24.2 Efficient Provision of a Public Good

Suppose Amy, Boris, and Carlos are housemates. They are considering buying a washer/dryer that costs $1,200. Once bought, they will all be able to use it: 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:

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.

Calculating social benefit from the public good

Suppose each individual’s benefit from $G$ minutes of fireworks, in terms of dollars, may be given by the utility function (Note: Another way to think of this is that they have a quasilinear utility function \(u(G,xrparen = 10G - \frac{1}{2}G^2 + x\)where $x$ is dollars spent on private consumption.) \(u_i(G) = 10G - \frac{1}{2}G^2\) Therefore their marginal benefit from another minute of fireworks – i.e., their MRS between fireworks and dollars of private consumption – is \(MRS_i = 10 - G\) Now, if fireworks were a private good, each consumer would set their own MRS equal to the price ratio, and the market demand would be the horizontal summation of each individual demand curve. But suppose fireworks cost 200 dollars 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 of the people in the town. So if there are 100 people in the town, the total social benefit to would be \(TSB(G) = 100u(G) = 1000G - 50G^2\) and the marginal social benefit of another minute of fireworks would be \(MSB(G) = 1000 - 100G\) 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 MRS’s.

Solving for the optimal quantity

We’re not quite done: to find the optimal quantity, we need to balance the benefits and costs of buying fireworks. Total social wefare may be given by \(\begin{aligned}W(G) &= TSB(G) - TSC(G)\\ &= 1000G - 50G^2 - 200G\end{aligned}\) To maximize this, we take the derivative and set it equal to zero, effectively equating the marginal social benefit and marginal social cost of fireworks: \(\begin{aligned} W^\prime(G) = 1000 - 100G - 200 &= 0\\1000 - 100G &= 200\\100G &= 800 \\G^\star &= 8 \end{aligned}\) 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 \(\sum_{i \in N} MRS_i(G) = MC(G)\) This is known as the Samuelson Condition. Essentially, it’s the same as the the roommates situation, in that the town should continue buying fireworks as long as the total marginal benenefit to the town is greater than the marginal cost.