<p>Mas-Colell, Whinston, and Green Proposition 6.C.1 (p. 187) says that for an expected utility maximizer with a Bernoulli utility function <span class="math-container">$u(\cdot)$</span> on amounts of money, saying that "the decision maker is risk averse" is equivalent to "<span class="math-container">$u(\cdot)$</span> is concave". </p>

<p>However, these other assumptions matter. A concave utility function isn't enough. Diminishing marginal utility in amounts of money doesn't have to imply risk aversion. For example, with gamble <span class="math-container">$(F)$</span> taking two values <span class="math-container">$W_L,\: P(W_L) = \theta$</span>  and <span class="math-container">$W_H&gt;W_L,\:P(W_H) = 1 - \theta$</span>, and a utility function <span class="math-container">$u{}$</span> with diminishing marginal utility in wealth, consider this person's welfare function over risky lotteries <span class="math-container">$(V)$</span>:
<span class="math-container">$$V(F, u(), \theta) = u(W_H)$$</span><br>
This person is not risk averse. They love risk (for example, they prefer a risky lottery to the expected value of that lottery). But, in situations without risk, they have diminishing marginal utility of wealth. </p>

CC BY-SA 4.0
2020-01-28T16:31:25.990Z
"these other assumptions" There are no assumptions in your answer that "these" could refer to.
Giskard
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2020-01-28T17:09:43.620Z
The other assumptions are that the agent is 1) an expected utility maximizer and 2) has a Bernoulli utility function.
BKay
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2020-01-28T17:43:23.993Z
Yes, of course. Sorry.
Giskard
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2020-01-28T16:09:08.647Z
2020-01-28T17:55:57.087Z
2020-01-28T17:55:57.087Z
BKay
<p>Diminishing marginal utility is a concept only in <a href="https://en.wikipedia.org/wiki/Cardinal_utility" rel="nofollow noreferrer">cardinal utility</a> theory rather than <a href="https://en.wikipedia.org/wiki/Ordinal_utility" rel="nofollow noreferrer">ordinal utility</a> theory. As diminishing marginal utility implies a concave shape of the utility function, does it necessarily mean that agents with diminishing marginal utility must be risk-averse (also has a function with a concave shape)? </p>

<p>Does cardinal utility theory assume the law of diminishing marginal utility in all of this theory? Could you have increasing marginal utility within cardinal utility theory?</p>

CC BY-SA 4.0
2020-01-28T19:28:28.197Z
I don't know if I fully understand the second part of your question. Are you asking if can have utility function with increasing marginal utility? I don't see why you couldn't theoretically have an agent with $e^{\beta \cdot W},\; \beta>0$ preferences. And quadratic utility preferences (which are common and standard in many applications) generally imply a region where marginal utility is increasing (though it may be at negative wealth levels.
BKay
CC BY-SA 4.0
2020-01-28T19:41:45.413Z
Thanks, so it is possible to have the increasing marginal utility in cardinal theory.
Aeeh
<microeconomics><utility><expected-utility><risk-aversion>
Diminishing mariginal utility and risk preferences
Aeeh
# Diminishing mariginal utility and risk preferences

(Asked by: username_0 on 2020-01-27)

Diminishing marginal utility is a concept only in [cardinal utility](https://en.wikipedia.org/wiki/Cardinal_utility) theory rather than [ordinal utility](https://en.wikipedia.org/wiki/Ordinal_utility) theory. As diminishing marginal utility implies a concave shape of the utility function, does it necessarily mean that agents with diminishing marginal utility must be risk-averse (also has a function with a concave shape)?

Does cardinal utility theory assume the law of diminishing marginal utility in all of this theory? Could you have increasing marginal utility within cardinal utility theory?

**Tags:** microeconomics, utility, expected-utility, risk-aversion
> **Comments:**
> **username_1**: I don't know if I fully understand the second part of your question. Are you asking if can have utility function with increasing marginal utility? I don't see why you couldn't theoretically have an agent with $e^{\beta \cdot W},\; \beta>0$ preferences. And quadratic utility preferences (which are common and standard in many applications) generally imply a region where marginal utility is increasing (though it may be at negative wealth levels.  
> **username_0**: Thanks, so it is possible to have the increasing marginal utility in cardinal theory.  

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## 1 answers

### Answer by username_1 on 2020-01-28. Score: 2 | Accepted answer

Mas-Colell, Whinston, and Green Proposition 6.C.1 (p. 187) says that for an expected utility maximizer with a Bernoulli utility function $u(\cdot)$ on amounts of money, saying that "the decision maker is risk averse" is equivalent to "$u(\cdot)$ is concave".

However, these other assumptions matter. A concave utility function isn't enough. Diminishing marginal utility in amounts of money doesn't have to imply risk aversion. For example, with gamble $(F)$ taking two values $W\_L,\: P(W\_L) = \theta$ and $W\_H>W\_L,\:P(W\_H) = 1 - \theta$, and a utility function $u{}$ with diminishing marginal utility in wealth, consider this person's welfare function over risky lotteries $(V)$:
$$V(F, u(), \theta) = u(W\_H)$$  
This person is not risk averse. They love risk (for example, they prefer a risky lottery to the expected value of that lottery). But, in situations without risk, they have diminishing marginal utility of wealth.

> **Comments:**
> **username_2**: "these other assumptions" There are no assumptions in your answer that "these" could refer to.  
> **username_1**: The other assumptions are that the agent is 1) an expected utility maximizer and 2) has a Bernoulli utility function.  
> **username_2**: Yes, of course. Sorry.  

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