Median Voter Theorem

Lecture 1 Part 2, Political Economics I

OSIPP, Osaka University

6 October, 2017

Masa Kudamatsu

What is Median Voter Theorem?

Elections induce politicians to propose

the policy favored by the median voter

Roughly speaking, the Theorem says:

The policy favored by the median voter

Example: the consumption tax rate in Japan

Survey of my undergrad class (9 students) last year

10%

Median

Basic Model

Known as the Downsian model of electoral competition

as it's first formalized by Downs (1957)

Here we follow the exposition by

Sections 3.2-3.3 of Persson and Tabellini (2000)

Players

Two candidates, each indexed by

P \in \{A, B\}

P \in \{A, B\}

A continuum of citizens, each indexed by

i

Candidates' preference

If they win the election, ego rent

R

If they lose the election, the payoff is normalized to zero

Denote candidate P's probability of winning by

p_P

p_P

Then candidate P's expected utility is

p_P R

p_P R

Candidates just want to be in office, not caring about policy

Often called office-seeking candidates

Candidate P chooses a policy to maximize

p_P R

p_P R

Candidates' action

Candidate P proposes a policy

g_P \in \mathbb{R}

g_P \in \mathbb{R}

e.g.

Tax rate

Government expenditure on education

NOTE: We assume candidates can commit to the proposed policy

This assumption is crucial for the Median Voter Theorem

Citizens' preference

The farther away the policy is from citizen i's ideal,

the smaller his/her payoff from that policy

Citizens have a single-peaked preference over policy g

i.e.

An example single-peaked preference

The farther away the policy is from the ideal

The lower the payoff

Citizens' preference (cont.)

W^i(g'') \leq W^i(g')

W^i(g'') \leq W^i(g')

for any \(g'\) and \(g''\) such that

g'' \leq g' \leq g^i

g'' \leq g' \leq g^i

g^i \leq g' \leq g''

g^i \leq g' \leq g''

Formally, denote citizen i's payoff function by

Citizen i's ideal policy is given by

g^i =\arg\max_g W^i(g)

g^i =\arg\max_g W^i(g)

W^i(g)

W^i(g)

W^i(g)

W^i(g)

represents a single-peaked preference if

W^i(g'') \leq W^i(g')

W^i(g'') \leq W^i(g')

for

g^i \leq g' \leq g''

g^i \leq g' \leq g''

g^i

g^i

g'

g''

g''

W^i(g'') \leq W^i(g')

W^i(g'') \leq W^i(g')

for

g^i

g^i

g'

g''

g''

g'' \leq g' \leq g^i

g'' \leq g' \leq g^i

NOTE: we will see how far we can relax this assumption

for the Median Voter Theorem to hold

Citizens' preference

Different citizens have different ideal policies

g^i

g^i

Citizens' preference

g^i

g^i

g^k

g^k

Different citizens have different ideal policies

Median Voter

Denote the median value of

g^i

g^i

g^M

g^M

Call the citizen whose ideal policy is

g^M

g^M

the median voter

indexed by

i=M

i=M

Citizens' action

Each citizen decides whether to vote for candidate A or B

NOTE: we assume no citizen abstains from voting

In Lecture 6, we discuss how we model voter turnout

Otherwise, we maintain this assumption throughout the course

(and also in most studies in political economics)

Timing of Events

Each citizen simultaneously decides whom to vote, candidate A or B

Each candidate simultaneously proposes policy

g_A, g_B

g_A, g_B

The candidate who obtains the majority implements the proposed policy. The payoff realizes.

Election Campaign

Election

Inauguration of a new government

Analysis

Subgame Perfect Nash Equilibrium

The model is an example of the extensive form game

Equilibrium concept: Subgame Perfect Nash Equilibrium

We can obtain the equilibrium by backward induction

See Chapters 7-8 of Tadelis (2013)

if this slide looks like Egyptian hieroglyphs

Backward induction

Timing of Events

Each citizen simultaneously decides whom to vote, candidate A or B

Each candidate simultaneously proposes policy

The candidate who obtains the majority implements the proposed policy. The payoff realizes.

Election Campaign

Election

Inauguration of a new government

Find the optimal behavior for the last player first

Citizen i's optimization

Vote for

W^i(g_A) > W^i(g_B)

W^i(g_A) > W^i(g_B)

A

B

<

Flip the coin

=

Vote for the candidate whose policy yields a higher payoff

To see which candidate obtains the majority of votes,

we can just consider the median voter's decision

g^i

g^i

g_A

g_A

g_B

g_B

Suppose candidates A and B propose the following policies

Since each citizen votes for the policy closer to their ideal...

Vote for B

\frac{g_A+g_B}{2}

\frac{g_A+g_B}{2}

Vote for A

To see which candidate obtains the majority of votes,

we can just consider the median voter's decision

g^i

g^i

g_A

g_A

g_B

g_B

If the median voter votes for B...

Vote for B

\frac{g_A+g_B}{2}

\frac{g_A+g_B}{2}

Vote for A

g^M

g^M

B gets the majority

More than 50% of votes

To see why the single-peaked preference matters...

W^M(g_A)>W^M(g_B)

W^M(g_A)>W^M(g_B)

Then all citizens with

g^i > g^M

g^i > g^M

also prefer to vote for A

Suppose

(1) Candidate A proposes a higher g:

g_A > g_B

g_A > g_B

(2) Median voter votes for A:

Then

g

g_A

g_A

g_B

g_B

g^M

g^M

by the single-peaked preference

g^i

g^i

So candidate A wins the majority by the definition of median

To see which candidate obtains the majority of votes,

we can just consider the median voter's decision

W^M(g_A) > W^M(g_B)

W^M(g_A) > W^M(g_B)

Then all citizens with

g^i < g^M

g^i < g^M

also prefer to vote for A

Suppose

(1) Candidate B proposes a higher g:

g_A < g_B

g_A < g_B

(2) Median voter votes for A:

Then

g

g_A

g_A

g_B

g_B

g^M

g^M

by the single-peaked preference

g^i

g^i

So candidate A wins the majority by the definition of median

To see which candidate obtains the majority of votes,

we can just consider the median voter's decision

W^M(g_A) < W^M(g_B)

W^M(g_A) < W^M(g_B)

Then all citizens with

g^i >g^M

g^i >g^M

also prefer to vote for B

Suppose

(1) Candidate B proposes a higher g:

g_A < g_B

g_A < g_B

(2) Median voter votes for B:

Then

g

g_A

g_A

g_B

g_B

g^M

g^M

by the single-peaked preference

g^i

g^i

So candidate B wins the majority by the definition of median

To see which candidate obtains the majority of votes,

we can just consider the median voter's decision

W^M(g_A) < W^M(g_B)

W^M(g_A) < W^M(g_B)

Then all citizens with

g^i < g^M

g^i < g^M

also prefer to vote for B

Suppose

(1) Candidate A proposes a higher g:

g_A > g_B

g_A > g_B

(2) Median voter votes for B:

Then

g

g_A

g_A

g_B

g_B

g^M

g^M

by the single-peaked preference

g^i

g^i

So candidate B wins the majority by the definition of median

Summing up the results so far

in terms of A's winning probability

(candidate A wins)

p_A = 1

p_A = 1

p_A

p_A

W^M(g_A)>W^M(g_B)

W^M(g_A)>W^M(g_B)

p_A = 0

p_A = 0

(candidate B wins)

<

p_A = \frac{1}{2}

p_A = \frac{1}{2}

=

Backward induction (cont.)

Timing of Events

Each citizen simultaneously decides whom to vote, candidate A or B

Each candidate simultaneously proposes policy

The candidate who obtains the majority implements the proposed policy. The payoff realizes.

Election Campaign

Election

Inauguration of a new government

Now find the optimal behavior of the first players

assuming they correctly predict the last players' behavior

Candidate A's optimization

To maximize

p_A

p_A

Propose

g_A

g_A

that is closer to

g_M

g_M

than

g_B

g_B

Candidate B's optimization

To maximize

1- p_A

1- p_A

Propose

g_B

g_B

that is closer to

g_M

g_M

than

g_A

g_A

Subgame Perfect Nash Equilirium

Both candidates propose the median voter's ideal policy

g_A^*=g_B^*=g^M

g_A^*=g_B^*=g^M

This is the prediction of the Median Voter Theorem

is sometimes called the Condorcet winner

g^M

g^M

A policy that defeats any other policy in a majority voting

Implications

Policy Convergence

Both candidates propose the same, centralist policy

Not too bad a prediction for Japan perhaps

In reality, however, there are situations of policy divergence

How to derive policy divergence as the equilibrium

has been a big research agenda in theoretical political economy

Robustness

of the basic model

Citizens' preference

The basic model imposes the single-peaked preference

This assumption can be relaxed to the single-crossing property

Gans and Smart (1996) show

Formal definition of the single-crossing property

Denote citizens' preference over policy by

W(q, \alpha^i)

W(q, \alpha^i)

Policy (e.g. tax rate)

q \in \mathbb{R}

q \in \mathbb{R}

Preference type (e.g. income)

\alpha \in \mathbb{R}

\alpha \in \mathbb{R}

The single-crossing property is satisfied if

q > q'

q > q'

W(q,\alpha) \geq W(q',\alpha) \Rightarrow W(q,\alpha') \geq W(q',\alpha')

W(q,\alpha) \geq W(q',\alpha) \Rightarrow W(q,\alpha') \geq W(q',\alpha')

for

\alpha > \alpha'

\alpha > \alpha'

and

q < q'

q < q'

, or for

\alpha < \alpha'

\alpha < \alpha'

and

Graphically, the single-crossing property holds if

\alpha

\alpha

W(q,\alpha)-W(q',\alpha)

W(q,\alpha)-W(q',\alpha)

0

Or if

\alpha

\alpha

W(q,\alpha)-W(q',\alpha)

W(q,\alpha)-W(q',\alpha)

0

See Example 1 (pp. 24-25) of Persson and Tabellini (2000)

for an example of the preference satisfying the single-crossing property but not single-peaked

Multi-dimensional policy

The basic model assumes one-dimensional policy space

For multi-dimensional policy space,

the Median Voter Theorem holds

if citizens' preference satisfies the following condition

(Grandmont 1978; see Section 2.2.2 of Persson and Tabellini 2000)

Citizens have intermediate preferences if their indirect utility function takes the form of

W(\mathbf{q}; \alpha^i) = J(\mathbf{q}) + K(\alpha^i)H(\mathbf{q})

W(\mathbf{q}; \alpha^i) = J(\mathbf{q}) + K(\alpha^i)H(\mathbf{q})

where q is the policy vector,

\alpha^i

\alpha^i

citizen i's type parameter,

K(\alpha^i)

K(\alpha^i)

monotonic in

\alpha^i

\alpha^i

In words, the intermediate preference allows

multi-dimensional policy space to be collapsed

into one-dimensional space

Multi-dimensional policy (cont.)

e.g.

Expenditure on education

Expenditure on health

Unemployment insurance

If a citizen prefers a higher spending for one policy,

she also prefers a higher spending for the others

Multi-dimensional policy (cont.)

Otherwise, the Downsian model has no equilibrium.

The most important example: the division of a pie

To analyze such policy-making, two approaches are now used

Probabilistic Voting Model

Legislative Bargaining Model

(Lecture 3)

(Lecture 5)

Multi-dimensional policy (cont.)

Candidates' preference

In the basic model, candidates earn "ego rent" if elected

What if candidates share the same preference as citizens'?

Wittman (1977) and Calvert (1985)

consider such an extension of the Downsian model

See Section 5.1 of Persson and Tabellini (2000) for detail

They show the Median Voter Theorem still holds

Candidates' commitment to electoral promise

In the basic model, candidates announce the policy before election

But the winning candidate may want to change their mind

See Section 5.2 of Persson and Tabellini (2000) for detail

Citizens won't be fooled by electoral promises

Alesina (1988) drops the commitment assumption

They vote for the candidate whose bliss point is closer to theirs

Then the candidate whose bliss point is closer to the median will win

Candidates' commitment to electoral promise

If the candidate whose bliss point is closer to the median will win...

Why not a citizen whose bliss point is even closer to the median won't run for office?

We need to consider which citizens run for office

The citizen-candidate model

(Lecture 2)

Summary

We cannot use the Median Voter Theorem when

The policy issue is multi-dimensional (e.g. division of a pie)

Citizens' preference deviates from the single-crossing property

Candidates cannot commit to their electoral promise

Applications

Despite all these limitations...

the Median Voter Theorem has been applied to many contexts

for its simplicity

Guembel and Sussman (2009 Restud) for a theory of sovereign debt

Median voter may prefer repaying debt

even without any penalty for default

Bolton and Roland (1997 QJE) for the break-up of a nation

The level of taxation is chosen by the median voter

of the whole nation or of the break-away territory

Jaimovich and Rebelo (2017 JPE) for the impact of taxation on growth

Among many other applications,

the most influential is perhaps Meltzer and Richard (1981)

They argue that

the franchise extension increases the size of government

This paper gave birth to so many theories and empirical studies

See Acemoglu et al. (2016) for the latest literature review

We follow Sections 3.1-3.3 of Persson and Tabellini (2000)

to illustrate their model as an example of how to incorporate the Downsian model of elections into a model of the economy

A model of public finance

Demography

A continuum of citizens with population size normalized to 1

Each citizen type is indexed by

i

Preference

Citizen of type i has the preference over consumption & public good

c^i + H(g)

c^i + H(g)

is a concave, increasing function

H(g)

H(g)

where

A model of public finance (cont.)

Endowment

Citizen of type i is endowed with income

y^i

y^i

Income distribution is given by a c.d.f.

F(y^i)

F(y^i)

Denote the median income as

y^m

y^m

F(y^m) = 1/2

F(y^m) = 1/2

i.e.

Empirically we observe

y^m < y \equiv E(y^i)

y^m < y \equiv E(y^i)

Japan's income distribution

in 2013

Source: Ministry of Health, Labour and Welfare

(国民生活基礎調査)

Image source: http://nbakki.hatenablog.com/entry/Distribution_of_Yearly_Household_Incomes_in_Japan_2013

A model of public finance (cont.)

Government

Collect tax at a common rate of income

\tau

\tau

Produce public good by the technology

g = \int_i \tau y^i dF = \tau y

g = \int_i \tau y^i dF = \tau y

Citizen i's disposable income is equal to their consumption

c^i =(1-\tau)y^i

c^i =(1-\tau)y^i

NOTE: Since the population size is 1,

GDP is the same as GDP per capita, both expressed as

y

There are two policy variables, tax and public good

A model of public finance (cont.)

Since the government budget is balanced

g = \tau y

g = \tau y

choosing the level of public good provision gives the unique tax rate

So there is effectively one single policy

g

Let's rewrite citizens' payoff as a function of

g

Citizen i' payoff as a function of public good

\equiv W^i(g)

\equiv W^i(g)

c^i + H(g) = (1-\tau)y^i + H(g)

c^i + H(g) = (1-\tau)y^i + H(g)

=\frac{y(1-\tau) y^i}{y} + H(g)

=\frac{y(1-\tau) y^i}{y} + H(g)

=(y-g)\frac{y^i}{y} + H(g)

=(y-g)\frac{y^i}{y} + H(g)

(by \(g=\tau y\))

Normative benchmark (First-best policy)

Social planner maximizes the sum of all citizens' payoffs

\int_iW^i(g)dF= \int_i \{(y-g)\frac{y_i}{y}+H(g)\}dF

\int_iW^i(g)dF= \int_i \{(y-g)\frac{y_i}{y}+H(g)\}dF

=\frac{y-g}{y}\int_i y^idF+H(g)

=\frac{y-g}{y}\int_i y^idF+H(g)

=(y-g)\frac{y}{y}+H(g)

=(y-g)\frac{y}{y}+H(g)

=W(g)

=W(g)

Social optimum is implicitly defined by

1=H_g(g^{FB})

1=H_g(g^{FB})

Endogenize the policy

Suppose the level of public good provision g is chosen by the following political process (i.e. the Downsian model)

Each citizen simultaneously decides whom to vote, candidate A or B

Each of the two candidates simultaneously proposes policy

The candidate who obtains the majority implements the proposed policy. The payoff realizes.

Election Campaign

Election

Inauguration of a new government

Analysis

Check whether citizens' preference satisfies the single-crossing property.

If so, the Median Voter Theorem holds:

the median voter's ideal policy will be the equilibrium policy proposed by both candidates

Policy space is one-dimensional

Single-crossing property

Citizen's preference type can be expressed by

y^i

y^i

W^i(g)=(y-g)\frac{y^i}{y} + H(g)

W^i(g)=(y-g)\frac{y^i}{y} + H(g)

We need to show

if \(g > g'\) and \( y^j > y^k \), or \(g < g'\) and \( y^j < y^k \)

W^j(g) \geq W^j(g') \Rightarrow W^k(g) \geq W^k(g')

W^j(g) \geq W^j(g') \Rightarrow W^k(g) \geq W^k(g')

Equilibrium policy

Payoff of citizens with median income

Since the Median Voter Theorem holds,

it's the median income earner's ideal policy

Maximizing this payoff w.r.t. g yields

W^m(g)=(y-g)\frac{y^m}{y} + H(g)

W^m(g)=(y-g)\frac{y^m}{y} + H(g)

y^m/y=H_g(g^*)

y^m/y=H_g(g^*)

Political distortion

y^m/y=H_g(g^*)

y^m/y=H_g(g^*)

Equilibrium policy is given by

Compare this to the first-best policy

1=H_g(g^{FB})

1=H_g(g^{FB})

Since

y^m < y

y^m < y

(empirically),

g^* > g^{FB}

g^* > g^{FB}

we have the over-provision of public goods:

Impact of franchise extension

Now suppose

only citizens with income higher than can vote

\overline{y}

\overline{y}

Suffrage used to be restricted by wealth in 19c Europe

In early 20c Japan

those paying more than 15 yen as taxes could vote

cf.

The Median Voter Theorem says

the median income earner among those franchised is decisive

Impact of franchise extension

Now suppose

only citizens with income higher than can vote

\overline{y}

\overline{y}

Then the equilibrium policy is given by

\overline{y^m}/y=H_g(\overline{g^*})

\overline{y^m}/y=H_g(\overline{g^*})

where

\overline{y^m}

\overline{y^m}

is the median income conditional on

y^i>\overline{y}

y^i>\overline{y}

Empirical predictions

y^m/y=H_g(g^*)

y^m/y=H_g(g^*)

Since

H_g(g)

H_g(g)

g

y^m/y

y^m/y

g^*

g^*

\overline{y^m}/y=H_g(\overline{g^*})

\overline{y^m}/y=H_g(\overline{g^*})

y^m < \overline{y^m}

y^m < \overline{y^m}

limited franchise

reduces public good provision

\overline{y^m}/y

\overline{y^m}/y

\overline{g^*}

\overline{g^*}

Evidence

There are so many studies exploring the relationship

between democracy and government expenditure

Some find no relationship; others find a positive association

But democracy may be mismeasured

Universal suffrage may not be actually enforced

Democracy is endogenous

More educated population demand

both more public good and more democracy

e.g.

Electronic voting in Brazil

Fujiwara (2015)

Paper-based voting effectively disenfranchises those illiterate

Image source: Figure 1 of Fujiwara (2015)

Electronic voting in Brazil

Fujiwara (2015)

Electronic voting was introduced in the mid-1990s

Image source: Figure 1 of Fujiwara (2015)

Electronic voting in Brazil

Fujiwara (2015)

If you enter the candidate number correctly...

Image source: Figure 1 of Fujiwara (2015)

Electronic voting in Brazil

Fujiwara (2015)

If you enter a wrong number...

Image source: Figure A1 of Fujiwara (2015)

Electronic voting in Brazil

Fujiwara (2015)

Brazil is a federation of 27 states

Every four years

each state holds elections for its legislature and governor

In 1994 elections, all states used paper-based voting

In 1998, municipalities with 40,500+ registered voters adopted electronic voting

In 2002, all used electronic voting

Electronic voting in Brazil

Fujiwara (2015)

In 1998, municipalities with 40,500+ registered voters adopted electronic voting

If you compare municipalities just above the 40,500 threshold

with those just below...

Any difference in outcomes can be interpreted

as the impact of electronic voting

Regression Discontinuity Design

# of valid votes as % of turnout

Image source: Figure 2 of Fujiwara (2015)

The 1996 elections were for state governments, not municipal

Image source: Figure 4 of Fujiwara (2015)

Policy outcomes

Policy outcomes are the same for municipalities within a state

We cannot use RDD for estimating the impact on policies

Instead...

Construct a state-level panel data

for every 4-year period from 1995-2006

Policy outcomes (cont.)

Obtain each state's share of registered voters in municipalities that introduced electronic voting in 1998, denoted by

Image source: Figure 5 of Fujiwara (2015)

S_i

S_i

If electronic voting increases spending for the illiterate

the 1998 elections should cause a positive association

S_i

S_i

g^{99-02}_i-g^{95-98}_i

g^{99-02}_i-g^{95-98}_i

0

1

Impact of EV

If electronic voting increases spending for the illiterate

After the 2002 elections, all municipalities reach the same level

S_i

S_i

0

1

Impact of EV

g^{03-06}_i-g^{95-98}_i

g^{03-06}_i-g^{95-98}_i

If electronic voting increases spending for the illiterate

So the comparison between 99-02 and 03-06 yields

S_i

S_i

0

1

Impact of EV

g^{03-06}_i-g^{99-02}_i

g^{03-06}_i-g^{99-02}_i

Identifying assumption

No omitted variable exhibits the same reversal of an association with policy changes

S_i

S_i

\Delta g_{i,94-98}

\Delta g_{i,94-98}

S_i

S_i

\Delta g_{i,98-02}

\Delta g_{i,98-02}

Estimation equations

\Delta g_{ir,94-98} =\alpha^{98} + \beta^{98} S_i + \gamma^{98}_r + \varepsilon_{ir,94-98}

\Delta g_{ir,94-98} =\alpha^{98} + \beta^{98} S_i + \gamma^{98}_r + \varepsilon_{ir,94-98}

\Delta g_{ir,98-02} =\alpha^{02} + \beta^{02} S_i + \gamma^{02}_r + \varepsilon_{ir,98-02}

\Delta g_{ir,98-02} =\alpha^{02} + \beta^{02} S_i + \gamma^{02}_r + \varepsilon_{ir,98-02}

Predictions

\beta^{98}+\beta^{02}=0

\beta^{98}+\beta^{02}=0

\beta^{98} (= -\beta^{02})

\beta^{98} (= -\beta^{02})

Impact of electronic voting is given by

Region

fixed

effects

Results on policy

\hat{\beta}^{98}

\hat{\beta}^{98}

\hat{\beta}^{02}

\hat{\beta}^{02}

\frac{\hat{\beta}^{98}+\hat{\beta}^{02}}{2}

\frac{\hat{\beta}^{98}+\hat{\beta}^{02}}{2}

Health care spending goes up by 34% of the sample mean

Sample

Mean

Results on infant health

\hat{\beta}^{98}

\hat{\beta}^{98}

\hat{\beta}^{02}

\hat{\beta}^{02}

\frac{\hat{\beta}^{98}+\hat{\beta}^{02}}{2}

\frac{\hat{\beta}^{98}+\hat{\beta}^{02}}{2}

Final thought

The Median Voter Theorem is perhaps still useful for

Compare democracy to non-democracy

Introduce politics into a complicated model of the economy

Its most important implication is the centripetal force in politics

To obtain the majority,

politicians want to propose a centrist policy

(if they can commit)