Question: Suppose an individual has initial wealth W. There is a chance of losing X with a probability \(\pi \). He can purchase insurance at an actuarially fair premium of \(\pi y\) for y dollars of coverage. The individual chooses the amount of coverage to maximize

Expected Utility \[=\pi U\left( A \right)+\left( 1-\pi \right)U\left( B \right)\]

where:

A= Individual’s wealth if he loses X but collects Y from the insurance company.

B = Individual does not lose X.

Note that in each case the individual has to pay the insurance premium \(\pi y\).

a) Derive the first order condition of utility maximization. Show that if U is concave, the SOC condition of utility maximization will be satisfied.

b) How much insurance will this person buy to protect against the loss of X? Does this depend on risk aversion of the individual? (Hint: what does concavity suggest about risk aversion).