It’s the 4th of July, and you need to decide how much money to spend on beer (b)
ID: 1137431 • Letter: I
Question
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)).
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 on beer.
(i) 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?
(ii) Is this efficient? Would you all be better or worse off if you agreed to each contribute $20 to fireworks?
Explanation / Answer
i) Now the total fireworks would be $20+ f
f=whatever you chose to spend yourself on fireworks
Beer consumption will be = 50-f
Maximizing 10sqrt(20+f)+50=f with derivative = 10/2qsrt(20+f)-1
Set this equal to zero,we get
5=sqrt(20+f), or 25=20+f, or f=5.
Therefore,when each of your friends are gonna spend $5 on fireworks,you would also choose to spend $5 on fireworks and the remaining $45 on the beer.
Utility would be = 10sqrt(25)+45 = 50+45 = 95.
ii) If everyone contributed $20 on fireworks then F=100 and everyone would be left with $30 for beer instead of $45
Everyone's utility would be =10sqrt(100)+30 =130
Since,by spending $20 on fireworks,everyone is getting higher utility,so,spending $5 per person on fireworks is not efficient.
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