Screening and Adverse Selection in Frictional Markets

Transcription

1 Screening and Adverse Selection in Frictional Markets Benjamin Lester Philadelphia Fed Venky Venkateswaran NYU Stern Ali Shourideh Carnegie Mellon University Ariel Zetlin-Jones Carnegie Mellon University Spring 2017 Disclaimer: The views expressed here do not necessarily reflect the views of the Federal Reserve Bank of Philadelphia. 1 / 51

2 Introduction Many important markets feature adverse selection (AS) and screening Examples: insurance, loans, some financial securities 2 / 51

3 Introduction Many important markets feature adverse selection (AS) and screening Examples: insurance, loans, some financial securities These markets are often also imperfectly competitive References 2 / 51

4 Introduction Many important markets feature adverse selection (AS) and screening Examples: insurance, loans, some financial securities These markets are often also imperfectly competitive References A model of AS, screening contracts & imperfect competition is lacking Quotes Large empirical literature Theory: restricted contracts or extremes (perfect/zero competition) 2 / 51

5 Introduction Many important markets feature adverse selection (AS) and screening Examples: insurance, loans, some financial securities These markets are often also imperfectly competitive References A model of AS, screening contracts & imperfect competition is lacking Quotes Large empirical literature Theory: restricted contracts or extremes (perfect/zero competition) Essential to answer some important questions, including... how does market structure affect contracts terms? estimates of AS? 2 / 51

6 Introduction Many important markets feature adverse selection (AS) and screening Examples: insurance, loans, some financial securities These markets are often also imperfectly competitive References A model of AS, screening contracts & imperfect competition is lacking Quotes Large empirical literature Theory: restricted contracts or extremes (perfect/zero competition) Essential to answer some important questions, including... how does market structure affect contracts terms? estimates of AS? will recent attempts to competition & transparency trade? welfare? 2 / 51

7 This Paper A tractable model of adverse selection, screening and imperfect competition 3 / 51

8 Key Ingredients A model of trade in assets with 4 / 51

9 Key Ingredients A model of trade in assets with Adverse Selection: sellers have private info about asset quality. 4 / 51

10 Key Ingredients A model of trade in assets with Adverse Selection: sellers have private info about asset quality. Screening: Uninformed buyers offer general menus of contracts. 4 / 51

11 Key Ingredients A model of trade in assets with Adverse Selection: sellers have private info about asset quality. Screening: Uninformed buyers offer general menus of contracts. Imperfect Comp: sellers either receive 1 or 2 offers (Burdett-Judd). 4 / 51

12 Preview of Results New techniques complete characterization of unique eqm No issues with existence, off-path beliefs. 5 / 51

13 Preview of Results New techniques complete characterization of unique eqm No issues with existence, off-path beliefs. Predictions for distribution of contracts offered/traded Equilibrium can be pooling, separating, or a mixture of both Separation when adverse selection (AS) severe, competition high Pooling when AS mild, competition low Identifying AS requires knowledge of market structure 5 / 51

14 Preview of Results New techniques complete characterization of unique eqm No issues with existence, off-path beliefs. Predictions for distribution of contracts offered/traded Equilibrium can be pooling, separating, or a mixture of both Separation when adverse selection (AS) severe, competition high Pooling when AS mild, competition low Identifying AS requires knowledge of market structure Effects of more competition & better info on trade volume, welfare AS severe: welfare -shaped with competition. Otherwise: decreasing. Low comp: welfare -shaped as transparency. Otherwise: decreasing. Competition interacts with IC constraints in non-monotonic fashion. competition/transparency desirable only when AS severe, competition low 5 / 51

15 Related Literature Empirical Chiappori-Salanie ( 00), Ivashina ( 09), Einav et al. ( 10, 12),... Adverse Selection and Screening Rothschild-Stiglitz ( 76), Dasgupta-Maskin ( 86), Rosenthal-Weiss ( 84), Bisin-Gottardi ( 06),... Mirrlees ( 71), Stiglitz ( 77),... Guerrieri-Shimer-Wright ( 10), Guerrieri-Shimer ( 14), Chang ( 14)... Imperfect Competition and Selection Burdett-Judd: Garrett, Gomes, and Maestri ( 14) Hotelling: V-B & S-M ( 99), Benabou-Tirole ( 14), Townsend-Zhorin ( 15), Weyl & co-authors... 6 / 51

16 Environment 7 / 51

17 Environment 2 buyers, large number of sellers Each seller has 1 unit of divisible good Good is of quality i {l, h} with probability µ i Seller: receives utility c i per unit of consumption. Buyer: receives utility v i per unit of consumption. 8 / 51

18 Environment 2 buyers, large number of sellers Each seller has 1 unit of divisible good Good is of quality i {l, h} with probability µ i Seller: receives utility c i per unit of consumption. Buyer: receives utility v i per unit of consumption. If seller gives up x units in exchange for a transfer t, payoffs are Seller: t + (1 x)c i Buyer: xv i t 8 / 51

19 Environment 2 buyers, large number of sellers Each seller has 1 unit of divisible good Good is of quality i {l, h} with probability µ i Seller: receives utility c i per unit of consumption. Buyer: receives utility v i per unit of consumption. If seller gives up x units in exchange for a transfer t, payoffs are Seller: t + (1 x)c i Buyer: xv i t Assumptions Gains from trade for both types: v h > c h and v l > c l Lemons assumption: v l < c h Adverse Selection: Only sellers know asset quality 8 / 51

20 Environment Screening Buyers post arbitrary menus of exclusive contracts General mechanisms + communication identical outcomes Proof 9 / 51

21 Environment Screening Buyers post arbitrary menus of exclusive contracts General mechanisms + communication identical outcomes Proof Imperfect competition Each seller receives 1 offer w/ prob 1 p & 2 offers w/ prob p From buyer s perspective, conditional on a match, Pr(seller has another offer): π = 2p 1 p+2p Can vary degree of competition with a single parameter, nesting extremes: p = π = 0: monopsony. p = π = 1: Bertrand/perfect competition. 9 / 51

22 Environment Screening Buyers post arbitrary menus of exclusive contracts General mechanisms + communication identical outcomes Proof Imperfect competition Each seller receives 1 offer w/ prob 1 p & 2 offers w/ prob p From buyer s perspective, conditional on a match, Pr(seller has another offer): π = 2p 1 p+2p Can vary degree of competition with a single parameter, nesting extremes: p = π = 0: monopsony. p = π = 1: Bertrand/perfect competition. Note that market is always fully covered under this formulation isolate effect of competition. (later: general setting where coverage also varies) 9 / 51

23 Applications Market for financial securities Buyers make offers to sellers (or issuers): price and quantity Sellers have private information about value Loan markets Lenders make offers to borrowers: loan size and interest rate Borrowers have private information about default risk Insurance markets Insurers make offers to potential customers: coverage and premium (Risk-averse) customers have private info about health/accident/death risk 10 / 51

24 Strategies buyer: offers menu of contracts sufficient to consider two contracts z {(x l, t l ), (x h, t h )} (IC i ) : t i + c i (1 x i ) t i + c i (1 x i ) i {l, h} 11 / 51

25 Strategies buyer: offers menu of contracts sufficient to consider two contracts z {(x l, t l ), (x h, t h )} (IC i ) : t i + c i (1 x i ) t i + c i (1 x i ) i {l, h} seller: chooses a contract from available menus 1 offer (captive seller): chooses (x i, t i ) by incentive compatibility 2 offers (non-captive seller): chooses (x i, t i ) or (x i, t i ) by χ i (z, z ) = < if t i + c i (1 x i ) = > t i + c i (1 x i ). 11 / 51

26 Equilibrium definition A symmetric equilibrium is a distribution Φ(z) such that almost all z satisfy, 1 Incentive compatibility: t i + c i (1 x i ) t i + c i (1 x i ) i {h, l} 2 Seller optimality: χ i (z, z ) maximizes her utility 3 Buyers optimality: z arg max z i {l,h} ] µ i [1 π + π χ i (z, z )Φ(dz ) (v i x i t i ) (1) z 12 / 51

27 Only Mixed Strategy Equilibrium for π (0, 1) Why? Suppose a pure strategy equilibrium exists. 1 Buyers make strictly positive profits from some type 2 Buyers compete for this type with probability π > 0 Therefore, Incentives to undercut Equilibrium necessarily features dispersion in menus 13 / 51

28 Characterization Strategy Equilibrium described by non-degenerate distribution in 4 dimensions Proceed in 4 steps 14 / 51

29 Characterization Strategy Equilibrium described by non-degenerate distribution in 4 dimensions Proceed in 4 steps 1. Show that menus can be summarized by a pair of utilities (u h, u l ) Reduces problem to distributions over 2 dimensions 14 / 51

30 Characterization Strategy Equilibrium described by non-degenerate distribution in 4 dimensions Proceed in 4 steps 1. Show that menus can be summarized by a pair of utilities (u h, u l ) Reduces problem to distributions over 2 dimensions 2. Show there is a strictly increasing map between u l and u h Menus rank-ordered (Strict Rank Preserving) Reduces problem to distribution in 1 dimension + a monotonic function 14 / 51

31 Characterization Strategy Equilibrium described by non-degenerate distribution in 4 dimensions Proceed in 4 steps 1. Show that menus can be summarized by a pair of utilities (u h, u l ) Reduces problem to distributions over 2 dimensions 2. Show there is a strictly increasing map between u l and u h Menus rank-ordered (Strict Rank Preserving) Reduces problem to distribution in 1 dimension + a monotonic function 3. Construct SRP equilibrium 14 / 51

32 Characterization Strategy Equilibrium described by non-degenerate distribution in 4 dimensions Proceed in 4 steps 1. Show that menus can be summarized by a pair of utilities (u h, u l ) Reduces problem to distributions over 2 dimensions 2. Show there is a strictly increasing map between u l and u h Menus rank-ordered (Strict Rank Preserving) Reduces problem to distribution in 1 dimension + a monotonic function 3. Construct SRP equilibrium 4. Show that constructed equilibrium is unique 14 / 51

33 A utility representation Result In all menus offered in equilibrium, the low types trades everything: x l = 1 IC l binds: t l = t h + c l (1 x h ) Reduces dimensionality of problem to distribution in 2 dimensions 15 / 51

34 A utility representation Result In all menus offered in equilibrium, the low types trades everything: x l = 1 IC l binds: t l = t h + c l (1 x h ) Reduces dimensionality of problem to distribution in 2 dimensions Result Equilibrium menus can be represented by (u h, u l ) with corresponding allocations t l = u l x h = 1 u h u l c h c l t h = u lc h u h c l c h c l 15 / 51

35 A utility representation Result In all menus offered in equilibrium, the low types trades everything: x l = 1 IC l binds: t l = t h + c l (1 x h ) Reduces dimensionality of problem to distribution in 2 dimensions Result Equilibrium menus can be represented by (u h, u l ) with corresponding allocations t l = u l x h = 1 u h u l c h c l t h = u lc h u h c l c h c l Since we must have 0 x h 1, c h c l u h u l 0 15 / 51

36 A utility representation We define the marginal distributions: F i (u i ) = z 1 [ t i + c i ( 1 x i ) ui ] dφ ( z ) i {h, l} 16 / 51

37 A utility representation We define the marginal distributions: F i (u i ) = z 1 [ t i + c i ( 1 x i ) ui ] dφ ( z ) i {h, l} Then, each buyer solves max u l c l, u h c h Π(u h, u l ) = max u l c l, u h c h i {l,h} s. t. c h c l u h u l 0 with Π l (u h, u l ) v l x l t l = v l u l µ i [1 π + πf i (u i )] Π i (u h, u l ) Π h (u h, u l ) v h x h t h = v h u h v h c l c h c l + u l v h c h c h c l 16 / 51

38 A utility representation We define the marginal distributions: F i (u i ) = z 1 [ t i + c i ( 1 x i ) ui ] dφ ( z ) i {h, l} Then, each buyer solves max u l c l, u h c h Π(u h, u l ) = max u l c l, u h c h i {l,h} s. t. c h c l u h u l 0 with Π l (u h, u l ) v l x l t l = v l u l µ i [1 π + πf i (u i )] Π i (u h, u l ) Π h (u h, u l ) v h x h t h = v h u h v h c l c h c l + u l v h c h c h c l Need to characterize the two interlinked distributions F l and F h. 16 / 51

39 Properties of Equilibrium Result F l and F h have connected support and are continuous. Except for a knife-edge case (see paper) Proof more involved than standard case because of interdependencies 17 / 51

40 Properties of Equilibrium Result F l and F h have connected support and are continuous. Except for a knife-edge case (see paper) Proof more involved than standard case because of interdependencies Result The profit function Π(u h, u l ) is strictly supermodular. Intuition: u l Π h stronger incentives to attract high types U h (u l ) argmax uh Π(u h, u l ) is weakly increasing 17 / 51

41 Properties of Equilibrium Result F l and F h have connected support and are continuous. Except for a knife-edge case (see paper) Proof more involved than standard case because of interdependencies Result The profit function Π(u h, u l ) is strictly supermodular. Intuition: u l Π h stronger incentives to attract high types U h (u l ) argmax uh Π(u h, u l ) is weakly increasing Theorem (Strict Rank Preserving) U h (u l ) is a strictly increasing function. Weakly increasing because of super-modularity Strictly increasing, not a correspondence because F l, F h well-behaved intuition 17 / 51

42 Strict Rank Preserving Equilibria Useful for characterization: Ranking of equilibrium menus identical across types Menus attract same fraction of both types F l (u l ) = F h (U h (u l )) Greatly simplifies our task: only have to find F l (u l ) and U h (u l ) 18 / 51

43 Strict Rank Preserving Equilibria Useful for characterization: Ranking of equilibrium menus identical across types Menus attract same fraction of both types F l (u l ) = F h (U h (u l )) Greatly simplifies our task: only have to find F l (u l ) and U h (u l ) Implications for outcomes: Terms of trade positively correlated across types Buyers don t specialize, trade with equal frequency across types 18 / 51

44 Constructing Equilibria 19 / 51

45 What We Already Know 1 µ h µ h µ h 0 0 (Monopsony) 1 π (Perfect Comp) 20 / 51

46 What We Already Know 1 µ h µ h µ h 0 0 (Monopsony) 1 π (Perfect Comp) Least Cost Separating Outcome (Rothschild- Stiglitz 76) Perfect comp and severe adverse selection Pure strategy separating eq. 20 / 51

47 What We Already Know 1 µ h Only Mixed Strategy Equilibria (Rosenthal- Weiss 84) µ h µ h 0 0 (Monopsony) 1 π (Perfect Comp) Perfect comp and mild adverse selection Mixed Strategy Eq. Least Cost Separating Outcome (Rothschild- Stiglitz 76) 20 / 51

48 What We Already Know 1 µ h Only Mixed Strategy Equilibria (Rosenthal- Weiss 84) µ h µ h No Trade With High Quality Seller (Stiglitz 77) 0 0 (Monopsony) 1 π (Perfect Comp) Least Cost Separating Outcome (Rothschild- Stiglitz 76) Monopsony and severe adverse selection No Trade with High Type 20 / 51

49 What We Already Know 1 µ h Pool High and Low Quality Sellers (Stiglitz 77) Only Mixed Strategy Equilibria (Rosenthal- Weiss 84) µ h µ h No Trade With High Quality Seller (Stiglitz 77) 0 0 (Monopsony) Monopsony and mild adverse selection Full Trade 1 π (Perfect Comp) Least Cost Separating Outcome (Rothschild- Stiglitz 76) 20 / 51

50 What We Already Know μ h 1 Cross Subsidizing Π l <0< < Π h Cross Subsidizing Πl < 0 < Π h μ h μ h No Cross Subsidizing: Π l, Π h (Monopsony) Cross Subsidizing or Not? π 1 (Perfect Comp) No Cross Subsidizing: Π l, Π h 0 20 / 51

51 Equilibrium Characterization Today: Construct equilibrium with µ h < µ explicitly Briefly describe equilibrium with µ h µ 21 / 51

52 Equilibrium Characterization Today: Construct equilibrium with µ h < µ explicitly Briefly describe equilibrium with µ h µ Terminology: Separating eqm: all contracts have u h > u l (i.e., x h < x l = 1) Pooling eqm: all contracts have u h = u l (i.e., x h <= x l = 1) Mixed eqm: some separating offers, some pooling. 21 / 51

53 Conjecture & Confirm: Equilibrium with µ h < µ h is Separating Remember the buyer s problem: Π(u h, u l ) = max u l c l, u h c h i {l,h} s. t. c h c l u h u l 0 with Π l (u h, u l ) v l x l t l = v l u l µ i [1 π + πf i (u i )] Π i (u h, u l ) Π h (u h, u l ) v h x h t h = v h u h v h c l c h c l + u l v h c h c h c l 22 / 51

54 Conjecture & Confirm: Equilibrium with µ h < µ h is Separating Marginal benefits vs costs of increasing u l µ l πf l (u l )Π l + (1 π + πf l (u l )) }{{} µ l }{{} MB: more low types trade MC + µ h v h c h c h c l }{{} MB: relaxed IC l = 0 23 / 51

55 Conjecture & Confirm: Equilibrium with µ h < µ h is Separating Marginal benefits vs costs of increasing u l µ l πf l (u l )Π l + (1 π + πf l (u l )) }{{} µ l }{{} MB: more low types trade MC Boundary condition + µ h v h c h c h c l }{{} MB: relaxed IC l = 0 F l (c l ) = 0 F l (ū l ) = 1 F l (u l ) 23 / 51

56 Conjecture & Confirm: Equilibrium with µ h < µ h is Separating Marginal benefits vs costs of increasing u l µ l πf l (u l )Π l + (1 π + πf l (u l )) }{{} µ l }{{} MB: more low types trade MC Boundary condition + µ h v h c h c h c l }{{} MB: relaxed IC l = 0 F l (c l ) = 0 F l (ū l ) = 1 F l (u l ) Equal profit condition [1 π + πf l (u l )] Π(U h, u l ) = Π = µ l (1 π)(v l c l ) U h (u l ) 23 / 51

57 Conjecture & Confirm: Equilibrium with µ h < µ h is Separating Marginal benefits vs costs of increasing u l µ l πf l (u l )Π l + (1 π + πf l (u l )) }{{} µ l }{{} MB: more low types trade MC Boundary condition + µ h v h c h c h c l }{{} MB: relaxed IC l = 0 F l (c l ) = 0 F l (ū l ) = 1 F l (u l ) Equal profit condition [1 π + πf l (u l )] Π(U h, u l ) = Π = µ l (1 π)(v l c l ) U h (u l ) These conditions are necessary (see paper for sufficiency). 23 / 51

58 Equilibrium No cross-subsidization Separation 24 / 51

59 Equilibrium Cross-subsidization equilibrium may feature: No cross-subsidization Separation 24 / 51

60 Equilibrium Cross-subsidization equilibrium may feature: Full Separation: 0 < x h < 1 a.e. Separation No cross-subsidization Separation 24 / 51

61 Equilibrium Pooling Cross-subsidization equilibrium may feature: Full Separation: 0 < x h < 1 a.e. Separation Full Pooling: x h = 1 a.e. No cross-subsidization Separation 24 / 51

62 Equilibrium Pooling Mix Cross-subsidization equilibrium may feature: Full Separation: 0 < x h < 1 a.e. Separation Full Pooling: x h = 1 a.e. Mix: pool below ū l, separate above No cross-subsidization Separation 24 / 51

63 Equilibrium Pooling Mix Cross-subsidization equilibrium may feature: Full Separation: 0 < x h < 1 a.e. Separation Full Pooling: x h = 1 a.e. Mix: pool below ū l, separate above No cross-subsidization Separation More competition (higher π ) less pooling gains to cream-skimming increase in π Milder adv sel (higher µ h ) more pooling increased incentives to trade with h 24 / 51

64 Equilibrium Pooling Mix Cross-subsidization equilibrium may feature: Full Separation: 0 < x h < 1 a.e. Separation Full Pooling: x h = 1 a.e. Mix: pool below ū l, separate above No cross-subsidization Separation More competition (higher π ) less pooling gains to cream-skimming increase in π Milder adv sel (higher µ h ) more pooling increased incentives to trade with h Theorem For every (π, µ h ) there is a unique equilibrium. 24 / 51

65 Context: Literature on Adverse Selection with Screening Most papers: competitive models with Bertrand-type structure Rothschild-Stiglitz, Riley,... Well-known problems with existence of equilibria 25 / 51

66 Context: Literature on Adverse Selection with Screening Most papers: competitive models with Bertrand-type structure Rothschild-Stiglitz, Riley,... Well-known problems with existence of equilibria One strand of lit: buyers have capacity constraints E.g. Gale, Guerrieri-Shimer-Wright... But what happens when a contract attracts more than 1 type? Requires a sampling rule Beliefs about rules for off-path offers? 25 / 51

67 Context: Literature on Adverse Selection with Screening Most papers: competitive models with Bertrand-type structure Rothschild-Stiglitz, Riley,... Well-known problems with existence of equilibria One strand of lit: buyers have capacity constraints E.g. Gale, Guerrieri-Shimer-Wright... But what happens when a contract attracts more than 1 type? Requires a sampling rule Beliefs about rules for off-path offers? This paper: 1 Varying degrees of competition 2 No capacity constraints 3 No need to separately specify off-path beliefs Meeting tech + equilibrium offer distribution pins down off-path payoffs 25 / 51

68 Implications 26 / 51

69 Positive Implications Dispersion in prices and quantities, across and within types 27 / 51

70 Positive Implications Dispersion in prices and quantities, across and within types SRP buyers don t target a specific type terms of each contract correlated across offers 27 / 51

71 Positive Implications Dispersion in prices and quantities, across and within types SRP buyers don t target a specific type terms of each contract correlated across offers Structure of eqm depends on distribution of asset quality (µ h ) determines structure of eqm: separating, mixed, or pooling (Burdett-Judd) 27 / 51

72 Positive Implications Dispersion in prices and quantities, across and within types SRP buyers don t target a specific type terms of each contract correlated across offers Structure of eqm depends on distribution of asset quality (µ h ) determines structure of eqm: separating, mixed, or pooling (Burdett-Judd) Effect of adverse selection on outcomes depends on trading frictions (π) need to know trading frictions to identify info frictions. 27 / 51

73 Normative Implications Are these policies desirable? Increase in competition - E.g. encouraging entry, price discovery - Implications for various interventions in lemons markets 28 / 51

74 Normative Implications Are these policies desirable? Increase in competition - E.g. encouraging entry, price discovery - Implications for various interventions in lemons markets Changes in information - E.g. credit scores, allowing principals to condition on more variables - Implications for restrictions/mandates 28 / 51

75 Welfare Criterion Utilitarian welfare: W = µ l v l + µ h [v h X h + c h (1 X h )] with X h ūl u l x h (u l ) d ˆF (u l ) Low type always trades fully so key is what happens to X h? 29 / 51

76 Welfare Criterion Utilitarian welfare: W = µ l v l + µ h [v h X h + c h (1 X h )] with X h ūl u l x h (u l ) d ˆF (u l ) Low type always trades fully so key is what happens to X h? Focus on severe adverse selection (µ h < µ h ), show all these policies Desirable or irrelevant at the extremes, i.e. π = 0 or π = 1 But, can be undesirable in the interior, esp. for π high 29 / 51

77 Welfare and Competition W (π) 0 1 π Result If µ h < µ h, W maximized at π (0, 1). Implications Taxing entry (or otherwise limiting buyer competition) may be desirable 30 / 51

78 (Quick) Intuition Key: interaction between competition and incentives. 31 / 51

79 (Quick) Intuition Key: interaction between competition and incentives. 1 All else equal, increasing competition for low types causes: buyers offer more utility to low types, which relaxes their IC constraint allows for more trade with high types, which increases W. 31 / 51

80 (Quick) Intuition Key: interaction between competition and incentives. 1 All else equal, increasing competition for low types causes: buyers offer more utility to low types, which relaxes their IC constraint allows for more trade with high types, which increases W. 2 All else equal, increasing competition for high types causes: buyers offer more utility to high types, which tightens IC constraint less trade with high types, which decreases W. 31 / 51

81 (Quick) Intuition Key: interaction between competition and incentives. 1 All else equal, increasing competition for low types causes: buyers offer more utility to low types, which relaxes their IC constraint allows for more trade with high types, which increases W. 2 All else equal, increasing competition for high types causes: buyers offer more utility to high types, which tightens IC constraint less trade with high types, which decreases W. Which effect dominates depends on relative profits (Π h /Π l ). 31 / 51

82 (Quick) Intuition Key: interaction between competition and incentives. 1 All else equal, increasing competition for low types causes: buyers offer more utility to low types, which relaxes their IC constraint allows for more trade with high types, which increases W. 2 All else equal, increasing competition for high types causes: buyers offer more utility to high types, which tightens IC constraint less trade with high types, which decreases W. Which effect dominates depends on relative profits (Π h /Π l ). First effect dominates when π is small (Π h /Π l small). Second effect dominates when π is large (Π h /Π l large). Details 31 / 51

83 Asset Purchases Asset purchases proposed to help markets suffering from adverse selection Similar: government option (insurance markets), FAFSA (student loans) 32 / 51

84 Asset Purchases Asset purchases proposed to help markets suffering from adverse selection Similar: government option (insurance markets), FAFSA (student loans) Lessons from lit with competitive markets (e.g., Tirole): 1 Can only W if government overpays for bad assets, loses money 2 But if government willing to do so, W for sure 32 / 51

85 Asset Purchases Asset purchases proposed to help markets suffering from adverse selection Similar: government option (insurance markets), FAFSA (student loans) Lessons from lit with competitive markets (e.g., Tirole): 1 Can only W if government overpays for bad assets, loses money 2 But if government willing to do so, W for sure Our model: neither result true when π < 1. Government losing money neither necessary nor sufficient for W 32 / 51

86 Asset Purchases Policy: Government will purchase any quantity at P [c l, v l ]. Can be mapped into an exogenous lower bound for u l F l(u l) 1 p, x x h(u l) π = 1 0 c l v l u l c l v l u l Government option never exercised, so cost to the government = 0. 1 Helpful for low π, P. 2 Harmful if π, P high enough. 33 / 51

87 What About More/Better Information? Examples Permitting insurance providers to discriminate based on observables Introducing credit scores in loan markets Requiring OTC market participants to disclose trades 34 / 51

88 What About More/Better Information? Examples Permitting insurance providers to discriminate based on observables Introducing credit scores in loan markets Requiring OTC market participants to disclose trades Allow principals to condition on more information 34 / 51

89 What About More/Better Information? Examples Permitting insurance providers to discriminate based on observables Introducing credit scores in loan markets Requiring OTC market participants to disclose trades Allow principals to condition on more information Can be mapped into a mean-preserving spread of µ h Need to compare E[W (µ h )] to W (E[µ h ]) Desirability is about the sign of W (µ h ) 34 / 51

90 What About More/Better Information? Examples Permitting insurance providers to discriminate based on observables Introducing credit scores in loan markets Requiring OTC market participants to disclose trades Allow principals to condition on more information Can be mapped into a mean-preserving spread of µ h Need to compare E[W (µ h )] to W (E[µ h ]) Desirability is about the sign of W (µ h ) Answer: desirability depends on (π, µ h ) Note: W is linear when π = 0 and π = 1 no effect on welfare 34 / 51

91 Desirability of information µ h < µ h :, W convex (concave) for low (high) π more info desirable in concentrated markets, undesirable otherwise µ h > µ h :, W is (weakly) concave for all π more info always undesirable 35 / 51

92 Robustness, Extensions, and Conclusion 36 / 51

93 Robustness 1 Endogenous π Details buyers choose advertising intensity at cost π Taxing this margin desirable when equilibrium π is high 2 Constrained efficiency Details A mechanism design approach µ h < µ h equilibrium is efficient 3 General meeting technologies Details Methodology extends to many buyers, arbitrary distribution over meetings Welfare effects of competition depend on strength of coverage effect 37 / 51

94 Other extensions (see paper) 1 Concave preferences: canonical insurance problem 2 Different levels of competition across types: π l π h 3 More than two types 4 Vertical/horizontal differentiation across buyers 5 Multi-dimensional heterogeneity across sellers 38 / 51

95 Conclusion Lots of interest in markets where asymmetric info is a significant concern Insurance, loans, CDS, / 51

96 Conclusion Lots of interest in markets where asymmetric info is a significant concern Insurance, loans, CDS,... Empirical: identifying adverse selection from terms/outcomes 39 / 51

97 Conclusion Lots of interest in markets where asymmetric info is a significant concern Insurance, loans, CDS,... Empirical: identifying adverse selection from terms/outcomes Theory: optimal intervention/regulation. 39 / 51

98 Conclusion Lots of interest in markets where asymmetric info is a significant concern Insurance, loans, CDS,... Empirical: identifying adverse selection from terms/outcomes Theory: optimal intervention/regulation. Existing literature either restricts contracts or assumes perfect comp 39 / 51

99 Conclusion Lots of interest in markets where asymmetric info is a significant concern Insurance, loans, CDS,... Empirical: identifying adverse selection from terms/outcomes Theory: optimal intervention/regulation. Existing literature either restricts contracts or assumes perfect comp This paper: 1 Tractable model w/ AS, imperfect comp, sophisticated contracts 2 Many testable implications 3 Novel normative implications: different from π = 1 case 39 / 51

100 Extra Stuff 40 / 51

101 Intuition Theorem U h (u l ) is a strictly increasing function. U h (u l ) û h û l1 û l2 u l Back to Properties of Equilibrium 41 / 51

102 Intuition Theorem U h (u l ) is a strictly increasing function. U h (u l ) Pr(u h = û h ) = û l2 û l1 f l (u l )du l û h û l1 û l2 u l Back to Properties of Equilibrium 41 / 51

103 Intuition Theorem U h (u l ) is a strictly increasing function. U h (u l ) û h2 û h1 û l u l Back to Properties of Equilibrium 41 / 51

104 Intuition Theorem U h (u l ) is a strictly increasing function. U h (u l ) û h2 û h1 Pr (u h = û l ) = û h2 û h1 f h (u h )du h û l u l Back to Properties of Equilibrium 41 / 51

105 Imperfect Competition Insurance markets Brown and Goolsbee (2002), Dafny (2010), Cabral et. al. (2014), Einav and Levin (2015)... Credit markets Ausubel (1991), Petersen and Rajan (1994), Calem and Mester (1995), Scharfstein and Sundaram (2013)... Financial markets Barclay et. al. (1999), Weston (2000),... Back to Introduction 42 / 51

106 Quotes Einav, Finkelstein, and Levin There has been much less progress on [...] models of insurance contracting that incorporate realistic market frictions. One challenge is to develop an appropriate conceptual framework. Even in stylized models of insurance markets with asymmetric information, characterizing competitive equilibrium can be challenging, and the challenge is compounded if one wants to allow for [...] market imperfections. Or, as Chiappori et al (2006) put it: there is a crying need for...models...devoted to the interaction between imperfect competition and adverse selection Back to Introduction 43 / 51

107 Why is Welfare Hump-Shaped in π? Because x h is hump-shaped in π and F l is shifting right. F l (u l ) x h (u l ) 1 0 c l v l u l c l v l u l 44 / 51

108 Why is Welfare Hump-Shaped in π? Because x h is hump-shaped in π and F l is shifting right. F l (u l ) 1 π x h (u l ) 0 c l v l u l c l v l u l 44 / 51

109 Why is Welfare Hump-Shaped in π? Because x h is hump-shaped in π and F l is shifting right. F l (u l ) 1 π x h (u l ) 0 c l v l u l c l v l u l 44 / 51

110 Why is Welfare Hump-Shaped in π? Because x h is hump-shaped in π and F l is shifting right. F l (u l ) 1 π x h (u l ) π = 1 0 c l v l u l c l v l u l 44 / 51

111 Why is x h (u l ) Hump-Shaped? x h(u l) U h(u l) u h 0 c h c l v l u l c l v l u l Two effects from competition: 1 buyers give more surplus to l type sellers. relaxes IC constraint x h 45 / 51

112 Why is x h (u l ) Hump-Shaped? x h(u l) U h(u l) u h 0 c h c l v l u l c l v l u l Two effects from competition: 1 buyers give more surplus to l type sellers. relaxes IC constraint x h 2 buyers give more surplus to h type sellers. tightens IC constraint x h 45 / 51

113 Why is x h (u l ) Hump-Shaped? x h(u l) U h(u l) u h 0 c h c l v l u l c l v l u l Two effects from competition: 1 buyers give more surplus to l type sellers. relaxes IC constraint x h 2 buyers give more surplus to h type sellers. tightens IC constraint x h Which dominates? Depends on whether U h(u l ) 1. i.e., whether buyers trying to attract more l or h. this depends on relative profits Π h Π l / 51

114 Severe Adverse Selection: Allocations Π j (u l ) U h (u l ) Π l u h Π h 0 c h c l v l u l c l v l u l Slope of U h determined by ratio of profits, Π h /Π l Back to Welfare 46 / 51

115 Severe Adverse Selection: Allocations Π j (u l ) U h (u l ) Π l u h Π h 0 c h c l v l u l c l v l u l Slope of U h determined by ratio of profits, Π h /Π l At low u l, Π h /Π l small, competition stronger for type-l, U h (u l ) < 1 At high u l, Π h /Π l large, competition stronger for type-h, U h (u l ) > 1 Back to Welfare 46 / 51

116 General Mechanisms A communication game between a seller and the buyer(s) she meets Buyers offer mechanisms that map seller s messages into an offer (x, t) - Deterministic and exclusive but otherwise unrestricted Seller sends a message to each buyer - Arbitrary message space (quality, contact with other buyer etc.) 47 / 51

117 General Mechanisms A communication game between a seller and the buyer(s) she meets Buyers offer mechanisms that map seller s messages into an offer (x, t) - Deterministic and exclusive but otherwise unrestricted Seller sends a message to each buyer - Arbitrary message space (quality, contact with other buyer etc.) Proposition Any equilibrium of the communication game can be achieved by a menu game. Proof: See Martimort and Stole (2002). Proposition In any menu, at most 2 contracts are chosen by some seller type in equilibrium. Proof: If type-j seller chooses 2 (or more) contracts in eq., they must yield same utility to seller AND same profit to buyer. Back 47 / 51

118 Constrained Efficiency: A mechanism design approach Types Seller: Quality, buyers matched with Buyer(s): Set of sellers matched with A direct mechanism: a map from reports to allocations, subject to Feasibility: Only matched agents can trade Incentive compatibility: Types reported truthfully Participation: Outside option is equilibrium described earlier Exclusivity: Each seller can trade with at most 1 buyer Proposition If µ < µ h, the equilibrium allocation is constrained efficient. Utilities are the same as in equilibrium (allocations might differ) Trade volume (or eq., utilitarian welfare) still maximized at interior π Back 48 / 51

119 General Meeting Technologies Large number of buyers and sellers (measure b and s resp.) Meeting technology: described by λ(α): Average number of offers sent by buyers P(n, α): Pr(a seller receives n offers) Q(n, α): Pr(offer received by seller with n 1 other offers) = np(n,α) λ(α) α: Summarizes frictions in matching Examples Poisson: λ(α) = α Geometric: λ(α) = P(n, α) = e α α n n! α 1 α P(n, α) = α n (1 α) For both, coverage (sellers with at least 1 offer) increases with α Back 49 / 51

120 General Meeting Technologies: Solution arg max u l,u h i {l,h} µ i [ n=1 Q(n)F n 1 i (u i ) ] Π i (u l, u h ) 50 / 51

121 General Meeting Technologies: Solution arg max u l,u h = arg max u l,u h i {l,h} i {l,h} µ i [ n=1 Q(n)F n 1 i (u i ) ] [ Q(1) µ i n =1 Q(n ) + n =2 Π i (u l, u h ) ] Q(n ) n =1 Q(n ) F n 1 i (u i ) Π i (u l, u h ) 50 / 51

122 General Meeting Technologies: Solution arg max u l,u h = arg max u l,u h = arg max u l,u h i {l,h} i {l,h} i {l,h} µ i [ n=1 Q(n)F n 1 i (u i ) ] [ Q(1) µ i n =1 Q(n ) + n =2 Π i (u l, u h ) ] Q(n ) n =1 Q(n ) F n 1 i (u i ) Π i (u l, u h ) µ i [1 π + πg i (u i )] Π i (u l, u h ) where π = 1 Q(1) n=1 Q(n) Characterization from baseline G i (u i ) (and therefore, F i ) Shape of W (α) depends on strength of coverage effect Hump-shaped for Poisson, always increasing for Geometric Back 50 / 51

123 General Meeting Technologies: Solution arg max u l,u h i {l,h} µ i [ n=1 Q(n)F n 1 i (u i ) ] Π i (u l, u h ) = arg max u l,u h i {l,h} µ i [1 π + πg i (u i )] Π i (u l, u h ) where π = 1 Q(1) n=1 Q(n) Characterization from baseline G i (u i ) (and therefore, F i ) Shape of W (α) depends on strength of coverage effect Hump-shaped for Poisson, always increasing for Geometric Back 50 / 51

124 Endogenizing π Buyer k also chooses ˆπ k : Pr(her offer reaches a seller) subject to cost C(ˆπ k ) max ˆπ k,u k l,uk h i {l,h} [ ( µ i ˆπ k 1 ˆπ k) ( )] ( ) + ˆπ k ˆπ k F k i ui k Π i ul k, uh k C(ˆπ k ), Optimality in a symmetric equilibrium Implications C (ˆπ ) = i {l,h} [ ( )] ( ) µ i 1 ˆπ + ˆπ F k i ui k Π i ul k, uh k. (2) Unique symmetric equilibrium (under regularity conditions on C) ˆπ increasing (decreasing) in µ h when µ h is less (greater) than µ h Welfare: taxing effort (advertising?) can be optimal if ˆπ sufficiently high Back 51 / 51