Public Choice – Part 2: „Direct Democracy“
Part 2
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy
University of Jena
Carl-Zeiss-Str. 3
07743 / Jena
room:
5.36
phone.:
03641-943257
email:
[email protected]
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
1
Part 2 „Direct Democracy“ – some subjects
What we are going to talk about:
• Costs of decision making (Mueller, 2003, chapter 4.3 and 4.4)
• Cyclical majorities (among others: Mueller, 2003, chapter 5.2)
• Agenda manipulation (Mueller, 2003, chapter 5.12)
• Tactical voting
• Median voter (among others: Mueller, 2003, chapter 5.3)
• Log-rolling (Mueller, 2003, chapter 5.9 ff.)
• Voting procedures
• Condorcet‘s Jury Theorem (Mueller, 2003, chapter 6.1)
• Clubs (Mueller, 2003, chapter 9.1 ff.)
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
2
Costs of decicion making
Question 1:
Characterize the different types of cost that come along with
collective decisions. Show a reasonable course of these cost
dependent on the necessary majority. What is the optimal
majority?
• Two general types of cost:
- External cost: borne by the “losers” of the decision (minority)
(e.g. decision: higher redistribution from “rich” to “poor”  The “rich” bear the external cost.)
- Direct cost of decision making: information and negotiating cost to achieve a majority
• The smaller the number of “losers”, the lower are – ceteris paribus – the external cost.
That means: the higher the necessary majority, the lower are the external cost of a decision.
• The higher the number of people who have to agree on the decision, the more complicated are
the negotiations. That means: the higher the necessary majority, the higher are the direct cost.
cost
Remark: A similar question could
refer on the kink at the 50
percent level or on the
optimal majority (perhaps
in consideration of
different kinds of issues).
Cex+ direct cost
direct cost
Cex
50%
Cmin
100%
necessary majority
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
3
Optimal majority – Severe decisions / big issues
Question 2:
What does theory tell us about the optimal majority for very
severe / weighty decisions?
• Big issues / severe decisions mean high external cost for the losers of the decision (minority).
• Big issues are for instance the constitution and human rights.
• Examples for extreme external cost for the losers of decisions on the constitutional level:
- minorities such as slaves
• Domination of the external cost of decisions can be shown by a very steep external-cost-curve.
• The optimal majority for these kinds of
decisions with respect to the expected
cost is the entire population.
cost
Cex+ direct cost
Cmin
Cex
• (Of course, one can imagine that it is
very difficult change the status quo
under unanimity rule.)
direct cost
50%
• Therefore, unanimity rule should be
chosen for very weighty decisions
(constitution, human rights etc.) in order
avoid dramatic losses for minorities.
100%
necessary majority
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
4
Cyclical majorities
Question 3:
Show how so called cyclical majorities can occur in polls under
the majority rule.
Question 4:
A flat-sharing community consisting of 3 students decided to
spend next Saturday together. The decision on the joint activity
has still to be made. There are three alternatives. A combination
of activities is not possible. The preferences of the three students
are as follows:
Priscilla
John
Victoria
1.
BBQ
Theater
Hiking
2.
Hiking
BBQ
Theater
3.
Theater
Hiking
BBQ
(a) What is the outcome of the voting?
(b) Suppose Priscilla knows the preferences of her mates and
can – as the “senior” of the flat – set the agenda.
What order would Priscilla choose?
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
5
Cyclical majorities
To question 3:
„Condorcet’s-Paradox“ (Marquis de Condorcet, 18th century):
• If there are more than two alternatives a collective decision can result in cyclical
majorities, although the individual sets of preferences are consistent.
(Transitivity: if A > B and B > C, than must A > C).
The collective set of preferences is not transitive: A > B; B > C and C > A
The result of the collective decision depends on the agenda
• The necessary condition for cyclical collective preferences are “multimodal”
preferences (not single peaked preferences)
• Not single peaktness means that – independent from the disposition of alternatives
– at least one member of the committee prefers the alternatives at the margins in
comparison with the one in the middle
• If the issue to be decided is scaled one-dimensional („small“ – „medium“ – „big“)
multimodality occurs if at least one voter prefers the “extreme” alternatives
Question: Give some reasonable examples for multimodal preferences.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
6
Cyclical majorities
To question 4:
Priscilla
John
Victoria
1.
BBQ
Theater
Hiking
2.
Hiking
BBQ
Theater
3.
Theater
Hiking
BBQ
Preference
(rank)
(a) The phenomenon of cyclical majorities occurs:
John
1.
• BBQ wins against Hiking
(P and J vs. V)
• Theater wins against BBQ
(J and V vs. P)
• Hiking wins against Theater (P and V vs. J)
2.
The outcome of the decision remains unclear and
depends solely on the agenda.
(b) If Priscilla knows the preferences of her mates
Priscilla
and can set the agenda she will choose the
following order:
Activity
Victoria
3.
BBQ
Hiking
Theater
1st round:
Hiking vs. Theater  hiking wins
2nd round:
BBQ vs. Hiking  BBQ wins
BBQ as “best alternative” because of the agenda setting
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
7
Agenda manipulation (example from the lecture)
x2
U P*
U J**
A‘‘
U V**
P = AP
S
A‘
J
V
A
U V*
U J*
x1
• Decision about the purchase of the
amount of two goods (X1 and X2)
• The circles (U i) represent utility levels of
the 3 students (indifference circles)
• The points P, J and V represent the
combinations with the highest utility for
Priscilla, John and Victoria
• Priscilla (the senior of the flat) is still the
agenda setter (and knows the preferences
of her mates).
• The starting point is S
• Priscilla chooses A as the first alternative
against S  A is closer to V and J than S
 A wins
• Now, Priscilla chooses A’ as the next
alternative against A  A’ is closer to P
and J than A  A’ wins
• The next alternative chosen by Priscilla is
A’’  A’’ is closer to P and V and wins
against A’ (By the way, this step is not
absolutely necessary.)
• In the next round, Priscilla chooses A P
which wins against A’’
• Priscilla now won’t choose any
alternative that is going to lose against A P
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
8
Tactical voting
Question 5:
Back to the example with the 3 students:.
Priscilla
John
Victoria
1.
BBQ
Theater
Hiking
2.
Hiking
BBQ
Theater
3.
Theater
Hiking
BBQ
Suppose no one of the three students could set the agenda (e.g.
agenda chosen by throwing dice). John knows the preferences of
his mates exactly. How is he going to vote in order to prevent the
hiking trip?
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
9
Tactical voting
Priscilla
John
Victoria
1.
BBQ
Theater
Hiking
2.
Hiking
BBQ
Theater
3.
Theater
Hiking
BBQ
• P and V prefer the hiking trip (which J wants to avoid) compared to the theater.
Therefore, John has to prevent that these two alternatives reach the final round.
• There are three possible orders of the collective decision:
1. 1st round BBQ vs. Hiking  BBQ wins  Hiking prevented o.k. 
2. 1st round Hiking vs. Theater  Hiking wins
 Hiking loses in the 2nd round against BBQ  Hiking prevented o.k. 
3. 1st round BBQ vs. Theater  Theater wins
 Theater loses in the 2nd round against Hiking  Hiking not prevented !!!
• If the decision in the first round is BBQ vs. Theater, John has to vote in favor of
the BBQ (against his actual preference!!!). So, the BBQ wins in the first round.
In the 2nd round the BBQ wins against the hiking-trip. Hiking prevented o.k. 
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
10
Median voter I
Question 6:
Our 3 student have decided to give a party in their flat. Now they
are going to make a decision about the number of guests. John
wants to celebrate the evening in a more intimate way with only 5
guests. Victoria wants to invite exactly 30 guests and Priscilla
wants to invite 50 guests. A convergence of the preferences
seems to be impossible. Therefore, a collective decision via
majority rule has to be made.
(a)
How many guests are going to be invited?
(b)
How would the decision be changed if the next-door student
flat (4 students) was involved in the collective decision and
two of the neighbors preferred 10 guests and the other two
wanted 50 guests?
(c)
What can be said about the allocative efficiency of the
collective decision?
(Remark: The hosts just provide their flat and the music, i.e.
BYOB (“bring your own beverages”)).
Underpin your answers with suitable charts.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
11
Median voter I
(a)
• One-dimensional issue (number of guests)
• No compromise  only 3 alternatives
• Clear cut preferences
(no multimodality)
- John (J):
utility
John
Victoria
Priscilla
5
- Victoria (V): 30
- Priscilla (P): 50
Who wins?
guests
5
30
50
• However the agenda looks like, Victoria / the alternative “30 guests” will win the
decision:
- 5 vs. 30: V and P vote down J  “30” wins
- 30 vs. 50: J and V vote down P  “30” wins
• 30 guests will be invited.
• Remark: The result won’t change if John’s preference was “at most 5 guests” and Priscilla’s
preference “at least 50 guests”. Only the shape of the utility functions would change.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
12
Median voter I
(b)
• 4 more voters (neighbors) with the following preferences:
- N1 und N2: „10 guests“
N1,2
utility
N3,4
Victora
- N3 und N4: „50 guests“
• Question: How does it change
the result?
John
Priscilla
• Answer: Nothing changes
• Proof:
5
10
30
50
guests
- 10 vs. 30: V, P, N3 and N4 vote down J, N1 and N2  “30” wins
- 30 vs. 50: J, N1, N2 and V vote down P, N3 and N4  “30” wins
• Victoria wins even if an even number of voters join the committee to her “left”
and her “right”.
• In general:
In the case of one-dimensional alternatives, and an uneven number of voters, the
preference of the median voter wins the election (if the preferences of the voters
are not multimodal). (Downs)
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
13
Median voter I
(c)
• “BYOB”:
- Costs of the party for the hosts are independent of the number of guests
- Only fixed costs, marginal costs are more or less zero
• Different utility functions of the hosts mean different “willingness to pay” for
additional guests.
• Marginal analysis to test the efficiency
(differentiation of the utility functions with respect to the number of guests)
• What kind of good is the number of guests?
• Number of guests is a public good (no rivalry in “consumption” and no exclusion).
• To find the optimal number of guests, the marginal willingness to pay of all members
of the committee have to be added (vertically) to get the collective willingness to pay.
• The optimal number of guests is reached if the collective marginal benefit equals
the marginal costs. (similar: Lindahl-Samuelson condition)
• Marginal costs are zero. Therefore optimal number of guests is reached if the
collective marginal utility is zero.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
14
Median voter I
MB = WP
 WP
MB
marginal benefit
WP
willingness to pay
MC
marginal costs
Gopt
optimal number of guests
WP(P)
WP(V)
WP(J)
MC = 0
guests
5
Gopt
30
50
• It would be pure chance if the outcome of the collective decision equals the optimal
solution ( WP = MC).
• In our example the result of the collective decision (30 guests) is not efficient because
not the sum of the marginal willingness to pay of all members of the committee decides,
but only Victoria’s utility function.
• The optimal / efficient number of guests is indeed lower than 30.
• Conclusion: Collective decisions under the majority rule do not guarantee efficient
solutions. A social optimum (maximum welfare) would be pure chance.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
15
Question 6 – Remark
• Please note, that in this very example negative marginal utilities occur because the individual
utilities of our three students reach their maximum level at a precise number of guests.
• If the number of guests invited exceeds the preferred level (e.g. 5 for John), the individual utility
is declining. This rather unorthodox example causes the illustrated slope of the total marginal
benefit / willingness to pay  WP on page 15.
• In cases where individuals do not receive a negative marginal utility from an extra unit of
consumption (which is the most common case you will find in most textbooks), the slope of the
total marginal benefit changes (“kinked”  WP ).
• The condition for the optimal provision / consumption of the public good
still holds  MB =  MC.
MB = WP
• Here is an example with positive (but constant) marginal costs and
three individuals (A, B and C):
MBC
 WP
 MC > 0
MBA
MBB
Xopt
MB
marginal benefit
WP
willingness to pay
MC
marginal costs
Xopt
optimal consumption level
X
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
16
Median voter II
Question 7:
How can the shift of two dominating candidates towards the
middle of the political spectrum be explained?
number of voters
L:
(+)
more left oriented candidate
R: rather conservative candidate
hw: „half way“ between the twocandidates
M: median voter
(-)
„left wing“
L
hw M
R
„right wing“
• Assumption: clear cut, one dimensional decision („left or right“) between two dominating
•
•
•
•
candidates
The candidate that attracts the median voter will win the election (50 percent plus – at least – one).
The median voter in a modern democracy is likely to have moderate preferences somewhere close to the middle
of the political spectrum. (normal distribution of voters as a stylized fact)
The candidate that is going to lose the election will “move” towards the median voter by modifying the political
program in favour of the preferences of the median voter. The other candidate is going to react the same way.
As a consequence, both candidates are “moving” towards the median voter who is supposed to be
somewhere in the middle of the political spectrum.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
17
Log-rolling I
Question 8:
Our three students are about to pep up their party with some
additional drinks. This, indeed, creates additional costs. Three
different drinks are worth considering, whereas the provision of
several drinks is possible, as well as none.
The net utilities of the students with respect to the different drinks
are structured as follows:
Priscilla
John
Victoria
Margarita
80
-20
-30
Whisky
-20
100
-50
Bordeaux
-10
-30
150
(a)
What is the optimal solution under the Kaldor-Hickscriterion?
(b)
Which drink(s) would be bought if the students decided about
“buying” and “not buying” on each drink?
(c)
What result would be likely if log-rolling was possible?
Justify your answers briefly.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
18
Log-rolling I
(a)
• Kaldor-Hicks: compensation (of „losers“) possible
Priscilla
John
Victoria
 individual
utility net
Margarita
80
-20
-30
30
Whisky
-20
100
-50
30
Bordeaux
-10
-30
150
110
• Sum of individual net utilities of the purchase every drink is positive, i.e. the
winner of the purchase could compensate the losers
• The purchase of all three drinks would be optimal.
(b)
• A binary decision (“purchase” vs. “not purchase”) on each of the drinks would turn
out in the rejection of all three drinks (P and J vote against the Bordeaux; J and
V vote against the Margarita; P an V vote against the Whisky).
• No one of the drinks would be purchased.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
19
Log-rolling I
(c)
• „Log-rolling“ = exchange of votes (more or less explicit) in voting coalitions
Priscilla
John
Victoria
Margarita
80
-20
-30
Whisky
-20
100
-50
Bordeaux
-10
-30
150
• Victoria could offer Priscilla to vote in favor of the margarita if Priscilla assured to
vote in favor of the Bordeaux.
• The net utility of Victoria would be 120. The net utility of Priscilla would be 70.
• The loser of this log-rolling would be John with a net loss of 50.
Question: Is this result stable?
• First, it is questionable if both partners of the coalition really stick to the agreement.
Why?
• Second, there is obviously a strong incentive for John to offer an exchange of votes
by himself. What could John do (besides leaving the flat ;-)?
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
20
Log-rolling I
• Prisoner‘s dilemma:
- Victoria votes in favor of the margarita  margarita is purchased
- Why should Priscilla stick on the agreement?
(Higher individual utility without the Bordeaux)
- It is very likely that Priscilla vote against the Bordeaux once the margarita has been put
through (higher net utility = incentive to defect).
- If V anticipates P’s behavior, she’ll vote against the margarita
 no agreement and log-rolling at all (legally binding contract hardly possible)
• But, the dilemma could be overcome if the two girls build up some sort of good reputation
(long-term coalition between the girls, also on other issues).
• But still, stability of the coalition questionable because counter strategies of John are likely:
- John offers P to vote in favor of the margarita if P vote against the Bordeaux
 Margarita is purchased; no Bordeaux and no Whisky
- Net utility of Priscilla raises by 10 up to 80
- Net loss of John is reduced by 30
- But, counter strategy of Victoria very likely  …  …
Conclusion: If log-rolling is possible, the exploitation of outsiders can occur.
But, no clear cut prediction about the outcome of collective decisions can be made
(because of bluffing, cheating, defecting and counter strategies), if log-rolling is
possible (which is very often the case).
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
21
Log-rolling II
Question 9:
3 countries are about to decide about different issues, to be
executed and financed collectively. The following matrix shows
the different issues and the net values of these issues for two
countries.
UK
Germany
France
Cut of agricultural subsidies
+ 50
-30
Introduction of stricter emissions thresholds
-20
-50
Increase in support of new technologies
-10
+50
Show how log-rolling processes can result in welfare losses.
In order to do so, choose reasonable individual net utilities for
Germany and illustrate a possible log-rolling process.
Discuss the results.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
22
Log-rolling II
UK
Germany
France
A: Cut of agricultural subsidies
+ 50
-20
-30
B: Introduction of stricter emissions
thresholds
-20
+30
-50
C: Increase in support of new
technologies
-10
-10
+50
Welfare net
0
- 40
+ 30
Log-rolling can be welfare improving as well as welfare deteriorating. The
final result depends on who is making a coalition with whom.
UK
Germany
France
A&B
+ 30
+ 10
-80
- 40
A&C
+ 40
- 30
+ 20
+ 30
B&C
- 30
+ 20
0
- 10
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
Welfare net
23
Log-rolling III
Question 10: Does log-rolling rather enhance or hinder democracy and does it
rather increase or reduce social welfare?
Advantages:
• Lacing of “political packages” could increase social welfare
 without log-rolling status quo (see again question 8)
• Log-rolling as exchange in line with market requirements  efficiency gains
• Attention to different intensities of preferences (e.g. “strong” preferences of minorities)
• Enhancing democratic principles (negotiations  compromises  majorities)?!
• Enforcement of necessary (and sometimes “hard”) reforms
Disadvantages:
•
•
•
•
•
Welfare losses (compared to status quo) possible (see question 9)
Stability of agreements questionable (cheating, bluffing…)
Negative perception of the des political “horsetrading” by the citizens
Dominance of well organized pressure groups (with sufficient capital) as a risk for democracy?!
Inefficient expansion of state budgets and activity in favor of strong (unproductive) interest
groups (oversupply of “public goods”, inefficient subsidies, overprotection etc.)
• Time-consuming and expensive negotiations about policy-packages
• Inconsistency of policies due to the combination of totally different issues
(e.g. taxes on cigarettes and the financing social security).
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
24
Voting procedures
Question 11: Give a brief overview of some voting procedures.
Unanimity rule:
• Every member of the committee has to agree upon the issue.
Majority rule:
• The alternative which is ranked first by more than one half of the voters wins.
• “Qualified majority” if more than 50% (plus one vote) is required to pass an issue, (e.g. two third
for changes in the constitution).
Majority rule, runoff vote:
• If one of the n alternatives receives more than half of the votes, this alternative is declared the
winner (decision is made).
• If no alternative receives more the 50%, a second vote is held between the two alternatives that
received the most votes in the first round. The alternative receiving the most votes in the second
round is the winner. (e.g. election of mayors or presidents in many countries)  example:
Plurality rule:
• The alternative which is ranked first by the largest number of voters wins.
• “The winner takes it all.”
Condorcet rule:
• The alternative which defeats all others in pairwise votes (under majority rule) wins.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
25
Voting procedures
Borda rule:
• Each of the n alternatives is given a score of 0 to (n-1) based on it’s rank in the preference
ordering of the voter.
• The alternative ranked first receives (n-1) points, the alternative ranked second receives (n-2)
points, and so on. The lowest-ranked alternative receives no point.
All points for each alternative are added up. The alternative with the highest score wins.
Approval voting:
• Each voter votes for k alternatives (1 ≤ k ≤ n) he / she ranks highest out of the n alternatives.
• Every voter can choose k independently.
• The alternatives with the most votes is declared the winner.
Hare system / Instant-runoff voting:
• Every voter calls a preference ordering including all alternatives.
• If no alternative reaches the absolute majority (50% plus one), the alternative with the least
“number 1” rankings is eliminated.
• One vote is transferred to the alternative ranked second in the preference ordering of a voter
who ranked the eliminated alternative first.
• The procedure can be repeated until one alternative gets the absolute majority or until only one
alternatives remains.
(The Coombs system is pretty similar. The alternative which is ranked last at most is excluded.)
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
26
„The Fatal Vote“ [Leininger, W. (1993)]
Question 12:
In 1991 the German Bundestag had to decide about the future location of
the parliament and the government. Three alternatives had been chosen in
advance:
A.
The government remains in Bonn and the parliament moves to Berlin.
B.
Both, the parliament and the government move to Berlin.
C.
Both, the parliament and the government stay in Bonn.
The preferences of the 657 (two not present) members of parliament (MdB)
could be discovered. The preference sets are structured as follows:
Profile 1: A > B > C (116 MdB)  ca. 17,6%
Profile 2: A > C > B (30 MdB)  ca. 4,5%
Profile 3: B > A > C (81 MdB)  ca. 12,3%
Profile 4: B > C > A (140 MdB)  ca. 21,3%
Profile 5: C > A > B (140 MdB)  ca. 21,3 %
Profile 6: C > B > A (150 MdB)  22,8%
What would have been the result of the collective decision under the
following voting procedures:
(a) majority rule (over all three alternatives); (b) plurality rule;
(c) majority rule with runoff vote, (d) Condorcet rule, (e) borda rule and
(f) approval voting?
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
27
„The Fatal Vote“
Profile 1: A > B > C (116)
Profile 3: B > A > C (81)
Profile 5: C > A > B (140)
Profile 2: A > C > B (30)
Profile 4: B > C > A (140)
Profile 6: C > B > A (150)
(a) Majority rule over all 3 alternatives (alternative with more than 50 percent wins):
• A: 116 (profile 1) + 30 (profile 2)
 146 = 22,22%
• B: 81 (profile 3) + 140 (profile 4)
 221 = 33,63%
• C: 140 (profile 5) + 150 (profile 6)  290 = 44,15%
 No absolute majority  no decision
(b) Plurality rule (alternative ranked first wins):
 C is obviously ranked first (290 votes) and is declared winner.
(c) Majority rule with a runoff vote:
• 1st round: no absolute majority, A is eliminated
• 2nd round: B vs. C:
- B: 116 (profile 1) + 81 (profile 3) + 140 (profile 4)  337
- C: 30 (profile 2) + 140 (profile 5) + 150 (profile 6)  320
 B wins under majority rule with runoff vote.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
28
„The Fatal Vote“
Profile 1: A > B > C (116)
Profile 3: B > A > C (81)
Profile 5: C > A > B (140)
Profile 2: A > C > B (30)
Profile 4: B > C > A (140)
Profile 6: C > B > A (150)
(d) Condorcet rule (pairwise vote):
• A vs. B:
- A: 116 + 30 + 140 = 286
- B: 81 + 140 + 150 = 371
 B wins
• B vs. C:
- B: 116 + 81 + 140 = 337
- C: 30 + 140 + 150 = 320
 B wins
• A vs. C:
- A: 116 + 30 + 81 = 227
- C: 140 + 140 + 150 = 430
 C wins
Which alternative is the Condorcet-winner?
 B is the “Condorcet-winner”. (no voting cycle in this case)
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
29
„The Fatal Vote“
Profile 1: A > B > C (116)
Profile 3: B > A > C (81)
Profile 5: C > A > B (140)
Profile 2: A > C > B (30)
Profile 4: B > C > A (140)
Profile 6: C > B > A (150)
(e) Borda rule (scores: n – 1, n – 2, …):
• A: 116 · (3 – 1) + 30 · (3 – 1) + 81 ∙ (3 – 2) + 140 ∙ (3 – 2) = 513
• B: 81 · 2 + 140 ∙ 2 + 116 + 150 = 708
• C: 140 ∙ 2 + 150 ∙ 2 + 30 + 140 = 750
 Under Borda rule C wins.
(f) Approval voting (Every voter can choose up to three alternatives):
• The results obviously depends on the specific behavior of the members of parliament.
• No clear cut prediction can be made.
• Example:
- No MdB votes for all three alternatives
- Those with profiles 4 and 6 only vote for their first rank
- Those with profiles 1, 2, 3 and 5 vote for the first and the second rank
What is the result in this example?
 A: 367 B: 337 C: 320  Under Approval voting (in this example) A wins.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
30
Condorcet‘s Jury Theorem
Question 13: What theorem gives strong support for a high number of “judges”
and therefore for referenda? Explain the nature of this theorem
and discuss the necessary assumption.
• Condorcet’s Jury Theorem
• Example: „guilty or not guilty“?!
pi
probability of the judge i to make the „correct“ decision
(assumption in this example: pi for all judges = 0,6)
Pn probability of the jury to make the correct decision under majority rule
a)
One judge:
Pn = P1 = 0,6
b)
Three judges A, B and C (intuitive example):
Correct decision if 2 or 3 judges are correct (majority)
Pn = P3 = 0,6 * 0,6 * 0,6 + 0,6 * 0,6 * 0,4 + 0,4 * 0,6 * 0,6 + 0,6 * 0,4 * 0,6
A, B, C correct
A, B correct
P3 = 0,216 + 3 * 0,144 = 0,648 > 0,6
B, C correct
A, C correct
P3 > P 1
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
31
Condorcet‘s Jury Theorem
c) 5 Judges
5!

 h
5 h
pn  p 5   
0,6

0,4

h  3  h! (5  h)! 
5
P5 = 0,3456 + 0,2592 + 0,07776 = 0,6824 > 0,648
h=3
d)
h=4
(h = (n+1)/2 = majority)
P5 > P 3
h=5
11 Judges: Pn = p11 = 0,753 > 0,6824
P11 > P5
51 Judges: Pn = p51 = 0,926 > 0,753
P51 > P11
more than 100 judges: pn = pretty close to 1

Given the necessary condition (pi > 0,5), the higher the number of judges, the higher the
probability of the jury to make the correct decision.

Necessary condition:
The individual probability of the judges to make the correct decision is larger than 0,5.
Is it realistic to assume this in reality?
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
32
Exit and voice in private markets
Question 14: How can individuals express their preferences on private markets
A. O. Hirschman (1970)?
Simply by their purchase pattern:
• Decision to purchase a product or not (“entry” or “exit”)
• Decision about the amount of the good to be purchased
• Changing the provider / the brand
• These decisions are determined by individual preferences and depend on price and quality
By “voice”:
• Complaints (e.g. customer service) or commendations
• “Sale or return” (return if the quality is not sufficient)
• Influence the price-quality nexus through bargaining
Necessary condition:
• Free entry and exit and a functioning price system (i.e. no monopolies)
• At least some information about the quality available
If these conditions are fulfilled, individuals can express their preferences effectively.
This is usually the case in private markets (private goods).
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
33
Clubs I
Question 15:
(a) Characterize club goods.
(b) How can the optimal size of a club be discovered if crowding
costs occur? Use a chart.
(c) How can individuals express their preferences if the formation of
clubs is possible? What are the necessary conditions to reach
global optimality?
(a) & (b)
• Club goods are characterized by non-rivalry in consumption and – in contrast to pure public
goods – the possibility to exclude individuals from the consumption / membership.
• Note that sometimes some sort of rivalry in consumption occurs (e.g. overcrowding).
MB
MC
F/3
F/4
F/8
MB
4
8
N*
marginal benefit (of an additional member)
MC
marginal costs (crowding costs)
N
number of club members
N*
optimal club size
• Club members share the fixed costs (F) of the
provision of the club good. No additional variable
costs until a certain club size is reached.
MC
3
MB
Club size N
(members)
• At a certain club size, (increasing) crowding costs
occur (positive and increasing MC).
• Optimal club size where MB = MC
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
34
Clubs I
(c)
• Different clubs are competing in order to attract members (to finance the provided pool of
club goods). Non-members are excluded.
• Individuals choose the clubs that meet their preferences at best (the combination of bundles of
provided goods, quality, membership fees etc.). Individuals “move” to the preferred club(s)
 “voting by feet”
• Advantages:
- Global efficiency is enhanced because different individual preferences can be satisfied.
- Competition between different clubs enhance the efficiency of the production of club goods.
- Individuals can be members in several clubs
• Necessary conditions to reach global optimality:
- Full mobility of citizens
- Precise information about all existing clubs (characteristics of the goods, price information)
- Availability of a range of clubs that satisfy all individual preferences
(Or, possibility for every individual whose desires are not satisfied to found new clubs.)
- No (substantial) economies of scale in the production of the public goods
- No (remarkable) spillovers across clubs public goods – the possibility to exclude
individuals from the consumption / membership.
- (No geographical restraints on individuals with respect to their earnings)
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
35
Clubs II
Question 16: Given a club that provides a public (club) good G (no fixed costs).
Each unit G costs pg. All n club members have the same utility
function U which depends on the consumption of a private good
X (price px) and the consumption of the public good G. All club
members have the same income Y. Every club member pays a
membership fee t, to finance the public good.
Prove that the Samuelson condition has to be satisfied to reach
an efficient provision of the public good.
Lagrangian method:
max!
s.t .
Maximize utility
U ( X ,G , n )
Y  px X  t
under the budget constraint.
L  U( X , G , n)  (Y  px X  t )
Lagrangian function
t  n  pg  G
Balanced budget of the community
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
36
Clubs II
p G

L  U ( X , G , n)    Y  p x X  G 
n 

Maximizing with respect to X, G, and n:
(1)
L U

 px  0
X X
U 1


X px
( 2)
L U pG


0
G G
n

( 3)
L U pGG

 2 0
n n
n
from (1) & ( 2)
U 1 U n
 

X px G pg
U n

G p g
U
pG

G
n

U
pX
X
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
Samuelson
condition:
ΣMRS = price ratio
q.e.d.
37
Direct democracy
Question 17: Discuss the following statements.
"On no question can a perfect unanimity be hoped." (Thomas Jefferson)
„Democracy and socialism have nothing in common but one word, equality.
But notice the difference: while democracy seeks equality in liberty,
socialism seeks equality in restraint and servitude.“ (Alexis de Tocqueville)
„In a democracy the poor will have more power than the rich, because there
are more of them, and the will of the majority is supreme.“ (Aristotle)
„In the strict sense of the term, a true democracy has never existed, and will
never exist. It is against natural order that the great number should govern
and that the few should be governed.“ (Jean-Jacques Rousseau)
Remarks:
• Some issues referring to these statements will be discussed in Part 3.
• In order to discuss the statements above one could think about some issues already presented in
this exercise and the lecture, such as costs of decision making, median voter, tactical voting,
cyclical majorities, interest groups, log-rolling, exit & voice etc.
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
38
Majority rule, runoff vote – Example „ presidental vote in Frnace“ April, May 2007
1st round, May, 7th 2007:
Candidate
M. Olivier BESANCENOT
no. of votes
Percent
1.498.581
4,08 %
Französische Präsidentschaftswahl
4%
2%
0%
19%
32%
Mme Marie-George BUFFET
M. François BAYROU
707.268
1,93
6.820.119
18,57
1%
2%
1%
Mme Ségolène ROYAL
9.500.112
25,87
M. Jean-Marie LE PEN
3.834.530
10,44
487.857
1,33
Mme Arlette LAGUILLER
M. Nicolas SARKOZY
11.448.663
31,18
2%
10%
1%
26%
M. Olivier BESANCENOT
Mme Marie-George BUFFET
M. Gérard SCHIVARDI
M. François BAYROU
M. José BOVÉ
Mme Dominique VOYNET
M. Philippe de VILLIERS
Mme Ségolène ROYAL
M. Frédéric NIHOUS
M. Jean-Marie LE PEN
Mme Arlette LAGUILLER
M. Nicolas SARKOZY
Second round
2nd round, May, 21st 2007:
Candidate
Mme Ségolène ROYAL
M. Nicolas SARKOZY
no. of votes
Percent
16.790.440
46,94 %
18.983.138
53,06 %
47%
53%
Mme Ségolène ROYAL
Dipl.-Volkswirt Sebastian Voll
Chair of Economic Policy-FSU Jena
WS 11/12- Exercise Public Choice / Part 2
M. Nicolas SARKOZY
39