Editing Arrow–Debreu model (section)

=== equilibrium vs "quasi-equilibrium" ===
The definition of market equilibrium assumes that every household performs utility maximization, subject to budget constraints. That is, <math display="block">\begin{cases}
\max_{x^i} u^i(x^i) \\
\langle p, x^i\rangle \leq M^i(p)
\end{cases}</math>The dual problem would be cost minimization subject to utility constraints. That is,<math display="block">\begin{cases}
u^i(x^i) \geq u^i_0\\
\min_{x^i} \langle p, x^i\rangle
\end{cases}</math>for some real number <math>u^i_0</math>. The [[duality gap]] between the two problems is nonnegative, and may be positive. Consequently, some authors study the dual problem and the properties of its "quasi-equilibrium"<ref>{{Cite book |last=Debreu |first=Gerard |url=https://books.google.com/books?id=QkX10epC46cC |title=Theory of Value: An Axiomatic Analysis of Economic Equilibrium |date=1959-01-01 |publisher=Yale University Press |isbn=978-0-300-01559-1 |language=en}}</ref> (or "compensated equilibrium"<ref name=":2">{{Cite book |last=Arrow |first=Kenneth J. |url=http://worldcat.org/oclc/817224321 |title=General competitive analysis |date=2007 |publisher=North-Holland |isbn=978-0-444-85497-1 |pages= |oclc=817224321}}</ref>). Every equilibrium is a quasi-equilibrium, but the converse is not necessarily true.<ref name=":2" />