REGIME SWITCHES AND THE FLEXIBLE PRICE MONETARY MODEL
Sarantis Kalyvitis
The analysis thus far has avoided the possibility that the behaviour of fundamentals might be
expected to change in the future.
In our previous examples if a variable was expected to grow it was expected to do so forever.
In some sense we focused only on long run behaviour and our earlier results can be thought of
as long run equilibrium.
Regime Switches and Short Run Dynamics
We examine the short run dynamics that occur when the behaviour of fundamentals is expected
to change.
Changes in the behaviour of fundamentals are referred to as regime switches.
1
How do we analyse a regime switch?
We have to make use of the following function:
Ae
t
 
λ
(1)
A is an arbitrary constant.
You do not need to know how this function is derived. But you do need to know how to find A.
Equation 1 can be defined so as to capture the effect of changes in the behaviour of future
fundamentals on the current exchange rate.
Recall that λ measures the effects of an expected depreciation on the current exchange rate.
Equation (1) also depends upon time. The date of the regime switch will matter.
The crucial aspect of the complementary function is the determination of A.
2
The solution method in general
Analysis of a regime switch proceeds as follows.
i. Since the behaviour of fundamentals changes there will exist at least two exchange rates that
correspond to the previous definition of long run equilibrium. Solve for these in the manner
already outlined.
ii. However the solution prior to the regime switch may no longer be valid. To allow for this
add equation (1) above to your "long run" equilibrium prior to the switch.
iii. Use the No-Arbitrage condition to solve for A. This requires that at the instant of the switch
the solution you have from the previous step is equal to the long equilibrium after the switch.
iv. Solve the last step for A. Substitute this back into step 2 graph and interpret.
3
An Example
Assumption: all the fundamentals are constant except for the money supply defined by:
m=m
t <T
(1)
m t = m + µt
t ≥T
(2)
Two different "long run" equilibrium solutions:
Up until time T we have:
st = m0 − φy − p ∗ + λi ∗
(3)
At and after time T:
st = + λµ − φy − p∗ + λi ∗ + m0 + µ(t − T )
(4)
Since this applies from t ≥ T at itself µ (T − T ) = 0 .
⇒ The money supply only starts to grow thereafter.
Question: Can the exchange rate follow (3) until T and then jump to (4)?
4
The key thing to note is that at T such a path entails a discrete jump depreciation of λµ .
Holders of domestic currency at T suffer capital losses equal to λµ × units of domestic currency
held. Graphically the path of the exchange rate is:
µ
st
λµ
T
Time
Can this be the correct solution?
If the "regime switch" is unpredictable then the solution is OK. Investors undergo a discrete
but unexpected capital loss.
If the regime switch is anticipated so too is the capital loss and the previous solution is not
sensible.
5
The No-Arbitrage Condition
Suppose that at time T, the exchange rate is not equal to the equilibrium after the regime switch.
Specifically at T the long run equilibrium value is,
1 Euro = 1 USD
But suppose in the instant prior to T the exchange rate is,
1.5 Euro = 1 USD
Anyone who buys Euro makes a profit.
Just prior to the switch 1USD converts to 1.5 Euro but at the time of the switch the exchange rate
jump appreciates so that I can re-convert this to 1Euro.
But buying the Euro causes it to appreciate before the switch!
Equivalently, if instead: 0.5 Euro = 1 USD
⇒ Anyone who sells Euro makes a profit.
Why? I can convert 0.5 Euro to 1USD to the switch but I can re-convert this to 1 Euro after the
switch. But selling the Euro causes it to depreciate before the switch.
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Conclusion: the exchange rate must equal the new equilibrium value at T. Graphically:
st
Jump Appreciation
Jump depreciation
T
Time
The no-arbitrage condition states that rational economic agents will eliminate all predictable
profits.
Alternatively the no-arbitrage condition effectively rules out such predictable jumps in the
path of the exchange rate.
7
Mathematical derivation
1
t
Add Ae λ to the long run solution prior to the regime switch.
st = m0 − φy − p + λi + Ae
∗
∗
1
λ
t
(5)
Use the no-arbitrage condition to define a boundary condition and solve for A.
At T:
sT [ pre − switch ] = sT [ post − switch ]
Put this all together we have,
1
∗
∗
m0 − φy − p + λi +
Aeλ
T
= λµ + m0 − φy − p∗ + λi ∗ + µ(T − T )
8
Cancellation leaves:
1
Ae λ
Since
T
= λµ
e a e −a = 1 we have that,
A = λµe
1
− T
λ
(6)
Substitute this back into (5) yields the solution for the exchange rate prior to the regime switch.
st = m0 − φy − p + λi + λµe
∗
∗
1
λ
[ t −T ]
(7)
For t ≤ T .
For t
≥T
(4) applies.
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Interpretation
Along our previous "long-run" equilibrium path speculators realise that at time T anyone holding
the domestic currency will suffer a capital loss at time T.
They will therefore be less willing to hold assets that are denominated in domestic currency. The
domestic interest rate (UIP) will rise and money demand will fall. The fall in money demand
depreciates the exchange rate prior to the regime switch.
The extent of this depends on how close the switch is and how sensitive money demand is to the
rising interest rate. Since t − T ≤ 0 if we examine final term in (7) we find that:
(
)
(i) when we do not take into account the regime switch, (7) is identical to the long-run
equilibrium when:
t − T → −∞ or λ = 0
(ii) relationship (7) is identical to the long-run equilibrium that applies after the regime switch
when either/or, t = T
or
λ →∞
or
t =T
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Possible path defined by (7)
st
T
Time
Results
Unpredictable events cause exchange rate jumps
The current exchange rate is affected by predictable changes in the behaviour of fundamentals
that occur in the future.
The strength of this affect increases the bigger is the change, the more sensitive money
demand is to the interest rate and the closer is the switch.
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