What does the Arrow-Pratt risk aversion measure mean in the deterministic case?

Question

What type of preferences that are not related to risk aversion can the Arrow-Pratt measure of absolute (or relative) risk aversion model?

So far, it seems to me that low RRA/ARA preferences imply that more is less better than for high RRA/ARA preferences (see example below). But:

1. is it always the case?
2. can RRA and ARA say more than that?

Example:

Agent A has the following utility: $U(Potatoes)=(Potatoes)^{0.01}$ (So degree of RRA=0.01). For this agent, eating at least one potato is enough to satisfy him and more is not better (or only to a very small extent).

Agent B has the following utility: $U(Potatoes)=(Potatoes)^{0.99}$ (So degree of RRA=0.99). For this agent, more potatoes is always better.

Answer 1

In the deterministic case, the utility function represents preferences over certain outcomes. It is purely ordinal, i.e. it is unique up to any increasing transformation. In particular, if there is only one good, then any increasing function represents the same preferences (more of the good is preferred to less). Both your example utility functions represent the same preferences over certain outcomes.

It is only in the case where the utility function is cardinal, as is the case when the utility function is used to represent preferences over lotteries (where the utility function is unique up to any increasing linear transformation), that the curvature (coefficient of risk aversion) becomes relevant.