Professor Rachel Kranton University of Maryland Econ 413 Fall Adverse Selection in Insurance Answers to Problems

Transcription

1 Professor Rachel Kranton University of Maryland Econ 413 Fall 2000 Adverse Selection in Insurance Answers to Problems 1. When the insurance industry is monopolized by a single rm, it will o er each type of consumer a policy that maximizes its pro ts from that consumer subject to the constraint that the consumer buy the policy. Graphically, the insurance companyreaches the highest possible iso-pro t line it can, given theconsumer s indi erence curve representing her expected utilitithout insurance. For a low risk consumer, we have: y s 41,209 Slope=-3 Π * 41,209 The expected utility for a low-risk consumer when she does not buy insurance is = 3 4 u(50;000) u(20;000)

2 = 3 4 (50;000) (20;000)1 2 = 203 She will earn exactly the same level of expected utilithen her net income when she is sick or well is (203) 2 = 41;209. This is the certainty equivalent for a low-risk person. This translates into an insurance policy R L = 50;000 = 50;000 41;209 = 8791 B L = y s 20;000 + R = 41;209 20; = 30;000 The insurance company maximizes pro ts by fulling insuring the individual. It o ers a policy that yields exactly the same income whether she is sick or well. Any policy that involves some uctuation in net income leaves the consumer facing some risk. The rm could earn further pro ts by insuring the consumer against this risk. Pro ts are maximized when the consumer faces no further risk. The insurance company s expected pro ts are L = (1 p)r L + p(r L B L ) = R L pb L = 8791 (1=4)(30;000) = 1291 These pro ts are equal to the risk premium associated with the low-risk consumer s situation. For the high-risk consumer we have: 2

3 y s Slope=-1/3 Π * The expected utility for a low-risk consumer when she does not buy insurance is = 1 4 u(50;000) u(20;000) = 1 4 (50;000) (20;000)1 2 ' 161:96 She will earn exactly the same level of expected utilithen her net income when she is sick or well is (161.96) 2 = 26;231. This is the certainty equivalent for a low-risk person. This net income translates into the following insurance policy: R H = 50;000 = 50;000 26;231 = 23;769 B H = y s 20;000 + R = 26;231 20; ;769 = 30;000 Notice that the high risk consumer pays a higher premium for the same bene ts. The insurance company maximizes pro ts by fulling insuring the consumer against risk and charges the highest 3

4 premium it can. The net income for the high risk consumer will be lower than the net income for the low risk consumer. The insurance company s expected pro ts are H = (1 p)r H + p(r H B H ) = R H pb H = 23;769 (3=4)(30;000) = 16;269 These pro ts are equal to the risk premium associated with the high-risk consumer s situation. (b) When the insurance industry is perfectly competitive, each rm must earn zero expected pro ts. The iso-pro t line for zero pro ts goes through the point (, ) which is the point where R = 0 and B = 0: The equilibrium insurance policy for each type of consumer will be where the iso-pro t line intersects the line. This point is where the consumer reaches the highest possible indi erence curve, subject to the constraint that insurance rms earn at least zero pro ts. For the low-risk consumer we have: y s 42,500 Slope=-3 Π=0 EU L 1 42,500 To nd the point on the line where rms earn zero pro ts, we set = y s and solve for y = = y s such that a rm earns zero pro ts. A rm s expected pro ts are: L = R L pb L = (50;000 ) (1=4)(y s 20;000 + R L ) = (50;000 ) (1=4)(y s 20; ;000 ) 4

5 Setting y s = = e have L = (50;000 ) (1=4)(y s 20; ;000 ) = (50;000 y) (1=4)y + (1=4)20;000 (1=4)50;000 + (1=4)y = 42;500 y Setting L = 0 gives us y = 42;500: This net income for the consumers, whether sick or well, translates back into an insurance policy of R L = 50;000 y = 50;000 42;500 = 7;500 B L = y 20;000 + R L = 42;500 20; ;500 = 30;000 Notice that the consumer is fully insured against the loss. If she were not fully insured against the loss, she would face uctuation in her income. There would be pro ts to be made fromselling her more insurance. An insurance companould enter the industry to sell the consumer such an insurance policy. There are no more pro ts to be made onlhen the consumer no longer faces anyrisk. When the insurance industry is perfectlycompetitive, the low riskconsumer pays a premium of $7500. This is lower than the premium of $8791 she would pay to a monopolist insurer. In the perfectly competitive case, the consumer earns all of the risk premium ($1291) associated with her situation. The consumer s expected net income is $42;500; which is exactly the expected valueof her income if she does notbuy insurance. The consumer does not lose any money in expected value terms frombuying the insurance, and she earns strictly greater expected utility. For the high-risk consumer we have: 5

6 y s Slope=-1/3 27,500 EU H 1 Π=0 27,500 have Solving for the insurance polichere y s = = y and expected pro ts are equal to zero,we H = (50;000 ) (3=4)(y s 20; ;000 ) = (50;000 y) (3=4)y + (3=4)20;000 (3=4)50;000 + (3=4)y = 27;500 y Setting H = 0 gives us y = 27;500: This net income for the consumers, whether sick or well, translates back into an insurance policy of R H = 50;000 y = 50;000 27;500 = 22;500 B H = y 20;000 + R H = 27;500 20; ;500 = 30;000 The consumer is fully insured against loss. She pays a premium of $22,500. This is a higher premium than for the low-risk consumer. The high risk consumer pays more for the same bene ts. This premium is lower than the premium the high risk consumer would pay under monopoly ($23;769). Under perfect competition, the consumer has the same expected income whether she buys insurance or does not buy insurance. She earns the full risk premium of her situation ($1269). 6

7 2. When the rm cannot observe whether a person is high risk or low risk, its expected pro ts are E = ¼ L (y L w;y L s ) + (1 ) ¼ H (y H w ;y H s ) where is the probability that a consumer is high risk, (1 ) is the probability that a consumer is high risk, ¼ L (y L w;y L s ) are the pro ts the rm earns from selling insurance to the low risk consumer as a function of y L w and y L s (derived from its policy designed for low-risk consumers), and ¼ H (y H w ;y H s ) are the pro ts the rm earns from selling insurance to the high risk consumer as a function of y H w and y H s (derived from its policy designed for high-risk consumers). The policies for the high risk and low risk consumer must satisfy two sets of conditions. The rst conditions are individual rationality conditions. Each consumer must earn at least as much utility from buying insurance as from not buying insurance. For the low-risk consumer, we have For the high risk consumer, we have 3 4 u(yl w) u(yl s ) 3 4 u(50;000) + 1 u(20;000) (IRL) u(yh w ) u(yh s ) 1 4 u(50;000) + 3 u(20;000) (IRH) 4 The second set of conditions are incentive compatibility conditions. The low risk consumer must earn higher utility from purchasing the policith y L w and y L s than from purchasing the policy with y H w and y H s. The high risk consumer must earn higher utility from purchasing the policy with y H w and y H s than from purchasing the policith y L w and y L s. For the low risk consumer, we have and for the high risk consumer, we have 3 4 u(yl w) u(yl s ) 3 4 u(yh w ) u(yh s ) (ICL) 1 4 u(yh w ) u(yh s ) 1 4 u(yl w) u(yl s ) (ICH) The following picture illustrates these conditions. The insurance policy A satis es IRH, and B satis es IRL. Both satisfy ICH and ICL. 7

8 y s A Π * B EUH 0 (b) The policies the rm will o er will depend on. Consider rst = 0. In this case, the rm will want to earn as high as pro ts as possible from the high risk consumers. That is, it will choose (y H w ;y H s ) to maximize ¼ H (y H w ;y H s ). Label this policy (y H w ;y H s ). The insurance company will (y L w;y L s ) so that the high risk consumers will buy (y H w ;y H s ) and not (y L w;y L s ). The rm can earn the highest pro ts possible from high risk consumers when high risk consumers no longer face any uctuation in income. That is, y H w = y H s. The policill lie on the and the high risk consumer will be just indi erent between buying the policy and not buying the policy. For = 0, we have the following: 8

9 y s A B π* H π L =0 The rm will o er A and B. The high risk consumers will choose A. A satis es both IRH and ICH; it gives them at least as high expected utility as not buying insurance and at least as high expected utility as B. The rm earns the highest pro ts it can from the high risk consumers. It earns zero pro ts from low risk consumers. Notice that high risk consumers are insured fully against loss. Low risk consumers, however, are not o ered any insurance. They still face losses. For 1 > > 0, the rmwilltrade o higher pro ts fromlow risk consumers and lower pro ts from high risk consumers. The rm o ers A and B to consumers. High risk consumers choose A and low risk consumers choose B. The rm increases its pro ts on low risk consumers from ¼ L = 0 to ¼ L > 0:. It earns less pro ts on high risk consumers, from ¼ H to ¼ H. This loss will be balanced by the gains in pro ts from the low risk consumers when is high enough. Note that the rmearns greater pro ts fromo ering A and B than fromjust o ering B. If it just o ered B, high risk consumers would purchase B. The loss in the rm s pro ts on high risk consumers would be greater; the iso-pro t line for the high risk consumer that goes through B is to the right of the line through A - and therefore represents a lower level of pro ts. 9

10 y s EU H 1 A π* H π H B π L π L =0 EU H 1 Notice that high risk consumers are still fully insured against loss. They face no uctuation in income. They pay a lower price for the bene ts than they did before, and earn greater utility. The high risk consumers are strictlybetter o. Low riskconsumers have less uctuation in their incomes, but they are just indi erent between buying insurance and not. For = 1, to maximize pro ts, the insurance companill earn the highest pro ts it can from low risk consumers. The companill o er B. Low risk consumers will purchase this policy, and the rm earns the highest pro ts it can. High risk consumers would be willing to buy B, and if there were high risk consumers, the companould lose money by selling to them. But at = 1; there are no high risk consumers in the population. 10

11 y s 41,209 B π* L π L =0 41,209 11