In the text we discussed two main conditions under which the median voters 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?
(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 pair wise elections when there is an even number of voters and the conditions of the median voter theorem are otherwise satisfied?
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?
(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 pair wise votes, which other policies could you implement assuming the first vote in the sequence needs to put up a policy against A?
(e) In our discussion of the “Anything-Can-Happen” theorem, we raised the possibility of single issue 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?
(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?
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 contributor 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?
(b)What shape does this function have with respect to the policy variable t?
(c) At what t does this function reach its maximum?
(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.
(e) What does this imply for the real shape of the individual’s preferences over the policy variable t assuming δ > I/4?