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*Bernheim, D. and M. Whinston (1986), Menu Auctions, Resource Allocation, and Economic Influence, Quarterly Journal of Economics 101(1), 1-32. [http://www.edegan.com/pdfs/Bernheim%20Whinston%20(1986)%20-%20Menu%20Auctions%20Resource%20Allocation%20and%20Economic%20Influence.pdf pdf]
Baron (2001) - Theories of Strategic Nonmarket Participation (view source)
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, 19:14, 29 September 2020no edit summary
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|Has article title=Theories of Strategic Nonmarket Participation
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*This page is part of a series under [[PHDBA279A]]
*This page is referenced in [[BPP Field Exam Papers]]
==Reference(s)==
Uses:
*Grossman, G. and E. Helpman (1994), Protection for Sale, American Economic Review 84, 833-50. [http://www.edegan.com/pdfs/Grossman%20Helpman%20(1994)%20-%20Protection%20for%20Sale.pdf pdf]
==Introduction==
A slightly simplified version of the model used now follows.
===Legislators and Interests===
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>.
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 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>.
===Resource Provision===
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. ===Making Legislators Indifferent===
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>.
===Total Resources===
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> ===Recruiting Votes===
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> ===Comparative Statics===
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>.
It is also possible to calculate when vote recruitment becomes too costly all together for the interest. This is covered in some detail in the paper, but loosely if <math>x > x^*(z_g,0)=\frac{8 \beta z_g}{4 \beta + \alpha}\,</math>, then the cost exceeds the gain.
==Interest group competition with an executive institution==
The following model is essential a two principal (the "Interests"), single agent (the "Executive"), common knowledge agency model. It is a direct application of the Bernheim and Whinston (1986) model, as implemented by Grossman and Helpman (1994).
===Interests (Principals) and the Executive (Agent)===
There are two interests <math>j = \{g,h\}\,</math> with ideal points <math>z_g > 0, \, z_h < 0</math> and support costs <math>c_j(x)\,</math>. The interests have additively seperable utility functions with an intensity factor <math>\beta_j\,</math>:
:<math>U_j=u_j(x)-c_j(x) \quad</math> where <math>u_j(x)=-\beta_j(x-z_j)^2\;\,\beta_j>0</math>
The executive choses a policy <math>x \in \mathbb{R}\,</math> and has an additively seperable utility function:
:<math>U_e=u_e(x)+c_g(x)+c_h(x)\quad</math>where <math>u_e(x)\,</math> represents the executives own policy preferences.
The status quo policy is taken to be <math>x>0\,</math> and the sequence of the game is a simultaneous move on support schedules by the principals followed by a choice of policy by the executive. The principals make menu offers, that is they state a resource contribution for each potential policy outcome, and these offers are binding. Once the executive has made the policy choice the contributions are transfered from the principals according the menu value of the chosen policy.
It should be noted that as a result of additive seperability in the utility functions with respect to the contributions, and that both principal(s) and agent value the seperable contribution identically, the contributions are transfers and the agent will maximize the joint surplus - this is proved below. Having two principals introduces competition which favors the agent in an enter/don't enter prisoner's dilemma game that occurs before this game and allows the agent to extract rents from both principals; however, we could correctly determing the outcome of this game by using a single representative principal whose utility function is the sum of the two principals, and then computing a standard principal-agent model.
===Conditions for an SPNE===
Baron (or Rui) define equilibrium as a triple <math>c_{g}^{\ast}(x), c_{h}^{\ast}(x), x^{\ast})</math> is defined as:
* <math>x^{\ast}\in\arg\max_{x}[u_{e}(x)+c_{g}^{\ast}(x)+c_{h}^{\ast}(x)]</math>.
* <math>c_{j}^{\ast}\in\arg\max_{c_{j}}[-\beta[x^{\ast}-z_{j}]^{2}-c_{j}^{\ast}(x^{\ast})]m j=g,h</math>.
* <math>c_{j}=\tau_{j}+u_{j}(x), j=h,g</math>, "Truth Telling."
Bernheim and Whinston (1984) provide four necessary and sufficient conditions for a sub-game perfect Nash equilibrium in this model. In the notion of the model, these are:
<math>\{c_j^*(x),x\}</math> is an SPNE iff:
:a) <math>c_j^*(x)\,</math> is feasible
:b) <math>x^*\,</math>maximizes<math>\quad u_e(x) + c_g^*(x) + c_h^*(x)</math>
:c) <math>\forall j (i \ne j) x^*\,</math>maximizes <math> \quad \{u_e(x) + c_j^*(x) + c_i^*(x)\} + \{u_j(x) - c_j^*(x)\} = \{u_e(x) + u_j(x) + c_i^*(x)\} </math>
:d) <math>\forall j (i \ne j) \exists x_j\,</math>s.t.<math>\quad u_e(x)+c_j^*(x)+c_i^*(x)\;</math> where <math>c_J^*(x) = 0 \quad \therefore x_j\,</math> maxes <math>\quad u_e(x) + c_i^*(x)\;</math>
===Local Truth Telling===
The first order conditions of (b) and (c) taken together imply:
:<math>c_J^*\prime(x^*) = u_J^*\prime(x^*)\quad</math>and therefore that the contribution schedules are locally truthful around <math>x^*</math>.
===Linear Contribution Schedules===
The contribution schedules are constructed as linear functions of the utility of the principals, specifically:
:<math>c_J^\tau(x,\tau_j) = max\{0,u_j(x) + \tau_j\}\quad</math> Note that the linear term is added not subtracted, but that it may be negative.
===Joint Surplus Maximization===
The principal's utility maximization and conditions (a) and (b) imply:
:<math>U_e(x^*) \ge U_e(x) \quad \forall x \in \mathbb{R}</math>
:<math>u_e(x^*) + c_g^*(x^*) + c_h^*(x^*) \ge u_e(x) + c_g^*(x) + c_h^*(x) \quad\forall x \in \mathbb{R}</math>
When both agents contribute we can substitute in the linear contribution schedules to get:
:<math>u_e(x^*) + u_g^*(x^*) + \tau_g + u_h^*(x^*) + \tau_h \ge u_e(x) + u_g^*(x) + \tau_g + u_h^*(x) + \tau_h \quad \forall x \in \mathbb{R}</math>
:<math>\therefore u_e(x^*) + u_g^*(x^*) + u_h^*(x^*) \ge u_e(x) + u_g^*(x) + u_h^*(x) \quad \forall x \in \mathbb{R}</math>
As <math>\widehat{U_e}(x) = u_e(x) + u_g(x) + u_h(x) \quad</math>, the principal's problem is to maximize the joint surplus.
===Solving For Linear Contribution Terms===
Using (d) we can note that if player <math>i</math> doesn't contribute then the agent choses:
:<math>x_j \in \arg \max u_e(x) + u_j(x)\quad</math>
Comparing this to the equilibrium where both players contribute and noting that for the agent <math>x^* \succsim x_g\,</math> and <math>x^* \succsim x_h\;</math>, and so it must be the case that <math>x_g , x_h\,</math> are off the equilibrium path.
Therefore the agent will choose <math>x^*\,</math> iff:
:<math>u_e(x^*) + c_g^*(x^*) + c_h^*(x^*) \ge u_e(x_g) + c_g^*(x_g) \quad</math> and likewise for <math>h</math>
We can solve for the other player's <math>\tau</math> by setting the inequality exact:
:<math>u_e(x^*) + c_g^*(x^*) + c_h^*(x^*) = u_e(x_g) + c_g^*(x_g) \,</math>
:<math>u_e(x^*) + u_g^*(x^*) + u_h^*(x^*) + \tau_g + \tau_h = u_e(x_g) + u_g^*(x_g) + \tau_g\,</math>
:<math>u_e(x^*) + u_g^*(x^*) + u_h^*(x^*) + \tau_h = u_e(x_g) + u_g^*(x_g)\,</math>
:<math> \tau_h = u_e(x_g) + u_g^*(x_g) - (u_e(x^*) + u_g^*(x^*) + u_h^*(x^*) )\,</math>
Likewise::<math>\tau_g = u_e(x_h) + u_h^*(x_h) - (u_e(x^*) + u_g^*(x^*) + u_h^*(x^*) )\,</math><math></math><math></math>
