A Country Cannot Grow on Machines Alone

In micro, one firm's production function settled that firm's choices. In macro, we add that function up to a whole country and run it for decades.

More ⚙️ machines raise output, but each one adds less. Worn-out machines, though, grow in a straight line with the pile. Growth stops where the two flows meet. That meeting point is the steady state in the formula below. Poor and rich countries both end up gathering at the same spot.

For example, picture a village that plants 🌱 seeds. At first, each extra seed makes the harvest jump. Once the field is crowded, extra seeds add only a little. Some seeds also rot every year. When new planting just matches the rotting, the harvest stops growing.

k_{t+1} = k_t + s\sqrt{k_t} - \delta k_t

k^* = (s/\delta)^2

y^* = s/\delta

Part of these learning paths

Examples & Exercises

A country doubles its saving rate from $s = 0.2$ to $s = 0.4$ (with $\delta = 0.1$ ).

What are the new steady-state capital and output, and does growth continue once it gets there?

Saving rate

0.4

, depreciation

0.1

— new steady state

k^* = 16

Show proof

Before, $k^* = (0.2 / 0.1)^2 = 4$ and $y^* = 0.2 / 0.1 = 2$ . Doubling the saving rate gives $k^* = (0.4 / 0.1)^2 = 16$ — four times as much capital — and $y^* = 0.4 / 0.1 = 4$ — twice the output.
On the way, saving beats depreciation every period so capital piles up and the economy grows — but the moment $k$ reaches $16$ the two flows are equal again and growth returns to $0$ . Saving changes the country's height (its level), never its permanent growth rate.

Two countries share the same saving rate $s = 0.25$ and depreciation $\delta = 0.1$ but start from capital $k_0 = 1$ and $k_0 = 25$ .

How fast does each grow in the first period, and where do they both end up?

Saving rate

0.25

, depreciation

0.1

— the shared steady state

k^* = 6.25

Show proof

The poor country: $k_1 = 1 + 0.25\sqrt{1} - 0.1 \cdot 1 = 1.15$ , a growth rate of $+15\%$ .
The rich country: $k_1 = 25 + 0.25\sqrt{25} - 0.1 \cdot 25 = 25 + 1.25 - 2.5 = 23.75$ , a growth rate of $-5\%$ .
With more capital, each new machine adds less output (diminishing returns) while more of the stock wears out.
Their steady state is the same: $k^* = (0.25 / 0.1)^2 = 6.25$ , $y^* = 2.5$ . Different starting points with the same saving rate and technology converge to the same height — and the poor country growing faster is exactly that convergence in action.

8. Growth: Solow and Beyond