I am stuck on the following exercise, related to preference relations and von-Neumann-Morgenstern utility function.
A farmer wants to dig a well in a square field $[0,1000]\times[0,1000]$. The preferences of the farmer on the possible locations are lexicographic, i.e:
- If $x_1<x_2$ then $(x_1,y_1)\prec(x_2,y_2)$ for all $y_1,y_2$.
- If $x_1=x_2=x$, then $(x,y_1)\prec(x,y_2)$ iff $y_1 < y_2$.
Initially, assume that the well location must have integer coordinates. Is there a preference relation on lotteries, that satisfies the von-Neumann-Morgenstern axioms, and extends the lexicographic preference relation? If so, what is a linear utility function that represents this relation?
I think the answer is yes, and a possible linear utility function is: $u(x,y)=100000x + y$.
Now, assume that the well location can have real coordinates. Prove that there is no linear utility function that represents the preference relation on lotteries. Which one of von-Neumann-Morgenstern axioms is violated by the preference relation on lotteries?
Here I am stuck. I don't understand why the utility function I suggested above doesn't work? And what axiom is violated here?