I know there can’t exist a usual Walrasian Equilibrium when both agents have the same lexicographic preferences:
If both agents had the preferences
$(x,y) \succeq (x’,y’) \iff:$ $x > x’ \text{ or } (x = x’ \text{ and } y \geq y’)$
Then for every $p_y > 0$, both agents would want to sell all their $y$-endowment to buy $x$, which is not feasible.
On the other hand, if $p_y = 0$, both agents would demand an infinite amount of $y$, which is also not feasible.
I just read on Wikipedia that there is/can be a general equilibrium with infinitesimal prices, on https://en.m.wikipedia.org/wiki/Lexicographic_preferences#:~:text=In%20economics%2C%20lexicographic%20preferences%20or,how%20much%20Y%20there%20is., in “Equilibrium in economies with lexicographic preferences”.
I don’t see why with infinitesimal prices, the above argument doesn’t apply.
I have never worked with infinitesimal prices.
How would I even propose an adequate General Equilibrium problem with infinitesimal utilities and prices, an how would I solve it?
