Appendix: Note on the Multiplier Relation
Assume that consumption, investment, imports, and exports are the only expenditures that are known to be functions of national disposable income and the functional form is linear in all cases, such that
where c, i, m, e, e′ are constants representing marginal propensities, and Ca, Ia, Ma, and Ea are parameters representing the autonomous elements in consumption, investment, imports, and exports, respectively, that cannot be explained by income movement. From the discussion in the text, national disposable income (neglecting the government sector) may be written as
where D is net donations received from abroad. When the above functional relations are substituted in this identity, the following is obtained:
provided c + i – m + (e + e′m) <1.
A further refinement may be introduced by taking into consideration the fact that the imports induced by a given dose of expenditure may be different according as expenditure has been directed toward consumption, investment, or exports. To bring this fact into the picture, imports may be rewritten as a function of the various components of income instead of a direct function of income itself, i.e.,
where m1, m2, and m3 are the coefficients, which are assumed to be constant, relating induced imports to consumption, investment, and exports, respectively, and Ma is the autonomous residue. The marginal propensity to import with respect to income as a whole, i.e., m in the previous equation, can be derived as follows:
When the new function for M is written into the national income identity,
The multiplier relation then becomes
If e′, i.e., “other countries’ propensity to spend back,” is considered negligible for certain countries, then this equation can be simplified to
where the multiplicand is the autonomous expenditures net of their import contents.
If more items of national expenditures are known to be functions of income, then the multiplier relation would become more complicated.
Since, in reality, the marginal propensities represented by small letters, c, i, m, e, etc., are more likely to be constant over a small range of variation of income than over its entire range, it will be safer to apply the multiplier formula only to determine the changes in the flow of income than to use it to determine the total level of income. When used in this less ambitious fashion, all the capital letters in the above formulae should be understood to stand for the increments of the different magnitudes which they represent respectively.
If in the practical application of the multiplier to balance of payments problems, changes in exports may be assumed as a first approximation to be entirely autonomous and changes in imports to be entirely induced, then the general multiplier formula may be reduced to the following:
in which the marginal propensity to export vanishes.
If in the case of a country with an over-all quantitative restriction of imports, or in the case of a country with a perfectly flexible exchange rate where imports and exports are instantaneously adjusted to each other by the flexible exchange rate, both exports or imports are to be regarded as autonomous magnitudes in our first approximation, then the multiplier formula is further simplified to
in which the marginal propensity to import also vanishes.
The derivation of the multiplier in the above manner, viz., from the national income identity and the functional relations between income and various induced expenditures, may give an entirely false impression that the multiplier operates instantaneously. In fact, the multiplier takes time to work out and the process can be properly shown only in a dynamic sequence analysis, which cannot be considered here.
Mr. S. C. Tsiang, economist in the Balance of Payments Division, Research Department, is a graduate of the London School of Economics, and was formerly Professor of Economics in the National Peking University and the National Taiwan University. He is the author of several articles in the Economic Journal and Economica.
Not including the donation contra-entries to imports and exports. For the effect of donations on income, see below.
National disposable income is merely the potential source of domestic expenditures. It has no effect upon domestic output, prices, and employment, unless it leads to expenditures on currently produced domestic goods and services. If it goes partly into hoardings, or is spent on imported goods and services, then to that extent it has no direct effect upon domestic output and prices. Therefore its variation is not a reliable measure of the actual expansion or contraction of domestic production and prices, as shall be shown in later examples. Only the variation of the national income at factor cost can accurately represent the increase or decrease of domestic output and prices (at factor cost), for national income at factor cost is necessarily identical with total expenditure upon currently produced domestic goods and services exclusive of indirect taxes.
If UNRRA relief is to be considered a direct donation to the people instead of to the government, then its place in social accounting may be shown as follows: The donation increases the transfer income of the people of the receiving country, and the increase in income is spent on additional consumption or investment goods which are imported from the donating country. Thus there is an additional item of domestic consumption or investment which offsets the import of relief goods. Consequently, there will be no change in the national income at factor cost of the receiving country, although national disposable income shows an increase.
See Mrs. J. Robinson, “The Foreign Exchanges,” Essays in the Theory of Employment (2nd edition, London, 1947), Part III, Chapter I, p. 135.
Private hoardings of gold are a borderline case. Private gold hoards may be regarded by the holder as a store of domestic value or as a foreign asset (a claim on foreigners). If they are regarded as foreign assets, changes in them become international capital movements, and should be recorded in the balance of payments. Since it is notoriously difficult to estimate the amount of or changes in private gold hoards, the Balance of Payments Manual treats them entirely as domestic assets (nonmonetary commodity), the increase or decrease of which, therefore, constitutes a domestic investment or disinvestment. One result of this expedient treatment is that withdrawal of the gold reserve of the central bank into private hoards of domestic residents is a conversion of international assets into domestic assets, and hence constitutes an “international capital movement” against a contra-entry of an “import” of commodity gold, although the whole transaction takes place at home. But since private gold hoards do not form the monetary basis of the country, except in countries where gold bullion is still a medium of exchange, not much harm is done.
See the Appendix for a further discussion of this multiplier relation.
See J. J. Polak, “The Foreign Trade Multiplier,” “Comment” by Gottfried Haberler, and “Joint Restatement” by Polak and Haberler, American Economic Review, December 1947. See also Fritz Machlup, International Trade and the National Income Multiplier (Philadelphia, 1943), and J. E. Meade, “National Income, National Expenditure and the Balance of Payments,” Economic Journal, December 1948 and March 1949.
What is described as an inflationary effect in this paper is merely a tendency to make the domestic money income (at factor cost) expand. It does not necessarily imply a tendency to make the domestic price level rise. For whether prices will rise or not depends not only on the effect of investment upon money income but also upon the effect of the investment, and the import that induced it, upon the current supply of goods (i.e., upon their period of turnover). For a discussion of this relation, see the author’s “Rehabilitation of Time Dimension of Investment in Macrodynamic Analysis,” Economica, August 1949. Such a discussion is beyond the scope of the present article.
Conceptually, this appears to be different from the “compensatory official financing” of the Balance of Payments Yearbook; for the latter includes part of the government donations on current account, whereas our concept strictly denotes capital movements on the part of the monetary authority to counterbalance excess demand or supply on the domestic foreign exchange market. In accounting, however, they would probably come to the same thing, if both were interpreted in the strict sense. This relation will be discussed later.
The classification here of capital transactions in the balance of payments into “compensatory capital movements” and “independent capital movements” is somewhat different from Professor Machlup’s classification of “induced” and “autonomous” capital movements in his International Trade and the National Income Multiplier (Philadelphia, 1943). Professor Machlup defines “induced capital movements” as those capital movements which are called into being by other changes in the international balance of payments, and “autonomous capital movements” as those which are not thus induced. The former would therefore include, besides the compensatory capital movements of the monetary authority, the speculative demand and supply of foreign exchange by dealers and speculators who buy and sell in response to other selling and buying on the foreign exchange market. He was wrong, however, in assuming that the induced capital movements are always equalizing or “accommodating.” As shown above, if the dealers’ and speculators’ expectation should become elastic, their induced purchases or sales of foreign exchange would be the opposite of “accommodating” and would themselves, as much as the autonomous capital movements, call for accommodating adjustments.
If the central bank itself observes a rigid reserve ratio between its current deposit liabilities and note issue on the one hand and its reserves of gold and foreign exchange on the other, as under the traditional gold standard system, then the multiple expansion of credit engendered by the compensatory capital export on the part of the central bank will be further magnified; for it would first induce the central bank itself to increase the money supply not merely by the amount of its foreign exchange or gold acquisition but by a multiple of it, and the multiple expansion of central bank credit would in turn set up a further multiple expansion of commercial bank credit.
At this limit, in the words of Hawtrey, it would rise practically to a vertical wall, whereas the rate of interest at which he can lend his own liquid funds or which he can obtain by investing them in other securities is more or less constant.
Since there is always the alternative to direct investment of lending the capital funds imported from abroad or buying securities on the domestic money market, the marginal lending rate is therefore an opportunity cost of his investment expenditures. If, however, it is the lower of the two, it is not the effective marginal cost.
In the case of direct investment by foreign capitalists, there will generally be an additional influence due to a simultaneous import of technical knowledge. The availability of technical “know how,” which had previously been inaccessible, will have the effect of raising the marginal efficiency schedule of investment because of its close complementarity with capital equipment, as described above.
Cf. the previous section on import surplus and independent capital imports for specific investment projects.