r/BehSciAsk - Behavioural Policy challenge: How well do people understand trade-offs and accept them? (n.d.). Reddit. Retrieved 9 September 2020, from https://www.reddit.com/r/BehSciAsk/comments/ii3n2x/behavioural_policy_challenge_how_well_do_people/

Galandra, C., Cerami, C., Santi, G., Dodich, A., Cappa, S., Vecchi, T., & Crespi, C. (2020). Covid-19 in mind: How job loss and health threatening events modulate risk-taking behaviours in real-life contexts [Preprint]. PsyArXiv. https://doi.org/10.31234/osf.io/5n942

A student asked

I want to ask why risk-averse has a decreasing marginal utility? Thank you.

Response:

If someone has a decreasing marginal utility of income and they maximise expected utility then they will be risk averse.

This is something that takes a long time to fully explain, and I try to give an explanation in the web-book and in lecture (and again in tomorrow's lecture).

One simple intuition.: Risk averse essentially means "I will never take any fair gamble".

E.g., "I'll never accept a bet with an equal chance of losing or gaining some amount X." How does diminishing MU of income explain this? If I have diminishing MU of income then my utility is increasing in income at a decreasing rate.

The first units of income (e.g., going from 0 income to 15k income) add more utility than the later units of income (e.g., going from 15k income to 30k income) , which adds more than even later increments (e.g., going from 30k to 45k), etc.

So "an equal chance of losing or gaining X" would not be attractive to such a person. Why not? Because relative to any point "losing X" reduces my utility more than "gaining X" increases it.

E.g., in the above example, if you started at 15K income you wouldn't want to have an equal chance of losing or gaining 15K in income. Having 0 income would be terrible, while having 30k income would be better, but not 'that much' better. As we said, the utility difference between 0 and 15K is much greater than the utility difference between 15k and 30k... because of the assumption of diminishing marginal utility. So it's better to have 15k for sure than to have a 50/50 chance of 0k or 30k.

The 'utility loss from losing 15k' is greater than the 'utility gain from gaining 15k'. As expected utility weights the utility of each outcome by its probability and sums these, in considering a 1/2 chance of losing 15k and a 1/2 chance of gaining 15k these probabilities weight equally, so I only need to consider "does the utility cost of losing 15k exceed the utility gain from gaining 15k" in this example. Because of diminishing MU, we know it does not. Nor does it for any "equal chance of losing or gaining some amount X". Thus this person is risk-averse.

I hope this helps. Looking at the 'utility of income' diagrams may also be helpful.