Abstract
Since block prices for water rise with usage, it is easy to assume that using less always makes each cubic metre cheaper. This paper counts, in a model tariff made of a basic charge and block prices that rise step by step, the price per cubic metre at each usage, and looks at where the cheapest usage falls. No new theorem or law is claimed.
First, using little makes each cubic metre more expensive, not less: at $5$ m³ it is $\mathbf{300.000000}$ yen, $\mathbf{1.714286}$ times the cheapest usage (Section 2).
Second, using a lot also makes it more expensive: $200.000000$ yen at $40$ m³ and $\mathbf{216.666667}$ yen at $60$ m³, $1.238095$ times the cheapest (Section 2).
Third, and this is the core. The cheapest usage falls on a block boundary: searching in steps of $0.01$ m³, the cheapest is $\mathbf{175.000000}$ yen at $20$ m³, above the $150$ yen of the left block and below the $200$ yen of the right (Section 2).
Fourth, the basic charge decides which boundary: with a basic charge of $400$ yen the cheapest is $140.000000$ yen at $10$ m³, with $2500$ yen it is $\mathbf{233.333333}$ yen at $30$ m³, and with $4000$ yen the price keeps falling up to $60$ m³, reaching $266.666667$ yen (Section 3).
Fifth, the U appears only with both a basic charge and rising blocks: with no basic charge the price per cubic metre only rose, and with a single block ($150$ yen) it only fell (Section 3).
Sixth, the separator is whether one asks about the total charge or divides by the cubic metres: the total only grows with use, but dividing sets the thinning of the basic charge against the rising blocks, and their balance falls on a block boundary (Section 4).
Keywords: water tariff, basic charge, block price, price per cubic metre, block boundary.
Scope (what is not claimed)
No new theorem or law is claimed: that water charges are set by a basic charge plus block prices by usage, that there are tariffs whose block prices rise with usage, and that an average price falls while the block price is below it and rises when above are all known in pricing and cost calculations. No real tariff is claimed: a basic charge of $1000$ yen and block prices of $100$, $150$, $200$ and $250$ yen every $10$ m³ are model values matching no water tariff. Other parts of the bill are not entered: sewer charges, basic charges by pipe size, tax, and billing periods are not included. Tariffs are not judged, which tariff is fair and how it affects saving water are not discussed. Relation to earlier papers: Paper 486 showed that the “exactly half” appearing in three fields is the vertex of a parabola, not physics. This paper stands beside it. There the best value came at the vertex of a smooth curve. Here the cheapest usage comes not at a smooth vertex but at a corner where the block price jumps. What is added is laying out the price per cubic metre at eight usages, measuring in steps of $0.01$ m³ that the cheapest usage falls on a block boundary, measuring that changing the basic charge moves that boundary, and placing the separator on total versus per cubic metre.
1. Notation and the Question
Setting: the monthly charge $C(u)$ is the basic charge $B=1000$ yen plus the usage $u$ m³ times the block prices. Block prices are $100$ yen for $0$ to $10$ m³, $150$ yen for $10$ to $20$ m³, $200$ yen for $20$ to $30$ m³, and $250$ yen beyond $30$ m³. The price per cubic metre is $C(u)/u$.
$$p(u)=\frac{C(u)}{u},\qquad \frac{dp}{du}=\frac{1}{u}\left(C'(u)-p(u)\right)$$$C'(u)$ is the block price at that usage. The average price $p$ falls while the block price is below $p$ and rises when above. Usage was moved from $1$ m³ to $60$ m³ in steps of $0.01$ m³ to find the cheapest usage.
Question: does using less always make each cubic metre cheaper?
2. Usage and Price per Cubic Metre
Computation 2.1: the charge and the price per cubic metre were computed at eight usages.
| usage (m³) | charge (yen) | per cubic metre (yen) |
|---|---|---|
| $5$ | $1500$ | $300.000000$ |
| $10$ | $2000$ | $200.000000$ |
| $15$ | $2750$ | $183.333333$ |
| $20$ | $3500$ | $175.000000$ |
| $25$ | $4500$ | $180.000000$ |
| $30$ | $5500$ | $183.333333$ |
| $40$ | $8000$ | $200.000000$ |
| $60$ | $13000$ | $216.666667$ |
Observation 2.2 this is the core. The price per cubic metre traces a U, and the cheapest usage falls on a block boundary. Searching in steps of $0.01$ m³, the cheapest usage is $20$ m³ at $175.000000$ yen: that value is above the left block price of $150$ yen and below the right block price of $200$ yen. Before the boundary the block price is below the average, so the average falls; past it the block price is above the average, so it rises. The price at $5$ m³ is $1.714286$ times the cheapest, and at $60$ m³ $1.238095$ times.
3. Basic Charge and the Cheapest Usage
Computation 3.1: keeping block prices, the basic charge was changed and the cheapest usage searched for.
| basic charge (yen) | cheapest usage (m³) | per cubic metre (yen) |
|---|---|---|
| $400$ | $10$ | $140.000000$ |
| $1000$ | $20$ | $175.000000$ |
| $2500$ | $30$ | $233.333333$ |
| $4000$ | still falling at the searched $60$ | $266.666667$ |
Observation 3.2 the basic charge decides which boundary. With basic charges of $400$, $1000$ and $2500$ yen, the cheapest usage moved to the block boundaries at $10$, $20$ and $30$ m³, and its price always lay above the left block price and below the right: with a basic charge of $4000$ yen the price was still falling at the searched $60$ m³, where it is $266.666667$ yen, above the top block price of $250$ yen. As controls, with no basic charge the price only rose, and with a single block ($150$ yen) it only fell; neither made a U.
4. The Separator
Observation 4.1 the separator is whether one asks about the total charge or divides by the cubic metres. Asking about the total, the charge only grows with use: dividing by the cubic metres sets the thinning of the basic charge over usage against the rising block prices. Their balance falls on the block boundary, where the block price jumps across the average.
Observation 4.2 placed beside Paper 486. 486 showed that the “exactly half” of three fields is the vertex of a parabola. There the best value came at the vertex of a smooth curve. Here the price curve has no vertex, and the cheapest usage comes at a corner where the block price jumps.
Observation 4.3 what is not said. Real water tariffs, sewer charges, basic charges by pipe size, tax, which tariff is fair, and effects on saving water are not treated. What is shown is that, in this model, the price per cubic metre traces a U, the cheapest usage falls on a block boundary, and the basic charge decides which.
5. Census and Closing
- $p(u)=C(u)/u$ (Section 1): per cubic metre.
- $\mathbf{300.000000}$ yen at $5$ m³ (Computation 2.1): using little is dear.
- $\mathbf{175.000000}$ yen at $20$ m³ (Observation 2.2): the core; a block boundary.
- between $150$ and $200$ yen (Observation 2.2): the average straddles the block prices.
- $\mathbf{216.666667}$ yen at $60$ m³ (Computation 2.1): using a lot is dear too.
- $1.714286$ and $1.238095$ times (Observation 2.2): the two ends of the U.
- $10$ m³ with a basic charge of $400$ yen (Computation 3.1): $140.000000$ yen.
- $30$ m³ with a basic charge of $2500$ yen (Computation 3.1): $\mathbf{233.333333}$ yen.
- no basic charge, and a single block (Observation 3.2): no U.
- total, or per cubic metre (Observation 4.1): the separator; Paper 486.
Closing: The price of water per cubic metre traces a U with usage; the cheapest usage falls on a block boundary, and the basic charge decides which. With a basic charge of $1000$ yen and rising block prices, the price was $300.000000$ yen at $5$ m³ and $216.666667$ yen at $60$ m³, and the cheapest was $175.000000$ yen at the block boundary of $20$ m³. There the average price lies between the left block’s $150$ yen and the right block’s $200$ yen. A basic charge of $400$ yen moved it to $10$ m³ and $2500$ yen to $30$ m³, and removing the basic charge or using a single block made the U disappear: the U comes from the thinning basic charge meeting the rising blocks. Placed among the earlier papers: beside Paper 486’s “the best value comes at a smooth vertex”, we placed a tariff whose cheapest usage comes at the corner of a block. To be honest: the basic charge and block prices are model values, and sewer charges and tax are not included.
References
- Yamagishi, Y. (2026). The “Exactly Half” in Three Fields Is the Vertex of a Parabola, Not Physics [Paper 486]. YOI: https://www.yamagishiyuuki.com/yoi/486; DOI: 10.5281/zenodo.22759952
Reference
Yamagishi, Y. (2026). Water’s Price per Cubic Metre Traces a U with Usage, Cheapest at a Block Boundary That the Basic Charge Chooses [Paper 897]. YOI: https://www.yamagishiyuuki.com/yoi/897
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 897. This paper is a preprint and has not been peer reviewed.