PLEASE TEACH THIS Cass. Sup-2. Consider the following sequential variant of the public goods game wpose that there are 2 consumers, 1 and 2. First consumer 1 chooses a quantity ₁ ≥ 0 toprovide of the public good. After observing 1's choice, 2 chooses a quantity 2 ≥ 0 to provide.When the price of the public good is p, 1's payoff is u₁(₁, 2) = a√x₁ + x₂-p²₁ where a > 0and 2's payoff is u2(x1, x2) = √√T1+T2 — px2.(a) Suppose that a = 1. Show that this game has a Nash equilibrium in which 1 contributesa positive amount.Solution: There are many such Nash equilibria. One is for 1 to contribute and for2 to contribute 0 regardless of how much 1 contributes.(b) Find all subgame perfect equilibria of this game for each (positive) value of a and p.Solution: Use backward induction. If 1 contributes ₁, then 2's optimal action is to con-tribute (1) max 7-2₁,0}. Given this strategy for 2, 1's payoff to contributing₁ is- px1a√₁-prThis payoff is maximized by choosing r1=0 if a < 2, x₁ = 0 or ₁ = 1/p² if a = 2, andx₁ = if a > 2. Therefore, the subgame perfect equilibria are as follows:if 1 ≤²otherwise.u(x₁, x₂(x₁)) =i. If a < 2, there is a unique SPE given by x₁ = 0 and x2(x₁) = max.{+ -21,0}.ii. If a = 2, there are two SPE, one given by x₁ = 0 and x₂(x₁) = max {-₁,0},{-1,0}.Ithe other given by x₁ = 1/p² and x₂(x₁) = maxiii. If a > 2, there is a unique SPE given by x₁ = and x₂(x₁) = max c{₁-2₁,0}.(c) How does the total public good provision in part (b) compare to the Nash equilibriumprovision when the consumers choose simultaneously?Solution: From class, total public good provision in the Nash equilibrium of the simul-taneous game isif a >1. In the sequential game, total provision isif a > 2. Total provision is never higher in theif a ≤ 1 andifa <2, or if a = 2, andsequential game, and is strictly lower if 1 < a < 2.

When the game is sequential consisting of two players , then the first player optimizes his utility…

c.ass. Sup- 2. Consider the following sequential variant of the public goods game w pose that there are 2 consumers, 1 and 2. First consumer 1 chooses a quantity ₁ ≥ 0 to provide of the public good. After observing 1's choice, 2 chooses a quantity 2 ≥ 0 to provide. When the price of the public good is p, 1's payoff is u₁(1, 2) = a√√x₁ + x₂-p²₁ where a > 0 and 2's payoff is u₂(x₁, x2) = √√x1 + x2 − px2. (a) Suppose that a = 1. Show that this game has a Nash equilibrium in which 1 contributes a positive amount. Solution: There are many such Nash equilibria. One is for 1 to contribute and for 2 to contribute 0 regardless of how much 1 contributes.

c.ass. Sup- 2. Consider the following sequential variant of the public goods game w pose that there are 2 consumers, 1 and 2. First consumer 1 chooses a quantity ₁ ≥ 0 to provide of the public good. After observing 1's choice, 2 chooses a quantity 2 ≥ 0 to provide. When the price of the public good is p, 1's payoff is u₁(1, 2) = a√√x₁ + x₂-p²₁ where a > 0 and 2's payoff is u₂(x₁, x2) = √√x1 + x2 − px2. (a) Suppose that a = 1. Show that this game has a Nash equilibrium in which 1 contributes a positive amount. Solution: There are many such Nash equilibria. One is for 1 to contribute and for 2 to contribute 0 regardless of how much 1 contributes.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.7P

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