1. Consider a von Neumann-Morgenstern individual whose utility function is logarithmic, i.e. u(w)=ln(w), where w is wealth. Given that the individual faces the prospect of gaining or losing an amount of wealth h with equal probability, determine the maximum insurance premium that the individual is prepared to pay.
Please see the attached word file for correct display of formulas and an illustration.
1. First of all let's be clear regarding the definitions:
Expected utility is the sum of utility in each outcome weighted by the probability.
Insurance premium - the amount of money that individual has to pay no matter what. In return, the insurance company guarantees coverage.
Logarithmic utility has two properties:
ln(a) + ln(b) = ln(a * b)
a * ln (b) = ln (ba)
In this problem, there are two possible outcomes: good (individual's wealth becomes equal to w + h) and bad (individual's wealth becomes equal to w - h), both outcomes have equal probability. Since probabilities of all outcomes have to add up to 1, we have that probability of each outcome is 0.5.
This allows us to ...
This solution explains how to find the maximum possible insurance premium that an individual would like to pay when he or she is facing uncertainty in her future wealth.
We are looking at a specific utility function - logarithmic utility, however the solution is detailed enough for an able student to apply the steps to another utility function. At the end a numerical example and a graphical illustration are provided.
This type of question is frequently used in intermediate and advanced microeconomics courses for economics, finance, business majors, as well as in other undergraduate and graduate programs.