S O L U T I O N S
28
Governments and Politics
Solutions for Microeconomics: An Intuitive
Approach with Calculus (International Ed.)
Apart from end-of-chapter exercises provided in the student Study Guide, these
solutions are provided for use by instructors. (End-of-Chapter exercises with
solutions in the student Study Guide are so marked in the textbook.)
The solutions may be shared by an instructor with his or her students at the
instructor’s discretion.
They may not be made publicly available.
If posted on a course web-site, the site must be password protected and for
use only by the students in the course.
Reproduction and/or distribution of the solutions beyond classroom use is
strictly prohibited.
In most colleges, it is a violation of the student honor code for a student to
share solutions to problems with peers that take the same class at a later date.
• Each end-of-chapter exercise begins on a new page. This is to facilitate maximum flexibility for instructors who may wish to share answers to some but
not all exercises with their students.
• If you are assigning only the A-parts of exercises in Microeconomics: An Intuitive Approach with Calculus, you may wish to instead use the solution set
created for the companion book Microeconomics: An Intuitive Approach.
• Solutions to Within-Chapter Exercises are provided in the student Study Guide.
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28.1 In Chapter 4, we considered different ways of thinking about single-peaked preferences over twodimensional issue spaces. We did so in particular in end-of-chapter exercise 4.11 which you can now
re-visit.
Answer: See the solutions for Chapter 4 exercises.
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28.2 In the text we discussed two main conditions under which the median voter’s favored policy is also
the Condorcet winner.
A: Review the definition of a Condorcet winner.
(a) What are the two conditions under which we can predict that the median voter’s position is
such a Condorcet winner?
Answer: A Condorcet winner is a policy that can beat any other policy in pair-wise voting.
The median voter’s ideal point is such a Condorcet winner as long as the policy issue is single
dimensional with voters’ preferences over the policy issue single peaked.
(b) Implicitly, we have assumed an odd number of voters (such that there exists a single median
voter). Can you predict a range of possible policies that cannot be beaten in pairwise elections
when there is an even number of voters and the conditions of the median voter theorem are
otherwise satisfied?
Answer: In the case of an even number of voters, the interval between the ideal points of the
two “median voters” contains all the policies that cannot be defeated by a policy outside the
interval. If two policies from within the interval are put up against each other in a pair-wise
vote, the outcome is a tie.
(c) Suppose that the issue space is two-dimensional — as in the case where we have to choose
spending levels on military and domestic priorities. Consider the following special case: All
voters have ideal points that lie on a downward sloping line in the two-dimensional space,
and voters become worse off as the distance between their ideal point and the actual policy
increases. Is there a Condorcet winner in this case?
Answer: Yes, in this case, the ideal point of the median voter (defined as the voter whose
ideal point on the line has half of all remaining ideal points to one side and the other half to
the other side) is a Condorcet winner. This is because no point on the line can beat it — and
any point off B off the line will get fewer votes than the point C that lies on the intersection
of the ideal points line and the line that passes through B and is perpendicular the ideal
points line.
(d) Revisit the “Anything-Can-Happen” theorem in the text. Suppose that the current policy A in
our two-dimensional policy space is equal to the ideal point of the “median voter” along the
line on which all ideal points lie. If you are an agenda setter and you can set up a sequence
of pairwise votes, which other policies could you implement assuming the first vote in the
sequence needs to put up a policy against A?
Answer: Since no alternative policy can defeat A, there is no way to end up at a policy other
than A with a sequence of pair-wise votes that starts with A against some other policy. Thus,
if the agenda setter has to start out with policy A, he cannot define an agenda that gets him
away from A.
(e) In our discussion of the “Anything-Can-Happen” theorem, we raised the possibility of singleissue committees as a mechanism for disciplining the political process (and limiting the set of
proposals that can come up for a vote in a full legislature). Is such structure necessary in our
special case of ideal points falling on the same line in the two-dimensional policy space?
Answer: No, this structure is not necessary now that we have a Condorcet winner. And if
the agenda setter sets up committees that bring forth proposals other than the Condorcet
winner — and if these proposals are put up against the status quo policy, the status quo
policy will always win if it is the Condorcet policy.
(f) In the more general case where we allow ideal points to lie anywhere, the agenda setter still
has some control over what policy alternative gets constructed in a structure induced equilibrium in which single-issue committees play a role. In real world legislatures, the ability
of the agenda setter to name members of committees is often constrained by seniority rules
that have emerged over time — i.e. rules that give certain “rights” to committee assignments
based on the length of service of a legislator. Can we think of such rules or norms as further
constraining the “Anything-Can-Happen” chaos of democratic decision-making?
Answer: Anything that limits an agenda setters ability to “stack” committees with those most
favorably to the agenda setters’ position naturally constrains the agenda setter’s ability to
manipulate the outcome. Thus, seniority rules do help to constrain the power of the agenda
setter.
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B: Consider a simple example of how single-peaked and non-single peaked tastes over policy might
naturally emerge in a case where there is only a single dimensional issue. A voter has preferences
that can be represented by the utility function u(x, y ) = x y where x is private consumption and y is
a public good. The only contributer to y is the government which employs a proportional tax rate t .
Suppose y = δt .
(a) Suppose individual has income I . Write his utility as a function of t , δ and I ?
Answer: Private consumption x is then (1 − t )I , and public good consumption is δt . We can
therefore express utility as
u(t ) = δt (1 − t )I .
(28.1)
(b) What shape does this function have with respect to the policy variable t ?
Answer: This is an example of a single peaked utility function. It’s peak occurs where the
derivative with respect to t is zero — i.e. where t = 0.5.
(c) At what t does this function reach its maximum?
Answer: As already indicated above, it reaches its maximum at t = 0.5.
(d) Suppose that an individual with income I can purchase a perfect substitute to y on the private
market at a price of 1 per unit. Determine, as a a function of I , at what level of t an individual
will be indifferent between purchasing the private substitute and consuming the public good.
Answer: If an individual consumes a private substitute z rather than the public good y ,
he still has to pay taxes — which implies that his budget constraint is x + z = (1 − t )I . To
determine how much z and x to purchase, he will solve
max xz subject to x + z = (1 − t )I .
x,z
(28.2)
This solves to give us
x=
(1 − t )I
(1 − t )I
and z =
.
2
2
(28.3)
This implies that the individual’s utility when he passes on the public good is
v(t ) =
(1 − t )I
2
µ
¶
(1 − t )I
(1 − t )2 I 2
=
.
2
4
(28.4)
The individual will therefore go with the private substitute if v(t ) > u(t ); i.e. if
(1 − t )2 I 2
> δt (1 − t )I
4
(28.5)
which solves to
t<
I
.
I + 4δ
(28.6)
(e) What does this imply for the real shape of the individual’s preferences over the policy variable
t assuming δ > I /4?
Answer: This implies that his preferences are
U (t ) = u(t ) = δt (1 − t )I for t ≥
= v(t ) =
I
I + 4δ
I
(1 − t )2 I 2
for t <
.
4
I + 4δ
(28.7)
Thus the utility over t has an initially downward sloping portion that becomes upward sloping and eventually again downward sloping.
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28.3 Everyday Exercise: Why Vote?: Voting is costly. If you vote in person, you have to find your polling
place and often stand in line until you get to the voting booth to vote. If you vote by absentee ballot, you
have to figure out how to get one and then be sure to mail it in. In both cases, you probably have to do
some work figuring out who the candidates and what the issues are.
A: Many people purposefully choose not to vote — and they often give the following reason: "I don’t
think it matters.” As we will see in this exercise, they might mean one of two things by this — and
they appear to be right in at least one sense.
(a) First, suppose we take the median voter model really seriously and believe it accurately predicts the position of the two candidates from which we choose. How might this justify the
excuse given by voters who don’t vote?
Answer: If we really believed the median voter model 100%, we would conclude that, in 2person elections, both people must be taking identical positions — and therefore “it does
not matter” which one we elect.
(b) In the real world, there are many frictions that keep the median voter model’s prediction from
fully coming to fruition. For instance, candidates might have to win party nominations first,
and then run in the general election — which means we tend to end up with right-of-center
and left-of-center candidates. In light of this, is it reasonable to think that the excuse given in
(a) is justified in the real world?
Answer: The “it does not matter because the two candidates are identical” excuse in part (a)
then does not work in the real world where real world factors cause candidates to ultimately
differ in their positions and in what they are likely to do if elected.
(c) Next, consider a different way in which the “it does not matter” statement might be meant:
Perhaps a voter recognizes that it matters which candidate wins (in the sense that the world
will change differently depending on which one wins), but she believes the candidate who
will win will almost certainly win whether any individual voter goes to the polls or not. Do
you think this is true in the real world?
Answer: It is very rare that an election ends up in a tie or is won by a 1-vote margin —
but that is required if one person’s vote is to actually make a difference in the outcome of
the election. It therefore seems almost certainly true that a voter’s vote will not change the
outcome of an election — and in that sense “does not matter”. (Some might argue that
it would matter if everyone took that view and stays home. While that is certainly true,
rational voters should know from observing past elections that not everyone will stay home
— and thus the argument continues to hold. My wife, a political science major, still hates
me for saying this.)
(d) In light of your answer to (c), might it be rational for many people not to vote?
Answer: Yes, if a voter rationally believes that her vote will almost certainly not change the
outcome of an election, and if it is costly to vote — even just a little bit costly — the marginal
benefit (of essentially zero) is not worth the marginal cost of voting.
(e) In the 2008 U.S. Presidential election, Barack Obama won by close to 10 million votes. In
what sense is the puzzle not so much why more people didn’t vote but rather why so many —
about 60% of eligible voters — did.
Answer: This follows straight from our answers to the previous part. In elections where 130
million people vote, one vote will almost certainly not make a difference in the outcome.
This suggests that the marginal benefit of voting (in terms of the probability of affecting the
outcome) is zero. So if there is any marginal cost to voting, it is surprising that so many
people still vote. In 2008, some people actually stood in line for 8 hours to vote — a large
marginal cost indeed given that the outcome was going to be the same whether any one of
the individuals in that 8 hour line had voted or not.
(f) Suppose we believe that governments are more effective the more voters engage in elections.
In what sense does this imply that voters have prisoners’ dilemma incentives that give rise to
free-riding?
Answer: This would mean that voting has a positive externality — i.e. by voting, I make the
government just a bit more effective even though I don’t change the outcome of the election. Even if that positive externality is tiny, multiply it by 300 million (i.e. the population
of the U.S.), and even a tiny externality adds up. For instance, if the positive externality is
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one hundredthousandth of a cent, it would add up to $30 per vote. We thus find ourselves
in a situation where we want effective government — and thus would like governments to
be elected by a sizable fraction of the electorate — but it is in our individual interest to not
vote since the marginal private benefit is zero and the marginal cost is positive. Put differently, our incentive is to free ride on the effectiveness of government that others produce by
voting. (In some countries, everyone is in fact required to vote or pay a fine — which is one
way to address the externality problem.)
(g) In Chapter 27, we suggested that one way charitable organizations overcome free-rider problems among potential donors is to find ways of eliciting within donors a “warm glow” from
giving. Can you think of an analogous explanation that can rationalize why so many people
vote in large elections?
Answer: From a very early age, children in democratic societies are typically told how important it is to vote. Thus, from an early age, we seem to attempt to shape preferences in
favor of voting so that we in essence produce a private benefit from voting (because not
voting makes us feel guilty.) On the day of elections in the U.S., voters get “I voted” stickers
that many wear all day. Much effort therefore goes into causing people to experience part of
the voting process as a private good — presumably in order to compensate for the free-rider
incentives that are otherwise present.
(h) Suppose that the voters who do not vote are those who are “disillusioned”. What positions
might two candidates take on the Hotelling interval [0,1] if the disillusioned voters (that do
not vote) are those that cannot find a candidate whose position is within 3/16ths of their ideal
point? Could we have an equilibrium where the extreme ends of the political spectrum do not
vote? Could we have one where the center does not vote?
Answer: In this case, any two positions that are at least 3/16ths away from the endpoints of
the interval can be an equilibrium with two candidates. This is because, no matter which
two points in the interval [3/16,3/16] the two candidate choose, they will split the votes
of those who vote. The farther apart they are, the more people will vote — but they will
split 50-50. (If one were to pick a position closer to the end-points, then the other could
win the election outright by picking a position within the [3/16,3/16] interval sufficiently
far from the other candidate.) It may therefore be that people on the extremes end up being
disillusioned and not voting — or it may be that people in the center don’t vote — depending
on where the two candidates stake out their position.
B: In the 2000 U.S. Presidential election, George W. Bush defeated Al Gore by 537 votes in the State of
Florida — and with those 537 votes won the election.
(a) The close margin in the 2000 election is often cited by politicians as evidence that you should
“not believe your vote does not matter”. I would argue that it shows the opposite: Even in close
elections, it is almost never the case that one vote counts. What do you think?
Answer: This is as close to a tied election as one is ever likely to see in a country of this size.
Yet even in this case, one vote more for the losing candidate would simply have meant that
George W. Bush would have won by 536 rather than 537 votes. There appears to be no way
to argue that any single voter’s behavior could have changed the outcome — and no reason
to question the presumption that the probability of a single voter affecting the outcome of
a large election is essentially zero.
(b) Ralph Nader, the Green Party candidate, received nearly 100,000 votes in the State of Florida
in the 2000 Presidential election. Many believe that, had Ralph Nader’s name not been on
the ballot, Al Gore would have won the State of Florida — and with it the Presidency. If so,
which one of Arrow’s axioms does this suggest is violated by the way the U.S. elects Presidents?
Explain.
Answer: The Independence of Irrelevant Alternatives (IIA) axiom states essentially that the
social ranking over a pair of alternatives should be independent of how those alternatives
are ranked relative to other alternatives. Yet the argument is that the inclusion of Nader on
the ballot caused a different social ranking of Bush and Gore than would have happened in
the absence of this “irrelevant” alternative.
(c) Some election systems require the winning candidate to win with at least 50% plus 1 votes —
and, if no candidate achieves this, require a run-off election between the top two candidates.
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Since this seems difficult to implement in the 50 state-wide elections that result in electoral
college votes that determine the winner of a U.S. Presidential election, some have proposed
a system of instant run-off voting. In such a system, voters rank the candidates from most
preferred to least preferred. In the first round of vote counting, each voter’s top ranked candidate is considered as having receive a vote from that voter, and if one candidate gets 50%
plus 1 votes, he is declared the winner. If no candidate receives that many votes, the election
authorities find which candidate received the lowest first round votes and then re-assigns that
candidate’s votes to the candidates that were ranked second by these voters. If one candidate
reaches 50% plus 1 votes, he wins — otherwise the election authorities repeat the exercise, this
time re-assigning the votes of the candidate who initially received the second to last number
of first place rankings. This continues until someone gets 50% plus 1 votes. Had the State of
Florida used this system in 2000, do you think the Presidential election outcome might have
been different?
Answer: If it is true that a sizable majority of Nader voters would have been Gore voters
had Nader not been on the ballot, these voters would presumable have been more likely
to rank Gore rather than Bush second in an instant run-off election. As a result, in the “instant run-off” where Nader’s first-round voters are reassigned based on their second choice,
Gore would have ended up with more additional votes than Bush — quite likely more than
537 (which is what he needed to win). At the same time, it is almost impossible to know
for sure whether this would have changed the election outcome — because the awareness
that votes would be counted differently might well have affected how candidates allocated
campaign resources and to what extent voters voted strategically. Nader might well have received more first round votes because many voters who preferred Nader to Gore might have
voted for Gore given they knew Nader had no chance of winning — but they also would have
ranked Gore second so that Gore would eventually have gotten those votes. But Gore might
have spent less time in Florida if he thought Nader was not as big a threat to his chances of
winning the state (because he would get most of the Nader voters in the second round) — in
which case Bush might have won by enough of a margin in the first round to win outright.
(d) Nader is often referred to as a “spoiler” — because of many people’s belief that he “spoiled” the
election outcome for Gore. True or False: It is much less likely that a third candidate plays the
role of spoiler in an instant run-off election — but it is still possible if the third candidate is
sufficiently strong.
Answer: The statement is true — in both its parts. First, it is much less likely that a candidate like Nader can spoil the election outcome for someone like Gore — because Gore will
ultimately get the Nader votes that would have otherwise come to Gore in the first round
(had Nader not been on the ballot). But it is still possible for the spoiler candidate to keep a
candidate that would have otherwise won in pair-wise voting from winning — if the spoiler
candidate is sufficiently strong to come in second but not with enough second round votes
to win the election. Consider, for instance, the example of the 1992 election where Bush
Sr. ran against Bill Clinton and a strong independent candidate, Ross Perot. At times, Perot
was polling upwards of 25-30% in national opinion polls (although he ultimately only won
19% because he dropped out and then came back into the race, citing some bizarre reasons.) Suppose Perot had won 30% of the vote with Bush winning 41% and Clinton winning
29%. This would have triggered Clinton being eliminated — with his second choice votes
being apportioned to Perot and Bush. Suppose a third of Clinton voters ranked Bush second and two thirds ranked Perot second. Then Bush would have been assigned one third
of 29% or 9.67% in additional overall votes — taking him to 50.67% and causing him to win
the election. But suppose that 75% of Perot voters preferred Clinton over Bush. Then, in a
head-to-head election between Bush and Clinton, Clinton would have gotten not only his
29% but also three quarters of Perot’s 30% — for a total of 51.5% and thus the Presidency.
Perot could thus have spoiled the election for Clinton — if he was sufficiently strong to get
into the top two, not sufficiently strong to win and if his voters were sufficiently tilted toward
Clinton.
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28.4 Everyday Application: “Winner-Take-All” Elections and the U.S. Electoral College: In the U.S., Presidential elections are not won by the candidate who wins the popular vote nationally. (If they were won in
this way, Al Gore would have become President in 2000.) Rather, each state is given a number of “electors”
equal to that state’s representation in the U.S. Congress. In almost all states, the candidate that gets the
most votes gets all the electors of that state — and the Presidency is won by the candidate who collects at
least 270 electoral college votes.1
A: Consider a simplified version of this system in which there are only 2 states, with state 1 more than
twice the size of state 2 and exactly twice the electoral college votes. Suppose all preferences are single
peaked along a “left/right” continuum. Let n i be the median voter’s ideal point in state i — with
n 1 < n 2 . In the event of a state-wide tie, assume the electoral college votes for the state are split.
(a) If the aim of two Presidential candidates is only to win, what position will they take in equilibrium?
Answer: There is no way to win the election while losing the vote in state 1 — which implies that the candidates must split the vote in state 1 in equilibrium. This implies that the
candidates either both pick the same position, or they pick positions opposite and equally
distant from n 1 . If the former is the case, then it must be that both candidates picked n 1
— otherwise one of the candidates can deviate and win the state 1 election (and with it the
whole thing). If the latter, then the candidate to the right of n 1 will win state 2 whose median
voter is to the right of n 1 — and, together with half the electoral votes in state 1, would win
the election. In that case, however, the other candidate could do better by simply picking
the median position in state 1 and thus winning state 1 (and with it the election). We are
therefore left with only one possibility: Both candidates choose n 1 — the median position
in state 1. They then tie in the election in both states (with presumably some slight noise
throwing the election one way or another).
(b) Suppose instead that there are four states, states 2 and 3 that are small (with 10% of the electoral votes each) and states 1 and 4 that are large (with 40% of the electoral college votes
each). Suppose further that the ideal points for median voters in each state are such that
n 1 < n 2 < n 3 < n 4 . What position to you now expect the candidates to take?
Answer: We would now expect both candidates to take the positions in the interval [n 2 ,n 3 ].
Suppose this is so: Then then either the two candidates pick different positions in the interval — in which case the one to the left will win states 1 and 2 and the one to the right
will win states 3 and 4, or they will pick the same position in the interval and split the vote
everywhere (in expectation). It can’t be that a candidate takes a position either to the left of
n 2 or to the right of n 3 — because then the other candidate can take the ideal point of the
median voter in one of the states and win both small states and a large state.
(c) Explain how this relates to the common observation that most of the U.S. Presidential election
actually takes place in a subset of states — often called “battle-ground states”, with the rest of
the country largely ignored by the candidates.
Answer: The above predicts that policy positioning will take place to target the states with
median voters that lie toward the center — with states that have median voters that lie closer
to the extremes taken for granted. In our example, states 2 and 3 are battleground states.
(d) In exercise 28.3, we suggested that one way to view the decision of whether or not to vote is
by comparing the marginal benefit of voting to the marginal cost. The marginal benefit of
voting includes the probability that one’s vote will determine the outcome of the election. If
this is a major consideration in people’s decision of whether to vote, how would you expect
voter participation in Presidential elections to differ across states?
Answer: Suppose candidate 1 settles closer to n 2 and candidate 2 settles closer to n 3 in
our example of part (b). Then a larger fraction of voters in state 1 will vote for candidate 1
than in state 2 (although candidate 1 will win both), and a larger fraction of voters in state
4 than in state 3 will vote for candidate 2 (although candidate 2 will win both). Since the
election is predicted to be tighter in states 2 and 3 than in states 1 and 4, the probability
that a vote will make the difference will be larger in states 2 and 3 than in states 1 and 4 (if
1 If no candidate gets 270 electoral college votes because of a 269-269 tie or because of 3 candidates
in the race, the U.S. House of Representatives decides the winning candidate. We will ignore this possibility here.
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there is some uncertainty.) We would therefore expect voter participation in states 2 and
3 — i.e. in the battleground states — to be higher. In the real world, for instance, there is
usually no doubt that Utah will vote for the Republican candidate and that Massachussetts
will vote for the Democratic candidate — but Missouri and Ohio can often go either way. We
would therefore expect voter participation to be larger in Missouri and Ohio than in Utah
and Massachussetts.
(e) The electoral college system gives each state 2 electors outright plus 1 elector for each representative that the state has in the House of Representatives (where representation in the House is
roughly in proportion to population). How would you evaluate the following statement: In
such a system, we would expect, all else being equal, disproportionately more resources spent
per voter in small states.
Answer: Consider a small state that is given only a single representative in the House of Representative — and a large state that is 31 times as large and thus given 31 representatives in
the House of Representatives. The electoral college system gives the first state 3 electors
and the second state 33 electors. Thus, while the population in state 2 is 31 times the population in state 1, it only gets 11 times as many electors. In a sense, the votes in state 1 are
therefore worth three times per vote what the votes in state 2 are worth — implying that,
in campaigns where resources are spent optimally, more will be spent per voter (in order to
spend the same per electoral vote) — all else being equal.
(f) Some states have considered switching from a state-wide winner-take-all system for electing
“electors” in Presidential races to a system in which electors from the state represent each candidate in proportion to the popular vote received in the state.2 Which of your answers would
be affected by such a change?
Answer: The two candidates would now find themselves in a pure median voter model —
thus taking the national median position. There would no longer be a reason to focus on
battleground sates — as votes in all states count, even if a candidate loses a state. There
would also be no further reason to expect either voting rates or candidate spending rates
per voter to differ across states. All this assumes that all states switch systems. If only one
state switches, it is more likely to experience a large change in how candidates treat it if it
was previously not a battleground state. (In the 2008 Presidential election, the Democratic
campaign, for instance, lavished attention on one congressional district in Nebraska because it had a chance of winning that district (and in fact did win that district) and because
Nebraska apportions all but 2 state-wide electors at the district level. Had Nebraska not split
its electoral votes by district, the Democratic candidate would have had no chance to win
any electoral college votes in Nebraska (which is a heavily Republican state in Presidential
elections) — and would thus have ignored the state.)
(g) Prior to running in the general election as either the Democratic or the Republican candidate,
a politician first needs to win a party’s presidential nomination. This is done mainly in earlier “primary” (or “caucus”) elections held in each state. In the Republican nomination fight,
almost all such primary elections are “winner-take-all” (like the electoral college system in
the general election), but on the Democratic side, most primaries allocate votes to each candidate proportionally. In which party would you expect more states to be ignored during the
nomination fight?
Answer: The Republican nomination battle typically ends earlier in part because of the
winner-takes-all feature — whoever wins some large states early gets large numbers of delegates that eventually elect the nominee. Democratic nomination fights can last much longer
(as in 2008 between Hilary Clinton and Barak Obama) — because no one gets a huge boost
from winning large states.
B: In exercise 28.3, we used the 2000 election and the controversy regarding Ralph Nader’s participation to suggest that the way we elect U.S. Presidents violates the spirit of Arrow’s IIA axiom. Is
2 Often such proposals envision winner-take-all elections at the House of Representatives District
level — which comes close to proportional allocation of electors in large states. The states of Maine and
Nebraska in fact allocate some of their electors in this way, and Nebraska was the only state in the 2008
election that therefore split its electoral vote.
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there any reason to believe that this would be less true if the U.S. switched to a proportional system
of electing its Presidents?
Answer: No, there is no reason that the same could not happen in a proportional system.
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28.5 Everyday Application: To Enter or Not to Enter the Race: Suppose there are three possible candidates
that might run for office, and each has to decide whether or not to enter the race. Assume the electorate’s
ideal points can be defined by the Hotelling line from Chapter 26 — i.e. the ideal points are uniformly
distributed on the interval [0,1].
A: Let πi denote the probability that candidate i will win the election. Suppose that the payoff to a
candidate jumping into the race is (πi − c) where c is the cost of running a campaign.
(a) How high must the probability of getting elected be in order for a candidate to get into the
race?
Answer: It must be that πi ≥ c.
(b) Consider the following model: In stage 1, three potential candidates decide simultaneously
whether or not to get into the race and pay the cost c. Then, in stage 2, they take positions on
the Hotelling line — with voters then choosing in an election where the candidate who gets
the most votes wins. True or False: If there is a Nash equilibrium in stage 2 of the game, it
must be that the probability of winning is the same for each candidate that entered the race
in stage 1.
Answer: True. If only one candidate entered, then the statement is trivially true. If two
entered, then we know the only equilibrium is one where they both take the median position
0.5 in the middle of the Hotelling line. If there are three candidates and the probabilities of
winning are not the same for all, then the candidate closest in position to the one whose
probability is highest can increase her probability of winning by moving slightly closer to
her opponent’s position.
(c) Suppose there is a Nash equilibrium in stage 2 regardless of how many of the three candidates
entered in stage 1. What determines whether there will be 1, 2 or 3 candidates running in the
election?
Answer: If there is a Nash equilibrium in stage 2, then three candidates will enter in stage
1 so long as c < 1/3 — because the probability of winning in stage 2 with three candidates
running is 1/3. If 1/3 < c < 1/2, only two candidates would enter in stage 1 — and if c > 1/2,
only a single candidate enters.
(d) Suppose that the probability of winning in stage 2 is a function of the number of candidates
that are running as well as the amount spent in the campaign, with candidates able to choose
different levels of c when they enter in stage 1 but facing an increasing marginal cost p(c) for
raising campaign cash. (The payoff for a candidate is therefore now (πi − p(c)).) In particular, suppose the following: Campaign spending matters only in cases where an election run
solely on issues would lead to a tie (in the sense that each candidate would win with equal
probability). In that case, whoever spent the most wins the election. What might you expect
the possible equilibria in stage 1 (where entry and campaign spending are determined simultaneously) to look like?
Answer: It could not be an equilibrium for candidates to spend different amounts — since
spending anything when your opponent spends more is a sure way to lose the election in
stage 2 where the probability of winning the election on issues alone can only result in a
tie. Thus, any candidate that enters must spend the same mount c ∗ . The fact that the
marginal cost of raising campaign cash is increasing then still permits different possibilities
for how many candidates might enter in the first stage — but the cost of the campaign is
now determined endogenously, with the payoff from running in the election π − p(c ∗ ) = 0.
(e) Suppose the incumbent is one of the potential candidates — and he decides whether to enter the race and how much to spend first. Can you in this case see a role for strategic entry
deterrence similar to what we developed for monopolists who are threatened by a potential
entrant?
Answer: Yes, there is now an analogous case for the incumbent if the incumbent gets to
announce his campaign war chest before the others decide whether or not to enter the race
(just as the monopolist has to be able to set quantity before the potential entrant decides
whether to enter). The incumbent then has to determine how costly it is to amass the war
chest necessary to limit the field of opponents — and can credibly commit to a level of
campaign spending because he gets to move first.
Governments and Politics
1056
(f) With the marginal cost of raising additional funds to build up a campaign war chest increasing, might the incumbent still allow entry of another candidate?
Answer: Yes, how much entry the incumbent allows depends on the ease with which he can
raise the campaign war chest.
B: Consider the existence of a Nash equilibrium in stage 2.
(a) What are two possible ways in which 3 candidates might take positions in the second stage of
our game such that your conclusion in A(b) holds?
Answer: If our conclusion from A(b) holds, then any Nash equilibrium with three candidates competing in stage 2 must have π1 = π2 = π3 . One way that this could happen is if all
candidates take the median position 0.5 — and thus end up in a three-way tie (in expectation). Another way for it to happen is for the candidates to locate at points 1/6, 1/2 and 5/6
— which would give each of them 1/3 of the vote.
(b) Can either of these be an equilibrium under the conditions specified in part A?
Answer: No. For the case where all three candidates take the position 0.5, one candidate
could move slightly to the left or right and increase his vote share to just under 1/2 — virtually assuring his election. For the case where the candidates take the positions 1/6, 1/2
and 5/6, the candidate at 1/6 could increase his vote share to close to one half by changing
his position to lie just to the left of 1/2 — again virtually assuring his election. (Something
similar applies to the candidate at 5/6).
(c) Suppose that, instead of voter ideal points being uniformly distributed on the Hotelling line,
one third of all voters hold the median voter position. How does your conclusion about the
existence of a stage 2 Nash equilibrium with three candidates change? Does your conclusion
from A(c) still hold?
Answer: If a third of the population holds the median position, then all candidates picking
the median position is an equilibrium in stage 2. This results in a probability of winning the
election equal to 1/3 for each of the three candidates. If any of the candidates deviated from
the median position, he would get less than 1/3 of a vote share. Our conclusion from A(c)
still holds.
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Governments and Politics
28.6 Everyday Application: Citizen Candidates: Whenever we have modeled political candidates who
stand for election, we have assumed that they care only about winning and are perfectly content to change
their position in whatever way maximizes the probability of winning. Now consider a different way of
thinking about political candidates: Suppose that the citizens again have uniformly distributed ideal
points on the Hotelling line [0,1]. Before any election is held, each citizen has to decide whether to pay
the cost c > 0 to run as a candidate — with the payoff from probability π of winning the election equal to
(π − c).
A: Assume candidates cannot change their position from their ideal point, and citizens who do not
become candidates get payoff equal to minus the distance of the winning candidate position x ∗ to
their own on [0,1]. The highest attainable payoff for a non-candidate is therefore 0. (Candidates that
lose get the same payoff as citizens who do not run — except that they also incur cost c from having
run.)
(a) For what range of c is the following an equilibrium: A citizen with the median position 0.5 is
the only candidate to enter the race and thus wins.
Answer: The cost of running for election has to be sufficiently large to keep any other citizen
from entering. The only other citizen who has a positive probability of winning an election
against the median citizen that is already a candidate is another candidate with the median
position of 0.5. Thus, for it to be an equilibrium for only one median citizen to be a candidate, it must be that another median citizen would do better by not entering and getting
a payoff of 0 (since the current candidate will implement the potential candidate’s favorite
position) rather than entering and winning with probability 1/2 — i.e. it must be that
1
−c ≤ 0
2
(28.8)
or simply c ≥ 0.5.
(b) How high does c have to be in order for the following to be a possible equilibrium: A citizen
with position 0.25 enters the race as the only candidate and therefore wins. How high must c
be for an equilibrium to have a citizen with position 0 be the only candidate to run (and thus
win)?
Answer: If the only candidate has taken position 0.25 and this is an equilibrium, it must be
that no citizen who can beat this candidate in an election gets a greater payoff from winning
than he does from simply allowing candidate 0.25 to win. The set of citizens that can beat
a candidate with position 0.25 is the set in the interval (0.25,0.75) — with any candidate in
this field getting a payoff of (1 − c) from running. The citizen that has the most to gain is
the citizen farthest from the current candidate’s position — i.e. the candidate that has the
most to gain is candidate with position of slightly below 0.75. His payoff if he does not run
is −(0.75 − 0.25) = −0.5. Thus, the only way a candidate with position 0.25 can be the only
candidate to run in equilibrium is if
1 − c ≤ −0.5
(28.9)
or c ≥ 1.5
Following the same reasoning, we can conclude that the only way that it can be an equilibrium for a single candidate with the extreme position 0 to run is if c ≥ 2.
(c) For what range of c will it be an equilibrium for 2 candidates with position 0.5 to compete in
the election?
Answer: If two median citizens compete in the election, the payoff to each of them is (0.5−c).
By not running, either one of them could get 0 payoff by simply enjoying the fact that the
other is implementing the favored policy. Thus, the payoff from staying in the election has
to be at least zero — i.e. 0.5 − c ≥ 0 or c ≤ 0.5. But it also has to be the case that no third
median candidate has an incentive to enter the race — i.e. (1/3) − c ≤ 0 or c ≥ 1/3. Thus, in
order for two median citizens to declare as candidates, it must be that
1
1
≤c ≤ .
3
2
(28.10)
Governments and Politics
1058
(d) For what range of c is it an equilibrium for two candidates with positions 0.25 and 0.75 to
compete?
Answer: First, it must be the case that neither candidate would do better by not running in
the election and simply accepting the opponent’s position. The two are 0.5 apart — which
means that the payoff from not running for either of them is −0.5. If both run, they will each
win with probability 0.5 — giving them payoff of (0.5−c). Thus, to be sure that neither drops
out of the election, we need
1
− c ≥ −0.5
2
(28.11)
or c ≤ 1. But it must also be the case that no other citizen has an incentive to join in as a
third candidate. There is, however, no citizen who can enter and have positive probability
of winning. A citizen with ideal point 0.5 would get 1/4th of the vote — those whose ideal
point lies in the interval (3/8, 5/8), with the other two candidates 3/8ths each. Any citizen
with ideal point less than 0.5 would hand the election to the candidate whose position is
0.75, and any citizen whose ideal point is greater than 0.5 would hand the election to the
candidate with ideal point 0.25.
(e) For what range of c is it an equilibrium for two candidates with positions 0 and 1 to compete?
Answer: First, we need to make sure c is not so large that either candidate has the incentive
to drop out of the election — i.e.
1
− c ≥ −1
2
(28.12)
or simply c ≤ 1.5. We also need to make sure that no other citizen has an incentive to enter. This incentive would be greatest for the citizen whose ideal point is farthest from the
extremes — i.e. the median citizen 0.5. In the absence of running, she has a payoff of -0.5.
If she runs, she will win the election (by getting 50% of the vote, with the other two splitting
the remaining 50% equally). Thus, if she runs, she gets payoff of (1 − c). We therefore need
1 − c ≤ −0.5
(28.13)
or simply c ≥ 1.5. Putting our two conditions together, the only way it can be an equilibrium
for the two extreme positions to be the only ones represented by candidates is if c = 1.5.
B: Consider the same set-up as in part A.
(a) Let x ∈ [0,0.5). For what range of c is it an equilibrium for a citizen with position x to be the
only candidate to run for office? Is your answer consistent with what you derived for A(b)?
Answer: For this to be an equilibrium, it must be that the citizen with the greatest distance
between her and x that can still beat candidate x does not have an incentive to enter the
race. The set of candidates that can enter the race and beat x is the set with ideal points in
the interval (x,(1 − x)), and the one that has the greatest incentive to enter is the one with
ideal point (1−x) (or just below that). If this citizen stays out of the election, she gets a payoff
of −(1 − x − x) = 2x − 1. If she enters the race, she gets payoff of (1 − c). Thus, for her to stay
out of the race, it must be that
1 − c ≤ 2x − 1
(28.14)
or c ≥ 2(1 − x). This is consistent with our answer to A(b) where we found that a single
candidate with x = 0.25 can be an equilibrium so long as c ≥ 1.5 = 2(1 − 0.25) and a single
candidate with position x = 0 can be an equilibrium so long as c ≥ 2 = 2(1 − 0).
(b) For what range of c is it an equilibrium for two candidates to compete — one taking position
x and the other taking the position (1 − x)? Is your answer consistent with your answers to
A(d) and A(e)?
Answer: First, we must be sure that neither candidate has an incentive to drop out. Given
that they have located in a way where both will get half the vote, the probability of winning
is 0.5 and the payoff from staying in the race is (0.5 − c). The payoff from dropping out
−(1 − x − x) = 2x − 1. Thus, for neither to have an incentive to drop out, it must be that
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Governments and Politics
2x − 1 ≤
1
−c
2
(28.15)
which implies c ≤ (1.5 − 2x). This is consistent with the condition derived in A(d) where
we concluded that it can be an equilibrium for two candidates with position x = 0.25 and
(1 − x) = 0.75 to be running in equilibrium so long as c ≤ 1.
But we must also make sure that no other citizen has an incentive to enter. It is not possible
for a citizen whose ideal point lies to the left of x or the right of (1 − x) to win an election. If
any third citizen can enter and win, it must be the median candidate 0.5. And in order for
the median citizen to be able to win, it must be that he can get at more than 1/3 of the vote —
with the other two splitting the remainder equally. The highest x for which this is possible is
x = 1/6 (and thus (1−x) = 5/6) — which implies we have to worry about the possibility of the
median citizen entering as a third candidate for x < 1/6. The payoff that the median citizen
gets from not entering is −(0.5 − x) = x − 0.5 while the payoff from entering as a candidate
when she can win (i.e. when x < 1/6) is (1 − c). Thus, in order for her not to enter, it must be
that
1 − c ≤ x − 0.5
(28.16)
or c ≥ 1.5. The two conditions combined imply that, for x ≤ 1/6, 1.5 ≤ c ≤ (1.5 − x) which
holds only for x = 0 with c = 1.5 — exactly what we derived for x = 0 in A(e). We therefore
conclude that we can have an equilibrium with two candidates running on x and (1 − x) so
long as c ≤ (1.5−x) when x ≥ 1/6 or when c = 1.5 when x = 0. But there is no x in the interval
(0,1/6) such that two candidates with positions x and (1 − x) can be an equilibrium.
(c) Let ǫ be arbitrarily close to zero. For what range of c will two candidates with positions (0.5−ǫ)
and (0.5+ǫ) be able to run against one another in equilibrium. What does this range converge
to as ǫ converges to zero.
Answer: For small ǫ, we don’t have to worry about a third candidate entering (as shown in
the previous part for x ≥ 1/6.) Thus, the only condition that has to hold is the condition that
neither candidate should have an incentive to drop out — i.e.
0.5 − c ≥ −2ǫ
(28.17)
or c ≤ 0.5 − 2ǫ. As ǫ goes to zero, this condition converges to c ≤ 0.5.
(d) How does the range you calculated in (c) compare to the range of c that makes it possible for
two candidates with position 0.5 to run against one another in equilibrium (as derived in
A(c))?
Answer: We concluded in A(c) that two median citizens running against one another can
be an equilibrium if 1/3 ≤ c ≤ 1/2, and we just concluded above that two candidates with
positions (0.5−ǫ) and (0.5+ǫ) can compete with one another in equilibrium so long as c ≤ 0.5
as ǫ converges to 0. The difference emerges from the lower end of the range that is 1/3 in one
case and 0 in the other. In the case where two candidates locate on symmetrically opposite
sides of 0.5 but close to 0.5, there is no way for a third citizen to enter and have any hope
of winning — but when both candidates locate precisely at 0.5, a third candidate can enter
at precisely 0.5 and have probability of 1/3 of winning. Thus, we have to have a campaign
cost sufficiently high to keep this from happening when the two candidates both take the
median position but we do not need that condition when they deviate even slightly from
the median position.
Governments and Politics
Voter 1
Voter 2
Voter 3
Voter 4
Voter 5
Total
Borda Count
A
B
C
5
4
3
5
4
3
5
4
3
1
2
5
1
2
4
17 16 18
1060
D
2
2
2
4
3
13
E
1
1
1
3
5
11
Table 28.1: Borda Count Implies C Wins and A comes in Second
Borda Count
A
B
C
Voter 1
4
3
2
Voter 2
4
3
2
Voter 3
4
3
2
Voter 4
1
2
4
Voter 5
1
2
3
Total 14 13 13
E
1
1
1
3
4
10
Table 28.2: Borda Count Implies A Wins and C comes in Second
28.7 Business and Policy Application: Voting with Points: Jean-Charles de Borda (1733-99), a contemporary of Condorcet in France, argued for a democratic system that deviates from our usual conception of
majority rule. The system works as follows: Suppose there are M proposals. Each voter is asked to rank
these — with the proposal ranked first by a voter given M points, the one ranked second given (M − 1)
points, and so forth.3 The points given to each proposal are then summed across all voters, and the top N
proposals are chosen — where N might be as low as 1. This voting method, known as the Borda Count is
used in a variety of corporate and academic settings as well as some political elections in countries around
the world.
A: Suppose there are 5 voters denoted 1 through 5, and there are 5 possible projects {A,B,C ,D,E} to
be ranked. Voters 1 through 3 rank the projects in alphabetical sequence (with A ranked highest).
Voter 4 ranks C highest, followed by D, E, B and finally A. Voter 5 ranks E highest, followed by C , D,
B and finally A.
(a) How does the Borda Count rank these? If only one can be implemented, which one will it be?
Answer: Table 28.1 presents the number of points assigned to each of the 5 alternatives by
the 5 voters and sums these in the last row. The Borda Count therefore ranks C highest,
followed by A, B , D and E.
(b) Suppose option D was withdrawn from consideration before the vote in which voters rank
the options. How does the Borda Count now rank the remaining projects? If only one can be
implemented, which one will it be?
Answer: The Borda Count is now re-done in Table 28.2 — with A (rather than C ) winning,
followed by B and C tied and E bringing up the rear.
(c) What if both D and E are withdrawn?
3 There exist other versions of Borda’s method — such as assigning (M − 1) points to the top ranked
choice and leaving zero for the last ranked. For purposes of this problem, define the method as it is
defined in the problem.
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Governments and Politics
Borda Count
A
B
Voter 1
3
2
Voter 2
3
2
Voter 3
3
2
Voter 4
1
2
Voter 5
1
2
Total 11 10
C
1
1
1
3
3
9
Table 28.3: Borda Count Implies A Wins and C comes in Third
Answer: Table 28.3 re-calculates the Borda Count with the initial winner C now coming in
third, and with A again winning.
(d) Suppose I get to decide which projects will be considered by the group and the group allows
me to use my discretion to eliminate projects that clearly do not have widespread support.
Will I be able to manipulate the outcome of the Borda Count by strategically picking which
projects to leave off?
Answer: Yes, as we have shown above. By leaving all proposals on the table, I know C will get
picked for sure, but by dropping D I can get A for sure instead (with C moving into second
place) and by dropping E as well I can get C to move into third place (with A still winning).
If I really want A implemented, I should therefore remove either D or both D and E from
the agenda — and if I want to make sure that C is last in line, I want to do the latter.
B: Arrow’s Theorem tells us that any non-dictatorial social choice function must violate at least one
of his remaining four axioms.
(a) Do you think the Borda Count violates Pareto Unanimity? What about Universal Domain or
Rationality?
Answer: Given the way the points are assigned, unanimously preferred choices will always
be chosen by the Borda count. The method does not restrict preferences — and it results in
a preference ranking that, by construction, does not give rise to intransitivities. Thus, none
of these axioms are violated.
(b) In what way do your results from part A of the exercise tell us something about whether the
Borda Count violates the Independence of Irrelevant Alternatives (IIA) axiom?
Answer: In part A, we showed that an agenda setter can eliminate alternatives that are
ranked 5th and 4th in the initial Borda count — and this can have an impact on what is
ranked first, second and third. Put differently, the inclusion or elimination of D and E affects how we rank A, B and C — but a social choice function that satisfies IIA would rank A,
B and C independent of what other options that are irrelevant are also included.
(c) Derive again the Borda Count ranking of the five projects in part A given the voter preferences
as described.
Answer: This is already done in A(a). It results in the Border Count ranking of
C ≻ A ≻ B ≻ D ≻ E.
(28.18)
(d) Suppose voter 4 changed his mind and now ranks B second and D fourth (rather than the
other way around). Suppose further that voter 5 similarly switches the position of B and D
in his preference ordering — and now ranks B third and D fourth. If a social choice function
satisfies IIA, which social rankings cannot be affected by this change in preferences?
Answer: If the social choice function represented by the Borda Count satisfied IIA, no social
ordering of a pair that does not involve B and/or D should change — because no individual
preferences over pairs involving A, C or E have changed. Similarly, we have not changed
how anyone feels over B and C — voter 4 still likes C better than B as does voter 5. Thus, the
Governments and Politics
Voter 1
Voter 2
Voter 3
Voter 4
Voter 5
Total
Borda Count
A
B
C
5
4
3
5
4
3
5
4
3
1
4
5
1
3
4
17 19 18
1062
D
2
2
2
2
2
10
E
1
1
1
3
5
11
Table 28.4: Borda Count Implies B Wins
social preference ordering over B and C should not change. There are a number of others
like this.
(e) How does the social ordering of the projects change under the Borda Count? Does the Borda
Count violate IIA?
Answer: This is calculated in Table 28.4 where the numbers that have changed from the
initial Table 28.1 are in bold. The Borda count ranking is now
B ≻ C ≻ A ≻ E ≻ D.
(28.19)
But we see that C was preferred by the Borda Count to A before the change and now A is
preferred to C . If the alternatives B and D were irrelevant to how the social preference order
ranks A and C , this should not happen. Note also that no voter changes their ranking of
B against C — yet the Borda count had C winning before and now has B winning. We see
again that the Borda Count violates IIA.
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Governments and Politics
28.8 Policy Application: Interest Groups, Transactions Costs and Vote Buying: Suppose that a legislature
has to vote for one of two mutually exclusive proposals — proposal A or B . Two interest groups are willing
to spend money on getting their preferred proposal implemented, with interest group 1 willing to pay up
to y A to get A implemented and interest group 2 willing to pay up to y B to get proposal B passed. Both
interest groups get payoff of zero if the opposing group’s project gets implemented. Legislators care first and
foremost about campaign contributions — and will vote for the proposal whose supporters contributed
more money, but they have a weak preference for project B in the sense that they will vote for B if they
received equal amounts from both interest groups.
A: To simplify the analysis, suppose that there are only three legislators. Suppose further that interest
group 1 makes its contribution first, followed by interest group 2.
(a) If y A = y B , will any campaign contributions be made in a subgame-perfect equilibrium?
Answer: In stage 2 when interest group 2 makes its decision, it will have observed interest
group 1’s contributions to the three legislators. Interest group 2 will make contributions
in stage 2 only if it can cause program B to be implemented — i.e. if it can surpass the
contributions made by interest group A for two of the three legislators. Thus, interest group
2 will give contributions equal to what A gave to the legislators that A gave the least to if
that amount is less than or equal to y B — otherwise it will not give anything and let A be
implemented. Interest group 1 knows this when it moves first — and the most interest group
1 is willing to give to get A implemented is y A . Suppose it gives the maximum it is willing
to give. Then it knows that interest group 2 will pick off the two legislators that received the
least from A — which will be less than y A and thus less than y B . Thus, there is no way that
interest group 1 can give an amount that it would be willing to give to get policy A and keep
interest group 2 from giving less than it would be willing to pay and still get B implemented.
As a result, interest group 1 gives nothing in stage 1 — which implies interest group 2 has to
give nothing in stage 2 to get B implemented. No campaign contributions are given in the
subgame perfect equilibrium.
(b) Suppose 1.5y B > y A > y B . Does your answer to (a) change?
Answer: Interest group 2’s strategy in stage 2 will be as it was in (a) — i.e. pick off the cheapest two legislators so long as the required sum of contributions is less than y B . Interest
group 1, it it contributes, must therefore find a way to make it just expensive enough for
interest group 2 to find it not worthwhile to pick off the cheapest two legislators in stage 2.
That means that the only way interest group 1 will make positive contributions in stage 1
is if the sum of the two lowest contributions it makes to legislators is just slightly above y B .
There is no sense in giving more extra contributions to the third legislator — which means,
if interest group 1 makes contributions, it will give equal amounts to the three legislators —
which implies the most it is willing to give to each legislator is y A /3. This will succeed so
long as y B < 2y A /3. But since y A < 1.5y B , interest group 1 still cannot give enough to keep
interest group 2 from getting B implemented. Thus, there will still be no contributions, and
project B gets implemented.
(c) Suppose y A > 1.5y B . What is the subgame-perfect equilibrium now?
Answer: Now interest group 1 places sufficient value on its favored project to succeed in
keeping interest group 2 from being able to get B implemented in stage 2. Interest group
1 will give (roughly) equal contributions to the 3 legislators — with each receiving at least
y B /2 and two of them receiving just slightly more (to make sure it is too expensive for interest group 2 to pick off the two cheapest). Thus, political contributions will equal approximately 1.5y B and project A will be implemented.
(d) Suppose that project B is extending milk price support programs while project A is eliminating such programs, and suppose that y A > 1.5y B because milk price support programs are
inefficient. Interest group 1 represents milk consumers and interest group 2 represents milk
producers. Which interest group do you think will find it easier to mobilize its members to
give the necessary funds to buy votes in the legislature?
Answer: This is a case of concentrated benefits and diffuse costs — milk producers are a
relatively small number of concentrated beneficiaries of project B , while milk consumers
are a large group of people who are slightly better off if A passes instead. It would therefore
be easier to organize the smaller number of concentrated beneficiaries of B rather than the
large number of diffuse beneficiaries of A.
Governments and Politics
1064
(e) Suppose y A > 3y B . It costs interest group 2 exactly $1 for every dollar in contributions to a
legislator, but — because of the transactions costs of organizing its members, it costs interest
group 1 an amount $c per $1 contributed to a legislator. How high does c need to be in order
for the inefficient project to be passed?
Answer: We know that interest group 1 will need to get approximately 1.5y B into the pockets
of legislators to get policy A implemented. It will be able to to do this so long as y A > 1.5y B c
or, since y A = 3y B , so long as 3y B > 1.5y B c — which solves to give us c < 2. Thus, if it costs
interest group 1 more than twice what it costs interest group 2 to raise money, it will not
be able to stop the inefficient project B even though its members would get three times the
benefit from A that members of interest group 2 get from passing B .
(f) How might the free rider problem be part of the transactions costs that affects interest group 1
disproportionately?
Answer: The larger the group membership, the bigger the free-rider problem faced by the
interest group that attempts to get members to contribute voluntarily.
B: Consider the problem faced by the interest groups in light of results derived in Chapter 27. In
particular, suppose that all members of interest group A have tastes u A (x, y ) = x α y (1−α) where x is
private consumption and y is a function of the likelihood that project A is implemented. Members
of interest group B similarly have tastes u B (x, y ) = x β y (1−β) where y is a function of the likelihood
that project B is implemented. Suppose that interest groups have successfully persuaded members
to believe y is equal to the sum of their contributions to the interest group. Everyone has income I ,
and there are N A members of interest group 1 and N B members of interest group 2.
(a) What is the equilibrium level of contributions to the two interest groups?
Answer: Using the result from exercise 27.1, we know that the equilibrium level of contributions to the two interest groups will be
yA =
N A (1 − α)I
α(N A − 1) + 1
and y B =
N B (1 − β)I
β(N B − 1) + 1
.
(28.20)
(b) Suppose again that B is a renewal of an inefficient government program with concentrated
benefits and diffuse costs — and A is the elimination of the program. What does this imply about the relationship between N A and N B ? What does it imply about the relationship
between α and β?
Answer: If the beneficiaries of B are concentrated and those of A are diffuse, it implies that
N A > N B and α > β.
(c) Suppose N A = 10,000, N B = 6, I = 1,000, α = 0.8 and β = 0.6. How much will each interest
group raise? How does your answer change if N A is 100,000 instead? What if it is 1,000,000?
Answer: Plugging these values into our results from (a), we get
y A = 250 and y B = 600.
(28.21)
Thus, when there are 10,000 in interest group 1, each contributes 2.5 cents while the 6 members of interest group 2 contribute $100 each. The aggregate result does not change as the
population of interest group 1 increases — all that happens is that individuals give even less,
with the total remaining at $250.
(d) Suppose that β is also 0.8 (and thus equal to α). If the vote-buying process is as described in
part A, will legislation B pass even though there are 1,000,000 members of interest group 1
and only 6 in interest group 2?
Answer: In this case, plugging in the new value for β, we get y B = 240 while y A is still 250.
Under the vote buying process in part A, that is not sufficient for interest group 1 to get A
implemented — and B will therefore still be implemented. Even though the benefits for a
member of interest group 1 are now defined the same as those for members of group 2, the
free rider problem is such that group 1 cannot raise substantially more than group 1 even
though there are 1,000,000 of them rather than just 6.
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(e) Finally, suppose that there is only a single beneficiary of B . How much will he contribute
when β = 0.8? What if β = 0.6? Within this example, can even one concentrated beneficiary
stop a project that benefits no one other than him?
Answer: The single beneficiary of B will contribute up to $200 if β = 0.8 and up to $400 if
β = 0.6. Thus, yes — in this example, even a single “concentrated” beneficiary can get his
policy implemented over the objection of all others.
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28.9 Policy Application: Political Coalitions and Public School Finance Policy: In this exercise, we consider some policy issues related to public support for schools — and the coalitions between income groups
that might form to determine the political equilibrium.
A: Throughout, suppose that individuals vote on only the single dimension of the issue at hand —
and consider a population that is modeled on the Hotelling line [0,1] with income increasing on
the line. (Thus, individual 0 has the lowest income and individual 1 has the highest income, with
individual 0.5 being the median income individual.)
(a) Consider first the case of public school funding in the absence of the existence of private school
alternatives. Do you think the usual median voter theorem might hold in this case — with the
public school funding level determined by the ideal point of the median income household?
Answer: While it depends on the nature of the tax system used to fund the public schools, it
is not unlikely that, given the tax system used to fund the schools, demand for public school
spending increases with household income. In that case, the ideal points for public spending would be increasing along the Hotelling line — with the median voter’s most preferred
public spending level representing the Condorcet winning policy.
(b) Next, suppose private schools compete with public schools, with private schools charging tuition and public schools funded by taxes paid by everyone. How does this change the politics
of public school funding?
Answer: Conditional on sending one’s children to private schools, one’s preferred tax — and
thus one’s preferred public school spending level — would drop to zero. Thus, the politics of
public school funding changes in the sense that those who choose private schools become
low demanders of public school quality.
(c) Some have argued that political debates on public school funding are driven by “the ends
against the middle”. In terms of our model, this means that the households on the ends of
the income distribution on the Hotelling line will form a coalition with one another — with
households in the middle forming the opposing coalition. What has to be true about who
disproportionately demands private schooling in order for this “ends against the middle” scenario to unfold?
Answer: For the “ends against the middle” scenario to unfold, it must be that demand for
private schools comes disproportionately from high income families who would favor high
public school spending in the absence of private schools but now favor low public school
spending.
(d) Assume that the set of private school students comes from high income households. What
would this model predict about the income level of the the new median voter?
Answer: This is illustrated in panel (a) Graph 28.1. The first line from 0 to 1 represents the
original Hotelling line with income and ideal levels of public school spending rising along
the line. As a result, 0.5 is the original median voter whose preferred public school spending
level is implemented in the absence of private schools. If then a private school opens and
draws the highest x income households away from the public system, that x segment of the
Hotelling line now moves to the other side of the original median voter in terms of its ideal
point for public good spending. As a result, the new median voter is n = 0.5 − x — which
implies the new median voter (in the presence of private schools) has less income than the
original median voter (in the absence of private schools).
(e) Consider two factors: First, the introduction of private schools causes a change in the income
level of the median voter, and second, we now have private school attending households that
pay taxes but do not use public schools. In light of this, can you tell whether per pupil public
school spending increases or decreases as private school markets attract less than half the
population? What if they attract more than half the population?
Answer: As we have seen, the median voter’s income will fall — which, all else being equal,
would imply a decrease in public school spending. But fewer kids attend public school and
thus every dollar in tax revenue results in a larger increase in per pupil spending. Thus, “all
else is not equal” — and the two forces point in opposite direction. This makes it impossible
to tell without further information whether per pupil public spending on education rises or
falls as more high income individuals go to private schools — at least so long as fewer than
50% do so. If more than 50% attend private schools, the median voter will be someone
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Graph 28.1: Median Voters for Public School Spending
who sends her child to private schools — which would cause us to predict a sharp drop
in public school funding (to zero, if we take the model completely literally). But if private
school attendance is less than 50%, you can think of public school attendees actually getting
a subsidy from private school attendees — so, although they would vote for less spending if
the price remained the same, they might vote for more given the implicit subsidy that allows
them to free ride on the tax payments of private school attending households.
(f) So far, we have treated public school financing without reference to the local nature of public
schools. In the U.S., public schools have traditionally been funded locally — with low income
households often constrained to live in public school districts that provide low quality. How
might this explain an “ends against the middle” coalition in favor of private school vouchers
(that provide public funds for households to pay private school tuition)?
Answer: In a system with widely varying school quality based on local income levels, it may
be the case that low income parents would be the first to switch to private schools if they
received a voucher — and would find this preferable to their current public school. High
income households already send their children to private schools — so vouchers would be
a pure income transfer to them (as they would not have to pay as much of their children’s
tuition bills anymore). Thus, there are two natural constituencies for private school vouchers: the poor who are constrained in the public system to the worst schools, and the rich
who already use private schools.
(g) In the 1970’s, California switched from local financing of public schools to state-wide (and
equalized) financing of its public schools. State-wide school spending appears to have declined as a result. Some have explained this by appealing to the fact that the income distribution is skewed to the left, with the statewide median income below the statewide mean
income. Suppose that local financing implies that each public school is funded by roughly
identical households (who have self-seleted into different districts as our Chapter 27 Tiebout
model would predict), while state financing implies that the public school spending level is
determined by the state median voter. Can you explain how the skewedness of the state income distribution can then explain the decline in state-wide public school spending as the
state switched from local to state financing?
Answer: Such a skewed income distribution is graphed in panel (b) of Graph 28.1 where I 1
is the median income level and I 2 is the mean income level. In a local system where income
types separate into distinct communities, there is broad agreement within the community
on how much to spend on the local school — because everyone has the same income. Thus,
each community funds its school in relation to the common income level of its residents —
with the average per pupil spending level in the state therefore approximately equal to the
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1068
income level that the average person would have chosen. When per pupil spending is determined in state-wide elections, however, the median income level will determine the spending level — and since the median is below the mean, we would expect the skewedness of the
income distribution to result in less overall public spending on education in the statewide
system than in the local system.
B: Suppose preferences over private consumption x, a public good y and leisure ℓ can be described
by the utility function u(x, y,ℓ) = x α y β ℓγ . Individuals are endowed with the same leisure amount
L, share the same preferences but have different wages. Until part (e), taxes are exogenous.
(a) Suppose a proportional wage tax t is used to fund the public good y and a tax rate t results in
public good level y = δt . Calculate the demand function for x and the labor supply function.
(Note: Since t is not under the control of individuals, neither t nor y are choice variables at
this point.)
Answer: The consumer then solves the problem
max x α y β ℓγ subject to x = (1 − t )w(L − ℓ).
x,ℓ
(28.22)
Solving this in the usual way, we get leisure demand of
γL
.
α+γ
(28.23)
µ
¶
αL
αL
and x ∗ = (1 − t )w
.
α+γ
α+γ
(28.24)
ℓ=
Subtracting this from L gives the labor supply l ∗ , and plugging it into the budget constraint
gives us x ∗ — which are
l∗ =
(b) Suppose instead that a per-capita tax T is used to fund the public good; i.e. everyone has to
pay an equal amount T . Suppose that a per-capita tax T results in public good level y = T .
Calculate the demand function for x and the labor supply function.
Answer: We now solve
max x α y β ℓγ subject to x = w(L − ℓ) − T.
x,ℓ
(28.25)
Solving this in the usual way, we get leisure demand
γ(wL − T )
.
α+γ
(28.26)
αwL − (2γ + α)T
αwL − γT
and x ∗ =
.
(α + γ)w
α+β
(28.27)
ℓ=
Subtracting from L gives labor supply l ∗ and substituting into the budget constraint gives
demand for x — i.e.
l∗ =
(c) True or False: Since the wage tax does not result in a distortion of the labor supply decision
while the per-capita tax does, the former has no deadweight loss while the latter does.
Answer: This is false. Efficiency losses from taxes happen from substitution effects — which
occur when taxes change opportunity costs and not when they do not. The wage tax changes
the price of leisure — and thus gives rise to substitution effects which happen to be masked
by an offsetting wealth effect in our example. But the substitution effect creates deadweight
loss. The lump sum tax T , on the other hand, does not give rise to any substitution effects
— even though it’s wealth effect causes a change in the optimal labor supply decision. But
the wealth effect does not cause a deadweight loss.
(d) Calculate the indirect utility function for part (a) (as a function of L, w and t ).
Answer: To get the indirect utility function, we plug ℓ∗ and x ∗ from (a) into the utility function and replace y with δt to get
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¶¸α
¶
µ
·
µ
γL γ
αL
(δt )β
V = (1 − t )w
α+β
α+γ
"
µ
¶(α+γ) #
αL
= (1 − t )α t β δβ w α
.
α+γ
(28.28)
(e) Now suppose that a vote is held to determine the wage tax t . What tax rate will be implemented under majority rule? (Hint: Use your result from (d) to determine the ideal point for
a voter.)
Answer: To determine a voter’s ideal tax rate t ∗ , all we have to do is maximize the indirect
utility function with respect to t . Taking the derivative of V with respect to t and setting it
to zero, we can then solve for voter’s optimal tax rate as
t∗ =
β
.
(α + β)
(28.29)
Since this optimal tax rate for our voter is not a function of wage (which is the only dimension on which voters differ), all voters agree that this is the optimal tax rate — i.e. all voters
have the same ideal point. This is because, although higher income voters demand more y
all else equal, they also have to pay more of a tax share — with the latter effect offsetting the
former.
(f) Suppose that y is per pupil spending on public education. What does this imply that δ is
(in terms of average population income I , number of taxpayers K and number of kids N in
school)?
Answer: Tax revenue from a tax rate t is then t K I — and per pupil spending is revenue
divided by N . Thus,
KI
when y = δt .
(28.30)
N
(g) Now suppose there exists a private school market that offers spending levels demanded by
those interested in opting out of public education (and assume that spending is all that matters in people’s evaluation of school quality). People attending private school no longer attend
public school but still have to pay taxes. Without doing any additional math, what are the
possible public school per pupil spending levels y that you think could emerge in a voting
equilibrium (assuming that public education is funded through a proportional wage tax)?
Who will go to what type of school?
Answer: As more children go to private school, N — the number of children going to public
school — falls. This implies that δ increases as more children go to private school. Our expression of t ∗ is independent of δ — which implies that those who remain in public schools
continue to favor the same tax rate as before, but those choosing private school will now favor a tax rate of zero. So long as the fraction going to private school is less than 0.5, however,
t ∗ remains the majority rule equilibrium — which implies that per pupil public spending
increases as private school markets expand. Those who attend private school will be the
richest households.
δ=
(h) Can you think of necessary and sufficient conditions for the introduction of a private school
market to result in a Pareto improvement in this model?
Answer: So long as less than half of the population goes to private school, the introduction
of private schools represents a Pareto improvement in this model. This is because per pupil
spending rises in public schools while tax rates remain the same — implying that public
school attending households are better off than they would be in the absence of private
schools. Private school attending households could choose to consume the higher levels of
public school spending at the same tax rates as well but choose instead to opt for private
school — which implies they are at least as well off as they would be in the public schools
that now spend more. Thus, private school attending households must also be better off
than they would be in the absence of a private school market. With everyone benefitting
from the introduction of private schools, the model therefore predicts a pareto improvement.
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1070
(i) In (e), you should have concluded that, under the proportional wage tax, everyone unanimously agrees on what the tax rate should be (when there are no private schools). Would the
same be true if schools were funded by a per-capita tax T ?
Answer: No. Under a per capita tax T , everyone pays the same amount (rather than the
same rate). This implies that ideal points will not all be the same in the absence of private
school market — with ideal points for T increasing with household income and the median
income household determining the level of T (and thus the level of per pupil spending) in
a voting equilibrium. Thus, only the median income voter would get his preferred level of
taxation.
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28.10 Policy Application: Government Competition, Leviathan and Benevolence: Suppose governments
can spend taxpayer resources on both public goods that have social benefits and political “rents” that are
private benefits for government officials. To the extent to which governments emphasize the latter over
the former, we have called them “Leviathan” — and this exercise investigates to what extent competition
between governments can restrain this Leviathan. To the extent to which governments emphasize the
former, we will call them “benevolent”. In part B of the exercise, we consider competition between such
benevolent governments.
A: Consider a collection N of local governments that can employ local property taxes to fund public
goods and local political rents. Suppose that local governments are pure Leviathans — i.e. they seek
only political rents. For simplicity, suppose also that all households are identical.
(a) Begin with a simple demand and supply (for housing) graph for one community. If a local
Leviathan government is a political monopolist in the sense that it faces no competitive pressures from other communities, how would it go about setting the tax rate that maximizes its
rents?
Answer: This is depicted in panel (a) of Graph 28.2 where the monopolist government can
simply impose taxes the way we modeled in Chapter 19. As a result of a tax t , price p r
paid by renters increases and price p s received by suppliers decreases in relation to the
relative elasticities of demand and supply. The rent-maximizing local government would
then increase the local tax rate until the shaded box of tax revenues is as large as it can
be. (At some point it will shrink as the quantity of house declines sufficiently to offset the
increased revenues on existing housing.)
Graph 28.2: Leviathan Local Governments
(b) Now consider the case where households are fully mobile across jurisdictional boundaries —
and thus choose to live where their utility is highest. In equilibrium, how must utility in any
jurisdiction i be related to utility in any other jurisdiction j ?
Answer: In equilibrium, it must be that the identical households get the same utility in all
jurisdictions.
(c) Suppose that the property tax is zero in all communities. Consider community i ’s Leviathan
mayor. If he raises t i above zero, and uses the revenues only for political rents, what will have
to be true about housing prices in community i after the tax is imposed (relative to before it
is imposed)? Can you demonstrate how this comes about? (Hint: Consider the competitive
pressure from household mobility.)
Answer: The difference between the mayor now and the monopolist local government in
(a) is that the major now knows that residents will move to other communities as the tax
increases until utility is again equalized across all communities. If community j is only
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one of many communities, then the impact of such mobility on other communities will
be negligible — which implies that the utility in community j cannot change as a result
of the tax. Since the tax is not used for anything that provides utility to residents, it must
therefore be that the local housing rental prices remain unchanged at p ∗ for residents. For
a tax t unilaterally imposed in jurisdiction j , demand for housing must therefore shift in
sufficiently to insure that the tax inclusive price remains p ∗ . This is depicted in panel (b) of
Graph 28.2 .
(d) True or False: So long as housing supply is not perfectly elastic, the Leviathan mayor in part
(c) will be able to raise property taxes to fund political rents.
Answer: We have just illustrated this for an upward sloping housing supply curve in panel
(b) of the graph. Were the housing supply perfectly elastic at p ∗ , the Leviathan would not
be able to impose a tax for rent seeking.
(e) Now consider all local governments setting some tax rate t and using revenues for political
rents. If t is very low, can a single community’s Leviathan’s mayor benefit from raising his tax
rate? If t is very high, can a single Leviathan mayor benefit from lowering his tax rate?
Answer: We have already shown that a single mayor is able to increase his rents when all
other tax rates are very low. (We did this in panel (b) of Graph 28.2.) In panel (c) of the graph
we illustrate the reverse — where tax rates are sufficiently high to have depressed housing
demand in all jurisdictions to a point where one community lowering their tax rate will
benefit from sufficient in-mobility of residents such that the lower rate causes an increase
in revenues. In the graph, area (a +b) is the initial tax revenue when the community charges
the same high rate as everyone else, and the rectangle (a + c) is the tax revenue after the
community unilaterally lowers its rate.
(f) Use your answer to (e) to argue that there must exist some level of Leviathan taxation across
competing communities that will be a Nash equilibrium.
Answer: We have shown that, when all communities charge low rates, a single community
can do better in terms of rent seeking by raising its tax — and when all communities charge
high rates, a single community can do better in terms of rent seeking by lowering its tax. It
must then be that somewhere in between “too low” and “too high” rates charged by everyone, there must be some t ∗ at which no community can do better by deviating given that
all other communities are charging the same rate. That’s the Nash equilibrium tax rate.
(g) Evaluate the following statement: “Unless housing supply is perfectly elastic, government
competition between Leviathan governments is not sufficient to eliminate political rents —
but it restrains the ability of Leviathan government to amass such rents.”
Answer: This is true — as we have shown. Competitive governments cannot get as much in
rents as monopoly governments, but they can get more than zero unless housing supply is
perfectly elastic.
(h) True or False: To the extent to which government behavior is characterized by rent-seeking,
greater competition between governments enhances efficiency.
Answer: This is true. All taxes to support rent seeking are inefficient (unless the taxes used
are themselves efficient and thus only transfer resources). Competition between local rentseeking governments limits the ability for such governments to seek rents — and thus enhances efficiency.
B: Next, consider the opposite type of government — i.e. one that is benevolent and raises taxes only
to the extent to which it can find worthwhile public goods to finance. Suppose again that there are
N such governments that use a local property tax to fund local public goods — and suppose that
all public benefits from such public goods are contained within each government’s jurisdictional
boundaries.
(a) Begin, as in A(a), by assuming that there is mobility of consumers across jurisdictions and
thus no government faces any competitive pressures. Will they produce the efficient level of
local public goods?
Answer: Assuming they have sufficient information to know what the efficient level is, the
answer is yes — there is nothing to prevent a benevolent government to do so.
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(b) Next, consider the competitive case. If the projects funded by local governments are truly local
public goods, in what sense are taxes imposed by benevolent governments offset by benefits
received?
Answer: If local governments buy efficient levels of truly local public goods with tax revenues, then consumers should be getting surplus — i.e. the value of the public goods is
higher than the tax payments received — and equal on the margin.
(c) Suppose other governments are charging low tax rates that result in inefficiently low levels of
public goods. If community i raises its tax rate and provides more public goods, will population increase or decrease in community i ? Will housing prices go up or down?
Answer: If local taxes are inefficiently low, then one community raising taxes allows that
community to provide surplus value higher than the tax payments made by consumers —
which implies that the community becomes more attractive and population should increase
— driving housing prices up.
(d) Consider an equilibrium with benevolent local governments providing efficient levels of local
public goods. Can any government raise property values by raising or lowering taxes? True
or False: Property value maximizing local governments behave like benevolent local governments.
Answer: In equilibrium, if all governments provide the efficient level of local public goods,
then any change on the local tax rate will either give up existing surplus (if the tax is lowered)
or create negative marginal surplus (if the tax is raised). Thus, any change in the tax rate will
make residents worse off — implying that housing prices must drop to compensate for this
in equilibrium. Thus, benevolent local governments that aim for efficiency indeed behave
as if they were trying to maximize local housing values.
(e) Suppose next that local property taxes are paid by both households and firms — but only
households benefit from local public goods (like schools). If firms are mobile, in what sense
does community i ’s decision to tax the property of firms give rise to a positive externality for
other communities?
Answer: The positive externality for other communities is the positive impact of firms moving out of community i to flee taxes.
(f) What does your answer to (e) imply about the spending levels by benevolent local governments
as competitive pressures increase in environments such as those described in (e)?
Answer: Any activity that gives rise to a positive externality will be underutilized — which
implies that communities will under-utilize taxes such as those described and thus spend
inefficiently little if they rely on such taxes. The would be better off switching to more
locally efficient taxes or having the federal government use the taxes that give rise to the
cross-border externalities and then have the revenues passed back to localities. (This was
discussed in an earlier end-of-chapter exercise in Chapter 20.)
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28.11 Policy Application: The Pork Barrel Commons and the “Law of 1/N ”: If you did not do these in
Chapter 27, you can now do end-of-chapter exercises 27.11 and 27.12 to explore the problem of porkbarrel projects and the “Law of 1/N .”
Answer: See our answer in the solutions to Chapter 27.