Homepage International Economics

* International Economics*, Robert A. Mundell, New York: Macmillan,
1968, pp. 43 - 53.

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The propositions established in Chapter 2 were derived from a model consisting of only two goods and two countries. The traditional use of this model in international trade literature is based on a belief that it suggests "theorems which may be seen to admit of extension to more concrete cases" ([12], p. 31). In this chapter general results are derived and it is shown that in at least one important case the propositions previously established for two countries hold for an arbitrary number of countries.

Suppose that there are *n* + 1 countries, and let all prices and balances
of payments be expressed in terms of the currency of country 0. If prices
(of export goods or currencies) are flexible, then the balance of each country
depends on all prices. The conditions of equilibrium can be written as follows:

where *alpha *is a parameter representing a particular
policy.^{2}
Note that only n equations are required, since we are dealing with
*normalized *prices; the balance of payments of country 0 follows from
*Cournot's
law*.^{3}

We wish to determine the effects of a shift in the policy a on the equilibrium
set of prices so we differentiate equations (1) with respect to *alpha*.
With appropriate choice of units of currencies or goods we get

where *b*_{ij} __=__
*deltaB*_{i} /
*deltaP*_{j} is the change in the balance of
the ith country due to a change in the price of the goods of the *j*th
country. Solving for *dP*_{i} / *d alpha*
we obtain

and where *Delta*_{ji}, is the cofactor of the
*j*th row and the *i*th column of *A*.

Equations (7) provide a general framework into which specific policy changes
can be introduced. But to evaluate any of the signs,
*dP _{i} / d alpha*, it is necessary to know the
values, or at least the signs, of two kinds of terms: the coefficients

To make progress some restriction on the signs of the elements
*b*_{ij} in the basic determinant *Delta*
is required. The most interesting special case, for present purposes, is
that where *b*_{ij} > 0 for *i not= j*.
This assumption means that an increase in the price of the exports of any
country, other prices being held constant, improves the balance of payments
of every other country; it also implies, by Cournot's law, that a rise in
the price level in one country worsens that country's balance of payments.
From this assumption flow two important deductions: (1) the system is stable
under the usual dynamic
postulates^{4}
([1], [19], and [79]) and (2) every ratio is negative ([68], pp. 49-51).
With this information and the additional assumption that goods are *net*
as well as *gross* substitutes, we can evaluate the direction of change
in the terms of trade resulting from many policy changes.

Suppose that output and expenditures in country 0, the numeraire country,
increase by *dX*_{0}*. At constant prices, assuming
no inferior goods, inhabitants of country 0 buy more of all goods, creating
a deficit in their own balance and a surplus in the balance of every other
country. The surplus in the balance of country *i* is
*m*_{i0}*dX*_{0}*,
where *m*_{i0} is the marginal propensity to spend in country
0 on the goods of the *i*th country. The typical term in equation (7)
is, therefore,

By substitution, then, we get the criterion for the change in the terms of trade of the growing country:

Every *m*_{j0} is positive in the absence of
inferior goods, and every *Delta _{ji} / Delta*
is negative if all exports are gross substitutes. Therefore,

Suppose that country 0 applies an undiscriminatory tariff equal to
*dt*_{0} on all imports. At constant foreign
prices the tariff-inclusive price of all imports in country 0 rises by
*dt*_{0}, and this induces a shift of demand
away from all foreign goods onto domestic goods. The change in demand for
the goods of the typical country *j* before redistribution of the tariff
proceeds is
-*eta*_{j0}*I*_{j0}*dt*_{0},
where *eta*_{j0} is the (positive) cross elasticity
of demand in country 0 for the goods of country *j* when the price of
the goods of country 0 falls relative to the price of the goods of country
*j*^{6};
and where *I*_{j0} is the initial level of imports
from country *j* to country 0. On the other hand, the redistribution
of the tariff proceeds raises the demand in country *i *for the goods
of country *j* by
*m*_{j0}*I*_{0} where
mj0 is the marginal propensity of country 0 to import from country *i*,
and *I*_{0} is the total level of imports into
country 0. The combined change in demand for *j*'s goods is, therefore,
(-*eta*_{j0}*I*_{j0}
+
*m*_{j0}*I*_{0})
*dt*_{0} =
*eta'*_{j0}*I*_{j0
}*dt*_{0}where
*eta'*_{j0} is the compensated cross elasticity
of demand since the income effect implicit within the price change when all
imports rise is also
*m*_{j0}*I*_{0
}*dt*_{0}. The typical term of the
general equation (7) then becomes *-delta B _{j}
/ delta t*

The elasticities are all defined to be positive if the goods are net substitutes,
so the conclusion is again unambiguous: *An increase in tariffs raises
the world price of the exports of the tariff-imposing country relative to
the prices of the exports of all other countries*.

The effects of a consumption tax on import goods are equivalent to the effects
of a tariff if there is no domestic production of these goods. If there is
*eta'*_{cj0}
*y*_{j0} where
*eta'*_{cj0} is the compensated cross elasticity
of consumer demand in 0 for the type of products exported by country *j*
when the price of country 0's goods falls, and
*y*_{j0} is the level of domestic consumption
of these products. The criterion for the change in the relative price of
the exports of the typical country *i* is, therefore,

which is negative since *eta'*_{cj0} is positive.
Thus *the terms of trade improve with respect to all countries as a result
of a tax on importable goods*. By similar reasoning it can be shown that
the terms of trade worsen as a result of an increased tax on the consumption
of export goods. And by changing signs we get the criteria for the effects
of subsidies on the consumption of import and export goods.

By similar analysis it can be shown that the typical coefficient of (7) resulting
from a tax on the production of exportable goods is
*epsilon'*_{pj0}*Y*_{jo},
where *epsilon'*_{pj0} is the elasticity of supply
in country 0 of the good exported by the *j*th country with respect
to a change in the price of the goods of the 0th country, and
*Y*_{jo} is the domestic production (that is,
the production in 0) of the good exported by the *j*th country. The
criterion is, therefore,

Thus if supply elasticities are all positive, *a tax on the production
of exportables improves the terms of trade of the taxing country with respect
to all other countries*. [Note that each term in (25) is zero if country
0 is completely specialized.] The same criterion applies, *mutatis
mutandis*, for production taxes on imported goods, or production subsidies.

Suppose that country 0 pays country *s* an annual tribute, or gift,
or loan (ignoring interest payments). Then expenditure in 0 decreases and
in *s* increases by the amount of the payment. The demand for all goods
in 0 decreases and in s increases if there are no inferior goods. At constant
prices the deficit created by these expenditure changes in the balance of
payments of the *j*th country is (*m*_{j0 }-
*m*_{js})
*dT*_{s0},where
*dT*_{s0} is the value of the transfer. The criterion
for the change in the price of exports in the receiving country relative
to the price of the exports of the paying country, is, therefore,

[In the term *j* = *s *the coefficient is
(*m*_{s0 }-
*m*_{ss}) =
(*m*_{s0} -
*c*_{s}) =
(*m*_{s0} + *m*_{s
}-1), where *c*_{s} and
*m*_{s} are, respectively, the marginal propensities
to consume and import in country *s*.] The unilateral payment, through
expenditure changes in the transferor and transferee, rearranges demand
throughout the world in a way which does not permit any a priori generalization,
a result which can be expected from the analysis of transfer between two
countries.^{7}

The foregoing account of policy changes in a system containing many national
units naturally leads to more complicated conclusions than in the case of
a system containing only two countries. Applying only the restrictions of
stability, we have not been able to show that two-country and multiple-country
systems lead to substantially the same conclusions. The difficulty lies in
relations of gross complementarity among the exports of the various countries.
Gross complementarity is consistent with stability in the multiple-country
system but inconsistent with stability in the two-country system. The two-country
model cannot therefore be expected to suggest "theorems which admit of extension
to more concrete cases" when gross complementarity is
involved.^{8}

However, when gross complementarity is absent -when all exports are gross
substitutes- there is a remarkable similarity between the conclusions of
the two models. Productivity, tariff, and tax changes move the terms of trade
in the same direction in the many-country system, as in the simpler system.
The explanation of this similarity lies in what may be called the law of
composition of countries. If all foreign exports are *perfect *substitutes
for each other, the foreign countries can be aggregated into a composite
country and called the "rest of the world"; in that case the conclusions
of the two models will agree both qualitatively and quantitatively. But if
foreign exports are only *imperfect* substitutes for each other, exact
results cannot be obtained by the use of a composite country. However, while
the *quantitative* conclusions of the two models will in this case differ,
the *qualitative* conclusions remain. There is a presumption, then,
that the use of two-country models will not be subject to serious error provided
all exports are gross substitutes.

Assume three countries, the United States (0), the United Kingdom
(I) and France (2) with the respective currency prices (expressed in dollars)
1, *P*_{1} and *P*_{2}. Choose units of pounds
and francs so that, at equilibrium, *P*_{1} and
*P*_{2} are each unity. (If equilibrium ratios are $1 = 1/2.80
pounds = 5 francs then 1/2.80 pounds and 5 francs become the British and
French currency units.) The two curves
*B*_{1}*B*_{1} and
*B*_{2}*B*_{2} trace the loci of pound-franc prices
that allow equilibrium in the British and French balances. At *Q* both
balances are in equilibrium implying, by Cournot's Law, equilibrium in the
United States balance. A third curve (not drawn) passing through *Q*,
*B*_{0}*B*_{0}, could be drawn to reflect the
pound-franc prices that will allow equilibrium in the United States
balance.

If all currencies are gross substitutes both curves have positive
slopes: This follows because appreciation of the franc must be associated
with appreciation of the pound to maintain equilibrium. Moreover,
*B*_{1}*B*_{1} has an elasticity greater than unity
and *B*_{2}*B*_{2} has an elasticity less than
unity, this follows because appreciation of the dollar is equivalent to
depreciation of the franc and pound in equal proportion; a movement along
the line *OQ *from *Q* to *Q'* must improve the French and
British balances and worsen the United States balance. Four quadrants can
then be identified: east and west of
*B*_{1}*B*_{1} there are deficits and surpluses,
respectively, in the British balance- and north and south of
*B*_{2}*B*_{2} there are deficits and surpluses
in the French balance. (The curve
*B*_{0}*B*_{0} (not drawn) would have a negative
slope and demarcate zones of surplus or deficit in the United States
balance.)

On the dynamic assumption that the dollar prices of pounds
and francs rise and fall in proportion to the disequilibrium in the British
and French balances, the arrows in each quadrant indicate the forces impelling
a return to equilibrium, and show that the equilibrium is stable. From the
disequilibrium position *A*, for example, the path may follow the broken
line AQ, becoming "trapped" in Quadrant I; or it may become trapped in Quadrant
III and hence move to equilibrium.

To determine geometrically the movement of the prices as a result of policy changes it is necessary to indicate the direction in which the two curves shift. If United States output and expenditure increase, more British and French goods are demanded at constant prices Improving both British and French balances; the two curves therefore shift away from the origin and the new equilibrium point moves to somewhere in Quadrant I Similarly, for the tariff and tax changes analyzed in the text both curves shift toward the origin with an unambiguous improvement in the United States terms of trade.

1 Adapted from: *Amer. Econ. Rev.*, 50, 68-110 (March,
1960).

2 Let *x*_{rs} and
*X*_{rs }be, respectively, consumption and production
of the *r*th good in the *s*th country, and let
*T*_{s} be the capital exports of country
*s*. Suppose that there are *n* countries and *m* + 1 goods.
Then a general model can be expressed by the following equations:

These are national "budget" equations, income-spending = lending, for each
country. If these equations are satisfied, each country is on its
*m*-dimensional offer curve

These are the market clearing equations, world supply = world demand, for every good. Notice that the excess supply function of the numéraire, commodity 0 (for example, gold), is omitted; if equations (2) are satisfied, then the last equation in (3) can be deduced from the others (or vice versa).

Demand for the *r*th good in the *s*th country depends on the level
of spending and the prices both expressed in terms of money, commodity 0.

*X*_{rs }= *X*_{rs
}(*P*_{1}, . . .,
*P*_{m}) (*r* = 0, 1, . . ., *m*;
*s* = 1, . . ., *n*). **(5)**

Supply of the *r*th good in the sth country depends on the prices.

Domestic expenditure is assumed to be linked to lending and other policy
variables. If we add to this system n equations of the form
*D*_{s} =
*D*_{s}(*T*_{s };
*alpha*), where *alpha* is any policy, then we have a system of
2(*mn* + *n*) + *m* + *n* equations and 2(*mn* +
*n*) + *m* + 2*n* unknowns. By specifying n parameters
representing the capital exports of each country, the system becomes determined.

In the text the *P*'s refer to the price level of each country's exports
(exchange rates constant), or the price of each country's currency (price
levels constant). If each country exports only one product, and the budget
equations (2) are satisfied, then the partial derivatives *delta
B*_{i} / *delta
P*_{j} __=
__*b*_{ij} [used in equation (6)] can be identified
with the word elasticity of demand and supply of the exports of the
*i*th country with respect to a change in the price of the exports of
the *j*th country, making use of the following relations:

*B*_{i }= value of exports - value of imports

= (foreign demand - domestic excess
supply)*P*_{i}

= *P*_{i}
(*x*_{i1}+ ... +
*x*_{in}) -
*P*_{i}*X*_{i},

and differentiating with respect to
*P*_{j}. Elasticities can then be formed by
multiplying and dividing by *X*_{i}, leaving
the typical term *delta B*_{i} / *delta
P*_{j} = *X*(*eta*_{ij} -
*epsilon*_{ij}). (This assumes that units are
chosen so that each *P*_{j }is initially 1.)

3 Cournot made extensive use of the proposition that the sum of all balances is necessarily zero ([9], chap. 3).

4 There are two approaches to the stability of international equilibrium. One approach is to treat the world economy like the domestic economy and postulate that the price of any good rises and falls in proportion to the excess demand and supply of that good. For example, the dynamic system may be written

where the summation is over countries. By linearizing (8) (retaining only linear terms of a Taylor series), and translating the resulting partial derivatives into demand and supply elasticities, we obtain

where the own elasticities of demand
(*eta*_{rr}) and supply
(*epsilon*_{rr}) in the world as a whole are
defined to be, normally, negative and positive, respectively. The linear
system can be stable only if the real parts of the roots of

are all negative (*delta*_{rq} is the Kronecker
delta). The theorem on gross substitutes states that if all cross elasticities
are positive (including the cross elasticities of demand for the numéraire
good), the system is stable.

The above system focuses attention on world markets for particular commodities.
Classical international trade theorists, on the other hand (with the exception
of F. D. Graham), emphasized the importance of disequilibrium in the balance
of payments, gold flows, and changes m the terms of trade. If we now let
*P*_{i }denote the world price of the exports
of country *i*, we have the following system that is more compatible
with the postulates of classical theory:

assuming that there are now *n* + 1 countries and that prices are expressed
in terms of the exports of country 0. Following the same procedure as above,
we find that stability requires that the real parts of the roots of the equation

must all be negative. Again, stability is assured if all
*b*_{ij} > 0 for *i not= j*. If prices
were all constant and exchange rates were all flexible, the system would
be stable if all currencies were gross substitutes.

Generally the systems (8) and (11) are fundamentally different. Goods may all be gross substitutes while some currencies are gross complements, and vice versa. The gap between the two systems narrows, however, when only one country produces each good: In that case an excess world demand for a good implies an excess demand for the currency of the country producing that good.

I have said that system (11) conforms more closely to the classical system than does system (8). This is not meant to imply that classical theorists would accept even as an approximation the rigid dynamic laws postulated. Consider, for example, the following passage from Marshall's privately circulated manuscript of 1879 ([52], pp. 19, 25):

. . . so that if we chose to assign to these horizontal and vertical forces any particular laws, we should obtain a differential equation for the motion of the exchange index . . . Such calculations might afford considerable scope to the ingenuity of those who devise mathematical problems, but . . . they would afford no aid to the economist.For the mathematical functions introduced into the original differential equations could not, in the present condition of our knowledge, be chosen so as to represent even approximately the economic forces which actually operate in the world... Whereas the use of mathematical analysis has been found to tempt men to expend their energy on the elaboration of minute and complex hypotheses, which have indeed some distant analogy to economic conditions, but which cannot properly be said to represent in any way economic laws.

5 Why has country 0, whose exports are numéraire,
been chosen as the growing country? Suppose instead that country *i*
grows by *dX*_{1}*. Then substituting in equation
(7) the terms

(the aggregate marginal propensity to import in country i), and the terms

But
*m _{i}Delta_{ii} / Delta
*is negative while all the other terms are positive. It would therefore
appear that no unambiguous result is possible, and that the method of treating
the change as occurring in the numeraire country is a special case.

Nevertheless, the economist's intuition tells him that in static analysis
the choice of numeraire cannot affect the ultimate change in *relative
*prices, and that if the terms of trade of the numeraire country deteriorate
when that country grows, the terms of trade of any other country must fall
when it grows. He may, therefore, conclude that *P*_{i
}must fall in equation (14) - that the negative term dominates.

This is, in fact, correct. Making use of the definition of the aggregate marginal propensity to import, that is,

The first term on the right side of (15) is clearly negative. The other terms
will be negative if the *principal* cofactor of *Delta* dominates
each of the other cofactors, that is, if |* Delta*_{ii} |*
* > |* Delta*_{ji} |* *(*j not= i*). But
by subtraction of the two cofactors it is easily shown that the resulting
(*n* - 1)th-order determinant *Delta*_{ii} -
*Delta*_{ji} , has all the characteristics of *Delta* (positive
off-diagonal elements and dominant negative diagonal elements) except that
its sign is opposite to the sign of *Delta*. (An analogous theorem has
been proved by Metzler in analysis of the matrix multiplier.) All the terms
on the right of (15) are therefore negative so *P*_{i
}must fall as a result of growth in country *i*.

This does not, however, conclude the proof that the replacement of country
0 by country *i* for the growing country leaves the essence of the
proposition unaltered. For *dP*_{i} /
*dX*_{i}* < 0 merely asserts that the ith
price falls relative to the 0th price. To complete the proof we must also
show that (*dP*_{i} /
*dX*_{i}* ) -
(*dP*_{j} /
*dX*_{i}* ) < 0 to establish that the
*i*th price falls in terms of *all *other prices.

Since

and again, except for the middle term on the right of (17), we are left with
an apparently ambiguous result. However, when we solve for *m*, from
(14) and substitute the result in (17). we get. after simplification,

The first term on the right of (18) is negative, since
*dP*_{i} /
*dX*_{i}* < 0 from (14), and its coefficient
is positive. The second term, however, is also negative since, by Jacobi's
ratio theorem,

which from Mosak's theorem, must have a sign opposite to
*Delta*_{ii}, which, in turn, has a sign opposite to *A*.
Thus, both terms on the right of (18)are negative, so
(*dP*_{i} /
*dX*_{i}* ) -
(*dP*_{j} /
*dX*_{i}* ) < 0 . It follows then that the
device employed in the text of using the numeraire country as that in which
the " policy change " occurred is a valid simplification.

The general implications of these mathematical manipulations are examined in greater detail in Chapter 7.

6 This term measures the percentage change in the imports
of country 0 from country *j* as a proportion of the percentage increase
in the price of goods of country 0, after the income effect implicit in the
elasticity has been canceled by the redistribution of the tariff proceeds.
This definition of the elasticity allows greater simplification than the
conventional definition because it exploits the Hicksian law of composite
commodities, which is applicable when the same rate of tariff is applied
to all commodities.

A more general formulation, of which equation (23) below is a special case,
is necessary to analyze the effects of discriminatory increases in tariff
rates in country 0. Let *eta*_{ij.0} be country
0's cross elasticity of demand for imports from country *i* with respect
to the (tariff-inclusive) price of imports from country *j*, and let
*dt*_{j0} be the change in country 0's tariff
on goods from country *j*. Then before taking account of the disbursement
of the tariff proceeds the effect on country *i*'s balance of payments
of a tariff in country 0 on the goods of country *j* is
*eta*_{ij.0}
*I*_{i0}*
dt*_{j0}, and the combined effects on country
*i*'s balance of tariffs applied to all imports is *Sigma
^{eta}_{j }*=

Against these terms, which measure the initial effect on country i's balance,
must be set the redistribution of the tariff proceeds, equal to *Sigma
^{eta}*

measures the effect of all the tariff changes on country *i*'s balance
of payments. The effect of the combined tariffs in country 0 on the price
of, say, country *k*'s exports is therefore

When the tariff rates are applied equally on all products we have
*dt*_{10} =
*dt*_{20} = ... =
*dt*_{n0} __=__
*dt*_{0} and (20) can be rewritten

But *Sigma
^{eta}*

which is equivalent to equation (23)

This section has been improved by H. G. Johnson's correction of my earlier treatment of income effects (see [35]).

7 The propositions in this section require minor modification if there are limiting cases such as infinite or zero elasticities, and zero or negative marginal propensities to spend

8 See Mosak [68] for an important generalization of the classical system and a discussion of complementarity. And see the Appendix to Part I of this book for an analysis of the general theoretical propositions associated with the presence or absence of homogeneity.

Literature Cited

[9] A. COURNOT, *Researches into the Mathematical Principles of the Theory
of Wealth, *translated by T. Bacon. New York: Macmillan, 1897.

[35] H. G. JOHNSON, "The Pure Theory of International Trade: Comment,"
*American Economic Review, *50, 721-722 (Sept. 1960).

[52] A. MARSHALL, *The Pure Theory of Foreign Trade. *London: 1879;
reprinted 1930.

[68] J. L. MOSAK, *General Equilibrium Theory in International Trade.
*Bloomington: Indiana University Press, 1944.

© Copyright Robert A. Mundell, 1968