how are the two definitions related?

function definitions

$u(x)$: this functions takes in a vector $x\in {R}^{n}$ and spits out a value $u(x)\in R$

$r(x)=\frac{-{u}^{\u2033}(x)}{{u}^{\prime}(x)}$ (risk coefficient)

$k(x)=\frac{|{u}^{\prime}(x)|}{(1+({u}^{\prime}(x){)}^{2}{)}^{3/2}}$

In class, we've been using utility functions that have a constant absolute risk version coefficient. This function is:

$u(x)=(\frac{4}{3})(1-(1/2{)}^{x/50})$

Thus, ${u}^{\prime}(x)=\frac{-(2\ast (1/2{)}^{x/50}\mathrm{ln}(1/2))}{75}$

${u}^{\u2033}(x)=\frac{-((1/2{)}^{x/50}log(1/2{)}^{2}}{1875}$

And we have:

$r(x)=\frac{-\mathrm{ln}(1/2)}{50}$

However, curvature is:

$k(x)=\frac{1}{((4(\frac{1}{2}{)}^{x/25}\frac{log(1/2{)}^{2}}{5625}+1{)}^{3/2}}=\frac{1}{(.00034166(\frac{1}{2}{)}^{x/50}+1{)}^{3/2}}$

So, clearly this line does not have constant curvature by the geometric definition. This is also obvious when looking at a graph of $u(x)$. So, I'm struggling to relate the two measures. Is Arrow-Pratt discussing the relationship between slope and curvature?

function definitions

$u(x)$: this functions takes in a vector $x\in {R}^{n}$ and spits out a value $u(x)\in R$

$r(x)=\frac{-{u}^{\u2033}(x)}{{u}^{\prime}(x)}$ (risk coefficient)

$k(x)=\frac{|{u}^{\prime}(x)|}{(1+({u}^{\prime}(x){)}^{2}{)}^{3/2}}$

In class, we've been using utility functions that have a constant absolute risk version coefficient. This function is:

$u(x)=(\frac{4}{3})(1-(1/2{)}^{x/50})$

Thus, ${u}^{\prime}(x)=\frac{-(2\ast (1/2{)}^{x/50}\mathrm{ln}(1/2))}{75}$

${u}^{\u2033}(x)=\frac{-((1/2{)}^{x/50}log(1/2{)}^{2}}{1875}$

And we have:

$r(x)=\frac{-\mathrm{ln}(1/2)}{50}$

However, curvature is:

$k(x)=\frac{1}{((4(\frac{1}{2}{)}^{x/25}\frac{log(1/2{)}^{2}}{5625}+1{)}^{3/2}}=\frac{1}{(.00034166(\frac{1}{2}{)}^{x/50}+1{)}^{3/2}}$

So, clearly this line does not have constant curvature by the geometric definition. This is also obvious when looking at a graph of $u(x)$. So, I'm struggling to relate the two measures. Is Arrow-Pratt discussing the relationship between slope and curvature?