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Section 4The Exchange Rate in the Long Run

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Content

• Objectives• Purchasing Power Parity• A Long-Run PPP Model• The Real Exchange Rate• Summary

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Objectives

• To understand the law of one price and purchasing power parity (PPP).

• To understand PPP as a theory of long-run exchange rate determination.

• To understand the real exchange rate.

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Purchasing Power Parity

• The Law of One Price: – Identical goods must be priced identically.

• Commodity Price Parity:– Goods in different countries must sell for the

same price, when their prices are expressed in the same currency.

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Purchasing Power Parity

• Commodity Price Parity:

Where Pkus is the USD price of commodity k in the U.S.,

Pkc is the CAD price of commodity k in Canada,

S(USD/CAD) is the price of CAD in USD.

Ck

USk PCADUSDSP )/(

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Purchasing Power Parity

• In the real world, CPP may not hold for the following reasons:

•  Transaction Costs:– Tariffs, transportation costs, insurance fees, and other

such costs mean that it may not be possible to make arbitrage profits even in the presence of price differences across countries.

• Nontraded Goods:– Several goods, such as services (haircuts) are

nontradable.

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Purchasing Power Parity

• Quotas:– Quotas and other such barriers to trade restrict

the ability to make arbitrage profits.

• Imperfect Competition:– Imperfect competition in commodity markets

may prevent prices from being equalized across countries. For example, price discrimination, entry costs, and menu costs would prevent CPP

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Purchasing Power Parity

• In an economy with many goods, purchasing power is defined in terms of a representative bundle of goods. It describes the number of baskets of goods you can buy. 

• The price level is the price of a particular basket of goods. The most common price index is the Consumer Price Index (CPI).

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Purchasing Power Parity

• Absolute Purchasing Power Parity:– Identical baskets of goods in different countries

must sell for the same price, when their prices are expressed in the same currency.

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Purchasing Power Parity

• Absolute Purchasing Power Parity:CUS PCADUSDSP )/(

Where Pus is the USD price of a basket of goods in the U.S.,

Pc is the CAD price of a basket of goods in Canada,

S(USD/CAD) is the price of CAD in USD

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Purchasing Power Parity

• In the real world, Absolute PPP may not hold for the following reasons: 

• Violations of CPP:– Absolute PPP is unlikely to hold if CPP is

violated. That is, if the individual goods in the representative consumption basket do not satisfy CPP, then PPP is likely to be violated.

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Purchasing Power Parity

• Differences in Baskets:– Absolute PPP will not hold if the composition

of the baskets differs across countries. For example, the presence of non-traded goods would make baskets different across countries

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Purchasing Power Parity

• The Relative PPP hypothesis states that the percentage change in the exchange rate reflects the difference between inflation at home and abroad. 

• Relative Purchasing Power Parity:

Ct

CT

t

TUSt

UST

P

P

CADUSDS

CADUSDS

P

P

)/(

)/(

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Purchasing Power Parity

• Relative PPP can also be expressed as:

• The linear version of relative PPP is:

 ttT

CTt

USTt SSS /)(,,

)1(]/)(1[1 ,,CTtttT

USTt SSS

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Purchasing Power Parity

• Where is the inflation rate:

ttTTt PPP /)(,

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Purchasing Power Parity

• Relative PPP states that inflation differences between countries should be reflected in percentage changes in the exchange rate. – For example, if the inflation rate is 5 % in the

US and 2 % in Canada, then the Canadian dollar should appreciate by 3 %.

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Purchasing Power Parity

– If prices in the US are rising faster than in Canada, Canada's exports are becoming relatively cheaper.

– This should attract importers, thereby increasing Canadian net exports and reducing US net exports.

– This should generate a relatively higher demand for CAD and promote an appreciation of the CAD against the USD.

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A Long-Run PPP Model

• The Monetary approach to the exchange rate• The Fundamental Equation:

– Price levels and the Demand for Money:

),(/ hhhh YiLMP

),(/ ffff YiLMP

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A Long-Run PPP Model

• Specific predictions from the monetary approach to long-run exchange rate determination:– Money supplies

• An increase in the USD money supply causes a proportional long-run depreciation of the USD against foreign currencies.

– Interest rates• A rise in the interest rate on USD assets causes a depreciation

of the USD against foreign currencies.

– Output levels• A rise in U.S. output causes an appreciation of the USD

against foreign currencies.

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A Long-Run PPP Model

– The international interest rate difference is the difference between expected national inflation rates.

– Recall UIP:

– Recall Relative PPP:

tteT

fTt

hTt SSSii /)(,,

tteT

fTt

hTt SSS /)(,,

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A Long-Run PPP Model

– So,

– And the interest differential is:

tteT

efTt

ehTt SSS /)(,,

efTt

ehTt

fTt

hTt ii ,,,,

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A Long-Run PPP Model

• The Real Interest Rate:

• The Fisher Relation (Real Interest Parity):

fTt

hTt rr ,,

)1/()1()1( ,,, TtTtTt ir

eTtTtTt ir ,,,

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A Long-Run PPP Model

• A simple numerical example.– Assume that:

• Pt = USD 100/Basket,

• PT = USD 101/Basket and

• it,T=0.03

• Today, period t– If I have USD xt = USD 100, I can buy 1 basket:

• B yt = USD xt / Pt = (USD 100)/(USD 100/Basket) = B 1

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A Long-Run PPP Model

– If invest, I get USD xT = USD 103:

• USD xT = (1+ it,T) USD xt = (1+0.03)USD 100 = USD 103

– If I have USD xt = USD 103, I can buy 1.02 basket:

• B yT = USD xT / PT = USD 103/USD 101/Basket = B 1.02

– Inflation is 1+t,T =1.01

• (1+t,T)=PT/Pt =(USD 101/B)/(USD 100/B) = 1.01

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A Long-Run PPP Model

• USD rate of return is 3 percent:– 1+ it,T =USD xT /USD xt = 1.03

• Real rate of return is 2 percent:– (1+rt,T) = B yT / B yt = b1.02/b1 = 1.02

• Thus, (1+rt,T) = (1+ it,T) Pt /PT = (1+ it,T)/ (1+t,T)

Or rt,T = it,T -t,T = 0.03 – 0.01 = 0.02

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A Long-Run PPP Model

• The Fisher Effect– A rise in a country’s expected inflation rate

generates an equal rise in the interest rate:

it,T = rt,T + et,T

– In the long run, inflation is a monetary phenomenon:

1+t,T = MT /Mt

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Slope = +

growth = +

growth =

M

Time

Slope = Slope =

t0

Slope = +

t0

t0

i2 = i1 +

i1

S

Time

i

Time P

Time

A Long-Run PPP Model

t0

The Fisher Effect

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A Long-Run PPP Model

– Jump in prices:

• Jump in interest rate for constant also force prices to jump.

– USD constantly depreciates and jumps because of absolute PPP.

),(/

),(//1

,

,,

tTtt

TTTTTtTTt YiLM

YiLMPP

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The Real Exchange Rate

• Recent empirical evidence suggests that– PPP does not hold in the short run.

– It is a good approximation for the currencies of countries experiencing hyperinflation.

– Prices of identical goods differ substantially across countries (the Economist and the Big Mac Index).

– PPP (or something like it) appears to hold in the long-run. That is, the real exchange rate is trend reverting.

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The Real Exchange Rate

• This empirical evidence also suggests that domestic and foreign baskets of goods on which price levels are computed are different across countries.

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The Real Exchange Rate

• The information summarized by PPP may still be useful to determine the competitive position of a particular country.

• The Real Exchange Rate summarizes this information. The Real Exchange Rate is:

US

C

P

PCADUSDSCUSQ

)/()/(

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The Real Exchange Rate

• The real exchange rate is the relative price of a foreign basket of goods in terms of a domestic basket of goods.– For example, assume that the price level is CAD 150

per basket of Canadian goods in Canada and USD 100 per basket of US goods in the US. Also assume that the exchange rate is USD 0.65/CAD.

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The Real Exchange Rate

– The real exchange rate is then:

– That is, the Canadian basket is cheaper than the American basket, which suggests that the American economy is somewhat less competitive than the Canadian economy.

– Another interpretation is that the costs of living are lower in Canada.

CUS

US

C

BBCUSQ

BUSD

BCADCADUSDCUSQ

/975.0)/(

/100

/150/65.0)/(

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The Real Exchange Rate

• Real depreciation of the USD– A rise in the real home price of a foreign basket

of goods.– A rise in the real USD/CAD exchange rate

• A fall in the purchasing power of a USD in Canada relative to the purchasing power of the USD in the U.S.

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The Real Exchange Rate

• The real exchange rate is the relative price of a foreign basket of goods. In the long run, this relative price is determined by the supply and demand for baskets of goods.

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The Real Exchange Rate

• These market forces can be summarized by:– An increase in world relative demand for

Canadian goods causes an increase in the price of the Canadian basket of goods and an increase in the real exchange rate.

– An increase in world relative supply of Canadian goods causes a reduction in the price of the Canadian basket of goods and a reduction in the real exchange rate.

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The Real Exchange Rate

• Nominal and Real Exchange Rates – The real exchange rate is mean reverting.

• The half life of changes in the real exchange rate is several years long.

• So, relative PPP holds in the long-run.

– The real and nominal exchange rates are highly correlated.

• This suggests that prices are sticky.

US

C

P

PCADUSDSCUSQ

)/()/(

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The Real Exchange Rate

• The real exchange rate and real interest parity.– The relative version of the real exchange rate is:

– or

USt

Ct

UST

CT

t

T

t

T

PP

PP

S

S

Q

Q

/

/

USTt

CTt

t

tT

t

tT

S

SS

Q

QQ,,

)(

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The Real Exchange Rate

• Uncovered Interest Parity

• Real Exchange Rate:

• Real Interest Rate:

t

teTC

TtUSTt S

SSii

,,

eUSTt

eCTt

t

teT

t

teT

S

SS

Q

QQ,,

)(

eTtTtTt ir ,,,

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The Real Exchange Rate

• Real Interest Parity (Part II)

• Uncovered Interest Parity

t

teTC

TtUSTt Q

QQrr

,,

t

teTC

TtUSTt S

SSii

,,

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The Real Exchange Rate

• The real interest parity condition states that differences in expected real interest rates depend on expected movements in the real exchange rate.

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The Real Exchange Rate

• An interpretation of real interest parity

– The real interest rate is a measure or the real return to capital. The higher that return, the higher the growth rate of the economy.

– So, if r>r*, the home economy is growing faster and must see its competitiveness improves. That is, the real exchange rate (the price of a foreign basket of goods) must rise.

t

teT

TtTt Q

QQrr

*

,,

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Summary

• Absolute PPP states that the purchasing power of any currency is the same in any country:

• Relative PPP predicts that percentage changes in exchange rates equal differences in national inflation

• The real interest rate is:eTtTtTt ir ,,,

ttTTtTt SSS /)(*,,

*PSP

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Summary

• The real exchange rate is the price of a foreign basket of goods:

• Real interest parity is:– If both UIP and PPP hold:

– If only UIP holds:

P

SPQ

*

*,, TtTt rr

t

teTC

TtUSTt Q

QQrr

,,

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Summary

• There are sizeable movements in the real exchange rate.– These movements are long-lasting, but not permanent.

– Thus, relative PPP holds in the long run.

• The real and nominal exchange rates are highly correlated.