This is for a HW question but I am not asking on how to solve the question, just a conceptual question.

According to my understanding,

  1. A Von-Neumann Utility function represents preferences over lotteries of monetary outcomes.
  2. A Bernoulli utility function represents preferences over sure amounts of money.

My homework question asks me to consider a VNM utility function and asks me to find restrictions on parameters for it to display risk aversion. I think this should say Bernoulli function instead.

All the definitions in MWG are defined in the context of Bernoulli not VNM i.e. Bernoulli utility needs to be concave ($u'(.)> 0$ and $u''(.)< 0$). But can I apply this definition to VNM and proceed the same way to find these restrictions, i.e. if Bernoulli is concave then VNM is concave and so I can find the restrictions same way?

  • $\begingroup$ Just a question: where did you read that Bernoulli utility is defined over sure amounts of money? A very nice book on utility functions and related concepts is written by Christian Gollier "The Economics of Risk and Time, 2001", it helped me a lot to understand the basic concepts. Perhaps it will help you to find the answers 😉 $\endgroup$
    – T123
    Mar 19, 2022 at 18:46
  • $\begingroup$ @T123 MWG pg 184 confirms this. Thanks for your suggestion. I' ll check it out. $\endgroup$
    – Kinno
    Mar 20, 2022 at 2:04

1 Answer 1


Terminology is not uniform; many authors call Bernoulli utility functions von Neumann - Morgenstern utility functions or just utility functions.

Expected utility is linear in lotteries and, therefore, trivially weakly concave if one interprets lotteries as elements of a suitable vector space.

Unless the function for which you should check parameters is a linear function of lotteries, it is probably supposed to represent a Bernoulli function and you can check for which parameters it is concave.

  • $\begingroup$ MWG places an emphasis on making that distinction, but I buy your vector space argument. Thanks! $\endgroup$
    – Kinno
    Mar 20, 2022 at 2:06

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