The ? [h (s ? p) ? ?] (7)

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:

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                           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??.