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And so the we have both that (as <math>U_e^\hat(x) = u_e(x) + u_g(x) + u_h(x) \quad</math>) the principal's problem is to maximize the joint surplus and that the contribution schedules are truthful.==Solving For Linear Contribution Terms===
Baron (2001) - Theories of Strategic Nonmarket Participation (view source)
Revision as of 19:14, 29 September 2020
, 19:14, 29 September 2020no edit summary
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|Has page=Baron (2001) - Theories of Strategic Nonmarket Participation
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|Has article title=Theories of Strategic Nonmarket Participation
|Has author=Baron
|Has year=2001
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*This page is part of a series under [[PHDBA279A]]
*This page is referenced in [[BPP Field Exam Papers]]
==Reference(s)==
: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.
:<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>
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_Jc_j^*(x),x\}</math> is an SPNE iff: :a) <math>c_Jc_j^*(x) is feasible\,</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.
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> \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>
