Phase Diagram

The phase diagram is the centrepiece of the Solow model. It turns the algebra from Section 3 into a picture, and it makes existence, uniqueness, and stability of the steady state obvious — no calculus required.

The diagram plots two curves against $k$ (capital per effective worker):

Actual investment $sf(k) = sk^\alpha$: the fraction of output saved and invested. This is concave — it bends toward the horizontal axis as $k$ grows.

Break-even investment $(n+g+\delta)k$: the amount needed to keep $k$ constant as capital depreciates and effective labour grows. This is a straight line through the origin.

Where they cross is the steady state $k^*$. The Inada conditions guarantee they cross exactly once at a positive $k^*$.

The sign of $\dot{k} = sf(k) - (n+g+\delta)k$ tells you the direction of motion.

**Left of $k^*$**: actual investment $sf(k)$ is above the break-even line, so $\dot{k} > 0$. Capital per effective worker rises — the economy moves right toward $k^*$.

**Right of $k^*$**: break-even investment is above actual investment, so $\dot{k} < 0$. Capital per effective worker falls — the economy moves left toward $k^*$.

**At $k^*$**: the curves meet, $\dot{k} = 0$, and the economy is on its balanced growth path.