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Diminishing marginal utility is a concept only in cardinal utility theory rather than 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?

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  • $\begingroup$ 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. $\endgroup$
    – BKay
    Jan 28, 2020 at 19:28
  • $\begingroup$ Thanks, so it is possible to have the increasing marginal utility in cardinal theory. $\endgroup$
    – Aeeh
    Jan 28, 2020 at 19:41

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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.

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    $\begingroup$ "these other assumptions" There are no assumptions in your answer that "these" could refer to. $\endgroup$
    – Giskard
    Jan 28, 2020 at 16:31
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    $\begingroup$ The other assumptions are that the agent is 1) an expected utility maximizer and 2) has a Bernoulli utility function. $\endgroup$
    – BKay
    Jan 28, 2020 at 17:09
  • $\begingroup$ Yes, of course. Sorry. $\endgroup$
    – Giskard
    Jan 28, 2020 at 17:43

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