Notation & Key Definitions

Aggregate variables

$Y$

Output — total GDP.

$K$

Capital — total physical capital stock.

$L$

Labour — total workers, equal to population and fully employed.

$A$

Technology — total factor productivity (TFP); labour-augmenting.

$C$

Consumption — total household consumption.

$I$

Investment — total spending on new capital.

Per effective worker

$k$

$K/(AL)$ — capital per effective worker; the state variable.

$y$

$Y/(AL) = f(k) = k^\alpha$ — output per effective worker.

$c$

$(1-s)\,y$ — consumption per effective worker.

$i$

$s\,y$ — investment per effective worker.

Parameters

$s$

Savings rate. $(0, 1)$, exogenous and constant.

$\delta$

Depreciation rate. $(0, 1)$; often $\approx 0.05$–$0.10$.

$n$

Population (labour) growth rate. $\geq 0$; often $\approx 0.01$–$0.02$.

$g$

Technological growth rate. $\geq 0$; often $\approx 0.01$–$0.02$.

$\alpha$

Capital's share of output. $(0, 1)$; often $\approx 1/3$.

Key concepts

Steady state

The point where $\dot{k} = 0$: actual investment $s f(k)$ exactly equals break-even investment $(n + g + \delta)k$, so capital per effective worker is constant.

Break-even investment

$(n + g + \delta)k$ — the investment needed just to hold $k$ steady as capital depreciates ($\delta$) and the effective labour force grows ($n + g$).

Balanced growth path

The long-run path on which per-effective-worker variables are constant while per-worker variables grow at $g$ and aggregates grow at $n + g$.

Golden Rule

The savings rate $s_{GR} = \alpha$ that maximises steady-state consumption per effective worker — not output, and not capital.

Conditional convergence

Economies with the same $s, n, g, \delta$ converge to a common steady state, so poorer ones grow faster. Absolute convergence (regardless of parameters) is not predicted.

Inada conditions

$f'(k) \to \infty$ as $k \to 0$ and $f'(k) \to 0$ as $k \to \infty$; with concavity they guarantee a unique, stable interior steady state.