Question

Consider a two-person exchange economy. Consumer 1 has utility function $u_1\left(x_1\right)=\sqrt{x_{11}}+\sqrt{x_{12}}$ and endowment $w_1=(2,0)$. Consumer 2 has utility function $u_2\left(x_2\right)=\min \left\{2 x_{21}, x_{22}\right\}$ and endowment $w_2=(2,4)$. Consumption sets $X_1=X_2=[0,10]^2$. Normalize prices so that $p_1+p_2=1$. (a) Verify that consumer 1's budget correspondence is not continuous and his demand correspondence is not uhc at $p=(0,1)$. (b) Does a Walrasian equilibrium exist for this economy? (c) Does your answer to (b) change if $u_2\left(x_2\right)=x_{22}$ ? (d) Does your answer to (b) change if $u_2\left(x_2\right)=x_{22}$ and $X_1=X_2=$ $[0,4]^2 ?$

   Consider a two-person exchange economy. Consumer 1 has utility function $u_1\left(x_1\right)=\sqrt{x_{11}}+\sqrt{x_{12}}$ and endowment $w_1=(2,0)$. Consumer 2 has utility function $u_2\left(x_2\right)=\min \left\{2 x_{21}, x_{22}\right\}$ and endowment $w_2=(2,4)$. Consumption sets $X_1=X_2=[0,10]^2$. Normalize prices so that $p_1+p_2=1$.
(a) Verify that consumer 1's budget correspondence is not continuous and his demand correspondence is not uhc at $p=(0,1)$.
(b) Does a Walrasian equilibrium exist for this economy?
(c) Does your answer to (b) change if $u_2\left(x_2\right)=x_{22}$ ?
(d) Does your answer to (b) change if $u_2\left(x_2\right)=x_{22}$ and $X_1=X_2=$ $[0,4]^2 ?$

Competitive Equilibrium: Theory and Applications

Bryan Ellickson 1st Edition

Chapter 7, Problem 12

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Step 1

Let's consider a sequence of prices $p^n=(\frac{1}{n}, 1-\frac{1}{n})$. For each $n$, consumer 1's budget set $B_1(p^n)$ is given by the set of all affordable consumption bundles at prices $p^n$. Since consumer 1's endowment is $w_1=(2,0)$, we can write the Show more…

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Consider a two-person exchange economy. Consumer 1 has utility function $u_1\left(x_1\right)=\sqrt{x_{11}}+\sqrt{x_{12}}$ and endowment $w_1=(2,0)$. Consumer 2 has utility function $u_2\left(x_2\right)=\min \left\{2 x_{21}, x_{22}\right\}$ and endowment $w_2=(2,4)$. Consumption sets $X_1=X_2=[0,10]^2$. Normalize prices so that $p_1+p_2=1$. (a) Verify that consumer 1's budget correspondence is not continuous and his demand correspondence is not uhc at $p=(0,1)$. (b) Does a Walrasian equilibrium exist for this economy? (c) Does your answer to (b) change if $u_2\left(x_2\right)=x_{22}$ ? (d) Does your answer to (b) change if $u_2\left(x_2\right)=x_{22}$ and $X_1=X_2=$ $[0,4]^2 ?$

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