# What is the risk aversion domain and how this could change in a dynamic market game?

Most of the market microstructure theory models assume a risk aversion coefficient, say $$\gamma$$ that is indexed with $$i$$ since any individual $$i$$ has her own $$\gamma_i$$ coefficient. Also, the inverse of risk aversion coefficient is defined as the so-called risk tolerance

$$\delta_i=\frac{1}{\gamma_i}$$

My question is, what is the domain of such a parameter (for example many assume that $$\gamma_i \in (1,+\infty)$$ so $$\delta_i\in(0,1)$$ and in case we work in a dynamic time model, should this parameter remain constant?

My question is, what is the domain of such a parameter

Risk aversion coefficient has actually domain $$(-\infty, \infty)$$. Since $$\gamma= -\frac{U''}{U'}$$ and this can be arbitrary positive negative or zero depending on exact utility function. To be more specific, $$\gamma$$ will be negative for risk loving person, 0 for risk neutral person and positive for risk averse person.

The risk tolerance is just reciprocal of the Arrow-Pratt measure of risk aversion $$\delta = -\frac{U'}{U''}$$. Depending on utility function it could be negative, positive or even zero.

The reason why most papers assume that $$\gamma$$ is non-negative is that empirically most people have aversion towards risk. Even though, some people might be risk loving or risk neutral, average person certainly is not risk neutral or risk loving so if you use any representative agent model you would assume representative agent is risk averse (unless you are specifically modeling something like gambling behavior or something like that).

Hence by making further assumptions on how reasonable utility function looks like you can further restrict the reasonable range of the coefficient. For example, if you assume that $$U'>0$$ and $$U''<0$$ then clearly $$\gamma > 0$$. There are further assumptions on utility that can restrict $$\gamma$$ to be above certain values.

dynamic time model, should this parameter remain constant?

This is hard to answer. On one hand it can be argued people's tastes, including risk appetite change over time. On other hand it is questionable how much they change over person's life time.

While making it dynamic might be more realistic its not apriori clear that this is necessary better modeling choice. You should do some thorough literature review on this if it exists. You could also try to make it dynamic and non-dynamic and compare results between the two models. If it qualitatively does not change anything you may just go with the constant version to simplify the model. But do not forget $$\gamma_i$$ is derived from utility function, so dynamic structure of $$\gamma$$ has to be consistent with the utility function.

• Yes, in case you have some specific utility function like exponential (CARA) utility function or mean variance utility function that are the most used in the literature, this risk aversion parameter is usually a constant parameter that belongs to $(0,+\infty)$. I do not know if there is any sophisticated part in literature where they solve for the utility function under specific conditions using differential equations theory. In such case there maybe examples where the parameter is time varying I guess...any reference would be appreciated though. Commented Jan 20, 2023 at 13:15
• @OliverQueen I do not know any such references (that sounds like a different question, so do not hesitate posting that in new question). However, just because nobody else did it does not mean you shouldn't try it.
– 1muflon1
Commented Jan 20, 2023 at 14:00
• is it a challenging topic for a PhD? :) I am 30plus...it is a bit difficult for me to apply in an Ivy League university...but maybe in some other university... who knows...I am thinking about this seriously... Commented Jan 20, 2023 at 14:03
• @OliverQueen I think having dynamic risk aversion will be challenging, but you can say that about any PhD topic. If something is easy yet interesting its almost certain someone else already did it long time ago. If the topic interests you, you should go for it.
– 1muflon1
Commented Jan 20, 2023 at 14:07