Marshall Learner Conditions

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ECON 3013/8015: International Economics - Honours and Graduate

Topic 2: Elasticities, Stability, and the Transfer Problem

Elasticity of the Offer Curve

The elasticity of a country’s offer curve is equal to the absolute value of its elasticity

of demand for imports, defined with respect to the relative price of the import good.

That is, letting M denote the volume of imports, X the volume of exports, and PM/PX

the price of the import good relative to the price of the export good, the elasticity of

the offer curve is given by

X

M

X

M P

P

P

Pd

M

dM

−=η

Since, at each point on the offer curve, the price ratio P M /P X is just equal to the

volume of exports offered divided by the volume of imports demanded, we can

replace the above with

M

X

M

X d

M

dM

−=η

Then, noting that, for small changes, the proportionate change in a ratio is

approximated by the proportionate change in the numerator minus the proportionate

change in the denominator, we can write

1.1

or +−=

−

−= X

M

dM

dX

M

dM

X

dX M

dM

η η

Now consider a point such as Q on the offer curve shown in Figure 1. dX /dM is the

reciprocal of the slope of the offer curve, BA/QA, while M / X is equal to QA/0A, so

we have

0B

A0 thereforeand

0A

B0

0A

0ABA1

0A

QA

QA

BA1==

+−=+−= η

η .

Note that this elasticity value will be greater than 1 at points along the positively

sloped (‘elastic’) segment of the offer curve (such as point Q in Figure 1), will be

equal to 1 at the point where the offer curve ‘bends backwards’ (if it does), and will

be less than one along any negatively sloped (‘inelastic’) segment of the offer curve.

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FIGURE 1Imports ( M )

0

R

B A

Q

Exports ( X )

Elasticities and Stability of Equilibrium: the Marshall-Lerner Condition

When a good is in excess demand in world markets, its relative price will increase.

Stability of the international trade equilibrium requires that this price change reduces

the excess demand. An unstable equilibrium would be one where a rise in a good’s

relative price increased excess demand for it, causing a further price rise, leading to

greater excess demand, etc.

In the offer curve diagram of Figure 2, point Q is a stable equilibrium. At a relative

price of wheat to cloth that is slightly less than the equilibrium terms of trade priceratio, such as is given by the slope of the line 0K, the foreign demand for wheat

imports, 0A, exceeds the home supply of wheat exports, 0B, so there is world excess

demand for wheat (and excess supply of cloth). This will cause the world price of

wheat to rise relative to the price of cloth, rotating the terms of trade line back

towards the equilibrium value, given by the slope of 0Q, and eliminating the excess

demand. In order to get this stable outcome, each offer curve needs to cut the other

from its ‘concave’ side. We discuss the required condition more formally shortly.

In the case shown in Figure 3, the equilibrium at point Q is unstable, with each offer

curve cutting the other from its ‘convex’ side. Here, excess demand for wheat occurs

at a relative price of wheat to cloth that is greater than the equilibrium value (forexample, with PW/PC as given by the slope of line 0J, the foreign demand for wheat

imports exceeds the home supply of wheat exports by the amount FG). Since this

excess demand will raise the world relative price of wheat, it will further increase

PW/PC above the equilibrium value. Thus, starting from an unstable equilibrium, any

small disturbance will result in a continuous movement away from that equilibrium

point. Note, however, that although the equilibrium at point Q is unstable, the extent

of movement away from this point in response to a shock is bounded by the existence

of two equilibria, at S and T, that are stable. Thus, the potential instability problem is

not that relative prices go on changing without limit but that there may be very large

changes in world prices before a stable equilibrium is reached. In principle, it would

be possible for there to be many intersections of two offer curves, alternating betweenstable and unstable equilibria.

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Cloth

(Home Imports)FIGURE 2

K

R*

Q

B A

Wheat (Home Exports)

R

0

Cloth

(Home Imports)FIGURE 3

T

R*

R

Q

F G

S

J

0Wheat (Home Exports)

The requirement for stability of the international trading equilibrium is the same

Marshal-Lerner condition that should be familiar from the study of open economy

macroeconomics. In that context, it is the condition required for a real exchange rate

appreciation (defined as an increase in the relative price of home to foreign goods) to

result in a reduction of net exports. The formal derivation is given below.

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Let p denote the terms of trade relative price cloth to wheat (PC/PW), M denote the

volume of home imports of cloth, and M* denote the volume of foreign imports of

wheat. The absolute home and foreign elasticities of import demand are then

( )

( ) p

pd

* M

*dM

p

dp

M

dM

1

1*and −=−= η η

For small changes,( )

( ) p

dp

p

pd −=

1

1, so

p

dp

M

dM

*

** =η

Foreign exports of cloth are equal to foreign imports of wheat multiplied by the

relative price of wheat (1/ p), so world excess demand for cloth is equal to M-M*/ p

and the stability condition that a rise in the relative price of cloth reduces excess

demand for that good is

( )0

*<−

dp

p M d

dp

dM (1)

At the international trade equilibrium, M*/ p = M so, dividing the numerators of (1) by

M (or M*/ p) and the denominators by p, we have

( ) ( )0

/

*/*

/

/<−

pdp

p M p M d

pdp

M dM

and hence

( ) ( )1

**or 0

/

**

/

/−<<

−−

pdp

M dM

pdp

M dM

pdp

pdp M dM

pdp

M dM

where the left-hand side is equal to -η and the right-hand side is equal to η *-1.Rearranging, we get the Marshall-Lerner condition for stability of equilibrium that

η + η * > 1

i.e. that the sum of the absolute values of the home and foreign elasticities of demand

for imports should be greater than unity.

Note that the elasticity of demand for imports can be decomposed into a pure

substitution elasticity (moving around a constant indifference curve) plus the marginal

propensity to consume the import good, which is also the marginal propensity to

import since variations in consumption can only come from variations in the volume

of imports. Denoting the absolute value of the substitution elasticity as c and themarginal propensity to import as m, we can write the Marshall-Lerner condition as

c + m + c* + m* > 1 or (c+c*) + m > 1-m* (2)

where 1-m* is the foreign marginal propensity to consume that country’s export good

(cloth). Note that c ≥ 0 and c* ≥ 0, since substitution (if any) can only be away from

cloth and towards wheat when the relative price of cloth increases. m and 1-m* may

be greater or less than zero, depending on whether cloth is a normal or inferior good

in consumption in the country in question.

The intuition for the result given by (2) is as follows. At equilibrium, home cloth

imports are equal to foreign cloth exports. A small increase in the relative price of

cloth (an improvement in the foreign terms of trade and a deterioration in the hometerms of trade) will, to a first order approximation, raise foreign real income and

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reduce home real income by the same amounts. If m > 1-m*, the fall in home income

will reduce home cloth consumption by a greater amount than the rise in foreign

income increases foreign cloth consumption, so overall world demand for cloth will

decrease even if the two substitution elasticities are both zero (if cloth and wheat are

perfect complements in both countries). If, however, m < 1-m*, the income effects of

the price change will tend to raise world cloth demand, by a greater amount the largeris the difference in the marginal propensities, and this effect will need to be offset by

a sufficiently high value of the combined substitution elasticities in order for world

excess demand to fall.

The Transfer Problem

Attention was first drawn to the transfer ‘problem’ by J.M.Keynes in discussing the

likely effects of the severe reparation payments that were imposed on Germany after

World War I. He argued that Germany would not only bear the direct cost of these

payments but would also suffer a terms of trade deterioration that would exacerbate

the adverse consequences for its population. This stimulated a debate, in particularwith Bertil Ohlin, over the question whether a country making a transfer, such as a

reparation payment or the payment of foreign aid, would suffer an additional (or

‘secondary’) burden due to a terms of trade deterioration. In the end, it turns out that

the analytical answer is relatively straightforward, but indeterminate, although many

argue that the usual presumption should still be that Keynes was right.

In our two country context, a transfer takes place when one country gives some of its

income to the other. The effect on the donor country’s terms of trade depends on

whether the transfer increases or reduces world demand for its export good. If world

demand for that good is increased, then its relative price will rise and the donor

country will get the ‘secondary benefit’ of an improvement in its terms of trade that

will partly offset the direct cost of the transfer. If world demand for the export goodfalls, then the terms of trade will deteriorate and the country will suffer the sort of

‘secondary burden’ about which Keynes was concerned.

The effect of the transfer on world demand for the export good depends on the

marginal propensities of the two countries to consume that good. Suppose that the

home country, which exports wheat, makes a transfer to the foreign country of an

amount that would, at original relative prices, purchase T units of wheat. Foreign

country income rises by this amount and its consumption of wheat increases by its

marginal propensity to consume wheat (its marginal propensity to import, m*) times

the increase in income. Home country income falls by the amount T and its

consumption of wheat decreases by its marginal propensity to consume wheat (oneminus its marginal propensity to consume cloth and, hence, 1-m) times the reduction

in income. Thus, the change in world demand for wheat, at constant prices, is

[m*-(1-m)]T . The world relative price of wheat will rise or fall according to whether

this is positive or negative. Thus, the donor country will get the secondary benefit of

a terms of trade improvement if

m* > (1 – m) so that m + m* > 1

and will suffer the secondary burden of a terms of trade deterioration if

m* < (1 – m) so that m + m* < 1

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Note, by referring back to (2), that the condition required for the donor country to

obtain a secondary benefit also guarantees stability of the international trade

equilibrium, although this condition is by no means necessary for stability.

Can it be Better to Give than to Receive?

Although there is a general presumption that countries will tend to have a higher

marginal propensity to consume their export goods than do foreigners (see Krugman

and Obstfeld, pp.101-102) so that a secondary burden is the more likely outcome, the

possibility that the donor country might gain a secondary benefit raises the question

whether, at least in principle, this could be sufficiently large that it outweighs the

direct cost of the transfer and leaves the donor country better off than in the absence

of the transfer.

In fact, with two countries (but not necessarily in a multi-country setting) and if the

two goods are not perfect complements, the donor must always be made worse off by

making a transfer – that is, the secondary benefit from a terms of trade improvement

cannot be large enough to offset the direct cost. To see this, we ask whether a termsof trade change that would be large enough to leave the donor country exactly as well

off as it was initially could be an equilibrium value for world prices. If the donor

country’s welfare were unchanged, the welfare of the recipient country would also be

unchanged (to a first order approximation for a small transfer). If neither country’s

well-being changes, there will be no income effects resulting from the combination of

transfer and terms of trade change, but the substitution effects due to the latter must

cause both countries to consume less of the donor country’s export good. With a

fixed world supply of that good, the lower demands must result in world excess

supply and, therefore, the price ratio that would be required to leave the donor country

with unchanged welfare cannot be an equilibrium value of world relative prices. The

equilibrium must involve a lower relative price of the donor’s export good.

FIGURE 4

C 1F

G

Q•

V

W

HWheat

C 1H

R T

S

F • E

0F

Cloth

0H

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The argument is illustrated in Figure 4, where the horizontal and vertical axes

measure the total world endowment of wheat and cloth, respectively, with the home

country’s endowment and consumption being measured from the origin 0H and the

foreign country’s endowment and consumption being measured from the origin 0F.

Suppose the initial endowment point is at E and that the two countries engage in trade,

with Home exporting wheat and Foreign exporting cloth, to arrive at the post-tradeequilibrium point Q, where Home is on its community indifference curve C 1

H and

Foreign is on its community indifference curve C 1F (these indifference curves have

been ‘pulled apart’ slightly from their point of tangency so as to make the diagram

clearer). Note that, if we included the two countries’ offer curves in the diagram, they

would have their origins at point E and intersect at point Q, with the equilibrium terms

of trade price ratio of wheat to cloth being given by the slope of the line EQ.

Now imagine that Home transfers the quantity of wheat EF to Foreign, so that the new

endowment point from which the countries trade lies at F. In order for Home to be as

well-off as at the original equilibrium, remaining on C 1H, the relative price of wheat

would need to rise so that it was equal to the slope of the line FG. At this terms oftrade, Foreign would also remain on C 1F (at least if the lump-sum transfer were

sufficiently small – we have made it relatively large in the diagram so that we can see

what is happening). However, both countries will have substituted away from

consumption of wheat, with Home’s consumption point moving to V and Foreign’s to

W. Thus, Home will be demanding only 0HR wheat and Foreign will be demanding

only 0FS wheat, leaving a total world excess supply of the amount RT. This excess

supply will force the relative price of wheat below that given by the slope of FG, so

the terms of trade can never improve sufficiently to make Home as well-off as before

the transfer.

Immiserising GrowthAlthough a country cannot gain by giving away some of its endowment and cannot

lose by being the recipient of a gift from another country, this is not to say that a

country cannot gain by destroying some of its endowment or lose as a result of having

some additional endowment bestowed upon it. The last case is referred to as

immiserising growth, where a country is made worse off by an expansion in its

production possibilities or, in our current case, by an exogenous increase in its

endowment.

The essential difference between the destruction of endowment and its transfer to

another country is that the former changes world supply, whereas the latter does not.

When the home country destroys some of its export good, this reduces total worldsupply of that good. So long as Home’s marginal propensity to consume the export

good is less than 1 (ie. the import good is strictly normal in consumption), home

consumption of the exportable will not fall by as much as world supply has been

reduced, so there will be excess world demand at unchanged prices and this will

necessarily improve the home country’s terms of trade. This change in the terms of

trade will reduce welfare in the foreign country.

Consider again the possibility that the terms of trade improve sufficiently that the

home country is exactly as well-off as initially. In Home, there will be a substitution

effect that reduces consumption of the export good while, in Foreign, both the

substitution and income effects will reduce consumption of that good. But worldsupply has also fallen and there is now no necessary presumption that these reductions

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2. The economics of the transfer problem draws attention to the fact that an

international transfer of purchasing power (such as a reparations payment or

the payment of foreign aid) may impose a "secondary" burden or benefit on

the transferor.

(a) Explain what is meant by a "secondary" benefit and state and explain

the conditions necessary for the transfer to produce such a benefit. If

both the home and the foreign country have identical, homothetic

preferences, what will be the effect of a transfer from the home country

to the foreign country on the terms of trade?

(b) In the two country model we have been examining, is it possible for

the secondary benefit to be so large that the transferor is actually made

better off by the transfer (and the transferee made worse off)?

3. Countries A and B are each endowed with quantities of cloth and wheat. At

the free trade equilibrium, A exports wheat and B exports cloth. Nowconsider each of the following changes to the initial setting.

(i) an exogenous increase in B’s endowment of wheat by Z units;

(ii) an exogenous reduction in A’s endowment of wheat by Z units; and

(iii) a transfer of purchasing power from A to B of an amount sufficient to

purchase Z units of wheat at initial prices.

Without any further information, to what extent is it possible to rank these

changes in terms of their effects on economic welfare in country A and B? If

you knew that country A would benefit from (ii) would that allow a more

definitive ranking?

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Marshall Learner Conditions