Given the expected inflation and the random shock to inflation, how to do we obtain the inflation at a given time?

Question

I am confused as to how in this example the following steps occurred:

$$\pi_1=\pi^*+0.5(-0.5(\pi_1-\pi^* ))+v_1$$

where $\pi_1$ is inflation (at time 1), $\pi^*$ is expected inflation, and $v_1$ is a random shock to inflation. How from that equation do we arrive at

$$\pi_1=\pi^*+0.8v_1$$

I don't understand how to get $0.8$ nor how the $v_1$ becomes a part of $0.8$.

Answer 1

I’ll assume you meant

$$\pi_1=\pi^*+0.5(-0.5(\pi_1-\pi^* ))+v_1$$ in which case this is just straightforward algebra. Multiply out the parentheses to get

$$\pi_1=\pi^*+0.25\pi^*-0.25\pi_1+v_1$$ then combine and transpose to get

$$1.25\pi_1=1.25+pi^*+v_1$$ and finally divide by 1.25 to get the desired result.