y is output, pi is inflation, pi^e is expected inflation, i is interest rate
a)
You can show this by showing that t period interest rate only t+2 period inflation. It doesn't influence t period or t+1 period inflation.
This way you show that t period inflation doesn't influence any t+i period inflation for i not equal to 2.
Now, we can write the Loss function as follows (by just opening the bracket):
To minimize, you'll have to differentiate the loss function with respect to t period interest rate. By properties of expectations, you have:
To get your final result, just differentiate the loss function.
Notice something. The derivative all other terms except the one that has t+2 period inflation, when differentiated with respect to t period interest rate, become 0. So whether you minimize the whole loss function, or whether you minimize only that term of the loss function that has t+2 period inflation, it won't make a difference. Your final result will be the same. Hence, you can write the optimization problem with just the term corresponding to t+2 period inflation (as has been written in part a of the question).
b) The optimal interest rate ensures that
Now, look at the final solution for t+2 period inflation in part a. You had:
Take expectations on both sides, and you'll get:
A central bank knows all the all the terms in the RHS of the expression for interest rate. This is because we've taken t period expectations- that means they used all the information available with them as of the end of period t. They know the actual inflation at time t: and their own target: . They know people's expectation of t period inflation set at t-1 period: . And they know the output level at t period: . So they will know exactly how much interest rate to set so as to ensure that .
c) From part a) you found that the optimization problem can be written as follows:
Now just use basic calculus to minimize this expression with respect to t period interest rate.
The final expression shows that the solution to the optimization problem requires that the two period ahead forecast of inflation equal the inflation target.
y is output, pi is inflation, pi^e is expected inflation, i is interest rate Consider the dynamic IS-AS model presented by Svensson (1999) where all the symbols have the usual meaning. The monetar...