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Signaling is the informed side taking actions to reduce (or maintain, depending on the private types) the information asymmetry. For example, high skill workers getting certifications to signal their productivity so as earn a higher wage. Or low skill workers trying to mimic the high-skilled's behavior as much as possible so that they cannot be separated.
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The statement does not seem to be true.
Define $u$ as $u(x) = -1$ if $x \leq 0$, $u(x) = 0$ if $x \in (0,1)$ and $u(x) = 1$ if $x \geq 1$. The set of points $x$ for which $u(x) \geq 0$ is $(0,\infty)$, which is not closed.
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This is an edit
By definition,for a convex subset $C\subseteq \mathbb{R}^n$ a convex function $f:C \to \mathbb{R}$ satisfy
$$ f((1-t)x+ty)\leq (1-t)f(x)+tf(y)$$
for every $x,y\in C$ and $t \in (0,1)$.
We can see if we let $x=(1,3)$ and $y=(3,1)$ and $t=\frac{1}{2}$ that
$$ u((1-t)x+ty)=u(2,2)=6 \not \leq 5=\frac{u(1,3)}{2}+\frac{u(3,1)}{2}=(1-t)u(x)+tu(...
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You need to differentiate with respect to Y, as you said! You substitute the phillips curve as you did (because it is a constraint for the central bank's optimization problem), then differentiate w.r.t $\bar{Y}$ (which I assume here stands for gap) and set this to 0 to obtain the optimality condition, which in turn will give you the monetary rule. Recall ...
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