Critical Mass Problems

A critical mass is considered to be the amount necessary to create a system that will sustain itself in perpetuity. Whenever one element is added to or substracted from this mass, the system begins to limitlessly deteriorate. These kinds of situations are ever present in society and the sciences.

For instance, think about an average seminar of students:

• Certain students will always participate
• Conversely, certain students will never participate
• The remainder of students may tend to participate more when others participate and less when others do not.

Below are two fairly more robust critical mass models.

The Used Car Market

It is well known that as soon as a new car is bought and driven off the lot, its value drops significantly. Walrasian economics cannot explain this phenomena. George Akerlof, however, developed a model that provides insight into this case.

The Model:

Cars can be divided into two categories and two subcategories:

• First, a car is either new or used
• Second, a car is either good or bad (a "lemon" or not)

The purchaser of a new car soon finds out whether or not his car is a lemon but, importantly, he is the only one who knows this.

Someone in the market for a used car knows that the market is comprised of both lemons and non-lemons. Knowing this, the used car purchaser seeks to bid no more than a weighted average of the price they'd pay for a lemon and the price they'd pay for a non-lemon using their best estimate of the populations for each.

Let: q_L=the estimated population of lemons p_L=price willing to pay for a lemon p_NL'=price willing to pay for non-lemon

Then,

Bid=q*p_L+(1-q)*p_NL

The important part to note is that the bid price will be lower than p_NL but higher than p_L. Owners of non-lemons will have less incentive to sell their cars while owners of lemons will have more.

The results fall out easily from here:

1. Some non-lemons owners will decide not to sell and accept a lower bid than the car is worth. This subsequently lowers the overall bid price.
2. The non-lemon owners who originally would sell observe the lower bid price and the growing lemon market and decide not to sell. This lowers the bid price.
3. This continues ad infinitum.

Broader Applications:

The Lemon Model can be generalized to markets with asymmetric information where the seller has more knowledge.

Examples include:

• Housing market
• Stock market
• Insurance

Tipping

Tipping in this model does not refer to adding gratuity to a bill after a meal (though a critical mass problem could feasibly be applied to such a situation). Rather, tipping is a term used to describe social migration of a society with two or more groups.

1960s Racial Example:

1. To start, whites dominate a given neighborhood or school district.
2. A few blacks move into the neighborhood. In response, a few whites leave.
3. The few whites who leave trigger a few more whites, who may have been fine with the white to black ratio after the few blacks initially moved in, to leave. Simultaneously, a few more blacks move into the neighborhood.
4. This trend continues until the neighborhood reaches a new equilibrium, which is either all blacks or a mix of blacks and whites who do not care about who lives around them.

Note: In this scenario, the whites are "tipping out" and the blacks are "tipping in." It is necessarily a inverse relationship.

This model can be applied on both more macro and more micro scales.

Home

---

Sources:
George A. Akerlof, The Market for "Lemons": Quality Uncertainty and the Market Mechanism, The Quarterly Journal of Economics, Vol. 84, No. 3 (Aug., 1970), pp. 488-500. Published by: The MIT Press Stable URL: http://www.jstor.org/stable/1879431

Thomas C. Schelling, Micromotives and Macrobehavior, New York: W.W. Norton & Company, 1978