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.