Please assist with the following problem:
Assume that the dollar loss, L, associated with being robbed by a mugger on the street is $3, and that being robbed occurs with a probability of p. Suppose an individual can influence p by exercising caution, but doing so will be costly. Let the cost, C, required to achieve probability p be given by C=4(1-p)^3. Assume that the individual is risk neutral.
1. If insurance is not available, what probability of loss will the agent choose?
Now, assume an insurance policy that pays the full amount of the loss (if it occurs) is now available at a total premium or price Z (i.e., the policy cost is Z, not ZL). Then,
2. If insurance has already been purchased, what probability of a loss will the agent choose? Given your answer to this, what premium must an insurance company charge in order to "break even?"
3. Using the answers obtained from part a and b, prove that a risk neutral agent will not purchase any insurance at the premium that must be charged in order for the insurance company to break even.