Benefits from signing a health insurance contract

Ž . perhaps national insurance without risk classification is more appropriate from an efficiency point of view.

2. Benefits from signing a health insurance contract

2.1. Basic assumptions The model is based on the following assumptions. Ž . Ž . There are two states of the world, health s s 1 and illness s s 2 . All individuals have the same state independent initial wealth endowment W . There is only one kind of disease which affects people accidentally and independently. The disease does not cause death but some kind of restriction in the quality of life 19 so Ž . Ž . Ž . Ž . that it reduces utility from U W to U W , where U W U W for all W. 1 2 1 2 Individuals are assumed to be risk-averse and have a strictly concave utility Ž X Ž . Y Ž . Ž . function in both states of the world where U W 0, U W - 0, U W ™0 s s s . for W ™0, ss1,2 . Moreover we assume that there exists only one kind of treatment, leading to full recovery at financial costs of T, which might well be larger than W. 20 Throughout the paper the expected utility of an insured person will be Ž . Ž abbreviated as EU p , a, r , where p is the personal illness probability where i i . 0 - p - 1 , a is the share in treatment cost that is refunded and r is the price of i one unit of refund, so that EU p , a, r s 1 y p U W y arT q p U W y T q a 1 y r T . 1 Ž . Ž . Ž . Ž . Ž . Ž . i i 1 i 1 Ž . Expected utility in case of no insurance will be written as EU p:,0 : EU p ,0 s 1 y p U W q p max U W ;U W y T 2 Ž . Ž . Ž . Ž . Ž . Ž . i i 1 i 2 1 Ž . In case of illness an uninsured person would be willing to pay at most R W for full recovery. 21 This implies U W s U W y R W . 3 Ž . Ž . Ž . Ž . 2 1 Ž . As U W ™0 for W™0 willingness-to-pay is smaller than wealth. s Initially, we assume that all individuals have the same illness risk p. There exists a fully competitive insurance market where firms compete in contracts. If information is distributed symmetrically, then everybody is offered a health 19 For example, in case of illness, the patient suffers from chronic pain. 20 The model is static and therefore limited. However, we know from the theory of adverse selection Ž . that for infinitely many periods, the first best can be obtained Dionne, 1983 . For finitely many periods, efficiency losses through adverse selection are reduced, but very similar to those in single period models. 21 Ž . A similar analysis has been carried out by Cook and Graham 1977 . Ž . 22 insurance contract that refunds all treatment costs a s 1 at a fair premium of pT. If we define B as the benefit from signing the insurance contract, we get: B s EU p,1, p y EU p,0 Ž . Ž . s U W y pT y 1 y p U W y pU W y min T ; R W 4 Ž . Ž . Ž . Ž . Ž . Ž . 1 1 1 Ž . If T - R W , B is nonnegative for all 0 - p - 1. In this case every risk-averse utility maximizing individual would sign a health insurance contract at the fair premium, and standard insurance market models can be applied. 2.2. Benefits from health insurance in case of treatment costs higher than willingness-to-pay In the introduction it has been asserted that in a considerable part of the health care sector treatment costs are higher than an uninsured person’s willingness to pay. In cases where T exceeds W this statement is trivial, and examples in health Ž . care are not hard to find e.g., in the areas of transplantation medicine or dialysis . Ž . Examples for situations with W T R W might be a visit to the health resort, treatment by the chief of staff, etc. In further analysis it will be shown that utility maximizing individuals may benefit from fair insurance against treatment costs higher than willingness-to-pay. Ž . Ž . If T R W , Eq. 4 simplifies to B s U W y pT y 1 y p U W y pU W y R W . 5 Ž . Ž . Ž . Ž . Ž . Ž . 1 1 1 B can be considered as a function of p, and now can adopt positive or negative values. With respect to p, B reaches a maximum 23 where X U W y pT s U W y U W y R W rT . 6 Ž . Ž . Ž . Ž . Ž . 1 1 1 An interior solution only exists if the derivative of B at p s 0 is positive: X U W - U W y U W y R W rT . 7 Ž . Ž . Ž . Ž . Ž . 1 1 1 This is the necessary condition for B to have a positive value. For a graphic analysis, see Fig. 1. An individual with illness probability p can reach point C on curve U at net wealth W y pT by signing an insurance contract with premium 1 pT. His expected utility in case of no insurance corresponds to the y-value of 22 This assumption corresponds to the observation that real health insurance companies in case of illness refund treatment costs and do not pay out money without treatment. The two main reasons are that treatment in may cases not only aims at cure but also prevention, and that results from treatment attempts often are part of the diagnosis. 23 2 2 Y Ž . 2 Since d Brd p sU W y pT T - 0, the second-order condition corresponds to a maximum. 1 Fig. 1. Benefits from signing an insurance contract. point D, lying on line A A X at the same net wealth W y pT. 24 The vertical 1 2 distance between C and D thus measures the benefit from signing the health Ž . insurance contract. If condition 7 holds, apart from A there is one more 1 Ž . X intersection point E between utility curve U and line A A . Accordingly, if p 1 1 2 Ž . Ž . is lower higher than some critical value p , B is positive negative and the crit Ž . individual will sign the health insurance contract stay without insurance , respec- tively. As an important result of these considerations we note that in case of treatment costs higher than willingness-to-pay, signing a fair insurance contract is advanta- Ž . geous if and only if condition 7 holds and p is smaller than p . Only low risks crit insure under a fair premium. This becomes clear if one considers that a fair premium from point of view of the insurer is unfair for the insured, as his willingness to pay is lower than treatment costs. On the other hand, insurance still eliminates any risk. For the high risks, the first effect dominates, so they do not Ž . buy insurance cover. While for low risks, if condition 7 holds, the second effect is more relevant. 25 24 w Ž .x X This can be seen by noting that the dotted line connects the points A W ; U W and A 1 1 2 w Ž .x Ž . W yT ; U W , and that it is cut by D in a ratio of p: 1y p . Expected utility without insurance is 2 Ž . Ž . Ž . Ž . calculated from EU p,0 s 1y p U W q pU W , which is the y-value of point D. 1 2 25 We thank one referee for pointing this out to us. 2.3. Efficiency effects of categorical discrimination in case of symmetric distribu- tion of information Having derived the demand for insurance under a fair premium, we now consider the effects of genetic testing on health insurance markets if the test results Ž . are available to the insurer. Hoy 1989 showed that there might be positive efficiency effects, if insurers could observe risk avoidance behavior and adjust premium rates accordingly. These effects will be left aside here. Assume that initially insureds are not informed about their risk types and Ž believe their illness probability to be as high as the average risk p where . 0 - p - 1 . Furthermore we assume that there exists a test to find out, if one’s risk type is either p or p . The test causes per capita costs of c utility units, and L H insurers are allowed to gather the test results from the insureds and use them for premium calculation. The ex ante per capita benefit G from taking the test can be calculated from G s q max EU p ,0 ;EU p ,1, p q 1 y q max EU p ,0 ; Ž . Ž . Ž . Ž . H H H H H L EU p ,1, p y max EU p,0 ; EU p,1, p y c. 8 Ž . Ž . Ž . Ž . L L where q is the probability of having a positive test result, such that H q p q 1 y q p s p. Ž . H H H L Ž . At first we assume p F p , and consequently p - p , so that uninformed crit L crit, individuals would choose to fully insure. This implies G s q max EU p ,0 ; EU p ,1, p Ž . Ž . H H H H q 1 y q EU p ,1, p y EU p,1, p y c. 9 Ž . Ž . Ž . Ž . H L L Ž . If p F p , the value of G is always negative, since EU p ,1, p H crit H H Ž . Ž Ž . q EU p ,1, p q 1 y q EU p ,1, p . Then genetic tests only lead to pre- H H H H L mium risk, an effect well known from the standard models. In case of p p , high risk types would not sign an insurance contract, H crit Ž . Ž . while low risk types would. As EU p ,0 EU p ,1, p , the benefit G then is H H H calculated from G s q EU p ,0 q 1 y q EU p ,1, p y EU p,1, p y c, 10 Ž . Ž . Ž . Ž . Ž . H H H L L which can be positive if costs of testing are low. An example is given in Fig. 2, where testing costs are assumed to be zero. Uninformed individuals will fully Ž . Ž . insure and reach an expected utility of EU p,1, p s U W y pT in point C. 1 After taking the test and learning their risk types, high-risk individuals will not Ž . Ž sign an insurance contract and have an expected utility of EU p ,0 s 1 y H . Ž . Ž Ž .. p U W q p U W y R W , which corresponds to the y-value of point H. H 1 H 1 Low risk individuals will become insured and reach point L. Ex ante expected utility of an uninformed person, if he learns his risk type, can be seen from the Fig. 2. Tests can be welfare improving. y-value of point C U , the point on line HL at net wealth W y pT. Thus, the benefit G from taking the test in this example is the vertical distance between the points C and C U . We can conclude that, in case of p p and zero information costs, G is H crit strictly positive if p G p p X , i.e., if an uninformed insured would reach a point crit w X Ž X .x within the segment EJ of the U utility curve, where J W y p T ; U W y p T is 1 1 the intersection point between line HL and the U utility curve. 26 1 Ž . If p p and consequently p p , uninformed individuals would choose crit H crit not to insure, and G can be calculated from G s q EU p ,0 q 1 y q max EU p ,0 ; Ž . Ž . Ž . H H H L EU p ,1, p y EU p,0 y c. 11 Ž . Ž . Ž . L L For zero information costs, G then is always nonnegative and strictly positive if p - p . In the example shown in Fig. 2, G in case of p p corresponds to L crit crit the vertical distance of lines HL and HE at the respective net wealth W y pT. 26 Accordingly, p X is defined as the minimum value for average illness probability, at which G is nonnegative if cs 0, and depending on the given values for p and p . This implies, that the number L H of high risks has to be sufficiently large for p to be larger than p X . With positive information costs no general statement is possible. If p p H crit and p - p , the ex ante benefits from taking the test can be positive or negative, L crit but are always negative if p F p X or p s 0. L To complete the analysis, we summarize the results of the relationship between the value of a test and the precision of the test in case of informational symmetry. 27 In contrast to standard models, where premium risk always reduces Ž . Ž . welfare, the value of a predictive genetic test can be positive if 1 members of Ž . the carrier group choose not to be insured, and 2 a negative test result does not imply that non-carriers are immune against the illness. This implies that the highest efficiency is obtained if members of the carrier group have a very large illness risk, while members of the non carrier group still have a non negligible illness risk.

3. Health insurance market equilibria under asymmetric information