AM I BEING TAKEN FOR A RIDE?
The formula behind the taximeter
In the film Manhattan, Woody Allen is in a taxi and quips to Diane Keaton, ‘You look so beautiful I can hardly keep my eyes on the meter.’ Taximeters have that sort of mesmerising effect on people. Yet, for all the hours people spend watching them tick upwards, few know the secret of how they work. Not even the drivers. When a sample of London cabbies was asked to explain the system, the response was typically, ‘Good question, guv’nor, I’ve often wondered about that myself.’
To the passenger crawling through the traffic, one thing certainly seems clear. Whether you are hurtling through the back streets or stuck at the lights, the meter keeps ticking over. It seems that the taxi driver can’t lose.
So what is the secret of the black box? Does the speed of your journey make any difference to the driver’s pay packet? Can devious cabbies squeeze higher fares out of passengers by playing the system? These questions have occurred to all of us at one time or another, as another pound clocks up on the meter.
The basic formula
The basic principle behind a taxi fare is simple enough. If you make a long journey, you should expect to pay more than you do on a short journey, and the taximeter, a device invented by Wilhelm Bruhn in 1896, charges you a rate for distance travelled. But what about heavy traffic or unexpected delays due to an accident? As far as taxis are concerned, a ‘long’ journey means a long time, as well as a long distance. To cover the driver for his work time spent sitting in a traffic jam or crawling through the rush hour, the taximeter also has a rate for the time spent on the journey.
So a taxi fare is calculated using a formula that charges you for your distance and your time. Actually, that’s a little misleading. It charges you for your distance or your time, but not both at the same time, as will become clear in a moment.
This distance-to-time approach is in fact the standard adopted internationally by taxis with meters, though it isn’t the only model they could have used. In fact, if you think about it, the principle of charging you for travelling slowly is the opposite to the way you are charged for a train journey. On a train journey you don’t pay any more for a journey that lasts twice as long. On the contrary, these days, because of the system of refunds, the longer your train journey takes the less you pay!
There is a general rule in economics that the better the product is, the more it costs. But, if you think of a taxi ride as being ‘get me there as quickly as possible’, then the economic law here is the opposite: the worse the product, the more you pay.
Buried away in small print that few people ever bother to read is the formula for calculating this taxi fare. At the time of writing, the formula for the basic London fare for daytime travel looked like this:
Those suspiciously precise figures of 189.3 metres and 40.8 seconds are enough to put anyone off trying to work out the fare for themselves, though this isn’t a deliberate ploy on the part of the taxi drivers to confuse passengers. The figure is actually set by the London authorities, and adjusted by an inflationary amount each year to try to keep driver earnings roughly fixed. If you go back far enough, there must have been a time when the distances and times were nice round numbers.
When the Secretary of State published the London Cab Order in 1934, the formula involved fractions rather than decimals. It looked like this:
3d (i.e. 3 old pennies) initial charge + 3d per 1⁄3 mile (if exceeding 51⁄3 miles per hour) or 3d per 31⁄2 minutes (if going at less than 51⁄3 mph).
This figure of 51⁄3 miles per hour was the estimated average speed of vehicles in London in 1934, perhaps kept low because of the relatively slow speed of horse-drawn vehicles, which were still reasonably common in the streets. What happened if your cab was travelling at exactly this average speed of 51⁄3 m.p.h.? It turns out that, if you travel 1⁄3 mile at 51⁄3 m.p.h., you will take exactly 31⁄2 minutes, the same duration as the time unit. In other words, however you measure it, the cabby got threepence for each unit of the journey if he travelled at the average speed.
Before investigating the mysterious taxi formula any further, here is a little quiz. Suppose you take a black cab from your local station to your home every day.
Quiz
1. Same distance, more time. If your taxi journey today takes a couple more minutes than your journey yesterday, is your fare:
(a) more than yesterday’s?
(b) less than yesterday’s?
(c) the same as yesterday’s?
2. Same time, longer distance. If your journey today diverts around some back streets, adding a few hundred yards to the journey, but takes exactly the same time as yesterday’s, is your fare:
(a) more than yesterday’s?
(b) less than yesterday’s?
(c) the same as yesterday’s?
Did you answer (a), (b) or (c) to either question? If you did, award yourself a point, because the answer to all of the options is ‘Possibly’. Just because a journey takes longer in time or distance, it doesn’t necessarily mean it will cost you more. Baffled? Read on…
How to calculate a city’s average speed
If you ever want to know the average speed of traffic in a city, check the taxi fares. The fare will be quoted as X pence per Y distance and X pence per Z time. Divide Y by Z and you have a good estimate of the average speed. For example, in London Y and Z are 189.3 metres per 40.8 seconds; 189.3 divided by 40.8 is 4.6 metres per second, or just over 10 m.p.h. In New York, Y/Z works out as 1⁄2-mile/90 seconds, or 20 m.p.h. Why does this trick work? Because taxi rates are set by modelling what the expected income will be on an average day in average traffic.
Does the formula work?
The principle of the taxi fare is easy enough, but what exactly does it mean? Here is the formula used until recently by all New York cabs. It’s the same principle as the London taxi, but the numbers are much simpler to handle:
$2 + 30c per 0.5 mile or 30c per 90 seconds (whichever is the higher)
One way to represent this formula is by plotting a graph:
When you cross the rectangle on the graph, the taximeter clocks up 30 cents. At speeds above 20 m.p.h. – the ‘critical speed’ – it is the distance unit that clocks, but for slower speeds, it is the time unit. Once the unit has clocked the 30 cents, the counters reset back at zero. At the corner of the rectangle (which is reached by travelling at exactly 20 m.p.h.) the distance and time units clock simultaneously, but only one charge of 30 cents is counted.
This critical speed (20 m.p.h. in New York, but only 10.4 m.p.h. in London) is an important part of taxi fares. If your taxi travels faster than this speed, then for a journey over a set distance, the actual cost is fixed because you are charged only for the distance travelled. Below this speed the overall journey fare increases. Another graph illustrates this quite well:
Towards zero speed, the cost of the journey soars upwards. In fact, if your taxi was parked permanently at a traffic light, the cost of your journey would head towards infinity – assuming you could be bothered to hang around that long, of course.
Notice how the curved part of the graph (for less than 20 m.p.h.) and the flat bit (over 20 m.p.h.) join each other. Smooth joins in graphs are a good way of avoiding fiddles – see page 132 for an example from everyday life where disjointed graphs encourage criminal activity.
So far, so good. The formula looks reasonably innocent and fair. But beware, there are hidden traps. The best way to illustrate this is with an example that you could more or less imagine happening in real life.
Suppose you are with a group of friends in New York, and you are travelling from your apartment to a restaurant. You can’t all squeeze into one cab, so you split into two. Both cabs leave at the same time, and arrive simultaneously at the hotel, having taken the same route. Imagine your annoyance when you discover that your friends’ fare was lower than yours. How can this be?
To illustrate why, let’s make the calculations simple. The cab journey is one mile which takes four minutes (240 seconds). To start with the meter reads $2, which will cover the first segment of the journey.
Your friends’ cab travelled at a steady speed for the whole journey – 15 m.p.h. Because this is lower than the critical speed of 20 m.p.h., the meter clocks for time rather than distance on this journey. After 90 seconds it clocks its next 30c, and after 180 seconds it clocks another 30c. Because the journey lasts only 240 seconds, the next unit of 90 seconds is never completed, so the total fare is $2.60.
Now to your journey. Although the start and end times of your journey were identical to your friends’, let’s suppose that for the first half-mile the driver of your cab put his foot down, hurtling along at 30 m.p.h. (that was your first minute). Then you were trapped behind a slow vehicle, and over the next half-mile you averaged only 10 m.p.h. Half a mile at 10 m.p.h. took three minutes.
This is how the taxi calculates your fare:
First half-mile – 30c for the distance covered (you are above the critical speed)
Second half-mile – 60c for the time taken (below the critical speed, two lots of 90 seconds)
Total cost of journey = $2 + 30c + 60c = $2.90.
In other words, you paid 30c more than your friends, a hike of over 10 per cent.
This example illustrates a quirk in taxi fares, namely that a journey on fast roads with frequent stops at traffic lights might well cost you more than a journey at a steady speed on slower roads. The longer the journey, the higher the discrepancy could be, and this discrepancy is possible in any taxi ride in any city, even if the standard taximeter has been perfectly calibrated. Because of this anomaly, it is possible to invent situations where the distance or the time of journey is bigger yet the fare is smaller (or vice versa). That is why the earlier quiz permitted every possible answer.
It would be hard for taxi drivers to exploit this deliberately, but, if they had a choice between smooth-flowing back streets or an expressway combined with slow exits, the latter might make them more money.
How does the cabbie maximise his income?
Although the taximeter can be squeezed for a few more quid here and there, this isn’t where the big margins are to be made as a cab driver. In the end, what he’s most interested in is pounds per hour. The best way of achieving this is to have nonstop work, and to complete each job as quickly as possible. The time rate on the meter effectively sets the minimum wage for the driver. As long as he has a passenger on board, he knows he is earning at least 30c per 90 seconds ($12 per hour) in New York, or 20p per 40.8 seconds (£17.60 per hour) in London. It may seem odd, by the way, to have a UK income that is double its US equivalent, but remember that this is only the minimum revenue, and among other things it probably reflects the higher fuel and training costs in London.
How taxes could learn from taxis
The taxi-fare calculation is just one example of a man-made pricing formula. Another example is the most disliked payment of all – taxation. Unlike the taxi graph, some tax graphs are not smooth. One example is the stamp duty on houses. In 1999 the Chancellor introduced new levels of stamp duty as follows:
|
Price of house: |
£0-£125K |
no tax pay |
|
£125K-£250K |
pay 1% on top of the price as stamp duty |
|
|
£251K-£500K |
pay 3% on top of the price as stamp duty |
|
|
£501K+ |
pay 4%… |
This percentage was to apply to the whole amount. So somebody buying a house for £250,000 would only pay 1%, or £2,500, while somebody buying a house for £251,000 had to pay a 3% tax, or £7,530. The graph of stamp duty paid looks like this:
Understandably, somebody buying a house in the region of £251,000 would much prefer it to cost £250,000 than £251,000 because of the sudden £5,000 hike in duty. A similar disproportionate effect happens when the price climbs from £500,000 to £501,000. The sudden hike in tax caused a distortion in the market, with sellers finding all sorts of methods of arranging the price to be just the right side of the boundary, some of which were more legal than others.
The moral from this is that graphs with sudden jumps can lead to corruption. That is why the smooth line of the graph of taxi fare against speed is a good one. It removes the incentive for a cabby to travel at a particular speed to receive a jump in income.
But what is the ideal journey for a taxi driver – one that brings in the most pounds per minute? It turns out that there are two candidates – the very short and the very long journey. In both cases, they are favoured by a figure built into the fare formula.
As soon as a passenger enters a cab, he owes the driver a hire charge, £1.20 in London and $2 in New York. That represents a fantastic return for only, maybe, ten seconds of effort, so, in terms of income per second, passengers who are in the cab for a short time represent the best return. In practice, though, no passengers ride for less than a minute. So a second factor needs to be taken into account – the tip. Tipping can represent a significant proportion of a taxi driver’s income. The most common sort of tip is to round up a fare to the nearest pound. This means that a cab fare of, say, £9.90, will often earn a total of £10, a miserly 1 per cent tip. The best fare to clock on the meter is probably something like £3.40. Chances are that the passenger will round this up to £4.00, a tip of nearly 20 per cent. Continuous £3.40 rides could bring the taxi driver something like £35 per hour. Nice work, if you can get it.
Ironically, high fares may also bring good tips. By the time somebody can afford a fare of, say, £42, they often have money to burn (their own or, better still, their employer’s), so, according to one cabby in Aberdeen at least, a cheery ‘Here’s fifty pounds, – keep the change’ is not out of the question. Another 20 per cent bonus. In addition, the distance rate on a taxi actually increases once the length of journey increases beyond a certain limit. This rule dates back to when taxis were pulled by horses and exhaustion was a consideration, but it also recognises that the further from the traffic hub a taxi is, the less chance it has of picking up a new ride very quickly.
According to Euclid, the shortest distance between two points is a straight line. But not according to a cabby. Because cities are laid out in grids, to travel from one point to another almost inevitably involves a zigzag route. There is almost always more than one ‘shortest distance’ between two points. For example, in this grid, the shortest distance from A to B is five blocks, and you should be able to find ten different ways of doing it (one is illustrated).
This way of working out distances has acquired a whole minibranch of mathematics known as taxicab geometry. It has all sorts of curiosities. For example, below, the points marked X are all the same taxicab distance (two units) from the central point. And what do you call a shape where everything on the perimeter is equidistant from the centre? A circle, of course. So, in the world of taxicab geometry, you can square a circle – a trick that has eluded Euclidean geometers for centuries.
Nonetheless, there are anomalies. A long journey from a remote location such as Heathrow airport to central London not only brings in a high rate of earnings, it also takes the driver straight to a hot pick-up zone. No wonder cab drivers love collecting newly arriving tourists.
In fact, whole mathematical models have been built to work out the optimal set of locations to send taxis to maximise the collection of passengers and income. There are also sophis -ticated models set up within the Department of Transport to look at traffic flows and establish what a fair fare should be.
As one cabby said, ‘Blimey, you’d never have thought there was so much maths in it. And to think, I had that Carol Vorderman in the back of my cab once.’