Parking Tariffs Aren't Hourly: They're a Step Function With a Receipt

Car park signs love a headline rate. 'From £2.20 per hour' sounds like a linear price: stay twice as long, pay twice as much. Then you reach the machine and find that one hour costs £2.20, two hours cost £4.00, three hours cost £6.50 and four hours cost £9.50. The hourly rate has not stayed still. It was never a rate at all. It was a step function.

That matters because most parking decisions are made at the margin: should I pay for one more hour, or risk the penalty? The advertised rate is a bad guide to that question. The true cost is a piecewise function, and once you write it down you can see where the extra money goes.

A typical tariff, written properly

Take a fictional but realistic council car park in a mid-sized English town. Call it Mill Lane. The tariff board says:

• 0–1 hour: £2.20
• 1–2 hours: £4.00
• 2–3 hours: £6.50
• 3–4 hours: £9.50
• 4–24 hours: £12.00
• Paying by app: 30p transaction fee

As a function of stay length t in hours, the tariff is:

C(t) = 2.20, if 0 < t ≤ 1 4.00, if 1 < t ≤ 2 6.50, if 2 < t ≤ 3 9.50, if 3 < t ≤ 4 12.00, if 4 < t ≤ 24

That is a piecewise function: several flat pieces joined at the hour marks. The app fee adds a constant:

C_app(t) = C(t) + 0.30

The effective hourly rate is C(t) / t. At one hour it is £2.20. At two hours it is £2.00. At three hours it is about £2.17. At four hours it is £2.38. The sign's 'from £2.20' is technically true, but only for the first hour. It is a minimum, not a rate.

Marginal cost is where the trap is

The number that should govern your decision is marginal cost: the extra charge for extending your stay. From 0 to 1 hour, the first hour costs £2.20. The second hour adds £1.80, from £2.20 to £4.00. The third adds £2.50. The fourth adds £3.00. The fifth to twenty-fourth adds £2.50 for a much longer block. Those marginal prices are not advertised, but they are the real price of saying 'I'll just stay a bit longer.'

Now suppose you need two hours and ten minutes. The two-hour ticket is £4.00; the three-hour ticket is £6.50. The extra ten minutes cost £2.50, an effective rate of £15 per hour for that sliver. If you buy the two-hour ticket and overstay, the penalty might be £50 or more. Suddenly the 'expensive' three-hour ticket is the cheaper expected choice. The algebra is:

Expected cost of underpaying = probability of overstay × penalty Extra cost of buying the next block = C(next block) − C(current block)

Buy the buffer when:

probability of overstay × penalty > extra block cost

If the penalty is £50 and the extra block is £2.50, you only need a 5% chance of overstaying for the buffer to be worth it. Most of us are more than 5% likely to be late.

Why the tariff is stepped

This is civic algebra, not a conspiracy. Councils and private operators use stepped tariffs for several reasons. Steps are easy to enforce and explain at a machine. They encourage short stays by making long stays relatively more expensive at the margin. They raise revenue without a single headline rate that looks high. And they can be adjusted in small political increments: 20p on the two-hour band is less noticeable than a broad hourly rise.

But the design has distributional effects. A driver who needs 61 minutes pays for two hours. A driver who needs 2 hours 5 minutes may pay for three. The person who can round their trip to the hour pays less per minute than the person who cannot. The sign hides that; the function reveals it.

App fees, rounding and the 'convenience' surcharge

Apps add another layer. A 30p fee sounds trivial, but it is a fixed charge per transaction, so it hits short stays hardest. One hour by app costs £2.50, an effective £2.50 per hour. Three hours costs £6.80, an effective £2.27 per hour. The app fee dilutes with stay length, which is one reason the 'cheapest' payment method depends on how long you park. Some apps also round up, add a text reminder fee, or charge a card surcharge. Put every fixed extra into the function as a constant a:

C_true(t) = C(t) + a

Then compare C_true(t) / t, not C(t) / t. The constant changes the ranking of short and long stays.

How to choose your stay length

Here is a practical routine:

1. Photograph the tariff board before you pay. You need the bands, not the headline.
2. Write the function for the hours you are considering.
3. Estimate your actual stay, then add a realistic buffer.
4. Price the next band up. Ask what the extra 30 or 60 minutes costs at the margin.
5. Compare that with the expected penalty for overstaying, and with the cost of a different car park.
6. Pay for the band that minimises expected total cost, not the one with the prettiest hourly rate.

For Mill Lane, a three-hour stay is the sweet spot for many errands: £6.50, or £6.80 by app, works out at about £2.17–£2.27 per hour. A four-hour stay jumps to £9.50, or £9.80 by app. The fourth hour costs £3.00 at the margin, which is more than the average rate suggests. If your meeting is likely to run to 3 hours 10 minutes, you are choosing between a £3.00 buffer and a £50 penalty. The buffer usually wins.

The rate you should quote

If you want a single number, quote the effective rate for your actual stay: C_true(t) / t. Do not quote the sign's 'from' rate unless you are staying exactly one hour and paying the cheapest way. The hourly rate is not exactly a lie; it is a minimum dressed up as a rate. In algebra terms, it is the value of a function at one point, presented as if the function were a straight line through the origin.

Parking is full of these small piecewise traps: the bus fare that changes at 9:30, the energy tariff that steps after a certain number of units, the council tax band that jumps at a valuation threshold. Once you start writing costs as functions, you stop comparing headlines and start comparing the thing you actually pay. At the meter, that is the difference between a cheap hour and an expensive ten minutes.