We’ve previously studied our consumer’s utility maximization problem under certainty; we now move on to consider utility maximization under uncertainty. We’ve just seen that given that the consumer’s utility function is concave he/she will always have a (latent) demand for insurance, i.e., if she can find someone willing to write an insurance contract, where the other person stands ready to take on the whole risk, she is always prepared to accept such a contract, if the price is not too high. We can also analyze this decision in an indifference curve diagram. In this case we, again, have to reinterpret what we put on the axes. On the vertical axis we measure c2 which now is the quantity of a consumption good consumed in state 2; and on the horizontal axis we measure c1 which is the quantity of the consumption good consumed in state 1. These goods are now called state−contingent goods since they are one and the same physical good, but consumed in quite different circumstances (e.g. an ice-cream cone, of a given brand, consumed when it is plus 30 degree centigrade is probably not consider by most people the same good as if it is consumed when it is minus 30 degrees). The indifference curves in this diagram show the consumer’s preferences over such state-contingent goods, or her state preferences.
In figure, point A is what the consumer can expect if she does not buy any insurance (her “endowment”). State 1 implies a high consumption level, and state 2 a low level, i.e., point A is below the dotted line which shows equal consumption in both states. The difference between the consumption levels can be viewed as the “loss” in state 2, compared to state 1. Since the utility function is concave (exhibits diminishing marginal utility), the marginal utility of an additional unit of consumption is higher in state 2 than in state 1. This implies that the consumer’s total utility increases if she can write an insurance contract where she sells a certain quantity of consumption goods in state 1, in exchange for a given amount of consumption goods in state 2. The consumer is “endowed” with Y1 amount of money in state 1 and Y2 in state 2. Without an insurance market our consumer would have to consume her endowment in each state, i.e., ci = Yi, for i = 1, 2, and her utility level is given by the indifference curve running through point A.
With an insurance market available we define “prices” for consumption good in state 1 as p1, and in state 2 as p2. Formally we write the consumer’s expected utility maximization problem as,
max π . u (c1) + (1 − π) . u (c2)s.t. : p1c1 + p2c2 ≤ p1Y1 + p2Y2,
The optimal solution involves setting the (expected) marginal rate of substitution equal to the price ratio, or,
(π.MU1)/[(1- π). MU2] = p1/p2
Note that if, p1/p2 = π/(1−π), we get,
MU1 = MU2,or,
c∗1 = c∗2,
i.e., the consumer writes contracts such that the consumption is completely equalized across states (point B), i.e., complete insurance, provided that the ratio of the “state-contingent” prices is equal to the ratio of the probabilities.
How should these “state-contingent” prices be interpreted? Well in reality all insurance contracts are sold via insurance companies which sell different quantities of insurance coverage charging a premium, q, per unit of coverage. Going back to figure, the consumer can move from point A to B by buying x amount of insurance coverage, at an insurance premium of q. In state 1 (the good state) the consumer pays a total insurance premium of q . x to the insurance provider, in state 2 (the bad state) the consumer receives x−q .x, or (1−q) x. The ratio of these two changes in consumption is,
Hence, it is as if the price of one unit of consumption good in the good state is q and the price in the bad state is 1 − q. From the perspective of the insurance company, its expected profits from this contract is,π. q . x − (1 − π) . (1 − q) . x = EΠ.
If the insurance market is perfectly competitive the expected profit will be equal to zero, and we say that the insurance premium is fair (or really a “fair game”). The ratio of the probabilities of the good and the bad state is then,
I.e., a fair insurance premium is such that the ratio of the net payout in the bad state (1 − q) per unit of insurance coverage, to the insurance premium (q), is equal to the ratio of the probabilities of the good to the bad state (the “odds ratio”). For example, if x = 1000, and π = 0.9, the odds ratio is 0.9/0.1 = 9, and the fair insurance premium is equal to, q . x = 100. Hence, the slope of the budget line in figure, with a perfectly competitive insurance industry offering fair insurance premiums, is equal to,
In reality insurance premiums cannot be completely fair and the price ratio will therefore be lower than the odds ratio [(p1/p2) < π(1−π)]. In figure below, we have added a price line with a flatter slope, and the solution to the utility maximization problem (point C) implies that the consumer will not buy complete insurance. For example, if the insurance companies need to put the premium equal to 0.12 to break even, the ratio [(1−q)/q] = 0.88/0.12 = 7.33 will be less than the odds ratio:
π/(1−π) = 9.
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