The new solution method (Riley  is to solve not for a single out come but for all possible outcomes.
Preferences are represented by an individuals marginal rate of substitution MRS(X,Y). This is the marginal willingness to trade away y for x.
Alex has a MRS of Ay(a)/x(a). Bev's MRS is By(b)/x(b). Below the case in which (A,B) = (2.1) is solved.
Given a price p for commodity x and 1 for commodity y Alex and Bev choose to consumer where MRS(X,Y) =p
The Walrasian equilibrium WE price ratio P is the price ratio that clears teh market
P = 2Y/X (*) and P = (1 -Y)/(1-X) (**) Cross multiplying and rearranging yields the following result (X-2)(Y+1) = - 2.
Then PX =2Y and P- PX = 1-Y Adding these equations, P=1-Y. Therefore Y=P-1. From (*) PX=2Y =2(P-1)
Pick any endowments. For example, (1,1) The prices are P and 1. The value of the endowment is therefore
1. Graphical example: Suppose that the initial allocation is at point X, where Jane has more apples than Kelvin does and Kelvin has more bananas than Jane does.
4. For existence and non-existence examples involving linear utilities, see Linear utility#Examples.
When there are indivisible items in the economy, it is common to assume that there is also money, which is divisible. The agents have quasilinear utility functions: their utility is the amount of money they have plus the utility from the bundle of items they hold.
A. Single item: Alice has a car which she values as 10. Bob has no car, and he values Alice's car as 20. A possible CE is: the price of the car is 15, Bob gets the car and pays 15 to Alice. This is an equilibrium because the market is cleared and both agents prefer their final bundle to their initial bundle. In fact, every price between 10 and 20 will be a CE price, with the same allocation. The same situation holds when the car is not initially held by Alice but rather in an auction in which both Alice and Bob are buyers: the car will go to Bob and the price will be anywhere between 10 and 20.
D. Using Walras' law and some algebra, it is possible to show that for this price vector, there is no excess demand in any product, i.e:
E. The desirability assumption implies that all products have strictly positive prices:
Algorithms for computing the market equilibrium are described in Market equilibrium computation.
In the examples above, a competitive equilibrium existed when the items were substitutes but not when the items were complements. This is not a coincidence.
For the computational problem of finding a competitive equilibrium in a special kind of a market, see Fisher market#indivisible.