Finding optimal Inflation - Walsh

Question

I am solving questions from Walsh and then verifying with a solutions manual. However, I keep solving a question and arriving at a slightly different answer than that suggested by the solution manual.

The solution given by the manual:

However, when I take the optimal $\pi^e$ and substitute into (121) I get

$$\pi^* = \frac{\pi^T[-1-\lambda]}{[-1-\lambda]} + \frac{\lambda k [ -1-\lambda]}{[-1-\lambda]} + \frac{\theta k [1+ \lambda]}{[-1-\lambda]} +e\frac{\lambda - \theta}{[-1-\lambda]}$$

and this simplifies to:

$$\pi^*= \pi^T + k(\lambda - \theta) - e\frac{\lambda + \theta}{[1 + \lambda]}$$

And so my issue is that I don't see how there is no $\lambda$ in the numerator of the fraction multiplying e at the end of the simplification.

Can anyone see my error? I must be missing something simple.

Answer 1

Edit: Roel Beetsma replied! He says he got the same derivation we basically got. So the solutions seem to be mistaken and we win! Also I'm removing the strikeout below.

Note that $$+ e\frac{\lambda - \theta}{[-1-\lambda]}$$

actually simplifies to

$$ - e\frac{\lambda - \theta}{[1 + \lambda]} $$

rather than with $\lambda + \theta$ in the numerator like you had.

So the solutions has

$$- e \frac{1 + \theta}{1 + \lambda}$$

instead of what we have above. The difference between the two is:

$$-e\frac{-1 - \lambda}{1 + \lambda} = e$$