Overlapping Generations Model

Overlapping generations model (commonly abbreviated as the OLG model or the Allais–Samuelson–Diamond model) is one of the two fundamental dynamic macroeconomic frameworks—alongside the infinitely lived representative-agent Ramsey–Cass–Koopmans model—in business and economics theory for analyzing life-cycle saving, capital accumulation, intergenerational fiscal policy, and monetary theory. Introduced by French Nobel laureate Maurice Allais in 1947 (Économie et Intérêt), formalized as a pure-exchange consumption-loan economy by American Nobel laureate Paul Samuelson in 1958, and extended to include neoclassical production and physical capital accumulation by American Nobel laureate Peter Diamond in 1965, the OLG model drops the fiction of a single immortal dynasty. Instead, new cohorts of finitely lived individuals are born in every discrete period t = 0, 1, 2, …, working while young and consuming their accumulated savings while old, so that at every date the young generation overlaps and trades with the preceding old generation.

Because an OLG economy runs over an infinite horizon with a continuous turnover of finite-lived agents, it possesses a double infinity of both dated commodities and distinct households. As Karl Shell (1971) demonstrated via his "Hotel Infinity" paradox, this demographic structure invalidates the standard finite-value aggregate budget constraint required by the First Fundamental Theorem of Welfare Economics in general equilibrium theory, allowing competitive equilibria in an OLG economy to fail Pareto efficiency through capital over-accumulation (dynamic inefficiency) and enabling intrinsically worthless fiat money or pay-as-you-go social security to improve welfare for every generation.

Demographics and Intertemporal Budget Constraints

In Samuelson's (1958) pure-exchange two-period OLG model, time is discrete (t = 0, 1, 2, …) and the population of the newborn young cohort N_t grows at constant rate n > −1 per period:

Each individual born at date t (Generation t) lives for two periods: when young in period t, the agent receives a perishable commodity endowment e₁ > 0, consumes c_1,t, and saves s_t = e₁ − c_1,t at gross real interest rate 1 + r_t+1; when old in period t + 1, the agent receives endowment e₂ ≥ 0 (typically 0 in retirement) and consumes c_2,t+1 = e₂ + (1 + r_t+1)s_t. Eliminating s_t yields the lifetime intertemporal budget constraint:

At date t = 0, there also exists an initial old generation (Generation −1) of size N₋₁ who live only during period 0 and seek to maximize old-age consumption c_2,0. Crucially, without an intergenerational social contract or durable store of value, young agents at date t cannot lend to old agents at date t (because the current old will be dead at t + 1 and cannot repay), nor can they lend to unborn future generations at t + 1 during period t.

The Biological Interest Rate and Fiat Money

Aggregate resource feasibility in period t requires N_t c_1,t + N_t−1 c_2,t = N_t e₁ + N_t−1 e₂. Dividing by N_t yields the stationary per-worker feasibility locus:

Comparing the slope of the social feasibility frontier (−(1 + n)) with the slope of the household budget line (−(1 + r)) led Samuelson to his celebrated Biological Interest Rate Theorem: a stationary competitive equilibrium is Pareto efficient if and only if the equilibrium real interest rate equals the biological rate of population growth (r* = n). If e₂ = 0 and no durable asset exists, autarky (c₁ = e₁, c₂ = 0, with implicit autarkic interest rate r^aut = −1 < n) is the unique commodity-market equilibrium, which is Pareto inefficient because every generation would be unambiguously better off transferring a fraction of its youth endowment to the current old in exchange for receiving a transfer scaled by (1 + n) when old. Introducing a fixed stock M of intrinsically unbacked fiat money gifted initially to the period-0 old—whose real value grows at the population growth rate P_t / P_t+1 = 1 + n—supports the Golden Rule allocation r* = n as a valued rational bubble.

Competitive Factor Markets and the Capital Transition Law

Peter Diamond (1965) embedded neoclassical production from the Solow–Swan model into Samuelson's demographic structure. Each young individual supplies 1 unit of inelastic labor at real wage w_t, divides wage income between first-period consumption c_1,t and physical capital saving s_t = w_t − c_1,t, and retires when old to consume the gross return on capital c_2,t+1 = (1 + r_t+1)s_t. Given a constant-returns-to-scale production function in intensive per-worker form y_t = f(k_t) with k_t ≡ K_t / N_t, competitive firms hire capital and labor up to their marginal products:

Assuming full capital depreciation per generation (δ = 1, or absorbing depreciation into f(k)), the entire capital stock in period t + 1 is financed exclusively out of the aggregate savings of the period-t young (K_t+1 = N_t s(w_t, r_t+1)). Dividing by N_t+1 = (1 + n)N_t yields Diamond's fundamental law of motion for capital per worker:

In the canonical benchmark with logarithmic lifetime utility U = ln(c_1,t) + β ln(c_2,t+1) (where income and substitution effects of r_t+1 via the Slutsky equation exactly offset each other) and Cobb–Douglas production f(k) = A k^α, the saving rate out of wages is constant at s_t = [β / (1 + β)] w_t, yielding an explicit nonlinear difference equation k_t+1 = [β(1 − α)A / ((1 + β)(1 + n))] k_t^α that converges monotonically to a unique positive steady state k*.

Unlike the infinitely lived Ramsey–Cass–Koopmans model—where the modified Golden Rule r* = ρ + θg > n + g guarantees that the steady-state capital stock is always strictly below the Golden Rule level k_gold (where f′(k_gold) − δ = n + g)—an OLG economy saves out of young labor income w(k) purely to provide for retirement consumption regardless of how low the return on capital falls. If household patience β is high or labor's income share 1 − α is large relative to α, the steady-state capital stock k* can exceed k_gold, driving the steady-state real interest rate below the economic growth rate (r* < n + g):

In a Pay-As-You-Go (PAYG) social security system, the government levies a lump-sum or payroll tax d on each young worker at date t and immediately transfers the revenue to the contemporary old generation, yielding a per-retiree benefit of b = (1 + n)d (an implicit internal rate of return equal to the population plus productivity growth rate n + g). In a Fully Funded (FF) system, each young cohort's contributions are invested in physical capital (as in modern portfolio theory), earning the market return r*. Although r* > n + g in a dynamically efficient economy implies that a Fully Funded system yields higher steady-state utility than a PAYG system, transitioning from an existing PAYG system to a Fully Funded system cannot be achieved as a Pareto improvement without hurting at least one transitional generation that must pay twice—funding both its own retirement capital and the promised benefits of the currently retired generation (Breyer, 1989).

How do economists test empirically whether actual economies are dynamically efficient or inefficient?

Andrew B. Abel, N. Gregory Mankiw, Lawrence H. Summers, and Richard J. Zeckhauser (1989) derived the AMSZ criterion for stochastic economies: an economy is dynamically efficient if and only if aggregate gross capital income (corporate profits, rental income, and interest payments) consistently exceeds aggregate gross domestic investment in every period, meaning the capital sector generates a net positive cash flow dividend to the household sector rather than acting as a net sink of resources. Empirical applications of the AMSZ criterion across OECD economies generally find that total capital income exceeds gross investment, although the safe government bond rate often lies below the GDP growth rate due to equity risk premia.

Why does Ricardian Equivalence fail in the Diamond overlapping generations model?

Robert Barro's (1974) Ricardian Equivalence theorem holds that shifting from tax financing to government debt financing leaves aggregate consumption and capital accumulation unchanged because taxpayers save the tax cut to pay the future tax liabilities. In the Diamond OLG model without operative altruistic bequests, a debt-financed tax cut for the current generation is repaid partially by future cohorts not yet born at date t; because current young and old households do not internalize the tax burden shifted onto unborn future generations, they treat government bonds as net wealth, raising current consumption and crowding out physical capital accumulation k_t+1.

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. Diamond, P. A. (1965). "National Debt in a Neoclassical Growth Model." American Economic Review, 55(5), 1126–1150. https://www.jstor.org/stable/1809231
3. Shell, K. (1971). "Notes on the Economics of Infinity." Journal of Political Economy, 79(5), 1002–1011. https://doi.org/10.1086/259811
4. Barro, R. J. (1974). "Are Government Bonds Net Wealth?" Journal of Political Economy, 82(6), 1095–1117. https://doi.org/10.1086/260266
5. 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/10.2307/2297762