Long-Term Contracting with Time-Inconsistent Agents ∗
Daniel Gottlieb and Xingtan Zhang
October 1, 2016
Abstract This paper studies contracting with agents who are time inconsistent when they cannot commit to long-term contracts.
The results crucially
depend on how aware they are of their time inconsistency.
We show
that being aware of one's time inconsistency makes contracting harder. Because partially naive agents overestimate the surplus from the contract, it is easier to sustain long-term contracting, which increases welfare. On the other hand, sophisticates are aware of their future temptation to drop contracts, making it harder to sustain long-term contracting. (JEL Code: D03, D81, D86)
1
Introduction
Many contractual relationships are limited by one-sided commitment, as consumers are allowed to terminate the agreements they have with rms or start new aggreements with another rm, but rms are contractually forced to serve consumers for the duration of the agreement. For example, with life insurance or annuities, policyholders are allowed to cancel their policies whenever they wish to, but rms cannot drop them.
Or, in many credit markets, borrowers can
prepay their debt but debtors cannot force them to repay before the contract is due. Accordingly, a recent literature has shown that rms can mitigate the lack of commitment on the customer side by oering front-loaded payment schedules, which lock consumers into their contracts (c.f, Dionne and Doherty [1994], Cochrane [1995], or Krueger and Uhlig [2006]). For example, Hendel and Lizzeri [2003] theoretically and empirically examine how life insurers mitigate reclassi cation risk by oering front-loaded policies.
But, while this literature as-
sumes that individuals are time consistent, many individuals are present biased ∗ Gottlieb is from Olin School of Business, Washington University in St. Louis (Email:
[email protected]). Zhang is from the Wharton School of Business of the Univerisity of Pennsylvania (Email:
[email protected]). We thank Jeremy Tobacman, Paul Heidhues, Botond Koszegi, Christian Gollier and seminar participants at Wharton BEPP for helpful comments.
1
(Laibson [1997]; O'Donoghue and Rabin [1999]; O'Donoghue and Rabin [2001]; Augenblick et al. [2015]). In this paper, we study how present bias aects long-term contracts when consumers cannot commit to long-term contracts.
Present-biased agents are
tempted to overconsume, making front-loaded contracts harder to sustain. So, intuitively, one would think that this bias would exacerbate the problem of lack of commitment. As we show here, the eect of present bias crucially depends on whether consumers are aware of their bias. Sophisticated consumers, who fully understand that they will be tempted to lapse, need substantial resources in order to sustain long-term contracting. For example, if their incomes are roughly constant over time, no long-term contract can be sustained. In contrast, it is much easier to sustain long-term contracts when consumers are partially naive. Because partially naive consumers overestimate their ability to avoid temptation, they overestimate the bene t of keeping a long-term contract. As a result, and somewhat paradoxically, those who are partially unaware of their self control problems are much more likely to keep their long-term contracts. In our competitive framework, this implies that partially naive consumers attain a higher welfare than sophisticates. Therefore, educating consumers about their self control problems would decrease welfare! This result is the opposite of the result under two-sided commitment (see e.g. Heidhues and K®szegi [2010]).
Related Literature This paper bridges the literature on dynamic risk-sharing and the literature on contracting with behavioral agents. In the dynamic risk-sharing literature, the most closely related papers are Krueger and Uhlig [2006] and Makarov and Plantin [2013]. Krueger and Uhlig [2006] shows that, depending on the parameters of the model, autarky, partial insurance, and full insurance can all be an equilibrium of his model. Makarov and Plantin [2013] show that, by introducing the assumption that households can terminate loan contracts at any point in time, their model can account for several features of the subprime market.
The main dierence between these
models and ours is that we assume that consumers are time inconsistent. As we show, time inconsistency leads to remarkably dierent results, especially when
1
consumers are not fully aware of it.
Our paper is also related to a recent literature on contracting with behavioral agents, carefully summarized in K®szegi [2014]. In particular, our paper builds on the credit card model of Heidhues and K®szegi [2010] by introducing the assumption that consumers cannot comit to long-term contracts and allowing for more than two periods.
1 Another branch of this literature studies repeated moral hazard without commitent (c.f., Chiappori et al. [1994] and Phelan [1995]). In a principal-agent model, Fudenberg et al. [1990] shows how the principal's incentive to provide insurance after the agent exerts eort can undermine incentives. Building on this result, Netzer and Scheuer [2010] show that, when a benevolent social planner cannot commit not to redistribute income, a competitive market can induce more e cient outcomes than the planner.
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The paper proceeds as follows. In Section 2, we describe the framework and the equilibrium concept. Then, we characterize the equilibrium (Section 3) and present the curse of awarenss results (Section 4).
Then, Section 5 concludes.
All proofs are in the appendix.
2
Model
2.1
Environment
We study a competitive market with many principals ( rms) and a nite number of agents (consumers).
T ≥ 3.
Time is discrete and nite:
t ∈ {1, ..., T } with w at time t.
Each agent has a constant deterministic endowment of
The assumption of a constant endowment is only made to simplify the exposition; as we show in the appendix, all our results generalize to non-constant endownments as well as stochastic endowments. Principals can freely save or borrow at an interest rate
R > 0.
Agents,
however, must rely on principals to transfer consumption across time. discounted pro ts of a principal who collects a stream of payments starting at time
s
The
{pt }t≥s
is
T X pt . Rt−s t=s Following O'Donoghue and Rabin [1999], we separately consider three possible preferences for agents: time consistent, partially naive, and sophisticated.
u(c), where u is twice continuously dieru(0) = 0, u0 (·) > 0, and u00 (·) < 0. time s, agents evaluate the stream of consumption {ct }t≥s according to X δ s−t u(cs ). u(ct ) + β
All of them have felicity functions entiable, At
s>t Agents are
time consistent
if
β = 1
and
time inconsistent
if
β < 1.
inconsistent agents must predict how they will choose in the future.
naive
Time-
Partially-
β , but believe that, in the βˆ ∈ (β, 1]. Sophisticated ˆ = β . So an agent parameter: β
agents have true time-consistency parameter
future, they will behave like agents with parameter agents perfectly know their time-consistency can be described by the pair
2.2
ˆ β). (β,
Market structure and equilibrium
In this economy, agents wish to smooth consumption across periods by contracting with pro t-maximizing principals. At any time
s,
Cs between a {pt (·)}t≥s , givpt (·) + ct (·) = w, ∀t ≥ s. a contract
principal and an agent speci es payment stream to the principal ing the agent a consumption stream
{ct (·)}t≥s ,
where
Payments can be an arbitrary function of any signals up to each period. In particular, it may be conditional on all possible actions by the agent (more on this
3
below). Principals oer contracts to maximize their pro ts and can commit to the contracts that they oer. Agents, however, are able to drop their contract in any period and replace it with a new one. We follow Heidhues and K®szegi [2010] in suppressing the strategic interactions between principals and agents and de ning an equilibrium directly in terms of the contacts that survive competitive pressure. Since there are many competing principals, each of them must make zero pro ts in a competitive equilibrium. Moreover, principals can oer contracts in any period. Therefore, in each period, an agent's outside option is endogenously determined by the best possible contract that they can obtain. As usual, we assume that there are no direct costs in leaving and signing new contracts. We are interested in whether long-term contracts can be supported in equilibrium.
There is no loss of generality in restricting attention to contracts in
which the agent never lapses (renegotiation proofness). To see this, consider a contract in which the agent lapses in some period, replacing it by a contract from another principal. Since the other principal must make zero pro ts from this new contract, the old principal could have replaced the terms of the old contract from this period on with the terms of the new contract, and the agent would be indierent between leaving or staying. Therefore, there is no loss of generality in imposing a renegotiation proof.
non-lapsing
constraint that requires contracts to be
This constraint requires that the agent's outside option
does not exceed the value from staying with the current principal, where the outside value is the best possible contract that other principals can provide.
De nition 1.
ˆ β) agents, a competitive equilibrium is (β, contracts Ct in each period t = 1, ..., T with the
For a market with
a pro le of principal's oered following properties: 1.
[Agent optimization] In each period
t,
the contract
Ct
maximizes the
agent's utility subject to the zero pro ts and the non-lapsing constraints. 2. [Non redundancy] All options in a contract aect the expectation or the behavior of the agent accepting the contract. Given the non-lapsing constraints, the rst-period contract
C1
the agent's equilibrium consumption. We therefore refer to it as the
contract.
determines
equilibrium
As in Heidhues and K®szegi [2010], the non-redundancy requirement
simpli es the analysis but does not aect our predictions about the outcomes and the welfare in equilibrium. By the non-redundancy condition, a partiallynaive agent has at most two options after any given history: choose the baseline option (corresponding to what he thought he would choose) and choose the alternative option (corresponding to what the principal knows that he will choose). For example, in the credit card model studied in Heidhues and K®szegi [2010], borrowers either repay their debt immediately (which is what they think they will choose) or they pay a fee to postpone the payment (which is what rms know that they will choose).
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Figure 1: Contracting with partially-naive agents
c3 (AA)
c4 (AA)
c3 (AB)
c4 (AB)
c3 (BA)
c4 (BA)
c3 (BB)
c4 (BB)
c2 (A)
c1
c2 (B)
2.3
Restating the problem
Consider rst the case of partially-naive agents. Since, in this case, agents may think they will choose something dierent from what they actually choose, a contract speci es payments conditional on the agent's actions. So, by the nonredundancy condition, we can restrict contracts to depend on the history of what the agent thinks he will choose in each period and what the agent actually chooses. We will refer to what the agent thinks he will choose as the baseline, which we will index by B . We will index the agent's actual choice by A. Figure 1 depicts the possible histories for
T = 4.
In period 1, the agent be-
(c1 , c2 (B), c3 (BB), c4 (BB)). However, at t = 2, because his true time-consistency parameter is β instead of βˆ, he chooses option A instead of B , which is what he previously thought he would choose. Then, he believes his consumption starting at t = 2 will be (c2 (A), c3 (AB), c4 (AB)). t 1 A time-t history h is a list of possible actions up to time t: h = ∅, h2 ∈ {A, B}, h3 ∈ {AA, AB, BA, BB}, etc. A contract speci es consumption t conditional on each possible history, i.e. C1 = {ct (h ) : 1 ≤ t ≤ T }. Notice that |ht | = 2t−1 for any t < T . However, since there are no actions after the last peT T −1 riod, there is no space for disagreement at t = T . Therefore, h = h . On the equilibrium path, the agent will always pick option A, but believes that he will E pick B in all future periods. Let c = {c1 , c2 (A), c3 (AA), · · · , cT (A · · · A)} deE note the agent's consumption on the equilibrium path, and let ct = ct (A · · · A) denote the consumption on the equilibrium path at period t.
lieves he will obtain the consumption stream given by
To understand the problem, imagine a principal contracting with an agent
5
who has an outside option with perceived utility
C1 = {ct (ht ) : 1 ≤ t ≤ T }
consumption pro le
u.
The principal oers a
that must satisfy the following
constraints. First, the agent must have a perceived utility (from the baseline alternatives) above
u.
Second, the agent must believe that he will choose the
This is called the perceived choice constraint (PCC). Third, the agent must end up picking A instead of B. This is the incentive constraint (IC). And lastly, the contract must satisfy the non-
baseline alternative in all future periods.
lapsing constraints. We summarize this result in the following lemma:
Lemma 2. Any equilibrium contract must maximize the agent's period-1 perceived utility subject to the zero-pro t, PCC, IC, and non-lapsing constraints. We say that the
market breaks down
in an equilibrium if the equilibrium
contract provides the same consumption as the endowment along the equilibrium
cE t = wt , ∀1 ≤ t ≤ T . We say that the equilibrium features a long-term contract if, at all t ≥ 2, the agent strictly prefers to stay in the time-1 contract
path:
than switch to another contract.
ˆ When, instead, agents know their true time-inconsistency parameter (β
β ),
=
they agree with principals about which future actions they will pick. There-
fore, options
A and B must coincide.
Since, in this case, there is only one history
up to each period, contracts now depend only on the period (C1
T })
3
= {ct : 1 ≤ t ≤
and the PCC and IC constraints automatically hold.
Charaterization of equilibrium contract
In this section, we study the equilibria in each of the three cases (time consistency, time inconsistency with and without sophistication) separately, obtaining conditions for markets to break down and for long-term contracts to be oered in equilibrium.
3.1
Time-consistent agents
We rst look at the benchmark case where agents are time consistent. equilibrium is characterized by the following program:
V1 = max {ct }
T X
δ t−1 u(ct )
t=1
subject to
T T X X ct w = , t−1 t−1 R R t=1 t=1 T X
δ t−τ u(ct ) ≥ Vτ , τ = 2, ..., T,
t=τ
6
The
where
Vτ
denotes the utility from the best possible contract the agent could re-
ceive at time
τ.
The rst constraint corresponds to the zero-pro ts constraint.
The second one is the non-lapsing constraint. To solve the program, we determined the outside options by backward induction, starting in the last period where
VT = u(w).
Such a procedure yields the following result:
Proposition 3. Suppose agents are time consistent.
1. If δR > 1, the equilibrium consumption grows over time and the equilibrium features a long-term contract. 2. If δR ≤ 1, the market breaks down and agents consume their endowments in each period: ct = w. So the long-term contract can be supported if and only if the interest rate
1 δ ). In this case, agents make an up-front payment similar to Hendel and Lizzeri [2003] that must be given up if they were to drop their is high enough (R
>
contracts.
3.2
Time-inconsistent agents
3.2.1 Sophisticated agents With sophisticated agents, the equilibrium program again maximizes the agent's period-1 utility subject to the zero-pro t and the non-lapsing constraints:
V1S =
max
(c1 ,··· ,cT )
u(c1 ) + β
T X
δ t−1 u(ct )
t=2
subject to
T T X X ct w = t−1 t−1 R R t=1 t=1
u(cτ ) + β
T X
δ t−τ u(ct ) ≥ VτS , τ = 2, ..., T,
t=τ +1 where
VτS
denotes the utility from the best possible contract the agent could
receive at time
τ.
As before,
VτS
is obtained by backward induction, yielding
the following results:
Proposition 4. With sophisticated time-inconsistent agents (β = βˆ < 1), there exists rT (β) and RT (β, δ) with 1 < rT (β) < (β, δ) such that: 1. If δR ≤ rT (β), the market breaks down and agents consume their endowment in each period ct = w. 2. If δR ≥ RT (β, δ), the equilibrium features a long-term contract and the long-term contract gives the same consumption as in the model with commitment.
7
rT (β)
Notice that the cuto
that determines when the market breaks down
is strictly greater than 1, showing that markets break down more often when agents are time inconsistent and sophisticated, so it is easier to sustain long-term contracts when agents are time consistent (we will present comparative statics results later). Since sophisticated agents understand their future temptation to drop a contract, they understand that it will be harder to avoid future lapses, making them less willing to front-load payments.
The interest rate must be
higher for sophisticated agents to be willing to sign long-term contracts than in the time-consistent case. When the interest rate is high enough, sophisticated agents will nd it optimal to save in order to consume more in the future. In this case, pre-payment is possible and long-term contracts are provided. The equilibrium contract is identical to the best contract from self 1's perspective without the non-lapsing constraints.
Even though dierent selves have dierent rates of intertempo-
ral substitution, which makes each self tempted to lapse and increase current consumption, the prepaid amount was large enough to ensure prevent lapsing.
¯ rT (β) < R < RT (β,δ) ), time-consistent agents δ δ receive the same long-term contract as when there is full commitment, but For intermediate interest rates (
sophisticated time-inconsistent agents do not.
3.2.2 Partially-naive agents We now consider agents who have a time-consistency parameter
β < 1 but βˆ ∈ (β, 1].
believe that, in the future, they will behave like agents with parameter
Recall that an equilibrium contract is a vector of consumption conditional on all histories equilibrium
C1 = {ct (ht ) : 1 ≤ t ≤ T }. The agent's consumption on E path is the vector c = {c1 , c2 (A), c3 (AA), · · · , cT (A · · · A)}.
We rst introduce an auxiliary problem. who has the following preference: consumption
{ct }t≥s
At time
the
Consider a market with agents,
s,
agents evaluate the stream of
according to
X
δ s−t u(cs ) + βδ T −t u(cT ).
t
β.
The auxiliary problem is given by
V1A = max {ct }
T −1 X
δ t−1 u(ct ) + βδ T −1 u(cT )
t=1
8
subject to
T T X X w ct = t−1 R Rt−1 t=1 t=1 T −1 X
δ t−τ u(ct ) + βδ T −τ u(cT ) ≥ VτA , ∀2 ≤ τ ≤ T
t=τ where
VτA
denotes the utility from the best possible contract the agent could
receive at time
τ.
Let
cA
be the maximizer for the auxiliary problem.
Proposition 5. In a market with partially-naive agents, the equilibrium path consumption is exactly same as in the auxiliary problem, i.e.,cE = cA .
To illustrate the proposition, we rst informally demonstrate the result for
T =3
and defer the proofs to the appendix. Suppose
T = 3,
by Lemma 2, the
problem is
max
c1 ,c2 (A),c2 (B),c3 (A),c3 (B)
u(c1 ) + β[δu(c2 (B)) + δ 2 u(c3 (B))]
subject to
1 1 c2 (A) c3 (A) + = w(1 + + 2 ) R R2 R R ˆ 3 (B)) ≥ u(c2 (A)) + βu(c ˆ 3 (A)) u(c2 (B)) + βu(c c1 +
(1) (PCC)
u(c2 (A)) + βu(c3 (A)) ≥ u(c2 (B)) + βu(c3 (B)) ˆ u(c (B)) + βu(c (B)) ≥ VˆN 2
3
(IC) (2)
2
c3 (B) ≥ w
(3)
u(c2 (A)) + βu(c3 (A)) ≥
V2N
(4)
c3 (A) ≥ w N
whereVt
is the outside option at time
outside option at time
t
(5)
t
from self
t's
from the perspective of self
perspective and
τ < t.
VˆtN
is the
The constraints (2)
- (5) are coming from the nonlapsing conditions. Also note that Heidhues and K®szegi [2010] solves the problem subjects to the rst three constraints because they have two-sided commitment. As in Heidhues and K®szegi [2010], the incentive constraint IC must be
c3 (B) to achieve a higher utility2 . Mathematically, u(c2 (A)) + βδu(c3 (A)) = u(c2 (B)) + βδu(c3 (B)). As we show in the appendix, self 1's perceived consumption at time 2 is 0, i.e. c2 (B) = 0. Since we assume u(0) = 0,we obtain
binding, as otherwise we can increase
u(c2 (A)) + βδu(c3 (A)) = βδu(c3 (B)) 2 By
increasing c3 (B), all constraints are still sati ed. 9
And the agent's objective becomes
u(c1 ) + β[δu(c2 (B)) + δ 2 u(c3 (B))] = u(c1 ) + δu(c2 (A)) + βδ 2 u(c3 (A)) which is identical to the objective in the auxiliary problem. We verify that (2) - (3) hold in the appendix. Thus, the maximization problem reduces to
max
c1 ,c2 (A),c2 (B),c3 (A),c3 (B)
u(c1 ) + δu(c2 (A)) + βδ 2 u(c3 (A))
subject to
1 c2 (A) c3 (A) 1 = w(1 + + 2 ) + R R2 R R u(c2 (A)) + βu(c3 (A)) ≥ V2
c1 +
(6) (7)
c3 (A) ≥ w
(8)
From here, we conclude that the equilibrium path consumption is exactly same as in the auxiliary problem.
By using backward induction to solve for
the auxiliary problem, we can nd the equilibrium contract for partially-naive agents.
Proposition 6. In a market with partially-naive agents, long-term contracts can be supported if and only if δR > 1. Proof.
Proof in the appendix.
So the cuto interest rate for supporting long-term contracts is
R > 1δ ,which
is the same condition in the case of time consistent agent.
Remark 7.
4
The equilibrium contract doesn't depend on
βˆ.
Curse of awareness
In this section, we compare the equilibrium contracts for time consistent agents and time inconsistent agents (sophisticated and partially-naive agents).
We
rst study the result for the market's break-down and then compare the welfare. We nd that being aware of one's time inconsistency makes contracting harder. Because partially naive agents overestimate the surplus from the contract, it is easier to sustain long-term contracting, which increases welfare. On the other hand, sophisticates are aware of their future temptation to drop contracts, making it harder to sustain long-term contracting.
curse of awareness. 4.1
We call the result
Market's break-down
4.1.1 Fixed discounting parameter Suppose the discounting parameter is xed at
δ
for time consitents agents and
time inconsistent agents. Combine our results on the market's break-down condition, we have the following result.
10
Proposition 8. Suppose 1 < δR < rT (β), then both time consistent agents and
partially-naive agents receive long-term contracts from principals but sophisticated agents only receive period-by-period contracts. In other words, when
δR ∈ (1, rT (β)),
unawareness of self-control actually
helps the market to provide the long-term contracts. To the best of our knowledge, our paper is the rst to show the value of unawareness in a contracting environment. Although time inconsistent agents are tempted to enjoy immediate consumption, agents who are unaware of self-control would over-predict the surplus of staying in the contract and thus be willing to be front-loaded in order to enjoy the over-predicted surplus of the long-term relationship.
For agents
who correctly predict future self-control problem, this channel doesn't exist and therefore prepayment scheme would not be possible.
4.1.2 Fixed
α
weighted patience
To compare between time consistent agents and time inconsistent agents, one common criticism is that the degree of patience are dierent when they both have the same discounting factors
δ.
One remedy is to x some weighted pa-
tience.
De nition 9. de ne
α
Suppose
α = (α1 , · · · , αT ),
where
αi > 0
and
P
αi = 1.
We
weighted patience to be
α1 + α2 βδ + · · · + αT βδ T −1 δ1 and time inconsistent δ2 . To x α weighted patience, we require
Suppose time consistent agents have a discounting of agents have a discouting parameter of that
α1 + α2 δ1 + · · · + αT δ1T −1 = α1 + α2 βδ2 + · · · + αT βδ2T −1 Since
β < 1,
it is easy to show that
δ1 < δ2 .
In other words, time inconsistent
agents need to have a higher discounting factor than time consistent agents so that they have similar degree of patience.
Then recall from the market's
break-down result, we have 1. For time consistent agents, the market breaks down when 2. For sophisticated agents, the market breaks down when
R<
3. For partially-naive agents, the market breaks down when Since
rT (β) > 1,
R<
1 δ1 ;
rT (β) δ2 ;
R<
1 δ2 .
the comparison between time consistent agents and sophisti-
cated agents is ambiguous and can vary depending on the model parameters.
rT (β) 1 1 1 δ2 < δ1 and δ2 < δ2 , we nd that it is easiest for partially-naive agents to sustain the market.
But since
11
4.2
Welfare
In this subsection, we study welfare.
Following the literature (e.g.
DellaVi-
gna and Malmendier [2004], O'Donoghue and Rabin [1999, 2001]), we use the agents' long-run time preferences as the relevant in the welfare analysis. Suppose
C (cC 1 , · · · , cT )
(cS1 , · · · , cST ) solves sophisN N ticated agents' problem, and (c1 , · · · , cT ) solves the equilibrium path problem 3 for partially-naive agents (or, equivalently, the auxiliary problem ). We denote solves time consistent agents' problem,
time consistent agents' welfare as
C T −1 UT := δ(u(cC u(cC 1 ) + δu(c2 ) + · · · + δ T )), sophisticated agents' welfare as
UTS := βδ(u(cS1 ) + δu(cS2 ) + · · · + δ T −1 u(cST )). Since partially-naive agents have a mistaken perception about their true timeconsistency parameter, we will attribute the correct parameter
β
in the welfare
comparison:
N T −1 UTN := βδ(u(cN u(cN 1 ) + δu(c2 ) + · · · + δ T )) By de nition, it is straightforward to see that
Lemma 10. Suppose cNT Proof.
βUT ≥ UTN
and
βUT ≥ UTS .
≥ cST , then UTN ≥ UTS .
Note that
UTN N T −1 = u(cN u(cN 1 ) + δu(c2 ) + · · · + δ T ) βδ N T −2 T −1 T −1 = u(cN u(cN u(cN u(cN 1 ) + δu(c2 ) + · · · + δ T −1 ) + βδ T ) + (1 − β)δ T ) ≥ u(cS1 ) + δu(cS2 ) + · · · + δ T −2 u(cST −1 ) + βδ T −1 u(cST ) + (1 − β)δ T −1 u(cST ) = u(cS1 ) + δu(cS2 ) + · · · + δ T −1 u(cST ) =
UTS , βδ
where the inequality comes from the fact that
cN
solves the auxiliary problem.
Corollary 11. Whenever the market breaks down for sophisticated agents, partially-naive agent obtains a higher utility. Proof.
If the market breaks down for sophisticated agents, it must be that
w. By the nonlapsing constraint, cN T N S nd UT ≥ UT . 3 Recall
≥ w.
Then applying previous lemma, we
N from Proposition 5, (cN 1 , · · · , cT ) solves the auxiliary problem.
12
cST =
When the market does not break down for sophisticated agents, the welfare comparison could be ambiguous. But as we let
T
go to in nity, we show that
partially-naive agents always receive a higher utility than sophisticated agents. We have the following asymptotic result.
Proposition 12. Suppose
u is bounded and δ < 1, then limT →+∞ (βUT − UTN ) = 0. Thus limT →+∞ UTN ≥ limT →+∞ UTS . But limT →+∞ (βUT − UTS ) is
not necessary equal to 0, so the inequality holds for some cases. Proof.
Proof in the appendix.
So asymptotically, being unaware of one's time inconsistency increases welfare. This result is the opposite of the result under two-sided commitment (see e.g. Heidhues and K®szegi [2010]).
5
Conclusion
In this paper, we study the role of time inconsistency in long-term contracting under one-sided commitment.
We show that if agents are sophisticated, it is
more di cult to have a long-term contract comparing to when agents are time consistent. If agents are partially naive, we show that the cuto for the market's break-down is exactly same as in the time consistent agents case. Our result suggest a surprising feature of awareness. It is possible that agents who are aware of self-control only get a short-term contract while agents who are unaware of self-control get a long-term contract. Our model suggest the channel that drives the curse of awareness result is that partially naive agents over-predict the value of long-term contract. Through this channel, partially-naive agents also obtain a higher utility than sophisticated agents asymptotically.
13
A
Omitted Proofs
Proof of Proposition 3
Proof.
The time consistent agent solves
1
c =
(c11 , c12 , · · ·
, c1T )
= arg max {ct }
T X
δ t−1 u(ct )
(9)
t=1
subject to
T T X X w ct = t−1 R Rt−1 t=1 t=1 T X
(10)
δ t−τ u(ct ) ≥ Vτ , ∀2 ≤ τ ≤ T
(11)
t=τ We show that the problem is equivalent to
c2 = (c21 , c22 , · · · , c2T ) = arg max {ct }
T X
δ t−1 u(ct )
(12)
t=1
subject to
T T X X w ct = t−1 t−1 R R t=1 t=1
(13)
T T X X ct w ≥ , ∀2 ≤ τ ≤ T t−τ R Rt−τ t=τ t=τ
(14)
It is clear that (11) implies (14) because otherwise agents could nd a better deal.
To show the reverse, i.e.
c2
satis es (11), we argue by contracdiction.
τ ≥ 2, Vτ >
Otherwise one of (11) is violated, say at
(c0τ , · · ·
, c0T ) solves the following program. (c0τ , · · · , c0T ) = arg max {ct }
T X
PT
t=τ
δ t−τ u(ct )
δ t−τ u(c2t ).
Suppose
(15)
t=τ
subject to
T T X X ct c2t = t−τ R Rt−τ t=τ t=τ T X
0
δ t−τ u(ct ) ≥ Vτ 0 , ∀τ + 1 ≤ τ 0 ≤ T
t=τ 0
14
(16)
(17)
PT PT PT c2t w t−τ u(ct ) ≥ Vτ > t=τ δ t−τ u(c2t ). t=τ Rt−τ ≥ t=τ Rt−τ , we have t=τ δ 3 2 2 0 0 Now consider the consumption stream c = (c1 , · · · , cτ −1 , cτ , · · · , cT ). It still 2 satis es both (13) and (14) but it achieves a higher objective than c , contra2 dicted to the optimality of c .
Since
PT
Now we can only need consider (13) and (14). Let the Lagrangian be
L=
T X
δ
t−1
u(ct ) −
λτ ≥ 0.
δ t−1 u0 (ct ) = δR ≤ 1. Then
Then
First consider
λτ
τ =1
t=1 where
T X
T T X X w ct − t−1 R Rt−1 t=τ t=τ
Pt
!
0 τ =1 λτ Rt−1 , or equivalently,u (ct )
=
Pt
τ =1 λτ (δR)t−1 .
u0 (c1 ) ≤ u0 (c2 ) ≤ · · · ≤ u0 (cT ) c1 ≥ c2 ≥ · · · ≥ cT . From the zero pro t condition, we then would cT ≤ w. We also must have cT ≥ w otherwise self T would leave the contract at the last period. So it must be the case that cT = w . Now we have c1 ≥ · · · ≥ cT −1 ≥ w . From the zero pro t condition cT −1 ≤ w . By the nonlapsing condition, we need to have cT −1 ≥ w . Similarly we conclude cT −1 = w. Apply same argument, we have c1 = · · · = cT = w. If δR > 1, we can solve the problem with the zero pro t constraints and
therefore, have
then verify that (14) holds automatically. Solving the problem with only the
c1 < c2 < · · · < cT . Notice that c1 < w because w ≤ c1 < c2 < · · · < cT , contradicted to the zero pro t condition. we have cT > w . Let ξ be the smallest index such that
zero pro t constraint gives otherwise, Similarly
c1 < · · · < cξ < w ≤ cξ+1 < · · · < cT It is clear that (14) holds strictly for
τ ≥ ξ +1.
Now consider
τ ≤ ξ,
we have
T X ct t−1 R t=τ
So (14) holds strictly for
=
τ −1 T X X ct ct − t−1 t−1 R R t=1 t=1
=
T τ −1 X X w ct − t−1 R Rt−1 t=1 t=1
>
T τ −1 X X w w − t−1 t−1 R R t=1 t=1
=
T X w t−1 R t=τ
τ ≤ ξ.
Thus, the equilibrium contract features increas-
ing consumption and the long-term contract is supported in this market.
15
Proof of Proposition 4
Proof.
Recall that sophisticated agents solve
V1S =
max
(c1 ,··· ,cT )
u(c1 ) + β
T X
δ t−1 u(ct )
t=2
subject to
T T X X w ct = t−1 t−1 R R t=1 t=1
u(cτ ) + β
T X
δ t−τ u(ct ) ≥ VτS , ∀2 ≤ τ ≤ T
t=τ +1 Denote the multiplier
λ1
for the zero-pro t constraint and
λτ
for the non lapsing
constraints. The Lagrangian condition implies that
u0 (c1 ) = λ1 λ1 − λ2 u0 (c2 ) βδu0 (c2 ) = R ··· λ1 βδ T −1 u0 (cT ) = T −1 − βδ T −2 λ2 + · · · + βδλT −1 + λT u0 (cT ) R c1 = · · · = cT = w. Then u0 (c1 ) = · · · = u0 (cT ). Denote r = δR and xi = λi+1 Ri+1 , ∀1 ≤ i ≤ T − 1. We can rewrite the conditions in matrix form as follows. 1 − βr 1 0 0 ··· 0 x1 2 βr 1 0 · · · 0 x2 1 − βr3 βr2 β 1 · · · 0 x3 = 1 − βr .. . . . . .. . . . . . . . . . . T −1 T −2 T −3 xT −1 1 − βr βr βr ··· 1
We rst look for the condition for the market's break-down, i.e.
For the market's break-down, we need to have
0, ∀1 ≤ i ≤ T − 1.
λτ > 0, ∀τ ≥ 2 , which impliesxi >
Inversed the lower-triangular matrix, one can nd that the
necessary and su cient condition is
1 ≥ βr + β(1 − β)r2 + · · · + β(1 − β)T −2 rT −1 The right-hand-side is an increasing function of When
r → ∞,LHS
So there exists
rT (β)
r.
When
r = 0,
t=0
β(1 − β)t <
∞ X
β(1 − β)t = β
t=0
16
LHS>RHS.
such that the 18 becomes a
equality. Also note
T −2 X
(18)
1 = 1. 1 − (1 − β)
rT (β) > 1.
So
We now look for conditions such that the market can provide a second best
C denote the contract determined by u0 (c∗1 ) = βδRu0 (c∗2 ) = · · · = P w P c∗t β(δR)T −1 u0 (c∗T ), Rt−1 = Rt−1 . Basically, we need nonlapsing constraints to hold for C. Note that the nonlapsing constraints are simultaneously monotonic ¯ such that if δR ≥ R ¯ , the contract C satis es in R, thus there exists a cuto R contract. Let
all the non lapsing constraints. In this case, the market provides a second best contract. So, if
¯, δR ≥ R
the long-term contract is provided and it is identical
to the optimal contract with commitment.
Proof of Proposition 5
Proof. for
We rst show all IC constraints are binding. Here we write down carefully
T = 4 case since the proof for general T
is essentially same but only lengthy.
We have two IC:
u(c2 (A)) + β[δu(c3 (AB)) + δ 2 u(c4 (AB))] ≥ u(c2 (B)) + β[δu(c3 (BB)) + δ 2 u(c4 (BB))] (19)
u(c3 (AA)) + βδu(c4 (AA)) ≥ u(c3 (AB)) + βδu(c4 (AB))
(20)
c4 (BB) c4 (BB) Then (19) must be binding because otherwise we can raise c4 (BB)
First notice that these two constraints give upper bound constraints for
c4 (AB). and c4 (AB).
and
In addition, no other constraints produce upper bound for
and the agent would receive a higher utility.
Substitute binding (19) to the
objective, we have
u(c1 ) + β[δu(c2 (B)) + δ 2 u(c3 (BB)) + δ 3 u(c4 (BB))] = u(c1 ) + δu(c2 (A)) + β[δ 2 u(c3 (AB)) + δ 3 u(c4 (AB))] + (β − 1)δu(c2 (B)) Now we have eliminated
c4 (BB).
By the same argument, (20) must be bind-
ing because otherwise we can raise
c4 (AB)
and achieve a higher utility.
By
substituting IC, the objective becomes
u(c1 )+δu(c2 )+· · ·+δ T −2 u(cT −1 )+βδ T −1 u(cT )+(β−1)[δu(c2 (B))+· · ·+δ T −2 u(cT −1 (A · · · AB))] Since
β < 1, we want to pick c2 (B), · · · , cT −1 (A · · · AB) as small as possible.
However, perceived path nonlapsing constraints give lower bound contraints for them. We next show that if
c2 (B) = · · · = cT −1 (A · · · AB) = 0,
erceived path nonlapsing constraints automatically hold.
17
then all the
Let
(ˆ ct , · · · , cˆT )
denote the maximizer for
VˆtN .
Then
T −t ˆ u(ct (A · · · AB)) + β(δu(c u(cT (A · · · AB · · · B)) t+1 (A · · · ABB)) + · · · + δ βˆ = (δu(ct+1 (A · · · ABB)) + · · · + δ T −t u(cT (A · · · AB · · · B)) β βˆ = (u(ct (A · · · AA)) + β(δu(ct+1 (A · · · AAB)) + · · · δ T −t u(cT (A · · · AAB · · · B)) β βˆ ≥ Vt (w) β βˆ ≥ (u(ˆ ct ) + β(δu(ˆ ct+1 ) + · · · + δ T −t u(ˆ cT )) β βˆ ˆ ct ) + β(δu(ˆ ct+1 ) + · · · + δ T −t u(ˆ cT )) = u(ˆ β ≥ Vˆt (w) All the perceived path nonlapsing constraints automatically hold. Thus there is eectively no lower bound constraints for
c2 (B), · · · , cT −1 (A · · · AB),
they must be equal to the minimum level of consumption, which is
therefore
0
by our
assumption. Recall
cE t denote
the equilibrium path consumption at time
t.
By substitut-
ing binding IC constraints, it is easy to see that equilibrium path nonlapsing constriants can be simpli ed as
E T −t N u(cE u(cE t ) + δu(ct+1 ) + · · · + βδ T ) ≥ Vt .
So, the problem reduces to
max u(c1 ) + δu(c2 ) + · · · + δ T −2 u(cT −1 ) + βδ T −1 u(cT ) ct
subject to
T T X X w ct = t−1 t−1 R R t=1 t=1
u(ct ) + δu(ct+1 ) + · · · + βδ T −t u(cT ) ≥ Vt , ∀2 ≤ t ≤ T Applying the same technique in Proposition 3, we can rewrite the nonlapsing constraints as
T T X X ct w ≥ , ∀2 ≤ τ ≤ T t−1 R Rt−1 t=τ t=τ
Thus, the problem is equivalent to the auxiliary problem.
Proof of Proposition 6
Proof.
Consider the auxiliary problem.
max u(c1 ) + δu(c2 ) + · · · + δ T −2 u(cT −1 ) + βδ T −1 u(cT ) ct
18
subject to
T T X X w ct = t−1 R Rt−1 t=1 t=1 T T X X ct w ≥ , ∀2 ≤ τ ≤ T t−1 t−1 R R t=τ t=τ Let the Lagrangian be
L=
T −1 X
δ
t−1
u(ct ) + βδ
T −1
u(cT ) −
λτ ≥ 0.
Pt
τ =1 λτ (δR)t−1
. If
λτ
τ =1
t=1 where
T X
If
T T X X w ct − t−1 R Rt−1 t=τ t=τ
1 ≤ t ≤ T −1, we have δ t−1 u0 (ct ) = 0
t = T, u (ct ) =
Pt
τ =1 λτ β(δR)t−1
>
Pt
τ =1 λτ (δR)t−1
. If
!
Pt
0 τ =1 λτ Rt−1 , or equivalently,u (ct )
δR ≤ 1,
then
u0 (c1 ) ≤ u0 (c2 ) ≤ · · · ≤ u0 (cT ), c1 ≥ c2 ≥ · · · ≥ cT . From the zero pro t condition, we then have cT ≥ w from self T's nonlapsing constraint. So it must be the case that cT = w . Now we have c1 ≥ · · · ≥ cT −1 ≥ w . From the zero pro t condition cT −1 ≤ w . By the nonlapsing condition, we need to have cT −1 ≥ w. Similarly we conclude cT −1 = w. Applying the same argument, we have c1 = · · · = cT = w . Now if δR > 1, consider the problem with the same objective function and
therefore,
cT ≤ w .
We also have
the zero pro t condition, but without the nonlapsing constraints.
c˜ = (˜ c1 , · · · , c˜T ) = arg max u(c1 ) + δu(c2 ) + · · · + δ T −2 u(cT −1 ) + βδ T −1 u(cT ) ct
subject to
T T X X ct w = t−1 R Rt−1 t=1 t=1
cT ≥ w u0 (˜ c1 ) = · · · = (δR)T −2 u0 (˜ cT −1 ) ≤ β(δR) u (˜ cT ). Since δR > 1, we have c˜1 < c˜2 < · · · < c˜T −1 . We next verify PT PT ct w that c ˜ satis es all the nonlapsing constraints: t=τ Rt−1 ≥ t=τ Rt−1 , ∀2 ≤ τ ≤ T , in which case, c˜ would be the optimal solution for the original problem and the long-term contract is supported in this market. To see that, let ξ be PT PT w t the largest index such that c ˜ξ < w. If τ ≥ ξ + 1, t=τ Rc˜t−1 ≥ t=τ Rt−1 . If τ ≤ ξ , we have Applying Lagrangian condition gives
T −1 0
19
=
T X c˜t t−1 R t=τ
=
T τ −1 X X c˜t c˜t − t−1 R Rt−1 t=1 t=1
=
T τ −1 X X w c˜t − t−1 R Rt−1 t=1 t=1
T τ −1 X X w w > − t−1 t−1 R R t=1 t=1
=
T X w t−1 R t=τ
c˜T > w, then the solution is given by u0 (˜ c1 ) = · · · = (δR)T −2 u0 (˜ cT −1 ) = P P T T c˜t w T −1 0 β(δR) u (˜ cT ) and t=1 Rt−1 = t=1 Rt−1 . If c ˜T = w, the problem can be further reduced to If
max u(c1 ) + δu(c2 ) + · · · + δ T −2 u(cT −1 ) ct
subject to
T −1 X t=1
T −1 X w ct = t−1 Rt−1 R t=1
Then the solution is determined by
PT −1
w Rt−1
, and
u0 (˜ c1 ) = · · · = (δR)T −2 u0 (˜ cT −1 ),
PT −1
c˜t t=1 Rt−1
c˜T = w.
t=1 In summary, long-term contracts can be supported if and only if
δR > 1.
Proof of Proposition 12
Proof. If δR ≤ 1, only period-by-period contracts can be supported. Then βUT = UTN . 0 C T −2 0 C Let's assume δR > 1. First notice that u (c1 ) = · · · = (δR) u (cT −1 ) = T −1 0 C (δR) u (cT ). It follws from the proof of Proposition 6 that u0 (cN 1 ) = ··· = T −1 0 N N 0 C 0 N (δR)T −2 u0 (cN u (cT ). If cC 1 − c1 > 0, u (c1 ) < u (c1 ), which T −1 ) ≤ β(δR) 0 C 0 N C N implies that u (ci ) < u (ci ) for all 1 ≤ i ≤ T − 1, i.e. ci − ci > 0. Since β < 1, C N we can similarly nd that cT − cT > 0. But this can't be true as it violates PT cCi −cN C N 0 C 0 N i i=1 Ri−1 = 0. So we must have c1 − c1 ≤ 0, u (c1 ) ≥ u (c1 ), which implies
20
=
that
0 N u0 (cC i ) ≥ u (ci )
1 βδ
for all
1 ≤ i ≤ T − 1,
i.e.
N cC i − ci ≤ 0.
Note that
lim (βUT − UTN )
T →+∞
T −1 X
= lim
T →+∞
N T −1 N δ i−1 [u(cC [u(cC i ) − u(ci )] + δ T ) − u(cT )]
i=1
N ≤ lim 0 + δ T −1 [u(cC T ) − u(cT )] T →+∞
=0 where the last step uses the fact that u is bounded and δ < 1. Recall that βUT − UTN ≥ 0 by the de nition. So limT →+∞ (βUT − UTN ) = 0. S To show βUT −UT does not converge to 0, it is su cient to give a counterexample. It is easy to verify that u(w) = −exp(−w) is such a counterexample.
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