Competitive equilibrium in an exchange economy with lexicographic preferences [closed]

Question

I am really stuck with this problem and not able to approach it. Any help will be much appreciated.

I tried to draw edgeworth box with initial endowment point. I do know that for competitive equilibrium, tangency condition must be met such that both utility curves are tangent at equilibrium point. But here, i am not able to identify utility curves. Please help.

Consider an exchange economy with two agents,1 and 2 and two goods X and Y. Agent 1's endowment is (0,10) and Agents 2's endowment is (11,0). Agent 1 strictly prefers bundle (a,b) to (c,d) if,either a>c or {a=c and b>d}. Agent 2 strictly prefers bundle (a,b) to (c,d) if min{a,b} > min{c,d}. For both agents, we say that bundle (a,b) is indifferent to bundle (c,d) if, neither (a,b) nor (c,d) is strictly preferred to each other.

Q. This exchange economy has: a)one competitive equilibrium allocation. b)two competitive equilibrium allocations. c)infinite number of equilibrium allocations. d) no competitive equilibrium allocations.

Answer : a

Answer 1

The preferences of agent $A$ cannot be represented by any utility function and the prefeences of $B$ not by a differentiable utility function, so forget calculus approaches.

Since $A$ has strictly monotone preferences, we must have $p_1>0$ and $p_2>0$ for every equilibrium. Also, $A$ is always willing to give up any amount of good $2$ to get more of good $1$. So he will spend his wealth on good $1$ with consumption of good 1 equal to $10 p_2/p_1$. Also, we know that $B$ must consume both goods in equal amounts, so you just have to plug the equality $x_1^B=x_2^B$ in the budget constraint (justification here). Then you have the demand for both goods and can calculate their excess demands. Since only relative prices matter, you can normalize by, say, setting $p_1=1$. Since both agents will spend their whole wealth, Walras' law applies, and in order to find an equilibrium, you just have to find prices under which the excess demand for one of the goods (your choice) is zero. If you find a unique solution, you have found a unique equilibrium (note that demand is unique given prices).