Chapter Questions
Show that $0 \in \operatorname{cl} \Delta_x P_i\left(x_i\right)$ if consumer $i$ is localiy nonsatiated at $x_i$.
Assuming that the commodity space $L$ is a TVS, explain why:(a) $\Delta_x P_i\left(x_i\right)$ is open in the relative topology of $\Delta_x X_i$ if and only if $P_i\left(x_i\right)$ is open in the relative topology of $X_i$.(b) $\operatorname{cl} P_i\left(x_i\right)-x_i=\operatorname{cl} \Delta_x P_i\left(x_i\right)$.
Using an Edgeworth box, illustrate how Theorem 8.5 can fail if strict upper contour sets are convex but $x_i \notin \mathrm{cl} P_i\left(x_i\right)$ for some consumer $i$.
Verify the claim made in proving Theorem 8.5: if $\Delta_{\tilde{w}} P_i\left(x_i\right)$ is open and the cheaper point property is satisfied, then $\Delta_{\tilde{w}} y_i \in \Delta_{\tilde{w}} P_i\left(x_i\right)$ implies that there exists a commodity bnndle $\Delta_{\tilde{w}} x_i^* \in \Delta_{\tilde{w}} P_i\left(x_i\right) \cap$ $H_o^{-}(p, 0)$.
Fill in the missing details in the proof of Lemma 8.6.
Prove Theorem 8.10.
Show that if $\succ_i$ is strictly convex, then demand $\phi_i$ is singleton-valued.
Using a diagram like Figure 8.8, illustrate the possibility of a core allocation where $\left|\inf p \cdot \Delta_w P_i\left(x_i\right)\right|=2 M=12$.
Compute the Anderson bound and sketch a diagram like Figure 8.8 for an economy with commodity space $\mathbf{R}^2$, separating price vector $p=(1 / 3,2 / 3)$, and three consumers with endowments $w_1=(2,10)$, $w_2=(6,7)$, and $w_3=(8,3)$.
Show that $(T, \mathcal{B}(T), \lambda)$ with $T=[0, \infty]$ and $\lambda$ Lebesgue measure is $\sigma$-finite (i.e., $T=\bigcup_{i=i}^{\infty} T_i$ with $\lambda\left(T_i\right)<\infty$ for each $T_i$.)
Is the probability space $(S, \mathcal{B}(S), \mu) \sigma$-finite?
Which of the followiug sequences is in $\ell_1$ ? in $\ell_2$ ? in $\ell_{\infty}$ ?(a) $x \in \operatorname{Map}\left(\mathbf{Z}_{+}, \mathbf{R}\right)$ such that $x(t)=t$ for all $t \in T$;(b) $x \in \operatorname{Map}\left(\mathbf{Z}_{+}, \mathbf{R}\right)$ such that $x(t)=1$ for all $t \in T$;(c) $x \in \operatorname{Map}\left(\mathbf{Z}_{+}, \mathbf{R}\right)$ such that $x(0)=1$ and $x(t)=1 / t$ for all $t>0$.
For which values of $r$ is the function on $T=[0, \infty]$ defined by $f: t \mapsto$ $t^r$ in $L_p(T, \mathcal{B}(T), \lambda)$ where $\lambda$ is Lebesgue measure?
What is the topological dual space of $\ell_2$ ? of $\ell_1$ ?
Show that if $L=\mathbf{R}^{\mathbf{m}}$, theu $\sigma(L, M)$ is the weakest topology such that $x \mapsto x_i$ is continuous for all $i=1, \ldots, m$.
Prove the First fundamental Theorem of welfare economics for the economy E described in Section 8.3.5. Does the proof require the equilibrium price functional p to be continuous?