Equilibrium Prices:

So far we have not introduced prices, or markets. If there are only two persons involved they will meet and presumably start a process of haggling about the terms of exchange of the two goods. The outcome of this process cannot be predicted, only that the outcome will likely occur inside the cross-hatched areas, which will define an exchange rate implicitly. However, let us assume that there are a lot of consumers of two types, one group of consumers with the same preferences as consumer 1, and another group with the same preferences as consumer 2. This makes it more likely that there will be one exchange rate between the two goods, which applies to all consumers, i.e., a market price ratio, p₁/p₂. In this case we write the budget constraints for the two consumers as,
 
Now, the question is how the prices are established? First we should note that we must choose one good to express the terms of exchange in. Let us choose good 2 and thus set p₂=1 (this means that one unit of good 2 cost one unit of good 2, not surprisingly). We can now look at how much of the goods that the two consumers want to sell or buy of the two goods as we vary the price of good 1.

In figure above I’ve drawn three price lines going through the endowment point. Let us start with the high price of good, p¹₁; in this case consumer1, who is a net seller of good 1, naturally wants to sell a lot of good 1in exchange for good 2, his best point is at C₁; consumer 2, who is a net demander of good 1, doesn’t want to buy very much, her best point is C₂. Since, consumer 1 wants to sell more than consumer 2 wants to buy at this price, we have an excess supply, and the price, p¹₁, is too high to be an equilibrium price. Turning now to the low price, p²₁, consumer 1 doesn’t want to sell very much (B₁), while consumer 2 wants to buy a great amount (B₂) of good 1; here the price is too low and there is an excess demand. The intermediate price, p^∗₁, is an equilibrium price, since at point A, the quantity that consumer 1 wants to sell is equal to the quantity that consumer 2 wants to buy of good 1. Note also that the equilibrium can be determined as the intersection of the two offer curves (OC₁ and OC₂) of the two consumers. We now proceed by defining a so called aggregate excess demand function for good 1 :

z₁(p₁) = (x¹₁ − ω¹₁) + (x²₁ − ω²₁).

At a general equilibrium the aggregate excess demand functions should be equal to zero (or non-positive) for all goods (z_i(p₁) = 0, i = 1, 2.). Using the prices in figure above, we have that,

z₁(p¹₁) < 0,
z₂(p¹₁) > 0,

and

z₁(p²₁) > 0,
z₂(p²₁) < 0,

while,

z₁(p^∗₁) = 0,
z₂(p^∗₁) = 0.

Here we also included the aggregate excess demand function for good 2, which we did not discuss before; however, in this two good case, it will always have the reverse sign compared to that for good 1 (unless we have an equilibrium).

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