# Diminishing mariginal utility and risk preferences

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?

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

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 $$(V)$$: $$V(F, u(), \theta) = u(W_H)$$