What is the value of central banks loss function

Question

Assignment:

1. (a) Suppose that the central bank has an inflation target of π^T = 2.5. Compute the loss L to the central bank when the actual inflation rate is 0, 1, 2.5, and 3. You don't have to show your work.

(b) When π T = 2.5, what is the value of π that minimizes the central bank's loss function given in equation (3)? You don't have to show your work.

2. (a) Rewrite the aggregate supply equation (2) to express the output gap y as a function of π and π^e. You don't have to show your work.

(b) Use your answer to part (a) to eliminate y from the IS equation (1) and solve for the real interest rate r as a function of π, π^e, and ε. You don't have to show your work.

(c) Suppose that the public expects that the inflation rate will equal the central bank's target (i.e., π^e = π^T ). Use the equation for the real interest rate that you derived in 2(b) to compute the appropriate real interest rate for each combination of π^T , π^e , and ε.

π^T       π^e    ε       r-

2.5     2.5   0
2.5     2.5   0.5
2.5     2.5   1
2.5     2.5  -0.5
2.5     2.5  -1

(d) Now, suppose that the public expects that the inflation rate will equal the central bank's target plus 1 percent. (i.e., π^e = π^T + 1). Use the equation for the real interest rate that you derived in 2(b) to compute the appropriate real interest rate for each combination of π^T , π^e, and ε.

π^T     π^e     ε       r

2.5   3.5    0
2.5   3.5    0.5
2.5   3.5    1
2.5   3.5   -0.5
2.5   3.5   -1

e) Compare your answers to part (d) with your answers to part (c). How does the increase in the expected inflation rate affect the  appropriate value of the real interest rate?