We had substantive posts this week on two of our recurring topics — economic efficiency and the foundations or arithmetic.

The former brought us the honor of an extended visit from Uwe Reinhardt, who, as far as I can tell, objects not to the concept of efficiency or to its usefulness, but to its name. But any crusade to change a well-established technical term is, I think, doomed to failure.

Efficiency, of course, is only one of the normative criteria in the economist’s arsenal. I pointed, for example, to an earlier post where I’d outlined a toy framework for evaluating some of the normative claims made by one of Professor Reinhardt’s Princeton colleagues. That toy framework employs a utilitarian criterion that goes beyond efficiency. It evaluates policies on the basis of “what an amnesiac would prefer”, which is very different than a pure efficiency criterion. This kind of analysis is perfectly standard in economics, so any allegation that we fixate exclusively on efficiency is a bum rap.

On the other hand, some fixation on efficiency can be an extremely valuable exercise, for reasons that I hope this week’s post made clear.

Re the foundations of arithmetic, I posted to dismiss the view that the natural numbers are fictitious. As one commenter pointed out, this was largely an attack on a straw man, because almost nobody believes otherwise. Indeed it was. This was intended as an educational post, not a contentious one, and attacking straw men can be a very effective form of education. When I teach students about continuous functions, I ask them to imagine a hostile party who insists that the function f(x) = x is not continuous, and we talk about how you could most effectively convince him otherwise. The hostile party is imaginary, but there’s a lot to be learned from thinking about how you’d refute him.

We also speculated on the defining idea of the next decade and the ideal reading list for a course on how economists view the world.

And then there was the probability problem: A woman has two children, one of whom is a boy born on a Tuesday. What is the probability they’re both boys? Several commenters explained the answer very clearly. In case you haven’t read the comments and don’t want me to give away the answer, I’ll just say that it’s greater than 45% but less than 49%. See the comments on the original post for the reason why.

We’re coming up on a long weekend, and I’m taking Labor Day off. I’ll see you Tuesday.

If there’s one thing I wish everybody understood about economics, it’s that wise resource allocation requires truly vast amounts of information, and that prices do an excellent job of summarizing that information. We led off the week by applying this principle to grocery shopping. A rather silly column in the New York Times had seemed to suggest that socially responsible shoppers should care about the energy costs of producing vegetables to the exclusion of all the other costs. The column was focusing, in other words, on the seen as opposed to the unseen. But the unseen costs of growing a tomato in one location rather than another are just as important as the obvious ones, and because they are unseen (and unseeable) the only feasible way to account for them is to look at prices. We followed up with a 25 year old application of exactly the same principle, this time to the problem of resource extraction.

We moved on to the perils of interpreting data, in this case with regard to the ingredients of a happy marriage. Then a look back to what the world of 1985 thought would constitue a marvelous future; we seem to have met expectations pretty well. And finally, we came in a sense full circle — from lamenting those focus single-mindedly on energy costs to the exclusion of all else to lamenting those who fault others for failing to focus single-mindedly on one political issue to the exclusion of all others.

I’ll be back next week with some thoughts on why we should care about economic efficiency, a little more on the foundations of arithmetic, and some surprises.

It was a week of mathematics here at The Big Questions. I am still reeling from the momentous events that inspired Monday’s post; we now know that the Internet has changed mathematics forever. On Friday, we celebrated the momentous achievenments of the new Fields Medalists.

In between, we began what will be an occasional series on the foundations of arithmetic. In Part I, we distinguished truth from provability. In Part II, we distinguished theories (that is, systems of axioms) from models (that is, the mathematical structures that the theories are intended to describe). A theory is a map; a model is the territory. In Part III we talked about consistency and stressed that it applies only to theories, not to models. A purported map of Nebraska can be inconsistent; Nebraska itself can’t be.

It turns out (a little surprisingly) that any consistent map must describe multiple territories. (That is, any consistent set of axioms must describe many mathematical structures — or in other words, any consistent theory must have many models.) (This assumes the map has enough detail to let us talk about addition and multiplication.) These territories—i.e. these mathematical structures, all look very different, even though they all conform to the map. Conclusion: No map can fully describe the territory. No set of axioms can fully describe the natural numbers.

I’ll continue this series sporadically, and eventually we’ll get into some controversial philosophical questions. So far we haven’t.

Speaking of controversy, I’ve increased the default font size for this blog. Tell me if you like it.

More posts than usual this week as I was motivated twice to add a mid-day post to my usual morning fare. As a result, I’m afraid Jeff Poggi’s remarkable sonnet to Darwin got less attention than it should have; I hope you’ll go back, read it, and spot the hidden Darwin references.

The mid-day posts were motivated by a pair of (in my opinion, of course) outrages — first Paul Krugman’s suggestion that if we control for education and a few other demographic factors, we can make a meaningful comparison of private and public sector wages, ignoring all the ways in which public and private sector jobs differ. (And ignoring, too, all the ways in which one college degree might differ from another.) I suggested that a better metric is the quit rate in each sector; some commenters rightfully pointed out that that’s also an insufficient statistic. I bet it still comes a lot closer than Krugman’s attempt, though.

The second outrage was the Administration’s willingness to act as the equivalent of a Mafia enforcer for firms who prefer not to compete with foreign labor. Some commenters asked how this differed from any other case of the American government enforcing American laws while asking the beneficiaries to contribute to the costs. That’s easy. This law, unlike, say, the laws against murder, has as its primary purpose the restraint of trade (as opposed to oh, say, the general welfare).

We talked about how to estimate the peak of the Laffer curve (answer—it’s at about the 70% marginal tax rate, though I indicated some reasons why it might be somewhat leftward of that), mused about the value of a good CEO, and gave new meaning to the phrase phone sex when we reported on the fact that iPhone users have many more lifetime sex partners than Android users.

Incidentally, those readers who thought the flashy iPhone pays off in the mating market can’t be right (or at least can’t have hit on the key story), because the effect holds even for 40 year olds, who surely did not acquire their iPhones until long after they’d acquired most of their sex partners.

And we noted in passing the announcement of a proof that P does not equal NP (where you can look here for a very rough idea of what this means). Over the course of the week, this developed into a story of, I think, monumental significance, which I will surely revisit early next week. See you then.