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# HW XXXXXXXXXXElectoral Competition XXXXXXXXXXCandidates Adams and Burr are running for public office. To campaign, each makes a binding promise regarding the level of effort ???? ≥ 0 or ???? ≥ 0 that...

HW XXXXXXXXXXElectoral Competition

are running for public office. To campaign,
each makes a binding promise regarding the level of effort ???? ≥ 0 or ???? ≥ 0 that
he will exert if elected. Voter utility is increasing in the effort exerted by the
winning candidate. That candidate receives a benefit of 100 representing the
perks of office (e.g. fame, influence, salary, etc.), and also bears a cost of ???? = ????2
or ???? = ????2 which increases (increasingly) in the effort he commits to exert.
What voter behavior and what candidate behavior do you predict will occur in
equili
ium, and why?
XXXXXXXXXXIn a certainty pool of politicians, half are high types (??) and half are low
types (??). An incumbent is drawn randomly from this pool and must implement
a policy ?? ∈ {??, ??}. A random variable ?? ∈ {??,??} denotes the right thing to do: if
?? = ?? and ?? = ?? or ?? = ?? and ?? = ?? then voters each receive utility 1;
Voters do not know what the right thing to do is (that is, they do not observe ??),
ut they think that ?? is more likely: ??(?? = ??) = .6 and ??(?? = ??) = .4. The
incumbent is better informed, because she observes a signal ?? ∈ {??,??} that is
informative of ??. If she is a high-type then her signal is accurate with probability
???? = 1. That is, ??(?? = ??|?? = ??) = ??(?? = ??|?? = ??) = 1. If she is a low-type
then her signal is only accurate with probability ??(??|??) = ??(??|??) = ???? = .8. In
that case, ??(??|??) = ??(??|??) = 1 − ???? = .2.
The notion of pandering is that the incumbent might implement policy ?? even
when she believes that the right thing to do is probably ??, so that voters will
elieve she is a high-type and reelect her. To explore this possibility, suppose
that voters believe the incumbent is doing what she honestly thinks is best (i.e.,
implementing ?? = ?? when she observes ?? = ?? and implementing ?? = ?? when
she observes ?? = ??), and answer the following.
a. (+4) If the incumbent is a high type, how likely is it that she will
implement policy ?? = ??? How likely is it that she will implement policy
?? = ???
. (+2) If the incumbent is a low type, how likely is it that she will
implement policy ?? = ??? ?? = ???
c. (+5) If voters observe that the incumbent implemented policy ?? = ??,
how likely is it that she was a high type? A low type?
d. (+2) If voters observe that the incumbent implemented policy ?? = ??,
how likely is it that she was a high type? A low type?
e. (+2) At the next election, voters can retain the incumbent or elect a
challenger, who will be high or low type with equal probability. If they
seek to maximize the probability of electing a high type, and the
incumbent implements ?? = ??, should they reelect her, or elect a
challenger? What if she implements ?? = ???
HW XXXXXXXXXXElectoral Competition

## Solution

Komalavalli answered on Mar 09 2022
Q1
In equili
ium the voter’s utility equals the candidate benefit.(e2A or e2A)
Maximum benefit that a voter can give with his effort is 100
e2A =CA = 100
eA = 50
Equili
ium is (100, 50)
Q2
a)
If incumbent is high type the likelihood of implementing the policy a is qH/x=a =0.6/1 =0.6
If incumbent is high type the likelihood of implementing the policy b is qH/x=b =0.4/1= 0.4
)
If incumbent is low type...
SOLUTION.PDF