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Eco 3320
Lecture Note # 9
Application of Phase Diagram: Open Macroeconomic Model
1. Introduction:
Dornbusch’s model of “Overshooting Exchange Rates” upon a change in any
exogenous variable, such as rf (foreign country’s interest rate).
2. Modification
1) AD would have one more item: Net Exports
Recall that AE = C+ I + G + X –M
Now we would like to endogenize the X-M or Net Exports.
The absolute foreign exchange rate e = domestic currency price of a unit of
foreign currency.
The Relative price level of the foreign country to the domestic country is defined
as
Pf
.

P
The relative foreign exchange rate is defined as
eP f
.

P
This measures how expensive, in domestic currency terms the foreign goods are
compared to the domestic goods. The larger the number, the more expensive the
foreign goods, and thus the more competitive the domestic goods are.
Its logarithmic values are: Taking log of the above, we get ln e  ln p f  ln P .
Now we may use the log data as the usual ones, and thus
e pf  p
(e  ln e; p f  ln P f ; p  ln P)
This relative foreign exchange rate affects the Exports minus Imports (=net
exports): An increase in the relative price of the foreign country to the domestic
country has a positive impact on the net exports. In the plain English, an increase
in the foreign price level to the domestic price level (in the same currency)
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Lecture Note # 9
encourages exports and discourages imports: The net exports are an increasing
function of the real foreign exchange rate.
2) International Investment
Uncovered Interest Rate Parity Theorem in the Long-run equilibrium
In brisk arbitrage and unblocked international capital flows, the long-run
equilibrium must witness the equality of the rates of return on the domestic and
foreign investment.
1
(1  r )  (1  r f )e , where e is the current FX rate, and e’ the FX rate
e
in the future.
As we know e  e  e, where e is the change in FX rates
between now and the future time, we can rewrite the above into
e  e
(1  r f )
e
e
(1  r )  (1  )(1  r )
e
Thus,
e 

f
(1  r )  1 
1  r 
e


(1  r ) 

e
By expanding the above equation, we get
1  r  1  r f  e  e  r f  0
 r  r f  e
//QED
If the domestic interest rate is higher than the foreign interest rate, this is due to
the fact that the FX rate will rise between now and the future: the domestic
currency will depreciate.
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Lecture Note # 9
3. System of Equation for an Open Economy:
To AD and AS functions, we will simply add the international investment
function, which is based on the Theorem of Uncovered Interest Parity.
The Uncovered Interest Parity Theorem states that the differences between the
domestic and foreign interest rates will lead to the future changes in foreign
exchange rates.
Suppose that the expected rate of inflation is equal to zero.
AD
Int’nal
Investment
AS
Y  a  b(e  p f  p)  hr ; IS Curve  (1)
m  p  ky  gr  u ; LM Curve
 (2)
r  r f  e; uncovered interest parity theorem  (3)
p  f ( y  y ); AS Curve (Philips)
 (4)
1) How to get the AD equations?
Y = f(C, I, G, X – M); ‘Holding C and I constant’;
 f (C , G, I (r ), NX (e  p f  p));
 a  h( r )  b(e  p f  p);
Relative (international) price levels
No change in the money market equation.
2) Explained.
3) The same as before: No expected inflation.
* An increase in the price level or P dot does not have to lead to an inflation
expectation if the increase in the price level is one-and-for-all, and thus is not a
sustained and continuous increase in the price level.
We do not want to solve this equation quantitatively, but we would like to examine
the equilibrium only qualitatively. The best way of doing so is by using the
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Eco 3320
Lecture Note # 9
differential equation method, which leads to the Phase Diagram. To do so, we have
to come up with a system of differential equations. In particular, we would like to
have a system of differential equations in terms of e and p, and their dot values.
4. How to get Differential Equations from above:
We can now adopt the simplifying assumptions of a=0, and b=g=k=1 now.
---------------------------(skip this if we do not have time)----------------------Step 1: Solve (1) and (2) for Y* and r*
r* 
k
m p
{a  b(e  p f  p)   }
g  kh
k k
Y *  a  b(e  p f  p)  h()
(1)*
→ (2)*
Step 2: Plug (1)* and (2)* into (3) and (4)
bk (e  p f )  (1  bk ) p  m  ka
e 
rf
hk  g
 bk (e  p f )  (h  gb) p  hm  ga

p  f 
 Y 
hk  g


How about simplifying the above system of equations:
Y  a  b(e  p f  p)  h( r ); b  0 But b  0

 0
=1
m  p  ky  gr
=1
=1
If you feel distracted due to the complexity of the equation, we can go all the way back to
the start of the question, and simplify the IS-LM curves and thus the AD curves by
assuming all the coefficients h=k=g = 1, and a=0.
---------------------(skip up this point)------------Thus, the Simplified AD curve is coming from:
Y  b(e  p f  p)  r : IS simplified. – (1)**
m  p  y  r : LM simplified.-(2)**
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Eco 3320
Lecture Note # 9
They will become the complete open macroeconomic model with the addition of the
following equations:
Int’nal
Investment
AS
r  r f  e; uncovered interest parity theorem  (3)
p  f ( y  y ); AS Curve (Philips)
 (4)
Let’s come up with the system of differential equations for this simplified AD curve:
Step 1: Solve for Y* and r*
b(e  p f  p)  r*  (Y *)  m  p  r *
2r*  b(e  p f  p)  (m  p)
r* 
b
1
( e  p f  p )  ( m  p ) ---(1)**
2
2
Y *  m  p  r* 
b
1
( e  p f  p )  ( m  p ) ---(2)**
2
2
Step 2: Substitute r* and Y* in (1)** and (2)** for r and y in equations (3), and
(4):
b
1
( e  p f  p )  ( m  p )  rf ---(3)*
2
2
1
b

p  f  ( e  p f  p )  ( m  p )  Y  ---(4)*
2
2

e  r  rf 
These are the differential equations.
5. Phase Diagram
1) For equation (3)*:
(1) Demarcation Line Step 1:
e  0
b
1
0  ( e  p f  p )  ( m  p )  rf
2
2
The slope of the e dot =0 curve is given by
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Eco 3320
de

dp

Lecture Note # 9
b 1

2 2   1  b  1
b
b
2
2) Trajectories:
de b
  0
de 
e  0
This means that as the value of e goes up, the value of e dot goes up too: As the
value of e goes up, the sign of e dot changes in the order of -, 0, and +.
 e  0
0  e 
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Eco 3320
2) From (4)*, we get
(1) Demarcation Line:
1
b

p  f  ( e  p f  p )  ( m  p )  Y 
2
2

p  0 ;
b
1
0  ( e  p f  p )  ( m  p )  Y
2
2
The slope of the p dot =0 curve is given by:
de

dp

b 1

2 2  0 .
b
2
p  0
e
p
(2) Trajectories:
p  0
e
dp
b 1
 f   (   )  0
dp
2 2
p  0
p  0
p
Lecture Note # 9
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Eco 3320
(3) Combining the above Two Trajectories:
We get the Phase Diagram as
p
0
e
0
6. Overshooting in the FX Market
What will happen if there is a change in an exogenous variable?
For instance, the foreign interest rate goes up.
1) When rf , the e  0 curve shifts up.
 e  r  rf  0 curve
b
1
( e  p f  p )  ( m  p )  rf  0
2
2
d e/d rf >0,
or intuitively, we can figure out.
2
 e  ......  rf  
b
Lecture Note # 9
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Lecture Note # 9
2) The economy goes to a new equilibrium by following the Saddle Path.
e
p  0
E2
E1
(∆ is ∆rf)
eg, US interest rate ↑
e  0
p
- When there occurs a disturbance (and this a dislocation from E1 ),
first, in the short-run where p is fixed, e↑ a lot;
Then, as times goes by, it follows the Saddle Path to E2 .
3) The time trend/ trace of the FX rate over time is as follows:
e
E2
E1
t
-
The peaked point is called ‘overshooting’ in the FOREX market upon a change in
exogenous variables.
(End).