Answer First
The Solow model converges to a steady state because diminishing marginal returns to capital make investment less productive as capital rises, while depreciation and dilution grow proportionally. Eventually, investment equals break‑even investment, and capital stops changing.
Problem Setup
Capital accumulation equation: \[ \dot{k} = s f(k) – (n + \delta) k. \] Steady state condition: \[ s f(k^*) = (n + \delta) k^*. \] Diminishing returns: \[ f'(k) > 0, \quad f”(k) < 0. \]
Step-by-Step Explanation
1. Investment rises with capital but at a decreasing rate
Because of diminishing marginal product of capital, each additional unit of capital produces less output and therefore less investment.
2. Depreciation and dilution rise proportionally with capital
Break‑even investment \((n + \delta)k\) grows linearly, eventually catching up to investment.
3. Convergence occurs where investment equals break‑even investment
At this point: \[ \dot{k} = 0. \] Capital per worker stops changing.
4. If capital is below steady state, it rises
Investment exceeds break‑even investment → capital accumulates.
5. If capital is above steady state, it falls
Break‑even investment exceeds investment → capital declines.
Intuition
Think of capital as water in a tank: inflow slows as the tank fills, while outflow rises proportionally. Eventually, inflow equals outflow, and the water level stabilizes.
Common Exam Mistakes
- Confusing steady state with zero output growth (output still grows with population).
- Thinking convergence requires identical countries (it requires identical parameters).
- Ignoring the role of diminishing returns.
- Mixing up capital per worker with total capital.
Final Summary
The Solow model converges to a steady state because diminishing returns reduce the productivity of capital while depreciation and dilution rise proportionally. These forces push the economy toward a stable level of capital per worker.
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