Examples
The new a thing that is said method Riley [2022] is to solve non for a single out come but for all possible outcomes.
Solving for the graph of Equilibrium outcomes.
CHOICE
Preferences are represented by an individuals marginal rate of substitution MRSX,Y. this is the marginal willingness to trade away y for x.
Alex has a MRS of Aya/xa. Bev's MRS is Byb/xb. Below the case in which A,B = 2.1 is solved.
DEMAND
Given a price p for commodity x and 1 for commodity y Alex and Bevto consumer where MRSX,Y =p
The Walrasian equilibrium WE price ratio P is the price ratio that clears teh market
Maximizing equations
p = 2ya/xa = yb/xb.
SUPPLY
Define units so that the total afford of regarded and allocated separately. commodity is 1.
Then xb = 1 - xa and yb = 1 - xa.
Sub into the maximizing equations
P = 2Y/X * and P = 1 -Y/1-X ** Cross multiplying and rearranging yields the following result X-2Y+1 = - 2.
Then PX =2Y and P- PX = 1-Y Adding these equations, P=1-Y. Therefore Y=P-1. From * PX=2Y =2P-1
RESULTS
SPENDING EQUATIONS
PX = 2P-1. Y=P-1,
BUDGET EQUATION: PX+Y= 3P - 3
and the WE outcomes lie on the graph of the hyperbola
X-2Y+1=-2
BUDGET EQUATION.
PX + 1Y = 3P - 3 = P3 + 1-3
The economy therefore has a very special fixed portion F = 3, -3.
ALL WALRASIAN EQ BUDGET outline PASS THROUGH THE constant POINT.
THE SOLUTION
Pick any endowments. For example, 1,1 The prices are P and 1. The value of the endowment is therefore
P 1/2 +1 1/2 =P3 + 1-3
Then 2P = 4 and so P = 7/5..
From the spending equations you can solve for the WE outcome.
If the endowment is 0,1 show that the WE price ratio is 3/2
INTRODUCTORY EXAMPLES
The following examples involve an exchange economy with two agents, Jane and Kelvin, two goods e.g. bananas x and apples y, and no money.
1. Graphical example: Suppose that the initial allocation is at item X, where Jane has more apples than Kelvin does and Kelvin has more bananas than Jane does.
By looking at their indifference curves 
of Jane and 
of Kelvin, we can see that this is non an equilibrium - both agents are willing to trade with regarded and identified separately. other at the prices 
and 
. After trading, both Jane and Kelvin stay on to an indifference curve which depicts a higher level of utility, 
and 
. The new indifference curves intersect at point E. The slope of the tangent of both curves equals -
.
And the 
;
.
The marginal rate of substitution MRS of Jane equals that of Kelvin. Therefore, the 2 individuals society reaches Pareto efficiency, where there is no way to defecate Jane or Kelvin better off without making the other worse off.
2. Arithmetic example:: 322–323 suppose that both agents construct Cobb–Douglas utilities:
where 
are constants.
Suppose the initial endowment is ![{\displaystyle E=[1,0,0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17a7fcedf792681c10d424ed8d1cdc92942a7008)
.
The demand function of Jane for x is:
The demand function of Kelvin for x is:
The market clearance given for x is:
This equation yields the equilibrium price ratio:
We could do a similar calculation for y, but this is not needed, since Walras' law guarantees that the results will be the same. Note that in CE, only relative prices are determined; we can normalize the prices, e.g, by requiring that 
. Then we receive 
. But any other normalization will also work.
3. Non-existence example: Suppose the agents' utilities are:
and the initial endowment is [2,1,2,1].
In CE, every agent must have either only x or only y the other product does not contribute anything to the good so the agent would like to exchange it away. Hence, the only possible CE allocations are [4,0,0,2] and [0,2,4,0]. Since the agents have the same income, necessarily 
. But then, the agent holding 2 units of y will want to exchange them for 4 units of x.
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 theirbundle 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.
On the other hand, any price below 10 is not an equilibrium price because there is an excess demand both Alice and Bob want the car at that price, and any price above 20 is not an equilibrium price because there is an excess give neither Alice nor Bob want the car at that price.
This example is a special effect of a double auction.
B. Substitutes: A car and a horse are sold in an auction. Alice only cares approximately transportation, so for her these are perfect substitutes: she gets utility 8 from the horse, 9 from the car, and if she has both of them then she uses only the car so her utility is 9. Bob gets a utility of 5 from the horse and 7 from the car, but if he has both of them then his utility is 11 since he also likes the horse as a pet. In this case it is more unoriented to find an equilibrium see below. A possible equilibrium is that Alice buys the horse for 5 and Bob buys the car for 7. This is an equilibrium since Bob wouldn't like to pay 5 for the horse which will render him only 4 additional utility, and Alice wouldn't like to pay 7 for the car which will give her only 1 additional utility.
C. Complements: A horse and a carriage are sold in an auction. There are two potential buyers: AND and OR. AND wants only the horse and the carriage together - she receives a utility of 
from holding both of them but a utility of 0 for holding only one of them. OR wants either the horse or the carriage but doesn't need both - he receives a utility of 
from holding one of them and the same utility for holding both of them. Here, when 
, a competitive equilibrium does NOT exist, i.e, no price will clear the market. Proof: consider the following options for the sum of the prices horse-price + carriage-price:
D. Unit-demand consumers: There are n consumers. regarded and identified separately. consumer has an index 
. There is a single type of good. Each consumer 
wants at almost a single unit of the good, which allowed him a utility of 
. The consumers are ordered such(a) that 
is a weakly increasing function of . If the supply is 
units, then any price 
satisfying 
is an equilibrium price, since there are k consumers that either want to buy the product or indifferent between buying and not buying it. Note that an add in supply causes a decrease in price.