econ.studio
Ramsey-Cass-Koopmans Model
Section 10 of 16
Section 10 - Long run

Steady State

A steady state is a fixed point of the dynamical system in Section 8: a pair (k∗,c∗)(k^*, c^*) with k˙=c˙=0\dot k = \dot c = 0. Setting each derivative to zero in turn gives two equations in two unknowns.

Two conditions, two unknowns

  1. Step 1
    c˙=0  ⟺  f′(k∗)−δ−ρ=0\dot c = 0 \;\Longleftrightarrow\; f'(k^*) - \delta - \rho = 0

    From the Euler equation (the c=0c = 0 trivial root is irrelevant).

  2. Step 2
      f′(k∗)=ρ+δ  \boxed{\;f'(k^*) = \rho + \delta\;}

    The modified golden rule. The net return on capital equals the household's pure rate of time preference.

  3. Step 3
    k˙=0  ⟺  c∗=f(k∗)−(n+δ)k∗\dot k = 0 \;\Longleftrightarrow\; c^* = f(k^*) - (n + \delta) k^*

    The capital accumulation equation at steady state.

Closed form under Cobb-Douglas

With f(k)=Akαf(k) = A k^\alpha and f′(k)=αAkα−1f'(k) = \alpha A k^{\alpha - 1}:

  1. Step 1
    αA(k∗)α−1=ρ+δ\alpha A (k^*)^{\alpha - 1} = \rho + \delta

    Substitute Cobb-Douglas into the modified golden rule.

  2. Step 2
      k∗=(αAρ+δ) ⁣1/(1−α)  \boxed{\;k^* = \left(\frac{\alpha A}{\rho + \delta}\right)^{\!1/(1-\alpha)}\;}

    Steady-state capital per worker.

  3. Step 3
    y∗=A(k∗)α=A1/(1−α)(αρ+δ) ⁣α/(1−α)y^* = A (k^*)^\alpha = A^{1/(1-\alpha)} \left(\frac{\alpha}{\rho + \delta}\right)^{\!\alpha/(1-\alpha)}

    Steady-state output per worker.

  4. Step 4
      c∗=y∗−(n+δ)k∗  \boxed{\;c^* = y^* - (n + \delta) k^*\;}

    Steady-state consumption per worker.

  5. Step 5
    i∗=(n+δ)k∗i^* = (n + \delta) k^*

    Steady-state investment per worker - just enough to offset dilution and depreciation.

The implied savings rate

Although there is no exogenous ss in RCK, the model does deliver an implied savings rate at the steady state:

s∗=i∗y∗=(n+δ)k∗f(k∗)  =  α⋅n+δρ+δ(Cobb-Douglas).s^* = \frac{i^*}{y^*} = \frac{(n + \delta) k^*}{f(k^*)}\;=\; \alpha \cdot \frac{n + \delta}{\rho + \delta}\quad \text{(Cobb-Douglas)}.
implied-savings
The optimal long-run savings rate is the capital share scaled by (n+δ)/(ρ+δ)(n + \delta)/(\rho + \delta). Since ρ>n\rho > n, the optimal s∗<αs^* < \alpha - always less than the golden rule savings rate.

Headline steady-state numbers

Move the sliders to see how each headline number responds. The metric below is live - it re-computes whenever a parameter changes.

No scalar found for key: steady_state_k
No scalar found for key: steady_state_y
No scalar found for key: steady_state_c
No scalar found for key: mpk_steady_state
QuantitySteady-state expression (Cobb-Douglas)
k∗k^*(αAρ+δ)1/(1−α)\left(\frac{\alpha A}{\rho + \delta}\right)^{1/(1-\alpha)}
y∗y^*A(k∗)αA (k^*)^\alpha
c∗c^*y∗−(n+δ)k∗y^* - (n + \delta) k^*
i∗i^*(n+δ)k∗(n + \delta) k^*
r∗r^* (net)ρ\rho
w∗w^*(1−α)A(k∗)α(1 - \alpha) A (k^*)^\alpha
s∗s^* (implied)α⋅n+δρ+δ\alpha \cdot \frac{n + \delta}{\rho + \delta}
The full long-run picture under Cobb-Douglas production.