# 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 ... May 25, 2016 at 20:34
• Which problem in Walsh is it? Maybe there's an errata somewhere for it. I got the same derivation as you. May 26, 2016 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, 2016 at 3:41
• 123, please share the knowledge (solution manual) ;) May 31, 2016 at 17:42
• Sure. Is there a way I can send you the file?
– 123
May 31, 2016 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. Jun 1, 2016 at 3:23
• Scorreeeeeeeeee Jun 1, 2016 at 16:15
• Do I get points for finding the error? Lol
– 123
Jun 1, 2016 at 23:29
• I can only give you the 5 with my upvote lol. Jun 2, 2016 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) Jun 2, 2016 at 15:43