Does decreasing marginal utility imply risk aversion?

Question

Unless I misunderstood something, seems like risk aversion and decreasing marginal utility is the same thing in the utility model, but intuitively, it seems entirely possible that an individual with no decreasing marginal utility is still risk averse.

For example, I can be exactly twice happier with 1000 dollars than only with 500 dollars. This implies constant marginal utility. With that, I can still be unwilling to enter a bet that pays me +1 or -1 with equal probability and generates an expected return of 0. It is the uncertainty that i do not like, irregardless of marginal utility.

Am i misunderstanding something? Can anyone please explain?

Answer 1

What you are misunderstanding, is that in expected utility theory, marginal utility is not an independent concept from "risk aversion", as the latter is defined in the context of that theory: "risk aversion" does not mean what it means in everyday language. Being "risk averse" does not mean for the theory "I dislike risk", because taken literally "disliking risk" would imply that "risk" is a separate entity, or an aspect of a situation, which produces negative utility.

A "risk averse" person is defined to be a person that has a strictly concave utility function (and so a function with decreasing 1st derivative).

PS: On another front, "being twice happier" reveals that you are considering cardinal utility, where quantitative comparisons between numeric utilities is considered to be meaningful. Be aware that the predominant paradigm in economics on the matter has been that of ordinal utility (this does not affect the mathematical properties and relations, only their interpretation).

Answer 2

Its one of those cases where math is clearer that words I think:

Yes, the definition of an agent with utility function $U(C)$ being risk averse is that $E[U(C)]<U(E[C])$ but this is true iff $U''(C)<0$, i.e. if U is concave, i.e. if U'(C) is decreasing.

How would you represent the fact that you are twice as happy with two units than with one unit, but that you don't like the uncertainty? You have that $U(A*C)=A*U(C)$ for any A which directly means $E[U(C)]=U(E[C])$. So you need to add something for you to dislike the uncertainty. You could add something like a 'habit' or an expected consumption $E[C]$, and say that you care not only about how much you consume but also about how close it was to what you expected.

[Edited to reflect correction suggested by @user1559897] Lets call this new utility function $V$ and define it as $V(C,E[C])=U(C)-b*(C-E[C])^2$. Now, if C could be 5 or 10, with E[C]=7.5, then you have that $V(10,10)=U(10)=2*U(5)=2*(V(5,5))$, which looks like risk neutrality, but at the same time, you have $E[V(C,7.5)]<V(7.5,7.5)=V(E[C],7.5)$, which looks like risk aversion! You could then state that, according to the definition, he is risk averse to variation in C after he forms the expectation $E[C]$, but risk neutral to variation in C before he forms that expectation.

Answer 3

This answer is closely related to the points raised by @Fix.B. and @AlecosPapadopoulos, which must be upvoted. But because @user1559897 still asks the question ''what sense does it make under the expected utility theory framework to unify the idea of risk averseness and diminishing marginal utility?'' let me try a different variant.

Let's define a risk-averse individual as an individual who, at any wealth level $w$, dislikes every lottery with an expected payoff of zero. So, according to this definition, an individual who is risk averse is not willing to enter a bet that pays her/him +1 or -1 with equal probability and generates an expected return of 0. This makes sense because the concavity of the relationship between wealth $w$ and utility $u$ is quite a natural assumption. It simply implies that the marginal utility of wealth is decreasing with wealth: one values a dollar increase in wealth more when one is poorer than when one is richer. Intuitively, the individual will care more about the \$1 loss than the \$1 gain. In contrast if the individual's marginal utility is constant there is no such argument and there is no reason to avoid a bet with an expected payoff of zero.

A very good reference is Economic and Financial Decisions under Risk by Louis Eeckhoudt, Christian Gollier, & Harris Schlesinger. The first chapter is freely available and discusses those notions based on Bernouilli's example of Sempronius.

Answer 4

This is quite an outdated question, but I saw it when I was posting another question about risk aversion. I'll try to answer it in an intuitive way for the ease of understanding.

You actually have a misperception here by saying you are "twice happier with 1000 dollars than only with 500 dollars". You are actually evaluating both the first \$500 and second \$500 at the point that you have \$0 in hand. At this point, you mistakenly think the second \$500 would give you the same utility/happiness as the first $500. However, when you have already received the first \$500 in your hand, the second \$500 becomes less valuable than you thought at the beginning.

Risk aversion is equivalent to having a concave utility function which implies a decreasing marginal utility. To illustrate this point intuitively, consider the following choice. You can choose between a sure amount of \$1000 and a lottery which gives you either \$800 or \$1200 with 50% chance. Which one would you choose? If you are risk-averse, you will choose the sure amount. The lottery actually gives you \$200 more than the sure amount with 50% chance and \$200 less with 50% chance also. Your choice of the sure amount \$1000 implies that the utility of getting \$200 more is less than the disutility of losing \$200 in absolute value, which in turn implies decreasing marginal utility.

Hope this helps.