Question

There are extremely high incidences of speeding in Bathlaham. The fine for speeding is currently $200, and it is estimated that the current probability of a speeding driver being caught and convicted is 5%. Studies show that most people stop speeding if the expected fine is at least $150.

(a) Provide an explanation for why speeding is so prevalent in Bathlaham.

(b) The local government of Bathlahem decides to do something to curtail speeding. Suppose that the government wants to increase the probability of catching speeding. How high must the probability of being caught and convicted be to discourage speeding at the current size of fine?

(c) Suppose that the government is to raise the fine while keeping the probability unchanged at 5%. How large must the fine be to discourage speeding?

(d) Suppose that it costs police resources of $10, 000 for every 1% increase in the probability of catching and convicting speeding, whereas increasing the fine by every $50 costs administrative expenses of $12, 000. At the same time, the court prevents local governments from charging a fine more than $500 for speeding. If the local government of Bathlaham wants to curtail speeding at the lowest cost possible and allowable, what will be the mix of fine and probability that it will choose? Explain your answer.

Please explain where you got numbers from, please and thank you!

Answer 1

(a)

Expected fine = $200 * 5% = $200 * 0.05 = $10

As, the expected fine is $10 which is less than critical limit of $150, the speeding is so prevalent in Bathlaham.

(b)

To curtail speeding, expected fine should be at least $150

$200 * p $\ge$ $150

=> p $\ge$ 150 / 200

=> p $\ge$ 0.75

The probability of being caught and convicted be to discourage speeding at the current size of fine is 0.75.

(c)

Let M be the fine amount.

M * 0.05 $\ge$ 150

M $\ge$ 150 / 0.05

M $\ge$ $3000

So, the fine should be $3000 to discourage speeding at the 5% caught and convicted probability.

(d)

To curtail speeding, expected fine should be at least $150

Let M be the increase in fine amount and p be the increase in the percentage probability of catching and convicting speeding to curtail speeding.

M * p = $150

=> p = 150 / M

Total cost C =  $12, 000 * (M / 50) + $10, 000 (100 *p)

= 240M + 1000000 * (150/M)

= 240M + 150000000/M

Differentiating, we get

dC / dM = 240 - 150000000/M² = 0

=> M² = 150000000 / 240 = 625000

=> M = $790.5694

p = 150 / M = 150 / 790.5694 = 0.1897

Thus, the fine should be $790.5694 and probability of catching and convicting speeding to curtail speeding be 0.1897.