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