1 / 21

More Insurance

More Insurance. How much insurance. We started talking about insurance. Question now is “how much?”

liona
Download Presentation

More Insurance

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. More Insurance

  2. How much insurance • We started talking about insurance. Question now is “how much?” • Recall that Elizabeth’s expected utility involves her wealth when ill, with probability of 0.05, or when healthy, with a probability of 0.95. If ill, her wealth will fall from $20,000 to $10,000. • We address Elizabeth’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 Elizabeth loses $10,000 if she is ill, it is a useful place to start.

  3. How much insurance • Suppose that Elizabeth must pay a 10% premium ($50) for her insurance or one dollar for every 10 that she is covered. Consider the following “worksheet” describing her wealth if she gets sick. Insurance Worksheet -- $500 coverage Original wealth $20,000 less Loss $10,000 Remainder $10,000 plus Insurance 500 Sum $10,500 less Premium 50 or New wealth $10,450

  4. How much insurance A • If she stays well, her 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 her. • His marginal cost is the expected marginal utility that the $50 premium costs her. • If Elizabeth is averse to risk, the marginal benefit (point A) of this insurance policy exceeds its marginal cost (point A). MB, MC A’ 500 Amount q

  5. How much insurance A • Should Elizabeth buy $1,000 of insurance rather than $500? Compare the MB of this next $500 increment to the marginal costs. • Since Elizabeth 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. MB from the second $500 increment is slightly smaller than for first $500 increment. MB1 is downward sloping, with her new marginal benefit at point B. • Similarly, since she 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 her MC1 is upward sloping. B MB, MC B A 500 1000 Amount q

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

  7. What if the premiums increase? Changes in Premiums How will the insurance decision change if premiums change? A higher premium, say 15%, rather than the 10% that used earlier. With the 15% premium ($75), Elizabeth faces the following calculation regarding a policy that provides $500 worth of coverage: Insurance Worksheet –Increased Premium Original wealth $20,000 less Loss $10,000 Remainder $10,000 plus Insurance 500 Sum $10,500 lessNewPremium75 orNew wealth $10,425

  8. How much insurance A • If she stays well, her wealth is $20,000 less the $75 premium, or $19,925. MB from the $500 from insurance is now $425, rather than the previous value of $450, so point C lies on curve MB2 below the previous marginal benefit curve MB1. • Similarly, marginal cost is the expected MU that the (new) $75 premium costs. This exceeds the previous cost in terms of foregone utility, so Clies on MC2 above the previous curve MC1. Again, fill in additional points, and find intersection of MB2 and MC2 at point Y. • Consumers react rationally to increased premiums by reducing their optimum coverage from q* to q**. B MC2 C MC1 MB, MC X Y C MB1 B MB2 A 500 1000 q** q* Amount q

  9. 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.

  10. Supply Assume that Elizabeth is buying insurance in a competitive market, and under perfect competition, all firms earn zero excess, or normal long run profits. Recall that Elizabeth faced a potential illness with a probability of 0.05 (one in twenty). She sought to buy insurance in blocks of $500, and at the outset, the insurer was charging $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

  11. 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.

  12. 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%.

  13. Supply 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.

  14. Some Algebra A little algebra can verify that the premiums must be tied directly to the probability of the claim. 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)

  15. Supply  = p + (t/q) So … 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) = 0.10. Hence in equilibrium  = (0.05 + 0.10) = 0.15. 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; 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.

  16. 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, Elizabeth will add coverage up to the point where her expected wealth will be the same whether she 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), Elizabeth’s wealth, if well, will be: Wealth (well) = $20,000 – cost of insurance, or: $20,000 – (premium )*(coverage q) Her wealth, if ill, will be: Wealth (ill) = $20,000 – loss + insurance payment – ins. 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).

  17. Put it all together Wealth (well) = $20,000 - q = $20,000 –$10,000 + q – q = Wealth (ill). Subtracting $20,000 - q from both sides, and rearranging terms yields: qopt = $10,000. The optimal level of coverage for Elizabeth to buy if she expects a loss of $10,000 is $10,000. This surprising result holds irrespective of her 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, Elizabeth will react to the higher price by buying less than $10,000 of coverage. That is, Elizabeth will insure for an amount that falls short of her expected loss if illness strikes.

  18. Gain from Insurance U = 10W – 0.25W2 Prob (Ill) = 0.3 Loss (Ill) = 15 E (Loss) = 0.3*15 = 4.5T E (U) = 0.3 U(5) + 0.7 U(20) = 83.25 Suppose you can buy a policy for 4.5T. Will you do it? What is your consumer surplus? A: Why?

  19. Moral Hazard Demand • 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 P1Q1. • Actuarially fair policy would cost 0.5*P1Q1. Price P1 Exp. Premium Q1 Quantity

  20. Moral Hazard Demand • What if demand was somewhat elastic? Price • If insurer charges 0.5*P1Q1, it will lose money. • Why, because expected payments are P1Q2. P1 Exp. Premium New Premium • What if insurer charges 0.5*P1Q2? • Customer may not buy insurance. Q1 Q2 Quantity

  21. Moral Hazard • Why? • Here, two dimensions to insurance • Premium against risk. Customer wishes to insure against this. • Extra resource cost due to moral hazard. The risk was P1Q1. Customer may not be willing to pay more to insure against that risk!

More Related