Health insurance when loss severities differ

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1 Health insurance when loss severities differ Franck Bien THEMA Université Paris X-Nanterre Preliminary version Abstract - With information asymetry between contracting parties, adverse selection may result. A separating equilibrium can achieve. When severity differences and health state are aconsidered, first best equilibrium can be attained. Equilibrium with positive profit, full coverage, and opposite Rothschild-Stiglitz cans be feasible. Key words - Adverse selection, Loss severity, Health insurance. Classification :D8,I1 200 avenue de la République Nanterre Cedex France ; mail :

4 Health insurance when loss severities differ 3 with p i the probability of illness, T the loss of health, D i medical spending and m the benefit of treatment. The health risk is mild so T + m =0 (2) We assume that utility is monotonically increasing and concave in health andwealthwhereu i and U ij denote the first and the second partial derivatives of U with respect to its i th and j th arguments respectively. So U 1 > 0,U 2 > 0,U 11 < 0,U 22 < 0. We will assume that absolute risk aversion of a financial risk is decreasing in respect to health and wealth. For a same wealth, a type L agent has a less risk aversion than a type H agent. Bad-health agent fears more financial risk because he wants to face medical spending. Disease probability p i, medical spending D i and initial health S i each take on two values in the population, i {L, H}, withs H <S L. The high-risk H or bad health s state and the low-risk L or good health s state are drawn, respectively with probability λ H and λ L,whereλ H + λ L =1. We assume that bad-health agents cost more in expectation than good-health agents. Hence, D L < D H.Threeconfigurations are possible : 1) < and D L <D H 2) < and D L <D H <D L 3) < and D L < D H. The parameters p i, D i, S i are common knowledge to the policyholders and to the insurers but the latter do not know the risk type of any individual. We assume that audit costs are too high for risk type verification to be feasible. When audit costs are high, incentive contracts are efficient. There exists a limited number 5 of insurance firms (N with N<m)that may potentially enter to the market. We assume that there are two type of insurers which differ from rationality. Compagnies which number is N c maximise their profit andmutualfirms (N m ) earn zero profit, with. A contract C (π,q) earns expected profit per customers of type i Π(C) =π p i q (3) We seek subgame perfect Nash equilibria of the following two-stage RS game. In the first stage insurance firms choose wether to enter the market and, if so, offer a single insurance policy to potential customers. In the second stage, individuals choose a single policy among those on offer. 5 because insurance reglementation.

5 Health insurance when loss severities differ 4 2 Self-selection constraints The characterization of equilibrium for the model proceed in several steps. First, we have to find which self-selection constraints are binding at equilibrium. Next, I establish for the three cases the specification of equilibrium. When information on individual risk characteristics is public, each insurance compagnies knows the risk type of each individual. The optimal individual contract (p i D i,d i ) yields full insurance coverage for each type of risk. Firms are constrained to earn zero expected profits. Under asymmetricalinformation,inthers model,fullinformationcompetitivecontracts are not adequate to allocate risk optimally. But, when loss severities differ, Doherty and Jung (1993) proved that first-best solutions became feasible. In this case, the self-selection constraints are not binding. If the self-selection is binding, one group of agent is rationed while other group is is fully insured. Under asymmetric information, self-selection of type i is not binding when i-risk types prefer (or are indifferent) full insurance under the i-risk contract to alternative contract offering full insurance to j-risk type. Thus, the constrainst is not binding when U(w p i D i,s i ) p i u(w p j D j + D j D i,s i )+(1 p j )u(w p j D j,s i ) (4) with (i, j) (L, H) 2. Lemma 1 The self-selection constraint of an i-risk agent is not binding when a) p i <p j b) p i >p j and S i < S i excepted for D H <D L with (i, j) (L, H) 2 and D H > D L Proof. Inequation (4) is only clear for H-agent for < and D L < D H.Itdoesnothold.Fortheotherscases,wehavetomakeanapproximation without loss of generality. We can show this lemma by using a Taylor expansion around (w p i D i ).Aftersimplifications 6, this inequation can be written : 0 ' (p i p j )D i + 1 ½ pi (1 p i )(D i D j ) 2 2 +(p i p j ) 2 Dj 2 ¾ u11 (w p i D i,s i ) u 1 (w p i D i,s i ) (5) 6 We denote that D H D L D H + D L = D L (1 ) D H (1 ).

6 Health insurance when loss severities differ 5 Two cases have to be considered. Case 1. If p i <p j this inequation is always true because u 11 < 0. So, lemma a) holds. Case 2. If p i >p j this inequation is more complex. Specifically, we have to define the coefficient of absolute risk aversion of a financial risk : R A (c, S i )= u 11(w p i D i,s i ) (6) u 1 (w p i D i,s i ) where c = w p i D i. Equation (5) can be simplified to : R A (c, S i ) ' 2(p i p j )D j pi (1 p i )(D i D j ) 2 +(p i p j ) 2 D 2 j ª (7) We can define a critical value R A (c, S i) at which the constrainst just binds. R A(c, S i ) This may be rewritten as 2(p i p j )D j pi (1 p i )(D i D j ) 2 +(p i p j ) 2 D 2 j ª (8) R A (c, S i ) >R A(c, S i ) (9) As,wehaveassumedthatthecoefficient of absolute risk aversion of a financial risk is declining in health, S i < S i implies that this last expression holds. So, lemma b) holds. We can now study our three cases : Case 1 : > and D H >D L. If i = L and j = H equation (5) is true because (p i p j ) < 0 and u 11 < 0. (lemma a). If i = H and j = L equation (5) is true if and only if R A (c, S H ) > R A (c, S H). Hence S H < S H. (lemma b). For low values S H, the two self-selection constrainst do not bind. Consequently, the first-best, fullinformation equilibrium will prevail. Case 2 : > and D L >D H > D L. If i = L and j = H equation (5) is true because (p i p j ) < 0 and u 11 < 0. (lemma a)

7 Health insurance when loss severities differ 6 If i = H and j = L equation (4) is wrong because D H <D L and D H > P L D L. The self-selection constraint of i-risk agent will always bind and a fullinformation equilibrium is precluded. Case 3 : > and D H >D L If i = H and j = L equation (5) is true because (p i p j ) < 0 and u 11 < 0. (lemma a). If i = L and j = H equation (5) is true if and only if R A (c, S L ) > R A (c, S L). Hence S L < S L. (lemma b). As case 1, for low values S L with S L >S H, full-information equilibrium will prevail. 3 Equilibrium in the assurance market When health state is unobservable, a difficulty arise : indifference curve may cross at twice. If separation equilibrium is possible, two cases appear. First, agents may prefer a policy with excessive premium (positive profit). Second, first-best contracts become feasible. We suppose that equilibrium exists 7. Our study focuses on the equilibrium type. Before characterizing equilibrium, we have to define equilibrium of this market. It is clear that the nature of equilibrium is function how individual firms anticipate the behavior of rivals. Each insurer takes the actions of its competitor as given. Definition 2 No contract in the equilibrium set makes negative expected profit. When indifference curve cross at twice, individual may prefer a policy with a smaller deductible and excessive premium. So, second property (there is no contract that can make a positive profit) of RS equilibrium is precluded. Under competition, firms are constrained to offer contract which yield the more utility to individual. Lemma 3 Insurance contract C i is the solution to : C i = argmax U i (C i ) subject to U j (C j ) U j (C i ) (10) Π(C i ) 0 (11) with {i, j} {H,L} 2 and i 6= j. 7 We assume that pooling equilibrium is not attractive for the two groups of risk.

8 Health insurance when loss severities differ 7 Proof. Under competition, firms offer the best contract to i-risk agent under the constrainst of positive contract (equation 11). But, this contract has to verify self-selection constraint (equation 10). If j-risk agents prefer the contract C i, this contract can make negative profit. When p i <p j a contract C i with coverage (q i )andwhichearnzeroprofit on i-risk type, makes negative profit if it is choosen by j-risk-type. Expected profit of insurance firm is : Π(C i )=λ j (p i q i p j q i )+λ i (p i q i p j q i ) (12) Π(C i ) < 0 because p i <p j. 3.1 < and D L <D H The characterization of equilibrium relies on the single-crossing property. As it is well known, an important issue in adverse selection models is whether the Spence-Mirrlees single crossing condition holds true. The equilibrium (if it exists) is separating if the slope of the S H type indifference curve is more steep than S L (in the plane q, π). So, for a policy (π,q)withπ premium and q coverage, we obtain : (1 ) u 1 (W π,s H ) u 1 (W π + q D H,S H ) < (1 ) u 1 (W π,s L ) u 1 (W π + q D L,S L ) (13) A necessary condition is : u 1 (W π,s H ) u 1 (W π + q D H,S H ) < u 1 (W π,s L ) u 1 (W π + q D L,S L ) (14) This last condition holds if u 1 (W π,s H ) u 1 (W π + q D L,S H ) < u 1 (W π,s L ) u 1 (W π + q D L,S L ) with R 1 = W π and R 2 = W π + q D L R 1 >R 2 implicates that u1(r,s H) u 1(R,S L is declining with R. ) Hence u 11(R, S L ) u 1 (R, S L ) < u 11(R, S H ) u 1 (R, S H ) (15) (16)

9 Health insurance when loss severities differ 8 H U H (S 2 H) U H (S 1 H ) l L U L (S L ) U L (S L ) D L D H q Figure 1: Equilibria when < and D L <D H This expression is true because we have assumed that the absolute risk aversion of a financial risk is declining with health (S H <S L ). So the Spence- Mirrlees single crossing condition holds true. The equilibrium (if it exists) is separating and agents obtain insurance at fair price. When S H = SH 1 < S H every agent prefers his contract of full information (noticed L, H cf figure 1). Thus, information asymmetrics do not always cause inefficiencies in insurance and related markets. All agents receive full insurance : H and L contracts. In this case, we find for health insurance the Doherty-Jung results. When S H = SH 2 S H, H-risk type agent prefers L-contract than H- contract because it offers more consumption in each state. So the low-risk contract (noticed l cf figure 1) isdefined by the self-selction of H-risk type. Agent in good health receive less than full insurance. The equilibrium is a standard Rothschild-Stiglitz equilibrium. Information s revelation is obtained by deductible. The H-risk type agent receives full insurance while L-risk obtains less than full insurance.this is stated in the following proposition. Proposition 4 When < and D L <D H,theequilibrium: (a) if S H > S H, is separating and type RS ;

10 Health insurance when loss severities differ 9 (b) if S H S H, coincides with the full information equilibrium. Proof. Lemma 1a indicates that self-selection constraint is never bind for the L-risk type. Lemma 1b implies that self-selection constraint of H-risk type is binding for high values of health state (S H > S H ). Two cases have to be considered. a) if S H > S H, H is obtained by maximising U H (C H ) subject to Π(C H )= 0, and L by maximising U L (C L ) subject to U H (C H )=U H (C L ) and Π(C L )=0 (lemma 3). We obtain C H =( D H,D H ) and C L =( q L,q L ) with q L < D L. So, H risk-type agent receives full insurance while L-risk type obtains less than full insurance. As the single crossing condition holds, equilibrium is separating and there is no contract outside the equilibrium set that can make a positive profit (for the proof, see RS). b) if S H S H, all self-selection constraint are never bindind (lemma 1). i-risk contract is obtained by maximising U i (C i ) subject to Π(C i )=0(lemma 3). We obtain C i =(p i D i,d i ). All agents obtain full insurance at fair price. In these two cases, insurance firms (compagnies or mutual firms) are indifferent between offering L and H contract. They offer one of these contracts. 3.2 < and D H <D L In this case, the L type agent s incentive constraint is not binding. However, the H-risk type agent s constraint is binding. So the L-contract is defined by this constraint. In the standard explanation of screening in insurance markets, firms use deductibles to sort bad and good risk. In the present health s context firms problems are more complicated. Preferences of agents can be double-crossing. Premiums which earn excessive profit arise in equilibrium when preferences are not single-crossing (Pannequin, 1992, Villeneuve, 1996 and Smart, 2000). It may be unfeasible for insurers to reduce premiums without violating incentive constraints. A preliminary step in describing equilibrium is to characterize the relative position of the slope of the indifference curves agents. To this end, let C =(π,q) be a contract. We have to compare the slope of the indifference curves, ie (1 ) u 1(W π,s H ) at (1 ) u 1(W π,s L ). u 1 (W π+q D H,S H ) u 1 (W π+q D L,S L ) u If for S L = S 1(W π,s H ) H < u 1(W π,s H ),thenitistruefor u 1(W π+q D H,S H ) u 1(W π+q D L,S H ) [u 1(R 2,S) u S L > S H because 1 (R 1,S)] < 0. The equilibrium is closed to standard S

11 Health insurance when loss severities differ 10 Rothschild-Stiglitz equilibrium. (1 p If for S L = S H ) u 1 (W π,s H ) H > (1 ) u 1 (W π,s H ),thenit u 1 (W π+q D H,S H ) u 1 (W π+q D L,S H ) does not always hold for S L >S H because the indifference curve of the H-risk type is declining with S H. Then three cases must be considered. 1) The equilibrium (if it exists) is separating with zero profit if the slope of the S H type indifference curve is more steep than S L for the L 1 -contract ( q L,q L )withq L <D H. The critical value of S H is defined when the slopes of the two type s agent are tangent with this contract. Set S H this value. It verifies the follow expression : (1 ) u 1 (W π,s H ) u 1 (W q L + q L D H,S H ) = (1 ) u 1 (W π,s L ) u 1 (W q L + q L D L,S L ) (17) Thus for S H S H the equilibrium is separating (cf figure 2a). 2) For S H <S H < S H, the indifferences curve are tangent for a deductible q 1 L with q L <q 1 L <D L. With this profitable L 2 -contract ( q 1 L + s, q1 L ) L- risk utility is higher than utility with the contrat ( q L,q L ) with s positive profit (cf figure 2b). Firms offer only profitable policy for L-risk and mutual firms policy with zero profit for H-risk. The equilibrium is separating and good-health agent pays an excessive premium for a deductible. When S H is incresing, the indifference s curve is less concave. The tangent point is shifted to the right. A critical value (S H ) is reached when 8 : (1 ) u 1 (W π, S H ) u 1 (W D L + D L D H, S H ) = (1 ) (18) 3) For S H S H,theslopeoftheH-risk indifference curve is higher than L slope for the L 3 -contract ( D L + s, D L ) (cf figure 2c). The tangent point is shifted to the right of D L. According to the lack of above-insurance the pooling contract is not reached. H-risk type can receive at most D H.Thus, every agent receive full insurance but good-health agent pays an excessive premium. Then Firms use excessive premium to sort bad and good health risk. 8 We assume that S H <S L.

12 Health insurance when loss severities differ 11 U H (S 1 H) U L (S L ) U H (S 2 H) U L (S L ) L 2 L 1 D H D L q D H D L q U L (S L ) U H (S 3 H) L 3 D H D L q Figure 2: Equilibria when < and D H <D L -Right:2a;Left:2b; Middle ; 2c The following proposition describes more precisely how the nature of equilibrium changes as health state increases. Proposition 5 When < and D H <D L,ifaseparatingequilibrium exists, then a bad-health agent obtains full insurance with fair price and (a) if S H S H, then good-health receives less than full insurance with fair price. (b) if S H <S H < S H, then good-health agent receives less than full insurance and pays an excessive premium. (c) if S L >S H S H, then good-health agent receives full insurance and pays an excessive premium. Proof. Lemma 1 indicates that the L type agent s incentive constraint is never binding and the H-risk type agent s constraint always binding. So H is

13 Health insurance when loss severities differ 12 obtained by maximising U H (C H ) subject to Π(C H )=0,andL by maximising U L (C L ) subject to U H (C H )=U H (C L ) and Π(C L ) 0 (lemma 3). We obtain C H =( D H,D H ). As the single crossing does not hold, Π(C L ) may be positive. Hence, the characterization of L contract depends on the relative position of the indifference curves. Three cases have to be considered. a) When S H = SH 1 with S1 H S H, the H-indifference curve is more concave. For L 1 -contract ( q L1,q L1 ) with q L1 <D L, the slope of the H- risk indifference curve is higher than L slope (cf. figure 2a). Equilibrium is separating with zero profit (Π(C L )=0). Insurance firms (compagnies or mutual firms) are indifferent between offering L and H contract. They offer one of those contracts. b) When S H = SH 2 with S H <SH 2 < S H,theH-indifference curve becomes less concave. For L 2 -contract (π H2,q L2 ), solution to lemma 3, with π H2 > q L2 and q L1 <q L2 <D L,theslopeoftheH-risk indifference curve is equal to the L-slope (cf. figure 2b). L type strictly prefers L 2 -contract, an incentive-compatible contract which lower deductible and higher premium than L 1 -contract. In this case, Π(C L ) > 0. Equilibrium is separating. Goodhealth agent receives less than full insurance and pays an excessive premium. Compagnies firms offer L-contract and mutual firm H-contract. c) When S H = SH 3 with S L >SH 3 S H,theslopeofH-indifference curve becomes less and less concave. For L 3 -contract (π L3,D L ),solutionto lemma 3, with π L3 > D L, the slope of the H-risk indifference curve is less than the L-slope (cf. figure 2c). L type strictly prefers L 3 -contract, an incentive-compatible contract which full coverage and higher premium than L 2 or L 1 -contract. In this case, Π(C L ) > 0. Equilibrium is separating. Every agent obtains full insurance but good-health agent and pays an excessive premium. Compagnies firms offer L-contract and mutual firm H-contract. To conlude, information s revelation is obtained by deductible and or excessive premium. The L-risk type agent receives less than full insurance and can pay an excessive premium. 3.3 < and D L <D H In this case, the H-risk type agent s constraint is never binding. However, the L-risk type agent s self-selection constraint can be binding. So the H- contract can be defined by this constraint. Can constrasted Rothschild- Stiglitz equilibrium appear?

14 Health insurance when loss severities differ 13 L H U H (S H ) U L (S L ) D L D H q Figure 3: First-best equilibrium In the present health s context firms problems are more complicated. First, equilibrium can be full-information. Second, rreferences of agents can be double-crossing. Excessive premiums can be arised in equilibrium. It may be unfeasible for insurers to reduce premiums without violating incentive constraints. Four equilibrium configurations are possible. When S H <S L <S L, any self-selection constraint is binding. Every agent prefers his contract of full information (noticed L, H cf figure 3). Thus, information asymmetrics do not always cause inefficiencies in insurance and related markets. All agents obtain full insurance. When S L >S L,theL-risk type agent s self-selection constraint is binding. A preliminary step in describing equilibrium is to characterize the relative position of the slope of the indifference curves agents. To this end, let C = (π,q) be a contract. We have to compare the slopes of the indifference curves, ie (1 ) u 1 (W π,s H ) at (1 ) u 1 (W π,s L ) u 1 (W π+q D H,S H ) u 1 (W π+q D L,S L.TheL-indifference curve ) depends on S L. It becomes less concave when S L increases. 1) We assume that the parameters (,, D H, D L and S H )arefixed like there exists a separating equilibrium exists. Thus, for H 1 -contract ( q H,q H ), (1 ) u 1 (W q H,S L ) u 1 (W q H + q H D L,S L ) < (1 ) u 1 (W q H,S H ) u 1 (W q H + q H D H,S H ) (19)

16 Health insurance when loss severities differ 15 U L (S 1 L ) U L (S L ) U H (S H ) U H (S H ) L L H 2 H 1 D L D H q D L D H q U H (S H ) U L (S L ) H 3 L D L D H q Figure 4: Equilibria when <P L and D L <D H -Left:4a;Right:4b; Middle : 4c. Proposition 6 When < and D L <D H,ifaseparatingequilibrium exists for S L, then a good-health agent obtains full insurance at fair price and (a) if SH < S L S L, then bad-health receives full insurance at fair price. (b) if S L <S L SL 1, then bad-health receives less than full insurance at fair price. (c) if SL 1 <S L < S L, then bad-health agent receives less than full insurance and pays an excessive premium. (d) if S L S L, then bad-health agent receives full insurance and pays an excessive premium. Proof. For (a) Lemma 1 indicates that self-selection constraint is never bind for the i-risk type. i-contract is obtained by maximising U i (C i ) subject to Π(C i )=0(lemma 3). We obtain C i =(p i D i,d i ) (cf. figure 3).

18 Health insurance when loss severities differ 17 agents are privately informed about their health and their expected losses, Rothschild-Stiglitz equilibrium can not be applied. The addition of severity losses difference and health change the nature of equilibrium in significant ways. First, our analysis shows the possibility that a first-best equilibrium could prevail. Second, premiums charged for insurance may exceed the actuarially fair rate that would be charged under full information. Third, when all agents obtain full insurance, excessive premiums can be necessary to screen for insured. Fourth, equilibruim can be constrated Rothschild-Stiglitz. Badhealth agent obtains less than full insurance while good-health agent receives full insurance.

19 Health insurance when loss severities differ 18 References Browne, M.J.(1992), Evidence of adverse selection in the individual health insurance, Journal of Risk and Insurance, vol 60, n 2, pp Cutler, D.M., Zeckhauser, R. (1997), Adverse selection in health insurance, NBER Working Paper, n Chiappori, P.A. (2000), Econometric models of insurance under asymmetric information, in G. Dionne, ed., Handbook of Insurance, North Holland. Doherty, N. and Jung, C. (1993), Adverse selection when loss severities differ : first-best and costly equilibria, The geneva Papers on Risk and Insurance, vol18, n 3, pp Marquis, M.S. (1992), Adverse selection with a multiple choice among health insurance : a simulation analysis, Journal of Health Economics, vol 11, pp Pannequin, F. (1992), Théoriedel assuranceetsécuritésociale,thèse de doctorat, Université de Paris. Rothschild, M.and Stiglitz, J. (1976), Equilibrium in the competitive insurance markets : an essay on the economics of imperfect information, Quaterly Journal of Economics, vol. 90, pp Selden, T.M. (1998), Risk adjustment for health insurance : theory and implications, Journal of risk and uncertainty, vol. 17, pp Smart, M. (2000), Competitive Insurance markets with two unobservables, International Economic Review, vol 41, n 1, pp Villeneuve, B. (1996), Essais en Economie de l Assurance, Thèse de doctorat, Ecole des Hautes Etudes en Science Sociale. Wolfe, J.R. and Goldeeris, J.H. (1991), Adverse selection, moral hazard and wealth effects in the medigap insurance market, Journal of Health Economics, vol10, n 4, pp

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