Answer First
Population growth dilutes capital because new workers must be equipped with capital. If investment does not grow fast enough to match population growth, capital per worker falls, reducing output per worker.
Problem Setup
Capital accumulation: \[ \dot{k} = s f(k) – (n + \delta)k. \] Dilution term: \[ nk. \] Steady state: \[ s f(k^*) = (n + \delta)k^*. \]
Step-by-Step Explanation
1. More workers require more capital
When population grows at rate n, the economy must invest nk just to keep capital per worker constant.
2. Dilution raises break‑even investment
Break‑even investment becomes: \[ (n + \delta)k. \] Higher n shifts this line upward.
3. Higher population growth lowers steady‑state capital
Because break‑even investment rises, the intersection with s f(k) occurs at a lower k*.
4. Output per worker falls with lower capital per worker
Lower k* → lower y* → lower consumption per worker.
5. Fast‑growing countries must save more to maintain capital
High‑n economies require higher saving rates to avoid falling capital per worker.
Intuition
Capital dilution is like sharing a fixed number of tools among more workers. Unless new tools are produced fast enough, each worker ends up with less.
Common Exam Mistakes
- Thinking population growth reduces total capital (it reduces capital per worker).
- Forgetting that dilution raises break‑even investment.
- Confusing transitional dynamics with steady‑state effects.
- Mixing up n with n + δ.
Final Summary
Population growth dilutes capital because new workers require capital to maintain productivity. Higher n raises break‑even investment and lowers steady‑state capital per worker, reducing output and consumption per worker.
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