Adverse Selection

1. Prerequisites Almost essential Risk Risk-taking Adverse Selection MICROECONOMICS Principles and Analysis Frank Cowell May 2004

2. The adverse selection problem • A key aspect of hidden information • Information relates to personal characteristics. • Hidden information about actions is dealt with under “moral hazard.” • Focus on the heterogeneity of agents on one side of the market. • Elementary model of a single seller with multiple buyers Jump to “Moral Hazard”

3. Overview... Adverse selection Principles Background and outline Monopoly problem Insurance

4. Key concepts (1) • Contract: • An agreement to provide specified good or service… • …in exchange for specified payment • Type of contract will depend on information available. • Fee schedule: • Set-up involving a menu of contracts • One party draws up the menu. • Allows some selection by other party • Again the type of fee schedule will depend on information available • Types: • Assume that hidden information is well structured • General shape of (e.g.) agents’ preferences is common knowledge • But there is heterogeneity as to (e.g.) intensity of preference • Correspond to different types of agents

5. Key concepts (2) • Adverse selection: • individuals faced with a contract… • …can choose to accept or reject . • Multiple contracts aimed at different types? • Then some individuals may choose the “wrong” one. • Screening: • Knowing this, other side of market seeks to respond. • Draw up contracts so that the various groups self-select the “right” ones. • “Adverse selection” and “Screening” are effectively equivalent labels. • Based on concept of Bayesian Nash equilibrium • Follow through a simplified version of the game

6. Screening: extensive-form game • "Nature" chooses a person's type • Probabilities are common knowledge • Firm may offer a contract, not knowing the type 0 p 1-p • Consumer chooses whether to accept contract [LOW] [HIGH] f f [OFFER] [OFFER] [NO] [NO] a b [reject] [accept] [reject] [accept]

7. Outline of the approach • Begin with monopolist serving a market • heterogeneous customers • differ in terms of taste for the product • other differences (income?) are unimportant • Easy to see what is going on • Main points can be established from case of just two customer types • Lessons from this are easily transferred to other contexts. • Examine these later

8. Overview... Adverse selection Principles A fee schedule to maximise profits from consumers with known tastes Monopoly problem • Exploitation: full information • Effect of hidden information • "Second-best" solution Insurance

9. Model structure • Monopoly produces good 1 using good 2 as input. • Constant marginal cost • Zero fixed cost • Good 1 cannot be resold. • The monopoly sells to heterogeneous customers • The firm wants to set up a system of payment. • A fee schedule • Some customer information might be concealed. • Imagine this information in the form of a parameter • Knowledge of each customer's parameter value would help the firm exploit the customer. Some fee schedules

10. Alternative fee schedules • A straight unit price • Two-part tariff 2 • Multi-part tariff Fee Revenue 3 • 1. F(x1) = px1 1 • 2. F(x1) = F0 + px1 x1 • 3. F(x1) = F0 + p′ x1, x1 ≤ x1 x1 = F0 + p′x1+ p′′ [x1x1], x1 > x1

11. Single customer type • Suppose there is just one type of customer • Income is y. • Utility is given by • U(x1, x2) = x2 + y(x1) where y(0) = 0. • Zero income effect for good 1 • Welfare can be measured by consumer surplus • Reservation utility level is u = U(y, 0) = y • Firm maximises profits subject to reservation constraint. • Monopoly position means firm can appropriate the surplus. • Can do this by imposing two-part tariff: • Fixed charge F0 . • Price p = c.

12. A two-part tariff F(x1) p x1 F0

13. * x1 Exploitative contract • Income • Preferences x2 • Reservation utility • Budget set arising from exploitative contract • Fixed charge • Optimal consumption y  F0 • By not participating consumer can get utility levelu=U(0,y) • Reservation indifference curve is given byU(x1,x2) = u u x1

14. Heterogeneous consumers • Two groups: a-types and b-types • Groups differ in their incomes and in their tastes. • Each group is internally homogeneous • Introduce the single-crossing condition: • imposes regularity on indifference curves; • makes it possible to compare the groups • easy to introduce “tailor-made” fee schedules

15. h x2 h x1 Two sets of preferences • a-type indifference curves • b-type indifference curves • Ua(x1a,x2a) = x2a + ta y(x1a) h=a • Ub(x1b,x2b) = x2b + tb y(x1b) • Single-crossing condition is satisfied h=b

16. Full-information contracts • Assume that there is full information about both types of consumer. • The firm knows • The preferences of both Alf and Bill • The incomes of both Alf and Bill • Therefore • Uses this information to design two-part tariffs... • Tailor-made for each type of consumer. • Forces each down to the reservation utility levels ua and ub • Outcome can be illustrated as follows.

17. b b x2 x1 a x2 b F0 *b *a x1 x1 a F0 a x1 Exploitation of two groups • Income of Alf and Bill • Preferences of Alf and Bill • Budget sets arising from exploitative contracts yb Alf Bill • Fixed charges  • Optimal consumptions ya  ua ub

18. Overview... Adverse selection Principles Key issues of the adverse-selection problem Monopoly problem • Exploitation: full information • Effect of hidden information • "Second-best" solution Insurance

19. Concealed information • Now suppose personal information is private • Can’t observe a customer’s taste characteristic • Can’t implement a fee-schedule conditioned on taste • Possibility of severe loss of profit to the firm. • The reason is that some individuals may masquerade: • High-valuation consumers can claim the contract appropriate to low-valuation consumers • Imitate the behaviour of low-valuation consumers • Enjoy a surplus by “hiding” amongst the others • Will this lead to an inefficient outcome?

20. b b x2 x1 b F0 *a *b x1 x1 a F0 b F00 Bill’s incentive to masquerade • Bill’s income and preferences • Contract intended for b-type • Contract intended for a-type • Bill would be better off with an a-type contract yb  • Now try cutting the fixed charge on a b-contract • To masquerade as an a-type Bill mimics Alf’s consumption • Bill finds the new b-contract at least as good as an a-contract

21. Insight from the masquerade • A “pooling contract” is not optimal • By cutting F0 for b-types sufficiently: • the b-types are at least as well off as if they had masqueraded as a-types • the firm makes higher profits • there is allocative efficiency • But this new situation is not a solution: • shows that masquerading is suboptimal • illustrates how to introduce incentive-compatibility • but we have not examined profit maximisation. • Requires separate modelling

22. Overview... Adverse selection Principles First look at a design problem Monopoly problem • Exploitation: full information • Effect of hidden information • "Second-best" solution Insurance

23. More on masquerading • Now consider profit maximisation • Build in informational constraints • Will act as additional side constraint on the optimisation problem • Because of this usually called a “second-best” approach • Another way of seeing that a pooling contracts not optimal • Let us examine the elements of an approach.

24. Elements of the approach • Choose F(•) to maximise profits • Subject to • Participation constraint • Incentive-compatibility constraint • The participation constraint is just as before • Can opt out and not consume at all • Incentive compatibility encapsulates the information problem • Individuals know that tastes cannot be observed • So they can select a contract “meant for” someone else • They will do so if it results in a utility gain • Run through logic of solution...

25. Logic of the solution: (1) • In principle we have a profit-maximisation problem subject to four constraints: • Low valuation a-types’ participation constraint • High valuation b-types’ participation constraint • a-types' incentive compatibility: must get as much utility as they would from a b-type contract. • b-types' incentive compatibility: must get as much utility as they would from an a-type contract. • But a-types have no incentive to masquerade • They would lose from a b-type contract • So a-type incentive-compatibility constraint is redundant • Can show this formally • Also b-types cannot be forced on to reservation utility level ub. • We’ve seen what happens: they would grab an a-type contract • So b-types’ participation constraint is redundant • So the problem can be simplified….

26. Logic of the solution: (2) • In practice we have a profit-maximisation problem subject to two constraints: • a-types’ participation constraint • b-types’ incentive compatibility: must get as much utility as they would from an a-type contract. • a-types can be kept on reservation utility level ua • There is an infinity of fee schedules that will do this. • b-types must be prevented from masquerading. • Do this by distorting upwards the unit price for the a-types • Force the b-types down to the indifference curve they could attain with an a-type contract. • Maximise profit from them by charging price=MC. • Check this in a diagram...

27. b b x2 x1 a x2 a a b x1 x1 x1 a x1 Second-best contracts • Income and preferences • Budget sets for full-information case • The a-type contract • b's utility with an a-type contract yb Alf Bill  • The b-type contract ya  ua

28. b a x1 x1 h x2 h x1 Second-best contracts (2) • Income and preferences for the two types • The a-type contract • b's utility with an a-type contract • The b-type contract yb  • The attainable set constructed by the firm • Implementing the contract with a multipart tariff. ya  • An a-type is forced down on to the reservation indifference curve. • A b-type gets the utility possible by masquerading as an a-type. ua • Multipart tariff has kink at x1. x1

29. a b x1 x1 x1 Second-best contracts (3) • Multipart tariff: firm’s viewpoint • An a-type choice • A b-type choice F(x1) • The cost function C(x1) • Profit on each a-type • Profit on each b-type F(•) C(•) Pb Pa x1

30. Second best: principles • b-types have to be made as well off as they could get by masquerading • So they have to keep some surplus • Full surplus can be extracted from a-types • High valuation b-type contract involves price = MC • "No Distortion at the Top" • Low-valuation a-types face higher price, lower fixed charge than under full information • they consume less than under full information • This acts to dissuade b-types.

31. Overview... Adverse selection Principles Heterogeneous risk types in an insurance market Monopoly problem Insurance

32. Adverse selection: competition • So far we have assumed an extreme form of market organisation • Power in the hands of a monopolist • Can draw up menu of contracts • Limited only by possibility of non-participation or masquerading • Suppose the monopoly can be broken • Free entry into the market • Numbers determined by zero-profit condition • What type of equilibrium will emerge? • Will there be an equilibrium? An important case

33. The insurance problem Jump to “Risk taking” • Apply the standard model of risk-taking. • The individual enjoys a random endowment • Has given wealth y • But faces a potential loss L. • Consider this as a prospectP0 with payoffs (y, y- L) • Individual’s preferences satisfy von-Neumann-Morgenstern axioms • Can use concept of expected utility • If p is probability of loss… • …lope of indifference curve where it crosses the 45º line is –[1–p]/p. • Competitive market means actuarially fair insurance • Slope of budget line given by –[1–p]/p... Graphical representation

34. _ _ ya ya A single risk type xBLUE • a-type indifference map • income and possible loss • actuarially expected income • actuarially fair insurance with premiumk. • a-type attainable set • • Slope is same on 45 line. • Gets “flatter” as p increases L-k 1- pa -  pa. A • Endowment point P0 has coordinates (y, y –L) P0 • y – L • Full insurance guarantees expected income k xRED 0 y

35. The insurance problem: types • An information problem can arise if there is heterogeneity of the insured persons. • Assume that heterogeneity concerns probability of loss • a-types: low risk, low demand for insurance • b-types: high risk, high demand for insurance • Types associated with risk rather than pure preference • Each individual is endowed with the prospect P0 . • Begin with full-information case

36. Efficient risk allocation xBLUE • Endowment point • a-type indifference curves • b-type indifference curves • Attainable set and equilibrium, a-types • Attainable set and equilibrium, b-types • P*a • pb > pa • • A b-type would prefer to get an a-type contract if it were possible P*b 1- pa -  pa. 1- pb -  pb. P0 • y – L xRED 0 y

37. Possibility of adverse selection • Indifference curves xBLUE • Endowment • a-type (low-risk) insurance contract • • b-type (high-risk) insurance contract • If Bill insures fully with an a-type contract • • If over-insurance were possible... • • (y, y- L) xRED 0

38. Pooling • Suppose the firm “pools” all customers • Same price offered for insurance to all. • Assume that a-types and b-types are in the proportions (g,1-g) • Pooled probability of loss is therefore • `p := gpa + [1-g] pb • pa <`p< pb • Can this be an equilibrium?

39. Pooling equilibrium? • Endowment &indiff curves xBLUE • Pure a-type, b-type contracts • Pooling contract, lowg. • Pooling contract, highg. • Pooling contract, intermediateg. • a-type’s choice with pooling • b-type’s unrestricted choice • b-type mimics an a-type • • A profitable contract preferred by a-types but not by b-types `P • • • A proposed pooling contract is always dominated by a separating contract • P0 xRED 0

40. Separating equilibrium? • Endowment &indiff curves xBLUE • Pure a-type, b-type contracts • b-type would like a pure a-type contract • Restrict a-types in their coverage • Then b-types take efficient contract • a-type’s and b-type’s preferred prospects to (Pa ,P*b) • ~ ^ • A pooled contract preferred by both a-types and b-type P • • P*b ~ Pa • Proposed separating contract might be dominated by a pooling contract. • This could happen if g were large enough • • P0 xRED 0

41. Insurance model: assessment • Why is the insurance case specially difficult? • Note the special role of parameters pa, pb. • As "type indicators" – shift the indifference curves. • As weights in the evaluation of profits. • The population composition affects profitability • Directly: expected profit on each contract written • Indirectly: through the masquerading process • As before, no pooling equilibrium • But may not be a separating equilibrium

42. What next? • Consider other fundamental models of information • Signalling models • also hidden personal characteristics... • ...but where the informed party moves first. • Moral hazard • hidden information about individual actions