NOTE-1287: A Mathematical Theory of Saving

Contribution

It solves the problem of optimal saving for a whole community over infinite time, by the calculus of variations, and states the solution as one rule. It shows how far the answer depends on the production function and the interest rate (hardly at all), and how it changes when utility is discounted. It then uses the same equations to ask what fixes the rate of interest. It is the founding text of optimal-growth theory and the reference for refusing to discount the future for being later.

Key insight

If the community can approach a finite maximum of enjoyment, Bliss, then any plan that never approaches it loses an infinite amount of enjoyment over endless time. So the problem becomes: minimize the total shortfall from Bliss. Changing the variable of integration from time to capital reduces it to minimizing an integrand at each level of capital. The rule follows: “rate of saving multiplied by marginal utility of consumption should always equal bliss minus actual rate of utility enjoyed” (p. 547). Keynes’s version: saving £1 less this year postpones the whole approach to Bliss by 1/z of a year, at a cost of that fraction of a year’s shortfall (pp. 547–548).

Assumptions

• No discounting of utility in Section I. “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). Discounting is reintroduced in II(c) and III.
• One community for ever. It does not change “either in numbers or in its capacity for enjoyment or in its aversion to labour” (p. 543), and enjoyments at different times are independent and additive.
• The same motives in every generation, “so that there is no chance of our savings being selfishly consumed by a subsequent generation; and that no misfortunes will occur to sweep away accumulations” (p. 544).
• No distribution. Total satisfaction is a function of total consumption and labour only (p. 543).
• A finite Bliss. “Economic causes alone could never give us more than a certain finite rate of enjoyment” (p. 545). Without it the integral need not converge.
• Homogeneous goods and labour; no technical progress except what accumulation itself brings (p. 543).

Key results

• The equations (pp. 544–546). Income f(a, c) from labour a and capital c; dc/dt + x = f(a, c) (1). Marginal disutility of labour equals its marginal product times the marginal utility of consumption, v(a) = (∂f/∂a)u(x) (2). Marginal utility falls at the rate of interest, d/dt u(x) = −(∂f/∂c)·u(x) (3).
• The rule (pp. 546–547). Integration gives dc/dt = {K − (U(x) − V(a))}/u(x) (4), and minimizing the shortfall from Bliss identifies K with B (5).
• Independence (p. 548). The rule “is altogether independent of the production function f(a, c), except in so far as this determines bliss”. How much to save out of a given income is independent of the interest rate, “unless this is actually zero”.
• How much (pp. 548–549). An illustrative scale (not data) gives total utility 2 to 8 for family incomes £150 to £5,000, with 8 as Bliss. Fitting a parabola gives marginal utility 1/60 at £200. A family on £500 should then save about £300, since 300 × 1/60 = 8 − 3. Ramsey calls this “greatly in excess of that which anyone would normally suggest”.
• What was neglected (p. 549).
  • Population growth, and inventions that raise Bliss, are reasons to save more. Inventions that make income cheaper are a reason to save less.
  • “The most serious factor neglected is the possibility of future wars and earthquakes destroying our accumulations.” These cannot be handled by a low rate of interest, since they destroy principal as well.
• Constant returns (II, pp. 549–551). With f = pa + rc, a tangent construction on the curve of utility of unearned income, W(y), tells how much unearned income to consume (Fig. 1, eq. 6). Marginal utility then falls as w(y) = Ae^(−rt) (7).
• A finite life (II(b), pp. 551–553). An individual with T years to live and a bequest c₃ to leave saves less than an everlasting community. Over a long life he saves first and then dissaves: “to begin with there is saving, and subsequently splashing”.
• Discounting (II(c), pp. 553–555).
  • Utility discounting is distinct from money discounting. Anyone who can borrow at r must value £1 now as £(1 + r) in a year, whatever his discount for utility.
  • At “about ¾ per cent”, utility now is worth twice that in a hundred years and four times that in two hundred.
  • A constant rate “is the only assumption we can make, without contradicting our fundamental hypothesis that successive generations are actuated by the same system of preferences”. With a higher rate for the first fifty years, our ranking of 2000 against 2050 would differ from that of the people alive in 2000.
  • For ρ < r, d/dt u(x) = −u(x)(r − ρ) (9). The solution is as before with a “modified utility”, the integral of w^(r/(r−ρ)). This translates “discounting of the future into a discounting of high incomes”, at a rate set by ρ/r alone (p. 554).
  • Special case w(y) = Dy^(−α), α > 1: a constant share of income is saved, printed as (r − ρ)/(r(α − 1) + ρ), or 1/(α − 1) if ρ = 0 (p. 555).
  • For ρ > r, marginal utility rises at ρ − r. Consumption falls toward subsistence, capital is exhausted and debt taken on to the limit of credit (p. 555).
• The rate of interest (III, pp. 555–559).
  • (α) If all discount at ρ, equilibrium needs ∂f/∂c = ρ. Out of equilibrium, which may last centuries, the rate is fixed by the demand for the existing capital stock and “may greatly exceed the reward ultimately necessary to induce abstinence” (p. 556). In a socialist state’s accounts the rate of interest would allocate existing capital, not guide how much to save (p. 557).
  • (β) People “are not so interested in their heirs as in themselves” (p. 557). If they ignore heirs, live off annuities and differ in ρ, capital has a supply price, that of the “marginal saver”. Those beyond the margin supply negatively, borrowing when young (p. 558).
  • (γ) Families that live for ever and differ in ρ: in equilibrium those with ρ < r are at Bliss and the rest at subsistence. Society divides into “the thrifty enjoying bliss and the improvident at the subsistence level” (p. 559).

Claims

id	claim	strength	support
C1	Without discounting, optimal saving makes saving times marginal utility of consumption equal to Bliss minus present net utility	strong under its assumptions: derived by the calculus of variations and by Keynes’s argument	§I, pp. 546–548
C2	The optimal saving out of a given income does not depend on the production function except through Bliss, nor on the interest rate unless it is zero	strong under its assumptions	p. 548
C3	The rule demands far more saving than is usual	weak: one invented utility scale	pp. 548–549
C4	With constant returns and a constant discount ρ < r, the rule holds for a modified utility, and the share saved depends on ρ/r; with ρ > r consumption falls to subsistence	strong under constant returns	§II(c), pp. 553–555
C5	Only a constant discount rate is consistent with successive generations sharing one system of preferences	moderate: argued from one example, not proved	p. 553
C6	Out of equilibrium the rate of interest is set by capital’s demand price and may far exceed the reward needed to induce abstinence	moderate: a diagram and an argument	§III(α), pp. 555–556
C7	Families that differ in discount rate end, in equilibrium, divided between Bliss and subsistence	strong in a stark model with a finite bliss income and a subsistence floor	§III(γ), pp. 558–559
C8	Discounting later enjoyments is ethically indefensible and comes from weakness of imagination	asserted, not argued	p. 543

Concepts

• Bliss (B): the maximum obtainable rate of net enjoyment, reached at a finite capital or approached as capital grows without limit; to be distinguished from the maximum conceivable rate (pp. 544–545).
• rate of discount for utility versus for money: the first a preference over when enjoyment comes; the second fixed at the interest rate for anyone who can borrow and lend (p. 553).
• modified utility: the integral of marginal utility raised to the power r/(r − ρ), which turns discounting of the future into discounting of high incomes (p. 554).
• earned and unearned income: pa and rc under constant returns (p. 550).
• splashing: dissaving late in a finite life (p. 553).
• marginal saver: the person whose discount rate equals the interest rate, who neither saves nor borrows except for old age (p. 558).

Connections

• Greaves and MacAskill (NOTE-1302). They name Ramsey among those who endorse a zero rate of pure time preference, the assumption their case calls “near-essential”. That is right of p. 543. It leaves out that Ramsey gives no argument there, and that he models discounting in the rest of the paper.
• Meyer, SEP (NOTE-1304). Its description, optimal savings without distributional concerns and no pure time discounting, matches p. 543.
• Thorstad (NOTE-1308) and Tarsney (NOTE-1247). Both discount the future for risk, not for time. Ramsey draws the same line in 1928: he rejects discounting for time, assumes no misfortunes, and names wars and earthquakes as “the most serious factor neglected”, which a low interest rate cannot capture (pp. 544, 549).
• Gardiner (LIT-1681), unread. The intergenerational storm is assumed away on p. 544: no chance “of our savings being selfishly consumed by a subsequent generation”.
• Hershfield (NOTE-1259). Ramsey’s diagnosis of discounting, “the weakness of the imagination”, is a psychological claim. Hershfield’s finding, that people who feel less connected to their future self, or see it less vividly, discount more, is a modern version for one life.
• Narveson (LIT-1769), unread. Population is fixed by assumption. Ramsey’s one remark on it, that growth is a reason to save more (p. 549), counts more future people as more to provide for, but says nothing about whether to create them.

Bearing on the record

• QUESTION-299. The question files Ramsey under the impersonal ground. Read, he fits it less neatly. These are inferences.

  • His unit is not future strangers but one community that “goes on for ever” with the same motives, and he writes its saving in the first person plural: “we shall be in 1 + 1/z year’s time exactly where we should have been” (p. 547). The zero discount is stated as the prudence of an everlasting “we”, not as a duty to others.
  • That “we” holds together only by assumption. Section II(c) needs successive generations “actuated by the same system of preferences” (p. 553), and p. 544 needs no generation to consume selfishly what earlier ones saved.
  • His reason against discounting is a diagnosis, not a principle: a failure of imagination. That is a claim the question’s psychological side can test. The record’s lay-judgement result, that most people judge extinction uniquely bad only when the lost future is made salient (Schubert et al., NOTE-1270), fits it.
  • When he describes how people do save (§III), they discount “for himself or his heirs” (p. 556) and are “not so interested in their heirs as in themselves” (p. 557). Concern falls off with distance from the self.

• The owner’s thesis (CLAIM-491). The founding statement of the zero-discount view is written in the thesis’s shape: a collective self extended through time, saving for itself. This is a modelling choice, not a claim about motive, and Ramsey offers it as what ought to be, not what is. His description of actual savers puts concern on a gradient, self first, heirs after, which is the scale reading the claim’s note on Hershfield already gives. A dated note is proposed for the claim:

  Ramsey (NOTE-1287), the source of the refusal to discount the future, states it as the saving of one community that “goes on for ever” with unchanged motives, in the first person plural. Its case for the far future is the prudence of a collective self extended in time, held together by the assumption that every generation shares one system of preferences. His actual savers are “not so interested in their heirs as in themselves”, and he blames discounting on “the weakness of the imagination”. Both fit the thesis’s reading of concern as scaled by the self’s reach, but neither is evidence about motive.

• The triviality objection (CLAIM-492). Not addressed. Ramsey’s “same system of preferences” is a criterion of a community’s identity over time, but it is a sameness of what is cared about, so it is not the independent criterion the objection asks for.

• No THEORY filed.

• No ML instruction.

Limitations

• The refusal to discount is asserted, in one clause, and the paper does not defend it.
• Strong assumptions. Fixed population, unchanging tastes and motives for ever, no catastrophes, no distribution, additive utility, a finite Bliss. Ramsey names several of these as the main things neglected (p. 549).
• The saving figure is illustrative. The utility scale on p. 548 is invented, “put forward merely as an illustration”.
• One community or individual at a time until §III; nothing on conflict between generations except by assumption.
• Typos and slips in the source.
  • p. 546: “Δx( + ∂f/∂c Δt)” drops the 1, for Δx(1 + ∂f/∂c Δt).
  • p. 548: “bliss minus actual rate of utility = 8 — = 5” drops the 3.
  • p. 551, eq. (8): “(Ux − V(a))” for (U(x) − V(a)).
  • p. 555: the denominator of the savings formula carries a stray “dy”.
  • p. 555: on this reading’s arithmetic, savings = y/(β − 1) makes (r − ρ)/(r(α − 1) + ρ) the ratio of saving to the unearned income consumed, not to unearned income as printed; as a share of unearned income it would be (r − ρ)/(rα). Checked once, not against later literature.
  • “9(a)” (p. 555) and “(9a)” (p. 554) for the same equation.

Open questions

• Did Ramsey argue for the refusal to discount anywhere else? The record holds none of his other work.
• Is “the weakness of the imagination” right? A study that made the far future vivid and measured the implied rate of pure time preference, apart from risk, would test it.