Download Fixed Exchange Rate Without Interest Parity

An Interest Rate Defense of a
Fixed Exchange Rate
From Flood and Jeanne
1/
Mt
  exp( rt )
St

S
Bt
*
t
rt  r   
St
St
(1)
(2)
M t  Dt  S t R
(3)
R 0
(4)
*
t
*
t
M = base-money, S = exchange rate, R* = international reserves,
B = world-wide private holding of domestic government debt,
r = domestic-currency interest rate, r* = foreign-currency
interest rate, D = CB domestic credit, s = lnS, m= lnM, b = lnB
/2
*
*

N t  rt ( N t  Dt )  r S t Rt  S t
N t  Bt  Dt
(5)
(6)
N =outstanding domestic nominal government bonds, B= worldwide private holding of domestic government debt, D= Central Bank
domestic credit, r(N-D) = net interest payment of consolidated
government, r*SR* = interest payments on international reserves,
  real level of taxes.
/3 An anticipated attack takes place when the shadow exchange
rate equals the fixed exchange rate. Under assumption that the
domestic credit grows at the same rate, g, before and after the
collapse,
if   0,
the collapse time T is the solution to
Money Supply
Money Demand
D0 exp( gT )

  exp(  (r  g )) (7)
S
DT
g = domestic credit growth
= rate of depreciation
/4
if   0,
Dynamics of the economy becomes more complicated,
but the results still hold for  small enough.
/5
Pegging the Interest Rate
Assume that interest rate is constant before and after
the attack but may jump at the time of the attack:
rt  r , t  T
rt  r , t  T
/6
Pre-Collapse Regime
The monetary authority intervenes in the domestic bond
market and in the foreign exchange market to fix the
exchange rate, and prices.
Asset market equilibrium is:
rt  r  
*
S t
0
St
Bt
_
S

N t  M t  SR
*
t
_
,
(8)
S
M t = money demand is
M  S  exp(  r )
constant , since r is
determined by the central
bank at the level: r  r , t  T .
Foreign reserves move to clear market.
t
/7
Pre-Collapse Debt Path
*

Nt  r ( Nt  M )  (r  r*)S Rt  S
from (8),
rt  r *  
Bt
_
S
SR 
*
t

N t  M t  S Rt*
_
,
S
( r  r*)S

N t  r * ( N t   )
 ( Nt  M )
/8

Nt  ( N0   ) exp( r * t )  
(9)
where
 (r  r*)  

 
  exp( r )  S
r *


(10)
/9
The variable  is strictly increasing with T and strictly decreasing
in . Hence, given N t , the level of nominal debt at any given
time t > 0, before the speculative attack, is strictly increasing in T and
strictly decreasing in the real tax receipts.
t  (0, T ),
N t
0
r
N t
0

(11)
Post-Collapse Regime
/10
Following the collapse, the economy settles immediately into
a real steady state, with constant real level of government debt,
N/S. Because, the primary surplus is constant and seignorage
is determined by the post-attack interest rate, the only
remaining balancing variable is the level of debt,
n  N S , which must satisfy:
n(r *  (n  m))  r m  
(12)
where
m   exp( r )
denotes the real quantity of money in the post-collapse regime.
/11
Equation (12) is a second-order polynomial equation in n,
whose unique positive root is a function of the interest rate
and fiscal receipts:
1  (r * m)
n   ( r ,  )  
2

(r * m) 4(r m   ) 



2



2
(13)
/12
If   0,
the interest parity disappear and  () takes the simple form:
r  exp( r )  
 (r ,  ) 
,
r*
Which is increasing in the level of tax receipts and in the postcollapse interest rate, when the economy is on the increasing
branch of the seignorage Laffer curve.
If   0,
/13
it remains true that

0

(14)

1
 0 iff r   
r

(15)
and
/14
The Collapse
The shadow exchange rate at time t is the value of the exchange
rate such that the exchange rate neither depreciate nor appreciate,
And the value of the outstanding domestic nominal government bonds
is unchanged
Nt
~   (r , )
S
~
S is proportional to the state variable N t .
The currency peg collapses if and when the shadow exchange rate
crosses the fixed peg, i.e., if there is a time T such that:
( N 0   )e
S
r *T

  (r ,  )
(16)
/15
The collapse is inevitable if nominal debt explodes, i.e., if
N 0    0.
In this case the time of collapse is given by
equation (16).
(1) It is always possible to raise the level of taxes such that
N 0    0, which prevents public debt from expanding.
(2) The time of the collapse is decreasing with the precollapse interest rate and increasing with the post-collapse
interest rate.
T
0
r
T
0
r
/16
Raising the interest rate before the attack and lowering it after the
attack hastens the collapse.
Intuition:
Raising the interest rate before the attack worsens the fiscal
imbalance, since it amounts to financing a stock of low-interestrate-bearing foreign assets by borrowing at a high interest rate. A
higher nominal interest rate is then reflected one-to-one in a higher
real rate of interest since the exchange rate and prices are fixed.
Higher interest rates after the collapse are associated with higher
seignorage revenues, allowing the government to service a larger
real debt in the steady state. After the collapse an increase in the
nominal rate of interest improves the fiscal situation by raising
seignorage revenues.