Perfect Competition in Markets with Adverse Selection Eduardo Azevedo and Daniel Gottlieb (Wharton) Presented at Frontiers of Economic Theory & Computer Science at the Becker Friedman Institute August 13, 2016 Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 1 / 51
Introduction Agenda Adverse selection is considered a first-order problem in many markets, which are already heavily regulated in complicated ways : Mandates, community rating, risk adjustment, differential subsidies, regulation of contract characteristics. All of these affect contract characteristics. This is a challenge to the standard models (Akerloff / Eivan Finkelstein and Cullen, and Rothschild and Stiglitz). Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 2 / 51
Introduction This paper Develops a price-taking model of adverse selection. Contract characteristics are endogenous. Consumers can be heterogeneous in more than one dimension. Equilibrium always exists. Basic idea: Start from broad set of potential contracts. Use the same logic as price-taking models (Akerlof and Einav-Finkelstein and Cullen) to determine both prices and which contracts are traded. Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 3 / 51
Determining prices p D ( p ) p ∗ AC ( q ) q
Determining which contracts are traded
Preview: Unintended Consequences 2.5 No Mandate Mandate 2 (Density) 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 Coverage
Model Outline 1 Model 2 Competitive Equilibrium 3 Application: Equilibrium Effects of a Mandate 4 Inefficiency and Policy Interventions Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 7 / 51
Model Model Consumers θ ∈ Θ, distributed according to a probability distribution µ . Contracts (or products) x ∈ X . Agent θ has utility U ( x , p , θ ) of buying x at a price p , and the cost is c ( x , θ ) ≥ 0 Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 8 / 51
Model Example we understand: Akerlof QJE 1970 Basic framework in Einav, Finkelstein and Cullen (2010), Hackman, Kolstad and Kowalski (2014), Handel, Hendel and Whinston (2014), Smetters and Scheuer (2014). Single, exogenous product: X = { 0 , 1 } . Quasilinear utility, U = u ( x , θ ) − p . Single product is often not realistic. No predictions on contract terms. In particular, the model is silent about intensive margin regulations. Yields useful predictions on pricing and efficiency. Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 9 / 51
Model Equilibria All that matters are willingness to pay and costs, u (1 , θ ) and c (1 , θ ). Can define demand D ( P ), and average cost AC ( Q ) curves. Equilibria are intersection of demand and average cost. p D ( p ) p ∗ AC ( q ) q Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 10 / 51
Model Toy example: Rothschild and Stiglitz QJE 1976 All consumers have same wealth, same risk preferences, and may suffer a loss of the same size. Only two types, who differ in their probability of a loss, Θ = { L , H } . Contracts specify % of loss covered, X = [0 , 1]. Even in this setting, equilibria do not necessarily exist. Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 11 / 51
Model Interesting example: Einav, et al. AER 2013 A model of health insurance. Higher dimensional heterogeneity of consumers: Loss distributions. Risk aversion. Moral hazard parameters. Will calibrate this model to illustrate ideas, with set of contracts X = [0 , 1] being % of coverage. Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 12 / 51
Model Interesting example: Einav, et al. AER 2013 A model of health insurance. Higher dimensional heterogeneity of consumers: Loss distributions. Risk aversion. Moral hazard parameters. Will calibrate this model to illustrate ideas, with set of contracts X = [0 , 1] being % of coverage. Assuming CARA preferences, u ( x , θ ) = x · M θ + x 2 2 · H θ + 1 2 x (2 − x ) · S 2 θ A θ , and c ( x , θ ) = x · M θ + x 2 · H θ . Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 12 / 51
Model Assumptions Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 13 / 51
Model Assumptions Simpler assumptions for the talk: 1 X and Θ are compact subsets of Euclidean space. 2 U ( x , p , θ ) = u ( x , θ ) − p , where u is Lipschitz in x . 3 u and c are continuous. Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 13 / 51
Model Prices and allocations A price is a measurable function p over X , price of contract x denoted p ( x ). An allocation α is a measure over Θ × X such that α | Θ = µ . Given ( p , α ), consumers are optimizing if, for ( x , θ ) with probability 1 according to α , for all x ′ ∈ X , u ( x , θ ) − p ( x ) ≥ u ( x ′ , θ ) − p ( x ′ ). Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 14 / 51
Model Prices and allocations A price is a measurable function p over X , price of contract x denoted p ( x ). An allocation α is a measure over Θ × X such that α | Θ = µ . Given ( p , α ), consumers are optimizing if, for ( x , θ ) with probability 1 according to α , for all x ′ ∈ X , u ( x , θ ) − p ( x ) ≥ u ( x ′ , θ ) − p ( x ′ ). Conditional moments are denoted as x , ˜ E x [ c ] = E [ c (˜ θ ) | α, ˜ x = x ]. Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 14 / 51
Equilibrium Outline 1 Model 2 Competitive Equilibrium 3 Application: Equilibrium Effects of a Mandate 4 Inefficiency and Policy Interventions Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 15 / 51
Equilibrium Weak equilibrium Definition A price-allocation pair ( p , α ) is weak equilibrium if 1 Consumers optimize. 2 All contracts make 0 profits, p ( x ) = E x [ c ] almost everywhere according to α . Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 16 / 51
Equilibrium Weak Equilibrium Example: Rothschild-Stiglitz X = [0 , 1] and Θ = { L , H } . H IC H L IC L p ( x ) x 0 1 Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 17 / 51
Equilibrium But there are many other weak equilibria X = [0 , 1] and Θ = { L , H } . H IC H p ( x ) ˜ L IC L p ( x ) x 0 1 Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 17 / 51
Equilibrium Definition: Perturbations A behavioral type x is an agent who always demands contract x , u ( x , x ) = ∞ , u ( x ′ , x ) = 0 if x ′ � = x , and c ( x , x ) = 0. Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 18 / 51
Equilibrium Definition: Perturbations A behavioral type x is an agent who always demands contract x , u ( x , x ) = ∞ , u ( x ′ , x ) = 0 if x ′ � = x , and c ( x , x ) = 0. A perturbation (¯ X , η ) is an economy with a finite set of contracts ¯ X ⊆ X , set of types Θ ∪ ¯ X , and distribution of types µ + η , where the support of η is ¯ X . We can define unrefined equilibria of perturbations because every perturbation is a particular case of the model. Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 18 / 51
Equilibrium of a Perturbation: Example H IC H L IC L x
Equilibrium of a Perturbation: Example $8,000 Equilibrium Prices Average Loss Parameter $6,000 ($) $4,000 $2,000 $0 0 0.2 0.4 0.6 0.8 1 Contract
Equilibrium Definition: Perturbations (continued) A sequence of perturbations (¯ X n , η n ) n ∈ N converges to the original economy if 1 Every point in X is the limit of a sequence ( x n ) n ∈ N with each x n ∈ ¯ X n . 2 The mass of behavioral types η n (¯ X n ) converges to 0. Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 21 / 51
Equilibrium Definition: Perturbations (continued) Consider a sequence of perturbations (¯ X n , η n ) n ∈ N converging to the original economy. A sequence of weak equilibria ( p n , α n ) n ∈ N converges to ( p ∗ , α ∗ ) if 1 The allocations α n ∈ ∆((Θ ∪ X ) × X ) converge to α ∗ weakly. 2 For every sequence ( x n ) n ∈ N , with each x n ∈ ¯ X n and limit x ∈ X , we have that p n ( x n ) converges to p ∗ ( x ). Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 22 / 51
Equilibrium Equilibrium Definition ( p ∗ , α ∗ ) is a competitive equilibrium if there exists a sequence of perturbations converging to the original economy with a sequence of weak equilibria that converges to ( p ∗ , α ∗ ). Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 23 / 51
Equilibrium: Example H IC H L IC L x
Equilibrium: Example H IC H L IC L x
Equilibrium: Example H IC H L IC L p ( x ) x
Equilibrium Existence Theorem A competitive equilibrium exists. Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 25 / 51
Equilibrium Proof Outline Step 1: Every perturbed economy has an equilibrium, by a standard fixed point argument. Step 2: Equilibrium prices in every perturbed economy are uniformly Lipschitz. Step 3: Every sequence of perturbations converging to the original economy has a convergent subsequence, and the limit is an equilibrium of the original economy. Eduardo Azevedo (Wharton) Adverse Selection August 13, 2016 26 / 51
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