# Does decreasing marginal utility imply risk aversion?

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?

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).

• I understand the definition of risk aversion you were quoting. However, it is not consistent with intuition. Intuitively, risk aversion corresponds to a fixed amount of risk premium one is willing to pay in return for a random payoff. Risk neutral pricing implies l risk premium is 0; the more risk averse one is, the higher the risk premium is. They is why I said I can have constant marginal utility, but still rejecting the 1/-1 bet because I am risk averse; I demand a positive risk premium. Intuitively, diminishing return is independent of risk aversion unless my understanding is off somewhere – user1559897 May 7 '16 at 1:07
• @user1559897 The fact that a theoretical definition is not "consistent with intuition" is hardly surprising. In the expected utility theory framework, the example in your post describes a risk-neutral individual (linear utility function, hence constant marginal utility), which is indifferent towards the bet you are describing. There is no "independent source of liking/disliking uncertainty" other than the utility function itself and its properties -so you cannot have constant marginal utility and be risk averse. If you want to have that combination, you have to create a different theory. – Alecos Papadopoulos May 7 '16 at 2:52
• Well, okay, but what sense does it make under the expected utility theory framework to unify the idea of risk averseness and diminishing marginal utility? I have read the formula and the definition like a hundred times, but when you plug in an expectation into a utility function, it is not calculating anything related to risk. An expectation is a fixed number; it does not represent any uncertainty/risk-averseness whatsoever. – user1559897 May 7 '16 at 3:50
• @user1559897 Utility functions describe preferences that have nothing to do with uncertainty directly. The term "risk averse" in EU theory is used to label a specific attitude in situations where structured risk is involved. It does not model "fundamental preferences" related to whether "experiencing risk/uncertainty" creates direct, psychological, "positive or negative" feelings. You may think that, then, it is a misnomer, and maybe it is. Still, it has proven a useful concept, but apparently is not what you hoped it would be by looking at the words used to describe it. – Alecos Papadopoulos May 7 '16 at 12:13

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.

• Your idea totally makes sense. However, I do not quite follow the equation V(X,E[C])=U(C)−b∗(C−E[C])2. When you take expectation of C, that indicates that C is a random variable. The term b∗(C−E[C])2 indicates that V(X, E[C]) returns a random variable instead a constant number. Is that a typo? Or am I misunderstanding something. Do you mean V(X, C)=U(X)−b∗(X−E[C])2 so V is really a function that maps a real number x and a random variable to a real number? – user1559897 May 6 '16 at 20:16
• Sorry, yes, you are right. It should all be C's, an they are random. – Fix.B. May 6 '16 at 20:25
• I really like the utility function you created. By treating C as a random variable, you are creating a random function of C essentially. Does this imply that the risk averseness we are discussing are an entirely different thing from the definition using diminishing returns? – user1559897 May 7 '16 at 4:28

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.

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.