Problem Set 9: Public Economics I

Due Saturday, November 29 at 11pm on Gradescope

Final Exam Question

I invite you to write a question for the final exam. Writing a truly remarkable problem is the only way to get an A+ in the class; it’s a differentiator that’s not available just from homeworks and exams. I will also use this as a “tie-breaker” if you’re right on the border between any two grades (e.g. A and A-, A- and B+).

So, here’s the assignment:

Look back on your previous exams and homeworks, and identify a question you got wrong (or struggled with). List the question, and say what you found confusing or difficult about it.

Create a multiple choice problem that addresses the central difficulty you had with the problem: that is, ask a question which requires really understanding the nuances of the problem.
Explain why the correct answer is correct.
Explain what error or misunderstanding would lead a student to select each of the incorrect answers (aka “distractors”). Specifically, use your (incorrect) answer to the question as one of the distractors.

My goal with this exercise is to help you review and reflect on questions you had difficulty with. I will grade it as follows:

3 points: the problem addresses the subject in a nuanced way, and the distractors clearly reflect a misunderstanding
2 points: the problem is straightforward, and the distractors are clearly wrong but at least make sense (they’re not just random numbers)
1 point: the problem is ambiguous and/or incorrect; the “correct” answer is not necessarily right and/or one or more of the “incorrect” answers are actually plausible/correct

I look forward to seeing what you come up with!

The Laffer Curve

Suppose supply and demand in a market are given by the equations $S(P_F) = P_F$ $D(P_C) = 120 - 2P_C$

What is the equilibrium price and quantity in the absence of any tax?

Find the equilibrium quantity, the price consumers pay, the price firms receive, and the government revenue if there is a tax of $t = 15$ per unit? What is the tax burden on consumers? On producers?

Now consider a more general tax of $t$ dollars per unit. Calculate the amount of the tax collected (i.e. the government revenue) as a function of $t$. Plot $G(t)$ in a graph with $t$ on the horizontal axis and $G$ on the vertical axis.

Arthur Laffer famously said that if taxes are too high, lowering the tax rate would actually increase government revenue! Is he right, given this model? If so, for what values of $t$ would lowering the tax rate actually increase revenue?

Elasticity and Tax Burden

Consider a market with supply and demand curves given by $S(P) = \frac{P}{2}$ $D(P) = 100 - 2P$

What is the equilibrium price in this market?

What is the price elasticity of supply at the equilibrium price? What is the price elasticity of demand at the equilibrium price?

Now suppose a tax of €10 per unit was imposed, to be paid by consumers; that is, demand would now be described by$D(P) = 100 - 2(P+10)$Solve for the new equilibrium price $P$, which is the price received by firms. Relative to your answer to (a), how much less do firms get per unit? How much more do consumers pay? (In other words, how is the tax burden borne by firms and consumers?)

Think about what elasticity means: that the more elastic demand or supply are, the more substitutes are available for consumers or firms respectively. With this in mind, how does your answer to (b) help explain your answer to (c)? In other words, how do the relative elasticities of demand and supply relate to the relative burden of the tax?

Efficient Production of a Good with an Externality

As we’ve done before, let’s model an economy “as if” it were a single producer and a single firm. The producer is a steel manufacturer with the cost function $c(q) = \tfrac{1}{2}q^2$ The steel company sells to a single consumer who has the following utility function (where Good 1 is steel and Good 2 is just money): $u(x_1, x_2) = 120x_1 - \tfrac{1}{2}x_1^2 + x_2$ In other words, the total benefit (in dollars) from $Q$ units of steel as $TB(Q) = 120Q - \tfrac{1}{2}Q^2$

What is the consumer’s marginal benefit function for steel, $MB(Q)$? (You can treat this as just the marginal utility from good 1, since the units of the quasilinear utility function are dollars.) Plot the marginal benefit curve. How do you see the total benefit of producing some $Q$ units in this diagram?

What is the producer’s (private) marginal cost of producing steel, $MC(Q)$? Plot the marginal cost curve. How do you see the total cost of producing some $Q$ units in this diagram?

Show that total surplus is maximized when society produces the amount $Q$ such that the marginal benefit to the consumers of the last unit produced equals the marginal cost to the producers: $MB(Q) = MPC(Q)$.

Plot the $MB$, $MPC$, $MSC$, and $MEC$ curves in a graph. How do you see the optimal quantities in this graph?

Market Failure with an Externality

In the previous problem, we found the socially optimal equilibrium without looking at prices or supply and demand functions. Now, imagine that steel is traded in a perfectly competitive market.

What is the consumer’s demand function for steel? What is the producer’s supply function? Show that the competitive market equilibrium is same as the socially optimal equilibrium found in the previous problem \textbf{without the externality}.

Of course, steel production does have the negative externality of emitting pollution, as in the previous problem. Because pollution is an externality, the producer’s marginal and total cost functions are unchanged: they do not internalize the bad outcomes from their pollution. Suppose the government steps in to curtail the pollution by imposing a constant per-unit tax $t$ known as a Pigouvian Tax. This means that consumers pay $p$ per unit and producers receive $p - t$. What is the equilibrium quantity as a function of $t$? How large of a tax should they impose to move the equilibrium from the competitive market equilibrium to the socially optimal equilibrium you found in part (d)?

With the government tax, the market is no longer operating at the competitive free market equilibrium. Why is this not inefficient? Provide some intuition.

A Steel Mill and a Fishery

Suppose there are two firms: a steel mill (firm 1) and a fishery (firm 2).

The steel mill sells its output at price of $p_1 = 160$, and has the cost function $c_1(q_1) = 8q_1^2$ The fishery is affected by firm 1’s pollution. Its cost function is given by $c_2(q_2 | q_1) = 10q_2^2 + 2q_1^2$ What level of tax on steel, $t$, would result in the socially efficient quantity of steel being produced?