Consider a Central Bank that maximizes the following utility function: Z-ky-ye)-(1)

where k is a positive constant. Its policy instrument is the growth rate of the money supply, w. Assume that the inflation target is a = 0. Explain this utility function and compare it with the loss function used in the chapter. (Hint: focus on how the central bank's utility rises with output. Is this central bank 'overambitious'?) Now assume that the central bank sets the money supply growth rate after economic agents have incorporated their expectations about inflation into their decision making, and thus faces a Phillips curve: === + ay-ye)- T=

(a) Assuming that agents have rational expectations, solve algebraically for the optimal inflation rate under discretion, i.e. find the inflation rate that the central bank will choose using its monetary policy instrument, M. (Hint: maximize utility with respect to y, having used the Phillips curve to substitute for y in the utility function; and used to substitute for T.)

(b) Suppose that, before private sector inflation expectations were formed, the central bank could commit to a particular rate of inflation. What would that rate be? Discuss.

(c) Now return to the case of discretion, and suppose that we extend the model to cover two periods. In other words, the central bank now cares about the sum of its loss functions in each period, i.e.

where the subscripts indicate the period.

Suppose also that in the first period, agents expect no inflation (0), while when the second period arrives agents expect that inflation will be equal to the rate that actually occurs in the first period (i.c. expectations are adaptive, so =). What will be the equilibrium rates of output and inflation in each period? Discuss your findings.

Total utility = [(ye) - (**)] + [K (1 = } Ye) - 2