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Topics in Health and Education Economics – class2

Topics in Health and Education Economics – class2. Matilde P. Machado matilde.machado@uc3m.es. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976). Summary: Shows the impact of imperfect information on the equilibrium outcome of a competitive insurance market.

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Topics in Health and Education Economics – class2

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  1. Topics in Health and Education Economics – class2 Matilde P. Machado matilde.machado@uc3m.es

  2. 2.2. Adverse Selection/Risk SelectionRothschild & Stiglitz (QJE,1976) Summary: • Shows the impact of imperfect information on the equilibrium outcome of a competitive insurance market. • Insurance companies offer insurance contracts that rely on a self-selection mechanism • High risk individuals cause an externality on low risk individuals • Everyone would be better off (or as well off) if risks were revealed ex-ante.

  3. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) Model I – the case of a single type of consumer: • There are two states of nature {no accident, accident}. • No insurance: wealth = {W,W-d} • p ≡ probability of accident occurring • a=(a1,a2) is the insurance vector where a1 is the premium paid by the consumer and a2 is the net compensation in case of accident, i.e. a2 = q- a1 where q is the insurance coverage. • With insurance: wealth = {W- a1,W-d+ a2}

  4. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) 1. Supply side of the insurance market: • Insurers are risk-neutral, they maximize expected profits • Perfect competition  Expected profit = 0 Expected profit of selling contract a to individuals with probability of accident = p Competitive relation between the net compensation and the premium

  5. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) It is well-known that at actuarial fair premiums risk averse individuals want to hire full insurance.

  6. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) Full insurance

  7. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) W2 Wealth after accident W1=W2 Full insurance combinations W2>W1 relevant W1>W2 45º W1 Wealth with no accident

  8. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) • All wealth combinations along EF are feasible (i.e. Expected profit=0). F is the feasible point of Complete insurance W2=W1 (Completely insured) W2 (with accident) Possible wealth combinations with competitive insurance, slope = - (1-p)/p W2>W1 F Wealth without insurance W1>W2 E W-d 45º W1 No accident W

  9. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) Derivation of EF line:

  10. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) Definition of Equilibrium: The equilibrium in a competitive insurance markets is a set of contracts such that: • No contract in the equilibrium set makes negative profits • There is no contract outside the set that, if offered, would make no-negative profits. We know that when the premium is actuarial fair, risk averse prefer F to the rest of the possible wealth combinations, i.e. complete insurance.

  11. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) 2. Demand Side of the Insurance Market • Individuals maximize their expected utility that depends only on their wealth • Individuals know p • Their utility function is state independent • Their expected utility function is:

  12. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) The slope of the indifference curve at F equals the slope of EF line. F is therefore the optimum point (highest i.c.) in EF. The slope of the indifference curve at W1=W2 is independent of U(.) and the same as the EF line. The tangency of the indifference curve to EF shows that individuals maximize their expected utility at F.

  13. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) The equilibrium: W2=W1 (total insurance) W2 F Optimal net compensation a*2 E W-d 45º W W1 Optimal premium=a*1

  14. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) For a*=(a*1, a*2) to be an equilibrium we need to check that it satisfies the two previous conditions which amount to: • BE=0 (The wealth combination must be along the EF line) • Any other insurance contract (a1,a2) that consumers may prefer has negative benefits <0. Those would be contracts that lead to wealth combinations in higher indifference curves and obviously those are lower premium for the same (or higher) compensation or higher compensations for the same (or lower) premium which yield negative expected profits.

  15. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) Model II – the case of two types of consumers: • Low Risk – probability of accident = pL • High Risk – probability of accident = pH • pH > pL • l ≡ proportion of high risks 0<l<1 • Individuals know their types and their probability of accident • Individuals only differ in their risk, the insurance company cannot distinguish them ex-ante, however the insurer knows the values of pH , pL and l.

  16. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) • The insurance company knows that for the same premium, high risks would like to hire more coverage. It will use this information to devise self-selection mechanisms. • Individuals cannot buy more than one insurance contract • There can only be two types of equilibrium: • Pooling – both types buy the same contract • Separating – Each type buys a different contract It can be shown that a Pooling Equilibrium never exists.

  17. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) Proof that a Pooling Equilibrium never exists. By contradiction, suppose a=(a1, a2) is a pooling equilibrium, in that case, it can be shown that Exp. Profits are a function of the average prob. of accident:

  18. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) The rate of substitution between the two states of nature can be derived for the high risks: And similarly for the low risks:

  19. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) So the difference of the rate of substitution for high and risks only depends on the probabilities: The indifference curve of the Low risks (for the equilibrium contract a) is steeper in absolute terms.

  20. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) Note that a must be in the EF curve so that the expected profit is =0 W2=W1 W2 a F Note that EF has now a slope that depends on the average prob. of accident UH W-d UL 45º W W1

  21. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) The existence of a contract b shows that a is not an eq. W2=W1 W2 a F bis preferred to a by the low risk and since it is in EF or even slightly above can be offered in themarket (because only low risks would buy it). UH W-d UL 45º W W1

  22. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) If an equilibrium exists it MUST be separating. They assume that there is no cross-subsidization, i.e. competition forces the insurance company to break even for every contract. The zero expected profit conditions are now given by: Which imply two different wealth combination lines from the initial E point.

  23. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) For the same premium the insurance company could offer a much higher net compensation to the low risks. W2=W1 W2 L Slope EL: H E 45º W1 Premium =a Measures competitive net compensation for high and low risks for a premium = a Slope EH:

  24. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) However H and L do not constitute an equilibrium Of all wealth combinations along EH, aH is the preferred one by the high risks. And of all those along EL, q is the preferred by the low risks. We know that Ep=0 if aH is sold to the high risks and q to the low risks. The problem is that q is also preferred to aH by the high risks since it means higher wealth in both states of nature. U’H W2 L q UH H aH E 45º

  25. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) And the insurer will have negative expected profits if it sells q to everyone. That is if all individuals buy the contractq then: W2 L q UH U’H aH E 45º

  26. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) The segment between EaL shows the set of wealth combinations that could be offered to the low risks at zero expected profit and are NOT preferred to aH by the high risks (incentive compatibility constraint) The set of candidates to an equilibrium is aH and contracts along EaL . Of all those along EaL , aL is the preferred by the low risk. So we will check under which conditions {aH ,aL} is an equilibrium. UL W2 L q UH aL aH E 45º

  27. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) To prove that {aH, aL} is indeed an equilibrium: The first condition is satisfied because the insurer has expected zero profit in both contracts. The second condition is the difficult one. The existence of equilibrium depends on the percentage of high risks, l. It turns out that if l is high enough there is an equilibrium, otherwise there isn’t.

  28. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) Suppose g is also offered. Then all individuals would prefer g to the eq. candidate. The question is can g be offered in the market? For {aH,aL} to be an equilibrium, the exp. profit from g should be <0. W2 L q UH g aL H aH E UL 45º

  29. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) For a low level of l, the line of possible wealth combinations achieved with contracts that are sold to both types and breakeven is near EL for example EF2. If l is high then the line is somewhere close to EH e.g. EF1. W2 F2 q UH g F1 aH E UL 45º For the high value of l (EF1) g cannot be offered at a profit so {aH,aL} is an equilibrium. For a low of l (EF2) there is no equilibrium in this competitive market.

  30. 2.2. Adverse Selection/Risk Selection Rothschild & Stiglitz (QJE,1976) Conclusions: • the incomplete information may cause a competitive market to have no equilibrium • The high risks are a negative externality on the low risks • Everyone would be at least as well off if everyone revealed their type

  31. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) This paper is nice because: • It is an application of R&S • It uses data!! (The British Household Panel Survey) • It tests adverse selection in a market where there are private health insurers on top of a National Health Service (NHS). • They find evidence of adverse selection in the private insurance market

  32. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) Setting: The British Private Health Insurance Market. • What’s especial? • There is a public system that covers all expenditures (no copayments with few exceptions e.g. dental) and everyone. • If people want to they can hire a private insurance. Private and Public are then substitutes for care. Everyone contributes to the public system through taxes. • Other systems such as most of the American is purely private (except for Medicare/Medicaid) • In other systems, the private is supplementary, people hire the private to cover for copayments and services not covered by the public (Belgium, France).

  33. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) • Model: • Individuals decide whether to take private insurance after observing their type and before becoming ill. • Individuals when become ill must choose where they want to be treated (private/public) and private and public services cannot be combined. • If an individual chooses the private service, the private insurer must cover the full cost. • Everyone contributes to the financing of the public health service regardless of whether he/she uses the public health services (PUB). • There are a large set of insurers (similar to R&S) • Individuals can be of one of two types {L,H}, 1>PH>PL>0 and they know their type • g≡ (1-l) proportion of low risks, 0< g<1

  34. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) • Model (timing): Individuals decide whether or not to take private insurance, and which insurer, conditional on the packages of all providers and their type Health authority chooses package Private insurers simultaneously make their packages conditional on the package of the HA

  35. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) • If an individual buys private insurance, he has double coverage and in case he gets sick must decide if he wants the public (PUB) or private treatment (PRI) • L0 ≡ loss if an individual gets sick and does not seek treatment • (Lpri,q) ≡ private contract (q ≡ premium, Lpri ≡ Loss i.e. L0-Lpri ≡coverage) • (Lpub,0) ≡ is the outside option of the individual offered by the public system (premium is paid through taxes). Assumption Lpub<L0 the public is effective.

  36. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) • W ≡ individual’s initial wealth net of taxes • If the individual does NOT take private insurance: • And he does not become ill: W • And he becomes ill : W-Lpub • If the individual TAKES private insurance: • And he does not become ill: W-q • And he becomes ill : W-q-Lpri Note: Private contracts where Lpri>Lpub are irrelevant because they are strictly dominated by the public package (the assumption implies that if ill the individual goes to the private system). The insurer commits to ensure the individual does not suffer a loss larger than Lpri

  37. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) • If the individual does NOT take private insurance: • Expected utility for probability p is: • If the individual TAKES private insurance: • Expected utility for probability p is: • The expected profit of a contract (L,q) is: Is the average probability of getting sick coverage

  38. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) • In the case of no-illness for a contract (L,q) and wealth w • In the case of illness : • The expected profit of a contract (L,q) is:

  39. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) • We have 2 zero-isoprofit curves depending on the individual’s type, with slopes: • The zero-isoprofit curves go through point E = (w,w-L0) (i.e. no insurance)

  40. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) 3.1. Symmetric Information • If there is NO public system, then the equilibrium would be just as in R&S with symmetric information, i.e. efficient contracts or complete coverage to both types i.e. {a*L,a*H} i.e. for J=H,L

  41. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) • In the presence of a public system the status-quo is not E=(w,w-L0) but P=(w,w-Lpub) W2=a Possible positions of the public package. If Lpub=L0 then E is the status quo with public coverage (i.e. it is as if the public system did not existed) L a*L W-pLL0 UH a*H W-pHL0 W-L0 E 45º W-pHL0 W-pLL0 W W1=n

  42. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) • Assumption 1: If all individuals of type J are indifferent between the public package P and the best private contract for them, all these individuals choose the public package.

  43. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) In the presence of P=(w,w-Lpub) some private contracts (in the symmetric information eq.) may not be attractive any more a=n a P3 L a*L W-pLL0 UH a*H W-pHL0 P2 W-L0 E P1 45º n W-pHL0 W-pLL0 If P=P1 then private market not affected, the public contract is NOT ACTIVE; If P=P2 a*H not attractive anymore; If P=P3 no private contract is attractive, the private market is NOT ACTIVE

  44. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) • The equilibrium will depend on P W2 L a*L W-pLL0 UH L0 a*H W-pHL0 H0 W-L0 E 45º W-pHL0 W-pLL0 Proposition 1: In case P is between E-H0 the equilibrium is {a*H,a*L}; if P lies between H0 and L0 the equilibrium is {P,a*L}, in case P lies strictly above L0 then only P exists

  45. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) So if both sectors are active (i.e. P is between H0 and L0) and there is no adverse selection the probability of illness among the ones with a private insurance ispL i.e. the low risk, lower than the average probability of illness.

  46. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) 3.2. Asymmetric Information • If there is NO public system, then the equilibrium would be just as in R&S with asymmetric information, i.e. efficient contracts or complete coverage to the high risks and less than full coverage to the low risks • We know that in R&S we need the proportion of high risks to be high enough so that there is equilibrium i.e. g≤g* • Assumption 2: assume g≤g*

  47. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) • The equilibrium will depend on P: W2 L W-pLL0 UH a*H H0 W-pHL0 W-L0 E L1 45º W-pHL0 W-pLL0 Again H0 is the public contract such that a high risk is indifferent between the public contract and the private contract a*H. L1 is the public contract such that the low-risk is indifferent between P and . Note that H0 is above L1 (Lemma 2)

  48. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) Case 1: P lies below L1  eq is P is not active Case 2-3: P coincides with L1 or is between L1 and H0 the equilibrium is {a*H,P} (assumption 2 is no longer necessary for the existence of the equilibrium.) Both Public and Private are ACTIVE. Case 4-5: P coincides with H0 or is above H0, in the equilibrium only the public system exists. If both sectors are active under adverse selection then the probability of illness of those who purchase private insurance is pH which is larger than the average probability of illness.

  49. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) • Their Main Theoretical Result: • When both markets are active i.e. the private and the public then, with perfect information (i.e. no adverse selection) only the low risk would buy private insurance • When both markets are active i.e. the private and the public then, with asymmetric information and therefore adverse selection only the high risks would buy private insurance. • i.e. the sign of the correlation between the probability of buying private insurance and individuals’ risk depends on whether there is adverse selection. This prediction can be tested.

  50. 2.2. Adverse Selection/Risk Selection Olivella, Vera-Hernández (WP06/02) Empirical Test: • In the UK the private system is substitute to the public one. • Everyone is entitled to the public system, and people pay it through taxes regardless of utilization • The private system offers better access, in particular negligible waiting times, in the model’s notation it is true that Lpub>Lpri • Data: Bristish Household Panel Survey (BHPS)

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