Preferences relation from utility function

Summary

Preferences relation from utility function

• Given: a utility function \(u=u\left( x_1,x_2 \right)\)
• Create a binary relation \(≽\) representing weak preferences from \(u\)
• Mathematically,

\[\left( x_1,x_2 \right)≽\left( y_1,y_2 \right)⟺u\left( x_1,x_2 \right)≥u\left( y_1,y_2 \right)\]

• This relation will then automatically satisfy the total order assumption.

Indifference curve from utility function

• Given: a utility function \(u=u\left( x_1,x_2 \right)\) and a utility level \(u_0\)
• The indifference curve for this utility level consists of all bundles \(\left( x_1,x_2 \right)\) such that \(u\left( x_1,x_2 \right)=u_0\) .

• The indifference curve for \(u_0\) is the level curve of the utility function, \(u\left( x_1,x_2 \right)=u_0\) .