A mathematical proof that in any expanding system, fixed-cost upgrades with positive efficiency gains are not optional luxuries — they are mathematical necessities.
The framework started as a question about when to research Bloodlines in Age of Empires II. It generalized into a theorem applicable to capital budgeting, software refactoring, education, and infrastructure investment.
Core Result#
An upgrade with efficiency multiplier G and fixed cost C becomes worthwhile when remaining resources exceed:
$$R^* = \frac{C \cdot G}{G - 1}$$As total resources grow without bound, the relative cost of any fixed-price upgrade collapses to zero. The gain scales linearly. The cost stays constant. The math is on your side.
Maximum Acceptable Cost#
Given an efficiency gain G, the maximum percentage of total resources you can spend before the upgrade becomes non-viable:
$$C = 100 \left(1 - \frac{1}{G}\right)$$| Efficiency Gain | Multiplier (G) | Max Cost (%) |
|---|---|---|
| 20% | 1.20 | 16.67% |
| 40% | 1.40 | 28.57% |
| 50% | 1.50 | 33.33% |
| 100% | 2.00 | 50.00% |
Even a 100% efficiency gain (doubling output) can never justify spending more than 50% of total resources.