It's the 4th of July, and you need to decide how much money to spend on beer (b) and how much to spend on fireworks (f). Beer and fireworks each have a unit price of $1, that is, $1 gets you one unit of either good. You have $50 to spend, and your utility function is

u(f, b) = 10 sqrt(f) + b

Recall that for any constants C and D, the derivative of C sqrt(D + x) is C / (2 sqrt(D + x)).

First, suppose you are spending the holiday alone.

(a) How much money will you choose to spend on fireworks, and how much on beer? (Since more of everything is better, your budget constraint will hold with equality, or f + b = 50; one way to solve this, then, is to plug in 50 - f for b, and find the value of f that maximizes u(f, 50 - f).)

(b) What is the efficient amount for you to spend on fireworks, and the efficient amount for you to spend on beer?

Next, suppose that you are spending the holiday with four of your friends. You can only drink the beer that you bought, but you get to enjoy the fireworks that everyone bought. That is, each of you gets utility

u = 10 sqrt(F) + b

where F is the total amount that the five of you collectively spent on fireworks, while b is still the amount that you individually spent beer.

(c) Suppose you are perfectly rational and selfish. If you anticipate that each of your friends is going to spend $5 on fireworks, how much would you choose to spend on fireworks and how much on beer? How much utility would this leave you with?

(d) Is this efficient? Would you all be better or worse off if you agreed to each contribute $20 to fireworks?

Finally, suppose you were spending the holiday at a public park, in a crowd of 500 people. Again, everyone's utility is 10 sqrt(F) + b, where F is now the amount that everyone in the crowd contributes to the fireworks show.

(e) If you anticipate that everyone else in the crowd is going to contribute 5 cents each to the fireworks display, how much would you choose to spend on fireworks and how much on beer? How much utility would this give you?

(f) Is this efficient? Would you be better or worse off if everyone at the park agreed to contribute $5 each toward fireworks? What if everyone at the park agreed to contribute $20 each?