Walrasian Price Adjustment and Stability in a Two-Good Economy
Summary
The document introduces a two-good exchange economy with two individuals and asks how a Walrasian auctioneer's price adjustments relate to equilibrium stability. The proposed tâtonnement rule raises a good's price when it has excess demand and lowers it when it has excess supply, subject to a nonnegative price constraint. The question focuses on why trade at prices that do not clear markets could alter endowments and lead to a different equilibrium.
The text does not provide an answer or a derivation of the claimed instability. It states that excess demand is rationed while excess supply remains unsold, then asks how this changes agents' endowments and why the original equilibrium would no longer be reached. Thus, it is useful mainly as a statement of the issue, not as a worked explanation or evidence for a general stability result. Any conclusion depends on the specific rationing and trading rules, which are left unspecified.
Key ideas
- The proposed auctioneer rule raises prices under excess demand and lowers them under excess supply.
- The setup assumes no trade at prices that fail to clear markets, preserving initial endowments during price adjustment.
- The question asks how off-equilibrium trade and rationing could alter endowments and subsequent equilibrium prices.
- No proof or answer is supplied, so the general stability claim remains unresolved in the document.
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# Stability of Equilibrium in 2 good exchange economy
# Stability of Equilibrium in 2 good exchange economy
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:
- 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 $B$.
- Suppose trade is done at non-equilibrium prices $(\tilde{p}_x, \tilde{p}_y)$, excess demand is rationed, excess supply is not sold.
- This leads to new endowments $((e^A)',(e^B)')$ and new equilibrium prices $(p_x',p_y')$.
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.