# Turning the Ratchet:
Dynamic Screening with Multiple Agents111We thank participants at various
seminars and conferences for helpful discussions and comments. All errors are
our own. Email: Ekmekci (mehmet.ekmekci@bc.edu); Maestri
(lucas.maestri@fgv.br); Wei (dwei10@ucsc.edu).

Mehmet Ekmekci Lucas Maestri Dong Wei Boston College FGV EPGE UC Santa Cruz

(May 2024)

###### Abstract

We study a dynamic contracting problem with multiple agents and a lack of
commitment. A principal who can only commit to one-period contracts wants to
screen efficient agents over time. Once an agent reveals his type, the
principal becomes tempted to revise contract terms, causing a “ratchet
effect.” Alterations of contracts are observable and, hence, whenever past
promises are not honored future information revelation stops. We provide a
necessary and sufficient condition under which the principal is able to foster
information revelation. When players are sufficiently patient, the agents’
private information is either never revealed or fully revealed in a sequential
manner. Optimal contracts entail high-powered incentives after an agent’s type
is initially disclosed, and rewards for information revelation disappear in
the long run.

Keywords: dynamic contracts, screening, limited commitment, ratchet effect
JEL classification: C73, D82, D86, J31

##  1 Introduction

In dynamic relationships with private information, information revelation is
often critical for improving allocative efficiency and welfare. For example,
in a worker-manager relationship, the worker’s ability is usually his private
information; obtaining this information would allow the manager to assign
different tasks based on the worker’s ability, which is beneficial for social
surplus and the firm’s profit. Nevertheless, the revelation of private
information can be difficult to incentivize, when the uninformed party lacks a
commitment to not exploiting such private information. This causes the well-
known “ratchet effect.” Indeed, if a worker expects his manager to implement
worse terms of trade (such as assigning him more work or reducing his wage
rate) after the manager knows his capability, the worker would be reluctant to
reveal such information in the first place.

In various organizations, however, a person in a supervising position is often
managing many subordinates at the same time. In such scenarios where one
principal engages with a large population of agents, the principal’s
commitment power may be partially restored, because if she breaks her promise
with one of the agents, there can be potential punishments imposed on her by
other agents; the linkage between these relationships could then push the
principal to be more restrained in each of them. In this paper, we study the
effect of a principal simultaneously contracting with multiple agents, and
whether and how that can alleviate the ratchet-effect problem. In particular,
we are interested in addressing the following questions. First, how should
dynamic incentives be designed in order to incentivize information revelation
over time? Relatedly, what are the qualitative features of the optimal
contract? Moreover, what determines whether any information is revealed in
equilibrium?

To answer these questions, we study a stylized model in which a principal
(she) interacts with the same mass of agents (he) over time. An agent can be
either the efficient (i.e., low-cost) type or the inefficient (i.e., high-
cost) type, which is his private information. In each period, an agent
produces an observable quantity, incurs a production cost, and collects some
payment from the principal. Meanwhile, in each period the principal tries to
induce a distinct subset of efficient agents to reveal their type. She does so
by promising a sequence of separating contracts to these agents while also
offering a pooling contract to those who wish to remain unrevealed. Free to
make any promise about the future, the principal can only commit to one-period
contracts for the current period, and her promise about what contract to offer
in any future period can be reconsidered and reneged on when the day arrives.
Nonetheless, reneging is not costless. We assume that the principal’s reneging
is publicly observable to all agents, and once it occurs, those agents who
haven’t yet revealed their private information will stop revealing altogether.
In this model, the principal’s fear for losing future information then becomes
a deterrent to the excessive exploitation of a low-cost agent after his type
is revealed. In turn, this gives each agent more incentives to reveal his type
in the first place, which potentially fosters information transmission and
alleviates the ratchet effect.

We say that a mechanism is revealing if it successfully induces some efficient
agents to reveal their type over time. Depending on the model parameters, an
optimal mechanism can be revealing or non-revealing. Our first main result
characterizes the qualitative features of an optimal mechanism when it is
revealing.222When an optimal mechanism is non-revealing, it is simply a static
pooling contract that extracts the full surplus from the inefficient type and
leaves some rent to the efficient type. Our second main result provides a
necessary and sufficient condition under which every optimal mechanism is
revealing when players are sufficiently patient.

When information revelation is feasible, in any optimal mechanism, every
period there is a distinct subset of efficient agents choosing to reveal their
type, and all efficient agents will reveal at some point. In other words,
information revelation is gradual and eventual. Gradual revelation ensures
that looking into the future from each period’s perspective, there is always
new information to arrive later. This gives the principal sustained incentives
to continue paying the promised information rent, instead of fully exploiting
every low-cost agent who has revealed his type. Eventual revelation is implied
by optimality, for otherwise the principal can make a higher profit in another
mechanism with more agents revealing in every period.

The pooling contract offered in each period (taken by all inefficient agents
and those efficient ones who wish to remain unrevealed) exhibits a simple and
intuitive dynamics. Specifically, each pooling contract asks the agent to
produce a quantity less than the inefficient type’s surplus-maximizing level
and remunerates him at the high cost of production;333Downward distortions of
pooling quantities are typical in standard screening models (i.e., an
inefficient type is never overworked). They are used to reduce the information
rent of the efficient type. as the efficient agents gradually drop out of the
pool, the demanded quantity converges to the surplus-maximizing level over
time.

On the other hand, once an efficient agent reveals his type, he will be
getting the separating contracts starting from the current period. This
sequence of contracts consists of two phases.444The sequence of separating
contracts an agent gets can differ depending on when he reveals his
efficiency. Nevertheless, these sequences of contracts all share the same
qualitative dynamics. In the first several periods after revelation, which we
call the “reward phase,” the agent is asked to produce greater than the
efficient type’s surplus-maximizing quantity and receives a payment equal to
the inefficient type’s high cost. During these periods, an efficient agent,
while working intensely, will be collecting positive rents. The demanded
quantity and offered payment in the contract will gradually decline during the
reward phase until the next phase, called the “exploitation phase,” is
reached. In the exploitation phase, the principal extracts the entire surplus
from an efficient agent by asking him to produce at his surplus-maximizing
level and paying him at the low cost of production.

The dynamics of the separating contracts are shaped by the interplay between
the principal’s limited commitment and the agent’s incentive compatibility.
First, the principal would like to frontload an efficient agent’s information
rent as much as possible. This is because by paying more today (with
commitment) and promising less in the future (without commitment), the
principal can deliver the same value to an efficient agent while relaxing her
non-reneging constraint.555In our model, although the principal could renege
on her promise when a future date comes, any implementable allocation requires
the principal to find it optimal to always keep her promise. Formally, this is
captured by a “non-reneging” constraint in each period. However, paying all
information rent to an agent in one period can be infeasible, because such a
high upfront payment may induce the inefficient agent to pretend to be
efficient, reap the high payment, and then exit from the relationship. To
hedge against this “take-the-money-and-run” strategy by the inefficient type,
the principal has to split the total information rent to an efficient agent
into the first several periods after his type is revealed, making the reward
phase nontrivial. Moreover, the upward distortion of the low-cost type’s
quantity is also driven by the principal’s incentive to frontload information
rent. Indeed, if his production is at or below the surplus-maximizing level,
the principal can increase the quantity and perturb the payment to further
relax her non-reneging constraint and/or increase the surplus.

Next, we derive a necessary and sufficient condition under which the agent’s
private information can be elicited. With limited commitment, the main
challenge in encouraging information revelation is to resolve the agent’s
concern—that the principal would not honor the promised information rent after
knowing his type. Naturally, our easy-to-check condition on the model
primitives involves a comparison between the principal’s benefit and cost of
keeping promises. From the principal’s perspective, the benefit of honoring
all promises (for one more period) is that more efficient agents will reveal
their type, and this additional information can be utilized later. Meanwhile,
the cost of keeping promises (for one more period) comes from two sources: i)
payment of information rent to those agents who are still in the reward phase
of their separating contracts, and ii) surplus loss due to the quantity
distortions in these contracts. We show that, when the players are
sufficiently patient, every optimal mechanism is revealing if and only if the
benefit of keeping promises outweighs its cost. This necessary and sufficient
condition uncovers a novel tradeoff between eliciting future information and
rewarding past disclosures; it also enables us to examine the effect of skill
heterogeneity on information revelation.

Our analysis highlights the key distinctions between contracting with one
agent and that with many agents. In the one-agent case, if the principal
breaks her promise, there is little punishment this single agent can impose on
the principal (other than perhaps quitting the relationship). In the many-
agent case, different agents could reveal their private information at
different points in time, and the communication among these agents would
publicize any mistreatment of an agent by the principal to all other agents.
Consequently, those who have not revealed their type could then impose a
collective punishment by stopping revealing information altogether. This much
harsher punishment induces the principal to honor the promised information
rent, which in turn gives the efficient agents more incentives to reveal their
type. This result suggests that, in organizations with commitment issues,
enhancing ties and facilitating (informal) communication between employees
could help overcome the ratchet effect and foster information transmission.

Our model also generates interesting testable predictions about the work and
wage dynamics in labor markets with a significant lack of commitment. In
particular, we predict the use of a dual-track contract system (pooling vs.
separating), a two-phase compensation scheme (reward vs. exploitation), and a
gradual decrease of work and wage stipulated by the separating contracts. As
previously explained, this comes as a consequence of the interaction between
the principal’s non-reneging constraint and the agent’s IC constraint. In
Chinese universities where the lack of commitment is often a genuine concern,
we find evidence of contracts that bear a remarkable resemblance to our
predictions. We discuss this example in detail at the end of Section 3.

Apart from contracting in labor markets, our model and insights extend
naturally to other related settings. For example, large retail corporations,
such as Walmart, engage with thousands of suppliers who possess valuable but
private information about production costs, etc. The ratchet effect
discourages these suppliers from revealing their true costs to the retailer,
as such disclosures might trigger worse terms of trade and erode their
profits. Our analysis offers a recipe to such retailers for leveraging the
interdependence between multiple relationships, fostering information
transmission, and improving profitability.

Related Literature. This paper contributes to the literature on ratchet
effect. The seminal works of Freixas et al. (1985) and Laffont and Tirole
(1988, 1993) first model the ratchet effect and show how it severely impairs
information revelation in dynamic relationships. Gerardi and Maestri (2020)
extend the model to infinite horizon and find that the unique equilibrium
outcome in the patient limit is pooling, so long as the probability of the
inefficient type is not too low.

The literature has identified several institutions that help mitigate the
ratchet effect. Kanemoto and MacLeod (1992) point out that competition for
second-hand workers can help firms with their commitment problems; Carmichael
and MacLeod (2000) find that a patient firm may have an incentive to honor its
promises to short-lived workers; when there are productivity shocks, Acharya
and Ortner (2017) show that the principal can progressively learn the agent’s
private information. By contrast, our paper demonstrates that the
interdependence between multiple relationships with long-lived workers could
restore the firm’s commitment power and relieve the ratchet effect. Indeed,
whenever information revelation is feasible, all private information will be
revealed eventually.

This paper also belongs to the broader literature on screening with limited
commitment. In the context of dynamic pricing, Hart and Tirole (1988) analyze
a model in which a principal rents over time an indivisible good to an agent
who is privately informed about his valuation; Skreta (2006, 2015) and Liu et
al. (2019) use a mechanism-design approach to study optimal auctions with
limited commitment; Bonatti (2011) studies optimal dynamic menu pricing when
past purchases can affect a consumer’s learning of product quality and thus
his future demands; Deb and Said (2015) demonstrate how delaying contracting
with high-valuation buyers can help a seller credibly maintain higher prices
in the future. More generally, Strulovici (2017) and Maestri (2017) examine
screening models with renegotiation opportunities; Bester and Strausz (2001)
and Doval and Skreta (2022) develop revelation principles for models involving
limited commitment and a single agent. Our paper differs from these studies by
analyzing the interaction between a principal and a continuum of agents, and
allowing for allocations that depend not only on each agent’s type but also on
his dynamic choices.

There is also a macroeconomics literature on optimal government policies with
a lack of commitment and a continuum of privately informed agents (see, for
example, Sleet and Yeltekin, 2008; Albanesi and Sleet, 2006; Acemoglu et al.,
2010; Scheuer and Wolitzky, 2016). These papers all assume that agents of the
same type receive the same allocation, while the principal in our model can
personalize the allocation to (ex-ante identical) agents who reveal their
private information at different points in time. A notable distinction is the
recent contribution of Golosov and Iovino (2021) who study a government’s
optimal provision of unemployment insurance. Apart from many other disparities
(e.g., in physical environments, the principal’s goals, and applications), the
two models differ in the nature of the agent’s private information: in their
setting, agents who do not have a job offer cannot pretend to have one; by
contrast, information is fully soft in our model, and the take-the-money-and-
run constraint of the inefficient agent leads to novel predictions about the
equilibrium dynamics.666Each of these assumptions fits the respective model
application. In the context of unemployment insurance, assuming that an agent
cannot fake a job offer makes much sense. In a manager-worker relationship,
however, information about skills (e.g., private production cost) can be
easily misrepresented both ways, and the take-the-money-and-run constraint of
the inefficient type is often of important concern (Laffont and Tirole, 1988,
1993). In our model with quasilinear preferences, if information were not
fully soft, the principal would be able to induce full separation in the first
period with a simple static contract.

Our paper is reminiscent of other contributions highlighting that interacting
with multiple agents help the principal keep her promises. In a moral-hazard
setting with complete information, Levin (2002) shows that through pooling
incentives the principal does better by making promises to several agents than
to only one. This differs from our model as there is no scope for information
revelation. Ausubel and Deneckere (1989) find that when selling a durable good
to a continuum of consumers, a patient seller can obtain a payoff arbitrarily
close to the monopoly profit in an equilibrium in which she slowly decreases
the price over time. In our model, the key tradeoff faced by the principal is
between reneging on the information rent promised in the past and eliciting
new information in the future. This tradeoff is only present in our paper
because in a durable-good setting a buyer leaves the market immediately after
he reveals his type (i.e., buys the good).

##  2 Model

###  2.1 Players, Types, and Payoffs

A principal (she) interacts with a continuum of agents (he) of measure $1$.
Time is discrete and both parties discount the future with a discount factor
$\delta\in\left(0,1\right)$. Each agent has a type
$\theta\in\left\\{L,H\right\\}$, which affects the agent’s cost of production
and is his private information. Initially, the measure of the low(-cost) type
is $\alpha_{0}\in(0,1)$ and the measure of the high(-cost) type is
$1-\alpha_{0}$.

In period $t$, if a type-$\theta$ agent produces quantity $q_{t}$ and receives
a payment of $x_{t}$, his payoff is

$x_{t}-C_{\theta}\left(q_{t}\right),$

where $C_{\theta}(q_{t})$ is the cost of production. We assume that the
quantity belongs to some compact set $[0,\bar{q}]$, and that
$C_{\theta}(\cdot)$ is smooth, strictly increasing, and convex, satisfying
$C_{\theta}\left(0\right)=0$ for every $\theta$ and
$C^{\prime}_{H}\left(q\right)>C^{\prime}_{L}\left(q\right)$ for every
$q\in[0,\bar{q}]$. This implies that $\Delta
C(\cdot)\mathrel{\mathop{\mathchar 58\relax}}=C_{H}(\cdot)-C_{L}(\cdot)$ is
nonnegative and strictly increasing. We further assume that $\Delta C$ is
concave, i.e., $\Delta C^{\prime\prime}(\cdot)\leq 0$.777The concavity of
$\Delta C$ ensures a cleaner statement of our main results. Our results still
hold qualitatively even without this assumption. See Section 5.2 for a
discussion. These assumptions are satisfied if, for example,
$C_{\theta}(q)=\theta q$.

The principal’s payoff from an agent is

$v(q_{t})-x_{t},$

where $v(\cdot)$ is strictly increasing, strictly concave, and differentiable,
satisfying $v(0)=0$.

In any period, the social surplus from type $\theta$ producing quantity $q$ is

$\pi_{\theta}(q)\mathrel{\mathop{\mathchar 58\relax}}=v(q)-C_{\theta}(q).$

For each $\theta\in\\{L,H\\}$, we define the surplus-maximizing quantity

$q_{\theta}^{\ast}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname*{arg\,max}_{q}\ \pi_{\theta}(q).$

We assume the solutions are interior, so that
$0<\pi_{H}(q_{H}^{\ast})<\pi_{L}(q_{L}^{\ast})$ and
$0<q_{H}^{*}<q_{L}^{*}<\bar{q}$.

If at any point an agent decides to irreversibly drop out of the relationship,
he will receive a payoff of $0$ as his outside option.

Throughout this paper, we maintain the following assumption that the initial
measure of the low-type agents is not too large.

######  Assumption 1.

$\alpha_{0}<\frac{\pi_{H}(q_{H}^{*})}{\pi_{L}(q_{L}^{*})}$.

This assumption guarantees that the principal should not simply ignore the
high types and extract full surplus from and only from the low types. See
Section 5.1 for a discussion on relaxing this assumption.

###  2.2 The Contracting Environment

We are interested in studying a contracting environment with three key
features:

  1. 1.

The principal can only commit to one-period contracts and may renege on her
past promises. However, any reneging is observable to all agents and could be
“punished” by the agents.

  2. 2.

Each agent has private information about his type, and any revelation of
private information has to be voluntary.

  3. 3.

Each agent can irreversibly leave the relationship in any period by refusing
the current contract.

To better understand the limited commitment assumption, note that in this
dynamic setting, the principal can not only specify a contract (typically
describing the amount of work and compensation) for the current period, but
also make promises about what contracts to offer in future periods. We assume
that any contract for the current period, once accepted, is binding and
enforceable. On the other hand, when a future period comes, the principal
could “renege” on her promise by presenting a contract that is different from
what she has promised to offer. This form of limited commitment can be caused
by various factors, such as the imperfect enforcement of long-term contracts,
the limited validity of verbal promises, and/or periodical leadership changes
within an organization.

#### Sequential Revelation Mechanisms

We focus our attention on a natural class of sequential revelation mechanisms
where low-cost agents can come forward to gradually reveal their efficiency
over time. In each period, any agent can announce that his type/cost is low,
and the contract each agent gets depends on whether he has made such an
announcement.

Formally, at the beginning of each period $t$, the principal proposes a
pooling contract $(q_{t}^{P},x_{t}^{P})$ and a sequence of separating
contracts
$\left\\{\left(q_{\tau}^{S}\left(t\right),x_{\tau}^{S}\left(t\right)\right)\right\\}_{\tau\geq
t}$. If an agent has not made any announcement (at $t$ or before), he will be
given the pooling contract $(q_{t}^{P},x_{t}^{P})$. If, in period $t$, an
agent announces that his type is low, then he will be given the separating
contract $\left(q_{t}^{S}\left(t\right),x_{t}^{S}\left(t\right)\right)$ in the
current period and be promised a sequence of separating contracts
$\left\\{\left(q_{\tau}^{S}\left(t\right),x_{\tau}^{S}\left(t\right)\right)\right\\}_{\tau>t}$
for all future periods $\tau>t$, where the $t$ in the parentheses indicates
the time of his revelation.888A contract $(q_{t},x_{t})$ in period $t$ should
be interpreted as a take-it-or-leave-it offer: An agent will be paid $x_{t}$
if he produces $q_{t}$ (or more), but he can also leave the relationship by
refusing the contract. We allow the sequence of separating contracts to depend
on the period in which the agent announces his (low) type; thus, low-type
agents revealing in two different periods may be promised two distinct
sequences of separating contracts.

From an agent’s perspective, in a sequential revelation mechanism, his choice
set in each period depends on whether he has made an announcement or not.
Specifically,

  * •

For an agent who has not made any announcement before $t$, in period $t$ he
can choose between: i) continuing to take the pooling contract (i.e.,
remaining silent), ii) starting to take the separating contract (i.e.,
revealing to be efficient), and iii) quitting the relationship.999We interpret
the agent’s choice within a menu as making a report (or remaining silent)
about his type.

  * •

For an agent who announced to be the low type at some $T<t$, in period $t$ he
can choose between: i) taking the separating contract
$(q_{t}^{S}(T),x_{t}^{S}(T))$ and ii) quitting the relationship.

Indeed, once an announcement is made, the agent is identified by the principal
and can no longer take the pooling contract in any future period.

From the principal’s perspective, because of her limited commitment, in each
period $t$ she can freely propose a contract
$(\tilde{q}_{t}^{S}(T),\tilde{x}_{t}^{S}(T))$ to agents who revealed at each
$T<t$. This contract can potentially differ from the one she promised to offer
$({q}_{t}^{S}(T),{x}_{t}^{S}(T))$, although in equilibrium they must be the
same.

In summary, the timeline within each period $t$ is as follows.

  1. 1.

The principal proposes a pooling contract $(q_{t}^{P},x_{t}^{P})$, a sequence
of separating contracts
$\left\\{\left(q_{\tau}^{S}\left(t\right),x_{\tau}^{S}\left(t\right)\right)\right\\}_{\tau\geq
t}$ (for those who wish to reveal at $t$), and
$(\tilde{q}_{t}^{S}(T),\tilde{x}_{t}^{S}(T))$ for each $T<t$ (for those who
revealed before $t$).

  2. 2(a).

If an agent always took the pooling contract prior to $t$, he can choose
between $(q_{t}^{P},x_{t}^{P})$, $(q_{t}^{S}(t),x_{t}^{S}(t))$, and quitting.

  3. 2(b).

If an agent took the separating contract $(q_{T}^{S}(T),x_{T}^{S}(T))$ at some
$T<t$, he can choose between $(\tilde{q}_{t}^{S}(T),\tilde{x}_{t}^{S}(T))$ and
quitting.

  4. 3.

The chosen contracts are enforced between the principal and each agent.

Sequential revelation mechanisms introduced above are suited to address our
key research questions—that whether and how the interdependence between
multiple relationships can facilitate information revelation over time. These
mechanisms focus particularly on the revelation of the efficient agent’s type,
as such information is most useful for the principal and creates critical
tensions between the two parties.101010In our model, knowing that an agent is
inefficient, compared to not knowing his type, does not change the principal’s
optimal contract with this agent. Nevertheless, whether or not the restriction
to this class of mechanisms is without loss remains an open question, as we
are not aware of a revelation principle applicable to this setting.111111Doval
and Skreta (2022) develop a revelation principle for screening models with
limited commitment. Their result does not apply to our setting, because we
have a continuum of agents and the contracts can be further personalized
within agents of the same type.

###  2.3 Implementable Allocations and the Principal’s Problem

#### Allocations

In general, an allocation is a collection of quantity and payment streams,
$\left\\{\left(q_{t}^{i},x_{t}^{i}\right)\right\\}_{t\geq 0,i\in[0,1]}$, which
describes the work and compensation of each agent in every period. In a
sequential revelation mechanism, because the contract each agent gets only
depends on his type and the time of revelation (if his type is low), an
allocation can be summarized by

$\left\\{\left(q_{t}^{P},x_{t}^{P}\right),\left(q^{S}_{\tau}(t),x^{S}_{\tau}(t)\right),r_{t}\right\\}_{t\geq
0,\tau\geq t},$ (1)

which specifies each agent’s contracts before and after revealing (that his
type is low), and the measure of low-cost agents, $r_{t}$, who reveal in each
period $t$.121212For any $t$ such that $r_{t}=0$, one can simply set
$\left(q^{S}_{\tau}(t),x^{S}_{\tau}(t)\right)=\left(q_{\tau}^{P},x_{\tau}^{P}\right)$
for all $\tau\geq t$, making the separating contracts starting from a non-
revealing period the same as the pooling contracts. Alternatively, one can
think $\left(q^{S}_{\tau}(t),x^{S}_{\tau}(t)\right)$ as defined only for $t$
such that $r_{t}>0$.,131313In an allocation described by (1), we implicitly
assume that each individual agent is using a pure strategy: all high-type
agents never choose to make a (false) announcement, and each low-type agent
reveals with probability one in some period. The revealing period may differ
across different low-type agents, thus $r_{t}$ can be positive in multiple
periods despite the fact that there is no randomization on the individual
level. Define $R_{t}\mathrel{\mathop{\mathchar 58\relax}}=\sum_{\tau<t}r_{t}$
to be the mass of agents who reveal before $t$. Since this paper focuses
solely on sequential revelation mechanisms, we only consider allocations that
can be represented by (1).

Given an allocation from a sequential revelation mechanism, in period $t$, the
continuation payoff of a type-$\theta$ agent if he were never to reveal is

$\displaystyle U_{\theta,t}^{P}\mathrel{\mathop{\mathchar
58\relax}}=\left(1-\delta\right)\sum_{\tau=t}^{\mathbb{\infty}}\delta^{\tau-t}\left[x_{\tau}^{P}-C_{\theta}\left(q_{\tau}^{P}\right)\right].$
(2)

The continuation payoff of a low-type agent who has revealed at $T\leq t$
(i.e., who starts to take the separating contract at $T$) is

$\displaystyle U_{L,t}^{S}\left(T\right)\mathrel{\mathop{\mathchar
58\relax}}=\left(1-\delta\right)\sum_{\tau=t}^{\mathbb{\infty}}\delta^{\tau-t}\left[x_{\tau}^{S}(T)-C_{L}\left(q^{S}_{\tau}(T)\right)\right].$
(3)

We define the continuation profit of the principal as

$\displaystyle\Pi_{t}\mathrel{\mathop{\mathchar 58\relax}}=$
$\displaystyle\left(1-\delta\right)\sum_{\tau=t}^{\mathbb{\infty}}\delta^{\tau-t}\left[\left(1-\alpha_{0}\right)+\left(\alpha_{0}-R_{\tau}-r_{\tau}\right)\right]\left[v(q_{\tau}^{P})-x_{\tau}^{P}\right]$
$\displaystyle+\left(1-\delta\right)\sum_{\tau=t}^{\infty}\delta^{\tau-t}\left[\sum_{T=0}^{\tau}r_{\tau}\left[v(q_{\tau}^{S}\left(T\right))-x_{\tau}^{S}\left(T\right)\right]\right],$
(4)

where the first term sums over the principal’s profit from the agents who take
the pooling contracts (which include all high types and some low types who
haven’t yet revealed), and the second term captures her profit from the agents
who have revealed at different times in the past.

#### Constraints and Implementable Allocations

In our environment with limited commitment, voluntary revelation, and
voluntary exit, an allocation can be implemented only if: i) the principal
never wants to renege on her promises; ii) information revelation is
incentive-compatible for the low types, while the high types never want to
make a false announcement; iii) the agents’ continuation payoffs never fall
below their outside options, on and off the equilibrium path. These
considerations lead to the following sets of constraints.

##### Non-reneging constraints:

$\Pi_{t}\geq
R_{t}\pi_{L}\left(q_{L}^{\ast}\right)+\left(1-R_{t}\right)\pi_{H}\left(q_{H}^{\ast}\right)\text{,
for all }t.$ (NR)

While the principal has to honor a one-period contract once it is
accepted,141414That is, if an agent is given a contract $(x_{t},q_{t})$ in
period $t$, then the payment $x_{t}$ must be made as long as $q_{t}$ is
produced. she may renege on her past promises at the beginning of a period by
presenting a contract that is different from what she has promised to offer.
We assume that the principal’s reneging is observable to all agents, which
makes it possible to punish such behavior. To understand the RHS, if the
principal were to renege on her previous promises in period $t$, the worst
punishment the agents can impose on the principal would be to cut off all
future information. In this case, the best the principal can do after reneging
is to extract the maximum surplus from the low-type agents who have revealed
and (because of Assumption 1) treat the unrevealed agents as if they were the
high type.151515Formally, the RHS of (NR) is the principal’s lowest possible
equilibrium payoff in the continuation game. To see this, note that the
principal can always guarantee herself such payoff by offering
$\left(q_{L}^{*},C_{L}(q_{L}^{*})\right)$ to each revealed agent and
$\left(q_{H}^{*},C_{H}(q_{H}^{*})\right)$ to each unrevealed agent in all
periods. Moreover, there is indeed an equilibrium that renders such payoff to
the principal.

We note that the RHS of (NR) is driven by two critical assumptions: i) that
the principal’s reneging on any agent is publicly observable to all agents;
and ii) that low-type agents who haven’t announced their type will react by
stopping revealing information altogether. Technically, these assumptions lead
to the worst punishment that agents as a group can impose on the principal.
Economically, assuming the principal’s reneging to be publicly observable
captures the idea that workers often talk/complain to each other especially
when they are mistreated by their manager. Meanwhile, the agent’s reaction to
reneging could come from the following reasoning: once a worker sees his
colleagues being mistreated by the manager, he will lose trust in the manager
and believe that he would be treated the same way if he were to reveal his
efficiency.161616This reaction by the agent is indeed sequentially rational if
the principal, upon reneging, would subsequently fully exploit anyone who
later reveals to be the low-cost type.

##### Incentive compatibility constraints:

For the low type,

$\displaystyle U^{S}_{L,t}(t)\geq\sup_{\mathbb{T}\geq
t}(1-\delta)\sum_{\tau=t}^{\mathbb{T}}\delta^{\tau-t}\left[x_{\tau}^{P}-C_{L}\left(q_{\tau}^{P}\right)\right]+\delta^{\mathbb{T}}U^{S}_{L,\mathbb{T}}(\mathbb{T}),\text{
for all $t$ such that }r_{t}>0,$ with equality if $0<r_{t}<\alpha-R_{t}$.
(IC-L)

(2.3) requires that, whenever a positive mass of agents makes the announcement
in period $t$, any low-type agent should prefer to take the separating
contracts beginning that period, rather than revealing at some optimal time
later (perhaps never). Moreover, if some but not all low-type agents reveal in
period $t$, then any low-type agent must be indifferent.

For the high type,

$U^{P}_{H,t}\geq U^{S}_{H,t}\mathrel{\mathop{\mathchar
58\relax}}=\sup_{\mathbb{T}\geq
t}\left(1-\delta\right)\sum_{\tau=t}^{\mathbb{T}}\delta^{\tau-t}\left[x_{\tau}^{S}\left(t\right)-C_{H}\left(q_{\tau}^{S}\left(t\right)\right)\right],\text{
for all }t.$ (IC-H)

(IC-H) insists that a high-type agent always prefers to take the pooling
contract, instead of falsely announcing his type in period $t$ followed by an
optimal quitting decision. This constraint hedges against the “take-the-money-
and-run” strategy, in which a high type may reap an attractive initial payment
in a separating contract and leave the relationship afterwards.

##### Participation constraints:

$\displaystyle U_{H,t}^{P}$ $\displaystyle\geq 0,\text{ for all }t.$ (IR-H)
$\displaystyle U_{L,\ t+s}^{S}\left(t\right)$ $\displaystyle\geq 0,\text{ for
all }t\text{ such that }r_{t}>0\text{ and for all }s\geq 0.$ (IR-L)

######  Definition 1.

An allocation,
$\left\\{\left(q_{t}^{P},x_{t}^{P}\right),\left(q^{S}_{\tau}(t),x^{S}_{\tau}(t)\right),r_{t}\right\\}_{t\geq
0,\tau\geq t}$, is implementable by a sequential revelation mechanism if it
satisfies (NR), (IC-H), (2.3), (IR-H), and (IR-L).

#### The Principal’s Problem

The principal aims to maximize her profit within the class of sequential
revelation mechanisms. That is, she looks for an implementable allocation that
maximizes her total discounted profit in period $0$:

$\displaystyle\begin{aligned}
\sup_{\left\\{\left(q_{t}^{P},x_{t}^{P}\right),\left(q^{S}_{\tau}(t),x^{S}_{\tau}(t)\right),r_{t}\right\\}_{t\geq
0,\tau\geq t}}&\Pi_{0}\\\ \text{s.t. }\eqref{eq:nonreneging},\
\eqref{eq:IC-H},\ \eqref{eq:IC-L},\ \eqref{eq:IR-H},&\text{ and
}\eqref{eq:IR-L}.\end{aligned}$ (5)

######  Theorem 1.

There is a solution to (5). That is, an optimal mechanism exists.

##  3 Properties of Optimal Mechanisms

We first define some terminology regarding the information revealed in a
sequential revelation mechanism.

######  Definition 2.

A sequential revelation mechanism is revealing if $r_{t}>0$ for some $t$. A
sequential revelation mechanism is non-revealing if $r_{t}=0$ for all $t$.

######  Definition 3.

For a revealing mechanism, we say that information revelation is gradual if
$r_{t}>0$ for all $t$, and that information revelation is eventual if
$\sum_{t=0}^{\infty}r_{t}=\alpha_{0}$.

Given Assumption 1, if the principal does not seek to induce any information
revelation, the best she can do is to offer the pooling contract
$\left(q_{H}^{*},C_{H}(q_{H}^{*})\right)$ in every period to all agents,
obtaining profit $\pi_{H}(q_{H}^{*})$. In this context, we are specifically
interested in the following questions: (When) Does an optimal mechanism
involve information revelation? And if so, what are the qualitative features
of the contracts offered to the agents? In this section, we characterize the
structure of optimal contracts, assuming that information revelation can be
fostered; in the next section, we provide a necessary and sufficient condition
for optimal mechanisms to be revealing.

In what follows, we assume that the maximal profit is strictly greater than
$\pi_{H}(q^{*}_{H})$ (thus non-revealing mechanisms are suboptimal),171717It
can be shown that when information revelation is possible, this condition
holds generically. and that the discount factor $\delta$ is greater than some
threshold $\bar{\delta}$ which we shall define later. Although there may be
multiple optimal mechanisms, our characterization will apply to every one of
them.

######  Lemma 1 (Frontloading of Payment/Rent to L).

In any optimal mechanism, the sequence of separating contracts
$\left\\{\left(q_{\tau}^{S}(t),x_{\tau}^{S}(t)\right)\right\\}_{\tau\geq t}$
for agents who reveal in period $t$ satisfies

$x_{\tau}^{S}(t)=\begin{cases}C_{H}(q_{\tau}^{S}(t)),&\text{ if
}t\leq\tau<t+s\\\ C_{L}(q_{\tau}^{S}(t))+\beta\Delta
C(q_{\tau}^{S}(t)),&\text{ if }\tau=t+s\\\ C_{L}(q_{\tau}^{S}(t)),&\text{ if
}\tau>t+s\\\ \end{cases}$ (6)

for some $s\geq 1$ and $\beta\in(0,1]$.

Lemma 1 indicates that the separating contracts entail two phases: a reward
phase and an exploitation phase. In the reward phase, the principal pays a
positive rent to a low-type agent for at least two periods once he reveals his
type. After the initial $s+1$ rent-paying periods, the exploitation phase
starts and the rent disappears forever. This implies that an agent’s
continuation payoff after revealing declines over time and reaches zero in a
finite number of periods. We call each period in the reward phase a debt
period, and we say that the principal is in debt with an agent in period $t$
if the agent’s continuation payoff is positive.

To understand Lemma 1, note that the principal can always keep an agent’s
present value while adjusting the timing of payment to compensate him as much
as possible in the initial periods following his type announcement. By
fulfilling more today and promising less in the future, such frontloading of
payment relaxes the principal’s non-reneging constraint (NR) and makes room
for a strict profit improvement.

However, there is a limit to how much rent can be paid to the low type in each
period: If the principal pays too much in the initial periods of the
separating contracts, the high type would be tempted to make a false
announcement, work only in the first period to reap the high payment, and then
exit the relationship; that is, (IC-H) would be violated. In any optimal
mechanism, the payoff of the high-type is always kept to be zero. This implies
that the principal can pay the low type at most $C_{H}(q_{\tau}^{S}(t))$ in
any period without inducing the high type to take the money and run. By doing
so, the principal can give at most $\Delta C(q_{\tau}^{S}(t))$ of rent to a
low type in a period. When the discount factor is sufficiently large, the rent
from each individual period matters little, so the principal has to use
multiple debt periods to deliver the low-type’s total information rent (i.e.,
$s+1\geq 2$).181818In any optimal mechanism, the low type’s total information
rent $U_{L,t}^{S}$ is equal to his payoff if he were never to reveal. This is
positive and bounded away from zero, because by taking the pooling contract a
low type gets paid $C_{H}(q_{t}^{P})$ in each period while only incurring a
cost of $C_{L}(q_{t}^{P})$.

######  Lemma 2 (Gradual and Eventual Information Revelation).

In any optimal mechanism, information revelation is gradual and eventual.

When an optimal mechanism is revealing, Lemma 2 indicates that in every period
there will be a positive measure of low-type agents revealing themselves, and
that every low-type agent will reveal his type in some period.

To see why information revelation must be gradual, note first that there
cannot be a final period of revelation after which no further revelation
occurs. Otherwise, the principal would not want to pay any information rent to
any low types after that final revealing period. But this is inconsistent with
the fact that the principal has to pay rent for at least two periods to the
agents who reveal in that final period. Moreover, there cannot be any gaps
between the periods with information revelation. Indeed, if $r_{t}$ and
$r_{t+2}$ are positive but $r_{t+1}=0$, then the principal’s non-reneging
constraint (NR) must be slack at $t+2$. This is because if the principal were
to renege, he would rather renege at the beginning of $t+1$ than at $t+2$, as
reneging one period earlier would save the rent she has to pay at $t+1$
without losing extra information (given that $r_{t+1}=0$); in other words, the
principal’s incentive to renege is strictly less at $t+2$ than at $t+1$. But
then, one can shift a small measure of agents who were supposed to reveal at
$t+2$ to reveal at $t+1$, without violating (NR) at $t+2$ (or any other
constraint).191919In general, by moving agents to reveal earlier, it will hurt
the non-reneging constraints in later periods because there is less future
information to lose. The slackness of (NR) at $t+2$ makes it possible to
induce earlier revelation without violating (NR). This variation successfully
induces some low-types agents to reveal earlier and strictly benefits the
principal.

Information revelation must also be eventual because if
$\sum_{t=0}^{\infty}r_{t}<\alpha_{0}$, one can create another implementable
allocation with the same set of contracts while scaling up the measure of the
revealing agents in every period. Doing so will not violate any constraint but
will induce more information to be revealed over time, thereby increasing the
principal’s profit.

######  Lemma 3 (Downward Distortion of H’s Quantity).

In any optimal mechanism, the sequence of pooling contracts
$\left\\{\left(q_{t}^{P},x_{t}^{P}\right)\right\\}_{t\geq 0}$ satisfies
$q_{t}^{P}<q_{H}^{*}$ for all $t$ and $\lim_{t\to\infty}q_{t}^{P}=q_{H}^{*}$.
Moreover, $x_{t}^{P}=C_{H}(q_{t}^{P})$.

Lemma 3 states that in any optimal mechanism, the high-type agents are asked
to produce less than their surplus-maximizing quantity in each period and the
quantity produced converges to the surplus-maximizing level in the long run.
This is an analogue of the standard distortion in screening models: the low-
type agents in our environment are given information rent to reveal their
type, and the total rent each of them gets depend on what he would have got if
he were never to reveal. In an optimal mechanism, the pooling contracts always
pay $C_{H}(q_{t}^{P})$ to give zero payoffs to the high types. This means that
by taking the pooling contract in period $t$, a low-type agent can obtain a
positive payoff of $\Delta C(q_{t}^{P})$ because he only incurs a cost of
$C_{L}(q_{t}^{P})$. Therefore, lowering $q_{t}^{P}$ from $q_{H}^{*}$ is
profitable, because it strictly reduces the rent having to be paid to the low
types while only having a second-order effect on the principal’s profit from
the high types. The convergence of $q_{t}^{P}$ to $q_{H}^{*}$ over time
follows from Lemma 2, as all low-type agents opt out of the pooling contracts
eventually.

######  Lemma 4 (Upward Distortion and Frontloading of L’s Quantity).

In any optimal mechanism, the sequence of separating contracts
$\left\\{\left(q_{\tau}^{S}(t),x_{\tau}^{S}(t)\right)\right\\}_{\tau\geq t}$
for agents who reveal in period $t$ satisfies: If $\\{t,t+1,...,t+s\\}$ is the
set of debt periods, then $q_{t}^{S}(t)>q_{t+1}^{S}(t)>...>q_{t+s}^{S}(t)\geq
q_{L}^{*}$ and $q_{\tau}^{S}(t)=q_{L}^{*}$ for all $\tau>t+s$.

Lemma 4 implies that the principal never asks a low-type agent to produce
below the surplus-maximizing level and will distort the quantities upward in
all (non-final) debt periods. Moreover, the quantities required in the
separating contracts are decreasing over time throughout the debt periods.

To understand the direction of distortion, note that if
$q_{\tau}^{S}(t)<q_{L}^{*}$, the principal can increase $q_{\tau}^{S}(t)$ and
$x_{\tau}^{S}(t)$ at the same time to keep the low-type agent’s payoff
unaffected while improving the surplus and her profit. Moreover, further
increasing $q_{\tau}^{S}(t)$ above $q_{L}^{*}$ in a debt period can help the
principal frontload information rent and thus relax her non-reneging
constraints in future periods.

The quantities in the separating contracts must be decreasing across the debt
periods, because otherwise if $q_{\tau}^{S}(t)\leq q_{\tau+1}^{S}(t)$ for a
debt period $\tau$, the principal can then increase
$\left(q_{\tau}^{S}(t),x_{\tau}^{S}(t)\right)$ by $(\varepsilon,\varepsilon
C_{H}^{\prime})$ and decrease
$\left(q_{\tau+1}^{S}(t),x_{\tau+1}^{S}(t)\right)$ by
$(\delta^{-1}\varepsilon,\delta^{-1}\varepsilon C_{H}^{\prime})$, leaving each
type’s payoff unchanged. This perturbation is profitable, as it leads to a
higher average surplus from the low types (due to the concavity of the surplus
function) and also allows the principal to frontload some information rent
from $\tau+1$ to $\tau$.202020Indeed, this perturbation increases the low
type’s payoff by $\varepsilon\Delta C^{\prime}$ at $\tau$ and decreases it by
$\delta^{-1}\varepsilon\Delta C^{\prime}$ at $\tau+1$.

Combining these results, the following theorem characterizes the qualitative
structure of the contracts offered in an optimal mechanism.

######  Theorem 2.

Suppose that non-revealing mechanisms are suboptimal. There exists a
$\bar{\delta}$ such that if $\delta>\bar{\delta}$, then in every optimal
mechanism,

  1. 1.

Information revelation is gradual and eventual;

  2. 2.

After a low-type agent reveals his type, both the amount of work demanded and
the payment he receives start high and decrease over time.

  3. 3.

A low-type agent is given positive rents for a finite number of periods
following his revelation, after which the principal extracts full surplus from
him.

  4. 4.

The amount of work demanded in the pooling contracts starts below the surplus-
maximizing level and converges to it over time.

Theorem 2 summarizes our testable predictions about the work and wage dynamics
in equilibrium. In particular, we predict the use of a dual-track contract
system (pooling vs. separating), and that the separating contracts involve a
reward phase followed by an exploitation phase, with decreasing work and wages
over time. This comes as a consequence of the interplay between the
principal’s non-reneging constraint and the (inefficient) agent’s IC
constraint.212121Our model assumes that each agent’s productivity is constant
over time. In reality, if a worker becomes more productive as he stays longer
in the relationship, that would be an important factor that drives up his wage
over time. In Chinese universities where long-term commitment is often
lacking, we find evidence of contracts that are strikingly similar to our
predictions. Specifically, the economics departments in a number of major
Chinese universities implement a dual-track system, with a traditional track
and a tenure track. The traditional track entails low research requirements
and low pay, corresponding to the pooling contracts in our model. On the other
hand, the tenure track starts with high publication requirements and high pay
in the pre-tenure period. After some form of “tenure” is granted, the research
requirements and often the salary will be lowered.222222Based on our
conversations with colleagues at several Chinese institutions, such contract
systems are/were used in Renmin University of China, Shanghai University of
Finance and Economics, Xiamen University, etc. As such, the contracts offered
in the tenure track closely resemble the separating contracts in our model,
featuring a two-phase compensation scheme and decreasing work and wages.
Although the observed salary cut following a promotion may appear
counterintuitive, our theory rationalizes it as an optimal payment arrangement
driven by the employer’s limited commitment and its associated frontloading
incentives.

##  4 When Is Information Revelation Feasible?

The analysis in the previous section presumes that the optimal mechanism is
revealing. We now turn to deriving conditions under which this presumption is
true. Our result will uncover a new tradeoff between fostering future
information revelation and rewarding past disclosures.

###  4.1 An Intuitive Derivation

Note that the key to overcoming the ratchet effect in our setting is to make
sure that the principal does not want to renege on paying the information rent
promised to the low-type agents. This is the premise for any low types to be
willing to reveal their private information.

We begin with some heuristic calculations to illustrate the main tradeoffs
faced by the principal. Consider a stationary auxiliary environment where: i)
a constant measure $r$ of low-cost agents reveal in each period; ii) the
principal does not distort the quantity demanded from the high-cost agents;
and iii) for $T$ debt periods after a low-type agent reveals, the principal
demands a fixed quantity $\tilde{q}_{L}\in[q_{L}^{*},\bar{q}]$ while paying
him $C_{H}(\tilde{q}_{L})$. These features mirror the key properties of an
optimal revealing mechanism (i.e., Lemmas 1 through 4), at least
asymptotically, while the stationarity of this auxiliary environment
simplifies the examination of the principal’s reneging incentives.

Specifically, in this stationary auxiliary environment, reneging is never
optimal for the principal if and only if waiting one more period to renege is
better than reneging right away. From the principal’s perspective, if he waits
one more period and then reneges, an additional measure $r$ of low-type agents
will reveal in the current period, enabling the principal to make an extra
profit of

$\displaystyle r\delta\left[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})\right]$

by exploiting them starting the next period.

On the other hand, waiting one more period to renege requires the principal to
pay the promised rent to those agents with whom she is still in debt. Because
in period $t$ the principal is in debt with agents who revealed at $t-1$,
$t-2$, …, $t-T$ and each of these agents gets paid $C_{H}(\tilde{q}_{L})$
while producing $\tilde{q}_{L}$, the total loss of profit from these agents in
the current period if the principal honors her promises is232323The loss of
profit per agent, $\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{L})$, is equal to the
information rent paid to a low-type agent, $\Delta C(\tilde{q}_{L})$, plus the
inefficiency loss due to quantity distortions,
$\pi_{L}(q_{L}^{*})-\pi_{L}(\tilde{q}_{L})$.

$(1-\delta)Tr\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{L})\right].$

Note further that the number of debt periods $T$ is endogenous as it is
determined by the number of periods needed to deliver a low-type’s total
information rent. Indeed, $T$ has to satisfy

$\displaystyle(1-\delta)\sum_{\tau=0}^{T-1}\delta^{\tau}\Delta
C(\tilde{q}_{L})=\Delta C(q_{H}^{*})$

where the RHS is the total information rent needed to induce a low-type to
reveal (equaling his payoff if he were to take the pooling contract forever)
and the LHS is the discounted sum of his rents from the $T$ debt periods after
revealing. When $\delta$ approaches one, this condition implies that242424To
see this, note that $(1-\delta)\sum_{\tau=0}^{T-1}\delta^{\tau}\Delta
C(\tilde{q}_{L})=\Delta C(q_{H}^{*})$ implies that $1-\delta^{T}=\frac{\Delta
C({q}_{H}^{*})}{\Delta C(\tilde{q}_{L})}$, and thus
$(-\ln\delta)T=\ln\left(\frac{\Delta C(\tilde{q}_{L})}{\Delta
C(\tilde{q}_{L})-\Delta C({q}_{H}^{*})}\right)$. Moreover, when $\delta$ is
close to $1$, $-\ln\delta$ is approximately $(1-\delta)$.

$\displaystyle(1-\delta)T\to\ln\left(\frac{\Delta C(\tilde{q}_{L})}{\Delta
C(\tilde{q}_{L})-\Delta C({q}_{H}^{*})}\right).$

Therefore, from the principal’s perspective, as $\delta$ approaches one,
waiting one more period to renege is strictly better than reneging right away
if and only if

$\displaystyle\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})>\ln\left(\frac{\Delta
C(\tilde{q}_{L})}{\Delta C(\tilde{q}_{L})-\Delta
C({q}_{H}^{*})}\right)\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{L})\right];$
(7)

that is, when the benefit of waiting outweighs the loss.

###  4.2 The Formal Result

While the above analysis focuses on an auxiliary environment, it turns out
that the same condition determines whether information revelation is feasible
in our original non-stationary setting. We have the following theorem.

######  Theorem 3.

  1. 1.

If for all $\tilde{q}_{L}\in[q_{L}^{*},\bar{q}]$,

$\displaystyle\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})<\ln\left(\frac{\Delta
C(\tilde{q}_{L})}{\Delta C(\tilde{q}_{L})-\Delta
C({q}_{H}^{*})}\right)\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{L})\right],$
(8)

then the (unique) optimal mechanism is non-revealing whenever $\delta$ is
sufficiently close to $1$.

  2. 2.

If for some $\tilde{q}_{L}\in[q_{L}^{*},\bar{q}]$,

$\displaystyle\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})>\ln\left(\frac{\Delta
C(\tilde{q}_{L})}{\Delta C(\tilde{q}_{L})-\Delta
C({q}_{H}^{*})}\right)\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{L})\right],$
(9)

then every optimal mechanism is revealing whenever $\delta$ is sufficiently
close to $1$.

Below we interpret this theorem economically and use it to analyze the effect
of skill heterogeneity on information revelation. For readers interested in
the technical details, we sketch the proof of this theorem in Section 4.4.

###  4.3 Economic Implications

#### Cost-Benefit Comparison

Economically, the LHS of condition (9) in Theorem 3 is equal to the maximum
gains in surplus after the principal obtains information about an agent’s
type, while the RHS measures the costs to the principal for inducing
separation of types. Specifically, to incentivize information revelation, the
principal needs to distort the low-type’s quantity (Lemma 4) and pay
information rent for a finite number of debt periods (Lemma 1). Based on the
previous calculations, $\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{L})$ measures
the principal’s profit loss per agent per debt period, and the multiplier,
$\ln\left(\frac{\Delta C(\tilde{q}_{L})}{\Delta C(\tilde{q}_{L})-\Delta
C({q}_{H}^{*})}\right)$, accounts for the discounted number of debt periods.
Theorem 3 says that an optimal mechanism is revealing if and only if the
social value of information exceeds the costs needed to incentivize a low-type
agent to voluntarily reveal his private information.

A useful observation from Theorem 3 is that, for an optimal mechanism to be
revealing, the discounted number of debt periods, $\ln\left(\frac{\Delta
C(\tilde{q}_{L})}{\Delta C(\tilde{q}_{L})-\Delta C({q}_{H}^{*})}\right)$, must
be less than $1$. Otherwise, it is impossible to use a smaller surplus gain
(per period) to cover a larger separation cost (per period).

######  Observation 1.

If an optimal mechanism is revealing, then $\ln\left(\frac{\Delta
C(\bar{q})}{\Delta C(\bar{q})-\Delta C({q}_{H}^{*})}\right)<1$.

This condition requires $\Delta C(q_{H}^{*})$, the low type’s total
information rent (i.e., his payoff if he were to take the pooling contract
forever), to be not too large, or $\Delta C(\bar{q})$, the maximal information
rent deliverable in a given period, to be sufficiently large.

Graphically, Figure 1 depicts the conditions in Theorem 3. The blue curves
plot the RHS of (9) as a function of $\tilde{q}_{L}\in[q_{L}^{*},\bar{q}]$,
and the yellow lines indicate the values of the LHS which are constant w.r.t.
$\tilde{q}_{L}$. Under the parameters of panels (a) and (b) (where
$\theta_{L}$ and $\theta_{H}$ are sufficiently distant from each other), there
exists some $\tilde{q}_{L}$ at which the yellow line is above the blue curve,
thus every optimal mechanism is revealing. In contrast, under the parameters
of panel (c) (where $\theta_{L}$ and $\theta_{H}$ are close), the yellow line
is always below the blue curve, so the optimal mechanism is non-revealing.

(a) $\theta_{L}=1.5,\ \theta_{H}=3$ (Revealing)

(b) $\theta_{L}=2,\ \theta_{H}=3$ (Revealing)

(c) $\theta_{L}=2.5,\ \theta_{H}=3$ (Non-Revealing)

Figure 1: Necessary and Sufficient Condition for Information Revelation. This
figure is plotted with the following functional forms and parameters:
$v(q)=2q^{1/2}$, $C_{\theta}(q)=\theta q$, $\bar{q}=1$, $\theta_{H}=3$,
$\theta_{L}=1.5$ for (a), $\theta_{L}=2$ for (b), and $\theta_{L}=2.5$ for
(c).

#### Skill Heterogeneity Facilitates Information Revelation

How does skill heterogeneity among agents affect information revelation? On
the one hand, greater skill heterogeneity increases the value of information
about an agent’s type. On the other hand, as the cost difference widens, the
low type can get a higher payoff by pretending the high type, so the principal
has to pay more information rent to induce separation of types. Figure 1
suggests that, overall, information revelation becomes easier when the agents’
skills are more heterogeneous, that is, when $\theta_{H}$ and $\theta_{L}$ are
further apart. We now formalize this prediction.

Let us first give some cardinal meaning to the agent’s type. To this end, we
assume

$\displaystyle C_{\theta}(q)=\theta q.$

We have the following result.

######  Proposition 1.

Fix all parameters except $\theta_{L}$ and $\delta$.252525This result also
requires $\bar{q}$ to be large enough (but fixed). See Appendix A.3 for
details. There exists $\bar{\theta}<\theta_{H}$ such that:

  * •

if $\theta_{L}>\bar{\theta}$, then the (unique) optimal mechanism is non-
revealing whenever $\delta$ is sufficiently close to $1$;

  * •

if $\theta_{L}<\bar{\theta}$, then every optimal mechanism is revealing
whenever $\delta$ is sufficiently close to $1$.

Proposition 1 indicates that information revelation is more likely when the
difference between the two types’ marginal costs gets larger, and that
separation becomes impossible if the difference in marginal costs is too
small. Indeed, when the gap in marginal costs vanishes, the value of
information is negligible and is thus unable to cover any cost for
incentivizing separation. On the other hand, if $\theta_{L}$ decreases (i.e.,
when the difference in marginal costs rises), both the surplus gains from
information revelation and the cost to induce separation will increase by
$q_{L}^{*}$. However, by Observation 1, the multiplier on the RHS of (9) is
less than $1$ whenever the condition is satisfied, making the surplus gains
increase at a higher rate. Therefore, if an optimal mechanism is revealing at
some $\bar{\theta}$, it remains so for any $\theta_{L}<\bar{\theta}$.

###  4.4 Sketch of Proof

Below we outline the key steps for proving Theorem 3, especially how they
relate to the principal’s tradeoffs in the auxiliary environment analyzed in
Section 4.1. The formal proof is in the appendix.

To establish the sufficiency of condition (9) for information revelation, note
that the biggest jump from the auxiliary environment to our model is the
constant measure $r$ of revealing agents in every period. This is apparently
infeasible in our model with a fixed mass of (low-type) agents. However, one
could set the measure of low-type agents revealing in period $t$ to be
$r_{0}\rho^{t}$. With $\rho$ close to (but less than) $1$ and $r_{0}$ close to
$0$, the sequence is almost constant while the sum can still equal
$\alpha_{0}$.262626Recall that in the auxiliary environment, condition (7) for
information revelation is independent of $r$. This gives us the freedom to
adjust $r_{0}$ without affecting the principal’s incentives. When condition
(9) holds, this construction allows us to obtain implementable allocations
where the principal’s incentives are almost identical to the auxiliary
environment.

Let us now turn to necessity. Assume that an optimal mechanism is revealing.
Then in every period, the principal should prefer to honor her promise for one
more period than reneging right away. As the quantity in the pooling contract
converges to $q_{H}^{*}$ in the long run (by Theorem 2), for each sufficiently
large $t\geq\tilde{t}$, we have

$r_{t}\delta\left[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})\right]\geq(1-\delta)\sum_{\tau=z(t)}^{t-1}r_{\tau}\left[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{t}^{S}(\tau))\right].$

The LHS is the benefit of honoring promise for one more period and then
reneging (relative to reneging right away), since information about an extra
$r_{t}$ measure of efficient agents will be revealed. The RHS represents the
cost, as the principal will forgo some profits from those agents revealing in
$\\{z(t),...,t-1\\}$ with whom she is still in debt.

Adding up these constraints from $\tilde{t}$ onward, we obtain

$\displaystyle\sum_{t=\tilde{t}}^{\infty}r_{t}\delta\left[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})\right]$
$\displaystyle\geq(1-\delta)\sum_{t=\tilde{t}}^{\infty}\sum_{\tau=z(t)}^{t-1}r_{\tau}\left[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{t}^{S}(\tau))\right]$
$\displaystyle\geq(1-\delta)\sum_{t=\tilde{t}}^{\infty}r_{t}\sum_{\tau=t+1}^{T(t)}\left[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{\tau}^{S}(t))\right]$
$\displaystyle\geq(1-\delta)\sum_{t=\tilde{t}}^{\infty}r_{t}\sum_{\tau=t+1}^{T(t)}\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{L,t})\right],$

where the second line follows from changing the order of summation and
dropping terms from periods before $\tilde{t}$, and the third line follows
from the concavity of $\pi$ and Jensen’s inequality, with $\tilde{q}_{L,t}$
defined as an “average” of $q_{\tau}^{S}(t)$ across the debt periods that
delivers the same total information rent.272727The formal proof also uses the
concavity of $\Delta C$ in this step to ensure the desired direction of the
inequality.

Rearranging the above inequality, we obtain

$\sum_{t=\tilde{t}}^{\infty}r_{t}\left\\{\delta\left[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})\right]-(1-\delta)(T(t)-t)\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{L,t})\right]\right\\}\geq
0.$

This means that for at least some $t$, we must have

$\delta\left[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})\right]\geq(1-\delta)(T(t)-t)\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{L,t})\right].$

Now the payoff comparison becomes very close to that in the auxiliary
environment. As $\delta\to 1$, calculations similar to Section 4.1 imply that
condition (8) must fail, establishing part (1) of Theorem 3.

##  5 Discussion

###  5.1 The Role of Assumption 1

We have assumed that
$\alpha_{0}<\ushort{\alpha}\equiv\frac{\pi_{H}(q_{H}^{\ast})}{\pi_{L}(q_{L}^{\ast})}$,
that is, the initial proportion of the efficient agents is not too large. This
assumption ensures that the inefficient agents are sufficiently important to
the principal, and in particular, at any history the worst possible
continuation equilibrium is obtained through the best pooling contract for the
principal.

On the other hand, it is easy to see that there exists
$\bar{\alpha}\in\left(0,1\right)$ such that if $\alpha_{0}>\bar{\alpha}$, the
optimal solution with full commitment is to make an offer that extracts the
entire surplus from the inefficient type, while the efficient type simply
leaves the relationship. Since this allocation does not entail time
inconsistency, it is also optimal in our setting with limited commitment.

What about $\alpha_{0}\in\left[\ushort{\alpha},\ \bar{\alpha}\right]?$ One can
show that our characterizations extend to a small region beyond these known
boundaries. In particular, there exist $\varepsilon_{1}>0$ and
$\varepsilon_{2}>0$ such that: when $\alpha\in\left[\ushort{\alpha},\
\ushort{\alpha}+\varepsilon_{1}\right]$, the optimal mechanism is revealing
whenever condition (9) in Theorem 3 is satisfied; when
$\alpha\in\left[\bar{\alpha}-\varepsilon_{2},\ \bar{\alpha}\right]$, it is
optimal for the principal to offer only one contract that excludes the
inefficient type. Unfortunately, figuring out the exact values of
$\varepsilon_{1}$ and $\varepsilon_{2}$, as well as what happens for priors in
$\left(\ushort{\alpha}+\varepsilon_{1},\ \bar{\alpha}-\varepsilon_{2}\right)$,
is a daunting task that requires a different set of techniques. We leave it as
a direction for future research.

###  5.2 The Role of the Concavity of $\Delta C$

The assumption that $\Delta C$ is concave (i.e., $C_{H}^{\prime\prime}\leq
C_{L}^{\prime\prime}$) is used to show that quantities in the separating
contracts are decreasing during reward phase (Lemma 4) and to prove Theorem 3.
To see the role of this assumption, note that an efficient agent’s total
information rent is determined by his payoff from not revealing, which is
captured by $\Delta C(q_{H})$, i.e., receiving a payment equal to the high
cost while incurring a low cost. Moreover, the maximum information rent
deliverable per period is constrained by the inefficient agent’s take-the-
money-and-run incentive, so an efficient agent can be compensated by at most
$C_{H}(q_{L})$ and enjoys a rent of $\Delta C(q_{L})$ per period. The
assumption $\Delta C^{\prime\prime}\leq 0$ then ensures that the principal’s
problem is concave in quantities, which substantially simplifies the analysis.

When $\Delta C$ is not concave, the principal’s problem may no longer be
concave. Nevertheless, Theorem 3 still holds with a slight modification: the
necessary and sufficient condition now involves an appropriate concavification
of the principal’s objective function. Furthermore, as $\delta$ goes to one,
we can show that the quantities in the separating contracts still decreases
over time in a probabilistic sense.

###  5.3 Replacements

We have followed the canonical models of ratchet effect by assuming that there
is a fixed pool of agents, but our results extend naturally to a model with
replacements. Consider an extension of our model in which the agents leave the
relationship at a constant rate $\lambda$ and are replaced by new ones over
time. One can show that whenever condition (9) from Theorem 3 holds, the
optimal stationary mechanism is revealing, and it will exhibit the same
qualitative structure as in Theorem 2. On the other hand, even if condition
(9) fails, information revelation is still possible when the replacement rate
$\lambda$ is sufficiently high. Intuitively, replacements make information
revelation easier because if an agent (with a known type) is to be replaced
very soon by another one (with an unknown type), the principal’s benefit from
exploiting this agent is rather limited, while the cost of losing future
information can still be substantial.

###  5.4 The Cardinality of the Set of Agents

In our model, the principal interacts with a continuum of agents. One may
wonder whether our results extend to environments with “fewer” agents.

First, if there is a fixed pool of finitely many agents, the unique
equilibrium outcome when players are sufficiently patient is pooling, i.e.,
information revelation is not feasible. To understand this result, note that
with finitely many agents, if there were information revelation, there would
be some point in the future when all agents who plan to reveal have revealed
their types. After the final revealing period, the principal would have no
future information to learn/lose, so she would fully exploit everyone, but
then no agent would be willing to reveal his private information in the final
revealing period, leading to unraveling.282828A similar argument goes through
even when agents can reveal probabilistically. With finitely many agents, the
expected number of agents revealing on the equilibrium path will get
arbitrarily small as time goes on; and when there is “almost” no future
information to arrive, the principal will again find it optimal to start fully
exploiting everyone who has revealed, but then no agent would reveal in the
previous period.

The distinction between finite and infinite agents in this model is
reminiscent of the fundamental difference between finitely and infinitely
repeated games. One of the key insights from this paper is that in order to
mitigate the principal’s temptation to renege, she needs to be deterred by the
negative consequence of losing future information. But to avoid unraveling,
such deterrence must be present in every period. Our modeling approach of
assuming a continuum of agent is one way to make this possible. Alternatively,
it could also be achieved in a model with countably many agents.292929This
would naturally arise in an environment where a principal interacts with
finitely many agents each period, with new agents arriving and replacing old
agents over time. Indeed, one can show that whenever information revelation is
feasible with a continuum of agents, there exists an environment with
countably many agents in which information revelation is also
feasible.303030Another setting where the cardinality of the set of agents or
agent types matters critically is in the context of durable goods monopoly.
Ausubel and Deneckere (1989) show that a monopolist can credibly commit to
gradually reducing prices over time by exploiting the punishment of returning
to a Coasean outcome. This approach has the potential to generate substantial
profits but is only viable when there are infinitely many types of buyers.

##  6 Concluding Remarks

We study how simultaneously contracting with many agents can alleviate a
principal’s commitment problem and facilitate information revelation in the
presence of the ratchet effect. The principal’s reneging on previous promises,
while tempting, will stop all future agents from revealing their private
information. Her fear for losing future information then becomes a deterrent
to the principal’s exploitation of the agents who have revealed their types in
the past. This in turn makes each agent more willing to reveal his type,
thereby mitigating the ratchet effect. We fully characterize the structure of
optimal contracts, and find that these contracts frontload information rent
and create distortions at all types of agents. We also provide a necessary and
sufficient condition that determines whether the agents’ private information
is never revealed or fully revealed in the long run; this condition uncovers a
novel tradeoff between fostering future information revelation and rewarding
past revelation.

Our analysis has focused exclusively on sequential revelation mechanisms.
While we believe these mechanisms are suited to address our main research
questions, without a formal revelation principle, one cannot rule out the
possibility that some other mechanisms may further improve the principal’s
profit. We leave the examination of mechanisms outside of this class and the
development of a more general revelation principle as interesting directions
for future research.

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##  Appendix A Proofs

###  A.1 Proof of Theorem 1

###### Proof of Theorem 1.

The principal can always choose to offer the pooling contract
$\left(q_{H}^{*},C_{H}(q_{H}^{*})\right)$ to all agents in every period. This
induces no information revelation and leads to a profit of
$\pi_{H}(q_{H}^{*})$. If there is no mechanism that generates a profit greater
than $\left(q_{H}^{*},C_{H}(q_{H}^{*})\right)$, we are done as this pooling
solution is optimal.

Now assume that there is a mechanism that yields more than
$\pi_{H}(q_{H}^{\ast}).$ Let $\pi^{\ast}$ be the supremum of the profit over
all such mechanisms. Clearly, $\pi^{*}$ is bounded by the commitment profit.
For every $n\in\mathbb{N},$ take an implementable allocation

$\mathscr{C}^{n}\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\left(q_{t,n}^{P},x_{t,n}^{P}\right),\left(q^{S}_{\tau,n}(t),x^{S}_{\tau,n}(t)\right),r_{t}\right\\}_{t\geq
0,\tau\geq t},$

such that

$\Pi\left(\mathscr{C}^{n}\right)>\pi^{\ast}-n^{-1}.$

By assumption, $q_{t,n}^{P},q^{S}_{\tau,n}(t)\in[0,\bar{q}]$ and
$r_{n}\in[0,1]$, and one can without loss set
$x_{t,n}^{P},x^{S}_{\tau,n}(t)\leq C_{H}(\bar{q})$ (see Lemma 1 and Lemma 3).
Thus the elements in $\mathscr{C}^{n}$ are uniformly bounded. Taking a
subsequence if necessary, $\mathscr{C}^{n}$ converges to an allocation
$\mathscr{C}^{\ast}$ in the product topology. As the profit
$\Pi\left(\cdot\right)$ is continuous in product topology,

$\Pi\left(\mathscr{C}^{\ast}\right)=\lim\Pi\left(\mathscr{C}^{n}\right)\geq\pi^{\ast},$

and hence $\Pi\left(\mathscr{C}^{\ast}\right)=\pi^{\ast}.$ Finally, because
all constraints in program (5) are weak inequalities, they are preserved in
the limit. So $\mathscr{C}^{\ast}$ is implementable and hence optimal. ∎

###  A.2 Proof of Theorem 2

In this section, we assume that the maximal profit is strictly greater than
$\pi_{H}(q^{*}_{H})$ (thus every optimal mechanism is revealing), and that the
discount factor $\delta$ is greater than some threshold $\bar{\delta}$ which
we shall define later.

For any implementable allocation
$\left\\{\left(q_{t}^{P},x_{t}^{P}\right),\left(q^{S}_{\tau}(t),x^{S}_{\tau}(t)\right),r_{t}\right\\}_{t\geq
0,\tau\geq t}$, let $\mathscr{R}$ denote the set of periods in which there is
revelation, i.e., $r_{t}>0$. Recall that $U_{L,t}^{S}(T)$, defined in (3), is
the continuation payoff of a low type at $t$ if he revealed in period $T\leq
t$, and

$U^{S}_{H,t}\mathrel{\mathop{\mathchar 58\relax}}=\sup_{\mathbb{T}\geq
t}\left(1-\delta\right)\sum_{\tau=t}^{\mathbb{T}}\delta^{\tau-t}\left[x_{\tau}^{S}\left(t\right)-C_{H}\left(q_{\tau}^{S}\left(t\right)\right)\right]$

is the high type’s payoff is he (falsely) reveals at $t$.

######  Claim 1.

Consider an implementable allocation
$\left\\{\left(q_{t}^{P},x_{t}^{P}\right),\left(q^{S}_{\tau}(t),x^{S}_{\tau}(t)\right),r_{t}\right\\}_{t\geq
0,\tau\geq t}$. Define

$\Delta\tilde{X}_{t}^{S}\equiv\frac{U_{H,t}^{S}}{1-\delta},$

and

$\tilde{x}_{\tau}^{S}(t)\equiv\begin{cases}C_{H}(q_{\tau}^{S}(t))+\Delta\tilde{X}_{t}^{S},&\text{
if }\tau=t,\\\ C_{H}(q_{\tau}^{S}(t)),&\text{ if }t<\tau<t+s\\\
C_{L}(q_{\tau}^{S}(t))+\beta\Delta C(q_{\tau}^{S}(t)),&\text{ if }\tau=t+s\\\
C_{L}(q_{\tau}^{S}(t)),&\text{ if }\tau>t+s\\\ \end{cases}$ (10)

where $s\geq 0$ and $\beta\in(0,1]$ are determined by

$U_{L,t}^{S}(t)=(1-\delta)\Delta\tilde{X}_{t}^{S}+(1-\delta)\sum_{\tau=t}^{t+s-1}\delta^{\tau-t}\Delta
C(q_{\tau}^{S}(t))+(1-\delta)\delta^{s}\beta\Delta C(q_{t+s}^{S}(t)).$

Then,
$\left\\{\left(q_{t}^{P},x_{t}^{P}\right),\left(q^{S}_{\tau}(t),\tilde{x}^{S}_{\tau}(t)\right),r_{t}\right\\}_{t\geq
0,\tau\geq t}$ is another implementable allocation that generates the same
profit for the principal. Moreover, if
$x_{\tau}^{S}(t)\neq\tilde{x}_{\tau}^{S}(t)$ for some $\tau\geq t$, then (NR)
is not always binding under the alternative allocation.

###### Proof.

By construction, the alternative allocation delivers the same total surplus
(because quantities are unchanged) and same payoffs to both types as the
original allocation does, so the principal’s profit, as well as (IC-H), (2.3),
(IR-H), and (IR-L), is unaltered.

We now argue that the alternative allocation (weakly) relaxes (NR). This is
true if for all $T\geq t$,

$(1-\delta)\sum_{\tau=t}^{T}\delta^{\tau-t}\tilde{x}_{\tau}^{S}(t)\geq(1-\delta)\sum_{\tau=t}^{T}\delta^{\tau-t}x_{\tau}^{S}(t).$
(11)

That is, if at any $T\geq t$ the principal has paid more (i.e., owes less) to
the low-type agent under the alternative allocation than she does under the
original allocation. Suppose toward a contradiction that there is $T^{*}\geq
t$ such that
$(1-\delta)\sum_{\tau=t}^{T}\delta^{\tau-t}\tilde{x}_{\tau}^{S}(t)<(1-\delta)\sum_{\tau=t}^{T}\delta^{\tau-t}x_{\tau}^{S}(t)$.

  * •

If $T^{*}\leq t+s$, then under the original allocation, the high type can get
a payoff greater than $(1-\delta)\Delta\tilde{X}_{t}^{S}$ by falsely revealing
at $t$ and quitting after period $T^{*}$, a contradiction to the definition of
$U_{H,t}^{S}$.

  * •

If $T^{*}>t+s$, because
$(1-\delta)\sum_{\tau=t}^{\infty}\delta^{\tau-t}\tilde{x}_{\tau}^{S}(t)=(1-\delta)\sum_{\tau=t}^{\infty}\delta^{\tau-t}x_{\tau}^{S}(t)$
by construction, we have
$(1-\delta)\sum_{\tau=T^{*}+1}^{\infty}\delta^{\tau-t}\tilde{x}_{\tau}^{S}(t)>(1-\delta)\sum_{\tau=T^{*}+1}^{\infty}\delta^{\tau-t}x_{\tau}^{S}(t)$.
But then,

$\displaystyle U_{L,T^{*}+1}(t)$
$\displaystyle=\left(1-\delta\right)\sum_{\tau=T^{*}+1}^{\mathbb{\infty}}\delta^{\tau-T^{*}-1}\left[x_{\tau}^{S}(t)-C_{L}\left(q^{S}_{\tau}(t)\right)\right]$
$\displaystyle<\left(1-\delta\right)\sum_{\tau=T^{*}+1}^{\mathbb{\infty}}\delta^{\tau-T^{*}-1}\left[\tilde{x}_{\tau}^{S}(t)-C_{L}\left(q^{S}_{\tau}(t)\right)\right]$
$\displaystyle=0,$

a contradiction (IR-L) at $T^{*}+1$.

Finally, if $x_{\tau}^{S}(t)\neq\tilde{x}_{\tau}^{S}(t)$ for some $\tau\geq
t$, let $\tilde{T}$ be the first period this occurs, so that
$x_{\tau}^{S}(t)=\tilde{x}_{\tau}^{S}(t)$ for $\tau<\tilde{T}$ and
$x_{\tilde{T}}^{S}(t)<\tilde{x}_{\tilde{T}}^{S}(t)$. Condition (11) then holds
strictly for $T=\tilde{T}$, implying that (NR) is slack at $\tilde{T}+1$ under
the alternative allocation. ∎

######  Claim 2.

There exists $\hat{\delta}\in\left(0,1\right)$ such that if $\delta>$
$\hat{\delta}$, then any implementable allocation with
$\mathscr{R}\neq\emptyset$ satisfies $\sup\mathscr{R}=\infty$, i.e., there
cannot be a last revelation period.

###### Proof.

Suppose that $\sup\mathscr{R}=T<\infty$. Because there is no information
revelation after $T$, (NR) requires that full exploitation of both types
happens after $T$, that is, for all $t>T$ we have $q_{t}^{P}=q_{H}^{*}$,
$x_{t}^{P}=C_{H}(q_{H}^{*})$, $q_{t}^{S}(T)=q_{L}^{*}$, and
$x_{t}^{S}(T)=C_{L}(q_{L}^{*})$.

To incentivize a low type to reveal at $T$, we need

$(1-\delta)[x_{T}^{S}(T)-C_{L}(q_{T}^{S}(T))]\geq(1-\delta)[x_{T}^{P}-C_{L}(x_{T}^{P})]+\delta\Delta
C(q_{H}^{*}),$

where the LHS is the low type’s payoff from revealing at $T$ and the RHS is
his payoff from taking the pooling contract forever. On the other hand, to
make sure that the high type does not want to falsely reveal at $T$ (and then
quit right after), we need

$(1-\delta)[x_{T}^{S}(T)-C_{H}(q_{T}^{S}(T))]\leq(1-\delta)[x_{T}^{P}-C_{H}(x_{T}^{P})].$

Combining these two inequalities, we have

$\displaystyle(1-\delta)[x_{T}^{S}(T)-C_{L}(q_{T}^{S}(T))]$
$\displaystyle\geq(1-\delta)[x_{T}^{P}-C_{L}(x_{T}^{P})]+\delta\Delta
C(q_{H}^{*})$
$\displaystyle=(1-\delta)[x_{T}^{P}-C_{H}(x_{T}^{P})]+(1-\delta)\Delta
C(x_{T}^{P})+\delta\Delta C(q_{H}^{*})$
$\displaystyle\geq(1-\delta)[x_{T}^{S}(T)-C_{H}(q_{T}^{S}(T))]+(1-\delta)\Delta
C(x_{T}^{P})+\delta\Delta C(q_{H}^{*})$
$\displaystyle=(1-\delta)[x_{T}^{S}(T)-C_{L}(q_{T}^{S}(T))]+(1-\delta)\Delta
C(x_{T}^{P})+\delta\Delta C(q_{H}^{*})$ $\displaystyle\quad\ -(1-\delta)\Delta
C(q_{T}^{S}(T)),$

which implies that

$(1-\delta)\Delta C(q_{T}^{S}(T))\geq\delta\Delta C(q_{H}^{*}).$

But because $\Delta C(q_{T}^{S}(T))\leq\Delta C(\bar{q})$, the above
inequality cannot hold if $\frac{1-\delta}{\delta}<\frac{\Delta
C(q_{H}^{*})}{\Delta C(\bar{q})}$, that is, if $\delta>\frac{\Delta
C(\bar{q})}{\Delta C(\bar{q})+\Delta C(q_{H}^{*})}\mathrel{\mathop{\mathchar
58\relax}}=\hat{\delta}$. ∎

We assume now that $\delta>\hat{\delta}$. Later we will further refine this
lower bound (see Claim 5).

######  Claim 3.

In any optimal mechanism, the low-type agent is indifferent between revealing
or not at every $t\in\mathscr{R}$.

###### Proof.

This follows immediately from Claim 2 and the presumption that the any optimal
mechanism is revealing. ∎

######  Claim 4.

In any optimal mechanism, the high type’s period-$0$ payoff satisfies
$U^{P}_{H,0}=0$. There exists an optimal mechanism with $U^{P}_{H,t}=0$ for
all $t\geq 0$, hence $x_{t}^{P}=C_{H}(q_{t}^{P})$ in every period. Moreover,
if $U^{P}_{H,t}>0$ at some $t$, then there exists an optimal allocation under
which (NR) is not always binding.

###### Proof.

Clearly $U_{H,0}=0$ for otherwise payments at $t=0$ can be uniformly decreased
by $\epsilon$.

Take any optimal allocation
$\left\\{\left(q_{t}^{P},x_{t}^{P}\right),\left(q^{S}_{\tau}(t),x^{S}_{\tau}(t)\right),r_{t}\right\\}_{t\geq
0,\tau\geq t}$. Consider an alternative allocation
$\left\\{\left(q_{t}^{P},\tilde{x}_{t}^{P}\right),\left(q^{S}_{\tau}(t),\tilde{x}^{S}_{\tau}(t)\right),r_{t}\right\\}_{t\geq
0,\tau\geq t}$ such that the pooling payment satisfies
$\tilde{x}_{t}^{P}=C_{H}(q_{t}^{P})$ for every $t$ and the separating payment
$\tilde{x}_{\tau}^{S}(t)$ is defined by (10) for every $t\in\mathscr{R}$ and
$\tau\geq t$ with $\Delta\tilde{X}_{t}^{S}=0$ and $s$ and $\beta$ being
determined by

$(1-\delta)\sum_{\tau=t}^{\infty}\delta^{\tau-t}\Delta
C(q_{\tau}^{P})=(1-\delta)\sum_{\tau=t}^{t+s-1}\delta^{\tau-t}\Delta
C(q_{\tau}^{S}(t))+(1-\delta)\delta^{s}\beta\Delta C(q_{t+s}^{S}(t)).$

Let $\tilde{U}_{\theta,t}^{S}$ and $\tilde{U}_{\theta,t}^{P}$ denote the
players’ payoffs under this alternative allocation. By construction,
$\tilde{U}_{H,t}^{P}=\tilde{U}_{H,t}^{S}=0$ for every $t$ and
$\tilde{U}_{L,t}^{P}=\tilde{U}_{L,t}^{S}(t)=(1-\delta)\sum_{\tau=t}^{\infty}\delta^{\tau-t}\Delta
C(q_{\tau}^{P})$ for every $t\in\mathscr{R}$, so (IC-H), (2.3), (IR-H), and
(IR-L) are satisfied.

We now show that the alternative allocation (weakly) relaxes (NR). By the
argument in the proof of Lemma 1, the variation of the separating payments
relaxes (NR). For the variation of the pooling payments, note that for any
$T\geq 0$ we must have

$\sum_{\tau=0}^{T}\delta^{\tau}\tilde{x}_{\tau}^{P}\geq\sum_{\tau=0}^{T}\delta^{\tau}x_{\tau}^{P},$
(12)

for otherwise if it were the case that
$\sum_{\tau=0}^{T}\delta^{\tau}x_{\tau}^{P}>\sum_{\tau=0}^{T}\delta^{\tau}\tilde{x}_{\tau}^{P}$,
then under the original allocation the high type would obtain a strictly
positive payoff by working until period $T$ and then quitting, a contradiction
to $U_{H,0}=0$. Therefore, at any $T\geq 0$ the principal has paid more (i.e.,
owes less) to the high-type agent under the alternative allocation than she
does under the original allocation, which further relaxes (NR).

Next we argue that the principal’s time-$0$ profit keeps the same under the
alternative allocation. First, $U_{H,0}^{P}=0=\tilde{U}_{H,0}^{P}$. Now
consider a low-type agent who reveals in period $t$ under the original
allocation. Because the original allocation is optimal, Claim 3 implies the
low type’s time-$0$ payoff can be attained by taking the pooling contract
forever, that is,

$U_{L,0}^{P}=U_{H,0}^{P}+\sum_{\tau=0}^{\infty}\delta^{\tau}\Delta
C(q_{\tau}^{P}).$

Meanwhile, by construction of the alternative allocation, the low type is
indifferent between when to reveal too, so the low type’s time-$0$ payoff is

$\tilde{U}_{L,0}^{P}=\sum_{\tau=0}^{\infty}\delta^{\tau}\Delta
C(q_{\tau}^{P}).$

Recall that $U_{H,0}^{P}=0$, so we have $U_{L,0}^{P}=\tilde{U}_{L,0}^{P}$. As
quantities (thus surplus) are unchanged, the principal’s time-$0$ profit
remains the same. Therefore, the alternative allocation is also optimal.

Finally, if $U_{H,t}>0$ for some $t$, then
$x_{\tau}^{P}\neq\tilde{x}_{\tau}^{P}$ for some $\tau$. Let $\tilde{T}$ be the
first period this occurs, so that $x_{\tau}^{P}=\tilde{x}_{\tau}^{P}$ for
$\tau<\tilde{T}$ and $x_{\tilde{T}}^{P}<\tilde{x}_{\tilde{T}}^{P}$. Condition
(12) then holds strictly for $T=\tilde{T}$, implying that (NR) is slack at
$\tilde{T}+1$ under the alternative optimal allocation. ∎

######  Definition 4.

Consider a low-type agent who reveals in period $t$ and is offered
$\\{q_{\tau}^{S}(t),x_{\tau}^{S}(t)\\}_{\tau\geq t}$. We say that the
principal is in debt with this agent in period $\tau$ if the agent’s
continuation payoff is strictly positive. Any such period is called a debt
period.

######  Claim 5.

There exists $\bar{\delta}\in(\hat{\delta},1)$ such that if
$\delta>\bar{\delta}$, any optimal allocation with separating payments given
by (10) satisfies $s\geq 1$ for every $t\in\mathscr{R}$. That is, it takes at
least two debt periods to reward a low type after his revelation.

###### Proof.

We prove a stronger claim: For every $N\in\mathbb{N}$, there exists
$\delta^{*}\in(\hat{\delta},1)$ such that if $\delta>\delta^{*}$ and
revelation happens in period $t$, then the reward of the low type takes at
least $N$ periods.

Assume toward a contradiction that we can find a sequence of optimal
allocations associated with $\delta_{n}\rightarrow 1$ and $M>0$ such that it
takes less than $M$ periods to reward the low type in some revealing period
with discount factor $\delta_{n}.$ Then by revealing the low type can get at
most $\left(1-\delta_{n}\right)M\Delta C\left(\bar{q}\right)$ which converges
to zero as $\delta_{n}\rightarrow 1.$ Because the low type is indifferent
between revealing or not (Claim 3), this implies that the low type’s payoff
from taking the pooling contract converges to zero. Hence, for every
$\epsilon>0$, the principal’s continuation payoff is bounded above by
$(1-\alpha_{0})\epsilon+\alpha_{0}\pi_{L}(q_{L}^{*})$. But the principal is
always able to guarantee herself a continuation payoff of
$\pi_{H}(q_{H}^{*})$, which is greater than $\alpha_{0}\pi_{L}(q_{L}^{*})$ by
Assumption 1, a contradiction. ∎

We assume for the reminder of the proof that $\delta>\bar{\delta}$.

######  Claim 6.

In any optimal mechanism, $q_{t}^{P}\leq q_{H}^{*}$ for every $t$, and
$q_{t}^{P}<q_{H}^{*}$ for every $t\in\mathscr{R}$.

###### Proof.

Take any optimal mechanism and, if necessary, apply the variation of payments
as in Claims 1 and 4. If $q_{t}^{P}>q_{H}^{*}$ at some $t$, then decreasing
$q_{t}^{P}$ to $q_{H}^{*}$ and $x_{t}^{P}$ to $C_{H}(q_{H}^{*})$ will increase
the surplus at $t$ and decrease the information rent to any low-type agents
who revealed before $t$. Moreover, if $r_{t}>0$ and $q_{t}^{P}=q_{H}^{*}$,
further decreasing $q_{t}^{P}$ by $\epsilon$ has no first-order effect on
surplus but can reduce the information rent to the low-type agents who reveal
at $t$ by $r_{t}(1-\delta)\Delta C^{\prime}(q_{H}^{*})\epsilon$. ∎

######  Claim 7.

In any optimal mechanism, $\mathscr{R}=\mathbb{Z}_{+}$. That is, information
revelation is gradual.

###### Proof.

Take any optimal mechanism and, if necessary, apply the variation of payments
as in Claims 1 and 4.

We first argue that $r_{0}>0$. If not, let $\tilde{t}=\min\mathscr{R}$. Note
that the principal’s profit is at most $\pi_{H}(q_{H}^{*})$ at each
$t\in\\{0,...,\tilde{t}-1\\}$. The principal’s total time-$0$ profit is given
by
$(1-\delta)\left[\sum_{\tau=0}^{\tilde{t}-1}\delta^{\tau}\pi_{H}(q_{\tau}^{P})+\delta^{\tilde{t}}\Pi_{\tilde{t}}\right]$,
which is presumed to be greater than $\pi_{H}(q_{H}^{*})$, so we must have
$\Pi_{\tilde{t}}>\Pi_{0}$. But then, the principal is strictly better off by
skipping the first $\tilde{t}$ periods and starting with the contracts from
time-$\tilde{t}$ onward.

Next we show that if $r_{t}>0$ then $r_{t+1}>0$. Suppose toward a
contradiction that $r_{t}>0$ but $r_{t+1}=0.$ Let $\hat{t}$ be the smallest
integer greater than $t$ such that $r_{\hat{t}}>0$. Such $\hat{t}$ exists
because $\sup\mathscr{R}=\infty$ by Claim 2. Hence $r_{\tau}=0$ for every
$\tau\in\\{t+1,...,\hat{t}-1\\}.$ Since $\delta>\bar{\delta}$, by Claim 5 the
principal must be paying debt at period $t+1$. But then, from the principal’s
perspective, her incentive to renege at $t+1$ is strictly stronger than that
at $\hat{t}$. Indeed, relative to reneging at $\hat{t}$, reneging at $t+1$
saves the principal from paying her time-$t+1$ debt while she loses nothing as
no one reveals between $t+1$ and $\hat{t}$. This implies that (NR) does not
bind at period $\hat{t}$, for otherwise the principal would find it optimal to
renege at $t+1$. Therefore we can implement a variation of the mechanism to
induce a fraction $\phi\in\left(0,1\right)$ of the agents that were supposed
to reveal at $\hat{t}$ to reveal at $\hat{t}-1$. This can be achieved by
offering them a sequence of separating contracts
$\\{\tilde{q}_{\tau}^{S}(\hat{t}-1),\tilde{x}_{\tau}^{S}(\hat{t}-1)\\}_{\tau\geq\hat{t}-1}$,
such that $\tilde{q}_{\hat{t}-1}^{S}(\hat{t}-1)=q_{L}^{*}$,
$\tilde{q}_{\tau}^{S}(\hat{t}-1)=q_{\tau}^{S}(\hat{t})$ for $\tau\geq\hat{t}$,
and the payments are given by (10). This is incentive-compatible, keeps the
agent indifferent, and increases the surplus, because before the variation
this fraction of low type agents only produce $q_{\hat{t}-1}^{P}$ at
$\hat{t}-1$, which is strictly less than $q_{L}^{*}$ by Claim 6. Thus this is
a profitable deviation and contradicts the optimality of the original
mechanism. ∎

######  Claim 8.

In any optimal mechanism, (NR) is binding in every period $t\in\\{1,2,...\\}$.
That is, the principal is always indifferent between reneging on past promises
or not.

###### Proof.

Take any optimal mechanism, and suppose toward a contradiction that (NR) is
slack in some period. Let $\hat{t}$ be the smallest period this happens. By
Claim 7, $r_{\hat{t}}>0$. As in the proof of Claim 7, one can construct a
sequence of separating contracts
$\\{\tilde{q}_{\tau}^{S}(\hat{t}-1),\tilde{x}_{\tau}^{S}(\hat{t}-1)\\}_{\tau\geq\hat{t}-1}$
to induce a fraction $\phi$ of the agents that were supposed to reveal at
$\hat{t}$ to reveal at $\hat{t}-1$. This variation increases the principal’s
profit by the previous argument. Note, however, that the principal is already
offering a sequence of separating contracts
$\\{{q}_{\tau}^{S}(\hat{t}-1),{x}_{\tau}^{S}(\hat{t}-1)\\}_{\tau\geq\hat{t}-1}$
for revealing at $\hat{t}-1$, so this variation will result in a randomization
between two separating offers starting $\hat{t}-1$, which is not allowed by
the definition of a sequential revelation mechanism. To construct a feasible
profitable deviation, we will replace
$\\{{q}_{\tau}^{S}(\hat{t}-1),{x}_{\tau}^{S}(\hat{t}-1)\\}_{\tau\geq\hat{t}-1}$
with a “convex combination” of
$\\{{q}_{\tau}^{S}(\hat{t}-1),{x}_{\tau}^{S}(\hat{t}-1)\\}_{\tau\geq\hat{t}-1}$
and
$\\{\tilde{q}_{\tau}^{S}(\hat{t}-1),\tilde{x}_{\tau}^{S}(\hat{t}-1)\\}_{\tau\geq\hat{t}-1}$.
This single offer will be taken by all of the $r_{\hat{t}-1}+\phi r_{\hat{t}}$
agents revealing at $\hat{t}-1$.

Specifically, let $(\chi_{\hat{t}-1},\chi_{\hat{t}},\chi_{\hat{t}+1},...)$ and
$(\tilde{\chi}_{\hat{t}-1},\tilde{\chi}_{\hat{t}},\tilde{\chi}_{\hat{t}+1},...)$
be the sequences of rents the low type gets from
$\\{{q}_{\tau}^{S}(\hat{t}-1),{x}_{\tau}^{S}(\hat{t}-1)\\}_{\tau\geq\hat{t}-1}$
and
$\\{\tilde{q}_{\tau}^{S}(\hat{t}-1),\tilde{x}_{\tau}^{S}(\hat{t}-1)\\}_{\tau\geq\hat{t}-1}$,
respectively. Note that the discounted sum of each sequence of rents is the
same. Define

$\hat{\chi}_{\tau}\mathrel{\mathop{\mathchar
58\relax}}=\frac{r_{\hat{t}-1}}{r_{\hat{t}-1}+\phi
r_{\hat{t}}}\chi_{\tau}+\frac{\phi r_{\hat{t}}}{r_{\hat{t}-1}+\phi
r_{\hat{t}}}\tilde{\chi}_{\tau}.$

Consider a new sequence of separating contracts
$\\{\hat{q}_{\tau}^{S}(\hat{t}-1),\hat{x}_{\tau}^{S}(\hat{t}-1)\\}_{\tau\geq\hat{t}-1}$
such that

$\hat{q}_{\tau}^{S}(\hat{t}-1)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}q_{L}^{*},&\text{ if }\hat{\chi}_{\tau}\leq\Delta
C(q_{L}^{*})\\\ \Delta C^{-1}(\hat{\chi}_{\tau}),&\text{ if
}\hat{\chi}_{\tau}>\Delta C(q_{L}^{*})\end{cases},$

and

$\hat{x}_{\tau}^{S}(\hat{t}-1)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}C_{L}(q_{L}^{*})+\hat{\chi}_{\tau},&\text{ if
}\hat{\chi}_{\tau}\leq\Delta C(q_{L}^{*})\\\
C_{H}(\hat{q}_{\tau}^{S}(\hat{t}-1)),&\text{ if }\hat{\chi}_{\tau}>\Delta
C(q_{L}^{*})\end{cases}.$

By construction, the low type’s payoffs from this new sequence of contracts
are $(\hat{\chi}_{\hat{t}-1},\hat{\chi}_{\hat{t}},\hat{\chi}_{\hat{t}+1},...)$
whose discounted sum is the same as the other two rent sequences.

Now we show that this new sequence of separating contracts increases the
principal profit against the randomization of the two contract sequences we
started with. Note that the principal’s profit in period $\tau$ is given by

$\Psi(\hat{\chi}_{\tau})=\begin{cases}\pi_{L}(q_{L}^{*})-\hat{\chi}_{\tau},&\text{
if }\hat{\chi}_{\tau}\leq\Delta C(q_{L}^{*})\\\
\pi_{H}(\hat{q}_{\tau}^{S}(\hat{t}-1)),&\text{ if }\hat{\chi}_{\tau}>\Delta
C(q_{L}^{*})\end{cases}.$

Using the concavity of $\pi_{H}(\cdot)$ and $\Delta C(\cdot)$, one can verify
that $\Psi(\cdot)$ is concave. This implies that for each $\tau\geq\hat{t}-1$,

$\Psi(\hat{\chi}_{\tau})\geq\frac{r_{\hat{t}-1}}{r_{\hat{t}-1}+\phi
r_{\hat{t}}}\Psi(\chi_{\tau})+\frac{\phi r_{\hat{t}}}{r_{\hat{t}-1}+\phi
r_{\hat{t}}}\Psi(\tilde{\chi}_{\tau}).$

So we have a profitable deviation. ∎

######  Corollary 1.

In any optimal mechanism, the separating payments are given by (10) and the
pooling payments satisfy $x_{t}^{P}=C_{H}(q_{t}^{P})$ for every $t$.

###### Proof.

This follows immediately from Claims 1, 4, and 8. ∎

###### Proof of Lemma 1.

This follows immediately from Corollary 1, Claim 4, and Claim 5. ∎

######  Claim 9.

In any optimal mechanism, $\sum_{t=0}^{\infty}r_{t}=\alpha_{0}$. That is,
information revelation is eventual.

###### Proof.

Take any optimal mechanism. Recall that $R_{t}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{\tau<t}r_{\tau}$. Suppose toward a contradiction that
$\lim_{t}R_{t}=\bar{R}<\alpha_{0}$. Let $V^{P}$ be the principal’s profit from
an agent who takes the pooling contract forever, and $V^{S}$ be the
principal’s average profit from the agents who reveal at some $t$. Remember we
assume $\pi_{H}(q_{H}^{*})<\Pi_{0}=(1-\bar{R})V^{P}+\bar{R}V^{S}$. By Claim 6,
$V^{P}<\pi_{H}(q_{H}^{*})$. This implies that

$V^{S}>\pi_{H}(q_{H}^{*})>V^{P}.$

Now consider an alternative allocation where all contracts remain the same
while the measure of agents revealing at $t$ is scaled up to
$\hat{r}_{t}=\frac{\alpha_{0}}{\bar{R}}r_{t}$ for every $t$. Clearly this
alternative allocation still satisfies (IC-H), (2.3), (IR-H), and (IR-L),
because all contracts are unaltered. Moreover, the principal’s profit under
the alternative allocation is $\alpha_{0}V^{S}+(1-\alpha_{0})V^{P}$, which is
greater than $\Pi_{0}$. This renders a valid profitable deviation and
contradicts the optimality of the original mechanism, if we can show that (NR)
remains satisfied.

To check (NR), let us fix any $t$. Claim 8 implies that under the original
optimal allocation the principal is indifferent between reneging at $t$ or
reneging at $t+1$. Consider three types of agents:

  * •

Those who are supposed to reveal at $t$, with measure $r_{t}$. If the
principal reneges at $t$, she gets $r_{t}\pi_{H}(q_{H}^{*})$ from them; if the
principal reneges at $t+1$, she gets
$r_{t}[(1-\delta)\pi_{H}(q_{t}^{S}(t))+\delta\pi_{L}(q_{L}^{*})]$ from them.

  * •

Those who revealed prior to $t$, with measure $R_{t}$. Let $B_{t}^{S}(\tau)$
be the rent the principal is suppose to pay at $t$ to an agent who revealed at
$\tau<t$. If the principal reneges at $t$, she gets $R_{t}\pi_{L}(q_{L}^{*})$
from them; if the principal reneges at $t+1$, she gets
$(1-\delta)\sum_{\tau<t}r_{\tau}[v(q_{t}^{S}(\tau))-B_{t}^{S}(\tau)]+\delta
R_{t}\pi_{L}(q_{L}^{*})$.

  * •

Those who are supposed to take the pooling contracts at $t$, with measure
$1-R_{t+1}$. If the principal reneges at $t$, she gets
$(1-R_{t+1})\pi_{H}(q_{H}^{*})$ from them; if the principal reneges at $t+1$,
she gets $(1-R_{t+1})[(1-\delta)\pi_{H}(q_{t}^{P})+\delta\pi_{H}(q_{H}^{*})]$.

Because the principal weakly prefers reneging at $t+1$ to reneging at $t$, we
have

$\displaystyle
r_{t}[(1-\delta)\pi_{H}(q_{t}^{S}(t))+\delta\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})]$
$\displaystyle+(1-\delta)\sum_{\tau<t}r_{\tau}[v(q_{t}^{S}(\tau))-B_{t}^{S}(\tau)-\pi_{L}(q_{L}^{*})]$
$\displaystyle+(1-R_{t+1})(1-\delta)[\pi_{H}(q_{t}^{P})-\pi_{H}(q_{H}^{*})]$
$\displaystyle\geq 0.$

As $\pi_{H}(q_{t}^{P})-\pi_{H}(q_{H}^{*})\leq 0$, the above inequality
continues to hold if every $r_{\tau}$ is scaled up the same factor (greater
than $1$). This means that (NR) is still satisfied at $t$. Because $t$ is
arbitrary, we are done. ∎

###### Proof of Lemma 2.

This follows immediately from Claims 7 and 9. ∎

######  Claim 10.

In any optimal mechanism, $\lim_{t\to\infty}q_{t}^{P}=q_{H}^{*}$.

###### Proof.

Notice that $r_{t}\to 0$. By Claim 8, at each $t$ the principal is indifferent
between reneging at $t$ and reneging at $t+1$. By reneging today, the
principal can save the information rents to those whom she is in debt with
and, more importantly, let the high type to produce $q_{H}^{*}$ today rather
than $q_{t}^{P}$. Therefore, relative to reneging at $t+1$, the principal’s
benefit of reneging at $t$ is bounded below by
$(1-\delta)(1-\alpha_{0})[\pi_{H}(q_{H}^{*})-\pi_{H}(q_{t}^{P})]$. On the
other hand, the cost of reneging at $t$ comes from not being able to exploit
the agents who are supposed to reveal at $t$. This is proportional to $r_{t}$
and converges to $0$ as $t\to\infty$. Therefore, for the principal to be
indifferent between reneging at $t$ or $t+1$, we must have
$(1-\delta)(1-\alpha_{0})[\pi_{H}(q_{H}^{*})-\pi_{H}(q_{t}^{P})]\to 0$, which
implies $\lim_{t}q_{t}^{P}=q_{H}^{*}$. ∎

###### Proof of Lemma 3.

This follows immediately from Claims 4, 6, 7, and 10. ∎

###### Proof of Lemma 4.

Take any optimal mechanism and fix any $t$.

First we argue that $q_{\tau}^{S}(t)\geq q_{L}^{*}$ for every $\tau\geq t$.
Suppose toward a contradiction that we can find $\tau\geq t$ such that
$q_{\tau}^{S}(t)<q_{L}^{*}$. The principal can then increase $q_{\tau}^{S}(t)$
to $q_{L}^{*}$ and increase $x_{\tau}^{S}(t)$ by
$C_{L}(q_{L}^{*})-C_{L}(q_{\tau}^{S}(t))$. This keeps the payoff of the low
type unchanged, relaxes the high type’s incentive compatibility, and increases
surplus, making it a profit deviation.

Next we show that $q_{\tau}^{S}(t)=q_{L}^{*}$ for all $\tau>t+s$, that is,
when the principal is no longer in debt with the agents who reveals at $t$.
Indeed, if $q_{\tau}^{S}(t)\neq q_{L}^{*}$, the principal can simply reset
$q_{\tau}^{S}(t)$ to $q_{L}^{*}$ and $x_{\tau}^{S}(t)$ to $C_{L}(q_{L}^{*})$.
This increases surplus, keeps the low type’s payoff unchanged, and maintains
the high type’s incentive compatibility,313131Note that the payment after the
debt periods is less than the high type’s cost, so the high type (even after
falsely revealing at $t$) would never stay in the separating contract till
then. making it a profitable deviation.

Now we prove that if $\tau$ and $\tau+1$ are debt periods then
$q_{\tau}^{S}(t)>q_{\tau+1}^{S}(t)$.

  * •

Suppose toward a contradiction that $q_{\tau}^{S}(t)<q_{\tau+1}^{S}(t)$. By
Lemma 1, $x_{\tau}^{S}(t)=C_{H}(q_{\tau}^{S}(t))$ and
$x_{\tau+1}^{S}(t)=C_{H}(q_{\tau+1}^{S}(t))$.323232This is true even if
$\tau+1$ is the last debt period. To see this, note that
$q_{\tau+1}^{S}(t)>q_{\tau}^{S}(t)\geq q_{L}^{*}$ by the contradiction
assumption. If it were the case that
$x_{\tau+1}^{S}(t)<C_{H}(q_{\tau+1}^{S}(t))$, then the principal could
decrease $q_{\tau+1}^{S}(t)$ and $x_{\tau+1}^{S}(t)$ at the same time to keep
the low type’s payoff unchanged and increase surplus. Importantly, this
variation would not affect the high type’s incentive compatibility because the
high type would never falsely reveal at $t$ and stay in the contract until
$\tau+1$. So this would be a profitable deviation. Now increase
$q_{\tau}^{S}(t)$ by $\epsilon$, decrease $q_{\tau+1}^{S}(t)$ by $\frac{\Delta
C^{\prime}(q_{\tau}^{S}(t))}{\Delta
C^{\prime}(q_{\tau+1}^{S}(t))}\delta^{-1}\epsilon$, and adjust payments
accordingly. By construction, this variation keeps the low type’s payoff
unchanged and frontloads some rent from $\tau+1$ to $\tau$ (thus relaxing (NR)
at $\tau+1$). Moreover, the change of surplus is given by

$\displaystyle(1-\delta)\epsilon\left[\pi_{H}^{\prime}(q_{\tau}^{S}(t))-\pi_{H}^{\prime}(q_{\tau+1}^{S}(t))\frac{\Delta
C^{\prime}(q_{\tau}^{S}(t))}{\Delta C^{\prime}(q_{\tau+1}^{S}(t))}\right],$

which is positive because i) $q_{L}^{*}\leq q_{\tau}^{S}(t)<q_{\tau+1}^{S}(t)$
and $\pi_{H}$ is strictly concave so that
$0\geq\pi_{H}^{\prime}(q_{\tau}^{S}(t))>\pi_{H}^{\prime}(q_{\tau+1}^{S}(t))$;
and ii) $\Delta C$ is concave so that $\frac{\Delta
C^{\prime}(q_{\tau}^{S}(t))}{\Delta C^{\prime}(q_{\tau+1}^{S}(t))}\geq 1$. So
this is a profitable deviation.

  * •

Suppose toward a contradiction that $q_{\tau}^{S}(t)=q_{\tau+1}^{S}(t)$. We
can apply the same variation as above, which keeps the agents’ payoffs and has
no first-order effect on surplus this time. But it still strictly relaxes (NR)
at $t+1$, thus a further variation based on Claim 8’s proof will render a
profitable deviation.

Therefore, $x_{\tau}^{S}(t)$ must be strictly decreasing in $\tau$ over the
debt periods and stay constant at $q_{L}^{*}$ after the debt periods. This
establishes Lemma 4. ∎

###### Proof of Theorem 2.

This theorem follows directly from Lemmas 1 through 4. ∎

###  A.3 Proof of Theorem 3 and Proposition 1

###### Proof of Part 1 of Theorem 3.

Suppose that (8) holds for all $\tilde{q}_{L}\in[q_{L}^{*},\bar{q}]$, that is,

$\displaystyle\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})<\ln\left(\frac{\Delta
C(\tilde{q}_{L})}{\Delta C(\tilde{q}_{L})-\Delta
C({q}_{H}^{*})}\right)\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{L})\right].$

Let us first prove that the best pooling mechanism is optimal when $\delta$ is
sufficiently close to $1$. Suppose (toward a contradiction) that for any
$\epsilon>0$, there is some $\delta>1-\epsilon$ such that every optimal
mechanism is revealing. We will construct a $\tilde{q}$ that violates the
above inequality.

Let us first fix a small $\epsilon$ and such a $\delta>1-\epsilon$. Take any
optimal mechanism. For every $t$, let $\\{z(t),...,t-1\\}$ be the set of
periods that the principal is still in debt with the agents revealing in these
periods. By Lemma 1, if an agent reveals in $\tau\leq t$, the principal’s
(flow) payoff from this agent in period $t$ is

$\pi_{t}^{S}(\tau)=\begin{cases}\pi_{H}(q_{t}^{S}(\tau)),&\text{ if
}x_{t}^{S}(\tau)=C_{H}(q_{t}^{S}(\tau))\\\
\pi_{H}(q_{t}^{S}(\tau))+(1-\beta)\Delta C(q_{t}^{S}(\tau)),&\text{ if
}x_{t}^{S}(\tau)=C_{L}(q_{t}^{S}(\tau))+\beta\Delta
C(q_{t}^{S}(\tau))\end{cases}.$

By Claim 8, the principal is indifferent between reneging at $t$ or reneging
at $t+1$. Following the analysis in the proof of Claim 9, the principal’s weak
preference for reneging at $t+1$ relative to reneging at $t$ tells us that

$\displaystyle
r_{t}[(1-\delta)\pi_{H}(q_{t}^{S}(t))+\delta\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})]$
$\displaystyle\geq$
$\displaystyle(1-\delta)\sum_{\tau=z(t)}^{t-1}r_{\tau}[\pi_{L}(q_{L}^{*})-\pi_{t}^{S}(\tau)]+(1-R_{t+1})(1-\delta)[\pi_{H}(q_{H}^{*})-\pi_{H}(q_{t}^{P})].$
(13)

Because $\pi_{H}(q_{H}^{*})\geq\pi_{H}(q_{t}^{S}(t))$ and
$\pi_{H}(q_{H}^{*})\geq\pi_{H}(q_{t}^{P})$, the above inequality implies that

$\displaystyle r_{t}\delta[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})]$
$\displaystyle\geq(1-\delta)\sum_{\tau=z(t)}^{t-1}r_{\tau}[\pi_{L}(q_{L}^{*})-\pi_{t}^{S}(\tau)].$

Summing from any $\tilde{t}$ onward, we have

$\displaystyle\sum_{t=\tilde{t}}^{\infty}r_{t}\delta[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})]$
$\displaystyle\geq(1-\delta)\sum_{t=\tilde{t}}^{\infty}\sum_{\tau=z(t)}^{t-1}r_{\tau}[\pi_{L}(q_{L}^{*})-\pi_{t}^{S}(\tau)]$
$\displaystyle\geq(1-\delta)\sum_{t=\tilde{t}}^{\infty}r_{t}\sum_{\tau=t+1}^{T(t)}\left[\pi_{L}(q_{L}^{*})-\pi_{\tau}^{S}(t)\right]$
$\displaystyle\geq(1-\delta)\sum_{t=\tilde{t}}^{\infty}r_{t}\sum_{\tau=t+1}^{T(t)-1}\left[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{\tau}^{S}(t))\right].$
(14)

The second line follows from changing the order of summation, dropping the
terms from periods before $\tilde{t}$, and letting $T(t)$ be the final debt
period for the agents revealing at $t$; the third line follows from the fact
$\pi_{\tau}^{S}(t)=\pi_{H}(q_{\tau}^{S}(t))$ in all debt periods except the
last one $T(t)$.

To further simply the RHS of (14), recall that the low types are indifferent
between revealing or not in every period (Claim 3). And because
$\lim_{t\to\infty}q_{t}^{P}=q_{H}^{*}$ (Claim 10), we know that for $t$
sufficiently large, we have

$(1-\delta)\sum_{\tau=t+1}^{T(t)-1}\delta^{\tau-t}\Delta
C(q_{\tau}^{S}(t))=\Delta C(q_{H}^{*})+O(\epsilon).$

Let $\tilde{q}_{t}\in[q_{L}^{*},\bar{q}]$ be such that

$(1-\delta)\sum_{\tau=t+1}^{T(t)-1}\delta^{\tau-t}\Delta
C(\tilde{q}_{t})=(1-\delta)\sum_{\tau=t+1}^{T(t)-1}\delta^{\tau-t}\Delta
C(q_{\tau}^{S}(t)).$

Because $\pi_{H}(\cdot)$ and $\Delta C(\cdot)$ are concave, we have

$\sum_{\tau=t+1}^{T(t)-1}\pi_{H}(\tilde{q}_{t})\geq\sum_{\tau=t+1}^{T(t)-1}\pi_{H}(q_{\tau}^{S}(t)).$

Thus (14) now becomes

$\displaystyle\sum_{t=\tilde{t}}^{\infty}r_{t}\delta[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})]$
$\displaystyle\geq(1-\delta)\sum_{t=\tilde{t}}^{\infty}r_{t}\sum_{\tau=t+1}^{T(t)-1}\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{t})\right]$
$\displaystyle=(1-\delta)\sum_{t=\tilde{t}}^{\infty}r_{t}[T(t)-t-1]\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{t})\right];$

that is,

$\sum_{t=\tilde{t}}^{\infty}r_{t}\left\\{\delta[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})]-(1-\delta)[T(t)-t-1]\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{t})\right]\right\\}\geq
0.$

So there must exist some $t$ and $\tilde{q}_{t}$ such that

$\delta[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})]\geq(1-\delta)[T(t)-t-1]\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q}_{t})\right],$
(15)

and

$(1-\delta)\sum_{\tau=t+1}^{T(t)-1}\delta^{\tau-t}\Delta
C(\tilde{q}_{t})=\Delta C(q_{H}^{*})+O(\epsilon).$ (16)

Now let us send $\epsilon$ to $0$, which also pushes $\delta$ to $1$ (remember
$\delta>1-\epsilon$). Taking a subsequence if necessary, let
$\tilde{q}\mathrel{\mathop{\mathchar 58\relax}}=\lim_{\epsilon\to 0}\
\tilde{q}_{t}$. By (16), we have

$\lim_{\epsilon\to 0}\ \delta^{T(t)-t-1}=\frac{\Delta C(\tilde{q})-\Delta
C(q_{H}^{*})}{\Delta C(\tilde{q})},$

which implies

$\lim_{\epsilon\to 0}\ -\ln\delta\left[T(t)-t-1\right]=\ln\left(\frac{\Delta
C(\tilde{q})}{\Delta C(\tilde{q})-\Delta C(q_{H}^{*})}\right).$

Note also that $\lim_{\epsilon\to 0}\frac{1-\delta}{-\ln\delta}=1$, so taking
the limit on both sides on (15) renders

$\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})\geq\ln\left(\frac{\Delta
C(\tilde{q})}{\Delta C(\tilde{q})-\Delta
C(q_{H}^{*})}\right)\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q})\right].$

Hence we have found a $\tilde{q}\in[q_{L}^{*},\bar{q}]$ that violates (8), a
contradiction.

Lastly, we argue that no optimal mechanism can be revealing when $\delta$ is
sufficiently large. Suppose (toward a contradiction) that there always is a
revealing mechanism that generates a profit of $\pi_{H}(q_{H}^{*})$ as
$\delta$ gets close to $1$. When proving Theorem 2, the presumption that
$\Pi_{0}>\pi_{H}(q_{H}^{*})$ is only used to show that $r_{0}>0$ in every
optimal mechanism. Even if $\Pi_{0}=\pi_{H}(q_{H}^{*})$, we can still find an
optimal mechanism with $r_{0}>0$,333333This mechanism can be constructed by
skipping the first several periods without information revelation. thus this
optimal mechanism satisfies all properties in Theorem 2 and Lemmas 1 through
4. But then, one can apply exactly the same argument prior to this paragraph
to reach the same contradiction, which establishes uniqueness in this case. ∎

###### Proof of Part 2 of Theorem 3.

Suppose that (9) holds for some $\tilde{q}_{L}\in[q_{L}^{*},\bar{q}]$. We will
construct an implementable allocation with information revelation that
generates a profit strictly greater than $\pi_{H}(q_{H}^{*})$.

Let $\tilde{q}\in[q_{L}^{*},\bar{q}]$ be such that

$\displaystyle\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})>\ln\left(\frac{\Delta
C(\tilde{q})}{\Delta C(\tilde{q})-\Delta
C({q}_{H}^{*})}\right)\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q})\right]+\epsilon,$
(17)

for some $\epsilon>0$. Consider the following allocation: For every $t$,

$\displaystyle q_{t}^{P}$ $\displaystyle=q_{H}^{*},$ $\displaystyle x_{t}^{P}$
$\displaystyle=C_{H}(q_{H}^{*}),$ $\displaystyle q_{\tau}^{S}(t)$
$\displaystyle=\begin{cases}\tilde{q},&\text{ if }t\leq\tau\leq t+T\\\
q_{L}^{*},&\text{ if }\tau>t+T\end{cases}$ $\displaystyle x_{\tau}^{S}(t)$
$\displaystyle=\begin{cases}C_{H}(\tilde{q}),&\text{ if }t\leq\tau<t+T\\\
C_{L}(\tilde{q})+\beta\Delta C(\tilde{q}),&\text{ if }\tau=t+T\\\
C_{L}(q_{L}^{*}),&\text{ if }\tau>t+T\end{cases}$ $\displaystyle r_{t}$
$\displaystyle=r_{0}\rho^{t},$

where $\rho$ is less than but close to $1$, $r_{0}=(1-\rho)\alpha_{0}$, $T$ is
the smallest integer that satisfies
$(1-\delta)\sum_{\tau=t}^{t+T}\delta^{\tau-t}\Delta C(\tilde{q})\geq\Delta
C(q_{H}^{*})$, and $\beta\in(0,1]$ is such that

$(1-\delta)\sum_{\tau=t}^{t+T-1}\delta^{\tau-t}\Delta
C(\tilde{q})+(1-\delta)\delta^{T}\beta\Delta C(\tilde{q})=\Delta
C(q_{H}^{*}).$ (18)

It is easy to verify that the constructed allocation satisfies (IR-H), (IR-L),
(IC-H), (2.3). We now show that when $\delta$ is sufficiently large, we can
always find some $\rho$ that keeps (NR) satisfied, so that the constructed
allocation is implementable. It suffices to ensure that in each period $t$,
the principal weakly prefers waiting one more period to renege than reneging
right away. Adapting condition (A.3) to the constructed allocation, we need

$\displaystyle
r_{t}[(1-\delta)\pi_{H}(\tilde{q})+\delta\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})]$
$\displaystyle\geq$
$\displaystyle(1-\delta)\sum_{\tau=t-T}^{t-1}r_{t}\rho^{\tau-t}[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q})]+(1-\delta)r_{t}\rho^{-T}(1-\beta)\Delta
C(\tilde{q}),$

or equivalently, canceling $r_{t}$ on both sides,

$\displaystyle(1-\delta)[\pi_{H}(\tilde{q})-\pi_{H}(q_{H}^{*})]+\delta[\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})]$
$\displaystyle\geq$
$\displaystyle(1-\delta)\sum_{\tau=t-T}^{t-1}\rho^{\tau-t}[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q})]+(1-\delta)\rho^{-T}(1-\beta)\Delta
C(\tilde{q}).$ (19)

When $\delta$ and $\rho$ both go to $1$, the LHS of (A.3) converges to
$\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})$, and the RHS of (A.3) converges to
$\ln\left(\frac{\Delta C(\tilde{q})}{\Delta C(\tilde{q})-\Delta
C({q}_{H}^{*})}\right)\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q})\right]$.343434Note
that $T$ is determined by (18), so it is independent of $\rho$. The limit of
the RHS of (A.3) is taken by first sending $\rho$ to $1$ for each given
$\delta$, rendering
$(1-\delta)T[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q})]+(1-\delta)(1-\beta)\Delta
C(\tilde{q})$, and then sending $\delta$ to $1$, rendering
$\ln\left(\frac{\Delta C(\tilde{q})}{\Delta C(\tilde{q})-\Delta
C({q}_{H}^{*})}\right)\left[\pi_{L}(q_{L}^{*})-\pi_{H}(\tilde{q})\right]$. But
because $\tilde{q}$ satisfies (17), this means that for each sufficiently
large $\delta$, one can find $\rho$ that makes (A.3) hold, thus (NR) is
satisfied.

We conclude the proof by showing that the principal’s profit from the
constructed allocation is strictly greater than $\pi_{H}(q_{H}^{*})$. Because
(NR) is satisfied, the principal’s total time-$0$ profit is weakly greater
than that from reneging at $t=1$, that is:

$\Pi_{0}\geq(1-r_{0})\pi_{H}(q_{H}^{*})+r_{0}\left[(1-\delta)\pi_{H}(\bar{q})+\delta\pi_{L}(q_{L}^{*})\right]>\pi_{H}(q_{H}^{*}),$

where the last inequality holds as long as
$\delta>\frac{\pi_{H}(q_{H}^{*})-\pi_{H}(\bar{q})}{\pi_{L}(q_{L}^{*})-\pi_{H}(\bar{q})}$.
Therefore, every optimal mechanism is revealing whenever $\delta$ is
sufficiently close to $1$. ∎

###### Proof of Proposition 1.

Let us first ignore the upper bound $\bar{q}$ on quantity. We first show that
there exists $\theta_{1}$ such that if $\theta_{L}<\theta_{1}$ then (9) is
satisfied by setting $\tilde{q}_{L}=q_{L}^{*}$. Notice that $q_{L}^{*}$ goes
to $\infty$ as $\theta_{L}$ goes to $0$. So we have

$\displaystyle\lim_{\theta_{L}\to 0}\
\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})-\ln\left(\frac{q_{L}^{*}}{q_{L}^{*}-q_{H}^{*}}\right)\left[\pi_{L}(q_{L}^{*})-\pi_{H}({q}_{L}^{*})\right]$
$\displaystyle=$ $\displaystyle\lim_{q_{L}^{*}\to\infty}\
\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})-\ln\left(\frac{q_{L}^{*}}{q_{L}^{*}-q_{H}^{*}}\right)\theta_{H}q_{L}^{*}$
$\displaystyle=$ $\displaystyle\lim_{q_{L}^{*}\to\infty}\
\pi_{L}(q_{L}^{*})-\pi_{H}(q_{H}^{*})-\theta_{H}q_{H}^{*}$ $\displaystyle=$
$\displaystyle\lim_{q_{L}^{*}\to\infty}\ \pi_{L}(q_{L}^{*})-v(q_{H}^{*})$
$\displaystyle=$ $\displaystyle\lim_{q_{L}^{*}\to\infty}\
v(q_{L}^{*})-v(q_{H}^{*})$ $\displaystyle>$ $\displaystyle 0.$

Therefore, there exists $\theta_{1}$ such that if $\theta_{L}<\theta_{1}$,
then (9) is satisfied by setting $\tilde{q}_{L}=q_{L}^{*}$. Fix any such
$\theta_{1}$. Let us assume that $\bar{q}>q_{L}^{*}(\theta_{1})$ and let
$\underline{\theta}$ be such that $q_{L}^{*}(\underline{\theta})=\bar{q}$. The
previous analysis implies that for every
$\theta_{L}\in(\underline{\theta},\theta_{1})$, (9) is satisfied by some
$\tilde{q}_{L}\in[q_{L}^{*},\bar{q}]$.

Next we show that there exists $\theta_{2}$ such that if
$\theta_{L}>\theta_{2}$ then (8) is satisfied by all
$\tilde{q}_{L}\in[q_{L}^{*},\bar{q}]$. Assume toward a contradiction that one
can find a sequence $\theta_{L}^{n}\to\theta_{H}$ and a sequence
$\tilde{q}_{L}^{n}\in[q_{L}^{*n},\bar{q}]$ such that

$\pi_{L}(q_{L}^{*n})-\pi_{H}(q_{H}^{*})\geq\ln\left(\frac{\tilde{q}_{L}^{n}}{\tilde{q}_{L}^{n}-{q}_{H}^{*}}\right)\left[\pi_{L}(q_{L}^{*n})-\pi_{H}(\tilde{q}_{L}^{n})\right],\
\forall n.$

Because $q_{L}^{n}\to q_{H}^{*}$, the LHS of the above inequality converges to
$0$. And because
$\ln\left(\frac{\tilde{q}_{L}^{n}}{\tilde{q}_{L}^{n}-{q}_{H}^{*}}\right)$ is
bounded below by $\ln\left(\frac{\bar{q}}{\bar{q}-{q}_{H}^{*}}\right)$, for
the inequality to hold we need $\tilde{q}_{L}^{n}\to q_{H}^{*}$. This implies
that

$\lim_{n}\
\ln\left(\frac{\tilde{q}_{L}^{n}}{\tilde{q}_{L}^{n}-{q}_{H}^{*}}\right)=\infty.$

On the other hand, because
$\pi_{L}(q_{L}^{*n})-\pi_{H}(q_{H}^{*})<\pi_{L}(q_{L}^{*n})-\pi_{H}(\tilde{q}_{L}^{n})$
for every $n$, we must have

$\limsup_{n}\
\ln\left(\frac{\tilde{q}_{L}^{n}}{\tilde{q}_{L}^{n}-{q}_{H}^{*}}\right)\leq
1,$

a contradiction.

Define

$\bar{\theta}\mathrel{\mathop{\mathchar 58\relax}}=\inf\left\\{\theta\ |\
\text{if $\theta_{L}>\theta$, \eqref{eq:nsconditionnonrevealing} is satisfied
by all }\tilde{q}_{L}\in[q_{L}^{*},\bar{q}]\right\\}.$

The previous analysis implies that $\bar{\theta}$ is well-defined and
satisfies $\theta_{1}\leq\bar{\theta}\leq\theta_{2}$. By Part 1 of Theorem 3,
if $\theta_{L}>\bar{\theta}$, then the optimal mechanism is non-revealing
whenever $\delta$ is sufficiently close to $1$.

Now fix any $\theta_{L}<\bar{\theta}$. Define

$\tilde{q}(\theta)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname*{arg\,min}_{\tilde{q}\in[q_{L}^{*}(\theta),\bar{q}]}\
\ln\left(\frac{\tilde{q}}{\tilde{q}-{q}_{H}^{*}}\right)\left[\pi_{L}(q_{L}^{*}({\theta}))-\pi_{H}(\tilde{q})\right].$

Let us argue that $\tilde{q}(\theta)>q_{L}^{*}(\theta)$ for every $\theta$.
Note that

$\displaystyle\frac{d}{d\tilde{q}}\left.\ln\left(\frac{\tilde{q}}{\tilde{q}-{q}_{H}^{*}}\right)\left[\pi_{L}(q_{L}^{*}({\theta}))-\pi_{H}(\tilde{q})\right]\right|_{\tilde{q}=q_{L}^{*}(\theta)}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{q_{L}^{*}(\theta)}-\frac{1}{q_{L}^{*}(\theta)-q_{H}^{*}}\right)(\theta_{H}-\theta_{L})q_{L}^{*}(\theta)+\ln\left(\frac{q_{L}^{*}(\theta)}{q_{L}^{*}(\theta)-{q}_{H}^{*}}\right)(\theta_{H}-\theta_{L})$
$\displaystyle=$
$\displaystyle\left[\ln\left(\frac{q_{L}^{*}(\theta)}{q_{L}^{*}(\theta)-{q}_{H}^{*}}\right)-\frac{q_{H}^{*}}{q_{L}^{*}(\theta)-q_{H}^{*}}\right](\theta_{H}-\theta_{L})$
$\displaystyle<$ $\displaystyle 0$

where the first equality follows from
$\pi_{H}(q_{L}^{*}(\theta))=\pi_{L}(q_{L}^{*}(\theta))-(\theta_{H}-\theta_{L})q_{L}^{*}(\theta)$
and $\pi_{L}^{\prime}(q_{L}^{*}(\theta))=0$, and the inequality follows
because $\ln(1+x)<x$ for all $x>0$. Hence $\tilde{q}=q_{L}^{*}(\theta)$ cannot
be the minimizer of the objective function. Next define

$G(\theta)\mathrel{\mathop{\mathchar
58\relax}}=\pi_{L}(q_{L}^{*}({\theta}))-\pi_{H}(q_{H}^{*})-\ln\left(\frac{\tilde{q}({\theta})}{\tilde{q}({\theta})-{q}_{H}^{*}}\right)\left[\pi_{L}(q_{L}^{*}({\theta}))-\pi_{H}(\tilde{q}({\theta}))\right].$

Note that

$\displaystyle G^{\prime}(\theta)$
$\displaystyle=-\left[1-\ln\left(\frac{\tilde{q}({\theta})}{\tilde{q}({\theta})-{q}_{H}^{*}}\right)\right]q_{L}^{*}(\theta)-\frac{d}{d\tilde{q}}\left[\ln\left(\frac{\tilde{q}}{\tilde{q}-{q}_{H}^{*}}\right)\left[\pi_{L}(q_{L}^{*}({\theta}))-\pi_{H}(\tilde{q})\right]\right]\frac{d\tilde{q}(\theta)}{d\theta}$
$\displaystyle=-\left[1-\ln\left(\frac{\tilde{q}({\theta})}{\tilde{q}({\theta})-{q}_{H}^{*}}\right)\right]q_{L}^{*}(\theta),$

where the second line follows from the envelope theorem which is applicable
because $\tilde{q}(\theta)>q_{L}^{*}(\theta)$. By definition of
$\bar{\theta}$, there exists $\tilde{\theta}\in(\theta_{L},\bar{\theta}]$ such
that $G(\tilde{\theta})\geq 0$. Remember $G(\tilde{\theta})\geq 0$ implies
$\ln\left(\frac{\tilde{q}(\tilde{\theta})}{\tilde{q}(\tilde{\theta})-{q}_{H}^{*}}\right)<1$,
so we have

$G^{\prime}(\tilde{\theta})<0.$

This argument can then be used to show that $G^{\prime}(\theta)<0$ for all
$\theta<\tilde{\theta}$, thus $G(\theta_{L})>G(\tilde{\theta})\geq 0$. Because
$\theta_{L}$ is arbitrary, applying Part 2 of Theorem 3, we conclude that if
$\theta_{L}<\bar{\theta}$, then every optimal mechanism is revealing whenever
$\delta$ is sufficiently close to $1$. ∎