This is a classic graduate/advanced undergraduate level problem in microeconomics/finance. It asks to calculate utility function that corresponds to a constant Arrow-Pratt measure of absolute risk-aversion.
Here's a problem from Mas-Colell, Whinston and Green textbook on Microeconomic Theory:
Given that the Arrow-Pratt measure of absolute risk-aversion is a constant, derive the corresponding form of the von Neumann-Morgenstern utility function.
Please see the attached file for detailed solutions. The text below is contained within the attached Word file along with the formulas:
Problem: Given that the Arrow-Pratt measure of absolute risk-aversion is a constant, derive the corresponding form of the von Neumann-Morgenstern utility funciton.
First of all let's be clear regarding the definitions:
Expected utility is the sum of utility in each outcome weighted by the probability.
Arrow-Pratt measure of absolute risk-aversion - the measure of how much an individual dislikes risk at different levels of wealth. For example, if you ...
This solution shows how to find the von Neumann-Morgenstern utility functions that displays constant measure of absolute risk-aversion (Arrow-Pratt measure) - CARA.
All of the mathematics required for this problem are presented in the beginning along with the necessary definitions. Note that the properties of integrals are stated without proof, so a student without a background in calculus will need to refer to the text to see the proof. Once the properties are given, the solution explains how to apply them correctly to arrive at the final solution. All the equations are beautifully typed using Microsoft equation editor.