So What Have You Been Maximizing Lately?

A while back, Brad DeLong referred to Ezra Klein’s review of Tyler Cowen’s book Discover Your Inner Economist. (Which I own but haven’t yet read; if it’s as interesting as the blog, I’m sure it will be great.) The question involves rational action in the face of substantial mark-ups on the price of wine in nice restaurants:

I did once try to convince Bob Hall at a restaurant in Palo Alto not to order wine: the fact that the wine would cost four times retail would, I said, depress me and lower my utility. Even though I wasn’t paying for it, I would still feel as though I was being cheated, and as I drank the wine that would depress me more than the wine would please me.

He had two responses: (i) “You really are crazy.” (ii) “Think, instead, that it’s coming straight out of the Hoover Institution endowment, and order two bottles.”

He is crazy, of course — crazy like an economist. I left a searingly brilliant riposte in the comment section of the post, which mysteriously never appeared. He will probably claim it was a software glitch or that I hit “Preview” instead of hitting “Post,” but I know better. What are you afraid of, Brad DeLong!?

Economists have a certain way of looking at the world, in which (to simplify quite a bit) people act rationally to maximize their utility. That sort of talk pushes physicists’ buttons, because maximizing functions is something we do all the time. I’m not deeply familiar with economics in any sense; everything I know about the subject comes from reading blogs. Any social science is much harder than physics, in the sense that constructing quantitative models that usefully describe the behavior of realistic systems is made enormously difficult by the inherent nonlinearities of human interactions. (“Ignoring friction” is the basis of 98% of physics, but nearly impossible in social sciences.) But I can’t help speculating, in a completely uninformed way, how economists could improve their modeling of human behavior. Anyone who actually knows something about economics is welcome to chime in to explain why all this is crazy (very possible), or perfectly well-known to all working economists (more likely), or good stuff that they will steal for their next paper (least likely). The freedom to speculate is what blogs are all about.

Utility is a map from the space of goods (or some space of outcomes) to the real numbers:

U: {goods} -> R

The utility function encapsulates preferences by measuring how happy I would be if I had those goods. If a set of goods A brings me greater utility than a set B, and I have to choose between them, it would be rational for me to choose A. Seems reasonable. But a number of issues arise when we put this kind of philosophy into practice. So here are those that occur to me, over the course of one plane ride across a couple of time zones.

• Utility is non-linear.

This one is so perfectly obvious that I’m sure everyone knows it; nevertheless, it’s what immediately popped into mind upon reading the wine story. We need to distinguish between two different senses of linear. One is that increasing the amount of goods leads to a proportional increase in utility: U(ax) = aU(x), where x is some collection of goods and a is a real number. Everyone really does know better than that; the notion of marginal utility captures the fact that eating five deep-fried sliders does not bring you five times the happiness that eating just one would bring you. (Likely it brings you less.)

But the other, closely related, sense of linearity is the ability to simply add together the utility associated with different kinds of goods: U(x+y) = U(x) + U(y), where x and y are different goods. In the real world, utility isn’t anything like that. It’s highly nonlinear; the presence of one good can dramatically affect the value placed on another one. I’m also pretty sure that absolutely every economist in the world must know this, and surely they use interesting non-linear utility functions when they write their microeconomics papers. But the temptation to approximate things as linear can lead, I suspect, to the kind of faulty reasoning that dissuades you from ordering wine in nice restaurants. Of course, you could have water with your meal, and then go home and have a glass of wine you bought yourself, thereby saving some money and presumably increasing your net utility. But having wine with dinner is simply a different experience than having the wine later, after you’ve returned home. There is, a physicist would say, strong coupling between the food, the wine, the atmosphere, and other aspects of the dining experience. And paying for that coupling might very well be worth it.

Physicists deal with this by working hard at isolating the correct set of variables which are (relatively) weakly-coupled, and dealing with the dynamics of those variables. It would be silly, for example, to worry about protons and neutrons if you were trying to understand chemistry — atoms and electrons are all you need. So the question is, is there an economic equivalent to the idea of an effective field theory?

• Utility is not a function of goods.

Another in the category of “surely all the economists in the world know this, but they don’t always act that way.” A classic (if tongue-in-cheek) example is provided by this proposal to cure the economic inefficiency of Halloween by giving out money instead of candy. After all, chances are small that the candy you collect will align perfectly with the candy you would most like to have. The logical conclusion of such reasoning is that nobody should ever buy a gift for anyone else; the recipient, knowing their own preferences, could always purchase equal or greater utility if they were just given the money directly.

But there is an intrinsic utility in gift-giving; we value a certain object for having received it on a special occasion from a loved one (or from a stranger while trick-or-treating), in addition to its inherent value. Now, one can try to account for this effect by introducing “having been given as a gift” as a kind of good in its own right, but that’s clearly a stopgap. Instead, it makes sense to expand the domain set on which the utility function is defined. For example, in addition to a set of goods, we include information about the path by which those goods came to us. Path-dependent utility could easily account for the difference between being given a meaningful gift and being handed the money to buy the same item ourselves. Best of all, there are clearly a number of fascinating technical problems to be solved concerning strategies for maximizing path-dependent utility. (Could we, for example, usefully approximate the space of paths by restricting attention to the tangent bundle of the space of goods?) Full employment for mathematical economists! Other interesting variables that could be added to the domain set on which utility is defined are left as exercises for the reader.

• People do not behave rationally.

This is the first objection everyone thinks of when they hear about rational-choice theory — rational behavior is a rare, precious subset of all human activity, not the norm that we should simply expect. And again, economists are perfectly aware of this, and incorporating “irrationality” into their models seems to be a growth business.

But I’d like to argue something a bit different — not simply that people don’t behave rationally, but that “rational” and “irrational” aren’t necessarily useful terms in which to think about behavior. After all, any kind of deterministic behavior — faced with equivalent circumstances, a certain person will always act the same way — can be modeled as the maximization of some function. But it might not be helpful to think of that function as utility, or as the act of maximizing it as the manifestation of rationality. If the job of science is to describe what happens in the world, then there is an empirical question about what function people go around maximizing, and figuring out that function is the beginning and end of our job. Slipping words like “rational” in there creates an impression, intentional or not, that maximizing utility is what we should be doing — a prescriptive claim rather than a descriptive one. It may, as a conceptually distinct issue, be a good thing to act in this particular way; but that’s a question of moral philosophy, not of economics.

• People don’t even behave deterministically.

If, given a set of goods (or circumstances more generally), a certain person will always act in a certain way, we can always describe such behavior as maximizing a function. But real people don’t act that way. At least, I know I don’t — when faced with a tough choice, I might go a certain way, but I can’t guarantee that I would always do the same thing if I were faced with the identical choice another hundred times. It may be that I would be a lot more deterministic if I knew everything about my microstate — the exact configuration of every neuron and chemical transmitter in my brain, if not every atom and photon — but I certainly don’t. There is an inherent randomness in decision-making, which we can choose to ascribe to the coarse-grained description that we necessarily use in talking about realistic situations, but is there one way or the other.

The upshot of which is, a full description of behavior needs to be cast not simply in terms of the most function-maximizing choice, but in a probability distribution over different choices. The evolution of such a distribution would be essentially governed by the same function (utility or whatever) that purportedly governs deterministic behavior, in the same way that the dynamics in Boltzmann’s equation is ultimately governed by Newton’s laws. The fun part is, you’d be making better use of the whole utility function, not just those special points at which it is maximized — just like the Feynman path integral established a way to make use of the entire classical action, not just those extremal points. I have no idea whether thinking in this way would be useful for addressing any real-world problems, but at the very least it should provide full employment for mathematical economists.

Okay, I bet that’s at least three or four Sveriges Riksbank Prizes in Economic Sciences in Memory of Alfred Nobel lurking in there somewhere. Get working, people!