Before we begin to formulate our assumptions in detail, let us try to understand the problem qualitatively. When we speed, we reach our destination faster, which is desirable. However, at the same time, we risk getting a speeding ticket, making our trip more expensive. There is a secondary cost associated with high speeds, which is related to lower fuel efficiency. (There is also a safety issue, which is hard to account for properly, and is hence ignored.)
Thus, this is a risk-reward problem. The reward (faster arrival) needs to be traded off against the risk (speeding ticket + lower gas mileage).
Let us take an engineering approach, and try to put some numbers together. Let us say that the speed limit is 65 mph, and that our gas mileage decreases with excess speed (beyond the speed limit, thus at 75mph the excess speed is 75 - 65 = 10 mph) as shown in green on the graph below. The graph also plots (on the right y-axis, blue line) the cost of a speeding ticket at the different excess speeds if one were to get caught.
Next we need some way of estimating the probability of getting caught. Now this probability will depend on (a) the excess speed, and (b) the duration of the trip. Drive too fast, and the odds of getting caught increase. Drive over long distances, and the odds of lucking out in spite of speeding decrease - since there are now "more opportunities" (read cop cars on the journey) to catch you in the act. Based on some speeding ticket data, we assume a probability of the following form:
We further assume that gas costs $3 per gallon, and that the amount of time that a speeding ticket takes to administer is 15 minutes. The length of the trip is assumed to be 100 miles. This gives us the following expected time and costs, for various levels of over-speeding.
The risk-reward picture may best be visualized as:
(1) If you are a low-risk person, then you probably want to drive 5-7 mph over the speed limit at most.
(2) If you are a high-risk person, then you are best off picking a speed somewhere along the sharp uprise.