260 N. Brooks J. of Economic Behavior Org. 44 2001 249–267
Even if the distribution of returns to trade improves i.e. the upper bound on w increases it will not necessarily be possible to support the interior equilibrium unless there is some
coordination among the individuals, since of course no single individual would have an incentive to unilaterally return to cooperative behavior when everyone else is deviating.
This might help explain why we might see two neighborhoods with similar economic characteristics but very different behavior patterns.
4. Comparative statics
This section considers what happens to the stable interior equilibrium when there are small changes in the distribution of returns to cooperative behavior w
i
.
Proposition 2. If ¯ w increases then the equilibrium w
∗
decreases at a decreasing rate, all else constant.
Proof. The derivative of the equilibrium solution of w
∗
as given in footnote 17 with respect to ¯
w is negative while the second derivative is positive. Figs. 1 and 2, which give the percentage of deviants in equilibrium, illustrate this rela-
tionship for a given set of parameters. This implies that as the distribution of returns to trade gets worse more individuals will choose to be “deviant” and at an increasing rate.
In these graphs, the lower bound on the distribution of the returns to trade has been as- sumed to be zero. This implies that as ¯
w is increasing both the average return and the variance of the returns are increasing. It is important to separate these two effects to determine which
effect is driving the comparative static results.
Fig. 2. Percentage of deviants.
N. Brooks J. of Economic Behavior Org. 44 2001 249–267 261
Proposition 3. If the average w
i
decreases, ceteris paribus, then the equilibrium percentage of deviant behavior will increase at an increasing rate. If the variance of the returns is
increased holding the average constant, then the equilibrium percentage of deviant behavior will increase at a decreasing rate.
Proof. To hold the variance constant substitute w = ¯ w − variance into the equilibrium
solution of w
∗
as given in footnote 17. The derivative of this with respect to ¯ w is negative
while the second derivative is positive. Similarly, to hold the average return constant substitute w = 2average w
i
− ¯ w into the
equilibrium solution of w
∗
. The derivative of this with respect to ¯ w is positive while the
second derivative is negative. When the average return in the community increases the interior stable equilibrium critical
return w
∗
decreases and thus the equilibrium percentage of deviants also decreases. It is important to note that the equilibrium percentage of deviants decreases non-linearly, thus
there will be a lower overall level of deviance if instead of having one community with a high average return and one community with a low average return there are two communities
with an average return in the middle. If the variance of the returns increases the equilibrium percentage of deviants increases at a decreasing rate.
These comparative static results of the stable interior equilibrium illustrate two points. First, the equilibrium percentage of deviant behavior decreases in response to a increase in
the average return to trade. This is consistent with evidence of the negative effect on behavior of concentrated poverty. Of particular interest is the non-linearity of this relationship, which
implies that the overall level of deviance can be reduced if different neighborhoods have a more equal income distribution. The second result illustrates the increase in the equilibrium
percentage of deviant behavior due to an increase in the mean preserving spread of returns. This implies that the equilibrium level of deviant behavior is a function of income inequality.
5. Discussion of theoretical results
