Median Voter Theorem Flashcards
Single peaked preferences
Every individual has a most preferred alternative
(Like a predetermined position on a linear ordering) e.g music volume
If we restrict preferences, which assumption is violated
Unrestricted domain - i.e universality - have to consider all individuals
Draw spatial representation of single peaked preferences pg 7
Identify bliss point
X is bliss point (most preferred)
Z and Z’ and Y and Y’ are welfare decreasing
Pg 8 diagram of no single peaked preference
Criteria for single peakedness
Individual preferences have to be smooth ie should prefer closer to ideal point rather than further apart.
Are single peaked preferences realistic
Yes in most economic situations, individual preferences are single peaked and smooth
Benefit of single peaked preferences
Condorcet winner always exists
Overlapping preferences diagram pg 12
Where is the win set?
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
Assumptions for MVT (3)
Single peaked prefs
Single dimension (only considering 1 policy)
All individuals vote sincerely
2 types of elections
Voting directly on policy
Select political bodies to represent
For 2nd, more likely to base policy to court the vote of the median voter, and voters likley to base choices on preferences over the policy. When may voters deviate from this
They could consider additional factors e.g ideology or sympathy e.g prefer a candidate to another
Main implication/result of MVT
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
Downsian police convergence theorem looked at why they converge.
What assumption needs to be made, and what happens
Assume median voter preference unannounced,
This means the party closest to median wins. 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
What is Downsian environment/opportunistic
Parties are only motivated by winning (do not care about policy, just want to win)
Payoff for parties
Probability of winning x benefit of winning
V = πa
π is probability of winning
a is benefit of winning
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
No motivation to deviate as expected return drops from half the benefits, to 0.
B)
Unstable - Deviation is beneficial - they 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
Given the payoff expression, what is the expected payoff for parties
V = πa
Since both pursue preference of MV, probability is 1/2
So payoff is 1/2a
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?
Poor prefer higher tax! The transfer is more beneficial to them, thus want a higher tax rate.
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
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
Σ tyi - δt²
(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
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?
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
We can also find the preferred policy of the median voter
What is their maximisation problem (similar to IO)
B) Then solve to find median voters optimal tax rate t
Max ui = Ci + T
Subject to budget constraint Ci <= (1-t)yi
B) use the indirect utility function and differentiate FOC, rearrange to find optimal t:
t*m = ybar - ym/2δ
t*m = ybar - ym/2δ
What does this show
Tax will increase as the difference between average income (ybar) and median income (ym) increases (more income inequality, more tax for redistribution!!!)
IRL do we observe higher or lower distributive tax compared to 1850.
Tax has increased: we have more redistribution
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
US is more inequal, and distributes less
compared to more equal EU countries, who distriubte more
