The Dornbusch

overshooting model shows the excessive fluctuation of the nominal exchange rate

in response to a change in the monetary supply. The model is situated on two central equations:

1) The uncovered

interest rate parity denoted by:

r = r* + ?s?

2) The money demand equation denoted by:

m – p = k? – lr

The equations of the model are:

All variables are

expressed in logs, R* and P* are

given.

Uncovered Interest Rate

Parity (UIRP):

r = r* +

?s? (1)

Exchange rate

expectations:

?s? = ? (? ? s),

? > 0 (2)

Demand for money:

m ? p

= k? ? lr

(3)

Aggregate demand (IS ? LM):

(P* is

normalised to 1)

yd = h (s ? p)

= h (q) (4)

Aggregate supply

(sticky prices):

?p = ?(yd

? ?)

(5)

The system can be

reduced to 2 equations only. Using equations (1) and (2), equation (3) becomes:

p = m ? k? + lr* ? l? (s ? ?) (6)

Using Equation (4),

Equation (5) becomes:

?p = ? h (s ? p) ? ? (7)

The model assumes that Aggregate demand is

determined by the standard of economy IS -LM mechanism.

Calculation of the UIRP

the UIRP say that the interest rate in domestic

economy should be equal to the foreign plus the depreciation rate of the

domestic currency.

The assumption is that financial market adjusts

instantaneously, and economic agents are risk neutral, therefore UIRP hold at

all time.

1 + r = (1 + r*) s?/s

If s?/s = 1+ ?s? it implies that

1 + r = (1 +r*)( 1+ ?s?)

1 + r = 1 + r* + ?s? + r*?s?

1?1 + r = r* + ?s? +

r*?s?

Since r*?s? is very negligible, we can omit it

unless we are in the situation of hyperinflation.

Therefore, the UIRP can be written as follow:

r = r* + ?s?

Expectation determination

In the long run, the exchange rate is assumed to be

at its steady state and deviate from its long run equilibrium level in the

short run.

Long run: s?

Short run: ?s? = ? (s? ? s)

? reflect the sensitivity of the market expectation (over

or undervaluation).

If s? s, we expect

that the exchange rate to converge downward in direction of the equilibrium.

If ? is large, the

exchange rate will converge more rapidly to its equilibrium value. The

convergence will be more rapid the further away the exchange rate is from

equilibrium at any moment.

This implies for UIRP:

r = r* +

?s?

r = r* + ? (? ? s)

Calculation of the Money Demand

We know that money demand is equal to m – p = ky –

lr, which is equal to money supply.

That means the condition Ms = Md should hold.

If we allow for inflation, this implies that Ms/p =

Md = ky – lr

Taken natural log of money supply side, we will

have:

m – p = ky – lr.

Problem 2. Solve the

model and derive the elements of the diagram which shows the dynamic adjustment

after an exogenous shock.

If we take for instance government expenditure as

the shock in the market that will lead to

an increase in income, therefore increase the

incentive to consume. This means an increase

in the short term of the aggregate demand in the

good market which can be denoted by:

yd = h (s – p) = h(q).

With demand increasing, there will be an adjustment

of the price to the short-term excess

demand which can be written down as follow:

?p = ?(yd ? ?).

This equation means the wider the gap between the

demand and output capacity is, the

greater the rate of inflation is. With the demand

increased due to the shock, prices will

automatically adjust to excess.

Solving the equation happen in the long run since

the model says that prices are sticky and

only adjust in the long haul. The model state that

if the economy was below or above its

normal equilibrium point, it is expected to return

or to move towards its equilibrium level as it

was before the shock. That means that in the long

run the rate of inflation will be equal to

zero and also the expected depreciation will be

equal to zero. This can be written as follow:

0 =

?p = ?(yd ? ?), yd

= ?, h (s – p) = ?

Consider a Monetary

Expansion

Long-Run Equilibrium

In the long run,

aggregate demand has to be equal aggregate supply, so output has to be equal to

?. Since, the rate of inflation is zero, we have:

yd = ?

Substitution

shows that:

?/h = ? – P? (8)

That is, in the long run, the only

variable that affects the real exchange rate is growth in capacity output.

Also, ?se

= 0 (that is, in the long run the expected change of inflection is equal to

zero).

Substituting into (2) yields r = r*. Substituting that into (6) shows:

P? = m – k? + lr* (9)

taking (8) & (9) together yields:

s? = (

– k) ? + m +lr* (10)

Since r* is

given, r = r* in the long-run (given that ?s? = 0), meaning

that both IS and LM return to their original positions;

In order for LM to

return to its original position following the increase in the money stock (Ms

> 0), there needs to be a proportional increase in the price level ?P >

0, It implies that aggregate demand shifts upward.

We know that IS curve

only depends on the real exchange rate, which means that the real exchange rate

has to return to its initial equilibrium. Given that Q = SP*/P with ?P* =

0, S should increase in the same proportion as P increases (depreciation).

Short-Run Equilibrium

m – p = ky – lr

p = m – ky + lr

(we remember from previously,

that goods markets adjust slowly while financial markets adjust

instantaneously). That means that:

An increase of the

money supply, shifts LM curve to the right, decreasing the real interest rate

which is known as the (“liquidity effect”);

A decrease in the

interest rate r induces a depreciation of domestic currency (s increases);

given UIRP, the decrease of the interest rate r, is related to an expected

future appreciation, i.e. a decrease in s, (r = r? +

?s?). In order to generate an expected appreciation, the exchange rate over

depreciates (“overshoots”);

With the exchange rate

depreciation, along with ?P = 0 in the short-run, it implies that an increase

of the real exchange rate increases the domestic competitiveness, and the IS

curve shifts to the right;

The shifts in the IS

and LM curves pushes aggregate demand which equals to the short-run aggregate

supply (horizontal). There is an increase in output.

The Dynamic Adjustment

towards the Long-Run

1)

Excess aggregate demand pushes up prices:

As a result of the

shock, changes in the real exchange rate leads to deficit or surplus of the

current account. With time unless another disturbance happens, the economy

moves back to its long run exchange rate as a result of changes in the nominal

exchange rate and the price level.

Note that long run

equilibrium is characterized by certain conditions:

Aggregate demand must

be equal to aggregate supply Ms = Md (no upward

or downward pressure on the price).

Domestic interest rate

must be equal to foreign interest rate with a static exchange rate, therefore

no expectation of depreciation or appreciation.

The real exchange rate

is at its long run level. That means there is no deficit or surplus of the

current account of the balance of payment.

The pre-shock

equilibrium is determined by point A, C, H and F in the graph with the P? and the nominal supply M?, a corresponding output y and a nominal exchange rate s??.

With the disturbance,

the LM curve moves to the right LM (M?/P?) at point b. The increase of money supply will increase

income, which leads to an increase of aggregate demand yd? with a relative low interest rate r? and an exchange rate s? (current account surplus).

Following the

disturbances:

(a) The increased price

level reduces real money supply so LM shifts back to its initial equilibrium.

The real interest rate rises to its initial equilibrium in r* so

the exchange rate appreciates;

(b) The increase in

prices, together with the currency appreciation, reduces competitive advantage,

so Q = SP*/P decreases and IS shifts back to its initial

level (current account surplus reduced);

2. We reach the initial

equilibrium but where:

(a) Prices are

increased (to the same proportion as ?Ms);

(b) The exchange rate

has depreciated (s increases);

(c) The real exchange

rate is the same as initially (and s increase in the same proportions.

Problem 3. Explain the dynamics in a diagram.

Money market equilibrium Goods market equilibrium

P – P? = – L?(s – ?) ?p = ?(?

– ?)

S

?p=0

s?? C

s??

………………….. B

MM(M?)

s?? A

MM(M?)

P

P??

P??

We start at point A. Let’s assume there

is an increase in money supply. The goods market equilibrium only depends on

the real exchange rate, so it is unaffected by the increase in money supply. On

the other hand, in the money market with a larger money stock, any given

exchange rate is consistent with a higher price level, so the MM curve shifts

upward to the right.

Since the economy is always on the MM curve and

prices are fixed in the short run, the exchange rate increase to ?? (a depreciation). In other words, both the real and nominal exchange

rate overshoot and the economy reaches point C.

In the long run, prices become more

flexible, so the economy move from point C to Point B, where the exchange rate

decreases to ??, and prices increase from P?? to P??.