Dynamic Contracting under Adverse Selection and Renegotiation Lucas Maestri FGV/EPGE January 30, 2017
Abstract In a repeated principal-agent model with adverse selection and renegotiation, a monopolist o¤ers long-term contracts to a consumer. The monopolist dynamically screens high-taste consumers. The possibility of renegotiation drastically reduces the monopolist’s ability to extract rents. We show that a version of the Coase conjecture holds for a nondurable-goods monopoly. As the parties become more patient, the allocation converges to an e¢ cient one. (JEL: D42, D82, D86).
1
Introduction
We study renegotiation in an in…nite horizon principal-agent model with adverse selection, considering a price-discriminating monopolist as the main interpretation. We assume that a monopolist faces a consumer who has private information about his marginal willingness to pay for a speci…c quality q 2 [0; 1] of a certain good. The consumer’s marginal valuation can be low (henceforth, low type) or high (henceforth, high type) and is persistent. The monopolist can o¤er long-term contracts, promising to deliver a particular I am very indebted to Larry Samuelson for his encouragement, support and guidance. I am also very grateful to Dino Gerardi and Johannes Hörner for their valuable advice. I thank Dirk Bergemann, Mehmet Ekmekci, Eduardo Faingold, Daniel Gottlieb, Vitor Luz, Rick Mans…eld, Giuseppe Moscarini, Philip Reny, Eduardo Rodrigues, Priscila Souza, Jean Tirole, Joel Watson and all the seminar participants at Columbia, Caltech, EPGE-FGV, FGV-SP, Insper, PUC-Rio, TSE, Yale and Washington University at Saint Louis. Any remaining errors are mine.
quality in each future period. The terms of the contract must be honored unless both parties agree to renegotiate. Consistent with the idea that the monopolist (as a principal) has the bargaining power, we select equilibria that maximize his pro…ts. Since the lack of commitment to not optimally exploit pro…table renegotiation opportunities is central to our model, we select equilibria that maximize the monopolist’s pro…ts after every history. The lack of commitment due to renegotiation raises several di¢ culties in the characterization of equilibrium outcomes. In their seminal paper, La¤ont and Tirole (1990) (henceforth, LT) overcome these challenges by focusing on a two-period horizon. Second-period allocations that leave no pro…table renegotiation opportunities are easily characterized. This enables the authors to solve for the most pro…table allocation subject to the secondperiod allocations satisfying this property. As the horizon increases beyond two periods, the set of future allocations that are immune to renegotiation becomes less tractable. In particular, this method seems unfruitful when the horizon is in…nite. This paper proceeds di¤erently. We show that all equilibria must satisfy sequential separating dynamics: along the screening process, either the monopolist becomes less convinced that the consumer has a high valuation or learns the consumer’s valuation. We show that the set of allocations that satisfy sequential separating dynamics leads to sharp predictions for the limit outcome as the parties become more patient. A central question in renegotiation is what happens as the parties become more patient and hence more willing to sacri…ce short-run payo¤s to in‡uence long-run outcomes. To study this question, LT assume that the weight of the …rst period is and the weight of the second is (1 ). They show that the equilibrium allocation is bounded away from the Pareto frontier for any 2 (0; 1) : This suggests that the monopolist can limit the information rent of the high type and preclude e¢ ciency in a fully dynamic model. To analyze this question, we begin by showing that the distortion of the quality consumed by the low type in a particular period is proportional to the probability that the high type chooses to reveal his private information in that period. Using this result, we obtain a lower bound on the amount of information rent of the high type. This lower bound increases as the parties become more patient and severely reduces the monopolist’s ability to extract rents from the high type by o¤ering an ine¢ cient allocation to the low type. In response, the monopolist chooses to o¤er menus inducing a fast separation of types. Accordingly, we show that a version of the Coase conjecture holds for the monopoly model even in the case of a nondurable good. The possibility of renegotiation decreases the monopolist’s ability to extract rents from high-type consumers by distorting the quality o¤ered to lowtype ones. Therefore as the parties become more patient, the allocation converges to the 2
most pro…table allocation for the monopolist among all individually-rational, incentivecompatible and e¢ cient ones.
1.1
Related Literature
This paper belongs to the literature on dynamic adverse selection and renegotiation. Hart and Tirole (1988) (henceforth, HT) study a …nite horizon monopoly model with two types of buyers under the assumption that quantities are in f0; 1g; so the monopolist either sells the good in a certain period or does not sell it. They show that the allocation converges to a pooling allocation as the parties become more patient and the number of periods of the relationship increases. The assumption of HT that quantities are binary is restrictive and excludes several important applications of price discrimination. First, it rules out applications in which it is e¢ cient to sell di¤erent positive quantities to di¤erent types. Second, it also excludes all applications in which the intensive margin trade-o¤ between e¢ ciency and rent extraction is important. LT build a model that allows for these trade-o¤s by extending the analysis of HT to the case in which there is a continuum of quantities, but restricting it to two periods. Our model departs from LT by assuming that there is an in…nite horizon. As in HT, the equilibrium becomes e¢ cient as the parties become more patient. Contrary to HT, the limit consumption of the consumer with the highest valuation is greater than the limit consumption of the consumer with lowest valuation. Contrary to LT, the allocation becomes e¢ cient as the parties become more patient.1 For a discussion of renegotiation in adverse selection models, see La¤ont and Tirole (1993).2 Strulovici (2015) studies non-cooperative foundations for renegotiation-proof contracts in adverse selection models. He shows that if the parties have several opportunities to bargain before a one-time trade decision, then all equilibria will be approximately e¢ cient. Our paper di¤ers because we restrict attention to equilibria in which there are no renegotiation opportunities left and analyze a model in which trade happens over time. This paper is also related to the literature on dynamic auctions without commitment. Skreta (2006, 2015) shows that when the horizon is …nite, a sequence of auctions with reserve prices are optimal dynamic mechanisms. Liu, Mierendor¤ and Shi (2014) characterize 1
See also Battaglini (2007), who extends the two-period model of La¤ont and Tirole (1990) to the case of changing types. 2 For an early contribution, see Dewatripont (1989). For important contributions on renegotiation under moral hazard, see Fudenberg and Tirole (1990) and Rey and Salanié (1996). See Strulovici (2011) for a model with ex-ante symmetric information and hidden actions. For renegotiation in repeated games, see Farrell and Maskin (1989) and Bernheim and Ray (1989).
3
optimal selling mechanisms in an in…nite horizon model. Their model di¤ers from ours because they assume that there are multiple buyers who purchase the object at most once. This paper is also related to the large literature on durable goods monopolists and the Coase conjecture. For seminal papers, see Stokey (1981), Fudenberg, Levine and Tirole (1985) and Gul, Sonnenschein and Wilson (1986). These papers di¤er from ours since they focus on durable goods. Section 2 develops the set up. Section 3 presents the equilibrium re…nement. Section 4 studies pro…t maximizing values. Section 5 presents our main results. Section 6 concludes. Omitted proofs are in the Appendices.
2
The Model
We study a dynamic principal-agent model with adverse selection and the possibility of renegotiation. We use the benchmark model of Mussa and Rosen (1978), recently explored by Battaglini (2005), to study long-term contracting. The model can have other interpretations, such as procurement or agency in the labor market. For concreteness, we interpret the model as the relationship between a monopolist and a consumer. A monopolist and a consumer meet at t = 0; 1; ::: In each period, the monopolist produces a quality q 2 [0; 1] of a non-durable good. Producing quality q costs c(q): We assume that c ( ) is a strictly increasing, strictly convex C 2 function with c(0) = 0 and minq2[0;1] c00 (q) > 0: A consumer has a type i 2 fL; Hg ; which identi…es his taste for the good. A low type is identi…ed by L and has the value L q for a quality q of the nondurable good. A high type is identi…ed by H and has the value H q for a quality q of the nondurable good (0 < L < H ): We write for H L : We write xt 2 < for the net transfer made from the consumer to the monopolist in period t: The monopolist’s realized pro…t is given by: (1
)
1 X
t
[xt
c (qt )] ;
t=0
and the type i 2 fL; Hg consumer’s realized rent is given by: (1
)
1 X
t
[ i qt
xt ] ;
(1)
t=0
where
2 (0; 1) is the common discount factor. Both players are risk-neutral expected
utility maximizers.
4
For i 2 fL; Hg ; we write
i (q)
:=
iq
c(q) for the surplus achieved by o¤ering quality
q to type i: We assume that the optimal quality for the low type is interior and that the optimal quality for the high type is the greatest available one: 0 < qL := arg max
q2[0;1]
A feasible contract in period t;
t;
L (q)
< qH := arg max
q2[0;1]
H (q)
= 1:
speci…es a transfer from the consumer to the mo-
nopolist, xt ; and a promise from the monopolist to the consumer for each future period: fq g
t
2 [0; 1]1 : Thus the set of feasible contracts in period t is n At := (xt ; fq g1=t ) : xt 2 <; fq g
t
o 2 [0; 1]1 :
The set of menus available in period t is the set of menus with two contracts: Mt = At At .
In the …rst period, the monopolist o¤ers to the consumer a menu m0 2 M0 : The consumer chooses an element from m0 [ f;g : If the consumer chooses ;; then the consumer makes
no payment to the monopolist, nothing is produced and period 0 ends. If the consumer chooses 0 = x0 ; fq g 0 2 m0 ; he pays x0 to the monopolist, consumes q0 , the period ends and he starts the next period endowed with promises fq g
1
:
Consider a period t > 0 in which the consumer starts with promises fq g x1t ; fq 1 g
t.
The
x2t ; fq 2 g
monopolist o¤ers a renegotiation menu mt = : The consumer t ; t either chooses one contract from that menu or rejects all contracts and remains with the promises fq g
t
: If the consumer chooses to remain with promises fq g
in the current period and the next period starts with promises fq g chooses a contract xjt ; fq j g t from the menu mt ; he monopolist, consumes qtj and the period ends. Then the fq j g t+1 : t
t
; he consumes qt
t+1 .
If the consumer
makes a transfer xjt 2 < to the next period starts with promises
A history of length t; h ; contains all the o¤ered menus in the previous periods and all
the contracts selected by the consumer in the previous periods. We set h0 = ;: We write
Ht for the set of histories of length t and H = [ 0 H for the set of all histories. M t A behavior strategy for the monopolist, M ; is a sequence ; where M t t (h ) is
a probability transition from Ht into Mt ; mapping a history ht into a (possibly random) menu. A behavior strategy for type i 2 fL; Hg consumer, t
t
i;
is a sequence f
i;t g ;
where
i;t
associates to each pair (h ; mt ) 2 H Mt a probability distribution over the set mt [ f;g (; stands for rejecting all contracts). We write = ( M ; L ; H ) for a strategy pro…le. We let p = fp(ht )ght 2H be the monopolist’s system of beliefs, where p(ht ) 2 [0; 1] represents the 5
probability that the monopolist assigns (at the history ht ) to the consumer being a high type. We assume that the initial prior is interior: p0 = p(h0 ) 2 (0; 1) : Given a strategy pro…le and a system of beliefs p, we let, for every history ht , V M (ht ; ( ; p)) denote the monopolist’s continuation pro…t at ht . Moreover, we write V M (ht ; ( ; p)) for the monopolist’s expected pro…t given ht : "1 X t ) E( ;p) [x V M ht ; ( ; p) : = (1 V M ht ; ( ; p) where E(
;p)
: = (1
) E(
;p)
"
=t 1 X
[x
=0
c (q )] j ht #
#
c (q )] j ht ;
[f j ht ] represents the conditional expected value (given ht ) of the random
variable f given the strategy pro…le
and the system of beliefs p: Analogously, for every
t
history h and for i 2 fL; Hg ; we let Vi (ht ; ( ; p)) denote the expected continuation rent at ht of type i and Vi (ht ; ( ; p)) the expected rent of type i given ht : "1 # X t Vi ht ; ( ; p) : = (1 ) E( ;p) [ iq x ] j ht Vi ht ; ( ; p)
: = (1
) E(
;p)
"
=t 1 X =0
[ iq
#
x ] j ht :
To simplify the notation, we omit the argument ( ; p) from the variables above whenever this omission leads to no confusion (for instance, we write V M (ht ) for V M (ht ; ( ; p))). For each event A; we write IA for its indicator function. For each (measurable) events A and B; we de…ne the probability of event A given B in the equilibrium ( ; p) by Pr(
;p)
(A j B) := E(
;p)
[IA j B] : For each set N , we write coN for its convex-hull. For
each element z of an Euclidian space, we write kzk for its Euclidian norm.
Remark 1 We make a few abuses of notation to simplify the exposition. First, we write pt for p(ht ) whenever this leads to no ambiguity. Second we often do not specify the period of the state and of the contract. For instance, we write (p; fq g) instead of pt ; fq g t ; which represents the state in period t: Also for simplicity, we write fqi g for the sequence fqi ; qi ; :::g and fq0 ; qg for the sequence fq0 ; q; q:::g:
6
3
Renegotiation Re…nement: Optimal Contracting with Imperfect Commitment Most of the literature that deals with renegotiation in screening models works with
…nite horizon models (e.g. HT (1988) and LT (1990)). Essentially these works explore the fact that last-period renegotiation-proof allocations are straightforward to characterize and then work backwards. This approach does not extend naturally to in…nite horizon models due to the absence of a last period in such models. In contrast with these studies, we propose a di¤erent solution concept that explores the recursive structure of our in…nite horizon model. Our approach has two components. First, it de…nes recursively for every state (belief and promises for future periods) the set of allocations that maximizes the monopolist’s payo¤. To solve for the optimal allocations for the monopolist for a given state, all players take into account that the continuation allocation starting in the next period solves a similar problem for the next-period state. Second, additionally to satisfying such optimality condition, we insist that our strategy pro…le and our system of beliefs ( ; p) constitute a perfect Bayesian equilibrium (Fudenberg and Tirole, 1991) Consider the state variables (p; fq g) 2 [0; 1]
[0; 1]1 ; where p represents the belief
held by the monopolist that the consumer is a high type, and fq g is the set of promises at
the beginning of the period. Interpret V M (p; fq g) 2 < as the monopolist’s continuation pro…t and (p; fq g) <2 as the set of possible continuation rents for both types, given the state variables. We impose that for each state (p; fq g); the monopolist maximizes his pro…t by choosing a solution to the problem below. We call values V M ;
satisfying the
four conditions below as pro…t maximizing: De…nition 1 V M ;
are pro…t maximizing values if:
I) They are K uniformly bounded for some constant K > that sup(p;fq
g)2[0;1] [0;1]1
V
M
(p; fq g) < K and sup(p;fq
II) For all (p; fq g) 2 [0; 1] V M (p; fq g) =
1
H:
there exists K >
g)2[0;1] [0;1]1
H
such
k (p; fq gk < K:
[0; 1] ; the program below achieves a maximum 8 2
max j j 2 m;aL ;aH ;(vL ;vH )j=1 ;(p1 ;p2 ) : j=1
j
ajL ; ajH
h
(1
h
) xj
c q0j
aji
i
+ V M pj ; q j
9 i=
(2)
0 and a1i + a2i = 1; where m = ((x1 ; fq 1 g) ; (x2 ; fq 2 g)) is a menu, ai = (a1i ; a2i ) ; with where aji is the probability that type i 2 fL; Hg chooses contract (xj ; fq j g) from menu m; 7
;
;
vij is a promised future expected rent to be delivered to consumer i if he chooses contract (xj ; fq j g), and accepted.
j
ajL ; ajH = pajH + (1
p)ajL is the probability that contract (xj ; fq j g) is
The maximization (2) is subject to the following constraints: Incentive compatibility: (1
)
k i q0
xk + vik
(1
)
for i 2 fL; Hg and for all k; j: Individual Rationality (IR) for the low type: h max (1
)
k
h
k L q0
h
j i q0
i i xk + vLk
i xj + vij
if aki > 0;
(3)
:
(4)
:
(5)
X
(1
)
Lq
X
(1
)
Hq
0
Individual Rationality (IR) for the high type: h max (1
)
k
h
k H q0
i i k xk + vH
0
Promise keeping: j vLj ; vH 2
Bayes rule: pj =
pajH j paH +(1
n o pj ; q j
if
p)ajL
>0
j
for all j:
(6)
ajL ; ajH > 0:
(7)
III) (vL ; vH ) 2 (p; fq g) if and only if there exists a (Borel-measurable) probability distribution over the argmax of (2) such that: vi =
for i = L; H:
Z
h max (1 k
)
h
k i q0
i i xk + vik d ;
IV) For any fq g [0; 1]1 ; let the correspondence fq g : [0; 1] (p; fq g) : This correspondence must have a closed graph. fq g (p) =
<2 be de…ned by
Naturally, if we want V M (p; fq g) to represent the monopolist’s pro…t when the game starts at the state (p; fq g); the value V M (p; fq g) must be bound. Indeed, it cannot exceed the (discounted) value for high type of consuming 1 in each future period:
take an arbitrary constant K >
H
and impose that sup(p;fq
8
g)2[0;1] [0;1]1
H:
Hence we
V M (p; fq g) < K.
Analogously we impose that sup(p;fq
g)2[0;1] [0;1]1
k (p; fq gk < K. This is the content of
part I of De…nition 1. To explain part II, we interpret condition (2) as the requirement that, for each state, we select a continuation equilibrium that maximizes the monopolist’s expected pro…t. For that, we select a menu m = ((x1 ; fq 1 g) ; (x2 ; fq 2 g)) with 2 contracts; a randomization
ai = (a1i ; a2i ) over the contracts of that menu by type i = L; H (where aji 2 [0; 1] denotes the probability that type i selects the contract (xj ; fq j g)); and a credible promised future
expected rent, vij ; delivered to consumer i if he chooses contract j.
The maximization program must satisfy four constraints. First, we have the incentive compatibility constraint (3). If type i chooses contract j 2 f1; 2g ; his expected rent can be divided into two elements: the current period’s rent (1 ) i q0j xj and his future expected rent vik : The constraint (3) requires that if type i chooses contract j with positive probability, then the expected rent from this choice must be weakly greater than the expected rent from choosing contract k 6= j: Second we have the individual rationality constraint for the low type (4) and for the high type (5). These constraints require that the rent of type i from accepting to renegotiate is weakly greater than the rent obtained from refusing to renegotiate and consuming his outside option (see the right-hand side of (4) and (5)). These constraints impose that if a period starts with promises fq g ; then the minimal utility that type i should expect is P P t (1 ) ) i q : Therefore, we let Ui fq g t : = (1 i q denote the resert
vation rent obtained by type i 2 fL; Hg when the state is fq g
t
2 [0; 1]1 :
Third, we have the promise keeping constraint (6). The set (p; fq g) <2 can be interpreted as the set of all credible continuation rents for each type for a period starting with the state (p; fq g): Thus, if contract (xj ; fq j g) gives rise to a next period state (pj ; fq j g),
constraint (6) insists that the continuation rent promised to each type of consumer in the j next period is credible when contract j is chosen: (vLj ; vH ) 2 (pj ; fq j g) : Fourth, we have Bayes’ rule (7). This constraint assures that the posterior upon the
acceptance of contract j must be consistent with Bayes’rule whenever possible. Part III of De…nition 1 de…nes the set of credible promises when the state is (p; fq g) :
(p; fq g): A continuation rent for each type of consumer (vL ; vH ) belongs to this set if and only if there exists a (mixed) solution to (2) subject to (3)-(7) which delivers these rents to the consumers. For part IV of De…nition 1, take any fq g 9
[0; 1]1 and de…ne the correspondence
fq g
: [0; 1]
<2 by
fq g (p)
:=
(p; fq g) : We impose that this correspondence has
a closed graph. This technical requirement guarantees that …xed-point pro…t-maximizing values V M ; are well behaved. As we will see in the next section, the requirements I -IV of De…nition I allow us to characterize pro…t maximizing values V M ;
without any
mention to equilibrium. De…nition 2 A pro…t-maximizing renegotiation equilibrium is a strategy pro…le
=
M
;
L;
and a system of beliefs p such that: a) ( ; p) is a perfect Bayesian equilibrium; b) There exists pro…t maximizing values V M ; the continuation play of ( ; p) is consistent with V M ;
such that at every history ht 2 H,
: That is, for every menu mt in
t the support of M t (h ); the menu mt ; the randomization of the consumer over its elements a = (a1 ; a2 ), the monopolist’s posterior after the acceptance of each contract p1 and p2 ; and
the respective continuation rents for the consumer (vL ; vH ) solve (2) subject to (3)-(7) for the state (p(ht ); fq g1=t ) (where fq g1=t are the promises at ht ). With a slight abuse of notation, we say that a menu, m = ((x1 ; fq 1 g) ; (x2 ; fq 2 g)) ; a
randomization over contracts, a = (a1 ; a2 ); a pair of posteriors (p1 ; p2 ) and promised rents
(vL ; vH ) are a plan, which we identify by P. A plan is feasible (for the state (p; fq g)) if it satis…es (3)-(7) (for the same state). A plan is optimal (for the state (p; fq g)) if it maximizes (2) over all feasible plans (for the same state).
De…nition 2 states that a pro…t-maximizing renegotiation equilibrium is a perfect Bayesian equilibrium (henceforth PBE) for which the play is consistent with a …xed-point V M ; . That is, take a history ht associated to the state (p(ht ); fq g). We insist that, for each menu
o¤ered with positive probability by the monopolist at ht , the menu, the randomization
over its elements, the monopolist’s posterior after the acceptance of each contract and the respective continuation rents for each type of consumer solve (2) subject to (3)-(7). First, program (2) insists that all contracts are always renegotiated. However, the monopolist can always o¤er one contract that is identical to the status-quo contract. Therefore this assumption is without loss of generality. Second, since there are two types, the argument in Bester and Strausz (2001) (pages 1094-1096) guarantees that when solving (2) subject to (3)-(7) we could always obtain a payo¤-equivalent solution for all players by restricting attention to menus in which only
10
H
two contracts are o¤ered. Hence, the …xed-point formulation above would lead to the same payo¤s even if more contracts were allowed. Third, the IR constraints (4) and (5) above impose that if a consumer rejects a renegotiation o¤er he can obtain the value of never renegotiating again. The latter value is the harshest punishment that could be imposed on a consumer. This raises the question of whether, after a deviation, one can …nd a continuation equilibrium that delivers the worstpossible rent to type i consumer when he rejects both o¤ers. We address this question when we prove equilibrium existence. In the equilibrium that we construct, the monopolist attributes probability one to the consumer being a high type after an o¤-path rejection of both contracts and never updates his belief in the future. In such continuation equilibrium, the monopolist insists on o¤ers that extract all surplus from the high type after every history and for which the low type payo¤ can be calculated by assuming that both contracts are rejected. Finally, it is assumed that all transfers are made in the …rst period. We make this assumption to ease notation: It is not di¢ cult to show that the same payo¤s are obtained if we assume that the contracts contain transfers for every future period too. Proposition 3 in Section 5.5 states that there exists an equilibrium satisfying our re…nement.
3.1
Additional De…nitions
For each (p; fq g) 2 [0; 1]
[0; 1]1 ; we de…ne
H
(p; fq g) as the set of credible rents
for the high type which are consistent with (p; fq g): Formally we de…ne: H (p; fq g) := ^ fvH : (vL ; vH ) 2 (p; fq g)g : We also UL (fq g) : vH 2 H (p; fq g)g : n de…ne H (p; fq g)2 := fvH o j j Consider a feasible plan P = m; aL ; aH ; vL ; vH j=1 ; (p1 ; p2 ) (for a state (p; fq g)). For each type i 2 fL; Hg and for each contract xjt ; fq j g 2 m; we write vi xjt ; qtj the rent of type i from contract
xjt ;
vi xjt ; qtj We write vH xjt ; qtj xjt ;
qtj
vL xjt ; qtj
qtj
for
:
:= (1
)
j i qt
xjt + vij :
for the high-type incremental rent of the contract
. We write vi (P) for the rent obtained by type i 2 fL; Hg from plan P :
vi (P) := maxj2f1;2g vi xjt ; qtj
: We call the di¤erence vH (P)
vL (P) the high-type
incremental rent from plan P: Analogously, we write V M (P) (resp. V M xjt ; qtj pro…t obtained by the monopolist from plan P (resp. contract xjt ; qtj ). 11
) for the
We say that two equilibria are outcome equivalent if they lead to the same distribution P1 t over consumption fq0 ; q1 ; :::g and discounted transfers t=0 xt for each type.
4
Pro…t Maximizing Values
This section presents the main properties of pro…t-maximizing values V M ; that rely only on De…nition 1. The next section uses these properties to study equilibrium allocations, usn ing De…nitions 1 and 2. For be an arbitrary plan, with
we add the superscript “F ” to plan P to emphasize that this is a feasible plan for this state. Similarly, write P for an arbitrary optimal plan for the state (p; fq g1=0 ): We show that optimal plans that satisfy our re…nement have a series of intuitive proper-
ties that hold in most contract-theory models. We review the most important ones below, relegating some properties of more technical nature to Appendix A. Property 1: The monopolist extracts all the surplus when he is sure about the consumer’s type. Lemma 3 in Appendix A shows that if V M ; L
(qL ) UL (fq g1=0 ) (resp. V M (1; fq g1=0 ) =
satisfy De…nition 1, then V M (0; fq g1=0 ) = H
(qH ) UH (fq g1=0 )) whenever the state
is (0; fq g1=0 ) (resp. (1; fq g1=0 )). Moreover, if p = 0 (resp. p = 1) and P is an optimal plan, then vL (P ) = UL (fq g1=0 ) (resp. vH (P ) = UH (fq g1=0 )). In other words, when
the monopolist is certain about the consumer’s type, the solutions to (2) are consistent with the monopolist implementing an e¢ cient allocation and leaving no rent for the consumer. Property 2: The high type always obtains a weakly higher rent than the low type. Lemma 4 in Appendix A shows that the rent of the high type is always weakly greater than the rent of the low type: For any state (p; fq g), if (vL ; vH ) 2 (p; fq g) ; then vH
o
j 2 the remainder of this section, let P = m; aL ; aH ; vLj ; vH ; (p1 ; p2 ) j=1 1 2 1 1 1 m = f(x1 ; fq g =0 ) ; (x2 ; fq g =0 )g : Given a state (p; fq g =0 ),
vL : This result is a direct consequence of the fact that the high type holds a higher
valuation for each quality level. Property 3: When the state starts with zero promises and puts positive probability on the low type, we may ignore the high-type IR (5). Moreover, the low-type IR (4) must bind.
12
Lemma 6 in Appendix A shows that when the state is (p; f0g) ; for some p 2 [0; 1);
we may ignore the high-type IR (5) when solving the monopolist’s problem. Moreover, the low-type IR (4) must bind. The reason behind this result is straightforward: Consider any feasible plan P F for the state (p; f0g): According to this plan, the low type selects a
contract with positive probability. Assume that this contract is (x1 ; fq 1 g1=0 ) : In this case, )( L q01 x1 ) + vL1 1 )( H q01 x1 ) + vH
since his IR holds, we have (1 (from Property 2), we obtain (1
1 0: Using H > L and vH vL1 0 and hence the high-type IR is
automatically satis…ed. On the other hand, if P is an optimal plan for the state (p; f0g)
and the low-type IR (4) does not bind, then one can …nd a small " > 0 and a feasible plan
P" that di¤ers from P only in that the transfer in every contract is increased by ": Clearly the plan P" leads to a higher pro…t than P ; contradicting the optimality of P : Property 4: We have (0;
qL ) 2
(0; f0g) and (
L qH ; H q H )
2
(1; fqH g) :
Property 4 follows from Lemma 7, which is stated and proved in Appendix A. Let us start explaining the …rst part of Property 4. Take the state (0; f0g) and assume that the
monopolist can …nd a feasible plan P F in which he extracts all the surplus of the current period from the low-type, L (qL ); and starts the next period with the same state with probability 1. This plan would lead to a pro…t equal to V M P F V
M
(0; f0g) : The feasibility of this plan would imply that V
hence
V M (0; f0g)
(1
)
L (qL )
M
= (1
(0; f0g)
) V
M
L (qL ) F
P
+
and
+ V M (0; f0g) ;
which would lead us to conclude that V M (0; f0g)
L (qL ):
Therefore this plan would leave
no rent to the low type and would implement an e¢ cient allocation. Therefore, if we could …nd such a plan then we would conclude that it is optimal. In fact, it is very easy to …nd such a plan. Consider the plan that o¤ers two (identical) contracts ( L qL ; fqL ; 0g) which lead to the same next-period state (0; f0g) and to the same next-period rents (0; vH ) 2 (0; f0g) ; where (0; vH ) is an arbitrary element of
(0; f0g) : Finally notice that if the
monopolist were to o¤er this plan in every period then the high type would obtain a rent
equal to qL . Thus we must have (0; qL ) 2 (0; f0g) : Let us now explain the second part of Property 4. Assume that the period starts with the state (1; fqH g) : If the monopolist were to commit to never renegotiate this contract,
he would extract all rent from the high type and implement an e¢ cient allocation. Therefore this commitment decision would be costless to him and it would lead to the rents 13
(
L qH ; H q H )
2
(1; fqH g) : Property 4 shows that the monopolist can credibly commit to
deliver these rents.
Property 5: If (vL ; vH ) 2
(0; fq g) ; then vH
vL
qL :
Property 5 follows from Lemma 14 in Appendix A. Notice that in the state (0; fq g)
the monopolist is sure that the consumer is a low type. By an argument similar to the one provided in the explanation of Property 4, the monopolist has access to optimal plans that implement the e¢ cient allocation and leave no rent to the low type. Therefore the low type should consume qL in each period, which implies that the high type can obtain an “information rent”at least as large as
5
qL by imitating the low type.
Equilibrium Analysis This section characterizes pro…t-maximizing renegotiation equilibria ( ; p): It imposes
De…nitions 1 and 2.
5.1
Commitment Allocation is not Implementable
The analysis starts by showing that the commitment allocation is not implementable. To implement the commitment allocation, the monopolist would have to learn the consumer’s type in the …rst period. Moreover, the monopolist would have to o¤er a quality, which we call q c ; strictly lower than the e¢ cient one to the low type in every period t 0 in order to extract more rent from the high type. However, once the monopolist learns that the consumer is a low type, there is an incentive to renegotiate and o¤er quality qL in every future period. This is exactly what he would do after learning the consumer’s type in every pro…t-maximizing renegotiation equilibrium as we explain below. Indeed, assume that there is an equilibrium in which the monopolist o¤ers the menu m0 = f
L;
Hg
inducing complete screening in the …rst period, where
i
is a contract
which is designed exclusively to type i 2 fL; Hg : If m0 led to the commitment solution,
the current-period quality of L would have to be q c . Let h11 be the history led by the acceptance of L in such equilibrium. In this case, the acceptance of the contract L leads
to a state of the form (0; fq g) and to rents (vL ; vH ) 2 vH
vL
qL and hence we must have
VH (h11 ) 14
(0; fq g) : Property 5 implies that
VL (h11 )
qL in such equilibrium.
Thus, from imitating the low type at h0 and choosing
L;
the high type would obtain a
rent at least as large as (1
)
qc +
VH (h11 )
VL h11
+ VL h0
(1
)
in such equilibrium, where the inequality used the fact that VL (h0 )
qc +
qL
0: We conclude that
the commitment allocation is not implementable as it leaves open pro…table renegotiation opportunities. Therefore, renegotiation necessarily reduces the monopolist’s pro…t.
5.2
An E¢ cient Incentive-Compatible Implementable Allocation
Our …ndings in the previous section raise the question of what kind of allocations are implementable. We will show that there exists a feasible plan for the initial state (p0 ; f0g)
that leads to the payo¤ for each player of the most pro…table allocation for the monopolist among all e¢ cient, individually-rational and incentive-compatible ones. f
Consider the following plan denoted by P ef : The monopolist o¤ers a menu m0 =
L;
Hg ;
in which the contract
L
= (
1
L qL ; fqL ; 0g)
is designed for the low type and
the contract H = (1 ) ( H qH qL ) ; fqH g is designed for the high type. The contract L leads to the next-period rents (0; qL ) 2 (0; f0g), while the contract H
leads to the next-period rents (
L qL ; H qH )
2
rents is guaranteed by Property 4. The contract
(1; fqH g) ; where the credibility of these L
leads to the next-period state (0; f0g)
and to the continuation pro…t L (qL ) for the monopolist according to Property 1. Using this property, we conclude that the acceptance of the contract H leads to the next-period continuation pro…t
H (qH )
UH (fqH g): It is easy to verify that this plan satisfy the
incentive-compatibility constraint (3) and the IR constraints (4) and (5). The plan P ef leads to the pro…t
V M P ef = p0 [
H (qH )
qL ] + (1
p0 )[
L (qL )]
(8)
for the monopolist, while the low type obtains the rent vL P ef = 0 and the high type obtains the rent vH P ef = qL : It is immediate to see that these payo¤s are obtained by the players in the most pro…table allocation for the monopolist among all e¢ cient, individually-rational and incentive-compatible ones.
15
5.3
Sequential Separating Dynamics
Consider a history ht associated to the state (pt ; fq g t ) with some pt 2 (0; 1). Assume that the monopolist o¤ers a menu mt : We say that a contract t 2 mt is a pooling contract if it leads to a posterior pt+1 < 1. We say that a contract it leads to the posterior 1:
t
2 mt is a revealing contract if
In the …rst step to understand the equilibrium allocations, we ask how the monopolist learns the consumer’s private information over time. A simple way that information is revealed is the following: either the posterior decreases, in which case the monopolist becomes more pessimistic about the consumer’s type, or the posterior jumps to 1, in which case the monopolist learns that the consumer is a high type. We refer to this belief evolution as sequential separating dynamics. Formally: De…nition 3 An equilibrium presents sequential separating dynamics if, for almost all histories associated with an interior belief, the monopolist o¤ers a menu containing only pooling contracts leading to (weakly) lower posteriors, and revealing contracts, i.e., contracts leading to the posterior 1. Assume that the current period’s belief is p 2 (0; 1) and the monopolist o¤ers a menu
with a pooling contract leading to a posterior p0 < p and a revealing contract (leading to the posterior 1). Since beliefs are martingales, one can calculate the probability that the pooling contract is accepted: 1 p (p; p0 ) := : (9) 1 p0 Proposition 1 justi…es the interest in equilibria presenting sequential separating dynamics. Proposition 1 Every pro…t-maximizing renegotiation equilibrium presents sequential separating dynamics. Proposition 1 shows that the monopolist o¤ers a menu with sequential separating dynamics in every period. The proof of Proposition 1 is by contradiction. If the result is false, we can …nd an on-path history ht of smallest length starting with the state (pt ; fq g)
in which the monopolist o¤ers a menu that does not present sequential separating dynamics. In this case, we can use Lemma 13 to show that there is a state (pt ; f0g), an optimal 16
plan Pt for that state which is not consistent with an equilibrium that presents sequen-
tial separating dynamics and a continuation equilibrium following that plan that satis…es De…nitions 1 and 2. The proof establishes a contradiction by showing that the plan Pt is
not optimal for the state (pt ; f0g) : The main challenge to …nd a better feasible plan for the state (pt ; f0g) is that we have little knowledge a priori about the endogenous objects
V M and . Therefore, it is hard to know the set of feasible plans and to compare the payo¤s from two elements of this set. To circumvent this di¢ culty, we use the fact that the continuation game is consistent with De…nition 2 to obtain information about feasible plans. Using this information, we are able to construct two feasible plans for the state (pt ; f0g) ; P~1 and P~2 ; and show that there exists a randomization among them that leads to
a strictly greater pro…t than V M (Pt ): n This will immediatelyoimply that at least one of them > V M (Pt ), which establishes leads to a strictly higher pro…t: max V M P~1 ; V M P~2 the desired contradiction.
What is the intuition behind Proposition 1? Sequential separation enables the monopolist to screen the high type faster in a …rst-order stochastic sense. Since it is e¢ cient to o¤er di¤erent qualities to di¤erent types, faster screening allows the monopolist to increase the e¢ ciency of the relationship without increasing the rent of each type. This increases the monopolist’s pro…t. Although our proof is considerably di¤erent, the property of sequential separating dynamics is also present in the contributions of HT (1988), LT (1990) and Battaglini (2007).
5.4
Optimality of Spot Pooling Contracts and RenegotiationProof Revealing Contracts
Assume that the monopolist o¤ers a contract (xt ; fq g1=t ) at a history ht :
Spot Contract. We say that a contract (xt ; fq g1=t ) is a spot contract if q = 0 for
every
> t:
Renegotiation-Proof Contract. We say that a contract (xt ; fq g1=t ) is renegotiation-
proof if the monopolist attributes probability 1 to one type upon its acceptance and if it o¤ers the e¢ cient quality to that type in every future period. Accordingly, we say that (xt ; fq g1=0 ) is a renegotiation-proof revealing contract it it leads to the posterior 1 and if
q = qH for every
t:
The information that every equilibrium presents sequential separating dynamics allows 17
the construction of an outcome equivalent equilibrium in which every pooling contract is a spot contract and every revealing contract is renegotiation-proof. Lemma 1 For every equilibrium, we can …nd an outcome equivalent equilibrium in which every pooling contract that is o¤ered on the equilibrium path is a spot contract and every revealing contract that is o¤ered on the equilibrium path is renegotiation-proof. The proof of Lemma 1 is relegated to Section 8.2 in Appendix B. Let us …rst explain why it is without loss to assume that pooling contracts are spot. Take a state (pt ; fq g1=t )
and notice that any feasible plan must deliver a rent at least as large as UL (fq g1=t )
to the low type and one at least as large as UH (fq g1=t ) to the high type. The di¤erence P t UH (fq g1=t ) UL (fq g1=t ) is increasing in (1 ) 1=t q and is thus minimized when 1 fq g =t = f0g : We use this observation to show that if the rents (vH ; vL ) are credible at
the state (pt ; fq g1=t ) ; (vH ; vL ) 2 0 state (pt ; f0g) ; (vH ; 0) 2
0 vH
(pt ; fq g1=t ) ; then we can …nd credible rents for the
(pt ; f0g) ; that lead to a lower incremental rent to the high type:
vH vL : Observe that the monopolist always pro…ts from leaving smaller incremental rents to the high type during the screening process as it allows him to charge more from revealing contracts. Therefore we can always …nd an optimal menu in which the pooling contract is spot. Now let us explain why it is without loss to assume that all revealing contracts are renegotiation-proof. As it is typical in contract-theory models, the constraint that guarantees that the low type does not want to imitate the high type does not bind. Therefore there is no reason to distort the allocation designed for the high type. It is then without loss to assume that the revealing contract is renegotiation proof3 . To simplify the exposition, we focus our attention on equilibria in which the monopolist only o¤ers spot pooling contract and renegotiation-proof revealing contracts for the remainder of this paper. Therefore take any on-path history ht associated to the state (pt ; f0g)
for some pt 2 (0; 1) : At that history, the monopolist o¤ers a menu mt with a pooling spot contract of the form xPt ; qtP ; 0 and a revealing contract of the form xR t ; fqH g . Using Property 3, we know that the low-type IR (4) binds and hence xPt = 3
P L qt :
There are outcome-equivalent equilibria in which the monopolist o¤ers pooling contracts that are not spot and revealing contracts that are not renegotiation proof. For instance, we can always construct an equilibrium in which the revealing contract is of the form (xR ; fq R g1=0 ) with q0R = qH and q R = qH " for > 0 (for some small " > 0). In such equilibria, after the acceptance of the revealing contract, the parties renegotiate in future period and the high type ends up consuming qH in every future period after the acceptance of the revealing contract.
18
5.5
E¢ ciency when
approaches one
Take an on-path history associated to an interior belief pt 2 (0; 1). According to our explanation in section (5.4), this history is associated to a state of the form (pt ; f0g) : We start showing that the revealing contract is accepted with positive probability. Indeed, if
the pooling contract were accepted with probability one then the monopolist payo¤ in that period would be bounded above by L (qL ): Therefore, since the next period would lead to the same state, we would conclude that V M (pt ; f0g) (1 ) L (qL ) + V M (pt ; f0g), implying V M (pt ; f0g)
L (qL ):
However, using our …ndings from section 5.2, there is a
feasible and e¢ cient plan for the state (pt ; f0g) that leads to the pro…t pt [
H (qH )
qL ] + (1
pt )[
L (qL )]
>
L (qL );
which would lead to a contradiction. We now provide a condition that links the pooling-contract quality qtP to the di¤erence between the current-period belief, pt ; and the posterior led by the acceptance of the pooling contract pt+1 : Indeed, for any interior quality qtP 2 (0; 1) ; the monopolist can construct
another feasible plan by changing this quality by an amount qtP 2 ( qtP ; 1 qtP ), simultaneously changing the pooling-contract transfer by L qtP and the revealing-contract transfer qtP is small, the pro…t from the
qtP to guarantee incentive compatibility. When
by
pooling contract changes approximately by
L
0
qtP
qtP : On the other hand, the revealing-
contract pro…t changes by qtP : Since the pooling contract is accepted with probability (pt ; pt+1 ) ; while the revealing contract is accepted with probability 1 (pt ; pt+1 ) ; a necessary condition for any positive quality is given by: 0 L
qtP =
1
(pt ; pt+1 ) (pt ; pt+1 )
=
pt pt+1 1 pt
(10)
:
Condition (10) allows us to link the ine¢ ciency of a pooling contract with the di¤erence between the prior and the posterior led by the acceptance of the pooling contract pt pt+1 that
0: Indeed, take " > 0 and notice that a necessary condition for pt+1
where we used4 pt
(1
pt )
0 L
(qL
")
(1
p0 )
0 L
(qL
")
qt1
pt+1 :=
< qL
" is
;
p0 . We conclude that the a large downwards distortion in the pooling-
contract quality is only possible when the pooling-contract acceptance also leads to a sig4
Because the equilibrium satis…es sequential separation dynamics.
19
ni…cant fall in the belief. However, for every # > 0; there is a maximal number of periods in which we may have pt+1 # along the equilibrium path. This leads to a bound on the number of periods in which the monopolist may o¤er pooling contracts containing large downward distortions. The payo¤ of this …nite number of periods becomes negligible as becomes close to one, which suggests that the expected discounted quality conditional on P the low type, (1 ) E( ;p) [ 1=0 q j L] ; cannot converge to a number lower than qL as converges to one. Since VL (ht ) 0 in every equilibrium and the high type can always follow the low type strategy, we have VH h0
(1
) E(
;p)
"
#
1 X
q jL ;
=0
qL as
which implies that the high-type payo¤ cannot converge to a value smaller than converges to one. Lemma 2 For any " > 0 there exists qL
" for any equilibrium ( ; p) :
2 (0; 1) such that if
then VH (h0 ; ( ; p))
>
Proof. Consider the belief evolution fpt g1 t=0 conditional on L and notice that the
sequential separation dynamics property implies that this sequence is decreasing. Let " > 0 and take := 2 " > 0: Notice that a necessary condition for qt < qL 0 (q L L
)
pt+1 1 pt
; where
pt+1 : Since p0
pt+1 := pt 0 (q
pt+1
) (1
L
p0 )
is:
pt for all t; we have :
(11)
Therefore, for (almost) all paths, we have the following lower bound for
1 P
(1
)
t
qt
t=0
conditional on L : 1 X
(1
t=0
)
t
"
If
pt+1 <(
+If
)
1 0
pt+1 (
(qL )
1 0
)(1 (qL
p0 )g
)(1
(qL p0 )g
) 0
#
:
Letting M ( ; p0 ) be the smallest positive integer among the upper bounds to the number of periods for which we can have
pt+1
(
)
1
0 L
(qL
) (1
p0 ) in any belief
evolution which satis…es sequential separating dynamics, we obtain the following lower M ( ;p0 ) " bound on the high type’s rent: M ( ;p0 ) qL 2 " qL Thus for all 2. n o M ( 1;p ) 0 max 1 2 " q ; 0 we have VH (h0 ) qL "; which proves the Lemma. L
20
The monopolist can always choose an allocation such that: a) each type consumes the e¢ cient quality in every period; b) the individual rationality constraint of the low type binds; and c) the high type is indi¤erent between his allocation and the low type’s one. Since the monopolist’s ability to extract rents from the high type by distorting the low-type allocation vanishes as
! 1; the monopolist cannot pro…t from taking a long real time to
screen the high type. Therefore the expected transfers and the expected quality of each type in each period converge to the values attained in the allocation satisfying a), b) and c). Thus, the possibility of renegotiation severely decreases the monopolist’s pro…ts. Proposition 2 Take a sequence of discount factors f n g1 n=1 ! 1 and an associated se-
quence of equilibria (
n ; pn ).
We have lim VL (h0 ; (
n ; pn ))
= 0; lim VH (h0 ; (
n ; pn ))
=
qL ; and lim E(
n ;pn )
"
(1
n)
1 X
t n
t=0
j
i
(qt )
i
#
(qi )j j i = 0 for i = L; H:
Proof. First notice that Lemma 6 in Appendix A implies VL (h0 ; (
n ; pn ))
= 0 for
0
all n because h is associated to the state (p0 ; f0g). Notice also that Lemma 2 implies P1 t lim inf VH (h0 ; ( n ; pn )) qL : Moreover, for i = L; H; we have E( n ;pn ) (1 n) t=0 n i (qt ) j i i
(qi ) for all n: Thus, using the quasilinearity of the environment, we have lim sup V M h0 ; (
n ; pn )
(1 p0 ) E( n ;pn ) (1 = lim sup +p0 E( n ;pn ) (1 (1 p0 ) L (qL ) + p0 [ H (qH )
n)
P1 n) t=0 P 1 t t=0
qL ] :
n
t n i
(qt ) j L VL (h0 ; ( n ; pn )) VH (h0 ; ( n ; pn )) i (qt ) j H
Therefore, if we could …nd a subsequence (without relabelling) in which either lim sup VH (h0 ; ( P1 t qL or lim inf E( n ;pn ) (1 n) t=0 n i (qt ) j i < i (qi ) for some i; then we would obtain V M (h0 ; ( n ; pn )) < V M P ef (see equation 8) for su¢ ciently large n, which leads to a contradiction.
It is interesting to use the …ndings above to compare the allocation obtained in our model with the one from the model of HT, in which the monopolist sells qualities in f0; 1g :
Under this unitary supply assumption, the authors show that the equilibrium outcome is the same whether trade opportunities are durable (selling) or subject to renegotiation (rental). Intuitively, when qt 2 f0; 1g the monopolist has only one device to screen the high type: 21
n ; pn ))
>
time. Hence, the well-understood Coasean forces drive down the monopolist’s pro…t when the parties become more patient. When qt 2 [0; 1] ; as in our model, the monopolist has two instruments to screen the high type: time and quality. Wang (1998) showed that when trade opportunities are durable (as in HT), these two instruments lead to the commitment pro…t. As shown here, the possibility of renegotiation drastically limits the monopolist’s pro…ts. The result that as the parties become more patient the equilibrium allocation becomes (approximately) e¢ cient can be interpreted as a version of the Coase conjecture for a nondurable-goods monopoly. One di¤erence between our result and the original version of the Coase conjecture is that in our model, the monopolist still makes more pro…ts from consumers with a higher valuation. Therefore, the lack of commitment reduces the monopolist’s market power only to the extent necessary to bring back e¢ ciency. All our results hold for every pro…t-maximizing renegotiation equilibrium. In the online appendix we prove Proposition 3, which shows that a pro…t-maximizing equilibrium exists. Proposition 3 There exists a pro…t-maximizing renegotiation equilibrium.
6
Concluding Remarks We studied a repeated principal-agent model with adverse selection and renegotiation.
We assumed that a monopolist o¤ers long-term contracts to a consumer. We showed that the monopolist dynamically screens high-taste consumers. We also showed that a version of the Coase conjecture holds for a nondurable-goods monopoly. As the parties become more patient, the allocation converges to the most pro…table allocation for the monopolist among all individually-rational, incentive-compatible and e¢ cient ones.
22
7
Appendix A: Characterization of Pro…t-Maximizing Values
7.1
Lemma 3
Lemma 3 Assume that the state is of the form (p; fq g1=0 ) for some p 2 f0; 1g : We have: Ai) If p = 0 we have V M (p; fq g1=0 ) = L (qL ) UL (fq g1=0 ): Aii) If p = 0 and P is an optimal plan then vL (P) = UL (fq g1 n=0 ):o 1
Aiii) If p = 0 and P is an optimal plan than any contract xj ; q j q0j
=0
probability o¤ ers = qL : Bi) If p = 1 we have V M (p; fq g1=0 ) = H (qH ) UH (fq g1=0 ): 1 Bii) If p = 1 and P is an optimal plan then vH (P) = UH (fq gn =0 ): o Biii) If p = 1 and P is an optimal plan than any contract xj ; q j
probability o¤ ers
q0j
1 =0
chosen with positive
chosen with positive
= qH :
Proof. We will prove Ai)- Aiii) (the proofs of Bi)-Biii) are analogous). Assume that the state is of the form (0; fq g1=0 ) : Assume that there exists an optimal plan P for this state that yields the rent vL (P ) to the low type. We …rst show that V M (0; fq g1=0 ) vL (P ) : The L (qL ) pro…t of the monopolist is: V M (0; fq g1=0 ) = a1L ; a1H ;
If min (1
) x1
a2L ; a2H c q01
2 X
h
ajL ; ajH
j=1
h ) xj
(1
c q0j
i
+ V M pj ; q j
i
:
(12)
> 0 assume without loss (relabelling if necessary) that + V M p1 ; q 1
(1
) x2
c q02
+ V M p2 ; q 2
:
Otherwise (relabelling the contracts if necessary) assume that a1L ; a1H = 1: In any case, Bayesrule implies p1 = 0: To simplify the notation below, we let x0 := x1 and q0 := q01 and notice that V M (0; fq g1=0 ) (1 ) [x0 c (q0 )] + V M 0; q 1 : Notice that the plan P con1 2 tained the promise vL1 ; vH 0; q 1 ; and (4) implies: vL (P ) := (1 ) [ L q0 x0 ] + vL1 : 1 From the de…nition of 0; q ; there exists a mixed-solution to the monopolist’s problem 1 : Thus there exists a pure solution P ~ = at the state 0; q 1 that yields the rents vL1 ; vH x ~1 ; q~1
; x ~2 ; q~2
j ;a ~L ; a ~H ; v~Lj ; v~H
2
j=1
; (~ p1 ; p~2 )
state 0; q 1 that gives rents (~ vL ; v~H ) such that v~L and proceeding analogous as above we have: V M 0; q 1 (1
)[
L q1
x1 ] +
(1 v~L1
vL1 :
23
from the monopolist’s problem at the vL1 : Using the optimality of P~ at 0; q 1
) x ~1
c(~ q11 ) + V M 0; q~1
(13) (14)
Letting x1 := x ~1 and q1 := q~11 and using (13) and (14) we obtain: (1 ) [x0 c (q0 )] + (1 ) [x1 c (q1 )]
V M (0; fq g1=0 ) vL (P )
(1
)[
L q0
x0 ] + (1
+
2
)[
V M 0; q~1 L q1
x1 ] +
2 1 vL :
Proceeding analogously and using the fact that V M and are bounded by K by assumption, PT 1 t M (1 ) t=0 [xt c (qt )] + T +1 K and vL (P ) for every T 2 N we haveV (0; fq g =0 ) PT P t T +1 T +1 (1 ) t=0 [ L qt xt ]+ K: Since K # 0 we conclude that limT !1 (1 ) Tt=0 t xt P limT !1 (1 ) Tt=0 t L qt vL (P ) ; which implies V M (0; fq g1=0 )
max (1 1
fq g
)
=0
1 X
t
L (qt )
vL (P ) =
L (qL )
vL (P ) :
(15)
t=0
Next, we will show that V M (0; fq g1=0 ) = L (qL ) UL q 1 ; proving Ai). For that we will construct a feasible plan P for the state (0; fq g1=0 ) : To construct the plan P ; take (vL ; vH ) 2 (0; fq g) and notice that the constraints (4) and (5) imply vi Ui (fq g1=0 ) for 1 1 i 2 fL; Hg : The plan P contains the contract x ; q that will be chosen exclusively by the 2 2 1 that is designed exclusively to the high low type (that is aL = 1) and the contract x ; q type (that is a2H = 1). Both contracts promise the future rents (vL ; vH ) 2 (0; fq g) : For the contract x1 ; q 1 ; we set x1 := L qL UL (fq g) and we set q01 := qL and q 1 := q 1 for > 0: For the contract x2 ; q 2 ; we set x2 := L qH UL (fq g) ; we set q02 := qH and q 2 := q 1 for > 0: Both contracts lead to the posterior 0 (notice that this is consistent with (7) because the contract x2 ; q 2 is accepted with probability 0). It is straightforward to check that the incentive-compatibility constraints (3) hold. Therefore to establish the feasibility of the plan P UL (fq g1=0 ) we must show that (4) and (5) hold. To see why (4) hold, we remark that vL implies (1 ) L q01 x1 + vL = (1 ) UL (fq g) + vL UL (fq g) : To check (5), notice that that vH UH (fq g1=0 ) implies (1
)
(1
)
= (1
2 L q0
0
x2 + vH
qH + (1
) @(1
)
X 0
) UL (fq g) + UH (fq g1=0 ) 1 (qH
From the feasibility of P ; we have V M (P ) least (1
) x1
c(q01 ) + V M (0; fq g1=0 ) = (1
which implies that V M (0; fq g1=0 ) that P is optimal and
L (qL )
V M (0; fq g1=0 ) =
q )A + UH (fq g1=0 )
UH (fq g1=0 ) :
V M (0; fq g1=0 ) and thus V M (0; fq g1=0 ) is at
)(
L (qL )
UL (fq g1=0 )) + V M (0; fq g1=0 ) ;
UL (fq g1=0 ) : Therefore from (15) we conclude L (qL )
24
UL (fq g1=0 ) :
(16)
which establishes Ai. Notice that this immediately implies that vL (P) = UL (fq g1=0 ) in any optimal plan P as if vL (P) > UL (fq g1=0 ) then (15) and (16) could not hold simultaneously. Thus we have Aii). n o Finally we prove Aiii). Take the optimal plan P and a contract xj ; q j that is accepted with positive probability. (7) implies that pj = 0: Applying Aii) to the plan P we have UL (fq g) = vL xj ; q j
(17)
= (1
) xj
j L q0
+ vLj
= (1
) xj
j L q0
+ UL
qj
>0
;
n o n o j where the last line used Aii) and (vLj ; vH )2 0; q j to conclude that vLj = UL q j : >0 >0 n o n o and using (17) to = L (qL ) UL q j Applying Ai) to conclude that V M 0; q j >0 >0 n o is equal to (1 ) L q0j + L (qL ) solve for xj ; we conclude that the pro…t from xj ; q j UL (fq g) : It follows from Ai) and the optimality of P that i h X ) L q0j + L (qL ) UL (fq g) : V M (P ) = L (qL ) UL (fq g1=0 ) = ajL ; ajH (1 j=1;2
Hence
7.2
ajL ; ajH > 0 implies q0j = qL :
Lemma 4
Lemma 4 For any state (p; fq g) ; if (vL ; vH ) 2
(p; fq g) then vH
vL :
Proof. Let A := sup fvL vH : (vL ; vH ) 2 (p; fq g) for some (p; fq g)g > 0: and assume towards a contradiction that5 A > 0: Take " 2 0; A(12 ) : Take (p; fq g) for which there is (vL ; vH ) 2 (p; fq g) such that vL vH > A ": Since (vL ; vH ) 2 (p; fq g) there is an optimal plan P for the state (p; fq g) for which vL (P) vH (P) > A ": Assume without loss that the low type chooses contract x1 ; q 1 with positive probability (according to the plan P) and thus vL (P) = (1 ) L q01 x10 + vL1 : Notice that the incentive-compatibility constraint for the high 1 ; and thus A type implies: vH (P) (1 ) H q01 x10 + vH " < vL (P) vH (P) A; which implies " A(1 ); a contradiction.
7.3
Lemma 5
Lemma 5 For any state (p; fq g) ; if (vL ; vH ) 2 5
Using I ) in De…nition 1 we know that A < 2K:
25
(p; fq g) then vH
vL +
:
Proof. Let A := sup fvH vL : (vL ; vH ) 2 (p; fq g) for some (p; fq g)g and assume towards a contradiction that6 A > : Take " 2 0; (1 ) A2 . Take (p; fq g) for which there is (vL ; vH ) 2 (p; fq g) such that vH vL > A ": There is an optimal plan P for the state (p; fq g) for which vH (P ) vL (P ) > A ": Assume without loss that the high type chooses the contract x1 ; q 1 with positive probability (according to the plan P ) and thus vH (P ) = (1
1 H q0
1 x1 + vH
vL (P ) + (1
)
where we have used the low-type IC (3) to conclude that vL (P ) Hence we have vH (P ) vL (P ) (1 ) + A and thus (1 A implies A < + ; contradicting A > : 2
(1 )
7.4
)
+ A; ) L q01 x1 + vL1 : + A > A ", which
Lemma 6
Lemma 6 Assume that the state is (p; f0g) for some p 2 [0; 1). Then, in any feasible plan P the high-type IR (5) is implied by the low-type IR (4). Furthermore, in any optimal plan P the low-type IR (4) binds. Proof. Consider the problem of solving (2) in which we ignore the high-type IR constraint (5) (it is only subject to (3), (4), (6) and (7)):Let P satisfy the constraints (3), (4), (6) and (7). We will show that (5) holds. For that assume w.l.o.g. that the low type chooses the contract 1 1 x1 ; q 1 with positive probability. Since vL1 ; vH 2 p1 ; q 1 and vH vL1 (Lemma 4) 1 and since (4) holds we have: 0
vL (P)
(1
)
1 H q0
1 x1 + vH
vH (P) ;
which implies vH (P) 0 and hence the constraint (5) holds. Next, assume that P is an optimal plan and low-type IR (4) does not bind: 0 < vL (P) : Consider the new plan P~ which di¤ers from P only in that the transfers from both contracts are uniformly increased by (1 ) 1 vL (P) : Trivially P~ leads to a higher pro…t, it satis…es (3), (4), (6) and (7) and from the …rst part of the Lemma it also satis…es (5). This contradicts the putative optimality of P:
7.5
Lemma 7
Lemma 7 We have (0;
qL ) 2
(0; f0g) and (
L qH ; H qH )
2
(1; fqH g) :
Proof. We only prove the Lemma for the state (0; f0g) (the proof for (1; fqH g) is analogous). Lemma 6 implies that in any optimal plan P~ for the state (p; f0g) the low-type IR (4) binds: vL P~ = 0: Therefore, there is (0; vH ) 2 (0; f0g) for some vH 0. Consider the plan P that contains two identical contracts ( L qL ; fqL ; 0g) associated to the promises (0; vH ): It suggests that both types randomize uniformly over these contracts and each 6
Using I ) in De…nition 1 we know that A < 2K:
26
contract leads to the posterior 0: It is straightforward to check that (3)-(7) hold and hence P is feasible at (0; f0g) and satis…es V M (P) = (1 ) L (qL ) + V M (0; f0g) : Notice also that (vL (P) ; vH (P)) = (0; (1 ) qL + vH ) On the other hand, Lemma 3 implies V M (0; f0g) = M (P) = L (qL ); which implies that P is optimal. Therefore (0; (1 ) qL + vH ) 2 L (qL ) and thus V (0; f0g) : And hence applying the same argument with (0; (1 ) qL + vH ) instead of (0; vH ) we conclude that 0; (1 ) (1 + ) qL + 2 vH 2 (0; f0g) : Proceeding inductively we have P for any T 2 N 0; (1 ) T=0 qL + T +1 vH 2 (0; f0g) and thus the closed-graph assumption in part IV of De…nition 1 implies (0; qL ) 2 (0; f0g) :
7.6
Lemma 8
Lemma 8 Let (p; fq g) be a state with p 2 (0; 1) and assume that there is an optimal plan P containing a pooling contract x1 ; q 1 leading to a posterior p1 p and to the next-period rents 1 ) and a revealing contract x2 ; q 2 . (vL1 ; vH i) There exists another optimal plan P~ containing the pooling contract x1 ; q 1 leading to 1 ) and a revealing contract of the form a posterior p1 < p and to the next-period rents (vL1 ; vH (xH ; fqH g) leading to the next-period rents ( L qH ; H qH ) 2 (1; fqH g) : The plan P~ and the plan P lead to the same payo¤ for all players. ii) If x2 ; q 2 is accepted with positive probability the quality q02 is equal to qH : Proof. We will use the optimal plan P to construct another optimal plan 2 j 1 = ; (p1 ; p2 ) at (p; fq g) such that v~L1 ; v~H P~ = x1 ; q 1 ; (xH ; fqH g) ; aL ; aH ; v~Lj ; v~H j=1
1 : We …rst construct P: ~ Then we show that it is feasible and …nally that it is optimal. vL1 ; vH 2 We use Lemma 7 to set v~L2 ; v~H := ( L qH ; H qH ) 2 (1; fqH g) : The transfers xH from (xH ; fqH g) are set to give the high type the same rent as in P : vH (P ) = (1 ) ( H qH xH ) + 1 2 ) [ H qH vH (P)] : Both types randomize in the same way in both plans v~H or xH = (1 so that contract x1 ; q 1 leads to the posterior p1 : By construction (4), (5), (6) and (7) hold. Trivially since the high type obtains the same rent from both contracts in both plans, the constraint (3) for his type holds. Therefore, in order to show that P~ is feasible we must show that the low-type IC (3) holds: If the low type purchases the contract x1 ; q 1 in the plan P~ he gets vL (P ) by construction. On the other hand, if he purchases (xH ; fqH g) he gets
(1
) (
L qH
~ xH ) + v~L2 = vH (P)
= vH (P)
:
Thus we invoke Lemma 5 to conclude (from the optimality of P) that vL (P ) vH (P ) and hence the low-type IC (3) holds. It is immediate to check that both plans give the same payo¤ to both types. Next we show that P~ is optimal. For that, it su¢ ces to show that the expected pro…t from the contract (xH ; fqH g) from the plan P~ is at least as large as the pro…t from x2 ; q 2 from the plan P: The expected pro…t from (xH ; fqH g) is (1 ) (xH c(qH )) + V M (1; fqH g) : Adding and subtracting the expression for vH (xH ; fqH g) ; the expected pro…t becomes (1
) (
H qH
c(qH )) +
2 V M (1; fqH g) + v~H
27
vH (P) =
H (qH )
vH (P):
(18)
~ is: (1 (from P)
The expected pro…t from x2 ; q 2
)
2 H q0
c(q02 ) +
h
V M 1; q 2
2 + v~H
1
2 vH (P): Using Bi) and Bii) from Lemma 3, the last expression is equal to (1 ) H (q0 ) + vH (P);which is weakly lower than (18), establishing i). Moreover, if q02 = 6 qH then H (qH ) the previous inequality is strict, which leads to a contradiction when x2 ; q 2 is accepted with positive probability. This establishes ii).
7.7
i
Lemma 9
Lemma 9 Take a state (p; f0g) with p 2 (0; 1) and consider a posterior p1 1 ) 2 (p ; q 1 (x10 ; q 1 ) and credible next-period rents (vL1 ; vH ) such that 1 0 1 (1
)
1 L q0
p, a contract
x10 + vL1 = 0:
(19)
There exists a feasible plan P F at the state (p; f0g) which contains the pooling contract q 1 ) and a revealing contract of the form (xH ; fqH g). Moreover, the contract (x10 ; q 1 ) leads to the posterior p1 and, for i 2 fL; Hg ; we have
(x10 ;
vi (P) = (1
1 i q0
)
x10 + vi1 :
Proof. We construct the feasible plan P F as follows. It contains the pooling contract 1 ) and the revealing contract (x ; fq g); q 1 ) leading to the posterior p1 and to the rents (vL1 ; vH H H which leads to next period rents ( L qH ; H qH ) and sets xH to make the high type indi¤erent between both contracts: (x10 ;
vH P F = (1
)
1 H q0
1 x10 + vH =
H
qH
(1
) xH :
1 vL1 and Notice that (19) implies (4). To see that (5) holds, notice that Lemma 4 implies vH 1 1 hence using (19) we conclude that if the high type chooses the contract (x0 ; q ) he gets at least 1 (1 ) q01 + (vH vL1 ) 0: The high type IC (3) holds by the de…nition of xH : Hence to check F the feasibility of P it su¢ ces to check that the low-type IC (3) holds. By construction, if the low qH : On the other ) ( L qH xH ) + L qH = vH (P) type chooses (xH ; fqH g) he gets (1 1 1 + v1 hand, if the low type chooses (x10 ; q 1 ) he gets (1 ) q x vH (P) qH ; L 0 0 L 1 1 where we used Lemma 5 to conclude that vH vL qH :
7.8
Lemma 10
Lemma 10 Let P be an optimal plan for the state
p; fq g
di¤ ers from P only in that all transfers are increased by (1 P=
x1 + (1
)
1
vL (P ) ; q 1
; x2 + (1
)
1
0
)
1
. Consider the plan P which vL (P ) :
vL (P ) ; q 2
j ; aL ; aH ; vLj ; vH
Then exactly one of the following holds: i) The plan P is optimal at (p; f0g) : ii) There exists an optimal plan P for the state (p; f0g) such that vH (P ) vH (P ) vL (P ) and V M (P ) > V M (P ) + vL (P ) :
28
2 j=1
; (p1 ; p2 )
vL (P ) <
Proof. Consider the following two problems: Problem La : Consists of …nding the best plan P~ =
x ~1 ; q~1
; x ~2 ; q~2
j ;a ~L ; a ~H ; v~Lj ; v~H
(maximizing the R.H.S. of (2)) given the prior p and subject to the following constraints: Incentivecompatibility (3), promise-keeping (6), Bayes-rule (7) and the constraint that guarantees that the low-type obtains at least the rent vL (P ) : vL P~ vL (P ) : In this problem we impose no IR constraint for the high type. Problem Lb : Consists of …nding the best plan P~ = x ~1 ; q~1 ; x ~2 ; q~2 ;a ~L ; a ~H ; v~j ; v~j L
H
2 j=1
2 j=1
; (~ p1 ; p~2 )
; (~ p1 ; p~2 )
(maximizing the R.H.S. of (2)) given the prior p and subject to the following constraints: Incentivecompatibility (3), promise-keeping (6), Bayes-rule (7) and the constraint that guarantees that the 0: In this problem we impose no IR constraint for low-type obtains at least the rent 0 : vL P~ the high type. Observation 1: Notice that Lemma 6 implies that the Problem Lb corresponds to the problem of …nding the optimal plan at the state (p; f0g). Observation 2: Notice that by assumption P is an optimal plan for the state p; fq g 0 and it satis…es all constraints of problem Problem La : Hence any solution to Problem La leads to a payo¤ weakly greater than V M (P) : Observation 3: Notice that a plan x ~1
(1
)
1
vL (P ) ; q~1
; x ~2
(1
)
is feasible for Problem La if and only if the plan
1
vL (P ) ; q~2 x ~1 ; q~1
j ;a ~L ; a ~H ; v~Lj ; v~H
2 j=1
; (~ p1 ; p~2 )
j ;a ~L ; a ~H ; v~Lj ; v~H
; x ~2 ; q~2
2 j=1
; (~ p1 ; p~2 )
is feasible for Problem Lb : Observation 4: P is feasible for problem Problem La and P is feasible for problem Problem Lb : First assume that ii) holds. Then V M (P ) > V M (P ) + vL (P ) = V M P ; which implies that P is not optimal at (p; f0g) :Therefore i) does not hold. Next, assume that i) does not hold: P is not optimal at at (p; f0g) : Therefore there exists an optimal plan P (p; f0g) such that
=
x1 ; q 1
; x2 ; q 2
j ; aL ; aH ; vLj ; vH
V M (P ) > V M P :
2
j=1
; (p1 ; p2 )
for the state (20)
Since V M P = V M (P ) + vL (P ) we have the second inequality in ii). Finally we prove that vH (P ) vL (P ) < vH (P ) vL (P ). Suppose towards a contradiction that vH (P ) vL (P ) vH (P ) vL (P ) : Observation 1 implies that P solves Lb and hence Lemma 6 implies vL (P ) = 0: Observation 3 implies that 9 8 < x1 (1 ) 1 vL (P) ; q 1 ; x2 (1 ) 1 vL (P) ; q 2 ; = P = 2 j : ; aL ; aH ; vLj ; vH ; (p1 ; p2 ) j=1
29
solves La and hence by construction vH (P ) vL (P ) = vH (P ) vL (P ) and vL (P ) = vL (P ) : Thus if vH (P ) vL (P ) vH (P ) vL (P ) we have vH (P ) vH (P ). Since P satis…es (5) at p; fq g 0 we conclude that P also satis…es (5) at p; fq g 0 and hence it is a feasible plan at V M (P
7.9
p; fq g
0
: Therefore the optimality of P
V M (P ) which implies V M (P )
)
at
p; fq g
0
implies
V M P ; contradicting (20).
Lemma 11
Lemma 11 Assume that (vL ; vH ) 2 (p; fq g) for some state (p; fq g) : There exists 3 optimal plans for the state (p; fq g) ; P1 ; P2 and P3 ; x1 (r); q 1 (r)
Pr = and (
1 2;
3)
; x2 (r); q 2 (r)
2 [0; 1]3 s.t.
P3
r=1
r
j ; aL (r); aH (r); vLj (r); vH (r)
= 1 and (vL ; vH ) =
P3
r=1
r
2 j=1
; (p1 (r); p2 (r)) ;
(vL (Pr ); vH (Pr )) :
Proof. Notice that (p; fq g) is a nonempty, convex and compact (see part IV of De…nition 1) subset of R2 . Thus from the Krein-Milman Theorem (Theorem 7.68 in Aliprantis and Border (2006)) (p; fq g) is the convex hull of its extreme points. From the Carathéodory Convexity Theorem (Theorem 5.32 in Aliprantis and Border (2006)) it can be written as a convex-combination of 3 (not necessarily di¤erent) extreme-point optimal plans.
7.10
Lemma 12
Lemma 12 Let P be an optimal plan for the state (p; f0g) with p 2 (0; 1). Assume that it contains a contract x1 ; q01 ; 0 leading to a posterior p1 p and a contract x2 ; fqH g leading to the posterior 1. If the contract x2 ; fqH g is accepted with positive probability then the rent obtained 1 := min by the high type from the contract x1 ; q01 ; 0 is set at the lowest level: vH H (p1 ; f0g): Proof. Assume that x2 ; fqH g is accepted with positive probability. We have V M (P ) =
(p; p1 ) (1 + (1
) x1
c(q01 ) + V M (p1 ; f0g) ) x2
(p; p1 )) (1
c(qH ) + V M (1; fqH g) :
From Lemma 6 we know that the high-type IR (5) can be ignored while the low-type IR (4) binds at the state (p; f0g): thus x1 = L q01 : Using this …nding, a lower bound to the rent of 1 : the high type is his rent from the contract x1 ; q01 ; 0 : vH x1 ; q01 ; 0 = (1 ) q01 + vH Lemma 9 implies that we can set the transfers x2 in such a way that this lower bound is achieved. Lemma 3 Bi) and Bii) imply that the pro…t from the contract x2 ; fqH g can be written as: 1 : Therefore V M (P ) equals (1 ) q01 vH H (qH ) (p; p1 ) (1
)
1 L (q0 )
+ V M (p1 ; f0g) + (1
(p; p1 ))
H (qH )
(1
)
1 ; the optimality of P implies v 1 := min Since (21) is strictly decreasing in vH H
30
q01
1 vH : (21)
H (p1 ; f0g):
7.11
Lemma 13
Two-period Outcomes: Take a state (p; fq g). An optimal plan P for this state leads to a distribution over current-period quality and transfers, future-period states and next-period continuation rents for the consumer: for j = 1; 2; with probability ajL ; ajH ; the plan P leads n o j j to the current-period transfer x ; current-period quality q0 ; the future-period state pj ; q j n o n o j and next-period promised-values vLj ; vH 2 pj ; q j : Now take for each state pj ; q j 2 p1 ; q 1
; p2 ; q 2
a distribution over optimal plans Gj such that: Z h i j j j j j j ~ ~ ~ vL ; vH = vL (P ); vH (P ) G dP :
As de…ned and explained below, this naturally yields a distribution over two-period outcomes. To the notation7 , we use Lemma 11 and assume that Gj is composed of 3 optimal plans: n simplify o j j j f1; 2; 3g P1j ; P2j ; P3j with probabilities 1 2 ; 3 . Let for (j; r) 2 f1; 2g Prj =
x1 (j; r); q 1 (j; r) ; x2 (j; r); q 2 (j; r) ; 2 s s aL (j; r); aH (j; r); (vL (j; r); vH (j; r))s=1 ; (p1 (j; r); p2 (j; r))
:
This procedure leads to a distribution over two-period outcomes: With probability ajL ; ajH ; n o is accepted in the …rst period. Following this event, with probability jr the contract xj ; q j the continuation plan Prj is selected in the next period. In this case, for s 2 f1; 2g ; with probability (asL (j; r); asH (j; r)) the contract (xs (j; r); fq s (j; r)g) is accepted. Hence, with probability ajL ; ajH
j r
(asL (j; r); asH (j; r)) we have the following outcome: two-period consumption:
q0j ; q1s (j; r) ; discounted transfers: xj + xs (j; r); two–periods ahead state: (ps (j; r); fq s (j; r)g) ; s (j; r)) : Note that the of de…nition of two-period outand two–periods ahead promises (vLs (j; r); vH comes does not distinguish the timing of the transfers over the two periods as long as they lead to the same (discounted) value. Lemma 13 Take a state (p; f0g) with p 2 (0; 1) and let P=
x1 ; q 1
; x2 ; fqH g
j ; aL ; aH ; vLj ; vH
2
j=1
; (p1 ; p2 )
be an optimal plan for that state.
Assume that P satis…es the high-type screening property: the contract x1 ; q 1 leads to a posterior p1 p and the contract x2 ; fqH g leads to the posterior 1 (p2 = 1). Assume also that the contract x2 ; fqH g leads to the next-period rents ( L qH ; H qH ) 2 (1; fqH g) : n o For j = 1; 2; let Gj be a distribution over optimal plans for the state pj ; q j such that 1 h i R j vLj ; vH = vL (P~ j ); vH (P~ j ) Gj dP~ j : 7
The general argument for the case in which Gj does not have a …nite support is analogous but requires heavier notation.
31
There exists an optimal plan P^ =
x ~1 ; q01 ; 0
; x2 ; fqH g
2
j ; aL ; aH ; v~Lj ; v~H
j=1
; (p1 ; p2 )
for the state (p; f0g) and distribution G1 and G2 of optimal plans for the states (p1 ; f0g) h and i R j j j j j ~ ~ (p2 ; fqH g) respectively such that for j = 1; 2 we have v~L ; v~H = vL (P ); vH (P ) G dP~ j :
Moreover, both triplets outcomes.
^ G1 ; G2 P;
and P; G1 ; G2 lead to the same distribution of two-period
Proof. We start with an observation that will be used throughout this lemma: The argument given in the proof of Lemma 8 implies that whenever the revealing contract is of the form x2 ; fqH g and leads to the next-period rents ( L qH ; H qH ) 2 (1; fqH g) then if the high-type IC (3) holds with equality the low type cannot pro…t from mimicking the high type and purchasing this contract x2 ; fqH g : To slightly simplify the exposition, we assume that G1 is a probability distribution over 3 optimal plans at the state8 p1 ; q 1 : Assume that for z 2 f1; 2; 3g ; G1 puts probability 1z 1 in the continuation plan Pz1 : Notice also that from Lemma 6 the low-type IR binds: 0 = (1
)
1 L q0
x1 +
3 X
1 z vL
z=1
Pz1 :
(22)
Furthermore, it follows that the transfers x2 should be set to make the high type indi¤erent between both contracts (1
)
H qH
2
x
+
H qH
= (1
1 H q0
)
1
x
+
3 X z=1
1 z vH
Pz1 :
(23)
Otherwise we could use increase x2 without violating any constraint (see the …rst observation of this lemma). Using Lemma 3 Bi) and the equations (22) and (23), the pro…ts from the plan P can be written as: " # 3 X 1 1 M (p; p1 ) (1 ) L (q01 ) + p1 ; q 1 (24) z vL Pz + V 1 + (1
(p; p1 ))
"
z=1
H
(qH )
(1
)
q01
3 X z=1
1 z
vH Pz1
vL Pz1
#
:
Observation 1: Notice that (24) is a linear function of the probability 1z that the plan Pz1 is chosen in the next period: F 11 ; 12 ; 13 = 11 F (1; 0; 0) + 12 F (0; 1; 0) + 13 F (0; 0; 1). Notice that we may assume w.l.o.g. that 1z 2 (0; 1) for all z: Let := f(1; 0; 0) ; (0; 1; 0) ; (0; 0; 1)g and let 8
Lemma 11 guarantees that this is w.l.o.g. in terms of payo¤s. The general argument requires replacing the summations below with integrals and leads to a somewhat heavier notation.
32
e := arg max 2 F ( ): Take 2 e and notice that F 11 ; 12 ; 13 inequality if e = 6 : Therefore if we show that there exists an optimal plan P^ =
x ~1 ; q01 ; 0
j ; aL ; aH ; v~Lj ; v~H
; x2 ; fqH g
V M P^ > F
1 1;
1 2;
j=1
; (p1 ; 1)
2 e then it will be proved that e =
at (p; f0g) such that V M P^ = F ( ) for all why, notice that if e 6=
2
F ( ) with strict
then there exists ^ 2
such that F ( ^ ) < V M P^
: To see
and therefore
1 3
= V M (P) ; which contradicts the optimality of P: Distribution over two-period outcomes led by P ( ) :Take 2 e and assume w.l.o.g. that = (1; 0; 0): Consider the following distribution over two-period outcomes led by P ( ) : At the state (p; f0g) the monopolist o¤ers the plan x1 ( ) ; q 1
P ( )=
j ; aL ; aH ; vLj ; ( ) ; vH ( )
; x2 ( ) ; fqH g
2 j=1
; (p1 ; 1) ;
where x1 ( ) and x2 ( ) are given by: 0 = (1 (1
)
H qH
)
x2 ( ) +
1 L q0 H qH
x1 ( ) + vL P11 : = (1
)
1 H q0
(25)
x1 ( ) + vH P11 ;
(26)
2 ( ) := ( q ; 1 1 1 ( ) := v (1; fqH g) : and vL2 ( ) ; vH where vL1 ( ) ; vH L H H qH ) 2 L P1 ; vH P1 We claim that the plan P ( ) is feasible at (p; f0g): Bayes rule (7) is trivially satis…ed. Since vL P11 ; vH P11 2 ; (6) is also satis…ed. The low-type IR is satis…ed by (25). p1 ; q 1 1 Since the state starts with promises f0g ; the high-type IR (5) is also immediately satis…ed (see Lemma 6). Moreover, notice that the high type is indi¤erent between both contracts by construction. Hence his IC (3) holds. Finally, the …rst observation of this lemma implies that the low-type IC (3) also holds. In summary, we have concluded that the plan P ( ) is feasible at (p; f0g). Next, we will show that it leads to pro…ts equal to F (1; 0; 0): To verify the last claim, consider the following distribution over two-period outcomes. Assume that at the state p1 ; q 1 the monopolist chooses the (optimal for that state) continuation 1
plan P11 and, at the state (1; fqH g) ; the distribution over continuation plans is the same as the one led by the acceptance x2 ; fqH g of at P. By construction, this distribution over two-period outcomes leads to the pro…t F (1; 0; 0): Therefore V M (P ( )) = F (1; 0; 0): Now we note that Observation 1 implies that e = : Possibility 1: The plan P~11 that di¤ ers from P11 in that the transfers in all contracts are increased by (1 ) 1 vL P11 is optimal at (p1 ; f0g) : 1 The optimality of P~11 implies that 0; vH P11 vL P11 2 (p1 ; f0g) : Let v~L1 ; v~H := 1 1 2 2 0; vH P1 vL P1 ; let v~L ; v~H := ( L qH ; H qH ). We claim that the plan P^ =
1 L q0 ;
q01 ; 0
; x2 ( ) ; fqH g
33
j ; aL ; aH ; v~Lj ; v~H
2 j=1
; (p1 ; 1)
(27)
is feasible at (p; f0g). Notice that the di¤erence between the plan P^ and the plan P ( ) is that the 1 ; while the plan plan P^ contains the pooling contract L q01 ; q01 ; 0 with the promises v~L1 ; v~H P ( ) contains the pooling contract x1 ; q 1 with the promises vL P11 ; vH P11 : Notice that the di¤erence between the current-period transfers from x1 ( ); q 1 and L q01 ; q01 ; 0 :(1 1 ) x1 ) 1 vL P11 ; is exactly o¤set by the (discounted by ) value from the L q0 = (1 di¤erence in transfers between P11 and P~11 . Therefore, it is straightforward to check that the ^ This immediately implies that if P~ 1 is chosen in feasibility of P ( ) implies the feasibility of P: 1 the next period at (p1 ; f0g) (and the distribution G2 is chosen at (1; fqH g) after the acceptance of x2 ( ) ; q 2 ) then we have the same distribution over two-period outcomes as the one led by P ( ): Possibility 2: The plan P~11 that di¤ ers from P11 in that the transfers in all contracts are increased by (1 ) 1 vL P11 is not optimal at (p1 ; f0g) : Therefore ii) in Lemma 10 applies and hence there exists an optimal plan P for the state (p1 ; f0g) such that vH (P ) vL (P ) < vH P11 vL P11 and V M (P ) > V M P11 + vL P11 : 2 := ( q ; 1 := (0; v (P ) ~2 vL (P )) 2 (p1 ; f0g); let v~L2 ; v~H Let v~L1 ; v~H L H H qH ) and set x H 2 1 1 to satisfy (1 ) H qH x ~ + H qH = (1 ) q0 + v~H : An argument analogous to the one showing the feasibility of the plan P ( ) above implies that the plan P^ =
1 L q0 ;
q01 ; 0
; x ~2 ; fqH g
j ; aL ; aH ; v~Lj ; v~H
2 j=1
; (p1 ; 1)
is feasible at (p1 ; f0g). Notice that since vH (P ) vL (P ) < vH P11 vL P11 the rent of the high type from the contract x1 ( ); q 1 in the plan P ( ) is strictly greater than the one from ^ Indeed (1 the contract L q01 ; q01 ; 0 in the plan P: ) H q01 x1 ( ) + vH P11 is equal to (1
)
1 H q0
+
vH P11
vL P11
> (1
)
1 H q0
+ [vH (P )
which implies that x ~2 > x2 ( ): Using this inequality, V M p1 ; q 1
1
vL (P )] ;
= V M P11 and V M (P ) >
V M P11 + vL P11 (from Lemma 10) we conclude that the payo¤ from P ( ) (p; p1 ) (1 + (1
1 L q0
)
(p; p1 )) (1
c q01
) (x2 ( )
+ vL P11 + V M P11
c (qH )) + V M (1; fqH g)
is strictly lower than (p; p1 ) (1 + (1
)
(p; p1 )) (1
1 L q0
c q01
) (~ x2
+ V M (P ) c (qH )) + V M (1; fqH g) ;
^ This shows that the plan P ( ) leads to a strictly lower pro…t than which is the payo¤ from P: ^ P; a contradiction. Thus we must have Possibility 1. Recall that we have shown that e = . Next, observe that since P^ is optimal then Lemma 1 12 implies that v~L1 ; v~H = (0; min H (p1 ; f0g)) ; which of course does not depend on the choe sen element 2 . It then follows that 0; vH Pz1 vL Pz1 = 0; vH Py1 vL Py1 for all
34
(x; y) 2 f1; 2; 3g2 : Clearly this implies that P^ is the same for all 2 : But then after the acceptance of theooptimal plan P^ in the …rst period the monopolist may randomize among the plans n 1 ~ ~ P1 ; P21 ; P~31 with probabilities ( 11 ; 12 ; 13 ) to generate the desired distribution over two-period outcomes, which proves the Lemma.
7.12
Lemma 14
Lemma 14 If (vL ; vH ) 2
(0; fq g) ; then vH
Proof. Let A := inf fvH
vL
qL :
vL : (vL ; vH ) 2
towards a contradiction that A <
(0; fq g) for some state (0; fq g)g and assume (1 )( qL A) : Take a state (0; f~ q g) for qL : Take " 2 0; 2
which there is (~ vL ; v~H ) 2 (0; f~ q g) such that v~H v~L < A + ": Therefore we can …nd an optimal plan P for the state (0; f~ q g) such that vH (P) vL (P) < A + ": Assume w.l.o.g. that the low with positive probability. Aiii) in Lemma 3 implies that type chooses the contract x1 ; q 1 1 2 q01 = qL : The Bayes’rule constraint (7) implies that p1 = 0 and thus vL1 ; vH
implies
1 vH
vL1
0; q 1
A: Therefore the incentive-compatibility constraint of the high type (3) implies
vH (P)
(1
)
H qL
1 x1 + vH
vL (P) + (1
)
qL + A;
which implies that vH (P) vL (P) (1 ) qL + A. Therefore we have A+" > (1 A; which implies " > (1 ) ( qL A) and leads to a contradiction.
8
>0
)
qL +
Appendix B
8.1
Proposition 1
We prove in the online appendix (Section 10)) that if we assume that there is an equilibrium that does not present sequential separating dynamics, we can …nd a state (pt ; f0g), an optimal plan Pt for that state which is not consistent with an equilibrium that presents sequential separating dynamics and a continuation equilibrium following that plan9 that satis…es De…nitions 1 and 2. Thus at ht the equilibrium play is led by the optimal plan Pt :=
j m(ht ); aL ht ; aH ht ; vLj ht ; vH ht
2 j=1
; (p1 ht ; p2 ht ) ;
n o n o o x1t ; q t;1 ; x2t ; q t;2 , that is not consistent with sequential separation n o dynamics. Hence, one contract, say x1t ; q t;1 , leads to a posterior p1t+1 < pt ; while the other
where10 m(ht ) =
n
9
It is possible to prove the following stronger result: we can construct another equilibrium containing an on-path history ht of length t associated with the state (pt ; f0g) (with pt 2 (0; 1)) in which the monopolist o¤ers a menu that violates sequential separating dynamics. However, the weaker result su¢ ces for a contradiction. 1 10 We write x1t ; q t;1 for x1t ; q t;1 =t to save on notation.
35
n o contract x2t ; q t;2 leads to the posterior p2t+1 2 (pt ; 1) : Write ht+1 for the t + 1 history reached j n o by the acceptance of x1t ; q t;1 for j = 1; 2: Notice that we can use Bayes’rule to …nd out the n o probability jt that contract xjt ; q t;j is accepted. Since pt = 1t p1t+1 + 2t p1t+1 and 1t + 2t = 1; we have
1 t
=
p2t+1 2 pt+1
pt p1t+1
and
2 t
=
pt p2t+1
p1t+1 p1t+1
:
We will show that there exists a feasible (not necessarily optimal) plan P~ at the state (pt ; f0g) ~ >V M (Pt ) = V M ht ; which is a contradiction. For that we will construct two such that V M (P) feasible plans at (pt ; f0g) ; P~1 and P~2 ; and show that there exists n a randomization among o them that M M M leads to a strictly greater pro…t than V (Pt ); implying max V P~1 ; V P~2 > V M (Pt ). n o The construction of P~1 ; P~2 and the randomization among them will use information from Pt : We remark that the putative optimality of Pt at (pt ; f0g) implies n othat the low-type IR (4) binds 1 with positive probability we (see Lemma 6) and hence since he chooses the contract xt ; q t;1 have: (1 ) L qtt;1 x1t + vL1 (ht ) = 0: (28) o n for a Notation Conventions for Contracts: i) For k = 1; 2; we will write xs ; q s;k contract that is o¤ ered in period s in the putative equilibrium play. ii) We will add the superscript R to a transfer to remind the reader that we are explicitly referring to a contract that leads to the posteriornone. iii) o When, while constructing the “alternative play”generated by the randomization ~ ~ between P1 ; P2 ; we use the same contract as used in the putative-play we will keep the same n o notation. For example, the alternative-play generated by the randomization between P~1 ; P~2 will n o that is o¤ ered in the equilibrium play. On the other hand, make use of the contract x1t ; q t;1 when we create a new contract that is not used in the putative-equilibrium play and we do not present explicitly itsntransfers, then we add the “ ” on the top of the transfers. For example, we o s;1 R ~s ; q for contracts o¤ ered in the “alternative play”. x ~s ; fqH g or x (for k 2 f1; 2g) for a putativeNotation Conventions for Histories: i) We write ht+s k play history leading to a posterior smaller than one. We write ht+s for a putative-play history ~ t+s (for k 2 f1; 2g) for an “alternative-play” history leading to the posterior one11 . ii) We write h k ~ t+s for an “alternative-play” history leading leading to a posterior smaller than one. We write h to the posterior one. n o Construction of P~1 : P~1 contains the pooling contract x1t ; q t;1 (the same from Pt ) and a revealing contract of the x ~R t ; fqH g : The consumer will randomize among the contracts n form o n o t;1 1 in such a way that xt ; q leads to the posterior p1t+1 (the same as the one from x1t ; q t;1 n o t;1 1 in Pt ), while the contract x ~R leads t ; fqH g leads to the posterior 1. The contract xt ; q
When a menu contains two contracts leading to posteriors smaller then one then we will use ht+s and 1 When a menu contains one contract leading to the posterior smaller than one and another equal to one then we will use ht+s and ht+s : 1 11
ht+s 2 :
36
1 ht to the next-period rents vL1 ht ; vH (also obtained from Pt ). The contract x ~R t ; fqH g leads to the next-period rents ( L qL ; H qH ) 2 (1; fqH g) (see Lemma 7). Using (28), Lemma n o9 1 implies that P~1 is feasible at (pt ; f0g) : Observe that conditional on P~1 the contract x ; q t;1 t
pt ; p1t+1
is accepted with probability : Construction of P~2 : Take an arbitrary12 (0; vH ) 2 (pt ; f0g). P~2 contains 2 identical contracts ( L qL ; fqL ; 0g) which promise the next-period rents (0; vH ) and lead to the same posterior pt : It is straightforward to check that P~2 satis…es (3), (4), (6) and (7). Since from Lemma 6 the constraint (5) is implied by the constraint (4) at the state (pt ; f0g), the plan P~2 is feasible at (pt ; f0g) : Probability t thatn theoplan P~1 is chosen: We choose t in such a way that the probability that the contract x1t ; q t;1 is chosen in the putative optimal plan Pt is the same as the probability that it is chosen under the randomization between the feasible plans P~1 and P~2 : p2t+1 pt p2t+1 p1t+1
=
t
1 pt 1 p1t+1
:
(29)
Since p1t+1 < pt < p2t+1 < 1 it follows that t 2 (0; 1) : To contradict the optimality of Pt ; we will show that M ~ M M t V M (P~1 ) + (1 (30) t t )V (P2 ) > V (Pt ) = V (h ): Notice that we can rewrite (30) as: h " pt ; p1t+1 (1 t
+(1
) x1t
pt ; p1t+1 ) (1
n o c(qtt;1 ) + V M p1t+1 ; q t;1 ) x ~R t
>t
c(qH ) + V M (1; fqH g)
i #
(31)
+ (1 ) L (qL ) + V M (pt ; f0g) t ) (1 i 3 h n o 2 t;1 1 (1 1 M p1 ; q t;1 ) x c(q ) + V t t t+1 h n o >t i 5 > 4 t;2 2 2 M + t (1 ) xt c(q ) + V p2t+1 ; q t;2 >t
Alternative nPlay: o We believe that it is helpful to see the ~ ~ domization over P1 ; P2 as an alternative play of the game. n o for the history led by the acceptance of x1t ; q t;1 (resp. n o starts with the state p1t+1 ; q t;1 (resp. (1; fqH g)). We >t
outcome obtained by the ran~ t+1 ) ~ t+1 (resp. h We write h 1 ~ t+1 x ~R t ; fqH g ) in period t (h1
~ t+1 for the history led write h 2
~ t+1 starts with the state (pt ; f0g)): For h ~ 2 by the acceptanceo of ( L qL ; fqL ; 0g) in period t (h 2 n ~ t+1 ; h ~ t+1 ; h ~ t+1 ; given ht ; let ~ ~ h t h be the probability of reaching the history h and let 1 2
~ VtM h
~ being be the expected continuation pro…t in period t conditional on the history h
12
Recall that Lemma 6 implies that if P is an optimal plan at the state (pt ; f0g) then the low-type IR binds. Thus (vL ; vH ) 2 (pt ; f0g) implies vL = 0:
37
~=h ~ t+1 : In this case reached. For example, take h 1 h n o i t;1 1 M 1 (1 ) xt c(q ) + V pt+1 ; q t;1 :
t
~ t+1 := h 1
~ t+1 := pt ; p1t+1 and VtM h 1
t
>t
t+1 Putative Equilibrium Play: For the putative equilibrium play histories h 2 ht+1 ; 1 ; h2 t+1 t+1 M 1 we de…ne t (h) andh Vt (h) analogously. For example, take h = h : In this case, h := t t 1i 1 n o
and VtM ht+1 := (1 1 t
) x1t
c(q t;1 ) + V M p1t+1 ; q t;1
>t
: Using the fact that
t
ht+1 = 1
~ t+1 (see 29) and V M ht+1 = V M h ~ t+1 ; (31) holds if and only if h t t 1 1 1 t
~ t+1 h
~ t+1 VtM h
+
t
~ t+1 V M h ~ t+1 > h t 2 2
t
ht+1 VtM ht+1 : 2 2
(32)
To better explain our argument, we …rst assume that the monopolist does not randomize over menus along the equilibrium play. In the appendix (see Step 3 in Section 10) we explain how to modify our argument to deal with case in which the monopolist randomizes. Our argument to under prove (32) depends on the continuation equilibrium chosen by the monopolist at history ht+1 2 the putative equilibrium play. We deal with Case 1 in the here. After a complete explanation of this case, we explain the main points from the general case. As it will be clear below, the explanation in the text considers a certain putative-play belief evolution. We relegate some missing details to the online appendix. o n leading to a Case 1: The monopolist o¤ ers a menu with a pooling contract x1t+1 ; q t+1;1 o n o n 1 ; q t+1;1 : The contract x at ht+1 posterior p1t+2 pt and a revealing contract x2t+1 ; q t+1;1 t+1 2 t+2 leads to the next-period rents vL (ht+2 1 ); vH (h1 ) t+1 Remark 1: We remind the reader n othat since the history h2 is associated to an optimal : We invoke Lemma 8 to …nd a payo¤ equivalent plan P ht+1 for the state p2t+1 ; q t;2 2 n>t o which di¤ ers from P ht+1 by replacing (if (for all players) plan for the state p2t+1 ; q t;2 2 >t o n by a revealing contract of the form xR necessary) the revealing contract x2t+1 ; q t+1;1 t+1 ; fqH g that leads to the next-period rents ( L qH ; H qH ) : This modi…cation does not change the value of VtM ht nor the incentives of the consumers. It is used because it slightly simpli…es the algebra below and it is kept for the remainder of this proof.n o Using the Remark above, the pro…t V M p2t+1 ; q t;2 is equal to >t
p2t+1 ; p1t+2 + 1
h
(1
p2t+1 ; p1t+2
) x1t+1 (1
t;1 c(qt+1 ) + V M p1t+2 ; q t+1;1
) xR t+1
c(qH ) :
>t+1
i
n o 1 ; q t+1;1 We write ht+2 for the history led by the acceptance of x and ht+2 the history t+1 1
t+1 t+2 t+2 led by the acceptance of xR ; we write t (h) for the t+1 ; fqH g at h2 : For h 2 h1 ; h t M t probability of reaching h from h and Vt h for the expected payo¤ at ht conditional on h
38
being reached. For example, take h = ht+2 1 : We have h VtM ht+2 := (1 ) x2t c(q t;2 ) + (1 ) x1t+1 1
t
ht+2 := 1
2 t
p2t+1 ; p1t+2 and
t+1;1 c(qt+1 ) +
2
V M p1t+2 ; q t+1;1
>t+1
i
:
(33) t+1 M ~ Alternative Play at h2 : Notice that V (pt ; f0g) is weakly greater than the pro…t from ~ t+1 by assuming any feasible plan at (pt ; f0g) : Therefore we calculate a lower bound to V M h t
2
that the monopolist plays according to a feasible play at (pt ; f0g): n o Feasible Plan at (pt ; f0g): The monopolist o¤ers a pooling contract x ~1t+1 ; q t+1;1 leading n o 1 ; q t+1;1 to the posterior p1t+2 and a revealing contract x ~R ; fq g : The acceptance of x ~ t+1 t+1 H t+2 leads to next-period rents vL (ht+2 ~1t+1 are set at 1 ); vH (h1 ) . The transfers x
1 L qt+1 + 1
vL (ht+2 1 );
while the transfers x ~R t+1 are set to make the high type indi¤erent between both contracts. We will show that this plan is feasible at (pt ; f0g) It is immediate to see that the constraints (7) n o and (6) t+1;1 1 hold. The low-type IR (4) holds because if he purchases the contract x ~t+1 ; q he gets 0 by construction. The high-type IR (5) follows from the low-type one because the state starts with the promises f0g (Lemma 7). The high-type IC (3) holds by construction For the low-type IC (3), notice that if the low type purchases x ~R (1 )x ~1t+1 ; which is t+1 ; fqH g he gets L qH equal to (1 )x ~1t+1 qH : (34) H qH The term in brackets in (34) is equal the ~R t+1 ; fqH g , which by construction o high-type rent from x n is equal to his rent from x ~1t+1 ; q t+1;1 (1
)
= (1
)
t+1;1 H qt+1
t+1;1 qt+1
: Thus (34) equals
x ~1t+1 + vH (ht+2 1 ) qH +
qH
vL (ht+2 1 )
vH (ht+2 1 )
qH
0;
where we used Lemma 5 to conclude that vH (ht+2 vL (ht+2 1 ) 1 ) n qH o 0: ~ t+2 ~ t+1 and h ~ t+2 for the history led by the acceptance of x at h ~1t+1 ; q t+1;1 We write h 2 1 o n t+2 : For h ~ 2 h ~ t+2 ; h ~ t+2 ; we write for the history led by the acceptance of x ~R t+1 ; fqH g at h 1 t
~ h
~ from ht in the alternative play and V M h ~ t for the for the probability of reaching h t ~ being reached. Notice that since we used a feasible but not expected payo¤ at ht conditional on h ~ t+1 V M h ~ t+1 is bounded below by necessarily optimal plan at (pt ; f0g) ; the expression h t
~ t+2 h 1
t
VtM
~ t+2 h 1 t
13
+
~ t+1 h
t
~ t+2 h
~ t+2 VtM h
~ t+1 VtM h
+
t
2
t
2
: Therefore to show (32) it su¢ ces to prove that
~ t+2 V M h ~ t+2 + h t 1 1
t
~ t+2 h
~ t+2 VtM h
(35)
t+2 VtM ht+2 + t ht+2 VtM ht+2 : > t h1 1 n o ~ 2 h ~ t+1 ; h ~ t+2 ; h ~ t+2 write p(h) ~ for the posterior at the alternative-play history13 h. ~ For h 1
~ t+1 ) = p(h ~ t+2 ) = 1 and p(h ~ t+2 ) = p(ht+2 ): We have p(h 1 1
39
By construction we have X X ~ t h = t+2 ~ fh ~ t+2 g ~ t+1 ;h ~ t+2 ;h h2fht+2 h2 g 1 ;h 1 Trivially this implies that
t
ht+2 = 1
X
t (h)
~ t+2 ~ t+1 ;h ~ t+2 ;h h 1
~ f h2
~ t+2 and h 1
t
t
ht+2
h2f
g =
X
~ p(h) ~ = h
t
t
~ t+2 h
+
t (h) p(h):
t+2 ht+2 1 ;h
t
~ t+1 h
g
:
~ t+2 : Notice that low type consumes q t;2 in Comparison between VtM ht+2 and VtM h t 1 1 t+1;1 t+2 period t and qt+1 in period t + 1 and expects the rents vL (ht+2 1 ) for period t + 2 under h1 : Since the low-type IR (4) binds under Pt at (pt ; f0g) ; we have:
0 = (1
)
t;2 L qt
t+1;1 L qt+1
x2t +
x1t+1
+
2
vL (ht+2 1 ):
(36)
t+1;1 in period t + 1 and expects On the other hand, the low type consumes qL in period t and qt+1 t+2 t+2 ~ t+2 by ~ the rents vL (h1 ) for period t + 2 under h1 : Since the low type obtains 0 rents under h 1 construction, we have:
0 = (1 Finally, notice that h ~ t+2 = (1 ) VtM h 1
)
L qL
L qL
L qL
c (qL ) +
t+1;1 L qt+1
+
c(q t;1 )
x ~1t+1
x ~1t+1
+
2
+
2
vL (ht+2 1 ):
(37)
V M p1t+2 ; q t+1;1
>t+1
Plugging (36) into (33) and (37) into (38) and subtracting (33) from (38) we have: ~ t+2 VtM h 1
= (1 VtM ht+2 1
and conclude: ~ t+2 Fact 1 If qtt;2 = qL then VtM h 1 Comparison between VtM ht+2
)
L (qL )
i
: (38)
t;2 L (qt )
~ t+2 > V M ht+2 : VtM ht+2 otherwise VtM h t 1 1 1 ~ t+2 and VtM h
: Notice that we have not assumed that
t+1 R the transfers xR make the high type t+1 at the revealing contract xt+1 ; fqH g o¤ered at h2 indi¤erent between both contracts14 . Therefore, only for the comparison between VtM ht+2 ~ t+2 ; we increase xR if necessary to guarantee the high type is indi¤erent between and V M h t
contracts15 .
both obtain:
t+1
In this case, using an argument absolutely analogous to the one above, we ~ t+2 VtM h
VtM ht+2
and conclude: ~ t+2 Fact 2 If qtt;2 = qL then VtM h
(1 VtM ht+2
)
L (qL )
t;2 L (qt )
~ t+2 otherwise VtM h
> VtM ht+2
:
14
This would necessarly be the case if p1t+2 > 0; but not necessarly true if p1t+2 = 0: 15 ~ t+2 This could only increase V M ht+2 and thus decrease the di¤erence V M h
40
V M ht+2
:
~ t+1 . First consider the history ht+2 : The Comparison between VtM ht+2 and VtM h assumption that the monopolist plays a pure strategy allows us to calculate the high type payo¤ at ht conditional on ht+2 being reached by: vH (Pt ) = (1
t;2 H qt
)
On the other hand we have: h VtM ht+2 = (1 ) x2t
x2t +
xR t+1
H qH
c qtt;2 + (1
) xR t+1
where we used Bi) in Lemma 3 to replace V M (1; fqH g) with straightforward algebra we have: VtM ht+2
= (1
)
t;2 H (qt )
+
H (qH )
+
2
H qH :
2
c(qH
(39)
i c(qH ) ;
(40)
c(qH ): Using (39), (40) and vH (Pt ) :
(41)
Next notice that16 ~ t+1 VtM h
)x ~R t
= (1
c(qH ):
(42)
On the other hand, by construction the high type obtains the rent vH (Pt ) if he chooses the ~ contract x ~R t ; fqH g from the plan P1 : Thus: vH (Pt ) =
)x ~R t +
(1
H qH :
(43)
vH (Pt ) :
(44)
From (42) and (43) we immediately have: ~ t+1 VtM h
=
H (qH )
Equations (41) and (44) imply ~ t+1 VtM h
VtM ht+2
~ t+1 Fact 3 If qtt;2 = qH then VtM h
= (1
)
VtM ht+2
h
H (qH )
~ t+1 otherwise VtM h
~ t+2 Facts 1,2 and 3 immediately imply that VtM h 1 ~ t+1 and VtM h
i t;2 (q ) : H t
(45) > VtM ht+2
~ t+2 ; VtM h VtM ht+2 1
:
VtM ht+2
~ t+1 ~ t+2 > V M ht+2 or V M h VtM ht+2 . Furthermore either VtM h t t 1 1
>
VtM ht+2 ; which implies17 (35). General Argument: : We conclude that if the monopolist intends to o¤er a menu with sequential separating dynamics in which the posteriors lie in the set [0; pt ] [ f1g in period t + 1 16
As above, we use Bi) from Lemma 3 to replace V M (1; fqH g) with hc(qH ): 17 ~ t+1 ~ t+1 VtM h Notice that we can rewrite (35) as: t h
t
~ t+2 h
h
~ t+2 VtM h
forward to check that
VtM ht+2 t
~ t+1 h
i
+
> 0 and
t t
~ t+2 h 1
h ~ t+2 VtM h 1
~ t+2 > 0 and h 1
41
t
~ t+2 VtM h 1 ~ t+2 h
0:
i
VtM ht+2
i
+
> 0: It is straight-
then he has a pro…table deviation in period t: However, it could be the case that the t+1 posterior from history ht+1 follows a di¤erent dynamic. We prove the result here under the assumption that 2 in every history reached by ht which is associated to an interior belief the monopolist o¤ers a menu with sequential separating dynamics. We explain how to extend the argument to encompass the possibility that the monopolist o¤ers menus which do not present sequential separating dynamics at histories reached by ht in the appendix. Hence, instead of assuming Case 1, assume that (see Remark 1 above): n o Case 2: The monopolist o¤ ers a menu with a pooling contract x1t+1 ; q t+1;1 leading to a t+1 t+2 R posterior p1t+2 > pt and anrevealing o contract xt+1 ; fqH g at h2 : Let h1 be the history led by the acceptance of x1t+1 ; q t+1;1 and ht+2 be the one led by xR t+1 ; fqH g : Using the same de…nition of t ; the R.H.S. of (32) is
ht+2 VtM ht+2 + 1 1
t
t
ht+2
VtM ht+2
:
(46)
~ t+1 (which In this case, we consider the following feasible plat at the deviating history h 2 is associated with the state (pt ; f0g)): Feasible Plan: The monopolist o¤ers a menu with two identical contracts ( L qL ; fqL ; 0g). Both contracts lead to the next-period state (pt ; f0g) and to some rents (0; vH ) 2 (pt ; f0g) : The feasibility of this plan at (pt ; f0g) implies that ~ t+2 for the history led by V M (pt ; f0g) (1 ) L (qL ) + V M (pt ; f0g). Therefore if we write h 1 this plan we have: ~ t+1 VtM h 2
~ t+2 := (1 VtM h 1
)
L (qL )
+
(1
)
L (qL )
+ V M (pt ; f0g) :
In this case, to show (32) it su¢ ces to prove that t
t
~ t+1 h
~ t+2 V M h ~ t+2 > h t 1 1
ht+2 VtM ht+2 + 1 1
VtM ht+2 : (47) t+1 t+2 t+2 ~ < Notice that since p(h1 ) > p(h2 ); the martingale property of beliefs implies that t h t+1 ~ h and hence (47) is equivalent to:
2 h 4
~ t+1 VtM h
t
~ t+1 h
t
+
ht+2 +
t
i
t
~ t+1 VtM h ht+2
+
t
t
~ t+2 V M h ~ t+2 h t 1 1
~ t+1 VtM h
t
t
ht+2
ht+2 VtM ht+2 1 1
VtM ht+2
3
5 > 0:
(48)
Notice now that the argument leading to Fact 3 above implies with strict inequality if qtt;2 6= qH : On the other hand, h i t+2 ~ t+1 + t t h t h h i t+2 ~ t+1 and 1 + pt t h t h t
VtM
~ t+1 h
VtM
ht+2
since beliefs are martingales we have ~ t+2 h 1
=
~ t+2 h 1
= p(ht+2 1 )
t
ht+2 1
(49) t
ht+2 : 1
Intuitively one can see the …rst line of (48) as the comparison between the pro…t led by the putative equilibrium pooling history ht+2 (associated with the posterior p(ht+2 1 1 )) with a lottery
42
~ t+1 ) and a pooling over an alternative history in which the high type is screened in period t (h history associated to the posterior pt in which the e¢ cient quality is o¤ered to the low type in periods t and t + 1: This lottery leads to the same expected posterior, but screens the high type earlier. This shows the main driving force of our argument. If an equilibrium does not present sequential separating dynamics then the monopolist’s payo¤ is bounded below by the payo¤ of a more e¢ cient play which gives the same rent to each type and in which the high type is screened earlier in a …rst-order stochastic sense. We must deal with two cases. Case A: Assume that continuation game starting at ht+2 leads to a history leading to a 1 posterior in [0; pt ] in …nite time. If at ht+2 the monopolist o¤ers a menu involving sequential separating dynamics in which 1 the pooling contract leads to a posterior p1t+3 > pt ; we apply the exactly same procedure. This is repeated until the …rst time that a posterior in the set [0; pt ] is reached under the putative equilibrium play. Since the arguments in all cases are similar, we consider the one which involves less notation and assume that: o n leading to a posteThe monopolist o¤ ers a menu with a pooling contract x1t+2 ; q t+2;1 t+3 t+2 R pt and a nrevealing o contract xt+2 ; fqH g at h1 : Let h1 be the history led by the and ht+3 be the one led by xR acceptance of x1t+2 ; q t+2;1 t+2 ; fqH g :
rior p1t+3
ht+2 VtM ht+2 1 1
+ ht+3 VtM ht+3 1 1
VtM ht+3 : ~ t+2 in Proceeding exactly as we did in Case 1 above, we can construct a lower bound to VtM h 1 In this case, we have
t
=
t
t
ht+3
t+2;1 which leads to which the monopolist o¤ers a pooling contract with the current period quality qt+2 t+3 18 x ); q the next-period state p(ht+3 h and a revealing contract of the form ~R t+2 ; fqH g . 1 1 t+3 t+3 ~ ~ the one by the second. Thus (48) holds if19 : Let h1 be the history led by the …rst and h1 i h t+3 M ~ t+3 V M h ~ t+3 (50) V h h t t t 1 1 1 i h ~ t+3 ~ t+3 VtM h VtM ht+3 + t h i h M t+3 t+2 M ~ t+1 ~ t+1 V h h + t h h V t t t i h ~ t+1 + t ht+2 VtM h VtM ht+2 > 0:
~ t+3 First assume that qtt;2 = qH : The same argument presented in Case 1 implies that VtM h 1
0; while all other terms in brackets are weakly positive. Similarly if ~ t+1 0 and VtM h
VtM ht+2
qtt;2
~ t+1 6= qH then VtM h
VtM ht+3
> 0; while the other terms in brackets are weakly positive.
18
As in Case 1, the transfers x ~R t+2 are set to make the high type indi¤erent between both contracts. R Moreover, the contract x ~t+2 ; fqH g leads to the next-period rents ( L qH ; H qH ): 19 ~ t+3 and that Using the martingale property of beliefs, it is immediate to see that i) t ht+3 = t h 1 1 t+2 ~ t+3 + ~ t+1 = ; ii) t ht+3 + t h t h t h which implies that the di¤erence in pro…ts from the alternative-play and the putative-equilibrium play is given by the following expression.
43
VtM ht+3 1
> >
Thus (50) holds. Case B: Assume that continuation game starting at ht+2 does not lead to a history leading 1 to a posterior in [0; pt ] in …nite time. In this case, applying the algorithm above, it su¢ ces to show that there exists T > 2 for which an expression similar to (48) is strictly positive: 2 h i PT t+T t+z ~ t+T V M h ~ t+T + ~ t+1 ~ t+1 h h h VtM h VtM ht+T t t t h1 t 1 1 1 z=2 t 4 PT t+z M t+1 M t+z ~ Vt h + z=2 t h Vt h (51) where we have used a similar notation to the one above: ~ t+T for the “alternative-play” history Notation Convention for Histories: We write h 1 that was reached by a menu with two contracts equal to ( L qL ; fqL ; 0g) in every period 2 is associated to a posterior in (p ; 1): ft; :::; t + T 1g : The putative-equilibrium history ht+T t 1 Finally, ht+z is a putative-equilibrium history in which the high type was screened in period t + z 1. Using the same argument as above, we obtain. ~ t+1 Fact 4: For all z 2 f2; :::; T g ; VtM h VtM ht+z with strict inequality if qtt;2 6= qH : For 2 ft; :::; t + T 1g ; write x (ht+T ) (resp. q (ht+T )) for the transfer (resp. quality) from 1 1 t+T period at history h1 : Using the fact that the low type (high type) obtains the rent 0 (resp. vH (Pt )) we have: t+T X1
0 =
=t t+T X1
vH (Pt ) =
(1
)
t
Lq
(ht+T ) 1
x (ht+T ) + 1
T
VL ht+T : 1
(1
)
t
Hq
) (ht+T 1
) + x (ht+T 1
T
: VH ht+T 1
=t
Moreover, the pro…t from h1t+T is: VtM h1t+T
=
t+T X1
(1
)
Since VL ht+T ; VH h1t+T 1
2
=t
i = L; H: Moreover V M h1t+T Recall also that
q(h1t+T )
=
qtt;2 :
x (ht+T ) 1
t
= VM
c q (ht+T ) 1
+
T
V M ht+T : 1
n o t+T p ht+T ; q h we have Vi ht+T 1 1 1 n o t+T t+T p h1 ; q h1 implies V M ht+T 1
K for K:
Using these observations and straightforward algebra we obtain
the following upper bounds for VtM ht+T 1
:
VtM ht+T 1
(1
)
L
qtt;2 +
L (qL )
VtM ht+T 1
(1
)
H
qtt;2 +
H
44
+2
T
(qH ) + 2
K: T
K
(52) vH (Pt ) :
(53)
3
5 > 0;
~ t+T Finally notice that VtM h 1 max
V M (pt ; f0g) ; j
Pt+T
1
=t
L (qL )j
(1
t
)
L (qL )
+
< K; we have:
~ t+T VtM h 1
L (qL )
2
T
T
V M (pt ; f0g) : Since
K:
(54)
Notice that we can rewrite (51) as: 2 h i 3 M ht+T ~ t+T V M h ~ t+T h V t t t 1 1 1 6 h ih i 7 PT 6 7 t+1 t+z M t+1 M ht+T ~ ~ h V h V 6 + t h 7: t t t 1 z=2 4 5 PT ~ t+1 VtM h + z=2 t ht+z VtM ht+z
(55)
First assume that qtt;2 = qH : In this case, using (52) and (54) to bound the …rst line of (55), 20 using (53) and (54) to bound the second and using Fact 4, we conclude that (55) is at least ! t+T (1 ~ h ) q q [ L ( L ) L ( H )] ) t( 1 log 4K t+T T ~ (1 ) [ L (q ) 4K > 0; for all T > h : L (q )] t
1
L H log t;2 that qt 6= iqH : Notice that Fact 4 implies that the third line of (51) is t;2 : Using the same argument as above to bound the other H qt
Next h assume (1 ) H (qH )
follows that (55) is at least ~ t+1 t h
(
log
)(1
)
[
H (qH ) 4K
log
8.2
t;2 H qt
(
t
)]
!
~ t+1 h
(1
)
h
H
(qH )
H
qtt;2
i
4K
T
at least lines, it
> 0; for all T >
:
Lemma 1
Lemma 1 For every equilibrium, we can …nd an outcome equivalent equilibrium in which every pooling contract that is o¤ ered on the equilibrium path is a spot contract and every revealing contract that is o¤ ered on the equilibrium path is renegotiation-proof. Proof. Consider a pro…t-maximizing renegotiation equilibrium ( ; p) : Take a menu m0 = 1 0 f 0 ; 20 g in the support of the monopolist’s behavior strategy M 0 (h ): For each one of its contract j j 1 0 2 m0 ; let hj be the history led by the acceptance of the contract 0 in the …rst period and let (pj1 ; fq j g 1 ) be the respective state. According to De…nitions 1 and 2, the menu m0 ; the randomization of the consumer over its elements, the posterior led by each contract and as the next-period future rents are consistent with an optimal plan P0 for the state (p0 ; f0g): Analogously, each menu in the support of the 1 M 1 monopolist’s behavior strategies M 1 (h1 ) and 1 (h2 ) (as well as the randomization of the consumer over its elements, the posterior led by each contract and as the next-period future rents) are consistent with optimal plans for the states (p11 ; fq 1 g 1 ) and (p21 ; fq 2 g 1 ) respectively. Applying Lemmas 8 and 13 we can …nd outcome-equivalent paths in which in every period t 2 f0; 1g every 20
The assumption that we have Case B implies that
45
t
~ t+T h 1
is constant for all T > 1:
pooling contract is a spot contract and every revealing contract is renegotiation proof. We will now claim that this outcome belongs to a PBE. Clearly, the monopolist does not have a pro…table deviation as he obtains exactly the same payo¤. Hence, to show that the consumer does not have a pro…table deviation it su¢ ces to show that a rejection of both contracts lead to the rents satisfying (4) and (5) with equality. This is indeed checked in our existence proof in Section 11.2.4. In summary, we have constructed a PBE (~ ; p~) in which in every period t 2 f0; 1g every pooling contract is a spot contract and every revealing contract is renegotiation proof. Hence, observe that only states of the form (p; f0g) for some p < 1 or (1; fqH g) are reached with positive probability in the second period. Therefore, the same argument allows us to construct an outcome equivalent PBE in which in every period t 2 f0; 1; 2g every pooling contract is a spot contract and every revealing contract is renegotiation proof. We claim that we can obtain an outcome equivalent PBE in which in every period t 2 Z+ every pooling contract is a spot contract and every revealing contract is renegotiation proof. Suppose towards a contradiction that this is not the case. Let T 2 N be the maximal number of periods for which this procedure can be applied. Find an outcome equivalent PBE ( ; p) for which this property holds every period t T and use the argument above to construct another PBE in which this property holds for every t T + 1: This contradicts the maximality of T and completes the proof.
9
References Battaglini, M., 2005. “Long-term contracting with Markovian consumers”. American
Economic Review. 95, 637–658. Battaglini, M., 2007. “Optimality and renegotiation in dynamic contracting”. Games and Economic Behavior 60 (2007) 213–246. Bernheim, B.D., and Ray, D., 1989. “Collective Dynamic Consistency in Repeated Games,” Games and Economic Behavior, 1, 295–326. Bester, H. and Strausz, R., 2001. “Contracting with Imperfect Commitment and the Revelation Principle: the Single Agent Case”. Econometrica, 69, 1077-1098. Dewatripont, M., 1989. “Renegotiation and information revelation over time: The case of optimal labor contracts”. Quarterly Journal of Economics 104, 589–619. Farrell, J., and Maskin, E., 1989. “Renegotiation in Repeated Games,” Games and Economic Behavior, 1, 327–360. Fudenberg, D., David K. L. and J. Tirole 1985. “In…nite-Horizon Models of Bargaining with One-Sided Incomplete Information”.Game Theoretic Models of Bargaining, Cambridge, England: Cambridge University Press.
46
Fudenberg, D., and Tirole, J., 1990. “Moral hazard and renegotiation in agency contracts”. Econometrica 58, 1279–1319. Gul, F., Sonnenschein, H. and Wilson, R, 1986. “Foundations of Dynamic Monopoly and the Coase conjecture”. Journal of Economic Theory 39, 155-190. Hart, O., Tirole, J., 1988. “Contract renegotiation and Coasian dynamics”. Review of Economic Studies. 55, 509–540. La¤ont, J.J., and Tirole, J., 1990. “Adverse selection and renegotiation in procurement”. Review of Economic Studies. 57, 597–625. La¤ont, J.J., and Tirole, J., 1993. “A Theory of Incentives in Procurement and Regulation”, MIT Press. Liu, Q., Mierendor¤, K., and Shi, S., 2014. “Auctions with Limited Commitment”. Working Paper. Mussa M., Rosen, S., 1978. “Monopoly and product quality”. Journal of Economic Theory, 18, 301-317. Rey, P., and Salanié, B., 1996. “Long-Term, Short-term and Renegotiation: On the Value of Commitment with Asymmetric Information.” Econometrica, 64, 1395-1414. Skreta, V., 2006. “Sequentially Optimal Mechanisms.” Review of Economic Studies, 73, 1085-1111. Skreta, V., 2015. “Optimal Auction Design under Non-Commitment.” forthcoming at the Journal of Economic Theory, 159, 854–890. Stokey, N.,1981. “Rational Expectations and Durable Goods Pricing”. The Bell Journal of Economics, 12, 112-128. Strulovici, B., 2015. “Contract Negotiation and the Coase conjecture”. Working paper. Wang, G, 1998. “Bargaining over a Menu of Wage Contracts”. Review of Economic Studies 65, 295-305.
47
10
Online Appendix I: Remaining Arguments of Proposition 1
Proof of Proposition 1: Step 1: Under the contradiction hypothesis, we can …nd a state (pt ; f0g), an optimal plan Pt for that state which is not consistent with sequential separating dynamics, and a continuation equilibrium following that plan that satis…es De…nitions 1 and 2. Assume towards a contradiction that the claim is false. Take an on-path history ht of minimal length in which the monopolist does not o¤er a menu presenting sequential separating dynamics. If t = 0 this step is obvious. Therefore assume that t > 0. Let fh0 ; :::; ht 1 ; ht g be the set of on-path histories that precede ht and include ht . Notice that at each one of these histories was reached by an optimal plan containing a contract leading to a lower posterior and another leading to the posterior 1. Using Lemma 8, it is easy to construct an outcome equivalent equilibrium in which every revealing contract was of the form x2t ; fqH g in every period t < t . Using Lemma 13, it is easy to construct an outcome equivalent equilibrium in which every pooling contract was a spot contract n every period t < t : Step 2: Argument for the case in which the monopolist o¤ ers menus which do not present sequential separating dynamics at histories reached by ht . Consider the history ht+1 2 : In the text, we presented a comparison of the payo¤ from this history with the payo¤ from a lottery (leading to the same expected posterior) over the alternative-play ~ t+1 (associated ~ t+1 (associated to the posterior pt ) and the alternative-play history h history h 2 to the posterior 1). We proceeded and built an algorithm that generated an alternative play for the case in which the putative-equilibrium play presented sequential separating dynamics in every period > t: Here we extend this algorithm for the general case. This algorithm will have the crucial property21 that to every putative-equilibrium history ht+k for which p ht+k 2 (pt ; 1) we associate a lottery (leading to the same expected posterior) over alternative-play histories in which the posterior belong to [0; pt ] [ f1g : Starting with the history ht+1 2 , there are four cases to consider. n o Case 1: The monopolist o¤ ers a menu with a pooling contract x1t+1 ; q t+1;1 leading to a n o o n 1 ; q t+1;1 posterior p1t+2 pt and a revealing contract x2t+1 ; q t+1;1 at ht+1 : The contract x t+1 2 t+2 leads to the next-period rents vL (ht+2 1 ); vH (h1 ) :
Case 2: The monopolist o¤ ers a menu with a pooling contract t+1 posterior p1t+2 > pt and a revealing contract xR t+1 ; fqH g at h2 :
Case 3: The monopolist o¤ ers a menu with a pooling contract
n o x1t+1 ; q t+1;1 leading to a n o x1t+1 ; q t+1;1
the history ht+2 and to the posterior p1t+2 > pt and another pooling contract 1 leading to the history ht+2 and to the posterior p2t+2 > pt at ht+1 2 2 : Case 4: The monopolist o¤ ers a menu with a pooling contract 21
This property was also true for the algorithm presented in the text.
1
leading to n o x2t+1 ; q t+1;2
n o x1t+1 ; q t+1;1
leading to
the history ht+2 and to the posterior p1t+2 1
pt and another pooling contract
leading to the history ht+2 and to the posterior p2t+2 2 (pt ; 1) at ht+1 2 2 : Cases 1 and 2 have been analyzed. Case 3: In this case, we have t+1 M t (h2 )Vt
ht+1 = 2
t+2 M t (h1 )Vt
ht+2 + 1
t+2 M t (h2 )Vt
n o x2t+1 ; q t+1;2
ht+2 : 2
(56)
~ t+1 : The monopolist o¤ers We consider the following feasible plan at the deviating history h 2 a menu with two identical contracts ( L qL ; fqL ; 0g). Both contracts lead to the next-period state ~ t+2 for the history led by this plan. A (pt ; f0g) and to some rents (0; vH ) 2 (pt ; f0g) : Write h 1 lower bound to the L.H.S. of (32) is: t
~ t+1 h
~ t+1 VtM h
+
~ t+2 V M h ~ t+1 : h t 1 1
t
(57)
Using the fact that beliefs are martingales, it is straightforward to check that there exists a ~ t+1 ) t (h ~ t+1 + ~ t+2 = (ht+2 ); b) 1+ unique ( ; ) 2 [0; 1]2 such that: a) h h t t t 1 1 (ht+2 ) t
t
(h~ t+2 1 )
pt = p(ht+2 1 ); c) (1
t+2 t (h1 )
(1 8 > > <
)
t
(h~ t+2 1 )
~ t+1 h
t
+(1
)
t
~ t+2 = h 1
t+2 t (h2 );
d) (1
)
t
(h~ t+1 ) t+2 t (h1 )
1+
pt = p(ht+2 2 ): Therefore, for (32) to hold it su¢ ces to show that:
t+2 t (h1 )
t+2 t (h1 )
> > : + t (ht+2 2 )
)
1
t
(h~ t+1 )
t+2 t (h1 )
(1
)
t
(h~ t+1 )
t+2 t (h2 )
VtM
~ t+1 h
t
+
~ t+1 VtM h
+
(h~ t+2 1 )
t+2 t (h1 )
(1
VtM
~ t+2 t h1 t+2 t (h2 ) )
(
)
~ t+2 h 1
VtM
~ t+2 VtM h 1
ht+2 1 VtM ht+2 2
9 > > = > > ;
> 0:
(58) The term in square brackets from …rst (second) line of (58) is a comparison between a lottery ~ t+1 and h ~ t+2 and a degenerate lottery that puts probability over the alternative-play histories h 1 (resp. ht+2 1 on the history ht+2 2 ). In each one of the comparisons, both lotteries have the same 1 mean posterior. We can thus proceed analogously in the next period. Case 4: In this case, we have t+1 M t (h2 )Vt
ht+1 = 2
t+2 M t (h1 )Vt
ht+2 + 1
t+2 M t (h2 )Vt
ht+2 : 2
(59)
~ t+1 : We consider the following feasible plat at the deviating history h 2 With probability , the monopolist o¤ers n o a menu similar to the one from Case 1: This menu contains a pooling contract x ~1t+1 ; q t+1;1 leading to the posterior p1t+2 and a revealing contract n o t+2 x ~R ~1t+1 ; q t+1;1 leads to next-period rents vL (ht+2 t+1 ; fqH g : The acceptance of x 1 ); vH (h1 ) . vL (ht+2 ~R t+1 are set to make 1 ); while the transfers x n o the high type indi¤erent between both contracts. The acceptance of x ~1t+1 ; q t+1;1 leads to the n o ~ t+2 ; while the acceptance of x ~ t+2 : history h ~1t+1 ; q t+1;1 leads to the history h 1 The transfers x ~1t+1 are set at
1 L qt+1
+
1
2
With probability (1 ) we consider a menu similar to the one from Case 3 above. The monopolist o¤ers a menu with two identical contracts ( L qL ; fqL ; 0g). Both contracts lead to the ~ t+2 for the history led next-period state (pt ; f0g) and to some rents (0; vH ) 2 (pt ; f0g) : Write h 2 by this plan. Using the martingale-property of beliefs, It is straightforward to show that if we let be deter1 pt ~ t+2 ~ t+1 ; we have 2 [0; 1] and t (ht+2 mined by t (ht+2 t (h2 ) 1 p1 1 ) = t (h1 ): Therefore, 1 ) = t+2
for (32) to hold it su¢ ces to show that 8 h i t+2 M h M ht+2 ~ t+2 > (h ) V V > t t t 1 1 1 > > 20 > < ~ t+1 ) t (h ~ t+1 VtM h t+2 6 B h ) ( t 2 > 6B + t ht+2 > 2 > ~ t+2 4@ (h~ t+2 ) > t (h2 ) ~ t+2 + ~ t+2 > + t ht+2 VtM h VtM h : t+2 2 t( 2 ) t (h2 )
1 C C A
VtM ht+2 2
9 > > > 3 > > = > 0: 7 7 > > 5 > > > ;
(60) Using the same argument as in Case 1, we conclude that the …rst term in square brackets in (60) is positive (with strict inequality if qtt;2 6= qL ). The second term in square brackets in (60) is a comparison between a lottery over the ~ t+1 ; h ~ t+2 and h ~ t+2 and a degenerate lottery that puts probability 1 alternative-play histories h 2 t+2 on the history h2 . In each one of the comparisons, both lotteries have the same mean posterior. We can thus proceed analogously in the next period.
Induction Argument Notice that in each one of Cases 2-4 we remained with the comparison of the pro…t from (where s could be 1 or 2) with a lottery over deviating-play a putative equilibrium history ht+2 s ~ t+1 (and if we had Case 4 histories which puts positive probability on the separating history h 1 t+2 t+2 ~ ~ also on a history h ) and a history hs which is associated to the belief pt : Therefore, if we do not have Case 1 in period t + 1 then we arrive in period t + 2 with a situation analogous to the one arrived in the beginning of period t + 1: Thus, we can apply the same algorithm in period t + 2: Inductively, for any T 2 N; we can apply the same algorithm in periods s = t + 2; t + 3; :::t + T: We stop the algorithm at the …rst time that a posterior pt+k reaches the set [0; pt ] [ f1g under the putative equilibrium play. The algorithm stops with a history ht+k with a posterior pt+k 2 [0; pt ] or with a revealing history ht+k : The remainder of the proof is analogous to the one from the benchmark case and is omitted for brevity. Step 3: We show how we extend the algorithm presented in Proposition 1 to the case in which the monopolist plays a mixed strategy. The deviation that we propose in period t is the same whether the monopolist randomizes or not on the putative equilibrium play. Therefore, it remains to show that (32) below holds: t
~ t+1 h
~ t+1 VtM h
+
t
~ t+1 > ~ t+1 V M h h t 2 2
t
ht+1 VtM ht+1 : 2 2
(61)
For every history ht+ and i 2 fL; Hg ; let Vi;t (ht+ ) be the expected pro…t of type i at ht conditional on the history ht+ being reached in period t + : Consider the history ht+1 2 . Notice that we are making no pure strategy assumption, thus we have to consider the possibility that the
3
monopolist randomizes over menus and that the expected realized rent of each type of consumer varies according to this randomization. A fact which will be key to our proof is that expected realized rents are martingales. Let G Pt+1 j ht+1 represent the distribution over optimal plans 2 Pt+1 at ht+1 . For i 2 fL; Hg ; we have: 2 h i Vi;t (ht+ ) = (1 ) i qtt;2 x2t+1 + EG vi (Pt+1 ) j ht+1 ; 2 where EG vi (Pt+1 ) j ht+1 is the expected value of vi (Pt+1 ) from the distribution G: We de…ne 2 for each Pt+1 the innovation in continuation rent for type i 2 fL; Hg by: i
Pt+1 j ht+1 := vi (Pt+1 ) 2
EG vi (Pt+1 ) j ht+1 : 2
Notice that 0 = EG
i
Pt+1 j ht+1 j ht+1 : 2 2
Summing martingales to realized payo¤s without changing their pected value. Applying (62) into (61) it su¢ ces to show that: i h 2 t+1 t+1 ~ t+1 ~ t+1 + VtM h t h H Pt+1 j h2 L Pt+1 j h2 6 EG 6 ~ t+1 ~ t+1 V M h + t h 4 t 2 2 t+1 t+1 M + L Pt+1 j ht+1 V t h2 t h2 2
(62) (ex-ante) ex-
3
7 7 > 0: 5
(63)
The L.H.S. of (63) can be interpreted as follows: when the monopolist’s mixture follows the t+1 optimal plan Pt+1 we transfer the amount L Pt+1 j ht+1 from the consumer to H Pt+1 j h2 2 t+1 ~ the monopolist at the alternative-play history h ; simultaneously we transfer L Pt+1 j ht+1 2 from the consumer to the monopolist at the putative-equilibrium history ht+1 2 : It is important to emphasize that this does not change (ex-ante) the pro…t of either the putative equilibrium play or the alternative play. Hence if the alternative play is more pro…table after this transformation we conclude that it must be more pro…table before it. for the menu o¤ered by the monopolist at ht+1 if he wants to Let us write mt+1 Pt+1 j ht+1 2 2 t+1 t+1 t+1 : Fact B1 below is straightinduce Pt+1 : Let us de…ne h2 (Pt+1 ) := h2 [ mt+1 Pt+1 j h2 forward: Fact B1 If L Pt+1 j ht+1 is transferred from the consumer to the monopolist when the 2 optimal plan Pt+1 is realized, the expected realized rent of the low type remains constant: VL;t ht+1 (Pt+1 ) 2
L
Pt+1 j ht+1 = 0; 2
(64)
(Pt+1 ) as the expected rent of type i 2 fL; Hg in period t conditional where we de…ne Vi;t ht+1 2 on the realization of the plan Pt+1 at ht+1 2 : t+1 The goal of transferring L Pt+1 j ht+1 from the consumer to the monopH Pt+1 j h2 2 olist in the deviating play associated with histories in which the monopolist holds the posterior 1
4
is to keep the high type’s rent the same under the alternative play and the putative equilibrium play (after transferring L Pt+1 j ht+1 to the monopolist). Notice that by construction we have 2 ~ t+1 VH;t h
+
H
Pt+1 j ht+1 2
L
Pt+1 j ht+1 2
= VH;t ht+1 (Pt+1 ) 2
L
Pt+1 j ht+1 ; 2
~ t+1 where we have de…ned VH;t h
as the expected rent of the high type in period t conditional on the alternative plan following the plan P~1 at ht : t+1 Fact B2 If the high type transfers L Pt+1 j ht+1 to the monopolist at H Pt+1 j h2 2 t+1 t+1 ~ h and he transfers L Pt+1 j h2 when the optimal plan Pt+1 is realized, then the high type obtains the same rent in both scenarios. Applying Fact B1 and Fact B2, the algorithm developed for the proof Proposition 1 can be easily extended to encompass mixed strategies.
11
Online Appendix II: Existence (NOT FOR PUBLICATION)
In Section 11.1, we construct values V M ; and check that they are pro…t-maximizing: That M is, we take the next-period values V ; and check that for every state (p; fq g) the solution to the monopolist’s problem (solving (2) s.t. (3)-(7)) yields V M (p; fq g) and that the mixture of solutions to the monopolist’s problem generates (p; fq g): In Section 11.2 we check that there exists a perfect Bayesian equilibrium ( ; p) that is consistent with V M ; according to De…nition 2.
11.1
Part 1: Construction of V M ;
In the …rst part of this proof, we construct a function V M : [0; 1) ! < and a correspondence R : [0; 1) <: We will later use V M (p) to represent the monopolist value function for the state (p; f0g). The correspondence R will be later used to construct (p; f0g) according to (p; f0g) := f(0; vH ) : vH 2 R(p)g :
11.1.1
STEP A: Construction of
VM ; R
We de…ne V M (0) := L (qL ) and R (0) := [ qL ; qH ] : 1 To construct V M (p) and R (p) for p 2 (0; 1) we build a sequence VnM ; Rn n=1 where VnM : [0; 1) ! < and Rn : [0; 1) <: We show that for every p 2 (0; 1) there exists n 2 N such M that n n implies Vn (p); Rn (p) = VnM (p); Rn (p) , which allows us to de…ne V M (p) ; R(p) = lim VnM (p); Rn (p) : n
5
Although we make no reference to equilibrium behavior at this point, the function V1M (de…ned below) corresponds to the monopolist’s pro…t from the most-pro…table incentive-compatible and individually rational allocation that screens the high type in one period and implements an e¢ cient allocation for every revealed type in the next period. R1 (also de…ned below) represents the M ;R rent delivered to the high type in that allocation. We then use VnM ; Rn to …nd Vn+1 n+1 ; M where Vn+1 is the monopolist’s pro…t from the most-pro…table, individually-rational and incentivecompatible allocation that screens the high-type in at most n + 1 periods and implementing an e¢ cient allocation for any revealed type and Rn+1 is the set of rents obtained by the high type in such allocations. Finally we comment that any solution for VnM ; Rn will generate one allocation to the high type and another to the low type. Any allocation that solves this problem will be incentivecompatible for the high type by construction and will give zero rents to the low-type. Hence it will give non-negative rents to the high type. Moreover, it will be monotonic and hence the low-type IC will be implied by the high-type one. The sequence VnM ; Rn will be constructed by de…ning V1M ; R1 and using VnM ; Rn to M ;R obtain Vn+1 n+1 : Step A1: Construction of V1M ; R1 : For each p 2 (0; 1) and q~ 2 [0; 1] we let ~) 1 (p; 0; q
= p[
H (qH )
(1
)
q~
qL ] + (1
p) [(1
)
q) L (~
+
L (qL )] ;
and de…ne: V1M (p) := maxq~2[0;1] Step Take
~) 1 (p; 0; q
R1 (p) := f(1
M ;R A2: Construction of Vn+1 n+1 : M 22 Vn ; Rn as given, de…ne vH;n (~ p) :=
~; q~) n+1 (p; p and
M Vn+1 (p)
: =
=
+ (1 max
p~2[0;p];~ q 2[0;1]
q~ +
qL : q~ 2 arg maxq
~)g : 1 (p; 0; q (65)
min Rn (p) and let
(p; p~) (1 ) (p; p~)) [ H (qH )
M (~ + Vn+1 p) (1 ) q~ vH;n (~ p)]
q) L (~
~; q~) : n+1 (p; p
We remark that if (~ p; q~) 2 arg maxp~2[0;p];~q2[0;1] 0 q) L (~
)
~; q~) n+1 (p; p p 1
p~ p~
22
then q~ > 0 implies
= 0:
(66)
Notice that R1 (p) is a function, while it is easy to show inductively that Rn (p) is an upperhemicontinuous correspondence, which guarantees that the minimum is well de…ned.
6
We de…ne Rn+1 (p) by: Rn+1 (p) = co f~ vH : v~H = (1
)
q~ + vH;n (~ p) and (~ p; q~) solve
~; q~)g : n+1 (p; p
The following Remark is straightforward and its proof is omitted. Remark 2 The correspondence Rn+1 (p) has a closed graph and is convex valued. Lemma 15 There exists p1 2 (0; 1) such that for all p 2 [0; p1 ] and p~ 2 (0; p) then maxq2[0;1] maxq2[0;1] n (p; 0; q ) for every n 2 N:
~; q ) n (p; p
Proof. Roughly, this shows that when p0 is small enough then it is optimal to take one period to screen the high type. For that, we build an upper bound to the monopolist’s payo¤ from allocations in which the high type is sequentially screened and the next-period’s posterior is p1 2 [0; p0 ]. This upper bound delivers the exact payo¤ from screening the high type in one period when p1 = 0: We proceed to show that when p0 is small then, when using this upper bound to assess the value from screening, it is optimal to set p1 = 0: This implies that the high type should be screened in one period. We will solve a relaxed problem which shows that for small beliefs the consumer is screened P in one period. We start de…ning the function r (p0 ) : = inf fp1 ;p2 ;:::g 1=0 (1 ) q (p ; p +1 ) subject to fp0 ; p1 ; :::g being decreasing and q (p ; p +1 ) implicitly de…ned by 0 (q (p L
;p
+1 ))
q (p ; p
p
= +1 )
=0
p +1 1 p if 0L (0)
if
0 (0) L p
>
p
p +1 1 p
:
p +1 1 p
(67)
The de…nition of (67) uses the optimality condition for the current-period quality as a function of the measure of high types that is screened in period t (see the explanation before equation (10)). The function r (p0 ) essentially provides a lower bound to the incremental rent to the high type. p00 Notice that r (p) 2 [0; ] ; hence it is well de…ned. Next take p00 such that 0L (0) = 1 p0 0
and assume that p0
0
p0 2
:=
p10 :
Thus (67) implies q (p0 ; p1 ) > 0:
It is straightforward to see that r (p0 ) is continuous. We will show that it is Lipschitz con1 m tinuous and hence absolutely continuous. Take a (double) sequence fpm 1 ; p2 ; :::gm=1 for which P1 r (p0 ) = limm!1 ) q pm ; pm+1 : Taking a subsequence if necessary we let p1 := =0 (1 limm!1 pm ) qL + r (p0 ) and thus 1 : Clearly, p1 < p0 , otherwise we have r (p0 ) = (1 r (p0 ) = qL ; but if we set p = 0 for all > 0 we get a smaller number. Therefore, we have r (p0 ) = (1 ) q (p0 ; p1 ) + r (p1 ) : It follows that for all " 2 0; p0 2 p1 we have r (p0 ") (1 ) q (p0 "; p1 ) + r (p1 ) and hence since r (p0 ") r (p0 ) we have 0
r (p0
") "
r(p0 )
(1
)
7
q (p0
"; p1 ) "
q (p0 ; p1 )
:
(68)
<
Since p0
p10 we have q (p0 ; p1 ) > 0; which allows us to di¤erentiate
w.r.t. p0 and obtain
@q(p0 ;p1 ) @p0
=
c00 (q(p0 ;p1 )) (
which shows that if we let K1 :=
1 p1 (1 p0 )2
0 (q (p ; p )) 0 1 L r(p0 ") r(p0 ) "
, implying 0
p0 p1 1 p0
= (1
)
( minq
)2
then K1 (1 ) is a Lipschitz constant. (1 ) minq c00 (q) Hence r is absolutely continuous and its derivative r0 exists [a.e.]. We can write r (p0 ) = Rp r (0) + 0 r0 (~ p) d~ p; with jr0 (~ p)j K1 (1 ) [a.e.]. Next we will consider a function which gives an upper bound to the monopolist’s pro…t in a period in which he starts with a prior p0 and sequentially screens the high type. 9 8 (1 p0 ) [(1 ) L (q (p0 ; p1 )) + L (qL )] > > > > = < p0 p1 ) q (p0 ; p1 ) r (p1 )] + 1 p1 [ H (qH ) (1 : (p0 ; p1 ) = i h > > > > p p 0 1 : + p0 [(1 ) L (q (p0 ; p1 )) + [ H (q ) r (p1 )]] ; 2 p10
H
1 p1
We payo¤ from is consistent with the following allocation: the low type consumes fq (p0 ; p1 ) ; qL g and obtains zero rents. The high type pools with the low type in the …rst period in such a way that the posterior falls to p1 in case of pooling. Hence the pooling probability is p10 pp11 : After pooling in the …rst period, the high type separates in the second period. His payo¤ is calculated to give him the minimal rents r (p1 ) from the second period on (so he obtains the expected rent (1 ) q (p0 ; p1 ) + r (p1 )). It follows that, for any p1 > 0; the payo¤ (p0 ; p1 ) is strictly greater than than the payo¤ obtained from separating the high type in n periods and, conditional on pooling in the …rst period, the posterior being p1 : Therefore we have, for each n 2 N : (p0 ; p1 ) > (p0 ; p1 ) =
n (p0 ; p1 ; q(p0 ; p1 ))
= maxq~ (p ; p ; q(p ; p )) = maxq~ n 0 1 0 1
~) ; n (p0 ; p1 ; q
if p1 > 0 (p ; p ; q ~ ) ; if p1 = 0: n 0 1
To prove the Lemma we must show that there exists p0 2 (0; 1) such that if p0 2 (0; p0 ) then 0 ;p1 ) 0 = arg maxp1 2[0;p0 ) (p0 ; p1 ) : For that it is enough to show that if p 2 (0; p0 ) then @ (p < 0. @p1 Let p1 be a point of di¤erentiability of r: We have 8 > > > <
@ (p0 ; p1 ) = > @p1 > > :
(1 1 p0 [1 p1 ]2
)
h
1 p0 1 p1
@ (p0 ;p1 ) @p1
0 (q (p ; p )) 0 1 L p0 r0 (p1 )
i
p0 p1 1 p1
@q(p0 ;p1 ) @p1
[ H (qH ) (1 ) q (p0 ; p1 ) r (p1 )] [(1 ) L (q (p0 ; p1 )) + [ H (qH ) r (p1 )]]
We work towards showing that (69) is negative. Step 1 The term j p0 r0 (p1 )j : From Step 1 we have j p0 rh0 (p1 )j (1 ) p0 K 1 : 1 p0 0 Step 2 The term (1 ) 1 p1 L (q (p0 ; p1 )) Notice that from (67) this term is 0:
8
p0 p1 1 p1
i
@q(p0 ;p1 ) @p1
:
9 > > > = > > > ;
:
(69)
c00 (q)
)2
(1
2
p10 )
;
Step 3 The term 1 [1 Noticing that than
p0 p1 ]2
1 p0 (1 p1 )2
1
Take p^0
[ H (qH ) (1 ) q (p0 ; p1 ) r (p1 )] [(1 ) L (q (p0 ; p1 )) + [ H (qH ) r (p1 )]]
p10 so that
(1
1 p10
p0 and that
(1
)
qH
0 H
qL
jq (^ p0 ; 0) 0R q
p^0 ) @
"Z
qL j < H
qL
0 H
2
L (q (p0 ; p1 ))
R qH
(s) ds
q L
q
(s) ds 0 (s)ds H
2
1
L (qL )
!
the term above is no more
p10 ; 0
qL
0 H
(s) ds
#
:
and 0R q
A
:
p^0 K1 > @
H
qL
4
1
A:
0 ;p1 ) Step 4 For p0 2 (0; p^0 ) and almost all p1 2 (0; p0 ) we have @ (p < 0. @p1 Using Steps 1 and 3 notice that 0 0R q 1 1 0R q 0 (s) ds H H @ (p0 ; p1 ) H qL q A p^0 K1 A < (1 < (1 ) @(1 p^0 ) @ )@ L @p1 2
0 H
4
(s) ds
1
A < 0:
This completes the proof: Lemma 16 establishes that there exists an increasing sequence of interior beliefs fpT g1 T =1 converging to 1 such that VnM (p) ; Rn (p) = VTM (p) ; RT (p) for all n T and p 2 [0; pT ] : Therefore for any p 2 (0; 1) we can …nd T 2 N such that p < pT , which establishes that the consumer is screened in at most T periods. 1 Lemma 16 There exists a sequence of beliefs fpT g1 such that limT !1 pT = 1 and T =1 2 (0; 1) M M Vn (p) ; Rn (p) = VT (p) ; RT (p) for all n T and p 2 [0; pT ] :
Proof. Step 0. Assume that for all t T there exists pT 2 (0; 1) such that for all p 2 (0; pT ) then the high type is screened in at most T periods: VnM (p) ; Rn (p) = VTM (p) ; RT (p) for all n T and p 2 [0; pT ] : Lemma 15 implies that this is true for T = 1: In Step 1 We show that there exists pT +1 > pT such that if p 2 0; pT +1 then VnM (p) ; Rn (p) = VTM+1 (p) ; RT +1 (p) for all n T + 1 and p 2 0; pT +1 : In Step 2 we show that limT !1 pT = 1: Step 1 If the induction step is false, for all " > 0 and for every m T; we can …nd n m, p0 2 [pT ; pT + "] and p1 2 arg maxp [maxq~ n+2 (p0 ; p; q~)] \ (pT ; pT + "]: In this case, write q(p0 ; p1 )
9
for arg maxq n+1 (p0 ; p1 ; q) : Take p2 2 arg maxp~ [maxq~ n+1 (p1 ; p~; q~)] such that vH;n+1 (p1 ) = (1 ) q(p1 ; p2 ) + vH;n (p2 ), where we write q(p1 ; p2 ) for arg maxq n+1 (p1 ; p2 ; q) : First assume towards a contradiction that for every " > 0; as we take m ! 1; we can …nd a sequence (nm )1 m for all m; such that the number of periods that it takes to m=1 ; with nm go from a prior in [pT ; pT + "] to a posterior in [0; pT ] increases without bound when maximizing nm +2 : Indeed, if this were the case, as " ! 0 and (then) m ! 1; we would conclude that VnMm +2 (p0 ) ! L (qL ) ; which is strictly smaller than p0 L (qL ) + (1 p0 )[ H (qH ) qL ] and hence we would obtain a strictly higher pro…t by setting p1 = 0 (for small " and large nm ). Therefore, we can …nd "1 > 0 such that whenever " 2 (0; "1 ] the number of periods that it takes to go from a prior in [pT ; pT + "] to a posterior in [0; pT ] is uniformly bounded. In this case, we M (p ) is equal to: may assume w.l.o.g. that p0 2 [pT ; pT + "] and p2 2 [0; pT ] : Hence Vn+2 0 (p0 ; p1 ) (1
)
+ (p0 ; p1 ) (1 + (1
L (q (p0 ; p1 ))
(p1 ; p2 )) [(1
(p0 ; p2 ))
H
(qH )
+ (1 )
(1
)
L (q (p1 ; p2 ))
L (q (p0 ; p1 ))
)
+ [
q (p0 ; p1 )
H
+
2
VnM (p2 )
(qH )
(1
(1
)
)
q (p1 ; p2 )
q (p1 ; p2 ) 2
vH;n (p2 )]]
vH;n (p) :
But notice that since p2 2 [0; pT ] the expression above is equal to (p0 ; p1 ) (1
)
+ (p0 ; p1 ) (1 + (1
L (q (p0 ; p1 ))
(p1 ; p2 )) [(1
(p0 ; p2 ))
H
(qH )
+ (1 )
(1
)
L (q (p1 ; p2 ))
L (q (p0 ; p1 ))
)
+ [
q (p0 ; p1 )
H
(1
+
2
(qH ) )
VTM (p2 ) (1
)
q (p1 ; p2 )
q (p1 ; p2 ) 2
vH;T (p2 )]]
vH;T (p) :
We will show that for " small this is less than: (p0 ; p2 ) (1 + (1
)
(p0 ; p2 )) [
L (q (p0 ; p2 )) H
(qH )
+ VTM (p2 )
(1
)
q (p0 ; p2 )
vH;T (p)] ;
so that the monopolist can pro…tably deviate by speeding-up the screening process: n+2 (p0 ; p1 ; q(p0 ; p1 )) < n+2 (p0 ; p2 ; q(p0 ; p2 )) ; which is a contradiction. Step 1.1 First Inequality. First, notice that since the monopolist can always screen the high R qH 0 type in one period we have VTM (p) L (qL ) + p q H (s) ds: Hence we have: L
(1
)
(1 (1
Take "2 2 0;
1 pT 2
) )[
+ (1
L (q (p0 ; p2 ))
+
L (q (p0 ; p1 ))
L (q (p0 ; p2 )) Z q H 0 ) pT H qL
VTM (p2 )
+ (1
(70)
)
L (q (p1 ; p2 ))
+
L (q (p0 ; p1 ))]
(s) ds:
such that if pT + "2
p0 > p1 > p2
10
pT then:
2
VTM (p2 )
L (q (p0 ; p2 ))
L (q (p0 ; p1 ))
pT
<
R qH q L
0 (s)ds H
2
!
: Hence (70) is at least: (1
)
pT
R qH
0 (s)ds H
q L
2
Step 1.2 Second Inequality. Next consider the term: (qH ) (1 ) q (p0 ; p2 ) vH;T (p2 ) (1 ) q (p0 ; p1 ) (1 ) q (p1 ; p2 ) = (1 ) [ q (p0 ; p1 ) q (p0 ; p2 )] (1 ) [vH;T (p2 ) q (p1 ; p2 )] : H
H
(qH )
2
vH;T (p)
Notice that vH;T (p2 ) qL and q (p0 ; p1 ) q (p0 ; p2 ), hence the term above is more than (1 ) [ qL q (p1 ; p2 )] : p0 ;~ p1 ) Next notice that there is K2 > 0 if pT + "1 p~0 > p~1 pT then @q(~ K2 . Thus the @ p~1 expression above is more than (1 ) K2 "2 : Step 1.3 Third Inequality. Finally consider the term H
(qH )
[(1 Since
(1 )
)
q (p0 ; p2 )
L (q (p0 ; p1 ))
+ [
vH;T (p2 ) H
(qH )
(1
)
q (p1 ; p2 )
vH;T (p2 )]] :
(qH ) > L (qL ) L (q (p0 ; p1 )) it is straightforward to show that there exists "3 2 such that if pT + "3 p0 > p1 > p2 pT then the above term is weakly positive. 0; Step 1.4 The result. Let " := min f"1 ; "2 ; "3 g and assume that pT + " p0 > p1 > p2 pT then the proposed deviation increases the pro…t by at least: 0 1 Rq pT q H 0H (s) ds L A (1 (p0 ; p2 ) (1 )@ (p0 ; p2 )) (1 ) K2 ": 2 H
1 pT 2
Next, notice that (p0 ; p2 ) (1 pT ) and (1 (p0 ; p2 )) the advantage of the deviation is: 2 0 1 Rq pT q H 0H (s) ds L A (1 ) 4(1 pT ) @ 2
Finally, take " 2 (0; ") such that whenever " !# " Rq
at least:
(1
pT )
pT
H
q L
4
0 (s)ds H
" 1 pT
1
" pT
"
"
: Thus a lower bound on 3
K2 "5 :
(71)
" the expression inside the brackets above is
> 0: Thus if pT + "
p0 > p 1 > p 2
pT we have a
contradiction, which proves this step. Step 2 Showing that lim pT = 1: Let pT be the supremum of all beliefs such that the high type is screened in at most T periods. Clearly fpT g is increasing and hence convergent. Assume that fpT g ! p < 1: By an argument analogous to the one in Step 1 we can …nd " > 0 such that if p 2 (p "; p + ") then the belief decreases by at least 2" which contradicts the assumption that p < 1 and concludes the proof.
11
!
:
STEP B: Using V M ; R to construct V M ;
11.1.2
:
De…nition 4 We say that V M ; are self-generating if: i) If P solves (2) subject to (3)-(7) for some state (p; fq g) then we have V M (P) = V M (p; fq g) and (vL (P) ; vH (P)) 2 (p; fq g): ii) For any state (p; fq g); (p; fq g) is convex and, for any extreme point (vL ; vH ) of (p; fq g); there exists P solving (2) subject to (3)-(7) for the state (p; fq g) such that (vL (P) ; vH (P)) = (vL ; vH ). Lemma 17 is immediate from De…nitions 1 and 4. Lemma 17 The values V M ; are pro…t-maximizing if and only if they satisfy the following conditions: i) They are uniformly bounded by some constant K > H ; ii) For all fq g ; the correspondence fq g has a closed graph; iii) They are self-generating. It is straightforward to see that the values V M ; that we will construct below will satisfy i) and ii) from Lemma 17. We will thus explain why the they satisfy iii).
STEP B.1: Construction of V M ; p = 0: We de…ne: V M (p; fq g) = and
(0; fq g)
:
L (qL )
for states (p; fq g) in which p 2 f0; 1g : Belief
UL (fq g)
(72)
= f(UL (fq g) ; vH ) : vH 2 [UL (fq g) +
qL ; UL (fq g) +
]g : (73)
Taking the next period-values (72) and (73) as given, to construct a feasible plan for the state (0; fq g) that delivers the pro…ts (72) and the rents (UL (fq g) ; UL (fq g) + qL ) we consider 1 the plan P1 that contains two contracts identical to L qL (1 ) UL (fq g) ; fqL ; 0g ; assume that both contracts are accepted with the same probability, lead to the posterior 0 and to the nextperiod rents (0; qL ) : It is straightforward to verify the plan is feasible and satis…es V M (P1 ) = UL (fq g) and (vL (P1 ) ; vH (P1 )) = (UL (fq g) ; UL (fq g) + qL ) : L (qL ) To deliver the rents (UL (fq g) ; UL (fq g) + ) consider the plan P2 that contains the con1 tracts 1 = L qL (1 ) UL (fq g) ; fqL ; 0g and 2 = L (1 ) 1 UL (fq g) ; f1; 0g : The contract 1 is accepted with probability 1 by the low-type and zero by the high type, leads to the posterior 0 and to the next-period rents (0; qL ) : The contract 1 is accepted with probability 0 by the low-type and 1 by the high type, leads to the posterior 0 and to the next-period rents (0; ) : It is straightforward to verify the plan is feasible and satis…es V M (P1 ) = L (qL ) UL (fq g) and (vL (P1 ) ; vH (P1 )) = (UL (fq g) ; UL (fq g) + ): Belief p = 1: We de…ne: V M (1; fq g) : =
H (qH )
UH (fq g)
(1; fq g) : = f(vL ; UH (fq g)) : vL 2 [UL (fq g) ; UH (fq g)]g :
12
(74) (75)
Take the next period-values (74) and (75) as given. We construct an optimal plan P that leads to the pro…ts (74) and yields the rents (UL (fq g) ; UH (fq g)) (resp. (UH (fq g) ; UH (fq g))). (The convexity of will allow us to generate any (vL ; vH ) such that vH = UH (fq g) and vL 2 [UL (fq g) ; UH (fq g)] in the current period by generating these extreme points). For that, assume that P contains that contract 1 = ( H (qH q0 ) ; fqH ; q1 ; q2 ; :::g) that leads to the next-period rents UL fq g 1 ; UH fq g 1 2 1; fq g 1 and the contract 2 = ( L q0 ; f0; q1 ; q2 ; :::g) (resp. H q0 ; f0; q1 ; q2 ; :::g)) that leads to the next-period rents 2 = ( UL fq g 1 ; UH fq g 1 2 1; fq g 1 (resp. UH fq g 1 ; UH fq g 1 2 1; fq g The contract 1 (resp. 2 ) is accepted with probability 1 by the high type (resp. low type). Moreover, both contracts lead to the posterior 1. It is straightforward to check that the plan leads to the pro…ts H (qH ) UH (fq g) and that (vL (P); vH (P)) = (UL (fq g) ; UH (fq g)) (resp. (UH (fq g) ; UH (fq g))).
STEP B.2: Construction of V M ; for states (p; fq g) in which p 2 (0; 1) and UH (fq g) UL (fq g) qL : Notice that the commitment allocation that delivers at least
the rent Ui (fq g) to type i is e¢ cient for this case. We de…ne: V M (p; fq g) : = p [
H (qH )
UH (fq g)] + (1
(p; fq g) : = f(UL (fq g); UH (fq g))g :
p)p [
L (qL )
UL (fq g)] :
(76) (77)
Take the next period-values (76) and (77) as given. We construct an optimal plan P that leads to the pro…ts (76) and yields the rents (UL (fq g) ; UH (fq g)) : For that, consider the plan P that contains the contracts 1 = (1 ) 1 (UL (fqL g) UL (fq g)) ; fqL g and 2 = (1 ) 1 (UH (fqH g) UH (fq g)) ; fqH g : The contract 1 is designed exclusively to the low type, leads to the posterior 0 and to the next-period rents (UL (fqL g) ; UL (fqL g) + qL ) 2 (0; fqL g) (see (73)). The contract 2 is designed exclusively to the high type, leads to the posterior 1 and to the next-period rents (UL (fqH g) ; UH (fqH g)) 2 (1; fqH g) (see (75)). It is straightforward to see that the plan P is feasible, leads to the pro…ts (76) and yields the rents (UL (fq g) ; UH (fq g)) :
STEP B.3: Construction of V M ; for states (p; fq g) in which p 2 (0; 1) and UH (fq g) UL (fq g) < qL : Notice that for these states the commitment allocation that
delivers at least the rent Ui (fq g) to type i is not e¢ cient. We use (74),(75), V M and R and to construct these values. Let us de…ne R : [0; 1) ! < by R(p) := qL if p = 0 and R(p) := M M min R(p) otherwise. Let us extend V to [0; 1) by V (0) = L (qL ): Recall also from (74) that V M (1; fqH g) = H (qH ) UH (fqH g): We de…ne:
13
1
).
8 <
9 =
UL (fq g + V M (~ p)) M p) (q ) (1 ) q ~ R(~ V (p; fq g) : = max H H (p; p~)) p~2[0;p];~ q 2[0;1] : (1 UL (fq g) p) + UL (fq g) UH (fq g) : s:t: : (1 ) q~ + R(~ (p; p~) (1
)
q) L (~
p) + UL (fq g) (UL (fq g) ; vH ) : vH = (1 ) q~ + R(~ (~ p; q~) solve (78)
(p; fq g) := co
; :
(78)
(79)
Notice that by construction V M (p; f0g) = V M (p) ; (p; f0g) = (0; R(p)) for all p 2 (0; 1) and (0; R(p)) 2 (p; f0g) for every p < 1: Assume that the next-period values and rents are de…ned by (72)-(79). p) + UL (fq g). We now show Take (~ p; q~) that solve (78) and let vH = (1 ) q~ + R(~ how to construct a plan P that delivers the pro…t V M (p; fq g) and the rents (UL (fq g) ; vH ) (convex-combinations generate (p; fq g)). The plan P contains the contracts:
1
=
2
=
~ Lq (1
(1 )
1
)
1
UL (fq g) ; f~ q ; 0g
UH (fqH g)
q~
(1
)
1
( R(~ p) + UL (fq g)) ; fqH g :
The low type chooses the contract 1 with probability one, while the high type randomizes in such a way that the acceptance of 1 leads to the posterior p~: The contract 1 (resp. 2 ) leads to the next-period rents (0; R(~ p)) 2 (~ p; f0g) (resp. (UL (fqH g) ; UH (fqH g)) 2 (1; fqH g)): Since p) + UL (fq g) UH (fq g) ; we see that P satis…es (5). It is straightforward to (1 ) q~ + R(~ verify that P also satis…es the remaining constraints and hence it is feasible. It is also immediate to see that V M (P) = V M (p; fq g) and that (vL (P); vH (P)) = (0; vH ) :
11.1.3
STEP C: Verifying that V M ;
are Self-Generating
STEP C.1: Veri…cation for states (p; fq g) in which p 2 f0; 1g : Notice that the
values in (72) and (74) are constructed in such a way that the monopolist implements an e¢ cient allocation and extracts all the rent from the type that he attributes probability one. This easily implies that if P is any feasible plan for the state (p; fq g) then V M (P) V M (p; fq g) : Moreover, if p = 0 (resp. p = 1) and vL (P) > UL (fq g) (resp. vH (P) > UH (fq g)) then we must have V M (P) < V M (p; fq g) : We now argue that if P is optimal for the state (0; fq g) then vH (P) 2 [UL (fq g) + qL ; UL (fq g) + To see why vH (P) qL + UL (fq g) ; notice that the low type must consume qL in any optimal plan for this state and use the low-type IR (4) and the IC constraint (3) that guarantees that the high type does not want to imitate the low type. On the other hand, using the IC constraint
14
qH ] :
(3) that guarantees that the high type would not imitate the low type, it is easy to show that vH (P) > UL (fq g) + qH implies vL (P) > UL (fq g) : Since we constructed in section (11.1.2) optimal plans that indeed deliver the two extreme points of (73), we conclude that (0; fq g) is indeed given by (73). We now argue that if P is optimal for the state (1; fq g) then vL (P) 2 [UL (fq g) ; UH (fq g)] : Trivially, if vL (P) > UH (fq g) then using the the IC constraint (3) that guarantees that the high type would not imitate the low type we would conclude that vH (P) > UH (fq g) and obtain a contradiction. On the other hand, vL (P) UL (fq g) follows from the IR constraint (4). As above, we conclude that (1; fq g) is indeed given by (75).
STEP C.2: Veri…cation for states (p; fq g) in which p 2 (0; 1) and UH (fq g) UL (fq g) qL : Notice that V M (p; fq g) achieves the value of the most pro…table allocation
for the monopolist among all the allocations are individually-rational for both types: (4) and (5). This easily implies that that the plan that we proposed in Section 11.1.2 is optimal. Moreover any plan P such that vi (P) > Ui (fq g) for some i 2 fL; Hg is not optimal and hence (p; fq g) := f(UL (fq g); UH (fq g))g :
STEP C.3: Veri…cation for states (p; fq g) in which p 2 (0; 1) and UH (fq g) UL (fq g) < qL : Notice we have constructed V M (p; fq g) and (p; fq g) by restricting
attention to feasible plans in which one contract, say 1 , is of the form (x1 ; fq1 ; 0g) and leads to a posterior p0 2 [0; p]; while the other, say 2 , is of the form (x2 ; fqH g) and leads to the posterior 1. Let V M and be given by (72)-(79). We claim that the payo¤s (V M (P) ; vL (P); vH (P)) generated by optimal plans P is spanned by the class of payo¤s generated by optimal plans that satisfy the restrictions above. First, take any feasible plan P for the state (p; fq g) such that one contract leads to a posterior p1 < p and the other contract leads to a posterior p2 2 (p; 1) : In this case, the argument from Proposition 1 implies that there exists a feasible plan P for the same state such that V M (P ) > V M (P) : Therefore, we can restrict attention to plans in which only posteriors in [0; p] [ f1g are reached. Next consider a plan P containing two contracts, 1 and 2 ; leading to the same posterior and accepted with positive probability. This plan is weakly dominated for the monopolist by the the feasible plan P~ that contains only a contract k 2 f 1 ; 2 g such that V M ( k ) V M j for j 6= k and a revealing contract that is accepted with probability zero. Moreover, we have vi P~ = vi (P) for i = L; H: Therefore, we conclude that it is without loss to restrict attentions to plans P containing one contract leading to a posterior in [0; p] and another to the posterior 1. Next consider a plan P containing a contract 1 leading to a posterior p0 p and a contract
15
leading to the posterior 1. Solve this problem ignoring the IC constraint (3) that guarantees that the low type does not choose the contract 2 ; while keeping the IC constraint that guarantees that the high type would not pro…t from choosing the contract 1 with probability one (as well as the other constraints). Then using (74) and (75), it follows that for any feasible plan P~ for this relaxed problem there is a payo¤ equivalent solution in which the contract 2 is of the form (x2 ; fqH g) and leads to the next-period rents (UL (fqH g) ; UH (fqH g)) 2 (1; fqH g) : However, in this case the low type would not pro…t from imitating the high type. Therefore the same set of payo¤s can be generated by assuming that 2 is of the form (x2 ; fqH g) : Finally, we must show that the same set of payo¤s can be generated by assuming that the contract that 1 (the one that leads to the posterior p0 p) is of the form (x1 ; fq1 ; 0g): First, take a plan and assume that vH (P) UL (fq g) + qL : In this case, consider the alternative plan 1 ~ ~ P containing the contract 1 = (qL (1 ) UL (fq g) ; fqL ; 0g) that is designed exclusively ~ qL ] ; fqH g) that is for the low type and the contract 2 = ((1 ) 1 [UH (fqH g) UL (fq g) ~ designed exclusively for the high type. The contract 1 leads to the next-period rents (0; qL ) 2 (0; f0g) ; while the contract ~ 2 leads to the next-period rents ( L qH ; H qH ) 2 (1; fqH g) : Using UH (fq g) UL (fq g) < qL ; it is straightforward to check that this plan is feasible and that it implements an e¢ cient allocation to both types. Moreover, the low type receives vL P~ = UL (fq g) ; while the high type receives vH P~ = UL (fq g) + qL : If vH (P) > ~ which implies that the plan P leads to a lower pro…t for UL (fq g) + qL ; then vH (P) > vH (P); ~ On the other hand, if vH (P) = UL (fq g) + the monopolist than the plan P: qL ; the e¢ ciency ~ of the plan P implies that both plans are payo¤ equivalent and hence lead to the same payo¤s for all players. Since the contract ~ is of the form (x1 ; fq1 ; 0g) the claim follows. Therefore, assume 2
1
that vH (P) < UL (fq g) + qL for the remainder of this proof. Assume that the plan P contains a contract 1 = (x1 ; fq 1 g) (with q 1 > 0 for some > 0) leading to a posterior p0 p and the contract 2 = (x2 ; fqH g) leading to the posterior one. First assume that p0 = 0: In this case, consider the plan P~ containing the contract ~ = (min q 0 ; q 1 1 L
(1
)
1
UL (fq g) ; fmin q10 ; qL ; 0g)
that is designed exclusively for the low type and the contract ~ = ((1 2
)
1
UH (fqH g)
min q10 ; qL
(1
)
1
qL ; fqH g)
that is designed exclusively for the high type. The contract ~ 1 leads to the next-period rents (0; qL ) 2 (0; f0g) ; while the contract ~ 2 leads to the next-period rents ( L qH ; H qH ) 2 (1; fqH g) : It is straightforward to verify that the plan P~ is feasible and that it leads to a weakly ~ < vi (P) for some i 2 fL; Hg : Henceforth greater pro…t than P; with strict inequality if vi (P) assume that p0 2 (0; p). Observation I will be useful.
16
0 ) 2 Observation I: Take (vL ; vH ) 2 (p0 ; fq g) for some p0 2 (0; 1) : There exists (0; vH 0 (p0 ; f0g) such that vH vH vL : To see why, …rst take (vL ; vH ) 2 (p0 ; fq g) such that UL (fq g) L qL : By construction 0 (vL ; vH ) 2 (p ; fq g) implies vH vL = UH (fq g) UL (fq g) L qL : On the other hand, by 0 0 0 < the de…nition of the program (78) we see that (0; vH ) 2 (p ; f0g) implies that vH L qL : Next, 0 take (vL ; vH ) 2 (p ; fq g) for UL (fq g) < L qL and notice that can rewrite the constraint of p) UH (fq g) UL (fq g) ; which is increasing in UL (fq g) : problem (78) as (1 ) q~ + R(~ Hence if plan P solves (78) when the state is f0g then either the plan P~ that di¤ers from P in that the transfers are uniformly increased by (1 )UL (fq g) solves (78) for the state (p0 ; fq g), which p) UH (fq g) UL (fq g) is violated. In implies the result, or the constraint (1 ) q~ + R(~ the latter case if P solves (78) then vH (P ) vL (P ) > vH (P); implying the result. Thus take a plan P that is feasible for the state (p; fq g) containing the contract 1 = (x1 ; fq 1 g) that leads to the posterior p0 2 (0; p) and the contract 2 = (x2 ; fqH g) that leads to the posterior one. First assume that the (1 ) q01 + R(p0 ) UH (fq g) UL (fq g) : In 1 ~ ~ ) 1 UL (fq g) ; fq01 ; 0g) this case, consider the plan P containing the contract 1 = ( L q0 (1 1 and ~ 2 = ((1 ) UH (fqH g) (1 ) q01 R(p0 ) UL (fq g) ; fqH g): The contract ~ 1 0 (resp. ~ 2 ) leads to the posterior p and to the next-period rents (0; R(p0 )) 2 (p0 ; f0g) (resp. 1 ~ vH (P): and ( L qL ; H qH ) 2 (1; fqH g)): Using Observation I, it is easy to show that vH (P) ~ vL (P): It is then easy to see that Moreover, since vL P~ = UL (fq g) ; we clearly have vL (P)
the acceptance of the contract ~ 1 (resp. ~ 2 ) from the plan P~ leads to a weakly higher pro…t than the acceptance of the contract 1 (resp. 2 ) from the plan P; with strict inequality whenever ~ < vL (P) (resp. vH (P) ~ < vH (P)). vL (P) Finally, assume that (1 ) minfq01 ; qL g + R(p0 ) < UH (fq g) UL (fq g) : In this case, let p be in…mum of all p~ such that (1 ) minfq01 ; qL g + R(~ p) UH (fq g) UL (fq g) : First assume that p = 0: In this case, since UH (fq g) UL (fq g) < qL we have q01 < qL : Let q~01 2 [q01 ; qL ) be de…ned by (1 ) q~01 + qL = UH (fq g) UL (fq g) : We can then construct a more pro…table plan containing the contracts ~ 1 = ( L q~01 (1 ) 1 UL (fq g) ; f~ q01 ; 0g) and ~ = ((1 ) 1 (UH (fqH g) UH (fq g)) ; fqH g) (with ~ 1 leading to the posterior 0). Next, 2 assume that p > 0 (and thus p < p). Since the correspondence p (p; f0g) is upper hemicontinuous and convex-valued, there exists (0; vH ) 2 (p; f0g) such that (1 ) q01 + vH = UH (fq g) UL (fq g) : In this case, consider the plan P~ containing the contracts ~ 1 = ( L minfq01 ; qL g (1 ) 1 UL (fq g) ; fminfq01 ; qL g; 0g) (leading to the posterior p and rents (0; vH ) 2 (p; f0g)) and ~ 2 = ((1 ) 1 (UH (fq g) UH (fq g)) ; fqH g) (leading to the posterior 1 and rents ( L qL ; H qH ) 2 (1; fqH g)). Notice that the contract ~ 2 is accepted (in the plan ~ with a higher probability than the contract 2 (in the plan P). Moreover, L (minfq 1 ; q g) P) 0 L 1 ~ vi (P) for i = L; H: Hence one can use an argument analogous to the one L q0 and vi (P) ~ > V M (P); which concludes the proof. from Proposition 1 to show that V M (P)
17
11.2
Part 2: Checking that there is a PBE consistent with V M ;
History h0 is associated with the state (p0 ; f0g) : We use V M ; to select a continuation equilibrium that maximizes the monopolist’s pro…t. This generates a revealing history h1 and a pooling history h1 leading to posterior p1 < p0 : Given the promised rents at h1 ; we may select another continuation equilibrium that maximizes the monopolist’s value and delivers those rents to the consumer. Proceeding inductively we guarantee that a continuation equilibrium that is consistent with V M ; is selected in each period. There are four points to verify.
11.2.1
For any o¤-path action from the monopolist we can …nd a posteriorupdate and a PBE in the continuation game
Consider the state (p; fq g): It is easy to …nd a continuation equilibrium after any o¤-path deviation when p 2 f0; 1g : We assume that p 2 (0; 1) and UL (fq g) < L qL (the case that UL (fq g) : x1 ; q 1 ; x2 ; q 2 L qL requires small modi…cations). The monopolist o¤ers the menu m = Without loss assume that: (1
)x1 + UL
q1
(1
)x2 + UL
q2
:
Possibility 1: (1 )x1 + UL q 1 < UL (fq g) Assume that the low-type consumer rejects both contracts with probability 1. In this case, the monopolist puts probability 1 on the consumer being a high type if the contract xi ; q i is accepted for i 2 f1; 2g. The low type (resp. high type) obtains a continuation utility equal to UL q i >0 resp: UH q i >0 if the contract i is accepted. Let xk ; q k (k 2 1; 2g) be such that: n o (1 )xk + UH qk (1 )xj + UH q j for j = 3 k: There are 2 cases. Case 1 For some vH 2
H
(1
p; fq g
>0
)xk + UH
we have n o qk (1
)
H q0
+ vH :
In this case both types reject both contracts with probability 1. Case 2 For all vH 2 H p; fq g >0 we have n o (1 )xk + UH qk > (1 ) H q0 + vH : There are two subcases. Case 2.1 (1
)xk + UH
n o qk
(1
)
18
H q0
+ [UL fq g
>0
+
qL ]:
(80)
In this case, the high type chooses the contract xk ; q k with probability 1. The monopolist puts probability 1 on the consumer being a low type conditional on both contracts being rejected. Inequality (80) shows that the high type has no pro…table deviation. This de…nes a continuation equilibrium in the continuation game. Case 2.2 n o (1 )xk + UH qk < (1 ) H q0 + [UL fq g >0 + qL ]: p0
As it is easily veri…ed from our construction that H has a closed graph. Hence there exists 0 2 2 (0; p) and vH H p; fq g >0 such that: n o 0 (1 )xk + UH qk = (1 ) H q0 + [UL fq g >0 + vH ]: (81)
Therefore, the high type randomizes between rejecting both contracts and accepting the contract xk ; q k in such a way that the monopolist updates his beliefs to p0 conditional on both contracts being rejected. Equation (81) shows that the high type has no pro…table deviation. This de…nes a continuation equilibrium in the continuation game. Possibility 2: (1 )x1 + UL q 1 UL (fq g) : This possibility is analogous to possibility 1, with the di¤erence that the low type accepts the contract x1 ; q 1 with probability 1.
11.2.2
For any o¤-path action from the consumer we can …nd a posteriorupdate and a PBE in the continuation game
Consider the state (p; fq g 0 ): By construction it would never be pro…table to deviate to a contract in the menu. Thus assume that the consumer rejects both contracts. We assume that the monopolist puts probability 1 on the consumer being a high type and the consumers obtain the continuation rents UL fq g >0 ; UH fq g >0 2 1; fq g >0 in the next period.
11.2.3
The monopolist does not have a pro…table deviation
Our construction in Section 11.2.1 implies that any o¤-path menu
n
x1 ; q 1
0
; x2 ; q 2
0
o
o¤ered at p; fq g 0 leads to a feasible continuation equilibrium at that state, which immediately implies that the monopolist has no pro…table deviation.
11.2.4
The consumer does not have a pro…table deviation
Consider the state (p; fq g 0 ): By construction, it would never be pro…table for a consumer to deviate to a contract in the menu. Thus assume that the consumer rejects both contracts. In this case, type i 2 fL; Hg obtains a rent equal to Ui fq g >0 ; which is at least this rent by accepting a contract from the menu (see (4) and (5)). Therefore the consumer has no pro…table deviation.
19
