9. Each of two firms has one job opening. Suppose that (for reasons not discussed here but relating to the value of filling each opening) the firms offer different wages: firm i = 1,2 offers the wage wi. Imagine two workers, each of whom can apply to only one firm. The workers simultaneously decide whether to apply to firm 1 or to firm 2. If only one worker applies to a given firm, that worker gets the job; if both workers apply to one firm, the firm hires one worker at random (probability 0.5 for hiring worker 1 and probability 0.5 of hiring worker 2) and the other worker is unemployed (which has a payoff of zero). Suppose workers are risk neutral i.e., u(w) = w. а. Construct the normal form game. (Hint: payoffs in the normal form game are expected utility payoffs) b. Solve for the pure strategy set of Nash equilibria of the workers normal-form game when 0.5w1< w2 < 2w1. At a pure strategy Nash equilibrium would a worker secure a job? Explain Find the mixed strategy Nash equilibrium (p1*,p2*) and verify that pi* E (0,1) for i = 1,2. (Hint: find the expected С. value for worker i = 1,2 for each A1 and A2 assuming the worker j chooses A1 with probability pi and A2 with probability 1- piand then equate these two expected values to find pi*) d. Solve for the set of Nash equilibria of the workers normal-form game when 0.5w1> w2. At equilibrium would a
9. Each of two firms has one job opening. Suppose that (for reasons not discussed here but relating to the value of filling each opening) the firms offer different wages: firm i = 1,2 offers the wage wi. Imagine two workers, each of whom can apply to only one firm. The workers simultaneously decide whether to apply to firm 1 or to firm 2. If only one worker applies to a given firm, that worker gets the job; if both workers apply to one firm, the firm hires one worker at random (probability 0.5 for hiring worker 1 and probability 0.5 of hiring worker 2) and the other worker is unemployed (which has a payoff of zero). Suppose workers are risk neutral i.e., u(w) = w. а. Construct the normal form game. (Hint: payoffs in the normal form game are expected utility payoffs) b. Solve for the pure strategy set of Nash equilibria of the workers normal-form game when 0.5w1< w2 < 2w1. At a pure strategy Nash equilibrium would a worker secure a job? Explain Find the mixed strategy Nash equilibrium (p1*,p2*) and verify that pi* E (0,1) for i = 1,2. (Hint: find the expected С. value for worker i = 1,2 for each A1 and A2 assuming the worker j chooses A1 with probability pi and A2 with probability 1- piand then equate these two expected values to find pi*) d. Solve for the set of Nash equilibria of the workers normal-form game when 0.5w1> w2. At equilibrium would a
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:9. Each of two firms has one job opening. Suppose that (for reasons not discussed here but relating to the value of
filling each opening) the firms offer different wages: firm i = 1,2 offers the wage wi. Imagine two workers, each of
whom can apply to only one firm. The workers simultaneously decide whether to apply to firm 1 or to firm 2. If only
one worker applies to a given firm, that worker gets the job; if both workers apply to one firm, the firm hires one
worker at random (probability 0.5 for hiring worker 1 and probability 0.5 of hiring worker 2) and the other worker is
unemployed (which has a payoff of zero). Suppose workers are risk neutral i.e., u(w) = w.
а.
Construct the normal form game. (Hint: payoffs in the normal form game are expected utility payoffs)
b. Solve for the pure strategy set of Nash equilibria of the workers normal-form game when
0.5w1< w2< 2w1. At a pure strategy Nash equilibrium would a worker secure a job? Explain
Find the mixed strategy Nash equilibrium (p1*,p2*) and verify that pi* E (0,1) for i= 1,2. (Hint: find the expected
С.
value for worker i = 1,2 for each A1 and A2 assuming the worker j chooses A1 with probability pi and A2 with
probability 1- piand then equate these two expected values to find pi*)
d. Solve for the set of Nash equilibria of the workers normal-form game when 0.5w1> w2. At equilibrium would a
worker secure a job? Explain
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