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I'm a little bit confused about the answers proposed in the solutions manual to the problem 11.15:

In the model of delegation analyzed in Section 11.7, suppose that the policymaker’s preferences are believed to be described by (11.59), with $a'> a$, when $\pi^e$ is determined. Is social welfare higher if these are actually the policymaker’s preferences, or if the policymaker’s preferences in fact match the social welfare function, (11.54)?

Important information of the problem are the equations:

$$L = \frac{1}{2}(y - y^*)^2 + \frac{1}{2}a(\pi\ - \pi^*)^2 \ ,\ \ \ y^* > \bar{y} \ , \ \ \ a>0. \ \ \ \ \ \ \ (11.54) $$

$$\pi^e = \pi^* + \frac{b}{a}(y^* - \bar{y})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (11.58) $$

$$L' = \frac{1}{2}(y - y^*)^2 + \frac{1}{2}a'(\pi\ - \pi^*)^2 \ ,\ y^* > \bar{y} \ , \ a'>0. \ \ \ \ \ \ \ (11.59) $$

The answer proposed in the Solutions Manual

Expected inflation, $\pi^e$, is determined by the public's beliefs. So both the $" a' " $policymaker and the $"a"$ policymaker face the same $\pi^e$, since in either case, the public believes it is facing an $" a' "$ policymaker. Thus both policymakers have the same choice set. The "a" policymaker makes her choice to maximize true social welfare, whereas the $" a' "$ policymaker makes her choice to maximize something else. Thus social welfare must be higher with the $"a"$ policymaker.

**My point is: considering $\pi^e = \pi$ and the fact that $a'>a$, which implies in $\frac{b}{a'} < \frac{b}{a}$, shouldn't the correct answer be the social welfare is higher with the $"a' "$ policymaker?

Reference: Romer, David. Advanced macroeconomics / David Romer. — 4th ed. (chapter 11)

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