Public goods in economics have been a contentious theoretical issue since Paul Samuelson introduced the concept in 1954. The main sources of contention are what real world things are public goods, and who should provide them. In this post I propose a new way of looking at goods that will shed light on why public goods have posed such a problem. In particular, I propose that there is an important distinction between physical goods and immaterial goods; that public goods can only be immaterial goods; and that this unique feature of public goods does not preclude the market to provide the “socially optimal level.

Introduction

Economists define a public good as something that is “non-rival” (meaning that one person’s consumption does not affect another person’s), and “non-excludable” (meaning that one person cannot stop another person from consuming the good.) Public goods are often contrasted with private goods, which are rival and excludable.

The implications are that public goods cannot be provided by a free market, because no one would have to pay for such a good, and so there would be so incentive to produce it. Therefore, the argument goes, the government ought to provide public goods.

An example of a private good is an apple. Imagine a world with just you, me, and an apple. If I take a bite out of the apple, there is now less apple for you to consume. That means it’s rival. If I put the apple in a locker to which only I know the combination, then again you are prevented from consuming the apple. This makes it excludable.

But what about examples of public goods? Many examples have been offered, most commonly roads, utilities like electricity, and national defense. But all of them are highly controversial. Roads, after all, were not invented by the government; and can be excluded by tolls and barriers. Utilities are also excludable; just try not paying your electric bill for a couple of months. And anyone who’s been in a warzone can attest that “national defense” is very often not very national.

These difficulties have led many economists to loosen the practical definition for a public good to “anything with a very high barrier cost.” But this seems like cheating.

So what’s the deal? Is the concept of public goods somehow flawed? Or is it merely a case of “works in theory, but not in practice”?

A New Look at Public Goods

In a working paper on this question, titled “An ‘Existence Proof’ of Public Goods’, my answer to both questions is no. The theory is fine. And there are practical applications. But as the scare quotes around the phrase ‘existence proof’ should clue you in, there is a twist to the story.

Namely, my thesis is that public goods must be immaterial (that is, non-physical) goods. The proof goes as follows:

Definition 1: A public good is a good that is both non-rival and non-excludable. In mathematical terms, let the total supply of a good X be X_T. Let x_i be individual i‘s consumption of X. A public good, therefore, is one such that

X_T = x_1 = x_2 = … = x_i, for all i. (Note that this is perfectly standard notation in the literature.) Definition 2: A private good is a good that is both rival and excludable. In mathematical terms, let the total supply of a good X be X_T. Let x_i be individual i‘s consumption of X. A public good, therefore, is one such that

X_T ≥ x_1 + x_2 + … + x_n, for n number of individuals. Definition 3: A physical good is a good that has a (non-zero) positive mass (or colloquially, weight). If a good is not physical, it is immaterial. The venerable Law of Conservation of Mass says that, in a closed system, mass cannot be created or destroyed. (Note for purists: you can replace “mass” with “energy”, and the essence of the argument remains. I chose mass because it is easier to conceive of than energy.) Mathematically, let M_T be the total mass of a good, and m_i be the mass of each individual component of that good. Therefore,

M_T = m_1 + m_2 + … + m_n, for n number of components. Call this “Equation 1”. If the consumption of a public good takes place in a closed system (such that the total supply of the good X_T is not increasing over time), then, for the Law of Conservation of Mass to hold, it must be the case that the mass of each individual’s consumption of the good is precisely equal to the mass of the total supply. Mathematically,

M_T = m_1 = m_2 = … = m_i, for all i. Call this “Equation 2”. But note that there are only two values for M_T and m_i that can satisfy both Equation 1 and Equation 2: either both M_T and m_i must equal infinity, or they must both equal zero. Since there is nothing within the universe that has infinite mass (the entirety of the observable universe has merely a finite mass), then the only plausible value for the mass of a public good is zero. By Definition 3, this means that public goods are immaterial. QED.

An example of an immaterial good is an idea. Infinitely many people can simultaneously “consume” the same idea (say, that a triangle has three sides) without it diminishing in quality or quantity. The same cannot be said of any physical good.

There are two implications from this analysis: first, all physical goods are in principle private goods; and that only immaterial goods are public goods. If something is physical, i.e., if it has mass, then it is by definition rival and is, in principle, excludable (using a physical barrier).

Second, in order to economize a public good, you must make it physical somehow. The word for this is reification. The idea of a story is reified into a book. The idea of a private club is reified into a physical clubhouse. The idea of a song is reified into a physical CD or a private concert. And so on.

Dealing with Objections

There are a few common objections to my proposal.

Objection 1: What about public art, like the Mona Lisa or the Eiffel Tower? Lots of people can simultaneously consume those without decreasing the “total mass”.

My response: Light is a physical object. It reacts with other physical objects; if you obstruct a light source with your hand it creates a shadow. As such, the light reflecting from art is not uniform. Looking at the Mona Lisa from 1 meter away is a different experience than looking at it from 100 meters away; or looking at it with 1 person standing directly in front of you. The total mass of the photons making their way to your eyes is rival and excludable.

Objection 2: What about services? Haircuts do not have mass, yet we pay for them.

My response: A haircut is not an abstract concept. It involves physical components: scissors, razors, and a person. All of these things individually are rival and excludable. You are paying for exclusive right to use them all at once.

Think of services as renting durable goods, like a car. A car has many components: seats, windows, a drivetrain. Because each of those components is physical, you can pay to have the exclusive right to use them all at once. Getting a haircut is the same thing.

Objection 3: This is too physicalist and objective for economics. Consumption is subjective. Public goods are about utility: your utility doesn’t depend on how many other people are consuming the good.

My response: putting aside the fact that the vast majority of authors use physical language when describing public goods, let’s assume this objection is true. Therefore, a better definition of a public good should be that “everyone derives the same utility from a public good”, or “no individual’s utility is lowered by another individual consuming the good.” The implication is that an individual is indifferent for any positive quantity of X. Mathematically, where u_i is individual i‘s utility, and X_T’ is a different quantity of X_T, and ~ represents indifference,

u_i(X_T) ~ u_i(X_T’), for all i, X_T, and X_T’ > 0.

Note that this definition violates the non-satiety assumption of indifference curves, as the consumer is indifferent between more or less of the goods. This is also counter to the law of diminishing marginal utility. But then the question becomes: why would a consumer be indifferent to consuming 1 unit or 1 billion? Why does this special class of goods exist?

One way to answer it would be to say “subjectivity”, and leave it at that. But I find this unsatisfactory. My answer is that consumption does have an objective element to it; and it only makes sense for a person to be indifferent between all levels of more and less if there is objectively no difference between more and less. For massless goods like ideas, it makes no sense of to talk about units. How many units does the Pythagorean Theorem have? Or the story of Aladdin?

Objective 4: This is too pedantic of an interpretation of public goods. The pure non-rival, non-excludable definition is only a limiting case. It’s enough for something to have a high transaction cost, such that the total supply is less than the socially optimal supply, for it to be a public good.

My response: pedantic or not, it is the textbook definition of a public good. And the reason that economists have given up looking for examples that try to fit this definition is because they have had the wrong framework. But the distinction between physical and immaterial goods clarifies this difficulty. The fact that all physical goods are, in principle, rival and excludable, means that if a physical good is not being provided, the working hypothesis should be that either (a) there is insufficient demand for such a good; or (b) there is a government regulation that is interfering with the market.

Conclusion

Public goods, as traditionally defined in physical terms, must be immaterial by the venerable Law of Conservation of Mass. Ideas, therefore, are public goods. But all other goods that have mass, or services that require physical things, are private goods. Public goods can be made private through reification.

The reason the working definition of public goods has been watered down is because until now, economists have had an inadequate framework for analyzing public goods. If a physical good is not being supplied at the “socially optimal level”, then the most likely causes are either (a) the demand for it is weak, or (b) there is government interference in its production.

Many authors have suggested there is some sort of “spectrum” for public goods; either from “pure” to “impure” public goods, or from “pure” public goods to “pure” private goods. I suggest that in light of the above arguments, it makes more sense to think about a Venn diagram: one set representing physical goods, and another set representing immaterial goods, with no intersection between them.