Why must risk averse be correlated with a concave utility function? Let my utility function $U: \mathbb{R}\to\mathbb{R}$ be arbitrary and suppose I am a risk averse person. Now suppose there was no risk involved with the attainment of money. Should should my utility function still be concave down? What if my love for money increased exponentially? 
Perhaps risk aversion is completely dependent upon my love for money such that if my utility function is exponential then I would naturally be risk seeking.
Thus no-one can have an "exponential love" for money and at the same time be risk averse because its rational to take risks given your utility for money is exponential.
Thanks for any direction or input
 A: Risk aversion is defined as having a lower expected utility from taking a lottery $L=(p_1,x_1;\dots;p_n,x_n)$ than the utility from taking the expected value of that lottery with certainty. Or mathematically,
\begin{equation}
EU=\sum_{i=1}^np_iu(x_i)<u\left(\sum_{i=1}^np_ix_i\right)=u(EV). 
\end{equation}
This defining condition of risk aversion turns out to be equivalent to requiring that $u(\cdot)$ be strictly concave because of Jensen's inequality. 
Unless you want to dispute the definition of risk aversion above, you'd have to live with its implications. 
In particular, suppose you have "exponential" love for money, say $u(x)=\mathrm e^x$. Then, we must have 
\begin{equation}
\frac12\mathrm e^1+\frac12\mathrm e^3>\mathrm e^2.
\end{equation}
Namely, you'd be willing to take a gamble that gives you equal chances of getting an ex post wealth of either $1$ or $3$ over a riskless option that guarantees a wealth level of $2$ (the expected value of the gamble). This behavior can hardly be squared with the usual understanding of risk aversion.  
