mercredi 13 décembre 2017

Math Problem - Car park revenue

A vacant lot has opened up next door to my work place that was turned into a temporary car park. When they first opened they were charging $18 per day and the place was empty for weeks. No one wanted to use it.

Eventually they dropped the price to $7 per day and within a couple of weeks the car park was full every day.

So now they are slowly putting up the prices, it went from $7 to $7.50 and now it is $8.00. I now see a few spaces in the car park that are there all day so the price increase was affecting take-up which would be expected.

I was wondering what the math would look like that works out the sweet spot, i.e. what is the price to park which will earn the most amount of money per day for the car park. To do this I have to make an assumption regarding how many people stop using the car park for every 50cent price increase and I think that function is nonlinear. So lets assume it is nx2. So if the price goes up my 50c then there is 1 new vacant space every day, increase of $1 is 2 vacant spaces, $1.50 is 4 spaces etc.

If we assume there are 200 spaces to begin with and $7 means it is always full and also assume that there is no initial lag such that the first initial increases in price has an immediate effect (i.e. the people who couldn't get a space when it was $7 don't turn up and take the extra spaces when it goes up to $8 due to new availability).

I can imagine a bell curve showing an increase in revenue as the price is increased because the lost revenue from the number of extra vacant spaces does not exceed the money they gain from putting up the price, but then at some point the loss of income from the number of vacant spaces will be greater than the extra amount they get from putting up the price. So I would need to work out the peak of the bell curve, the point just before revenue starts going down with another price hike.

How would I approach this mathematically (BTW, this is not a course work problem, I was just trying to figure it out on my way home the other night after noticing the extra spaces in the car park after the last price hike, but my math is rusty and I was never that good anyway).

Any takers?


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