econ.studio
Ramsey-Cass-Koopmans Model
Section 9 of 16
Section 9 - Closing the model

Transversality Condition & the Complete System

The Euler equation and the capital accumulation equation form a system of two ODEs. Solutions form a two-parameter family. The initial condition k(0)=k0k(0) = k_0 fixes one parameter. We need *one more* boundary condition to pin down c(0)c(0). That condition is the **transversality condition** (TVC).

Statement of the TVC

lim⁡t→∞e−(ρ−n)t μ(t) k(t)=0.\lim_{t \to \infty} e^{-(\rho - n) t}\, \mu(t)\, k(t) = 0.
tvc

Using μ=u′(c)=c−θ\mu = u'(c) = c^{-\theta} for CRRA, this is equivalent to:

lim⁡t→∞e−(ρ−n)t c(t)−θ k(t)=0.\lim_{t \to \infty} e^{-(\rho - n) t}\, c(t)^{-\theta}\, k(t) = 0.
Discounted shadow value of the terminal capital stock must vanish - no useful capital can be left lying around at infinity.

Why the TVC is needed

Consider what happens to a candidate solution that violates the TVC:

Candidate pathWhat it impliesWhy it is suboptimal
Asymptotic capital exceeds k∗k^*Household holds capital whose present value is strictly positive at infinity.Household could consume that capital today and gain utility without giving anything up - strictly improving.
Asymptotic capital depletes to 0Consumption falls to zero asymptotically.Inada conditions force u′(c)→∞u'(c) \to \infty, so marginal utility is unbounded - violates the no-Ponzi condition for the household.
Asymptotic capital equals k∗k^* exactlyCapital and consumption converge to (k∗,c∗)(k^*, c^*).TVC is satisfied; this is the **saddle path**.
Only the saddle path satisfies the TVC.

The complete system

Putting everything together:

  1. Step 1
    k˙=f(k)−c−(n+δ)k\dot k = f(k) - c - (n + \delta) k

    Capital accumulation (resource constraint).

  2. Step 2
    c˙=cθ(f′(k)−δ−ρ)\dot c = \frac{c}{\theta}\bigl(f'(k) - \delta - \rho\bigr)

    Euler equation.

  3. Step 3
    k(0)=k0    givenk(0) = k_0\;\; \text{given}

    Initial condition on the state.

  4. Step 4
    lim⁡t→∞e−(ρ−n)tc(t)−θk(t)=0\lim_{t \to \infty} e^{-(\rho - n) t} c(t)^{-\theta} k(t) = 0

    Transversality condition.

Existence and uniqueness

Under the assumptions of Section 4, the system has a unique steady state (k∗,c∗)∈R+2(k^*, c^*) \in \mathbb{R}^2_+ that is a **saddle point**: the Jacobian has one positive and one negative eigenvalue. The set of trajectories that satisfy the TVC is a one-dimensional manifold - the **stable manifold** of (k∗,c∗)(k^*, c^*). For any initial k0>0k_0 > 0, there is a unique c0c_0 on that manifold. This is the saddle path.