CURRENCY CRISES WITH
SELF-FULFILLING
EXPECTATIONS
Maurice Obstfeld and Morris and
Shin
Speculative Attacks: NonUniqueness
• Government commits a finite stock of reserves—
R—to defend the domestic currency.
• Two private traders play; each one has initially a
domestic money endowment = 6.
• A fixed cost (=1) of selling the money
endowments and purchasing foreign currency.
• A depreciation of 50% if attack successful.
• If both individuals successfully attack, they split
R evenly between them.
• (a) A high-Reserve Game,
• (b) A Low-Reserve Game,
• (c) An Intermediate-reserve game.
A High-Reserve Game: R=20: The
Payoff
Matrix
Hold
Sell
0,0
0, -1
Hold
Nash Equilibrium
-1, 0
-1, -1
Sell
(note: private-sector speculative buying power is 6+6=12)
A Low-Reserve Game: R=6 Payoff
Matrix
Hold
Sell
Hold
Sell
0, 2
0,0
1/2, ½
2, 0
Nash Equilibrium
(Note: speculative buying power =6+6=12)
Hold
Sell
An Intermediate-Reserve Game:
R=10:
Payoff
Matrix
Hold
Sell
0,0
First Nash Equilibrium
0, -1
3/2, 3/2
-1, 0
Second Nash
Equilibrium
Note: the speculative buying power = 6+6=12)
I. Common Knowledge
• Needs two to successfully attack.
• Agent’s strategy: Attack, A, or Not Attack,
N.
• Attack involves fixed cost=1
• Post-attack exchange-rate depreciation=X.
• If X<1, no agent attacks; if X>1, there are
multiple equilibria.
Payoff matrix:
Agent 2
A
Agent 1
N
A
X-1, X-1 -1, 0
N
0, -1
0,0
Estimating Fundamentals and
Second Guessing of Others
Imagine that a large population of agents have access to
public or private information on the underlying fundamentals. Each one
aims to take action appropriate to the underlying
State. But they also engage in race to second-guess the
actions of other individuals. Thus decision makers are interested
Parties in the actions of others.
Public information has attributes that make
It a double-edged instrument. On the one hand it conveys information
on the underlying fundamentals; on the other hand it serves as a focal
point for the beliefs of the group as a whole. Public information
serves as a coordinating device.
Morris and Shin (AER 2003).
X is random: each agent receives
some private information (noisy
signal)
• Signal Z is distributed
uniformly around the
true value of X.
• Z is the best estimate
of the true value of X.
• Z lies between
X

And
X 
.
Both the distributions of X and Z are common knowledge.
Each agent’s expected payoff when using Attack strategy is Z-1,
IF the other agent attacks, as well; or -1, if the other agent
does not attack.
If A is chosen, expected payoff :
Probability{other agent attacks}{Z-1}+
(1- Probability{other agent attacks){-1}
= Probability{other agent attacks}{Z}-1
For this payoff to be positive we need two elements:
(1) The signal Z must be sufficiently high;
(2) The probability that the other agent attacks must
be sufficiently high.
Estimating probabilities
• If one agent receives a
signal Z=1+y, y small,
she will attack if she
believes that
the other agent attacks
with a probability close
to 100%. BUT:
• If agent receives a
signal Z=1+y, she
estimate the true X=1+y.
This implies that she
believes that the other
agent receives signal
below 1 with a
probability of 50% and
above 1 with 50%
she will decide not to
attack.
Cut Off Z*
Since the expected payoff of attacking is
prob(other attacks)*Z - 1
at Z* you are indifferent between attacking or not attacking, This implies
prob(other
attacks | Z*) times (Z*) = 1
But symmetry implies other party only attacks if his Z (call Z2) is
greater
than or equal to Z*.
so,
p(other attacks|Z*) p(Z2>Z*|Z*)
But knowing that your
signal is Z* only tells you that it is 50-50 that their signal is higher
than yours so p(other attacks)=.5
implying
.5Z*=1 so Z*=2
•
•
A slightly higher signal, Z=1+2y
will not be enough to generate
positive payoff.
At some point we can find a
cutoff, Z=Z*
at which the agent rationally
switch from N
to A.
Since the expected payoff of attacking
is p(other attacks)*Z - 1 and we
know
at Z* you are indifferent between
attacking and not this implies
p(other
attacks | Z*) times (Z*) = 1
but symmetry implies other party
only attacks if his Z (call Z2)is
greater
than or equal to Z* so p(other
attacks|Z*) p(Z2>Z*|Z*) but knowing
that your
signal is Z* only tells you that it is
50-50 that their signal is higher
than yours so p(other attacks)=.5
meaning .5Z*=1 so Z*=2
Result: Unique, fundamentals-based explanation for the
for a speculative attack.
Unemployment-Depreciation Model
of Currency Crisis (Barro-Gordon
Type)
• Government
Objective:
Min ( y  y*) 2   2  c( )
Expectations-augmented Phillips Curve
_
y  y   (   )  u
e

=the change in the exchange rate(+ or -)
y* =output target
y =natural output
 e =domestic price setters expectations of based on lagged information
u =i.i.d., zero-mean shock

Assume:
y*  y
Dynamic inconsistency

Government can choose after observing u; unlike price setters.
Any depreciation (  0 )has a cost
Any appreciation has a cost :c(low u) for appreciation, or c(high u)
for depreciation. If u is between low u and high u the fixed rate is
Maintained.
Given u<low u, or u> high u the government chooses:
 ( y *  y  u )   2 e
 (u )   
2
 
With an output level:
 2 ( y *  y )  u  e
y (u )  y  y 
2  
I. Values Policy loss function (Ignoring the fixed cost):
(u ) 
V
flex
V
fixed

 
2
( y *  y  u   )
e 2
(u )  ( y *  y  u   )
e 2
2. Solving for high u and low u:
V flex (u )  c(u )  V fixed ,
V flex (u )  c(u )  V fixed .
u  [  ,  ]
, uniform
distribution.
,
Rational expectations of next period
given the price expectation  e
is:
E   E  / u  u  Pr u  u 

 
 E  / u  u Pr u  u

 0  Pr u  u  u


E 
E   E  / u  u  Pr u  u 

 
 E  / u  u Pr u  u

 0  Pr u  u  u


45 degrees
Three Equilibria

e