In the congressional district described by the previous problem, let us investigate what will happen if the two candidates do not care about the number of votes that they get but only about the amount of campaign contributions that they receive. Therefore each candidate chooses his ideological location in such a way as to maximize the amount of campaign contributions he receives, given the position of the other.* Let us define a left-wing extremist as a voter whose political views lie to the left of the leftmost candidate, a right-wing extremist as a voter whose political views lie to the right of the rightmost candidate, and a moderate voter as one whose political views lie between the positions of the two candidates. Assume that each extremist voter contributes to the candidate whose position is closest to his or her own views and that moderate voters make no campaign contributions. The number of dollars that an extremist voter contributes to his or her favorite candidate is proportional to the distance between the two candidates. Specifically, we assume that there is some constant C such that if the left-wing candidate is located at x and the right-wing candidate is located at y, then total campaign contributions received by the left-wing candidate will be $Cx(y − x) and total campaign contributions received by the right-wing candidate will be $C(1 − y)(y − x).
(a) If the right-wing candidate is located at y, the contribution maximizing position for the left-wing candidate is x = ______ If the left-wing candidate is located at x, the contribution-maximizing position for the right-wing candidate is y = ______.
(b) Solve for the unique pair of ideological positions for the two candidates such that each takes the position that maximizes his campaign contributions given the position of the other. ______
(c) Suppose that in addition to collecting contributions from extremists on their side, candidates can also collect campaign contributions from moderates whose views are closer to their position than to that of their rivals’ position. Suppose that moderates, like extremists, contribute to their preferred candidate and that they contribute in proportion to the difference between their own ideological distance from their less-preferred candidate and their ideological distance from their more-preferred candidate. Show that in this case the unique positions in which the leftand right-wing candidates are each maximizing their campaign contributions, given the position of the other candidate, occurs where x = ¼ and y = 3/4. ______.