Section 6

Golden Rule Equilibrium

A higher savings rate raises $k^*$ and $y^*$ , but does not necessarily make people better off. The Golden Rule asks: what savings rate maximises steady-state consumption per effective worker?

Derivation

Step 1
$c^* = f(k^*) - (n+g+\delta)k^*$
Steady-state consumption: $c^* = f(k^*) - (n+g+\delta)k^*$ . Differentiating with respect to $k^*$ and setting to zero:
Step 2
$f'(k^*_{GR}) = n + g + \delta$
Step 3
$k^*_{GR} = \left(\frac{\alpha}{n+g+\delta}\right)^{\frac{1}{1-\alpha}}$
For Cobb-Douglas ( $f'(k) = \alpha k^{\alpha-1}$ ):
Step 4
$s_{GR} = \alpha$
The Golden Rule savings rate: $s_{GR} = \alpha$ . For Cobb-Douglas, the optimal savings rate equals capital's share of income.

Golden Rule savings rate

\alpha

For Cobb-Douglas, the optimal savings rate equals capital's share of income.

Policy implications

If $f'(k^*) > n+g+\delta$ (below the Golden Rule): raising $s$ increases long-run consumption, but the current generation bears a short-run cost.

If $f'(k^*) < n+g+\delta$ (above the Golden Rule): dynamically inefficient. Reducing $s$ raises consumption for every generation simultaneously.

Most developed economies appear below or near the Golden Rule (positive real interest rates exceeding $n+g$ ).