Interactions between the Multiplier Analysis and the Principle of Acceleration

Samuelson opens by warning that the “multiplier” analysis of government deficit spending, while illuminating, risks hardening into a dogma that obscures important subsidiary relations (p. 75). His key clarifying point is that the multiplier in its usual sense does not give the ratio of total induced income to the original amount of government money spent – it gives the ratio of the total increase in national income to the total amount of investment, governmental and private combined, treating effects on private investment as a separate, often neglected channel. Building on a “new model sequence” that Alvin Hansen had already devised to combine the multiplier with the acceleration principle, Samuelson’s contribution is to give that combined model a rigorous algebraic treatment – deriving, from first principles, the complete catalogue of qualitatively different income paths it can produce (p. 75).

National income in period t, Y_t, is defined as the sum of three components: government expenditure (g_t), consumption expenditure (C_t), and induced private investment (I_t) (p. 76). Consumption in the current period is assumed equal to the marginal propensity to consume, a, times the previous period’s income (C_t = aY_{t-1}); induced investment is assumed proportional, via the relation b, to the increase in consumption from the previous period to the current one (I_t = b[C_t - C_{t-1}]); and government expenditure is held constant at one dollar per period (g_t = 1). Substituting these assumptions into the income identity yields a single second-order linear difference equation for Y_t in terms of Y_{t-1} and Y_{t-2} (p. 76).

With the marginal propensity to consume set at one-half and the relation set at unity, Samuelson works the model forward period by period starting from a government expenditure of one dollar in period 1, showing income rising to a peak by the third period, falling to a trough by the seventh, rising to a peak again by the eleventh, and so on – an oscillatory pattern that, Samuelson stresses, “could not occur in the conventional model sequences” without the accelerator term (Table 1 and accompanying text, pp. 75-76). Chart 1 breaks the total income in each period into its government, consumption, and private-investment components, visually showing private investment turning negative in some periods – interpreted as meaning investment for the system as a whole is lower in that period than it otherwise would have been, not that literal disinvestment occurs (Table 1, note, p. 76).

Table 2 reruns the model for several different combinations of the marginal propensity to consume and the relation, showing that seemingly small changes in these two coefficients produce qualitatively very different income paths – from no oscillation, to undamped regular oscillation, to explosive oscillation, to non-oscillatory explosive growth approaching a compound interest rate (p. 76). Samuelson notes this raises a natural worry: if minor changes in assumptions produce such different behavior, how can one be confident that other, untried coefficient values would not produce still further new types of behavior, especially over longer horizons? (p. 76).

Rather than trying to check further cases arithmetically – which cannot cover all possible coefficient values or compute an endless number of terms – Samuelson turns to “comparatively simple algebraic analysis” of the difference equation itself, which can establish all possible qualitative types of behavior at once (p. 76). Because the equation is a standard second-order linear difference equation with constant coefficients, its solution is fully characterized by the roots of an associated characteristic quadratic in a and b; Samuelson states that the qualitative behavior of the whole model sequence is governed by whether these roots are real or complex, and whether they are greater or less than unity in absolute value (p. 77, footnote 1).

The full field of possible values of the marginal propensity to consume and the relation divides into four regions with qualitatively different behavior under a constant, continuing level of government expenditure (pp. 77-78). In Region A (relatively small values of the relation), national income approaches asymptotically a fixed multiple, 1/(1-a), of the constant government expenditure, with no oscillation; a single impulse or a finite burst of expenditure dies out gradually back toward the original income level. In Region B, a constant level of expenditure produces damped oscillatory movements that gradually converge on the same asymptote. In Region C, the oscillations around that asymptote are instead explosive, growing larger without bound. In Region D (large values of both the marginal propensity to consume and the relation), income growth is no longer oscillatory at all but explosive and monotonic, eventually approaching a compound-interest rate of growth (pp. 77-78).

Within Region D, Samuelson notes that a single impulse of net investment sends the system upward to infinity at a compound interest rate of growth, while, by symmetry, a single infinitesimal unit of disinvestment sends the system ever downward at an increasing rate – a highly unstable configuration that he identifies as the case most closely resembling pure pump-priming, in which the total increase in national income bears no finite ratio to the original stimulus (p. 78). He treats this as an extreme, cautionary case illustrating how far the combined multiplier-accelerator mechanism can diverge from the well-behaved convergence of the simple multiplier.

Samuelson explicitly flags that the model treats the marginal propensity to consume and the relation as constants, whereas in an actual economy both would change with the level of income, so the representation is strictly a marginal analysis intended for studying small oscillations rather than a complete theory of income determination (p. 78). He also notes in a footnote that the formal mathematical structure of his difference-equation approach is identical to model sequences already used by Lundberg and by Tinbergen in their dynamic business-cycle work, positioning this note as a simple, tractable entry point into that broader mathematical literature (p. 78).

Samuelson ends by pushing back against the view that mathematical methods make economic theory more abstract and remote from reality; he argues instead that, properly used, they liberate economists to entertain and analyze more realistic and more complicated hypotheses than purely verbal reasoning could handle (p. 78) – a methodological point illustrated by the paper itself, since the four-region classification could not have been established by numerical example alone.