Question

17. Insurance is a financial contract in which the policyholder (i.e., customer) is compensated if a pre-specified adverse event (e.g., accident) occurs, in exchange for premiums paid to the insurer (i.e., insurance company) by the policyholder. Consider an individual who owns a house. The value of the house is ??. When there is fire, the value of the house decreases to ??−??. The probability of fire is ??. The utility function of the house owner is ??= ln(??) where ?? is her wealth. Thus, her expected utility without insurance can be written as ??(??)=??ln(??−??)+(1−??)ln(??). Suppose that an insurance company offers fire insurance. The policyholder can choose the coverage level ?? of the insurance. Coverage is the maximum amount of compensation that a policyholder can receive if an adverse event occurs. The house owner considers the insurance. She will receive ?? from the insurance company if there is fire. In return, she should pay insurance premium of ?? whether there is fire or not. The expected profit of the insurance company is ??= (1−??)??+??(??−??).

(a) Suppose also that the insurance company is in a perfectly competitive market. If a company’s profit is higher than the others, then the company can offer a lower premium to customers. By offering a low premium, the company can win the competition. However, a lowered premium implies a lower profit. Companies cannot survive under negative profits. Thus, in a perfectly competitive market, the expected profit of a company approaches zero in equilibrium. What is the insurance premium in equilibrium?

(b) Under the condition of (a), what is the expected utility of the house owner with insurance?

(c) Suppose also that the house owner maximizes her expected utility. Under the condition of (a), which coverage level should she choose?

Answer 1

Part (a)

The expected profit of the insurance company is ? = (1 − ?) x ? + ? x (? − ?)

In a perfectly competitive market, the expected profit of a company approaches zero in equilibrium.

Hence, ?= (1 − ?) x ? + ? x (? − ?) = 0

Hence, r − ? x ? + ? x ? − ? x ? = 0

Hence, insurance premium in equilibrium, ? = ??

Part (b)

E (U) = p x ln (W - F) + (1 - p) x ln (W)

In case there is a fire, I is the insurance received and premium r is paid in any case.

Hence, the expected utility function will get revised to:

E (U) = p x ln (W - F + I - r) + (1 - p) x ln (W - r)

r = pI

Hence, E (U) = p x ln (W - F + I - pI) + (1 - p) x ln (W - pI)

Differentiate, E(U) w.r.t I and equate it to zero

Recall: d[ln(x)] / dx = 1 / x

d[E(U)]/dI = p x (1 - p) / (W - F + I - pI) - p x (1 - p) / (W - pI) = 0

Hence, p x (1 - p) x [ 1 / (W - F + I - pI) - 1 / (W - pI)] = 0

Hence, W - F + I - pI = W - pI

Hence, I - F = 0

Hence, I = F

Hence, she should choose coverage level, I = F = loss in value of house on fire