EC476 Contracts and Organizations, Part III: Lecture 2 Leonardo - - PowerPoint PPT Presentation

EC476 Contracts and Organizations, Part III: Lecture 2

Leonardo Felli

32L.G.06

19 January 2015

Moral Hazard:

◮ Consider the contractual relationship between two agents (a

principal and an agent)

◮ The principal hires the agent to perform a task. ◮ The agent chooses his effort intensity, a, which affects the

• utcome of the task, q.

◮ The principal only cares about the outcome, but effort is

costly for the agent, hence the principal has to compensate the agent for incurring the cost of effort.

◮ Effort is observable only to the agent, (it is the agent’s private

information).

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 2 / 48

General Setup

◮ Assume that the outcome of the task can take only two

values: q ∈ {0, 1}.

◮ We assume that when q = 1 the task is successful and when

q = 0 the task is a failure.

◮ The probability of success is:

P{q = 1|a} = p(a), p′(·) > 0, p′′(·) < 0. where p(0) = 0, lim

a→∞ p(a) = 1, and p′(0) > 1. ◮ The principal’s preferences are represented by:

V (q − w), V ′(·) > 0, V ′′(·) ≤ 0 where w is the transfer to the agent.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 3 / 48

General Setup (2)

◮ The agent’s preferences are represented by the utility function

separable in income and effort: U(w) − φ(a), U′(·) > 0, U′′(·) ≤ 0 where φ′(·) > 0, φ′′(·) ≥ 0.

◮ For convenience we take

φ(a) = a and we normalize the agent’s outside option: U = 0.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 4 / 48

General Setup (3)

Assume that:

◮ a is chosen by the agent before uncertainty is realized; ◮ a is only observed by the agent. It is his private information. ◮ q is verifiable information (observable to all agents involved in

the contract Court included).

◮ the transfer w can only be contingent on the verifiable

information q.

◮ q is not in a one-to-one relation with the effort a.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 5 / 48

First Best Benchmark

◮ The contract theory literature defines the first best world as a

world where there are no frictions.

◮ In the current setting this implies that the contract offered by

the principal can be contingent on the effort a.

◮ In other words, the effort a is verifiable (observable to all

agents involved in the contract Court included).

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 6 / 48

First Best Contract

◮ The first best contract is obtained as the solution to the

problem: max

a,wi

p(a) V (1 − w1) + (1 − p(a)) V (−w0) s.t. p(a) U(w1) + (1 − p(a)) U(w0) ≥ a

◮ The optimal pair of transfers w∗ 1 and w∗ 0 are such that the

following FOC (Borch optimal risk-sharing rule) are satisfied: V ′(1 − w∗

1 )

U′(w∗

1 )

= V ′(−w∗

0 )

U′(w∗

0 )

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 7 / 48

First Best Contract (2)

◮ These transfers are paid only if the effort level coincides with

a∗ that satisfies the following FOC: p′(a∗) V (1 − w∗

1 ) − V (−w∗ 0 )

V ′(1 − w∗

1 )

+ U(w∗

1 ) − U(w∗ 0 )

U′(w∗

1 )

• =

1 U′(w∗

1 ) ◮ Finally the agent’s expected utility coincides with the outside

• ption:

p(a∗) U(w∗

1 ) + (1 − p(a∗)) U(w∗ 0 ) = a∗

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 8 / 48

First Best Contract – Risk Neutrality

◮ If the principal is risk neutral:

V (x) = x

◮ Then the conditions above become:

w∗

1 = w∗ 0 = w∗

and U(w∗) = a∗, p′(a∗) = 1 U′(w∗)

◮ If, instead, the agent is risk neutral:

U(x) = x

◮ Then the optimum entails:

w∗

1 − w∗ 0 = 1,

p′(a∗) = 1.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 9 / 48

Second Best Contract

◮ If a is not verifiable then, for every w1 and w0, a is determined

so that: max

a

p(a) U(w1) + (1 − p(a)) U(w0) − a (1)

◮ The latter is the agent’s incentive problem. ◮ Only the agent controls a and hence incentives for the agent

to choose the principal’s desired level of a have to be induced through the contingent trasfer w(q).

◮ In other words, the second best contract can be contingent

• nly on q.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 10 / 48

Second Best Contract (2)

◮ The second best contract can be obtained as the solution to

the problem: max

ˆ a,wi

p(ˆ a) V (1 − w1) + (1 − p(ˆ a)) V (−w0) s.t. p(ˆ a) U(w1) + (1 − p(ˆ a)) U(w0) ≥ ˆ a ˆ a ∈ arg max

a

p(a) U(w1) + (1 − p(a)) U(w0) − a

◮ The first constraint is known as the agent’s individual

rationality constraint,

◮ The second constraint is known as the agent’s incentive

compatibility constraint.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 11 / 48

Second Best Contract (3)

◮ The FOC of the incentive compatibility constraint are:

p′(ˆ a) [U(w1) − U(w0)] = 1 (2)

◮ A first observation: from this condition it is clear that full

insurance: w1 = w0 leads to no incentives: p(0) = 0

◮ Assumptions on p(·) imply that the solution to this condition

is unique for any pair (w0, w1).

◮ We can replace the agent’s (IC) by the set of FOC in (2). ◮ In general replacing the (IC) constraint with the FOC of the

agent’s effort choice problem is not a valid approach.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 12 / 48

Risk Neutral Agent

◮ Consider now the case in which the agent is risk neutral:

U(x) = x we have seen that first best optimality requires p′(a∗) = 1

◮ In this case the FOC of the (IC) constraint becomes:

p′(ˆ a)(w1 − w0) = 1

◮ Therefore setting

w1 − w0 = 1 leads to the first best allocation: optimal risk sharing and

• ptimal incentives.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 13 / 48

Risk Neutral Agent (2)

◮ The reason is that:

◮ optimal risk sharing requires that the agent bears all the risk in

the environment,

◮ optimal incentives requires that the agent is residual claimant.

◮ This is achieved by selling the activity to the agent at a fix

price −w0 > 0 so that the risk averse principal receives full insurance.

◮ Notice that in this case we need the agent to have deep

enough pockets: when the outcome is q = 0 the agent’s payoff is w0 < 0.

◮ The agent must be willing to incur a loss with a strictly

positive probability.

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Resource Constrained Agent

◮ It is often natural to assume that the agent has no resources

to put in the activity.

◮ This implies a resource constraint: wi ≥ 0. ◮ In this case the problem becomes:

max

ˆ a,wi

p(ˆ a) V (1 − w1) + (1 − p(ˆ a)) V (−w0) s.t. p(ˆ a) w1 + (1 − p(ˆ a)) w0 ≥ ˆ a p′(ˆ a)(w1 − w0) = 1 wi ≥ 0 ∀i ∈ {0, 1}

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 15 / 48

Resource Constrained Agent (2)

◮ In the situation in which the agent is resource constrained not

all the risk can be transferred to the agent: the constraint wi ≥ 0 will be binding for the transfer w0: w0 = 0

◮ It is still possible to create first best incentives but for this

purpose the agent’s needs to be rewarded.

◮ If w1 − w0 = 1 then the agent’s payoff is:

p(a∗) − a∗ > 0 since p′(0) > 1 and p′(a∗) = 1.

◮ In other words the (IR) constraint is not binding. This is not

necessarily optimal for the principal.

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Resource Constrained Agent (3)

◮ In particular, if we assume that the principal is risk neutral as

well: V (x) = x then the principal’s problem is: max

ˆ a,wi

p(ˆ a) (1 − w1) − (1 − p(ˆ a)) w0 s.t. p(ˆ a) w1 + (1 − p(ˆ a)) w0 ≥ ˆ a p′(ˆ a)(w1 − w0) = 1 wi ≥ 0 ∀i ∈ {0, 1}

◮ The solution implies that

w0 = 0, w1 = 1 p′(ˆ a)

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 17 / 48

Resource Constrained Agent (4)

◮ Moreover, ˆ

a solves the constrained problem: max

ˆ a

p(ˆ a) (1 − w1) s.t. p′(ˆ a) w1 = 1

• r

p′(ˆ a) = 1 − p(ˆ a) p′′(ˆ a) (p′(ˆ a))2

◮ Given that p′′(·) < 0 then we conclude:

ˆ a < a∗.

◮ The resource constraint implies a second best level of effort.

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Resource Constrained Agent (5)

◮ The principal trades off the lower effort choice by the agent

against the higher compensation that the agents needs to provide for the first best level of effort.

◮ However the agent still gets a strictly positive payoff:

p(ˆ a) p′(ˆ a) − ˆ a > 0

◮ Indeed, by Taylor expansion we can show that there exists

ξ ∈ (0, ˆ a) such that p(ˆ a) − p′(ˆ a) ˆ a = −p′′(ξ) ˆ a2 2 > 0

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 19 / 48

Risk Averse Principal and Agent: First Best Contract

◮ Recall now that the first best contract with risk averse

principal and agent led to the Borch optimal risk-sharing rule: V ′(1 − w∗

1 )

U′(w∗

1 )

= V ′(−w∗

0 )

U′(w∗

0 ) ◮ and the optimal effort level a∗ that satisfies:

p′(a∗) {V (1 − w1) − V (−w0) + λ[U(w1) − U(w0)]} = λ

• r

p′(a∗) V (1 − w∗

1 ) − V (−w∗ 0 )

V ′(1 − w∗

1 )

+ U(w∗

1 ) − U(w∗ 0 )

U′(w∗

1 )

• =

1 U′(w∗

1 )

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 20 / 48

Risk Averse Principal and Agent: Second Best Contract

◮ Consider now the second best contract, this is the solution to

the following problem max

ˆ a,wi

p(ˆ a) V (1 − w1) + (1 − p(ˆ a)) V (−w0) s.t. p(ˆ a) U(w1) + (1 − p(ˆ a)) U(w0) ≥ ˆ a p′(ˆ a) [U(w1) − U(w0)] = 1

◮ Let λ and µ be the lagrange multipliers of the (IR) and (IC)

constraints, respectively.

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Risk Averse Principal and Agent: Second Best Contract (2)

◮ The FOC with respect to w1 and w0 imply:

V ′(1 − w1) U′(w1) = λ + µ p′(ˆ a) p(ˆ a) and V ′(−w0) U′(w0) = λ − µ p′(ˆ a) 1 − p(ˆ a)

◮ Clearly for µ = 0 we get back Borch rule, however in general

µ > 0: optimal insurance is distorted.

◮ Since V ′′(·) < 0 and U′′(·) < 0 the agent faces in equilibrium

more risk than he would face in the absence of moral hazard: w1 > w∗

1 ,

w0 < w∗

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Risk Averse Principal and Agent: Second Best Contract (3)

◮ The second best level of effort ˜

a is such that: p′(˜ a) {V (1 − w1) − V (−w0) + λ[U(w1) − U(w0)]} = λ − µ p′′(˜ a) [U(w1) − U(w0)]

◮ Recall that the first best level of effort a∗ is such that:

p′(a∗) {V (1 − w1) − V (−w0) + λ[U(w1) − U(w0)]} = λ

◮ We therefore can conclude that if the (IC) constraint is

binding — µ > 0 — then in equilibrium the agent under-invests: ˜ a < a∗

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 23 / 48

Adverse Selection

◮ Static Adverse Selection problem: one principal facing one

agent who has private information on his type (preferences, intrinsic productivity).

◮ We consider the simple monopolist pricing model: a

transaction between a buyer (the agent) and a seller (the principal).

◮ The seller sets the terms of the contract (tioli from the

principal to the agent).

◮ The seller does not know how much the buyer is willing to pay

for the commodity.

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Static Adverse Selection: Setup

◮ The buyer’s preferences are represented by:

U(q, T, θi) = q p(x, θi) dx − T

◮ T total transfer from the buyer to the seller, ◮ θi preference characteristics of the buyer, ◮ p(x, θi) inverse (Marshallian) demand curve of the buyer. ◮ A special and convenient case is:

U(q, T, θi) = θi u(q) − T where u′(·) > 0, u′′(·) < 0, u(0) = 0, lim

q→0 u′(q) = +∞.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 25 / 48

Static Adverse Selection: Setup (2)

◮ The seller is risk neutral, the unit’s cost of production is c > 0

and her profit for selling q units in exchange for T is: Π = T − c q

◮ Question: what is the profit maximizing set of pairs (T, q)

the seller will be able to induce the buyer to choose (price discriminating monopolist)?

◮ Assume that:

θi ∈ {θL, θH} and λ = Pr{θi = θL}

◮ Let U be the buyer’s outside option, normalized: U = 0.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 26 / 48

Static Adverse Selection: First best

◮ Assume that the seller is perfectly informed on each buyer’s

type θi.

◮ The contract is then (T ∗ i , q∗ i ), for i ∈ {L, H} ◮ The seller’s problem is:

max

Ti,qi

Ti − c qi s.t. θi u(qi) − Ti ≥ 0

◮ The constraint is known as the individual rationality (IR)

constraint of the agent.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 27 / 48

Static Adverse Selection: First best (2)

◮ The solution is such that:

θi u′(q∗

i ) = c,

∀i ∈ {L, H} and T ∗

i = θi u(q∗ i ),

∀i ∈ {L, H}

◮ The seller chooses a quantity q∗ i so that marginal utility

equals marginal cost (efficiency),

◮ extracts the consumer’s total willingness to pay by means of

the transfer T ∗

i . ◮ The seller’s total expected profit:

λ (T ∗

L − c q∗ L) + (1 − λ) (T ∗ H − c q∗ H)

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 28 / 48

Static Adverse Selection: Second best:

◮ If the seller cannot observe the buyer’s type then she has to

• ffer the same contract to both types.

◮ In other words the seller may offer to the agent (whatever his

type) a set of choices {(TL, qL), (TH, qH)}

◮ The problem is that the contract space is potentially very

large: the set of functions T(q), of all shapes and features.

◮ Fortunately, the Revelation Principle simplifies the search for

the best contract from the principal’s perspective.

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Revelation Principle:

◮ General procedure: we transform a problem we cannot solve

(contract space not well defined) in a problem in which all possible contracts are well defined and simple to manage.

◮ Each agent i observes its own preference characteristic: θi. ◮ If the principal has all the bargaining power he chooses the

mechanism (from the set of all possible games) which has the best equilibrium from her view point (mechanism design).

◮ The principal is a (Stackelberg) leader, she selects the game

the agents will play so that the equilibrium of the agent’s subgame is the best one from her viewpoint.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 30 / 48

Centralize the Mechanism:

◮ The first step consists in moving from the problem where the

agent chooses the price (transfer) and quantity to a communication mechanism.

◮ A communication mechanism is a mechanism where the agent

reports his private information ˆ θ (the agent of course can lie) and the principal associates this report with a transfer T(ˆ θi) and quantity q(ˆ θi).

◮ Clearly there is no loss in generality in this first step. ◮ The revelation principle (Green and Laffont 1977, Myerson

1979, Harris and Townsend 1981, Dasgupta, Hammond and Maskin 1979) identifies the set of mechanisms among which the principal selects.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 31 / 48

Using Revelation Principle

◮ The second step consists in focussing on an equilibrium of the

direct revelation mechanism (communication game) where the agent reports his true type.

◮ Revelation Mechanism guarantees that there exists no loss in

generality in focussing on this type of equilibrium: direct revelation mechanism with truth-telling.

◮ The equilibrium is selected so as to maximize the principal’s

(mechanism designer’s) payoff.

Result (Revelation Mechanism)

Every equilibrium of the communication game (indirect revelation game) corresponds to an equilibrium of the direct relegation game where it is optimal for each participating party to tell the truth about his type.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 32 / 48

Timing of revelation game:

◮ The principal selects the contract: the communication game

• ffered to the agent.

◮ The agent decides whether to participate (accept the contract

• r not).

◮ The agent sends his message ˆ

θi to the principal.

◮ The principal implements the allocation associated with the

message received (T(ˆ θi), q(ˆ θi)).

◮ We focus on the communication games where it is optimal for

the agent to tell the truth: ˆ θi = θi.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 33 / 48

The Static Adverse Selection Problem:

Step 1: By revelation principle the principal’s problem identifies the direct revelation mechanism (Ti, qi) = (T(q(ˆ θi)), q(ˆ θi)), i ∈ {L, H} that solves the problem: max

Ti,qi

λ (TL − c qL) + (1 − λ)(TH − c qH) s.t. θH u(qH) − TH ≥ θH u(qL) − TL θL u(qL) − TL ≥ θL u(qH) − TH θH u(qH) − TH ≥ 0 θL u(qL) − TL ≥ 0

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 34 / 48

The Static Adverse Selection Problem (2):

◮ Two constraints guarantee individual rationality (IR), in other

words they guarantee that both type of agents are willing to accept the contract offered by the principal: θH u(qH) − TH ≥ 0 θL u(qL) − TL ≥ 0

◮ Two constraints guarantee incentive compatibility (IC), in

• ther words they guarantee that the contract offered by the

principal is such that in equilibrium both types of agent report the truth about their type: θH u(qH) − TH ≥ θH u(qL) − TL θL u(qL) − TL ≥ θL u(qH) − TH

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 35 / 48

The Static Adverse Selection Problem (3):

Step 2: The individual rationality constraint of the type H will not bind at the optimum. Indeed since θH > θL: θH u(qH) − TH ≥ θH u(qL) − TL > θL u(qL) − TL ≥ 0 Step 3: Solve the relaxed problem that ignores the (ICL) constraint. To select which constraint to omit consider the two (IC) constraints at the first best optimum: θH u(q∗

H) − T ∗ H = 0,

θH u(q∗

L) − T ∗ L = (θH − θL) u(q∗ L) > 0

clearly θH u(q∗

H) − T ∗ H < θH u(q∗ L) − T ∗ L

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 36 / 48

The Static Adverse Selection Problem (4):

While θL u(q∗

L) − T ∗ L = 0,

θL u(q∗

H) − T ∗ H = (θL − θH) u(q∗ H) < 0

clearly θL u(q∗

L) − T ∗ L > θL u(q∗ H) − T ∗ H

Therefore the key (IC) constraint is the one of the H-type. The reason why the (IC) constraint of only one type of agent binds is Spence-Mirrlees Single Crossing Condition: ∂ ∂θ

• − ∂U/∂q

∂U/∂T

• = u′(q) > 0

Marginal utility of consumption (relative to the marginal utility of money) rises with θ. This is key to be able to separate the two types.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 37 / 48

The Static Adverse Selection Problem (5):

Step 4: Notice that the relaxed problem is such that both constraints bind at the optimum: max

Ti,qi

λ (TL − c qL) + (1 − λ)(TH − c qH) s.t. θH u(qH) − TH ≥ θH u(qL) − TL θL u(qL) − TL ≥ 0 Proof: If (ICH) does not bind then the principal can raise TH without affecting (IRL), while improving the maximand. In other words the (ICH) is binding.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 38 / 48

The Static Adverse Selection Problem (6):

Solve the binding (ICH) for TH: TH = θH(u(qH) − u(qL)) + TL Substituting in the maximand, the relaxed problem becomes: max

Ti,qi

λ (TL − c qL) + (1 − λ)[(θH(u(qH) − u(qL)) + TL − c qH] s.t. θL u(qL) − TL ≥ 0 The maximand is monotonic increasing in TL while (IRL) is monotonic decreasing in TL. Hence, at the optimum (IRL) must be binding.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 39 / 48

The Static Adverse Selection Problem (7):

Step 5: Solve the binding (IRL) for TL and substitute them into the maximand. We get: max

qi

λ [θL u(qL) − c qL] + + (1 − λ) [θH u(qH) − (θH − θL) u(qL) − c qH]

• r

max

qi

[λ θL − (1 − λ) (θH − θL)] u(qL)−λ c qL + + (1 − λ) [θH u(qH) − c qH] The second best contract (q∗∗

i , T ∗∗ i ) is then the solution to the

unconstraint maximization problem above.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 40 / 48

The Static Adverse Selection Problem (8):

To characterize the solution we distinguish two cases. Case 1: [λ θL − (1 − λ) (θH − θL)] ≤ 0 In this case the slope of the maximand with respect to qL is strictly negative for every qL ≥ 0: [λ θL − (1 − λ) (θH − θL)] u′(qL) − λ c < 0 Therefore the principal chooses qL at a corner: q∗∗

L = 0,

T ∗∗

L

= 0

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 41 / 48

The Static Adverse Selection Problem (9):

In other words the principal decides not to serve the type θL of the agent. The principal then serves only the type θH of the agent: q∗∗

H = q∗ H,

T ∗∗

H = T ∗ H

Recall that in this case the (ICL) constraint we omitted is satisfied since: θL u(q∗

H) − T ∗ H < 0

In other words, the type θL agent is strictly better off by announcing the truth about his type.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 42 / 48

The Static Adverse Selection Problem (10):

Case 2: [λ θL − (1 − λ) (θH − θL)] > 0 In this case the optimal contract (q∗∗

i , T ∗∗ i ) is such that:

1) it satisfies efficiency at the top: q∗∗

H = q∗ H

θHu′(q∗∗

H ) = c

In other words, according to the optimal contract the θH agent receives the efficient quantity q∗

H.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 43 / 48

The Static Adverse Selection Problem (11):

2) it safisfies inefficiency at the bottom: q∗∗

L < q∗ L

θLu′(q∗∗

L ) = c

• λ θL

λ θL − (1 − λ) (θH − θL)

• > c

In other words, according to the optimal contract the θL agent receives an inefficiently low quantity q∗∗

L .

3) inefficient premium to the top type: T ∗∗

H < T ∗ H

T ∗∗

H = θH u(q∗∗ H ) − (θH − θL) u(q∗∗ L ) < θH u(q∗ H) = T ∗ H

In other words, according to the optimal contract not all the consumer surplus θH u(q∗

H) is extracted from the θH agent.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 44 / 48

The Static Adverse Selection Problem (12):

4) it prescribes an efficient transfer for the bottom type: T ∗∗

L

< T ∗

L

T ∗∗

L

= θL u(q∗∗

L )

In other words, according to the optimal contract all the consumer surplus θL u(q∗∗

L ) is extracted from the θL agent:

perfect price discrimination. Notice that from the first two conditions we conclude q∗∗

L < q∗∗ H

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 45 / 48

The Static Adverse Selection Problem (13):

Step 6: We still need to check that the omitted constraint (ICL) holds. This is indeed the case: θL u(q∗∗

L ) − T ∗∗ L

≥ θL u(q∗∗

H ) − T ∗∗ H

Since we do know that θH > θL and θH [u(q∗∗

H ) − u(q∗∗ L )] = T ∗∗ H − T ∗∗ L

> 0

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 46 / 48

Taxation Principle:

◮ The result obtained can be re-interpreted in terms of the

taxation principle.

◮ The principal offers a menu of (two) two-part tariff contracts:

{(q∗∗

H , T ∗∗ H ), (q∗∗ L , T ∗∗ L )} ◮ These contracts are such that the L-type agent self-selects in

choosing the contract (q∗∗

L , T ∗∗ L ), ◮ While the H-type agent self-selects in choosing the contract

(q∗∗

H , T ∗∗ H ).

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 47 / 48

Quantity Discounts

◮ This re-interpretation corresponds to a realistic indirect

mechanism.

◮ An alternative indirect mechanism that is quite frequently

• bserved in real life is the following:

◮ The good is offered at the price T ∗∗

L ,

◮ If the consumer is willing to buy any quantity in excess of q∗∗

L

then he is offered a discount in the amount of (T ∗∗

H − T ∗∗ L ),

◮ the balk quantities offered are either q∗∗

L

• r q∗∗

H .

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 48 / 48