Changes

A slightly simplified version of the model used now follows.

Legislators have ideal points: <math>z \backsim U \left [ - \frac{1}{2},\frac{1}{2} \right ]</math> with the median legislator's ideal point denoted <math>\, z_m = 0</math>.

:<math>\quad U \left( w , z \right ) = -\alpha(w-z) + r_w, \quad \alpha>0</math> where <math>\, \alpha</math> represents the intensity of preferences.

The Interest seeks <math>x > 0,\quad x \ge y</math> where <math>\, y</math> is the status quo and the Agenda is <math>\, A=\{x,y\}</math>.

A necessary condition for nonmarket action is that <math>\frac{(x+y)}{2} > z_m \,</math>.

We can also consider the indifferent voter <math>z_i</math> and note that this votes will be inactive if <math>z_i \le z_m </math> and active if <math>\, z_i > z_m</math>.

A legislator has an absolute-value policy plus resource contribution based utility function. That is a legislator will vote for <math>x</math> over <math>y</math> iif:

:Case 3: <math>x \le z: \quad r_x=-\alpha (x-y)</math> obtained by noting that both of the absolute values are negative and rearranging.

For simplicity consider the case where <math>z_m < y < \frac{x+y}{2} </math>. Putting these points on a line divides the line into four regions. The resource provision required to make a legislator indifferent in each region is:

Note that the legislators with ideal points <math>z_m > \frac{x+y}{2}</math> always vote for the interest's policy and there is no need to contribute resources to them. Likewise in it unnecessary to contribute to legislators below the median, at least if there is no uncertainty of types and a majority rule is in place (etc). Further more the resources needed are decreasing in <math>z</math> for <math>z \in (z_m,\frac{x+y}{2}]</math>, so interests must provide more resources to more strongly opposed legislators, and are strictly increasing in <math>x</math>.

The total resources required are:

:<math>R = \int_0^{\frac{x+y}{2}} r^*</math>

:<math>\quad r^*=\frac{\alpha}{4}\left ( x+y \right )^2 - \alpha y^2</math>

:<math> \quad r^*=\frac{\alpha}{4}\left ( x+y \right )^2</math>

Now suppose that the interest has a utility function described by:

:<math>U_g(w,z_g) = =\beta (w - z_g)^2 -R(x,y)\,, \quad</math>

where<math>z_g \ge x\,</math> is the interest's ideal point and <math>\beta > 0\,</math> is the strength of the interest's preferences.

For <math>y \le 0\,</math>, the interest will recruit votes iff:

:<math>z_g \ge z_g^+(x,y) \equiv \frac{x+y}{2} \left (1+\frac{\alpha(x+y)^2}{4 \beta (x-y)}\right) - \frac{\alpha y^2}{2 \beta (x-y)}</math>

A crucial contribution of this model is that it allows some basic comparative statics. Examination of the effects of changes in exogenous parameters for the case where <math>y \le 0\,</math> shows that: