Problem Set 10: When Markets Fail: Public Goods

Due Wednesday, June 3 at 11pm on Gradescope

A Grand Finale

The residents of a town all love fireworks, but disagree on how much. In particular, if $G$ is the number of minutes of fireworks,

2000 people receive a total benefit of $TB(G) = 5\ln G$
2500 people receive a total benefit of $TB(G) = 4\ln G$
3500 people receive a total benefit of $TB(G) = 2\ln G$

Fireworks cost $1000 per minute.

What is the optimal length of the fireworks show for the town?

If the citizens of the town agreed that everyone should contribute some equal amount $g$, and voted on what that amount should be, how long would the resulting fireworks show be?

Maximum Fun

Stanford cares deeply about the total amount of fun on campus, and wants to maximize total fun had by everyone. However, large parties can get out of hand.

Suppose there’s one and only one party on campus. If $x$ people go to the party, each partygoer has $800 - x$ units of fun, so the total amount of fun had by partygoers is $800x - x^2$. Anyone who doesn’t go to the party hangs out with friends in their room and has 400 units of fun. There are $N > 800$ students on campus.

If people are free to go to the party or not (and every student will of course choose the option with the most fun) how many people will go to the party?

What is the globally fun-maximizing number of people to attend the party, $x^*$? (Remember, fun is had both at the party and not.)

Stanford decides to limit the number of people who can attend the party by making them earn a wristband to allow entry. They earn the wristband by performing community service. Each hour of community service costs the student 100 units of fun. How many hours of community service should Stanford require if they want $x^*$ people to attend the party?

For questions 10.3-10.6, you may hand in solutions to questions 2, 3, 7, and 8 from the Fall 2025 final exam, which you may find on Canvas. However, if you’ve already reached 100 homework points, there is no need to do so; indeed, it would be a lot of grading work for the TA’s, so please don’t!