Lecture 23: Public Goods and Common Resources

In the last class we analyzed the problem of externalities: situations in which one agent’s consumption or production of a good impacted someone else. For example, producing steel can generate a negative externality in the form of pollution, or planting flowers in your front yard can generate a positive externality by improving the quality of life in your neighborhood. In this context, market equilibrium – in which every agent equates their private marginal benefits and marginal costs – will fail to reach the point at which the social marginal benefit is just offset by the social marginal cost.

However, up to now we have been concerned only with one type of good – namely, private goods. A private good is one which has two key attributes:

• It is rival: one person consuming the good impinges on others’ ability to consume the same good. For example, an apple is rival because one person eating it prevents another person from eating it. It’s just physics; two people can’t eat the same apple.
• It is excludable: one can legally (and technically) be prevented from consuming the good. An apple is excludable because property rights are well established over apples. If I buy an apple, it’s my apple; you’re not allowed to take my apple, that would be stealing.

To a certain extent, goods with externalities start to push the boundaries of these definitions. If you plant flowers in your front yard, it’s not that your consumption of the flowers has the positive externality; it’s that the flowers themselves are something of a nonrival good, insofar as they may be enjoyed by others. (If you put up a fence so that nobody else could see them, then you enjoying your flowers would no longer have the positive external effect.) Conversely, one way of thinking about the steel mill polluting a river, which then impacts a fishery, is that the river itself is a rival good, but one that the steel mill uses at zero cost.

Good that are either nonrival or nonexcludable may be categorized in three ways:

Club goods

Club goods are excludable but not rival: for example, think about the case of satellite radio.

• Satellite radio is nonrival because one person listening to it doesn’t prevent another person from listening to it as well: the satellite signal is broadcast all over the planet, regardless of how many people are receiving it.
• However, satellite radio is excludable because you need an active code to receive the signal. A satellite company can encrypt the signal, and only give the ability to decrypt it to people who pay for their service.

Common resources

Common resources are rival but not excludable: for example, think about a small fishing pond on public land.

• Let’s assume the pond is rival because the more people fish on the pond, the less the fish are able to reproduce – so one person fishing reduces the fish caught by everyone else.
• However, the pond is nonexcludable because it’s on public land. Everyone is allowed to go fish in the pond.

Public goods

Public goods are neither rival nor excludable. Standard examples of public goods include national defense or, more locally, a fireworks show in a public park.

• A fireworks show is nonrival because one person watching the show doesn’t prevent someone else from watching it as well.
• It’s also nonexcludable if it’s held in a public park where everyone is allowed to go.

Caveats

None of the above categories has a hard boundary, and indeed the same good may be rival in some circumstances and nonrival in others: for example, a freeway in Los Angeles is generally nonrival at 4am when there’s no traffic, but very much rival at 4pm when there’s gridlock. Likewise, even some public parks have gates and controlled entry, making them excludable at certain times of the day. But thinking about rivalry and excludability is a useful starting point for analyzing how the nature of a good can result in predictably inefficient outcomes.

With that said: let’s now look three canonical models that illustrate the fundamental problems around public goods and common resources.

• We’ll start by looking at the “tragedy of the commons,” which delves into why common resources may be overused
• We’ll then look at a discrete choice of whether to provide a public good
• Finally, we’ll look at a continuous choice of how much of a public good to provide

In each of these cases, we’ll also consider how the “public” might go about collectively choosing how to solve the problem – and how the nature of collective choice leads to problems of its own.

Model 1: The Tragedy of the Commons

A common resource is rival but non-excludable. Because it’s rival, one person’s use of a common resource affects others’ enjoyment of it. But because it’s non-excludable, there is a risk that too many people make use of it, resulting in a worse outcome for everyone.

A classic example of the tragedy of the commons is the problem of overfishing. Sustainable fishing means leaving enough fish to reproduce, so that fish stocks can remain constant or grow from season to season. If too many fish are caught, especially before they reach the age of reproduction, a fishery can collapse.

Consider a town of 35 people. Each person can choose to fish in the town lake or to hunt in the nearby forest. If $L$ people choose to fish, the total number of fish caught on the lake is given by $f(L) = 40L - L^2$.

Suppose that fish sell for $€10$ each, and each person who hunts always gets $€100$ of meat.

Efficient use of the commons

What is the optimal division of labor in this town? Total revenue, as a function of the number of fisherpeople $L$, is given by \(TR(L) = \overbrace{40L - L^2}^\text{total fish} \times \overbrace{10}^\text{price per fish} + \overbrace{(35 - L)}^\text{number of hunters} \times \overbrace{100}^\text{each hunter's payoff}\) or \(TR(L) = \overbrace{400L - 10L^2}^\text{total revenue from fish} + \overbrace{3,500 - 100L}^\text{total revenue from hunting}\) The change in profit when society devotes another person to fishing rather than hunting is \(MR(L) = \overbrace{400 - 20L}^\text{marginal revenue from fish} - \overbrace{100}^\text{opportunity cost from one less hunter}\) Setting this equal to zero yields $L = 15$. Under this “social planner’s optimum,” the total number of fish caught will be \(f(15)=40L-L^2 = 600 - 225 = 375\) and total revenue from the lake will be $375 \times 10 = 3,750$. Since 15 people will fish, the other 20 will choose to hunt and earn 100 each, so total revenue for the town will be 5,750.

Competitive Equilibrium

Suppose each fisherperson who chooses to fish on the lake receives the average amount of fish caught; so fishing has a payoff to each fisherperson of \(\pi_F(L) = AR(L) = \frac{(40L - L^2)\text{ total fish}\times €10\text{ per fish}}{L\text{ fisherpeople}} = 400 - 10L\) It’s easy to see that the more people fish on the lake, the fewer fish are caught by each person. If people can freely choose whether to fish or hunt, the equilibrium payoff to fishing ($\pi_F$, above) and hunting ($\pi_L = 100$) must be the same – otherwise, people would switch from the less-profitable task to the more-profitable one. In this “free entry” equilibrium, therefore, we would have \(\begin{aligned}\pi_F &= \pi_H\\\\ 400 - 10L &= 100\\\\ L &= 30\end{aligned}\) Hence the total number of fish caught will be \(f(30)=40L-L^2 = 1200 - 900 = 300\) and total revenue from the lake will be $300 \times 10 = 3,000$. Since 30 people will choose to fish, the other 5 will choose to hunt and earn 100 each, so total revenue for the town will be 3,500 – much less than before!

Analysis of the Problem

What’s happening here? Each person who fishes exerts a negative externality on the others who are already fishing, by decreasing the average catch for everyone by 10. When there are 15 people fishing, therefore, each fisherperson gets a total benefit of $400 - 10\times 15 = 250$, but the marginal fisherperson generates a negative externality of $€150$; so the total benefit of 250 is balanced by the opportunity cost of 100 (what they could have gotten by hunting) and the marginal external cost of 150. By contrast, the 30th person to choose fishing earns a private benefit of 100, which just offsets their opportunity cost of hunting; but also generates a whopping $€300$ negative externality on their fellow fisherpeople; so the 30th fisherperson actually decreases the town’s overall revenue by $€300$!

Potential Solutions

One solution for this problem is to have fishing permits. Suppose the village issued fifteen fishing permits and sold them for price $c$. In this case, the benefit to fishing would be \(\pi_F(L|c) = 400 - 10L - c\) As we found before, when $L = 15$ each fisherperson would earn a private benefit of $€250$ from fishing; so they would be willing to pay at most $\overline c = 150$ for the permit.

An alternative solution would be to have a sort of Pigovian tax. In this case, the market price of fish would be $p - t$, so if fish sold for $€10$ each, we would have \(\pi_F(L|t) = (40 - L)\times(10 - t) = 400 - 10L - (40-L)t\) Again, at the optimal number of $L = 15$, this would be equal to $250 - 25t$; equating this to the opportunity cost of hunting, we would therefore need to set a tax of $t = 6$.

Political problems and public choice

The above analysis is perhaps a bit too sunny. For example, we blithely asserted that the village could issue the right number of permits, and sell them for a price $c$. How does the village know for sure how many permits is the right number? Who should get the permits? And what’s likely to ensure that they pay the correct amount $c$?

In reality, when you artificially constrain the number of people who can fish, you create something of value: a fishing permit. How you distribute those permits can raise some thorny issues. History is rife with local leaders distributing things of value – not only permits, but also things like government contracts – to friends and allies, and freezing out opponents. If a single person is in charge of distributing fishing permits, for example, it can create a situation in which resources are expended to curry favor with that individual, leading to political corruption. So an economic solution – creating scarcity where there was too little scarcity before – can easily create a political problem.

Even voting on how to allocate the permits could raise potential problems. But to see how voting itself can be a problem, let’s turn to our next topic: the provision of a public good.

Model 2: When to provide a public good

There are two types choices around public goods we might consider: a discrete (either/or) choice, like whether to build a community park; and a continuous choice, like how long of a fireworks show to put on in that community park.

Let’s start with a discrete choice, and to keep things simple, we’ll consider the smallest possible society: just three housemates, named Amy, Boris, and Carlos. They are considering buying a washer/dryer that costs €1,200. Once bought, they will all be able to use it freely: so within the context of their house, the washer/dryer will be a public good.

Their individual willingness to pay is:

Since the total willingness to pay for the washer/dryer is €350 + €150 + €800 = €1,300, which is greater than the €1,200 cost, it’s efficient for them to buy the good. That’s because there is a possible split of the cost which would make buying the washer/dryer a Pareto improvement over the status quo: for example, suppose Amy pays €325, Boris pays €125, and Carlos pays €750. Then they’re all better off:

An interesting question that might arise is whether the three roommates would, in fact, buy the washer/dryer? This depends on the rules they’ve established for social choice: that is, how they as a society make economics choices that affect them all.

Suppose that when they all moved in together, the three roommates had agreed that any expenses would be split evenly, and that all decisions on what to buy would be made by majority rule. This seems like a fairly innocuous and straightforward way of doing things, but we can immediately see there’s a problem: if they split the cost of the €1,200 washer/dryer evenly, each would have to pay €400 – but only one of them (Carlos) values it at €400 or more! Since Amy and Boris would vote against the purchase, the public good wouldn’t be provided, even though it would be efficient for them to do so.

Lest you think that this only works in one direction, suppose their valuations of the good were as follows:

Now, their total willingness to pay is just €1,100, which is less than the €1,200 price, so it’s inefficient for them to buy the washer/dryer. However, two of the three now value it at more than €400 – so a majority would vote to provide the good!

Majority rule and the median voter theorem

The above analysis is actually an example of an important theorem in public choice theory, which lies at the intersection of economics and political science: the median voter theorem. The Wikipedia article on this is actually excellent, and has the following definition:

If voters and candidates are distributed along a one-dimensional spectrum and voters have single-peaked preferences, any voting method that is compatible with majority-rule will elect the candidate preferred by the median voter.

There’s a lot to unpack there – more than we have time for in this class – but in this case, note that Amy has the median valuation of the washer/dryer in each case. Since in either scenario, Boris isn’t willing to pay the €400 for his share, and Carlos is willing to pay €400, whoever Amy sides with will get two out of three votes. So, if Amy values the washer/dryer at less than €400, they won’t buy it, while if she values it at more than €400, they will – in each case, regardless of whether it’s actually efficient for them to do so.

Model 3: How much of a public good to provide

While some public goods, like roommates buying a washer/dryer, present an “either/or” question, other public goods present a “how much” question: for example, how many teachers should a public school hire, or how much should a country spend on its military?

To model this, let’s think of a simple quantity decision: how many minutes of fireworks to have in a town fireworks show. Let $G$ be the total quantity of the public good provided – that is, the total number of minutes of fireworks. Let’s assume that each minute of fireworks costs 200 dollars.

Suppose that everyone in a town has preferences over the length of the fireworks show $G$ and their own private consumption $x_i$, measured in dollars, is given by \(u_i(G,x_i) = a_i \sqrt{G} + x_i\) where $a_i$ is a personal preference parameter. Note that this is a quasilinear utility function, so we can think of this as saying that the total benefit of the good for person $i$, measured in dollars, is \(TB_i(G) = a_i \sqrt{G}\) Therefore their marginal benefit from another minute of fireworks – i.e., their MRS between fireworks and dollars of private consumption – is \(MB_i(G) = {a_i \over 2 \sqrt{G}}\) To be slightly more specific: let’s assume there are two types of people:

• There are $N_1 = 100$ people who really like fireworks: $a_1 = 12$, so their total benefit from a fireworks show of length $G$ is $TB_1(G) = 12 \sqrt{G}$.
• There are $N_2 = 200$ people who kinda love fireworks: $a_2 = 4$, so their total benefit from a fireworks show of length $G$ is $TB_2(G) = 4 \sqrt{G}$.
• Finally, assume that each minute of fireworks costs €200.

Now, if fireworks were a private good, each consumer would set their own marginal benefit equal to the marginal cost they would have to pay (or equivalently, set their MRS equal to the price ratio), and the market demand would be the horizontal summation of each individual demand curve. In such a case, for every price, we would add up the amount demanded by each person to get the total quantity demanded in the market:

But suppose fireworks cost €200 per minute – then no individual would have enough individual benefit to buy even a single minute of fireworks!

However, remember that each minute of fireworks brings utility to all the people in the town. So a fireworks show of length $G$ brings a total social benefit of the town of \(\begin{aligned} TSB(G) &= N_1 \times TB_1(G) + N_2 \times TB_2(G)\\ &= 100 \times 12\sqrt{G} + 200 \times 4\sqrt{G} \end{aligned}\) Therefore the marginal social benefit of another minute of fireworks would be \(MSB(G) = 100 \times {6 \over \sqrt{G}} + 200 \times {2 \over \sqrt{G}} = {1000 \over \sqrt{G}}\) Note that this is the sum of the individual marginal rates of substitution of all the people in the town: in other words, the willingness of the town as a whole to pay for fireworks is the sum of each individual’s willingness to pay. Therefore, instead of horizontally summing individual demand curves, as we would for a private good, we find the overall marginal benefit by vertically summing the individual MB’s:

We’re now in a position to solve for the optimal quantity. As with externalities, the socially optimal length of a fireworks show is the one that sets marginal social benefit equal to the marginal cost: in this case, €200 per minute. Hence, the value of $G$ which maximizes social welfare sets \(\underbrace{100 \times {6 \over \sqrt{G}} + 200 \times {2 \over \sqrt{G}}}_\text{Marginal social benefit} = \underbrace{200}_\text{MC}\) which solves to $G^\star = 25$, as you can see if you drag the quantity in the graph above to 25.

More generally, if $N$ is the set of all individuals who will enjoy the public good, and each individual $i \in N$ has a marginal rate of substitution $MRS_i(G)$ between the public good and dollars spent on their own private consumption, then the optimal quantity of the public good is established by equating \(\sum_{i \in N} MB_i(G) = MC(G)\) This is known as the Samuelson condition. ⊕ A technical note: this is only the Samuelson condition for the case of quasilinear utility; more generally, we would use the MRS, not the MB. But for our purposes this is sufficient. Essentially, it’s the same as the roommates situation, in that the town should continue buying fireworks as long as the total marginal benefit to the town is greater than the marginal cost.

Public choice

In each of the stories above, we calculated the efficient outcome: the three roommates should buy the washer/dryer, and the town should have a 25-minute firework show. And, if the world were run by all-knowing economists who were universally trusted by everyone to make all their decisions for them, we might be done.

We do not, however, live in such a world.

Let’s turn now to the fireworks show. Suppose we assume that each citizen has €100, and that of this citizen $i$ contributes some amount $g_i \ge 0$ to the fireworks show. Therefore, the amount of money they have left over for private consumption is \(x_i = 100 - g_i\) So we could in fact write their utility function over the length of the fireworks show, $G$, and their own contribution $g_i$, as \(u_i(G,g_i) = a_i \sqrt{G} + 100 - g_i\) Let’s think about a few mechanisms for funding the show.

Voluntary contributions

If contributing were voluntary, how much would each citizen contribute to the fireworks show? Depressingly, the answer is: usually, nothing. To see why, put yourself in the shoes of someone who really loves fireworks, and assume the rest of the town had contributed €200, enough for a one-minute fireworks show. How much money should you contribute? Well, if you contributed $g_i$, the total contributions would be $200 + g_i$, so the length of the fireworks show would be \(G = 1 + {g_i \over 200}\) and your utility, as a function of $g_i$, would be \(u_i(g_i) = a_i \sqrt {1 + {g_i \over 200}} + 100 - g_i\) If you were to take the derivative of this, set it equal to zero, and solve for the optimal contribution, you would get \(g_i^\star = {a_i^2 \over 800} - 200\) You would have to have $a_i > 400$ to even contribute a penny! And in this town, even the people who really loved fireworks only had $a_i = 12$. So relying on voluntary contributions isn’t going to get us very far.

This is known as the “free rider” problem: if you can enjoy the show without paying for it, and especially if your individual contribution would have a negligent effect on the quantity of the good provided, you face a strong incentive to contribute nothing and just “free ride” off of everyone else’s contribution.

Paying a tax based on how much you like fireworks

Now suppose that, having realized that relying on everyone to contribute voluntarily would result in no fireworks show at all, they propose that everyone pay their fair share for a 25-minute fireworks show. That show would cost €5000, and bring a total benefit of $TB_1(25) = 12\sqrt{5} = 60$ to each person who really loves fireworks, and a total benefit of $TB_2(25) = 4\sqrt{5} = 20$ to each person who kinda likes fireworks.

One possible way to raise the €5000 would be to charge everyone half their valuation: that is, charge the 100 people who love fireworks €30 each, and the 200 who kinda like fireworks €10 each. This would fully fund the fireworks show, and be fair insofar as everyone is paying half their valuation.

…but would it work? Even if you know that 1/3 of the people love fireworks, does the “government” know who they are? And what would each person say, if asked which “type” they are? Since the amount each individual gives doesn’t affect the overall total very much, everyone would have an incentive to portray themselves as someone who only kinda likes fireworks. And if everyone contributes just €10, they’d only get a 15-minute show.

This illustrates a fundamental problem with public choice: how to get people to truthfully reveal how much they value something. We’ll get into this a bit more in Econ 51, but if you’re interested in this kind of problem I highly recommend taking Paul Milgrom’s Econ 136 course.

Voting on a tax rate

Having abandoned the idea of voluntary contributions, and then having given up on people paying in accordance with how much they value the good, the town decides that everyone will pay the same amount $g$, and that everyone should vote on what that amount is.

Here we can get into what the median voter theorem means when it talks about “single-peaked preferences.” Let’s say that the payoff to a citizen with preference parameter $a_i$ who has to contribute $g_i$ is their total benefit minus their contribution: that is, \(u_i(G,g_i) = a_i\sqrt{G} - g_i\) Now, if all $N = 300$ citizens contribute some amount $g$, the total amount raised for fireworks would be $Ng = 300g$; and since fireworks cost $p = 200$ per minute, the length of the fireworks show would be \(G(g) = {Ng \over p} = {300g \over 200} = {3g \over 2} \text{ minutes}\) Therefore, preferences over $g$ for citizen $i$ are \(u_i(g) = a_i\sqrt{3g \over 2} - g\) This reaches its maximum when $u_i^\prime(g) = 0$: \(\begin{aligned} {a_i \over 2}\sqrt{3 \over 2g} - 1 &= 0\\ a_i \sqrt{3 \over 8} &= \sqrt{g}\\ g^\star(a_i) &= {3a^2 \over 8} \end{aligned}\) If we plot this out, we can see that each citizen has a “peak” at their own personal preferred policy $g^\star$:

You can use the slider to confirm that for those who really love fireworks $(a_i = 12)$, their preferred contribution is $g^\star = 54$, meaning each citizen would contribute €54, and the fireworks show would be 81 minutes long! On the other hand, for those who only kinda like fireworks, their preferred contribution is $g^\star = 6$, implying a 9-minute fireworks show.

If the town were to vote on this, the median voter theorem says that the preferences of the person with the median preferences would prevail; in this case, since there are more people who kinda like fireworks than those who really love fireworks, a candidate suggesting a 9-minute fireworks show and a low tax rate of $g = 6$ per person would win in a landslide.

It’s worth noting, though, that this isn’t the optimal length of 25 minutes we found before! That’s because the social optimum takes everyone’s preferences into account, while the preferences of the median voter are just that – the preferences of the median voter.

Arrow’s Impossibility Theorem

So, we’ve shown that problems of public goods and common resources are solvable – but not necessarily by majority rule. OK, you say, but majority rule is only one among many ways a society could make a collective choice. Maybe some other mechanism could do better?

The answer is one of the most depressing, tantalizing, and infuriating results in all of economics:

No.

But this has been enough reading for today, so we’ll talk about that in class. But if you’re curious, search up Arrow’s Impossibility Theorem…

Conclusion

We spent the first seven weeks of this class looking at how individual agents (consumers and firms) make decisions that only affect them. We spent weeks 8 and 9 looking at how markets converge to equilibrium, and how that equilibrium may or may not be efficient. In this lecture we’ve looked at a third kind of decision: decisions made by collectives about things that affect all of them. In doing so, we got a glimpse into the thorny problem of public choice.

While the view of the efficiency of markets we got a few weeks ago was comforting, this analysis is as frustrating as it is important. We need to be able to make choices as a society – even the most capitalistic and libertarian societies have some form of governance. But making those choices is hard in any world where all the members of the society aren’t on the same page about absolutely everything. In general, it will be impossible to find a first-best solution: that is, there is no mechanism which will guarantee that the collective body always makes the most efficient choice. So, we have to settle for two main takeaways:

• In public goods problems, there generally is a solution available to an omniscient social planner
• In public choice problems, collective decision making won’t generally get you there

Next time, we’ll wrap up by showing one last reason we might not get to efficiency: market power.

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