Median Voter Theorem (2 ways to show)

Question 1

Single peaked preferences

Answer 1

Every individual has a most preferred alternative

(Like a predetermined position on a linear ordering) e.g music volume

Question 2

If we restrict preferences, which assumption is violated

Answer 2

Unrestricted domain - i.e universality - have to consider all individuals

Question 3

Draw spatial representation of single peaked preferences pg 7

Identify bliss point

Answer 3

Utility Y axis
X axis Is whatever being measured

X is bliss point (most preferred)
Z and Z’ and Y and Y’ are welfare decreasing

Question 4

Pg 8 diagram of no single peaked preference

Answer 4

Preferences are not smooth! PTO see what smooth should be

Question 5

Criteria for single peakedness

Answer 5

Individual preferences have to be smooth ie should prefer closer to ideal point, further is welfare-decreasing

Question 6

Are single peaked preferences realistic

Answer 6

Yes in most economic situations, individual preferences are single peaked and smooth

Question 7

Benefit of single peaked preferences

Answer 7

Condorcet winner always exists and coincides with the median voter’s bliss point!

Question 8

Overlapping preferences diagram pg 12

Where is the win set?

Answer 8

Each line presents different individual

Red horizontal line shows all positions red prefers over X
Green horizontal line shows all positions along line preferred over X
Black horizontal line shows all preferred positions over X

So thus win set is between X and Y - anywhere along that line will be preferred by all 3 individuals compared to initial point X

Question 9

Assumptions for MVT (3)

Answer 9

Single peaked prefs
Single dimension (only considering 1 policy)
All individuals vote sincerely

Question 10

Main implication/result of MVT

Answer 10

Convergence towards the median voter - parties adopt the same policy towards the preference of the median voter.

a1=a2=Xm
Xm is preference of median voter

Question 11

Downsian police convergence theorem looked at why they converge.

Suppose median voter preference unannounced, what happens?

Answer 11

This means the party closest to median wins. This is not stable, as other party has incentive to move closer to Qm to win. Keep converging and competing it towards the centre till both policies are the same. Decide winner by fair coin (50/50)

Question 12

What is Downsian environment/opportunistic

Answer 12

Parties are only motivated by winning (do not care about policy, just want to win)

Question 13

Payoff for parties

Answer 13

Probability of winning x benefit of winning
V = πa
π is probability of winning
a is benefit of winning

Question 14

So the result is converged policies i.e α1=α2=Xm is a stable nash equilibrium.

Proof is as follows:
Reason 1: If both parties choose median voter position, no motivation to deviate. Why?

B) Reason 2: If both choose the same position but not median voter position, what happens?

C) Reason 3: if both choose different positions and neither of median voter position what happens

Answer 14

No motivation to deviate as expected return drops from half the benefits (since 50/50 fair coin), to 0.

B)
Unstable - Deviation is beneficial - currently have 50/50 chance of winning (tie) , so can change outcome from a tie (in that position) to a win (by moving closer to the median voter)

C) incentive to deviate too - can change outcome from a loss to a win

Question 15

Given the payoff expression, what is the expected payoff for parties

Answer 15

V = πa

Since both pursue preference of MV, probability is 1/2

So payoff is 1/2a

Question 16

Applied example of MVT: redistributive taxation. Assume a income tax rate t and gov provides lump-sum transfer T for everyone. Assume costs of tax per person (for gov) are δt². Who wants the high tax rate? Poor or rich?

Answer 16

Poor prefer higher tax! The transfer is more beneficial to them, thus want a higher tax rate.

Question 17

Model set up
Agent maximises utility
Ui = Ci + T
Utility is consumption + transfer

What is the budget constraint and government budget constraint

B) What is lump sum transfer per person (hint: linked to government budget constraint

Answer 17

Ci <= (1-t)yi
Consumption has to be less/equal to disposable income i.e the budget constraint

Government budget constraint is just their income - costs
R = Σ tyi - nδt²
(Sum of tax revenue from each individual - costs of taxing them

B)
T is the lump sum - which is just government budget divided by number of individuals (n)
T = R/n = tybar - δt²

Question 18

IMPORTANT FLASHCARD:
Find indirect utility function by subbing in lump sum (T) and Ci (our budget constraint) into the utility function U = Ci + T

B) If we then differentiate FIRST AND SECOND ORDER with respect to T, what does this help prove us?

Answer 18

Wi = (1-t)yi + tybar - δt²

B) differentiate twice get us
-2δ<0
Negative so proves there is maximum I.E SINGLE-PEAKEDNESS

THUS MVT APPLIES, WE WILL SEE CONVEGENCE

Question 19

Now we have established singlepeakedness, we can find median voters bliss point for tax

How

Answer 19

Use the FOC of the indirect utility function rearrange to find optimal t:

-yi + ybar - 2δt = 0 rearranges to

ti = ybar - ym/2δ
Since voter i is the median voter as established, we replace i with m!
t*m = ybar - ym/2δ

Question 20

t*m = ybar - ym/2δ

What does this show

Answer 20

Tax will increase as the difference between average income (ybar) and median income (ym) increases (more income inequality, more tax for redistribution!!!)

Question 21

IRL do we observe higher or lower distributive tax compared to 1850.

Answer 21

Tax has increased: we have more redistribution

Question 22

So Meltzer and richards predict higher income inequality (ybar - ym) leads to more redistibution (higher t)

However little evidence of greater inequality increasing redistribution. Example

Answer 22

US is more inequal, and distributes less

compared to more equal EU countries, who distriubte more

Question 23

Do all citizens have equal influence on policy: reason for yes and no

Answer 23

Yes - in MVT, every voter is equal

No: if one dollar one vote. then sometimes privileged group can dominate: when income of this group increases, aggregate redistributive policies tild towards this group’s preferred policies

Question 24

Karabarbounis results on whether one dollar one vote exists i.e political influence increases with income

Answer 24

They find
As rich get wealthier, less redistribution
As middle class becomes wealthier, less redistribution
Key point: As poor gets wealthier, more redistribution

thus true, poor people vote and policy reflects their preference of wanting more redistirbution!political influence is increasing with income

Question 25

MVT drawbacks (5)

Answer 25

Requires odd number to ensure majority (if tie: theorem doesn’t help determine which will win) One dimensional - IRL consider multiple policies Only assumes 2 parties - with 3, no stable equilibrium since always incentive to move in response to opponents positions Assumes parties only care about winning, irl may consider ideology and own views rather than primarily courting for the median voter preference Assumes voters only base votes on policy

Question 26

Do policies reflect median voter iRL?

Answer 26

Mixed Yes in policy where median voter can credibly understand and care about public policy e.g education,health etc But in other policy areas e.g trade policy (MVT performs poorly, median voter may not care/understand)

Question 27

Real life example of MVT: Miller

Answer 27

Saw universal womens suffrage (right to vote) in US in 1920 Median voter bliss point adjusted to reflect women’s preferences (who care more about health expenditure) correspondingly increase public health spending, esp hygiene campaigns, child mortality fell 8-15%

Question 28

so if question asks about MVT, 2 good ways to show proof of it

Answer 28

Use payoff one, and the 3 scenarios FC14 Optimal tax (nδt² thing)

Question 29

What does condorcet paradox likelihood increase with (2)

Answer 29

Number of voters and alternatives

Question 30

Why is condorcet paradox bad (implications, mock q1c) (2)

Answer 30

As intransitive cycle means voting is unstable, people need to find was to break ties e.g repeat votes, external interventions etc which reduce legitimacy of outcomes (E.g seminar 1 q1c shows since no condorcet winner, they settle through voting agenda. This method also shows how different agenda sequences have different winners, exemplifying the less legitimacy!) Agenda manipulation - agents don’t vote truthfully and instead strategically

Question 31

What does condorcet paradox violate from Arrows impossibility theorem

Answer 31

Rationality - since means are cyclic ie intransitive preferences !