Cheat sheet

Solow–Swan: Equations & Formulas

One-page reference for the Solow–Swan growth model: the production function, the capital-accumulation equation, steady-state and Golden-Rule formulas, convergence speed, notation, and the mistakes examiners look for.

Symbol key

Notation & Key Definitions

The key notation for the equation, followed by the handful of concepts.

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.

Formula sheet

Core Equations

Production

Y = F(K, AL) = K^\alpha (AL)^{1-\alpha}, \quad \alpha \in (0,1)

Aggregate Cobb–Douglas with labour-augmenting (Harrod-neutral) technology.

y = f(k) = k^\alpha, \qquad k \equiv \frac{K}{AL}, \quad y \equiv \frac{Y}{AL}

Intensive form: output and capital per effective worker.

r = f'(k) = \alpha k^{\alpha - 1}, \qquad w = f(k) - k f'(k) = (1-\alpha) k^\alpha

Competitive factor prices: each factor earns its marginal product.

Capital accumulation

I = sY, \qquad C = (1-s)Y

Goods-market clearing: output is split between investment and consumption.

\dot{K} = I - \delta K = sY - \delta K

Aggregate capital stock evolves as gross investment less depreciation.

\dot{k} = s f(k) - (n + g + \delta)\,k = s k^\alpha - (n + g + \delta)\,k

The fundamental equation of motion

The single equation that governs the entire dynamics of the model.

\underbrace{s k^\alpha}_{\text{actual investment}} \;\;\text{vs}\;\; \underbrace{(n + g + \delta)\,k}_{\text{break-even investment}}

Break-even investment is what is needed to hold

k

constant against depreciation, population growth, and technological growth.

Steady state ( $\dot{k} = 0$ )

k^* = \left(\frac{s}{n + g + \delta}\right)^{\frac{1}{1-\alpha}}

Steady-state capital per effective worker

y^* = (k^*)^\alpha = \left(\frac{s}{n + g + \delta}\right)^{\frac{\alpha}{1-\alpha}}

Steady-state output per effective worker

c^* = (1-s)\,y^*, \qquad \frac{k^*}{y^*} = \frac{s}{n + g + \delta}

Steady-state consumption and the capital–output ratio.

Golden Rule

f'(k^*_{GR}) = n + g + \delta

The savings rate that maximises steady-state consumption sets the net marginal product of capital to

n + g + \delta

k^*_{GR} = \left(\frac{\alpha}{n + g + \delta}\right)^{\frac{1}{1-\alpha}}, \qquad s_{GR} = \alpha

For Cobb–Douglas, the Golden-Rule savings rate equals capital's share of income.

Convergence

|\lambda| = (1-\alpha)(n + g + \delta)

Speed of convergence

Rate at which the gap to

k^*

closes (from linearising

\dot{k}

around

k^*

t_{1/2} = \frac{\ln 2}{|\lambda|} = \frac{0.693}{(1-\alpha)(n + g + \delta)}

Half-life of convergence

With

\alpha = 1/3

n = 0.01

g = 0.02

\delta = 0.05

|\lambda| \approx 5.3\%

/yr and

t_{1/2} \approx 13

years.

Growth rates at the steady state

Variable	Long-run growth rate
$k,\; y,\; c,\; i$ (per effective worker)	$0$
$Y/L,\; K/L,\; C/L$ (per worker)	$g$
$Y,\; K,\; C$ (aggregate)	$n + g$

Only technological progress

g

drives long-run growth in output per worker. The savings rate sets the level of the balanced growth path, not its slope.

Pitfalls