econ.studio
Solow–Swan Growth Model
Section 6 of 9
Section 6

Golden Rule Equilibrium

A higher savings rate raises k∗k^* and y∗y^*. But saving more means consuming less today. So more saving does not automatically make people better off.

The Golden Rule asks: which savings rate ss maximises steady-state consumption per effective worker c∗c^*? The answer is elegant: sGR=αs_{GR} = \alpha, capital's share of income.

Geometric intuition

In steady state, c∗=f(k∗)−(n+g+δ)k∗c^* = f(k^*) - (n+g+\delta)k^*. This is the vertical gap between the production curve and the break-even line. The chart below shows this gap.

Golden Rule: maximising steady-state consumption

Consumption c∗=f(k∗)−(n+g+δ)k∗c^* = f(k^*) - (n+g+\delta)k^* is the vertical gap between the production curve and the break-even line. The gap is widest at kGR∗k^*_{GR}, where f′(k)=n+g+δf'(k) = n+g+\delta (the two curves are parallel). Push capital past that point and all extra output is eaten by extra maintenance.

The gap is widest where the production curve is parallel to the break-even line — that is, where f′(k∗)=n+g+δf'(k^*) = n+g+\delta. This is the Golden Rule capital stock kGR∗k^*_{GR}. Push capital past it and all extra output is consumed by extra depreciation and labour-force growth.

Derivation

  1. Step 1
    c∗=f(k∗)−(n+g+δ)k∗c^* = f(k^*) - (n+g+\delta)k^*

    Steady-state consumption equals output minus the investment needed to hold k∗k^* constant. We maximise this over the choice of k∗k^* (equivalently, over ss).

  2. Step 2
    f′(kGR∗)=n+g+δf'(k^*_{GR}) = n + g + \delta

    First-order condition: differentiate with respect to k∗k^* and set to zero. The slope of ff must equal the slope of the break-even line.

  3. Step 3
    kGR∗=(αn+g+δ)11−αk^*_{GR} = \left(\frac{\alpha}{n+g+\delta}\right)^{\frac{1}{1-\alpha}}

    For Cobb–Douglas f′(k)=αkα−1f'(k) = \alpha k^{\alpha-1}, solving αkα−1=n+g+δ\alpha k^{\alpha-1} = n+g+\delta gives the Golden Rule capital stock.

  4. Step 4
    sGR=αs_{GR} = \alpha

    Compare kGR∗k^*_{GR} with the general formula k∗=(s/(n+g+δ))1/(1−α)k^* = (s/(n+g+\delta))^{1/(1-\alpha)}. Setting them equal gives s=αs = \alpha. The Golden Rule savings rate equals capital's share of income.

Golden Rule savings rate
sGR=αs_{GR} = \alpha
For Cobb–Douglas, the savings rate that maximises c∗c^* equals capital's share. With α≈1/3\alpha \approx 1/3, sGR≈33%s_{GR} \approx 33\%.

Policy implications

Below the Golden Rule (f′(k∗)>n+g+δf'(k^*) > n+g+\delta, equivalently s<αs < \alpha): the economy is dynamically efficient. Raising ss increases long-run consumption, but the current generation bears a short-run cost (they must consume less now to build up k∗k^*).

Above the Golden Rule (f′(k∗)<n+g+δf'(k^*) < n+g+\delta, equivalently s>αs > \alpha): the economy is dynamically inefficient. Too much is being invested. Reducing ss raises consumption for every generation simultaneously — a Pareto improvement. This corresponds to a real interest rate below the growth rate.

Most developed economies appear below or near the Golden Rule (positive real interest rates exceeding n+gn+g), so they are dynamically efficient.