Download DMF model and exchange rate overshooting

DMF model and exchange rate
overshooting
Lecture 1, MSc Open Economy
Macroeconomics, Birmingham, Autumn
2015
Tony Yates
Motivation
• Dornbusch (1976) writing shortly after demise
(1973) of fixed exchange rate system agreed at
Bretton Woods
• Era of exchange rate volatility [that continued
of course]; fluctuations seemed to exceed
what was sensible from fundamentals
• ‘Overshooting’ provided the answer.
RER=changes in real purchasing power. Note less volatile in Gold Standard and
Bretton-Woods periods.
Source: ‘Real exchange rate volatility….’, IMF wp, 1998, by Hong Liang.
Source: ‘Real exchange rate volatility….’, IMF wp, 1998, by Hong Liang
The DMF model equations
it1 i e t1 e t
Uncovered interest parity
m t p t it1 y t
Money demand
q e p  p
y dt y 
e t p  p t q
,  0
Defn of the real exchange rate
Aggregate demand. q_bar is RER
when ag demand=natural rate.


p t1 p t 
y dt y
p t1 p t 
Open economy Phillips Curve

p t e t p t q t
p_tilda is price that would emerge if
output=natural rate.
Strategy to derive and study the
overshooting result
• Algebraic:
– Use model equations to form two difference
equations, in q and e.
– Solve them to find their steady states in terms of
exogenous money stock.
– Show that when we change money, e movement
initially exceeds eventual new steady state.
• ‘Intuitive’ [in quotes because even this is quite
mathematical.]
Getting our difference equation in q


p t1 p t 
et1 p t1 q t1 
e t p t q t 
p t1 p t 
y dt ye t1 e t
q t1 q t1 q t 
q t q
Defn of the change in the eq’m
price
Philips curve with this defn
substituted in, noting that p_star
and rer eq’m ,q_bar, are constant
Philips curve now with aggregate
demand and defn of rer
substituted in.
Getting our difference equation in e
p  y i 0
Simplifying asssumptions…
m t p t 
e t1 e t 
q t q
…substituted into money demand eq
…using defn of real er.
m t e t q t 
e t1 e t 
q t q
e t1  et 

1
qt


q
mt

Rearranged to give us our difference
equation in e
Now we can solve these two difference equations to get steady state values of e
for different values of exogenous money, m. But let’s try to see how overshooting
comes about using some economic intuition.
Intuition for overshooting: proof by
contradiction
• Let’s imagine an increase in the money stock
from m to m’
• Then suppose that there is no overshooting, so
that the short run value for e [while prices are
fixed] is the same as the long run value for e.
• Then study our model equations and see that this
assumption violates the equation that imposes
UIP.
• Hence overshooting must happen.
Overshooting intuition: 1
m t p t it1 y t
Money demand equation tells us REAL
balances will rise by m-m’ if money stock rises
from m to m’, since prices are fixed.

m m
From the aggregate demand curve below,
this is how much m-m’ increases aggregate
demand. ..
..because in LR we know e rises 1 for 1
with m, so we substitute m-m’ for e in
here to get the short run change in output.
Noting all other terms in the equation are
fixed.
y dt y 
e t p  p t q
Overshooting intuition: 2
m t p t it1 y t

m m
Take the money demand curve again, and
substitute in the change in demand we just
worked out, for y.
And this is what you get for the change in
money demand.
Because phi*delta<1, this means that money
demand is rising less than money supply,
which rises by m’-m.
In order for the money market to clear, this
means that the interest rate has to fall next
period in order to clear the market for
money. Yet this contradicts our model….
Overshooting intuition: 3
it1 i e t1 e t
Take the UIP condition, and rearrange it.
it1 i e t1 e t
If i_t+1 falls, this means that the LHS is –ve. Why?
Because i) we assume i=i_star in a world of stable
exchange rates. And i_star is fixed.
This means the RHS is –ive.
That means that e_t+1<e_t, which contradicts what
we said about e jumping to its new steady state
straight away.
If we work back from here, then we need that e
initially jumps HIGHER than its steady state, then
falls.
Assumptions required to get
overshooting
• Phi*delta<1 [?]
• Prices sticky for one period [ok]
• Perfect foresight rational expectations [in fin
mkts, maybe, but prob not]
• Particular conjectured functional forms for
money demand [this one ok], aggregate demand
[contestable], Phillips Curve [likewise].
• No microfoundations to assess whether these are
possible worlds.
• Empirically controversial model components.
Word on next lecture
• Controversial model with clear hypothesis:
money supply changes cause volatility in e.
• We will look at time series VAR techniques for
testing this and related models.
• Relies on deducing monetary shocks from long
run neutrality in how they affect real things.
• Then measuring contribution of monetary
shocks to everything.
Deriving overshooting analytically
• Solve our two difference equations, essentially
by repeated forward substitution.
• Substitute the solution for q into the solution
for e, which will be in terms of m.
• Then study dynamics of e in response to
change from m to m’, and prove conditions
under which e overshoots.
The two difference equations for e and
q
q t1 q 
1 

q t q
et
e t1   

1
qt

q
mt
 
Solve e equation first, by forward
substitution
et
e t1   


1
qt

q
mt
 
First, unpack the
difference term…..
1
mt
e t q  1
e t1 q1 
q t q1


1 
m

e t1 q

q t q t
1 
1 
1 


1 
1 
m



e t2 q

q t1 q t1 

q t q m t
1  1 
1 
1 
1 
1 
e t q 
Then use the equation for e_t-q to substitute out for future terms. Here’s one
step, but we do it over and over until we spot the pattern…..
Solving forward the difference
equation for e_t

e t q 
1
1



1st m s
1
1


st
 st

qs
1

q
st
What we get by repeatedly substituting out for the e_t+n terms…
e t q m
1
1


 st

qs
1

q
st
Equation for e_t simplifies when we impose constant money supply m_bar.
Solving for e_t in terms of m and q.
q s q 
1 st 
q t q
e t q m
1
1 
qt

q
st
1
e t q m 1
q t q
Solution for q_t also got by repeated
forward substitution.
 st
st

1




1

Substitute this expression
for q_t into equation for
e_t, and then simplify….
….and we get this.
Deriving the overshooting result
• Now we have an equation for e solely in terms
of q and m.
• q is constant.
• So we can substitute in different values for m
and see what happens.
• Specifically, we derive the analytical condition
for overshooting.
• The lecture notes do this for you, but, next
slides recap and take you through it.
Deriving overshooting from our e
equation
e t m q
e m q
q 0 e 0 m
Initial steady state for e given by this [note p=m_bar]
New steady state, with higher money stock, rises to
e’.
Initially prices don’t move, so q_0 given by this
expression. [we substitute p_0=m_bar into the defn
of the real exchange rate…]
Finally, the condition for overshooting!
1
q 0 m m q 1
q 0 q
This is our solved difference eq for e, but having substituted out for e_0
on the right hand side.
Now we solve for q_0, then, given that solution, we find out what e_0 is,
and look to see what conditions make e_0>e’, the new long run value for
e with the higher money stock.
e 0 m 
m
1 
1

m

q
This is what we get, after a
few basic algebra steps.
And a few more steps yield this condition for overshooting.
We are done!! Not so hard, really!
Recap
• DMF model consisted of:
– Money demand relation
– Open economy Phillips Curve
– Aggregate demand equation
– Assumption of perfect foresight RE
Two strategies for deriving the
overshooting result
• First was to prove by contradiction
– We supposed that the short and long run e were
the same after an increase of m to m’
– And we realised this contradicted the UIP
condition
• Second was to solve algebraically, forming
difference equations in e and q, solving them
forward by repeated substitution.
Overshooting: so what? Why do we
care?
• DMF provided a rational for exchange rate
volatility seen in floating ex rate regimes.
• Suggests a route whereby we can get
significant welfare costs from unwarranted
monetary policy shocks.
• Shows the strengths and weaknesses of oldstyle macroeconomic RE modelling.
• S: powerful results. W: where do these
curves come from?
Next lecture: time series analysis and
overshooting
• DMF pointed the way to famous time series
VAR work on overshooting and the nonneutrality of monetary shocks on the real
exchange rate and output.
• Clarida and Gali.
• This is the [hard!] topic for the next lecture.
• Enjoy the exercises [!], and remember, that
they are mostly much harder than the
examined material.