Section 5

Notation & Variables

Quantities are written in per-worker terms unless otherwise marked. Aggregate counterparts use upper case. Time arguments are suppressed in derivations when no ambiguity arises.

Aggregates

Symbol	Name	Description
$K(t)$	Capital stock	Aggregate productive capital.
$L(t)$	Labour	Population: $L(t) = L_0 e^{n t}$ .
$Y(t)$	Output	$Y = F(K, L)$ with CRS.
$C(t)$	Aggregate consumption	$= c L$ .
$I(t)$	Aggregate investment	$= \dot K + \delta K$ .

Per-worker quantities

Symbol	Name	Definition
$k$	Capital per worker	$k = K/L$ .
$y$	Output per worker	$y = f(k)$ , with $f(k) = F(k, 1)$ .
$c$	Consumption per worker	$c = C/L$ .
$a$	Asset holdings per worker	$a = A/L$ ; in equilibrium $a = k$ .
$w$	Real wage	$w = f(k) - k f'(k)$ .
$r$	Real interest rate	$r = f'(k) - \delta$ .

Parameters

Symbol	Name	Role
$\alpha$	Capital share	Cobb-Douglas exponent on $K$ .
$A$	Total factor productivity	Scale of output.
$\delta$	Depreciation rate	Capital lost per unit time.
$n$	Population growth	Per-capita dilution.
$\rho$	Pure rate of time preference	Household impatience.
$\theta$	Inverse intertemporal elasticity of substitution	Curvature of $u$ ; risk aversion under uncertainty. $1/\theta$ = EIS.

Endogenous objects (derived)

$k^*$: Steady-state capital per worker. Solution of $f'(k^*) = \rho + \delta$ .
$c^*$: Steady-state consumption per worker. $c^* = f(k^*) - (n + \delta) k^*$ .
$k_{GR}$: Golden-rule capital. Solution of $f'(k_{GR}) = n + \delta$ .
$\mu(t)$: Current-value co-state on capital - the shadow price of one extra unit of $k$ .
$\lambda^-$: Stable eigenvalue of the linearized system. Convergence rate $|\lambda^-|$ ; half-life $\ln 2 / |\lambda^-|$ .