# How to find the General Equilibria allowing for infinitesimal prices?

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?

• The main hurdle is learning the non-standard analysis needed to formulate the problem. But the basic idea is that if everyone has the same preferences and the price for the second good is positive but infinitely smaller than the price for the first good, then you cannot buy more of the first good by selling the second good and neither can you get more of the second good for free. Commented May 13, 2023 at 13:44
• @MichaelGreinecker So the idea is that now noone would want to trade as they can't get the second good without sacrificing the first one, and they also wouldn't want to sell the second good as it wouldn't be enough to buy any of the first good. Since noone wants to trade, the economy is in Equilibrium. Commented May 15, 2023 at 14:32
• Would love a mathematical formulation of the problem if anyone here knows how to. Commented May 15, 2023 at 14:32
• As I wrote, you just have to learn non-standard analysis. Commented May 15, 2023 at 14:46