Dongguk Generalized Tipping Model Discussion

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. 2 attachmentsSlide 1 of 2attachment_1attachment_1attachment_2attachment_2.slider-slide > img { width: 100%; display: block; }
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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.