Editing Arrow–Debreu model (section)

=== Imposing an artificial restriction ===
The functions <math> D^i(p), S^j(p)</math> are not necessarily well-defined for all price vectors <math>p</math>. For example, if producer 1 is capable of transforming <math>t</math> units of commodity 1 into <math>\sqrt{(t+1)^2-1}</math> units of commodity 2, and we have <math>p_1 / p_2 < 1</math>, then the producer can create plans with infinite profit, thus <math>\Pi^j(p) = +\infty</math>, and <math>S^j(p)</math> is undefined.

Consequently, we define "'''restricted market'''" to be the same market, except there is a universal upper bound <math>C</math>, such that every producer is required to use a production plan <math>\|y^j\| \leq C</math>. Each household is required to use a consumption plan <math>\|x^i\| \leq C</math>. Denote the corresponding quantities on the restricted market with a tilde. So, for example, <math>\tilde Z(p)</math> is the excess demand function on the restricted market.<ref>The restricted market technique is described in (Starr 2011), Section 18.2. The technique was used in the original publication by Arrow and Debreu (1954).</ref>

<math>C</math> is chosen to be "large enough" for the economy so that the restriction is not in effect under equilibrium conditions (see next section). In detail, <math>C</math> is chosen to be large enough such that:

* For any consumption plan <math>x</math> such that <math>x \succeq 0, \|x\| = C</math>, the plan is so "extravagant" that even if all the producers coordinate, they would still fall short of meeting the demand.
* For any list of production plans for the economy <math>(y^j\in PPS^j)_{j\in J}</math>, if <math>\sum_{j\in J} y^j + r \succeq 0</math>, then <math>\|y^j\| < C</math>for each <math>j\in J</math>. In other words, for any attainable production plan under the given endowment <math>r</math>, each producer's individual production plan must lie strictly within the restriction.

Each requirement is satisfiable.

* Define the set of '''attainable aggregate production plans''' to be <math>PPS_r = \left\{\sum_{j\in J} y^j : y^j \in PPS^j \text{ for each } j\in J, \text{ and }\sum_{j\in J} y^j + r \succeq 0\right\}</math>, then under the assumptions for the producers given above (especially the "no arbitrarily large free lunch" assumption), <math>PPS_r</math> is bounded for any <math>r \succeq 0</math> (proof omitted). Thus the first requirement is satisfiable.
* Define the set of '''attainable individual production plans''' to be <math>PPS_r^j := \{y^j \in PPS^j: y^j\text{ is a part of some attainable production plan under endowment }r\}</math>then under the assumptions for the producers given above (especially the "no arbitrarily large transformations" assumption), <math>PPS_r^j</math> is bounded for any <math>j\in J, r \succeq 0</math> (proof omitted). Thus the second requirement is satisfiable.

The two requirements together imply that the restriction is not a real restriction when the production plans and consumption plans are "[[Interior solution (optimization)|interior]]" to the restriction.

* At any price vector <math>p</math>, if <math>\|\tilde S^j(p)\| < C</math>, then <math>S^j(p) </math> exists and is equal to <math>\tilde S^j(p) </math>. In other words, if the production plan of a restricted producer is interior to the artificial restriction, then the unrestricted producer would choose the same production plan. This is proved by exploiting the second requirement on <math>C</math>.
* If all <math>S^j(p) = \tilde S^j(p)</math>, then the restricted and unrestricted households have the same budget. Now, if we also have <math>\|\tilde D^i(p)\| < C</math>, then <math>D^i(p) </math> exists and is equal to <math>\tilde D^i(p) </math>. In other words, if the consumption plan of a restricted household is interior to the artificial restriction, then the unrestricted household would choose the same consumption plan. This is proved by exploiting the first requirement on <math>C</math>.

These two propositions imply that equilibria for the restricted market are equilibria for the unrestricted market:{{Math theorem
| name = Theorem
| note = 
| math_statement =If <math>p</math> is an equilibrium price vector for the restricted market, then it is also an equilibrium price vector for the unrestricted market. Furthermore, we have <math>\tilde D^i(p) = D^i(p), \tilde S^j(p) = S^j(p)</math>.
}}