This provides the loss function and parameters
Setup: The RBA loss function is: $$L(u,\pi)=u^2+2\pi^2$$
Subject to the expectations augmented Phillips curve:$$u-\overline{u}=-4(\pi-\mathbb{E}(\pi))$$ (a) Suppose $\overline{u}$ is 5%. Calculate inflation under discretion.
These photos provides the solution Part (a) working 'backwards', first solve RBA problem. Plugging the Phillips curve into the loss function gives: $$L=\lbrack\overline{u}-4(\pi-\mathbb{E}(\pi))\rbrack^2+2\pi^2$$
The first order condition for a minimum is : $${{dL}\over {d\pi}} = 2\lbrack \overline{u}-4(\pi+\mathbb{E}(\pi))\rbrack(-4)+4\pi=0$$ $$\iff \boxed{8\lbrack\overline{u}-4(\pi-\mathbb{E}(\pi))\rbrack=4\pi}$$ Solve for $\pi$ in terms of $\mathbb{E}(\pi)$ and $\overline{u}$. Which solves for: Part(a) cont: this gives $$\pi=(2/9)\overline{u}+(8/9)\mathbb{E}(\pi)$$ [Last image blurry][3]
Although this may be simple algebra, can someone please help walk me through the complete derivation and how that answer is ascertained? I do not understand how to find the derivative outside of the brackets and would appreciate some guidance of where to begin and how that solution is ascertained! Thank you for your time. [3]: https://i.stack.imgur.com/YwB7K.png
