Tuesday, 15 April 2014

The Alchian-Allen Theorem Holds for Randomizing Irrational Actors

In a recent blog post commemorating the 100th anniversary of the birth of Armen Alchian, I discussed his Theorem with William Allen. In this post, I would like to connect to the Theorem some insights on irrational behaviour explored by Gary Becker in the early 1960's.
A big theme with Armen Alchian was to assume extremely little doing economics. Alchian believed people to be rational; for instance, on page 437 of Exchange and Production (1983), he and William Allen posit that wars occur because "people are reasonable and act in accord with their interest". But if he could avoid assuming rational behaviour, he did not hesitate to do so. So one might say that it was in Alchianesque fashion that Gary Becker showed that agents need not be rational for demand curves to slope downward. For example*, if people simply randomize, demand curves will retain the familiar slope.
Assume as Becker did that consumers make random purchasing decisions. I have added transportation costs on the graph below, which pull the budget line towards the origin and change its slope to approach -1. I have marked the mid-points on the pre- and post-transportation cost budget lines, so that - if random behaviour results in consumers distributing themselves uniformly along the lines - the mid-point is both the mean and the median in the distributions. It is evident from just eye-balling the graph that the proportion of the high-quality good increases when transportation costs are added; the dashed lines reveal the average quantities demanded before and after transportation costs were added, and while low-quality goods were clearly most popular before, the shares are more nearly equal after (click to enlarge).

Random behaviour played a big theme in Alchian's famous article 'Uncertainty, Evolution and Economic Theory', so I wanted to highlight the connection with the Theorem here.

*"For example", because Becker also showed that if consumers are characterized by inertia - so that they would simply pick the shortest route to the new budget line from their point on the old budget line - demand curves retain their traditional slope, too. However, for this sort of irrational behaviour, the Alchian-Allen Theorem will not hold, although this point is irrelevant since there was no price less transportation costs with an associated quantity demanded from which consumers receiving exported goods could move inertly.

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