The Efficient Provision of Public Goods. Explanation and Overview

We've been discussing how public goods can lead to a market failure because of the free-rider problem which basically says that why would people want to pay for a public good if they believe it's going to be provided anyways and there's no way to exclude them from enjoying that good. So the question is if we're going to have the government come in if the government is going to play the role and provide the public good and then tax people or something so for it to fund it, the question is well what is the efficient quantity, what is this socially optimal the socially efficient quantity of that public good? In question how do we determine? Let's say for example that we're talking about streetlights, so we're thinking of installing some streetlights in a neighborhood. We're trying to figure out how many should we set up, what is the socially efficient number of streetlights. Well, the rule is that we want to do the number of streetlights where the marginal social benefit is equal to the marginal social cost.

Now you might be thinking hey we've got an issue here, people don't buy public goods. They're not like private goods it's not like we're just going to ask people in the neighborhood "Why don't you go out and buy your own streetlights" So how do we figure out what is the total marginal social benefit? How do we figure that out and what we could do? One way we could do it is we could basically ask each individual and figure out hypothetically what would be the marginal benefit to each individual. How much would they pay for one additional unit one additional streetlight under different scenarios? So say if the quantity is "X" how much would person A pay for one additional streetlight and so then we could put them together once we know that the preferences of the people we can put together a demand curve for each person who lives in that neighborhood. Then we can sum those demand curves together what's basically called the collective demand curve and then we map out that collective demand curve and that's going to be the marginal social benefit and then where that's equal to the marginal social cost, where these two curves intersect that's going to be the efficient quantity of the public good in question so for example streetlights

So let's just say I just want to plot this out for you so it's a little less abstract. We're using the example of streetlights and so let's say these numbers are in thousands of dollars here I have no idea how much it is for a streetlight but let's say it's thousands of dollars. Let's say that the marginal cost curve, we have two marginal cost curves and we've got two people in this neighborhood. So we've got person 1 their demand curve is this orange line and now I'll show you how I've plotted it out for you. So at a quantity of 0, at the origin there this person their marginal benefit from an additional streetlight would be 6. Now if they were at let's say we're at a quantity of 1 then person 1 their marginal benefit of an additional streetlight would be 5 and so that's how I plotted out all these lines here. I just basically went and said "What would be the marginal benefit at different quantities." and when you get to where there are 6 streetlights at that point for person 1 the marginal benefit of inner street light is zero, so what we've got person 2 here is the red line.

So to come up with the collective demand curve that's like to think about the total marginal benefit for all of society of this society of two people here this neighborhood. What we do is we add each together so that might look a little hard to do but if you think about it let's go with zero quantity, at zero we add the marginal benefit of each person. So for person 2 it would be 3 is the marginal benefit and then for person 1 it would be 6, so (6 + 3) is going to give us 9. So our first point will be 9. When we have zero streetlights the marginal benefit of providing additional streetlights is 9. Now for 1 streetlight, the marginal benefit would be (2 + 5) which would be the total, so that would be 7. Then if we were at 2 the marginal benefit of one more would be, 1 for person 2 and then 4 for person 1. So (4 + 1) is 5. So we will be at point 5. Now when we get to 3 at this point gets a little weird because person 2 no longer has any marginal benefit now. So it's basically the curve is just going to come in with person 1 because the total marginal benefit is just person 1 now which is 3. Then with 4 streetlights, the marginal benefit would be 2, and then with 5 streetlights, it would be 1. When we get 6 streetlights, it would be 0. Neither of these two people is going to get any more marginal benefit from an additional streetlight. So we can go and we can map this out. This new curve is the collective demand curve and you could think of that as the marginal benefit.

How do we determine the optimal socially efficient level of this public good of streetlights? So we're going to do it where the total marginal benefit equals the marginal cost. So the marginal benefit equals the marginal cost that's our socially efficient level and so we see we got our marginal cost curve and then here's our marginal benefit curve also known as the collective demand curve. So where the two curves intersect that's where marginal benefit equals marginal cost and we say "Well, what is the quantity there?" the quantity at that point is 2. So what that means is that the socially efficient amount of streetlights, the amount where the collective marginal benefit equals the marginal cost for this neighborhood would be 2. So that means that the socially efficient level of streetlights would be 2.