Question

If firms offer actuarially fair insurance, how much insurance will a consumer demand given loss ?? with probability ??. At the optimum, what is the marginal rate of substitution between income in the bad state and income in the good state with respect to probability ???

Answer 1

The answer to this question on the price of insurance - the premium you'd have to pay. Let's say this price is r, for $1 worth of insurance, so for $x of insurance, you'd be paying $rx as a premium.

For insurance to be actuarially fair, the insurance company should have zero expected profits. We can set up their problem as under:

With probability p, the insurance company must pay $x, while receiving $rx in premiums. With probability (1-p), they pay nothing, and continue to receive $rx in premiums. So their expected profit is:

p(rx - x) + (1-p)rx

If this equals zero, we have: px(r-1) + (1-p)rx = 0

Dividing throughout by x, we get: pr - p + r - pr = 0

i.e. p = r.

This is called a fair premium since the insurance company breaks even on average. If a risk averse consumer can buy insurance at a fair price the consumer will fully insure.

If p >r then, it is partial insurance (or no insurance).
If p < r then, it is over insurance.

In practice, the premium may not equal the probability of the accident since the risk premium may be higher for some individuals than others. Hence the company may charge premium accordingly.

The consumer may purchase insurance after evaluating his expected utility in the good state and the bad state.

U (cb; cg) = pu (cb) + (1-p) u(cg);

A risk averse investor may want to equalize your wealth across all circumstances – irrespective of the state.

(where p = ?)