Section 1

Introduction

The Ramsey-Cass-Koopmans (RCK) model is the canonical model of **optimal growth**. It keeps Solow's production technology and capital accumulation but replaces Solow's exogenous savings rate with the consumption-savings decision of a forward-looking representative household. The path of capital is no longer driven by an arbitrary parameter $s$ - it is the solution to a dynamic optimization problem.

Three results follow from this change of perspective: a sharp characterization of the steady state in terms of preferences ( $f'(k^*) = \rho + \delta$ , the *modified golden rule*), a unique **saddle path** along which consumption co-moves with capital, and the **transversality condition** that pins down the initial level of consumption $c_0$ .

What this model answers

How should savings respond to changes in productivity, impatience, or the desire to smooth consumption over time?
Why does optimal capital accumulation fall *short* of the level that would maximise long-run consumption?
What determines the speed of convergence to the steady state - and why does that speed depend on the curvature of preferences?
When is a competitive equilibrium with infinitely-lived agents Pareto efficient (spoiler: under RCK assumptions, always)?

At a glance - RCK versus Solow

Solow-Swan: Savings rate $s$ exogenous. Steady state at $sf(k^*) = (n+g+\delta) k^*$ . Welfare may exceed or fall short of the golden rule depending on $s$ .
Ramsey-Cass-Koopmans: Savings chosen optimally by a household with discount rate $\rho$ and intertemporal elasticity $1/\theta$ . Steady state at $f'(k^*) = \rho + \delta$ . Optimal capital is always *below* the golden rule (modified golden rule).

Roadmap for this chapter

Historical context: Ramsey 1928, Cass 1965, Koopmans 1965.
First principles: why endogenising the savings rate matters.
Assumptions and variables.
The household problem - preferences, budget, intertemporal trade-off.
The firm problem - production and factor prices.
Hamiltonian and the Euler equation.
Transversality condition and the complete system.
Steady state and the modified golden rule.
Phase diagram, linearization, and the saddle path.
Comparative statics in $\rho$ , $\theta$ , $n$ , $\delta$ , $\alpha$ .
Applications, critiques, and extensions.

\max_{\{c_t\}_{t\geq 0}} \int_0^{\infty} e^{-\rho t}\, u(c_t)\, dt \quad \text{subject to} \quad \dot k = f(k) - c - (n + \delta) k

rck-problem

The full Ramsey problem in one line. Everything in this chapter is either setting it up or solving it.