When Do Two Price Plans Switch Places?

1. The Question: Which Plan Is Cheaper?

Mobile phone and subscription services often offer a low-base-fee plan alongside another plan with a lower usage charge. Consider these fictional mobile plans:

Light Plan looks better for light use. As usage increases, however, Flat Plan should eventually become cheaper. Where does that reversal happen?

2. Price Formula

Let monthly data usage be $x$ GB and the total price be $y$ yen.

Light Plan

Flat Plan

Both have the linear form $y=ax+b$. The intercept $b$ is the basic fee, while the slope $a$ is the extra charge for each additional gigabyte.

3. Comparison

Light Plan is cheaper at 5GB, Flat Plan is cheaper at 15GB, and both cost ¥3,000 at 10GB.

4. Intersection: The Equal-Price Point

At the boundary where the cheaper plan changes, the two prices are equal.

At 10GB, both plans cost ¥3,000. This is the intersection of the two linear functions.

5. General Formula

Write two plans as follows:

Here $a,c$ are basic fees and $b,d$ are per-unit charges. Their equal-price point is

If $b=d$, the lines have the same slope. Unless their basic fees are also equal, they are parallel and the cheaper plan never changes.

6. Subscription Example

Suppose a facility offers Basic Plan at ¥500 per month plus ¥300 per visit, and Unlimited Plan at ¥2,500 per month.

The mathematical intersection is about 6.67 visits, but there is no such thing as 0.67 of a visit. In practice, Unlimited Plan becomes cheaper from the seventh visit. A mathematical result must be interpreted in the units of the real situation.

7. Real-World Conditions

Real plans may include tiers, price caps, rounding, temporary discounts, taxes, cancellation fees, or minimum terms. Such prices may not fit one straight line.

Even so, separating the basic fee from the usage-dependent fee is a useful first step toward seeing how the comparison works.

8. Dr.WataWata's Insight