Chapter 3: Exchange rates I: the monetary approach in the
long run
Goals: Understanding how the exchange rates are
determined in the long run.
In microeconomics law of one price (LOOP) states that in the
absence of trade frictions, and under the conditions of free
competition and price flexibility, identical goods sold in
different locations must sell for the same prices when the prices
are expressed in a common currency.
𝑔
𝑔
Let 𝑃𝐸𝑈𝑅 denote the euro price of goods g sold at Eurozone, 𝑃𝑈𝑆
the dollar price of goods g sold at USA, and 𝐸$/€ the spot
exchange rate of US dollar against euro. Mathematically, LOOP
means
𝑔
𝑔
𝑃𝑈𝑆 = 𝑃𝐸𝑈𝑅 𝐸$/€
So if we use dollar as the common currency, the dollar prices of
the same goods g at two locations should be equal. Otherwise
there are arbitrage opportunities, and arbitrage will ensure that
LOOP holds eventually.
Exercise: please re-express LOOP if we use euro as the common
currency.
If we define the relative price of goods g as
1
𝑔
𝑔
𝑞𝑈𝑆/𝐸𝑈𝑅
=
𝑃𝐸𝑈𝑅 𝐸$/€
𝑔
𝑃𝑈𝑆
Then LOOP also implies that
𝑔
𝑞𝑈𝑆/𝐸𝑈𝑅 = 1
𝑔
The relative price 𝑞𝑈𝑆/𝐸𝑈𝑅 measures how many units of the US
goods can be exchanged for one unit of the same goods in
Eurozone. There is arbitrage opportunity if
𝑔
𝑔
𝑔
𝑞𝑈𝑆/𝐸𝑈𝑅 ≠ 1 (or equivalently, 𝑃𝑈𝑆 ≠ 𝑃𝐸𝑈𝑅 𝐸$/€ ).
Exercise
𝑔
Show how arbitrage works when 𝑞𝑈𝑆/𝐸𝑈𝑅 > 1.
𝑔
𝑎
𝑔
Step 1: 𝑃𝐸𝑈𝑅 𝐸$/€ (> 𝑜𝑟 <) 𝑃𝑈𝑆 (Hint: 𝑏 > 𝑐 ⇒ 𝑎 >
𝑏𝑐 𝑖𝑓 𝑏 > 0)
So the goods g is costlier in (Eurozone or USA)
Step 2: people will buy g in (Eurozone or USA) and sell it in
(Eurozone or USA)
𝑔
𝑔
Step 3: 𝑃𝐸𝑈𝑅 will go (up or down); 𝑃𝑈𝑆 will go (up or down)
𝑔
Step 4: 𝑞𝑈𝑆/𝐸𝑈𝑅 will converge to __________ assuming 𝐸$/€ is
fixed.
LOOP also implies a formula for the exchange rate:
2
𝑔
𝐸$/€ =
𝑃𝑈𝑆
𝑔
𝑃𝐸𝑈𝑅
So if there is only one goods g, then LOOP implies that the
exchange rate equals the price ratio.
The principle of purchasing power parity (PPP) is the
macroeconomic counterpart to the microeconomic LOOP. PPP
states that all baskets of goods have the same prices when
expressed in a common currency. Mathematically,
𝑃𝑈𝑆 = 𝑃𝐸𝑈𝑅 𝐸$/€
where 𝑃𝑈𝑆 denotes the price of a basket of US goods (or price
level), and 𝑃𝐸𝑈𝑅 the price level in Eurozone.
The rationale is that if LOOP holds for each goods, then PPP
must hold because the baskets in two locations contain the same
(tradable) goods with the same weights.
It is called purchasing power parity because it implies the a
currency have the same purchasing powers at different locations.
To see this, note
1 US dollar can buy ____________ US baskets
Using the exchange rate 𝐸$/€ , 1 US dollar can be converted to
__________ euros and that amount of euros can
buy_____________ Eurozone baskets.
3
Are the numbers of baskets bought with one US dollar the same
at two locations? Why?
Let’s define the real exchange rate of US dollar against euro as
𝑞𝑈𝑆/𝐸𝑈𝑅
𝑃𝐸𝑈𝑅 𝐸$/€
=
𝑃𝑈𝑆
The real exchange rates measure how many US baskets of goods
can be exchanged for one Eurozone basket. It is the
macroeconomic counterpart of relative price.
To see this, note
You need ___________ euro to buy one Eurozone basket.
You can convert that amount of euro to ________ dollars, and
buy ______ US baskets.
Discuss: what is the difference between (nominal) exchange rate
and real exchange rate.
PPP implies that
𝑞𝑈𝑆/𝐸𝑈𝑅 = 1
Exercise: please show how arbitrage works if 𝑞𝑈𝑆/𝐸𝑈𝑅 < 1
Finally, PPP implies that
𝑃
𝐸$/€ = 𝑃 𝑈𝑆
𝐸𝑈𝑅
(3-1) Absolute PPP
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So the spot exchange rate equals the ratio of price levels.
For example, if a basket of goods costs $460 in US and the same
basket costs €400 in Eurozone, the PPP would predict an
exchange rate of 𝐸$/€ = _______________
If we plug in the expected future price levels in (3-1) then we
𝑒
can obtain the expected future spot exchange rate 𝐸$/€
.
The PPP we talk about so far is absolute purchasing power
parity. Absolute PPP implies relative PPP, which says that if (31) holds, then
∆𝐸$/€,t
𝐸$/€,t
≡
𝐸$/€,t+1 −𝐸$/€,t
𝐸$/€,t
≈ 𝜋𝑈𝑆 − 𝜋𝐸𝑈𝑅 (3-2) Relative PPP
where ∆ denotes time difference, 𝐸$/€,t the spot rate at time t,
𝐸$/€,t+1 the spot rate at time t+1,
𝐸$/€,t+1 −𝐸$/€,t
𝐸$/€,t
the rate of
depreciation of the nominal exchange rate, and
∆𝑃𝑡 𝑃𝑡+1 − 𝑃𝑡
𝜋=
=
𝑃𝑡
𝑃𝑡
the inflation (rate). So relative PPP means that the rate of
depreciation of the nominal exchange rate equals the difference
between the inflation rates of two countries.
Proof for (3-2) (optional)
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From calculus, the quotient rule of differentials says (see, e.g.,
page 187 of Chiang and Wainwright’s Fundamental Methods of
Mathematical Economics, 4th edition)
𝑥
𝑑 ( ) = ____________________
𝑦
So
𝑥
𝑑 (𝑦 )
𝑥 = ____________________
(𝑦 )
Then according to equation (3-1),
𝑑𝐸$/€
𝐸$/€
= _____________________________
The final result follows if we approximate the differential (𝑑) by
difference (Δ).
In practice, we can use (3-2) to forecast exchange rate in long
run.
Exercise: please show how to modify (3-2) if absolute PPP does
not hold (Hint: use the definition of real exchange rate, and read
SIDE BAR on page 73).
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Discuss Figure 3-2 and 3-3: the empirical evidence for PPP
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A simple monetary model of the exchange rate
In macroeconomic course you learn the quantity theory of
money:
𝑀𝑑 = 𝐿̅𝑃𝑌,
where 𝑀𝑑 denotes the aggregate money demand, 𝐿̅ denotes a
constant that measures how much demand for liquidity is
generated for each dollar of nominal income (the inverse of 𝐿̅ is
the income velocity of money), 𝑃 the price level, and Y the real
income.
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When money market is in equilibrium, the demand and supply
of money are equal:
𝑀𝑑 = 𝑀
where 𝑀 denotes the money supply. Together we get a formula
for the price level:
𝑀
𝐿̅𝑌
𝑃=
The above formula implies that
𝜋 = 𝜇−𝑔
where 𝜋 denotes the inflation rates, 𝜇 the growth rates of money
supply, and 𝑔 the growth rates of real income.
Proof:
If we add subscripts to index countries, the absolute PPP (3-1)
implies that
𝑃
𝐸$/€ = 𝑃 𝑈𝑆 =
𝐸𝑈𝑅
𝑀𝑈𝑆
̅
𝐿𝑈𝑆 𝑌𝑈𝑆
𝑀𝐸𝑈𝑅
̅𝐸𝑈𝑅 𝑌𝐸𝑈𝑅
𝐿
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(3-3)
Equation (3-3) is the fundamental equation of the monetary
approach to exchange rates.
Moreover, (3-3) implies that
𝐸$/€,t+1 −𝐸$/€,t
𝐸$/€,t
≈ 𝜋𝑈𝑆,𝑡 − 𝜋𝐸𝑈𝑅,𝑡 = (𝜇𝑈𝑆,𝑡 − 𝜇𝐸𝑈𝑅,𝑡 ) −
(𝑔𝑈𝑆,𝑡 − 𝑔𝐸𝑈𝑅,𝑡 )
(3-6)
So the expected rate of depreciation equals the differential in
growth rates in money supply minus the differential in growth
rates in real income. (3-6) can be used when forecasting the
future spot rate.
Reminder
(3-6) is based on the assumptions that both the absolute PPP (31) and quantity theory hold.
Example
Case 1: suppose 𝜇𝐸𝑈𝑅 = 0, 𝑔𝐸𝑈𝑅 = 0, 𝑔𝑈𝑆 = 0, 𝑀𝑡−1 =
̅ , 𝑀𝑡 = 1.1𝑀
̅ , 𝑀𝑡+𝑗 = 𝑀𝑡+1 , 𝑗 ≥ 1
𝑀
So there is one-time 10% unexpected increase in money supply
at time t
Let 𝑃𝑡−1 = 𝑃̅, 𝑃𝑡 = _______________, 𝑃𝑡+𝑗 =
______________________, 𝑗 ≥ 1
̅ 𝐸$/€,t = _________, 𝐸$/€,t+j = ____________ 𝑗 ≥ 1
Let 𝐸$/€,t−1 = 𝐸,
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Conclusion: a one-time increase in money supply causes a onetime depreciation.
Case 2: suppose 𝜇𝐸𝑈𝑅 = 0, 𝑔𝐸𝑈𝑅 = 0, 𝑔𝑈𝑆 = 0 but the US
money supply grows at a steady fixed rate 𝜇. At time t the
growth rate of money growth changes to 𝜇 + ∆𝜇
Let 𝜋𝑡−1 = 𝜋̅, 𝜋𝑡 = _______________, 𝜋𝑡+𝑗 =
______________________, 𝑗 ≥ 1
Let 𝜆 denote the rate of depreciation of 𝐸$/€ , then (3-6) can be
rewritten as
𝜆 = 𝜋𝑈𝑆,𝑡 − 𝜋𝐸𝑈𝑅,𝑡 = (𝜇𝑈𝑆,𝑡 − 𝜇𝐸𝑈𝑅,𝑡 ) − (𝑔𝑈𝑆,𝑡 − 𝑔𝐸𝑈𝑅,𝑡 )
6`)
(3-
𝜆𝑡−1 = 𝜆̅, 𝜆𝑡 = _______________, 𝜆𝑡+𝑗 = ______________________, 𝑗
≥1
In conclusion, all else equal, increases in the rate of money
supply growth should equal increases in the rate of inflation and
the rate of exchange rate depreciation.
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A general long-run model for exchange rates and interest rates
The general model starts with a modified money demand:
𝑀𝑑 = 𝐿(𝑖)𝑃𝑌,
where letter 𝑖 denotes nominal interest rate, which is the
opportunity cost of holding money. Everything else equal, when
nominal interest rate goes up, people tend to hold less money
(and more interest-bearing assets). So 𝐿(𝑖) is a decreasing
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function of 𝑖. When the money market is in equilibrium, the
following equation holds:
𝑀
𝑃
where
𝑀
𝑃
= 𝐿(𝑖)𝑌,
is called real money balance.
The relative PPP implies that
𝑒
Δ𝐸$/€
𝑒
𝑒
= 𝜋𝑈𝑆
− 𝜋𝐸𝑈𝑅
𝐸$/€
Furthermore, UIP implies that
𝑒
Δ𝐸$/€
𝐸$/€
= 𝑖𝑈𝑆 − 𝑖𝐸𝑈𝑅
So if both relative PPP and UIP hold, we can derive an
important equation as follows:
𝑒
𝑒
𝑖𝑈𝑆 − 𝑖𝐸𝑈𝑅 = 𝜋𝑈𝑆
− 𝜋𝐸𝑈𝑅
(3-8)
which is called the Fisher effect.
By definition, the real interest rate, 𝑟, is the inflation-adjusted
return on an interest-bearing asset. Mathematically,
𝑟 =𝑖−𝜋
So the Fisher equation implies real interest parity
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𝑒
𝑒
𝑟𝑈𝑆
= 𝑟𝐸𝑈𝑅
14
(3-9)
Finally, the fundamental equation under the general model is
𝑃
𝐸$/€ = 𝑃 𝑈𝑆 =
𝐸𝑈𝑅
𝑀𝑈𝑆
𝐿𝑈𝑆 (𝑖𝑈𝑆 )𝑌𝑈𝑆
𝑀𝐸𝑈𝑅
𝐿𝐸𝑈𝑅 (𝑖𝐸𝑈𝑅 )𝑌𝐸𝑈𝑅
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(3-10)
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