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Hello I was wondering if someone could help me with the following question relating to the stability of an equilibrium in a two good exchange economy and the Walrasian auctioneer.

The setting is the following: we consider an exchange economy with two individuals $A$ and $B$ and two goods $x$ and $y$ and consider price vectors that differ from the equilibrium price vector.

A reasonable rule is that if there is current excess demand for commodity $x$, i.e. $E_x>0$ then the auctioneer ought to put $p_x$ up a bit; if there is excess supply ($E_x<0$) then if the price is not already 0, it ought to come down a bit, i.e. $E_x(\textbf{p}) > 0 \implies \uparrow p_x$, $E_x(\textbf{p}) <0 \implies \downarrow p_x$.

No trade takes place if the prices called out by the auctioneer are non-market clearing prices. If this were not the case, and trade would take place at non-market-clearing (out-of-equilibrium prices) then actual wealth in the economy would change.

This is the statement I have trouble understanding. My lecture notes claim the following:

  1. suppose that with initial allocation $(e^A,e^B)$ equilibrium prices are $(p_x,p_y)$. (here $e^A=(e_x^A,e_y^A)$ denotes the initial endowment of $A$, and similarly for $B$.

  2. Suppose trade is done at non-equilibrium prices $(\tilde{p}_x, \tilde{p}_y)$, excess demand is rationed, excess supply is not sold.

  3. This leads to new endowments $((e^A)',(e^B)')$ and new equilibrium prices $(p_x',p_y')$.

  4. The equilibrium $(p_x,p_y)$ will not be reached in this process.

I have trouble seeing how step 3 and 4 follow and was hoping someone could explain this to me.

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