A Mathematical Theory of Saving

Ramsey opens by stating the answer before deriving it: “The rate of saving multiplied by the marginal utility of money should always be equal to the amount by which the total net rate of enjoyment of utility falls short of the maximum possible rate of enjoyment” (p. 543). The rest of the paper is devoted to making this precise and to establishing the “conditions of surprising generality” under which it holds.

The community is assumed to persist forever unchanged in numbers, tastes, and aversion to labour; enjoyments and sacrifices at different dates can be calculated independently and summed; and no new inventions occur except those conditioned solely by the accumulation of wealth itself (p. 543). Distributional questions within the community are set aside, and different kinds of goods and labour are treated as homogeneous. Ramsey singles out one assumption for special emphasis: “it is assumed that we do not discount later enjoyments in comparison with earlier ones, a practice which is ethically indefensible and arises merely from the weakness of the imagination” (p. 543) – though he immediately notes that Section II will reintroduce a constant rate of discount to see how the conclusions change. He also assumes away permanent indebtedness to foreigners, selfish consumption of savings by later generations, and catastrophic destruction of accumulated wealth (p. 544).

Bliss (B) is the maximum rate of enjoyment, U(x)-V(a), that a community’s economic circumstances allow it to obtain, given a fixed capital stock c held constant – a quantity that generally rises with capital up to a point (p. 544). Ramsey distinguishes two ways the rise can stop: a finite capital stock may already exhaust the possible gains in income or leisure, or the community may already be at the maximum conceivable rate of enjoyment; failing either, the rate of enjoyment either rises to infinity as capital grows (a possibility Ramsey dismisses as economically impossible) or approaches a finite limit asymptotically – this limit is what he calls Bliss (p. 545). The argument for why a community must always save enough to reach or approach Bliss is that only in this way can the amount by which enjoyment falls short of Bliss, summed throughout time, be kept finite – “if it should be possible to reach bliss or approach it indefinitely, this will be infinitely more desirable than any other course of action,” and reaching it is always possible “since by setting aside a small sum each year we can in time increase our capital to any desired extent” (p. 545).

Equation (2) equates the marginal disutility of labour v(a) to the marginal efficiency of labour (partial f/partial a) times the marginal utility of consumption u(x) – the standard labour-supply condition (p. 546). Equation (3) equates the advantage of an increment of consumption now against postponing it briefly, showing that the marginal utility of consumption u(x) falls at a proportionate rate equal to the current rate of interest (partial f/partial c) – so that, since income and hence consumption keep rising over time, “x continually increases unless and until either a or u(x) vanishes, in which case… bliss must have been attained” (p. 546).

Combining the savings identity with equations (2) and (3) and integrating, Ramsey obtains equation (4): f(a,c) - x = {B - [U(x)-V(a)]} / u(x) – rate of saving equals (Bliss minus current utility) divided by the marginal utility of consumption, i.e., “rate of saving multiplied by marginal utility of consumption should always equal bliss minus actual rate of utility enjoyed” (p. 547). Ramsey also derives the identical result via the calculus of variations by minimizing the integral of the shortfall from Bliss over all time, after changing the variable of integration from time to capital (p. 547). Keynes’s simpler derivation, which Ramsey credits explicitly, compares spending an extra pound now (worth u(x), the marginal utility of money) against the sacrifice of instead saving it: saving one pound less this year postpones reaching Bliss by a small increment of time dt, at a cost of dt times (B minus current utility); equating this sacrifice to u(x) reproduces the same rule (pp. 547-548). Ramsey notes, however, that “this simple reasoning cannot be applied when we take account of time-discounting,” which is why he retains the more general calculus-of-variations derivation for the extensions in Section II (p. 548).

Ramsey states that the rule “is altogether independent of the production function, except in so far as this determines bliss,” and that “the amount we should save out of a given income is entirely independent of the present rate of interest, unless this is actually zero” (p. 548). Using an illustrative utility schedule (a table mapping family income to a numerical utility index, with Bliss set at a utility of 8), Ramsey calculates that a family income of 500 pounds should see roughly 300 pounds saved – a rate “greatly in excess of that which anyone would normally suggest” (pp. 548-549). He also notes two considerations pulling in opposite directions that the model neglects: prospective population growth and the prospect that future inventions might raise the Bliss level both argue for saving more, while the prospect that future inventions might make a given income easier to obtain argues for saving less; the “most serious factor neglected,” in his view, is “the possibility of future wars and earthquakes destroying our accumulations” (p. 549).

*With earned income pa and unearned income rc treated separately, Ramsey defines “unearned income available for consumption” y = x - pa and its total and marginal utilities W(y) and w(y), and shows the savings rule becomes B - W(y) = (rc-y)w(y), i.e., the point (rc, B) lies on the tangent, at y, to the curve z=W(y) (p. 550). This yields a graphical rule (Figure 1): draw the tangent from the point (rc,B) to the curve z=W(y); the abscissa of the point of contact gives the amount y that should be consumed, and rc-y should be saved – “of course y may be negative, which would mean that not only would the whole unearned income be saved, but part of the earned income also” (p. 550). Equation (7), w(y) = A*e^(-rt), further shows how the marginal utility of unearned income declines exponentially at the rate of interest over time, letting Ramsey compute the time required to accumulate a target capital stock from an initial one (p. 551).

For an individual living only T years and wishing to leave a specified bequest c3, the same tangency logic applies but the constant K in the savings condition is no longer equal to Bliss and must be determined jointly with the initial and terminal points (pp. 551-552). Depending on the length of the horizon T and the relationship between the initial and bequest capital levels, Ramsey shows there can be either continuous saving throughout life (“this happens when T is small”) or a two-phase pattern of saving followed by “splashing” (dissaving) before death, when T is large (p. 552-553) – an early formal treatment of the life-cycle saving pattern.

With a constant utility discount rate p and constant interest rate r, Ramsey shows (equation 9) that if p is less than r, the analysis goes through exactly as before but with the marginal utility of unearned income raised to the power r/(r-p) – a “modified utility” – so the savings rule and its independence from the interest rate (when p=0) survive in modified form: “the main conclusion of section I is thus confirmed” for the special case p=0 (p. 554). For a community with a marginal utility of the constant-elasticity form w(y)=Dy^(-a), the constant proportion of unearned income saved works out to (r-p) / [r(a-1)+p] (p. 555). If instead the interest rate is smaller than the discount rate on utility, the qualitative picture reverses entirely: “the marginal utility of consumption will rise at a rate p-r, and consumption will fall towards the barest subsistence level,” with capital exhausted and debt accumulated up to the point where the community can just service the interest on it (p. 555).

Ramsey argues the rate of interest functions asymmetrically: it is a demand price for the whole existing stock of capital, but a supply price only for the flow of new saving, not for a quantity of capital (p. 556). Graphically, the observed interest rate is set by the intersection of the capital-demand curve (r = partial f/partial c) with a temporary vertical supply curve at the currently given capital stock c0, while the “ultimate” supply curve r=p only governs how fast c0 moves toward its long-run value; consequently “the rate of interest is governed primarily by the demand price, and may greatly exceed the reward ultimately necessary to induce abstinence” (p. 557). Ramsey notes the same logic applies to a Socialist state’s accounting rate of interest, whose role would be to guide the efficient use of existing capital rather than to signal how much income should be saved (p. 557).

Assuming each family lives forever, discounts future utility at its own constant rate, and that labour is fixed so that income is a function of capital alone (with the interest rate equal to the marginal product of capital, f’(c)), Ramsey shows that in equilibrium the community divides into exactly two classes: those whose discount rate is below the equilibrium interest rate must have already attained Bliss (income x1), while everyone else must be driven down to the bare subsistence income x2, since anyone in between would still be adjusting their consumption according to equation (9a) (pp. 558-559). “In such a case, therefore, equilibrium would be attained by a division of society into two classes, the thrifty enjoying bliss and the improvident at the subsistence level” (p. 559) – there is no equilibrium in which the whole community converges on some common intermediate standard of living.