Parking, Fireworks, and Making Sense of Prices

Why do universities have persistent shortages of parking? Why do states allow you to purchase fireworks, but not set them off?

This week, I’d like to discuss two very different topics: state firework policies and university parking. Although the topics themselves seem to have nothing to do with one another, I think that the price theoretic aspects of these topics have something very important in common. Allow me to set the stage.

Perhaps it is different now, but when I lived in Michigan, there was a combination of two laws that really confused me. On the one hand, it was legal to buy and sell fireworks. On the other hand, it was illegal to set off fireworks. Violations were subject to a fine. (In fairness, this isn’t completely true. There were certain days in which one was exempt from fines. I believe that New Year’s, Memorial Day, and Independence Day celebrations were exempt from fines.) It seemed contradictory to allow people to buy and sell fireworks, but that lighting off the fireworks on most days of the year resulted in a fine. Why allow people to buy something and then fine them when they consume it?

Another thing that I’ve long found perplexing is the parking situation on university campuses. I’ve spent a lot of years on university campuses and the one thing these campuses have in common is that there never seems to be enough parking to satisfy demand. This is such a problem that universities often have meetings about how to solve the parking problem that involve discussions about whether to build a new parking garage, but rarely about price. One would think if the objective was to solve an excess demand problem, the obvious solution would be to increase the price. Yet, somehow this never comes up in the meetings.

What is going on here? My conjecture is that both of these examples represent applications of something like a two-part tariff.

In price theory, a two-part tariff is a type of price discrimination. This particular type of price discrimination refers to a scenario in which the price of a good includes two parts: a lump-sum amount and a per-unit price. This type of arrangement applies to a variety of circumstances. Some major retailers like Costco or Sam’s Club are known for selling large quantities of goods at lower prices per unit than their competitors. However, they also charge a fixed annual fee to be a member and receive that lower per-unit pricing. Another example would be to think about bars that have a cover charge. The cover charge allows you to get in the door, but then they charge you for each individual drink you order.

Okay, but how does this relate to the examples that I started the post with?

University Parking

Let’s start by thinking about university parking.

One thing about university parking is that there always seems to be excess demand. This would seem to suggest that parking passes are priced below their market-clearing level. However, that doesn’t really make sense. Why would the university do this?

For example, imagine that the university has a fixed number of parking spaces available. In addition, suppose for simplicity that each person with a parking pass comes to campus every day and demands one parking spot. It would make sense to charge a price that is just high enough that the number of people who buy a parking pass is equal to the number of spots on campus. Charging a higher price will result in empty parking spaces and foregone revenue. Charging a price below this amount will lead to excess demand. There will be more people on campus each day who want to park than there are parking spots available.

Of course, if there are more cars on campus than there are spots to park them, these cars will have to park somewhere. Universities tend to have a common way of dealing with that problem: parking tickets.

Herein lies the solution to the puzzle of pricing for parking. The university is price discriminating when it comes to parking in order to maximize revenue. The university price discriminates via a two-part tariff: a fixed entry fee (the parking pass) plus per-unit pricing (parking tickets).

Let’s think about how this might work. Let’s imagine that there are a fixed number of of parking spots, denoted by Q_s. Furthermore, the inverse demand curve for parking spots is given as p = a - bQ.

Let p* denote the price that clears the market such that

\(p^* = a - bQ_s\)

Now suppose that the university sets the price equal to p_u such that

\(p_u = a - bQ_u\)

Note that by lowering the price of a parking permit from p to p_u, the university has the following foregone revenue:

\((p^* - p_u)Q_s - p_u(Q_u - Q_s)\)

where the first term is the lost revenue from lowering the price at the previous quantity and the second term is the increase in revenue from increasing the quantity sold at the lower price.

The excess demand for parking is a source of parking tickets. Suppose that there is some probability, $\lambda$, that the people parking illegally get caught and that enforcement is costless. Those caught parking illegally must pay a fee f. It follows that the university will generate revenue from parking tickets equal to

\(\lambda f (Q_u - Q_s)\)

The university benefits from the two-part tariff if the revenue from parking tickets exceeds the foregone revenue from the lower price of the parking pass:

\((\lambda f + p_u) (Q_u - Q_s) > (p^* - p_u)Q_s\)

Given the demand curve, this simplifies to

\((\lambda f + p_u) > bQ_s\)

Recall from the definition of the price elasticity of demand, evaluated at the market-clearing price, we have:

\(\varepsilon = \frac{dQ}{dp}\frac{p}{Q} = -\frac{1}{b} \frac{p^*}{Q_s} \implies |\varepsilon| = \frac{1}{b} \frac{p^*}{Q_s} \implies b = \frac{1}{|\varepsilon|} \frac{p^*}{Q_s}\)

Thus, in order for the university to generate more revenue from the two-part tariff than from the market-clearing price for parking permits, the university simply needs to set the expected fine such that

\(\lambda f > \left(\frac{1}{|\varepsilon|} - 1\right)p^* + \delta\)

where $\delta$ is the difference between with market-clearing price and the actual price of the permits. What this shows is that if the fines and/or the probability of getting caught is sufficiently high, then the university is actually better off with the two-part tariff. The necessary magnitude of the fine depends on the elasticity of demand for permits and the magnitude of the discount. All else equal, the bigger the discount, the bigger the necessary fine. Furthermore, as demand becomes more inelastic, the magnitude of the fine has to get bigger. The intuition is easy to understand. The university is selling a number of permits that is equal to the quantity demanded at the price it sets. When demand is inelastic, they lose revenue from reducing the price of permits because the percentage decrease in the price is bigger than the percentage increase in the quantity demanded. Thus, one needs a bigger fine per violation to make up for the lost revenue from permit sales.

As a result, it is possible for the lump-sum permit and the per-violation fines can maximize revenue. The basic idea here is that the university is able to price discriminate. Those who have a higher willingness to pay will not only buy the permit, but they will continue to park on campus even when there are no spots available and incur the fine whereas those with a lower willingness to pay will tend to arrive earlier to make sure that they get a spot.

Interestingly, this also seems to have an added benefit for the university. If parking is in short supply, high parking permit prices might draw the ire of students and their parents. High permit prices might make it prohibitively costly to park on campus. However, by reducing the permit prices and instituting fines for violations, the cost of parking on campus is now endogenous to the behavior of the driver. Parents are therefore likely to blame their children for the high cost of parking due to repeated violations instead of the university since the fines could have been avoided with better preparation. This tendency of parents to blame their child for the high cost of parking will be true even if the university has itself engineered excess demand with lower permit prices, which increases the likelihood of violations.

Fireworks

When it comes to the fireworks example, our analysis should consider the objective of the state government. Why would they allow you to buy fireworks, but fine you for lighting them? Aren’t they sending mixed signals?

Suppose that their objective is to raise the maximum amount of revenue possible. One way for the state to generate revenue would be for the state to charge a tax on every unit of fireworks that they sell. However, there are limits to how much tax revenue can be raised.

For example, consider the case in which all consumers are identical and the supply and demand curves are linear. In the absence of the tax, quantity supplied is equal to quantity demanded and the market clears at some equilibrium price. In this case, the total surplus from trade (consumer surplus + producer surplus) is maximized.

When the government levies a tax per unit of fireworks sold, this drives a wedge between the price that consumers pay and the price that sellers receive that is equal to the size of the tax. Because of this wedge, both consumer surplus and producer surplus are lost. The distribution of the foregone surplus depends on the relative price elasticities of supply and demand. For our purposes, we don’t need to worry about the burden of the tax. The more important point is that the foregone surplus is exactly equal to the amount of tax revenue raised plus the deadweight loss.

Tax revenue is easy to understand. The tax revenue generated is the tax per unit multiplied by the number of units sold given the tax. Tax revenue increases in proportion to the tax.

The deadweight loss is the residual. It is the total loss in surplus to consumers and producers minus the revenue that went to the government. Given this definition, it is not hard to understand why it is referred to as a deadweight loss. It is the foregone surplus that is lost and goes to no one. It is the value of the gains from trade that are simply foregone because of the tax.

What is particularly important about the deadweight loss is its relationship to the tax rate. With linear supply and demand curves, we know that the deadweight loss can be illustrated graphically as a triangle. What we know about triangles and this triangle in particular can actually generate an important insight about the costs of taxation.

Recall the following. The area of a rectangle is (Base x Height)/2. The base of the triangle in this example is equal to the size of the tax. The height of the triangle is the change in the quantity traded because of the tax. We can then use the definition of elasticity to derive a precise equation for the deadweight loss of a tax as a function of the tax itself. Allow me to explain.

Note that the price elasticity of demand is defined as follows:

\(\varepsilon = \frac{dQ}{dP}\frac{p}{Q}\)

Solve this expression for dQ and note that dP = t, where t is the tax. It follows that

\(dQ = \frac{Q}{p} \varepsilon t\)

Now, using our definition of the area of a triangle, we have:

\(DWL = \frac{1}{2} \times Base \times Height = \frac{1}{2} dQ t = \frac{1}{2} \frac{Q}{p} \varepsilon t^2\)

Note the important insight here. Tax revenue is proportional to t, but the deadweight loss is proportional to t^2. What this means as a practical matter is that the deadweight loss from taxation grows much faster as the size of the tax grows than tax revenue grows. This is where the Laffer curve comes from. For low levels of taxes, an increase in the tax will tend to increase revenue. However, as the magnitude of the tax gets larger, the deadweight loss grows faster, which implies that tax revenue will eventually start to decline.

Now, think about this from the perspective of the government. Imagine that their objective is to collect as much revenue as possible from fireworks. What they would want to do is choose the size of the tax that maximizes the amount of revenue that the tax generates. In the state of Michigan, they collect the normal 6 percent sales tax, but also collect an additional 6 percent tax that applies to fireworks.

Nonetheless, it is important to note that when choosing the tax to maximize revenue, there is still consumer surplus and producer surplus to be captured. If you are a government and your objective is solely to maximize tax revenue, you would also like to figure out a way to collect all of that surplus as well. Of course, you know that you cannot vary the size of the tax to generate more revenue because it would distort behavior in the market. If the state has already set the tax rate at the height of the Laffer curve, then this would be self-defeated.

All is not lost for this revenue-maximizing government. What it could do is institute a two-part tariff. The government could combine a lump-sum tax with the per unit tax. Given that the per unit tax maximizes revenue, the government could levy a lump-sum tax on consumers less than or equal to the remaining consumer surplus and a lump-sum tax on producers less than or equal to the remaining producer surplus.

Again, the state of Michigan appears to do something like this. The fine for setting off fireworks is the same no matter the quantity of fireworks that have been lit. One way to interpret the fine is as a lump-sum tax to collect at least some of the remaining consumer surplus. Michigan also levies a lump-sum license fee on retailers who sell fireworks. In order to sell fireworks, retailers effectively have to pay an entry fee. This is a flat fee that is again independent of the number of fireworks sold.

Concluding Thoughts

When I lived in Michigan, the rules on fireworks were initially confusing to me. Why would the state allow people to purchase them, but then turn around and fine people for actually consuming them? Furthermore, one persistent problem on college campuses is that universities tend to lack adequate parking. Yet, despite the excess demand, there is little discussion of raising the price of a parking permit.

Hopefully, what I have shown in this post is that both of these examples can be explained through some very basic price theory. If the objective of the state and of the university is to maximize revenue, then something like a two-part tariff can be optimal. In the case of the university, this is a straightforward application of price discrimination. The two-part tariff allows them to charge more to those with a higher willingness to pay. For the state, this is a way for the state to capture as much revenue as possible from a sometimes dangerous recreational activity.