Econ 50: Economic Analysis I

Problem Set 10: Public Economics II

Due Tuesday, December 9 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,

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?

Long Run Competitive Equilibrium

Suppose the market demand for a particular product is given by \(D(p) = 360-10p.\) Each firm in this market has a total cost function\(c(q) = 72 + \frac{1}{2}q^2\)where the fixed cost of 72 includes all opportunity costs, including the opportunity cost of switching industries.

Suppose there are $N_F = 10$ firms. What is the equilibrium price? How much does each firm produce?

Suppose the market is in short-run equilibrium as you found in (a). Draw the MR, MC, AR, and AC curves for a typical firm in this industry. Indicate the quantity produced, and shade the area representing profit or loss. Will firms enter or exit this industry, or is it in long-run equilibrium?

In the long run, with entry and exit, what is the equilibrium price and market quantity in this industry? How much does each firm produce?

Suppose the market is in long-run equilibrium as you found in (c). Draw the MR, MC, AR, and AC curves for a typical firm in this industry. Place a point at the equilibrium quantity produced by the firm, and price for which it is sold.

What is the long-run equilibrium number of firms in this industry?

Monopsony

Consider a firm with the production function \(f(L,K) = LK\) It has $\overline K = 3$, and sells its output in a perfectly competitive market at price $p = 4$.

The firm has market power in the labor market: specifically, if it sets wage $w$, it will hire \(L(w) = w^a\) Note that all your answers below will be in terms of $a$.

What is the firm’s total cost of hiring $L$ workers, $TC(L)$? What is its firm’s marginal cost of the $L^{th}$ worker, $MC_L(L)$? (Hint: to answer this, think about what wage the firm must set if it wants to hire $L$ workers…)

As we showed in class, the firm will maximize its profits by setting $MRP_L = MC_L(L)$. What amount of labor maximizes its profits, and what wage rate will it offer its workers? (Remember: $MRP_L = p \times MP_L$.)

Find the wage rate offered if $a = 0.5$, $a = 1$, and $a = 2$. Does the wage rate increase with $a$, or decrease?