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.