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Question 4: Julia, Kevin, and the Federal Election In this question we study the economics behind the Labor party's choice of Kevin Rudd over Julia Gillard to face off against Tony Abbott in this year's federal election. Consider a costly rent-seeking model where prior to the federal election ("period the Labor party chooses either Gillard or Rudd to be their leader. Given their choice, the Labor leader faces off against Abbott in the federal election ("period Having run some opinion polls, Labor finds Gillard is becoming unpopular among the electorate and will have a tougher time winning the election over Abbott. In particular, assume that if Gillard faces Abbott in the election, that if Gillard spends rent-seeking effort r g and Abbott spends rent-seeking effort rA, then the probabilities that Gillard and Abbott win the election are: 0.5rg where pg is the probability that Gillard wins and pa is the probability that Abbott wins. If instead Rudd faces Abbott in the election and spends rr rent seeking effort, then the probabilities that Rudd and Abbott win the election are given by: 0.75rr where pr is the probability that Rudd wins and pa is the probability that Abbott wins. If Gillard/ Rudd/ Abbott spend rent-seeking effort rg/rr/ra, assume that it costs them rg/rr/ra regardless of the outcome of the election.2 Also assume throughout that rent-seeking effort lewls must be non-negative. Hint: You may find it helpful to use the quadratic formula in answering question 4.3. In particular, for a quadratic equation aa:2 + bx + e = 0 (where a, b, e are constants) the roots can be found with the quadratic formula: a: 4.1 Derive the first order condition that characterises Gillard's optimal rent seeking effort lewl r* as a function of Abbott's rent seeking level ra. 4.2 Derive the first order condition that characterises Rudd's optimal rent seeking effort level r; as a function of Abbott's rent seeking level ra. 4.3 Suppose that Abbott does not have any economists on his political staff and that he simply shouts out before Labor makes their Gillard/Rudd choice that he does not. care who Labor chooses to face him and that he will commit to choose ra 0.25 regardless of Labor's choice. Find Gillard's and Rudd's optimal rent-seeking levels, r* and r; , given Abbott's choice of ra = 0.25. 4.4 Using your results from 4.3, show that Labor should choose Rudd as their party's leader if they want to maximise the probability that they win the federal election. 2+2+4+2=10 Marks 2So, for example, Abbot's pmvoffs when facing Gillard would be