Sticky-Price Exchange Rate Models
Obstfeld and Rogo¤, Chapter 9
1
Empirical Motivation for Sticky Prices
Variability of the real exchange rate EP
P
the nominal exchange rate (E )
is very similar to variability of
The real exchange rate is much more volatile over ‡oating rate periods
Engel demonstrated that the relative price of similar goods between the
US and Canada is much more volatile than the relative price of di¤erent
goods within a country
Why would a country deliberately depreciate its currency to shift demand
toward their goods if prices immediately adjusted, eliminating the e¤ect?
1.1
Mundell-Fleming-Dornbusch model
1.2
Equations of the model
Small letters denote logarithms with the exception of the interest rate
Uncovered interest rate parity with …xed foreign interest rate
it+1 = i + et+1
et
Money demand
mt
pt =
it+1 + yt
Real exchange rate ‡uctuates in the short run and has a …xed long-run
value (…xed foreign price level at unity implying p = 0)
q = qt = et + p
pt
Demand is increasing in the real exchange rate relative to its long-run value
ytd = y + (et + p
ytd
y = (qt
pt
q)
q)
Phillips Curve describing price adjustment
pt+1
pt =
ytd
y + p~t+1
p~t
– prices rise in response to excess demand and to in‡ation which would
occur in full equilibrium
– p~t is de…ned as the price level that would prevail if the goods market
cleared
p~t
et + p
q
1.3
Solution of the Model with Phase Diagrams
Write the model as two di¤erence equations in the real and nominal exchange rates, qt and et
Di¤erence equation in qt
– Substitute excess demand into the Phillips Curve
pt+1
pt =
(qt
q ) + p~t+1
p~t
– Use equilibrium price to substitute for equilibrium in‡ation
p~t+1
p~t = et+1
et
– Substitute into the in‡ation equation
pt+1
pt =
(qt
q ) + et+1
et
– Solve for the rate of change of the real exchange rate
pt+1
et+1
qt+1
(pt
et) =
qt =
(qt
(qt
q)
q)
Di¤erence equation in et
– Money demand equals money supply
mt
pt =
it+1 + yt
– Simplify by setting
y=i =0
Substitute de…nition of real exchange rate for price
pt = et
qt
Interest rate parity for the interest rate
it = et+1
et
And aggregate demand for output
yt = (qt
q)
to yield money demand as
mt
et + qt =
(et+1
et) +
(qt
q)
– Solve for the rate of change of the nominal exchange rate
et+1
et =
1
[ mt + et
qt +
(qt
q )]
Phase diagram with e on vertical axis and q on horizontal
qt+1 = qt+1
et+1 = et+1
–
et =
1
qt =
[ mt + et
(qt
q)
qt +
(qt
q )]
qt+1 = 0 requires
qt = q
qt+1 = 0 is vertical at qt = q
When qt increases,
qt+1 falls
@ qt+1
=
@qt
<0
implying that arrows of motion point towards
qt+1 = 0
–
et+1 = 0 requires
et = mt + qt
Slope of
(qt
q)
et+1 = 0 is 1
Assume that money is constant and assume that output is not too
responsive to relative price so that
1
such that the slope of
When et increases,
>0
et+1 = 0 is positive
et+1 increases
@ et+1
1
= >0
@et
implying that arrows of motion point away from
et+1 = 0
– System has an upward sloping saddlepath and a steady state with
e=m+q
Comparative Statics: Unanticipated increase in mt
–
et+1 = 0 curve shifts vertically upwards by the increase in the money
supply
– new saddlepath shifts upwards to go through the new steady state
– system must jump to the saddlepath immediately or dynamics will not
take it to the steady state
– price cannot jump, but the exchange rate can
– an increase in et increases et and qt by equal quantities, moving the
system along a 45 degree line from the initial equlibrium toward the
saddlepath
– the exchange rate overshoots its long-run equilbrium value, and therefore begins falling after the initial increase
Intuition for overshooting
– money market equilibrium
mt
pt =
it+1 + yt
– increase in the money supply with prices …xed requires an increase in
money demand
– if output is not too responsive, get increase in money demand with a
fall in the nominal interest rate
– interest rate parity implies
it+1 = i + et+1
et
therefore, for the nominal interest rate to fall et must rise by more
than et+1 rises, that is, overshooting
– if output is more responsive, the nominal interest rate could rise and
overshooting would not occur
– implies that exchange rates are forecastable
1.4
Analytical Solution
Solve equation for real exchange rate …rst
qt+1
qt+1
qt =
q = (1
(qt
q)
) (qt
q)
– Solve forward
qt+2
q = (1
) (qt+1
qs = q + (1
)2 (qt
q ) = (1
)s t (qt
q)
q)
Solve equation for nominal exchange rate
et =
1+
et+1 +
1
1+
[mt + qt (1
)+
q]
– Subtract q
et
q=
1+
(et+1
q) +
1
1+
[mt + (qt
q ) (1
)]
– Solve forward
et+1
q=
1+
(et+2
q) +
1
1+
[mt+1 + (qt+1
q ) (1
)]
– Substitute for (et+1
et
q =
1+
+
1+
1
+
1+
=
1+
1
+
1+
1
+
1+
q)
"
1+
"
1
1+
[mt+1 + (qt+1
(et+2
[mt + (qt
"
1+
q)
(et+2
[mt + (qt
!2
#
q ) (1
q ) (1
)]
#
)]
#
)]
q)
q ) (1
[mt+1 + (qt+1
)]
q ) (1
et
1
q =
1+
1
X
s=t 1 +
+ lim
T !1
1+
!s t
!T
[ms + (qs
et+T
q ) (1
)]
q
– Setting the limit equal to zero places the system on the saddlepath by
ruling out speculative bubbles
– Set mt = m and substitute for qs
for the saddlepath
q from above yields the equation
et
1
q = m+
1+
(1
= m+
1+
(1
= m+
1+
–
!s t
h
1
X
s=t 1 +
1
X
)
(qt q )
s=t
)
(qt
s t
(1
)
(1
1+
)
(qt
q ) (1
!s t
i
)
q)
Saddlepath relates the value of the current nominal exchange rate
to the current real exchange rate
Substitute for (qt
es
q ) to solve for the exchange rate going forward
q =m+
(1
1+
)
(1
)s t (qt
q)
Solve for jumps in et and qt due to the increase in m
det
(1
=1+
dm
1+
) dqt
dm
– recognize that
det
dqt
=
dm
dm
– substitute
det
(1
=1+
dm
1+
det
1+
=
dm
+
– yields overshooting if
<1
) det
dm
For e¤ect on interest rate need e¤ect on et+1
det+1
(1
=1+
dm
) (1
1+
) dqt
dm
– Substitute
dqt
1+
det
=
=
dm
dm
+
det+1
(1
=1+
dm
) (1
+
)
– Subtracting
det+1
dm
det
(1
) (1
)
= 1+
dm
+
1 + (1
) (1
=
+
(1
)
<0
=
+
(1 +
)
)
Consider model with monetary growth
ms = mt +
(s
t)
– Substitute into solution for exchange rate
1
et q =
1+
1
1+
1
X
s=t 1 +
1
X
s=t 1 +
et
!s t
[ mt +
!s t
q = mt +
[m +
+
(s
t) + (qs
(s
t)] = m +
(1
1+
)
(qt
q)
q ) (1
)]
– compute e¤ect on et of increase in money growth
det
=
d
(1
+
1+
det
=
d
(1 +
+
) det
d
)
– solve generally for es
es
– substitute for qs
es
q = mt +
q = ms +
(1
+
1+
)
(1
1+
)
(qs
q)
q and for ms
(s
t) +
+
(1
)s t (qt
q)
– solve for et+1
et+1
q = mt +
+
+
det+1
= 1+
d
+
= 1+
+
(1
1+
(1
)
(1
) (1
1+
(1
) (1
+
) (qt
) det
d
)
q)
– subtracting e¤ect of
det+1
d
on future and current exchange rates
det
= 1+
d
= 1+
= 1+
=
=
+
[
(1
) (1
+
1
+
+
[
1]
+
+
(1 +
+
(1
+
(1
+
)
)
]
+
) (1
+
)
(1 +
+
(1
)
) (1
+
)
>0
– implying that the interest rate rises when money growth increases
)
1.5
Real shocks
Increase in q
e=m+q
Exchange rate adjusts immediately to long-run equilibrium value
– No need for price to adjust so no delay and no overshooting
1.6
Correlations between money shocks, real interest rates
and nominal interest rates
Permanent increase in money supply
– Future exchange rate (et+1) rises less than current exchange rate (et)
implying the nominal interest rate falls from IRP
it = i + et+1
et
Increase in money growth rate
– Future exchange rate rises more than current exchange rate implying
the nominal interest rate rises
– Fisher relation
Real interest rate equals the nominal rate plus the rate of depreciation of
the real exchange rate
it
(pt+1
pt) = i + et+1
= i + qt+1
et
(pt+1
pt )
qt
– Solution for qt+1
qt+1
– Solving for qt+1
q = (1
) (qt
q)
qt
qt+1
qt =
(qt
q)
– Real interest rate is high when the real exchange rate is low
it
(pt+1
pt ) = i
(qt
q)
Empirical evidence
– Strong dollar in 1980’s when US nominal interest rates were high
– Countries which adopt tight monetary policy, raising nominal interest
rates, experience an increase in real interest rates and currency appreciation
– Empirically, do not get tight relationship between the real exchange
rate and real interest rate over short horizons
– Monetary model of the nominal exchange rate
Meese and Rogo¤ show that a random walk model predicts better
Asset price prediction is generally di¢ cult and exchange rates are
asset prices
More recent empirical work is more supportive of monetary model as
a long-run equilibrium relationship
2
2.1
Gold Standard
Assumptions
Two-country model
Money supply in each country is proportional to gold
Mt = ZtGt
Mt =
–
is money multiplier
Zt Gt
– Zt is …xed domestic-currency price of gold
– Gt is quantity of gold
– superscript * denotes foreign country
World supply of gold is …xed
Gt + Gt = G
2.2
Prices satisfy PPP
Purchase red wine from France under the gold standard
– sell $1 for Z1 units of gold
sell Z1 units of gold for ZZ francs
trade ZZ francs for P1 units of wine
Alternatively, purchase identical quality red wine from California
– sell $1 for P1 units of wine
Assumption of identical wine, arbitrage requires that the $1 purchase the
same quantities of wine
1
Z 1
=
P
Z P
Z
1
P
=
=
P
Z
E
– Relative gold prices determine relative price levels and the nominal
exchange rate
2.3
Distribution of gold (relative money supplies) is endogenous
Money demand for each country in logs
mt
pt = yt
it+1
mt
pt = yt
it+1
Subtract foreign from domestic
mt
mt
(pt
pt ) = (yt
yt )
(it+1
it+1)
– money di¤erence
mt
mt =
+ zt
zt + gt
gt
– PPP
pt
pt = z t
zt
– IRP with …xed exchange rates
it+1
it+1 = 0
+ gt
gt = (yt
– Substituting
yt )
Solve for distribution of gold
gt
gt = (yt
yt )
(
)
– money market equilibrium determines the world distribution of gold and
therefore the world distribution of money supplies
– gold ‡ows into countries with relative higher money demand (higher
income) and with a relatively lower gold multiplier ( )
need more gold to meet money demand to make up for the smaller
multiplier
2.4
Great Depression
Began in the US with bank failures reducing money multiplier,
– Gold ‡ows into US and out of the rest of the world
– Tight money in US, due to bank failures, was transmitted to ROW
through gold standard
Hypothesis: if money has real e¤ects, then countries which abandoned the
gold standard more quickly should have recovered from the Great Depression more quickly
– Veri…ed
– Considered one of the major pieces of evidence for real e¤ects of money
3
Choice between Fixed and Flexible Exchange
Rates
3.1
Real E¤ects of Money
If money has no real e¤ects, both systems are equivalent
Choice can depend on why and how money has real e¤ects
3.2
Keynesian Sticky Prices
Model
– Money demand with an error
mt
pt = yt
it+1 + t
– Demand for goods with an error
ytd = y + (et + p
– Let prices be …xed in the short run
Fixed exchange rates
pt
qt)
it+1 + vt
– Money demand disturbances do not a¤ect interest rate and are not
transmitted to goods market
Money demand disturbances only a¤ect endogenous quantity of money
– Aggregate demand disturbances directly a¤ect aggregate demand without any o¤set from the exchange rate
Flexible exchange rates
– Increase in aggregate demand appreciates the exchange rate (et falls)
o¤setting the e¤ect of the disturbance on output
– Increase in money demand a¤ects the interest rate, transmitting the
e¤ect of the disturbance to the goods market.
Choice between …xed and ‡exible exchange rates depends on where most
disturbances originate
– money market ) …xed rates are better
– goods market ) ‡exible rates are better
– optimal ‡exibility could be based on fraction of disturbances originating
in each market
3.3
Other Considerations
Flexible exchange rates are volatile and imply volatile real exchange rates
– Volatility could increase uncertainty, thereby reducing trade, and the
gains from trade
– Countries which borrow in foreign currency could see large ‡uctuations
in indebtedness
Fixed exchange rates require that the pegging country give up independent
monetary policy
– Currency crisis when country needs to regain independent monetary
policy
Country could have trouble maintaining low in‡ation under ‡exible exchange rates and independent monetary policy.
– Fixed rates might provide discipline
4
4.1
Optimum Currency Areas (Mundell)
Monetary Union
Adopt a common currency
Common monetary policy
Equivalently, irrevocably …xed exchange rates
4.2
Bene…ts
Reduction in transactions costs for trades within the monetary union
– Increased trade and increased gains from trade
– Gains from real and …nancial integration
Reduction in volatility of real exchange rates due to …xed nominal exchange
rate
Credibility for a monetary policy with low in‡ation
Eliminate currency crises and associated speculation (really?)
4.3
Costs
Each country loses independent monetary policy
– Loss is more serious
the greater the real e¤ects of money perhaps due to sticky prices
the less mobile is labor across countries
– Consider a switch in demand from country A to country B
Labor mobility
If labor is mobile, labor moves from country A to country B
If not, and if prices are sticky, employment falls in country A and
rises in country B
Price ‡exibility
If prices are ‡exible, prices rise in country A relative to country B
and output and employment are unchanged
Monetary policy can substitute for labor mobility and price ‡exibility
Increase in money supply in country A raises demand
Decrease in money supply in country B lowers demand
– Southern European countries in 2012
Stimulate economies if they could use expansionary monetary policy
Depreciate independent currencies and switch demand toward their
goods
Each country loses right to determine its own reliance on seigniorage as a
method of …nancing government spending
– Each receives a share of seigniorage determined by the union monetary
authority
– For countries with previously high in‡ation, represents a reduction in
revenue, to which they must adjust by raising other taxes or reducing
spending
Financial crisis in a single country shared by others
5
5.1
Second Generation Currency Crises
Models with Multiple Equilibria
One equilibrium in which exchange rate remains …xed
Another in which …xed exchange rate fails
System which prevails depends on beliefs by agents
Role for speculators’ beliefs to determine whether …xed rate succeeds or
fails
Monetary expansion increases domestic in‡ation and output
5.2
Model
Government loss function
L = (yt
y
k )2 +
2
+ c ( t)
– Government wants output higher than full employment (y ) by k
– Government cares about deviations of output from goal and about
in‡ation
– Loss is quadratic in deviations
– Political cost of abandoning the …xed exchange rate
If country depreciates, c ( t) = c
If country appreciates, c ( t) = c~
– In‡ation is initially zero under …xed rates
Output depends on Lucas in‡ation surprises with a disturbance zt
yt = y + t
e
t
zt
5.3
Monetary Policy
One-shot game
– Government chooses t after observing et and zt
Solve for minimum loss conditional on abandoning …xed exchange rate and
choosing t to minimize loss
– Loss with output substituted
L=( t
e
t
zt
k)2 +
2
t
+ c ( t)
– Minimize with respect to t
@L
= 2( t
@ t
e
t
t=
k+
zt
e
t
t)
=0
+ zt + k
1+
– When et = 0; consistent with con…dence in the …xed exchange rate,
and zt takes on its mean value of zero, the incentive to in‡ate depends
on a positive value for k
– Minimum loss under ‡exible exchange rates is
Lf lex
=
=
e
t
1+
+ zt + k
1+
e
t
zt
k
!2
( et + zt + k)2 + c ( t)
+
e
t
+ zt + k
1+
!2
+ c ( t)
Loss under …xed exchange rates sets t = 0
Lf ix = ( et + zt + k)2
Compare the loss under maintaining the …xed rate with the loss from abandoning
Lf ix
Lf lex = ( et + zt + k)2
=
1
1+
"
1+
( et + zt + k)2
#
( et + zt + k)2 + c ( t)
c ( t)
– If the cost of abandoning the …xed rate were zero, the country would
generally choose ‡exible rates due to k > 0
– Incentive to abandon the …xed rate depends on
Preferences towards in‡ation,
The more tolerant the country is toward in‡ation, the lower the ;
and the more willing it is to abandon the …xed rate
Economic fundamentals, zt
Demand is low when zt is high
Cost of maintaining the …xed exchange rate is higher in recession
Expectations of in‡ation depend on the cost of maintaining the …xed
rate
When demand is low, cost are higher
Therefore, et will be high when zt is high
Multiple equilibria
– Solve for shock values which create an exchange rate crisis
Raise exchange rate (devalue) if
1
1+
z
( et + zt + k)2 > c
z = [c (1 +
1
)] 2
k
e
Reduce exchange rate (revalue) if
1
1+
z
z~ =
( et + zt + k)2 > c~
[~
c (1 +
1
)] 2
k
e
Retain …xed rate if
z 2 [~
z; z]
Monetary authorities defend the …xed exchange rate for all but large
(in absolute value) shocks
– Expectations must be rational
E = E [ j z < z~] Pr (z < z~) + E [ j z > z ] Pr (z > z )
Assume a uniform distribution for z on [ Z; Z ]
In‡ation is given by
t=
e
t
+ zt + k
1+
Expected in‡ation
E =
e
t
+ k E [z j z < z~] Pr (z < z~) + E [z j z > z ] Pr (z > z )
+
1+
1+
very non-linear function and can equal e at multiple values
multiple equilibria possible only for extreme values of z
– when z takes on moderate values, do not get possiblity of multiple
equilibria