Question Description

I’m working on a economics test / quiz prep and need an explanation and answer to help me learn.

Recall the generalized tipping model discussed in the exercise of Section 7. We assumed

?? ? ?? > 1 in the lecture note, but now assume ?? ? ?? < 1.
(a) (5 points) Is there equilibrium where no one participate the protest? State yes or
no explicitly. Explain the reason. (b) (5 points) Suppose that all players make a decision independently and
simultaneously. Is there any equilibrium where ?? players participate depending
on the parameter values even with ?? ? ?? < 1? State yes or no explicitly. If yes,
show the condition on the parameter values for an existence of such
equilibrium, and explain the reason. If no, explain the reason.*The attached picture is the generalized tipping model discussed in the exercise of Section 7.
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Question: Consider the following generalized tipping model. There are n individuals,

indexed by i (i.e., i = 1,2,…,N), where N is arbitrarily large. The benefit from challenging

is b>0 for all citizens. Note that this is not the benefit from democratization. An individual

gets b when they participate the protest regardless of the outcome. The example of such

benefits is satisfaction by showing their righteous indignation to the public. The cost of

challenging is idiosyncratic, though related to the number of citizens who challenge. In

particular, for citizen i, the cost of challenging is

Y

i +

M

when M individuals challenge where the parameter y > 0. The idiosyncratic component of

the cost (i) may represent psychological factors or the opportunity cost of time spent

challenging. The common component (y/M) can be interpreted as follows: Any citizen who

challenges faces the possibility of being punished by the regime, where the probability of

punishment is inversely proportional to the number of citizens who challenge. In the

following questions, assume b-y > 1.

(i)

(ii)

n

n

Suppose that no one participate, which means that all citizens get the payoff of 0. If

Citizen 1 (i = 1) deviates by participating, his/her utility becomes b-1- y. Since

we assumed b-y >1, b-1-7>0. Therefore, at least one citizen has an

incentive to deviate, so “no one participates” is not an equilibrium.

Suppose that Citizen 1 to n participate (so there are n participants). Then, Citizen

n’s utility is b- n – If it is positive, that is,

ben

*20

> 0

(1)

These n participants have an incentive to participate. If not, some of them do not

have such an incentive. Therefore, the equilibrium number of participants n* should

satisfy

V

b-n* > 0 and b – (n* +1) y, so (2) is

satisfied. Therefore, individuals 1 to b – 1 can get positive utility by participating/ As

a result, the number of citizens who challenge is n* = b 1.

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GENERALIZED TIPPING MODEL

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