Rationing By Racing

Using price theory to explain a land rush

One of the critical insights of price theory is that there is always another margin on which people can optimize. This point is important. A common view of non-economists is that markets aren’t always the appropriate means of allocating resources. Economists, however, are trained to ask the question: compared to what? A price theorist recognizes that when a money price is precluded from allocating resources, some other allocation mechanism will emerge. People will optimize on another margin.

As I have written about previously, Yoram Barzel developed a theory of rationing by waiting. The basic idea is that sometimes price is precluded from being used to allocate resources. One alternative to using prices is to allocate through a first-come, first-served mechanism. Under these conditions, people might show up and form a line in order to obtain the good. A natural question that we might ask is whether price theory can tell us anything about the resulting allocation.

The answer is yes. People have a willingness to pay for a good. The fact that a money price is precluded from allocating the good doesn’t change this. If the good is allocated on a first-come, first-served basis, then the willingness to pay can be measured in terms of time. If a person is willing to pay $50 for the good and their wage is $25 per hour, then they should be willing to wait in line for two hours. The length of the line and thus how long a given person will have to wait will be determined by the willingness to pay (measured in terms of time) of the marginal buyer.

Things can get complicated here when it comes to the distribution of the final allocation since people generally have different costs associated with their time. The distribution of the good might change depending on the mechanism. For example, we might get a different allocation of the good when it is rationed by time than when money prices are posted. For example, someone with a high willingness to pay, but also a high opportunity cost of time might be willing to wait less time than someone who had a lower willingness to pay in terms of money prices but a low opportunity cost of time.

Secondary markets can provide an additional complication. Those with a high willingness to pay in terms of money, but with a high opportunity cost of their time might hire people with a low opportunity cost of time to wait in line on their behalf.

Of course, a line isn’t the only way to ration in this scenario. Rationing with a line suggests that the good being purchased is something where access to the good in limited in some way. An example might be something like gasoline. One arrives at the gas station and there are a limited number of gas pumps. Since one cannot put gas into his or her car without the use of the gas pump, a line must form to allocate the gasoline. When price ceilings on gasoline were imposed, lines at gas stations were frequently observed.

However, first-come, first-served looks differently if access to the good is not limited in some way such that the desired good can simply be claimed upon arrival. An example of this would be land. If land was to be allocated on a first-come, first-served basis, the first person to show up on a particular plot of land would be able to claim that land. Subsequent arrivals would have to claim some other plot of land or might find no land left to claim.

Generally speaking, we don’t allocate land in a first-come, first-served manner. Typically, there is an owner of the land and property rights are transferred within standard market interactions. But there was a time in the U.S. when property was, in fact, allocated this way. I’m referring, of course, to homesteading.

Beginning with the Homestead Act of 1862, the U.S. federal government encouraged westward migration by allowing people to claim federal land as their own. U.S. citizens could claim 160 acre plots of land in exchange for living on and cultivating the land for at least 5 years (there was also a filing fee).

In this case, the first-come, first-served looks much less like waiting in line and much more like a race. In a race, people optimize on speed. A race on foot will be won by the fastest runner. However, people need not be limited to running. Those with the highest willingness to pay would be expected to invest in capital to get them to their destination faster.

Barzel’s framework can nonetheless help us to understand the allocation mechanism. People have a willingness to pay for this land. However, the land must simply be claimed and maintained. The consumer surplus of the claimant is therefore equal to the willingness to pay for the land less the cost of investments in capital necessary to generate enough speed to get there first. If the process is competitive, we should assume that the consumer surplus is zero for the marginal claimant. Put differently, the price of the land is the cost of acquiring speed for the marginal buyer.

This generates testable predictions. Higher quality land has greater value. As a result, people should be wiling to spend more resources to produce the speed necessary to get to the land first. Thus, one prediction is that the land with the highest value should be claimed the fastest.

At the same time, since allocation is based on the speed with which people arrive at their desired plot of land, the cost of acquiring speed is crucial. The lower the cost of acquiring speed, all else equal, the quicker that land should be claimed.

Doug Allen and Bryan Leonard set out to test the predictions of Barzel’s model for the Oklahoma land rush. To do so, they first separate the different plots into two groups: the three areas in Oklahoma with the highest quality land and the three areas in Oklahoma with the lowest quality land. They then estimate survival functions that measure the probability that land is still available by a particular date. When comparing the survival functions, they find that the survival function for the higher quality land lies below the survival function for the lower quality land. This suggests that the probability that the higher quality land was still available by a particular date was below the probability that the lower quality land was still available for all possible dates. In other words, the higher quality land was claimed faster than the low quality land.

Their subsequent regression analysis shows that land with better soil quality, more rainfall, and flatter land all reduced the amount of time before the land was claimed. This confirms that higher quality land was claimed sooner.

Land closer to the “starting point” of the race and land closer to railroads also led to land being taken sooner. Both shorter distances and the railroad reduce the cost of the “race” to claim the land. Thus, one would expect that people would reach this land faster and claim it. These results confirm the prediction that the lower the cost of speed acquisition, the faster the land will be claimed.

Overall, the key lesson here is that one can remove price from the allocation process, but some other allocation mechanism will emerge. Sometimes that allocation mechanism might resemble waiting in line. At other times, the allocation mechanism might resemble a race. Despite the absence of price, price theory will still help us to determine how goods are allocated. We can use the same tools that we typically use to study how prices allocate resources to similarly determine how long people will be willing to wait in line and how much people will be willing to invest to acquire speed. No matter the mechanism, price theory can be your guide.