Answer First
The Golden Rule capital level maximizes consumption because it occurs where the marginal product of capital equals the break‑even investment rate. At this point, the economy saves just enough to maintain the capital stock without over‑saving or under‑saving.
Problem Setup
Steady‑state consumption per worker: \[ c^* = f(k^*) – (n + \delta)k^*. \] Golden Rule condition: \[ MPK = f'(k_{GR}) = n + \delta. \] Break‑even investment: \[ (n + \delta)k. \]
Step-by-Step Explanation
1. Consumption equals output minus break‑even investment
To maximize consumption, the economy must choose the capital level that maximizes: \[ f(k) – (n + \delta)k. \]
2. The derivative of consumption with respect to capital gives the Golden Rule
Setting the derivative equal to zero: \[ f'(k_{GR}) = n + \delta. \] This is the condition for maximizing steady‑state consumption.
3. If capital is below the Golden Rule, more saving raises consumption
MPK > n + δ → capital is too low → increasing k raises consumption.
4. If capital is above the Golden Rule, saving too much reduces consumption
MPK < n + δ → capital is too high → maintaining it requires excessive saving.
5. The Golden Rule is a normative benchmark
It describes the consumption‑maximizing steady state, not necessarily the one the economy reaches on its own.
Intuition
The Golden Rule balances the benefit of more capital (higher output) against the cost of maintaining it (higher depreciation and dilution). The optimal point is where these forces are equal.
Common Exam Mistakes
- Confusing Golden Rule with the steady state the economy naturally reaches.
- Thinking Golden Rule maximizes output (it maximizes consumption).
- Forgetting that MPK must be compared to n + δ, not δ alone.
- Mixing up capital per worker with total capital.
Final Summary
The Golden Rule capital level maximizes steady‑state consumption because it equates the marginal product of capital with the break‑even investment rate. Below this point, more capital raises consumption; above it, more capital reduces consumption.
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