Screening and Adverse Selection in Frictional Markets
Benjamin Lester
Philadelphia Fed
Ali Shourideh
Wharton
Venky Venkateswaran
Ariel Zetlin-Jones
NYU Stern
Carnegie-Mellon
March 2015
1 / 31
Introduction
Many markets feature adverse selection and imperfect competition
• Examples: insurance, loans, financial securities
• Screening, e.g. through differential coverage, loan amounts, trade sizes
2 / 31
Introduction
Many markets feature adverse selection and imperfect competition
• Examples: insurance, loans, financial securities
• Screening, e.g. through differential coverage, loan amounts, trade sizes
A unified theoretical framework is lacking
• Large empirical literature (and some theory)
• But typically restricts contracts and/or assumes perfect competition
2 / 31
Introduction
Many markets feature adverse selection and imperfect competition
• Examples: insurance, loans, financial securities
• Screening, e.g. through differential coverage, loan amounts, trade sizes
A unified theoretical framework is lacking
• Large empirical literature (and some theory)
• But typically restricts contracts and/or assumes perfect competition
This project: A tractable model that combines all three elements
• Complete characterization of the unique equilibrium
• Testable predictions for distribution of contracts
• Policy experiments: changes in competition, transparency
2 / 31
Environment
A seller with 1 divisible good, type i ∈ {h, `} with prob. µi
• privately observed value ci ,
• payoff from an allocation (x, t):
ch > c`
ci (1 − x) + t
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Environment
A seller with 1 divisible good, type i ∈ {h, `} with prob. µi
• privately observed value ci ,
• payoff from an allocation (x, t):
ch > c`
ci (1 − x) + t
Two buyers with value vi
• payoff from (x, t):
vi x − t
• gains from trade for both types: vi > ci
• ‘lemons’ assumption: vl < ch
3 / 31
Environment
A seller with 1 divisible good, type i ∈ {h, `} with prob. µi
• privately observed value ci ,
• payoff from an allocation (x, t):
ch > c`
ci (1 − x) + t
Two buyers with value vi
• payoff from (x, t):
vi x − t
• gains from trade for both types: vi > ci
• ‘lemons’ assumption: vl < ch
Search frictions
• buyers post arbitrary menus of exclusive contracts
• seller receives 1 offer w.p. 1 − π and both w.p. π
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Strategies
buyers: by the revelation principle, sufficient to consider
• menus with two contracts
(ICj ) :
z ≡ {(xl , tl ), (xh , th )}
tj + cj (1 − xj ) ≥ t−j + cj (1 − x−j )
j ∈ {h, l}
4 / 31
Strategies
buyers: by the revelation principle, sufficient to consider
• menus with two contracts
(ICj ) :
z ≡ {(xl , tl ), (xh , th )}
tj + cj (1 − xj ) ≥ t−j + cj (1 − x−j )
j ∈ {h, l}
seller: chooses a contract from available menus
• optimality with 2 menus
χi (z, z0 ) =




0




1
2 


 1 

if




<




ti + ci (1 − xi )
ti0 + ci (1 − xi0 ).
=




 > 
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Equilibrium definition
A symmetric equilibrium is a distribution Φ(z) such that almost all z satisfy,
1
Incentive compatibility:
tj + cj (1 − xj ) ≥ t−j + cj (1 − x−j )
2
j ∈ {h, l}
Seller optimality:
χi (z, z0 ) maximizes her utility
3
Buyers optimality:
z ∈ arg max
X
z
i∈{l,h}
χi (z, z )Φ(dz ) ,
Z
0
µi (vi xi − ti ) 1 − π + π
0
(1)
z0
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Characterization
No pure strategy eq. for π ∈ (0, 1) → mixing over multiple dimensions !
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Characterization
No pure strategy eq. for π ∈ (0, 1) → mixing over multiple dimensions !
Proceed in 4 steps
1. Show that menus can be summarized by a pair of utilities (uh , ul )
2. Prove that equilibrium is strictly rank-preserving (SRP)
3. Construct SRP equilibrium
4. Show that constructed equilibrium is unique
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A utility representation
Result
In all menus offered in equilibrium,
• the low types trades everything:
• ICl binds:
xl = 1
tl = th + cl (1 − xh )
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A utility representation
Result
In all menus offered in equilibrium,
• the low types trades everything:
• ICl binds:
xl = 1
tl = th + cl (1 − xh )
Result
Equilibrium menus can be represented by (uh , ul ) with corresponding allocations
tl = ul
xh = 1 −
uh − ul
ch − cl
th =
ul ch − uh cl
ch − cl
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A utility representation
Result
In all menus offered in equilibrium,
• the low types trades everything:
• ICl binds:
xl = 1
tl = th + cl (1 − xh )
Result
Equilibrium menus can be represented by (uh , ul ) with corresponding allocations
tl = ul
xh = 1 −
uh − ul
ch − cl
th =
ul ch − uh cl
ch − cl
Since we must have 0 ≤ xh ≤ 1,
ch − cl ≥ uh − ul ≥ 0
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A utility representation
Marginal distributions
Z
Fj (uj ) =
1 tj0 + cj 1 − xj0 ≤ uj dΦ z0
j ∈ {h, l}
z0
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A utility representation
Marginal distributions
Z
Fj (uj ) =
1 tj0 + cj 1 − xj0 ≤ uj dΦ z0
j ∈ {h, l}
z0
Then, each buyer solves
Π(uh , ul )
=
max
ul ≥cl , uh ≥ch
X
µj [1 − π + πFj (uj )] Πj (uh , ul )
j∈{l,h}
ch − cl ≥ uh − ul ≥ 0
s. t.
with Πl (uh , ul )
≡
vl xl − tl = vl − ul
Πh (uh , ul )
≡
vh xh − th = vh − uh
vh − ch
vh − cl
+ ul
ch − cl
ch − cl
8 / 31
A utility representation
Marginal distributions
Z
Fj (uj ) =
1 tj0 + cj 1 − xj0 ≤ uj dΦ z0
j ∈ {h, l}
z0
Then, each buyer solves
Π(uh , ul )
=
max
ul ≥cl , uh ≥ch
X
µj [1 − π + πFj (uj )] Πj (uh , ul )
j∈{l,h}
ch − cl ≥ uh − ul ≥ 0
s. t.
with Πl (uh , ul )
≡
vl xl − tl = vl − ul
Πh (uh , ul )
≡
vh xh − th = vh − uh
vh − ch
vh − cl
+ ul
ch − cl
ch − cl
Need to characterize the two linked distributions Fl and Fh !
8 / 31
Some useful results
Result
Fl and Fh have connected support and are continuous.
• Except for a knife-edge case (see paper)
• Proof more involved than standard case because of interdependencies
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Some useful results
Result
Fl and Fh 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(uh , ul ) is strictly supermodular.
• Intuition: ul ↑ ⇒ Πh ↑ ⇒ stronger incentives to attract high types
•
⇒ Uh (ul ) ≡ argmaxuh Π(uh , ul )
is weakly increasing
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Strict Rank Preserving
Violates no−mass−points in Fh
Uh
Uh
Violates no−mass−points in Fl
ul
ul
Theorem
Uh (ul ) is a strictly increasing function.
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Strict Rank Preserving
Violates no−mass−points in Fh
Uh
Uh
Violates no−mass−points in Fl
ul
ul
Theorem
Uh (ul ) is a strictly increasing function.
• Rank ordering of equilibrium menus identical across types
• Menus attract same fraction of both types Fl (ul ) = Fh (Uh (ul ))
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Strict Rank Preserving
Theorem
Uh (ul ) is a strictly increasing function.
Fl (ul ) = Fh (Uh (ul ))
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Equilibria: The two limit cases
Monopsony: π = 0
• µh < µ̄h ⇒ Sep. with xh = 0
• µh ≥ µ̄h ⇒
and Πl > Πh = 0
Pooling with xh = xl = 1 and Πh > 0 > Πl
12 / 31
Equilibria: The two limit cases
Monopsony: π = 0
• µh < µ̄h ⇒ Sep. with xh = 0
• µh ≥ µ̄h ⇒
and Πl > Πh = 0
Pooling with xh = xl = 1 and Πh > 0 > Πl
Bertrand: π = 1
• µh < µ̄h ⇒
Sep. with xh < 1, Πh = Πl = 0
• µh < µ̄h ⇒
Sep. with xh < 1, Π = 0, but Πh > 0 > Πl
Intuition: Higher µh
⇒
Relaxing IC l more attractive
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Types of equilibria
Cross-subsidization
No cross-subsidization
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Types of equilibria
High µh
Cross-subsidization
• Πh > 0 > Πl
• All separating, all pooling or a mix
Low µh
• Πl , Πh ≥ 0
No cross-subsidization
• All separating, Uh (ul ) 6= ul
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No cross-subsidization: Characterization
Marginal benefits vs costs of increasing ul
πfl (ul ) Πl
µh vh − ch
+
1 − π + πFl (ul )
1 − µh ch − cl
|
{z
}
|
{z
}
MB of more low types
=
1
|{z}
MC
MB of relaxing ICl
14 / 31
No cross-subsidization: Characterization
Marginal benefits vs costs of increasing ul
πfl (ul ) Πl
µh vh − ch
+
1 − π + πFl (ul )
1 − µh ch − cl
|
{z
}
|
{z
}
MB of more low types
=
1
|{z}
MC
MB of relaxing ICl
Boundary conditions
Fl (cl ) = 0
Fl (ūl ) = 1
→
Fl (ul )
Equal profit condition
[1 − π + πFl (ul )] Π(Uh , ul ) = Π
→
Uh (ul )
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No cross-subsidization: Distributions, π = 0.2
15 / 31
No cross-subsidization: Distributions, π = 0.5
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No cross-subsidization: Distributions, π = 0.9
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No cross-subsidization: Allocations
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No cross-subsidization: Allocations
Uh (ul ) − ul
ch − cl
xh (ul )
=
1−
xh0 (ul )
>
0 ⇔ Uh0 (ul ) > 1
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No cross-subsidization: π = 0.2
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No cross-subsidization: π = 0.5
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No cross-subsidization: π = 0.9
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No cross-subsidization: Welfare
Z
W = (1 − µh )(vl − cl ) + µh (vh − ch )
xh (ul )d F̂ (ul )
23 / 31
No cross-subsidization: Welfare
Z
W = (1 − µh )(vl − cl ) + µh (vh − ch )
0
π
xh (ul )d F̂ (ul )
1
23 / 31
No cross-subsidization: Welfare
Z
W = (1 − µh )(vl − cl ) + µh (vh − ch )
0
π
xh (ul )d F̂ (ul )
1
Intuition:
Profits change rapidly close to ul = vl
⇒ Uh0 (ul ) > 1 (because it has a bigger effect on profits) ⇒ xh0 < 0
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No cross-subsidization: Price vs quantity (conditional)
ph
π = 0.2
Correlation < 0 for suff. high π
π = 0.5
ph
A strategy to infer competitiveness ?
ph
π = 0.95
xh
24 / 31
No cross-subsidization: Policies
Do the following policies improve welfare ?
• Allowing insurance providers to discriminate based on observables
• Introducing credit scores in loan markets
• Requiring OTC market participants to disclose trades
25 / 31
No cross-subsidization: Policies
Do the following policies improve welfare ?
• Allowing insurance providers to discriminate based on observables
• Introducing credit scores in loan markets
• Requiring OTC market participants to disclose trades
Answer: Depends on π !
25 / 31
No cross-subsidization: Policies
Do the following policies improve welfare ?
• Allowing insurance providers to discriminate based on observables
• Introducing credit scores in loan markets
• Requiring OTC market participants to disclose trades
Answer: Depends on π !
• Can be mapped into a mean-preserving spread of µh
• W is linear when π = 0 and π = 1 ⇒ no effect on welfare
• W is concave when π is high ⇒ bad for welfare
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Cross-subsidization
No cross-subsidization
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Cross-subsidization
No cross-subsidization
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Cross-subsidization: Separating
All equilibrium menus
• cross-subsidize Πl < 0
• separate (except the best
one) Uh (ul ) > ul
No cross-subsidization
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Cross-subsidization: Pooling
All equilibrium menus
• cross-subsidize Πl < 0
• pool Uh (ul ) = ul
No cross-subsidization
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Cross-subsidization: Semi-Pooling
Equilibrium menus
• cross-subsidize Πl < 0
• separate at the high end,
Uh (ul ) > ul iff ul > ûl
No cross-subsidization
• and pool at the low end,
Uh (ul ) = ul iff ul ≤ ûl
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Conclusion
Methodological contribution
• Imperfect competition and adverse selection with optimal contracts
• Rich predictions for the distribution of observed trades
Substantive insights
• Depending on parameters, pooling and/or separating menus in equilibrium
• Competition, transparency can be bad for welfare
Work in progress
• Generalize to N types, curved utility
• Non-exclusive trading
31 / 31