Course 3 - Adverse selection

Types of Information Asymmetry

• Agent’s Characteristics (Adverse Selection): The agent (buyer) has private information about their own characteristics (e.g., willingness to pay).

• Agent’s Actions (Moral Hazard): The agent’s actions are not observable by the principal (seller), leading to issues in incentivizing the agent effectively.

Buyer-Seller Situation

Seller not know whether the buyer has a high / low willingness to pay => This uncertainty affects pricing strategies & contracts offered

Principal-agent Relationship

Principal (seller) designs a contract to manage agent's (buyer's) behavior based on their private information.

Principal must consider agent's constraints & ensure participation

Goal is to incentivize the agent to reveal their type through the contract

Utility Function of Buyer

Reflects how much buyer values the good based on their type (θ) => Higher θ indicates higher valuation

u(q,T,θ) = θv(q) − T

• q = quantity/quality of good purchased

• T = amount paid to seller

• v(q) = value derived from the quantity q, with specific properties of function indicating diminishing returns

Seller’s Profit Function

Calculated as:

π = T − cq

• c = marginal cost of providing the good

First Best Outcome : NO Information Asymmetry

Seller can :

• perfectly price discriminate &

• maximize profits

by offering tailored contracts to each buyer type

First Best Outcome : WITH Information Asymmetry - Linear Pricing

• Seller ≠ distinguish between buyer types

• Offer a single linear contract (T = Pq)

u(q,T,θ) = θ_iv(q) − T= θ_iv(q) − Pq

Buyer - Problem

Buyer chooses q based on their private valuations => leading to inefficiencies (H types may not fully reveal their willingness to pay)

=> BUT = optimal for seller to serve both types

Single price = charged per unit of good / service

=> not fully extract consumer surplus / account for variations in consumer valuations

First Best Outcome : WITH Information Asymmetry - Two-Part Tariffs

A method of contract design that allows seller to extract more surplus by:

• charging a fixed fee + variable cost (tailored to different buyer types)

Mechanism helps firms capture more consumer surplus while ensuring participation, but it complicates the incentive structure

Second Best Outcome : WITH Information Asymmetry - Nonlinear Pricing

With adverse selection, seller cannot achieve the first-best outcome & must design contracts that accommodate IA (information asymmetry)

Profit = β [T(qL​) − c⋅qL​] + (1−β) [T(qH​) − c⋅qH​]

• T(q) = total payment from consumers based on q consumed

• c = cost of producing the good

• β = weight given to low types versus high types

Prices vary with the quantity consumed (e.g., quantity discounts). This can be used to tailor prices to different consumer segments, optimizing revenue but requiring complex incentive considerations to ensure truthful reporting of private information.

High types enjoy informational rents due to their private information, leading to inefficiencies in the market.

Second Best Outcome : 4 Constraints

Incentive Constraint for High Type (ICH):

θHv(qH) − TH ≥ θHv(qL) − TL

=> H must receive higher utility from choosing contract designed for them compared to the one for L.

Incentive Constraint for Low Type (ICL):

θLv(qL) − TL ≥ θLv(qH) − TH

=> Ensures that L prefers their own contract over H’s contract

Individual Rationality Constraints (IRH):

θHv(qH) − TH ≥ 0

=> Ensures that H prefers own contract

Individual Rationality Constraints (IRL):

θLv(qL) − TL ≥ 0

=> Ensures that L prefers own contract

Second Best Outcome : Steps analysis

1. Apply the Revelation Principle: 

Simplify the analysis by considering only the contracts designed specifically for each type, thereby focusing on [qL,T(qL)] & [qH,T(qH)]

2. Identify Binding Constraints:

In optimal pricing, some constraints will bind at the solution, meaning they will be equalities rather than inequalities.

3. Relax the Problem:

the seller's profit-maximization problem without the non-binding constraints.

4. Eliminate T_L​ and T_H​:

Substitute the constraints back into the profit function to express the maximization purely in terms of quantities.

5. FOC:

Derive the conditions necessary for optimality (where marginal benefits equal marginal costs):

• For High Type: θHv′(qH) = c

• For Low Type: θLv′(qL) = c / (1−(1−β) / (β⋅(θH − θL) / θL))

These conditions allow the seller to determine the optimal quantities to provide for each type.

First-Best Outcome vs. Second-Best Outcome

First-Best Outcome:

This occurs in a world with perfect information, where :

• all agents have complete knowledge of their valuations & the cost structures

• Resources = allocated efficiently, maximizing total surplus without any distortion.

Second-Best Outcome:

This refers to scenarios with private information & market imperfections.

• Achieving efficiency = impossible due to constraints

• Allocation = still optimized but may lead to some inefficiencies (informational rents & deadweight losses)

Key results of these models

• Informational Rent:

  This refers to the surplus that the informed party retains due to their private information.

  → For example, in a market where buyers know their valuations, sellers may have to leave some surplus for buyers to induce participation.

• Inefficient Consumption:

  When agents do not reveal their true valuations, it can lead to misallocation of resources.

  → For instance, if a buyer under-reports their valuation, they might consume less than the socially optimal quantity, leading to deadweight loss.