Show all your working. Explain your answers.

Suppose a nation has a large number of voters whose population is normalized to one. The government imposes a flat tax 7 and transforms the tax revenue into public goods. Let g be the level of public goods provided by the government. Then T = g. All voters have the same income y = 1. However, voters differ in their preferences for private consumption related to public goods. Voter i's utility function is u, 2 01,0 +1119, where c = 1 r is voter i's private consumption; and parameter a,- 6 [1,9] captures the voters' preference for private consumption. 30% of the population has a, = 1; 10% of the population has a, = 9. For the rest of the population, a, is distributed uniformly over interval [1, 9]. Voters cast their votes to decide the tax rate r for the nation. Is there a tax rate 7 that is a Condorcet winner? If so, what value of r is the Condorcet winner? Show your reasoning and calculation