Samuelson Consumption-Loan Model

The Samuelson consumption-loan model is an overlapping-generations (OLG) framework introduced by Paul Samuelson in the 1958 Journal of Political Economy article "An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money". It describes an economy in which people live for two periods, work when young and retire when old, and in which no one is alive at both the beginning and the end of time. Samuelson used it to show that a competitive market economy may fail to deliver Pareto efficiency, contrary to the impression conveyed by the standard general equilibrium theory of Arrow and Debreu, and that money can be valued even when it has no intrinsic worth. The model is the basis of modern OLG macroeconomics and monetary theory, including the search-based Lagos–Wright monetary model.

Time is discrete and extends indefinitely. In each period a new generation of N identical people is born. Each person receives an endowment of a single perishable consumption good: w₁ when young and w₂ when old, with w₁ > w₂ in the leading example, so that the person would like to move consumption from youth to old age. Preferences are represented by a utility function u(c₁, c₂) that is increasing and concave. Population grows at rate n, so each generation is (1 + n) times larger than the previous one. The good cannot be stored.

There is no trade between generations unless a mechanism is available. Without one, each agent consumes the endowment in each period (autarky), which is a competitive equilibrium. In autarky, the marginal rate of substitution between old and young consumption differs from one, since the agent would like to save but cannot.

Samuelson identified a benchmark rate of return. Consider a social arrangement in which each young person transfers one unit of the good to the old person alive at the same time, in return for the promise that the next generation will do the same. Each young person gives up one unit and, in the next period, receives (1 + n) units from the larger cohort that follows. The implied return is the growth rate of the population, n. Samuelson called this the biological rate of interest. It is the rate at which the economy can transform current goods into future goods through the population structure, and it is the rate that would prevail in a golden-rule steady state.

In an economy without capital, the market rate of interest r that arises in autarky depends on the marginal rate of substitution at the endowment point. If autarky interest rate r is below n, then a chain of transfers from young to old makes everyone better off: each generation gives up a little in youth and receives more than it would have saved at the market rate. This is a Pareto improvement over autarky, so the competitive equilibrium is inefficient. The inefficiency arises because there is an infinite sequence of agents, which allows the first-welfare theorem, which relies on a finite number of goods and traders, to fail.

The competitive market cannot create the chain of transfers on its own, since no one can enforce a promise by the unborn. Samuelson's second contribution was to show that money can play the role of the missing institution. Suppose that old agents hold a stock of fiat money, which is an intrinsically useless object. A young agent exchanges some of the endowment for money from an old person and later, when old, trades the money to the next young generation for goods. Provided each generation expects the next to accept money, the arrangement is self-fulfilling, and money has positive value.

In equilibrium, the real return on money equals the rate of growth of the population, so money implements the biological interest rate if the money supply is fixed. This outcome is better than autarky in the case where r < n. The model therefore gives a precise justification for money as a store of value and a means of exchange across time. At the same time, it has a second equilibrium in which money is worthless, so the model displays Nash equilibria that depend on expectations, the property later exploited by Kareken and Wallace to show that exchange rates between currencies are indeterminate. Introducing growth of the money stock produces inflation, linking the model to the study of inflation and monetary policy, and social security and public debt can serve a similar function, which links it to fiscal policy.

Several authors extended or anticipated the analysis.

• Maurice Allais (1947) had described a similar overlapping-generations structure in Économie et intérêt and derived the same biological-interest-rate result. Allais is therefore often credited as a co-originator, and Samuelson acknowledged his priority in later years.
• Peter Diamond (1965) added production, capital, and a neoclassical technology to the OLG framework. He showed that when the competitive capital stock exceeds the golden-rule level, the economy is dynamically inefficient: the interest rate is below the growth rate, and a reduction of capital would raise consumption of every generation. Government debt can increase welfare by crowding out excess capital.
• David Gale (1973) gave a systematic analysis of the pure exchange case. He classified economies into the "classical" case, in which the autarky interest rate exceeds the growth rate and the competitive equilibrium is efficient, and the "Samuelson" case, in which it does not. He showed that in the Samuelson case, stationary monetary equilibria exist.
• Abel, Mankiw, Summers, and Zeckhauser (1989) tested for dynamic inefficiency in US and other advanced-economy data by comparing capital income with investment, and concluded that these economies were dynamically efficient. The finding implies that the classical case is the relevant one for most advanced economies, while the Samuelson case may remain relevant in some emerging economies.

The model makes the question of efficiency a matter of the interaction between demographics and the structure of markets. A transfer scheme is Pareto-improving in the Samuelson case because it makes use of the "last generation never arrives" feature: since the economy has no terminal date, every generation can be a net recipient. A pay-as-you-go pension has the same logic, with a return equal to the growth rate of the wage bill, which explains why such systems are justified when the interest rate is low and challenged when it is high. The problem of designing institutions in which agents cannot commit to future behavior relates to mechanism design, and it is also a case of the missing markets that arise from information asymmetry between living and unborn agents.

Critics point to the model's reliance on a two-period life, which makes the periods very long, and to its predictions that are sensitive to expectations. Real monetary economies have a variety of assets and frictions, and the model is silent on the medium of exchange function in a short-period setting. The later search-theoretic and Lagos–Wright approaches address some of this by deriving money's role from trading frictions. Still, the Samuelson consumption-loan model established that interest rates, money, and public debt can be studied with the same tools, and that competitive equilibrium does not guarantee efficiency in dynamic economies with infinite horizons.

1. Samuelson, P. A. (1958). An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money. Journal of Political Economy, 66(6), 467–482. https://doi.org/10.1086/258100
2. Gale, D. (1973). Pure Exchange Equilibrium of Dynamic Economic Models. Journal of Economic Theory, 6(1), 12–36. https://doi.org/10.1016/0022-0531(73)90041-090041-0)
3. Abel, A. B., Mankiw, N. G., Summers, L. H., & Zeckhauser, R. J. (1989). Assessing Dynamic Efficiency: Theory and Evidence. The Review of Economic Studies, 56(1), 1–19. https://doi.org/10.2307/2297746
4. Diamond, P. A. (1965). National Debt in a Neoclassical Growth Model. American Economic Review, 55(5), 1126–1150. https://www.jstor.org/stable/1809231