Paul A. Samuelson | Macro Paper Warehouse

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

This 1958 Journal of Political Economy paper by Paul Samuelson builds a deliberately stripped-down model of a world in which goods cannot be stored or invested – nothing “keeps” – so people can only shift consumption across their lifetime by trading with other, differently-aged people currently alive, an arrangement he calls the consumption-loan model. Assuming three-period overlapping lifetimes (working, working, retired) and a population that may be stationary or growing at a constant rate, Samuelson shows that if such consumption loans clear competitively period by period, the interest rate that would maximize a representative person’s lifetime welfare exactly equals the population’s biological growth rate, so that in a stationary population the socially optimal interest rate is exactly zero. He then proves, in an “impossibility theorem,” that this social optimum can never actually be reached by a genuinely free, decentralized market relying only on voluntary bilateral trade between generations, because a young lender’s eventual repayment must come from someone who was never party to the original exchange; in a worked numerical example the free market instead settles permanently on a substantially negative real interest rate, leaving every generation worse off than the optimum. Samuelson identifies two escapes from this market failure: an explicit Hobbes-Rousseau social contract that guarantees support for the aged by drawing on the yet-unborn (a forerunner of social security), or the spontaneous emergence of a durable, intrinsically worthless money that successive generations agree to accept and pass on, whose real value can adjust as population changes so that its return replicates the optimal biological rate. The paper’s larger claim is that this reframes one function of money – not as a mere convenience for barter, but as a social compact that a purely competitive, atomistic market cannot generate on its own.

Interactions between the Multiplier Analysis and the Principle of Acceleration

This short 1939 note by Paul Samuelson, written during his time as a member of the Society of Fellows at Harvard at the suggestion of Alvin Hansen, formalizes Hansen’s combination of Keynes’s income-expenditure multiplier with the accelerator principle of induced private investment. Samuelson sets up national income in each period as the sum of a constant level of government deficit spending, consumption equal to a fixed fraction (the marginal propensity to consume, a) of the previous period’s income, and induced private investment equal to a coefficient (the “relation,” b) times the change in consumption between the previous two periods. Working through a numerical example (a = 1/2, b = 1, tabulated period by period) and then a table of alternative coefficient values, he shows that adding this accelerator term to the plain multiplier can turn an otherwise smoothly convergent income sequence into one that oscillates, and that whether the resulting path damps out, repeats indefinitely, explodes in oscillation, or instead grows or shrinks monotonically without any oscillation at all depends purely on the numerical values of a and b. He derives the exact boundaries between four such qualitatively distinct regions algebraically from the roots of the underlying difference equation’s characteristic quadratic and displays them as a chart in (a, b) space, so that the conventional multiplier (recovered as the special case b = 0) appears as one special case of a more general family in which cyclical fluctuations can arise purely from the mechanical interaction of consumption and investment lags, without any outside shock. Samuelson closes by flagging that the whole analysis is explicitly marginal – it treats a and b as constants even though they would actually shift with the level of income – and by noting that the formal structure of his model sequence parallels contemporaneous dynamic work by Lundberg and Tinbergen.