# Solution

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```					                                        PC4 corrected

Exercise #1
1. In a fixed exchange-rate regime, PPP involves a constant price level (Eq. (3)) and UIP
involves a constant interest rate (Eq. (4)). To keep the price level constant while
having a constant interest rate, money supply must remain constant (Eq. (1)). Hence,
official reserves must decrease if domestic credit increases, and vice versa (Eq. (2)).

In a free-floating regime, official reserves remain constant. Hence, money supply, the
price level and the nominal exchange rate move in parallel to (1-)dt, for a given
interest rate. Eq. (4) will then lead to a differential equation between the exchange rate
and its variation.

2. In a fixed exchange-rate regime, we have it = i* and pt = p* + e : : it and pt are
constant. Money supply must then remain constant: (1-)(d0 +  t) +  rt = mt = cst =
(1-)d0 +  r0. Hence:

1 
rt  r0           t


The (fixed) exchange rate is e such as:

e  pt  mt   i *  (1   )d 0  r0   i *

e   r0   i *

The regime is sustainable as long as rt > 0. Official reserves are depleted in:

      r0
T
1  

Reserve depletion comes sooner the higher credit growth . If the central bank can
borrow reserves from other central banks (cf. ERM, Chiang Mai) or on international
capital markets, the regime can last longer than T because official reserves can be
negative.

3. When reserves have been depleted (rt = 0), money supply shrinks to: mt =(1-)t.
From Eq. (1) and PPP, we get: mt = et –  it. Using UIP (Eq. (4)), we get the
differential equation:

et –  (i* +êt) = (1 - ) t

Or, equivalently:

et –  êt = i* + (1 - ) t                      (1)

1
The general solution of this first order linear difference equation is of the form:

et  a  bt  c exp( t /  )
c
Indeed, derivating the above equation with respect to time one gets: et  b 
ˆ                    exp( t /  ) ,

    c            
hence: et  et  et    b  exp(t /  )   et  b  c exp(t /  ) .
ˆ                          
                
Noting that the above postulated solution form also implies that: et  c exp( t /  )  a  bt , one
gets: et  et  a  bt  b  (a  b)  bt (2)
ˆ

Identifying terms between equations (1) and (2) one obtains b = (1-) and a =b + i*= (1-
) + i* A general solution to the differential equation is:

et =i* + (1 - ) +(1- )t+c exp(t/

A particular solution is obtained by assuming c = 0, namely:

~
et  a  bt   (1   )    i *  (1   ) t

This ensures that the solution is not explosive. The shadow exchange rate depreciates
continuously at rate (1 - ).

Another way of finding the solution is to derivate the following expression with respect to t:

et –  êt = i* + (1 - ) t

We get:            êt –  dêt/dt= (1-)

An obvious solution is êt = (1-). A specific solution then is:
~
e t = (1-) + i* + (1-)t
~                       ~ˆ                           ~     ˆ
~
The solution without second term is e t = exp(t/) [because et  exp(t /  ) /  , hence et  et ].
This solution can be discarded because it is explosive (at an exponential rate) when t goes to
infinity (speculative bubble). Hence, there is a single solution (the specific solution written
above).
~
The shadow exchange rate e t can be written as a function of the fixed exchange rate e :

~
et  e   (1   )   r0  (1   ) t  e   (1   )   r0  (1   )d t

~
The shadow exchange rate e t crosses the fixed exchange rate e at time T’ such as:

    r0
T'                T 
1  

2
T’ < T. At time T’, a speculator could make riskless profit by selling the domestic currency.
Remaining reserves would be instantaneously depleted. The amount of reserves at time T’ is:

1             1 
rT '  r0           T '               0
               

At time T’, reserves jump from rT’ to zero and the exchange rate starts a new path where it
depreciates at rate (1-).

At time T’, domestic credit is:

1 
d T '  d 0  T '            r0  


Relationship between domestic credit and the nominal exchange rate

e
~
et  e   (1   )   r0  (1   )d t

e =  r0 +  i*

d

dT’=r0/(1-) -

3
Exercise #2
1. The government has an incentive to devalue to curb the unemployment rate, but this will
entail a fixed cost c. It decides to devalue if the gain in terms of lower unemployment exceeds
the cost in terms of foregone reputation. It is a policy decision, not an obligation triggered by
reserve depletion. However, a speculative attack makes the defense of the peg more costly,
hence if is an incentive to devalue.

2. No expected devaluation: e = 0.

-     If the government does not devalue, its loss is: L0  ut 1 2
0

-     If the government devalue, its loss is: L0  ut 1   2  c
d

It devalues if Ld  L0 , i.e. if  < -with
0    0                             
c
 2 u t 1
d
This condition is more easily satisfied if the devaluation cost c is small and if inherited
unemployment is high. summarizes economic ‘fundamentals’.

3. Expected devaluation: e = .

-     If the government does not devalue, its loss is: L0  u1   2
d

-     If the government devalue, its loss is: Ld  u 1   c
d           2

It devalues if Ld  L0d , i.e. if   d .
d

4.
Devaluation whatever          Devaluation if expected                  No devaluation
expecations.         devaluation, otherwise no                 whatever expectations.
devaluation.


-                                          

4

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