I think I understand the fact that when marginal utilities of the same function are equal (a consequence of the actuarially fair insurance), the independent variables in it must be equal -- right? But what it is the reason in this for a consumer being risk averse? What a $u''<0$ changes in comparison to a $u">0$ condition?
Edit: Example found here
"As a risk-averse consumer, you would want to choose a value of $x$ so as to maximize expected utility, i.e.
Given actuarially fair insurance, where $p = r$, you would solve: $\max \left[pu(w - px - L + x) + (1-p)u(w - px)\right]$, since in case of an accident, you total wealth would be $w$, less the loss suffered due to the accident, less the premium paid, and adding the amount received from the insurance company.
Differentiating with respect to $x$, and setting the result equal to zero, we get the first-order necessary condition as: $(1-p)pu'(w - px - L + x) - p(1-p)u'(w - px) = 0$,
which gives us: $u'(w - px - L + x) = u'(w - px)$
Risk-aversion implies $u'' < 0$, so that equality of the marginal utilities of wealth implies equality of the wealth levels, i.e.
$w - px - L + x = w - px$,
so we must have $x = L$.
So, given actuarially fair insurance, you would choose to fully insure your car. Since you're risk-averse, you'd aim to equalize your wealth across all circumstances - whether or not you have an accident.
However, if $p$ and $r$ are not equal, we will have $x < L$; you would under-insure. How much you'd underinsure would depend on the how much greater $r$ was than $p$."
Now, how the condition $u''<0$ changes anything to reach the result expressed above?