stability properties, Steady states.
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JEL Classification: C61 F31, F41
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A New & Simple Model of Currency Crisis: Bifurcations and the Emergence of a Bad Equilibrium
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Introduction
Countries that maintain pegged exchange rates face currency and financial crises, which force them to devalue their currencies, or switch from pegged to flexible exchange rates regimes. It is widely recognized now that concerted speculative attacks on fixed, or ‘fixed
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but adjustable,’ exchange rate regimes is a common feature that characterizes the landscape of modern foreign exchange markets (see Flood and Kramer, 1996 among many). Exchange market and banking crises in Latin America, Asia and Europe have renewed interests in the economics of speculative attacks. Economic models of speculative attacks have been around for some time and are useful in understanding the dynamics of forces that gather momentum to result in a real-world speculative attack on a
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currency. There are two dominant types of models that characterize the formal analysis: first generation models interpret speculative attacks as the natural and the anticipated demise of an inconsistent policy regime (see Agenor, Bhandari and Flood, 1992 for a
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review). This approach highlights a conflict between objectives for nominal exchange rate stability and a monetary/fiscal policy that is inconsistent with this objective (Krugman, 1979). As a result it is possible to predict precisely the timing of a crisis that depends on the evolving economic fundamentals and their effects on the official foreign
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exchange reserve. Crises are due to bad policies. Confidence in the first generation models wanes due to the subsequent observation that, in many cases, the underlying policy conflict seems missing. As examples, observers cite the cases of ERM crisis in 1992 and the Mexican crisis of 1994 when crises were not preceded by policy conflicts.
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These observations have inspired second generation models that explain currency and financial crises in terms of shifts in private expectations (Obtsfeld, 1986). These models explain speculative attacks in terms of the fundamentals identified in first generation
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models, but the fundamentals are themselves sensitive to shifts in private expectations about the future. It is not possible to predict the timing of a crisis, because the crises are driven by unexplained but self-fulfilling shifts in private expectations. As a result, the crisis – according to second generation models - is due to bad luck. A self-fulfilling
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speculative attack is one in which holders of a currency sell it because they expect it to depreciate, with the outcome that their sales lead to a depreciation of the currency in question. The upshot is that there are two equalibria that second generation models highlight: in the first equilibrium a fixed exchange rate is sustainable if it does not get attacked since current policies are not fundamentally inconsistent with the survival of the fixed exchange rate. In the second equilibrium, the exchange rate regime is to be
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abandoned because an attack causes a modification of policies so that these policies become inconsistent with the new exchange rate regime. It is only recently that we know that the likelihood of a self-fulfilling attack is in part dependent on the economic
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fundamentals (Kenen, 1996). Speculative attacks can be self-fulfilling but they don’t take place randomly. They are not always induced by bad news about the economy. They are often driven by bad news about the government – news that motivates market participants to ask if the government can handle skillfully bad news about the economy.
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This paper exploits developments in bifurcation theory to show how exchange
rates could shift from stability to instability that can, in turn, trigger crises. From standard models we develop two feedback mechanisms: first, it is argued that the dynamics of
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fundamentals determines the time path of exchange rates. Secondly, it is also argued that the time profile of fundamentals is driven by expectations about exchange rates and the exchange rate dynamics. Here we introduce a new idea that agents hold beliefs, which
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deviate from rational expectations. In this sense agents are modeled as irrational (Caplan, 2003). One can model this behavior as a case rational irrationality (Caplan, 2001) such that agents make a rational trade-off between wealth and irrationality. How reasonable is this assumption of departure from rationality? It is now well recognized that rational
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agents find it extremely difficult to forecast foreign exchange rates (Meese and Rogoff, 1983). Experimental work of Smith (1991) has established that boundedly rational is the norm than the exception. A recent study by Sonnemans et al. (2004) also lent support to bounded rationality in the context of a cobweb economy. Foreign exchange markets also tend to be highly volatile and this excess volatility causes two types of problems: first, this excess volatility poses a problem for correctly forecasting the time profile of
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exchange rates. Secondly, the excess volatility signifies the presence of speculative bubbles that can engender misspecifications of fundamentals and consequently a serious overshooting problem (see Elhood, Ahmed and Rosser, 1999; Frankel and Rose, 1995;
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Flood and Hodrick 1990). Speculative bubbles that arise under floating rates are analogous to speculative attacks that occur in fixed exchange rates regimes (Frankel, 1996, pp. 154). Typically econometric evidence has failed to explain exchange-rate movements by fundamentals on a short-term basis (Frankel and Rose, 1995). One
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possibility is that exchange-rate movements are driven by non-fundamentals: the exchange rate goes up in each period because traders expect it to go up further the next period. Even if the exchange rate gets further and further away from the value
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underpinned by economic fundamentals – each individual trader knows a contrarian action will make him lose money. It is now widely accepted that short-term expectations of exchange rates are not formed by looking far into the future (Frankel, 1996), thus these
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expectations are far removed from rational expectations.
The spirit of this paper is akin to the state space analysis (see Gangopadhyay, 1997 and Smith, 2004). The mutual feedbacks between exchange rates and economic fundamentals are examined when economic agents don’t entertain rational expectations
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regarding exchange rates. The paper closely follows the logic of Lyapunov (1892) to examine the link between the eigenvalues of the Jacobian matrix of partial derivatives of the proposed dynamical system at an equilibrium point and the local stability of that equilibrium point. It is well recognized that a sufficient condition for local stability is that the real parts of the eigenvalues for the Jacobian matrix be negative. Thus, an eigenvalue with a zero real part is the point that is the borderline between stability and instability.
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This method is applicable to the case of continuous time model. In discrete time models (as in this paper) the stability conditions are explained in terms of the modulus of the aforementioned eigenvalues. By examining the modulus of eigenvalues, this paper
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thereby offers an alternative explanation of exchange rate crises due to a regime-shift from a stable to an unstable equilibrium - even when there is no prima facie case for ‘bad policy’ or ‘bad luck.’ The plan of the paper is as follows: Section 2 develops the baseline model and Section 3 extends this model to the case of quasi-rational agency. Section 4
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concludes.
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2.1 Dynamics of Money Supply & Expectations We postulate the demand for money function to be:
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MD/P=Ze-i
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2. A Simple Model
(1a)
MD: Demand for nominal balances, P: Price level, i: domestic interest rate, Z: National Income. Our model of monetary dynamics is similar to the standard model as highlighted
of money (M) is given as: M=R+F
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in Barnett and Kwag (2005). We have suppressed the output in our equations. The supply
(1b)
R: Domestic assets of central bank, F: Foreign reserves held by central bank The purchasing power parity (PPP) dictates: P=SP*
(1c)
S: nominal exchange rate, P*: Foreign price.
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The uncovered interest rate parity (UIP) calls forth: i=i*+[(S*-S)/S]
(1d)
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i*: Foreign interest rate, S*: Expected/ ex ante exchange rate From the money market equilibrium condition, after substituting (1c) and (1d), we
know
M/(SP*)=Ze-(i*-1+S*/S)
(2a)
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Taking log of (2a) and simplification yield:
Where
Log S =Log (R+F)+(S*/S)+V
(2b)
V=(i*-Log P*--Log Z)
(2c)
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(2005), pp. 4. We now date equation (2b) that yields: Log (Rt+Ft)=Log St-Log St*+K
(2d)
Log (Rt-1+Ft-1)=Log St-1-Log St-1*+K
Subtracting (2e) from (2d) yields: rt= -Et+Et* where r=Log (R+F), E=Log S, E*=(S*/S).
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That is,
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Our system is essentially analogous to equations (1), (2) and (3) of Barnett and Kwag
(2e)
(3a)
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Equation (3a) is a simplification and not a precise scientific determination of exchange rates by any measure. In all strands of monetarist approach, the determination of exchange rate is predicated upon the demand for and the supply of money. Any fluctuations in the variables that affect money demand and money supply will also impact on exchange rates (see Barnett and Kwag, 2005). In equation (3a) we have introduced the subjective expectation of agents about the movement/change in exchange rate as a factor
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that also affects exchange rate. However, this introduction of subjective expectations is undertaken at the expense of scientific rigour since changes in exchange rates, subjective
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expectations and monetary variables are not traced back to the initial period and we instead focus upon an arbitrary starting point at date t-1. This is one of two inconsistencies. The second inconsistency is that Et measures the actual change in exchange rate while Et* measures the expected depreciation, or appreciation, in the exchange rate. However, it is well-recognised that the monetarist approach has many
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other inconsistencies: as an example, the simple-sum monetary aggregates as a stock of money supply and short-terms interest rates as the opportunity cost of holding money balances are inconsistent with the aggregation and index number theory (see Barnett and
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Kwag, 1995). The consequences of these simplifications and inconsistencies in the traditional monetarist models are far-reaching, which came to be known in the literature as the “Barnett Critique” (see Barnett and Binner, 2004; Barnett and Wu, 2005; Barnett,
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2007).
It is also important to discuss the relevance of uncovered interest parity (UIP), as introduced by equation (1d), which brings subjective expectations in our simple model. There is a consensus among economists that the UIP works as a consistent theoretical
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apparatus that is inconsistent with the available data: it predicts that countries with high interest rates should have depreciating currencies while such currencies have displayed appreciation in reality. This is known as the forward premium puzzle in the literature (see Macdonald and Taylor, 1992). The empirical rejection of the UIP theory is often justified on the ground that the theory assumes agents to entertain rational expectations while, in reality, agents have irrational expectations (see Frankel and Froot, 1990; Mark and Wu,
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1998). The introduction of subjective forecasts in our simple model will allow us to consider the impacts of rational irrationality for the dynamics of exchange rates. The role of expectations is also highlighted in McCallum (1994) in resolving the forward premium
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puzzle when agents form expectations about the impact of government policy on the exchange rate. By using an expanded dataset Christensen (2000) found empirical support for the role of expectations as articulated by McCallum (1994). Thus, our model is a very small step towards introducing subjective forecasts and human irrationality into the
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determination of exchange rates, which is a major challenge for modern financial economics.
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2.2 Model of Determination of Exchange Rate From the standard monetary model of exchange rates we know: Et=rt+[()/(1+-)]rt
(3b)
fundamentals rt as: rt rt 1 (rt 1 rt 2 ) t
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The above determination of exchange rate as an asset price is hinged on the dynamics of
(3b')
0 1 and denotes shocks with mean zero.
rt-1=k1 We then have
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Assuming rt-1 to be a constant for simplification
Et=[(+1)/(1+-)]rt -k1 [()/(1+-)]
(3b'')
(3c)
It is instructive to note that the dynamic behaviour that we propose in this paper is relevant only for the following type of decision-making: a decision-maker at date t
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considers relevant past variables at date t-1 as a datum. The decision-maker does not take into account how these variables behaved before date t-1. The decision-maker forms
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expectations of the relevant variables only at date t+1. The decision-maker is neither endowed with the entire history of past variables and nor interested in predicting the future variables beyond date t+1. As a result, the predictions and economic outcomes do not turn on the pivot of rational expectations. At any point in time economic agents take
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actions by looking at the immediate past and considering the impact of their action on relevant variables in the immediate future. Economic agents are simply myopic in either directions of the arrow of time. The modeller also follows this decision-making arrangement in describing the relevant dynamics.
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rt= -Et+Et*
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As a result, the dynamical system is thus reduced to two equations: (3a)
Et=[(+1)/(1+-)]rt - k1 [()/(1+-)]
(3c)
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What we are proposing is two-fold: first, we posit that the fundamentals (mainly money supply) evolve according to equation (3a). Both actual and anticipated appreciation / depreciation of the currency impinge on the dynamics of the fundamentals. Secondly, we state that the dynamics of exchange rate is influenced by the evolution of the
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fundamentals. Only at a long-run equilibrium, these two dynamic equations mesh in to produce the monetary model of exchange rate determination in which all expectations are correct and fulfilled. This idea that the monetary model holds in a long-run equilibrium has been widely accepted as Rapach and Wohar (2004) write: “…the core of the monetary model is unlikely to hold at each point in time, and therefore the monetary model should be viewed as a long-run, or steady-state model of exchange
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rate determination.” (pp. 841). We now introduce the question of determining the expected appreciation/depreciation, or
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change in exchange rates, E*.
2.3. Expectations of Changes in Exchange Rates: Rational Irrationality Instead of assuming self-confirming expectations, as recommended by Engel (1996), we
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assume
Et*=H(Et, rt)
(3d)
Equation (3d) subsumes that expectations are not self-fulfilling. However, the expected change in exchange rate is partly explained by the actual change in exchange rate and the
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actual change in fundamentals. This is in breach of the rational expectations hypothesis of
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Engel (1996) that assumes Et*=Et with a random error term.
The idea is that agents hold beliefs that deviate from rational expectations. In this
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sense agents are modeled as irrational (Caplan, 2003). One can model this behavior as a case rational irrationality (Caplan, 2001) such that agents make a rational trade-off between wealth and irrationality. It can be shown that agents will have zero irrationality when the price of irrationality is significantly high. However, these agents will hold
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irrational beliefs as long as the price of irrationality is small in comparison with the benefit of reaching rational expectations. Thus these agents will exhibit large systematic biases when irrationality is cheap. The upshot is that each agent believes that the exchange rate is driven by fundamentals and other factors so that a self-confirming individual forecast is not economically meaningful. The cost of so doing outweighs the benefit.
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The only restriction that we impose is that both partial derivatives of H(.) are nonzero. (3a) and (3c) will be reduced to:
(3a*)
Et=[(1+)/(1+-)]rt -[()/(1+-)]k1
(3c)
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rt= -Et+H(Et,rt)
By construction:
rt+1= rt+rt
(4a)
rt= rt-1+rt-1
(4b)
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(4a)-(4b) yields:
rt+1=2rt-rt-1 =-2Et+2H(Et,rt))-rt-1
(4c) (4c')
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Similarly it can be calculated that
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Et+1=2[(1+)/()] H(Et,rt)+ rt-1
(4d)
rt-1=k1
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We simplify the calculation by assuming
(4e)
Representing X=r and Y =E will give equation (4c') and (4d) as: Xt+1=-2Yt+2H(Xt, Yt))- k1 Yt+1=[2(1+)/()]H(Xt, Yt)+k1
(5a) (5b)
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Since , k1are both constant the Jacobian evaluated at the fixed point gives us: -2[H/X]
2[-1+(H/Y)]
[2(1+)/()](H/X)
[2(1+)/()](H/Y)
J=
(5c)
Let us call H/X=HX and H/Y=HY to yield(5c) as
2HX
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J=
[2(1+)/()]HX
2(HY-1) (5c') [2(1+)/()]HY
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The characteristic polynomial is given by: 2-m+n=0
(5d)
where
m=Tr J= [2(1+)HY+2HX)]/
(6a)
and
n=Det J=[4 (1+)HX]/
(6b)
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We now know:
(1)=1-m+n=0
(6c)
(-1)=1+m+n=0
(6d)
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We draw these lines in Diagram 1 that will give us the triangle of stability. We also know: T=m=Tr J=[2(1+)HY+2HX)]/
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D=n=Det J=[4 (1+)HX]/
(7a)
(7b)
2.4 Stability Property of the Steady State:
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Suppose HX is positive, then the sufficient condition that D>2 is 2(1+)/()>(1/HX)
(7c)
If the inequality (7c) holds, the steady state is characterized by the stability property such that the combination of T and D will be located outside the triangle ABC. In that case the steady state will be intrinsically unstable. As (7c) suggests the combination of T and D will be above the line BC. This steady state will therefore be a source that is highly
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unstable. Note that the system is stable if HX=0 and HY=0.
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Diagram 1: Stability/Instability of the Steady State
D
+1
C
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-2
B
T O
A
+2
-1
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If (7c) holds, then the steady state will be either an unstable spiral, or an unstable node. Does the system diverge from the steady state or return to it rapidly when perturbed? This
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gives us a qualitative understanding of whether or not the system converges to the steady state and if it is sensitive to the slight perturbation of parameters. The stability of the exchange rate regime is of great importance since a system which is prone to instability can easily lead to the destruction of critical resources such as foreign exchange reserves –
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hence such a system must be managed with a great deal of care1.
Another possibility is that the initial combination of (T, D) is on the line BC and 2(1+)/()=(1/HX). The equilibrium is stable. Now a small increase in HX, or in ,or a small decline in will have profound qualitative change in the steady state as the equilibrium will become unstable. This change in the stability property of the steady state is popular as Hopf bifurcation (see Lorenz, 1993 for a complete treatment of Hopf
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bifurcation in economic systems). We now propose a new definition of currency crisis in the light of these results. If
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the steady state is intrinsically unstable, or bifurcates into the zone of instability with changes in parameters, the exchange rate mechanism faces crisis in the sense that a slight perturbation will take the system further and further away from the steady state in real time. The divergence from the steady state may cause a progressive loss of foreign
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exchange reserves and ultimately an exhaustion of these reserves and a collapse of the 1
I am grateful to the referees for raising an important question about the modus operandi of managing a system with fragility. In Gangopadhyay and Gangopadhyay (2008), it is constructively argued that policy/decision-making in the context of a fragile system is beset with irreversibility and, hence, there is an option value in waiting for the arrival of better, or finer, information before choosing an optimal policy. In other words, Gangopadhyay and Gangopadhyay (2008) established that policy makers should actively seek better, or finer, information before acting and, also, there is a virtue in a sequential and partial adjustment in the choice variable.
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system becomes inevitable. It is important to note that a collapse is not triggered by a bad
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policy, or bad luck, but by an unstable steady state – a bad equilibrium. In this sense the results offer a new explanation of currency crisis: a crisis that erupts even when there is
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no evidence of bad policy, or of multiple equilibria (bad luck).
3. An Extension with Heterogeneous Agents
In this section we consider a scenario in which some agents are fully rational while the
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rest are ‘rationally irrational.’ Models with heterogeneous interacting agents have proven to be quite successful in modeling the behavior of speculative markets. For instance, they are able to replicate the statistical properties of financial prices quite well (see Day and Huang, 1990; de Grauwe et al., 1993; Brock and Hommes, 1997; Lux and Marchesi, 2000 and Westerhoff, 2004). The role of heterogeneity in the context of currency crises has been examined in Arifovic and Masson (2004). The evolution of expectations has
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been the focus of Arifovic and Masson (2004). As the previous section an agent, who is rationally irrational will form his belief
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by equation (3d). On the other hand, a rational agent predicts the relevant variables by forming expectations rationally and assuming everybody-else is doing so. From equation (3a), the rationality of expectation requires that expectations are self-fulfilling while Et is given by equation (3c):
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rte= Et (-1)
(8a)
Substituting (3c) into (8a) yields: rte= [(-1)k1]/[(+ -1]
Similarly, the rational expectations of Et is given as
(8b)
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Ete= [k1]/[(+ -1]
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(8d)
Suppose w percentage of agents are rationally irrational, 1-w percentage of them are fully rational. This is an arbitrary mix underlining the heterogeneity of agents. This kind of
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modeling has been called quasi-rational models by Thaler (1991). If the actual variables are now a weighted average of the expectations of these two types of agents, then the dynamics (5a)-(5b) can be re-written as:
(9a)
Yt+1=[2(1+)/()]H(Xt, Yt)+k1+(1-w)[ Ete]
(9b)
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Xt+1=w[-2Yt+2H(Xt, Yt))- k1] +(1-w) [rte]
The sufficient condition for unstable fixed point (7c) will be altered to 2w2(1+)/()>(1/HX)
(9c)
It is evident that the co-existence of fully-rational agents lends a huge stability to the system since w<1. There is a critical value of w, w*, w*=Square root {()/[2HX(1+)]}
(9d)
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If w>w*, then the equilibrium is inherently unstable and the conclusion of the previous section is still tenable. However, if there is a sufficiently large number of fully rational
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agents in the system (if w
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paribus, can trigger serious instability to a previously stable system.
4. Concluding Comments First generation models of currency crises highlight a conflict between objectives for nominal exchange rate stability and a monetary/fiscal policy that is inconsistent with
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these objectives. Currency crises are simply due to bad policies. Confidence in the first generation models wanes due to the subsequent observation that, in many cases, the underlying policy conflict seems missing. These observations have inspired second
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generation models that explain currency and financial crises in terms of shifts in private expectations. These models explain speculative attacks in terms of the fundamentals identified in first generation models, but the fundamentals are themselves sensitive to shifts in private expectations about the future. It is not possible to fix the timing of a
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crisis, because the crises are driven by unexplained but self-fulfilling shifts in private expectations. As a result, the currency crisis according to second generation models is due to bad luck.
What we have proposed in this paper is three-fold: first, we posit that the time path of fundamentals (mainly money supply) is influenced by both actual and anticipated appreciation / depreciation of the currency. Secondly, we argue that the dynamics of
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exchange rate is influenced by the evolution of these fundamentals. Finally, we assume that agents don’t hold rational expectations. Instead we posit that the expected change in exchange rate is partly explained by the actual change in exchange rate and the actual
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change in fundamentals. One may note that upon an introduction of rational expectations and at a steady state, the above dynamics will collapse into the much-vaunted monetarist model of determination of exchange rates. However, since the system is out of steady state and agents don’t hold rational
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expectations, it begs a question of whether the above system converges on a stable steady state. We derive an inequality that offers an answer to this question. If this inequality holds the steady state will be intrinsically unstable. We offer a new definition of currency
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crisis in the light of this instability. If the steady state is intrinsically unstable, or bifurcates into the zone of instability with changes in relevant parameters, the exchange rate mechanism faces crisis in the sense that a slight perturbation will take the system
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further and further away from the steady state in real time. The divergence from the steady state may cause a progressive loss of foreign exchange reserves and ultimately an exhaustion of these reserves and a collapse of the system becomes inevitable. It is important to note that this collapse is not triggered by a bad policy, or bad luck, but by an
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unstable steady state – a bad equilibrium. In this sense the results offer a new explanation of currency crisis: a crisis that erupts due to a bad equilibrium even when there is no evidence of bad policy, or of multiple equilibria (bad luck).
Finally we consider a model with heterogeneous agents some of who are fully rational. The introduction of heterogeneity lends more stability to the system. However, it fails to get rid off the possibility of a bad equilibrium. The existence of an unstable
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equilibrium is linked to the heterogeneity coefficient and other parameters of the system.
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Barnett, W. A. and C. H. Kwag (2005). Exchange Rate Determination from Monetary Fundamentals: An Aggregation Theoretic Approach. Working Paper, No. 200513, Department of Economics, University of Kansas. in Econometrics, Elsevier: Amsterdam.
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Flood, R. P. and C. Kramer (1996). Economic Models of Speculative Attacks and the Drachma Crisis of May 1994. Open Economies Review, 7, pp. 591-600.
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The Manchester School, Volume LXV, No. 5, pp. 534-568.
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Lorenz HW. (1993) Bifurcation Theory and Economic Dynamics. In: Nonlinear Dynamical Economics and Chaotic Motion. Springer, Berlin, Heidelberg Lux, T. and Marchesi, M. (2000). Volatility Clustering in Financial Markets: A Micro-
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of Mathematical Studies, 17, Princeton University Press Princeton. MacDonald, R. and M.P. Taylor (1992). Exchange Rate Economics: A Survey. In: IMF Staff Papers, 39, No. 1, March, pp. 1-57.
Mark, N.C., Wu, Y.(1998). Rethinking Deviations from Uncovered Interest Parity: The Role of Covariance Risk and Noise, The Economic Journal,108, pp1686–1706.
McCallum, B.T.(1994). A Reconsideration of the Uncovered Interest Parity Relationship,
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Meese, R. A. and K. Rogoff (1983). Empirical Exchange Rate Models of the 1970s: Do They Fit out of Sample? Journal of International Economics, 14, pp. 2-14.
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Rapach, D. E. and M. E. Wohar (2004). Testing the Monetary Model of Exchange Rate Determination: A Closer Look at Panels. Journal of International Money and Finance, 23(6), pp. 841-865.
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a Heterogeneous Cobweb Economy: A Strategy Experiment on Expectation Formation. Journal of Economic Behavior and Organization, 54, pp. 453-481. Thaler, R. (1991). Quasi Rational Economics, Russell Sage Foundation, New York. Westeroff, F. (2004). Multi Asset Market Dynamics. Macroeconomic Dynamics, 8, pp.
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PII: DOI: Reference:
S0378-4371(19)31625-5 https://doi.org/10.1016/j.physa.2019.122860 PHYSA 122860
To appear in:
Physica A
Received date : 11 July 2018 Revised date : 12 July 2019 Please cite this article as: P. Gangopadhyay, A new & simple model of currency crisis: Bifurcations and the emergence of a bad equilibrium, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.122860. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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*Highlights (for review)
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The Highlights: This paper examines currency crisis in a dynamic setting with agents who don’t hold rational expectations. In a simple and new setting the paper highlights the fragility of the foreign exchange market: it demonstrates the possibility of a collapse of the currency due to a progressive loss of reserves. It is important to note that a collapse is not triggered by the two known sources - a bad policy, or bad luck, but due to a regime-shift from a stable to an unstable and unique steady state – bad equilibrium. The paper hence proffers a new explanation of currency crisis: a crisis that erupts even when there is no evidence of bad policy, or of multiple equilibria (bad luck). The model is further extended to a case of heterogeneous agents.
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*Revised Manuscript with Annotation Click here to view linked References
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A New & Simple Model of Currency Crisis: Bifurcations and the Emergence of a Bad Equilibrium
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Partha Gangopadhyay, University of Western Sydney, NSW, Australia.
Abstract: This paper considers the question of currency crisis in a dynamic setting in which agents don’t hold rational expectations. Under a sufficient condition the paper shows the possibility of a collapse of the currency due to a progressive loss of reserves. It
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is important to note that a collapse is not triggered by a bad policy, or bad luck, but due to a regime-shift from a stable to an unstable and unique steady state – bad equilibrium. In this sense the paper offers a new explanation of currency crisis: a crisis that erupts even when there is no evidence of bad policy, or of multiple equilibria (bad luck). The model is further extended to a case of heterogeneous agents.
Keyword: Bifurcations, Currency crisis, Dynamics of fundamentals, Rational
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irrationality, Stability properties, Steady states.
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JEL Classification: C61 F31, F41
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A New & Simple Model of Currency Crisis: Bifurcations and the Emergence of a Bad Equilibrium
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Introduction
Countries that maintain pegged exchange rates face currency and financial crises, which force them to devalue their currencies, or switch from pegged to flexible exchange rates regimes. It is widely recognized now that concerted speculative attacks on fixed, or ‘fixed
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but adjustable,’ exchange rate regimes is a common feature that characterizes the landscape of modern foreign exchange markets (see Flood and Kramer, 1996 among many). Exchange market and banking crises in Latin America, Asia and Europe have renewed interests in the economics of speculative attacks. Economic models of speculative attacks have been around for some time and are useful in understanding the dynamics of forces that gather momentum to result in a real-world speculative attack on a
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currency. There are two dominant types of models that characterize the formal analysis: first generation models interpret speculative attacks as the natural and the anticipated demise of an inconsistent policy regime (see Agenor, Bhandari and Flood, 1992 for a
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review). This approach highlights a conflict between objectives for nominal exchange rate stability and a monetary/fiscal policy that is inconsistent with this objective (Krugman, 1979). As a result it is possible to predict precisely the timing of a crisis that depends on the evolving economic fundamentals and their effects on the official foreign
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exchange reserve. Crises are due to bad policies. Confidence in the first generation models wanes due to the subsequent observation that, in many cases, the underlying policy conflict seems missing. As examples, observers cite the cases of ERM crisis in 1992 and the Mexican crisis of 1994 when crises were not preceded by policy conflicts.
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These observations have inspired second generation models that explain currency and financial crises in terms of shifts in private expectations (Obtsfeld, 1986). These models explain speculative attacks in terms of the fundamentals identified in first generation
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models, but the fundamentals are themselves sensitive to shifts in private expectations about the future. It is not possible to predict the timing of a crisis, because the crises are driven by unexplained but self-fulfilling shifts in private expectations. As a result, the crisis – according to second generation models - is due to bad luck. A self-fulfilling
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speculative attack is one in which holders of a currency sell it because they expect it to depreciate, with the outcome that their sales lead to a depreciation of the currency in question. The upshot is that there are two equalibria that second generation models highlight: in the first equilibrium a fixed exchange rate is sustainable if it does not get attacked since current policies are not fundamentally inconsistent with the survival of the fixed exchange rate. In the second equilibrium, the exchange rate regime is to be
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abandoned because an attack causes a modification of policies so that these policies become inconsistent with the new exchange rate regime. It is only recently that we know that the likelihood of a self-fulfilling attack is in part dependent on the economic
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fundamentals (Kenen, 1996). Speculative attacks can be self-fulfilling but they don’t take place randomly. They are not always induced by bad news about the economy. They are often driven by bad news about the government – news that motivates market participants to ask if the government can handle skillfully bad news about the economy.
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This paper exploits developments in bifurcation theory to show how exchange
rates could shift from stability to instability that can, in turn, trigger crises. From standard models we develop two feedback mechanisms: first, it is argued that the dynamics of
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fundamentals determines the time path of exchange rates. Secondly, it is also argued that the time profile of fundamentals is driven by expectations about exchange rates and the exchange rate dynamics. Here we introduce a new idea that agents hold beliefs, which
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deviate from rational expectations. In this sense agents are modeled as irrational (Caplan, 2003). One can model this behavior as a case rational irrationality (Caplan, 2001) such that agents make a rational trade-off between wealth and irrationality. How reasonable is this assumption of departure from rationality? It is now well recognized that rational
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agents find it extremely difficult to forecast foreign exchange rates (Meese and Rogoff, 1983). Experimental work of Smith (1991) has established that boundedly rational is the norm than the exception. A recent study by Sonnemans et al. (2004) also lent support to bounded rationality in the context of a cobweb economy. Foreign exchange markets also tend to be highly volatile and this excess volatility causes two types of problems: first, this excess volatility poses a problem for correctly forecasting the time profile of
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exchange rates. Secondly, the excess volatility signifies the presence of speculative bubbles that can engender misspecifications of fundamentals and consequently a serious overshooting problem (see Elhood, Ahmed and Rosser, 1999; Frankel and Rose, 1995;
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Flood and Hodrick 1990). Speculative bubbles that arise under floating rates are analogous to speculative attacks that occur in fixed exchange rates regimes (Frankel, 1996, pp. 154). Typically econometric evidence has failed to explain exchange-rate movements by fundamentals on a short-term basis (Frankel and Rose, 1995). One
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possibility is that exchange-rate movements are driven by non-fundamentals: the exchange rate goes up in each period because traders expect it to go up further the next period. Even if the exchange rate gets further and further away from the value
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underpinned by economic fundamentals – each individual trader knows a contrarian action will make him lose money. It is now widely accepted that short-term expectations of exchange rates are not formed by looking far into the future (Frankel, 1996), thus these
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expectations are far removed from rational expectations.
The spirit of this paper is akin to the state space analysis (see Gangopadhyay, 1997 and Smith, 2004). The mutual feedbacks between exchange rates and economic fundamentals are examined when economic agents don’t entertain rational expectations
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regarding exchange rates. The paper closely follows the logic of Lyapunov (1892) to examine the link between the eigenvalues of the Jacobian matrix of partial derivatives of the proposed dynamical system at an equilibrium point and the local stability of that equilibrium point. It is well recognized that a sufficient condition for local stability is that the real parts of the eigenvalues for the Jacobian matrix be negative. Thus, an eigenvalue with a zero real part is the point that is the borderline between stability and instability.
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This method is applicable to the case of continuous time model. In discrete time models (as in this paper) the stability conditions are explained in terms of the modulus of the aforementioned eigenvalues. By examining the modulus of eigenvalues, this paper
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thereby offers an alternative explanation of exchange rate crises due to a regime-shift from a stable to an unstable equilibrium - even when there is no prima facie case for ‘bad policy’ or ‘bad luck.’ The plan of the paper is as follows: Section 2 develops the baseline model and Section 3 extends this model to the case of quasi-rational agency. Section 4
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concludes.
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2.1 Dynamics of Money Supply & Expectations We postulate the demand for money function to be:
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MD/P=Ze-i
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2. A Simple Model
(1a)
MD: Demand for nominal balances, P: Price level, i: domestic interest rate, Z: National Income. Our model of monetary dynamics is similar to the standard model as highlighted
of money (M) is given as: M=R+F
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in Barnett and Kwag (2005). We have suppressed the output in our equations. The supply
(1b)
R: Domestic assets of central bank, F: Foreign reserves held by central bank The purchasing power parity (PPP) dictates: P=SP*
(1c)
S: nominal exchange rate, P*: Foreign price.
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The uncovered interest rate parity (UIP) calls forth: i=i*+[(S*-S)/S]
(1d)
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i*: Foreign interest rate, S*: Expected/ ex ante exchange rate From the money market equilibrium condition, after substituting (1c) and (1d), we
know
M/(SP*)=Ze-(i*-1+S*/S)
(2a)
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Taking log of (2a) and simplification yield:
Where
Log S =Log (R+F)+(S*/S)+V
(2b)
V=(i*-Log P*--Log Z)
(2c)
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(2005), pp. 4. We now date equation (2b) that yields: Log (Rt+Ft)=Log St-Log St*+K
(2d)
Log (Rt-1+Ft-1)=Log St-1-Log St-1*+K
Subtracting (2e) from (2d) yields: rt= -Et+Et* where r=Log (R+F), E=Log S, E*=(S*/S).
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That is,
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Our system is essentially analogous to equations (1), (2) and (3) of Barnett and Kwag
(2e)
(3a)
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Equation (3a) is a simplification and not a precise scientific determination of exchange rates by any measure. In all strands of monetarist approach, the determination of exchange rate is predicated upon the demand for and the supply of money. Any fluctuations in the variables that affect money demand and money supply will also impact on exchange rates (see Barnett and Kwag, 2005). In equation (3a) we have introduced the subjective expectation of agents about the movement/change in exchange rate as a factor
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that also affects exchange rate. However, this introduction of subjective expectations is undertaken at the expense of scientific rigour since changes in exchange rates, subjective
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expectations and monetary variables are not traced back to the initial period and we instead focus upon an arbitrary starting point at date t-1. This is one of two inconsistencies. The second inconsistency is that Et measures the actual change in exchange rate while Et* measures the expected depreciation, or appreciation, in the exchange rate. However, it is well-recognised that the monetarist approach has many
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other inconsistencies: as an example, the simple-sum monetary aggregates as a stock of money supply and short-terms interest rates as the opportunity cost of holding money balances are inconsistent with the aggregation and index number theory (see Barnett and
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Kwag, 1995). The consequences of these simplifications and inconsistencies in the traditional monetarist models are far-reaching, which came to be known in the literature as the “Barnett Critique” (see Barnett and Binner, 2004; Barnett and Wu, 2005; Barnett,
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2007).
It is also important to discuss the relevance of uncovered interest parity (UIP), as introduced by equation (1d), which brings subjective expectations in our simple model. There is a consensus among economists that the UIP works as a consistent theoretical
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apparatus that is inconsistent with the available data: it predicts that countries with high interest rates should have depreciating currencies while such currencies have displayed appreciation in reality. This is known as the forward premium puzzle in the literature (see Macdonald and Taylor, 1992). The empirical rejection of the UIP theory is often justified on the ground that the theory assumes agents to entertain rational expectations while, in reality, agents have irrational expectations (see Frankel and Froot, 1990; Mark and Wu,
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1998). The introduction of subjective forecasts in our simple model will allow us to consider the impacts of rational irrationality for the dynamics of exchange rates. The role of expectations is also highlighted in McCallum (1994) in resolving the forward premium
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puzzle when agents form expectations about the impact of government policy on the exchange rate. By using an expanded dataset Christensen (2000) found empirical support for the role of expectations as articulated by McCallum (1994). Thus, our model is a very small step towards introducing subjective forecasts and human irrationality into the
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determination of exchange rates, which is a major challenge for modern financial economics.
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2.2 Model of Determination of Exchange Rate From the standard monetary model of exchange rates we know: Et=rt+[()/(1+-)]rt
(3b)
fundamentals rt as: rt rt 1 (rt 1 rt 2 ) t
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The above determination of exchange rate as an asset price is hinged on the dynamics of
(3b')
0 1 and denotes shocks with mean zero.
rt-1=k1 We then have
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Assuming rt-1 to be a constant for simplification
Et=[(+1)/(1+-)]rt -k1 [()/(1+-)]
(3b'')
(3c)
It is instructive to note that the dynamic behaviour that we propose in this paper is relevant only for the following type of decision-making: a decision-maker at date t
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considers relevant past variables at date t-1 as a datum. The decision-maker does not take into account how these variables behaved before date t-1. The decision-maker forms
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expectations of the relevant variables only at date t+1. The decision-maker is neither endowed with the entire history of past variables and nor interested in predicting the future variables beyond date t+1. As a result, the predictions and economic outcomes do not turn on the pivot of rational expectations. At any point in time economic agents take
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actions by looking at the immediate past and considering the impact of their action on relevant variables in the immediate future. Economic agents are simply myopic in either directions of the arrow of time. The modeller also follows this decision-making arrangement in describing the relevant dynamics.
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rt= -Et+Et*
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As a result, the dynamical system is thus reduced to two equations: (3a)
Et=[(+1)/(1+-)]rt - k1 [()/(1+-)]
(3c)
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What we are proposing is two-fold: first, we posit that the fundamentals (mainly money supply) evolve according to equation (3a). Both actual and anticipated appreciation / depreciation of the currency impinge on the dynamics of the fundamentals. Secondly, we state that the dynamics of exchange rate is influenced by the evolution of the
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fundamentals. Only at a long-run equilibrium, these two dynamic equations mesh in to produce the monetary model of exchange rate determination in which all expectations are correct and fulfilled. This idea that the monetary model holds in a long-run equilibrium has been widely accepted as Rapach and Wohar (2004) write: “…the core of the monetary model is unlikely to hold at each point in time, and therefore the monetary model should be viewed as a long-run, or steady-state model of exchange
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rate determination.” (pp. 841). We now introduce the question of determining the expected appreciation/depreciation, or
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change in exchange rates, E*.
2.3. Expectations of Changes in Exchange Rates: Rational Irrationality Instead of assuming self-confirming expectations, as recommended by Engel (1996), we
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assume
Et*=H(Et, rt)
(3d)
Equation (3d) subsumes that expectations are not self-fulfilling. However, the expected change in exchange rate is partly explained by the actual change in exchange rate and the
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actual change in fundamentals. This is in breach of the rational expectations hypothesis of
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Engel (1996) that assumes Et*=Et with a random error term.
The idea is that agents hold beliefs that deviate from rational expectations. In this
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sense agents are modeled as irrational (Caplan, 2003). One can model this behavior as a case rational irrationality (Caplan, 2001) such that agents make a rational trade-off between wealth and irrationality. It can be shown that agents will have zero irrationality when the price of irrationality is significantly high. However, these agents will hold
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irrational beliefs as long as the price of irrationality is small in comparison with the benefit of reaching rational expectations. Thus these agents will exhibit large systematic biases when irrationality is cheap. The upshot is that each agent believes that the exchange rate is driven by fundamentals and other factors so that a self-confirming individual forecast is not economically meaningful. The cost of so doing outweighs the benefit.
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The only restriction that we impose is that both partial derivatives of H(.) are nonzero. (3a) and (3c) will be reduced to:
(3a*)
Et=[(1+)/(1+-)]rt -[()/(1+-)]k1
(3c)
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rt= -Et+H(Et,rt)
By construction:
rt+1= rt+rt
(4a)
rt= rt-1+rt-1
(4b)
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(4a)-(4b) yields:
rt+1=2rt-rt-1 =-2Et+2H(Et,rt))-rt-1
(4c) (4c')
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Similarly it can be calculated that
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Et+1=2[(1+)/()] H(Et,rt)+ rt-1
(4d)
rt-1=k1
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We simplify the calculation by assuming
(4e)
Representing X=r and Y =E will give equation (4c') and (4d) as: Xt+1=-2Yt+2H(Xt, Yt))- k1 Yt+1=[2(1+)/()]H(Xt, Yt)+k1
(5a) (5b)
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Since , k1are both constant the Jacobian evaluated at the fixed point gives us: -2[H/X]
2[-1+(H/Y)]
[2(1+)/()](H/X)
[2(1+)/()](H/Y)
J=
(5c)
Let us call H/X=HX and H/Y=HY to yield(5c) as
2HX
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J=
[2(1+)/()]HX
2(HY-1) (5c') [2(1+)/()]HY
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The characteristic polynomial is given by: 2-m+n=0
(5d)
where
m=Tr J= [2(1+)HY+2HX)]/
(6a)
and
n=Det J=[4 (1+)HX]/
(6b)
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We now know:
(1)=1-m+n=0
(6c)
(-1)=1+m+n=0
(6d)
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We draw these lines in Diagram 1 that will give us the triangle of stability. We also know: T=m=Tr J=[2(1+)HY+2HX)]/
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D=n=Det J=[4 (1+)HX]/
(7a)
(7b)
2.4 Stability Property of the Steady State:
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Suppose HX is positive, then the sufficient condition that D>2 is 2(1+)/()>(1/HX)
(7c)
If the inequality (7c) holds, the steady state is characterized by the stability property such that the combination of T and D will be located outside the triangle ABC. In that case the steady state will be intrinsically unstable. As (7c) suggests the combination of T and D will be above the line BC. This steady state will therefore be a source that is highly
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unstable. Note that the system is stable if HX=0 and HY=0.
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Diagram 1: Stability/Instability of the Steady State
D
+1
C
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-2
B
T O
A
+2
-1
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If (7c) holds, then the steady state will be either an unstable spiral, or an unstable node. Does the system diverge from the steady state or return to it rapidly when perturbed? This
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gives us a qualitative understanding of whether or not the system converges to the steady state and if it is sensitive to the slight perturbation of parameters. The stability of the exchange rate regime is of great importance since a system which is prone to instability can easily lead to the destruction of critical resources such as foreign exchange reserves –
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hence such a system must be managed with a great deal of care1.
Another possibility is that the initial combination of (T, D) is on the line BC and 2(1+)/()=(1/HX). The equilibrium is stable. Now a small increase in HX, or in ,or a small decline in will have profound qualitative change in the steady state as the equilibrium will become unstable. This change in the stability property of the steady state is popular as Hopf bifurcation (see Lorenz, 1993 for a complete treatment of Hopf
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bifurcation in economic systems). We now propose a new definition of currency crisis in the light of these results. If
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the steady state is intrinsically unstable, or bifurcates into the zone of instability with changes in parameters, the exchange rate mechanism faces crisis in the sense that a slight perturbation will take the system further and further away from the steady state in real time. The divergence from the steady state may cause a progressive loss of foreign
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exchange reserves and ultimately an exhaustion of these reserves and a collapse of the 1
I am grateful to the referees for raising an important question about the modus operandi of managing a system with fragility. In Gangopadhyay and Gangopadhyay (2008), it is constructively argued that policy/decision-making in the context of a fragile system is beset with irreversibility and, hence, there is an option value in waiting for the arrival of better, or finer, information before choosing an optimal policy. In other words, Gangopadhyay and Gangopadhyay (2008) established that policy makers should actively seek better, or finer, information before acting and, also, there is a virtue in a sequential and partial adjustment in the choice variable.
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system becomes inevitable. It is important to note that a collapse is not triggered by a bad
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policy, or bad luck, but by an unstable steady state – a bad equilibrium. In this sense the results offer a new explanation of currency crisis: a crisis that erupts even when there is
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no evidence of bad policy, or of multiple equilibria (bad luck).
3. An Extension with Heterogeneous Agents
In this section we consider a scenario in which some agents are fully rational while the
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rest are ‘rationally irrational.’ Models with heterogeneous interacting agents have proven to be quite successful in modeling the behavior of speculative markets. For instance, they are able to replicate the statistical properties of financial prices quite well (see Day and Huang, 1990; de Grauwe et al., 1993; Brock and Hommes, 1997; Lux and Marchesi, 2000 and Westerhoff, 2004). The role of heterogeneity in the context of currency crises has been examined in Arifovic and Masson (2004). The evolution of expectations has
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been the focus of Arifovic and Masson (2004). As the previous section an agent, who is rationally irrational will form his belief
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by equation (3d). On the other hand, a rational agent predicts the relevant variables by forming expectations rationally and assuming everybody-else is doing so. From equation (3a), the rationality of expectation requires that expectations are self-fulfilling while Et is given by equation (3c):
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rte= Et (-1)
(8a)
Substituting (3c) into (8a) yields: rte= [(-1)k1]/[(+ -1]
Similarly, the rational expectations of Et is given as
(8b)
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Ete= [k1]/[(+ -1]
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(8d)
Suppose w percentage of agents are rationally irrational, 1-w percentage of them are fully rational. This is an arbitrary mix underlining the heterogeneity of agents. This kind of
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modeling has been called quasi-rational models by Thaler (1991). If the actual variables are now a weighted average of the expectations of these two types of agents, then the dynamics (5a)-(5b) can be re-written as:
(9a)
Yt+1=[2(1+)/()]H(Xt, Yt)+k1+(1-w)[ Ete]
(9b)
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Xt+1=w[-2Yt+2H(Xt, Yt))- k1] +(1-w) [rte]
The sufficient condition for unstable fixed point (7c) will be altered to 2w2(1+)/()>(1/HX)
(9c)
It is evident that the co-existence of fully-rational agents lends a huge stability to the system since w<1. There is a critical value of w, w*, w*=Square root {()/[2HX(1+)]}
(9d)
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If w>w*, then the equilibrium is inherently unstable and the conclusion of the previous section is still tenable. However, if there is a sufficiently large number of fully rational
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agents in the system (if w
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paribus, can trigger serious instability to a previously stable system.
4. Concluding Comments First generation models of currency crises highlight a conflict between objectives for nominal exchange rate stability and a monetary/fiscal policy that is inconsistent with
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these objectives. Currency crises are simply due to bad policies. Confidence in the first generation models wanes due to the subsequent observation that, in many cases, the underlying policy conflict seems missing. These observations have inspired second
p ro
generation models that explain currency and financial crises in terms of shifts in private expectations. These models explain speculative attacks in terms of the fundamentals identified in first generation models, but the fundamentals are themselves sensitive to shifts in private expectations about the future. It is not possible to fix the timing of a
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crisis, because the crises are driven by unexplained but self-fulfilling shifts in private expectations. As a result, the currency crisis according to second generation models is due to bad luck.
What we have proposed in this paper is three-fold: first, we posit that the time path of fundamentals (mainly money supply) is influenced by both actual and anticipated appreciation / depreciation of the currency. Secondly, we argue that the dynamics of
al
exchange rate is influenced by the evolution of these fundamentals. Finally, we assume that agents don’t hold rational expectations. Instead we posit that the expected change in exchange rate is partly explained by the actual change in exchange rate and the actual
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change in fundamentals. One may note that upon an introduction of rational expectations and at a steady state, the above dynamics will collapse into the much-vaunted monetarist model of determination of exchange rates. However, since the system is out of steady state and agents don’t hold rational
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expectations, it begs a question of whether the above system converges on a stable steady state. We derive an inequality that offers an answer to this question. If this inequality holds the steady state will be intrinsically unstable. We offer a new definition of currency
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crisis in the light of this instability. If the steady state is intrinsically unstable, or bifurcates into the zone of instability with changes in relevant parameters, the exchange rate mechanism faces crisis in the sense that a slight perturbation will take the system
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further and further away from the steady state in real time. The divergence from the steady state may cause a progressive loss of foreign exchange reserves and ultimately an exhaustion of these reserves and a collapse of the system becomes inevitable. It is important to note that this collapse is not triggered by a bad policy, or bad luck, but by an
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unstable steady state – a bad equilibrium. In this sense the results offer a new explanation of currency crisis: a crisis that erupts due to a bad equilibrium even when there is no evidence of bad policy, or of multiple equilibria (bad luck).
Finally we consider a model with heterogeneous agents some of who are fully rational. The introduction of heterogeneity lends more stability to the system. However, it fails to get rid off the possibility of a bad equilibrium. The existence of an unstable
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equilibrium is linked to the heterogeneity coefficient and other parameters of the system.
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