3 equation model (Carlin and Soskice, 2005)

Assume that the IS curve of an economy is given by the following equation:

yt = 100 - 4r_t-1;

The economy is currently in the full-employment (long-run) equilibrium where yt = ye = 80 and inflation is at the target level: _t = * = 2. The short-run Phillips curve is given by

_t = _t-1 + 1/10 (yt - ye).

The Central Bank selects monetary policy to minimize the following loss function:

L_t = (yt - ye)² + (_t - * )² ;

(a) What is the value of the stabilizing interest rate (r_s) in this equilibrium?

(b) Derive the monetary policy rule/monetary response function in terms of the parameter .

(c) If the economy receives a temporary inflation shock that shifts the short-run Phillips curve upward to _t = 4 + _t-1 + 1/10 (yt - ye) (the shock only occurs one period), graphically show the new short-run Phillips curve and derive/calculate y, r, and for the following two periods if =2 and if =5.

(d) Now assume instead that there is a permanent shock that hits the IS curve and it causes it to shift to yt = 120 - 4r_t-1. Calculate the new level of output and the rate of inflation directly after the IS shock (assuming r_s has not changed due to the time lag). Further, derive/calculate y, r, and for the following two periods if =2 and if =5.

(e) Graphically illustrate how the economy returns to a stable equilibrium.