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Baron (2001) - Theories of Strategic Nonmarket Participation (view source)
Revision as of 23:14, 28 October 2009
, 23:14, 28 October 2009no edit summary
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:
<math>-\alpha \|x-z\| + r_x \ge -\alpha \|y-z\| \quad</math> - eq(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 lhs 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_x=0</math>
<math>z \in (z_m,y] \quad r_x=\alpha (x-y)</math>
<math>z \in (y,frac{x+y}{2}] \quad r_x=2 \alpha \left (\frac{x+y}{2} + z \right )</math>
<math>z \in (frac{x+y}{2},\infty] \quad r_x=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>.
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