Glossary

Steady State

A steady state is a point in a dynamic model where the state variables stop changing — their time derivatives vanish: $\dot{k} = 0$ , $\dot{y} = 0$ . In the Solow-Swan model, steady-state capital per worker $k^*$ solves $sf(k^*) = (n + \delta)k^*$ : gross investment equals break-even investment. With labour-augmenting technological progress at rate $g$ , the condition becomes $sf(k^*) = (n + g + \delta)k^*$ , and per-capita output grows at rate $g$ in steady state — not zero.

Think of $k^*$ as the long-run resting point the economy is pulled toward. Above $k^*$ , depreciation and population growth erode capital faster than saving replenishes it, so $k$ falls. Below $k^*$ , investment outpaces break-even losses and $k$ rises. The steady state is what the economy converges to, not where it is now. A higher saving rate $s$ raises the level of $k^*$ permanently but cannot raise the long-run growth rate of output per worker — that stays at $g$ . This level-vs-growth result is one of the model's signatures.

sf(k^*) = (n + \delta)\,k^*

steady-state-condition

Break-even condition: saving

sf(k^*)

equals the investment needed to offset depreciation

\delta k^*

and equip new workers

n k^*

. For Cobb-Douglas

f(k) = k^\alpha

, this yields

k^* = \bigl(s/(n+\delta)\bigr)^{1/(1-\alpha)}

Balanced growth path: The long-run path on which per-effective-worker variables are constant, per-worker variables grow at $g$ , and aggregates grow at $n + g$ .
Break-even investment: $(n + g + \delta)k$ — the investment per effective worker needed to hold $k$ constant as capital depreciates at $\delta$ and effective labour grows at $n + g$ .
Golden-rule capital stock: The $k^*$ that maximises steady-state consumption per effective worker. For Cobb-Douglas it requires saving rate $s_{GR} = \alpha$ .
Transitional dynamics: The path of $k$ converging toward $k^*$ from any $k_0 > 0$ . Growth is fastest far below the steady state and slows as the economy approaches it.