Homework 1 - QUESTION 1 - Efficient Allocation

I own a plot of land in northern Wisconsin, on which I vacation. My next-door neighbor wants my land as well, to expand his own house. The value of the land to me is $150,000. (That is, I would be indifferent between keeping the house, and losing the house but having $150,000.) The value of the land to my neighbor is $275,000.

Which of the following are Kaldor-Hicks improvements?

(i) My neighbor buys my land for $200,000

(ii) My neighbor buys my land for $275,000

(iii) My neighbor pressures the town council to force me to sell him the land for $125,000

(iv) My neighbor, while drunk, buys my land for $350,000

QUESTION 2 - Efficient Production

Upon graduation, you realize you have three options: you can take a job as a computer programmer, take a job as a coal miner, or remain unemployed and play video games all day.

As a computer programmer, you would create value of $2,000 per week for your employer, and receive disutility of $1,000 per week relative to playing video games. (To put it another way, at a wage of $1,000 a week, you would be exactly indifferent between taking the programming job and staying home.)

As a coal miner, you would create value of $3,000 per week, and receive disutility of $1,800 a week (mining coal is less fun than an office job, which is still less fun than playing video games).

(a) What is the efficient use of your time?

(b) Assume that both the programming and coal mining industries are perfectly competitive when it comes to hiring, so either firm would offer you a wage equal to the value they get from you. Which job would you end up taking?

Now suppose that mining coal also creates an externality. If you work as a coal miner, more coal gets dug up and eventually burned, leading to more pollution. Suppose that the total externality associated with you working as a coal miner is $400 per week, and is spread over a large number of people. (In particular, the coal company does not have to pay for it.)

(c) What is the efficient use of your time?

(d) If you are perfectly rational and selfish, which job will you end up taking?

QUESTION 3 - Public and Private Goods on the Fourth of July

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 on 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 (499 others plus yourself). 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?