<p>Quadratic utility is given by
<span class="math-container">$$u(w) = w - b w^2$$</span>
which has derivative
<span class="math-container">$$u'(w) = 1- 2b w$$</span>
such that for high levels of <span class="math-container">$w, u'(w)&lt;0$</span>. That is, the utility is not everywhere increasing. This may be weird because even people with high wealth should prefer more to less.
The second derivative is
<span class="math-container">$$u'(w) = -2b$$</span>
such that absolute risk aversion is
<span class="math-container">$$\frac{- u''(w)}{u'(w)} = \frac{ 2b}{1- 2b w},$$</span>
which is increasing in wealth. This contradicts evidence that wealthier people take more financial risks instead of less.</p>

CC BY-SA 4.0
2020-11-26T09:21:26.897Z
2020-11-26T09:21:26.897Z
Bayesian
<p>I am taking macro class this fall. One of the problems asks whether decreasing absolute risk-aversion and ever-increasing consumption are two unattractive implications of the quadractic utility function. Could anyone please help with this?</p>

CC BY-SA 4.0
2020-11-26T08:23:53.327Z
I think quadratic utility is associated with *increasing* absolute risk-aversion. The assumption being that there is reduced risk-taking from wealthier folk, because the marginal utility on conducting risk is decreasing.
EB3112
<macroeconomics><microeconomics><utility><preferences><portfolio-theory>
Quadratic utility: monotonicity and risk aversion
user30845
# Quadratic utility: monotonicity and risk aversion

(Asked by: username_0 on 2020-11-26)

I am taking macro class this fall. One of the problems asks whether decreasing absolute risk-aversion and ever-increasing consumption are two unattractive implications of the quadractic utility function. Could anyone please help with this?

**Tags:** macroeconomics, microeconomics, utility, preferences, portfolio-theory

> **Comments:**
> **username_1**: I think quadratic utility is associated with *increasing* absolute risk-aversion. The assumption being that there is reduced risk-taking from wealthier folk, because the marginal utility on conducting risk is decreasing.  

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## 1 answers

### Answer by username_2 on 2020-11-26. Score: 7 | Accepted answer

Quadratic utility is given by
$$u(w) = w - b w^2$$
which has derivative
$$u'(w) = 1- 2b w$$
such that for high levels of $w, u'(w)<0$. That is, the utility is not everywhere increasing. This may be weird because even people with high wealth should prefer more to less.
The second derivative is
$$u'(w) = -2b$$
such that absolute risk aversion is
$$\frac{- u''(w)}{u'(w)} = \frac{ 2b}{1- 2b w},$$
which is increasing in wealth. This contradicts evidence that wealthier people take more financial risks instead of less.


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