Damian Dovarganes/Associated Press

Have I tragically underestimated the ability of foreign cellphone companies to come up with devilishly complex pricing plans and the skill of economists to convert those schemes into pat theories that can be illustrated with neat charts and graphs?

The many people who have written me over the last two days about my Sunday article on the method behind the madness of cellphone pricing say I have.

The article began quoting Barry Nalebuff, a Yale economist, saying cellphone pricing plans were “weird.” Alex Kaufman, a doctoral candidate in economics at Harvard, wrote to chastise me for my “characterization of economists as throwing up their hands in confusion over this problem.”

Indeed, he pointed me to the work of Michael Grubb, now a professor at M.I.T.’s Sloan School of Business. I had spoken to Mr. Grubb for the article, in fact, on the recommendation of Mr. Nalebuff. His research seemed interesting, but a bit too involved to describe in the article, given how many other topics I had to cover.

Mr. Grubb told me that as a grad student he too was perplexed by pricing plans with big bundles of included minutes and high charges for extra minutes, what economists call three-part tariffs. After investigation, he wrote a paper arguing that phone companies were exploiting overconfidence by consumers about their ability to predict how many minutes they would use. He summarized this in an e-mail message to me:

Three-part tariffs optimally exploit overconfident consumers because overconfident consumers both underestimate the likelihood of very high usage, and the need to pay high overage charges, and underestimate the likelihood of very low usage, and the likelihood of not getting a refund for included “free” minutes.

Consumers certainly do make bad predictions about their own behavior. Studies show that people buy gym memberships, rather than paying per gym use, because they believe they will work out more than they really do. And when I wrote about the credit card industry, I saw studies that showed people thought they paid off their bills in full, and thus avoided interest charges, far more often than they actually did.

That’s not the only factor here, though. Andrew M. Odlyzko, a mathematics professor at the University of Minnesota, wrote me with a very interesting paper he co-wrote in 1979 which argued that consumers often preferred flat-rate pricing to paying by usage even if it would save money. It cited data from phone companies that showed half the people who chose local phone plans with unlimited calling would have saved money choosing a pay-per-call plan. The paper listed three reasons:

(i)Insurance: It provides protection against sudden large bills. (What happens if my son comes back from college,and starts talking to his girlfriend around the clock?)

(ii) Overestimate of usage: Customers typically overestimate how much they use a service, with the ratio of their estimate to actual usage following a log-normal distribution.

(iii) Hassle factor: In a per-use situation, consumers keep worrying whether each call is worth the money it costs, and it has been observed that their usage goes down. A flat-rate plan allows them not to worry as to whether that call to their in-laws is really worth $0.05 per minute.

The second factor there matches up with Mr. Grubb’s idea that consumers are overconfident of their ability to predict how to save money. The cellphone executives I spoke to believed that the desire for stable prices, encouraged by their pain-inducing overage charges, was a better explanation for consumer behavior.

On another topic, several readers outside of the United States chastised me for oversimplifying cellphone pricing structures in the rest of the developed world. (And I didn’t even talk about emerging economies like India, where wireless service is much cheaper, mainly because of lower costs.)

One important difference is that in most other countries, the person who calls a cellphone pays the bill. Receiving wireless calls is free. This seems very attractive to consumers, especially when rates are high, and it helped explain the rapid deployment of cellphones in Europe and Asia.

But as it has developed, the caller-pays system has inhibited the use of cellphones. The rate to call a cellphone in many countries can be quite high and can vary depending on the carrier of the person you are calling (which you may not know). That means that calling one cellphone from another can lead to nasty charges. This, some say, explains why people in caller-pays countries talk on cellphones far less than people in the United States.

The article also didn’t make it clear that carriers in many countries offer subsidized handsets for people willing to commit to a service contract. Indeed, such plans are becoming more common in many countries as people try to afford expensive smartphones. But it is also true that it is far, far more common in many countries for people to buy phones at retail stores and then to buy prepaid blocks of minutes from carriers as they need them.

Spend a minute looking at the various options available in Britain, and you wouldn’t say that things are any less complicated there. Here is the page, from Carphone Warehouse, a large retailer, for the iPhone 3GS and for the BlackBerry Curve 8520.

One footnote I left out: A few people asked for citations for the effective average price of 5 cents a minute and 1 cent a text message I cited. The voice statistic came from the CTIA, a wireless industry trade group. The text figure came from a study of monthly cellphone bills conducted by Nielsen.