# Representing these lexicographic preferences by non-continuous utility function?

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 ?

Because now

$U(R_1)>U(R_2)$ <-> $R_1 \succ R_2$

$U(R_1)=U(R_2)$ <-> $R_1 \sim R_2$

So, is it finally possible to represent lexicographic preferences by anon-continuous utility function?

Thanks a lot for help!

Felix

• Could you be more specific on the dimensions of R1? If it is single dimensional, I have no clue how you are trying to construct lexicographic preferences here. – HRSE Feb 11 '16 at 3:38

As @HRSE pointed out, your terminology is a bit misleading, and these preferences are not exactly what we refer to when we use the word 'lexicographic'. You might be tempted to consider every $R>0$ as a two-dimensional vector $R=(\mathbb{1}_{R \leq Q},R)$ and to apply the lexicographic order on these vectors, but I would say it is a bit unnatural since the two dimensions of the vector are not independent (you cannot vary the second freely without affecting the first).