CURRENCY CRISES WITH SELF-FULFILLING EXPECTATIONS

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CURRENCY CRISES WITH SELF-FULFILLING EXPECTATIONS

• Maurice Obstfeld and Morris and Shin

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Speculative Attacks Non-Uniqueness

• Government commits a finite stock of
  reservesRto 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.

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A High-Reserve Game R20 The Payoff Matrix
Hold
Sell
0,0 Nash Equilibrium 0, -1
-1, 0 -1, -1
Hold
Sell
(note private-sector speculative buying power
is 6612)
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A Low-Reserve Game R6 Payoff Matrix
Hold
Sell
0,0 0, 2
2, 0 1/2, ½ Nash Equilibrium
Hold
Sell
(Note speculative buying power 6612)
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An Intermediate-Reserve Game R10 Payoff Matrix
Hold
Sell
0,0 First Nash Equilibrium 0, -1
-1, 0 3/2, 3/2 Second Nash Equilibrium
Hold
Sell
Note the speculative buying power 6612)
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I. Common Knowledge

• Needs two to successfully attack.
• Agents strategy Attack, A, or Not Attack, N.
• Attack involves fixed cost1
• Post-attack exchange-rate depreciationX.
• If Xlt1, no agent attacks if Xgt1, there are
  multiple equilibria.

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Payoff matrix
Agent 2

A N
Agent 1 A X-1, X-1 -1, 0
N 0, -1 0,0
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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).
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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

And
.
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Both the distributions of X and Z are common
knowledge. Each agents 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
Probabilityother agent attacksZ-1 (1-
Probabilityother agent attacks)-1
Probabilityother agent attacksZ-1

• For this payoff to be positive we need two
  elements
• The signal Z must be sufficiently high
• The probability that the other agent attacks must
• be sufficiently high.

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Estimating probabilities

• If one agent receives a signal Z1y, 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 Z1y, she estimate
  the true X1y. 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.

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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(otherattacks Z) times (Z) 1
• But symmetry implies other party only attacks if
  his Z (call Z2) is greaterthan or equal to Z.
• so,
• p(other attacksZ) p(Z2gtZZ)
• But knowing that yoursignal is Z only tells you
  that it is 50-50 that their signal is higherthan
  yours so p(other attacks).5implying
• .5Z1 so Z2

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• A slightly higher signal, Z12y will not be
  enough to generate positive payoff.
• At some point we can find a cutoff, ZZ
• at which the agent rationally switch from N
• to A.

• Since the expected payoff of attacking is p(other
  attacks)Z - 1 and we knowat Z you are
  indifferent between attacking and not this
  implies p(otherattacks Z) times (Z) 1but
  symmetry implies other party only attacks if his
  Z (call Z2)is greaterthan or equal to Z so
  p(other attacksZ) p(Z2gtZZ) but knowing that
  yoursignal is Z only tells you that it is 50-50
  that their signal is higherthan yours so p(other
  attacks).5meaning .5Z1 so Z2
• Result Unique, fundamentals-based explanation
  for the
• for a speculative attack.

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Unemployment-Depreciation Model of Currency
Crisis (Barro-Gordon Type)

• Government Objective

Expectations-augmented Phillips Curve
the change in the exchange rate( or -) output
target natural output domestic price setters
expectations of based on lagged
information i.i.d., zero-mean shock
y
u
Assume
Dynamic inconsistency
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Government can choose after observing u
unlike price setters. Any depreciation ( )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 ultlow u, or ugt high u the
government chooses
With an output level
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I. Values Policy loss function (Ignoring the
fixed cost)
2. Solving for high u and low u
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, uniform distribution.
Rational expectations of next period ,
given the price expectation is
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45 degrees
Three Equilibria