Introduction

The richest countries are roughly 30–60 times wealthier than the poorest, measured in income per person. That gap is not an accident of last year's policy. It has compounded over decades.

The chart below shows why growth rates are everything. A country growing at 2% instead of 1% ends up about 2.7 times richer after a century. The same logic, applied for generations, produces the 30–60× gaps we see today.

The model tracks one thing: how a country's capital per effective worker evolves over time. Two forces act on it simultaneously:

Investment adds to capital. A fraction $s$ of output is saved and invested each period, so investment per effective worker is $sf(k)$.

Break-even investment drains capital. To keep $k$ constant, the economy must replace worn-out capital (rate $\delta$) and equip the growing effective labour force (rate $n + g$). The total drain is $(n + g + \delta)k$.

When investment exceeds the drain, $k$ rises. When it falls short, $k$ falls. Eventually the two forces balance at a unique steady state $k^*$. That balance point is the long-run destination of the economy.

The model shows that saving more raises the level of output per worker — but it does not raise the long-run growth rate. Growth in the long run comes only from technological progress $g$.

More surprisingly, the model's own math shows that saving rates alone cannot explain the 30–60× income gaps across countries. The elasticity of steady-state income with respect to the saving rate is only $\alpha / (1-\alpha) \approx \tfrac{1}{2}$. A 10× income gap would require implausibly large saving-rate differences. The action must be in technology $A$, not just capital.