Consider an individual who owns a warehouse that is subject to a fire danger. Suppose that the warehouse, if it burns at all, suffers damage that has a uniform distribution over the range from $10,000 to $190,000. The owner of this warehouse can take precautions against the event of fire -for simplicity, we imagine that the owner can either "take care" or "be negligent" The owner s decision whether to take care or be negligent affects the probability whether there is a fire at all: If she takes care, the probability a fire occurs 1r1, while if she is negligent, the probability of a fire is 1r2 > 1r1• The extent of damage is independent of the action the owner takes, if there is a fire.
The owners preferences are described by a von Neumann-Morgenstern utility function U that depends on the level of damages incurred in a fire (if any) and the actions she takes to prevent a fire. We write her utility function as U(K - L - E) where K is some constant, L is the size of her loss, if any, and E = 0 if she is negligent and E = $10, 000 if she takes care. Note that if there is no fire, her utility will be U(K - E).
Assume that U is concave.
This individual has the opportunity to insure against her loss by fire. A typical insurance contract will take the following form: The owner pays a premium P up front; if the owner sustains a loss of size L, then the contract specifies an amount a.(L) that is returned to the owner. In this case, the owner's utility is U(K - L + a.(L) - E - P) if she has a fire with loss at level L and U(K - E -P) if she has no fire. (We implicitly assume that a.(O) = 0.)
The insurance companies with which she might deal are all regulated by government authorities, and they are required to offer insurance at pre miums that on average will net zero profit. That is, the premium P must be set to equal the expected payout on the insurance policy. (If you want to make the problem more realistic, you can change this. Assume instead that the insurance company is regulated so that some fixed small percent age of the premium it charges is taken as profit. This will, however, make some of the answers obtained below less clean than in the formulation we gave.) The owner can deal with at most one of these insurance companies.
(a) Suppose that insurance companies are compelled to offer complete coverage: a(L) = L. What will be the results in terms of the insurance policy that is offered and the level of care the warehouse owner will take?
(b) Suppose insurance companies can offer insurance that insures only a portion of the loss incurred: a(L) = YL for some Y 1. Show by means of a numerical example that this might make the warehouse owner better off than if the insurance companies were compelled to offer full coverage.
(c) Suppose insurance companies can offer insurance that insures the full loss but with a deductable: a(L) = max{L - D,O} for some fixed de
ductable D. Show by means of a numerical example that this might make the warehouse owner better off than if the insurance companies were com pelled to offer full coverage.
(d) Consider the problem of designing an insurance policy of the sort given above {where there is a fixed premium and a payment given back to the owner in the event of a fire in an amount that depends on the losses sustained) that is "optimal'' in the sense that it makes the warehouse owner as well off as possible, given the regulatory constraint that the insurance company must break even on average. How would you go about finding this optimal policy? Can you say anything about the function a(L) at the optimal policy? For example, does a (.L) rise with L?
(e) Suppose that the size of the loss sustained by the warehouse owner, given there is a fire, depends on the care she did or didn't take. More specifically, suppose that she is more likely to sustain a larger loss, given she has a fire at all, if she is negligent than if she takes due care. How does this affect your answers to part (d)?