Existence of a competitive equilibrium
The Arrow–Debreu model shows that a CE exists in every exchange economy with divisible goods satisfying the following conditions:
The proof good in several steps.: 319–322
A. For concreteness, assume that there are 
agents and 
divisible goods. Normalize the prices such(a) that their sum is 1: 
. Then, the space of all possible prices is the 
-dimensional unit simplex in 
. We call this simplex the price simplex.
B. permit 
be the excess demand function. This is a function of the price vector 
when the initial endowment 
is kept constant:
It is so-called that, when the agents defecate strictly convex preferences, the Marshallian demand function is continuous. Hence, is also a non-stop function of .
C. Define the coming after or as a result of. function from the price simplex to itself:
This is a non-stop function, so by the Brouwer fixed-point theorem there is a price vector 
such(a) that:
so,
D. Using Walras' law and some algebra, this is the possible to show that for this price vector, there is no excess demand in any product, i.e:
E. The desirability given implies that all products have strictly positive prices:
By Walras' law, 
. But this implies that the inequality above must be an equality:
This means that is a price vector of a competitive equilibrium.
Note that Linear utilities#Existence of competitive equilibrium.
Algorithms for computing the market equilibrium are remanded in Market equilibrium computation.
In the examples above, a competitive equilibrium existed when the items were substitutes but non when the items were complements. This is not a coincidence.
Given a benefit function on two goods X and Y, say that the goods are weakly gross-substitute GS whether they are either Independent goods or gross substitute goods, but not Complementary goods. This means that 
. I.e., if the price of Y increases, then the demand for X either supports constant or increases, but does not decrease.
A utility function is called GS if, according to this utility function, all pairs of different goods are GS. With a GS utility function, if an agent has a demand generation at a given price vector, and the prices of some items increase, then the agent has a demand bracket which includes all the items whose price remained constant. He may decide that he doesn't want an module which has become more exensive; he may also decide that he wants another unit instead a substitute; but he may not decide that he doesn't want a third item whose price hasn't changed.