a risk lover agent preferences and the preference of risk natural agent may be the same

Question

Consider two lotteries $N$ and $M$. Agent $i$ is risk-averse and prefers $N$. Agent $j$ is risk-neutral and prefers $M$. Would any risk-loving agent $k$ also prefer $M$? That is, would $j$ and $k$ have the same preferences in this scenario?

My attempt:

For example, I can easily show that a risk averse agent can behave as if it is risk natural. I can show this on indifference curves by using the equal marginal rate of substitution.

Then I consider and follow the same way to demonstrate a risk lover agent behave as if risk natural agent by using MRS. But I cannot a result that does make sense.

But I know and assume that I need to use MRS and indifference curve. After that point, I am glad if you give any help. Thanks a lot!

Answer 1

Another way of looking at this problem is to consider the means and variances of the lotteries.

• A risk averse agent (RA) likes high mean and low variance
• A risk neutral agent (RN) likes high mean and is indifferent to changes in variance
• A risk loving agent (RL) likes high mean and high variance

From the fact that RN chooses $M$ over $N$, we known that the mean of $M$ is higher than the mean of $N$, or $E(M)>E(N)$.

The fact that RA chooses $N$ over $M$ despite the latter having a higher mean must imply that $N$ has a much lower variance than $M$.

Thus, given the features of RL's preference (she likes high mean and high variance), the obvious choice is therefore $M$.

Answer 2

Don't commit the cardinal mistake of equating preferences with choices.

In the context of Expected Utility Theory, the fact that a risk-averse agent ($RA$) would choose $N$ over $M$ implies that

$$E[u_{RA}(N)] > E[u_{RA}(M)]$$

The fact that a risk-neutral agent ($RN$) could choose $M$ over $N$ implies that

$$E[u_{RN}(N)] < E[u_{RN}(M)] \implies E(N) < E(M)$$

A risk-lover is a person that could choose $M$ over $N$ even if $E(N) > E(M)$. So here that the opposite inequality holds, it is certain that it will also choose $M$ over $N$, given also the fact that there appears to be no other lotteries to choose from.

But never say that the risk-neutral and the risk-lover have "same preferences". They don't -it just happens, given the available lotteries, that they choose the same lottery, even though they have different preferences.