### economics of public goods

I'm sure lots of people more competent in economics have worked this out before, but I want to work it out for myself.

Scenario: there's something that lots of people (for simplicity, let's assume everybody in a given town) wants, which by its nature will benefit either everybody (if it's built/bought) or nobody (if it isn't). For example, a fire engine, a water treatment plant, an additional police officer, an emergency room, etc. However, the town in question is in New Hampshire, and extremely suspicious of taxes and big government, so they want to build/buy the thing with voluntary donations. Under what circumstances will it actually happen?

Let's say the desired item costs $X, and that its utility to each person in town is $Y. (Simplifying assumption: everybody in town values it the same. We'll get back to this.) At first blush, one might think it's rational for each person to donate $Y, so it'll happen iff X ≤ NY (where N is the number of potential donors). And that is indeed what would happen if

If P is the probability that "my donation will make the difference", then the rational amount to donate is $PY, so the desired item will be built/bought iff X ≤ NPY.

Now, where does P come from? Well, a lot of information could go into estimating it, but it seems reasonable to assume that it's inversely proportional to N, cancelling the N in the above inequality. Which means that it's no easier to buy a fire engine for a town of 10,000 than for a town of 100; the purchase decision depends only on the total cost and the utility to each individual. Odd.

Of course, what matters isn't the

But regardless of where P comes from, if P is (say) 1%, then the thing won't be purchased by rational voluntary donors unless its total utility is at least 100 times its cost.

Looking at things from the outside of our idealized town, we know that P=1, because everybody in town values the thing equally, so either everybody will contribute or nobody will. That's a very unrealistic assumption; to be more realistic, we would assume some sort of distribution function on the value various people place on the thing in question. Let's say Y

More realistically, different people with estimate P differently; say P

Scenario: there's something that lots of people (for simplicity, let's assume everybody in a given town) wants, which by its nature will benefit either everybody (if it's built/bought) or nobody (if it isn't). For example, a fire engine, a water treatment plant, an additional police officer, an emergency room, etc. However, the town in question is in New Hampshire, and extremely suspicious of taxes and big government, so they want to build/buy the thing with voluntary donations. Under what circumstances will it actually happen?

Let's say the desired item costs $X, and that its utility to each person in town is $Y. (Simplifying assumption: everybody in town values it the same. We'll get back to this.) At first blush, one might think it's rational for each person to donate $Y, so it'll happen iff X ≤ NY (where N is the number of potential donors). And that is indeed what would happen if

*one*person were making*one*decision: the total utility of the thing is NY, so it's rational to buy it if it costs less than that. But that's not what happens with a bunch of people making their own decisions, because people like getting something for nothing. The typical donor reasons about three possibilities:- it won't happen without my donation, and it will happen with my donation, so it's rational for me to donate $Y.
- it won't happen without my donation, but my donation won't make a difference, so I'll get nothing for my donation, so it's rational for me to donate $0.
- it will happen even without my donation, so again I get nothing for my donation, so it's rational for me to donate $0.

If P is the probability that "my donation will make the difference", then the rational amount to donate is $PY, so the desired item will be built/bought iff X ≤ NPY.

Now, where does P come from? Well, a lot of information could go into estimating it, but it seems reasonable to assume that it's inversely proportional to N, cancelling the N in the above inequality. Which means that it's no easier to buy a fire engine for a town of 10,000 than for a town of 100; the purchase decision depends only on the total cost and the utility to each individual. Odd.

Of course, what matters isn't the

*real*probability, but people's*perceptions*of the probability. And real people don't deal with very small probabilities very well, often treating them as much larger than they really are. This would mean P as perceived is proportional not to 1/N but to 1/sqrt(N) or something like that, in which case a town of 10,000*would*have an easier time raising funds for a fire engine than a town of 100.But regardless of where P comes from, if P is (say) 1%, then the thing won't be purchased by rational voluntary donors unless its total utility is at least 100 times its cost.

Looking at things from the outside of our idealized town, we know that P=1, because everybody in town values the thing equally, so either everybody will contribute or nobody will. That's a very unrealistic assumption; to be more realistic, we would assume some sort of distribution function on the value various people place on the thing in question. Let's say Y

_{k}is the value that person k places on it. Still assuming P is fixed (which doesn't make much sense), the amount raised in donations from rational people is Σ_{i=1}^{N}(P * Y_{i}) = P times N times the average value people place on it.More realistically, different people with estimate P differently; say P

_{k}is the probability assigned by person k. Now it's Σ_{i=1}^{N}(P_{i}* Y_{i}) . If people are at all sensible, P_{k}should be an increasing function of Y_{k}(since the more you donate, the more likely your donation is to make a difference).
(Deleted comment)hudebnikindividual decisions to donate. A tax is imposed oneveryonein the town (perhaps equally, perhaps in proportion to income or property value or something, but in some sense "fairly") orno-one, so there's no risk of my donation making no difference; in other words, P=1.Which means we're back to the "one decision, one decision-maker" situation, and the total utility only needs to be greater than the cost in order for it to happen. In other words, doing this through general taxation leads to a Pareto-optimal allocation of resources, and doing it through voluntary contributions doesn't. (In case anybody didn't notice, the previous sentence is heresy.)

Consider a unionized workforce and a non-unionized workforce. The former makes a single decision (do we accept the contract or not?), while the latter makes a bunch of independent decisions (do I accept this offer or not?). In this case the "desirable good" (from the workers' perspective) is some hoped-for employer concession (safer working conditions, longer lunch hours, etc.). The likelihood (P) of any one employee persuading the employer to offer this concession "or I'll quit" is very small, but the employer might well offer a concession in order to keep the entire workforce. Again, the total utility of the concession to the workers may be large, but if each one negotiates individually, and few or none of them insist on it, it won't happen. Again, the union negotiation leads to a Pareto-optimal decision (do it iff the utility exceeds the cost), and the bunch of individual negotiations don't.

osewalrusThere are several possible reasons for this, some of which are testable. For example, if we assumed that in a smaller community individuals are better able to assess he value and to police free riders, then we would expect more such projects to succeed in smaller towns. Alternatively, I might derive a disproportionate benefit and therefore contribute more. For example, do people who have suffered from fires contribute more to buy fire equipment because they disproportionately value the availability of such equipment? Alternatively, does the sale of "naming rights" work because the purchaser of the naming right values the advertising?

hudebnikthe morepeople donate,the betterthe results will be, rather thanif enoughpeople donate, then suddenly the results will be very good. In other words, they persuade people that the probability P of their donation making a (perhaps small, but measurable) difference is large.Your point about small communities policing free riders is important too; I was consciously leaving that out.

And yes, one assumes that people who value the public good more highly are more likely to donate, and at higher levels.