17
Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 1 of 17 Problem Set #1 - Answers 1. Use indifference curves and a curved transformation curve to illustrate a free trade equilibrium for a country facing an exogenous international price. Then show what happens if that exogenous price changes in the direction of raising the relative price of the country’s exported good. Such a change is customarily called an “improvement” in the country’s “terms of trade.” Is this terminology necessarily appropriate? As shown at the right for a country that initially exports good X, an exogenous rise in the price of X steepens the price line. If the country is behaving optimally in all other respects, so that it is on its community indifference curve after the price change, then the country as a whole is definitely made better off. This is shown by its reaching a higher indifference curve, but it could also be inferred from the fact that the new price line, which constitutes the country’s aggregate budget line, passes outside the initial consumption point, C, making it possible to consume more of both goods. The terminology “improvement in the terms of trade” is therefore appropriate. However, it should not be thought that any change in relative prices in favor of the country’s export good is necessarily beneficial for it. In the example here, the price change was exogenous. One could easily construct a case where, if the country were large enough in world markets to influence world prices, a change in something other than prices (such as a fall in the country’s resources) could cause it to supply less to the world market. This would then cause an increase in the world price, and while the price change alone could be thought of as beneficial, it could still be true that the country would lose from the combined effects of the loss in resources and the induced “improvement” in the terms of trade. Y X P P’ C’ C

Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

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Page 1: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 1 of 17

Problem Set #1 - Answers 1. Use indifference curves and a curved transformation curve to illustrate a free trade

equilibrium for a country facing an exogenous international price. Then show what happens if that exogenous price changes in the direction of raising the relative price of the country’s exported good. Such a change is customarily called an “improvement” in the country’s “terms of trade.” Is this terminology necessarily appropriate?

As shown at the right for a country that initially exports good X, an exogenous rise in the price of X steepens the price line. If the country is behaving optimally in all other respects, so that it is on its community indifference curve after the price change, then the country as a whole is definitely made better off. This is shown by its reaching a higher indifference curve, but it could also be inferred from the fact that the new price line, which constitutes the country’s aggregate budget line, passes outside the initial consumption point, C, making it possible to consume more of both goods.

The terminology “improvement in the terms of trade” is therefore appropriate. However, it should not be thought that any change in relative prices in favor of the country’s export good is necessarily beneficial for it. In the example here, the price change was exogenous. One could easily construct a case where, if the country were large enough in world markets to influence world prices, a change in something other than prices (such as a fall in the country’s resources) could cause it to supply less to the world market. This would then cause an increase in the world price, and while the price change alone could be thought of as beneficial, it could still be true that the country would lose from the combined effects of the loss in resources and the induced “improvement” in the terms of trade.

Y

X

P P’

C’

C

Page 2: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 2 of 17 2. The diagram below shows a free-trade equilibrium between two countries, A and B.

Technological change now causes country A to be able to produce exactly 10% more output of good X than it could before, for any given input of resources.

a) Determine what this will do to the two countries’ transformation curves, to their offer curves, and therefore to the world equilibrium prices.

The change in technology does nothing to Country B’s transformation curve, but it stretches A’s to the right by 10%, as shown in the broken curve at the right. (The shift shown is somewhat more than 10%, to make it easier to see.) At initial prices (shown by the parallel price line tangent to the new transformation curve), output of X rises and output of Y falls (since the expansion flattens the transformation curve by 10% at the point where Y is unchanged). Consumption moves to the new tangency of the price line with an indifference curve, but without assumptions about preferences this could entail a fall in demand for one of the goods if it is an inferior good. The case shown has demand rising for both, in which case desired trade expands, since production of Y falls while consumption of Y rises. In general, if good Y is sufficiently inferior, its consumption could fall by more than production and desired trade would contract.

What happens to the offer curves can be inferred from this change in trade at constant prices. Country B’s offer curve is unchanged, while Country A’s expands outward from the origin if good Y is a normal good, as drawn, but could contract inward if Y is sufficiently inferior.

This in turn causes the world price of good X to change, falling in the case shown of Y a normal good, but possibly rising if Y is sufficiently inferior for the A offer curve to contract.

Y

X

Country A

p0

p0

Y

X

A

A’

B

p0 p1

Page 3: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17

b) Without further assumptions about preferences, are there any of these variables the direction of change in which cannot be determined unambiguously?

As already indicated, the direction of shift of country A’s offer curve is ambiguous, as is therefore the direction of change in the world relative price.

c) If you assume that preferences are homothetic, can any of these ambiguities be resolved?

Yes. With homothetic preferences (or even just with good Y a normal good, which is weaker than homotheticity), country A’s offer curve must expand as shown and the world relative price of A’s export good, X, must fall.

d) Show the new equilibrium in both countries for the case of homothetic preferences.

Since we know that the world price of X falls, we can be sure that country B’s output of X falls and its consumption of both goods rises (again, due to homotheticity). Thus B’s imports of X must rise, although its exports of Y may rise or fall. The case drawn has Country A still reaching a higher indifference curve than it did before the expansion, but this is not necessary. Country A could lose from its own expansion, the case of Bhagwati’s “Immizerizing growth.” (See Bhagwati, Jagdish. 1958. "Immiserizing Growth: A Geometrical Note," Review of Economic Studies 25, (June), pp. 201-205.)

Y

X

Y

X

Country A Country B

Page 4: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 4 of 17 3. Consider a country engaged in production and trade of an arbitrary number of goods.

Assume that there are no transport costs and that producers are competitive profit maximizers. Assuming that trade is free, show that if the vector of world prices, pw, changes by a vector Δpw, then the vector of the resulting changes in outputs, ΔX, will be nonnegatively correlated with Δpw (assuming that pw is always normalized to lie on the unit simplex).

From maximization

FX Xp Xp w0w ∈∀≥ 00

and similarly p1

w 1X ≥ p1wX ∀X ∈ F

where

p1

w = p0w + Δ wp

1X = 0X + ΔX

Letting X = 1X in the first inequality and X = 0X in the second, we can add them together to get

p0

w 0X + p1w 1X ≥ p0

w 1X + p1w 0X

or

p0

w ( 0X - 1X ) + p1w ( 1X - 0X ) ≥ 0

(p1w - p0

w )( 1X - 0X ) ≥ 0 Δ wp ΔX ≥ 0

If p0w and p1

w are both on the unit simplex (which means that the elements of both vectors add up to one), then the sample mean of the elements inΔ wp is

Page 5: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 5 of 17

Mean(Δ wp ) = 1n∑Δ j

wp

= 1n∑(p1

wj - p0

wj )

= 1n∑ p1

wj - ∑ p0

wj

#$ %&

= 1n[1 - 1] = 0

Therefore the correlation betweenΔ wp , andΔX has the same sign as their inner product:

Sign[Cor(Δ wp ,ΔX)] = Sign[Δ wp ΔX] ≥ 0

(See Deardorff, “The Correlation Between Price and Output Changes When There Are Many Goods,” Journal of International Economics 10, August 1980, pp. 441-43.)

4. a) Derive and draw the transformation curves of the two economies whose endowments and technologies are those described below. Each has a fixed endowment of labor – its only factor of production – and can produce two goods, X and Y, using the following constant amounts of labor per unit of output:

Per-unit labor requirement for producing

Endowment of Labor

X

Y

Country A 60 1 2

Country B 120 2 3

Page 6: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 6 of 17

b) How much of goods X and Y will be produced and consumed in autarky in these two countries, and what will be their relative prices, assuming that demanders always insist on consuming them in fixed proportions of 1 unit of X for each unit of Y?

A: X=Y

X + 2Y = 60 3X = 60

X = Y = 20 pX /pY = 2

B: X = Y 2X + 3Y = 120

5X = 120 X = Y = 24

pX/pY = 2/3

c) Derive and draw the world transformation curve.

Y

X

Country A Y

X

Country B

60 60

30 40

Page 7: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 7 of 17

d) Derive the free trade equilibrium relative price of X and Y, plus the equilibrium quantities of the goods produced, consumed, and traded.

From the picture in part (c), Y=X requires producing in the upper segment of the world transformation curve. Thus with free trade

XpYp = 12

and outputs in country B are B

X = 0, BY = 40

World outputs: W

X = WY = A

X = AY + 40

AX + 2 A

Y = 60 W

X + 2( WX - 40) = 60

3 WX = 60 + 80 = 140 W

X = WY = 140

3 = 46 2

3

∴ AX = 46 2

3, A

Y = 46 23

-40 = 6 23

A’s income in units of Y = 30:

AX = AY , 12

AX + AY = 30

32

AX = 30, AX = 23

30 = 20, AY = 30 - 10 = 20

∴ AX = AY = 20

B’s income in units of Y = 40:

Y

X 60

70

120

40

60

Y=X

Page 8: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 8 of 17

BX = BY , 12

BX + BY = 40

32

BX = 40, BX = 23

40 = 26 23

= BY

∴ BX = BY = 26 23

Trade:

A exports 26 23

units of X.

B exports 13 13

of Y.

5. In the Dornbusch, Fischer, and Samuelson Ricardian model with transport costs, suppose that there is an increase in the cost of transportation, and suppose also that other conditions happen to be such that the wage ratio, ω, remains unchanged as a result. Determine the effects of this change on

a) the pattern of trade and specialization (which goods each country produces, exports, and imports,

b) the domestic prices of goods in both countries, and

c) real wages (the ratio of the wage to an arbitrary index of domestic prices). Hint: for the real wage, see if you can show that the nominal wage either rises or falls relative to the prices of all goods. If so, then the exact price index does not matter.

Page 9: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 9 of 17

The initial equilibrium is shown below by the solid curves for gA, (1/g)A, and the associated zA and zB. The initial wage ratio is ω0, with A producing [0,zA) and B producing (zB,1]. A exports [0,zB), B exports (zA,1], and (zB,zA) are not traded.

a. When the cost of transportation increases, the fraction of each

good that survives being traded, g, falls from g to g′. The curves shift to the positions shown by the dashed curves g′A and (1–g′)A. Each country now produces a larger range of goods, [0,z′A) and (z′B,1] respectively, while each country exports a smaller range of goods, [0,z′B) and (z′A,1].

b. With nominal wages unchanged, the prices of domestically produced goods in each country are unchanged, while the prices of goods that continue to be imported rise by the increase in transport costs. The prices of goods that were previously imported but no longer are also rise but by less than the increase in transport costs.

c. Since nominal wages are unchanged, and since all prices in each country either remain unchanged or rise, the real wage in each country falls.

gA

Ag)/1(

Ag ʹ′

Ag )/1( ʹ′

Az Bz Bzʹ′ Azʹ′ 1 0

Page 10: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 10 of 17 6. In the Edgeworth Box below, point E is one point on the efficiency locus for the two

industries, X and Y, whose representative isoquants are as shown. Assuming that the production functions for X and Y are linearly homogeneous, derive the rest of the efficiency locus.

Ans: By expanding and contracting the two isoquants radially with respect to their origins, you can find other points on the efficiency locus. Note that the resulting locus, OXAEBOY, is not concave to the diagonal. You should check, however, that the resulting transformation curve, though piecewise linear, is concave to its origin.

X=X0

Y=Y0

E

OX L

T

OY

X=X0

Y=Y0

E

OY

OX L

T

A

B

Page 11: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 11 of 17 7. The following are the equations used by Jones (1965) to determine factor-price

changes from commodity-price changes: XTXLX prw ˆˆˆ =+θθ

YTYLY prw ˆˆˆ =+θθ

where θLi +θTi =1 , i = X, Y. Solve these equations for w and r in terms of pXˆ and pYˆ . From your solution, show that if good X is relatively labor intensive, so that LXθ > LYθ , then a rise in Xp relative to Yp will raise w relative to both prices and

reduce r relative to both prices. Ans:

( ) ( )

( ) ( )

( )YXLYLX

TXX

LYLX

YXTXXTXTY

LYLX

YTXXTY

LXLYLYLX

YTXXTY

TXLYTYLX

YTXXTY

TYLY

TXLX

TYY

TXX

ppp

ppppp

pppppp

w

ˆˆˆ

ˆˆˆˆˆ

11ˆˆˆˆˆ

ˆ

ˆ

−−

+=

−+−=

−=

−−−

−=

−==

θθθ

θθθθθ

θθθθ

θθθθθθ

θθθθθθ

θθ

θθ

θ

θ

(1)

or

( ) ( )

( ) YYXTXTY

TY

TXTY

YTXTYYXTY

ppp

pppw

ˆˆˆ

ˆˆˆˆ

+−−

=

−+−=

θθθ

θθθθθ

(2)

If 0ˆˆ >− YX pp and LYLX θθ > (and hence TXTY θθ > ) , then From (1): Xpw ˆˆ > From (2): Ypw ˆˆ > Similarly

( ) ( )

( )YXLYLX

LYY

LYLX

YXLYYLYLX

LYLX

XLYYLX

TYLY

TXLX

YLY

XLX

ppp

ppppppp

r

ˆˆˆ

ˆˆˆˆˆˆˆ

ˆ

−−

−=

−−−=

−==

θθθ

θθθθθ

θθθθ

θθ

θθ

θ

θ

(3)

Page 12: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 12 of 17 or

( ) ( )

( ) XYXLYLX

LX

LYLX

XLYLXXYLX

ppp

pppr

ˆˆˆ

ˆˆˆˆ

+−−

−=

−+−=

θθθ

θθθθθ

(4)

If 0ˆˆ >− YX pp and LYLX θθ > , then From (3): Ypr ˆˆ < From (4): Xpr ˆˆ < 8. Use the Lerner (unit-value-isoquant) diagram to work out the effects of a price

change in the 2×2 Heckscher-Ohlin model as follows. Consider a small country that faces prices x

0p and y0p for goods X and Y. Suppose that its endowments of labor

and land are such that, at these prices, it produces exactly $2 worth of each good. Work out what happens when the price of good X rises by 20%, to x

1p , the price of good Y remaining constant. Assume that good X is the relatively labor-intensive good. Determine, if possible, the direction of change in the following variables:

i) The nominal wage. ii) The ratio of the nominal wage to the price of good X. iii) The rental price of land. iv) The ratio of land to labor used in producing good X. v) The ratio of land to labor used in producing good Y. vi) The average ratio of land to labor employed in both industries

together. vii) The output of good X. viii) The output of good Y.

Page 13: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 13 of 17 Ans: First draw the Lerner diagram for the initial situation.(Draw it big enough to see what’s going on.)

L

T

O

0/1 XpX =

00 /2 XpX =

0/1 YpY =

00 /2 YpY =

),( TL 0Yt

0Xt

Page 14: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 14 of 17 Now construct the new equilibrium with the price of X 20% higher: 01 2.1 XX pp = . This shifts the unit-value X-isoquant approximately 20% in toward the origin (actually by 1/6, though I’ve drawn it shifting more, for clarity). From that construct the new common tangent, the rays for 1

Xt and 1Yt , and the parallelogram showing the allocation of factors

that will maintain full employment of each. Isoquants through these allocations indicate the new outputs X1 and Y1.

From this figure you can read the results for parts (iv)-(viii). The ratio of land to labor has increased in both industries, although the average ratio of land to labor employed has not changed (since it must continue to equal the unchanged endowment). That is possible because more of both factors is employed in the X industry and less of both in the Y industry, resulting in a larger output of X and a smaller output of Y.

L

T

O

0/1 XpX =

00 /2 XpX =

0/1 YpY =

00 /2 YpY =

),( TL 1Yt

0Xt

1/1 XpX =

0Yt

1Xt

1Y

1X

Page 15: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 15 of 17 To get effects on factor prices, use the intercepts of the isocost lines to measure their reciprocals in nominal terms – that is, in units of the currency in which the prices have been expressed. From these it follows immediately that the nominal wage has risen and the nominal rental on land has fallen. This is also a fall in the real rental, since one price is unchanged and the other has risen.

To find the effect on the real wage, construct a wage, w~, which is above w0 by just the percentage of the price increase. This is done by constructing the dotted line parallel to the initial isocost line but tangent to the new unit-value X-isoquant. It is then clear that the new wage, w1is above w~ and has therefore risen not only relative to the constant price of Y but relative also to the increased price of X. Thus the real wage of labor is increased.

L

T

O

0/1 XpX =

00 /2 XpX =

0/1 YpY =

00 /2 YpY =

),( TL 1Yt

0Xt

1/1 XpX =

0Yt

1Xt

1Y

1X

1/w0 1/w1

1/r0

1/r1

1/w~

Page 16: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 16 of 17 9. The Lerner diagram below shows an initial equilibrium, in the 2×2 Heckscher-Ohlin

model, for a small-open economy facing fixed world prices of goods X and Y, X0p

and Y0p . Its initial endowments of the two factors, unskilled labor U and skilled labor

S, are shown by point E. Suppose now that some unskilled workers become skilled, moving the endowment point first to E’ and then to E’’. Determine the effects of these changes on outputs of both goods and on factor prices.

0/1 XpX =

0/1 YpY =

U

S

O

E

E′

E′′

0Yt

0/1 YpY =

U

S

O

E

E′

E′′

0/1 XpX =

0Xt

1Yt

X0

X1 X2=0

Y0

Y1

Y2

1/w0 1/w2

1/s2

1/s0

Page 17: Problem Set #1 - Answers - alandear/courses/641/ps/ps1a.pdf · Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 3 of 17 b) Without further assumptions about preferences,

Econ 641 Alan Deardorff Fall Term 2014 Problem Set #1 - Answers Page 17 of 17 Constructing parallelograms from points E and E′, this initial allocation and the allocation for endowment E′ are both found inside the diversification cone, i.e., between rays 0

Xt and 0Yt . For endowment E′′, since it is outside the cone, only good Y is produced

using the expanded version of the Y isoquant through E′′. Using the distance from the origin along rays 0

Xt and 0Yt to measure output, one sees that output of X falls from X0 to

X1 and then to X2=0. Output of Y rises from Y0 to Y1 to Y2.

The wage of unskilled labor, w, is initially w0 and remains there when the endowment moves to E′. But when the endowment moves outside the cone to E′′, then the unskilled wage rises to w2. Likewise, the wage of skilled labor, s, remains at s0 for E and E′, but then falls to s2 at E′′.