econ.studio
Ramsey-Cass-Koopmans Model
Section 12 of 16
Section 12 - Welfare

The Modified Golden Rule

The golden rule capital stock kGRk_{GR} is the level of kk that maximises steady-state consumption per worker. It is a Solow-era benchmark: with exogenous ss, the planner could choose any ss, and sGRs_{GR} corresponds to the maximum of the k˙=0\dot k = 0 curve.

Deriving the golden rule

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

    Steady-state consumption as a function of steady-state capital.

  2. Step 2
    dc∗dk∗=f′(k∗)−(n+δ)=!0\frac{d c^*}{d k^*} = f'(k^*) - (n + \delta) \stackrel{!}{=} 0

    First-order condition for c∗c^* to be maximised at kGRk_{GR}.

  3. Step 3
      f′(kGR)=n+δ  \boxed{\;f'(k_{GR}) = n + \delta\;}

    Golden rule: net marginal product of capital equals the dilution rate.

The modified golden rule (RCK steady state)

  f′(k∗)=ρ+δ  \boxed{\;f'(k^*) = \rho + \delta\;}
Modified golden rule: net MPK equals the household's pure rate of time preference.

Comparing the two

Since ρ>n\rho > n (assumption P4), the marginal product at k∗k^* exceeds the marginal product at kGRk_{GR}, and by diminishing returns:

f′(k∗)>f′(kGR)  ⟺  k∗<kGR.f'(k^*) > f'(k_{GR}) \;\Longleftrightarrow\; k^* < k_{GR}.
The RCK steady state is to the left of the golden rule - capital is under-accumulated relative to the consumption-maximising level.
QuantityGolden ruleRCK (modified golden rule)
Defining conditionf′(kGR)=n+δf'(k_{GR}) = n + \deltaf′(k∗)=ρ+δf'(k^*) = \rho + \delta
Maximises what?Steady-state consumptionLifetime discounted utility
Compatible with optimization?Only if ρ=n\rho = nYes, by construction
Position in phase planePeak of k˙=0\dot k = 0 curveStrictly left of the peak
Long-run interest rater=nr = nr=ρr = \rho
Implied savings rate (CD)sGR=αs_{GR} = \alphas∗=α⋅n+δρ+δs^* = \alpha \cdot \frac{n + \delta}{\rho + \delta}
Golden rule maximises a snapshot; modified golden rule maximises a present-discounted integral.

Why optimal saving is below the golden rule

It is tempting to think the modified golden rule is worse than the golden rule because consumption is lower. But that confuses two different welfare criteria:

Dynamic efficiency

A steady state with k>kGRk > k_{GR} would be dynamically inefficient: the economy could permanently consume more by cutting capital. RCK rules this out endogenously - the household would never voluntarily over-save. No optimising representative agent ever over-accumulates capital relative to the golden rule.

Live comparison

No scalar found for key: steady_state_k
No scalar found for key: golden_rule_k
No scalar found for key: dynamic_efficiency_gap