Voting for Public Goods
(Fisher, Ch. 3)
© Allen C. Goodman 2009
Bread and Schools
60
Prefers
Schools
Schools
• Suppose that we
live in a suburb.
• Suppose there are
10 residents. Each
one earns $60,000.
• They can spend it
on bread, or
schools.
Prefers
Bread
Bread
60
Bread and Schools
60
Prefers
Schools
Schools
• They have to pick
a tax level that
each one of them
will pay.
• If they decide on
$4,000, each will
pay $4,000.
Prefers
Bread
Bread
60
Bread and Schools
• Let’s add a few
more “identical”
people.
s1 > s 2 > s 3 > s 4 > s 5
60
s1
• We have five
possible levels of
“schools”
Schools
s2
s3
s4
s5
Bread
60
Bread and Schools
• We have five different
levels of taxes.
Schools
• Alternatively, individuals
1-5 are willing to give up 60
different amounts of
s1
bread to get school
s2
resources.
t1 > t2 > t3 > t4 > t5
s3
s4
s5
Bread
60
How do we decide?
1
2
3
4
Schools
• Consider a politician.
She has to win an
60
election, and she has to
s1
get enough votes by
s2
promising the right
amount of school
s3
resources
s4
• Suppose she promises s5.
Person 5 is happy (he
s5
didn’t want much). But
everyone else wanted
more. So politician loses
election 4-1.
Role
Playing
2 Candidates
5
Bread
60
How do we decide?
1
60
s1
2
3
s2
Schools
• Suppose she promises s4.
Persons 1, 2, and 3 are
happier because they’re
getting closer to what they
want. But she’ll still lose
3-2.
• Suppose she now promises
s3. She’ll win the election
because Persons 1 and 2
are happier yet, and Person
3 is happiest, he’s getting
exactly what he wants.
4
s3
5
s4
s5
Bread
60
If you don’t believe me ...
1
60
s1
2
3
s2
Schools
• Suppose another politician
promises s2. Person 3
won’t be happy anymore
because you’re providing
MORE school resources
than he wants … so he’ll
vote against it.
• KEY POINT !!! The
median voter is decisive.
Eq’m school will be at s3.
Each voter will pay 60 - b3
in taxes and get s3.
4
s3
5
s4
s5
b3
Bread
60
Median Voter and Optimum
• We’ve seen that we will agree on some
particular amount, by majority vote.
• Is this amount economically optimal, based
on the rule that we developed in the
previous lecture?
• Why or why not!
$ spent
Single Peaked Preferences
• Suppose you had 3
people with indicated
preferences.
• Suppose they’re not
single-peaked, i.e.
• If they are single
peaked, there is a
clearly defined
“winner” and the
individual gets less
satisfaction if he/she
moves away from it.
Person
A
First
Choice
4000
Second
Choice
Third
Choice
Less U
Even less U
6000
8000
Less U
B
6000
8000
C
8000
4000
Level 1
Level 2
Level 3
4000
6000
8000
Even less U
4000
6000
What about
this?
Another Way to Look at It
UA
UB
Single
peak
4
6
8
Spending
UC
Single
peak
4
6
8
Spending
?
?
4
6
8
Spending
4,000 v. 6,000
UA
UB
Single
peak
UC
Single
peak
?
?
Preferred!
4
6
8
4
6
NOT
Preferred!
8
4
6
8
6,000 v. 8,000
UA
UB
Single
peak
UC
Single
peak
?
?
Preferred!
Preferred
4
6
8
4
6
8
NOT!
4
6
8
4,000 v. 8,000
UA
UB
Single
peak
UC
Single
peak
?
?
Preferred!
NOT
4
6
8
4
6
8
NOT!
4
6
8
So?
• $4,000 beats $6,000
• $6,000 beats $8,000, BUT
• $8,000 beats $4,000!
Choices
aren’t
transitive
Why does this happen?
• Person C does not have single-peaked
preferences, but is rather an extremist! …
alternatively he/she HATES mediocrity.
• C becomes less and less happy until
spending gets very low.
• C is unhappy with moderate positions.
How much should we worry?
• If we believe in
standard demand
theory, we shouldn’t
worry too much
about the singlepeaked problem
• Why?
Price in $
Demand theory is
broadly consistent
with Single-Peaked
preferences
• At Q*, we have
optimal amount of Q.
• As we go further
away in either
direction, happiness ↓.
Quantity
Q*
Median Voter Theorem
• If voters’ preferences are single-peaked,
• if the choice to be made by voting is
represented along a single continuum,
• if all alternatives are voted on, and
• if voters act on their true preferences,
• THEN the choice selected by majority vote
is the median of the desired outcomes.
What does median voter model say?
• If you have some number of jurisdictions,
one can argue that the levels of schools, fire
protection, police protection are broadly
consistent with consumer preferences.
• Is it perfect?
– No, not all citizens vote.
– If there are a lot of issues, the same citizen is
not likely to be the median voter on every issue.
Median and Optimum
• Median may be close to optimum as
determined by theoretical criteria.
• BUT, it may be either greater than or less
than the optimum.
• By the very nature of the equilibrium, some
substantial number might want more … and
some other substantial number might want
less.
Other Models
Price ($)
Ev is preferred by
median voter
If referendum is
rejected, Expenditures
revert to Er.
• Some (generally
conservative) economists
propose a model in which
government officials wish
to:
Loss of Consumer
Surplus of C1.
C1
– Maximize the amount of
government spending.
– Remain in office.
• How does this work?
Expenditure
Er
Ev
Other Models
Price ($)
Ev is preferred by
median voter
If referendum is
rejected, Expenditures
revert to Er.
• Instead, government
officials, in trying to
maximize expenditures,
pick as much as EM. Why?
• How does this work?
Loss of Consumer
Surplus of C1.
C1
C1
C1
C1
Expenditure
Er
Ev
EM
Comments
• Median voter model seems to explain multiple suburbs.
• Does the monopoly model explain central cities?
– How does one explain the Detroit Schools?
– How does one explain the Detroit City Council?
• These are NOT unique to Detroit.
• Many large cities have these problems.
• Are they unique to large cities?
Comments
• Median voter model seems to address consumer
preferences, BUT a large group of people are dissatisfied.
• Monopoly models may appeal to some voters, BUT they
don’t explain how a government that continually does not
satisfy median voter stays in power.
• Michigan’s Headlee Amendment makes it very difficult to
raise taxes. Suggests that at least a lot of Michigan voters
are concerned about governmental powers.
NEXT!
Chapter 4