Speculative attacks and defenses as wars of attrition
Kevin Griera1, Shu Linb
a
University of Oklahoma, Department of Economics, 729 Elm Ave., Norman OK 73019
b
University of Colorado Denver, Department of Economics, Campus Box 181, Denver
CO 80217
ABSTRACT
This paper extends 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 (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.
JEL Classifications: F3
Key Words: Currency Crisis, Speculative Attacks
1
Corresponding author. Tel.: +1 405 325 3748; fax: +1 405 325 5842. Email address: angus
@ou.edu (K. Grier).
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.2 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.3
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
2
See Krugman (1979) and Flood and Garber (1984).
3
That is to say, there are only either successful defenses or no defense in these models.
1
asymmetric information.4 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, 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.5 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
4
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.
5
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 Handerson (1978), and one of the models in Obstfeld (1994) follows that of Barro and Gordon
(1983).
2
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
3
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.6 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); (2) explain the systematic variation in the
duration of a defense. Unfortunately, the second generation models of Obstfeld (1994,
1996) cannot explain these facts.7
6
7
It is also the case that there are both long and short successful defenses and long and short unsuccessful defenses.
There are other recent studies in the literature also trying to explain these facts. For example, taking a global-game
4
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,
yt  y   (   e )  u
(1)
where   0 . y is the natural output level,   e e 1 the change in the exchange rate
(price of foreign currency in terms of domestic currency), and  e is the expected
devaluation based on lagged information. u is a one time shock. The government
chooses  after observing  e and u .
The government cares about output and exchange rate stability. Its loss function is
L  max[ 0, ( y *  y)]  ZVg
(2)
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.8 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
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.
8
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.
5
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 V g can be considered as the fixed cost of devaluation or the value of the
peg to the government. V g 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 V g . The probability density function
and the cumulative distribution function of V g are assumed to be f (V g ) and F (V g ) with
the support of [Vg ,Vg ] .
Given (1) and (2), if it is costless to change the exchange rate ( V g  0 ), the
government would set
   e  ( y *  y  u) / 
(3)
to minimize its loss function, and its loss would be
L flex  0
(4)
To maintain the peg (   0 ), the government’s loss is instead
L fix  y *  y  u   e
(5)
Therefore, the government’s flow cost of maintaining the peg is
C g  L fix  L flex  y *  y  u   e
(6)
while its winning prize of successfully maintaining the peg is V g , which is the fixed cost
associated with a devaluation.
If we assume perfect capital mobility and normalize the foreign nominal interest rate
6
to zero, uncovered interest rate parity implies that the nominal rate of return on domestic
currency denominated assets should equal expected rate of devaluation,  e . 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, C s , therefore, is just  e . 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  k , k  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 , k ] .
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
7
will be a costly battle (war of attrition) between the two parties.9 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 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
EU g (Tg )  C g Tg prob[Ts  Tg ]  prob[Ts  Tg ]E[(V g  C g Ts ) | Ts  Tg ]
(7)
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
9
There are also other equilibria in which one party concedes immediately in our model.
8
on having a longer optimal concession time. The government’s objective is to find an
optimal concession time T g to maximize its payoff. Similarly, speculators also choose an
optimal time of concession to maximize their payoff
EU s (Ts )  C s Ts prob[Tg  Ts ]  prob[Tg  Ts ]E[(Vs  C s Tg ) | Tg  Ts ]
(8)
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 ' (Vi )  0 and Ti ' (Ci )  0 , i  g , s .
(See Appendix 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   g (V g ) and Ts   s (Vs ) .10 In this
equilibrium, if the government behaves according to  g (V g ) , speculators will find it
optimal to concede according to  s (Vs ) and vice versa.
1
Define i   i (Ti )  Vi , i  g , s . LEMMA 1 implies that
prob[Ts  Tg ]  H ( s (Tg ))
(9)
prob[Tg  Ts ]  F ( g (Ts ))
(10)
Substituting (9) and (10) into (7) and (8), respectively, we obtain
10
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 (2004) for details.
9
 s (Tg )
EU g (Tg )  C g Tg [1  H ( s (Tg ))]  
k
 g (Ts )
EU s (Ts )  C s Ts [1  F ( g (Ts ))]  
Vg
[V g  C g  s (Vs )]h(Vs )dVs
(11)
[Vs  C s  g (V g )] f (V g )dV g
(12)
Differentiating (11) with respect to Tg and setting it equal to zero, we obtain the first
order condition for the government’s optimization problem
C g  Vg [
h( s (Tg ))
1  H ( s (Tg ))
] s ' (Tg )
(13)
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.
f ( g (Ts ))
C s  Vs [
] g ' (Ts )
1  F ( g (Ts ))
(14)
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  e 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  E[ | Tg  Ts ] prob[Tg  Ts ]   e
(15)
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
10
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
11
parameters in the model. We assume that both V g and k have an exponential distribution
with   1 (therefore, Vs  k has an exponential distribution with   1 /  ), which
implies that
h( s (Tg ))
1  H ( s (Tg ))

1

and
f ( g (Ts ))
1  F ( g (Ts ))
 1 . Using (13) and (14) and i (0)  0 ,
we can obtain each party’s optimal time of concession
Tg  kVg /( y *  y  u   e )
(16)
Ts  k[ e  ( y *  y  u) /  ]Vg /  e
(17)
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 y , the size of the adverse shock u , and the slope of the
Phillips curve  . Substituting (16) and (17) into (15) and using the exponential
distribution assumption, we have
Vg ( y *  y  u  e ) 2
[( y  y  u ) /  ]  [  ( y  y  u ) /  ]e
*
e
 e
*
0
(18)
Given the non-linear feature of (18), there can be multiple solutions. For example,
assuming y *  y ,   0.8 , u  0.1 , V g  1 , Figure 1 illustrates that there are two
solutions:  1e  0.0685 and  2e  4.1663 .
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 equations (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
12
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  Ts ) with a longer duration while the
second attack has a shorter duration and a successful outcome ( Tg  Ts ) with an eventual
devaluation of   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 ( C g ) with the cost of abandoning the peg ( V g ).
The equilibria can be solved by using the rational expectation condition
E  E[ | Vg  C g ] prob[Vg  C g ]   e
(19)
Substituting equations (3) and (6) into equation (19) and using the assumptions of
exponential distribution of V g and y *  y ,   0.8 , u  0.1 , we find two numerical
e
e
solutions:  1  0.0148 and  2  4.3463 , which are shown in Figure 2.
The multiple equilibria we obtain imply that speculative attacks can again be selffulfilling in this new setup. Moreover, since there is still uncertainty about the value of
the peg to the government (
assuming
Vg  1
Vg
), we can also explain failed attacks. For example,
, it can be verified that the first attack is a failed attempt (
the second attack is a successful one (
C g  Vg
C g  Vg
), and
，   4.4713 ). However, without
modeling the speculators’ side, this new setup can not explain the costly battles between
13
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).11 Therefore, the comparison of these
three different setups show 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.
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
11
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.
14
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.
15
Appendix
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
prob[Ts  Tg ]  K (Tg ) and prob[Tg  Ts ]  G (Ts )
(A1)
Cross-twice differentiating (7) with respect to T g and V g and using (A1), we obtain
d 2 EU g (Tg )
dTg dVg
 k (Tg )  0
(A2)
Cross-twice differentiating (7) with respect to T g and C g and using (A1), we have
d 2 EU g (Tg )
dTg dC g
 [1  K (Tg )]  0
(A3)
Similarly, we can show that
d 2 EU s (Ts )
 g (Ts )  0
dTs dVs
(A4)
d 2 EU s (Ts )
 [1  G(Ts )]  0
dTs dC s
(A5)
Equations (A2)-(A5) imply that, when the other party is acting optimally, dEU i / dTi is
increasing in Vi but decreasing in C i . Therefore, each party’s optimal time of concession
time Ti is monotonically increasing in Vi but decreasing in C i , i  g , s .
16
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17
Fig. 1. Multiple solutions of equation (18)
18
Fig. 2. Multiple solutions of equation (19)
19