# Does the Arrow-Pratt measure of absolute risk aversion have a lower bound with DARA utlity?

Say I have the following utility function:

$$u(x,w)=f(w+x)$$

This utility exhibits decreasing absolute risk aversion ($$f'>0$$, $$f''<0$$ and $$f'''>0$$). $$x\in R_+$$ is the control variable and $$w\in R_+$$ is an exogenous parameter.

Is there a parametric form of $$f$$ for which $$A(x)>\frac{1}{x}$$?

Where $$A(x)=\frac{-f''}{f'}$$ is the Arrow-Pratt measure of absolute risk-aversion. I cannot manage to find such a function, which makes me believe that maybe the Arrow-Pratt measure has a lower bound when the utility is DARA.

Many thanks!

Ok, I think the answer is no. There does not exist a utility function satisfying DARA such that $$A(x)>\frac{1}{x}$$ for all $$x\in R_+$$.
To prove that, let's assume such a function exists. Then, there must exists $$a\in R_+$$ such that: $$\frac{-f''}{f'}>\frac{1}{x}\Leftrightarrow\frac{-f''}{f'}-\frac{1}{x}-a=0$$ If we rearrange this function a bit by taking $$y=x+w$$, we have: $$-\frac{f''(y)}{f'(y)}=\frac{1+a(y-w)}{y-w}\qquad\text{(1)}$$ Expression (1) is a second-order nonlinear ordinary differential equation. Solving it (here pardon my laziness, I used Wolfram) gives: $$f(y)=\int_1^{y}exp\bigg(\int_{1}^{\phi}\frac{a(\rho-w)+1}{w-\rho}d\rho\bigg)c_1d\phi+c_2$$ Taking the derivative of $$f$$ we get: $$f'(y)=exp\bigg(\int_1^{y}\frac{a(\rho-w)+1}{w-\rho}d\rho\bigg)c_1$$ So we do have the condition $$f'>0$$ fulfilled as long as $$c_1>0$$. Taking the second-order derivative of $$f$$, we get: $$f''(y)=\frac{a(y-w)+1}{w-y}exp\bigg(\int_1^{y}\frac{a(\rho-w)+1}{w-\rho}d\rho\bigg)c_1$$ This expression is negative if and only if: $$\frac{a(y-w)+1}{w-y}>0$$ Using the fact that $$x=y-w$$: $$-\frac{1+ax}{x}>0\Leftrightarrow x<-\frac{1}{a}$$ We have a contradiction -- if I did not make any mistake in the computations -- there does not exist a DARA utility function where $$A(x)>\frac{1}{x}$$, for all $$x\in R_+$$.