Model overview
Diamond Overlapping Generations Model
Agents live two periods. They work and save when young, consume their savings when old. Capital accumulation is driven entirely by the savings of the young cohort. Because each generation cares only about its own life-cycle utility — not the welfare of all future generations — the resulting equilibrium can over-accumulate capital relative to the golden rule, producing the famous *dynamic inefficiency* result and a Pareto-improving role for pay-as-you-go social security.
Navigate the learning sections below, then move into the interactive model once you want to experiment with parameters.
Introduction
Diamond's overlapping generations (OLG) model replaces the infinitely-lived household of Ramsey–Cass–Koopmans with cohorts that live only two periods. This single change generates rich new dynamics: capital accumulation is driven by the savings of the young, the steady state need not be efficient, and government policy (social security, public debt) acquires real bite.
Historical Context
The OLG framework was introduced by Paul Samuelson in 1958 to study money as a store of value. Peter Diamond's 1965 paper turned it into the modern workhorse of growth theory with production and capital.
First Principles
Finite lifetimes break the equivalence between individual optimisation and social welfare maximisation. We trace the chain from 'agents care only about their own two periods' to 'the steady state can be inefficient'.
Assumptions
We collect the full list of modelling assumptions, separated into demographic, preference, technology, and market-structure groups.
Variables and Notation
A single reference table for every symbol used in the derivations. We separate aggregate quantities, per-young-worker quantities, and primitive parameters.
The Household Problem
A young agent at $t$ chooses how to split wage income $w_t$ between consumption today and savings, which become consumption tomorrow at the (anticipated) gross return $1 + r_{t+1}$.
The Firm Problem
Firms rent capital and labour each period in competitive markets. Factor prices are pinned down by marginal products. With Cobb–Douglas and full depreciation, $w_t$ and $1 + r_t$ are simple closed-form functions of $k_t$.
Derivations
We derive the closed-form savings function under CRRA utility, then specialise to log utility ($\theta = 1$), which yields the remarkable result that the savings rate is independent of the interest rate.
Capital Accumulation
The savings of the young today *are* the capital of tomorrow. Combined with the factor-price equations, this produces a one-dimensional non-linear dynamical system in $k_t$ — the heart of the model.
Steady State
At the interior steady state $k^*$, every per-young-worker quantity is constant. We derive closed forms for $k^*$, $y^*$, $w^*$, $r^*$, $c_1^*$, $c_2^*$, and the implied savings rate, all under log utility and Cobb–Douglas with full depreciation.
Dynamic Inefficiency
The Diamond OLG steady state can lie above the golden-rule capital level. When it does, the economy is *dynamically inefficient* — reducing $k^*$ would raise consumption for every generation. This is the model's signature departure from Ramsey–Cass–Koopmans.
Comparative Statics
How changing the parameters affects the Diamond OLG steady state.
Applications
Applying Diamond OLG to public debt and Social Security models.
Critiques and Extensions
Critiques of the basic Diamond OLG setup.