European Journal of Political Economy 25 (2009) 540–546
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European Journal of Political Economy
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e j p e
Speculative attacks and defenses as wars of attrition
Kevin Grier a,⁎, Shu Lin b
a
b
Department of Economics, University of Oklahoma, 729 Elm Avenue, Norman, OK 73019, United States
Department of Economics, University of Colorado Denver, Campus Box 181, Denver, CO 80217, United States
a r t i c l e
i n f o
Article history:
Received 28 March 2007
Received in revised form 9 June 2009
Accepted 16 June 2009
Available online 25 June 2009
JEL classification:
F3
Keywords:
Currency crisis
Speculative attacks
a b s t r a c t
This paper extends the currency crisis models in Obstfeld [Obstfeld, M., 1994. The logic of
currency crises. Cahiers Economiques et Monetaries], [Obstfeld, M., 1996. Models of currency
crises with self-fulfilling features. European Economic Review 40, 1037–1047] by modeling both
the government's side and speculators' side and introducing uncertainty about each party's
payoff. We argue that a speculative attack (defense) can be well modeled as a war of attrition
between the government and speculators under asymmetric information. We then solve for a
pure strategy, weakly perfect, Bayesian rational expectation equilibrium in which each party's
time until concession depends on her benefits from winning and her costs of fighting. Using this
model, we are able to explain important facts of currency crises. First, the model shows that
failed defenses (attacks) can be ex-ante rational for governments (speculators). Second, the
model predicts systematic variations in the durations of defenses. Finally, we also show that
currency crises can be self-fulfilling in the sense that there exist multiple rational expectation
equilibria in our model.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
Theoretical studies of currency crises originate from Krugman's (1979) seminal paper. In a Krugman style first-generation crisis,
the government does not act strategically.1 It responds passively to a speculative attack on the currency peg. As has been noted, the
passive government assumption seems to be unrealistic because, in practice, governments often defend pegs aggressively. In
second-generation currency crisis models (Obstfeld, 1994, 1996), the government responds actively to a speculative attack. It
makes a strategic decision between maintaining and abandoning the peg by comparing the costs each action imposes on its
welfare. Another important feature of the second-generation models is that they predict that currency crises can be self-fulfilling.
The second-generation models make important contributions to the literature, but there is still room for improvement. For
example, these models only focus on modeling the government's behavior while ignoring speculators' involvement in an attack.
Furthermore, the government's decision to abandon the peg is immediate in these models, which contradicts the fact that some
speculative attacks (defenses) can have long durations. Finally, these models cannot explain the defenses observed in practice.2
In this paper, we extend the currency crisis models in Obstfeld (1994, 1996) by modeling both the government's side and
speculators' side and introducing uncertainty about each party's payoff. We argue that a speculative attack and defense can be well
modeled as a war of attrition game between speculators and the government under asymmetric information.3 In our model,
maintaining the peg has value to the government while the collapse of the peg has value to speculators. The true values are private
information. In addition, a defense by the government is assumed to be costly to both sides. In a pure strategy, weakly perfect,
⁎ Corresponding author. Tel.: + 1 405 325 3748; fax: + 1 405 325 5842.
E-mail address: [email protected] (K. Grier).
1
See Krugman (1979) and Flood and Garber (1984).
2
That is to say, there are only either successful defenses or no defense in these models.
3
Our paper builds on an observation made by Bensaid and Jeanne (1997), who state that a defense during a speculative attack is a war of attrition between the
two sides. However, they do not develop this statement in the paper.
0176-2680/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.ejpoleco.2009.06.003
K. Grier, S. Lin / European Journal of Political Economy 25 (2009) 540–546
541
Bayesian rational expectation equilibrium of the game, each party's optimal concession time depends on its reward for winning
and its flow waiting cost. The higher (lower) one's value for winning (waiting cost) is, the later one concedes. Of course, she who
concedes last wins the war!
Our modeling choice is motivated by some stylized facts of speculative attacks and defenses to be discussed in Section 2.
Though the war of attrition game is standard in applied game theory and has been widely used in other settings, we are able to use
it here to make significant contributions to the currency crisis literature.4 In particular, our model has the following features: 1) the
model shows that an ex-ante rational attack (defense) can be either successful or unsuccessful. 2) We show that, in a pure strategy
weak perfect Bayesian rational expectation equilibrium, an agent's optimal time until conceding depends on her value of winning
and her flow waiting cost. Therefore, the model makes explicit predictions about variations in the duration of defenses. 3) The
model also shares a common feature of previous second-generation models in that currency crises can be self-fulfilling. There exist
multiple rational expectation equilbria, and the government has no control of which equilibrium will realize.
The rest of this paper is organized as follows: Section 2 provides some stylized facts about speculative attacks (defenses).
Section 3 discusses the setup of the model. In Section 4, we solve for a pure strategy weak perfect Bayesian rational expectation
equilibrium. A numerical example is provided in Section 5 to illustrate the main implications of the model. Section 6 concludes.
2. Some stylized facts of speculative attacks and defenses
2.1. There are both successful defenses and unsuccessful defenses
This is well illustrated by two well-known historical examples: Hong Kong's successful defense during the 1997 Asian crisis and
Sweden's failed defense in the 1992 EMS crisis. Both monetary authorities had fought fiercely to defend their currencies. In Hong
Kong, the Hong Kong Government (HKMA) raised the overnight HIBOR to 280% on Oct. 23 1997 to defend its currency board
system. It also raised interest rates, though to a much less substantial level, during the next two waves of speculative attacks in
January and June 1998. In August 1998, the HKMA mounted operations in both the stock market and the money market to
counteract speculators' activities. The currency board system was eventually successfully defended. In the Swedish case, the
Riksbank initially took a very tough stance when the speculative attacks first hit the Krona peg in August 1992. The first wave of
speculative attack was defeated. Yet when a new wave of attacks came two months later, the Riksbank surprisingly floated the
Krona shortly after a minimal defense. As these examples illustrate, an acceptable theoretical model must be able to explain this
uncertain outcome of defenses and, perhaps more importantly, it must be able to admit a failed defense or attack as being ex-ante
rational. In our view, this requires incorporating asymmetric information into the theoretical analysis.
2.2. The duration of the battle between the government and speculators varies widely in different cases
For example, soon after the Asian financial turmoil hit Indonesia in July 1997, its monetary authority widened the Rupiah
trading band from 8% to 12%. On August 14, the monetary authority further replaced the managed floating exchange rate regime
with a freely floating regime. The attack on the Rupiah lasted about a month. On the other hand, the battle between speculators and
Hong Kong described above lasted about a full year, during which the HK dollar experienced four separate waves of attacks.5
Therefore, in our view, an acceptable theoretical model should allow for systematic variations of the duration of speculative attacks
and defenses.
In summary, given these stylized facts of attacks and defenses, an adequate model must be able to (1) explain the possible
outcomes of attacks (defenses) and allow ex-ante rational unsuccessful attacks (defenses); and (2) explain the systematic
variation in the duration of a defense. Unfortunately, the second-generation models of Obstfeld (1994, 1996) cannot explain these
facts.6
3. Setup of the model
In this section, we model a speculative attack and defense as a war of attrition game between the government and speculators
under asymmetric information. We consider an economy in which there is a domestic government and a group of homogenous
speculators. We model the government side of the economy following Obstfeld (1996).
Assuming that purchasing power parity holds, output in the economy is determined by an expectation-augmented Phillips
curve relation,
e
yt = ȳ + αðε−ε Þ−u
ð1Þ
4
There is a long tradition of borrowing models in the literature. For example, the model in Krugman (1979) is based on the model of Salant and Henderson
(1978), and one of the models in Obstfeld (1994) follows that of Barro and Gordon (1983).
5
It is also the case that there are both long and short successful defenses and long and short unsuccessful defenses.
6
There are other recent studies in the literature also trying to explain these facts. For example, taking a global-game approach, Angeletos et al. (2007a) analyze
a static game that can explain failed attacks. Angeletos et al. (2007b) study a dynamic global-game that can allow for varying durations of attacks. Our simple
model is able to explain both facts.
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K. Grier, S. Lin / European Journal of Political Economy 25 (2009) 540–546
where α N 0. y̅ is the natural output level, ε = e − e− 1 is the change in the exchange rate (price of foreign currency in terms of
domestic currency), and εe is the expected devaluation based on lagged information. u is a one time shock. The government
chooses ε after observing εe and u.
The government cares about output and exchange rate stability. Its loss function is
L = max½0; ðy*−yÞ + ZVg
ð2Þ
We assume y* ≥ y ̅ so that the government faces a dynamic inconsistency problem as discussed in Barro and Gordon (1983).
Following Obstfeld (1994), we rule out the possibility of a revaluation.7 Assuming that u is non-negative, we have a simple linear,
instead of quadratic, loss function. The first term captures the loss due to deviations from the optimal output level y*. The lower
bound zero ensures that the government will not try to achieve an output level higher than y*. The second term is the cost of
devaluation. Z is a dummy variable that takes the value of one when there is devaluation and zero otherwise, and Vg can be
considered as the fixed cost of devaluation or the value of the peg to the government. Vg is the government's private information, so
speculators do not know the relative weight of the cost associated with devaluation in the government's loss function. Speculators
only know the distribution of Vg. The probability density function and the cumulative distribution function of Vg are assumed to be
f(Vg) and F(Vg) with the support of ½Vg ; Vg .
P
Given (1) and (2), if it is costless to change the exchange rate (Vg = 0), the government would set
e
ε = ε + ðy*− ȳ + uÞ = α
ð3Þ
to minimize its loss function, and its loss would be
L
flex
ð4Þ
=0
To maintain the peg (ε = 0), the government's loss is instead
L
fix
e
= y*− ȳ + u + αε
ð5Þ
Therefore, the government's flow cost of maintaining the peg is
fix
Cg = L −L
flex
e
= y*− ȳ + u + αε
ð6Þ
while its winning prize of successfully maintaining the peg is Vg, which is the fixed cost associated with a devaluation.
If we assume perfect capital mobility and normalize the foreign nominal interest rate to zero, uncovered interest rate parity
implies that the nominal rate of return on domestic currency denominated assets should equal expected rate of devaluation, εe.
That is to say, to defend the peg, the government has to raise the interest rate by the same size as the expected rate of devaluation.
Speculators short-sell domestic currency denominated assets in a speculative attack. Their flow cost (opportunity cost of selling
domestic currency denominated assets for foreign currency denominated assets) of attacking the peg, Cs, therefore, is just εe. On
the other hand, we assume the winning prize for the speculator is proportional to the size of the government's desired devaluation
under flexible exchange rate, Vs = kε, k N 0, if the attack turns out to be successful. The parameter k is assumed to be speculators'
private information. The probability
density function and the cumulative distribution function of Vs, are assumed to be h(Vs) and
_
H(Vs) with the support of [k_ε, k ε].
In Obstfeld (1996), the speculators' side of the game is not modeled. In addition, there is no uncertainty about each party's payoff.
The outcome of a speculative attack in Obstfeld's model is, therefore, simply determined by the government's comparison of its cost
and benefit of defense, and the government's devaluation decision is immediate. There will not be any lasting battle between the two
parties. Furthermore, since the government faces no uncertainty, the model in Obstfeld (1996) rules out any possibility for
unsuccessful defenses, which are often observed in practice. This is not the case in our model. In our model, given each party's winning
prize unknown to her rival, there will be a costly battle (war of attrition) between the two parties.8 The outcome of a speculative attack
in our model depends not only on the government's benefits and costs but the speculators' benefits and costs as well. We shall also
show that, since each party faces uncertainty, our model justifies ex-ante rational unsuccessful defenses (attacks).
4. Solving the model
In a speculative attack, speculators take a costly position gambling that the government will devalue the currency, while the
government also bears a cost due to output loss. During the battle, each party updates its knowledge about the distribution of its
rival's winning prize at any point of time based on available information. The battle will keep going until one party realizes her rival
has more at stake and concedes. The model is solved by deriving each party's optimal concession time Ti, i = g, s, at which her flow
7
Also, unlike Obstfeld (1996), we assume that the loss to the government is independent of the size of the devaluation. These assumptions are made to make
the model tractable. They are not crucial to our results.
8
There are also other equilibria in which one party concedes immediately in our model.
K. Grier, S. Lin / European Journal of Political Economy 25 (2009) 540–546
543
waiting cost is equal to her value to winning times the probability that the other party will concede at the next instant, along with a
rational expectation condition.
If we ignore discounting, the government's payoff is
EUg ðTg Þ = −Cg Tg prob½Ts ≥Tg + prob½Ts bTg E½ðVg −Cg Ts ÞjTs bTg :
ð7Þ
The first term on the right hand side is the government's expected total waiting cost, which is equal to its total waiting cost
times the probability that it concedes earlier than the speculator. The second term on the right hand side is the government's
expected benefit, which is equal to the probability that the speculator concedes first times the differential between the
government's winning prize and its total waiting cost conditional on having a longer optimal concession time. The government's
objective is to find an optimal concession time Tg to maximize its payoff. Similarly, speculators also choose an optimal time of
concession to maximize their payoff
EUs ðTs Þ = −Cs Ts prob½Tg ≥Ts + prob½Tg bTs E½ðVs −Cs Tg Þ jTg bTs :
ð8Þ
To solve the model, we first need to show that each party's optimal time of concession is monotonic in its winning prize and
flow waiting cost.
Lemma 1. Ti′(V i) N 0 and Ti′(Ci) b 0, i = g, s. (See Appendix A for proof.)
Based on Lemma 1, we solve this game by deriving a pure strategy weak perfect Bayesian equilibrium in which one party's
concession behavior is described by a strictly increasing function of its winning prize: Tg =βg(V g) and Ts =βs(V s).9 In this equilibrium,
if the government behaves according to βg(V g), speculators will find it optimal to concede according to βs(V s) and vice versa.
1
Define ϕi = β−
(Ti) = V i, i = g, s. Lemma 1 implies that
i
prob½Ts bTg = Hðϕs ðTg ÞÞ
ð9Þ
prob½Tg bTs = Fðϕg ðTs ÞÞ:
ð10Þ
Substituting (9) and (10) into (7) and (8), respectively, we obtain
ϕ ðTg Þ
EUg ðTg Þ = −Cg Tg ½1−Hðϕs ðTg ÞÞ + ∫kεs
¯
ϕ ðTs Þ
EUs ðTs Þ = −Cs Ts ½1−Fðϕg ðTs ÞÞ + ∫Vgg
½Vg −Cg βs ðVs ÞhðVs ÞdVs
ð11Þ
½Vs −Cs βg ðVg Þ f ðVg ÞdVg :
ð12Þ
¯
Differentiating (11) with respect to Tg and setting it equal to zero, we obtain the first order condition for the government's
optimization problem
"
#
hðϕs ðTg ÞÞ
0
ϕ ðT Þ:
1−Hðϕs ðTg ÞÞ s g
Cg = Vg
ð13Þ
Similarly, the first order condition for speculators' optimization problem can be derived by differentiating (12) with respect to Ts
and then setting it equal to zero.
"
Cs = Vs
#
f ðϕg ðTs ÞÞ
0
ϕ ðT Þ:
1−Fðϕg ðTs ÞÞ g s
ð14Þ
The two first order conditions state that concession occurs when the marginal cost of waiting equals the expected marginal
benefit from waiting. The left hand side of each condition is the flow cost of waiting one more instant to concede. The right hand
side of each condition is the expected gain from waiting another instant to concede, which is the product of the winning prize and
the conditional probability that the other party concedes in the next instant.
The above first order conditions are derived taking εe as given. In a rational expectation equilibrium, speculators use all
available information to form their devaluation expectations. Their expectations are based on the following equation
e
Eε = E½εjTg bTs prob½Tg bTs = ε :
ð15Þ
9
In fact, in the model there is a continuum of equilibria. The continuum is obtained by picking either one of the two sides in the war of attrition and letting an
(arbitrary) interval of the higher waiting cost (lower winning prize) of that side to concede immediately at time 0. By varying the size of the conceding interval,
one can get the continuum. That is, the equilibrium we consider is obtained by using implicitly the selection criterion that no side concedes at time 0. See
Martinelli and Escorza (2007) for details.
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K. Grier, S. Lin / European Journal of Political Economy 25 (2009) 540–546
The solution of the model can be obtained by solving the three-equation system (13), (14), and (15). Although no analytical
solutions can be obtained without making specific assumptions about the distributions of winning prizes, we can still obtain the
following implications of the model.
First, Lemma 1 shows that each party's optimal time of concession decreases with its waiting cost but increases with its winning
prize. The model thus implies that the durations of speculative attacks will vary systematically across specific circumstances.
Second, notice that the equilibrium of the game is such that, even though attacking (defending) is ex-ante rational, the loser of the
war will experience regret. Therefore, our model justifies rational unsuccessful defenses (attacks). Finally, similar to Obstfeld
(1996), currency crises can also be self-fulfilling in our model. There can be multiple rational expectation equilibria in our model
(besides those mentioned in footnote 9). The equilibrium expected devaluation rates can be found as solutions to the above threeequation system, and we cannot rule out the possibility for multiple equilibria. The existence of multiple equilbiria means that
speculative attacks can be self-fulfilling, and the government is powerless in choosing a small devaluation expectation or a large
devaluation expectation, and in turn the outcome of the battle.
5. A numerical example
In this section, we use a simple numerical example to illustrate the implications our model as well as the importance of
modeling both the government's side and speculators' side and introducing uncertainties of each party's payoff.
We first solve for equilibria (besides those mentioned in footnote 9) of our model by making assumptions about the
distributions of each party's winning prizes and values of parameters in the model. We assume that both V g and k have an
exponential distribution with λ = 1(therefore, V s = kε has an exponential distribution with λ = 1/ε), which implies that
hðϕs ðTg ÞÞ
f ðϕg ðTs ÞÞ
1
1−Hðϕs ðTg ÞÞ = ε and 1−Fðϕg ðTs ÞÞ = 1. Using (13) and (14) and ϕi(0) = 0, we can obtain each party's optimal time of concession
e
Tg = kVg = ðy*− ȳ + u + αε Þ
e
ð16Þ
e
Ts = k½ε + ðy*− ȳ + uÞ = αVg = ε :
ð17Þ
It is obvious that each party's optimal time of concession is increasing in her winning prize but decreasing in her waiting cost,
which are further determined by fundamentals, such as the gap between y* and ȳ, the size of the adverse shock u, and the slope of
the Phillips curve α. Substituting (16) and (17) into (15) and using the exponential distribution assumption, we have
e
*
−Vg ðy − ȳ + u + αεe Þ2
αεe
½ðy*− ȳ + uÞ = α−½ε + ðy*− ȳ + uÞ = αe
= 0:
ð18Þ
Given the non-linear feature of (18), there can be multiple solutions. For example, assuming y* = ȳ, α = 0.8, u = 0.1, V g = 1,
Fig. 1 illustrates that there are two solutions: ε1e = 0.0685 and ε2e = 4.1663.
Fig. 1. Multiple solutions of Eq. (18).
K. Grier, S. Lin / European Journal of Political Economy 25 (2009) 540–546
545
Fig. 2. Multiple solutions of Eq. (19).
These two solutions are associated with two speculative attacks (and defenses). The first attack is based on a small devaluation
expectation, and the second one is based on a large devaluation expectation. Therefore, speculative attacks can be self-fulfilling in
our models. Furthermore, using Eqs. (16) and (17), we can also determine the duration and the outcome of each battle. The
duration of a battle is equal to min[Tg, Ts], and the outcome of an attack depends on which party has a longer optimal time of
concession. It is easy to verify that the first speculative attack (the one based on the small devaluation expectation) turns out to be
a failed attack (Tg N Ts) with a longer duration while the second attack has a shorter duration and a successful outcome (Tg b Ts) with
an eventual devaluation of ε = 4.2913. The example clearly shows that our model allows for failed attacks (defenses) and predicts
systematic variations in the durations of attacks (defenses).
Next, we are also interested to learn about the equilibria that would arise if we do not explicitly model the speculators' side but
still assume V g to be the government's private information. In this case, the government determines the outcome of an attack by
simply comparing the cost of maintaining the peg (Cg) with the cost of abandoning the peg (V g). The equilibria can be solved by
using the rational expectation condition
e
Eε = E½εjVg bCg prob½Vg bCg = ε :
19
Substituting Eqs. (3) and (6) into Eq. (19) and using the assumptions of exponential distribution of V g and y* = ȳ, α = 0.8,
u = 0.1, we find two numerical solutions: ε1e = 0.0148 and ε2e = 4.3463, which are shown in Fig. 2.
The multiple equilibria we obtain imply that speculative attacks can again be self-fulfilling in this new setup. Moreover, since
there is still uncertainty about the value of the peg to the government (V g), we can also explain failed attacks. For example,
assuming V g = 1, it can be verified that the first attack is a failed attempt (Cg b V g), and the second attack is a successful one
(Cg N V g, ε = 4.4713). However, without modeling the speculators' side, this new setup cannot explain the costly battles between
the two parties we often observe in practice. The outcome of an attack is completely determined by the government. As a result,
failed defenses are not allowed in this new setup. There will be either a successful defense (as in the first case) or no defense (as in
the second case). Furthermore, this new setup does not predict any systematic variations in the durations of attacks (defenses)
either. The outcomes of these attacks are revealed immediately.
Finally, we can compare the results from the above two setups with those in Obstfeld (1994, 1996). In Obstfeld's models, the
speculators' side is ignored. In addition, there are no uncertainties about each party's payoff. As a result, the models in Obstfeld
(1994, 1996) cannot justify failed attacks (or defenses), nor can they explain the systematic variations in the durations of attacks
(defenses).10 Therefore, the comparison of these three different setups shows that, in order to generate failed attacks (defenses)
and explain the systematic variations in the durations of currency crises, it is crucial to model both the government's side and the
speculators' side and to introduce uncertainties about each party's payoff.
10
There is another important difference between our model and those in Obstfeld (1994, 1996). In our model, multiple equilibria arise because of randomness
of the government's cost of devaluation. In Obstfeld's models, multiple equilibria are due to random supply shocks.
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K. Grier, S. Lin / European Journal of Political Economy 25 (2009) 540–546
6. Conclusion
This paper extends the currency crisis model in Obstfeld (1994, 1996) by modeling both the government's side and speculators'
side and introducing uncertainties of each party's payoff. We argue that a speculative attack (defense) can be well modeled as a
war of attrition between the government and speculators under asymmetric information. We show that, in a pure strategy weak
perfect Bayesian rational expectation equilibrium, each party's fighting time strictly increases with its value to winning and
decreases with its waiting costs, which suggests that the durations of speculative attacks will vary systematically across specific
circumstances. Our model is also able to justify the ex-ante rationality of unsuccessful defenses (attacks). In addition, we show that
currency crises can be self-fulfilling in the sense that there exist multiple rational expectation equilibria in our model. The
existence of multiple rational expectation equilibria implies that the government has no control of which equilibrium is going to
realize.
Appendix A
Proof of Lemma 1. Define K(T) (G(T)) as the cumulative distribution function of speculators' (the government's) optimal time of
concession such that
prob½Ts bTg = KðTg Þ and prob½Tg bTs = GðTs Þ:
ðA1Þ
Cross-twice differentiating (7) with respect to T g and V g and using (A1), we obtain
2
d EUg ðTg Þ
= kðTg Þ N 0:
dTg dVg
ðA2Þ
Cross-twice differentiating (7) with respect to T g and Cg and using (A1), we have
d2 EUg ðTg Þ
= −½1−KðTg Þb0:
dTg dCg
ðA3Þ
Similarly, we can show that
d2 EUs ðTs Þ
= gðTs Þ N 0
dTs dVs
ðA4Þ
d2 EUs ðTs Þ
= −½1−GðTs Þb0:
dTs dCs
ðA5Þ
Eqs. (A2)–(A5) imply that, when the other party is acting optimally, dEUi/dTi is increasing in Vi but decreasing in Ci. Therefore,
each party's optimal time of concession time Ti is monotonically increasing in Vi but decreasing in Ci, i = g, s.
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