An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money

Samuelson sets out to give “a complete general equilibrium solution to the determination of the time-shape of interest rates,” and flags immediately that doing so exactly requires determining all interest rates between now and the end of time, since “every finite time period points beyond itself” (p. 467). His stated target is a case that neither Böhm-Bawerk nor Fisher had solved: a perfect, perfectly certain capital market with no possibility of storage or productive investment, so that whatever theory of interest emerges must come purely from the intertemporal trading of consumption among people of different ages, not from capital’s “net productivity.”

Samuelson assumes each person lives three equal periods – two working periods in which he produces exactly one unit of the (single, non-storable) consumption good, and a third, retirement period in which he produces nothing – and that literally nothing keeps from one period to the next, ruling out any trade with “Mother Nature” (i.e., no storage, no productive investment) (p. 468). Population is either exactly stationary (a constant number of births B every period) or growing/declining at a constant rate m, and each generation’s tastes are described by the same ordinal utility function U(C₁, C₂, C₃) (p. 469).

Each person, facing market discount rates R_t (between period t and t+1) and R_{t+1}, chooses a lifetime consumption plan (C₁, C₂, C₃) to maximize utility subject to a budget constraint equating the discounted value of lifetime consumption to the discounted value of lifetime production; aggregating each generation’s resulting “net saving” functions S₁, S₂, S₃ and requiring total net saving across all currently living generations to sum to zero in every period gives the fundamental market-clearing equation (Eqs. 1-5, pp. 469-470). Because every equation for a finite stretch of time still contains interest rates from before and after that stretch, Samuelson notes “we never seem to get enough equations” – lengthening the time period considered always adds as many new unknowns as new equations (p. 470).

Samuelson shows algebraically that R = 1 (i.e., a zero interest rate) is always one solution to the market-clearing equation in a stationary population, following directly from the individual budget identity (p. 471). He offers a common-sense reading: in a stationary system everyone goes through the same life-cycle, so giving goods to an older person today is “figuratively giving over goods to yourself when old,” and since a chocolate transferred physically today is literally a chocolate, the “transfer through time” must occur at a one-for-one rate – independent of whether individuals have any subjective preference for present over future consumption (p. 471).

Samuelson’s theorem states that “every geometrically growing consumption-loan economy has an equilibrium market rate of interest exactly equal to its biological percentage growth rate” – that is, R = 1/(1+m), or equivalently i = m, where m is the population’s per-period growth rate (p. 472). A population growing at 15 percent per period thus has an equilibrium interest rate of 0.15; a shrinking population (as he notes was true of Sweden or Ireland at the time) would have a negative equilibrium interest rate. He offers a “common-sense” explanation – in a growing population there are more young workers per retiree than in the stationary case, so the aged can be supported more generously, the excess showing up as positive interest – but explicitly calls this explanation less than “entirely convincing,” since competitive markets have no reason to “teleologically respect the old” (p. 473).

Yes: solving directly for the allocation that maximizes a representative person’s lifetime utility subject to the true per-capita resource constraint of a population growing at rate m yields exactly the same consumption pattern as the competitive equilibrium at R = 1/(1+m) (Eqs. 10-13, pp. 472-473). Samuelson treats this identity between the biological market-clearing rate and the welfare-maximizing rate as a demonstrated theorem, not an assumption, for utility functions with the usual quasi-concavity.

With only two life-periods (work and retirement), Samuelson shows algebraically that the market-clearing equations still admit the “biological” solution R = 1/(1+m), yet this cannot be the true competitive equilibrium, because with only one older generation to bargain with, “there can be no voluntary saving in a two-period world”: the economically correct solution is S₁ = 0 = S₂ with R = +∞ (an interest rate of −100 percent) (p. 474). He resolves the contradiction by noting that no worker can ever find someone younger than himself in a two-period world to be bribed into future repayment, so the mathematically “impeccable” biological solution is, in this stark case, “economically nonsense.”

Samuelson’s impossibility theorem shows that even though the three-period model’s equations admit S₁ ≡ 0 ≡ S₂ ≡ S₃ (the biologically optimal solution) as a valid mathematical root, this can never be the economically relevant free-market outcome, because any chain of voluntary trades that makes a young lender (A) support an old person (B) must eventually repay A out of someone (C) who was never party to any of the exchanges that made A’s initial gift possible – and no such C can be compelled to pay (pp. 475-477). Tracing this logic rigorously, Samuelson concludes the free market’s actual equilibrium path can never even approach R = 1 (the optimum in the stationary case) over any span of time, however long.

Using logarithmic utility U = log C₁ + log C₂ + log C₃ (the case of pure symmetry, with no systematic time preference), Samuelson derives a recursive equation for the equilibrium interest rate sequence and finds that, starting from any initial conditions, the free competitive market asymptotically approaches R ≈ 3.297, corresponding to a market interest rate of i = −2.297/3.297 ≈ −70 percent per period – meaning “consumption loans lose about two-thirds of their principal in one period” (Eqs. 16-21, pp. 477-478). This is a specific, calculated instance of a market permanently worse off than the biologically optimal (here, zero-interest) stationary solution, and Samuelson stresses the local stability analysis (Eq. 20) confirms this negative-rate outcome, not the optimum, is what the market actually converges to.

Samuelson shows that if mankind enters into an enforced “Hobbes-Rousseau social contract” in which the young are guaranteed retirement subsistence in return for supporting the current aged today – with that guarantee itself enforced by a “draft on the yet-unborn” – the social optimum can be achieved within a single lifetime, since the resulting equations reduce to the same optimality conditions derived earlier (pp. 479-480). He is explicit that this requires social coercion rather than pure self-interest, since an unenforced social norm (“the Golden Rule or Kant’s Categorical Imperative”) is not self-enforcing when a single individual can gain by disobeying it.

In his conclusion, Samuelson argues that introducing a durable but intrinsically worthless unit of money – accepted purely “by a grand consensus,” officially or by custom – lets the young and middle-aged hold something to carry into retirement even though nothing physical keeps, and, crucially, “without legislating social security or entering into elaborate social compacts,” society moves from the non-optimal negative-interest-rate configuration to the optimal biological-interest-rate configuration (pp. 481-482). With a stationary population and fixed money stock, the price level stabilizes and the real interest rate on money settles at zero (the stationary optimum); with population growing at rate m, a fixed money stock implies prices falling at rate m, so that each dollar saved earns a real return of exactly m – “just what the biological social-optimality configuration calls for.” Samuelson frames this as revealing a function of money as “a social compact” whose value being expected to hold (or move in a specific way) is itself an unwritten social agreement, not a natural market outcome (“On what tablets is that injunction written?”) (p. 482).

Several. The whole analysis explicitly rules out Böhm-Bawerk’s first and third classical causes of interest – technological net productivity of “roundabout” production is assumed away by fiat (nothing keeps, no investment), and the “expecting to be poorer/richer in the future” cause is if anything reversed, since retirement consumption is lower than working consumption by construction (p. 474, “Common-Sense Explanation” section, and footnote 16, p. 479). Böhm’s second cause (systematic subjective time preference) is “soft-pedaled” rather than ruled out, and Samuelson notes in a footnote that a colleague’s (T. Ophir’s) unpublished work shows systematic time preference would alter the equilibrium pattern (p. 479, n. 16). The negative-70-percent numerical result is tied to the specific logarithmic-utility example and Samuelson is explicit that other utility functions would give a less extreme (but still negative, and still below the biological optimum) asymptotic rate (p. 478, n. 14). Finally, Samuelson flags as an open question of his own model the “infinity paradox”: whether cutting off a hypothetically finite-lived human race (e.g., after one million generations) at some terminal date, and how the market treats that terminal generation, might undercut the whole analysis – a difficulty he says “arises from the ‘infinity’ aspect of our model” and does not fully resolve (p. 480, n. 19).