Before introducing Walrasian Equilibrium, we need to define a concept which is Excess Demand.
where is the Endowment. Excess demand is positive (negative) if consumers demand greater (less) quantity of a good than they supply via their Endowment. Thus in equilibrium, we must have .
The intuition of this concept is that imagine there exists a Walrasian Auctioneer, who announces a price vector . All buyers and sellers in this exchange economy are “price takers”. If in the economy under this price, there are consumers that want to buy but could not, i.e., , then the auctioneer would increase the price of this good to reduce demand, and may decrease other goods’ price to increase demand. This process continues until the market clears, i.e., .
We won’t focus too much on the proof of the existence of Walrasian Equilibrium.