Section 5

Variables and Notation

Aggregate variables

Symbol	Meaning
$L_t$	Size of the young cohort at time $t$ (= labour supply).
$L_{t-1}$	Size of the old cohort at time $t$ .
$K_t$	Aggregate capital stock at the start of period $t$ .
$Y_t$	Aggregate output produced in period $t$ .
$I_t$	Aggregate investment in period $t$ ( $= K_{t+1} - (1-\delta) K_t$ ).
$S_t$	Aggregate savings of the young cohort at $t$ ( $= L_t s_t$ ).
$C_t$	Aggregate consumption: $L_t c_{1,t} + L_{t-1} c_{2,t}$ .
$w_t$	Real wage per unit of labour at $t$ .
$r_t$	Net real interest rate paid on assets held into period $t$ .
$1 + r_t$	Gross real return; in the no-depreciation accounting equals $f'(k_t)$ when $\delta = 1$ .

All aggregate variables are indexed by the date

t

Per-young-worker variables

Lower-case symbols denote per-young-worker quantities, i.e. divided by $L_t$ — the size of the *young* cohort. Many textbooks divide by total population instead; the choice does not change steady-state results but does change the algebra. We stick with per-young-worker because it gives the cleanest law of motion for capital.

Symbol	Definition	Meaning
$k_t$	$K_t / L_t$	Capital per young worker.
$y_t$	$Y_t / L_t = A k_t^{\alpha}$	Output per young worker.
$c_{1,t}$	consumption of a young agent at $t$	First-period consumption.
$c_{2,t+1}$	consumption of an old agent at $t+1$	Second-period consumption of the cohort born at $t$ .
$s_t$	$w_t - c_{1,t}$	Savings of a young agent at $t$ .

Per-young-worker variables — the working variables of the model.

Parameters

Symbol	Default	Bounds	Interpretation
$\alpha$	0.33	$(0, 1)$	Capital share in Cobb–Douglas.
$\beta$	0.6	$(0, 1)$	Discount factor between life-periods. With a 30-year period and $\rho \approx 0.02$ /yr, $\beta = (1.02)^{-30} \approx 0.55$ . We default to 0.6.
$\theta$	1.0	$(0, \infty)$	Coefficient of relative risk aversion / inverse EIS. $\theta = 1$ ⇒ log utility (closed form).
$n$	0.5	$\ge 0$	Population growth per period (30 years). Annualised $\sim 1.4\%$ .
$\delta$	1.0	$(0, 1]$	Depreciation per period. Default 1 (full).
$A$	1.0	$> 0$	TFP level.
$k_0$	0.1	$> 0$	Initial capital per young worker.

Primitive parameters of the model.

The unusual size of $n$ (0.5) reflects that one period in this model is a generation, not a year. An annualised population growth of about 1.4 percent compounded over 30 years gives $(1.014)^{30} \approx 1.5$ , so $n \approx 0.5$ per period. Likewise $\beta = 0.6$ comes from an annualised pure discount rate near 2 percent. Comparative statics are *qualitatively* the same as in continuous time, but the *magnitudes* belong to the generational time scale.