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And so the we have both that (as As <math>\widehat{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.
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
|Has bibtex key=
|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>
===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>
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>
===Solving For Linear Contribution Terms===
:<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>
