Sequential Nonlinear Pricing of Experience Goods With Network

Effects

Dawen Meng ∗ Guoqiang Tian†

October 8, 2018

Abstract

We study a nonlinear pricing problem of experience goods in dynamic environments

where consumers’ private information evolves stochastically over time, and their present

types and/or quitting costs are affected by their past consumptions. Our model sheds some

light on the pricing of addictive goods as well as goods with bandwagon and/or snob effects.

Also, to obtain a dynamically implementable contract, we provide a new ironing technique.

In standard contract model, to guarantee the static incentive-compatibility (IC) condition,

the principal needs to set an allocation monotonic in the agent’s type. If the contract ob-

tained from the relaxed problem is not everywhere monotonic, a horizontal ironing technique

is adopted to revise it and a contract with bunching intervals is thus obtained. In our dy-

namic setup, however, monotonic condition is not sufficient for implementability. The slope

of allocation with respect to type needs to be sufficiently larger. So we adopt a sloping

ironing technique to obtain a perfect sorting contract. Moreover, consumers in this paper

are assumed to be connected via a social network. In contrast to the existing literatures of

network game, this paper focuses on a principal-agent model on the network. We discuss

how network structure impacts the form of optimal contracts in various environments.

Keywords: nonlinear pricing, experience goods, dynamic contract, social network, mecha-

nism design

∗E-mail: devinmeng@hotmail.com, School of Economics, Shanghai University of Finance and Economics and

Key Laboratory of Mathematical Economics (SHUFE). Financial support from the National Science Foundation

of China (NSFC-71301094) is gratefully acknowledged.†tian@ecomail.tamu.edu, Department of Economics, Texas A&M University. Financial supports from the

National Science Foundation of China (NSFC-71371117) and the Key Laboratory of Mathematical Economics

(SUFE) at Ministry of Education of China is gratefully acknowledged.

1

1 Introduction

Nelson(1970) classifies products into search and experience goods according to the point in the

purchasing process when consumers can accurately assess a product’s value and thus know their

own preferences. Search goods (like clothes) are products for which a prospective buyer knows

his real preference, either by inspection or by recommendation, prior to making the purchase

decision. Experience goods (like foods) refer to the products or services whose attributes can

only be ascertained upon consumption (the proof of the pudding is in the eating). The accurate

preference for these goods is unknown to a consumer himself until his actual consumption is

finished.1 In a typical nonlinear pricing problem (See, for example, Mussa and Rosen (1978),

Maskin and Riley (1984)), buyers are assumed to know their own preferences from the outset

of game, the seller needs to construct a static nonlinear pricing contract to elicit truthtelling

and to maximize her own payoff. For experience goods, however, a dynamic contracting model

needs to be adopted since a buyer’s private information is revealed gradually to himself.

In addition to the stochastically evolving preferences, experience goods also exhibit some

other differences in comparison to search goods. When consuming experience good, a consumer’s

future preference and withdrawal cost may be affected by his current assumption and/or that

of others. Typical examples are addictive goods. People get addicted to some healthy goods

or activities, such as delicious food, fascinating book, music, sport, etc. The more you have

partaken of them, the more you want to partake in the future. In contrast, some other goods or

activities (say, alcohol, cocaine, cigarettes, antibiotic and gambling, etc.) have harmful addictive

effect. A user find that over time they have to take increasingly large amount to achieve the same

efficacy as a result of repeated use, but it is painful to abstain from these consumptions because

the quitting cost may also increase with one’s past consumption. In the environment with

multiple users, interpersonal interactions between behaviors and utilities are ubiquitous. These

interactions may either be contemporaneous or be intertemporal. The network effect arises when

a user’s payoff increases with the consumptions of his neighbours who choose the same product

simultaneously. The bandwagon (resp. snob) effect arises when people’s future preference for

a commodity increases (resp. decreases) as the consumptions of other users increase. In this

paper, we try to explain the monopoly pricing of experience goods encompassing these effects.

Our research contributes to the previous studies on dynamic mechanism design and complex

1Darby and Karni (1973) introduces the term credence goods and add this type to Nelson’s classification.

Credence goods are products or services, such as complex automobile repairs or medical services, whose accurate

value is unknown to the common users even after consumption due to their lacking of sufficient technical expertise.

2

social network. We enrich the existing literatures in several dimensions.

First, we allow for the dependence of an agent’s private information on the past allocations

of his own and/or that of others. In typical dynamic mechanism design settings, an agent’s type

is assumed to evolve stochastically over time, and is affected by some noisy terms. Kakade et al.

(2013) and Pavan et al. (2014) are two notable exceptions. Kakade et al. (2013) assume that an

agent’s valuation is a function of her own type and the seller’s past and current actions. Pavan

et al. (2014) allow for both serial correlation of an agent’s information and the dependence of

this information on his own past allocations. But they both assume off the impact of the other

agents’ past allocations on the distribution of an agent’s current type. We do this work in the

present paper by combining the usual direct effect of a change in an agent’s report on his current

allocation, with the novel indirect effects stemming from the induced change in the distribution

of his future type, and the cross effect stemming from the change in the distribution of other

agents’ future types.

Second, we give a novel ironing technique–sloping ironing technique–to obtain a dynamically

implementable contract. The optimal static mechanism design problem typically includes three

steps. First, identifying necessary/envelop conditions for incentive compatibility that permit one

to express information rents as a function of allocations and express the principal’s objective

as virtual surplus. Second, optimizing the principal’s objective with respect to all possible

allocations, including those that are potentially not incentive compatible. Third, verifying that

the allocation obtained from the relaxed program is monotone with respect to types so that the

allocation and a transfer/rent guaranteeing all the local constraints constitute a fully incentive-

compatible and individually-rational mechanism. The monotonicity/implementability condition

holds when certain appropriate primitive conditions on the distribution of types are satisfied. In

some special case where the monotonicity condition does not hold everywhere, one must resort

to a horizontal “ironing” method to derive the optimal allocation, which flattens parts of the

benchmark contract and leads to bunching for different types. In the dynamic contracting model,

however, we find that the slope of allocation with respect to type needs to be sufficiently large

to guarantee the dynamic incentive compatibility condition. So we need to iron the allocation

rule obtained from the relaxed problem with a slope line and thus obtain a perfectly sorting

contract.

Third, we incorporate the network effect into a dynamic contracting model by assuming

that an agent’s utility depends on the activities undertaken by other agents with whom he is

directly linked to. The theory of “games on networks/graphs” aims either at modeling network

3

formation or at determining the equilibrium of a game under given network structure. Every

agent perfectly knows the structure of the network and the payoffs of others. In contrast, we

discuss a contract model on the network in which the preference of each buyer is unobservable to

the seller and other buyers. In our model, a principal offers a sequentially incentive compatible

contract to a cohorts of interlinked and privately informed agents. The central problem is

to discuss the impact of network structure on the optimal selling mechanism provided by the

principal.

1.1 Related literatures

This paper is related to two bodies of work. One deals with dynamic mechanism design. The

last two decades have witnessed significant interest in extending the theory of mechanism design

to dynamic environments. Two objectives in the dynamic mechanism design are maximizing

the aggregate welfare of all parties (efficiency) and maximizing the revenue of the principal

(optimality). The first strand of studies investigates how to implement dynamically efficien-

t (surplus-maximizing) allocations in settings where the agents’ types change over time thus

extends the Vickrey-Clarke-Groves (VCG) and d’Aspremont-Gérard-Varet (AGV) results from

static to dynamic settings (see, among many others, Bergemann and Välimäki (2010), Athey

and Segal (2013)). Another strand of literature, to which our research belong, focuses on the

design of optimal (revenue-maximizing) mechanisms in various dynamic settings. This strand of

literature goes back to the pioneering work of Baron and Besanko (1984). They characterize the

optimal contract in a two-period model using an “informativeness measure” of initial-period pri-

vate information on future types. Besanko (1985) gives the optimal infinite-horizon mechanism

for an agent whose type follows a first-order autoregressive process over a continuum of states.

Courty and Li (2000) studies a two-period monopolistic price discrimination model where con-

sumers are uncertain about their subsequent demand but obtain an initial signal affecting their

valuation in the sense of first-order stochastic dominance or mean-preserving spread. Battaglini

(2005) considers the revenue-maximizing long-term contract of a monopolist in a model with

an infinite time horizon when types of the buyer changes in a two-state Markovian fashion over

time. Esö and Szentes (2007) extends Courty and Li (2000) to a multi-agent setting. They

analyse a situation where a monopolist is selling an indivisible good to buyers who don’t know

their accurate valuations. The seller can control, but cannot observe, certain additional signals

that affect the buyers’ valuations. They show that in the optimal mechanism the seller releases

all the information she can, and her expected revenue is as high as it would be if she could

4

observe the part of the information controlled by her that is “new” to the buyers. Board (2007)

is the first to consider a multi-agent environment with infinite time horizon. He extends the

analysis of Esö and Szentes (2007) to a setting where the timing of the allocation is endogenous,

so that the principal is selling options. Krahmer and Strausz (2011) study an optimal two-period

procurement contract with pre-project investigations. They discuss the distortions of informa-

tion acquisition in the joint presence of adverse selection and moral hazard. Boleslavsky and

Said (2013) characterize the optimal long-term selling contract in a setting where the buyer is

privately informed about both the value and the stochastic evolving process of his type. Kakade

et al. (2013) consider a problem of designing optimal selling mechanisms in dynamic settings

where a buyer’s valuation is affected by the history of both his own private information and the

seller’s actions. Under separable conditions, they present an optimal virtual-pivot mechanism,

which combines ideas based on the “virtual valuation” formulation of Myerson (1981) and the

dynamic “pivot” mechanism proposed by Bergemann and Välimäki (2010) for maximizing social

welfare. Pavan et al. (2014) extend the traditional first-order approach to dynamic environ-

ment and thus offers a novel methodology for studying dynamic mechanism problem in quite

general settings. They first give a dynamic version of envelope theorem that characterizes local

incentive-compatibility constraints, then show that this dynamic envelop formula, paired with an

appropriate dynamic integral monotonicity condition, guarantees global incentive compatibility.

Another related body of work focuses on complex social network. Over the past decades, a

large body of research in sociology, mathematics, physics, computer science and more recently

in economics, has studied the importance of social networks in different activities. Theoretical

analyses in economics focus mainly on how network structure relates to behavior. A strand of re-

search pays attention to the impact of strategic behavior on network structure and thus explains

the endogenous mechanisms of network formation. In this regard, see, among others, Myerson

(1977, 1991), Jackson and Wolinsky (1996), Bala and Goyal (2000), Liljeros et al. (2001), Akerlof

and Kranton (2002), Bearman, Moody and Stovel (2004), Jackson and Rogers (2007), Golub

and Livne (2010), for detailed discussions. Another strand of literature explores the converse

problem. It takes the network as given and study how structural properties of a network (say,

degree distribution, homophily, eigenvalues, clustering, and the centralities of nodes) impact the

behaviors of the agents in the network.2 Jackson and Yariv (2011) discuss how the equilibrium

points of a diffusion process and their stability depend on the degree distribution of the network.

2Jackson et al.(2017) divides network characteristics into two categories: (i) “macro” measures, which capture

the characteristics of the network as a whole, and (ii) “micro” measures, which are about characteristics of

individual nodes.

5

Currarini et al.(2009), Bramoullé et al. (2012), Golub and Jackson(2012), among many others,

examine how the outcome of game depends on network homophily: the tendency of agents to

associate with other agents who have similar characteristics. Coleman(1988), Newman(2003),

Jackson, Rodriguez-Barraquer, and Tan (2012), discuss, in various settings, the role of clustering

coefficient in determining the equilibrium of a game. Bramoullé et al. (2014) explore the impact

of network absorptive capacity, measured by the lowest eigenvalue of a graph, on properties

(uniqueness, interiorness, and stability) of game equilibria. In a public goods provision game,

Elliott and Golub(2017) discuss the role of the largest eigenvalue and eigenvector centrality of

a graph on the efficiency of outcome. Ballester et al. (2006) (hereafter BCZ(2006)) consider a

network game in which each agent reaps complementarities from all his direct network peers.

They find that the network game has a unique Nash equilibrium where each agent’s strategy

is proportional to his Katz-Bonacich centrality measure. Also, they propose a new measure

of prominence of players inside a network –inter-centrality measure–to identity the key player,

whose isolation from the rest of the group leads to the largest reduction of aggregate activity.

Following the seminal work of Ballester et al. (2006), a vast literature has investigated key-

player determination in network game. (say, among many others, Calvó-Armengol et al. (2009),

Ballester et al. (2010), Ballester and Zenou (2014), de Mart́ı and Zenou (2015), Zhou and Chen

(2015)).

To our knowledge, none of the existing researches incorporates dynamic mechanism design

and complex social network, which are the two primary components of this paper. The rest of

paper is organized as follows. In section 2, we discuss the one-agent case; in section 3, a model

of multiple agents is presented; section 4 concludes.

2 one-agent case

Preferences, information, and mechanisms. We consider an adverse selection model involving a

seller and a buyer interacting for two periods. In each period, the seller offers certain amoun-

t/quality of goods to the buyer at certain cost and collect transfers from him. x (resp. X)

represents either quantity of quality of good provided at the first (resp. the second) period, t

(resp. T ) denotes the transfer charged by the principal at the first (resp. the second) period.

C(x) = x2/2 is the principal’s cost of production. When x and X denote quantity, they are

both nonnegative; whilst when representing quality, x and X are allowed to be negative. The

sign of quality variables indicates only the direction toward which some features (say, sweet vs.

spicy, black vs. white, minimalism vs. luxury) of a product are intensified.

6

We denote by γ (resp. α) the buyer’s type in the first (resp. the second) period, then the

buyer obtains a net benefit of γx − t at the first period and αX − T at the second period. γ

is drawn from a distribution supported on [γ, γ] with cumulative distribution function H(·) and

density h = H ′. We assume that h is strictly positive and differentiable on [γ, γ] and satisfies

the monotone hazard rate property:

d

dγ

1−H(γ)h(γ)

6 0 6 ddγ

H(γ)

h(γ). (1)

Customers of experience goods do not fully know their preferences for the products until they

gain some experience with them. We assume that a buyer’s current preference is determined

by his past preference and an exogenous random shock, and may also be affected by his past

consumption x: α = γ+kx+ε, k ∈ [−1, 1]. ε is independent with γ and is distributed according

to the c.d.f G(·) and density function g(·). The distribution of α conditional on γ is therefore

given by c.d.f G(α− γ − kx) and density g(α− γ − kx).

Parameter k captures the sensitivity of current preference to past consumption. If k > 0, a

customer is growing fonder of products that they have purchased in the past due to some habitual

reasons. So a positive coefficient k is an indication of consumption inertia or persistence. If

k < 0, given levels of consumption are less satisfying when past consumption has been greater.

For example, patient often develops tolerance to some prescription medications (such as pain

killer, antibiotic and almost all kinds of psychoactive drugs) due to constant exposure to them.

After repeated use, one has to increase the dosage taken or switch to more potent ways of

taking these drugs, such as snorting or vein injecting, to achieve the same efficacy as before.

So a negative coefficient k is an indication of tolerance. γ and α are observable only to the

buyer, while their distributions are common knowledge. The seller offers a direct contract menu

C = {x(γ), t(γ);X(γ, α), T (γ, α)} to the buyer specifying his quantities or qualities consumed

and payments charged in each period contingent on the history of reports submitted. A contract

is incentive compatible when it induces truthtelling in every period. Truthful report in period 2

requires:

IC2 : U(γ, α) ≡ αX(γ, α)− T (γ, α)− U0(x(γ)) > αX(γ, α̂)− T (γ, α̂)− U0(x(γ)), ∀γ, α, α̂. (2)

The subjective satisfaction of an agent in the second period comes from the psychological com-

parison between utility bestowed by the contract and the reservation utility bestowed by outside

option, though he actually has no option to exit the mechanism in the second period.3 The

3In the present paper, consumers are assumed to have full commitment over the entire time horizon. That is,

a consumer can’t leave the relationship once the long-term contract is signed at the outset of the game.

7

agents’ date-2 reservation utility upon initial consumption x is U0(x). If x and X represent

quality, the reservation utility is normalized to zero, i.e., U0(x) ≡ 0. If x and X are quantities,

however, this reservation utility is negative and depends on his past consumption. Notable ex-

amples are addictive goods such as illegal drugs. After chronically high levels of exposure to

an addictive stimulus, users find it increasingly difficult to cease consumption. We impose the

following assumptions on the agent’s quitting cost.

Assumption 1 U0(0) = 0, U′0(x) < 0, U

′′0 (x) > 0, limx→0 U

′0(x) = −∞, limx→+∞ U ′0(x) =

0, ∀x ∈ R+.

This assumption guarantees an interior solution and is ubiquitous in practical applications. It

shows that as the stock of past consumption increases, the quitting cost rises, but at a decreasing

rate. A daily smoker is much more painful than an intermittent smoker when he abstains from

smoking; while a two-pack a day smoker may suffer nearly as much as a three-pack a day smoker

when quitting cold-turkey.

Both the principal and the agent are patient and do not discount future payoffs. Let U(γ, γ̂)

denote the continuation utility of type γ if he accepts the contract and reports γ̂:

Ũ(γ, γ̂) ≡ γx(γ̂)− t(γ̂) +∫ +∞−∞

[αX(γ̂, α)− T (γ̂, α)− U0(x(γ̂))] dG(α− γ − kx(γ̂)).

We use U(γ) ≡ Ũ(γ, γ) to denote the truthful continuation payoff. To ensure the agent’s partic-

ipation and truthful report in the first period, we impose the following constraints:

IC1 : U(γ) > Ũ(γ, γ̂), (3)

IR : U(γ) > 0. (4)

As is standard in the sequential screening literature (see, for example, Krähmer and Strausz

(2011)), we require individual rationality from an ex-ante perspective only. Once an agent

enters into a long-term contractual arrangements in the first period, he has no option to exit

the mechanism in the second period. When the agent is of type γ, the principal obtains payoff

W (γ) ≡ γx(γ)−C(x(γ))+∫ +∞−∞

[αX(γ, α)− C(X(γ, α))] dG(α−γ−kx(γ))−U0(x(γ))−U(γ).

From an ex-ante perspective, the principal’s optimization problem, referred to as P, can be

represented as

[P] : maxC

∫ γγW (γ)dH(γ), s.t. : IC1, IC2, IR

Applying the well-known approach for solving static screening problems, the second period

incentive compatibility constraint IC2 is equivalent to the necessary/envelope condition (EN2)

and the sufficient/implementabiluity condition (IM2).

8

Lemma 1 For any γ ∈ [γ, γ], ∃ a transfer T (γ, α) such that IC2 holds if and only if

EN2 : X(γ, α)is non-decreasing in α (5)

IM2 :∂U(γ, α)

∂α= X(γ, α) (6)

The proof of this Lemma is standard and is thus omitted. The characterization of IC2 in terms

of monotonicity and envelope conditions allows us to eliminate transfers from the principal’s

problem and to consider optimization problem with respect to allocations only.

In contrast, the first period incentive compatible constraints (IC1) cannot be characterized

in terms of envelope condition and monotonicity conditions of the allocation x(γ). The reason

is that the agent’s continuation payoff includes an expectation over his future type, which is

affected by the current allocation. So it depends on the whole schedule of allocations instead of

a single, type specific allocation only. The next lemma gives a dynamic version of envelope and

implementability conditions.

Lemma 2 Under EN2 if IC1 holds, then

EN1 : U ′(γ) = x(γ) +∫ +∞−∞

X(γ, α)dG(α− γ − kx(γ)) (7)

IM1 : x′(γ) +

∂2Γ

∂γ∂γ̂(γ, γ) > 0. (8)

Conversely, if EN1 and condition

IM ′1 : x′(γ̂) +

∂2Γ

∂γ∂γ̂(γ, γ̂) > 0,∀γ, γ̂ (9)

hold, then there exist a transfer t(γ) such that IC1 is satisfied, where

Γ(γ, γ̂) ≡∫ +∞−∞

[αX(γ̂, α)− T (γ̂, α)− U0(x(γ̂))] dG(α− γ − kx(γ̂)).

Proof. See appendix.

EN1 follows by a standard envelope argument. A key problem in contract theory is whether

the solution to a relaxed problem that imposes only envelope condition actually satisfies the

global incentive compatibility constraint. In static problems, this simply amounts to checking

a monotonicity condition, which in turn is guaranteed by a monotone hazard. In a dynamic

setup, however, the first-period incentive constraints, in general, cannot be characterized in

terms of monotonicity. In the sequel, we will distinguish two cases where x and X represent,

respectively, quantities and qualities. In the quantity case, it is shown that condition IM ′1 is

implied by the familiar monotonicity condition typically considered in the static contract model.

So the solution to the relaxed problem is indeed the solution to the original problem. In the

9

quality case, however, it is shown that IM1 is equivalent to IM′1 and is stronger than the usual

monotonicity condition, it thus cannot be satisfied automatically. In this case, one must resort

to an ironing technique to derive the optimal allocation.

2.1 quantity case

If x andX denote quantities, they are both nonnegative and thus U(γ) is increasing in γ. Clearly,

the participation constraint of the lowest type must be binding at the optimum. Integrating by

parts and setting U(γ) = 0, we rearrange the seller’s objective as:

W ≡∫ γγ

(γ + θ(γ))x(γ)− x2(γ)/2− U0(x(γ))+∫ +∞−∞

[(α+ θ(γ))X(γ, α)−X2(γ, α)/2

]dG(α− γ − kx(γ))

dH(γ), (10)where θ(γ) ≡ H(γ)−1h(γ) . We can now maximize this expression pointwise with respect to (x,X)

over R2+ to obtain the optimal quantities.

Proposition 1 If x and X denote quantities, Assumption 1 holds, k ∈ [−1, 1], then

X∗(γ, α) = max {α+ θ(γ), 0} , (11)

x∗(γ) is given implicitly by

γ + θ(γ)− x− U ′0(x) + k∫ +∞−θ(γ)−γ−kx

[ε+ (θ(γ) + γ + kx)] dG(ε) = 0. (12)

Proof. See appendix.

Corollary 1 Suppose that the good is harmful addictive (i.e., k ∈ [−1, 0]), and γ > 0, γ <

1/h(γ),

• if −U ′0(x∗(γ)) + k∫ +∞−γ−kx∗(γ)[ε + γ + kx

∗(γ)]dG(ε) > 0, then x∗(γ) > xSB(γ), with strict

inequality for γ < γ(Figure 1a);

• if −U ′0(x∗(γ))+k∫ +∞−γ−kx∗(γ)[ε+γ+kx

∗(γ)]dG(ε) < 0, then ∃γ̂ ∈ (γ, γ), such that x∗(γ) >

xSB(γ), ∀γ ∈ [γ, γ̂), x∗(γ) < xSB(γ), ∀γ ∈ (γ̂, γ](Figure 1b).

Proof. See appendix.

Harmful addictive behavior is usually assumed to involve both “reinforcement” and “toler-

ance” effects. Reinforcement effect means that greater past consumption increases the desire

for present consumption. Tolerance effect suggests that a given level of consumption yields less

satisfaction as cumulative past consumption increases. Becker and Murphy (1988) shows that

10

(a) case 1

(b) case 2

Figure 1. Comparison between x∗(γ) and xSB(γ)

reinforcement effect exists if an increase in past use raises the marginal utility of current con-

sumption. In our model, however, we interpret reinforcement effect as a withdrawal cost which

increases with the previous consumption. Our assumption makes sense since quitting costs is

a critical feature for harmful addiction. More importantly, there is no clear evidence that a

heavy smoker derives greater gratification from every additional cigarette than a lighter smoker.

But compared to a lighter user, a heavy user suffers obviously more severe physical and mental

discomforts upon the cessation or interruption of consumption.

Reinforcement effect suggests the seller to increase his initial supply since a heavy user is

more likely to succumb to the good in the future. Tolerance effect, however, suggests the seller

to reduce his first-period supply, since consumers experience progressively lower sensitivity and

thus lower desire of buying upon a high dose of initial use. Overall, the comparison between the

initial supply x∗(γ) and the second-best quantity xSB(γ) hinges on the tradeoff between these

opposing effects. For consumers at the low-end of spectrum, consumptions are very low so that

reinforcement effect measured by the marginal decrease to reservation utility is relatively large.

The initial supply is therefore higher than its second-best value. For high-demanders, we get

ambiguous results: if reinforcement effect dominates, initial supply of an addictive good may be

higher than its second-best value; if tolerance effect dominates, the opposite holds. In Corollary

1, the reinforcement effect of the highest type γ is measured by −U0(x∗(γ)), while the tolerance

effect is measured by −k∫ +∞−∞ X

∗(γ, α)dG(α−γ−kx∗(γ)) = −k∫ +∞−γ−kx∗(γ)[ε+γ+kx

∗(γ)]dG(ε).

So distortions of x∗(γ) relative to xSB(γ) is determined by the comparisons between these two

terms. The negative impact of tolerance effect on initial supply is ubiquitous in pharmaceutical

industry. In the presence strong tolerance effect, a company has little incentive to develop new

11

antibiotics or psychoactive drugs because this leads to loss of efficacy of the currently available

products. Increasing R&D investment may re-amplify the effects of their products, however this

may accelerate tolerance, further reducing the drug’s effects and the firm’s profit.

2.2 quality case

In this subsection, we assume that x and X represent qualities. The analysis parallels the

earlier subsection, except that there is no withdrawal cost (i.e., U0(x) ≡ 0), and both x and X

are allowed to be negative. In this case, U ′(γ) = x(γ) +∫ +∞−∞ X(γ, α)dG(α − γ − kx(γ)) may

now be either positive or negative. The agent is tempted to understate his private information

for some of its realizations, and to overstate it for others, countervailing incentives problem may

therefore arise.

Proposition 2 Suppose that γ < 0 < γ, x and X represent qualities of good, then the optimal

contract entails:

x∗(γ) =

γ+

H(γ)h(γ)

1−k if γ ∈ [γ, γ∗]

γ−x̄opt2 if γ ∈ [γ

∗, γ∗∗]

γ− 1−H(γ)h(γ)

1−k if γ ∈ [γ∗∗, γ]

,

X∗(γ, α) =

α+ H(γ)h(γ) if γ ∈ [γ, γ

∗]

α+ (1−k)(γ−x̄opt)

2 − γ if γ ∈ [γ∗, γ∗∗]

α− 1−H(γ)h(γ) if γ ∈ [γ∗∗, γ]

,

where

x̄opt =

xl if W (xl) > W (xh)

xh if W (xl) < W (xh)

xl or xh if W (xl) =W (xh)

,

W (x) ≡ 11− k

∫ γ∗(x)γ

[γ + H(γ)h(γ)

]2dH(γ) +

∫ γ∗∗(x)γ∗(x)

[(1−k)(γ−x)

2

]2dH(γ)

+∫ γγ∗∗(x)

[γ − 1−H(γ)h(γ)

]2dH(γ) + 1−k2 V ar(ε)

,xl = max{γ1, x̄∗}, xh = min{γ2, x̄∗∗}, x̄∗ = −1+k1−kγ, x̄

∗∗ = −1+k1−kγ, γ1 and γ2 are determined by

γ1 +H(γ1)h(γ1)

= γ2 − 1−H(γ2)h(γ2) = 0. γ∗(x) is given implicitly by γ + H(γ)h(γ) =

1−k2 (γ − x), γ

∗∗(x) is

given by γ − 1−H(γ)h(γ) =1−k2 (γ − x), γ

∗ = γ∗(x̄opt), γ∗∗ = γ∗∗(x̄opt).

Proof. See appendix.

Figure 2 illustrates how x̄opt and x∗(γ) are determined in different setups. In standard

contract literatures, countervailing incentives problem exists when the information rent function

12

Slope=1/2

(a) γ1 6 x̄∗ 6 x̄∗∗ 6 γ2,W (x̄∗∗) > W (x̄∗)

Slope=1/2

(b) γ1 6 x̄∗ 6 x̄∗∗ 6 γ2,W (x̄∗∗) < W (x̄∗)

Slope=1/2

(c) x̄∗ 6 γ1 6 x̄∗∗ 6 γ2,W (x̄∗∗) < W (γ1)

Slope=1/2

(d) x̄∗ 6 γ1 6 x̄∗∗ 6 γ2,W (x̄∗∗) > W (γ1)

Slope=1/2

(e) γ1 6 x̄∗ 6 γ2 6 x̄∗∗,W (x̄∗) > W (γ2)

Slope=1/2

(f) γ1 6 x̄∗ 6 γ2 6 x̄∗∗,W (x̄∗) < W (γ2)

Slope=1/2

(g) x̄∗ 6 γ1 6 γ2 6 x̄∗∗,W (γ1) > W (γ2)

Slope=1/2

(h) x̄∗ 6 γ1 6 γ2 6 x̄∗∗,W (γ1) < W (γ2)

Figure 2. the optimal qualities x∗(γ) (the red solid curve) in different setups

13

U(γ) is U−shaped. When U(γ) is increasing (resp. decreasing) with the agent’s type γ, he

has an incentive to understate (resp. overstate) his private information, the optimal contract

will prescribe an allocation below (resp. above) the first-best level. For intermediate type

realizations, the countervailing incentives compel the principal to set an allocation that does not

vary with γ, hence, the equilibrium involves an interval of pooling. The neglected monotonicity

condition is met provided the monotone hazard rate property (1) is satisfied; in cases when (1)

does not hold everywhere in the entire type space, one must resort to the “ironing” technique

to derive the optimal allocation. In our dynamic contracting setup, however, we need to adopt

an ironing technique to get an implementable allocation even when (1) is satisfied. The only

difference is that, we now “iron” the allocation using a sloping line rather than perform the

traditional horizontal ironing procedure. We next illustrate the result of Proposition 2 with a

numerical example.

An numerical example. Suppose that γ is uniformly distributed on [γ, γ], then γ1 = γ/2, γ2 =

γ/2, γ∗(x) =2γ−(1−k)x

3+k , γ∗∗(x) = 2γ−(1−k)x3+k ,

W (x̄) =2(1− k)(3 + k)2

x̄2 −(γ + γ)(1− k)

(3 + k)2x̄+

(γ2 + γγ + γ2)(k2 + 2k + 5)

3(3 + k)2(1− k)+

1

2V ar(ε).

The optimal ironing parameter x̄opt depends on parameters k and t ≡ γ/γ:

• if γ2γ 6 −1+k1−k 6 −

1+k1−k

γ

γ 612 , γ/γ > −1, x̄opt = x̄∗ ≡ −

1+k1−kγ;

• if γ2γ 6 −1+k1−k 6 −

1+k1−k

γ

γ 612 , γ/γ 6 −1, x̄opt = x̄∗∗ ≡ −

1+k1−kγ;

• if γ2γ 6 −1+k1−k 6

12 6 −

1+k1−k

γ

γ , x̄opt = γ2 ≡ γ/2;

• if −1+k1−k 6γ

2γ 6 −1+k1−k

γ

γ 6γ2γ , x̄

opt = γ1 ≡ γ/2;

• if −1+k1−k 6γ

2γ 612 6 −

1+k1−k

γ

γ , x̄opt = γ1 ≡ γ/2 or γ2 ≡ γ/2.

All the above-mentioned cases are depicted in the following Figure 3.

3 multi-agent case

In this section, we extend the above model to multi-agent environment. I ≡ {1, · · · , N} denotes

the set of interlinked agents, where N > 1. We keep track of social connections in this network

by its symmetric adjacency matrix Σ ≡ [σij ], where σij = σji = 1 if i and j are directly linked,

and σij = 0 otherwise. We also set σii = 0, ∀i ∈ I. Each agent has a utility function:

ui(γi,x, ti,Σ) = γixi +1

2

∑j ̸=i

σijxixj − ti

14

xopt

=

x*= -

1 +k

1 -k

xopt

=

x**

= -1 +k

1 -k

xopt

= 2 =1

2

xopt

= 1 =1

2

xopt

= 1 =1

2 or 2 =

1

2

-1.0 -0.5 0.0 0.5 1.0

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

k

t

Figure 3. optimal choice of x̄ for uniform distribution t ≡ γ/γ

depending on his own consumption xi and those of others xj , j ̸= i. γi denotes the individual

preference and ti is the transfer. The principal provides a vector x of either quantities or qualities

to N agents at a quadratic cost C(x) = x′Cx/2, where C ≡ [cij ] is a symmetric positive definite

matrix. Supplying goods of certain quantities or qualities to N consumers can be regarded

as engaging in multiple tasks. The diagonal elements cii reflect the task-specific cost, and the

off-diagonal elements cij represent the technological interactions between tasks. Tasks i and j

are technologically substitute (resp. complementary, independent) if cij > 0, (resp. < 0,= 0).

Agent i’s future type αi depends on his current type γi, current consumptions of all agents x

and a noisy term ϵi:

αi = γi +∑j∈I

aijxi + ϵi, ∀i ∈ I.

Let ai ≡ (ai1, · · · , aiN )⊤, A ≡ [aij ], γ ≡ (γ1, · · · , γN )⊤, α ≡ (α1, · · · , αN )⊤, ϵ ≡ (ϵ1, · · · , ϵN )⊤.

The agents’ type γi, i ∈ I are drawn in an i.i.d. manner from a distribution with density h(·)

and c.d.f. H(·) supported on [γ, γ]. For each i, the monotone hazard rate properties

d

dγi

[1−H(γi)h(γi)

]6 0 6 d

dγi

[H(γi)

h(γi)

]hold. ϵi, i ∈ I are also i.i.d. with density g(·), c.d.f. G(·) on (−∞,+∞), and have zero mean

and unit variance E(ϵi) = 0, V ar(ϵi) = 1, ∀i ∈ I. For any pair (i, j), γi and ϵj are statistically

independent. Analogous to the one-agent case, aii captures the the effect of a consumer’s

present consumption on his future willingness to pay due to habitual persistence or tolerance

15

effect depending on its sign. The off-diagonal element aij , i ̸= j captures the impact of agent j’s

current consumption on agent i’s future type. If aij > 0, a bandwagon effect is observed since a

consumer’s preference for a good increases as the consumption of his peer increases. If aij < 0,

a snob effect is observed, since a consumer’s preference for a good becomes stronger as the

consumption of others decreases. The bandwagon effect is a psychological phenomenon in which

people chooses a commodity primarily because other people are doing so. This phenomenon is

ubiquitous during bull markets and the growth of asset bubbles. On the contrary, snob effect

refers to the desire to possess a unique commodity having a prestige value. The smaller is the

number of people owning a good, the stronger is a consumer’s desire of possessing it. Notice

that matrix A is not necessarily symmetric. As commonly observed, the current behavior of a

superstar may affect the future appetite of his/her fans but not vice versa.

Let xi and Xi denote consumptions of, respectively, the first and second periods, ti and Ti

are the associated transfers. The principal offers and commits to a direct mechanism

C ≡ {xi(γ), ti(γ), Xi(γ, α), Ti(γ, α)}i∈I

stipulating consumptions and transfers contingent on history of reports up to the time consid-

ered. Still, U0(xi(γ)) represents the reservation utility of agent i upon initial consumption xi(γ),

it satisfies Assumption 1. To economize notations, we represent the consumptions of the first and

second periods by vectors x(γ) ≡ (x1(γ), · · · , xN (γ)) and X(γ, α) ≡ (X1(γ, α), · · · , XN (γ, α)).

ui(γ) ≡ γixi(γ)+12∑

j ̸=i σijxi(γ)xj(γ)−ti(γ) and Ui(γ, α) ≡ αiXi(γ, α)+12

∑i̸=j σijXi(γ, α)Xj(γ, α)−

Ti(γ, α)−U0(xi(γ)) denote the instantaneous net information rents of, respectively, the first and

the second periods. Ûi(γ, αi) ≡ Eα−i [Ui(γ, α)] and Ui(γi) ≡ Eγ−iui(γ) + Eγ−iEαUi(γ, α) repre-

sent the agent’s net expected payoffs conditional on information sets available at the start of,

respectively, the second and the first periods. Then, we have

Ûi(γ, αi) = αiX̂i(γ, αi)− T̂i(γ, αi) +1

2

∑j ̸=i

σijX̂ij(γi, γ−i, αi)− U0(xi(γi, γ−i))

Ui(γi) =

γix̂i(γi) +

12

∑j ̸=i σij x̂ij(γi)− t̂i(γi)

+∫γ−i

[∫αiÛi(γ, αi)g

(αi − γi − a⊤i x(γ)

)dαi

]∏j ̸=i h(γj)dγ−i

,

16

where

x̂i(γi) ≡∫γ−i

xi(γ)∏j ̸=i

h(γj)dγ−i,

t̂i(γi) ≡∫γ−i

ti(γ)∏j ̸=i

h(γj)dγ−i

x̂ij(γi) ≡∫γ−i

xi(γ)xj(γ)∏j ̸=i

h(γj)dγ−i,

X̂i(γ, αi) ≡∫α−i

Xi(γ, α)∏j ̸=i

g(αj − γj − a⊤j x(γ)

)dα−i,

T̂i(γ, αi) ≡∫α−i

Ti(γ, α)∏j ̸=i

g(αj − γj − a⊤j x(γ)

)dα−i,

X̂ij(γ, αi) ≡∫α−i

Xi(γ, α)Xj(γ, α)∏j ̸=i

g(αj − γj − a⊤j x(γ)

)dα−i.

The principal’s ex-ante payoff could be represented as

W ≡∫[γ,γ]N

W (γ)

(N∏i=1

h(γi)

)dγ1 · · · dγN ,

where,

W (γ) =

N∑i=1

ti(γ)− C(x) +∫RN

[N∑i=1

Ti(γ, α)− C(X)

]N∏i=1

g(αi − γi − a⊤i x(γ)

)dα1 · · · dαN

=

γ⊤x(γ)− 12x⊤(γ)(C − Σ)x(γ)

+∫α

α⊤X(γ, α)−12X

⊤(γ, α)(C − Σ)X(γ, α)

∏Ni=1 g

(αi − γi − a⊤i x(γ)

)dα

−∑N

i=1 U0(xi(γ))−∑N

i=1 Ui(γi)

. (13)

The incentive compatibility constraints in periods 2 and 1 can be written as

ICi2 :

αiX̂i(γ, αi)− T̂i(γ, αi) + 12σijX̂ij(γ, αi)

>

αiX̂i(γ, α̂i)− T̂i(γ, α̂i) + 12σijX̂ij(γ, α̂i)

,∀γ, αi, α̂i

ICi1 :

γix̂i(γi)− t̂i(γi) + 12

∑j ̸=i σij x̂ij(γi) + Γi(γi, γi)

>

γix̂i(γ̂i)− t̂i(γ̂i) + 12∑

j ̸=i σij x̂ij(γ̂i) + Γi(γ̂i, γi),

, ∀γi, γ̂i,where,

Γi(γ̂i, γi) ≡∫γ−i

∫αi

Ûi(γ̂i, γ−i, αi)g(αi − γi − a⊤i x(γ̂i, γ−i)

)dαi

∏j ̸=i

h(γj)dγ−i.

17

To guarantee the agent’s participation in the contract, the following individually rational con-

straint needs to be met:

IRi : Ui(γi) ≡ γix̂i(γi)− t̂i(γi) +1

2

∑j ̸=i

σij x̂ij(γi) + Γi(γi, γi) > 0. (14)

As is standard in the sequential screening literature, we require individual rationality from an

ex ante perspective only.

Lemma 3 ICit holds if and only if the following envelop and implementability conditions ENit ,

IM it , ∀i ∈ I, ∀t ∈ {1, 2} hold:

EN i2 :∂Ûi(γ, αi)

∂αi= X̂i(γ, αi), a.e. (15)

IM i2 : X̂i(γ, αi) is nondecreasing in αi (16)

EN i1 : U ′i(γi) = x̂i(γi) +∂Γi∂γi

(γi, γi) (17)

IM i1 : x̂′i(γi) +

∂2Γi∂γi∂γ̂i

(γi, γi) > 0. (18)

Proof. See appendix.

3.1 quantity case

This subsection discusses the optimal selling contract in settings where X and x represent

quantities. Note that

∂Γi∂γi

(γ̂i, γi) =

∫γ−i

∫αi

[−Ûi(γ̂i, γ−i, αi)g′(αi − γi − a⊤i x(γ̂i, γ−i))

]dαi

∏j ̸=i

h(γj)dγ−i

=

∫γ−i

∫αi

−Ûi(γ̂i, γ−i, αi)g(αi − γi − a⊤i x(γ̂i, γ−i))∣∣∣+∞−∞+g(αi − γi − a⊤i x(γ̂i, γ−i))X̂i(γ̂i, γ−i, αi)

dαi∏j ̸=i

h(γj)dγ−i

=

∫γ−i

∫α

[Xi(γ̂i, γ−i, α)

N∏i=1

g(αi − γi − a⊤i x(γ̂i, γ−i))

]dα∏j ̸=i

h(γj)dγ−i. (19)

The second equality follows from integration by parts and EN i2, the third equality follows from

limαi→−∞

Ûi(γ̂i, γ−i, αi)g(αi − γi − a⊤i x(γ̂i, γ−i))

= limαi→+∞

Ûi(γ̂i, γ−i, αi)g(αi − γi − a⊤i x(γ̂i, γ−i)) = 0.

EN i1 then takes the following form

U ′i(γi) =∫γ−i

[xi(γ) +

∫αXi(γ, α)

∏i∈I

g(αi − γi − a⊤i x(γ)

)dα

]∏j ̸=i

h(γi)dγ−i. (20)

18

Since x and X are all nonnegative, the individual rationality constraint IRi1 binds at the lowest

type, i.e., Ui(γ) = 0. Using integration by parts, we obtain∫ γγ

Ui(γi)dH(γi) =∫ γγ

1−H(γi)h(γi)

U ′i(γi)dH(γi)

=

∫γ

∫α

[1−H(γi)h(γi)

xi(γ) +1−H(γi)h(γi)

Xi(γ, α)

] N∏i=1

g

αi − γi−a⊤i x(γ)

N∏i=1

h(γi)dαdγ.

Inserting this expression into the principal’s objective (13) allows us to rewrite the optimization

problem as:

[P1] : maxx(γ)∈RN+ ,X(γ,α)∈RN+

W, s.t. : IM it , ∀t ∈ {1, 2}, i ∈ I,

where

W ≡∫[γ,γ]N

[γ + θ(γ)]⊤x(γ)− 12x⊤(γ)(C − Σ)x(γ)

+∫α

(α+ θ(γ))⊤X(γ, α)−12X

⊤(γ, α)(C − Σ)X(γ, α)

∏Ni=1 g

αi − γi−a⊤i x(γ)

dα−∑N

i=1 U0(xi(γ))

∏i∈I

h(γi)dγ,

θ(γ) ≡ (θ1(γ1), · · · , θN (γN ))⊤, θi(γi) ≡ H(γi)−1h(γi) .

For expositional convenience, we introduce a few new notations and definitions. Let Co(B) ≡

{x ∈ RN |∃λ ∈ RN+ ,∋ x = Bλ} be the convex polyhedral cone spanned by columns of matrix

B ∈ RN×N . For any subset K ⊆ I ≡ {1, · · · , N}, B(K) denotes a matrix obtained from

B by replacing the ith column bi with −ei for all i ∈ I\K, where ei = (0, · · · , 1, · · · , 0)⊤ is

a N−dimentional vector with 1 in the ith entry and zeros in all other entries. BKK is the

principal submatrix of B containing all rows and columns indexed by k ∈ K, xK ≡ (xi)i∈K and

x−K ≡ (xi)i∈I\K are the sub-vectors of x indexed by entries in K and I\K, respectively. A

matrix A ∈ Rn×n is said to be an M−matrix if it has nonpositive off-diagonal entries and an

entry-wise nonnegative inverse, i.e., Aij 6 0,∀i ̸= j, A−1 > 0. 4 AN denotes the class of N ×N

matrices with positive diagonal elements and nonpositive off-diagonal elements, i.e., aii > 0 for

each i and aij 6 0 whenever i ̸= j; MN denotes the class of N ×N M−matrices; RN×N+ denotes

the class of N×N element-wise nonnegative matrices; M−1N ⊂ RN×N+ denotes the class of N×N

inverse M−matrices; PN denotes the class of N ×N positive definite matrices.

Proposition 3 Suppose that λmin(C−Σ) >√λmax(AA⊤), then the optimal sequential contract

entails:

4Throughout this paper, A ≡ [aij ] > B ≡ [bij ] means that aij > bij ,∀i, j; A > B means A > B and A ̸= B;

A ≽ (resp. ≻)B means that A−B is positive simidefinite (resp. definite).

19

• the second-period quantity X∗(γ, α) given by

X∗K(γ, α)X∗−K(γ, α)

= [(C − Σ)KK ]−1[αK + θK(γK)]

0

if α+ θ(γ) ∈ Co[(C − Σ)(K)]

K ̸= ∅

0 if α+ θ(γ) ∈ Co[(C − Σ)(∅)]

;

(21)

• the first-period quantity x∗(γ) given implicitly by

[C − Σ−A⊤V(x, γ)A

]x−

[I +A⊤V(x, γ)A

](γ + θ(γ)) +∇

[∑i∈I

U0(xi)

]= 0, (22)

where ∇ denotes the gradient operator, VK(x, γ) ≡∫SK(x,γ)

∏i∈I g(ϵi)dϵ, SK(x, γ) ≡

Co[(C−Σ)(K)]−{γ+θ(γ)+Ax} is convex cone obtained by shifting the vertex of Co[(C−

Σ)(K)] to point −γ −Ax− θ(γ),

V(x, γ) ≡∑

K∈2I\∅

VK(x, γ)

[(C − Σ)KK ]−1 00 0

;• the principal’s payoff

W ∗ = Eγ[[γ + θ(γ)]⊤x∗(γ)− 1

2[x∗(γ)]⊤(C − Σ)x∗(γ) + ρ(x∗(γ), γ)

],

where

ρ(x∗(γ), γ) =1

2

∑K∈2I\∅

∫Co[(C−Σ)(K)]

ν⊤

[(C − Σ)KK ]−1 00 0

ν N∏i=1

g

νi − γi−θi(γi)− a⊤i x∗(γ)

dν.Proof. See appendix.

The following Figure 4 depicts the optimal quantities in the two-agent case. When α+θ(γ) ∈

Co[(C −Σ)(1, 2)], X∗(γ, α) = (C −Σ)−1[α+ θ(γ)]; when α+ θ(γ) ∈ Co[(C −Σ)(i)], X∗i (γ, α) =αi−θi(γi)

cii, X∗j (γ, α) = 0, ∀i ∈ {1, 2}, ∀j ̸= i; when α+ θ(γ) ∈ Co[(C − Σ)(∅)], X∗(γ, α) = 0.

Corollary 2 Suppose that λmin(C − Σ) >√λmax(AA⊤), cij 6 σij , ∀i ̸= j, A ∈ RN×N+ , then

x∗(γ) > xSB(γ), ∀γ ∈ [γ, γ]N .

Proof. See appendix.

The result in this Corollary is stronger than those of Milgrom and Shannon (1994), who

merely assume supermodularity and increasing differences. Their weaker assumptions ensure

only that the set of optimal choices is non-decreasing in the exogenous parameters. In our

model, however, an interior solution in dynamic setting guaranteed by Assumption 1 cannot be

a solution in static setting, so we obtain a strict comparison between xSB(γ) and x∗(γ). In the

20

Co[(C - ) (1, 2)]Co[(C - ) (2)]

Co[(C - ) (!)] Co[(C - ) (1)]

" +# ($)

" +# ($)

" +# ($)

X2* % c21 -&21c22 -&22 '

X1* % c11 -&11c12 -&12 '

"2 +#2 ($2 )c22 -&22

% c21 -&21c22 -&22 '

"1 +#1 ($1 )c11 -&11

% c11 -&11c21 -&22 '

(

(

-10 -5 0 5 10

-10

-5

0

5

10

Figure 4. X∗(γ, α) for N = 2

multi-agent dynamic contracting environment, when the strategic complementarity between any

pair of agents outweighs the technological substitutability between them (i.e., cij 6 σij , ∀i ̸= j),

reinforcement effect (U ′0(xi) < 0,∀i), consumption inertia effect (aii > 0, ∀i ∈ I) and bandwagon

effect (aij > 0,∀i ̸= j) are aligned with each other. They all require an upward distortion of the

date-1 quantities relative to the static contract, so we have xSB(γ) < x∗(γ).

Corollary 3 If A ∈ RN×N+ , cij 6 σij , ∀i ̸= j, λmin(C −Σ) >√λmax(AA⊤)+ 1, then adding an

additional link between agents i and j will increase the date-2 quantities X∗(γ, α) and strictly

increase the date-1 quantities x∗(γ).

Proof. See appendix.

This corollary shows that when any pair of agents are complementary in both technologi-

cal and strategic sense (cij 6 0, σij > 0, ∀i ̸= j), or the strategic complementarity outweighs

the technological substitutability between them (0 < cij < σij , ∀i ̸= j), consumption inertia and

bandwagon effects prevails but are not too strong in the economy (aij > 0, ∀i, j,√λmax(AA⊤) <

λmin(C − Σ) − 1), the denser is the network, the larger quantities will be provided by the

seller. As a limiting case, a complete network (σij = 1,∀i ̸= j) has the highest volumes of

transactions in both periods. Notice that the first-period quantities increase strictly when an

additional link is built, while the second-period quantities increase weakly since corner solutions

may arise in the optimization problem of date two. The following figures 5(a) and 5(b) depict

the comparison between X∗(γ, α, Σ̂) and X∗(γ, α,Σ) for two-agent case.−→OA = C·1 − Σ·1 ≡

21

(c11, c21 − σ21)⊤,−−→OA′ = C·2 − Σ·2 ≡ (c12 − σ12, c22)⊤,

−−→OB = C·1 − Σ̂·1 ≡ (c11, c21 − σ21 − 1)⊤,

−−→OB′ = C·2 − Σ̂·2 ≡ (c12 − σ12 − 1, c22)⊤, In Figure 5(a),

−−→OO1 = α + θ(γ) ∈ Co[(C −

Σ)(1, 2)], X∗1 (γ, α,Σ) = |OC|/|OA| < X∗1 (γ, α, Σ̂) = |OF |/|OB|, X∗2 (γ, α,Σ) = |OC ′|/|OA′| <

X∗2 (γ, α, Σ̂) = |OF ′|/|OB′|. In Figure 5(b),−−→OO2 = α+θ(γ) ∈ Co[(C−Σ)(1)]

∩Co[(C−Σ̂)(1, 2)],

X∗1 (γ, α,Σ) = |OE|/|OA| < X∗1 (γ, α, Σ̂) = |OC|/|OB|, X∗2 (γ, α,Σ) = 0 < X∗2 (γ, α, Σ̂) =

|OD|/|OB′|.−−→OO3 = α + θ(γ) ∈ Co[(C − Σ)(2)]

∩Co[(C − Σ̂)(1, 2)], X∗1 (γ, α,Σ) = 0 <

X∗1 (γ, α, Σ̂) = |OD′|/|OB|, X∗2 (γ, α,Σ) = |OE′|/|OA′| < X∗2 (γ, α, Σ̂) = |OC ′|/|OB′|.−−→OO4 =

α + θ(γ) ∈ Co[(C − Σ̂)(2)], X∗1 (γ, α,Σ) = X∗1 (γ, α, Σ̂) = 0, X∗2 (γ, α,Σ) = |OG′|/|OA′| =

X∗2 (γ, α, Σ̂) = |OF ′|/|OB′|.−−→OO5 = α + θ(γ) ∈ Co[(C − Σ̂)(1)], X∗2 (γ, α,Σ) = X∗2 (γ, α, Σ̂) = 0,

X∗1 (γ, α,Σ) = |OG|/|OA| = X∗1 (γ, α, Σ̂) = |OF |/|OB|.−−→OO6 = α + θ(γ) ∈ Co[(C − Σ̂)(∅)],

X∗i (γ, α,Σ) = X∗i (γ, α, Σ̂) = 0, ∀i ∈ {1, 2}.

(a) α+ θ(γ) ∈ Co[(C − Σ)(1, 2)]

(b) α+ θ(γ) ∈ R2\Co[(C − Σ)(1, 2)]

Figure 5. X∗i (γ, α, Σ̂) > X∗i (γ, α,Σ), ∀i ∈ I

3.2 quality case

When x and X represent qualities of goods, the principal’s optimization problem is the same as

that in the quantity case, except that the choice variables are now allowed to be negative and

there is no reinforcement effect, i.e., U0(xi) = 0, ∀i ∈ I. The principal’s optimization problem

is then represented as

[P] maxx(γ)∈RN ,X(γ,α)∈RN

∫[γ,γ]N W (γ)

∏i∈I h(γi)dγ −

∑i∈I∫ γγ Ui(γi)h(γi)dγi,

s.t. : IRi(14), EN i1(20), IMi2(16), IM

i1(18),∀i ∈ I,

22

where

W (γ) ≡

γ⊤x(γ)− 12x

⊤(γ)(C − Σ)x(γ)

+∫α

α⊤X(γ, α)−12X

⊤(γ, α)(C − Σ)X(γ, α)

∏Ni=1 g

αi − γi−a⊤i x(γ)

dα .

We need to introduce some new notations to facilitate our further analysis. Let ∆ ≡ (C −

Σ)−1, Q ≡ [C − Σ − A⊤(C − Σ)−1A]−1[I +A⊤(C − Σ)−1

], M ≡ [I + (C − Σ)−1A][C − Σ −

A⊤(C −Σ)−1A]−1[I +A⊤(C −Σ)−1]. γ∗i (ci) and γ∗∗(ci) are given implicitly by H(γ∗i )/h(γ∗i ) =

−Mii(γ∗i − ci)/(Mii +∆ii) and [H(γ∗∗i ) − 1]/h(γ∗∗i ) = −Mii(γ∗∗i − ci)/(Mii +∆ii), Aij denotes

the (i, j) element of matrix A. u(c) ≡ (u1(c1), · · · , uN (cN ))⊤, Ω(c) ≡ diag[ω1(c1), · · · , ωN (cN )],

Ŵ (c|E) ≡ [u⊤(c)(M +∆)u(c) + Tr(M +∆)Ω(c) + Tr∆]/2, where

ui(ci) =

∫ γγ γidH(γi) + ∫ γ∗i (ci)γ H(γi)h(γi) dH(γi)−∫ γ∗∗i (ci)γ∗i (ci)

Mii(γi−ci)Mii+∆ii

dH(γi) +∫ γγ∗∗i (ci)

H(γi)−1h(γi)

dH(γi)

,ωi(ci) =

∫ γ∗i (ci)γ [γi + H(γi)h(γi) ]2 dH(γi) + ∫ γ∗∗i (ci)γ∗i (ci) [γi − Mii(γi−ci)Mii+∆ii ]2 dH(γi)+∫ γγ∗∗i (ci)

[γi − 1−H(γi)h(γi)

]2dH(γi)− [ui(ci)]2

.γi(c−i) and γi(c−i) are determined, respectively, by∑

j ̸=iQijuj(cj) +Qii

[γi+H(γ

i)

h(γi)

]= 0

and ∑j ̸=i

Qijuj(cj) +Qii

[γi −

1−H(γi)h(γi)

]= 0.

ti(c−i) ≡ γi(c−i) + [(Mii +∆ii)H(γi(c−i))]/[Miih(γi(c−i))], ti(c−i) ≡ γi(c−i)− (Mii +∆ii)[1−

H(γi(c−i))]/[Miih(γi(c−i))], ci(c−i) = max{ti(c−i), γ}, ci(c−i) = min{ti(c−i), γ}. D(c) ≡∏i∈I [ci(c−i), ci(c−i)]. Let E ≡ {C,A,Σ, γ, γ,H(·)} denote the economic environment encom-

passing all characteristics of agents. ϕ(c|E) ≡ argmaxc̃∈D(c) Ŵ (c̃|E) denotes the set of maximiz-

ers of Ŵ (·) in environment E given vector c, F(E) ≡ {c|c ∈ ϕ(c|E)} is the set of fixed points of

correspondence ϕ(·) in environment E .

Proposition 4 Suppose that λmin(C − Σ) >√λmax(AA⊤), Qii > 0, ∀i ∈ I, F(E) ̸= ∅, then

the optimal contract entails:

• qualities of the first and second periods given by

x(γ) = Q[γ + ϑ(γ, c∗)] (23)

X(γ, α) = (C − Σ)−1[α+ ϑ(γ, c∗)] (24)

23

where ϑ(γ, c∗) = (ϑ1(γ1, c∗1), · · · , ϑN (γN , c∗N ))⊤,

ϑi(γi, c∗i ) =

H(γi)h(γi)

if γi ∈ [γ, γ∗i (c∗i )]

−Mii(γi−c∗i )

Mii+∆iiif (γ∗i (c

∗i ), γ

∗∗i (c

∗i )]

H(γi)−1h(γi)

if γi ∈ (γ∗∗i (c∗i ), γ]

,

c∗ = argmaxc̃∈F(E) Ŵ (c̃|E).

• a payoff obtained by the principal:

Ŵ (c∗|E) = 12u⊤(c∗)[M + (C − Σ)−1]u(c∗) + 1

2Tr[M + (C − Σ)−1]Ω(c∗) + 1

2Tr(C − Σ)−1.

Proof. See appendix.

A numerical example. Suppose γ is uniformly distributed on [γ, γ], then

γ∗i (ci) =ciMii + (Mii +∆ii)γ

2Mii +∆ii

γ∗∗i (ci) =ciMii + (Mii +∆ii)γ

2Mii +∆ii

ai(ci) =Mii

2Mii +∆ii

(ci −

γ + γ

2

)ui(ci) =

2Miici + (∆ii +Mii)(γ + γ)

2(∆ii + 2Mii)

γi(c−i) =

1

2

γ − ∑j ̸=iQij(γ+γ

2 +Mjj

∆jj+2Mjj

(cj −

γ+γ

2

))Qii

γi(c−i) =

1

2

γ − ∑j ̸=iQij(γ+γ

2 +Mjj

∆jj+2Mjj

(cj −

γ+γ

2

))Qii

ti(c−i) = −∆ii2Mii

γ − 2Mii +∆ii2Mii

∑j ̸=iQij

[γ+γ

2 +Mjj

(cj−

γ+γ

2

)∆jj+2Mjj

]Qii

ti(c−i) = −∆ii2Mii

γ − 2Mii +∆ii2Mii

∑j ̸=iQij

[γ+γ

2 +Mjj

(cj−

γ+γ

2

)∆jj+2Mjj

]Qii

ci(c−i) ≡ max{ti(c−i), γ}

ci(c−i) ≡ min{ti(c−i), γ}

ωi(ci) =(ci − γ)(ci − γ)M2ii

(∆ii + 2Mii)2 +

(2∆iiMii +∆

2ii + 5M

2ii

)(γ − γ)2

12 (∆ii + 2Mii)2

Since the objective Ŵ (c|E) is now convex in c, an optimal solution must be attained in a vertex

of the hyperrectangle feasible region D(c). We denote by V(D(c)) the set of vertexes of D(c).

24

Table 1. F(E) = (−4.98, 0.25)(γ, γ) = (−10, 20) (c1, c2) Ŵ (c1(c2), c2(c1)) Ŵ (c1(c2), c2(c1)) Ŵ (c1(c2), c2(c1)) Ŵ (c1(c2), c2(c1)) c1(c2) = c1

c2(c1) = c2

(−3.95,−1.42) 87.01 87.31 86.95 87.03 c1(c2) = c1

c2(c1) = c2

(−4.98, 0.25) 87.57 90.10∗ 86.64 88.13 c1(c2) = c1

c2(c1) = c2

(−1.31,−2.16) 87.49 86.76 87.44 87.23 c1(c2) = c1

c2(c1) = c2

(2.35,−0.49) 87.87 88.59 86.95 87.38

When N = 2, the possible maximizer is chosen among points A(c1(c2), c1(c2)), B(c1(c2), c2(c1)),

C(c1(c2), c1(c2)), D(c1(c2), c1(c2)). In Tables 1 to 3, blue numbers denote the largest row value,

a superscript star identifies a diagonal element which is the highest in its row, red numbers

denote the fixed points in F(E). F(E) may be a singleton, an empty set or have more than one

elements.

• Case (i). γ = −10, γ = 20,

A =

2 12 3

, C = 8 −1

−1 6

,Σ = 0 1

1 0

,F(E) = (−4.98, 0.25) (Table 1), Figure 6 depicts the corresponding optimal qualities;

• case (ii). γ = −40, γ = 40,

A =

2 12 3

, C = 8 −1

−1 6

,Σ = 0 1

1 0

,F(E) = ∅ (Table 2);

• case (iii). γ = −30, γ = 30,

A =

0 00 0

, C = 12 2

2 17

,Σ = 0 1

1 0

,then F(E) = {(15.460, 15.644), (−15.460,−15.644)} (Table 3).

Corollary 4 Suppose that H(·) is uniform distributions supported on symmetric interval [−γ, γ],

there is no intertemporal effect on consumption, i.e., A = 0, matrix Σ̃ ≡ C − Σ satisfies the

following property:

25

Table 2. F(E) = ∅(γ, γ) = (−40, 40), (c1, c2) Ŵ (c1(c2), c2(c1)) Ŵ (c1(c2), c2(c1)) Ŵ (c1(c2), c2(c1)) Ŵ (c1(c2), c2(c1)) c1(c2) = c1

c2(c1) = c2

(−2.137,−1.236) 435.66 438.58 435.71 446.66 c1(c2) = c1

c2(c1) = c2

(−4.886, 3.211) 439.42 437.83 433.27 433.94 c1(c2) = c1

c2(c1) = c2

(4.886,−3.211) 439.21 443.82 439.34 457.20 c1(c2) = c1

c2(c1) = c2

(2.137, 1.236) 441.69 436.28 435.63 437.70

Table 3. F(E) = {(15.460, 15.644), (−15.460,−15.644)}(γ, γ) = (−30, 30) (c1, c2) Ŵ (c1(c2), c2(c1)) Ŵ (c1(c2), c2(c1)) Ŵ (c1(c2), c2(c1)) Ŵ (c1(c2), c2(c1)) c1(c2) = c1

c2(c1) = c2

(−15.460,−15.644) 31.279∗ 31.279 31.279 30.294 c1(c2) = c1

c2(c1) = c2

(−14.577, 14.393) 30.786 30.786 31.771 30.786 c1(c2) = c1

c2(c1) = c2

(14.577,−14.393) 30.786 31.771 30.786 30.786 c1(c2) = c1

c2(c1) = c2

(15.460, 15.644) 30.294 31.279 31.279 31.279∗

26

(a) x1(γ1, γ2)

-10 -5 5 10 15 20

5

10

(b) x̂1(γ1)

(c) x2(γ1, γ2)

-10 -5 5 10 15 20

-10

-5

5

10

15

20

(d) x̂2(γ̂2)

Figure 6. Optimal qualities under parameters given in case (i).

27

• Σ̃ is element-wise nonnegative and positive definite, i.e., Σ̃ ∈ RN+∩

PN ;

• all Schur complements 5 of order 2 are nonnegative;

• matrix ∆ ≡ Σ̃−1 is diagonally dominant of its rows, i.e., ∆ii >∑

j ̸=i |∆ij |, ∀i ∈ I.

Then the ironing vector is c∗ = (∆+ ∆̂)−1∆̂v∗γ, v∗ = 1 or −1, the principal obtains a payoff

Ŵ ∗ = 1⊤[α∆̂(∆ + ∆̂)−1∆̂ + β∆̂

]1,

where 1 denotes the all-ones vector, ∆̂ ≡ diag(∆11, · · · ,∆NN ) is a diagonal matrix composed of

diagonal elements of ∆, α = γ2/9, β = 5γ2/27 + 1/2.

Proof. See appendix.

We next discuss the determination of the key agent, i.e., the agent who, once removed, leads

to the highest aggregate quality reduction. In a game-theoretic setup, Ballester, Calvó-Armengol

and Zenou (2006) propose a measure of network centrality, the so-called inter-centrality measure

and show that the key player in a network game is, precisely, the individual with the highest inter-

centrality. This index takes into account both the direct impact on his own activity, following

the removal of a player, and the indirect impact on the activities of others. In the spirit of BCZ,

we investigate the key player problem in a principal-agent model. Suppose that the principal

intends to isolate one player from the rest of the network to maximally reduce the total quality

provided. Which player should she target? We construct a new index, which depends on both

the inter-centrality measure given by BCZ(2006) and on the diagonally adjusted inter-centrality

measure given in this paper, to identity the key agent. The ranking of agents according to this

index needs not always to coincide with the ranking induced by their inter-centralities. We

denote by

Σ−i ≡

Σ−i,−i 00⊤ 0

the new adjacency matrix, obtained from Σ by setting to zeros all of its ith row and ith column

entries, ∆−i = (C − Σ−i)−1.5Let M be an n× n matrix partitioned as follows

M =

M11 M12M21 M22

where Mij ∈ Rni×nj , i, j ∈ {1, 2}, n = n1 + n2, The matrix, M22/M11 ≡ M22 −M21M−111 M12, is called the Schur

Complement of M22 in M .

28

Definition 1 Consider a network g with adjacency matrix Σ and a scalar λ such that M(λ,Σ) =

[I − λΣ]−1 is well-defined and positive semidefinite. The vector of Bonacich centralities of

parameter λ in g is: b(λ,Σ) = [I − λΣ]−11.

Definition 2 Consider a network g with adjacency matrix Σ and a scalar λ such that M(λ,Σ) =

[I−λΣ]−1 is well-defined and positive semidefinite. The inter-centrality of player i of parameter

λ in g is:

ci(λ,Σ) =b2i (λ,Σ)

Mii(λ,Σ)

Definition 3 Consider a network g with adjacency matrix Σ and scalars (λ, ℓ) such that ma-

trices

M(λ,Σ) ≡ [I − λΣ]−1 (25)

M̂(λ, ℓ,Σ) ≡

I − λΣ+[diag

(M(λ,Σ)− λℓb(λ,Σ)b

⊤(λ,Σ)

1 + λℓ1⊤M(λ,Σ)1

)]−1−1

(26)

are well-defined and positive semidefinite. The diagonally-adjusted inter-centrality of player i of

parameter (λ, ℓ) in g is:

ĉi(λ, ℓ,Σ) =b̂2i (λ, ℓ,Σ)

M̂ii(λ, ℓ,Σ),

where b̂(λ, ℓ,Σ) = M̂(λ, ℓ,Σ)1.

Corollary 5 Suppose that γi, i ∈ I are independent and identically distributed (i.i.d.) on sym-

metric interval [−γ, γ], A = 0, Cii = s,∀i ∈ I, Cij = ℓ,∀i ̸= j, s > ℓ > 1, s − ℓ > λmax(Σ),

ℓ > 11−T , T < 16, where T ≡ 1⊤[(s − ℓ)I − Σ]−11, then the key agent in quality is i∗ =

argmaxi∈I ρi (λ∗,Σ), where λ∗ = 1s−ℓ , ρi(λ, ℓ,Σ) ≡ ĉi(λ, ℓ,Σ) + δi(λ,Σ) + υ(λ, ℓ, ci(λ,Σ)),

υ(λ, ℓ, x) ≡ −1 + λℓ

[1⊤M(λ,Σ)1 − x

]2 + λℓ+ 2λℓ [1⊤M(λ,Σ)1 − x]

δi(λ∗, ℓ,Σ) ≡ 1⊤

IN−1 − λ∗Σ−i,−i

+

diag (IN−1 − λ∗Σ−i,−i)−1 − λ∗ℓb−i(λ∗,Σ)b⊤−i(λ∗,Σ)1+λ∗ℓ1⊤M(λ∗,Σ)1

+(λ∗)2(IN−1−λ∗Σ−i,−i)−1Σ−i,iΣi,−i(IN−1−λ∗Σ−i,−i)−1

1−(λ∗)2Σi,−i(IN−1−λ∗Σ−i,−i)−1Σ−i,i

−1

−1

−IN−1 − λ∗Σ−i,−i +diag

(IN−1 − λ∗Σ−i,−i)−1−λ

∗ℓb−i(λ∗,Σ−i)b⊤−i(λ

∗,Σ−i)

1+λ∗ℓ1⊤M(λ∗,Σ−i)1

−1

−1

1

6Notice that this condition holds for sufficiently large s− ℓ.

29

Proof. See appendix.

With complete information, the network structure is irrelevant to the aggregate absolute

quality, i.e., 1⊤Ex(γ) = 1⊤EX(γ, α) = 1⊤∆E(γ) = 0, so all agents are identical from an ex ante

perspective. In asymmetric information environments, however, the expectation of aggregate

quality depends on the ironing vector c, which are interdependent across agents. So the network

structure matters. The index for identifying the key agent with respect to quality is decomposed

into three items, i.e., ρi(λ∗, ℓ,Σ) ≡ ĉi(λ∗, ℓ,Σ) + δi(λ∗,Σ) + υ(λ∗, ℓ, ci(λ∗,Σ)). Similar to the

discussions in network game literature, each agent’s quality in the optimal contract is affected by

the location of all other agents in the network but distant neighbors have less impact due to the

decay factor λ∗, which scales down the relative weight of longer paths. In our model, the decay

factor depends on how the agent-specific costs (s) dominate the technological substitutabilities

between different agents (ℓ). When s− ℓ is sufficiently large, the impact of other nodes is very

weak, and is greatly discounted by distance. So ĉi(λ∗, ℓ,Σ) is ordinally equivalent to ci(λ

∗,Σ)

and thus is equivalent to υ(λ∗, ℓ, ci(λ∗,Σ)) since υ(λ∗, ℓ, ·) is strictly increasing. Also, δi(λ∗,Σ)

is negligibly small. Therefore, the rankings induced by ρi(λ∗, ℓ,Σ) and ci(λ

∗,Σ) are aligned.

When s− ℓ is relatively small, however, the ranking in terms of ρi(λ∗, ℓ,Σ) does not necessarily

corresponds to a ranking in terms of ci(λ∗,Σ). Tables (4) and (5) computes ci, ρi and degrees

of vertexes di for different values of (s, ℓ) in a network with eleven players, red numbers denote

the highest row value. Figures (7) and (8) identify the key agent with red nodes.

i 1 2 3 4 5 6 7 8 9 10 11

ρi 0.07666 0.09635 0.09620 0.07725 0.09620 0.09635 0.09635 0.09620 0.07725 0.09620 0.09635

ci 1.42860 1.53741 1.54218 1.42816 1.54218 1.53741 1.53741 1.54218 1.42816 1.54218 1.53741

di 4 5 5 4 5 5 5 5 4 5 5

Table 4. s = 45, ℓ = 20

i 1 2 3 4 5 6 7 8 9 10 11

ρi 0.02204 0.02756 0.02757 0.02205 0.02757 0.02756 0.02756 0.2757 0.02205 0.02757 0.02756

ci 1.10877 1.13613 1.13648 1.10876 1.13648 1.13613 1.13613 1.13648 1.10876 1.13648 1.13613

di 4 5 5 4 5 5 5 5 4 5 5

Table 5. s = 100, ℓ = 20

Corollaries 4 and 5 assume that there is no intertemporal impacts of consumptions on pref-

erences, but the technological interactions among consumers still exist. The following corollary

30

1

2

3

4

5

67

8

9

10

11

Figure 7. key player for s = 45, ℓ = 20

3

5

8

10

1

2

4

6

7

9

11

Figure 8. key player for s = 100, ℓ = 20

discusses the determination of optimal ironing vector from a different perspective. It assumes

off any technological interactions but allows for intertemporal effects.

Corollary 6 Suppose that γi is independent and identically distributed on symmetric interval

[−γ, γ], N > 3, C = sI, Aij = k, ∀i, j, Σ is a complete graph, s > N − 1, −(s+1N − 1

)< k < −1

then the optimal ironing vector is c∗ = κ(±1), where

κ ≡ γ(s+ 2−N)[s+ 2 + k −N(1 + k)][N − (s+ 1)][N − (s+ 2)] + k[N2 −N(2 + s) + 2(s+ 1)]

[1 + s− (1 + k)N ]3 + k + 2s− (1 + k)N

.

Proof. See appendix.

4 Conclusion

In this paper, we study a nonlinear pricing problem of experience goods in dynamic environ-

ments where consumers’s preferences evolve stochastically over time. Our analysis encompasses

both single and multiple agents cases. In each case, the seller is allowed to provide goods of

either certain quantities or certain qualities to buyers. Our work contributes to and extend the

existing literatures on dynamic mechanism design in the following aspects. First, we assume

that consumers’ present preferences and quitting costs are affected by their past consumptions.

Such an extension would shed light on the pricing of some addictive goods. We find the distor-

tion of their initial supply hinges on the tradeoffs between reinforcement and tolerant effects.

31

Second, we show how a “sloping ironing” technique is used to obtain an optimal quality con-

tract satisfying the dynamic incentive compatibility condition. In the standard static screening

model, allocations in the optimal contract are required to be monotonic in the agent’s type to

guarantee the incentive compatibility condition. In the case where the solution to the relaxed

problem is not everywhere monotonic, an ironing technique need to be adopted to obtain a con-

tract with bunching intervals. In our dynamic setup, however, the traditional ironing method

does not work, since the dynamic implementability condition requires the allocation to increase

in the type sufficiently fast. So we adopt a slopping ironing technique to obtain a perfect sorting

contract. Third, consumers in this paper are assumed to distributed on a network with local

payoff complementarities. In contrast to the existing literatures on network games, we focus on a

network contracting model. We analyze the problem of optimal mechanism design given certain

network structure and discuss further the impact of network structure on the form of contract.

In the quantity case, we discuss the change of contract when two separate nodes are bridged by

an additional link. In quality case, we discuss the problem of key player determination. The

key player corresponds to the node whose isolation from the rest of network leads to the largest

decrease in aggregate allocation. We construct a new index for identifying the key player on

aggregate quality, which may be unaligned with the traditional intercentraliy measure given by

BCZ(2006).

Appendix

Proof of Lemma 2

IC1 suggests

γx(γ)− t(γ) + Γ(γ, γ) > γx(γ̂)− t(γ̂) + Γ(γ, γ̂) (27)

γ̂x(γ̂)− t(γ̂) + Γ(γ̂, γ̂) > γ̂x(γ)− t(γ) + Γ(γ̂, γ). (28)

(27) and (28) imply

x(γ̂) +Γ(γ, γ̂)− Γ(γ̂, γ̂)

γ − γ̂≶ U(γ)− U(γ̂)

γ − γ̂≶ x(γ) + Γ(γ, γ)− Γ(γ̂, γ)

γ − γ̂, (29)

and

x(γ)− x(γ̂)γ − γ̂

+

Γ(γ,γ)−Γ(γ,γ̂)γ−γ̂ −

Γ(γ̂,γ)−Γ(γ̂,γ̂)γ−γ̂

γ − γ̂> 0. (30)

Letting γ̂ converge to γ, we have

EN1 : U ′(γ) = x(γ) + Γγ(γ, γ) (31)

IM1 : x′(γ) + Γγ,γ̂(γ, γ) > 0. (32)

32

Applying limε→−∞ g(ε) = limε→+∞ g(ε) = 07, EN2 : ∂U(γ, α)/∂α = X(γ, α) and integration

by parts technique, we have

∂Γ

∂γ(γ, γ̂) = −

∫ +∞−∞

U(γ̂, α)dg(α− γ − kx(γ̂))

= − U(γ̂, α)g(α− γ − kx(γ̂))|+∞−∞ +∫ +∞−∞

∂U

∂α(γ̂, α)dG(α− γ − kx(γ̂))

=

∫ +∞−∞

X(γ̂, α)dG(α− γ − kx(γ̂)).

Hence, EN1 takes the form

EN1 : U ′(γ) = x(γ) +∫ +∞−∞

X(γ, α)dG(α− γ − kx(γ)). (33)

Conversely, if EN1 and IM′1 : x

′(γ) + Γγ,γ̂(γ, γ̂) > 0, ∀γ, γ̂ hold, we have:

U(γ)− Ũ(γ, γ̂) = U(γ, γ)− Ũ(γ̂, γ̂) + Ũ(γ̂, γ̂)− Ũ(γ, γ̂)

=

∫ γγ̂

U ′(s)ds−∫ γγ̂

∂Ũ∂γ

(s, γ̂)ds

=

∫ γγ̂

[x(s) +

∂Γ

∂γ(s, s)

]−[x(γ̂) +

∂Γ

∂γ(s, γ̂)

]ds

=

∫ γγ̂

[∫ sγ̂

(x′(t) +

∂2Γ

∂γ∂γ̂(s, t)

)dt

]ds > 0, ∀γ, γ̂.

So IC1 holds.

Proof of Proposition 1

The expressions of x∗(γ) and X∗(γ, α) are obtained directly by maximizing (13) pointwise sub-

ject to nonnegative constraints. Notice that X∗(γ, α) is of a truncated form, whereas x∗(γ)

attains an interior solution, which is guaranteed by Assumption 1. IM2 is obvious. To guar-

antee IC1, we need to verify conditions IM′1. From Γγ(γ, γ̂) =

∫ +∞−∞ X

∗(γ̂, α)dG(α − γ −

kx∗(γ̂)) =∫ +∞−θ(γ̂)−γ−kx∗(γ̂) (ε+ γ + kx

∗(γ̂) + θ(γ̂)) dG(ε), we have Γγ,γ̂(γ, γ̂) = [1 − G(−θ(γ̂) −

γ − kx∗(γ̂))][kx∗′(γ̂) + θ′(γ̂)], where θ(γ) ≡ H(γ)−1h(γ) . (12) implies

dx∗(γ)

dγ=

[1 + θ′(γ)] {1 + k [1−G(−θ(γ)− γ − kx(γ))]}1− k2 [1−G(−θ(γ)− γ − kx(γ))] + U ′′0 (x)

. (34)

Since θ′(γ) > 0, k ∈ [−1, 1], U ′′0 (x) > 0, we have x∗′(γ) > 0. Therefore, IM ′1 : x∗′(γ̂)+Γγ,γ̂(γ, γ̂) =

{1 + [1−G (−θ(γ̂)− γ − kx∗(γ̂))] k}x∗′(γ̂) + [1−G (−θ(γ̂)− γ − kx∗(γ̂))]θ′(γ̂) > 0 holds.7Otherwise,

∫ +∞−∞ g(ε)dε will be infinitely large.

33

Proof of Corollary 1

• Curves x∗(γ) and xSB(γ) intersect at most once on [γ, γ]. Suppose that there exist two

points γ1, γ2 ∈ [γ, γ], such that x∗(γ1) = γ1+ θ(γ1), x∗(γ2) = γ2+ θ(γ2), then by the mean

value theorem, there exist γ∗ ∈ (γ1, γ2) such that x∗′(γ∗) = 1+ θ′(γ∗). It follows from (34)

that

k(1 + k) [1−G (−θ(γ∗)− γ∗ − kx∗(γ∗))] = U ′′0 (x∗(γ∗)) (35)

The left-hand side of (35) is negative, while the right-hand side is positive since k ∈ [−1, 0],

U ′′0 (x) > 0, ∀x, a contradiction. Therefore, x∗(γ) intersects γ + θ(γ) and thus xSB(γ) =

max{γ + θ(γ), 0} at most once.

• γ > 0 > γ − 1/h(γ) implies xSB(γ) = 0 < x∗(γ) and

x∗(γ)− xSB(γ) = −U ′0(x∗(γ)) + k∫ +∞−γ−kx∗(γ)

[ε+ γ + kx∗(γ)] dG(ε).

Therefore, if −U ′0(x∗(γ)) + k∫ +∞−γ−kx∗(γ) [ε+ γ + kx

∗(γ)] dG(ε) > 0, x∗(γ) > xSB(γ), ∀γ,

with strict inequality for γ < γ. If −U ′0(x∗(γ)) + k∫ +∞−γ−kx∗(γ) [ε+ γ + kx

∗(γ)] dG(ε) < 0,

we have x∗(γ) < xSB(γ). The Intermediate Value Theorem suggests that: ∃γ̂ ∈ (γ, γ)

such that x∗(γ̂) = xSB(γ̂) and x∗(γ) < (resp. >)xSB(γ), ∀γ > (resp. 0, EN1 : U ′(γ) = x(γ) +∫ +∞−∞

X(γ, α)dG (α− γ − kx(γ)) .

Incorporating IR and EN1 into the principal’s objective function, we construct the Hamil-

ton and Lagrangian functions as follows:

H(x,X,U , λ) =

γx(γ)−x2(γ)

2 − U(γ)+∫ +∞−∞

[αX(γ, α)− X

2(γ,α)2

]dG (α− γ − kx(γ))

h(γ)+λ(γ)

{x(γ) +

∫ +∞−∞

X(γ, α)dG (α− γ − kx(γ))},

L (x,X,U , λ, τ) = H+ τU ,

34

x,X are control variables, U is the state variable, λ(γ)is the costate variable of EN1,

τ(γ) is the multiplier of constraint IR. By Pontryagin Maximum Principle, the following

conditions are necessary and sufficient for optimality:

– maximality condition:

(x,X) ∈ argmaxx,X

H(x,X,U , λ); (36)

– costate equation:

−λ′(γ) = ∂L∂U

= −h(γ) + τ(γ); (37)

– state equation:

U ′(γ) = x(γ) +∫ +∞−∞

X(γ, α)dG (α− γ − kx(γ)) (38)

– complementary slackness:

τ(γ)U(γ) = 0, τ(γ) > 0,U(γ) > 0; (39)

– transversality condition:

λ(γ) = λ(γ) = 0. (40)

(36) implies

x∗(γ) =γ + λ(γ)h(λ)

1− k, (41)

X∗(γ, α) = α+λ(γ)

h(λ). (42)

(37) and (40) implies that λ(γ) = H(γ) when γ is close to γ, λ(γ) = H(γ)−1 when γ is close

to γ. Substituting (41) and (42) into (38), we get U ′(γ) = x∗(γ) +∫ +∞−∞ X

∗(γ, α)dG(α −

γ − kx∗(γ)) = 2x∗(γ). Therefore, U(γ) is increasing (resp. decreasing, constant) when

x∗(γ) > (resp. 0. IR can therefore be binding on a nondegenerate interval only if

λ̂′(γ) 6 h(γ).

Our strategy to solve the relaxed problem is to conjecture a solution and verify that it

satisfies the sufficient conditions. The critical part is to construct the right solution for

35

λ(γ). Consider the following schedule, illustrated in Figure 9,

λ∗(γ) =

H(γ)− 1 if λ̂(γ) 6 H(γ)− 1

−γh(γ) if H(γ)− 1 < λ̂(γ) 6 H(γ)

H(γ) if λ̂(γ) > H(γ)

In order for λ∗(γ) to satisfy the sufficient conditions, we need to check that λ̂′(γ) 6 h(γ)

whenever H(γ)− 1 < λ̂(γ) 6 H(γ). If h′(γ) > 0, then −γh(γ)h′(γ) 6 H(γ)h′(γ). Hence,

−γh′(γ)/h(γ) 6 H(γ)h′(γ)/h2(γ) 6 1, the last inequality follows from the monotone

hazard rate condition d[H(γ)/h(γ)]/dγ > 0. If h′(γ) < 0, then −γh(γ)h′(γ) < [H(γ) −

1]h′(γ). Hence, −γh′(γ)/h(γ) < [H(γ) − 1]h′(γ)/h2(γ) 6 1, the last inequality follows

from the monotone hazard rate condition d[(1 − H(γ))/h(γ)]/dγ 6 0. In both cases, we

have −γh′(γ) 6 h(γ) 6 2h(γ). It follows that λ̂′(γ) = −h(γ) − γh′(γ) 6 h(γ). The

corresponding optimal solution to [Pr] is depicted in Figure 10, where γ1 and γ2 are given

by

γ1 +H(γ1)

h(γ1)= γ2 −

1−H(γ2)h(γ2)

= 0.

Figure 9. The optimal costate variable λ∗(γ)

• Checking the implementability conditions. IM2 is obvious. Since

∂2Γ

∂γ∂γ̂(γ, γ̂) =

∂2Γ

∂γ∂γ̂(γ̂, γ̂)

=∂

∂γ̂

∫ +∞−∞

X∗(γ̂, α)dG(α− γ − kx∗(γ̂))

= kx∗′(γ̂) +λ∗(γ̂)

h(γ̂)

∣∣∣∣′γ̂

,

IM1 (8) and IM′1 (9) are equivalent. So we need only to verify condition

IM1 : x∗′(γ) +

∂2Γ

∂γ∂γ̂(γ, γ) = (1 + k)x∗′(γ) +

λ∗(γ)

h(γ)

∣∣∣∣′γ

= 2x∗′(γ)− 1 > 0.

36

B C

A

D

Figure 10. The optimal solution to the relaxed problem

Since k ∈ [−1, 1], we have x∗′(γ) > 1/2,∀γ ∈ [γ, γ1)∪(γ1, γ]. Obviously, IM1 fails for

γ ∈ [γ1, γ2]. So we need to iron curve ABCD in Figure 10 by a line ℓ(γ) = (γ − x)/2.

Denoting by λ̃(γ, x) ≡ [(1− k)(γ − x)/2− γ]h(γ), we have the ironed costate variable:

λ∗(γ, x) =

H(γ)− 1 if λ̃(γ, x) 6 H(γ)− 1

λ̃(γ, x) if H(γ)− 1 < λ̃(γ, x) 6 H(γ)

H(γ) if λ̃(γ, x) > H(γ)

Representing the principal’s expected payoff as a function of x, we get

W (x) =

∫ γγ

maxx

[(γ +

λ∗(γ, x)

h(γ)

)x− x

2

2+ Π(x)

]dH(γ) (43)

37

where,

Π(x) = E[maxX

[(α+

λ∗(γ, x)

h(γ)

)X − X

2

2

]∣∣∣∣ γ, x]=

1

2E

[(α+

λ∗(γ, x)

h(γ)

)2∣∣∣∣∣x]

=1

2

[E(α+

λ∗(γ, x)

h(γ)

∣∣∣∣x)]2 + 12V ar[α+

λ∗(γ, x)

h(γ)

∣∣∣∣x]=

1

2

[kx+ γ +

λ∗(γ, x)

h(γ)

]2+

1

2V ar(ε).

We simplify (43) further to

W (x̄) =

∫ γγ

1

1− k

(γ +

λ∗(γ, x)

h(γ)

)2dH(γ) +

1

2V ar(ε)

=1

1− k

∫ γ∗(x̄)γ

[γ + H(γ)h(γ)

]2dH(γ)

+∫ γ∗∗(x̄)γ∗(x̄)

[(1−k)(γ−x̄)

2

]2dH(γ)

+∫ γγ∗∗(x̄)

[γ − 1−H(γ)h(γ)

]2dH(γ)

+1

2V ar(ε),

(44)

where γ∗(γ) and γ∗∗(γ) are determined, respectively, by λ̃(γ, x) = H(γ) and λ̃(γ, x) =

H(γ) − 1. The critical work is to determine the optimal ironing parameter xopt =

argmaxx̄W (x̄). It follows from (44) that: W′(x) = (1 − k)

∫ γ∗∗(x)γ∗(x) (x − γ)dH(γ)/2,

W ′′(x) = (1−k)[(x−γ∗∗(x))dγ∗∗/dx− (x−γ∗(x))dγ∗/dx]/2+(1−k)[H(γ∗∗)−H(γ∗)]/2.

γ∗∗(x) > γ∗(x), x ∈ [γ∗, γ∗∗], dγ∗/dx < 0, dγ∗∗/dx < 0 imply that W ′′(x) > 0. Figure 11

shows that points A(γ, γ/(1−k)) and C(γ2, 0) (resp. D(γ, γ/(1−k)) and B(γ1, 0)) lie below

(resp. above) line ℓ(γ) = γ−x2 . Therefore, xl ≡ max{γ1,−1+k1−kγ} 6 x̄ 6 min{γ2,−

1+k1−kγ} ≡

xh. The convex function W (x) may attain its maximum over interval [xl, xh] either at xl

or at xh, depending on the comparison between W (xl) and W (xh).

Proof of Lemma 3

The fact that ICi2 is equivalent to ENi2 and IM

i2 is obtained by carrying over the standard

approach for solving static screening problem to the second period. The first and second order

conditions for ICi1 are, respectively,

γix̂′i(γi)− t̂′i(γi) +

1

2

∑j ̸=i

σij x̂′ij(γi) +

∂Γi∂γ̂i

(γi, γi) = 0, (45)

and

γix̂′′i (γi)− t̂′′i (γi) +

1

2

∑j ̸=i

σij x̂′′ij(γi) +

∂2Γi∂γ̂2i

(γi, γi) 6 0. (46)

38

B C

A

D

Figure 11. The optimal solution to the original problem

Since U ′i(γi) = γix̂′i(γi)− t̂′i(γi)+ 12∑

j ̸=i σij x̂′ij(γi)+

∂Γi∂γ̂i

(γi, γi)+ x̂i(γi)+∂Γi∂γi

(γi, γi), (45) implies

EN i1. Differentiating (45) with respect to γi yields

γix̂′′i (γi)− t̂′′i (γi) +

1

2

∑j ̸=i

σij x̂′′ij(γi) +

∂2Γi∂γ̂2i

(γi, γi) + x′i(γ) +

∂2Γi∂γ̂i∂γi

(γi, γi) = 0,

so we get IM i1 from (46).

Proof of Proposition 3

Before giving the formal proof, we need to prepare some notations and lemmas. A vector η ∈ RN

and a positive definite matrix Σ ∈ RN×N are partitioned as follows:

η =

ηSη−S

,Σ = ΣSS ΣS−S

Σ−SS Σ−S−S

S ⊆ I ≡ {1, · · · , N}, −S ≡ I\S, |S| denotes the cardinality of set S. ΠS(ηS ,y) ≡ η⊤S y −12y

⊤ΣSSy, y∗S(ηS) = argmaxy∈R|S|+

ΠS(ηS ,y) , Π∗S(ηS) = maxy∈R|S|+

ΠS(ηS ,y). IAS(ηS) ≡{i ∈ S| E⊤i y∗S(ηS) > 0

}denotes the set of inactive constraints at optimum. We drop the sub-

script S when S = I to write Π(η,y), Π∗(η), y∗(η) and IA(η).8 It’s clear that a full quadratic

program attains a maximum at least as large as that of the subproblem, i.e., Π∗T (ηT ) 6 Π∗S(ηS)

whenever T ⊆ S. M(K) ≡{η ∈ RN |IA(η) = K

}represents the set of parameters where the

full quadratic program has active constraints set K at its optimum.

8Please notice the difference between y∗K|I(η) representing subvector of the solution to the full problem, and

y∗K(ηK) representing the solution to a subproblem.

39

Lemma 4 If ∆ ≡ Σ−SSy∗S(ηS)− η−S > 0, then y∗(η) = (y∗S(ηS),0).

Proof. Since y∗S ≡ y∗S(ηS) is solution to program [PS ] : maxy∈R|S|+[η⊤S y−

12y

⊤ΣSSy], there

exist KKT multipliers λS such that:

ηS − ΣSSy∗S + λS = 0 (47)

λS > 0,y∗S > 0 (48)

(y∗S)⊤λS = 0. (49)

If furthermore ∆ ≡ Σ−SSy∗S − η−S > 0, then vector y∗ ≡ (y∗S ,0) and multipliers λ ≡ (λS ,∆)

satisfy the following KKT system ηSη−S

− ΣSS ΣS−S

Σ−SS Σ−S−S

y∗S0

+ λS

∆

= 0

0

y∗ > 0, λ > 0, (y∗)⊤λ = 0.

So, (y∗S ,0) = argmaxy∈RN+

[α⊤y− 12y

⊤Σy].

Lemma 5 If ΣjKy∗K(ηK)− ηj < 0 for some j ∈ I\K, then Π∗K∪{j}(ηK , ηj) > Π∗K(ηK).

Proof. Let A ≡ IAK(ηK),−A ≡ K\A, y∗A ≡ y∗A|K(ηK) = y∗A(ηA) = Σ

−1AAηA ∈ R

|A|++.

∇yΠK∪{j}

ηA

η−A

ηj

,

y∗A

0

0

=

ηA

η−A

ηj

−

ΣAAy∗A

Σ−AAy∗A

ΣjAy∗A

=

0

η−A − Σ−AAy∗Aηj − ΣjAy∗A

,

where ∇y(η,y) is the gradient operator with respect to y. For ∆yA ∈ R|A|+ , we have

y∗A

0

0

+

∆yA

0