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

Generalization of the Classical Model 1

Robert A. Mundell

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 bij = deltaBi / deltaPj is the change in the balance of the ith country due to a change in the price of the goods of the jth country. Solving for dPi / d alpha we obtain

and where Deltaji, is the cofactor of the jth row and the ith column of A.

Equations (7) provide a general framework into which specific policy changes can be introduced. But to evaluate any of the signs, dPi / d alpha, it is necessary to know the values, or at least the signs, of two kinds of terms: the coefficients delta Bj / delta alpha and the ratios Deltaji / Delta. Now, the coefficients delta Bj / delta alpha describe the change in the balances of payments arising from the policy change at constant prices; to evaluate these coefficients, then, we can apply the method of comparative statics. The ratios Deltaji / Delta, on the other hand, indicate the interactions of price changes in multiple markets, and the effectiveness of price changes in relieving the initial disequilibrium. In the two-country case these signs were determined by the stability conditions; in the multiple-country case, however, it is easily shown that some of the ratios may be positive while others are negative without conflicting with the conditions of stability. It appears, then, that in evaluating equations (7) we will be left with some positive and some negative terms, and no general presumption about the sign of dPi / d alpha.

To make progress some restriction on the signs of the elements bij in the basic determinant Delta is required. The most interesting special case, for present purposes, is that where bij > 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 postulates4 ([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.

Productivity Changes

Suppose that output and expenditures in country 0, the numeraire country, increase by dX0*. 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 mi0dX0*, where mi0 is the marginal propensity to spend in country 0 on the goods of the ith 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 mj0 is positive in the absence of inferior goods, and every Deltaji / Delta is negative if all exports are gross substitutes. Therefore, an improvement in productivity unambiguously worsens the commodity terms of trade of the growing country: The prices of the exports of every other country rise relative to the price of the exports of the growing country.5

Tariff Changes

Suppose that country 0 applies an undiscriminatory tariff equal to dt0 on all imports. At constant foreign prices the tariff-inclusive price of all imports in country 0 rises by dt0, 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 -etaj0Ij0dt0, where etaj0 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 j6; and where Ij0 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 mj0I0 where mj0 is the marginal propensity of country 0 to import from country i, and I0 is the total level of imports into country 0. The combined change in demand for j's goods is, therefore, (-etaj0Ij0 + mj0I0) dt0 = eta'j0Ij0 dt0where eta'j0 is the compensated cross elasticity of demand since the income effect implicit within the price change when all imports rise is also mj0I0 dt0. The typical term of the general equation (7) then becomes -delta Bj  / delta t0 = eta'j0Ij0 , so that the criterion for the change in the terms of trade of country 0 is

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.

Consumption and Production Tax Changes

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 yj0 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 yj0 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'pj0Yjo, where epsilon'pj0 is the elasticity of supply in country 0 of the good exported by the jth country with respect to a change in the price of the goods of the 0th country, and Yjo is the domestic production (that is, the production in 0) of the good exported by the jth 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.

Unilateral Payments

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 jth country is (mj0 - mjs) dTs0,where dTs0 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 (ms0 - mss) = (ms0 - cs) = (ms0 + ms -1), where cs and ms 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

Comparison of Two-Country and Multiple-Country Models

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.

Figure 3-1.

Assume three countries, the United States (0), the United Kingdom (I) and France (2) with the respective currency prices (expressed in dollars) 1, P1 and P2. Choose units of pounds and francs so that, at equilibrium, P1 and P2 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 B1B1 and B2B2 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, B0B0, 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, B1B1 has an elasticity greater than unity and B2B2 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 B1B1 there are deficits and surpluses, respectively, in the British balance- and north and south of B2B2 there are deficits and surpluses in the French balance. (The curve B0B0 (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.


Notes

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

2 Let xrs and Xrs be, respectively, consumption and production of the rth good in the sth country, and let Ts 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 rth good in the sth country depends on the level of spending and the prices both expressed in terms of money, commodity 0.

Xrs = Xrs (P1, . . ., Pm) (r = 0, 1, . . ., m; s = 1, . . ., n).  (5)

Supply of the rth 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 Ds = Ds(Ts ; alpha), where alpha is any policy, then we have a system of 2(mn + n) + m + n equations and 2(mn + n) + m + 2n 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 Bi / delta Pj  = bij [used in equation (6)] can be identified with the word elasticity of demand and supply of the exports of the ith country with respect to a change in the price of the exports of the jth country, making use of the following relations:

Bi = value of exports - value of imports

    = (foreign demand - domestic excess supply)Pi

    = Pi (xi1+ ... + xin) - PiXi,

and differentiating with respect to Pj. Elasticities can then be formed by multiplying and dividing by Xi, leaving the typical term delta Bi / delta Pj = X(etaij - epsilonij). (This assumes that units are chosen so that each Pj 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 (etarr) and supply (epsilonrr) 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 (deltarq 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 Pi 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 bij > 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 dX1*. Then substituting in equation (7) the terms

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

But miDeltaii / 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 Pi 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 | Deltaii |  > | Deltaji | (j not= i). But by subtraction of the two cofactors it is easily shown that the resulting (n - 1)th-order determinant Deltaii - Deltaji , 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 Pi 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 dPi / dXi* < 0 merely asserts that the ith price falls relative to the 0th price. To complete the proof we must also show that (dPi / dXi* ) - (dPj / dXi* ) < 0 to establish that the ith 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 dPi / dXi* < 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 Deltaii, which, in turn, has a sign opposite to A. Thus, both terms on the right of (18)are negative, so (dPi / dXi* ) - (dPj / dXi* ) < 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 etaij.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 dtj0 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 etaij.0 Ii0 dtj0, and the combined effects on country i's balance of tariffs applied to all imports is Sigma etaj = 1 etaif.0Ii0dtj0 . This term can be split into income and substitution effects to get

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 etaj = 1 Ij0 dtj0, which by itself increases country i's balance by Sigma etaj = 1 mi0Ij0dtj0 . The redistribution of the tariff proceeds therefore exactly cancels the income effect and we are left with the compensated elasticities. Thus

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 dt10 = dt20 = ... = dtn0 = dt0 and (20) can be rewritten

But Sigma etaj  = eta'j.0 , the sum of the compensated elasticities of demand in 0 for i's imports when all import prices rise is equivalent to the compensated elasticity of demand for i's imports when the price of the goods of the 0th country falls, that is,  Sigma etaj  = 1 eta'ij.0  = eta't0 so where eta't0 is the elasticity of demand of i's imports as defined in the text. Hence (21) can be rewritten

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