Man, sometimes I can’t believe I do this work for free. Metrolinx owes me big-time. First, I showed them how to save face, save the train, and provide public transit. Now, I’ll tell them how much they should charge to make the most money.
Metrolinx gave free rides on the UPX this weekend—doubtless, of course, they did it to boost the dismal reputation of the train in the face of withering criticism from the press and politicians. Doing so, however, gave us a little bit of data to use. We found out how many people would take the train if it were free. The answer is depressing: only about 10,000.
With that data point and the knowledge that 2200 people take the train daily when the average fare is about $22, we can find the revenue-maximizing price! Economists, follow along in the footnotes!
That best price to charge is, according to my calculations, $14. At that price, 5030 people would ride the UPX. You could get more or fewer people on, as Metrolinx has showed, but the total revenue would add up to less.
Those 5030 people would bring in revenue of $70420. Unfortunately, it costs about $186,000 to run the UPX every day, so it will still lose gob-smacking amounts. This, however, will surprise nobody except for Metrolinx: the Auditor General and SNC Lavelin already told them there was no money in this business.
We have two points (10000, 0) and (2200, 22), so we can calculate the slope of the ridership function: it’s rise/run, which is 7800/-22, or -355. We can find the whole function easily because Metrolinx gave us the y intercept: it’s y=10000-355x, where y is ridership and x is price.
The revenue for the train is x•y. We know y, so that’s x•(10000-355x). Revenue=10000x-355x².
We then need to find the maximum of that function, and that means taking the derivative, which is 10000-710x. We need to find where the slope is 0, so we say: 0=10000-710x and solve for x, which gives us 14. $14, then, is the revenue maximizing price.
Of course, there are a lot of assumptions here, most notably that the riders this weekend were actually going to the airport and not just joyriding–a very big assumption. I’ve also assumed that the relationship between price and ridership is linear, although the result is fairly impervious to mild nonlinearity.