Contract Negotiation and Screening with Persistent Information∗
Bruno Strulovici
Northwestern University
January 12, 2017

1

Introduction

Whether it concerns financial bailouts, tax schemes, social security regulations, labor contracts,
debt refinancing, or international agreements, renegotiation of social and corporate agreements is
a pervasive aspect of economic activity. Modeling renegotiation is challenging, however, and many
economists have focused their analysis on renegotiation-proof contracts which are, loosely speaking,contracts which cannot be Pareto improved following any history, and which are aimed to capture
the main effects of renegotiation.1
While they offer a parsimonious approach to capture renegotiation, renegotiation-proof contracts raise conceptual issues, particularly when some parties hold private information. First,
renegotiation-proofness is generally not an intrinsic property of contracts.2 Instead, it depends on
the negotiating parties’ beliefs and on the set of contracts which the parties view as credible alternatives to the current contract. The expression “renegotiation-proof contract” is thus misleading,because the qualification really concerns pairs of contracts and beliefs. To illustrate this point,
suppose that a principal, P, has signed a contract with an agent whose type, L or H, is private, andthat P is given an opportunity to renegotiate the contract. Also assume that the initial contract,
C, is efficient conditional on facing L, but inefficient if P is facing H. In this situation, P should
renegotiate the contract only if he assigns a high enough probability to facing H. In some cases∗

I am particularly grateful to Alessandro Pavan for numerous conversations on the paper’s topic. I also benefited

from comments by Faruk Gul, Larry Samuelson, Zhiguo He, and seminar audiences at Princeton University, theUniversity of Rochester, UCSD, Caltech, Oxford, PSE, EPFL, the LSE, the University of Essex, and the 2013 North
American Winter Meetings of the Econometric Society. This research is supported by an NSF Grant (No. 1151410) anda fellowship from the Alfred P. Sloan Foundation. The hospitality of Princeton University’s Economics Department,
where part of this paper was developed, is gratefully acknowledged.
1
Strulovici (2017) provides an explicit foundation for such contracts for environments, like the one of this paper,in which one of the party has private information and the other party has the bargaining power.
2
See, e.g., Hart and Tirole (1988) and, more recently, Maestri (2015).

(see, e.g., Hart and Tirole (1988)), the principal may prefer to abstain from completely learning
the agent’s type to avoid subsequent commitment problems, and some inefficient contracts can berenegotiation-proof.
These observations raise another issue with renegotiation-proofness, which may be best understood with three periods. Whether a contract is renegotiation-proof in the second period depends
on P’s belief in that period, as was just discussed. This belief, in turn, depends on the agent’s strategy in period 1. For example, suppose that L and H both choose some contract C in period 1 withpositive probability. Whether C’s continuation in period 2 (and, hence, C itself) is renegotiationproof depends on P’s belief in period 2 which, by Bayesian updating, depends on the probabilities
with which each type of the agent have chosen C. Therefore, whether C is renegotiation-proof
depends not only on P’s initial belief about the agent, but also on the agent’s strategy in period
1, which itself depends on the continuation of C: renegotiation-proofness is an equilibrium concept,
even when the horizon is finite.
Offsetting this complexity, renegotiation-proofness also implies a form of interim efficiency thatcan significantly simplify the concept. Indeed, because contracts that are efficient for some type of
the agent are typically inefficient for other types, renegotiation generates a separating force. When
this force is sufficiently strong, it can entirely resolve the issues mentioned earlier. Suppose, forexample, that contract efficiency in the last period guarantees that the agent’s types are always
separated in that period. Viewed from the penultimate period, regardless of the agent’s strategy
and the principal’s current belief, the agent’s types will be separated in the next period even,for that matter, if his type is persistent. This property reduces the set of contracts that need
to be considered in the penultimate period, and may be used to show that the efficient contracts
among those must also be separating in that period. In many cases, analyzed in later sections,an induction argument may be used that renegotiation-proof must always be separating. Thus, bypushing towards efficiency at all periods, renegotiation-proofness can force the principal to learn
the type of the agent. This is the separating power of renegotiation.
With more periods, characterizing the set of renegotiation-proof contracts can be particularly
challenging. To be renegotiation-proof, a contract must be interim efficient, i.e., sustain the comparison with alternative contracts whose current-period specification and continuations may bothdiffer from the current contract and, crucially, whose continuations have to be renegotiation-proof.
When assessing the interim efficiency of a given contract, the principal must thus consider perturbations of the contract that preserve renegotiation-proofness of the continuation contract. To
address this difficulty, the paper introduces a concept of RP-preserving perturbations. The paper
will provide a general specification in which these perturbations constitute a group, in the algebraic
sense of the term. If one can characterize this group through its multiplication, as in the several

2

examples described here, then can exploit interim efficiency to show that RP contracts must be
separating.

2

Setting: Contracting with Persistent Private Information

We consider a dynamic principal-agent contracting model between a risk neutral principal and arisk-averse agent. There are T < ∞ periods.3 The private type θt of the agent lies in some typespace Θ ⊂ R and evolves according to some Markov process.
Given a stochastic consumption stream {yt }t≤T , the agent receives an expected utility" T#Xu(θt , yt )U (y) = E
t=1

where the period utility u is defined on Θ × Y and Y is an interval of R. For expositional simplicity,
the discount rate is set to zero for both parties.4
The agent receives his consumption stream from the principal, whose objective is to minimize
the expected cost
" T#
X
Q(y) = Eyt ,t=1

subject to providing the agent with a lifetime expected utility greater than or equal to
U (y) ≥ w.
The principal’s marginal cost is thus assumed to be constant across periods and normalized to 1.5
A contract is defined recursively as follows: in period T , a contract is simply a transfer yT ∈ Y.
Let CT denote the set of such contracts. In period t < T , a contract consists of a transfer yt forthat period and of a finite menu Mt+1 ⊂ Ct+1 of continuation contracts at period t + 1. The set of
period-t contracts is denoted Ct .
The timing of the game is as follows:• In period 1, the principal proposes an initial menu M1 of contracts in C1 .
• The agent learns his type θ1 and picks a contract C1 = (y1 , M2 ) from M1 .
• The principal gives y1 to the agent, and the game moves to period 2.
3

Focusing on a finite horizon gets rid of conceptual difficulties that arise with the definition of renegotiation-proof

contracts.
4
The analysis easily extends to a common positive discount rate for the principal and the agent.
5
The analysis can be extended to the case of arbitrary deterministic time-varying cost functions. Furthermore,
Atkeson and Lucas (1992) observe that the general equilibrium price process obtained to decentralize the central
planner’s problem is typically constant for the simple parametric specifications used in the present paper.

3• The agent learns his type θ2 for that period, and picks an item C2 = (y2 , M3 ) from M2• y2 is paid, etc.
The distinction between the terms “menu” and “contract” is for expositional clarity. A “menu”
really corresponds to a contract at the beginning of some period, whereas the term “contract” isreserved for end-of-period contracts, after the agent has made his choice.
For simplicity, both the agent and the principal use pure strategies. This restriction is without
loss of generality for the examples considered in this paper.6
A menu M1 is feasible if there is a choice strategy for the agent that yields an expected utility
greater than or equal to the promised utility w, given the principal’s prior over the agent’s initial
type θ1 .
A menu M1 induces a sequence of choices by the agent. Let Ht (M1 ) denote the set of possible
choices induced M1 until period t − 1, and denote by ht a typical element (“history”) of Ht (M1 ).
Regardless of the agent’s types in those periods, a given history ht implies a menu Mt in period t,
as specified by M1 .
A menu M1 is separating if there exists an optimal strategy for the agent such that i) for any
period t and choice history ht , each type θt chooses a distinct item from the menu Mt implied by
ht , resulting in a transfer yt that depends on the agent’s type.
That is, given the menu and the agent’s best response, with probability one the payment thatthe agent receives in any period t < T always depends on his current type.7
Let ∆(Θ) denote the set of probability distributions over Θ. For any t ≤ T and contract C ∈ Ct ,let Qδ (C) and Uθ (C) denote the expected cost and expected utility corresponding to C from periodt to period T , when the principal’s belief about θt is δ ∈ ∆(Θ) and the type of the agent is θ.
Because of persistence, the expected cost of a particular contract typically depends on thecurrent type of the agent. Indeed, the agent’s current type affect the distribution of his future
type, which determines which contracts he will choose from future menus, and hence how costly agiven menu will turn out to be for the principal. This point is developed in detail in later sections.
6

In these examples, renegotiation-proof contracts are shown to be either fully separating or fully pooling. The

definitions below easily extend to allow mixing by the agent. When the agent’s utility is multiplicatively separable inθ and y, as in this paper, ruling out randomization by the principal is without loss because the principal can always
reduce his cost by replacing lotteries by certainty equivalents without affecting incentives.
7
This definition allows the agent to be indifferent between different items in the menu, but requires him to play aseparating strategy in equilibrium. When indifferent between several strategies, the agent is assumed to choose theone that minimizes the principal’s cost.

4

2.1

Renegotiation-Proof Contracts

We now define the concept of renegotiation-proof contracts. Whether a contract is renegotiationproof generally depends on the principal’s belief about the current type of the agent. It is intuitive,
for example, that a contract that is efficient for some type θ but not for some other type θ̂ should
be renegotiation-proof if the principal assigns probability 1 to θ, but not if he assigns probability 1
to θ̂. For each t, Rt ⊂ Ct × ∆ will denote the set of period-t contract-belief pairs (C, δ) such that
C is renegotiation-proof given belief δ.
Because the horizon is finite, renegotiation-proof contracts can be defined by backward induction
on t. In period T , all contracts yT ∈ Y are declared renegotiation-proof, for any belief δ. Intuitively,
there is nothing to renegotiate in period T , since any transfer yT is Pareto efficient. Thus, RT =CT × ∆. Similarly, all menus M ⊂ Y in period T are declared renegotiation-proof for any belief δ.
If δ denotes the distribution of θt , let δ̂ denote the distribution of θt+1 , obtained by the transition
probabilities of the stochastic process {θt }Tt=1 .
A period-t contract C = (yt , Mt+1 ) is said to have an RP-continuation given δ if the menu Mt+1
is renegotiation proof given δ̂.
The definition of renegotiation-proof contracts for periods t < T involves two components. First,
the continuation menu of a contract must be renegotiation-proof, given posterior beliefs. Second,the contract should not be improvable by a menu of contracts whose continuations are themselves
renegotiation-proof.
Definition 1 For t < T , a period-t contract C ∈ Ct is renegotiation-proof given belief δ ∈ ∆(Θ)
if there does not exist a partition (Θ1 , . . . , Θm ) of Θ(δ) and contracts {Ci ∈ Ct }mi=1 such thati) Each θ ∈ Θi weakly prefers Ci to C and to Cj for all j:Uθ (Ci ) ≥ max{Uθ (C), max{Uθ (Cj )}}.
jii) The principal’s expected cost is reduced by this proposal:mXδ(Θi )Qδi (Ci ) < Qδ (C),

(1)

i=1

where δi is the conditional distribution δ(·|Θi ) = δ(·)/δ(Θi ) defined over Θi .
iii) For each i, Ci has an RP continuation given δi .
To close this inductive definition of renegotiation-proofness, we need to define renegotiationproof menus.

5Definition 2 A period-t menu Mt = {Ci } is renegotiation-proof if there is an optimal choicestrategy σ A for the agent over {t, . . . , T } given Mt such that the belief δi generated under thatstrategy by the choice Ci is such that Ci is renegotiation-proof given δi , for all Ci ’s that are chosenwith positive probability under σ A .8
While the formal definition of renegotiation-proofness is somewhat complicated, there are twosituations in which a contract is easily shown not to be renegotiation-proof.
Definition 3 Given a period t, contract C ∈ Ct , and belief δ, say that Ĉ is a Pareto improvement over C given δ, if for each θ ∈ Θ(δ),
Uθ (Ĉ) ≥ Uθ (C) and Qδ (Ĉ) < Qδ (C).
The following result immediately follows from the definition of renegotiation-proofness.
Proposition 1 Given a period t, contract C ∈ Ct , and belief δ, suppose that there exists Ĉ ∈ Ct
that is a Pareto improvement of C such that Ĉ is has an RP continuation given δ. Then C is notrenegotiation-proof given δ.
Proof.

To contradict renegotiation-proofness of C, take the partition with a single element

Θ1 = Θ(δ) and associated contract C1 = Ĉ. Notice that the LHS of (1) corresponds to Qδ (Ĉ). 
Another important case is when there exist separating contracts, one for each type θ in Θ(δ)
that are incentive compatible and reduce the cost of the principal. Abusing notation, let Qθ (C)denote the expected cost to the principal when the belief is degenerate, putting all mass on type θ.
Proposition 2 Given a period t, contract C ∈ Ct , and belief δ, suppose that there exist contracts{Cθ }θ∈Θ(δ) in Ct such that, for all θ ∈ ∆(θ),• Uθ (Cθ ) ≥ max{Uθ (C), Uθ (Cθ0 )}• Qθ (Cθ ) ≤ Qθ (C)• Cθ has renegotiation-proof continuations given δθwith at least one strict inequality among the cost inequalities. Then, C is not renegotiation-proof
given δ.
Proof.

Take the partition {Θθ = {θ}}θ∈Θ(δ) . To show (1), notice that each term of the LHS

is weakly smaller than the corresponding term entering the computation of Qδ , with at least one
strict inequality.
8This definition concerns only in-equilibrium choices.

6

Finally, a simple case in which a contract is renegotiation-proof is when the principal knows thetype of the agent, and the contract is Pareto efficient. Precisely, given a type θ and period t. For
any θ, let δθ denote the degenerate (Dirac) distribution on Θ such that δθ (θ) = 1 and δθ (θ0 ) = 0
for all θ0 6= θ.
Say that C ∈ Ct is θ-efficient if it has an RP continuation given δθ and there there does notexist any contract Ĉ ∈ Ct that has an RP continuation given δθ and is such that• Uθ (Ĉ) ≥ Uθ (C), and• Qθ (Ĉ) < Qθ (C).
Proposition 3 A period-t contract C ∈ Ct is renegotiation-proof given δθ if and only if it is θefficient.
Proof. With δ = δθ , the only feasible partition of Θ(δ) in Definition 1 is the partition with a single
element, equal to θ. Therefore, the definitions of renegotiation-proofness given δθ and θ-efficiency
coincide.

3Renegotiation Proofness of Continuation Contracts for Parametric Utility Functions

When trading off the cost of the current subsidy with future ones, the principal must take into
account the renegotiation proofness constraint for continuation contracts. This section specializes
the analysis to a class of parametric utility functions, for which the analysis of renegotiation-proof
contracts is more tractable.
An affine transformation group of Y is a map υ : Y × I → Y for some open interval I of R
such that υ(y, ρ) = υ1 (ρ)y + υ2 (ρ), where υ1 is strictly positive, υ1 and υ2 are strictly increasing
and differentiable, there exists ρ̄ such that υ(y, ρ̄) = y for all y, and for any ρ ∈ I, there exists ρ̃(denoted ρ−1 , the “inverse” of ρ) such that υ(υ(y, ρ), ρ̃) = y.
Condition 1 (Affine Transformation Group) There exist increasing functions a : R++ →R++ and b : R++ × Θ → R and a transformation group υ : Y × I → Y such that, for each θ ∈ Θ,y ∈ Y, and ρ ∈ I,u(θ, υ(y, ρ)) = a(ρ)u(θ, y) + b(ρ, θ).
Condition 1 is satisfied by the logarithmic, power, and exponential utility specifications:• If u(θ, y) = θy 1−γ /(1 − γ) for γ 6= 1, then u(θ, ρy) = ρ1−γ u(θ, y),7• If u(θ, y) = θ log(y), then u(θ, ρy) = u(θ, y) + θ log(ρ),• If u(θ, y) = − exp(−(θ + y)), then u(θ, y + ρ) = exp(−ρ)u(θ, y).
For any period t and continuation contract C, let C(ρ) denote the contract which has the samemenu structure as C, but provides the subsidy yτ (ρ) = υ(yτ , ρ) in period τ ≥ t whenever C provides
the subsidy yτ at that period. The transformation M (ρ) is defined analogously.
Theorem 1 (“Clockwork” Principle) Suppose that u satisfies Condition 1. Then, for any t,a period t contract C is renegotiation-proof if and only if the contract C(ρ) is renegotiation-proof,
and a period t menu M is renegotiation-proof if and only if M (ρ) is.
Theorem 1 provides a way of building credible perturbations of a given continuation contract,
and thus of analyzing meaningful tradeoffs between current and future subsidies. See Section 7.1
for a proof.

4

T Periods and Binary Types: Taste Shock Example and CostWeighted Euler Equation

This section focuses on the case of binary types: Θ = {L, H} with L < H. The type of the agent
is persistent and mean-reverting: for any t ∈ {1, . . . , T − 1},
0 < P r(θt+1 = H|θt = L) < P r(θt+1 = H|θt = H) < 1.
For tractability, the environment is further narrowed down to a taste shock model with constantrelative risk aversion:u(θ, y) = θ

y 1−γ
1−γ

or
u(θ, y) = θ log ywhich corresponds to γ = 1.
The properties of renegotiation-proof contracts are summarized here. For any history ht andtype θ, let yt (θ|ht ) denote the transfer received by type θ in period t after history ht .
Theorem 2 Any renegotiation-proof menu M is separating and has the following properties after
any history ht :• Ordered Allocation
yt (L, ht ) < yt (H, ht ).
8• Ordered Marginal Utility For any histuy (L, yt (L, ht )) < uy (H, yt (H, ht )).
• Cost-Weighted Euler Equation For all t̃ > t,uy (θt , yt (θt , ht )) = E α̂ [uy (θt̃ , yt̃ )|θt , ht ]for some transition probability distribution α̂ that assigns more weight, relative to the true
distribution α, to type realizations that entail higher costs to the principal.9• Gradual Reward/Punishment
yt+2 (θt+1 = θt+2 = L, ht ) > yt+1 (θt+1 = L, ht )yt+2 (θt+1 = θt+2 = H, ht ) < yt+1 (θt+1 = H, ht )4.1

Two Periods

To gain some intuition about the result, we start by considering the case of 2 periods. With T = 2,C1 reduces to Y 2 : a contract in period 1 simply consists of an immediate y1 and of a promisedtransfer y2 in the second and last period.
Renegotiation-proof menus are separatingWith two periods, the cost of a contract C is simply Q(C) = y1 +y2 and is therefore independentof the agent’s type. Moreover, the definition of renegotiation-proofness for that case reduces to i)
and ii) of Definition 1: iii) is always satisfied. Now consider any non-degenerate belief δ on Θ.
The problem can be visualized in the contract space Y 2 with y1 and y2 as coordinates. Isocost
curves of the principal are straight lines, and isolutility curves of the agent, where the utility is given
by Uθ (y1 , y2 ) = u(θ1 , y1 ) + E θ1 u(θ2 , y2 ), define convex upper level sets. For C to be renegotiationproof the isocost curve of the principal must be tangent to the isoutility curve of both types of theagent. Otherwise, there must be a new contract that strictly reduces the cost of the principal andis accepted by at least one type of the agent, contradicting Definition 1.
However, the single-crossing property assumed on the agent’s utility guarantees that such situation can never occur. Indeed, ex post efficiency of the low type implies thatuy (L, y1 ) = E[uy (θ2 , y2 )|L].

(2)

Because θ2 ≥ L and uy is increasing in θ and decreasing in y, this is only possible if y1 < y2 . Bya similar argument, ex post efficiency of the high type implies that y1 > y2 , showing the claimed
impossibility.
9

See Equations (9) and (10) for the definition of α̂.

9If, for example, y1 > y2 , the principal can propose a new allocation (y10 , y20 ) with y10 < y20
which provides the high type with the same expected utility at an efficient level, and reduces thecost of the principal. This shows that shows that (y1 , y2 ) is not renegotiation-proof. Thus, any
renegotiation-proof contract is separating.
This shows that any renegotiation-proof contract must put all weight on a single type of theagent. As a result, any renegotiation-proof menu must be separating.
Now consider the allocations y H = (y1H , y2H ) and y L = (y1L , y2L ) provided to each type of theagent (one may assume without loss of generality, in light of the separation, that each type picks asingle contract). Renegotiation-proofness implies that these allocations must be ex post efficient, by
Proposition 3. As argued above, this implies that y1H > y2H and that y1L < y2L . Moreover, incentive
compatibility requires that either y1H ≤ y1L and y2H ≥ y2L or vice versa, with both inequalities strict
if one is strict. Combining these observations proves that y1H > y1L and that y2H < y2L . To rank
marginal utilities, observe thatuy (y1H , H) = E H uy (y2H , θ2 )(3)

> E H uy (y2L , θ2 )(4)

≥ E L uy (y2L , θ2 )(5)

= uy (y1L , L)

(6)

where the equalities come from ex post efficiency, and the strict inequality comes from the fact thaty2H < y2L and that u is strictly concave in y.
Because the marginal utility of H is greater than for L, the IC constraint of the low type must
be binding for the optimal renegotiation-proof contract. Otherwise, the principal could reduce hiscost by increasing the consumption of H and reduce that of L.
This is intuitive: if he could abstract from incentive compatibility constraints, the principal
would always want to give more to the high type in all periods, because a high type values the good
more today and, by persistence, is more likely to value it more in the future. The only reason why
such allocation is infeasible is that it violates the IC constraint of the low type.
For CRRA utility, the set of contracts Cθ that are efficient for θ form on a straight line characterized by (2), and the IC constraint for L defines another, downward sloping, curve line linking thetwo contracts. As the promised utility w varies, the allocation (y1H , y2H , y1L , y2L ) is scaled accordingly.
In particular, the prior δ about the agent’s type, and the promised utility w do not affect theshape of the optimal renegotiation-proof menu. They only affect the scaling factor of that menu.

10

4.2

Induction for T Periods

The proof works by induction on number T of remaining period, for T ≥ 2. Precisely, we consider
the following induction hypothesis:
Induction Hypothesis Any renegotiation-proof menu M with T remaining periods has thefollowing properties.
i) The menu is separating and the contracts CH and CL chosen by each type of the agent satisfy
y1H > y1L in period 1.
ii) The marginal utilities satisfy uy (H, y1H ) > uy (L, y1L ) anduy (θ, y1θ )
1
= 2
2
θ
Uρ (θ, ρ )Cρ (θ, ρθ )for θ = {H, L}, where C 2 (θ, ρ) and U 2 (θ, ρ) respectively denote the expected cost and utility
from Period 2 onwards when the agent’s type in Period 1 is θ, as a function of the scaling
factor ρ for continuation contracts at period 2, and ρθ is the scaling factor for the period 2
continuation menu contained in Cθ .
iii) There exists a two-contract menu MT such that any optimal renegotiation-proof menu M is
proportional to MT : for any prior δ about θ1 and any promised utility w, there exists ρ ≥ 0such that M = MT (ρ).
The hypothesis holds for T = 2. Consider now the case of T + 1 ≥ 3 periods, appending aperiod “0” to T periods indexed by 1, . . . , T (this notation simplifies the application of the induction
hypothesis). Any renegotiation-proof contract consists of a quantity y0 for the current period anda pair of contracts CH , CL ∈ C1 such that Cθ is renegotiation-proof given δθ , and which satisfy
properties i) and ii) of the hypothesis.
Ruling out Pooling Contracts
Suppose that in period 0, some contract C = (ȳ0 , M1 = {CH , CL }) is renegotiation-proof given
some non-degenerate belief δ. This implies that in period 1, the agent will, irrespective of his typein period 0, choose CH in period 1 if θ1 = H, and CL otherwise, by the induction hypothesis. In
particular, the agent will not engage in multiple deviations: by the induction hypothesis, whatever
his type turns out to be in period 1, will be revealed in that period. Moreover, Theorem 1 implies
that the same will hold if the continuation menu M1 is replaced by its scaled version M1 (ρ) for any
factor ρ > 0.10
The argument for ruling out pooling contracts, provided in the appendix (Section 7.2), is basedon trading off small perturbations of ρ around ρ̄ = 1 with perturbations of y0 around ȳ0 .
10More precisely, υ1 (ρ) = ρ, υ2 (ρ) = 0, a(ρ) = ρ1−γ and b(ρ, θ) = θ log ρ1γ=1 .

11

Properties of Renegotiation-Proof Contracts
Because any renegotiation-proof menu is separating in period 0, it must consist of two contracts(y0H , M1H ) and (y0L , M1L ), respectively chosen by θ0 = H and θ0 = L. We now derive some propertiesof those contracts and verify that the induction hypothesis is satisfied with the additional periodt = 0.
First, notice that, conditional on facing type θ in period 0, the menu M1θ must be the cheapest
way of providing type θ with a given continuation utility (otherwise, the contract is not renegotiation
proof). From iii) of the induction hypothesis, this implies that M1H and M1L are scalings of the same
contract MT = (CTH , CTL ), regardless of the particular continuation utilities that these contracts
provide, and regardless of the probability distribution for θ1 that each of the two types in period 0
H
LL
generate. Therefore, M1θ = MT (ρθ ) for some ρH , ρL ≥ 0. Let cH
T = QH (CT ) and cT = QL (CT )denote the expected cost implied by menu MT conditional on facing types H and L, respectively,in period 1.
From Proposition 3, efficiency requires that the marginal utility ratio for each type of the agent
be equal to corresponding marginal cost ratio of the principal, in the (y0 , ρ) space. Otherwise, theprincipal can strictly reduce his cost and preserve the utility of the agent by perturbing the scaling
factor ρθ and y0θ of the type who violates this inequality. Replicating a computation of Section 7.2
for allocations of each type, this implies thatuy (H, y0H ) = α̂H uy (H, y1HH ) + (1 − α̂H )uy (L, y1HL )(7)uy (L, y0L ) = α̂L uy (H, y1LH ) + (1 − α̂L )uy (L, y1LL )

(8)

and

where, in the right-hand side, the first (second) superscript indicates the type of the agent inperiod 0 (period 1), and
α̂H =

αH cH
T
.
H
αH cT + (1 − αH )cL
T

(9)

α̂L =

αL cH
T
.
H
αL cT + (1 − αL )cL
T

(10)

and

Also notice that α̂H > α̂L because the function α 7→ αc1 /(αc1 + (1 − α)c2 ) is increasing in α
for all c1 , c2 > 0.
By the induction hypothesis, we have y1HH > y1HL , y1LH > y1LL , uy (H, y1HH ) > uy (L, y1HL )and uy (H, y1LH ) > uy (L, y1LL ). Equation (7) then implies that uy (H, y0H ) < uy (H, y1HH ) and, byconcavity of u, thaty0H > y1HH > y1HL .

12

Similarly, we havey0L < y1LL < y1LH .
This, along with incentive compatibility in period 0, implies that y1LH > y1HH (otherwise the hightype’s allocation would Pareto dominate the low type’s, since continuation contracts are all scalingsof one another) and that y0H > y0L .
Finally, notice thatuy (y0H , H) = α̂H uy (y1HH , H) + (1 − α̂H )uy (y1HL , L)
> α̂L uy (y1HH , H) + (1 − α̂L )uy (y1HL , L)
> α̂L uy (y1LH , H) + (1 − α̂L )uy (y1LL , L)
= uy (y0L , L),
where the equalities come from (7) and (8), the first inequality comes from the fact that α̂H > α̂Land uy (y1HH , H) > uy (y1HL , L), and the second inequality comes from the fact that y1LL > y1HL
and y1LH > y1HH (as observed earlier, these last two inequalities are equivalent, since continuation
contracts are scalings of one another, and must both hold for incentive compatibility).
Renegotiation-proofness implies that the menu M0 = ((y0H , ρH ), (y0L , ρL )) satisfies efficiency
conditions for H and L . For the menu to be optimal, the IC constraint of the low type must
be binding: since H’s marginal utility is higher than L’s, the principal could gain by transferring
resources from L to H, if it were not for L’s IC constraint. Taken together, these three conditions
imply that the optimal renegotiation-proof menu is pinned down in the (y0 , ρ) space, up to a single
parameter, which is determined by the promised expected utility w.
This shows that the shape of M0 is completely independent of the principal’s prior δ about θ0
and of the promised utility w. Those parameters only affect the scaling level of M0 , and concludes
the verification of the induction hypothesis for T + 1 periods.

5

Income Shock with Constant Absolute Risk Aversion (Arbitrary
Number of Types and Periods)

When u(θ, y) = − exp(−θ − y) (the CARA coefficient is set to 1 for simplicity, the analysis is
identical for other values of that coefficient), the relevant transformation υ(y, ρ) is the translation
υ(y, ρ) = y + ρ, with ρ ∈ R. To interpret θ as the income of the agent, we must reverse theconvention of earlier sections that marginal utility is increasing in the type of the agent. In thissection, a higher income agent has a lower marginal utility for additional consumption.
The agent’s utility is still multiplicatively separable in θ and y. To fix ideas and to covercommon specifications of the income process, we assume that the type space Θ is equal to R. The
13

analysis can be easily adapted to the case of finite or bounded intervals type spaces in R. Meanreversion is generalized as follows:exp(−θ)Definition 4 (Mean Reversion) The type of the agent is mean-reverting if E[exp(−θis2 )|θ1 =θ]increasing in θ.
As is easily checked, when there are only two types, this definition reduces to the fact that the typeof the agent can switch to the other type with positive probability.
The analysis still works by backward induction. With two periods, renegotiation-proof menus
must be separating, and the agent’s bundle (y1θ , y2θ ) is such that y1θ (resp. y2θ ) is decreasing (increasing) in the agent’s first-period income θ, the marginal utility uy (θ, y1θ ) still satisfies
uy (θ, y1θ ) = E[uy (θ2 , y2θ )|θ1 = θ],and this marginal utility is decreasing in θ.
Intuitively, this means that individuals with a low income in the first period receive more fromthe principal in that period, but not so much that their total consumption in that period exceeds
the total consumption of individuals with high incomes. This intuition continues to hold with more
periods, as established by Theorem 3 below.
Moreover, the optimal renegotiation-proof contract is entirely pinned down up to a single translation parameter, which is then adjusted to match the promised utility w, given the principal’s prior
about the agent’s type. The argument for this is proved more generally in the induction hypothesis
below and is thus omitted.
The analysis for more periods is simpler than in the CRRA case, because the marginal costCρ (θ, ρ) of increasing future consumption by the translation amount ρ is now independent of thetype of the agent and of the contract considered, as it is simply equal to the number of remaining
periods (the translation amount, ρ, being added to all yt ’s, regardless of the preceding histories).
The induction hypothesis is modified as follows:
Induction Hypothesis
With T periods to go, indexed by 1, . . . , T , the following holds:i) Any period-1 renegotiation-proof menu is separating and y1θ is decreasing in θ.
ii) The marginal utilities uy (θ, y1θ ) are decreasing in θ and define martingale processes:uy (θ, y1θ ) = E[uy (θt , yt )|θ1 = θ]iii) There exists a menu MT such that any optimal renegotiation-proof menu M is a translationof MT : for any prior δ about θ1 and any promised utility w, there exists ρ ∈ R such that
M = MT + ρ.
14

Suppose that the hypothesis holds for T , and append a period 0. The first step is to showthat renegotiation proof menus are separating in period 0. We make the following assumption(Monotone Likelihood Ratio Property, “MLRP”) on the transition probability.
Definition 5 (MLRP) i) For any state θ0 = θ, the distribution of θ1 conditional on θ0 has full
00

0

support on Θ and has a density αθ (·). ii) for any θ0 < θ00 , the ratio αθ (θ)/αθ (θ) is increasing in θ.
Now consider a contract (y0 , M1 ) that is renegotiation proof given some belief δ, and supposeby contradiction that δ is non-degenerate. By the induction hypothesis, M1 is associated with atranslation parameter ρ̄ of the reference menu MT . For all types θ choosing the same item, theequality by the utility and cost marginal rates of substitutionqUθ =uy (θ, y0 )1
= = qCVρ (θ, ρ̄)TbecomesZαθ (θ0 )gθ0 fθ0 dθ0 ,gθ f0 =
Θ

0

where gθ = exp(−θ) is decreasing in θ, f0 = − exp(−y0 ), fθ0 = − exp(−y1θ ) is increasing in θ0 , by
the induction hypothesis. This implies thatZβθ f0 =λθ (θ0 )f (θ0 )dθ0

(11)

Θ

where βθ = gθ /Dθ , λθ (θ0 ) = αθ (θ0 )gθ0 /Dθ , and Dθ =θ 0θ0Θ α (θ )gθ . Notice that λ (·) is positive andRsums up to one, for each θ: it forms a family of probability distributions on Θ.
For any θ00 > θ0 ,00

00

αθ (θ) Dθ0λθ (θ)=.
λθ0 (θ)αθ0 (θ) Dθ00
Therefore, the family {λθ }θ of density functions inherits the MLRP from the family {αθ }θ .11
R
Since βθ is decreasing in θ (by the definition of mean reversion) and Θ λθ (θ0 )f (θ0 )dθ0 is increasing in θ by the MLRP and the fact that f is increasing in its argument, one θ can satisfy
Equation (11), showing that any renegotiation-proof contract must be separating.
This means that each type θ receives a distinct contract (y0θ , ρθ ), where ρθ is the translationlevel compared to MT . Moreover, the fact that qUθ = 1/T and the induction hypothesis immediately
implies that martingale property for the marginal utility and, hence, for the period-utility of theagent.
11

It is easy to show by counter examples that the distributions λθ are not FOSD ordered even if αθ are FOSD

ordered. Assuming first-order stochastic dominance for the transition probability distributions therefore seems insufficient for the results of this section.

15

0

00

Now consider two types θ0 < θ00 . We will show that for any RP menu, we have y0θ > y0θ . For
0

00

0

00

the contracts to be incentive compatible, we must have either A) y0θ > y0θ and ρθ < ρθ , or B)
0

00

0

00

y0θ ≤ y0θ and ρθ ≥ ρθ .
Repeating the previous computations we get the equationsZ
00
θ0λθ (θ)fθθ dθ
βθ0 f0 =

(12)

Θ

and
00

βθ00 f0θ =

Z

00

00

λθ (θ)fθθ dθ

(13)

Θ
0

0

0

0

where f0θ = − exp(−y0θ ) and fθθ = − exp(−yθθ ), with similar notations for θ00 .
0

00

0

Since continuation contracts are translations of each other, we have fθθ = exp(ρθ − ρθ )fθθ
00

00

0

for all θ. Suppose by contradiction that we are in case B). In that case, exp(ρθ − ρθ ) < 1,
R
0
00
00
and the RHS of (12) is less than Θ λθ (θ)fθθ dθ. Moreover, because fθθ is increasing in θ and
0

00

00

λθ ≺M LRP λθ we conclude that the RHS of (12) is less than the RHS of (13). However, y0θ ≥ y0θ

0and θ00 > θ0 imply that the LHS of (12) is strictly greater than the LHS of (13), showing that one
the equations must be violated.
Therefore, y0θ is decreasing in θ and, by incentive compatibility, ρθ is increasing in θ.
We now turn to the ranking of marginal utilities. Let y1 (θ) denote the period 1 subsidy of typeθ under the reference contract MT , of which all continuation contracts are translations. We have

uy



00
θ , y0θ

00



Z

=
=
<
≤
=



00
00
αθ (θ)uy θ, y1 (θ) + ρθ dθ
Θ
Z


00
0
00
0
exp(ρθ − ρθ )αθ (θ)uy θ, y1 (θ) + ρθ dθ
Θ
Z


0
θ00
α (θ)uy θ, y1 (θ) + ρθ dθ
ZΘ


0
0
αθ (θ)uy θ, y1 (θ) + ρθ dθΘ0
uy θ0 , y0θ

The first and last equalities come from the Euler equation. The second equality comes fromthe translation property of continuation contracts. The strict inequality comes from the fact,00

0established earlier, that ρθ > ρθ , and the weak inequality comes from the MLRP property for the
distributions αθ and the fact that the induction hypothesis, which implies that that uy (θ, y1 (θ)) is
decreasing in θ.
There remains to show that optimal renegotiation-proof contracts are entirely pinned down, up
to a single parameter. This is shown by exploiting the efficiency condition and incentive compatibility of the contracts to derive an ordinary differential equation for the contract. This is left for
the appendix.
16

Theorem 3 With exponential utility, mean reversion, and the MLRP property, the following holds• Any renegotiation-proof contract is separating,• The standard Euler equation is satisfied,• After any common history ht , ytθ and uy (θ, ytθ ) are both decreasing in θ,• The optimal renegotiation-proof menu is entirely pinned down up to a translation parameter.
The mean reversion and MLRP conditions are trivially satisfied when the agent’s type is i.i.d.
over time, so the agent’s utility is a martingale. This immediately implies the following result.
Corollary 1 When the agent’s type is i.i.d., renegotiation-proof contracts do not cause immiserization.
More generally, the mean reversion Definition 4 is satisfied for any mean reverting AR(1) process:ifθt+1 = κθt + (1 − κ)θ̄ + εt+1
with κ ∈ (0, 1) and the shocks εt are independently distributed, thenexp(−θt )= ct exp(−(1 − κ)θt ),E exp(−θt+1 )where ct = 1/E(exp((1 − κ)θ̄ + εt+1 )). The ratio is therefore decreasing in θt .
It is also easy to check that the MLRP condition is also satisfied when the shocks εt ’s arenormally distributed.
Thus, Theorem 3 shows that renegotiation-proof contracts are separating for CARA utility
when θ follows an AR(1) process with Gaussian noise.
Corollary 2 If εt ’s are normally distributed, RP contracts are separating.

6

General Utility Functions and Type SpaceIn ProgressThis section considers more general utility functions and any finite, totally ordered type space
Θ, restricting the analysis to two periods. Following the income shock convention of Section 6, theutiltiy function u is assume to be submodular uy (θ, y) is decreasing in θ. For simplicity, attention
is restricted to lottery-free contracts.12
As before, renegotiation-proofness implies the following result.
12

Lotteries are suboptimal for multiplicatively separable utilities. It is conjectured that they are also suboptimalin the income shock case when ū exhibits decreasing absolute risk aversion.

17

Proposition 4 Any outcome (δ, y) of the renegotiation stage satisfies the following property: forall θ ∈ Θ(δ),
uy (θ1 , y1 )= 1.
E θ1 uy (θ2 , y2 )The intuition for Proposition 4 is the following: if the equality were violated, the principal could
always reduce his cost by propose a small perturbation of the bundle. The following single crossing
property reflects the intuition, present in earlier sections, that strong enough mean reversion implies
that renegotiation proof contracts are separating:Condition 2 (Single Crossing) The utility function u and type process {θt }t=1,2 satisfy the single crossing property if for any (y1 , y2 ) ∈ Y 2 , the normalized marginal rate of substitution function
θ 7→ ρ(θ) = E θ [uy (θ2 , y2 )]/uy (θ, y1 ) − 1
has the following single crossing property: there exists a threshold θ(y1 , y2 ) such that ρ(θ) ≥ (>)0
if and only if θ ≥ (>)θ(y1 , y2 ).
While Condition 2 is readily checked, it mixes properties of the type process with properties of
the utility function. Simple distinct conditions on the type process and the agent’s utility function
implying Condition 2 are provided later in the section.
Theorem 4 If u and θ satisfy the single crossing property, then any renegotiation-proof contract
is separating. Moreover, y1 (θ) is decreasing in θ, and y2 (θ) is increasing in θ.
Proof. Following any message m1 and bundle (y1 , y2 ), Condition 2 implies that at most one type
can satisfy ex post efficiency, by Proposition 4. To show monotonicity, consider two types θ < θ0 ,with respective bundles (y1 , y2 ) and (y10 , y20 ). Condition 2 implies that0uy (θ0 , y1 ) < E θ uy (θ20 , y2 )If y10 ≥ y1 , incentive compatibility implies that y20 < y2 . This implies, by concavity of u, that0uy (θ0 , y10 ) ≤ uy (θ0 , y1 ) < E θ uy (θ20 , y2 ) ≤ uy (θ0 , y20 )which violate ex post efficiency for θ0 , by Proposition 4.
To obtain an ordering of marginal utilities, the following persistence condition is assumed.
Definition 6 (Persistence) The type of the agent is persistent if the distribution of θ2 given θ
is increasing in the sense of first order stochastic dominance, i.e., for any θ̃ ∈ Θ,
P r(θ2 ≥ θ̃|θ1 = θ)is increasing in θ.
18

Theorem 5 If u and θ satisfy the single crossing and θ is persistent, then uy (y1 (θ), θ) is decreasingin θ.
Proof.

0

For any θ < θ0 ,uy (θ0 , y10 ) = E θ uy (θ2 , y20 ) < E θ uy (θ2 , y2 ) = uy (θ, y1 ), where the inequality

comes from the fact that y2 < y20 , submodularity of u, and FOSD of the distributions for θ2 .

6.1

An income-shock example with an AR(1) process

Suppose that θt is the income of the agent at time t and that yt is a subsidy (or a tax, depending
on its sign) from the principal to the agent. Also suppose that u(y, θ) = log(y + θ) or that1−γu(y, θ) = (y+θ)
1−γ.

The agent’s income follows an AR(1) process: θ2 = (1−κ)+κθ1 +ε2 , for κ ∈ (0, 1), where ε2 is arandom variable independently distributed from θ1 , and which is bounded below (this boundedness
assumption is necessary to guarantee the expected utility is finite for y high enough). The following
result is proved in the appendix.
Proposition 5 u and θ satisfy the single-crossing condition (Condition 2).

6.2

Multiplicatively Separable Utility

Suppose now thatu(θ, y) = g(θ)ū(y),where g is a positive and strictly decreasing in θ for the order on Θ, so that u is strictly submodular.
In the taste shock application, g(θ) = θ and the order on Θ is the opposite of the usual order on R
(i.e, θ ≺Θ θ0 ⇔ θ0 < θ). In the income shock application with constant absolute risk aversion,u(θ, y) = − exp(−(θ + y)), so g(θ) = exp(−θ) and ū(y) = − exp(−y). The order on Θ is the usual
order on R.
For each θ ∈ Θ, let β(θ) = E[g(θ2 )|θ1 = θ]. By persistence (Definition 6), β(θ) is decreasing inθ if g is decreasing.
Definition 7 (Mean Reversion) {θt }t=1,2 is mean reverting if β(θ)/g(θ) is increasing in θ
To understand this definition, recall that a higher type is associated with a lower value of g. Meanreversion means that the expected value of g(θ2 ) conditional on θ1 , while decreasing in θ1 , does notdecrease as fast at g(θ1 ) itself.
Proposition 6 (Single Crossing) u and θ satisfy single crossing.

19

Proof. Fix any y1 and y2 . We haveE θ uy (y2 , θ2 )/uy (y1 , θ) − 1 =ū0 (y2 ) β(θ)− 1.
ū0 (y1 ) θThis ratio is strictly increasing in θ, by mean reversion, and the single-crossing property is satisfied.

7

Appendix

7.1

Proof of Theorem 1

It suffices to show that, when Condition 1 holds, comparisons between any two continuation contracts C, C 0 from the agent and the principal’s perspective are equivalent to comparisons between
their transformations C(ρ) and C 0 (ρ), and strategy comparisons for the agent and the principal
under contract C are the same as under contract C(ρ). The expected utility of an agent under contract C following a given strategy σA for the entire game, and taking the principal’s renegotiation
strategy as given, is equal toUθσA (C) = E θ,σA

" T
X

#
u(yτ , θτ ) ,τ =t

while it is equal toUθσA (C(ρ)) = E θ,σA

" T
X

#
u(υ(yτ , ρ)θτ ) = a(ρ)UθσA (C) + E θ

" T
X

#
b(ρ, θτ ) ,

τ =t

τ =t

under the contract C(ρ) and the same strategies. Since the second term of the right-hand side
is independent of the contract, and a(ρ) > 0, this shows that any strategy of the agent that is
optimal for C is optimal for C(ρ), and that any comparison of contracts C, C 0 is equivalent to thecomparison of contracts C(ρ) and C 0 (ρ). Similarly, the principal expected cost under C and given
a renegotiation strategy σP and belief δ, taking the agent’s strategy as given, is equal to
" T
#
X
σP
Qδ (C) = E δ,σP
yτ ,
τ =t

while it is equal toQσδ P (C(ρ)) = E δ,σP

" T
X

#
υ(yτ , ρ) = υ1 (ρ)Qσδ P (C) − (T − t + 1)υ2 (ρ)τ =tunder the contract C(ρ) and the same strategies, which shows the result.

20

7.2

Proof of Theorem 2

Let qUθ denote the marginal utility ratio for type θ between changes in y0 and changes in ρ, evaluatedθ denote the marginal cost ratio for the principal between
at the initial contract. Similarly, let qC

similar changes. The proof that pooling contracts are not renegotiation-proof works by comparing
these different ratios.
Step 1 We start by showing that the inequality qUL < qUH holds, which is the single crossing
property in the (y0 , ρ) space. Equivalently,uy (H, ȳ0 )uy (L, ȳ0 )<,Vρ (L, ρ̄)Vρ (H, ρ̄)
where V (θ, ρ) = E[

(14)

PT

t=1 u(θt , ρyt )|θ0 = θ] is the expected continuation utility of the agent when

his period-0 type is θ. We haveVρ (θ, ρ̄) = E θ0 =θ

TX

yt uy (θt , yt ).

t=1

Since uy (θ, y) = θy −γ , (14) is equivalent to
#
" T
#
" T
X θt 1−γX θt 1−γ
y
> EH
y
EL
L t
H t

(15)

t=1

t=1

Suppose first that γ = 1. We have L ≤ θt ≤ H for all t, and these inequalities are strict with positive
probability. Therefore, the left-hand side of (15) is strictly greater than 1, while its right-hand sideis strictly less than 1, proving the inequality.
For γ < 1, (15) is equivalent, dividing by (1 − γ) > 0, to
W (H)
W (L)
>
,
L
H
where W (θ) = V (θ, 1) is the expected utility of type θ from period 1 onwards under his optimal
strategy. One possible strategy for L is to mimic the strategy of H. In that case his expectedutility is equal toW (L; z) = E

L

T
X
t=1

θt

zt1−γ
1−γ

where {zt } denotes the subsidy stream generated by mimicking the strategy of H. Starting from
θ0 = L, the type process {θt }t satisfies θt H/L ≥ H for all t, and with a strict inequality withpositive probability. Because yt1−γ /(1 − γ) > 0, we obtainT

X z 1−γH
W (θL; z) > E H
θt t= W (H).
L
1−γ
t=1

21

This shows that H/LW (L) > W (H).
For γ > 1, dividing (15) by (1 − γ) < 0 yields
W (L)
W (H)
<
.
L
H
Repeating the previous argument, but this time having H mimic L, yields the desired inequality.
H < q H or q L > q L , there is a Pareto improvement to the initial
Step 2 We now show that if qC
UCU
H < q H . Then, the principal can reduce his cost and improve
contract. Consider the first case, qCU

H’s utility by increasing y0 and reducing ρ, towards a contract that is strictly suboptimal for the
L. In the second case, the principal can improve L’s utility and strictly reduce his cost by reducing
y0 and increasing ρ, without attracting the high type. In either case, the principal strictly reduces
his cost, preserving incentive compatibility and keeping renegotiation-proof continuations.
H ≥ q H and q L ≤ q L . We will show that this case is ruled
Step 3 The remaining case is that qC
UCU

out by the induction hypothesis. Let wθ denotes the expected utility of an agent with type θ inperiod 1 under the menu M1 .
For γ 6= 1, the continuation utility in period 1 of an agent of type θ1 = H is given as a functionof ρ bywH ρ1−γ = u(H, y1H ρ) + U 2 (H, ρ),
where we recall from the statement of the induction hypothesis that U 2 (θ, ρ) denotes the expectedutility from Period 2 onwards when the agent’s type in Period 1 is θ, as a function of the scaling
factor ρ for continuation contracts at period 2. Similarly, if θ1 = L,wL ρ1−γ = u(L, y1L ρ) + U 2 (L, ρ).
For γ > 1, wH , wL are strictly negative (since u(θ, y) = θy 1−γ /(1 − γ) < 0).
Differentiating these inequalities with respect to ρ and evaluating the result at ρ = ρ̄ = 1, one
obtains(1 − γ)wθ = y1θ uy (H, y1θ ) + Uρ2 (θ, ρ̄).
By the induction hypothesis, this implies that
(1 − γ)wθ = (y1θ + Cρ2 (θ, 1))uy (θ, y1θ ),where we recall that C 2 (ρ, θ) denotes the expected cost from period 2 onwards when the agent’s
type in period 1 is θ, as a function of the scaling factor ρ for continuation contracts at period 2.
Moreover, because V (θ, ρ) = αθ ρ1−γ wH + (1 − αθ )ρ1−γ wL , we obtain the following formula for
qUθ :
qUθ =

uy (θ, ȳ0 )αθ uy (H, y1H )(y1H + Cρ2 (H, 1)) + (1 − αθ )uy (L, y1L )(y1L + Cρ2 (L, 1))
22

Similarly, the expected cost for the principal as a function of the agent’s type θ in period 1 isgiven by cθ ρ = ρy1θ + C 2 (ρ, θ). Differentiating with respect to ρ and evaluating the derivative at
ρ = 1 yields cθ = y1θ + Cρ2 (ρ̄, θ).
H and q L ≥ q L . Combining the previous results, we get
Now consider the case in which qUH ≤ qCUCuy (ȳ0 , θL ) ≥ α̂L uy (y1L , L) + (1 − α̂L )uy (y1H , H)

(16)uy (ȳ0 , θH ) ≤ α̂H uy (y1L , H) + (1 − α̂H )uy (y1H , H),

(17)

and

where

αH y1H + Cρ2 (H, ρ̄)
αH cH


=
.
αH cH + (1 − αH )cL
y1H + Cρ2 (H, ρ̄) + (1 − αH ) y1L + Cρ2 (L, ρ̄)(18)
αL y1H + Cρ2 (H, ρ̄)
αL cH


.
α̂ =
=
αL cH + (1 − αL )cL
αL y1H + Cρ2 (H, ρ̄) + (1 − αL ) y1L + Cρ2 (L, ρ̄)

(19)

α̂H =
and

αH

L

Since uy (y1L , L) < uy (y1H , H) (by the induction hypothesis), (16) implies that uy (ȳ0 , L) > uy (y1L , L)
and, hence, that ȳ0 < y1L < y1H . However, (17) also implies that uy (ȳ0 , H) < uy (y1H , H) and thusthat ȳ0 > y1H > y1L , a contradiction.
It is easily checked that the same computations obtain for the logarithmic case γ = 1.

7.3

Almost Uniqueness of Renegotiation-Proof Contracts for CARA utility
functions

To fully verify the induction hypothesis for Section 5, there remains to verify that optimal renegotiationproof menus are pinned down up to a single parameter. We have already established that any
renegotiation proof menu is separating, and characterized by a family {(y0θ , ρθ )}θ∈R with y0θ (resp.
ρθ ) decreasing (increasing) in θ. Let U (θ) denote the continuation utility of a type θ0 = θ under
the reference continuation menu MT .
By incentive compatibility, we must have
n
o
θ0
θ0
θ ∈ arg max−
exp(−θ)exp(−y)+exp(−ρ
)U
(θ)
.
0
0
θ

Local incentive compatibility is characterized by the first-order condition1300exp(−θ) exp(−y0θ )
13

dρθ
dy0θ
θ
(θ)
−
exp(−ρ
)U(θ)(θ) = 0
dθ0
dθ0

(20)

By monotonicity, y0 and ρ are a.e. differentiable in θ. Incentive compatibility also implies that y0 and ρ arecontinuous in θ.

23

Moreover, because the agent’s isoutility curves satisfy the single crossing property in the (y0 , ρ)space (as established in the main text when showing that pooling contracts are impossible), thelocal IC constraint is sufficient for global incentive compatibility.
Equation (20) defines a parametric equation for (y0θ , ρθ ) in the (y0 , ρ) space.
In addition, we know that renegotiation-proof contracts must be efficient, which means thatany point (y0 , ρ) of the curve covered by the renegotiation-proof contract is associated with theefficiency line of exactly one type θ = τ (y0 , ρ).
Plugging this efficiency relation into (20), and using the parametric relation dρ/dy0 = (dρ/dθ)/(dy0 /dθ),
we get the ordinary differential equation14
exp(ρ − y0 ) exp(−τ (y0 , ρ))
dρ
=
.
dy0
U (τ (y0 , ρ))(21)This pins down the renegotiation-proof contract up to a single parameter, which is a translation coefficient. Indeed, it is easy to check that if the parametric curve (y0 (θ), ρ(θ)) satisfies Equation (20),then so does the curve (y0 (θ) + ρ̂, ρ(θ) + ρ̂) for any ρ̂ ∈ R, and that the set of efficient allocations(y0 , ρ) for any given type θ forms a 45 degree line. Equivalently, if any function y0 7→ ρ(y0 ) satisfies
the ODE (21), then so does the function ρρ̂ : y0 7→ ρ(y0 − ρ̂) + ρ̂, for ρ̂ ∈ R.
Proof of Corollary 2
For any θ0 , the density of θ1 ’s distribution is given byαθ0 (θ1 ) = φ(θ1 − (κθ0 + (1 − κ)θ̄)),
where φ is the density of a standard Gaussian distribution. This implies that


00

1
αθ (θ)
0
2
00
2
= exp
(θ − (κθ + (1 − κ)θ̄)) − (θ − (κθ + (1 − κ)θ̄)).
2αθ0 (θ)Therefore, it suffices to show that (θ − (κθ0 + (1 − κ)θ̄))2 − (θ − (κθ00 + (1 − κ)θ̄))2 is nondecreasingin θ, for θ0 < θ00 . That expression is equal to(θ − κθ0 )2 − (θ − κθ00 )2 + 2(1 − κ)κθ̄(θ00 − θ0 )The second term is independent of θ, while the first term is equal to 2θκ(θ00 − θ0 ) + κ2 (θ02 − θ002 ),and is therefore increasing in θ for θ0 < θ00 , proving the claim.

14

The RHS is negative, since U is. The solution y0 7→ ρ(y0 ) of the ODE is therefore downward sloping.

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