User:Indnwkybrd/sandbox
type | from | gates/delay | construction |
---|---|---|---|
XOR | NAND | fewer | ![]() |
XOR | NAND | more | ![]() |
XOR | NOR | less | ![]() |
XOR | NOR | more | ![]() |
XNOR | NAND | less | ![]() |
XNOR | NAND | more | ![]() |
XNOR | NOR | fewer | ![]() |
XNOR | NOR | more | ![]() |
The separating hyperplane theorem has a central application in mathematical microeconomics. If you've studied it or taken some micro courses, then you might recognize this diagram:
If not: this is an Edgeworth box, a visualization tool used in economics. It models a simple case of general equilibrium theory, a pure exchange economy with 2 agents O and A (apparently, also known as Octavio and Abby), and 2 goods X and Y. This particular example abstracts away from production of the goods, in order to focus on "pure exchange"; hence, the quantity of each good in overall economy is constant at ΩX and ΩY. Each agent starts with a part of this overall quantity which is his/her endowment. In vector form, O has endowment (ωX, ωY), and A has (ΩX - ωX, ΩY - ωY). The goods can both be subdivided into any real-valued amount.
What we are interested in, is whether O and A might like to barter some of their endowments with one another, in a way that makes both of them feel more satisfied with making a deal than with not making one. Therefore, the last piece of the model is a pair of preference relations, which quantify the notion of "more satisfied": two binary relations on the set of possible have-able quantities of goods.
utility functions: one for each agent, UO and UA, to quantify the notion of "more satisfied". The utility function is
Side note: you do not really need to assume that UO and UA exist a priori as such; you can show their existence as a consequence of some binary
makes each of them more satisfied with making a deal than s/he would have been with not making one.
Therefore we need one more element of the model: a pair of utility functions UO
The point prominently marked by the arrows,
https://web.stanford.edu/~jdlevin/Econ%20202/General%20Equilibrium.pdf