A Mathematical Theory of Saving

Frank Ramsey’s 1928 paper asks how much of its income a nation ought to save, and derives a rule – that the rate of saving times the marginal utility of consumption should equal the gap between attainable “Bliss” and the community’s actual current rate of enjoyment – valid, he shows, “under conditions of surprising generality.” The setup assumes a community that persists forever without changing in numbers or tastes, whose enjoyments and sacrifices at different times can be added independently, and which – crucially – does not discount later enjoyments merely because they are later, a practice Ramsey calls “ethically indefensible” and traceable only to “the weakness of the imagination” (though Section II relaxes this to allow a constant positive discount rate). Denoting consumption x(t), labour a(t), and capital c(t), with income f(a,c) satisfying the accounting identity that savings plus consumption equal income, and given utility of consumption U(x) and disutility of labour V(a), Ramsey defines “Bliss” (B) as the maximum obtainable rate of net enjoyment U(x)-V(a), which the community either reaches in finite time or approaches asymptotically forever; because only reaching or approaching Bliss keeps the cumulative shortfall from Bliss, summed over all time, finite, the paper argues the community is bound to save enough to do so. Solving the resulting calculus-of-variations problem (jointly with an optimal labour-supply condition equating the marginal disutility of labour to the marginal efficiency of labour times the marginal utility of consumption) yields the headline rule: the rate of saving times the marginal utility of consumption should always equal Bliss minus the actual rate of utility enjoyed – a result Ramsey also derives, via a suggestion from Keynes, by a much simpler direct argument comparing the loss from postponing consumption by an infinitesimal interval. The rule’s most striking feature, Ramsey notes, is that it is independent of the production function except through Bliss, and independent of the current rate of interest (when the future is not discounted) except where that rate is exactly zero; a numerical illustration using an assumed utility schedule implies saving roughly three-fifths of income at a family income of 500 pounds, “greatly in excess of that which anyone would normally suggest.” Section II specializes to a linear income function f(a,c) = pa + rc (constant wage and interest rates) to give a graphical solution, extend the analysis to an individual with a finite lifetime who wishes to leave a bequest, and rework the rule under constant time-discounting of future utility – showing the discounted version depends only on the ratio of the discount rate to the interest rate, and that if the interest rate is smaller than the discount rate, consumption is driven toward bare subsistence and debt accumulates without limit. Section III turns to how the interest rate itself is determined, showing that out of equilibrium the interest rate behaves as a demand price for the whole stock of capital but a supply price for the flow of new saving, so it can substantially exceed what would ultimately be needed to induce thrift; and that when different individuals apply different constant discount rates, a stationary equilibrium divides the community into a class that reaches Bliss and a class driven down to bare subsistence, rather than settling on some common intermediate standard.

F. P. Ramsey | Macro Paper Warehouse

A Mathematical Theory of Saving