DMF model and exchange rate overshooting

Lecture 1, MSc Open Economy Macroeconomics, Birmingham, Autumn

2015Tony 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

Motivation and context - recap

• Nominal exchange rate very volatile• Leads to volatility in the real exchange rate• That real exchange rate volatility is considered

harmful and costly.

The DMF model equationsit 1 i et 1 et

m t p t it 1 yt

q e p p

ytd y et p pt q, 0

p t 1 p t ytd y p t 1 p t

Uncovered interest parity

Money demand

Defn of the real exchange rate

Aggregate demand. q_bar is RER when ag demand=natural rate.

Open economy Phillips Curve

p_tilda is price that would emerge if output=natural rate.

p t et pt q t

Explaining the model equations

• Let’s take each equation and explain what it says and where it comes from.

• We’ll also go through the equilibrium concepts, the letters with ‘bars’ over them.

Explaining money demand

m t p t it 1 yt

LHS, m-p= real balances. The amount of goods you can buy with the money in your pocket.

These said to i) fall as the opportunity cost of holding them [the interest rate on a bank account, say] rises; ii) increase as expenditure increases.

This is the same as the demand for any other good that you might eat, which has a complement. So if the price goes up, you eat less; and if you expect to eat more of the complementary good, you will eat more of this one.

Explaining UIP

• UIP=uncovered interest parity

it 1 i et 1 et

A gap between interest rates at home and abroad must be compensated for by an expected change in the exchange rate in the future.

If interest rates are lower than abroad, it must be that at the end of the period we expect to be able to buy more foreign currency with our home currency.

If this wasn’t the case, people would sell the home asset, driving the home currency down, which would then mean it was expected to go up later!

This condition very controversial in empirical macro. See eg Meese and Rogoff on reading list for later lecture.

Explaining the RER

• RER=real exchange rate:

q e p p

RER would be 1 if:

the exchange rate e was such that when you sold a Big Macg in China, for Yuan,

then swapped Yuan for dollars at a Travel Bureau

Then used the dollars to buy a Big Mac in New York, you ended up with the same amount of money [0!] as you started with.

Explaining RER/ctd…

• We will study this in detail later in the course, but….

• RER might not be 1 because of:– Transport costs– Non-traded inputs– Barriers to arbitrage, like tarriffs, lack of

information, costs of search….– See the Economist’s ‘Big Mac’ index!:– http://www.economist.com/content/big-mac-index

Explaining agg demand

ytd y et p pt q, 0

In the long run, y_d given by the natural rate of output, y_bar.

That’s the output governed by population, capital, working hours, technology, and frictions in the labour market.

In the short run, if something makes foreign goods cheaper [if the real exchange rate rises above its long run q_bar]…

Consumers will feel richer and want to buy more.

Explaining the OE Philips Curve

p t 1 p t ytd y p t 1 p t

Just as each market has a demand and a supply curve, so for the whole economy.

This is the supply curve for the whole, open economy.

Firms will produce in the long run the natural rate of output y_bar.

But in the short run if demand rises above the natural rate, they will produce more, but this will lead to an expected increase in prices [see the LHS].

They will also produce more if there is an expected increase in the equilibrium price….. So let’s look at what that eqm price is.

Explaining the eq’m price

p t et pt q t

Equilibrium price:

The price that you would get if the real exchange rate were at its equilibrium level q_bar.

So if this were to be expected to rise, then firms would be induced to produce more.

What have we done?

• Explained why DMF model was constructed, what it was for.

• Explained the DMF equations one by one.• Now let’s derive the overshooting result.

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.]

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 yt

ytd y et p pt q

m m

Money demand equation tells us REAL balances will rise by m-m’ if money stock rises from m to m’, since prices are fixed.

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.

Overshooting intuition: 2

m t p t it 1 yt

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 et 1 et

it 1 i et 1 et

Take the UIP condition, and rearrange it.

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.

Recap on how we got overshooting

• We said what happens if money supply increases from m to m’?

• We supposed that exchange rate e rises straight away to its new long run level.

• We used that assumption to figure out how much aggregate demand would rise…

• And we saw because of delta<1, that ag demand rises less than the change in e,m.

Recap on overshooting

• We then fed the aggregate demand increase into the money demand equation to see how much the demand for real balances would rise.

• This was even smaller still, since phi<1.• We realised that this meant there would have

to be an expected fall in interest rates to encourage real balance demand to rise by enough to soak up the extra money supply.

Recap on overshooting

• But , for UIP to hold…• this expected fall in the interest rate would

mean that the exchange rate would have to be expected to RISE next period.

• This contradicts what we said at the outset, which is that the ex rate moves straight to its long run level.

• So there must be overshooting.

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

• Derive 2 difference equations in e and q• Solve our two difference equations 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.

Getting our difference equation in q

p t 1 p t et 1 p t 1 q t 1 et p t q t

p t 1 p t ytd y et 1 et

qt 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

m t p t et 1 et qt q

m t et q t et 1 et qt q

et 1 e t 1 q t

qm t

Simplifying asssumptions…

…substituted into money demand eq…using defn of real er.

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.

The two difference equations for e and q

q t 1 q 1 q t q

et 1 e t 1 q t

qm t

Solve e equation first, by forward substitution

et 1 e t 1 q t

qm t

et q 1 et 1 q 1

1 q t q m t1

et q 1 et 1 q 1

1 q t q m t1

1

1 et 2 q 1 1 q t 1 q m t 1

1 1 1 q t q m t

1

First, unpack the difference term…..

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

et q 11

s t

1 s tm s

1 1

s t

1

s tqs q

et q m 1 1

s t

1

s tqs q

What we get by repeatedly substituting out for the e_t+n terms…

Equation for e_t simplifies when we impose constant money supply m_bar.

Solving for e_t in terms of m and q.

qs q 1 s tq t q

e t q m 1 1 q t q

s t

1

s t1 s t

et q m 1 1 q t q

Solution for q_t also got by repeated forward substitution.

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

et m q

e m q

q0 e0 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!

q0 m m q 1 1 q0 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.

e0 m m m 1 q This is what we get, after a few

basic algebra steps.

1 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 old-style 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 non-neutrality 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.