Finding optimal Inflation - Walsh

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.

• Found almost like you with a $\lambda - \theta$ for the last numerator. No sign of the $1+\theta$ . The solution seems weird as the factor on the "e" is $\lambda - \theta$ in equation (121) and I don't see how it can turn in what is here given as a solution ... – Alexis L. May 25 '16 at 20:34
• Which problem in Walsh is it? Maybe there's an errata somewhere for it. I got the same derivation as you. – Kitsune Cavalry May 26 '16 at 2:49
• #14, chapter 7. It is page 328 of 3rd edition. I think it is #11 of chapter 8 for the second edition. – 123 May 26 '16 at 3:41
• 123, please share the knowledge (solution manual) ;) – An old man in the sea. May 31 '16 at 17:42
• Sure. Is there a way I can send you the file? – 123 May 31 '16 at 21:35

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:

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

• Mods, feel free to delete this if you feel it is a non-answer. I did email the original people who did the derivation (Beetsma and Jensen) that Walsh got from their paper, and linked them back here though. – Kitsune Cavalry Jun 1 '16 at 3:23
• Scorreeeeeeeeee – Kitsune Cavalry Jun 1 '16 at 16:15
• Do I get points for finding the error? Lol – 123 Jun 1 '16 at 23:29
• I can only give you the 5 with my upvote lol. – Kitsune Cavalry Jun 2 '16 at 2:33
• Sorry 123 but your math were wrong so points go to Kitsune ! (However I had risen the minus sign before so technically I should attribute points to me lol) – Alexis L. Jun 2 '16 at 15:43