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International Economics

M.Sc. in Management Engineering Politecnico di Milano

December 1, 2017

Contents 1 International Trade 2

1.1 Ricardo: technological change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Ricardo: three countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Ricardo: a new good . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Ricardo: three goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Heckscher-Ohlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Heckscher-Ohlin: Stolper-Saumuelson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.7 Commercial Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.8 Commercial Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.9 Commercial Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.10 Commercial Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 International Finance 19 2.1 IS-LM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Mundell-Fleming: increase in i∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Mundell-Fleming: expected depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Mundell-Fleming: expected appreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Mundell-Fleming: decrease in money supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6 Dornbusch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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1 International Trade

1.1 Ricardo: technological change In Honduras and Guatemala two goods are being produced, coconut (C) and textile (T ), using exclusively labor as a production input. Technology has constant returns to scale and is characterized in the following table:

Honduras Guatemala units of labor for 1kg of coconut 4 3 units of labor for 1mt of textile 8 2 total units of labor available 1200 200

Consumers in both countries share the same preferences, which are described by the following utility function:

U = C 3 4T

1 4

where C and T denote the quantities of coconut and textile consumed, respectively. Answer to the following questions:

1. Which country have the absolute and comparative advantage in the production of each good?

2. Determine the equilibrium production/consumption in autarky.

3. Let open the two countries to free trade. What is the pattern of specialization? What is the free-trade relative price?

4. Quantify the benefits of free trade in terms of the variation in collective utility before and after free trade, i.e. the welfare variation.

5. Suppose that in Honduras a group of management engineers improve the production technology in both sectors, allowing to produce 1kg of coconut and 1mt of textile with 3 and 6 units of labor respectively. What happens to the pattern of specialization? And to welfare? Discuss.

Solution.

1. Guatemala has an absolute advantage in the production of both goods. Guatemala has a comparative advantage in the production of textile, while Honduras in the production of coconut. Indeed,

aHC aHT

= 4

8 <

3

2 = aGC aGT

where aij denotes the unit labor requirement for the production of good j in country i.

2. The problem for country i is

max C,T

U = C 3 4T

1 4

sub. L = aiCC + a i TT

which delivers the following system of equations{ 3TC =

aiC aiT

L = aiCC + a i TT

.

2

Figure 1: PPF: Honduras and Guatemala

Notice that 3T/C is the marginal rate of substitution, MRSC,T = (∂U/∂C) / (∂U/∂T ), while the ratio aiC/a

i T is the marginal rate of transformation, MRTC,T . The latter is the slope of the production-

possibility frontier, which is the constraint; displayed in figure 1. The autarky equilibrium relative price, PC/PT , is equal to the slope of the production possibility-frontier, since, under perfect competition, the price of good j is equal to its marginal cost of production, Pj = MCj = w · aij, where w is the real wage rate. Therefore, to obtain equilibrium quantities (consumed and produced), solve the system for Guatemala: {

C = 50

T = 25 , PC PT

= 3

2 , UGaut = 42.04

and for Honduras: { C = 225

T = 37.5 , PC PT

= 1

2 , UHaut = 143.76.

3. The relative demand schedule is obtained as the equality between the marginal rate of substitution and the relative price:

3 T

C =

PC PT

.

This demand function holds for both countries, as consumers share the same preferences. It can also be obtained by solving the consumer’s problem, with a generic budget constraint (e.g. PCC + PTT = I). The relative supply schedule can be found graphically, in the space (C/T, PC/PT ); see figure 2. The free- trade equilibrium is given by their intersection, which happens to be in C/T = 3 at a free-trade relative price of (PC/PT )

FT = 1. The pattern of specialization dictates that the two countries fully specialize in

the production of the goods in which they have their comparative advantage. Then we have to find the equilibrium consumption levels under free-trade. For each country, we have to maximize consumption subject to the budget constraint: the equilibrium production valued at the free-trade relative price. In

3

Figure 2: Free-trade equilibrium: relative demand and relative supply.

other words, the consumption-possibility frontier widens above the production-possibility frontier; see figure 3. For instance, in Guatemala we have:

max C,T

U = C 3 4T

1 4

sub. C = 100− T.

Therefore, equilibrium production and consumption in Guatemala are:{ Tprod = 100

Cprod = 0

{ Tcons = 25

Ccons = 75

while in Honduras they are: { Tprod = 0

Cprod = 300

{ Tcons = 75

Ccons = 225 .

4. The welfare is higher in free-trade than in autarky. Indeed,

UGft = 56.99

UHft = 170.96.

5. The comparative advantage in Honduras remains unchanged, with aHC /a H T = 1/2. The world relative

demand remains unchanged as well. However, the world relative supply must be modified: the full- specialization point now moves to the right, at C/T =

( LH/aHC

) / ( LG/aGT

) = 4. The resulting free-

trade equilibrium relative price is then (PC/PT ) FT ′

= 3/4; see figure 4. In Guatemala, production and

4

Figure 3: Free-trade equilibrium: production-possibility vs. consumption-possibility.

consumption are now: { T ′

prod = 100

C ′

prod = 0

{ T ′

cons = 25

C ′

cons = 100

while in Honduras are: { T ′

prod = 0

C ′

prod = 400

{ T ′

cons = 75

C ′

cons = 300 .

Notice that Honduras suffers a negative shock in the terms of trade, i.e. Pexport/Pimport. Yet, welfare is still higher in free-trade than in autarky (which is always true) and than in the previous free-trade situation (which is not always true!).

1.2 Ricardo: three countries Consider a world economy made up by three countries: A, B, and C. Each country can produce two goods, food (F ) and vodka (V ), using labor as the unique production input. Labor endowments and production technologies are characterized in the following table:

country A country B country C units of labor for 1 unit of food (F ) 1 3/4 1 units of labor for 1 unit of vodka (V ) 5 1 1/2

total units of labor available 10 6 4

Consumers in both countries share the same preferences, which are described by the following utility function:

U = F 3 5V

2 5

where F and V denote the quantities of food and vodka consumed, respectively. Answer to the following questions:

5

Figure 4: Free-trade equilibrium: relative demand and relative supply.

1. Determine the equilibrium production and the relative price of food in autarky.

2. Let open the two countries to free trade. What is the pattern of specialization? Determine consumption and production in free trade, as well as the free-trade relative price of food.

3. Suppose that, in country A, the production technology in the vodka sector improves until is equal to that in country C. What happens to the pattern of specialization? And to the relative price of cheese?

Solution.

1. The chain of comparative advantage,

aAF aAV

= 1

5 < aBF aBV

= 3

4 < aCF aCV

= 2,

implies that country A has a comparative advantage in the production of food, country C in the produc- tion of vodka, while country B stands in the middle. The autarky equilibrium relative price are equal to the ratio of unit labor requirements, because of perfect competition; that is,(

PF PV

) A

= 1

5 ,

( PF PV

) B

= 3

4 ,

( PF PV

) A

= 2.

The production-possibiliy frontiers are depicted in figure 5. The autarky equilibrium is found by solving, for each country i = {A,B,C},{

MRSF,V = aiF aiV

Li = aiFF + a i V V

⇒

{ FA = 6

V A = 4/5 ,

{ FB = 24/5

V B = 12/5 ,

{ FC = 12/5

V C = 16/5 .

6

Figure 5: Production-possibility frontiers.

2. The free-trade equilibrium relative price is (PF /PV )ft = 3/4, which is found by the intersection of the relative world demand,

3

2

V

F =

PF PV

,

and the relative world supply, which can be found graphically, by construction; see figure 6. The pattern of specialization requires that FA+B+C/V A+B+C = 2, which implies that A and C fully specialize in the production of goods F and V , in quantities FAprod = 10 and V

C prod = 8 respectively, while country

B produces both goods, in quantities FBprod = 36/5 and V B prod = 3/5. Instead, consumption is found by

using the consumption possibilities frontier (which is given by the valuation of home production at the world relative price) and is equal to{

FAcons = 6

V Acons = 3 ,

{ FBcons = 24/5

V Bcons = 12/5 ,

{ FCcons = 32/5

V Ccons = 16/5 .

Notice that country B enjoys the same level of consumption as in autarky; indeed, the relative price didn’t changed when moving to free trade.

3. After the technological improvement, country A becomes effectively identical to country C. The world relative supply does change: nobody produces food anymore at price 1/5 and the full specialization point moves to FA+B+C/V A+B+C = 2/7. The resulting free-trade equilibrium relative price is (PF /PV )ft′ = 2, at total relative quantity F/V = 3/4.

1.3 Ricardo: a new good (This exercise is taken from the midterm exam of the a.y. 2015/16.)

Consider two economies, NATO (N) and ASIA (A). They both produce two goods, food (F) and machines (M). Production requires only labor, according to the following technology, with constant returns to scale:

NATO N ASIA A units of labor for one unit of food (F ) 4 16

units of labor for one unit of machines (M) 8 12 total units of labor available 200 480

7

Figure 6: Free-trade equilibrium: relative demand and relative supply.

Consumers in both countries share the same preferences, described by the following utility function:

U = F 3 5M

2 5

where F and M are the quantities of food and machines consumed, respectively. Answer the following questions:

1. Is there any country holding an absolute advantage in the production of both goods? And who has the comparative advantage in the production of machines?

2. Find the autarky equilibrium in both countries: production, consumption, and the autarky relative price of machines.

3. Find the free trade equilibrium: production and consumption, in both countries, and the world relative price of machines.

4. From now on, assume that wages are given: w = 5 in Nato and w = 2 in Asia. Then, introduce another tradable good, services (S), whose production requires 3 and 15 units of labor in NATO and ASIA respectively. Moreover, assume that all the labor endowment in both countries is still employed in the production of the other two goods, F and M, and there is perfect competition in the factor (labor) market. In which country services are going to be produced?

Solution.

1. NATO has an absolute advantage on both goods, while ASIA has a comparative advantage in the pro- duction of machines.

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2. In autarky, consumption and production coincide. The problem is to maximize utility given the resource constraint of each country. The resulting optimality condition for NATO is

3M

2F =

1

2 ,

while for ASIA is

3M

2F =

4

3 .

Use the resource constraints, 200 = 4F + 8M and 480 = 16F + 12M for NATO and ASIA respectively, to obtain the equilibrium quantities: FN = 30,MN = 10 and FA = 18,MA = 16. The equilibrium price is the ratio of unit labor requirements: PM/PF = 2 in NATO and PM/PF = 3/4 in ASIA.

3. The world relative demand is

3M

2F =

PF PM

The relative supply is instead derived graphically, with only one step in F/M = 5/4. The equilibrium is exactly at that step, with a relative price of PF /PM = 6/5. NATO is completely specialized in food production while ASIA in machines production, for total quantities of FN = 50 and MA = 40. To find consumption, we first need to derive the countries’ budget constraints, by valuing production at world prices: FN = 50 − 56MN and MA = 40 −

6 5FA. Use the optimality condition for consumption,

3M 2F =

PF PM

= 56 , to obtain: { FN = 30

MN = 24 and

{ FA = 20

MA = 16

4. We can derive prices as P = aw, given the prevailing wages. Therefore, services have a price of PNS = 3 · 5 = 15 in NATO while a price of PAS = 15 · 2 = 30 in ASIA. Hence, services are going to be produced in NATO. (One can also check that, at these prices, the pattern of specialization of food and machines is consistent with point (3).)

1.4 Ricardo: three goods Consider a world economy made up by two countries: Alfa and Beta. Each country can produce three goods: trucks (T ), vests (V ), and magazines (M). Labor is the only factor of production and unit labor requirements are defined in the following table:

trucks (T ) vests (V ) magazines (M) Alfa 8 3 5 Beta 32 6 18

We take as given the wage rate in both countries: 27 in Alfa and 9 in Beta, per unit of labor. Answer to the following questions:

1. Determine the equilibrium production and commercial specialization in free trade, for both countries.

2. Graphically depict the relationship between the relative labor supply and the relative wage. If, for instance, labor supply in Beta increases, what happens to specialization? And to the relative wage?

3. Introduce transportation costs that are 50% of the production costs. What happens to trade flows? What is the maximum level of transportation costs that still allows trade between the two countries?

9

Solution.

1. The chain of comparative advantage is

aαT

aβT =

8

32 < aαM

aβM =

5

18 < aαV

aβV =

3

6 .

Specialization depends on the relative wage, wα/wβ = 27/9. In particular, the rule is that a good is produced in a country if and only if it is cheaper than in the other country; that is, good j is produced in country i if and only if

wiaij < w −ia−ij ,

where “−i” denotes the country that is not i. Therefore, country α produces both M and T , while country β produces V .

2. The equilibrium relative wage depends on the supply and demand of labor. We can plot them in the diagram L/L∗ vs. w/w∗, where “∗” denotes the foreign country, which is β. The relative supply of labor is a vertical line, which is given. The relative demand of labor can be formally derived from the demand of goods (but we do not do it!); in general, it is a downward sloping relationship, as increasing the relative wage can only be consistent with a diminishing relative labor supply (so that the home country becomes more specialized and its wage rises). Therefore, increasing the labor supply of β shifts the relative labor supply (L/L∗) to the left and increases the relative wage (w/w∗).

3. With a tariff t = 50%, country α cannot export anymore (since country β would rather produce the goods by itself !), while country β could still export vests (since country α would be indifferent between importing or producing internally – but for trade to be balanced, no trade will in fact occur in equilibrium!). The maximum tariff, for each good, is obtained by equating the export price to the autarky price,

P expαT = P β T ⇒ w

αaαT (1 + t ∗ T ) = w

βaβT ⇒ t ∗ T = 33.3%

P expαM = P β M ⇒ w

αaαM (1 + t ∗ M ) = w

βaβM ⇒ t ∗ M = 20%

P expβ V = P

α V ⇒ wβa

β V (1 + t

∗ V ) = w

αaαV ⇒ t∗V = 50%.

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1.5 Heckscher-Ohlin The small open economy Slavonia, where labor and capital are used to produce goods X and Y , is charac- terized by the following transformation curve:

Y 2

144 + X2

64 = 1.

Consumers in both countries share the same preferences, which are described by the following utility function:

U = X 1 2Y

1 2 .

1. Find the equilibrium relative price of X and production levels in autarky.

2. Suppose that the international relative price of X is 3. Find the equilibrium levels of production and consumption in a free-trade environment. Find also the change in welfare.

Solution.

1. Slavonia’s problem is as follows:

max X,Y

U = X 1 2Y

1 2

sub. Y 2

144 + X2

64 = 1

which can be solved by standard optimization techniques. That is, the Lagrangian is

L = X 1 2Y

1 2 + λ

( 1− Y

2

144 − X

2

64

) and then we need to solve the following system of equations

∂L ∂X = 0 ∂L ∂Y = 0 ∂L ∂λ = 0

⇒

1 2X − 12Y

1 2 = λ 264X

1 2X

1 2Y −

1 2 = λ 2144Y

Y 2

144 + X2

64 = 1

which, by dividing the first with the second line, becomes MRSX,Y︷︸︸︷ Y

X =

MRTX,Y︷ ︸︸ ︷ 144

64

X

Y Y 2

144 + X2

64 = 1

⇒

{ X = 4

√ 2

Y = 6 √

2

The MRSX,Y is the marginal rate of substitution, which comes from the utility function and is defined as the ratio between the marginal utility of good X and the marginal utility of good Y . The MRTX,Y is instead the marginal rate of transformation, which comes from the production-possibility frontier (or also transformation curve) and is defined as the marginal opportunity costs of good X in terms of good Y . In equilibrium, they must be equal. Furthermore, in autarky, they are also equal to the equilibrium autarky relative price PX/PY , which can be obtained by evaluating the MRSX,Y (or MRTX,Y ) at the equilibrium quantities; that is, PX/PY = 3/2.

11

2. Since the free-trade relative price of X is higher than in autarky, Slavonia will export good X. To solve for the equilibrium, start first from production (and only then move to consumption). The producers’ problem is

max XP ,YP

PXXP + PY YP

sub. Y 2P 144

+ X2P 64

= 1

which yields the following system 14464 XPYP = ( PX PY

)FT = 3

Y 2P 144 +

X2P 64 = 1

⇒

{ XP = 16/

√ 5

YP = 12/ √

5

where subscript P denotes “production”. Then, we can value production at international prices to get the country’s budget constraint: first, impose passage for (XP , YP ) and, second, impose slope of (PX/PY )

FT = 3, to get

Y︸︷︷︸ 12√ 5

= I

PY − PX PY︸︷︷︸ 3

· X︸︷︷︸ 16√ 5

⇒ Y = 60/ √

5− 3X.

In turn, the consumers’ problem is

max XC ,YC

X 1 2

CY 1 2

C

sub. YC = 60√

5 − 3XC

which yields the following system YCXC = ( PX PY

)FT = 3

YC = 60√ 5 − 3XC

⇒

{ XC = 10/

√ 5

YC = 30/ √

5

The resulting change in welfare can be obtained from the difference in overall utility, moving from autarky to free trade,

UAUT = X 1 2Y

1 2 = 6.928

UFT = X 1 2

CY 1 2

C = 17.32

1.6 Heckscher-Ohlin: Stolper-Saumuelson Consider a world economy made up by two countries: α and β. Production technology is identical across countries and employs labor (L) and capital (K), which are perfectly mobile within countries but not across countries, in order to produce either trucks (T ) or bread (B). Furthermore, α is characterized by the following transformation curve:

T 2

196 + B2

81 = 1,

while preferences are described by the following utility function:

U = T 1 2B

1 2 .

12

1. Determine the equilibrium production and the relative price of bread in autarky in α.

2. Suppose that bread production is intensive in labor, while trucks production is intensive in capital. Also, suppose that α is relatively more endowed with labor than β. Can the international relative price of bread be equal to 2 when the two countries engage in free trade? What would be the equilibrium production in α, given that relative price? And equilibrium consumption?

3. In α, capitalists are opposing free trade policies and are asking for duties on imports. Are they acting in their own interest, rightfully? And what about the interest of the whole country?

Solution.

1. In autarky, we maximize utility subject to the transformation curve constraint. We obtain the following system, which delivers equilibrium consumption and production:{

B T =

81 196

T B

T 2

196 + B2

81 = 1 ⇒

{ B = 9√

2

T = 14√ 2

with an equilibrium price of PB/PT = 14/9. This is obtained by imposing the equilibrium solution into MRSB,T = T/B.

2. Yes, this would be in accordance with Heckscher-Ohlin’s theorem: a labor-abundant country will export the labor-intensive good. The free-trade equilibrium yields production of{

PB PT

= 2 = 19681 B T

T 2

196 + B2

81 = 1 ⇒

{ BP =

81√ 130

TP = 98√ 130

while consumption of { BC TC

= 12 TC =

260√ 130 − 2BC

⇒

{ BC =

65√ 130

TC = 130√ 130

3. Capitalists would oppose free-trade, rightfully so. See Stolper-Samuleson’s theorem.

13

1.7 Commercial Policies Argue whether the following statement is either true, false, or uncertain:

In a perfectly competitive market with a perfectly inelastic supply curve the state introduces a specific tax on consumers i.e. a tax on each unit of the good purchased. In this environment the effective burden of the tax falls entirely on producers and not on consumers.

Solution. True. The price paid by the consumers (inclusive of tax) after the introduction of the tax coincides with the price paid before the tax. Differently, the net price perceived by the producers decreases by an amount equal to the tax with respect to the price before the introduction of the tax. The quantity of the good exchanged is not affected though.

1.8 Commercial Policies The demand for hotel rooms in Mallorca is given by YD = 10−p (where p denotes the price of a hotel room). The supply of hotel rooms is given by YS = p.

1. Determine the equilibrium price and sales for this market both graphically and analytically. Label your graph carefully.

2. Mallorca’s prime minister plans to introduce a specific tax t = 2 on each hotel room on each unit produced (i.e. a tax on producers). Determine the new (after tax) equilibrium quantity. How much will consumers have to pay for a hotel room after the introduction of the tax? What will be the price received by producers for each unit supplied?

3. Calculate the tax revenue of the government.

4. Calculate the deadweight loss generated by the tax.

Solution.

1. See figure 7. By imposing equality between supply and demand, YS = YD, obtain:

p = 10− p ⇒ p∗ = 5, Y ∗ = 5.

2. In any new equilibrium, it must be true that

pS + t = pD

where pS is the price received by producers, while pD is the price paid by consumers. We can then substitute our demand and supply equations to find the equilibrium,

YS = pS → pS YD = 10− pD → pD

YS + 2 = 10− YD ⇒ Y ∗∗ = 4

where the prices are: pD = 6 and pS = 4. The gap between the two is exactly the tax.

14

Figure 7: Market equilibrium for hotel rooms.

3. Government revenues (which we denote as government’s surplus: GS) are computed as the tax rate multiplied by the equilibrium:

GS = t · Y ∗∗ = 2 · 4 = 8.

4. The deadweight loss is the sum of any loss that arises from departing from the equilibrium under perfect competition and no distorsions. It is the sum of the changes of: consumers’ surplus (CS), producers’ surplus (PS), and government’s surplus:

DWL = ∆CS + ∆PS −GS =

= t · (Y ∗∗ − Y ∗)

2 =

2(5− 4) 2

= 1.

1.9 Commercial Policies The market for bananas in countries Home and Foreign is characterized by the following demand and supply schedules:

DH = 120− 20PH SH = 40 + 20PH

DF = 60− 20PF SF = 20 + 20PF

1. Derive the schedules of import demand and export supply of bananas (from the excess demand and excess supply schedules) and the equilibrium price and quantity that occurs in the world market.

2. Suppose that Home country introduces an ad valorem duty of 50% on imports of bananas. Show graphically and analytically the new equilibrium in the world market.

15

Figure 8: Market equilibrium: Home, Foreign, and World.

(a) What will be Home’s tax revenue from this maneuvre?

(b) How does total surplus change?

3. Suppose instead that Foreign country helps its exporters with a subsidy of 1$ for any exported banana. Show graphically and analytically the effects of such policy.

Solution.

1. In autarky, market equilibrium occurs when supply equals demand. That is, equilibrium prices are

DH = SH → PH = 2 DF = SF → PF = 1

so that, for an international price 1 < PW < 2, Home will be an importer while Foreign will be an exporter. The import and export schedules are defined as excess demand and excess supply,

IMPH = DH − SH = 80− 40PW

EXPF = SF −DF = −40 + 40PW

and the world market equilibrium occurs when they are equal,

IMPH = EXPF → PW = 3/2,

with an associated traded quantity of QW = 20. See figure 8.

2. The introduction of the duty t creates a wedge between the international markets’ price perceived in Home country (PW,Ht ) and the one perceived in Foreign country (P

W,F t ), so that

PW,Ht = (1 + t)P W,F t .

We can use this fact to modify the import and export demand schedules as follows:

IMPHt = 80− 40 (1 + t)P W,F t

EXPFt = −40 + 40P W,F t

The equilibrium yields PW,Ft = 1.2 and P W,H t = (1 + t)P

W,F t = 1.8, with an associated traded quantity

of QWt = 8.

16

(a) The tax revenue, which is the government surplus, can be computed as:

GSt = t · PW,Ft ·QWt = 0.5 · 1.2 · 8 = 4.8.

(b) The total surplus is the sum of consumers, producers, and government surpluses. In the ex ante equilibrium (with no duty), consumers and producers surpluses were

CS =

( 6− 32

) 90

2 = 202.5

PS =

( 3 2 + 2

) 70

2 = 122.5

for a total surplus of TS = CS + PS = 325. Now, in the ex post equilibrium (with the duty), consumers and producers surpluses are

CSt = (6− 1.8) 84

2 = 176.4

PSt = (1.8 + 2) 76

2 = 144.4

for a total surplus of TSt = CSt + PSt + GSt = 325.6. Thus, Home country does benefit from the duty: the gain in government revenus is enough to offset the deadweight losses. In general, the effect on welfare of a duty is ambiguous when trading countries are of similar sizes, i.e. any country can affect world prices: who gains and who loses depends on the price elasticities of imports and exports schedules.

3. First of all, with free trade, the price perceived by consumers in both countries must be the same: PWs . This is also the price at which Home producers will sell their products. However, Foreign producers face different incentives: if they sell in Foreign country, they get PWs , but if they sell in Home country (i.e. export), they get PWs + 1, since the subsidy only applies to exports. Thus, Foreign producers will export as much as they can, before starting to sell goods in their country; on the other side, Home producers will satisfy the residual demand in their country and then they will export in Foreign country. Thus, we write the import demand and export supply schedules,

IMPHs = D H ( PWs

) − SH

( PWs

) =80− 40PWs

EXPFs = S F ( PWs + 1

) −DF

( PWs

) =− 20 + 40PWs ,

which result in an equilibrium price of PWs = 1.25 and equilibrium quantity of EXPs = IMPs = 30. These are net quantities, however. In fact, Foreign producers are actually exporting all their production:

EXPFs,gross = S F s = 20 + 20

( PWs + 1

) = 65,

Can Home demand absorb all Foreign exports? Yes. Indeed, Home demand is: DH ( PWs

) = 95.

Therefore, Foreign produces will export all their production to Home, while Home producers will supply goods both in the Home market (to satisfy the residual Home demand, equal to 30) and in the Foreign market (to satisfy all Foreign demand, equal to DF

( PWs

) = 35).

1.10 Commercial Policies Sri Lanka is a small open economy that produces cocoa beans. Being small, it takes the cocoa beans price as given, in world markets: PW = 12$. Producers in Sri Lanka have the following supply schedule

S = 50 + 15P

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while the demand schedule of consumers is

D = 400− 10P.

1. Determine the free trade equilibrium e the change in welfare with respect to autarky.

2. Suppose that Sri Lanka introduces a duty on imports of 5$ per unit. Determine the effects on interna- tional trade.

3. To improve the international competitiveness of its cocoa beans producers, the Sri Lankan government is evaluating the following policies: (i) to introduce a subsidy on production of 5$ per unit, or (ii) to introduce a subsidy on exports of 5$ per unit. Which of the two policies will be more effective in increasing exports? And which one would be preferred by Sri Lankan consumers?

Solution.

1. In autarky, P = 14 and Q = 260. In free trade, given PW = 12, Sri Lanka imports cocoa beans: IMP = 50. The increase in welfare is the area of the triangle that represents the difference between the increase in consumers’ surplus and the decrease in producers’ surplus: ∆TS = 0.5 (50 · 2) = 50.

2. The duty increases the domestic import price to P = 17. But, at this price, consumers would buy from domestic producers only, who guarantee the autarky price. Therefore, this duty is prohibitive, since it shuts down international trade.

3. A subsidy creates a wedge between the price Ps perceived by consumers and the price P ps /P es perceived by producers/exporters. The production subsidy (i) modifies the whole domestic supply schedule, shifting it downwards,

Si = 50 + 15P ps

= 50 + 15 (Ps + 5) = 125 + 15Ps

which would imply an autarky price of Ps = 11. Consumers would be happy, but producers have another option: to produce and sell in world markets at PW = 12, getting the subsidy in both cases. Thus, the equilibrium price must be the world price, with demand Di

( PW

) = 280 and supply Si

( P ps = P

W + 5 )

= 305, making Sri Lanka a net exporter of cocoa beans, with EXP i = 25. Producers receive P ps = PW+5 = 17 for any unit they sell and the government pays GSi = −s · Si = −1525$. Consider now the export subsidy (ii). The world price PW gives to producers the option to sell at P es = P

W + 5 = 17 in world markets. Hence, they will supply Sii (P es ) = 305, exporting all of these units. Consumers instead can buy in world markets at PW , with demand Dii

( PW

) = 280. Sri Lanka

becomes again a net exporter, with EXP ii = 25. The government surplus is GSii = −s ·Sii = −1525$, since all the producers are exporting. That is, the policies (i) and (ii) are in practice identical.

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2 International Finance

2.1 IS-LM Consider a closed economy that is modeled through the followings relationships:

C = 80 + cYd

I = 200 + dY

G = 220

Ld = kY − hi Ls = 220

where the coefficients are c, d, k, d = 0.25, Ld is the real money demand and Ls = Ms/P is the real money supply, i.e. the nominal money supply Ms divided by the price level P , which is assumed constant. The variable Yd is disposable income, i.e. income net of taxes and transfers. Assume that there are no taxes and transfers, T = TR = 0. Answer to the following questions:

1. Find the equilibrium of the economy, i.e. the couple E ≡ (i, Y ), and represent it in the IS-LM diagram.

2. Give an example of a couple (i, Y ) that is characterized by excess demand for money and by excess supply of goods. Describe the adjustment mechanism towards the equilibrium.

3. Starting from the equilibrium E derived in point (1), suppose that the government increases public expenditures by ∆G = 30. Find the new equilibrium E′, providing a graphical illustration and a brief written description of the process.

4. Consider instead an increase in (real) money supply: what happens to equilibrium E?

Solution. First of all, consider the goods market. The level of aggregate demand is

AD = C + I +G

= 80 + 0.25Yd + 200 + 0.25Y + 220

= 500 + 0.5Y

while aggregate supply is Y . In equilibrium, demand determines supply, i.e. Y = AD, which delivers the IS curve:

Y = 1000 (1)

which in fact is a vertical curve, since we are assuming that any of the components of aggregate demand does not depend on the interest rate.

Second, consider the money market. In equilibrium, demand and supply are equal, i.e. Ld = Ls. This delivers the LM curve:

i = Y − 880. (2)

1. The equilibrium of the economy is that value of (i, Y ) such that both the IS and the LM are satisfied. Therefore, {

IS : Y = 1000

LM : i = Y − 880 ⇒ E =

( iE , Y E

) = (120, 1000) .

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2. Any point above (or to the right of) the IS curve is characterized by excess supply of goods, since we need to decrease the interest rate or to decrease income to obtain the equilibrium in the goods market1. Similarly, any point above (or to the left of) the LM curve is characterized by excess supply of money, since we need to decrease the interest rate or increase income to obtain the equilibrium in the money market. Therefore, e.g. the couple (100, 1100) is characterized by both excess demand for money and excess supply of goods, since it is below the LM curve and to the right of the IS curve. There is no adjustment process, since the economy directly jumps to the equilibrium point E. However, in general (that is, when AD depends on the interest rate), the adjustment process would be as follows: the economy jumps immediately on the LM curve, so as to restore the equilibrium in the money market, at the initial level of income but with an higher interest rate. Then, slowly, the economy will move along the LM curve, with an higher interest rate that depresses aggregate demand, thereby reducing it, and eventually the economy will reach the equilibrium E.

3. Given the change in government expenditures, we have that

G′ = G+ ∆G = 250

and the new level of aggregate demand is AD = 530 + 0.5Y . Accordingly, the IS curve modifies to IS’, shifting to the right. To find the new equilibrium, we solve the system:{

IS′ : Y = 1060

LM : i = Y − 880 ⇒ E′ = (180, 1060) .

4. An expansionary monetary policy shifts the LM curve to the right. This induces a decrease in the interest rate. However, since the IS curve is vertical, income remains at its previous level.

2.2 Mundell-Fleming: increase in i∗

Consider a small open economy in a fixed exchange rate regime. Assume that capitals can costlessly and freely move between the economy and the rest of the world, whose variables are denoted by “∗”. Assume the following relationships hold:

C = 100 + 0.8Yd

I = 200− 200i G = 500

T = tY

X = 220R

Q = 0.1Y/R

Ld = 0.2Y − 200i Ls = 400

where t = 0.25 is the tax rate, X and Q are export and imports respectively, and R is the real exchange rate. The domestic price level is P = 20, while the international price level is P ∗ = 10. The nominal exchange rate is e = 2, in terms of domestic for foreign currency, and the international interest rate is i∗ = 10%. Answer to the following questions:

1. Find the internal equilibrium, Ei = ( ii, Y i

) .

1Notice that equilibrium in the goods market requires: Y = AD (Y, i). Thus, any chance in Y affects both supply and demand, but the former more than the latter.

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2. Compute the required change in real money supply ∆Ls to reach the external equilibrium E = (i, Y ).

3. What is the amount of capital inflows CF that occurs at the external equilibrium E?

4. Start from the external equilibrium E and suppose that the economy moves to a flexible exchange regime. What happens to the nominal exchange rate following an increase in the international interest rate to i∗new = 15%?

Solution.

1. The internal equilibrium is defined as the equilibrium that would prevail without imposing the BP con- straint2. Therefore, we only need to solve the system between the IS and LM curves. The real exchange rate is

R = eP ∗

P = 1

while disposable income is Yd = (1− t)Y . Hence, by imposing that supply equals demand in the goods market,

Y = AD = C + I +G+ (X −Q) = (100 + 0.6Y ) + (200− 200i) + (500) + (220− 0.1Y ) = 1020 + 0.5Y − 200i

we obtain the IS curve:

Y = 2040− 400i. (3)

Then, by imposing that supply equals demand in the money market, Ls = Ld, obtain the LM curve:

Y = 2000 + 1000i. (4)

Their combination gives the internal equilibrium,{ IS : Y = 2040− 400i LM : Y = 2000 + 1000i

⇒ Ei = (2.86%, 2028.57) .

2. The external equilibrium is the one prevailing with free capital mobility, i.e. by adding the BP curve, which requires i = i∗. Since the equilibrium interest rate of the internal equilibrium is below the inter- national interest rate, i.e. ii < i∗, there is an excess supply of domestic currency, since everyone wants to invest in international markets. But the central bank is committed to a fixed exchange regime and so, as long as it has sufficient foreign exchange reserves3, it participates in the foreign exchange market by buying domestic currency, thereby supporting its price. That is, domestic money supply will shrink, moving the LM to the left, eventually reaching the external equilibrium E, given by the intersection of the IS and BP curves, i.e.{

IS : Y = 2040− 400i BP : i = 0.1

⇒ E = (10%, 2000) .

2For instance, this corresponds to a situation in which the economy imposes strong capital controls. This is an example of the trilemma, known in the international economics literature as the impossibility of achieving simultaneously all of the following goals: (i) control of domestic monetary policy, (ii) control of the exchange rate, and (iii) freedom of capital mobility. In our example, the central bank is renouncing to goal (iii).

3Think about the opposite situation: ii > i∗. Would foreign exchange reserves be a constraint for the central bank? Is there, in fact, an asymmetry?

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To find the change in real money supply ∆Ls, we can use the LM curve to find the real money supply Ls that support the external equilibrium,

Ls = Ld (E) = 0.2 · 2000− 200 · 0.1 = 380,

which delivers ∆Ls = 380− 400 = −20. That is, the central bank must reduce the money supply by 20 units, in real terms, which also corresponds to 400 units in nominal terms.

3. The balance of payments identity requires that the current account (which is the trade balance) and the capital inflows sums to zero, i.e.

BP = CA+ CF = 0.

Therefore, we can compute the current account,

CA = X −Q = 220− 0.1 · 2000 = 20,

and use it to impute the required level of capital inflows:

CF = −CA = −20.

Remember that a positive current account corresponds to an excess of domestic savings over investment, CA = S − I. That is, the economy is moving its saving abroad, i.e. it is exporting capitals, and indeed the level of capital inflows is negative4.

4. In a flexible exchange rate regime, the central bank will keep real money supply unchanged, at Ls = 380. The new equilibrium E′, with the new interest rate i∗ = 15%, will then be given by the intersection of The LM and BP curves, i.e.{

LM : 380 = 0.2Y − 200i BP : i = 0.15

⇒ E′ = (15%, 2050) .

To back up the new nominal exchange rate, we need to go through the IS curve. Indeed, the real exchange rate that supports the new equilibrium E′ is such that it clears the goods market,

2050 = AD (E′)

= (100 + 0.6 · 2050) + (200− 200 · 0.15) + (500) + (

220R− 0.12050 R

) = 2000 + 220R− 205

R

which can be rewritten as

220R2 − 50R− 205 = 0. (5)

Solving equation (5) yields R = 1.0856, as the other solution is not admissible, being negative. There- fore, the new nominal exchange rate is

e = R PP∗ = 2.17.

4Does capital flow to developing countries? This is a long-standing puzzle in international economics. Indeed, developing countries are thought to have a higher return on capital, since it is relatively scarce. However, capital does not flow to them, quite the opposite.

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2.3 Mundell-Fleming: expected depreciation Consider a small open economy in a flexible exchange rate regime, with perfect capital mobility, and charac- terized by the following relationships:

C = 800 + 0.8Yd

I = 100− 200i G = 1120

T = tY

X = 400R

Q = 0.2Y/R

Ld = 0.2Y − 200i Ls = 2 ·MB

where t = 0.25 is the tax rate, X and Q are export and imports respectively, R is the real exchange rate, and MB = 390 is the real monetary base (currency plus reserves). The domestic price level is P = 2, the international price level is P ∗ = 1 and the international interest rate is i∗ = 0.1. Answer to the following questions:

1. Find the equilibrium E = (i, Y ) of the economy.

2. Suppose the economy is considering to enter in a fixed exchange rate regime. What is the level of the nominal exchange rate that allows the equilibrium in the balance of payments?

3. Suppose now that international investors fear that the domestic currency will depreciate by 10%. What happens to the equilibrium in case:

(a) the economy maintains a fixed exchange regime; or

(b) the economy suddenly devalues its currency by 10%, thereby keeping it fixed.

Solution.

1. Given that the economy is in a flexible exchange rate regime, the LM does not change but it is the IS (through the exchange rate) that bears all the adjustment. We can find the equilibrium with the intersection of the LM and BP curves, i.e.{

LM : 780 = 0.2Y − 200i BP : i = 0.1

⇒ E = (0.1, 4000) .

2. To back up the nominal exchange rate, we use the IS:

Y = (800 + 0.8 (1− 0.25)Y ) + (100− 200i) + (1120) + (

400R− 0.2Y R

) = 2020 + 0.6Y − 200i+ 400R− 0.2Y

R

that delivers R = 1 when (i, Y ) are at their equilibrium values. This corresponds to a nominal exchange rate of

e = 2,

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which guarantees the equilibrium in the balance of payments, i.e. there are no infinite inflows or outflows of capitals. In particular, the current account (or trade balance) is equal to

CA = 400− 0.2 · 4000 = −400,

which means that the economy is a net importer, i.e. the economy is receiving a net inflow of foreign capitals.

3. The fear of depreciation5 translates into a new BP condition,

i = i∗ + E [ė] = 20%.

(a) To maintain the exchange rate peg, at e = 2, the central bank has to accommodate the foreign exchange market, by buying domestic currency (and reducing the domestic monetary base) to compensate for an excess supply of domestic currency. That is, the LM shifts to the left and the new equilibrium E′ will be found by the intersection of the IS and BP, i.e.{

IS : Y = 2020 + 0.6Y − 200i+ 400− 0.2Y BP : i = 0.2

⇒ E′ = (0.2, 3′966.67) .

(b) Assume now that the central bank devalues the exchange rate, so that e′ = 2.2 and R′ = 1.1, and thereafter maintains the peg at the new level. Moreover, assume that this maneuvre eliminates the expectations of further devaluations. First, the IS curve shifts to the right, because of the devaluated currency, which improves net export. Second, also the LM curve has to shift to the right, otherwise there would be an excess demand of domestic currency. Therefore, the new equilibrium E′′ will be given by the intersection of the IS and BP, i.e.{

IS : Y = 2020 + 0.6Y − 200i+ 400 · 1.1− 0.2 Y1.1 BP : i = 0.1

⇒ E′′ = (0.1, 4′192) ,

while the level of the money supply can be found from the LM curve, by imposing the equilibrium values for Y and i, i.e.

Lsnew = 0.2Y − 200i = 818.4,

which implies an increase in the monetary base of ∆MB = 19.2. 5Let α be the (subjective) probability that the government will devalue the currency, to a new level e1 from the current level

e0. Then, the expected percentage depreciation is equal to: E [ė] = (e′ − e0) /e0, where e′ = αe1 + (1− α) e0.

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2.4 Mundell-Fleming: expected appreciation Consider a small open economy in a flexible exchange rate regime, with perfect capital mobility, and charac- terized by the following relationships:

C = 720 + 0.8Yd

I = 400− 200i G = 500

T = tY

X = 400R

Q = 0.2Y (5−R) Ld = 0.4Y − 400i Ls = 1560

where t = 0.25 is the tax rate and the other variables are defined as usual. The domestic price level is P = 8, the international price level is P ∗ = 4 and the international interest rate is i∗ = 0.1. Answer to the following question.

1. Find the equilibrium and the associated nominal exchange rate.

2. Suppose that international investors hold the expectation of an appreciation of 2.5% of the domestic currency. What happens to the equilibrium?

Solution.

1. The equilibrium is given by the intersection of the LM and BP, i.e.{ LM : 1560 = 0.4Y − 400i BP : i = 0.1

⇒ E = (0.1, 4′000) .

To determine the exchange rate, use the IS curve,

Y = 720 + 0.8 (1− 0.25)Y + 400− 200i+ 500 + 400R− 0.2Y (5−R) ,

which gives R = 2.5 and in turn e = RP/P ∗ = 5 when evaluated at the equilibrium values in E.

2. The new BP curve is

i′ = i∗ + E [ė] = 7.5%.

Since the economy starts from equilibrium E, with i = 0.1 > 0.075 = i′, there is now an excess demand of domestic currency. The central bank does not control the exchange rate; hence, the domestic currency effectively appreciates. The new equilibrium is given by the intersection of the LM and BP, i.e.{

LM : 1560 = 0.4Y − 400i′

BP : i′ = 0.075 ⇒ E′ = (0.075, 3′975) ,

which is characterized by a lower level of income, despite the lower interest rate. That is, the decrease in net export (induced by the currency appreciation) must have been greater than the increase in investment (induced by the lower interest rate). Solving for the exchange rate, using the IS curve, obtain R′ = 2.47 and in turn e′ = 4.94.

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