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 if you were to make a short journey. But what about heavy traffic or unexpected delays due to traffic? 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. However, this can be a little misleading.  It charges you for your distance or your time, but not both at the same time. I will go into this later. There is a general rule in economics that the better the product is, the more you pay. However, in a taxi ride, the idea is to “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 have the time or interest to read is the formula for calculating the taxi fare. The current rates in 2007 are an £1 and 20p per 189.3 metres or 40.8 seconds. These values seems suspiciously precise, and they seem enough to put anybody off trying to calculate the fare for themselves, but this wasn’t put in deliberately by taxi drivers. The figure is set by London authorities, and adjusted yearly according to inflation to try and keep driver earnings roughly fixed. When taxis were first invented, these figures were round numbers. So blame inflation.

How 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 by all New York cabs. I’m using this example instead of London ones, as the figures are much easier to work with: $2 + 30c per 0.5 mile or 30c per 90 seconds (whichever is 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 miles per hour – the “critical speed” it is the unit that clocks, but for slower speeds, it is the time unit. Once the unit has clocked 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. The 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.  Towards zero speeds, the cost of the journey soars upwards.

In fact, if your taxi was parked permanently at a traffic light and you hadn’t run off already, the cost of your journey would head towards infinity.  Notice how the curved part of the graph (for less than 20 mph) and the flat bit (over 20 mph) join each other. Smooth joins in graphs are a good way of avoiding fiddles.            However, things are not always this simple. 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, only to discover that your friends’ fares were 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 pf 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 $2.60c. Now to your journey. Although the start and ends times of your journey were identical to your friends, let’s suppose that for the first half mile the driver of your cab was driving super fast, hurtling at the massive speed of… 30 mph (that was your first minute). Then you were trapped behind a slow vehicle, and over the next half mile you averaged only 10 mpg. Half a mile at 10 mph takes 3 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 over 10% more than your friends. This example illustrates a quirk in taxi fares, namely that a journey on fast roads with frequent stops at traffic lights night 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 the journey is bigger yet the fare is smaller.  That is why the earlier quiz permitted every possible answer. It would be hard for taxi drivers to exploit this deliberately, but, if you are ever offered between the smooth-flowing back streets or an expressway combined with slow exits, it is the latter that will cost you more money.

How can a cabbie maximise his income?
Although the taximeter can be squeezed for a few more quid here and there, it isn’t exactly where the big margins are made as a cab driver. In the end, what he’s most interested about is pounds per hour. The best way of achieving this is to have non-stop 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 20p per 40.8 seconds (£17.60 an hour) in London. But what is the ideal journey for a taxi driver – one that brings in the most pounds per minute? It turns out that thee are two candidates –the very short and the very long journey. In both cases, they are favoured by a figure built on the fare formula.  As soon as a passenger enters the cab, he owes the driver a hire charge and $2 in New York. That represents a fantastic return for only something like 10 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 in a cab 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% tip. The best fare to clock on the meter is probably something like £3.40. Chances are that the passenger will round this tip up to £40, a tip of nearly 20%. Continuous £3.40 rides could bring the taxi driver something like £35 an hour.  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 (usually their employer’s), so often paying £50 and not waiting around for change is not uncommon. Another 20% bonus. In addition, the distance rate on a taxi actually increases once the length of a 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.  Interestingly, 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 have also been simulation models to try and predict the average earnings for a cab driver parked outside Heathrow all day. Basically, there is a lot of Maths in a simple cab ride.

http://robbenraccoon.wordpress.com/2007/07/20/the-formula-behind-the-taximeter/