Question

Consider the dynamic IS-AS model presented by Svensson (1999) where all the symbols have the usual meaning. The monetary authority set its policy instrument it to minimize the following loss function min Et 2 (a) Bxplain why and how the intertemporal minimization problem above can be transformed in the fol- lowing simple specification [5 marks (b) Derive the optimal interest rate rule [10 marks] (c) Show that the solution of the model implies that the two period ahead forecast of inflation πt+2 is equal to the inflation target π* [5 marks]

y is output, pi is inflation, pi^e is expected inflation, i is interest rate

Answer 1

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.

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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):

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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.

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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 $E_t(\pi_{t+2})=\pi^{*}$

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: $\pi_t$ and their own target: $\pi^{*}$. They know people's expectation of t period inflation set at t-1 period: $\pi^{e}_t$. And they know the output level at t period: $y_t$. So they will know exactly how much interest rate to set so as to ensure that $E_t(\pi_{t+2})=\pi^{*}$ .

c) From part a) you found that the optimization problem can be written as follows:

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Now just use basic calculus to minimize this expression with respect to t period interest rate.

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ACa dit 2

The final expression shows that the solution to the optimization problem requires that the two period ahead forecast of inflation equal the inflation target.