I am stumped as to how to explain the following using optimization as a guiding principle. Powerball managers are contemplating reducing the chance of winning this lottery in response to declining demand brought about by too many jackpots depleting the pool from which a fraction is drawn to pay out said jackpots.
Here's the back-of-the-envelope way that I think this problem through. The price of a Powerball ticket is its face value minus the expected value of any winnings. This is a positive value, which decreasing the probability of winning is tantamount to increasing. The inverse demand by individuals follows the law of demand, with price being negatively related to payouts, F, as in Q = a - bF - cP. The Powerball managers face marginal costs that are approximately zero, and fixed costs equal to the payouts, F. Their goal is to maximize profits, which are PQ - F, by choosing P (since they have a perfectly elastic supply). Profit maximization entails setting marginal revenue equal to marginal cost (which is zero here), and solving for P. Revenue is PQ = (a-bF-cP)P, and marginal revenue is then a-bF-2cP. Solving for the optimal price yields P = (a-bF)/2c. So, the optimal response to an increase in payouts is to lower the price of a ticket.
As I age I increasingly tend to follow the Chicago philosophy that human actions are the result of optimal behavior and that it sometimes takes economists a while to figure out why the behavior is in fact optimal. So, where'd I go wrong?
With a degree in business administration you could try to fix the Powerball system yourself.