The theory of Incentives 2

Chapter 2. The rent extraction-efficiency trade-off

1. Intro

(p) lack of time/lack of ability + bounded rationality → [delegation/contract (a) → Incentive → benefitting/conflicting benefits] ← asymmetric information/private knowledge (a) (e.g. cost of task, tech used, level of matching) === adverse selection

==> information rent/cost (+ technological cost of performing task) => distortions in trade (volume) =>> (p) trades-off allocative efficiency against costly information rent (given up to (a) to induce information revelation)

Delegation of task in a one-shot relationship:
== (p) (a) cannot rely on repetition of relationship to achieve efficient trades == short-term relationship is only regulated by a contract (under legal framework ← sometimes unrealistic)

==>>> (p) design contractual offer to (a) to realize optimal [rent extraction-efficiency] trade-off:
1) describe the [ set of allocations (output + distribution of gains) ] that (p) can achieve under [ a) info gap/asy info (a set of incentive compatibility constraints) + b) participation constraints ] == incentive feasible allocations
2) optimize the principal’s objective function within the set of incentive feasible allocations
== the optimal second-best contract (a distortion in the volume of trade away from the first-best) giving up some information rents to the most efficient agents

2. The basic model

Agent is informed about its type θ:
– efficient agent (θ_): C(q, θ_) = θ_q + F  /probabilities ν
– inefficient agent (¯θ):  C(q, ¯θ) = ¯θq + F  /probabilities 1 − ν
– ∆θ = ¯θ − θ_ > 0 = the spread of uncertainty on the agent’s marginal cost

A = {(q, t) : q ∈ IR+, t ∈ IR}
– q = quantity produced
– t = value received by the agent
– observable and verifiable by a third party

Time sequence:
(a) discovers his type θ (asymmetric info) → (p) offers a contract → (a) accepts or refuses the contract → the contract is executed

3. The complete information optimal contract

complete information efficient production levels: (p) marginal value = (a) marginal cost
S’ (q_∗ ) = θ_
S’ (¯q∗ ) = ¯θ

Social value:
W_∗ = S(q_∗) − θ_q_∗ − F ≥
W¯ ∗ = S(¯q∗) − ¯θq¯∗ − F ≥ 0
=> q_∗ > q¯∗

agent’s participation constraints (outside opportunity utility=0):
t_ − θ_q_ ≥ 0
t¯− ¯θq¯ ≥ 0
=> take-it-or-leave-it-offers:
t¯∗ = ¯θq¯∗
t_ ∗ = θ_q_∗
(delegation is costless for (p), and (p) could carry the task himself if cost function is the same)

V=S(q) – t   → strictly concave indifference curve of (p) is tangent to
U = t – θq   → zero rent iso-utility curve of the corresponding type (a)
( (p) having all bargaining power => V¯∗ = W¯∗ /  V_∗ = W_∗ )

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4. Incentive feasible menu of contracts

marginal cost θ is (a) private information => (p) offers the menu of contracts {(t_∗, q_∗); (t ¯∗, q¯∗)}, hoping θ_ choose former and ¯θ choose later

### => Offering the menu (A∗, B∗) fails to have the agents self-selecting properly within this menu => efficient type mimics the inefficient one and selects also contract B∗ to get a positive utility rather than 0
=> such {(t_∗, q_∗); (t ¯∗, q¯∗)} is not incentive compatible

=>> a) incentive compatibility constraints: allocations weakly preferred to each other by agents
t_ − θ_q_ ≥ t¯ − θ_q¯
t¯ − ¯θq¯ ≥ t_ − ¯θq_

=>> b) participation constraints: yield to each type at least its outside opportunity level
t_ − θ_q_ ≥ 0
t¯ − ¯θq¯ ≥ 0

a) + b) = incentive feasible allocations 

Special cases 1) bunching or pooling contracts:
t_ = t¯, q_ = ¯q  and both types of agents accept this contract
→ Incentive compatibility is thus easy to satisfy, but at the cost of an obvious loss of flexibility in allocations  = only the participation constraints of inefficient agent matter now ↓

Special cases 2) shut-down of the least efficient type:
t  − θ_q ≥ 0 , 0 ≥ t − ¯θq =>  only the efficient type accepts the non-zero contract (t, q)
= the cost of such a contract may be an excessive screening of less efficient types

# monotonicity constraint (not exist under complete information):
q_ ≥ q¯  = Any  (q, q¯) which can be reached by an incentive compatible contract, must satisfy this implementability condition

5. Information rents

the utility level that a θ_agent would get by mimicking a ¯θ agent:
t ¯− θ_q¯ = t ¯− ¯θq¯+ ∆θq¯ = U¯ + ∆θq¯
=> even if  U¯ = t ¯− ¯θq¯ = 0, the θ_agent benefits from an information rent = ∆θq¯ from mimic
=>> the principal must give up a positive info rent to a θ_agent ( from (a) info advantage)
==>>> (p) problem is to determine the smartest way to give up such a rent (to choose incentive feasible contract)

information rent (= utility without mimics):
U_ = t_ −  θ_q_
U¯ = t¯ − ¯θq¯

6. The optimization program of the principal

max{(t¯,q¯);(t_,q_)} [ ν (S(q_) − t_) + (1 − ν) (S(¯q) − t¯) ]  ==
max{(U¯,q¯);(U_,q_)} [ ν ( S(q_) − θ_q_ ) + (1 − ν)(S(¯q) − ¯θq¯) −  (νU_ + (1 − ν)U¯) ]
=   max [ expected allocative efficiency/social value  −  expected info rent ]

outputs →  impact on allocative efficiency and the overall gains from trade.
information rents →  distributive impact of asymmetric information.

The principal is ready to accept some distortions away from efficiency to decrease the agent’s information rent:

incentive constraints:
U_ ≥ U¯ + ∆θq¯, U¯ ≥ U_ − ∆θq_

participation constraints:
U_ ≥ 0, U¯ ≥ 0

=> SB ( second-best) the solution to this problem

7. The rent extraction-efficiency trade-off

both constraints must be binding at the optimum of the principal’s problem (P)

U_ = ∆θq¯  , U¯ = 0

→ max {(q_,q¯)} [ ν ( S(q_) − θ_q_ ) + (1 − ν) ( S(¯q) − ¯θq¯ )  − ν∆θq¯ ]

# Compared with the full information setting, asymmetric information alters the (p)’s optimization simply by the subtraction of the expected rent given up to the efficient type
# the inefficient type gets no rent
# the efficient type gets the information rent that he could obtain anyway by mimicking the inefficient type → rent depends only on the level of production requested from this inefficient type

=> S’ (q_SB) = θ or q_SB = q∗ 
=> (1 − ν) ( S’ (¯qSB) − ¯θ ) = ν∆θ  => S’ (¯qSB) = ¯θ + [ν/(1 − ν)] ∆θ
=>> trade-off between efficiency (expected marginal efficiency cost) and rent extraction (expected marginal cost of the rent brought)
=>>> No output distortion for the efficient type (v.s. the first-best), q_SB = q_∗ / A downward output distortion for the inefficient type, q¯SB < q¯∗
=>>> information rent: USB = ∆θq¯ SB

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Shut-down Policy:
If FOC has no positive solution of ¯q, ¯q=0, q_SB=q_* => no rent, but significant inefficiency as ¯θ not work
=> ν∆θq¯ SB ≥ (1 − ν) ( S(¯qSB) − ¯θq¯ SB)
=> expected cost of the efficient type’s rent due to inefficient one ≥ expected benefit from transacting with the inefficient type at the second-best level of output

=>> the occurrence of shut-down can also be interpreted as saying that the principal has, on top of the agent’s production, another choice variable to solve the screening problem

8. The theory of the firm under asymmetric information

When the delegation of task occurs within the firm, because of asymmetric information, the firm does not maximize the social value of trade (profit)
← allocative efficiency is only part objective, the allocation of resources within the firm remains constrained optimal once considering informational constraints

Leibenstein (1966): management failures within the largest firms ← the most likely to suffer from significant internal informational problems

Williamson (1975): transaction cost that impede transactions → “informational impactedness” as an important source of (allocative) inefficiency
transaction costs may be mitigated by the choice of convenient organizational forms → various organizational forms generate different degrees and costs of asymmetric information between partners => theory of vertical integration

(??? opportunistic behaviour still occurs under perfect information. ↓ That’s whyHart argue the basic problem is property rights. Probability of opportunism exist as long as there are benefits to bargain, which means the only way to avoid opportunism without buying property rights is that make each participants realize that the other has no benefit to bargain under the perspective of long-term profits = a nash equilibrium)

Arrow (1975): an upstream firm may want to integrate backward and acquire a downstream supplier to reduce the extent of asymmetric information between → limitation: vertical integration improves information as exogenous =>> debate
(proponents) Williamson (1985)) and (opponents) Grossman and Hart (1986): prefer to see information structures being derived from the property rights associated with different organizational forms

# those inefficiencies do not call for any public policy ← any benevolent policy maker when correcting these inefficiencies would face the same informational constraints as (p) + Nevertheless, the policy-maker might want to implement different trade-offs between efficiency and rent extraction for redistribution

9. Asymmetric Information and marginal cost pricing

(p) as consumers, (a) as a firm producing a consumption good, under complete info → (marginal utility of consumption) price = marginal cost

under asymmetric information
=> efficient firm (θ_): price equates marginal cost
=> inefficient firm (¯θ): p(¯θ) = ¯θ + [ν/(1 − ν)] ∆θ
=>> generalized marginal cost = traditional marginal cost of ¯θ + information cost
=>> p(¯θ) is higher than marginal cost to decrease the quantity ¯q produced by the inefficient firm +  reduce the efficient firm’s information rent

(??????? so the consumer will pay higher price for good produced by inefficient firm than efficient one?)

10. The revelation principle

↑ restricted (p) to offer a menu of contracts, one for each type → a) a more complex contract? b) communication device used to transmit information to (p)?  = not the case
=> Revelation Principle:  any allocation rule a(θ) obtained with a mechanism (M, g˜(·)) can also be implemented with a direct and truthful revelation mechanism
direct: a mapping g(·) from Θ to A which writes as g(θ)=(q(θ), t(θ)) for all θ belonging to Θ

truthful: direct revelation mechanism satisfies incentive compatibility constraints
→ simplification of contract theory as restricting the analysis to a simple and well defined family of functions [the truthful direct revelation mechanisms]

(revelation principle is from game theory and is a fundamental principle of mechanism design)

11. A more general utility function for the agent

Spence-Mirrlees condition: Cθq  a constant sign condition / different types of agent have indifference curves which cross each other at most once / a more efficient type is also more efficient at margin

incentive and participation constraints as the same:
U_ = t_ − C(q_, θ_) ≥ t ¯− C(¯q, θ_),
U¯ = t ¯− C(¯q, ¯θ) ≥ t_ − C(q_, ¯θ)
U_ = t_ − C(q_, θ_) ≥ 0
U¯ = t ¯− C(¯q, ¯θ) ≥ 0

Spence-Mirrlees condition guarantees that only the efficient type’s incentive constraint has to be taken into account

11.1 The optimal contract

U_ ≥ U¯ + Φ(¯q)
Φ(¯q) = C(¯q, ¯θ) − C(¯q, θ_)
U¯ ≥ 0
→ U_ = Φ(¯q)
Since Φ0 (·) > 0, reducing the inefficient agent’s output reduces also, as in Section 6, the efficient agent’s information rent

q_SB = q_∗ ,  S’ (q_∗ ) = Cq(q_∗ , θ_)  / no output distortion for efficient
q¯SB < q¯∗ ,  S’ (¯q∗ ) = Cq(¯q∗ , ¯θ)  / a downward output distortion for inefficient
→ S’ (¯qSB) = Cq(¯qSB, ¯θ) + [ν/ (1 − ν)] Φ’ (¯qSB)
→ U_SB = Φ(¯qSB) / information rent
→  t_SB = C(q_∗, θ_) + Φ(¯qSB) / t ¯SB = C(¯qSB, ¯θ)

11.2 Non-responsiveness

S(q, θ): common value model where the agent’s type directly affects the principal’s utility function
usual assumptions: a positive and decreasing marginal value + Sqθ(q, θ) > 1  the marginal gross value of trade for (p) increases sharply with the agent’s type

Sq(q_∗, θ_) = θ_  /  Sq(¯q∗, ¯θ) = ¯θ
Sqθ(q, θ) > 1 => q_∗ < q¯∗
→ not satisfy the monotonicity condition implied by incentive compatibility (Guesnerie & Laffont (1984): a phenomenon of non-responsiveness)
→ strong conflict makes screening of types quite difficult: Sq(¯qSB, ¯θ) = ¯θ + [ν/(1 − ν)] ∆θ ← when ν small, ¯qSB close to  ¯q∗ > q_SB=q_∗ → violate monotonicity condition q_SB ≥ q¯SB
=> forces (p) to use a pooling allocation: distort ¯q∗ down to q_∗ to decrease the θ-type’s information rent to contract D while still preserving incentive compatibility ← better by moving along the zero iso-utility line of a ¯θ-type  best pooling allocation:

max {(q,t)}  νS(q , θ_) + (1 − ν)S(q, ¯θ) − t
t − ¯θq ≥ 0
q < q¯∗  ← Sqθ(q, θ) > 0

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11.3 More than two goods

q = (q1,… ,qn), C(q, θ) with C(·) strictly convex in q, S(q) with S(·) strictly concave in q
U_ = Φ(q),  Φ(q) = C(q, ¯θ) − C(q, θ_)

q_SB = q∗ , Sqi (q_∗ ) = Cqi (q_∗ , θ_) for all i in {1,… ,n}
¯qSB ?? q¯∗ , Sqi (¯qSB) = Cqi (¯qSB, ¯θ) + [ν/(1 − ν)] Φqi (¯qSB), for all i in {1,… ,n}
→ without specifying further the value and cost functions, hard to compare second-best outputs above with the first-best outputs / may ¯qSBi > ¯q∗i for a subset of indices i

C(q_, ¯θ) − C(¯q, ¯θ) ≥ C(q_, θ_) − C(¯q, θ_), for all implementable pairs (¯q, q)
→this incentive compatibility condition satisfied if the Spence-Mirrlees conditions Cqiθ(·) > 0 + if monotonicity conditions ¯qi < q_i for all i (however, may ¯qSBi > q_SBi = ¯q∗i )

12. Ex ante versus ex post participation constraints

↑ contracts offered at the interim stage
/ sometimes, (p) (a) contract also at ex ante stage: before (a) discovers his type
optimal contract under various assumptions about risk aversion of (p) and (a)

12.1 Risk neutrality

(a) ex ante participation constraint: νU_ + (1 − ν)U¯ ≥ 0.

an example of ex post incentive compatible + ex ante participation constraint:
U_∗ = (1 − ν)∆θq¯∗ > 0 and U¯∗ = −ν∆θq¯∗ < 0 [← U_ ≥ U¯ + ∆θq¯ and U¯ ≥ U_ − ∆θq_]
→ with such a rent distribution, the optimal contract implements the first-best outputs costlessly for (p)
=>> (a) is rewarded when efficient and punished when inefficient => there must be some risk on the distribution of information rents to induce information revelation, but this risk is costless for the principal because of the (a)’s risk neutrality
(such an ex ante contract requires a strong ability of the Court of Justice to enforce contracts)

(in that case of non-responsiveness, even under ex ante contracting and risk neutrality, some inefficiency still arises)

more leeayway U_ and U¯ → incentive constraints hold + ex ante participation constraint an equality:
{(t_∗, q_∗); (t¯∗, q¯∗)} where t_∗ = S(q_∗)−T and t¯∗ = S(¯q∗)−T with T being a lump-sum payment → incentive compatible constrain:
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participation constrain: T = ν(S(q_∗) − θ_q_∗) + (1 − ν)(S(¯q∗) − ¯θq¯∗)
=>> implementation of the first-best outcome amounts to have (p) selling the benefit of the relationship to risk neutral (a) for a fixed up-front payment T → (a) will benefit from the full value of the good and will trade-off the value of any production against its cost = (a) is residual claimant for the firm’s profit (???)

# Harris and Raviv (1979): firm as a mechanism allocating resources at the ex ante stage → first best allocation remains implementable when the firm has a strong ability to enforce contracts
# Loeb and Magat (1979): regulation as a p-a problem with adverse selection → asymmetric info is not an obstacle to the implementation of marginal cost pricing if the regulated firm accepts the regulatory contract at the ex ante stage

12.2 Risk aversion

[1] A risk adverse (a)

↑ first-best is feasible with risk neutrality → but (a) is subject to risk→ such a risk is obviously costly if (a) is risk averse

a risk averse agent with a Von Neuman-Morgenstern utility function u(·):
ex ante participation constraint: νu(U_) + (1 − ν)u(U¯) ≥ 0
(p): max {(U¯,  q¯);(U_,q_)} [ν(S(q_) − θ_q_ − U_) + (1 − ν)(S(¯q) − ¯θq¯− U¯)]
incentive compatible: U_ ≥ U¯ + ∆θq¯

=>  q_SB = q_∗ / no output distortion for efficient
=> q¯SB < q¯∗ / a downward output distortion for inefficient, with
S'(¯qSB) = ¯θ + [ν (u'(U¯SB) − u'(U_SB)) ] / [νu'(U_SB) + (1 − ν)u'(U¯SB) ] * ∆θ
=> U_SB > 0 > U¯SB
→ to insure the participation of risk averse (a), (p) must also pay a risk premium for wedge between U_ and U¯ → reducing this premium calls for a downward reduction in the inefficient type’s output so that the risk is lower = (a)’s risk aversion leads the (p) to weaken the incentives(+ outputs)
(when zero risk aversion → section 12.1 / when infinitely risk adverse → section 2)

constant absolute risk aversion utility function u(x) = [1−exp(−rx)] / r
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# Salani´e (1990): the case of a continuum of types → pooling for the least efficient types occurs when risk aversion is large enough
# Laffont & Rochet (1998): a similar phenomenon with ex post participation constraints when a regulator (p) maximizes ex ante social welfare with a risk averse firm

[2] A risk adverse (p)

a risk averse principal with utility function v(·) on S(q) − t:
S(q_∗) − θ_q_∗ − U_∗ = S(¯q∗) − ¯θq¯∗ − U¯∗
νU_∗ + (1 − ν)U¯∗ = 0

=> U_∗ = (1 − ν) [ S(q_∗ ) − θ_q_∗ − (S(¯q∗) − ¯θq¯∗) ]
=> U¯∗ = −ν [ S(q_∗ ) − θ_q_∗ − (S(¯q∗ ) − ¯θq¯∗) ]

=> U_∗ − U¯∗ = S(q_∗ ) − θ_q_∗ − (S(¯q∗) − ¯θq¯∗) > ∆θq¯∗
=> U¯∗ − U_∗ =S(¯q∗) − ¯θq¯∗   − (S(q_∗ ) − θ_q_∗) > −∆θq_∗
→ satisfies both types’ incentive compatibility constraints 
=>> when (p) is risk averse over S(q) − t and contracting takes place ex ante, the optimal incentive feasible contract implements the first-best outcome (???)

<= the same in the case of lump-sum payment T
by making the risk neutral (a) residual claimant for the value of trade, ex ante contracting allows the risk averse (p) to implement the first-best outcome despite the informational problem

max {(U¯, q¯),(U_,q_)}  [ν v(S(q_) − θ_q_ − U_) + (1 − ν) v(S(¯q) − ¯θq¯− U¯)]
U_ ≥ U¯ + ∆θq¯  //  U¯ ≥ 0
=> q_SB = q_∗ / no distortion for efficient just as in the case of risk neutrality
=> ¯qSB < q¯∗ / a downward distortion of inefficient, with
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=> by increasing ¯q above its value with risk neutrality, the risk averse (p) reduces the difference between V_SB and V¯SB → gives insurance and increases ex ante payoff

# Risk aversion (p) is quite natural in some contexts: a local regulator with a limited budget (Lewis&Sappington (1995): regulation of public utilities) or a specialized bank dealing with relatively correlated projects may be insufficiently diversified to become completely risk neutral

13. Commitment

↑ (a) incentive problem with private info ← (p) a distribution of rents inducing information revelation + some allocative inefficiency designed at reducing the cost of this revelation ← 2 assumptions/imperfections:
[1] Court of Justice can perfectly enforce the contract
[2] neither renegotiating nor reneging on the contract is a feasible alternative

[2.1] renegotiating a contract

a limited commitment that should benefit both (p) and (a) = a Pareto improvement
→ after revelation by selecting, (p) can propose a renegotiation to raise allocative inefficiency on the inefficient (a) output = ¯qSB → q¯∗, t¯∗ = ¯θq¯∗ 
→ hardens the incentive compatibility constraint of the efficient (a) → no truthful revelation at all => a fundamental trade-off between raising efficiency ex post and hardening ex ante incentives

not a problem: 1) increasing productive capacity can be costly; 2) using indirect mechanism that led the agent choose the output → no scope for renegotiation since one-shot relationship

[2.2] reneging on a contract 

either (p) or (a) may breach the contract and thus renege on his previous contractual obligation:
1) once (a) has revealed type by selecting, (p) may propose the complete information contract which extracts all rents → this breach of contract should be anticipated by (a) and thus interfere with truthful revelation in the first place
2) (a) may want to renege on a contract with a negative ex post utility level (Section 12.1) → the threat of (a) reneging at the ex ante stage forces (a) participation to be written in ex post terms

14. Stochastic Mechanisms *

the optimal mechanism cannot be stochastic. To show this result suppose, on the contrary, that this mechanism is stochastic. A random direct revelation mechanism is then a probability measure on the set of possible transfers and outputs, which is conditional on the agent’s report of his type. Lett {(˜q_, U_˜ ); (˜q¯,  U˜¯)} be such a random stochastic mechanism. We can replace this stochastic mechanism by the deterministic mechanism constructed with the expectations of those variables, namely E(˜q¯), E(˜q_), E(U_˜ ) and E(U ˜¯)

15. informative signals to improve contracting *

how information exogenous to the relationship (observation in similar one + monitoring and auditing)  can help (p) to improve the contract designed with (a) by somewhat filling the information gap

15.1 ex post verifiable signal

15.2 ex ante non-verifiable signal

15.3 more or less favorable distribution of types

16. Contract Theory at Work

[1] Regulation

Baron&Myerson (1982): Regulating a Monopolist with Unknown Costs
(p) a regulator → (decide how the firm’s regulated price/quantity and subsidy/tax should be determined, as functions of some cost report from the firm) → maximizes
[1] a linear social welfare function of the consumers’ surplus S(q) − t |(t= qP(q)) ; [2] a regulated monopoly’s profit U = t − θq , with weight α<1
→ V = S(q) − θq − (1 − α)U
α<1  → socially costly to give up a rent to the firm
+ incentive and participation constraints → firm has nonnegative profit and has no incentive to misrepresent its costs

(The Revelation Principle: Without any loss of generality, the regulator may be restricted to regulatory policies which require the firm to report its cost parameter and which give the firm no incentive to lie →To see why it is true:
suppose that the regulator chose some general regulatory policy, not of the form described in the proposition. For each possible value of θ, let Φ(θ) be the cost report that the firm would submit if its true cost parameter were θ. That is, Φ(θ) maximizes the firm’s expected profit, when it is confronted with this regulatory policy and its true cost parameter is θ. Now consider the following new regulatory policy: ask the firm to report its cost parameter θ; then compute Φ(θ); and then enforce the regulations that would have been enforced in the original regulatory policy if Φ(θ) had been reported there. It is easy to see that the firm never has any incentive to lie to the regulator in the new policy. (Otherwise it would have had some incentive to lie to itself in the originally given policy.))

=> qSB = q∗ for the efficient type // a downward distortion for the inefficient type 
→ S’ (¯qSB) = ¯θ + [ν/(1 − ν)] (1 − α)∆θ

a higher value of α → reduces the output distortion ← regulator is less concerned by the distribution of rents within society

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Laffont and Tirole (1993): government intervention under asymmetric information ← a comprehensive view of theory + implications for designing real regulatory institutions

[2] Nonlinear Pricing of a Monopoly

Maskin&Riley (1984): Monopoly with incomplete information
p: seller of a private good, utility function V = t – cq
a: buyer, U = θu(q) − t  ←  θ of each buyer = {θ_, ¯θ} with respective probabilities 1 − ν | ν

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¯θ is the efficient type → binding constraints are U¯ ≥ U_ + ∆θu(q_) | U_ ≥ 0

=> ¯qSB = ¯q∗ | θ¯u'(q¯∗ ) = c
=> a downward distortion of the low valuation agent’s output:  q_SB < q_∗ 
[ θ_ − (ν/1 − ν) ∆θ ]  u’ (q_SB) = c

Tirole (1988) + Varian (1988) + Wilson (1993): reviews on literature on nonlinear pricing

[3] Quality and Price Discrimination

Mussa&Rosen (1978):  Monopoly and product quality
(a) buy a commodity with quality q + vertically differentiated with their preferences for the good
→ U = θq − t with θ in Θ = {θ_, ¯θ} with respective probabilities 1 − ν and ν
(p): V = t − C(q)
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¯θ is efficient type
=> ¯qSB = ¯q∗ | θ¯ = C’ (¯q∗)

=> a downward distortion of the low valuation agent’s output:  q_SB < q_∗ 
θ_ = C’ (q_SB) + [ν/(1 − ν)] ∆θ

the spectrum of qualities (¯q, q_)  larger under asymmetric information than under complete information → incentive of (p) to put a low quality good on the market ← a well documented phenomenon in the industrial organization literature
=> damaging its own goods may be part of optimal selling strategy when screening of the consumers’ willingness to pay for quality is important

[4] Financial Contracts

Freixas&Laffont (1990):  Optimal banking contracts
(p): a lender who provides a loan with size k to a borrower, utility function V = t − Rk, R = risk free interest rate
(a): V = θf(k) − t, f(k) = return on capital, t = borrower’s repayment to (p), θ = a productivity shock in Θ = {θ_, ¯θ} with 1 − ν and ν
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¯θ is high productivity/efficient type
=> ¯kSB = ¯ k∗ | ¯θf'(¯ k∗) = R
=> a downward distortion of the low productivity borrower’s loan: k_SB < k_∗
[ θ_ − (ν/1 − ν) ∆θ ]  f’ (k_SB) = R

Screening borrowers according to the size of their loans → rationing for the low productivity firms ← Freixas&Rochet (1999, Chapter 5)

[5] Labor Contracts

Green&Khan (1983) + Hart (1983): Optimal labour contracts under asymmetric information:
(p): a union who provides its labor force l to a firm, assuming that it has full bargaining power, utility function defined on consumption and labor → V = v(t − ψ(l)), ψ(·)=disutility of labor (increasing & convex)
(a): firm makes a profit: U = θf(l)−t, θ = a productivity shock in {θ_, ¯θ} with 1 − ν, ν
++ firm’s boundaries are determined before the realization of the shock and contracting takes place ex ante ++ a risk averse (p) and a risk neutral (a)

=> risk averse union will propose a contract to the risk neutral firm → provides full insurance and implements first-best levels/efficient employments l¯∗ and l_∗
=>  ¯θf'(¯l∗) = ψ'(¯l∗) | θf'(l_∗ ) = ψ'(l_∗)

more difficult case: a utility function exhibiting an income effect: V = v(t) − ψ(l)
→ first-best optimal contract would still require efficient employment in both state
→ can also equating the worker’s marginal utility of income across states: t_∗ = t¯∗
=> ν(¯θf(¯l∗ ) − t¯∗ ) + (1 − ν)(θf(l_∗ ) − t_∗ )=0.
=> t¯∗ = t_∗ = ν ¯θf(¯l∗) + (1 − ν)θf(l_∗ ) = E(θf(l∗))スクリーンショット 2017-08-18 15.41.16スクリーンショット 2017-08-18 15.42.05

 

optimal overemployment has often been critized ← worker’s utility function assume that labor is a normal good → more general preferences: underemployment can instead be obtained as an optimal solution to the asymmetric information problem ← Hart& Holmstrom (1987)

 

A case of general model:

我们假设有一个RPG世界,有个国王要一种魔兽肉。他找来勇者帮他去下副本打魔兽,一个魔兽稳定掉一个魔兽肉。

现在市面上有两个勇者,一个50级,一个10级。打魔兽需要买血瓶,一个2金币=2G。lv50打5个魔兽消耗一个血瓶,lv10打一个消耗一个。

国王的问题就是颁布任务选择奖励让勇者去打魔兽。

在完全信息下,国王分别找来lv50的0.4G一个发任务,找来lv10的2G一个发任务。数量到国王的边际效用下降到每个勇者边际成本。

对了以上还要假设这个RPG世界就国王一个人发任务,不去下副本就无聊死,所以机会成本=0。

所以国王还要lv10干嘛?全部让lv50打啊。

好吧我们现在假设副本有种障毒,导致进副本有时间限制,而魔兽是定时刷的,导致lv50最多刷20头就要回城了,而且再也进不来这个副本了。而国王需要30头的野兽,或者说在30头时候边际收益递减到2G一个。

那我们就要把lv10也利用起来了。20头任务给lv50,8G,10头任务给lv10,20G,一共28G。完美。

完美个屁。问题来了,lv50不是傻的,由于国王不知道勇者等级,lv50说自己lv10,要接10头任务,2G打一头,其实只花两瓶药4G,白赚6G爽翻天。

国王说神他么你lv10,市面上有就一个lv10,有个人在骗朕,找出来砍了,我要重新定个发任务方针。

核心是让lv50说谎也没有好处。所以直接给他加xG到20头任务,让这个x=他说谎做lv10任务能赚到的钱,这样宁愿多放几个无双的lv50就乖乖去刷20头任务了。

我们假设效用是一个单调递减函数 S(q)= 62 – q*2
国王最大化:max (S(q1)- 0.4*q1 -x)+(S(q2)- 2*q2)
说谎没好处:x = q2*2 – q2*0.4

带进去求个导:S'(q1)=0.4, 因为上限20到不了边际效用,q1=20。没变
S’(q2)= 2+1.6=3.6 (靠忘记设计整数了,不管了)q2 = 29.2-20 = 9.2,  x= 14.72G

简单来讲,国王现在计策成了发两个任务,一个杀9.2头拿18.4G,一个杀20头拿8+14.72=22.72G。这样就算lv50冒充lv10,也和做20头任务拿一样的钱。国王一共花41.12G得到29.2个魔兽楼。虽然多花了13.12块,但按照原来那种方案,lv50冒充,最后国王花了40块钱也只有20个魔兽肉。14.72G是付给q1的信息租,其中13.12G是国王新拿出来的,1.6是从q2少赚到的那个0.8个魔兽肉1.6G,是一个从first best的偏离。

这样故事终于完美了。

完美个屁。同样是private information,凭什么国王会知道勇者的边际成本,都全知全能到知道边际成本了,还不知道勇者的等级,你在逗我吧、、、两个任务的单价差完全不相当,现实中有这种事吗。没有,为什么,因为这样的话没有lv50会去领任务。市面上只剩lv10的,睿智的对边际成本无所不知的国王不得不为了20只魔兽肉花40G,而lv10领到任务只会去找lv50,lv50花8块钱买药,剩下的32块钱和lv10bargain一下分了,这才完美呢。

算了,这个问题暂且不表。我们加入v这个概率元素。

仍然是打魔兽,只不过这次是市面上有一堆勇者,40%是lv10的,60%是lv50的。

现在有个勇者来到国王面前应征了,但是国王不知道勇者的level,也不知道他会不会说谎,所以国王应该怎么给出奖励呢。重新写一下函数

max  0.6*(S(q1) – 0.4*q1 -x1)+  0.4*(S(q2) – 2*q2 – x2)
x1 = q2*2 – q2*0.4, x2 = 0

q1 = 20
S’(q2)= (0.6*1.6 + 0.4*2) / 0.4 = 2+ (0.6/0.4)*1.6 = 4.4
q2 = 28.8 – 20 = 8.8, x = 14.08

所以只要对市面上勇者的概率的估计是正确的,想要最大化期望效用的国王应该拿出两个方案给勇者选择,杀20只魔兽奖励8+14.08=22.08G,杀8.8只魔兽奖励17.6G,这样如果来的是lv50勇者,就会选择第一种方案,如果来的是lv10勇者就会选第二种。注意x1=q2*1.6,国王减少给lv10勇者的任务中的杀怪数量(整体效率efficiency)以换来更低的信息租(information rent)。而市面上lv50勇者的可能越大,q2就越低,国王就越不用付出信息租。相反市面上lv10勇者的可能越大,国王就要给出越高的面向lv50的奖励。

所以大概这个RPG世界,菜鸡相安无事,高手不死不休。

 

A case of Quality and Price Discrimination:

用常见的机票class来做案例。企业面临问题是,要怎么分别控制business class和economy class的quality和他们的差¯q – q_,从而使得利润最大。

U = θq − t , θ_等于穷逼的边际效用=12,概率=1 − ν=80% |  ¯θ等于富人的边际效用=20,概率=ν=20% | t=票价

企业利润 = t – C(q),假设C(q) = q^2

企业max  ν(t¯ – C(¯q) ) + (1-ν) (t_ – C(q_) )
=> max  ν(θ¯q¯ – C(¯q) ) + (1-ν) (θ_q_ – C(q_) ) – (νU¯  + (1-ν)U_ )
=> max  20%(20q¯ – ¯q^2 ) + 80% (12q_ – q_^2) – (20%U¯  + 80%U_ )

需要满足:
U_ ≥  θ_q¯ − t¯ = U¯ − ∆θq¯  |穷逼不会去买bc
U¯ ≥  θ¯q_ − t_ = U_ + ∆θq_  |富人不会去买ec
U_ ≥ 0   =    θ_q_  ≥  t_           |穷逼不会嫌ec太贵
U¯ ≥ 0   =    θ¯q¯  ≥  t¯           |富人不会嫌bc太贵

=> 假设乘飞机相对刚性,不用给穷逼任何超过效用,U_ = 0,  θ_q_=  t_ ,而穷逼肯定不会去买贵的bc,所以只需满足 U¯ =  ∆θq_ 和 θ_q_  ≥  t_

=> max  20%(20q¯ – ¯q^2 ) + 80% (12q_ – q_^2) – 80%∆θq_
=>  ¯q = 10 富人的bc质量设在边际效用等于边际成本,t¯ = θ¯q¯  – ∆θq_ = 200 – 16 = 184
=> 穷逼: 12 – 2q_ – ∆θ = 0, q_ = 2, t = θ_q_ = 24
=> 如果富人买ec票,U¯ = 20*2 -24 = 16

如果不存在富人的时候
=> 企业max     t_ – C(q_)  =  θ_q_ – C(q_)
=> q_ = 6, t_ = 72

如果不存在穷逼的时候
=> q¯ = 10 , t¯ = θ¯q¯ = 200

所以航空公司为了招揽富人的bc客户,会大幅降低穷逼坐ec的体验。注意因为θ_ = C’ (q_SB) + [ν/(1 − ν)] ∆θ,当v越大也就是富人的比率越大,或者∆θ越大也就是富人与穷逼的边际效用差越大时,ec的质量就会越差,虽然价格始终是质量*边际效用。另外要注意当边际效用差太大之后q_可能<0,意味着screening直接把穷人挤出去了。

这可能就是为什么当美国70年代末放开航空管制之后(包括票价管制),票价大幅下降,服务质量也随着大幅下降的一个原因了。

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