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Baron (2001) - Theories of Strategic Nonmarket Participation (view source)
Revision as of 22:26, 29 October 2009
, 22:26, 29 October 2009→Vote Recruitment in Client Politics
The utility function of legislators is additively-seperable with a term representing their constituent's preferences and a term for the resources provided to them by the client:
:<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 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:
:<math>-\alpha \left|x-z\right| + r_x \ge -\alpha \left|y-z\right| \quadqquad</math> - eqequation (1)
Note that a legislator votes on his (using male for the agent) induced preferences, not on whether they are pivotal. However, in equilibrium the pivotal votes are recruited.
The resources that must be provided to a legislator to swing his vote (essentially <math>U(y,z)-U(x,z)</math>) are calculated according to equation (1) above for three different cases (locations of z).
:Case 1: <math>z \le y: \quad r_x=\alpha (x-y)</math> obtained by noting that <math>x \ge y</math> and that both of the absolute values are positive and rearranging.
:Case 2: <math>y \le z \le x: \quad r_x= 2 \alpha \left (\frac{x+y}{2} + z \right )</math> obtained by noting that the LHS absolute value in equation (1) is positive, whereas the RHS value is negative.
: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:
:<math>z \in [-\infty,z_m]\,, \quad r^*=0</math>
:<math>z \in (z_m,y]\,, \quad r^*=\alpha (x-y)</math>
:<math>z \in (y,\frac{x+y}{2}]\,, \quad r^*=2 \alpha \left (\frac{x+y}{2} + z \right )</math>
:<math>z \in (\frac{x+y}{2},\infty]\,, \quad r^*=0</math>
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>
In the case where <math>y > 0\,, <math>::<math>\quad r^*=\frac{\alpha}{4}\left ( x+y \right )^2 - \alpha y^2</math>
In the case where <math>y \le 0\,, <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>-\Beta (x - z_g)^2 - \frac{\alpha}{4}\left ( x+y \right )^2 \ge \beta (y - z_g)^2</math>
:Or: <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)</math>
Therefore if the agenda is exogenous, the interest will recruit if and only if <math>z_g</math> is to the right of the midpoint by the recruitment factor <math>\frac{\alpha(x+y)^2}{4 \beta (x-y)}</math>; that is the interest must be extreme in its interests by this factor to undertake recruitment.
Likewise one can calculate the upper limit of the range <math>z_g^+\,</math> for when <math>y > 0\,</math>. The interest will then 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:
:<math>z_g^-(x,y)\,</math> is strictly decreasing in <math>\beta\,</math>: With more intense interests there is a smaller centralist set.
:<math>z_g^-(x,y)\,</math> is strictly increasing in <math>\alpha\,</math>: With more intense legislator preferences there is a larger centralist set.
:<math>z_g^-(x,y)\,</math> is strictly increasing in <math>x\,</math>: A more extreme alternative leads moderate interests not to try to change the policy.
Also as <math>x \uparrow</math> the #votes recruited <math>\downarrow</math>, and as <math>x \uparrow</math> the cost of recruiting a vote <math>\uparrow</math>.
