EQUILIBRIUM CONDITIONS IN THE OPEN
ECONOMY AND THE MONETARY MODEL OF
EXCHANGE RATE DETERMINATION
Sarantis Kalyvitis
The Nominal Exchange Rate
Definition: The exchange rate is the relative price of different
currencies
An increase in S means that it takes more units of
domestic currency to purchase a unit of foreign currency.
This is called depreciation (devaluation).
A fall in S is called an appreciation (revaluation).
The Real Exchange Rate
Definition: The relative price of goods in different countries is
the Real Exchange Rate (and is an index of
competitiveness!).
S t Pt ∗
Qt =
Pt
A real depreciation (improvement in competitiveness) can
occur if:
The domestic price level falls
The foreign price level rises
The nominal exchange rate depreciates
1
Equilibrium in International Goods Markets:
Arbitrage and Purchasing Power Parity
S t Pt ∗
Qt =
=1
Pt
In equilibrium,
Taking logs we get:
⇒
q = st + pt∗ − pt = ln(1) = 0
st + pt∗ = pt
Purchasing Power Parity (PPP or Law of One Price)
PPP is an equilibrium condition that applies to the goods market.
Very important note:
PPP does depend on the rapid adjustment of prices in response to
arbitrage opportunities!
If PPP would apply in the very short run then given a constant
foreign price level fluctuations in the nominal exchange rate
lead to equivalent variations in the price level.
⇒ The real exchange rate is constant and the nominal side of the
economy does not affect the real side (under continuous PPP,
money does not affect output).
This short-run adjustment is not supported by the evidence
though!
BP/US relative prices
0,4
2
1,5
0,35
0,3
0,25
1
1975 1980 1985 1990 1995
0,2
1,40
1,20
1,00
0,80
0,60
0,40
0,20
0,00
BP/USD relative prices
BP/USD exchange rate
12
10
8
6
4
2
0
18
20
18
40
18
60
18
80
19
00
19
20
19
40
19
60
19
80
2,5
BP/USD exchange rate
0,45
2
A simple application in PPP
Suppose we differentiate PPP with respect to time,
ds dp dp ∗
=
−
dt dt dt
Since PPP is a log equation the coefficients (they are all 1) are
elasticities (written as):
ε t = π t − π t∗ Relative Purchasing Power Parity (PPP)
ε
where t = rate of currency depreciation and
domestic inflation
πt =
rate of
To maintain competitiveness the exchange rate must depreciate at a
rate equal to the inflation differential between the two countries (by
the way, what are the implications for monetary union?)
3
Equilibrium in International Financial Market:
Uncovered Interest Parity
Question: How do private investors decide how much to invest in each
currency assuming domestic and foreign assets are
perfect substitutes?
Answer: Choose the asset with the highest expected rate of return.
The return on domestic assets equals the (nominal) domestic
interest rate i on those assets
The expected return on foreign assets in domestic currency
∗
equals the i t = foreign interest rate plus the expected capital
gain caused by expected exchange rate depreciation.
For example at its low the Euro traded at:
1 Euro = 0.9 USDollars
It is currently worth:
1 Euro = 0.95 USDollars
⇒ Had an investor used USDs to buy Euros at the lower rate and sold
at the higher rate, he would have gained 5 cents (capital gain). In
.95 − .90
× 100 ≈ 5.6%
percentage terms:
.90
Total
return
it∗ +
from
foreign
Et (S t +1 ) − S t
St
assets
equals
in
discrete
time:
If the assets are perfect substitutes then if:
it > it∗ +
E (S t +1 ) − S t
St
⇒ Investors can make expected profits by borrowing abroad and then
investing in domestic assets. (Note: since the investor makes a
profit for each unit of investment the capital flow toward domestic
assets is potentially infinite).
4
Similarly if: i
t
< it∗ +
Et (St +1 ) − St
St
⇒ Only foreign currency assets will be held in international portfolios
and the capital flow towards foreign assets is infinite.
Equilibrium requires the expected return on foreign and domestic
assets is equal.
Uncovered Interest Parity
it = it∗ +
Et (S t +1 ) − S t
St
UIP is a fundamental parity condition that applies to financial
markets in an open economy when domestic and foreign assets are
perfect substitutes.
UIP is an equilibrium condition that applies to financial markets.
Very important notes:
UIP depends on the expected exchange rate which is
unobservable!
If foreign assets carry some extra risk then UIP needs to be
modified to include a risk premium.
Any test on the UIP is a joint test on the UIP and the
hypothesis on the formation of expectations.
∆se = (i - i* )
⇒ Uncovered Interest Parity:
Expected Depreciation = Interest Differential
The domestic interest rate must be higher (lower) than the foreign
interest rate by an amount equal to the expected depreciation
(appreciation) of the domestic currency.
5
Foreign Exchange Market Equilibrium
The exchange rate is determined in the foreign exchange market.
Demand for foreign currency:
Domestic and Foreign Public agents require foreign currency for:
imports and exports
portfolio investment
foreign direct investment
(for instance, domestic importers require foreign currency to buy
foreign goods)
These agents can be treated as price takers.
Supply for foreign currency: The central bank is a monopoly
supplier of domestic currency.
The Balance Sheet of the Central Bank
ASSETS (claims held by the CB on foreign or domestic entities)
Domestic government bonds (Domestic Credit)
Foreign Currency (Reserves)
Also: Foreign currency denominated bonds, Gold
LIABILITIES
Monetary base (amount of currency it has issued and is held by the
public)
MB =D+R
MB = Monetary Base
D = Domestic Credit
R = Reserves
6
Monetary Policy
Suppose that the central bank sells one of its assets for domestic
money.
If it sells a foreign (domestic) asset reserves (domestic credit) falls.
On the other hand, MB also falls.
⇒ The money supply falls via the appropriate multiplier.
(If it buys an asset from the public or the government the money
supply expands.)
Note: The Central Bank raises the money supply by:
Step 1: buying government bonds from the public and thus
increasing the monetary base
Step 2: the money supply increases through the money multiplier
Step 3: the demand for government bond has risen, which drives
their price up and the interest rate down.
Exchange Rate Regimes
If the central bank does not intervene in the foreign exchange
market (i.e. it does not use the amount of foreign reserves to
influence the price of the foreign currency) the exchange rate is said
to be in a floating rate regime.
If the central bank intervenes via its reserves to keep the exchange
rate constant, there is a fixed exchange rate regime.
Intermediate cases: managed floating, target zones
Fixed vs. Floating Exchange Rate Regimes: A Comparison
Floating systems
Fixed systems
- stability in trade (growth)
- flexibility for national policies
- absence of speculation
- no need for reserves
- absence of ‘bubbles’
- immediate adjustment
- no imported inflation
- no excess volatility
⇒ STABILITY
⇒ INDEPENDENCE
7
Monetary approach to exchange rate determination
95% of daily transactions in the foreign exchange market are caused by financial flows
resulting from portfolio diversification.
The exchange rate is the relative price of different currencies. Therefore the demand and
supply for currencies will determine the exchange rate.
⇒ To build a model of a small open economy we need to consider the goods market, the
financial market and the money market, as well as the relationship between them.
Assumptions
Goods Market Equilibrium
PPP condition is the equilibrium condition that applies to the goods market and holds
continuously (very strong assumption!) – this can be modified later on by assuming
sticky prices, i.e. short-run rigidity in the goods market.
Financial Market Equilibrium
UIP condition is the equilibrium condition that applies to the financial market and holds
continuously– this is easier to believe as it deals with nominal variables that can be
adjusted instantaneously.
The Money Market
The exchange rate is the price of domestic money relative to foreign money: the money market
is central to the determination of the exchange rate.
Ms =Md
The money supply is determined by the central bank and is exogenous to the model.
Money demand depends upon (standard assumptions):
Real Money Demand: As the price level increases, aggregate money demand increases
roughly by the same amount.
Opportunity Cost of Money: The economy has three financial assets: domestic and foreign
bonds, and domestic money. By assumption the bonds are perfect substitutes. Bonds
represent interest bearing illiquid assets whilst money is a liquid asset bearing no interest. As
the interest rate rises a greater fraction of aggregate wealth is held in the form of bonds and
less in the form of money.
Transactions: As individuals become wealthier in real terms they purchase more goods. Thus,
as real GDP rises the demand for money increases.
1
Money demand equation
mt − pt = φyt − λit
(all variables in logs except of the interest rate)
φ = elasticity of real money demand with respect to output.
λ = semi-elasticity of real money demand with respect to the interest rate.
Typical values are φ = 1 and λ = .2 , which imply that persistent inflation is always caused
by expansion of the money supply.
The four fundamental equilibrium (PPP, UIP, Money demand equation, Money market
equilibrium) equations can be combined to form a –monetary- model of the open economy.
Additional assumptions:
The foreign money market matters too, because the exchange rate between any two
currencies depends on the demand and supply for both currencies.
The domestic economy is small, so it cannot affect foreign variables, which are exogenous.
There is perfect foresight, i.e. the future behaviour of all the variables in the above system of
equations is known with certainty (obviously not very realistic!).
2
Flexible Price Monetary Model
Financial Markets (Perfect Substitutes)
it =
it∗
•
+ Et ( s t )
(1)
Goods Markets (One good world, flexible prices)
st = pt − pt∗
(2)
Domestic Money Market
mt − pt = φyt − λit
(3)
Foreign Money Market
mt∗ − pt∗ = φyt∗ − λit∗
(4)
3
What are we interested in?
Question 1: What are the determinants of the exchange rate?
Question 2: What is the relationship between real GDP and the nominal variables such as the
money supply and output?
The answer to question (2) is that there is no relationship at all, because the PPP condition
is assumed to hold at all times and the real exchange rate is constant.
⇒ The real exchange rate is the relative price of identical products expressed in a single
currency. If this is not affected by variations in the domestic money supply then money
cannot affect output.
Notice that the assumption of PPP is contradicted by the data. It is even difficult to show
that PPP holds in the long run. (Reasons: differentiated products and/or sticky prices).
⇒ Relaxing PPP is therefore likely to give effects of nominal on real variables. Still,
although we know the model is hard to believe it still provides a useful vehicle for
analysing many issues.
4
Solving the model
∗
Solve for pt and pt from (3) and (4) and then substitute into (2). Then use (1) to replace the
interest differential to obtain:
 ds 
st = mt − mt∗ − φ ( yt − yt∗ ) + λEt  t 
 dt 
(4)
An alternative is to simply ignore the foreign money market and write (4) as,
 dst 
st = mt − φyt − p + λi + λEt  
 dt 
∗
t
∗
t
(5)
The exchange rate depends upon four variables plus the expected rate of depreciation.
Question: What causes the exchange rate to depreciate?
The exchange rate will depreciate if the money supply rises, real GDP falls, the foreign price
level falls or the foreign interest rate rises.
It is reasonable to suppose that the expected rate of depreciation depends upon the four
variables that precede it in equation (5). These are called "fundamentals" (fundt).
5
Method of Solution
Step 1. Write the exchange rate as,
 ds 
st = fundt + λEt  
 dt 
∗
∗
where the "fundamentals" are fundt = mt − φyt − pt + λit
(1)
(2)
Step 2. Specify the behaviour of fundamentals appropriately in (2). For example, if a variable
such as the money supply is growing over time this must be explicitly recognised.
Step 3. Postulate that:
 d ( fund ) 
 ds 
Et   = Et 
=?
 dt

 dt 
Substitute the results from steps 2 and 3 back into 1.
6
Example 1: constant fundamentals
 dst 
st = mt − φyt − p + λi + λEt  
 dt 
∗
t
∗
t
Denote the fact that the fundamentals are constant by:
f t = f = m − φy − p ∗ + λ i ∗
Given that none of the fundamental determinants of the exchange rate are changing or indeed
expected then it is reasonable to suppose that:
 ds 
Et  t  = 0
 dt 
So that the solution for the exchange rate is:
s = m − φy − p ∗ + λ i ∗
7
Example 2: A constant rise in the money supply
Now suppose that the money supply grows at the rate
be constant.
µ.
All other fundamentals are assumed to
If a variable X(t) grows at a constant rate µ then the log of X(t) will equal after time t:
x(t ) = x(0) + µt
where X(0) is the value at time 0 and µt the growth that has since occurred.
The money supply at time t is equal to,
mt = m0 + µt
(8)
All other fundamentals are constant. Because the money supply is growing over time it is
easy to infer that the exchange rate should depreciate. But by how much?
8
Money Market Equilibrium
The depreciation reflects the excess money supply but if there is an excess money supply the
money market cannot be in equilibrium.
The depreciation of the exchange rate cures this. Since PPP holds prices rise at the same rate as
the currency.
ds dp dp∗
= −
dt dt dt
Foreign inflation is constant by assumption ⇒ ds = dp
dt
dt
As the exchange rate depreciates aggregate demand rises. However because prices are perfectly
flexible prices instantly rise at the same rate µ .
⇒ money demand increases at the same rate the money supply rises ⇒
•
•
•
mt = p t = s t
If the domestic money supply is expanded by 10% the nominal exchange rate depreciates by
10% and the rate of inflation is 10%. The real exchange rate is constant.
9
The rate of currency depreciation
∗
∗
Step 2 implies that: f t = m0 + µt − φy − p + λi
The long-run solution implies that:
ds •
= st = µ
dt
Given pt , yt , it money demand is constant so rate of money growth of µ creates an excess supply
of domestic currency and an expected exchange rate depreciation of rate µ.
st = m0 + µt − φy − p ∗ + λi ∗ + λµ
10
Why is there a jump depreciation of λµ ?
Consider the last example in terms of our original equations:
it =
it∗
•
+ Et ( s t )
st = pt − pt∗
mt − pt = φyt − λit
With perfect foresight the exchange rate depreciation is common knowledge.
⇒ A rise in the rate of expected depreciation raises the return on foreign assets.
⇒ To maintain UIP the domestic interest rate must rise by the same amount.
A once-off increase in the domestic interest rate creates a downward jump in money demand
and a discrete depreciation of the nominal exchange rate:
 ds 
Et   = µ ⇒ ∆i = µ ⇒ ∆md = − λµ ⇒ ∆s = λµ
 dt 
11
Summary of Results
The real and nominal sectors are by assumption independent (classical ‘dichotomy’).
This is a consequence of price flexibility through PPP.
Changes in the variables comprising the fundamentals affect the current exchange rate.
The monetary model may be a good theory of long-run exchange rate determination
(depending on the validity of PPP), but is very unlikely to be confirmed in the short
run.
The expected exchange rate matters for the determination of the current exchange rate
(‘forward-looking’ variable).
12