Jump to content

User:Indnwkybrd/sandbox

From Wikipedia, the free encyclopedia
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