Gabrito2a Exercicios Resolvidos

5.3 Lobbying in a representative democracy\n\nThis problem is based on Besley and Coate (1997 and 1999). Consider a model with N citizens, E = [L, N]. They derive and assure that the median voter, yj, still votes for their preferred candidate among yj and yj'. 2. The reason is that, if they changed their preferences, if voters were least preferred alternatives in pairs would no longer be equilibria. The reason is that, with extreme platforms, a third candidate could enter the middle and win the election.\n\nThe solution is found numerically: yj = 1.64, yj' = 0.61. All pairs (yj, yj') such that yj' in the interval (0.64, 2) are equilibria. Chapter 5\nPartisan politicians\n\n5.1 Probabilistic voting with outcome-seeking politicians\n\nAssume that the indirect utility function of tax policy is described by wi = - (τ - τi)².\n\nThere are two political parties, one with preferred tax rate 0, the other with preferred tax rate 1. The parties can commit to a party platform that will be implemented should the party win the election. The political parties are uncertain about the most preferred tax rate τm of the median voter and assign a uniform probability distribution between (1/2 - a) and (1/2 + a) to τm. The parameter a ∈ (0,1). The parties are exclusively policy motivated. Let τ0 and τ1 be the policies proposed by parties 0 and 1 respectively.\n\na. Show that the parties will never choose their bliss points and will never converge completely.\n\nb. Solve for the equilibrium policies given that the equilibrium is symmetric, that is τ0 = 1-τ1. Discuss how the equilibrium policies depend on the level of uncertainty as described by a.\n\nc. Show that the equilibrium must be of the form τ0 = 1 - τ1.\n\nAnswer\na. Let p0 be the probability that party 0 wins the election. Party 0 maximizes\n\nmax τ0 -p0τ0² - (1 - p0)τ1².\n\nThe first order condition to this problem is \ndp0/dτ0 (τ0² - τ1²) - 2p0τ0 = 0. 44 CHAPTER 5. PARTISAN POLITICIANS\n\na. What policy would a winning candidate with income yj implement?\n\nb. Suppose that ε = √2 - √3 - √(1/4). In what region must the status quo policy, τ, lie in order for the equilibrium with only the citizen with median income as candidate to exist. Do there exist other one-candidate equilibria?\n\nc. Characterize the two-candidate equilibria.\n\nd. Describe what would happen in a two-candidate equilibrium if the median candidate would enter. How would the set of possible equilibria change if voters voted sincerely, that is voted for the candidate that gave them highest utility?\n\nAnswer\na. The most preferred tax rate of candidate with income yj is the solution to\n\nmax τ (√(1 - τ)yj + √τy).\n\nThis is solved by τi = yj / (yj + y).\n\nwhich is the policies that a candidate with income yj would implement.\n\nb. The median and mean incomes both equal 1, and the most preferred tax rate of the median income citizen is thus 1/2. For the citizen with median income as stand candidate, it must be the case that w™(τm) - w™(τ) ≥ ε, or equivalently\n\n√(τ/2) + √(1/2) - (√((1 - τ) + √τ) ≥ √2 - √(3/4).\n\nThis holds for τ ≤ 0.25 and τ ≥ 0.75.\n\nYes, there exist other one-candidate equilibria. Suppose another candidate runs. If the candidate is sufficiently close to the median, then all of the other candidates that would beat him are so close to his own position that they do not find it worthwhile to run. CHAPTER 5. PARTISAN POLITICIANS\n\nBy raising the tax level above its most preferred level 0, party 0 increases the probability that it will win the election and implements its platform. On the other hand by raising its proposed tax level, party 0 gets to implement a less preferred policy if it wins the election. If the parties had converged completely, then there would be no incentive for the party to increase its probability of winning the election but the incentive to move closer to its ideal point would still be present. Therefore, total convergence can not be an equilibrium. If the parties choose their most preferred points, then they would have no incentives to move closer to their ideal points, while they would have an incentive to increase their probability of winning the election. Therefore this could not be an equilibrium.\n\nb. The probability that party 0 will win the election is p_0 = Pr( τ_m < τ_0 + τ_1 )/2 = 1/2(τ_0 + τ_1 - 1/2 + a) = 1/2 τ_0 - (1 - τ_1)/4a\n\nInserting the expression for p_0 in the first order condition of party 0 yields\n\nτ_0 = 1/(2 + 4a)\n\nand similarly for party 1\n\nτ_1 = 1/(2 + 4a);\n\nTax platforms and uncertainty\n\nAs uncertainty grows, the competition to win the election becomes less sharp and the parties move away from the expected median voter position and closer to their own ideal points. 5.2 The citizen-candidate model\n\nConsider, a society, inhabited by a continuum of citizens with incomes distributed uniformly between 0 and 1. Each citizen i has private consumption c and a publicly provided private good g that an\nis: c_i = (1 - τ_i) y_i.\n\nConsider the following timing: (1) Any citizen may enter as a political candidate at cost; (2) An election is held among the candidates; the candidate who gets plurality wins the election, and it is resolved by the\nwinning candidate selects a tax rate τ_i if there are no candidates, then a default tax rate θ is implemented.\n\nThe public good g is financed through a proportional income tax τ, and the government budget constraint is τy / g where y is average income. Private consumption is given by c_i = (1 - τ)y_i.