Question 13 macroeconomics

13. Consider a government with preferences represented by the loss function L = 12 + au2 where it is the inflation rate and u is the unemployment rate. The coefficient a indicates the strength of the government's preference for low and stable unem- ployment relative to low and stable inflation. The Phillips curve is U = Un - b(7 - 7#) where " denotes inflation expectations, Un is the natural rate of unemployment, and b is a positive constant. In this question, assume monetary policy is able to control the current inflation rate 7. Suppose that the government is not restricted by any past commitments (the gov- ernment acts with discretion). This means inflation expectations me are always taken as given when monetary policy is chosen. (a) [3 marks] Derive the first-order condition for the inflation rate i that mini- mizes the loss function subject to the Phillips curve for given me. Assume expectations are formed rationally and everyone has perfect information about the workings of the economy and the government's incentives. (b) [2 marks] Derive the equilibrium inflation rate and unemployment rate. (c) [2 marks] Carefully explain the sense in which the monetary policy in parts (a) and (b) has an 'inflation bias'. Suppose a new government is elected with a/2 in place of a in its loss function. (d) [3 marks] Assuming that expectations are formed rationally and everyone has perfect information about the new government's intentions, find the new equilibrium values of the inflation and unemployment rates. What is the 'sac- rifice ratio' (cumulated increase in the unemployment rate/permanent reduc- tion in the inflation rate) in this case? (e) [3 marks] Now suppose the credibility of the new government is doubted initially. Specifically, suppose expected inflation lies halfway between the inflation rates found in parts (b) and (d). Derive the resulting inflation and unemployment rates in this case. Assuming no-one doubts the government's intentions after its first year in office, calculate the sacrifice ratio. Now assume inflation expectations are formed adaptively, that is, this year's ex- pected inflation is equal to last year's actual inflation rate. Before the new govern- ment was elected, the inflation rate was equal to the value from part (b). (f) [3 marks] Compare the inflation and unemployment outcomes in the new government's first year to parts (d) and (e). What happens in the long run? (g) [2 marks] Calculate the sacrifice ratio in the case of adaptive expectations and compare to your findings in parts (d) and (e). (h) [2 marks] Would it be possible to reduce the sacrifice ratio by committing to do all the long-run adjustment of inflation from part (f) in one year