Existence of competitive equilibrium between max utility function and min utility function

Question

u1(x1,y1)=max(x1,y1) ω1=(0.2,0.2);
u2(x2,y2)=min(x2,y2) ω2=(0.8,0.8)

The utility functions for two individuals and their endowments are given above for a two person two good economy. My question is that is there any competitive vector of prices that gives rise to competitive equilibrium in the economy? I used the Edgeworth box analysis and determined that the indifference curves for both the individuals will coincide completely and will be L shaped for individual 2 and an inverted L shape for individual 1. However, individual 1 will always look for a boundary optimum while the optimum for individual 2 will be at the kink of the L shaped indifference curve.

I assumed good 2 to be the numeraire good with price = 1, and assumed the price of good 1 is simply p. So income of individual 1 = 0.2p+0.2 and income of individual 2 = 0.8p+0.8. I also inspected three cases

CASE 1

When p>1. Demand for good 1 by individual 1 will be zero, because good 2 is cheaper. Demand for good 1 by individual 2 = (0.8p+0.8)/(1+p). I equated demand and supply, and found that the price was coming out to be negative, which is an impossibility so I ruled out this case.

CASE 2

When p<1. Demand for good 1 by individual 1 = (0.2p+0.2)/p and by individual 2 = (0.8p+0.8)/(1+p). I equated demand and supply, and found that the price was coming out to be negative, which is an impossibility so I ruled out this case too.

CASE 3

When p=1. Individual 1 will either demand zero units or all 0.4 units that he can afford at that price. Individual 2's demand, meanwhile, will be 0.8 units of good 1. I got stuck here. It seems apparently that a competitive equilibrium does not exist. Am I wrong somewhere?

Answer 1

Indeed, there is no competitive eq. To see this case-by-case analysis is the best way forward.

If either price is 0, consumer 1 will demand an infinite amount, markets will not clear. If prices are both positive, consumer 1 will only ever demand 1 of the 2 goods. But, consumer 2 demands an equal amount of each good is such any such case. No allocation is compatible with both these demands.

The failure lies in the fact that the demand functions (in particular of consumer 1) are not continuous. This happens because 1 has a convex utility function.