Quiz, week 8

Consider the Bellman equation about looking for a job: $$V(w) = \max \{ \frac{w}{1 - β}, b + β Σ_{i=1}^n p_i V(w'_i)\}.$$ ($w$ means the wage offer, $b$ is unemployment benefits, and $p_i$ is the probability of being offered $w'_i$ tomorrow.) What is the domain of the corresponding Bellman operator?

Suppose the corresponding Bellman operator is a contraction. Is it continuous?

If the domain of a contraction is not complete, can it have two fixed points?

Suppose $f$ is a contraction of degree $β$ on $(X, d)$ Does this imply $g(x) = f(f(x))$ is a contraction?

Pick any point $x_0$ inside the space $(X, d)$. Is the function $f : X \rightarrow X$ defined by $f(x) = x_0$ a contraction?

Suppose $f : X \rightarrow X$ is a contraction on $(X, d)$. Does this imply that $X$ is bounded?

Consider the function $f(x) = x$ on the space $(X, d)$. Is $f$ a contraction?