Steady State and Dynamics

The steady state is the long-run equilibrium where $\dot{k} = 0$. Capital per effective worker is constant, and the economy travels along a balanced growth path.

At steady state $\dot{k} = 0$ with $f(k) = k^\alpha$:

Existence: the Inada conditions guarantee the investment and break-even curves always intersect at some positive steady state.

Uniqueness: the concavity of the production function ensures exactly one interior steady state.

Global stability: for all $k_0 > 0$, the economy converges to $k^*$.

In steady state, output per worker $Y/L = A \cdot y^*$ grows at rate $g$, driven entirely by technological progress. Without technology ($g=0$), there is no long-run growth in per-worker output.

$k$, $y$, $c$, $i$ per effective worker: all constant (grow at 0%).

$Y/L$, $K/L$, $C/L$ per worker: grow at rate $g$.

$Y$, $K$, $C$ aggregate: grow at rate $n + g$.

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

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

Absolute convergence: poor countries grow faster regardless of structural parameters. The basic Solow model does not necessarily predict this.

Conditional convergence: poor countries grow faster given the same $s$, $n$, $g$, $\delta$. Each country converges to its own steady state. Cross-country regressions consistently find conditional but not absolute convergence.