Reference no: EM13968648
PART -1:
SECTION A
1. Choose four out of the six terms below. All terms are marked equally. For each one of the terms you choose, write a short explanation of what the term means and how it relates to what we studied in lectures.
a. Political Parties.
b. Political Economy.
c. Redistributive Politics.
d. Independence of Irrelevant Alternatives.
e. Strategic Voting.
f. Citizen-Candidate Model.
SECTION B
Answer three questions from the five below. All questions are marked equally. All sub questions are equally marked.
2. In this question we analyze the citizen-candidate model with approval voting. Approval voting is a voting system in which voters can cast up to two votes to two candidates that they like the best and the candidate with the highest number of votes wins the election (in case of a tie, each candidate is chosen with equal probability). In particular, we will assume that voters cast one votexto their preferred candidate when there are two candidates running, and that they cast two votes, one for each of their two favourite candidates, when there are more than two candidates. As in the citizen-candidate model studied in lectures, assume that individuals have ideal points that are uniformly distributed on [-1,1]. An individual with ideal point x has a utility from policy y given by -Ix-yl. Moreover, any individual that decides to run pays a cost c and gains a benefit of b if he wins the election. If no candidate runs for office, all individuals experience a utility of -∞
a) Assume that 2c<b<3c. Can you find an equilibrium in which two individuals are running who have ideal points which are different and symmetrically located around the median? If yes, find it, if not prove that there doesn't exist such an equilibrium.
b) Assume that 2c<b<3c. Find an equilibrium in which two individuals run.
c) Assume that c<b<2c. Is there an equilibrium with two candidates running?
d) Assume now that voters have only one vote to cast and a simple majority determines the winning candidate. Assume that c<b<2c. Is there an equilibrium with two candidates running? If yes find it, if not prove that there isn't one.
3. In this question we analyze how individual political preferences might be derived from an economic model. Assume that individuals only care about their income, y. The income of an individual depends on social circumstances, in, e.g., whether his parents were rich or poor and on the effort the individual exerts, ei. In particular,
yi= Πi + θei
where θ is the relative importance of effort to social circumstances. Exerting effort is costly; in particular the cost of effort is given by c(ei)=(( ei2)/2)
In this model, we assume that politics is about choosing the tax rate. The society must choose a tax rate, t, such that the after-tax utility of individuals is given by,
U(t, ei) = (1-t)yi + T(t)-c(ei)
where T is a lump sum transfer of all tax revenues, i.e., T(t)=t y^(t), where y^(t) is the average (before tax) income.
Assume that there are two individuals in society, individuals 1 and 2 with parameters Π1 < Π2.
a) Given a tax rate t and taking average income at 9(t) as a given, what level of effort will individual i choose to exert?
b) What is the individual's (indirect) utility from a tax rate t?
c) Is the indirect utility you found in (b) single peaked?
Note that in this model, the ideal point of agent i can be written as t*F-a(yi- 9(t)) where a>0.
t.i = -a(yi - y^(t))
where a> 0
d) Is it possible that t.2 >0?
4. Consider the following model of parties. Assume that two parties, L and R, compete in an election. Both parties want to maximise their vote shares. A party that chooses a platform y in [-1,1] influences the voters' beliefs about what policy will ensue if this party wins. In particular, assume that the mean of the policy voters expect is y and that the variance is σ(y)=1-|y|. Voters, who have quadratic loss utilities, only care about the mean and the variance of policy. In particular, voters have ideal points distributed uniformly on [-1,1], and a voter with ideal policy x, has an expected utility of -p(x-y)2-(1-p)σ(y) from a party offering platform y. Here the parameter p is in [0,1] and it measures the relative weight voters attach to expected policy rather than its variance.
a) Assume that p=1. What are the equilibria of the electoral competition game between the parties?
b) Assume that p=0. What are the equilibria of the electoral competition game between the parties?
c) Will the equilibria you found in (b) survive when p is very small?
d) In this question, the beliefs about the policies that would ensue if a party with a particular platform were assumed rather than modelled. Explain how such a connection can arise in political parties in practice. In general, would you expect Q(y) to decrease in y?
5. Suppose that f is the aggregation rule that a group of four individuals, 1,2,3 and 4, uses to find out their social preference over three alternatives, a,b and c.
a) Give an example of an aggregation rule that satisfies Pareto efficiency and non-dictatorship. Does it satisfy IIA as well? Explain your answer.
b) Suppose that f is Pareto efficient. Find an example of a preference profile for which alternative c will be ranked second in the social ranking. Explain your answer.
The following table summarises the 2008 preferences of the four individuals, 1,2,3 and 4, about the three alternatives, a,b and c.
2008 preferences
1 2 3 4
best a b a a
middle b a b c
worst c c c b
c) True or false: Pareto efficiency implies that under f alternative c must be ranked third in the social ranking of alternatives. Explain your answer.
d) Suppose that in 2008 f ranked alternative 'a' first. In 2009 the preferences of individuals was changed to:
2009 preferences
1 2 3 4
best b b a a
middle a a b c
worst c c c b
what can you say about the ranking of alternative 'a' in 2009 given that f satisfies IIA?
6. Suppose that two candidates, A and B, are competing in an election and both want to win the election. The policy space is a set of discrete policies, P, in which p is a typical policy.
There are n voters each with a utility function Ui(.) defined over P. (note that these preferences do not have to be single peaked). Assume that no voter is indifferent between any two policies.
Assume that there exists a unique policy p* that Strictly Pareto dominates all other policies in P, i.e., for any p in P, p ≠ p, Ui(p) < Ui(p.) for all i = 1,...,n.
Each voter can decide whether to go and vote or not, and if he goes to vote, who to vote for. The cost of going to vote, c, is strictly positive but very small. If no voter votes, policy is chosen randomly with an equal probability for each platform.
a) Suppose voter i goes to vote in equilibrium and is facing platforms pA and pa. Which platform/candidate will the voter vote for if he never uses weakly dominated actions?
b) Suppose the voters are facing platforms pA and pa such that pA ≠ pa. Show that it must be that some voters go and vote in equilibrium (remember that the cost of going to vote is very small).
c) Suppose the voters are facing platforms pA and pa such that pA = pa. In equilibrium, how many voters will go to vote?
d) Find the set of equilibria in this game. How many voters go to vote in any equilibrium?
e) Discuss the normative predictions of this model. In your discussion take into account both the policies chosen and the costs that voters accrue in going to vote.
PART -2:
SECTION A
1. For each one of the terms you choose, write a short explanation of what the term means and how it relates to what we studied in lectures.
a. The citizen-candidate model.
b. Political Economy.
c. Duverger's law.
d. Credible platform.
e. strumental voting.
Arrow's Impossibility Theorem.
SECTION B
2. In this question we analyze the citizen-candidate model with negative plurality. Negative plurality is a voting system in which voters cast a negative vote to one candidate and the candidates with the least number of negative votes wins the election (in case of a tie, each candidate is chosen with equal probability). As in the citizen-candidate model studied in lectures, assume that individuals have ideal points that are uniformly distributed on [-1,1]. An individual with ideal point x has a utility from policy y given by -Ix-yl. Moreover, any individual that decides to run, pays a cost c and gains a benefit of b if he wins the election. If no candidate runs for office, all individuals experience a utility of -03. Finally, we assume that voters vote sincerely; they cast their negative vote to the candidate they least prefer.
a. Assume that b<c. Is there an equilibrium in which one individual is running that has an ideal point that is not the median? If yes, find it, if not, prove that such an equilibrium doesn't exist.
b. You might have noticed that the analysis in (a) is similar to the analysis we did in lectures when the voting system was simple plurality. Discuss why this is the case.
c. Assume that 2c<b<3c. Can you find an equilibrium in which two individuals are running who have ideal points which are symmetrically located around the median? If yes, find it, if not, prove that there doesn't exist such an equilibrium.
d. Assume that 2c<b<3c. Find an equilibrium in which two individuals run.
e. Assume that 3c<b<4c. Find an equilibrium in which three individuals run.
3. In this question we analyze the role of political parties. Consider a society with two groups, the rich (R) and the poor (P). The poor make up two thirds of the population and the rich a third. The two groups have conflicting views regarding redistribution, the poor want full redistribution and the rich want no redistribution.
a. Suppose that each group has a representative that could, at a cost, run for office. The winning representative always chooses his groups' ideal policy. If costs of running are not too high, what would be the resulting policy of this citizen-candidate model?
Suppose that the young poor, consisting of a little less than half of the poor have suddenly become politically organized. They bring about the issue of Education which they want to be financed by the government. This group has a representative as well. The new configuration of society is as follows. There are now three Groups, YP (the young poor), OP (the old poor) and R (the rich). There are two policy dimensions: Redistribution and education. Therefore, the society needs to decide how many taxes to collect (x), and given the taxes how much revenue to allocate to education (y). The preferences of the three groups are given by
Ui(x,y)=-ai(ix-x)2-(1- ai )( iy -y)2, for i=YP,OP,R and al in [0,1].
Furthermore assume that the ideal points, (ix, iy), of the groups are: (0,0) for group R, (1,0) for group OP and (1,1) for group YP. i.e., group R are in favour of no taxation (and therefore no education or redistribution), group OP are in favour of full redistribution and group YP are in favor of full taxation towards the funding of education. Assume that group R prefer the ideal point of group OP to that of VP and that group YP prefer the ideal point of OP to that of group R.
b. If the group representatives play the citizen candidate game, what would be the policy implemented in this society?
c. Suppose that ayp=0 (i.e., the young poor care only about education) and that aR=1. If the young poor and the rich form a party together, what are the platforms that they could credibly commit to? Do they have a credible platform that would win the election?
d. Suppose that ayp =0 and that aR =0. Could the rich and young poor credibly commit to a platform that would win the election?
e. Interpret the young poor as a minority group and discuss what the analysis in (a)-(d) implies about the political power of minority groups.
4. Suppose that two opponents are trying to bribe a committee to decide in their favour. Opponent L wants policy I to be implemented and Opponent R wants policy r. Opponent L has BL pounds to spend and R has BR. There are n (odd) committee members to be bought, each willing to vote in the direction of the highest bidder (if the bids are equal, or nil, they randomly pick a vote). The decision in the committee is taken by majority vote. The opponents, each in his turn, make offers to pay committee members for their votes. Suppose that R moves first and that L moves second.
a. Suppose that R has allocated all his money by dividing it equally among ((n+1)12) committee members. With how much money can L win the committee decision?
b. Suppose that R has allocated all his money by distributing it equally among all members of the committee. With how much money can L win the committee decision?
c. Can R do something better than what he does in (b)?
d. As compared with L, how much more money does R need to have to win the committee over?
e. Now consider the case in which both opponents move simultaneously and suppose they have the same sum of money at their disposal. This game is called the "Colonel Blotto" game. Show that the game has no Nash equilibrium.
f. One can interpret the model in (e) as a model of the electoral competition over redistribution. Discuss this possibility (e.g., try to think what BL and BR are, who the committee members are and what kind of a redistribution scheme the model allows the government to implement).
5. Three people, 1, 2 and 3, have to decide how they will decide things in the future. Suppose that there are two periods, 0 and 1. In period 1, the three individuals will have to choose between alternatives 'a' and 'b'. In this period they will have preferences regarding these two alternatives; each will have a favourite policy that if implemented will provide him with a utility of 1 and a least preferred alternative that would provide him with a utility of zero.
Suppose that the individuals consider three decision rules: Unanimity A in which only if all individuals vote for *a' then it is chosen and otherwise 'b' is chosen. Unanimity B, in which only if all individuals vote for 'b' then it is chosen and otherwise 'a' is chosen. Majority Rule, in which the option with the majority of votes is chosen. In what follows, the individuals will choose the rules to determine policy either in period 1, when they know their preferences or in period 0 when they don't know their preferences in period 1. In period 0, each individual believes that his preferred policy in period 1 will be 'a' with probability half. Each individual also believes that the others' preferences are determined in a similar fashion, but independently of each other. The status quo policy, which is implemented in the absence of any decision, is policy 'a'. Decisions on rules are always taken with consensus; if there is no rule that has consensus (i.e., for which no agent prefers another rule), the status quo remains.
a. Suppose that in period 1, individuals 1 and 2 prefer 'a' and 3 prefers 'b'. Suppose they hadn't chosen a rule in period 0. Does any of the rules have consensus?
b. Show that if no rule was chosen in period 0, then with any configuration of preferences- except when all three individuals prefer 'b'- alternative 'a' will be implemented.
c. What is the expected utility of an individual at period 0, when no rule is chosen at period 0.
d. Suppose Unanimity A is chosen at period 0. Compute the expected utility of an individual at period 0.
e. Show that majority rule chosen at period 0 maximizes the expected utility of individuals. Is this utility larger than the one computed at (c)?
f. Interpret the model and results as they relate to the study of political institutions.
6. One possible explanation to the observed patters of turnout in elections might be the efforts exerted by parties to persuade voters to come and vote. Parties have many resources and so are able to mobilize large numbers of people. As parties have large incentives to win the election and as they always expect to be pivotal in the election, they don't suffer from the usual problems with models of instrumental voters. In this question we try and investigate this possibility. Suppose that two political parties, L and R, compete to win an election. The probability of winning the election depends on the efforts of the two parties to bring out the vote. Each party i
can spend effort el in [0,k] on encouraging favourable voters to vote and the probability of party L winning is given by:
Pr(L winsleL, es)=f(eL, eR) The corresponding probability for Party R is: Pr(R wins' eL, eR)=1-Pr(L wins' eL, eR)=1-f (eL, eR)
a. Consider the following three specifications for the function f(eL, es). Among them, choose the specification that you think is most suitable for the application we have in mind. Explain your choice.
f(eL, eR) = eL + es f(eL, eR) = eL x es f(eL, eR) = eL + eR)
b. We will actually work with the specification f(eL, eR) = Trei../( TreL + es), where Tr>1. Interpret the parameter 1T as a bias towards one of the parties. In favour of what party is this bias? Provide a possible explanation for the origin of such a bias.
To close the model we assume that effort is costly for the party; party i pays ((ei2)/2) for effort level ei. The party enjoys a benefit of b from winning the election and receives zero otherwise.
c. Suppose that Party L expects Party R to exert effort eR. Write the Party's expected utility as a function of its planned effort level et.
You are fortunate as I have taken liberty and computed the Nash equilibrium for you. In the Nash equilibrium
eL = eR =(41-r)/(Tr+1)
d. Let us interpret the sum of efforts exerted by the parties as a measure which is proportional to turnout in the election. Compute this sum for Tr=1 and Tr=4. Is this effect of 1T on turnout consistent with empirical observations about turnout? Provide an intuition for this result.
e. Calculate the equilibrium probability that party L wins the election. How does it change when Tr increases?
f. Discuss the voting paradox and whether this model goes some way towards solving it.