An ongoing debate in my hometown just hit the "newspaper" again, provoking another round of arguments over the city's decision to install an expensive work of public art next to the newly renovated City Hall. The sculpture ended up costing the city $750,000, and many people are taken aback by the price. (There's a smaller sub-issue caused by about 20% of the money going to a German design artists, but I won't address that here.)
A few of the anonymous comments on the story provide a good picture of the main complaint:
Benjamin: "This is a testimony to the condescending city government and their attitude toward the hard working citizens of Ann Arbor. They reduce our services (fire/police, leaf pick-up, parks etc.) and then they waste our money on this "art". How in the world was there room in the budget for this ahead of other city services?"
f4phantomll: "How many potholes can you fill for $750,000? How many cops or firefighters can you rehire for $250,000? Sheesh."
GoBlue: "I'm embarrassed and ashamed. What does this say about us as a city? Our economy is in shambles, what is crucial is ignored (bridge for example), key safety elements (police and fire) being reduced and council uses our tax dollars for this and could not even use local artists, to add insult to injury. Our city has sunk to new lows."
AlwaysAmazed: "How many hungry children and homeless families could this money have helped? You've got to be kidding me!"
grye: "We should put one of these on every street corner, then get rid of all the city employees, police officers, firemen, shut down all the utilities, no garbage pickup. Get rid of everything so we can sit on our collective behinds and admire expensive art. Who needs anything else? And with such art and no city services, millions will flock to this city to enjoy the art and wish they could move here because it is such a wonderful place.
Is our mayor and city council ever going to get it?"
mun: "$750,000 could have paid for how many police officers, how many fire fighters, how many school buses?"
The root of all of these complaints is that the money spent on this project could have been put to better use in other ways: that is, there are other ways the money could have been spent that would give us a higher ratio of utility to cost.
The problem with this kind of simplistic cost-benefit analysis, though, is that it implicitly assumes that the utility of a certain scalable choice is linear in the chosen magnitude of the choice. That is, if we are calculating the utility of hiring x firefighters, the assumption seems to be that U(x) = mx. That translates to each additional firefighter giving us m units of utility. We could have a similar equation for the utility of spending $y on public art, and have that U'(y) = m'x. (Throughout this discussion I'll make the simplifying assumption that the money is being spent most efficiently on each project independently: the best firefighters are being hired for their salary, and the best art is being purchased for its cost. Altering that assumption would lead to a different, though also interesting, issue.)
On that model, it makes sense to ask, like 'mun' did, how many firefighters could have been hired for the cost of the public art. If we had that number, we could do a pretty simple comparison between U(x) (with x being the number of firefighters) and U'(750,000). Whichever utility is greater is where we should spend our money.
The problem is that this is a horribly inaccurate approximation of what our utility functions should look like. This approximation not only leads to slightly incorrect answers (as is inevitable with any approximation), but it actually causes us to ask the wrong question. To see why, consider the following simple example:
You have $2, and you go to a nearby store, that sells two items: loaves of bread and bottles of water. Each loaf and each bottle cost $2, and for whatever reason saving the money isn't an option. It's logical to assume that, faced with this choice an individual will perform an approximate evaluation of the utilities of a loaf of bread versus a bottle of water, and pick the one that gives a higher utility. Assume that, for some underlying reason, that ends up being the loaf of bread.
Now, hold everything in the example constant, but assume that you actually have $4. The linear utility model tells us that you'll buy two loaves of bread — but does that make sense? Of course not. The right answer is that we don't know what you'll do. It could well be that you're extremely hungry and not at all thirsty, so two loaves of bread is more attractive than one loaf and one bottle of water. But maybe you were somewhat hungry and also a bit thirsty. If you can only buy one thing, you'll get the loaf...but that one loaf is enough to satiate you, and so with the second $2 you'll buy a bottle of water. The reason is that the utility of the second loaf of bread is not equal to the utility of the first loaf of bread. As you get more food, additional food has a much smaller marginal benefit.
Returning to the debate over city services versus public art, it's reasonable to think that both of those will have a utility function that follows a similar style to the one at right. A city with no investment in public art, for instance, can see a marked increase in quality of living from a relatively small amount of investment in improving the appearance of the area. Once the city looks nice, however, additional installations will have less and less of an effect.
Similar logic applies to hiring firefighters. A city with no firefighters is basically screwed, so hiring that first team is very, very useful. But given that there are only a certain number of people that can effectively help in any given emergency, the increases in utility fall off as you hire more and more firefighters. The first team gives you a certain percentage chance of putting out a fire. The second team gives you (probably) the same percentage chance of putting out a fire, but only if two fires happen at once. So on and so forth.
After that excessive explanation, we can finally return to the angry local commentators. Why, they moan, are we paying for this public art when we could hire more firefighters? The city obviously didn't compare the value of a single art installation to that of x firefighters.
They probably didn't, but such a comparison would also be the wrong one — or, at best, a nonsensical one. 'AlwaysAmazed' shouldn't be interested in just how many families could be fed; he should be interested in how many families could be fed and how many already are. 'mun' shouldn't be concerned with a straight translation between the amount of money spent and the cost of buses, firefighters, or police. He should be concerned with that, and how many buses, firefighters, and police officers Ann Arbor already has. Correct cost-benefit analysis can't compare different choices for marginal spending decisions without also considering the current state of the world. If the current levels of spending were actually as irrelevant as the angry people would have you believe from their questions, then we would end up spending all of our money on that single, perfect, utility maximizing project.