A Model of Commitment Strengthening in a Fixed Exchange Rate Regime

Y. Stephen Chiu

HKIMR and CUHK

August 2001

Introduction

• Currency attacks may arise from neither bad fundamentals nor irresponsible domestic policies, but simply from a time consistency problem

• The time consistency problem--Ex ante, the government prefers to keep the peg. But upon a large scale attack, it prefers to abandon the peg.

• One remedy is to increase the devaluation cost the government incurs should the latter scenario arises. This mitigates the time consistency problem, and speculators will become less aggressive. Based on this idea, Chan and Chen (1999) and Miller (1998) proposed HK$ put option/structured notes to deal with currency crisis in HK in 1998.

• This paper builds up a theoretical model so that such a proposal can be evaluated.

Model

• Stage 1: The fundamentals are sound, but bad fundamentals in the future cannot be completely ignored. Hence, the government attempts to strengthen its commitment to the peg (to choose D).

• Stage 2: the fundamentals are realized. Upon observing a private noisy signal x, each speculator decides whether to short sell one unit of local currency.

• Stage 3: Observing the fundamentals and the scale of attack, the government decides to defend or not. In the latter case, the currency will simply go afloat.

What is commitment strengthening?

• Any means that increases the costs incurred by the government/policy makers upon a devaluation. Examples:

– Fixed rate being written down in the constitution

– compensation to the central banker contingent on whether the peg is maintained

– a voluntary transfer, upon a devaluation, to the Easter Island government to conserve its sculpture

– loans in US$ to authorized institutions giving the latter the option to repay in local currency at the pre-devalued fixed rate

• Version 1: a generic commitment device; no income is received by the government ex ante

• version 2: HK$put option/structured notes where the contract holders need to pay the market clearing price for the contracts.

Basic Results• (Ver 1) Commitment strengthening (CS) is always welfare worsening

– if the government has a low credit line, or

– if the marginal cost of defense is negligible.

• (Ver 1) CS is more likely to be beneficial if the prior probability of devaluation is low; otherwise, D must be large enough to be welfare improving.

• (Ver 2) When the CS is made via a market mechanism, it is generally beneficial. The weaker the government, the greater the desire to have CS.

• (Ver 2) When the government’s preferences are uncertain, the public’s belief plays an important role. There exist one pooling equilibrium at D>0 and one at D=0. The latter equilibrium happens when the public holds the beliefs that the government which uses CS is more likely to be weak.

Details of the model• G(): cumulative function where - < < • F( |x): cumulative function conditional on observation x

• R() = e*-k() 0 for all : a speculator’s capital gain of from one unit of short selling

• t: cost of short selling one unit of domestic currency, assumed independent of D

• a continuun of speculators: a measure of one

• Stage 3: At state , the government’s welfare is equal to – v-c(,,D) - FR(D) if the peg is kept– -L(,D) - FR(D) otherwise

where L(,0)=0; one example of L(,D) is DR()

Suppose speculators’ actions remain the same upon an increase change in D. Devaluation will occur at a smaller probability.

a(,D=0)

a(,D>0)

Scale of attack that makes the government indifferent between defense and not

No devaluationdevaluation

s(k, )

Unique equilibrium in the attack game

• At stage 2: (Morris-Shin) For D=0, as long as signals x are fairly precise, there is always a unique equilibrium, where each speculator adopts the same strategy: attack iff the observed signal xk*. k* satisfies

• Remark: same strategy does not imply same action.

• It turns out that for D>0 the unique equilibrium result still holds true.

a(,D=0)a(,D>0)

No devaluationdevaluation

s(k, )

CS may be welfare improving

a(,D=0)

a(,D>0)

Welfare worsening

Welfare worsening

Credit line

Stage 1 problem

• At stage 1, government’s objective to maximize

Results (no income ex ante)

• Commitment strengthening is beneficial only if the cutoff *(D) is reduced.

• Commitment strengthening is welfare worsening– if the government has a small credit line, or

– if the marginal cost of defense is negligible.

• Commitment strengthening is more likely to be beneficial if the prior probability of devaluation is low; otherwise, D must be large enough to be welfare improving.

Version 2: Structured notes

• Note assume that L( ,D)=DR( ).

• The commitment is made through the sale of insurance contract via market mechanism. As long as the government’s expected capital loss due to devaluation is correctly reflected in the price/premium of the contract, commitment strengthening (starting from D=0) is always beneficial.

• Simulation results: The optimal D* is increasing in b and d, and decreasing in v. Hence, the weaker the government, the greater desire it is to have commitment strengthening.

Simulation Studies

• A specification: Both and x conditional on are normally distributed;

• R( )=0.5-(1/)arctan so that R() is decreasing and 0 < R() < 1.

• c(,,D) = b2+d/(1-R()).

• a government is weaker, the larger b and d are, and the smaller v is

• Credit line is infinite

Optimal Level of Commitment

2.5

3.5

4.5

5.5

6 7 8 9 10

v

Opt

imal

D

b=5, d=1

b=4, d=1

b=3, d=1

b=5, d=1.5

b=4, d=1.5

b=3, d=1.5

Incomplete Information Game

• What happens if the market assessment of devaluation risk differs from the government’s own assessment?

• There are two types of government: strong and weak

• A modified game

– stage 0: two types of government: strong and weak. Nature decides the true type. The prior belief s is commonly known.

– Stage 1: each type i chooses Di, I=S,W. The price of option determined, and updated belief s

u(D) is formed.

– Stages 2 & 3: the same as in the previous game

Some results

• The weak type wants to mimic a strong type in order to reduce the aggressiveness of speculators. However, by mimicking the strong type, it also will receive a smaller premium.

• Assume that the belief about the type of government remains unchanged.

• The strong government will always find it beneficial to increase D (starting from D=0). The weak government’s welfare, however, may decrease with D when s is high.

• Given a low D, a government’s welfare is increasing in s.

government welfare vs commitment

2.5

2.6

2.7

2.8

2.9

3

3.1

commitment

wel

fare

ww(pis=0)

ww(pis=1)

ww(pis=0.5)

ww(pis=0.1)

Equilibrium

• Assumption: D[0,Dm]

• There always exists a pooling equilibrium where both types choose Dm.

• There exists a * such that a pooling equilibrium at D=0 exists if and only

if *. (The public holds a pessimistic view that only weak government will use commitment strengthening. Therefore, no government will choose D>0.)

• There does not exist naturally separating (semi-separating) equilibrium where the strong type chooses Ds>0 while the weak type chooses Dw=0.

• There does not exist unnaturally separating (semi-separating) equilibrium where the strong type chooses Ds=0 while the weak type chooses Dw>0.

Concluding Remarks

• There is a strong force of pooling!! In general, except for the pooling at D=0 case, strong government issuing structured notes is beneficial.

• Shall we adopt the policy of structured notes?

• Extensions:– allowing speculators to buy the structured notes

– does structured notes issuing facilitate coordination among speculators

– explicitly allowing large players in the model

– allowing residents to sell local currency