Market failures: Public goods

[This essay is intended for students in my law and economics and my microeconomics class (for law students). Read this essay one time through, ignoring the queries that are interspersed. Go back to the queries after you feel you have understood the essay. The queries test whether you have understood enough to develop a richer understanding of the topic of the essay (public goods). These instructions will also apply to future essays.

I have bolded certain words. These are terms that you should know going forward. If you do not understand them, ask your favorite foundational AI model about them.

I will post the python notebook that generates all the figures in this essay in case you are interested in reproducing them.]

In a previous essay (https://anupmalani.substack.com/p/when-are-markets-efficient-and-when), I discussed the assumptions under which a market equilibrium would be (socially) efficient. One of these assumptions was that goods being exchanged are private goods. I want to unpack that a bit in this essay by giving an example that illustrates why markets lead to inadequate production of so-called public goods.

Taxonomy of goods

In that prior essay, I presented the following taxonomy of goods based on whether they are excludable and rivalrous. By excludable, I mean that a producer can stop a customer from consuming a good (unless they pay the price that the producer demands). By rivalrous, I mean that, if one customer consumes a unit of the good, another customer cannot also consume that unit of the good. A good example of a good that is both excludable and rivalrous is a banana. A grocery store can stop you from taking a banana without paying for it. And if you eat a specific banana, I cannot also eat that banana. A banana is what we call a private good.

There are goods that are not excludable, some that are not rivalrous, and some that are neither. Examples are given in the table above. Goods that are not excludable include national defense and fish in the sea. It is hard for the US government to stop a person in Ohio from consuming (benefiting from) national defense even if that person does not pay taxes to fund defense. Likewise, because there is no international government that can forcibly stop ships from fishing in international waters, fish swimming in the ocean are largely non-excludable. Goods that are non-rivalrous include national defense and a music concert. If the US military deters a foreign country from lobbing missiles at the US, you may benefit (assuming you live in the US). But the fact that you benefit does not stop your neighbor, also in the US, from benefiting from that deterrence. Likewise, if you and I both attend a Bruno Mars concert, the fact that you are enjoying his (live) performance, does not limit my ability to enjoy that concert.¹ Things that are rivalrous, but not excludable, are called common pool resources (e.g., fish in the sea). Things that are not excludable, but are rivalrous, are called club goods (e.g., a concert in a private venue). Things that are neither excludable nor rivalrous are called public goods (e.g., national defense for persons living in the US).

Why is this taxonomy useful?

Why do we care whether goods are excludable or rivalrous? Producers care if goods are excludable. If they are not, then producers cannot ensure that customers pay before they consume a good. And without pay, producers have less revenue. Since profit is revenue minus costs, non-excludable goods are less profitable. This gives us our initial intuition for why non-excludable goods might not be adequately supplied.

We care about whether a good is rivalrous because that affects what the aggregate demand for the good is, and how we derive or determine that aggregate demand. This requires us to revisit the question of how to aggregate goods. We do that in the next section. And then we explain why non-rivalrous goods are undersupplied in a market.

Revisiting aggregate demand

Recall that in our discussion of demand and equilibrium, we discussed how to aggregate demand. Back then we were focusing on private goods, i.e., goods that were rivalrous and excludable. You should have drawn two lessons from that discussion. First, the total amount of private goods that are demanded when the good has a price p is the sum of the quantity that each person individually would buy at that price. So if there were just 2 people in the market, you and me, aggregate demand at price p would be the amount you demand plus the amount I demand. If we were to plot quantity on the vertical axis and price on the horizontal axis, this logic implies aggregate demand would be the vertically sum of our individual demand curves. Second, because we typically plot demand curves with price (rather than quantity) on the vertical axis and quantity on the horizontal axis, aggregate demand is instead plotted as the horizontal sum of individual demand curves. To repeat, aggregate demand for private goods is the horizontal sum of individual demands.

Before we switch to aggregate demand for non-rivalrous goods, let’s take a detour into the demand for another private good: a sheep. Why? Because sheep have an interesting feature: they bundle two separate products: wool and mutton (their meat). This means that the willingness to pay for a sheep is the vertical sum of the willingness to pay for the wool and for the mutton, even when price is on the y axis. See the figure below. Now, we are still talking about demand for an individual sheep, not aggregate demand for sheep. But if I knew aggregate demand for mutton and aggregate demand for wool.² Then I would vertically sum the aggregate demand for wool and aggregate demand for mutton to get the aggregate demand for sheep. Why? Because the production of sheep satisfies both the demand for wool and the demand for mutton. Abusing terminology a bit, I might say production of wool is not rivalrous with the production of mutton.

Now let’s consider a truly non-rivalrous good, like a music concert (for people inside the concert venue) or national defense (for people inside the country). The difference between this example and the sheep example is that both consumers get the exact same thing with the concert or defense, while in the sheep example one consumer got wool and the other mutton.

If Bruno Mars were to sing some songs for us live at a local park, he could entertain both you and I at the same time and with the same effort. In the figure below, I show both your individual demand and my individual demand for Bruno Mars songs. (The y axis is price, the x axis is quantity measured in the number of songs sung live.) I have made my individual demand D1 somewhat higher than your demand D2.

Because Bruno Mars can entertain both of us with the same songs, aggregate demand is not the horizontal sum of our individual demands. Rather, it is the vertical sum of our demand. Even though price is on the vertical axis. Just like in our sheep example. To summarize: because Bruno Mars songs are non-rivalrous, aggregate demand is the vertical rather than horizontal sum of our individual demands. (If those songs were a rivalrous good, aggregate demand would be the horizontal sum.)

Supply of a public good

Let’s examine Bruno Mars’ supply curve. We said he can entertain multiple people with the same effort. Specifically, it costs him the same amount to sing for 2 people as 1 person. The only thing that takes greater effort is singing more songs. So, in the figure above, we drew his cost or supply function as increasing in quantity. But that supply can satisfy both our demands.

Before we discuss how much Bruno Mars will actually supply in a market, let’s consider what the efficient level of supply would be. The efficient number of songs for Bruno Mars to sing is the quantity q* where aggregate demand is equal to aggregate supply. That would generate a consumer surplus equal to the difference between the aggregate demand curve and price p* (to the left of q*), and producer surplus equal to the area below p* and above the supply curve (to the left of q*).

Aside: Economists will sometimes phrase efficient supply of a public good differently. They will say that each individual demand curve is an individual’s marginal valuation MV. Then aggregate demand is the sum of individual marginal valuations at each quantity MV(q) = MV1(q) + MV2(q) + …. Finally, they will say that the efficient quantity of production is the quantity where the marginal cost of production MC(q) is equal to the sum of aggregate marginal valuations at that quantity MV(q).

But what will Bruno Mars actually supply in a private market? It depends on what price he is able to charge. To answer that, we need to keep in mind our assumption that his concert is non-rivalrous. Meaning, if he starts singing in a local park, he cannot stop me from coming by listening. Nor can he stop you from coming by and listening. He can put up a sign that says, “Pay me $p please”, but cannot make you or I pay. The only thing he can control is the number of songs he sings.

So the amount that Bruno Mars gets for singing depends on how much we would rationally pay him. As you consider your answer, keep in mind that I am also assuming we are rational. We don’t pay out of a sense of guilt. And if you think that is unreasonable, consider the typical behavior of people walking by a street performer. The image below, artificially generated by DALL-E, should give you a hint. The amount of money in a typical street performer’s tip jar case is probably less than what the performer could have earned working as a barista, i.e., less than the value (thus opportunity cost) of the performer’s time.

The answer is that Bruno Mars will get zero revenue. I won’t pay because I benefit when you pay. And you don’t pay because you would benefit if I paid. Each of us prefers to be the free rider who benefits without cost because the other person pays. If both of us behave this way, then we get an equilibrium where neither of us pay.

Knowing this, what is Bruno Mars likely to do? Not sing. Revenue of zero doesn’t even cover the cost of one song. So Bruno Mars will not sing even one song. Want evidence? Ask yourself: how often do stars give free concerts? Rarely. Instead, what they do is sing songs in private venues so that can charge you for entry, and thus to hear their songs. I.e., they sing behind walls (or paywalls when they stream music), so that they can convert their product from one that is non-rivalrous to one that is rivalrous.

The punchline is that, when there is a non-rivalrous and non-excludable good, i.e., a public good, the market leads to underproduction of the good relative to what is efficient. When costs are positive, that may mean no production at all.

Query: What happens to the supply curve, and thus public goods production, when the producers gets utility from supplying the good? E.g., a street performer enjoys making people smile or the attention from passers by? How does that affect the supply of non-rivalrous street music?

Query: Why do people pay street performers, even though they can get the entertainment for free? What (additional) good are the tippers getting, aside from the street performer’s music? Is the “warm glow” from financially supporting a local artist rivalrous? How do you account for this in the way you would re-draw your individual demand curve for the street performance?

Query: How are governments able to solve the public goods problem associated with, e.g., the production of national defense?

Query: Imagine a private court system, e.g., arbitration. Would it produce the optimal amount of precedents? Does this explain why arbitrators might not public opinions, even if parties had no privacy objections to them doing so?

1. If you are seated ahead of me and stand up to block my view, you may impede my ability to enjoy that concert. First, don’t do that. It’s not cool. Second, yes, there are ways that concert goers can interfere with other attendees’ enjoyment of the concert But they can also enjoy it without doing so.

2. And aggregate demand for wool (mutton) would still be derived by horizontally summing individual demands for wool (mutton).