Curvature and the Arrow Pratt Absolute Risk Coefficient

Question

So I'm in my first year of grad school, and I'm taking a decision analysis course. One of the topics we're covering is risk aversion, and with that comes discussion of the Arrow Pratt Absolute Risk Aversion coefficient. I know that this coefficient is supposed to be a measure of the curvature of an individual's utility function; however, using the little bit I remember from differential geometry, the Arrow Pratt coefficient definitely is not equal to the standard geometric definition of curvature.
My question is: 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 $u(x) \in R$
$r(x)= \frac{-u''(x)}{u'(x)}$ (risk coefficient)
$k(x) = \frac{|u'(x)|}{(1+(u'(x))^{2})^{3/2}}$

In class, we've been using utility functions that have a constant absolute risk version coefficient. So, I used the one given in class when toying around in Matlab. This function is:
$u(x)=(\frac 4 3)(1-(1/2)^{x/50})$
Thus, $u'(x)= \frac {-(2*(1/2)^{x/50}\ln(1/2))}{75}$
$u''(x)= \frac{-((1/2)^{x/50}log(1/2)^2}{1875}$

And we have:
$r(x) = \frac{-\ln(1/2)}{50}$

However, curvature is:
$k(x) = \frac{1}{((4(\frac12)^{x/25}\frac{log(1/2)^2}{5625} + 1)^{3/2}} = \frac{1}{(.00034166(\frac12)^{x/50}+1)^{3/2}}$

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? Could someone help elucidate this.

Thanks!

Answer 1

Following @Michael, I do not think the Arrow-Pratt coefficient was constructed with the differential geometry curvature notion in mind. Rather, it was coined as a way to encompass some characteristics of the utility function which have economic interpretations.

In the case of expected utility theory, economists want to identify features of the utility function which determines the level of risk aversion. Intuitively, one sees that risk-aversion depends to some extent on the "curvature" or the " degree of concavity" of the Bernoulli utility function. The question is which notion/measure of "curvature" or "degree of concavity" does the job of being a sensible measure of risk aversion.

One possible way to formalize this is to say that for a given lottery $F$, agent $1$ is more risk averse than agent $2$ if the amount of certain money which leaves $1$ indifferent with getting the outcome of $F$ is lower than that of $2$. This can be written $C_1(F) < C_2(F)$, where $C$ stands for "certainty equivalent". Now what Pratt theorem shows is that the following are equivalent (see here for a proof) :

1) $C_1(F) < C_2(F)$

2) $u_1 = g \circ u_2$ for some concave function g

3) $r_1 \geq r_2$ everywhere.

From the equivalence between 1) and 3), one sees that the the Arrow-Pratt coefficient is a sound measure of risk aversion as conceptualized in terms of certainty equivalence.

The equivalence between 1) and 2) also indicates that the differential geometry measure of curvature is not great as a measure of risk aversion. Take a function $u(x)$ with a constant curvature (in the differential geometry sense) and apply transformations $g_n (t) := t^{1/n}$ to it. At the limit, the function $g_n\big(u(x)\big)$ will have zero curvature almost everywhere, although it displays infinite risk aversion.

As far as the constance of the coefficient is concerned, we have a similar story. A constant $r$ has an interpretation in risk aversion terms : the risk premium, the additional certain $\pi$ that would leave an agent indifferent with lottery $[x - h ; x + h]$ does not depend on $x$, the current wealth of the agent, as pointed out by @Michael (approximatively, as $h$ tends to zero, see here for a derivation). Again, constant curvature does not guarantee that this kind of property holds.

Finally, the equivalence between 2) and 3) tells us that it would probably be more appropriate to speak of the Arrow-Pratt coefficient as a measure of concavity than as a measure of curvature. If one is satisfied with the idea that the concave transformation of a function $u$ makes $u$ "more concave", then we see that the "more concave" (partial) ordering is represented by the Arrow-Pratt coefficient.

All this also indicate that economists do not think about the notion of curvature in differential geometric terms. I think it is fair to say that most economists identify the notion of curvature with that of "degree of concavity" as defined above. See for instance the most used microeconomic theory text book, namely Mas-Collel, Whinston and Green, on page 190:

" It seems logical that the degree of risk aversion be related to the curvature of $u(\cdot)$.[...] One possible measure of curvature of the Bernoulli utility function $u(\cdot)$ at $x$ is $u''(\cdot)$. However this is not and adequate measure because it is not invariant to positive linear transformations of the utility function. To make it invariant, the simplest modification is to use $u''(x)/u'(x)$. If we change sign so as to have a positive number for an increasing and concave $u(\cdot)$, we get the Arrow-Pratt measure"

Answer 2

More than two years later, I stumble upon Pratt's own argument for why the curvature is not a good measure of risk-aversion, which echoes the above citation from Mas-Collel, Whinston and Green:

"It is clear from the foregoing that for this purpose $r(x) = - u''(x)/u'(x)$ can be considered a measure of the concavity of u at the point $x$. A case might perhaps be made for using instead some one-to-one function of $r(x)$, but it should be noted that $u''(x)$ or $- u''(x)$ is not in itself a meaningful measure of concavity in utility theory, nor is the curvature (reciprocal of the signed radius of the tangent circle) $u''(x)(1 + [u'(x)]2)-3/2$. Multiplying $u$ by a positive constant, for example, does not alter behavior but does alter $u''$ and the curvature."

(Pratt, John W. "Risk aversion in the small and in the large." Foundations of Insurance Economics. Springer Netherlands, 1992. 83-98.)

Answer 3

A good answer by @Martin Van der Linden, but there is an additional piece of economic intuition that may help that @Michael mentioned in comments, because a related problem comes up when thinking about the slope of demand and supply curves. The solution is to think not of slope, but of elasticity, which is the ratio of percentage changes. So $\frac{\frac{du'(x)}{u'}}{\frac{dx}{x}}$ becomes $\frac{u''(x)x}{u'(x)}$ and toss in a minus sign and you get the Arrow-Pratt measure of Relative Risk Aversion. You can likewise get the measure of Absolute risk aversion from $\frac{d}{dx}log\ u'(x).$