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Question 9 Private agents form rational expectations of inflation, but policymakers move after and may deliver surprise inflation. Unexpected inflation may come with two benefits. The first is to lower unemploy- ment according to a Phillips curve: U = UN - a(1 - *') where as usual " denotes the natural rate of unemployment, * is inflation, " is expected inflation, and o > 0 is a slope parameter. The second is that it lowers the real value of the public debt that the government must pay for. Consider a simple linear relation capturing this: b =6-6 - 7(x - *") so that debt can fall from its initial level b because of unexpected inflation with sensitivity parameter > 0, or because of sovereign default 6. Society's loss function is: #2 Mu - UN + k)2 0b2 2 2 + 85 Society dislikes: inflation; unemployment above a desired rate that is k > 0 below the natural rate; debt above zero; sovereign defaults as they endanger financial stability. The parameters A, e, and @ are all positive. (a) Is there time inconsistency in this economy and why? Is there an inflation bias and why? How does the inflation bias vary with k and why? How does the inflation bias vary with & and why? (b) Fiscal policy chooses how much to default on the debt, f. Because it takes long to have default approved, by the time it happens people know about it. Consider first the case where the fiscal authority cares only about the losses from debt: 2 - + 86, and the fiscal authority chooses a before the public forms its expectation. Solve for equilibrium value of & chosen by an optimizing fiscal authority. How does it vary with & and why? (C) Solve for equilibrium inflation. How does it vary with society's aversion to sovereign default and why