Question

1. For a utility function u(x) the measure of Absolute Risk Aversion is defined as Alca) = uchun Consider the utility function u(x)=1-e-axi where a is some positive parameter. Show that this utility function is for risk-averse consumer (concave utility/negative second derivative). Show that this utility function exhibits Constant Absolute Risk Aversion. Find the value of this constant.

Answer 1

For the given utility function : U'(x) = ae^{​​​​​(-ax)}.

U"(x) = -a×a×e^(-ax)  ​​​​​​= -(^{​​​​​​}a^​​​2 e^(-ax))

It's clear that the second derivative of the utility function is negative . Hence the consumer is risk averse.

Further A(x) = - (-( a^{​​​​​​2} e^(-ax)) )/ ae^{​​​​​(-ax)} = a

This implies that the utility function exhibits Constant Absolute Risk Aversion equal to a.