# Quadratic utility: monotonicity and risk aversion

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?

• 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 Nov 26 '20 at 8:23

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.