Consider the following model of demand for insurance (identical to that studied
ID: 1196292 • Letter: C
Question
Consider the following model of demand for insurance (identical to that studied in class). Risk-averse individuals maximize expected utility of wealth where wealth is random due to a loss L, that occurs with probability . Insurance carries a premium, P = p · I, where p is the price per dollar of insurance and I is the dollar amount of insurance purchased. Full insurance implies I = L while a deductible implies I < L. Please answer the following: (a) It is assumed that insurance companies operate in a perfectly competitive market. What does that imply for their profits? What does that imply for the value of the insurance premium P? (b) Let wealth in the no-loss state be denoted W1 while wealth in the loss state is denoted W2. What is the rate at which agents can transfer wealth between the loss and the no-loss state via the purchase of insurance? (c) What is the slope of individuals’ indifference curves, i.e. what is the rate at which agents are willing to transfer wealth between the loss and the no-loss state? (Yet another way to put the same question: What is the individuals’ marginal rate of substitution between wealth in the two states of the world?) (d) Show that, in this scenario, individuals will choose to purchase full insurance against the loss. (e) Illustrate your results for parts (a)-(d) in a graph in the (W1, W2)-space.
Explanation / Answer
Suppose that the rational preference relation % on the space of lotteries satisfies the reduction axiom, continuity and independence. Then, there exists a function of the expected utility form that represents %. That is, we can assign a number un to each outcome n = 1,..., N such that for any two lotteries L = (p1,..., pN ) and L = (p 1 ,..., p N ) we have L % L if and only if X N n=1 unpn X N n=1 unp n .
A DM is called risk averse (or said to exhibit risk aversion) if, for any lottery F(·), the degenerate lottery that yields the amount R xdF(x) with certainty is at least as good as the lottery F(·) itself. If for any F(·) the DM is indifferent between these two lotteries, we say that he is said to be risk neutral. We say that he is strictly risk averse if indifference holds only when the two lotteries are the same (when F(·) is degenerate). Opposite relations refer to risk loving
Bernoulli utility fns of risk averse agents in [x, u(x)] space are concave; the more concave they are, the more risk-averse is the agent risk loving agents are strictly convex. This is a direct consequence of Jensen’s inequality; for a real continuous concave fn u(·) we have Z + u(x)dF(x) u Z + xdF(x) . Hence the expected utility is smaller than the utility of an expected value.
A useful concept for the analysis of risk aversion: Definition: Given a Bernoulli utility function u(·), the certainty equivalent of F(·), denoted c(F, u) is the amount of money for which the individual is indifferent between the gamble F(·) and the certain amount c(F, u); that is, u(c(F, u)) = Z + u(x)dF(x). The DM is risk averse if and only if c(F, u) Z xdF(x) for all F(·).