Section 4

Steady State and Dynamics

The steady state is the long-run destination: the value $k^*$ at which $\dot{k} = 0$ . Capital per effective worker stops changing, and the economy travels along a balanced growth path.

Solving for the steady state

Set $\dot{k} = 0$ in the law of motion $\dot{k} = sk^\alpha - (n+g+\delta)k$ . Solving for $k$ :

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

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

c^* = (1-s)\,y^*

No scalar found for key: steady_state_k

Existence, uniqueness, and stability

These three properties follow directly from the shape of $f(k)$ . See the phase diagram in Section 5 for the visual argument.

Existence: the Inada conditions force $sf(k) > (n+g+\delta)k$ near $k=0$ and $sf(k) < (n+g+\delta)k$ for large $k$ . By continuity, they must cross at least once.

Uniqueness: the concavity of $f(k)$ means $sf(k)$ bends away from the linear break-even line. There is exactly one crossing point.

Global stability: for any $k_0 > 0$ , the sign of $\dot{k}$ always points toward $k^*$ . No matter where the economy starts, it converges.

Capital–output ratio

\dfrac{s}{n+g+\delta}

k^* / y^* = s/(n+g+\delta)

. Constant in steady state — a Kaldor stylised fact.

The balanced growth path

At steady state, what is growing and what is constant? The table below answers this. It matches the Kaldor stylised facts — the empirical regularities that a good growth model should reproduce.

Variable	Growth rate in steady state	Kaldor fact?
$k$ , $y$ , $c$ , $i$ (per effective worker)	$0$	Yes — capital-output ratio is stable
$Y/L$ , $K/L$ , $C/L$ (per worker)	$g$	Yes — living standards grow steadily
$Y$ , $K$ , $C$ (aggregate)	$n + g$	Yes — total output grows
Factor shares ( $\alpha$ , $1-\alpha$ )	$0$	Yes — shares are roughly constant

Without technology (

g = 0

) there is no long-run growth in living standards. The model tells us technology matters; it does not explain where technology comes from.

Transition dynamics: what happens after a rise in $s$

Suppose the economy is at $k^*$ and the savings rate permanently rises from $s$ to $s' > s$ . What happens?

Immediately: the investment curve $sf(k)$ shifts up. At the old $k^*$ , actual investment now exceeds break-even investment, so $\dot{k} > 0$ .

During the transition: $k$ rises toward the new, higher $k^{*\prime}$ . Output per effective worker $y = k^\alpha$ rises too. Growth is temporarily positive — but it slows as $k$ approaches $k^{*\prime}$ .

In the new steady state: $k$ and $y$ are permanently higher. But the long-run growth rate of $y$ is back to zero (or $g$ per worker). This is the key lesson: saving more shifts the level of output, not the long-run growth rate.

Speed of convergence

Linearising $\dot{k}$ around $k^*$ gives the convergence speed:

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

With $\alpha = 1/3$ , $n = 0.01$ , $g = 0.02$ , $\delta = 0.05$ : $|\lambda| \approx 5.3\%$ per year.

Half-life of convergence

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

Absolute versus conditional convergence

Absolute convergence: all countries converge to the same steady state, so poor countries always grow faster. The basic Solow model does not predict this — it says countries converge to their own steady state.

Conditional convergence: a country grows faster the further it is below its own $k^*$ . Given the same $s$ , $n$ , $g$ , $\delta$ , poorer countries grow faster. Cross-country regressions consistently find conditional but not absolute convergence.