Consider the time inconsistency model without economic shocks. Assume that the social cost of inflationso that the marginal social cost of inflation. Assume that the (unemployment reduction) benefit of surprised inflation isso that the marginal benefit of surprised inflation. Also assume the Phillips curve is given bywhere(8%) is the natural rate of unemployment.

1. Suppose. Show graphically (in the MC and MB graph) and algebraically that it is optimal for the monetary authority to implement some positive inflation policy. How does such "cheating" outcome compare to a "zero-inflation rule" outcome? Which one is more socially desirable? Explain.
2. For any given expected inflation, use the marginal condition to derive the policy response function of the government? Graph the policy response function. Find the discretionary outcome that satisfies the marginal condition as well as the rational expectation condition. Solve for it algebraically, as well as show it graphically in the graph of the policy response function.
3. Compare the "zero-inflation rule" outcome and the "discretionary" outcome in (b), which one is better?Explain.
4. Suppose before time t₀, the economy operates under discretion without any commitment device. After t₀, however, the Central Bank Independence and nominal anchor make the zero-inflation rule credible and as a result inflation outcome follows such rule. Draw the log P vs time graph from some time before t₀and some time after t₀.