PLEASE CHECK THIS HOW TO SOLVE show steps EXPLAIN BASICS AND CONCEPTS b) how do you know each cases Consider the following sequential variant of the public goods game we studied in class. Sup-pose 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 x2 > 0 to provide.When the price of the public good is p, 1's payoff is u₁(x1, x2) = a√√x1 + x2−pr₁ where a > 0and 2's payoff is u2(x1, x₂) = √√x1 + x2 - рx₂.(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 contribute2 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 x1, then 2's optimal action is to con-tribute x₂(1): = max {-2₁,0}. Given this strategy for 2, 1's payoff to contributing1 isu(x₁, x₂(x1)) =2p-px1a√₁ - priand forif x1 ≤ 4²otherwise.ii. If a = 2, there are two SPE, one given by x1the other given by x1 = 1/p² and x2(x1) = maxiii. If a > 2, there is a unique SPE given by x1 =This payoff is marimized by choosing r₁ = 0 if a < 2, x₁ = 0 or x₁ = 1/p² if a = 2, and= if a > 2. Therefore, the subgame perfect equilibria are as follows:x1 =i. If a < 2, there is a unique SPE given by x1 = 0 and x2(x1) = max-: {−₁,0}.= 0 and x₂(x1) = max < {²-₁,0},x{-x₁,0}.and x₂(x1) = max = {2-²1,0}.

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

Consider the following sequential variant of the public goods game we studied in class. Sup- 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 x₂ > 0 to provide. When the price of the public good is p, 1's payoff is u₁(x₁, x2) = a√x₁+x2−px₁ where a > 0 and 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 contributes a positive amount.

Consider the following sequential variant of the public goods game we studied in class. Sup- 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 x₂ > 0 to provide. When the price of the public good is p, 1's payoff is u₁(x₁, x2) = a√x₁+x2−px₁ where a > 0 and 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 contributes a positive amount.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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