Figure 2. Equilibrium Responses
The first condition requires that every individual choose a best response, and the second
guarantees that these responses generate the equilibrium quantity of activism. The third con-
dition ensures that at the going compensation rate the equilibrium amount of activists will be
supplied.
An equilibrium is a collection (Ah, sh, Am, sm) such that (Ah, sh) is an equilibrium response to
Am, and (Am, sm) is an equilibrium response to Ah.
The following background result sets the stage for the rest of the paper.
PROPOSITION 1. There exists a unique equilibrium.
We are going to go over the proof of existence, not for the sake of technicalities but because
the argument reveals a lot about the shape of the equilibrium response functions and therefore
about the observations that follow subsequently.
We start by showing that there is a unique equilibrium best response. So in the discussion
that initially follows, Am is simply given.
To begin with, construct a “demand curve” for activists by finding the choice of “desired” Ah
for each sh . Consult (3). Fix values of sh and Ah on the left-hand side of (3). This yields a value
for individual contributions for each type z; call it r(z, sh, Ah, Am). It is the value of r that solves
(3). Add this up over all types z to obtain an aggregate value of “desired contributions”: call
it C(sh , Ah , Am). Notice that C is strictly decreasing in Ah as long as it is positive, so there is a
unique value of Ah (for each sh) such that C(sh, Ah, Am) = shAh. We have found a point on the
“demand curve”.
Now raise sh. Because C is declining in sh, the new value of “desired” Ah — the new value
that equates C(sh, Ah, Am) and shAh — must be strictly lower, as long as it is positive to start
with.
Figure 2 superimposes this demand curve on the earlier supply curve for activists, given by
Figure 1. There is a unique intersection of the two curves, and this determines the equilibrium
response (Ah, sh) to Am.