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
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.
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.
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?