Section 8 - Derivation

Hamiltonian & Optimality Conditions

The household problem is an infinite-horizon optimal control problem. We solve it using **Pontryagin's maximum principle** with a current-value Hamiltonian. The mechanics are routine; the result - the Euler equation - is the core content of the chapter.

Setting up the current-value Hamiltonian

Recall the problem (using the equilibrium-substituted budget constraint, since we will solve under market clearing):

\max_{\{c\}} \int_0^\infty e^{-(\rho - n) t}\, u(c)\, dt\quad \text{s.t.}\quad \dot k = f(k) - c - (n + \delta) k.

State: $k$ (capital per worker). Control: $c$ (consumption per worker). Co-state (current-value): $\mu$ (shadow value of one extra unit of $k$ ). The current-value Hamiltonian is:

\mathcal{H}(k, c, \mu) = u(c) + \mu \bigl[f(k) - c - (n + \delta) k\bigr].

current-value-H

The current-value Hamiltonian - utility flow plus the shadow value of investment.

First-order conditions

Pontryagin's maximum principle requires:

Step 1
$\textbf{(i)}\quad \frac{\partial \mathcal{H}}{\partial c} = 0 \;\Longrightarrow\; u'(c) = \mu$
The current marginal utility of consumption equals the shadow value of capital.
Step 2
$\textbf{(ii)}\quad \dot \mu = (\rho - n) \mu - \frac{\partial \mathcal{H}}{\partial k}= (\rho - n) \mu - \mu \bigl[f'(k) - (n + \delta)\bigr]$
The co-state equation. Notice the discount rate is the *effective* rate $\rho - n$ , matching the time-discount in the objective.
Step 3
$\textbf{(iii)}\quad \dot k = f(k) - c - (n + \delta) k$
State equation - the capital accumulation identity.
Step 4
$\textbf{(iv)}\quad \lim_{t \to \infty} e^{-(\rho - n) t} \mu(t) k(t) = 0$
The transversality condition - covered next section.

From the FOCs to the Euler equation

Differentiate condition (i) with respect to time and divide by $\mu$ :

Step 1
$u'(c) = \mu \;\;\xrightarrow{\;\;\frac{d}{dt}\;\;}\;\; u''(c)\, \dot c = \dot \mu$
Differentiate the FOC for $c$ .
Step 2
$\frac{u''(c)\, \dot c}{u'(c)} = \frac{\dot \mu}{\mu}$
Divide by $u'(c) = \mu$ .
Step 3
$\frac{\dot \mu}{\mu} = (\rho - n) - \bigl[f'(k) - (n + \delta)\bigr]= \rho + \delta - f'(k)$
From co-state equation (ii), divided by $\mu$ .
Step 4
$\frac{u''(c)\, \dot c}{u'(c)} = \rho + \delta - f'(k)$
Combine the two preceding lines.
Step 5
$\text{Define the curvature of $u$: } \theta(c) := -\frac{c\, u''(c)}{u'(c)}.$
For CRRA, $\theta(c) = \theta$ - a constant.
Step 6
$-\theta\, \frac{\dot c}{c} = \rho + \delta - f'(k)$
Substitute the CRRA curvature.
Step 7
$\boxed{\;\frac{\dot c}{c} = \frac{1}{\theta}\bigl(f'(k) - \delta - \rho\bigr)\;}$
The **Euler equation** (Keynes-Ramsey rule). Consumption grows when the net return on capital exceeds the discount rate; the intertemporal elasticity $1/\theta$ scales the response.

What the Euler equation says

Three regimes, each with a clean economic interpretation:

The Euler equation pairs with the capital accumulation equation to give a **two-dimensional dynamical system** in $(k, c)$ :

\dot k = f(k) - c - (n + \delta) k, \qquad \dot c = \frac{c}{\theta}\bigl(f'(k) - \delta - \rho\bigr).

full-system

The complete RCK dynamics. Solve subject to

k(0) = k_0

and the transversality condition.