Editing Arrow–Debreu model (section)

=== [[Fundamental theorems of welfare economics]] ===
In welfare economics, one possible concern is finding a [[Pareto efficiency|Pareto-optimal]] plan for the economy.

Intuitively, one can consider the problem of welfare economics to be the problem faced by a master planner for the whole economy: given starting endowment <math>r</math> for the entire society, the planner must pick a feasible master plan of production and consumption plans <math>((x^i)_{i\in I}, (y^j)_{j\in J})</math>. The master planner has a wide freedom in choosing the master plan, but any reasonable planner should agree that, if someone's utility can be increased, while everyone else's is not decreased, then it is a better plan. That is, the Pareto ordering should be followed.

Define the '''Pareto ordering''' on the set of all plans <math>((x^i)_{i\in I}, (y^j)_{j\in J})</math> by <math>((x^i)_{i\in I}, (y^j)_{j\in J}) \succeq((x'^i)_{i\in I}, (y'^j)_{j\in J})</math> iff <math>x^i \succeq^i x'^i</math> for all <math>i\in I</math>.

Then, we say that a plan is Pareto-efficient with respect to a starting endowment <math>r</math>, iff it is feasible, and there does not exist another feasible plan that is strictly better in Pareto ordering.

In general, there are a whole continuum of Pareto-efficient plans for each starting endowment <math>r</math>. 

With the set up, we have two fundamental theorems of welfare economics:<ref>(Starr 2011), Chapter 19</ref>

{{Math theorem|name=First fundamental theorem of welfare economics|note=|math_statement= Any market equilibrium state is Pareto-efficient.}}

{{Math proof|title=Proof sketch|proof=  
The price hyperplane separates the attainable productions and the Pareto-better consumptions. That is, the hyperplane <math>\langle p^*, q\rangle = \langle p^*, D(p^*)\rangle</math> separates <math>r + PPS_r</math> and <math>U_{++}</math>, where <math>U_{++}</math> is the set of all <math>\sum_{i\in I} x'^i</math>, such that <math>\forall i\in I, x'^i\in CPS^i, x'^i \succeq^i x^i</math>, and <math>\exists i\in I, x'^i \succ^i x^i</math>. 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.  
}}

{{Math theorem
| name = Second fundamental theorem of welfare economics
| note = 
| math_statement = For any total endowment <math>r</math>, and any Pareto-efficient state achievable using that endowment, there exists a distribution of endowments <math>\{r^i\}_{i\in I}</math> and private ownerships <math>\{\alpha^{i,j}\}_{i\in I, j\in J}</math> of the producers, such that the given state is a market equilibrium state for some price vector <math>p\in \R_{++}^N</math>.
}}Proof idea: any Pareto-optimal consumption plan is [[Hyperplane separation theorem|separated by a hyperplane]] from the set of attainable consumption plans. The slope of the hyperplane would be the equilibrium prices. Verify that under such prices, each producer and household would find the given state optimal. Verify that Walras's law holds, and so the expenditures match income plus profit, and so it is possible to provide each household with exactly the necessary budget.
{{Math proof|title=Proof|proof=

Since the state is attainable, we have <math>\sum_{i\in I}x^i \preceq \sum_{j\in J}y^j + r</math>. The equality does not necessarily hold, so we define the set of attainable aggregate consumptions <math>V := \{r + y - z: y \in PPS, z \succeq 0\}</math>. Any aggregate consumption bundle in <math>V</math> is attainable, and any outside is not.

Find the market price <math>p</math>.

: Define <math>U_{++}</math> to be the set of all <math>\sum_{i\in I} x'^i</math>, such that <math>\forall i\in I, x'^i\in CPS^i, x'^i \succeq^i x^i</math>, and <math>\exists i\in I, x'^i \succ^i x^i</math>. That is, it is the set of aggregates of all possible consumption plans that are strictly Pareto-better. Since each <math>CPS^i</math> is convex, and each preference is convex, the set <math>U_{++}</math> is also convex.

: Now, since the state is Pareto-optimal, the set <math>U_{++}</math> must be unattainable with the given endowment. That is, <math>U_{++}</math> is disjoint from <math>V</math>. Since both sets are convex, there exists a separating hyperplane between them.

: Let the hyperplane be defined by <math>\langle p, q\rangle = c</math>, where <math>p\in \R^N, p\neq 0</math>, and <math>c= \sum_{i\in I}\langle p, x^i\rangle</math>. The sign is chosen such that <math>\langle p, U_{++}\rangle \geq c</math> and <math>\langle p, r+PPS\rangle \leq c</math>.

Claim: <math>p \succ 0</math>.

: Suppose not, then there exists some <math>n\in 1:N</math> such that <math>p_n <0</math>. Then <math>\langle p, r + 0 - k e_n\rangle > c</math> if <math>k</math> is large enough, but we also have <math>r + 0 - k e_n\in V</math>, contradiction.

We have by construction <math>\langle p, \sum_{i\in I}x^i\rangle = c</math>, and <math>\langle p, V\rangle  \leq c</math>. Now we claim: <math>\langle p, U_{++}\rangle > c</math>.

: For each household <math>i</math>, let <math>U_+^i(x^i)</math> be the set of consumption plans for <math>i</math> that are at least as good as <math>x^i</math>, and <math>U_{++}^i(x^i)</math> be the set of consumption plans for <math>i</math> that are strictly better than <math>x^i</math>.

: By local nonsatiation of <math>\succeq^i</math>, the closed half-space <math>\langle p, q\rangle \geq \langle p, x^i\rangle</math> contains <math>U_+^i(x^i)</math>.

: By continuity of <math>\succeq^i</math>, the open half-space <math>\langle p, q\rangle > \langle p, x^i\rangle</math> contains <math>U_{++}^i(x^i)</math>.

: Adding them up, we find that the open half-space <math>\langle p, q\rangle > c</math> contains <math>U_{++}</math>.

Claim (Walras's law): <math>\langle p, r + \sum_j y^j\rangle =c =\langle p, \sum_i x^i\rangle</math>

: Since the production is attainable, we have <math>r + \sum_j y^j \succeq \sum_i x^i</math>, and since <math>p\succ 0</math>, we have <math>\langle p, r + \sum_j y^j\rangle \geq \langle p, \sum_i x^i\rangle</math>.

: By construction of the separating hyperplane, we also have <math>\langle p, r + \sum_j y^j\rangle \leq c =\langle p, \sum_i x^i\rangle</math>, thus we have an equality.

Claim: at price <math>p</math>, each producer <math>j</math> maximizes profit at <math>y^j</math>,

: If there exists some production plan <math>y'^j</math> such that one producer can reach higher profit <math>\langle p, y'^j\rangle > \langle p, y^j\rangle</math>, then

: <math display="block">\langle p, r\rangle+ \sum_{j\in J}\langle p, y'^j\rangle >\langle p, r\rangle+  \sum_{j\in J}\langle p, y^j\rangle = c</math>

: but then we would have a point in <math>r+PPS</math> on the other side of the separating hyperplane, violating our construction.

Claim: at price <math>p</math> and budget <math>\langle p, x^i\rangle</math>, household <math>i</math> maximizes utility at <math>x^i</math>.

: Otherwise, there exists some <math>x'^i</math> such that <math>x'^i \succ^i x^i</math> and <math>\langle p, x'^i\rangle \leq \langle p, x^i\rangle</math>. Then, consider aggregate consumption bundle <math>q' := \sum_{i'\in I, i' \neq i}x^i + x'^i</math>. It is in <math>U_{++}</math>, but also satisfies <math>\langle p, q'\rangle \leq \sum \langle p, x^i\rangle = c</math>. But this contradicts previous claim that <math>\langle p, U_{++}\rangle > c</math>.

By Walras's law, the aggregate endowment income and profit exactly equals aggregate expenditure. It remains to distribute them such that each household <math>i</math> obtains exactly <math>\langle p, x^i\rangle</math> 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.
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