Suppose the economy is described by the system:

= 1 + ( ),(1)

= ( ),(2)

= ( ),(3)

where > 0, > 0, > 0, represents equilibrium output, represents the equilibrium real interest rate, and represents the inflation target. Equation (3) is a special case of what is known as a Taylor rule and describes a rule for conducting monetary policy.

question: Suppose that the central bank's loss function is given by:

= ( )² + ( )²,

where > 0 represents the relative weight assigned to stabilizing inflation (about its target). In light of the central bank's loss function, determine the optimal choice of and show whether this optimal value satisfies > 0. Be sure to show all your working.