## Tuesday, April 26, 2011

### The Alchian-Allen controversy

Down in the Bahamas, a discussion got started in the hotel room with Jesse and Kyle regarding an example used to illustrate the Alchian-Allen theorem in Walter Williams’s class. The Alchian-Allen theorem states that if a flat cost is added to any two goods, one low-quality and one high-quality, people will choose more of the high-quality goods relative to the low-quality ones. The flat cost added to each good makes the number of low-quality goods that one must give up to get the high quality good lower, lowering the relative cost of the high-quality good.

The example that Walter Williams gave in class was that one would expect that couples with children would be more likely to go on nice dates to the theater than a cheap one to the movies, because the couples with children must pay a flat fee to a baby-sitter regardless of their choice of date.

The objection was raised in class (by Sam, I believe) that the baby-sitter should not enter into the decision making calculus because at the point where the couple makes the decision of which date to go on the babysitter is a sunk cost. The couple with children, therefore, would be just as likely to choose the movie over the theater as the couple without children.

Walter Williams, then, was wrong, if the problem is framed as it is above. William framed it in such a way that he would be right – that there is a sorting effect due to the flat cost which makes the couple who wants to go to the theater more likely to pay for a babysitter, with the end result being that you see more couples with children at the theater than at the movies. There’s also the possibility that both prof. Boettke and Charity-Joy suggested, that couples make the choice of a high-quality or a low-quality date before hiring the baby-sitter, and don’t change their plan once they make the choice. Which saves Prof. Williams’s example as well.

That misses the main point of controversy though, which is this: if someone makes a plan to do one thing over an alternative because of a flat cost involved, do they change their plans if they pay the flat fee before making the choice between alternatives?

Option 1: Once the babysitter cost is paid, the couple with children will choose the same way they would have if they were childless. The A-A effect does not apply, except potentially in the selection effect.

Option 2: (my argument) At the point in time after the babysitter is paid but before the couple actually goes to either the theater or the movie, the couple still takes into account the flat cost because going to either the movie or the theater requires that the flat cost be paid again. So the cost, after the baby-sitter is paid, of going to the theater is the price of the theater ticket and the foregone movie, and the cost of going to the movie is the price of tickets and the foregone theater trip. The value of both the foregone theater trip and the foregone movie include the price of the babysitter because either choice in the future necessitates the hiring of a babysitter. Thus, even once the babysitter is paid this time, the movie and the theater still cost the same.

My intuition comes from the claim that on a trip to Maine, tourists will more likely choose expensive lobster than cheap lobster because in order to get expensive Maine lobster ever again one has to take a trip to Maine, so even though the current trip to Maine is a sunk cost the A-A effect still applies.

I’m interested in getting this resolved so that I can stop thinking about it.