# Utility function and preferences?

This might be more of an economical question, but perhaps some of you might've come across it before as math is involved;

A person P prefers bundle (x1,x2) over (y1,y2) if (x1*x2) > (y1*y2).

What is then the utility function? And how would I plot the indifference curves, and what can I say about the preferences? Rational, complete, transitive?

I have a hard time grasping the concept of a utility function in general.. it seems so... arbitrary?

The utility function is $U(x_1,x_2)=x_1\cdot x_2$
To plot the indifference curves, you have to solve $U(x_1,x_2)=x_1\cdot x_2$ for $x_2$, where $U(x_1,x_2)$ becomes $\overline U$. $\overline U$ is now a parameter, where you can plug in several values for each indifference curve.
complete: Every combination of $(x_1,x_2)$ can be compared. This is true, because you have functional relation between $U, x_1$ and $x_2$. For every combination of $(x_1,x_2)$ you have an unique value for $U$.
transitive: If $U(x_1,x_2) \succeq U(y_1,y_2)$ and $U(y_1,y_2) \succeq U(z_1,z_2)$, then it has to be $U(x_1,x_2) \succeq U(z_1,z_2)$. This can be shown with a relation between $x_i,y_i$ and $z_i, \ \forall \ i=1,2$