Long run trade off between inflation and output

Question

In "The incredible Volcker Disinflation" by Goodfriend and King (2005) we are presented with the NK Pricing equation

$\pi_t = E_t\pi_{t+1} + h(y_t-y_t^*)$

the author then mentions that there is no "long run trade-off" in the equation since "output is at capacity when current and expected future inflation are equal".

Can anybody elucidate the meaning of "long run trade-off" and the following sentence?

Answer 1

Note that if $\pi_t = E_t\pi_{t+1}$ (current and expected future inflation are equal) then

$$\pi_t = E_t\pi_{t+1} + h(y_t-y_t^*) \implies 0 = h(y_t - y_t^*) \implies y_t = y_t^*$$

So as the output gap tends to zero in the long run, inflation will perfectly determine next period's inflation, so monetary policy that affects inflation in the short run cannot affect output in the long run. Thus there is no tradeoff between inflation and output in the long run.