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, $u\left( x_1,x_2 \right)=u_0$ .