I am trying to build an economic model which charges people for road use depending on the demand for the road that they are on. Assumptions are as follows:
- every car is tracked so distance traveled is always known
- the speed limit is known for all road sections
- the average speed of cars on any given section of road is known
I need a formula which achieves the following:
- increases the cost per meter for each additional car on a section of road in a way that is proportional to the decrease in average speed that the additional car has caused for all cars.
- works the same at various speed limits
If there are any other variables or data points that are needed then please assume that they are available and list them...
My apologies if the above is not possible - I may have confused mathematician with magician but any help would be greatly appreciated!
Many thanks
I'm not sure I really get the problem but naively I get this solution:
For each section $i$, let $sl_i$ be the speed limit and $as_i(t)$ be the average speed at time $t$ on section $i$.
For each car let $c:\mathbb{R}\to \mathbb{R}$ be the function such that: $c(t)=0$ if the car is not in any section of the road. $c(t)=sl_i-as_i(t)$ if the car is on section $i$ at time $t$.
For any car that leave the road at time $t$ charge $\int_0^t c(x)dx$.
Is that what you are looking for? Or did I miss something?