Model overview
Ramsey-Cass-Koopmans Model
Optimal growth with a representative dynastic household. Endogenous savings, saddle-path dynamics, and the modified golden rule.
Navigate the learning sections below, then move into the interactive model once you want to experiment with parameters.
Introduction
Where the Ramsey-Cass-Koopmans model sits in growth theory and what it adds to the Solow-Swan picture.
Historical Context
From Frank Ramsey's 1928 paper to the Cass-Koopmans general-equilibrium formulation of 1965, and how RCK became the backbone of modern macroeconomics.
First Principles - Why Endogenise Savings
What goes wrong when the savings rate is just an exogenous number, and what we gain by making it the answer to an optimization problem.
Assumptions
Every assumption used in the derivation, grouped by which part of the model it constrains.
Notation & Variables
The notation used throughout the chapter, grouped by stocks, flows, and parameters.
The Household Problem
Preferences, the per-worker budget constraint, the no-Ponzi condition, and the full statement of the household's optimization problem.
The Firm Problem & Market Clearing
Where factor prices come from, and how household assets equal aggregate capital.
Hamiltonian & Optimality Conditions
Set up the current-value Hamiltonian, take first-order conditions, and derive the Euler equation - the heart of the RCK model.
Transversality Condition & the Complete System
Why $k_0$ is not enough - and how the transversality condition delivers a unique solution.
Steady State
Solve $\dot c = \dot k = 0$ for the steady-state capital and consumption per worker, and compute the closed-form expressions under Cobb-Douglas.
Phase Diagram & Saddle-Path Dynamics
Visualise the system in $(k, c)$ space, identify the zero-locus curves, linearize around the steady state, and read off the saddle path.
The Modified Golden Rule
Why the optimal long-run capital stock falls short of the consumption-maximising level - and what that says about dynamic efficiency.
Comparative Statics
How $k^*$, $c^*$, and the speed of convergence respond to changes in each structural parameter, derived from the steady-state equations.
Applications, Critiques & Extensions
Where RCK has been used, the standard objections, and the modern extensions that grew out of those objections.