econ.studio
Ramsey-Cass-Koopmans Model
Section 13 of 16
Section 13 - Perturbing the parameters

Comparative Statics

Comparative statics in RCK is mostly an exercise in implicit differentiation of f′(k∗)=ρ+δf'(k^*) = \rho + \delta and c∗=f(k∗)−(n+δ)k∗c^* = f(k^*) - (n + \delta) k^*. The signs of the resulting derivatives are clean and intuitive.

A reference table

ParameterEffect on k∗k^*Effect on c∗c^*Effect on r∗=ρr^* = \rhoSpeed of convergence
ρ\rho (discount rate) ^vv^ one-for-oneFaster
θ\theta (curvature) ^000Slower
nn (population growth) ^0v0Slower (smaller ρ−n\rho - n)
δ\delta (depreciation) ^vv (ambiguous in CD)0Faster
α\alpha (capital share) ^^^0Slower
AA (TFP) ^^^0Same rate, higher level
Comparative statics at the steady state. θ\theta does not move the steady state - it only changes *how fast* the economy gets there.

Derivation - change in ρ\rho

Take f′(k∗)=ρ+δf'(k^*) = \rho + \delta and totally differentiate with respect to ρ\rho (holding other parameters fixed):

  1. Step 1
    f′′(k∗) dk∗dρ=1f''(k^*)\, \frac{dk^*}{d\rho} = 1

    Implicit differentiation.

  2. Step 2
    dk∗dρ=1f′′(k∗)<0\frac{dk^*}{d\rho} = \frac{1}{f''(k^*)} < 0

    Since f′′<0f'' < 0, more impatience ⇒\Rightarrow less capital.

  3. Step 3
    dc∗dρ=[f′(k∗)−(n+δ)] dk∗dρ=(ρ−n) dk∗dρ<0\frac{dc^*}{d\rho} = \bigl[f'(k^*) - (n + \delta)\bigr]\, \frac{dk^*}{d\rho} = (\rho - n)\, \frac{dk^*}{d\rho} < 0

    Since ρ>n\rho > n and dk∗/dρ<0dk^*/d\rho < 0. Both directly and indirectly, impatience lowers long-run consumption.

Derivation - change in θ\theta

θ\theta does not appear in either steady-state equation, so the steady state does not move. But θ\theta enters det⁡J\det J and therefore the eigenvalues:

  1. Step 1
    det⁡J=c∗θ f′′(k∗)(negative)\det J = \frac{c^*}{\theta}\, f''(k^*) \quad \text{(negative)}

    Larger θ⇒∣det⁡J∣\theta \Rightarrow |\det J| smaller.

  2. Step 2
    λ−=(ρ−n)−(ρ−n)2−4det⁡J2\lambda^- = \frac{(\rho - n) - \sqrt{(\rho - n)^2 - 4 \det J}}{2}

    Eigenvalue formula.

  3. Step 3
    θ↑  ⇒  ∣det⁡J∣↓  ⇒  ∣λ−∣↓\theta \uparrow \;\Rightarrow\; |\det J| \downarrow \;\Rightarrow\; |\lambda^-| \downarrow

    Convergence becomes slower.

Derivation - change in nn

Population growth does not enter the modified golden rule f′(k∗)=ρ+δf'(k^*) = \rho + \delta, so k∗k^* does not move. But nn enters c∗c^* directly:

  1. Step 1
    dc∗dn=−k∗<0\frac{dc^*}{dn} = -k^* < 0

    Faster population growth dilutes capital faster; less consumption left over after replenishing kk.

  2. Step 2
    dr∗dn=0\frac{dr^*}{dn} = 0

    Interest rate pins down at ρ\rho, independent of nn.

Phase diagram under perturbation

Raising ρ\rho slides the vertical c˙=0\dot c = 0 locus to the *left* and pulls the saddle path with it. The economy gravitates to a smaller k∗k^*. Drag the ρ\rho slider in the live plot below to watch.

Phase diagram (responds to $\rho$, $\theta$)