Section 6 - Household

The Household Problem

The representative dynastic household chooses a consumption path $\{c(t)\}_{t\geq 0}$ to maximise its discounted lifetime utility. Three components must be specified: the objective, the asset accumulation equation, and the boundary conditions.

Objective

U_0 = \int_0^\infty e^{-\rho t}\, u(c(t))\, L(t)\, dt

objective

Family-wide utility. Each living member at time

t

has utility

u(c(t))

; the family discounts at rate

\rho

Normalising $L_0 = 1$ and using $L(t) = e^{n t}$ , this can be written in per-member form:

U_0 = \int_0^\infty e^{-(\rho - n)\, t}\, u(c(t))\, dt.

The *effective discount rate* is

\rho - n

, not

\rho

u(c) = \begin{cases} \dfrac{c^{1-\theta} - 1}{1 - \theta} & \theta \neq 1 \\[4pt] \log c & \theta = 1 \end{cases}

Per-period CRRA utility.

The budget constraint

Let $a(t)$ denote per-worker asset holdings. The household earns wage $w(t)$ on its unit of labour and interest $r(t)$ on its assets, consumes $c(t)$ , and accumulates the remainder. Aggregate assets evolve as $\dot A = r A + w L - c L$ . Dividing by $L$ and using $\dot a = \dot A / L - n\, a$ :

Step 1
$A(t) = a(t)\, L(t),\quad \dot A = \dot a L + a \dot L = (\dot a + n a) L$
From the definition of $a$ and Leibniz.
Step 2
$\dot A = r A + w L - c L \;\Longleftrightarrow\; (\dot a + n a) L = (r a + w - c) L$
Substitute the aggregate accumulation equation.
Step 3
$\boxed{\;\dot a(t) = \bigl(r(t) - n\bigr) a(t) + w(t) - c(t)\;}$
The per-worker budget constraint. Per-capita dilution shows up as $-n a$ .

The no-Ponzi-game condition

Without a restriction on debt, the household could roll over interest payments forever and consume infinitely. The **no-Ponzi condition** forbids this:

\lim_{t \to \infty} a(t)\, \exp\!\left(-\int_0^t [r(s) - n]\\, ds\right) \geq 0.

no-ponzi

Discounted asset holdings cannot diverge to

-\infty

- the present value of any debt must be (weakly) zero in the limit.

The full statement

\max_{\{c(t)\}}\; \int_0^\infty e^{-(\rho - n) t}\, u(c(t))\, dt\quad \text{s.t.}\quad \dot a = (r - n) a + w - c,\; a(0) = a_0,\; \lim_{t\to\infty} a(t) e^{-\int_0^t(r-n)ds} \geq 0.

household-problem-full