Banded rates

Package: TABLES_PACKAGE. Registered by createEngine(); for a slimmer engine, register it explicitly (see choosing packages).

Many charges are not one rate on the whole amount. An income tax charges nothing on the first part of an income, one rate on the next part, and a higher rate on what is left. A sales commission pays more on sales past a target. An electricity tariff charges one price for the first hundred units and another for the rest. Each of these is a banded (or tiered, or progressive) rate: the amount is cut into bands, and each band has its own rate.

The part that trips people up is that a higher band only charges the part of the amount inside it. Earning past a threshold does not move the whole income to the higher rate. That is why banded arithmetic is fiddly by hand, and why the engine works it out for you from a table you write.

Writing the bands as a table

A band table has one row per band. The first column says where the band starts, and the last column says its rate. Each band runs from its own start up to where the next one starts, and the last band runs on without end.

| from   | rate |
| ------ | ---- |
| 0      | 0%   |
| 10,000 | 20%  |
| 40,000 | 40%  |

45,000 through bands above                // 8,000
45,000 through bands above / 45,000 as %  // 17.78%
column "rate" for 45,000 in bands above   // 40.00%

Read it as three bands: nothing on the first 10,000, 20% on the 30,000 from 10,000 to 40,000, and 40% on everything past 40,000. On 45,000 that is 0, plus 6,000, plus 2,000 on the last 5,000, which is 8,000.

The engine assumes no bands at all. You write them, so the same form covers any country’s income tax, any commission scheme and any tiered price, and a schedule that changes next year is a table you edit.

The total across the bands

<amount> through bands above is the total charged across every band the amount reaches, each part at its own band’s rate. through the bands reads the same, and above is optional, since the nearest table above is the one read.

The whole expression before through bands is the amount, so 40,000 + 5,000 through bands above is the total on 45,000. Arithmetic after it applies to the total: dividing by the amount, as above, gives the overall share charged, 17.78%, which is well below the 40% top rate the amount reached.

The band an amount falls in

column "rate" for 45,000 in bands above reads one cell from the band the amount falls in, rather than totalling across them. For a tax table that is the rate on the next unit earned (the marginal rate); for a postage table it is the price for a parcel of that weight. It is the table lookup form with in bands added, and in bands is what makes it match the band rather than an exact label.

| weight (kg) | price  |
| ----------- | ------ |
| 0           | $4.50  |
| 2           | $7.95  |
| 10          | $14.00 |

column "price" for 3.5 in bands above   // $7.95
column "price" for 2 in bands above     // $7.95
column "price" for 12 in bands above    // $14.00

An amount exactly on a band’s start belongs to that band, so 2 kg is charged at the band that starts at 2. Any column can be read, not only the last.

Money, shares and prices

Amounts and band starts can be money. A plain amount takes the table’s currency, and money stays exact to the penny:

| sales from | commission |
| ---------- | ---------- |
| $0         | 5%         |
| $50,000    | 8%         |
| $100,000   | 10%        |

$120,000 through bands above                     // $8,500.00
120000 through bands above                       // $8,500.00
column "commission" for $120,000 in bands above  // 10.00%

The rate column can hold three kinds of rate, and each is charged on the part of the amount inside its band:

• A percentage charges a share of that part, as a tax or a commission does.
• A price ($0.18) charges that much for each unit of the part, as a tiered tariff does. The amount is then a count, such as units used, and the total is in the price’s currency.
• A plain number multiplies the part, so 0.2 works as 20% does.

| kWh | price |
| --- | ----- |
| 0   | $0.12 |
| 100 | $0.18 |
| 500 | $0.25 |

350 through bands above                 // $57.00
column "price" for 350 in bands above   // $0.18

That is 100 units at $0.12 and 250 at $0.18. A plain 0 can sit among percentages or prices, since a zero rate is zero whichever way it is read.

When bands are refused

A table that does not say clearly what its bands are is refused, with the line to fix, rather than read one way and answered with a number that might be wrong.

| up to  | rate |
| ------ | ---- |
| 10,000 | 0%   |
| 40,000 | 20%  |

45,000 through bands above   // ERROR: The first column is headed "up to", which reads as where each band ends. Bands are read by where each one starts: head the column "from" and write each band's starting point, the first being 0.

A table written by where each band ends reads one row out if taken as starts, which would charge every part at the wrong rate, so a first column headed up to, to, under, max and the like is refused. Write the starts instead, the first being 0.

| from   | rate |
| ------ | ---- |
| 10,000 | 20%  |
| 40,000 | 40%  |

45,000 through bands above                // ERROR: The first band starts at 10,000, so the part of an amount below it falls in no band. Add a band from 0, at 0% if nothing is charged there.
column "rate" for 5,000 in bands above    // ERROR: 5,000 falls below the first band, which starts at 10,000.

A total needs a first band from 0, so that every part of the amount falls in one. A band that charges nothing is written with a 0% rate rather than left out, which keeps the table saying what it means.

| from   | rate |
| ------ | ---- |
| 0      | 0    |
| 10,000 | 20   |
| 40,000 | 40%  |

45,000 through bands above   // ERROR: Line 4's rate 20 is a plain number among percentages: write 20% if a percentage is meant.

A plain 20 among percentages is far more likely a missing % than a rate of twenty times the amount, so it is refused rather than charged. Starts that do not rise down the table, a start that is not a number or an amount of money, a missing or unreadable rate, rates that mix percentages and prices, an amount in a different currency from the bands, and an amount below zero are refused the same way.

What banded rates do not do

• Rules beyond the bands are not modelled. An allowance that shrinks as income rises, a threshold that depends on something other than the amount, or a flat fee charged per band are outside what a table of starts and rates says. The payroll forms carry the full UK rules for England, Wales and Northern Ireland, taper included.
• The first column is where each band starts, and the last is its rate. Columns between them, such as a band’s name, are ignored by the total.
• Amounts are plain numbers or money. An amount with a unit such as 3 kg is refused rather than compared with a table that does not state its unit; write the number alone.
• Only the nearest table above is read, as with table columns.

Like the other cross-line forms, banded rates only work inside a document, since they read a table elsewhere in the note.