I have the following preferences and I am wondering whether I could represent them by a utility function:
$R_1 \succ R_2$ <-> $R_1\leq Q$ and $R_2 > Q$,
$R_1 \succ R_2$ <-> $R_1 \leq Q$ and $R_2 \leq Q$ and $R_1>R_2$,
$R_1 \sim R_2$ <-> $R_1 > Q$ and $R_2 > Q$
R is positive and real-valued, Q is kind of the "threshold" of R for the first criterion (below Q everything prefered to above Q). These are lexicographic preferences, right?
Can't I represent these preferences by the utility function:
$U(R)=a+bR$ for $R\leq Q$ and positive a,b
$U(R)=0$ otherwise ?
$U(R_1)>U(R_2)$ <-> $R_1 \succ R_2$
$U(R_1)=U(R_2)$ <-> $R_1 \sim R_2$
So, is it finally possible to represent lexicographic preferences by anon-continuous utility function?
Thanks a lot for help!