More Insurance

How much insurance We started talking about insurance. Question now is “how much?” Recall that John’s expected utility involves his wealth when ill, with probability of 0.05, or when healthy, with a probability of If ill, his wealth will fall from $20,000 to $10,000. We address John’s optimal purchase by using the concepts of marginal benefits and marginal costs. Consider first a policy that provides insurance covering losses up to $500. Although it would hardly seem worth buying a $500 insurance policy when John loses $10,000 if he is ill, it is a useful place to start.

How much insurance Suppose that John must pay a 10% premium ($50) for his insurance or one dollar for every 10 that he is covered. Consider the following “worksheet” describing his wealth if he gets sick. Insurance Worksheet -- $500 coverage Original wealth$20,000 lessLoss$10,000 Remainder$10,000 plusInsurance 500 Sum$10,500 lessPremium 50 or New wealth $10,450

If he stays well, his wealth is $20,000 less the $50 premium, or $19,950. His marginal benefit from the $500 from insurance is the expected marginal utility that the additional $450 brings him. His marginal cost is the expected marginal utility that the $50 premium costs him. If John is averse to risk, the marginal benefit (point A) of this insurance policy exceeds its marginal cost (point A). How much insurance Amount q MB, MC 500 A A’

Should John buy $1,000 of insurance rather than $500? Compare the MB of this next $500 increment to the marginal costs. Since John is slightly wealthier than before, if ill (starting at $10,450 rather than $10,000) the marginal utility from an additional $450 of wealth (calculated as before) will be slightly smaller than before. Hence MB from the second $500 increment is slightly smaller than for first $500 increment. MB 1 is downward sloping, with his new marginal benefit at point B. Similarly, since he is a little less wealthy than before if well, an additional $50 in premiums will cost a little more in foregone (marginal) utility than the first increment at point B. Thus his MC 1 is upward sloping. How much insurance Amount q MB, MC 500 A A’ B B’ 1000

Continuing this exercise, we get to equilibrium point X, where marginal benefits equal marginal costs. How much insurance Amount q MB, MC 500 A A’ B B’ 1000 MC MB X q*

What if the premiums increase? Changes in Premiums How will the insurance decision change if premiums change? Consider first the impact of a higher premium, say 15%, rather than the 10% that we used earlier. With the 15% premium ($75), John faces the following calculation regarding a policy that provides $500 worth of coverage: Insurance Worksheet –Increased Premium Original wealth$20,000 lessLoss$10,000 Remainder$10,000 plusInsurance 500 Sum$10,500 lessNew Premium 75 or New wealth $10,425

If he stays well, his wealth is $20,000 less the $75 premium, or $19,925. The MB from the $500 from insurance is now $425, rather than the previous value of $450, so point C lies on curve MB 2 below the previous marginal benefit curve MB 1. We fill in additional points, with the higher premium. Similarly, the marginal cost is the expected MU that the (new) $75 premium costs. This exceeds the previous cost in terms of fore- gone utility, so C lies on MC 2 above the previous curve MC 1. Again, fill in additional points on this curve, and find the intersection of MB 2 and MC 2 at point Y. The resulting analysis suggests that consumers react rationally to increased premiums by reducing their optimum coverage from q* to q**. How much insurance Amount q MB, MC 500 A A’ B B’ 1000 MC 1 MB 1 X C C’ q* MB 2 MC 2 Y q**

The Supply of Insurance In the last example, we assumed a 10% premium rate. Ultimately, to determine the amount of coverage an individual will buy, we must know how insurers determine the premium. To do this, we develop a model of the insurance market, beginning again with the club that insures its members against illness. The officers of the club do not know, nor necessarily care, who will file a claim. All that is necessary for the club to function as an insurer is that revenues cover costs. In practice, insurance companies will incur administrative and other expenses that must also be covered by premiums.

Supply Let’s assume that John is buying insurance in a competitive market, and under perfect competition, all firms earn zero excess, or normal long run profits. Recall that John faced a potential illness with a probability of 0.05 (one in twenty). He sought to buy insurance in blocks of $500, and at the outset, the insurer was charging him $50 for each block of coverage, or an insurance premium of 0.10 ($50 as a fraction of $500). Assume also that it costs the insurer $5 to process each insurance application. The insurer’s profits per policy are: Profits = Total revenue – Total costs

Supply Profits = Total revenue – Total costs The firm’s revenues are $50 per policy. What are its costs? For 95% of the policies, the costs are $5, because the insured do not get sick, and hence need not collect insurance. The only costs are the $5 processing costs per policy. The costs for the other 5% of the policies are $505, which consist of the $500 payment to those who are ill plus the processing costs of $5.

Supply Thus the profits per policy are: Profits = $50 – (Probability of illness * costs if ill) – (Probability of no illness * costs if no illness) Profits = $50 – (0.05 * $505) – (0.95*$5) Profits (premium = 10%) = $50 (revenues) - $30 (costs) = $20. These are positive profits, implying that another firm (also incurring costs of $5 to process each policy) might enter the market and charge a premium of, say 8%. The cost side of the equation would remain the same but the revenues, which equal the premium fraction multiplied by the amount of insurance, would fall, here, to $40. Hence profits fall to: Profits (premium = 8%) = $40 – $25.25 – 4.75 = $10, still positive. It is easy to see that entry will continue until the premium has fallen to $0.06 per dollar of insurance, or 6%, which would provide revenues of $30, offset by the $30 in costs, to give zero profits.

A little algebra can verify that the premiums must be tied directly to the probability of the claim. Quite simply, for the firm, the revenue per policy is  q, where  is the premium, in fractional terms. The cost per policy in terms of pay-out is the probability of pay-out p, multiplied by the amount q, plus a processing cost t. So: Profit =  q – (pq + t) =  q – pq – t With perfect competition, profits = 0, so: 0 =  q – pq – t. We can solve for the competitive premium  as:  = p + (t/q) Supply

 = p + (t/q) This expression shows that the competitive value of  equals the probability of illness p, plus the processing (sometimes called loading) costs as a percentage of policy value q, or t/q. If loading costs are 10 percent of the policy value q then (t/q) = Hence in equilibrium  = ( ) = The premium for each dollar of insurance q is 15 cents. If insurers charge less, they will not have enough money to pay claims. If they charge more, there will be excess profits in the business, and other firms will bid down rates in perfectly competitive markets. Previously, in the discussion on the bearing of risk, we considered insurance policies that would compensate the individual against the loss based solely on the probability of the event’s occurring. Such rates are referred to as actuarially fair rates. We see that the actuarially fair rates correspond to the rates in which the loading costs t approach 0 as a percentage of insurance coverage. Supply

Put it all together Knowing that premium  equals p under perfect competition (with no loading costs), we can now solve for the optimal coverage against any expected loss. It can be shown that to maximize utility, John will add coverage up to the point where his expected wealth will be the same whether he is ill or well. In the earliest example, the particular illness occurred with a probability of 0.05, and incurred a loss of $10,000. In a competitive insurance market (ignoring loading costs), John’s wealth, if well, will be: Wealth (well) = $20,000 – cost of insurance, or: $20,000 – (premium  )*(coverage q) His wealth, if ill, will be: Wealth (ill) = $20,000 – loss + insurance payment – insurance premium $20,000 –$10,000 loss + (coverage q) – (insurance premium  q). Equating the two, yields: Wealth (well) = $20,000 -  q = $20,000 –$10,000 + q –  q = Wealth (ill).

Wealth (well) = $20,000 -  q = $20,000 –$10,000 + q –  q = Wealth (ill). Subtracting $20,000 -  q from both sides, and rearranging terms yields: q opt = $10,000. The optimal level of coverage for John to buy if he expects a loss of $10,000 is $10,000. This surprising result holds irrespective of his wealth, and irrespective of the probability of the event. From our previous discussion, however, if the equilibrium price of insurance (premium  ) exceeds the probability of the event p, John will react to the higher price by buying less than $10,000 of coverage. That is, John will insure for an amount that falls short of his expected loss if illness strikes. Put it all together

Moral Hazard With health insurance, the amount of expenditures may depend on whether you have insurance. Suppose that probability of illness is 0.5. Suppose demand for care (if sick) is P 1 Q 1. Actuarially fair policy would cost 0.5*P 1 Q 1. Quantity Price Demand Q1Q1 P1P1 Exp. Premium

Moral Hazard What if demand was somewhat elastic? Quantity Price Demand Q1Q1 P1P1 Exp. Premium If insurer charges 0.5*P 1 Q 1, it will lose money. Why, because expected payments are P 1 Q 2. Q2Q2 What if insurer charges 0.5*P 1 Q 2 ? New Premium Customer may not buy insurance.

Moral Hazard Why? Two dimensions to insurance –Premium against risk. Customer wishes to insure against this. –Extra resource cost due to moral hazard. The risk was P 1 Q 1. Customer may not be willing to pay more to insure against that risk!