The Foreign Trade Multiplier: A Restatement

J. J. Polak and Gottfried Haberler’s December 1947 article is a joint concluding statement reconciling the positions advanced in two preceding papers. Its six numbered propositions clarify when a foreign-trade multiplier explains the evolution of national income and when it merely expresses an accounting relationship. The governing distinction is methodological: an equation that is necessarily true at a given moment need not explain how income changes over time.

Much confusion concerning the multiplier (and not only the foreign trade multiplier) has arisen from inadequate distinction between definitional or instantaneous, and causal or serial relationships.

The authors treat the multiplier’s principal contribution as causal analysis. A useful multiplier should incorporate as many parameters as can reasonably be assumed constant, making the stability of behavioral relationships central to its explanatory force. The restatement thus reconciles the earlier positions by distinguishing different uses of their formulas, rather than simply declaring one equation correct and the other incorrect. The formulas themselves belong to the preceding discussion and are not reproduced here; this article supplies their conditions of applicability.

The second and third propositions contrast a multiplier whose multiplicand is exports, (X), with one whose multiplicand is the export surplus, (X-M). The export-based formula can explain a sequence of income adjustments when exports remain constant and imports change as a constant function of income, (Y). This does not require exports to remain fixed indefinitely. Moderate fluctuations are compatible with the explanation, provided income has time to adjust to changing export levels. Constancy is therefore a condition for tracing an adjustment process, not a claim that the external stimulus never changes.

The export-surplus formula faces a stricter condition: the surplus itself must remain constant. The authors argue that this requires both constant (M_a) and (q=0). Certain government policies might produce that configuration, but it is unlikely; whenever (q>0), the formula lacks serial applicability. Their preference for the export-based formulation concerns its greater usefulness as a causal model, not the logical falsity of the alternative.

Formula (1) is, of course, always true in the definitional, instantaneous sense. But as such it is rather useless and must be handled with care in order to prevent wrong interpretations.

This warning makes the article’s central conceptual move explicit. Definitions cannot themselves be refuted, yet treating them as causal explanations can generate false conclusions. The authors invoke the quantity theory as a familiar instance of that danger. What matters is not merely whether an equation holds, but whether its terms retain the meanings and stability required by the proposed explanation.

The fourth proposition develops this point through the propensity to consume, (c). In an instantaneous identity, it must be defined as current consumption divided by current income, (C_t/Y_t). A sequential model may instead relate current consumption to previous-period income, (C_t/Y_{t-1}). Which ratio is more stable is an empirical question, although the authors expect stability, if present, more plausibly in the lagged relationship.

At any rate it should be observed that if in a sequence model (as constructed by Machlup, Metzler and others) the lagged c is assumed to be constant, the instantaneous c cannot be constant, but has to change from period to period.

Consequently, a constant behavioral coefficient in a sequential model does not justify holding the corresponding instantaneous multiplier constant. Both the multiplier and its multiplicand may vary during adjustment.

The final propositions delimit what can be concluded about particular changes. An equal simultaneous increase in exports and (M_a) yields no general prediction for income: acceleration effects and changes in consumption and investment propensities require further information. A separate limiting case—(M_a=0), (c=1), and imports approaching exports asymptotically—is referred back to the first paper’s result. The restatement’s enduring relevance lies in this disciplined separation of accounting identities, temporal mechanisms, and empirical assumptions: multiplier reasoning becomes explanatory only when the conditions connecting them are made explicit.