$[$Proof using algebra not explanation$]$ Consider the "two$-$by$-$two" pure exchange economy discussed in class.

Denote the consumption bundle for each individual $i=\{1,2\}$ as $x_i=(x_1^i,x_2^i).$

Suppose endowments are ${}_i=(1,1)$ for each $i.$ Further, suppose preferences are

characterized by the following utility functions:

$U^1(x_1)=$min$\{x_1^1,x_2^1\}$

$U^2(x_2)=$max$\{x_1^2,x_2^2\}$

Characterize the equilibrium allocation$(s)x_i^{*}$ in this economy, if any.