Politics through the Lens of Economics
Lecture 2: Median Voter Theorem
Masayuki Kudamatsu
11 October, 2017
Did you pick a policy?
(for your written assignment)
How to make a case in writing
"Five Paragraph Essay" (taught in US high schools)
1 Motivation & Main Idea
5 Conclusion
4 Deal with counter arguments
3 Cases for Main Idea
2 Background
Motivation & Main Idea
Main idea: Statement that you will prove to be the case
"Median Voter Theorem cannot explain why the gun control hasn't been adopted in the United States."
Motivation: Reasons why we should care about your statement
1st paragraph
"The lack of the gun control in the U.S. causes the high murder rate in the country..." (then cite statistics etc.)
Better to be short, at most 2 paragraphs
e.g.
e.g.
Background
Any information that will repeatedly be used in the arguments below
Description of political institutions in the U.S.
History of (lack of) gun control in the U.S.
2nd paragraph
DO NOT include unnecessary information in this paragraph
e.g.
Better to be short unless it's really necessary to provide lots of information beforehand
Cases for Main Idea
Reasons why the main idea is correct
"Opinion polls suggest the vast majority of Americans are in favour of gun control." (and then cite statistics)
3rd paragraph
e.g.
Use more than one paragraph if there are many reasons
Deal with counter-arguments
Reasons why the counter-arguments are wrong
"The majority of Americans may not trust in government to enforce gun control. But ..."
4th paragraph
e.g.
Talk to people to learn the list of counter-arguments
Use more than one paragraph if there are many counter-arguments
It's okay to admit the limitation of your argument
Conclusion
Restate the main idea
"For all the reasons described above, the Median Voter Theorem cannot explain why Americans have failed to adopt the gun control.
5th paragraph
e.g.
Then add a statement for future
"We need a better theory of politics so that we will be able to find a solution to stop Americans from dying by gunshots."
e.g.
A great textbook for written assignments
Read pages 24-108 twice
First, before starting to write
Second, when done with the first draft
The book also discusses how to write
Novels (pp. 110-183)
Blog posts (pp. 186-242)
Motivational Examples
for
Median Voter Theorem
Motivational example #1: Japan
Similar economic policies between LDP and DPJ in the mid-2000s.
Figure 3 of Kubo (2016)
Motivational example #2: UK
"Third Way" by Tony Blair (Prime Minister 1997-2007)
Image sources:
Motivational example #3: USA
Donald Trump advocates for big government
(Talking about healthcare) Everybody’s got to be covered. This is an un-Republican thing for me to say... Everybody’s going to be taken care of much better than they’re taken care of now. ...the government’s gonna pay for it.
Image/quote source: thefederalist.com/2016/01/27/5-times-trump-told-you-he-loves-big-government/
Median Voter Theorem says
these are all politicians' optimal behaviour
Today's Road Map
What is Median Voter Theorem?
Evidence for the theorem
Limitations of the theorem
Graphic proof of the theorem
Three assumptions for the theorem
Today's Road Map
What is Median Voter Theorem?
Evidence for the theorem
Limitations of the theorem
Graphic proof of the theorem
Three assumptions for the theorem
("Model of the World")
(Optimization + Equilibirum)
(Empiricism)
What is Median Voter Theorem?
We break down this question into three:
What is Median?
What is Median Voter?
What is Median Voter Theorem?
Median
50% below
50% above
Median Voter
The voter whose ideal policy takes the median value
II
Let's find the median voter in this class room.
What is your ideal consumption tax rate?
Value-added tax rate across OECD countries in 2015
VAT tax revenue as % of total tax revenue in 2014
Source: OECD (2017) "Goods and services as percentage of total taxation" Government At A Glance 2017
Median Voter Theorem
Median voter's ideal policy is implemented
Voters care about one single policy
Two candidates compete
Winner keeps electoral promise
In a model of the world where:
Today's Road Map
What is Median Voter Theorem?
Three assumptions for the theorem
Graphic proof of the theorem
Limitations of the theorem
Evidence for the theorem
Voters care about one single policy
Two candidates compete
Winner keeps electoral promise
There are so many policy issues
But an opinion on one policy often predicts the one on another
abortion
marijuana
same-sex marriage
e.g.
We can bundle policies into a few
Social: Conservative vs Liberal
Economic: Small vs Big government
Diplomatic: Hawk vs Dove
Assumption #1
Voters care about one single policy
Two candidates compete
Winner keeps electoral promise
But do voters care about
only one bundle of policies?
Very strong assumption
But simple enough as a starting point
Assumption #1
Voters care about one single policy
Two candidates compete
Winner keeps electoral promise
Quite common for plurality elections
e.g. Osaka gubernational election in 2015
Assumption #2
Voters care about one single policy
Two candidates compete
Winner keeps electoral promise
But not plausible for proportional representation electoral system
Proportional representation
Plurality
Mixed
Assumption #2
Voters care about one single policy
Two candidates compete
Winner keeps electoral promise
Assumption #3
Perhaps unrealistic assumption
Voters care about one single policy
Two candidates compete
Winner keeps electoral promise
Plausible if breaking promise means defeat in next election
Assumption #3
Image source: https://blogs.yahoo.co.jp/nisekoannnuburi/26318040.html
Today's Road Map
What is Median Voter Theorem?
Three assumptions for the theorem
Graphic proof of the theorem
Limitations of the theorem
Evidence for the theorem
To see why the theorem holds,
use the two defining features of economics:
optimization & equilibrium
Look at voters' optimization
Figure out which candidate wins
Consider candidate's optimization
Check whether it's the equilibrium
Step 1
Step 2
Step 3
Step 4
Voter's
ideal
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Voter's optimization
Choose the candidate whose promise is closer to their ideal policy
Step 1
Voter's
ideal
Policy
Candidate A's promise
Candidate B's promise
Vote for B
Choose the candidate whose promise is closer to their ideal policy
Voter's optimization
Step 1
Voter's
ideal
Policy
Candidate A's promise
Candidate B's promise
Vote for B
Choose the candidate whose promise is closer to their ideal policy
Voter's optimization
Step 1
Voter's
ideal
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Choose the candidate whose promise is closer to their ideal policy
Voter's optimization
Step 1
Choose the candidate whose promise is closer to their ideal policy
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Voter's optimization
Step 1
This is each voter's best affordable option
Which candidate wins?
Depends on where the median voter is
50% below
50% above
Median
Step 2
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
Candidate A wins
>50%
Case 1
Which candidate wins?
Step 2
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
Candidate B wins
>50%
Case 2
Which candidate wins?
Step 2
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
Candidate B wins
>50%
Case 3
Which candidate wins?
Step 2
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
Candidate A wins
>50%
Case 4
Which candidate wins?
Step 2
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
Either candidate wins
with 50% of the chance
Case 5
Which candidate wins?
Step 2
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
Winner: the candidate
the median voter votes for
>50%
Which candidate wins?
Step 2
Summary
Candidate's optimization
Consider Case 1: A is closer to median
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
Candidate A wins
>50%
Step 3
But B is better off by choosing x instead
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
>50%
x
Candidate's optimization
Step 3
We're now in case 2: B is closer to median
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
Candidate B wins
>50%
Candidate's optimization
Step 3
A is then better off by choosing x instead.
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
>50%
x
Candidate's optimization
Step 3
We're back to Case 1: A is closer to median
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
Candidate A wins
>50%
Candidate's optimization
Step 3
Policy
Candidate A's promise
Candidate B's promise
Median
Voter
Both candidates want to be closer to median than the other
Candidate's optimization
Step 3
A wants to pick x to be closer to median than B
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
>50%
x
Candidate's optimization
Step 3
How about Case 3?
B wants to pick x to be closer to median than A
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
>50%
x
Candidate's optimization
Step 3
In Case 4
In Case 5, each wants to be closer to median to win for sure
Policy
Candidate A's promise
Candidate B's promise
Vote for A
Vote for B
Median
Voter
Candidate's optimization
Step 3
Policy
Candidate A's promise
Candidate B's promise
Median
Voter
Both candidates want to be closer to median than the other
Candidate's optimization
Step 3
In all cases
What is the equilibrium?
Step 4
Remember what the equilibrium is?
What is the equilibrium?
Candidates change their policy when neither picks median
Policy
A
B
Median
Voter
Step 4
Candidate picks median if the other already picks median
Policy
A
Median
Voter
B
What is the equilibrium?
Step 4
When both promise median...
Policy
Every voter is indifferent between A & B
A & B
Median
Voter
What is the equilibrium?
Step 4
Policy
Median
Voter
Every voter votes randomly
A & B
When both promise median...
What is the equilibrium?
Step 4
Policy
Median
Voter
Each candidate wins 50% of the chance
A & B
When both promise median...
What is the equilibrium?
Step 4
Policy
Median
Voter
Each candidate wins 50% of the chance
A & B
When both promise median...
Neither A nor B benefits from changing their choice
What is the equilibrium?
Step 4
Policy
Median
Voter
Each candidate wins 50% of the chance
A & B
This is the equilibrium!
When both promise median...
Neither A nor B benefits from changing their choice
Policy
Median
Voter
A & B
This is the equilibrium!
Winning candidate will then implement
median voter's ideal policy
Policy
Median
Voter
Each candidate wins 50% of the chance
A & B
Equilibrium is stark, not very realistic
Policy
Candidate A's promise
Candidate B's promise
Median
Voter
But the main lesson is the centripetal force in politics
Economics is sometimes a bit like a fairy tale
Unrealistic, but a lesson relevant for reality
Today's Road Map
What is Median Voter Theorem?
Three assumptions for the theorem
Graphic proof of the theorem
Limitations of the theorem
Evidence for the theorem
Limitations of Median Voter Theorem
Important to understand for you to figure out
whether the Theorem can explain the policy of your choice
for term paper
Limitation of Median Voter Theorem #1
e.g. economic policy + foreign policy
We take up this issue in Lecture 6: Probabilistic Voting Model
When voters care about more than one policy
Hawk
Dove
Small govt.
Big govt.
When politicians cannot commit to electoral promise
Limitation of Median Voter Theorem #2
Plausible if politicians have their own ideal policy
With no commitment,
Winning candidate will be the one
whose ideal policy is closest to median
Limitation of Median Voter Theorem #2
Policy
A wins!
B
Vote for A
Vote for B
Median
Voter
>50%
Then why the median voter doesn't become a candidate?
We will look at this issue in Lecture 4: Citizen-candidate Model
Limitation of Median Voter Theorem #2
Policy
A
B
Median
Voter
("I will run for office!")
Limitation of Median Voter Theorem #3
e.g. Tokyo gubernatorial election in 2016
image source: headlines.yahoo.co.jp/hl?a=20160719-00010001-bfj-soci
When there are more than two candidates
Policy
Median
Voter
A & B & C
Suppose all three parties promise median voter's ideal policy
Limitation of Median Voter Theorem #3
When there are more than two candidates
Policy
Median
Voter
A & B
Any party (say, C) wants to deviate a little
C
Vote for C
(almost 50%)
Vote for A or B
(each gets 25%)
Limitation of Median Voter Theorem #3
When there are more than two candidates
We need to consider what determines # of candidates
When there are more than two candidates
Limitation of Median Voter Theorem #3
We will look at this issue in Lecture 4: Citizen-candidate Model
As we see
economists start with a simple (if unrealistic) model of the world
Using this simple model as the benchmark
economists modify the model to understand the world further
image source: babsreviews.com/online-marketing-this-week.html
Today's Road Map
What is Median Voter Theorem?
Three assumptions for the theorem
Graphic proof of the theorem
Limitations of the theorem
Evidence for the theorem
Remember the 3rd defining feature of economics?
Evidence for Median Voter Theorem
Collect many observations to see if we get the graph below
median voter's ideal policy
politician's policy
Evidence for Median Voter Theorem
We do see this graph for homogeneous districts in U.S. (Gerber and Lewis 2004)
median voter's ideal policy
politician's policy
Evidence for Median Voter Theorem
But that's not enough because...
Correlation
between A & B
A causes B
B causes A
C causes A & B
Evidence for Median Voter Theorem
Voters' ideal policy may change in response to politician's choice
Correlation
between A & B
A causes B
B causes A
C causes A & B
Evidence for Median Voter Theorem
Both voters and politicians may respond to common shock
Correlation
between A & B
A causes B
B causes A
C causes A & B
Evidence for Median Voter Theorem
We need an exogenous change in median voter's ideal policy
Correlation
between A & B
A causes B
B causes A
C causes A & B
Women's suffrage
in US states
in early 20th century
Women care more about children
Female suffrage will move the median voter's ideal policy more child-friendly
Natural experiment:
Spending on child health
Median
before
women's suffrage
Men
Women
Spending on child health
Median
after
women's suffrage
Median
before
women's suffrage
Year of Women's Suffrage across US States
Figure I of Miller (2008)
Municipal health spending rose after women's suffrage
Figure II of Miller (2008)
Hygiene campaign spending
Child mortaliry dropped after women's suffrage
Figure IV of Miller (2008)
(accounting for 10% of the decline during 1900-1930)
Summary
Policy
A
B
Median
Voter
Politicians tend to pick the policy closer to median
Next lecture
We will apply the Median Voter Theorem to understand...
Income Redistribution
image source: whitewraithe.files.wordpress.com/2014/01/richpoor.jpg
Next lecture
We will apply the Median Voter Theorem to understand...
Democratization
image source: en.wikipedia.org/wiki/Liberty_Leading_the_People
Medium Voter Theorem
Chapter 1 of Gelbach (2013)
Evidence for Medium Voter Theorem
This lecture is based on the following academic articles and books:
Politics through the Lens of Economics: Lecture 2 - Median Voter Theorem
By Masayuki Kudamatsu
Politics through the Lens of Economics: Lecture 2 - Median Voter Theorem
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