risk aversion and the law of diminishing marginal utility

Question

I see many plots where x is wealth and y is utility. If a person is risk averse, he has a concave line on the plot. If the person is risk neutral, her line on the plot is straight.

On the other hand, I see from lecture notes and other websites relating the concept risk averse to risk to the concept of diminishing marginal utility of consumption (such as eating more cookies). This is where I get confused. I would be happy to earn 100 dollars now, but I imagine if I have one billion dollars, I wouldn't care much about bending down to pick up a 100 dollar bill. I don't see how this has anything to do with my risk appetite. I simply don't care for more money when I already have enough money to buy everything I want, the same way I don't care for another cookie when I am already stuffed. In this scenario, even if I am a risk neutral person, my line on the plot should still be concave. And this seems to contradict with the description above.

So my question is, what is causing the contradiction here? What am I missing?

Here is one of my guess: The examples I've seen are always based on conditions that higher expected return bears higher risk, so maybe while the x-axis is wealth, it implies higher risk. But then, higher wealth shouldn't necessarily mean higher risk, because a person can work at a higher paying job and gain guaranteed wealth.

This problem is especially confusing when looking at absolute and relative risk aversion. No term in the formula is risk! There is only wealth. Yet the end result is a measurement for risk aversion.

This post is similar to my question. But it doesn't have an accepted answer yet. And the top voted answer currently says risk averse is defined as such, and that it is not the risk averse we think of in our everyday language. But the example in wikipedia and other lecture notes/websites I read usually explains it as the intuitive risk averse in everyday language, so I don't think it answers my question.

Answer 1

If your preferences over lotteries satisfy the vNM-axioms, i.e. if you are an expected utility maximizer, then you cannot be risk averse and at the same time not have diminishing marginal utility of wealth. After all, risk means that you could end up with different wealth levels and therefore different utility levels, and thus you compare your expected utility with the utility of your certain current wealth level whenever you decide whether or not to buy a lottery ticket.

If you would bend down now to pick up a 100 dollar bill, but not if you already were a billionaire, then this simply means that your utility of wealth function is relatively steep at a low wealth level and relatively flat at a high wealth level. Thus, its slope is decreasing. But this is just diminishing marginal utility of wealth, or concavity.

In much the same way, if you compare a 50:50 lottery with a low and a high wealth outcome with the certainty of a medium wealth level, then going for the certain medium wealth level means that at the medium wealth level the graph of your utility function is above the straight line connecting the low wealth and the high wealth points. Again, this is just concavity. So these two concepts, distinct as they seem, cannot be separated.

Answer 2

Risk aversion means that, when faced with a risky alternative and a sure alternative whose value is equal to the expected value of the risky one, the sure alternative is weakly preferred to the risky one.

For example, between a lottery of winning \$1 or \$5 with equal probability and a \$3 cash award with certainty, a risk averse person would choose the \$3 over the lottery.

Diminishing marginal utility matters here because, relative to the expected value of the lottery, i.e. \$3, the marginal utility from getting an extra \$2 (to a total of \$5) is lower than the marginal disutility from getting \$2 less (to a total of \$1). Intuitively, risk means sometimes you gain and sometimes you lose (relative to the expected value). If you have diminishing marginal utility, the marginal losses will always outweigh the marginal gains. Therefore, you will choose the expected value (with certainty) over the lottery itself. This choice, by the definition stated at the beginning, classifies you as a risk averter.

Answer 3

I used to ask my students if they preferred getting \$4 with probability 1 or \$10 with probability 1/2 and \$0 otherwise. Most students went for the gamble. When I asked for their reasoning they said it had a higher expected value. So I asked the same question with \$4 million and \$10 million (and $0). In this case most students opted for certainty; they hadn't considered before that the small amounts were small relative to their current assets, while the millions were not. You can argue that the per dollar marginal utility of additional \$4 and an extra \$6 on top of that is almost identical. Yet \$4 million will have a large impact on most people's life while an extra \$6 million will not; the per dollar marginal utility of these extra \$6 million dollars is much lower than that of the first 4 million.