PerfectCompetitioninMarketswithAdverseSelectionPerfectCompetitioninMarketswithAdverseSelection...

Perfect Competition in Markets with Adverse Selection

Eduardo Azevedo and Daniel Gottlieb(Wharton)

Presented at Frontiers of Economic Theory & Computer Science at the Becker FriedmanInstitute

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 Finkelsteinand Cullen, and Rothschild and Stiglitz).


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 andEinav-Finkelstein and Cullen) to determine both prices and whichcontracts are traded.


Determining prices

q

p

AC(q)

D(p)

p∗

Determining which contracts are traded

Preview: Unintended Consequences

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

Coverage

(Density)

No MandateMandate

Model

Outline

1 Model2 Competitive Equilibrium3 Application: Equilibrium Effects of a Mandate4 Inefficiency and Policy Interventions


Model

Model

Consumers θ ∈ Θ, distributed according to a probability distributionµ.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


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.


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.

q

p

AC(q)

D(p)

p∗


Model

Toy example: Rothschild and Stiglitz QJE 1976

All consumers have same wealth, same risk preferences, and maysuffer 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.


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 contractsX = [0, 1] being % of coverage.

Assuming CARA preferences,

u(x , θ) = x ·Mθ + x2

2 · Hθ + 12x(2− x) · S2

θAθ, and

c(x , θ) = x ·Mθ + x2 · Hθ.


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 contractsX = [0, 1] being % of coverage.Assuming CARA preferences,

u(x , θ) = x ·Mθ + x2

2 · Hθ + 12x(2− x) · S2

θAθ, and

c(x , θ) = x ·Mθ + x2 · Hθ.


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.


Model

Prices and allocations

A price is a measurable function p over X , price of contract xdenoted 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 probability1 according to α, for all x ′ ∈ X ,

u(x , θ)− p(x) ≥ u(x ′, θ)− p(x ′).

Conditional moments are denoted as

Ex [c] = E[c(x̃ , θ̃)|α, x̃ = x ].


Equilibrium

Outline



Equilibrium

Weak equilibrium

DefinitionA price-allocation pair (p, α) is weak equilibrium if

1 Consumers optimize.2 All contracts make 0 profits,

p(x) = Ex [c]

almost everywhere according to α.


Equilibrium

Weak Equilibrium Example: Rothschild-Stiglitz

X = [0, 1] and Θ = {L,H}.

x0

p(x)

1

ICL

ICH

L

H


Equilibrium

But there are many other weak equilibria

X = [0, 1] and Θ = {L,H}.

x0 1

ICL

ICH

L

H

p̃(x)

p(x)


Equilibrium

Definition: Perturbations

A behavioral type x is an agent who always demands contract x ,u(x , x) =∞, u(x ′, x) = 0 if x ′ 6= x , and c(x , x) = 0.

A perturbation (X̄ , η) is an economy with a finite set of contracts

X̄ ⊆ X ,

set of typesΘ ∪ X̄ ,

and distribution of typesµ+ η,

where the support of η is X̄ .We can define unrefined equilibria of perturbations because everyperturbation is a particular case of the model.


Equilibrium of a Perturbation: Example

H

x

ICL

ICH

L

Equilibrium of a Perturbation: Example

0 0.2 0.4 0.6 0.8 1$0

$2,000

$4,000

$6,000

$8,000

Contract

($)

Equilibrium PricesAverage Loss Parameter

Equilibrium

Definition: Perturbations (continued)

A sequence of perturbations (X̄n, ηn)n∈N converges to the originaleconomy if

1 Every point in X is the limit of a sequence (xn)n∈N with eachxn ∈ X̄n.

2 The mass of behavioral types ηn(X̄n) converges to 0.


Equilibrium

Definition: Perturbations (continued)

Consider a sequence of perturbations (X̄n, ηn)n∈N converging to theoriginal economy. A sequence of weak equilibria (pn, αn)n∈N converges to(p∗, α∗) if

1 The allocations αn ∈ ∆((Θ ∪ X )× X ) converge to α∗ weakly.2 For every sequence (xn)n∈N, with each xn ∈ X̄n and limit x ∈ X , we

have that pn(xn) converges to p∗(x).


Equilibrium

Equilibrium

Definition(p∗, α∗) is a competitive equilibrium if there exists a sequence ofperturbations converging to the original economy with a sequence of weakequilibria that converges to (p∗, α∗).


Equilibrium: Example

H

x

ICL

ICH

L


x

ICL

ICH

L

H


H

x

ICL

ICH

L

p(x)

Equilibrium

Existence

TheoremA competitive equilibrium exists.


Equilibrium

Proof Outline

Step 1: Every perturbed economy has an equilibrium, by a standardfixed point argument.Step 2: Equilibrium prices in every perturbed economy areuniformly Lipschitz.Step 3: Every sequence of perturbations converging to the originaleconomy has a convergent subsequence, and the limit is anequilibrium of the original economy.


Equilibrium

Equilibrium properties

Proposition1 Every equilibrium is a weak equilibrium.2 Equilibrium prices are continuous and almost everywhere

differentiable.3 For every contract x with strictly positive equilibrium price there

exists a consumer θ who is indifferent between her current contractand x. Moreover,

c(x , θ) ≥ p(x).


Equilibrium

Strategic Foundations

The paper shows that the competitive model is a particular limitingcase of Bertrand competition with differentiated products.

In particular, this means that competitive models (Rothschild andStiglitz, Akerlof, Riley) are limiting cases of the differentiated-productsmodels used in empirical IO (Starc, Veiga and Weyl).Key assumptions: Many, small firms, with a small degree ofdifferentiation.


Equilibrium

Bertrand Game (definitions)

Fix a perturbation (E , X̄ , η).n firms selling differentiated varieties of each product x .Logit shares S(P, p, x , θ) equal to

eσ·(u(x ,θ)−P)

eσ·(u(x ,θ)−P) + (n − 1) · eσ·(u(x ,θ)−p(x)) +∑

x ′ 6=x n · eσ·(u(x ′,θ)−p(x ′)) .

Profits

Π(P, p, x) =∫θS(P, p, x , θ) · (P − c(x , θ)) d(µ+ η)

if firms produce less than scale q̄ or −∞ if the firm produces more.


Equilibrium

Bertrand Game (result)

PropositionThere exists a constant K such that, if

1n < q̄ < K,

then an equilibrium exists. Moreover, profits per unit sold are lower than

2σ.

Can show that with fixed small scale and large number of firms, aselasticities go to infinity equilibria converge to the perfectlycompetitive outcome.Bottom line: perfect competition is the limit of a Bertrand game withmany, small, and undifferentiated firms.


Equilibrium

Literature

Other GE notions: Gale Restuds (1992), Dubey and Geanakoplos QJE(2002).In one-dimensional case, coincides with some standard notions fromthe signaling and screening literatures:

Rothschild and Stiglitz, when their equilibrium exists.Riley Ecma (1979) reactive equilibrium.Banks and Sobel Ecma (1987) D1.Bisin and Gottardi JPE (2006) EPT equilibrium.

But differs from notions that allow firms to cross-subsidize contracts:Wilson JET (1977) - Miyazaki BJE (1977) anticipatory equilibrium.Netzer and Scheuer IER (2014).

Veiga and Weyl (2014)Complementary to our work, many similar comparative statics.Key differences are imperfect competition and product variety (tomatosauce).


Calibration

Outline



Calibration

Calibration: Einav et al. health insurance model

u(x , θ) = x ·Mθ + x2

2 · Hθ + 12x(2− x) · S2

θAθ, and

c(x , θ) = x ·Mθ + x2 · Hθ.

A H M SMean 1.5E-5 1,330 4,340 24,474

Log covarianceA 0.25 -0.01 -0.12 0H σ2

log H = 0.28 -0.03 0M 0.20 0S 0.25


Equilibrium prices and adverse selection

0 0.2 0.4 0.6 0.8 1$0

$2,000

$4,000

$6,000

$8,000

Contract

($)

Equilibrium PricesAverage Loss Parameter

Equilibrium demand profile

$1,000 $10,000 $100,00010

−6

10−5

10−4

Average Loss, Mθ

Ris

k A

ve

rsio

n,

Aθ

0.2

0.4

0.6

0.8

1

Simulating a Mandate

0 0.2 0.4 0.6 0.8 1$0

$2,000

$4,000

$6,000

$8,000

Contract

($)

No Mandate − PricesNo Mandate − LossesMandate − PricesMandate − Losses

Unintended Consequences

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

Coverage

(Density)

No MandateMandate

85% of consumers purchase minimum coverage, up from 80% before the mandate.

Calibration

Theoretical Results

Consider an economy with mandated coverage [m + dm, 1], andequilibria (pdm, αdm).We will derive comparative statics with respect to dm.Denote (p0, α0) as (p, α).See the paper for necessary regularity conditions.Define the intensive margin selection coefficient as

SI(x) = ∂xEx [c]− Ex [mc].


Calibration

Theoretical Results

PropositionThe change in the prices of minimum coverage is

limx→m

∂dmpdm(x)|dm=0 = −SI(m) + ξ,

where the error term ξ is small if g(m)/G(m) is small.

PropositionWhenever

SI(m) 6= 0,

there are consumers who change their decisions beyond the direct effectsof the mandate.


Optimal Regulation

Outline



Informal Example

Will cover main ideas behind optimal regulation in a simple informalexample.Let X = [0, 1].

Assume that consumers only adjust in the “intensive margin” whenprices change.

A benevolent government can regulate menus and prices, but has thesame information as the firms.Kaldor-Hicks efficiency, no excess burden of public funds.

Optimal Regulation

Equilibrium Inefficiency

Starting point for regulation: equilibria are inefficient.Definitions:

Marginal costmc(x , θ) = ∂xc(x , θ).

Marginal utilitymu(x , θ) = ∂xu(x , θ).


Equilibrium is InefficientPrivate optimum where marginal utility equals p′.

x

p′

muθ

xeq

Equilibrium is InefficientBut social optimum where marginal utility equals marginal cost.

x

p′

muθ

mcθ

xeq xeff

Equilibrium is InefficientTwo distortions: Ex [mc] 6= p′ and mcθ 6= Ex [mc].

Ex[mc]

x

p′

muθ

mcθ

xeq xeff

Equilibrium is InefficientSources: adverse / advantageous selection and multidimensional heterogeneity.

Ex[mc]

x

p′

muθ

mcθSI

xeq xeff

Optimal Regulation

Optimal Regulation

Effectively, any regulation can be implemented setting p(x).It is insightful to write a formula for the per unit subsidy thegovernment must give firms,

p(x) + t(x) = Ex [c].

Will now find necessary conditions for optimum by perturbing a priceschedule. This is an old trick in optimal tax theory that has seen arevival since the 2000s.Increase p′ by dp′ in the interval x + dx .


Perturbation

0

p(x)

1¯x x x+dx

Perturbation

0

p(x)

1¯x x x+dx

p̃(x)

Optimal Regulation

Perturbation (continued)

Denote intensive margin elasticity as ε(x , θ).In an optimal price schedule, it must be that

Ex [ε · (mu −mc)] = 0.

We have

0 = Ex [p′ −mc] · Ex [ε] + Covx [−mc, ε]= (SI − t ′) · Ex [ε]− Covx [mc, ε].


Optimal Regulation

Consequences

Optimal regulation is a modified risk adjustment formula: risk adjustmentplus covariance term:

t ′(x) = SI(x)− Covx [ε,mc]Ex [ε]


Regulation Example

0 0.2 0.4 0.6 0.8 1$0

$2,000

$4,000

$6,000

$8,000

Contract

($)

No Mandate − PricesNo Mandate − LossesMandate − PricesMandate − Losses

Regulation Example

0 0.2 0.4 0.6 0.8 1$0

$2,000

$4,000

$6,000

$8,000

Contract

($)

Equilibrium PricesEquilibrium LossesOptimum PricesOptimum Losses

Optimal Regulation

Consequences

The mandate raises welfare by $127 per consumer.Optimal regulation increases it by $279.A simple policy like the mandate can increase efficiency.But also has important unintended consequences: with adverseselection mandates subsidize low-quality coverage.Optimal policy also addresses selection in the intensive margin.


Optimal Regulation

Conclusion

Key idea is to apply the supply and demand approach from theone-contract model to a more general case.Gives a simple model to explain what contracts are traded, and effectsof policy.Standard policies have important unintended consequences, andregulation should also address selection on the intensive margin.

Thank You!


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