lecture three: adverse selection - hkuccfour/eo3.pdf · lecture three: adverse selection cheng chen...

Lecture Three: Adverse Selection

Cheng Chen

School of Economics and Finance

The University of Hong Kong

(Cheng Chen (HKU)) Econ 6006 1 / 14

Introduction

Motivation

Examples:I The lemon market (Akerlof, 1970).I The insurance market.

I Dropouts of classes after the exam.I Others?

Why do we care about it?I Information asymmetry is everywhere.I Neoclassical economics and frictions. (Pareto improvement and the

mechanism design).I Market e�ciency and welfare (e.g., market shutdown).

Introduction

Motivation

Examples:I The lemon market (Akerlof, 1970).I The insurance market.I Dropouts of classes after the exam.I Others?

Introduction

Motivation

Why do we care about it?I Information asymmetry is everywhere.

I Neoclassical economics and frictions. (Pareto improvement and themechanism design).

I Market e�ciency and welfare (e.g., market shutdown).

Introduction

Motivation

Introduction Preliminaries (Section 1.3.1)

An Example

Adverse selection: Hidden information.A bilateral contracting: an employer (the principal) and an employee(the agent).

I The employee has one unit of time.I Output equals θi (1− li ), where i ∈ H, L (θH > θL two types).I The employee knows his type, while the employer does not.

I The employer's utility is

U(αθi (1− li )− ti ),

and the employee's utility is

u(θi li + ti ) ≥ u(θi ).

Assume that ex post output is unobservable and α > 1.

No asymmetric information→ θi = ti and li = 0 for all i (First-Best orFB). (Derivation?)

Not incentive compatible: L type guy wants to mimic H type guy.

(Why?)

An Example

I The employee has one unit of time.I Output equals θi (1− li ), where i ∈ H, L (θH > θL two types).I The employee knows his type, while the employer does not.I The employer's utility is

U(αθi (1− li )− ti ),

(Why?)

An Example

U(αθi (1− li )− ti ),

(Why?)

An Example

U(αθi (1− li )− ti ),

(Why?)

Mathematical Formulation

Maximize the principal's payo� under constraints:

maxlj ,tj

pLU(αθL(1− lL)− tL) + pHU(αθH(1− lH)− tH)

s.t. u(θLlL + tL) ≥ u(θL),

u(θH lH + tH) ≥ u(θH),

u(θH lH + tH) ≥ u(θH lL + tL),

u(θLlL + tL) ≥ u(θLlH + tH).

Revelation Principle.

First two inequalities: Individual Rationality (IR) or Participation

Constraints (PC).

Last two inequalities: Incentive Compatibility (IR) constraints.

Mathematical Formulation

Maximize the principal's payo� under constraints:

maxlj ,tj

pLU(αθL(1− lL)− tL) + pHU(αθH(1− lH)− tH)

s.t. u(θLlL + tL) ≥ u(θL),

u(θH lH + tH) ≥ u(θH),

u(θH lH + tH) ≥ u(θH lL + tL),

u(θLlL + tL) ≥ u(θLlH + tH).

Revelation Principle.

First two inequalities: Individual Rationality (IR) or Participation

Constraints (PC).

Last two inequalities: Incentive Compatibility (IR) constraints.

Basic Principles

One option: lH = lL = 0 and tH = tL = θH (a simple contract).I L type agent: information rents (monopoly power).I H type agent: no allocative distortion.

In general, the tradeo� between information rents and allocativee�ciency.

I The agent that has incentive to mimic (L type): information rents.I The agent that is mimicked (H type): allocative distortion.

How to solve the problem?I If the uninformed party makes the contract: screening. (i.e., Mirrlees

(1971), Myerson (1979), Stiglitz and Weiss (1981))I If the informed party makes the contract: signalling. (i.e., Spence

(1973, 1974))

Basic Principles

(1973, 1974))

Basic Principles

(1973, 1974))

Screening: Two Types (Section 2.1) Theory

Two Types: Setup

Classical references: Mussa and Rosen (1978) and Maskin and Riley

(1984a)

A bilateral contracting: a seller (the principal) and a buyer (theagent).

I The seller's payo�: π = T − cq.I The buyer's payo�: u(θi , q,T ) = θiv(q)− T ≥ u where i ∈ {H, L}.I We have θH > θL.I Asymmetric information: seller does not know θi . Prob(θ = θL) = β.

FB (no information friction):

maxTi ,qi

Ti − cqi

s.t. θiv(qi )− Ti ≥ u.

Solution (type-speci�c two-part tari�s): θiv′(qi ) = c and

θiv(qi ) = Ti + u.

Two Types: Setup

(1984a)

maxTi ,qi

Ti − cqi

θiv(qi ) = Ti + u.

Two Types: Setup

(1984a)

maxTi ,qi

Ti − cqi

θiv(qi ) = Ti + u.

Incentive CompatibilityHowever, this is not incentive-compatible for H type. I.e., H type

wants to mimic L type:

TL = θLv(qL)− u

If H type takes L type's contract, he receives

θHv(qL)− TL = (θH − θL)v(qL) + u > u.

On the other hand, payment to H type is

TH = θHv(qH)− u

Incentive compatible for L type. If L type takes H type's contract, he

receives

θLv(qH)− TH = (θL − θH)v(qH) + u < u.

Intuition: Transfer to employer is too much for H type, since its

willingness to pay is high.

TL = θLv(qL)− u

TH = θHv(qH)− u

receives

willingness to pay is high.

TL = θLv(qL)− u

TH = θHv(qH)− u

receives

willingness to pay is high.(Cheng Chen (HKU)) Econ 6006 7 / 14

Two Types: Linear Pricing

Normalize u = 0.

Suppose it is optimal to sell to both types of agents.

Linear pricing: (q, P) (one type of contract)

q: quantity; P : unit price.

From F.O.C. θiv′(q) = P , we derive demand function: qi = Di (P).

Buyer's payo�:

Si (P) = θiv [Di (q)]− PDi (P).

D(P) = βDL(P) + (1− β)DH(P)

S(P) = βSL(P) + (1− β)SH(P).

Normalize u = 0.

Buyer's payo�:

D(P) = βDL(P) + (1− β)DH(P)

S(P) = βSL(P) + (1− β)SH(P).

Normalize u = 0.

Buyer's payo�:

D(P) = βDL(P) + (1− β)DH(P)

S(P) = βSL(P) + (1− β)SH(P).

Normalize u = 0.

Buyer's payo�:

D(P) = βDL(P) + (1− β)DH(P)

S(P) = βSL(P) + (1− β)SH(P).

Normalize u = 0.

Buyer's payo�:

D(P) = βDL(P) + (1− β)DH(P)

S(P) = βSL(P) + (1− β)SH(P).

Two Types: Linear Pricing (Cont.)

Maximization problem for seller:

(P − c)D(P),

where θiv′(Di (P)) = P .

Solution:

Pm = c − D(P)

D ′(P).

Rents for both types of buyers:

SL(P) > 0; SH(P) > 0.

Ine�ciently low consumption:

θiv′(q) = P > c.

(P − c)D(P),

Solution:

Pm = c − D(P)

D ′(P).

SL(P) > 0; SH(P) > 0.

θiv′(q) = P > c.

(P − c)D(P),

Solution:

Pm = c − D(P)

D ′(P).

SL(P) > 0; SH(P) > 0.

θiv′(q) = P > c.

Two Types: Single Two-Part Tari�

Single two-part tari�: (Z , P) (one type of contract)

Z : �xed fee. P : unit price.I H type always participates.

Z + (P − c)D(P),

where Z = SL(P) and SL(P) = θLv [DL(P)]− PDL(P): buyer's netsurplus.

I Solution:Pd = c − D(P)+S′L(P)

D′ (P)

and S′L(P) = −D

′L(P) > 0.

Extract all rents from L type and leave some rents to H type.

Single two-part tari� is better than linear pricing, and

Pm > Pd > Pc = c .

Z + (P − c)D(P),

D′ (P)

and S′L(P) = −D

′L(P) > 0.

Pm > Pd > Pc = c .

Z + (P − c)D(P),

D′ (P)

and S′L(P) = −D

′L(P) > 0.

Pm > Pd > Pc = c .

Graphical Representation

Two Types: IC Contracts

Second-Best outcome (SB) and nonlinear pricing (remember the

revelation principle):

maxTi ,qi

β[T (qL)− cqL] + (1− β)[T (qH)− cqH ]

s.t. θHv(qH)− TH ≥ θHv(qL)− TL, (ICH)

θLv(qL)− TL ≥ θLv(qH)− TH , (ICL)

θHv(qH)− TH ≥ 0, (IRH)

θLv(qL)− TL ≥ 0. (IRL)

First two inequalities: IRs.

Last two inequalities: ICs.

Two Types: IC Contracts

Second-Best outcome (SB) and nonlinear pricing (remember the

revelation principle):

maxTi ,qi

β[T (qL)− cqL] + (1− β)[T (qH)− cqH ]

s.t. θHv(qH)− TH ≥ θHv(qL)− TL, (ICH)

θLv(qL)− TL ≥ θLv(qH)− TH , (ICL)

θHv(qH)− TH ≥ 0, (IRH)

θLv(qL)− TL ≥ 0. (IRL)

First two inequalities: IRs.

Last two inequalities: ICs.

IC Contracts: Two Binding Constraints

IRH is redundant, because

θHv(qH)− TH ≥ θHv(qL)− TL > θLv(qL)− TL ≥ 0.

IRH is redundant → ICH holds with equality.I If not, principal can increase tH until ICH holds with equality.

ICH + ICL → qH ≥ qL (monotonicity):

θH [v(qH)− v(qL)] ≥ TH − TL ≥ θL[v(qH)− v(qL)]→ qH ≥ qL.

ICH holds with equality + (qH ≥ qL) → ICL is redundant.

θH [v(qH)− v(qL)] = TH − TL

→ θL[v(qH)− v(qL)] ≤ TH − TL

→ θLv(qL)− TL ≥ θLv(qH)− TH .

ICL is redundant → IRL should hold with equality.I If not, principal can increase tL until IRL holds with equality.

IC Contracts: Simpli�ed Problem

Reduced problem:

maxqL,qH

β[θLv(qL)− cqL] + (1− β)[θHv(qH)− (θH − θL)v(qL)− cqH ],

where (θH − θL)v(qL) captures information rents.

Solution:I No distortion at the top: θHv

′(q∗H ) = c .

I Downward distortion at the bottom (due to information rents):

θLv′(q∗L) =

1−ββ

θH−θLθL

) > c .

Tradeo� between allocative e�ciency and rent reduction.

IC Contracts: Simpli�ed Problem

Reduced problem:

maxqL,qH

β[θLv(qL)− cqL] + (1− β)[θHv(qH)− (θH − θL)v(qL)− cqH ],

where (θH − θL)v(qL) captures information rents.

Solution:I No distortion at the top: θHv

′(q∗H ) = c .

I Downward distortion at the bottom (due to information rents):

θLv′(q∗L) =

1−ββ

θH−θLθL

) > c .

Tradeo� between allocative e�ciency and rent reduction.

lecture three: adverse selection - hkuccfour/eo3.pdf · lecture three: adverse selection cheng chen...

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