International Finance (WS09/10)Problem Set 2: Solution

Prof. Dr. Gerhard Illing, Jin Cao

January 7, 2010

1. Mundell-Fleming model

Consider a small open economy which is characterized by the following equations

1. Goods market equilibrium

Y = C(Y − T ) + I(Y, r) + G + Nx(Y,Y∗, ε); (1)

2. Money market equilibrium

MP

= YL(i); (2)

3. International capital market equilibrium

st = E[st+1] + i∗t − it. (3)

The price level is assumed to be fixed and normalized to be P = P∗. Also assume that∂Nx∂ε > 0. For the expectation on the future exchange rate, distinguish between two cases:

1. Constant expectation, i.e. E[st+1] = constant = se with st = se + i∗t − it;

2. Static expectation, i.e. E[st+1] = st with i∗t = it.

a) Explain graphically the difference in the slope of the IS curve with and without integrated

exchange rate adjustment.

b) Explain graphically the effects of an expansionary monetary policy under the two as-

sumptions about the expectation on the future exchange rate with flexible / fixed exchange rate

regimes.

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c) Explain graphically the effects of an expansionary fiscal policy under the two assumptions

about the expectation on the future exchange rate with flexible / fixed exchange rate regimes.

RBlanchard & Illing (2009), K 20, or Romer (2006), C 5.3, The open econ-

omy.

Answer:

a) For a small open economy, in the goods market equilibrium (1)

Y = C(Y − T ) + I(Y, r) + G + Nx(Y,Y∗, ε)

= C(Y − T ) + I(Y, i) + G + Nx(Y,Y∗, se + i∗ − i),

now a fall in the domestic (real) interest rate i leads to 

  

 

 

      

Figure 1: IS curve with (black line) and without (blue line) integrated exchange rate adjustment(under the constant expectation on the future exchange rate)

1. A decrease in investment spending, and, as a result, to a decrease in the demand for do-

mestic goods, hence to a decrease in output. This is the typical response in the traditional

IS − LM model;

2. In addition, the fall in the interest rate leads to depreciation of the domestic currency,

hence a shift in foreign demand towards domestic goods, and, as a result, to an increase in

net exports (given that Marshall-Lerner condition holds, such that ∂Nx∂ε > 0). The increase

in net exports increases the demand for domestic goods. This leads, through the multiplier,

to a rise in output as well.

Therefore, the impact of interest rate on output is stronger here than that in the traditional IS −

LM model, as Fig. 1 shows.

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Figure 2: The effects of an expansionary monetary policy under the constant expectation on thefuture exchange rate with flexible exchange rate regime (note that the blue line denotes the IScurve without integrated exchange rate adjustment)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

   

 

 

 

   

 

Figure 3: The effects of an expansionary monetary policy under the constant expectation on thefuture exchange rate with fixed exchange rate regime

b) See Fig. 2, 3, 4, and 5.

c) See Fig. 6, 7, 8, and 9.

2. Inflation and exchange rate regimes

Consider a two-country model with flexible exchange rates. The domestic loss function is

L = τ2 + aπ2 and the foreign one is L∗ = τ∗2 + aπ∗2, in which τ / τ∗ is the domestic / foreign

tax rate and π / π∗ is the domestic / foreign inflation rate. The domestic / foreign government

spending g / g∗ (with g = g∗) is exogenously given, which is to be financed by taxation and

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Figure 4: The effects of an expansionary monetary policy under the static expectation on thefuture exchange rate with flexible exchange rate regime

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

   

 

 

 

   

 

Figure 5: The effects of an expansionary monetary policy under the static expectation on thefuture exchange rate with fixed exchange rate regime

money creation, i.e. g = τ + µ / g∗ = τ∗ + µ∗ in which µ / µ∗ denotes the growth rate of money

supply at home / in foreign country. Assume that the relative purchasing power parity holds, and

economic growth is neglected.

a) Determine the optimal money growth rate at home and abroad under flexible exchange

rate regime. How about the exchange rate?

Assume the fixed exchange rate regime and symmetric intervention reactions for the follow-

ing questions.

b) Find the home country’s reaction function in response to the money growth abroad, and

vice versa.

c) Compute the Nash equilibrium, and show it graphically that social welfare gains under

cooperation.

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Figure 6: The effects of an expansionary fiscal policy under the constant expectation on thefuture exchange rate with flexible exchange rate regime (note that the blue line denotes the IScurve without integrated exchange rate adjustment)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

   

 

 

  

 

Figure 7: The effects of an expansionary fiscal policy under the constant expectation on thefuture exchange rate with fixed exchange rate regime

Now assume that the foreign country unilaterally pegs its currency to the home country’s,

and it’s entirely responsible for fixing its exchange rate.

d) Determine the money supply growth rates for both countries, and compare the welfare to

the case with symmetric intervention reactions.

Now assume that there are N countries with fixed exchange rates and identical loss functions,

Li = τ2i + aπ2

i , and budget constraints, gi = τi + µi.

e) Show that with symmetric intervention reactions the Nash equilibrium is µ = NN+a g.

RIlling (1997), K 10.3.

Answer:

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Figure 8: The effects of an expansionary fiscal policy under the static expectation on the futureexchange rate with flexible exchange rate regime 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 9: The effects of an expansionary fiscal policy under the static expectation on the futureexchange rate with fixed exchange rate regime

a) For home country, the policy maker’s problem is to

minµ,τ

L = aπ2 + τ2,

s.t. π = µ,

g = τ + µ.

By the first order condition, the optimal monetary policy is

µ =g

1 + a,

i.e. π =g

1+a , τ =ag

1+a . The same procedure applies for the foreign country, so it’s easily seen that

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π∗ = µ∗ =g∗

1+a . As for the exchange rate, SS = π − π∗ = 0 when g = g∗. The social loss is

L = aπ2 + τ2 =a

1 + ag2.

b) Under fixed exchange rate and relative PPP, SS = π − π∗ = 0, which requires π = π∗.

Suppose that the weight of foreign country in global economy is α, then each country’s inflation

rate is the linear combination of the two countries’ money growth rates, i.e.

π = π∗ = αµ∗ + (1 − α)µ.

By symmetry, α = 12 , i.e. π = π∗ =

µ+µ∗

2 . Now the home country’s problem becomes

minµ,τ

L = aπ2 + τ2,

s.t. π =µ + µ∗

2,

g = τ + µ.

By the first order condition,

µ =g − a

4µ∗

1 + a4.

The same procedure applies for the foreign country, so it’s easily seen that µ∗ =g∗− a

4µ

1+ a4

.

c) Solve the two equations above to get the non-cooperative solution

µN = µ∗N =g

1 + a2,

πN = π∗N =g

1 + a2.

Further, the tax rate is τN = aπ2 . The social loss is

LN = aπ2N + τ2

N =a(a + 4)(2 + a)2 g2.

Social welfare gains under cooperation, which is easily seen from the figure we drew in the

class.

d) When the foreign country unilaterally pegs its currency to the home country’s, its mon-

etary policy will be determined by the home country. Therefore, the foreign country’s reaction

function to home money growth rate µ is simply µ∗ = µ, and the inflation rates for both countries

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are π = π∗ =µ+µ∗

2 = µ. The equilibrium precisely replicates the cooperative solution.

The social loss in non-cooperative solution is higher than that in cooperative solution only if

LN > La(a + 4)(2 + a)2 g2 >

a1 + a

g2

a(2 + a)2 > 0.

Obviously this is true whenever a > 0.

e) With N symmetric countries, each country with

Li = aπ2i + τ2

i ,

gi = τi + µi.

The inflation rate for country i is determined by the weighted money growth rates of all countries.

By symmetry, the weight for each country is 1N , i.e.

πi =

∑Nj=1 µ j

N.

Now country i’s problem becomes

minµi,τi

Li = aπ2i + τ2

i ,

s.t. πi =

∑Nj=1 µ j

N,

gi = τi + µi = g.

By the first order condition,

g − µi =aN

∑N

j=1 µ j

N

.By symmetry, ∀i, j ∈ 1, . . . ,N, µi = µ j = µ. Therefore

µ =N

a + Ng.

It’s easily seen that µ increases with N, i.e. the problem of negative externality becomes more

severe with more countries. In other words, a larger share of one country’s inflation can be

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exported with more foreign countries.

3. Purchasing power parity

a) Explain the absolute and relative purchasing power parity. Which one is the more stringent

concept? What are the assumptions for the absolute purchasing power parity?

b) Is the relative purchasing power parity met in Figure 10? Provide your reasons.

Übung Währungstheorie

WS 2008/09 2 - 3 Julia Bersch

9. Kaufkraftparität

a) Erklären Sie die absolute und die relative Kaufkraftparität. Welches ist das strengere Kon-

zept? Welche Annahmen müssen erfüllt sein, damit die absolute Kaufkraftparität gilt?

b) Ist die relative Kaufkraftparität in der folgenden Grafik erfüllt? Welche Erklärung könnte es

für die gleichlaufende Entwicklung von nominalem und realem Wechselkurs geben?

Nominale und reale Wechselkurse, Deutschland und USA, 1975-1998:

c) Die inländische Inflationsrate sei π = 0,2 und die ausländische Inflationsrate π* = 0,4. Wie

verändert sich der nominale Wechselkurs bei Gültigkeit der absoluten bzw. relativen Kauf-

kraftparität?

Literaturhinweis: Hallwood und MacDonald (2000), Kap.7

Figure 10: The nominal and real exchange rates, Germany and USA, 1975–1998.

c) Now assume that the domestic inflation rate is π = 0.2 and the foreign is π∗ = 0.4. What

happens to the nominal exchange rate given that the absolute / relative purchasing power parity

holds?

RHallwood and MacDonald (2000), C 7.

Answer:

a) The purchasing power parity (PPP) theory arises from the concept of the law of one

price, which states that in competitive markets free of transportation costs and official barriers

to trade (such as tariffs), identical goods sold in different countries must sell for the same price

when their prices are expressed in termns of the same currency. The law applies to individual

commodities, while PPP applies to the general price level which is a composite of the prices of

all the commodities that enter into the reference basket.

The purchasing power parity theory assumes a stable relationship between the domestic

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price level P, foreign price level P∗, and nominal exchange rate s. It assumes that the real

exchange rate ε remains constant over time, ε = P∗SP = c, i.e. the percentage change in the

nominal exchange rate between two currencies over any period equals the difference between

the percentage changes in national price levels — the relative PPP. From ε = P∗SP = c one can

easily see that εε = S

S + P∗P∗ −

PP = 0, which implies that S

S = π − π∗.

The absolute purchasing power parity requires that ε = 1, which is a more stringent concept.

Assumptions for the absolute purchasing power parity:

• All goods are tradable;

• Homogeneous goods;

• No trade barriers like tariffs, transaction / transportation costs;

• No capital flows;

• Full employment;

• No price rigidities;

• Perfect competition in the international commodity market;

• Consumers have no preference for local goods.

• ... ...

The absolute purchasing power parity need not be met if any of the assumptions are not

fulfilled, e.g.

• Not all goods which are contained in the price index are internationally tradable;

• The tradable goods are not homogeneous and thus difficult to be compared;

• Consumers may have preferences for domestic goods;

• For price indices in different countries, the weights of commodities differ;

• ... ...

b) The relative purchasing power parity is not met in the graph. It would be satisfied only

when the real exchange rate is constant.

c) The relation SS = π−π∗ works for both absolute and relative PPP, therefore S

S = 0.2−0.4 =

−0.2, i.e. home currency appreciate by 20%.

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4. Balassa-Samuelson effect

Consider two countries A and B, each producing both tradable and non-tradable goods. La-

bor is the only factor used for production, which is homogenous and completely mobile between

the two production sectors in one country. The production functions are linear and the produc-

tivity is 1 in country B’s both sectors. In country A, the productivity of the tradable good is twice

as high as the non-tradable one.

a) Assume that in both countries the price index is the geometric mean of the prices for the

tradable and non-tradable goods, i.e. P = PαT P1−αN , as well as P∗ = P∗αT P∗1−αN . Show that the

real exchange rate is greater than 1 given that the absolute purchasing power parity holds for the

tradable good, andP∗NP∗T

> PNPT

.

b) Determine the relations between wages and goods prices in both countries. Assume that

there’s neither transportation cost nor tariff for the tradable good.

c) Assume that in both countries consumers spend half of their income on tradable and non-

tradable goods respectively. Again, the price index is the geometric mean of the prices for the

tradable and non-tradable goods. Show that the real exchange rate of country A is less than 1.

d) Would the result qualitatively change, if the expenditure shares of tradable good differ in

the two countries?

e) How does the real exchange rate change, ceteris paribus, if (i) the productivity of tradable

good in country A rises, or (ii) the productivity of non-tradable good in country A rises?

RHallwood and MacDonald (2000), C 7.

Answer:

a) The absolute purchasing power parity holds for the tradable good implies that

P∗T SPT

= 1.

The real exchange rate is determined as

ε =P∗S

P

=P∗αT P∗1−αN S

PαT P1−αN

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=

P∗NP∗TPNPT

1−α

.

Therefore, ε > 1 implies thatP∗NP∗T

> PNPT

.

b) The production functions for both sectors in both countries are

YT = aT LT ,

YN = aN LN ,

Y∗T = a∗T L∗T ,

Y∗N = a∗N L∗N .

The firm’s problem for each sector of each country is to

maxLi

Πi = PiaiLi − wiLi,∀i ∈ {T,N, ∗T, ∗N}.

The first order condition requires that

wT = PT aT ,

wN = PNaN ,

w∗T = P∗T a∗T ,

w∗N = P∗Na∗N .

Further, that labor is homogenous and completely mobile between the two production sectors in

one country implies that

wT = wN = w = PT aT = PNaN ,

w∗T = w∗N = w∗ = P∗T a∗T = P∗Na∗N .

With absolute PPP for tradeable goods, P∗T SPT

= 1,

w = PT aT = P∗T S aT =w∗

a∗TS aT = S aT w∗

when a∗T = 1, i.e. under the same currency (S w∗), home’s wage level is aT times higher than the

foreign level.

Now let’s have a look at the goods prices. For tradeable ones, the relation is pinned down by

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the absolute PPP P∗T SPT

= 1. For the non-tradeables,

PN =waN

=S aT w∗

aN=

aT

aNS P∗Na∗N =

aT

aNS P∗N

when a∗N = 1, under the same currency (S P∗N), home’s non-tradeable good is aTaN

times more

expensive than the foreign ones. This implies that the price level for domestic non-tradeable

goods is determined by the productivities of both sectors at home, or, the relative productivity

difference.

c) That consumers spend half of their income on tradable and non-tradable goods respec-

tively implies that tradable and non-tradable goods account for half respectively in the commod-

ity baskets of both countries, i.e. α = 12 .

Given that a∗T = a∗N = 1 and aT = 2aN , one can easily see that P∗N = w∗ = P∗T as well asPTPN

=aNaT

= 12 . The real exchange rate of the home country is thus

ε =P∗T SPT

P∗1−αN P1−αT

P∗1−αT P1−αN

=

(12

)1−0.5

≈ 0.7

< 1.

d) Suppose that the share of home country’s income on tradable remains α but the share for

the foreign country is β , α now. The real exchange rate is thus determined as

ε =P∗S

P

=

(P∗NP∗T

)1−β

(PNPT

)1−α .

Apply the numbers, one can easily see that

ε =

(P∗NP∗T

)1−β

(PNPT

)1−α

=1

21−α

= 0.7,

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the result doesn’t change becauseP∗NP∗T

= 1.

e) Again, starting from

ε =P∗T SPT

P∗1−αN P1−αT

P∗1−αT P1−αN

=P∗T SPT

(a∗T aN

a∗NaT

)1−α

one can easily see that

1. If the productivity of tradable good in country A, aT , rises, ε is going to fall — apprecia-

tion;

2. If the productivity of non-tradable good in country A, aN , rises, ε is going to rise —

depreciation.

References

B, O. G. I (2009): Makrookonomie (5. Auflage). Munchen: Pearson.

H, P. R. MD (2000): International Money and Finance (3rd Ed.). Oxford:

Blackwell.

I, G. (1997): Theorie der Geldpolitik. Heidelberg: Springer Verlag.

R, D. (2006): Advanced Macroeconomics (3rd Ed.). Boston: McGraw-Hill Irwin.

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