| dbp:mathStatement
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- For any strictly positive endowment distribution , and any strictly positive price vector , define the excess demand as before.
If on all ,
* is well-defined,
* is differentiable,
* has ,
then for generically any endowment distribution , there are only finitely many equilibria . (en)
- For any total endowment , and any Pareto-efficient state achievable using that endowment, there exists a distribution of endowments and private ownerships of the producers, such that the given state is a market equilibrium state for some price vector . (en)
- Any core state is equitable. (en)
- Any market equilibrium state is Pareto-efficient. (en)
- If is a polygonal cone, or if every has nonempty interior with respect to , then is the set of market equilibria for the original economy. (en)
- Market equilibria are core states. (en)
- If is an equilibrium price vector for the restricted market, then it is also an equilibrium price vector for the unrestricted market. Furthermore, we have . (en)
- An equilibrium price vector exists for the unrestricted market, at which point the unrestricted market satisfies Walras's law. (en)
- An equilibrium price vector exists for the restricted market, at which point the restricted market satisfies Walras's law. (en)
- satisfies weak Walras's law: For all ,
and if , then for some . (en)
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| dbp:proof
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- We use the "underdog coalition".
Consider a core state . Define average distributions .
It is attainable, so we have
Suppose there exist any inequality, that is, some , then by convexity of preferences, we have , where is the worst-treated household of type .
Now define the "underdog coalition" consisting of the worst-treated household of each type, and they propose to distribute according to . This is Pareto-better for the coalition, and since is conic, we also have , so the plan is attainable. Contradiction. (en)
- Since the state is attainable, we have . The equality does not necessarily hold, so we define the set of attainable aggregate consumptions . Any aggregate consumption bundle in is attainable, and any outside is not.
Find the market price .
: Define to be the set of all , such that , and . That is, it is the set of aggregates of all possible consumption plans that are strictly Pareto-better. Since each is convex, and each preference is convex, the set is also convex.
: Now, since the state is Pareto-optimal, the set must be unattainable with the given endowment. That is, is disjoint from . Since both sets are convex, there exists a separating hyperplane between them.
: Let the hyperplane be defined by , where , and . The sign is chosen such that and .
Claim: .
: Suppose not, then there exists some such that . Then if is large enough, but we also have , contradiction.
We have by construction , and . Now we claim: .
: For each household , let be the set of consumption plans for that are at least as good as , and be the set of consumption plans for that are strictly better than .
: By local nonsatiation of , the closed half-space contains .
: By continuity of , the open half-space contains .
: Adding them up, we find that the open half-space contains .
Claim :
: Since the production is attainable, we have , and since , we have .
: By construction of the separating hyperplane, we also have , thus we have an equality.
Claim: at price , each producer maximizes profit at ,
: If there exists some production plan such that one producer can reach higher profit , then
:
: but then we would have a point in on the other side of the separating hyperplane, violating our construction.
Claim: at price and budget , household maximizes utility at .
: Otherwise, there exists some such that and . Then, consider aggregate consumption bundle . It is in , but also satisfies . But this contradicts previous claim that .
By Walras's law, the aggregate endowment income and profit exactly equals aggregate expenditure. It remains to distribute them such that each household obtains exactly as its budget. This is trivial.
: Here is a greedy algorithm to do it: first distribute all endowment of commodity 1 to household 1. If household 1 can reach its budget before distributing all of it, then move on to household 2. Otherwise, start distributing all endowment of commodity 2, etc. Similarly for ownerships of producers. (en)
- Define the "equilibrium manifold" as the set of solutions to . By Walras's law, one of the constraints is redundant. By assumptions that has rank , no more constraints are redundant. Thus the equilibrium manifold has dimension , which is equal to the space of all distributions of strictly positive endowments .
By continuity of , the projection is closed. Thus by Sard's theorem, the projection from the equilibrium manifold to is critical on only a set of measure 0. It remains to check that the preimage of the projection is generically not just discrete, but also finite. (en)
- The price hyperplane separates the attainable productions and the Pareto-better consumptions. That is, the hyperplane separates and , where is the set of all , such that , and . That is, it is the set of aggregates of all possible consumption plans that are strictly Pareto-better.
The attainable productions are on the lower side of the price hyperplane, while the Pareto-better consumptions are strictly on the upper side of the price hyperplane. Thus any Pareto-better plan is not attainable.
* Any Pareto-better consumption plan must cost at least as much for every household, and cost more for at least one household.
* Any attainable production plan must profit at most as much for every producer. (en)
- Define the price hyperplane . Since it's a supporting hyperplane of , and is a convex cone, the price hyperplane passes the origin. Thus .
Since is the total profit, and every producer can at least make zero profit , this means that the profit is exactly zero for every producer. Consequently, every household's budget is exactly from selling endowment.
By utility maximization, every household is already doing as much as it could. Consequently, we have .
In particular, for any coalition , and any production plan that is Pareto-better, we have
and consequently, the point lies above the price hyperplane, making it unattainable. (en)
- If total excess demand value is exactly zero, then every household has spent all their budget. Else, some household is restricted to spend only part of their budget. Therefore, that household's consumption bundle is on the boundary of the restriction, that is, . We have chosen to be so large that even if all the producers coordinate, they would still fall short of meeting the demand. Consequently there exists some commodity such that (en)
- By definition of equilibrium, if is an equilibrium price vector for the restricted market, then at that point, the restricted market satisfies Walras's law.
is continuous since all are continuous.
Define a function on the price simplex, where is a fixed positive constant.
By the weak Walras law, this function is well-defined. By Brouwer's fixed-point theorem, it has a fixed point. By the weak Walras law, this fixed point is a market equilibrium. (en)
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