Optimal growth with endogenous saving

Ramsey-Cass-Koopmans Model

An optimal growth model where households choose consumption and saving over time, replacing the fixed savings rule from Solow with the Euler equation.

Macroeconomics Growth Intermediate EasyEcon / Marimo Growth, business cycles, and open economy
Focus

Euler equation, steady state, and optimal paths

Trace the saddle path through the (k, c) phase plane instantly, fire permanent-shock experiments, then open the notebook for the discrete-time shooting derivation.

Interactive diagram

Ramsey

Drag a slider — every curve and number responds instantly.

The (k, c) phase plane and its saddle path Capital per worker is on the horizontal axis and consumption per worker on the vertical axis. The hump-shaped locus marks where capital is unchanging; the vertical locus marks where consumption is unchanging. They cross at the steady state, and the saddle path is the unique trajectory along which the economy converges to it. When an experiment is selected the old loci appear dimmed, consumption jumps vertically onto the new saddle path, and the economy transitions along it to the new steady state. Capital per worker k Consumption per worker c k̇ = 0ċ = 0saddle path k* c*
k̇ = 0 ċ = 0 Saddle path Shock (jump + transition) Old economy (dimmed)

How to read this

Unlike Solow's fixed saving rule, Ramsey households choose consumption at every instant. The phase plane shows both state and choice at once: the hump (k̇ = 0) collects the points where capital is exactly maintained, and the vertical line (ċ = 0) sits at the capital stock whose return just compensates impatience — the modified golden rule f′(k*) = δ + ρ. Their crossing is the steady state.

Almost every path through this plane flies off to ruin — over-consuming into zero capital or over-saving into zero consumption. Exactly one trajectory threads the needle: the saddle path. Wherever the economy's capital starts, optimal consumption jumps straight onto this curve and rides it home. The small arrows show which way the current pushes in each of the four regions.

The experiment dropdown freezes today's economy, applies a permanent surprise shock, and redraws the new loci with the old ones dimmed. Consumption — the jump variable — leaps vertically onto the new saddle path; capital — the state variable — then adjusts slowly along it. The G ↑ experiment is the punchline: consumption absorbs the entire shock instantly and there are no transition dynamics at all.

Macroeconomic growth and cycles

Step 2 of 5

What to explore

Change parameters and watch the model adjust.

  • Capital share, discount factor, risk aversion, depreciation, and TFP
  • Initial capital and horizon for the optimal transition path

Core ideas

Interpret the mechanics before you chase the graphs.

  • The Euler equation governs consumption growth across time.
  • The steady state balances impatience against the marginal product of capital.
  • The shooting algorithm finds an initial consumption level that satisfies the model's terminal condition.

Learning goals

What this model should help students internalize.

  • Link the Euler equation to optimal consumption growth and steady-state capital.
  • Interpret shooting methods for saddle-path stable dynamics.
  • Compare endogenous saving to the fixed-savings Solow benchmark.

Prerequisites

Concepts to review before diving in.

  • Solow model intuition
  • Basic familiarity with optimisation and intertemporal choice
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Ramsey notebook

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Euler equation, steady state, and optimal paths

Trace the saddle path through the (k, c) phase plane instantly, fire permanent-shock experiments, then open the notebook for the discrete-time shooting derivation.

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