Waste and the Value of Time

Concerning yesterday’s post, Beware Free Apples, a number of people wrote to me along the following lines, "some people have a low value of time, these people will be the ones who will be attracted to the giveaway so the time spent waiting in line is not as wasteful as you suggest."  Surprisingly, this plausible analysis is not correct or at least seriously incomplete.

To see why suppose that the giveaway were held in a poor country.  Would the waste be any less?  No.  Everyone in line would have a low value of time but for precisely this reason the waiting time would increase and the total waste would not change.

So long as there are more people with a low value of time than there are iBooks the waste will be complete.  What does make a difference is diversity.  If there are a few low value people and lots of high value people then the low value people can earn a rent.  A direct analogy is to gold mines.  If there are a lot of low cost gold mines then the price of gold is low and none earn a rent.  If there are just a few low cost gold mines and many high cost gold mines then the price of gold will be high; the marginal mine will just break even and the low cost mines will earn a rent.  To earn a rent there must be a scarcity – scarce land, scarce mines, or scarce low time-value people.

Suppose that we have a continuum of high to low-value types.  We can say immediately that "The total price for the marginal consumer will tend to rise so that it equals the marginal value of the good."  In other words, the marginal consumer will do only slightly better than if he were to buy the good at the market price (if he were to do much better then by continuity there is another consumer willing to outbid him by waiting in line a bit longer.)  Thus on the margin dissipation is complete.  What about the infra-marginal consumers?

The infra-marginal consumers will earn a rent but given some plausible assumptions about the distribution of types it’s surprising how little difference this makes to total dissipation.  I did some very basic calculations in Mathematica assuming that the value of time is Normally distributed with mean $15 and sd $4.  Under these assumptions the "first" person in line has a value of time of only $.84 per hour.  Nevertheless, the total rent dissipation is 80 percent of what it would be if everyone had a value of time of $11.63, the value of time of the marginal consumer.  Some brief experiments suggest that this sort of result if quite robust.  Specifics will depend on the exact distribution assumed.  Here is a pdf
and here is the Mathematica Notebook if anyone wants to generalize. 


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