Caveat Archos

One of the interesting trends in economic history over the last 40 years or so consists in the rise and fall of the union. For example, in America we have seen the percentage of the population that belongs to a union fall from 25% to around 10% between 1970 and today, while the percentage was even higher pre-1970. In countries such as Sweden, however, we’ve seen exactly the opposite trend: levels of 60% to nearly 90% over the same time period. To a certain extent legal institutions have contributed to these numbers, but can we model the dynamics of union density in any way that may enlighten us as to the underlying psychological reasons behind its divergence? That is, within firms how do people react to union membership?

This was part of my first problem set for my Microeconomics course with Sam Bowles:

1. Suppose the costs of being a union member are represented by c,  1>c>0
2. Suppose the benefits of being a union member are represented by “b” and these benefits are increasing in the number of union members
3. Suppose “d” represents union density (normalized so 1>d>0)
4. Finally, y, 1>y>0  is the strength of conformist feeling, or a “solidarity coefficient”

Now we can model the utility of a union member as U_m = b - c + y(d-0.5) and the utility of a nonmember as U_n = b + y(0.5-d).

For example, the utility of a member is derived from benefits minus costs (b-c) plus the “solidarity benefits” of a member IF d>0.5. We are assuming here that being around a lot of nonmembers will be uncomfortable for the member and may induce him to switch (and vice versa). The utility of a nonmember is derived from benefits (and no costs, of course) plus the “solidarity” or “conformity” benefits of not having many members around, i.e. IF d<0.5.

Now we want to model dynamics. So assume that members and nonmembers occasionally meet and talk about being in a union or not being in a union, and through this conversation they start to compare utilities.  The probability that a member will meet a nonmember is simply d(1-d) and based on this probability they will compare utilities and evaluate (U_m-U_n). Hence we can say that over any two periods, the change in the amount of members will be governed by the equation:

(change in d) = d(1-d)(U_m-U_n) = d(1-d)(-c +2y(d-0.5))

It would be nice to know if there are any times there will be no change in d, that is, (change in d)=0. This occurs at d=1, d=0, and d=(c+y)/(2y). So, two values of d that will cause no change are when there are all members and all nonmembers. If we are at either of those points, is it possible to move away from them given, say, a few people in an all-member group decide nonmembership is best, or vice versa? Can this happen?

This question is equivalent to asking if the equilibrium points above are stable. We know from calculus that equilibrium points will be stable if the derivative of the function at that point is negative.  Taking the derivative of (change in d) and evaluating it at d=0, d=1 we obtain:

if d=0, y+c>0 will give stability. but both y and c were assumed to be greater than zero, so this will always hold.

if d=1, y>c. y is the strength of conformity or “solidarity” while c are the costs of being a member. So basically what this is saying is that complete membership will be stable if and only if the strength of conformity outweighs the costs of being a member.

While the whole story is obviously not being told here, it does give some interesting insight into worker behavior through a more psychological perspective of why people join unions.  We took an evolutionary approach by modeling how d would change if some people met others and talked about membership/nonmembership and found that we could shed some light on union density.