<p>Terminology is not uniform; many authors call Bernoulli utility functions von Neumann - Morgenstern utility functions or just utility functions.</p>
<p>Expected utility is linear in lotteries and, therefore, trivially weakly concave if one interprets lotteries as elements of a suitable vector space.</p>
<p>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.</p>

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2022-03-20T02:06:34.280Z
MWG places an emphasis on making that distinction, but I buy your vector space argument. Thanks!
Kinno
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2022-03-19T17:21:44.067Z
2022-03-19T17:21:44.067Z
Michael Greinecker
<p>This is for a HW question but I am not asking on how to solve the question, just a conceptual question.</p>
<p>According to my understanding,</p>
<ol>
<li>A Von-Neumann Utility function represents preferences over lotteries of monetary outcomes.</li>
<li>A Bernoulli utility function represents preferences over sure amounts of money.</li>
</ol>
<p>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.</p>
<p>All the definitions in MWG are defined in the context of Bernoulli not VNM i.e. Bernoulli utility needs to be concave (<span class="math-container">$u'(.)&gt; 0$</span> and <span class="math-container">$u''(.)&lt; 0$</span>). 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?</p>

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2022-03-19T18:46:52.417Z
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 
T123
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2022-03-20T02:04:49.480Z
@T123 MWG pg 184 confirms this. Thanks for your suggestion. I' ll check it out.
Kinno
<microeconomics><uncertainty>
Von-Neumann vs Bernoulli Utility? Do properties of Bernoulli translate to VNM?
Kinno
# Von-Neumann vs Bernoulli Utility? Do properties of Bernoulli translate to VNM?

(Asked by: username_0 on 2022-03-19)

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?

**Tags:** microeconomics, uncertainty

> **Comments:**
> **username_1**: 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   
> **username_0**: @username_1 MWG pg 184 confirms this. Thanks for your suggestion. I' ll check it out.  

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## 1 answers

### Answer by username_2 on 2022-03-19. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_0**: MWG places an emphasis on making that distinction, but I buy your vector space argument. Thanks!  

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