Dynamic De…ned-Contribution Pension Design with Adverse Selection and Moral Hazard Tsz-Nga Wong Washington University in Saint Louis First Version: Mar 2009, This Version: Sept 2011
Abstract We study voluntary de…ned-contribution pension contracts with the incentive problem of early retirement and low contributions over time. Agents are free to retire, quit a plan and choose between plans. The ‡uctuation of labor productivity throughout working life and the length of working life are private information. The optimal contract can be implemented through transfers (sometimes negative) and contribution deductions from agents’ pensions over time. The optimal contract features a punishment phase, an accumulation phase and a retirement phase. We …nd that the amount of pension is higher under the optimal contract than under laissez faire. Working agents enjoy higher consumption, contribute less, and retire later. The result is robust to di¤erent environments. Keywords: De…ned-Contribution Pension, Dynamic Adverse Selection and Moral Hazard, Retirement, Risk-Neutral Measure
I am grateful to Rody Manuelli for guidance on continuous-time models. I would like to thank comments by seminar participants of various conferences and institutions, in particular, Costa Azariadis, Marcus Berliant, Kim Sau Chung, Hugo Hopenhayn, Boyan Jovanovich, James Mirrlees, Chris Sleet, Ping Wang, Ivan Werning, Steve Williamson and John Zhu on earlier versions. Financial support by McDonnell International Scholars Academy and by the Center for Research in Economics and Strategy (CRES), in the Olin Business School, Washington University in St Louis is gratefully acknowledged. Correspondence: Department of Economics, Washington University in Saint Louis, MO, USA, 63130. Tel: 314-935-5670, E-mail:
[email protected]. Web: sites.googlesite.com/site/tszngawong
1
Introduction
The last three decades have witnessed a drastic change in pension plans: from de…nedbene…t pension to de…ned-contribution pension (DC). This paper1 studies voluntary DC plans with asymmetric information. A DC plan is a contract that speci…es the level of pension based on the history of contribution and the date of retirement. Among pension designer’s concern in providing pension are that paticipants to DC plans might retire too early to collect the pension, or contribute too low. In particular, we focuse on the incentive-insurance trade-o¤ in DC plans, which is di¤erent from those found in standard dynamic contract problems. The insurance part comes as the follows. The economy consists of heterogenous agents, who are di¤erent in the length of working life and the path of labor productivity, like skills and physical strength, during one’s working life. These characterize the life-cycle of an agent. The shock to one’s length of working life determines his retirement date and pension. Shocks to one’s labor productivity a¤ect his incomes and hence contributions to pension plans. These shocks call for the insurance to improve agents’welfare. On the other hand, one’s life-cycle is also private information. A DC mechanism consists of many DC plans, and each plan speci…es an allocation to target the agent with a particular pattern of life-cycle. The asymmetric information of life-cycle creates a special form of adverse selection and moral hazard problem. Agents of the type of long working life can game the mechanism by retiring earlier than their natural ends of working life. Agents of the type of high labor productivity can game the mechanism by consuming more privately and contributing less publicly. This is can be done by switching to plans with low contributions, which are supposed to target agents with low labor productivity. The presence of asymmetric information undermines the scope of insurance provided by a mechanism. 1
One may be surprising that, despite its importance, many issues of de…ned-contribution pension mechanism regarding uncertain life-cycle of labor productivities and its information structure are still open questions and remain unexplored. For example, Diamond (2009) comments that, "In particular, I think we have done too little study of the issues around tax-favored retirement savings accounts, studies that need to recognize uncertainty in future earnings ... and earnings opportunities...."
1
This paper studies a minimal form of dynamic, voluntary, DC mechanism. The level of pension depends only on the contribution history and the retirement time (the minimal observation made by the pension designer). To feature such a form of mechanism, we explicitly model an information structure in which agents’consumption, the retirement shock and labor productivity shocks are private information. The mechanism provides di¤erent pension plans for heterogenous agents. Agents choose whether to keep working or to retire before the retirement shock hits. The pension designer cannot distinguish whether an agent retires due to his own choice or due to the retirement shock. Once the agent retires, a pension plan provides a lump sum of pension according to the mechanism committed at the beginning of working life. Agents can quit the plan any time to take an exogenous outside option. We also extend to our analysis to environments with continuum labor choice and with observable incomes. Our qualitative results are robust to di¤erent environments. We focus on the economy with a benevolent pension designer, who maximizes agents’lifetime utilities with feasibility and incentive constraints. It can also be interpreted as the typical …rm in the competitive pension fund market where …rms competes with each other by providing DC pension contracts, so the set of equilibrium contracts coincides with the optimal DC mechanism. The benevolent pension deisgner can also be interpreted as the government. The extension where the pension designer also observes agents’income captures some features of social security system in practice. This paper also serves as a positive theory of pension. There are several stylized features of the DC plans. First majority of the DC plans are voluntary and run by the private sector. Agents who are eligible to participate must actively choose to do so. The private sector provides a variety of plans with di¤erent levels of pension and contribution arrangement. Second, withdrawals from DC plans (cash out) are common. This attributes to the fact that most DC plans have relatively modest or low balance. See United States Government Accountability O¢ ce (2007). Third, the retirement time and participation rates to DC plans are various amongst agents. According to Survey of Consumer Finance (2007), the participation rates range from 34% to 89% across the income clusters, and from 67% to 85% across the age clusters. 2
Our results demonstrates that these stylized features can be explained by studying a DC mechanism with asymmetric information. The analysis and results of this paper are developed as follows. First, to support any mechanism to be incentive compatible, proposition 1 characterizes the set of incentive-compatible recommended actions as a function of promised continuation utility. The evolution of agent’s promised continuation utility in turn depends on recommendations. These can be characterized tractably because we formulate the agent’s problem in a "risk-neutral world", in which the observed contributions following a Brownian motion. Then, we set up the Hamilton-Jacob-Bellman (HJB) equation which corresponds to the pension designer’s problem, treating the promised continuation utility as the state variable. Proposition 2 characterizes the optimal mechanism. The optimal mechanism features three phases, namely a punishment phase, an accumulation phase and a retirement phase. Then, we show that the pension balance provided by the optimal mechanism can be expressed in a distortion form. Proposition 3 characterizes the optimal dynamic transfers and contribution deduction rates over time. Then, to identify the source of welfare improvement, proposition 4 compares the optimal mechanism with laissez faire. To see how shocks a¤ect the incentive problems and allocations, proposition 5 provides the comparative statics with respect to di¤erent shock parameters. Then, we compare the optimal mechanism with the …rst best, in which there is no asymmetric information, to understand the cost of providing incentives. Finally, to check the robustness of our results, we extend to environments with continuum labor choice and with observable income.
2
Related Literature
Our model follows the seminar work of Diamond and Mirrlees (1978, 1982), and other. Diamond and Mirrlees (1978) study an economy without discount in continuous time, with constant labor productivity, and with private information of the retirement shock only. They show that it is desirable to have agents indi¤erent on the date of retirement. However, results for general life-cycle features remain unknown. To apply the analysis to a dynamic, voluntary, DC mechanism, we extend to a general 3
environment with discount, general utility function and general income processes. In fact, we show under the optimal mechanism, agents strictly prefer working to retirement during a punishment phase and an accumulation phase. The presence of private labor productivity shocks implies, interestingly, the moral hazard problem of the private retirement shock is not always binding. On the other hand, there is a literature on dynamic optimal taxation which emphasizes the importance of private labor productivity shocks. See Kocherlakota (2005), Albanesi and Sleet (2006) and Golosov and Tsyvinski (2006, 2007). The optimal taxation scheme to some degree sheds light on the optimal DC mechanism, if we approximate pension accumulation as individual saving without retirement. There is also a strain of this literature focuses on e¢ cient allocations in dynamic, private information economies, for example Atkeson and Lucas (1992), Cole and Kocherlakota (2001) and Golosov, Kocherlakota and Tsyvinski (2003). These papers have focused, almost exclusively on …xed horizon of agent, …nite or in…nite, rather than endogenous timeframe of agents’activities which allows voluntary retirement. Endogenous retirement essentially implies that the mechanism designer has to deal with the incentive problem of retirement. The mechanism designer also has to determines the length of the mechanism. These are beyond the scope of the above papers. Since solving for dynamic contracting problem is notoriously complicated, it is common that the analysis is up to providing some …rst order conditions of constrained optimality. One direction in the literature is to focus on the optimal stationary action contract, such as Holmstrom and Milgrom (1987), DeMarzo and Sannikov (2006), and Biais et al (2007). Another solution is to study an environment under which the optimal contract has some simple form, e.g. Golosov and Tsyvinski (2007). But the expenses are that either we have to give up to characterize dynamic features of the optimal contract, or to focus on some restricted applications. Yet another direction is to attack the general problem under a continuous-time framework, notable examples are Sannikov (2008) and Williams (2011). Also see Holmstrom and Milgrom (1987), DeMarzo and Sannikov (2006), Zhang (2009), Piskorski (2010) and Farhi and Werning (2010) for other applications. See Biais et al (2010) for an application with Poisson processes. One justi…cation is that the continuous4
time model is a good approximation of discrete-time model. See Biais et al (2007). The continuous-time model allows us to use powerful tools of stochastic calculus. See Yong and Zhou (1999), Karatzas and Shreve (2000 KS), and Ma and Yong (2007). The elegant applications of stochastic calculus in Sannikov (2008) and Williams (2011) are important technical references for our paper. This fruitful modeling strategy motivates us to study DC mechanisms in a continuous-time model. Because of our purpose to study DC mechanisms, our model is di¤erent from the principal-agent models such as Sannikov (2008) and Williams (2011), in term of preference, information structure and agents’ actions. Their frameworks do not incorporate the Poisson shock, so the optimal contract outlined in their models cannot capture the incentive problem of the retirement. There is no direct motive of retirement and pension in their models, but there nevertheless is an event when zero e¤ort is recommended. This event can be interpreted as retirement. Here to capture the private information of the retirement shock, we explicitly model it with a Poisson process. Our model captures the adverse selection problem of early retirement: able agents may game the mechanism by collecting the pension early. This incentive problem is absent in their environment. We also focus on the implementation of the optimal mechanism, welfare implications and the comparison with laissez faire, which are beyond the concern of standard principal-agent problem. One can interpret the accumulation of pension under a DC plan as a optimal saving problem with the uncertainty of income and life-cycle. It connects with the literature of precautionary saving such as Huggett (2004) and many other. It also connects with the literature of optimal consumption and portfolio choice with uncertainty, such as Merton (1971) and many other. From the agents’point of view, one crucual di¤erence is that the path-dependence of the pension/ saving. In the standard saving problem, any two paths of saving with the same prevent value lead to the same amount of pension at a retirement date. This is not true for a DC plan. As show later, under the optimal mechanism, a path of contributions with more frontloaded contributions features a higher level of pension. In our paper, it is our goal to study how the optimal path-dependence pension provides incentives and insurance.
5
3
Model
Time is continuous and goes forever. The economy is populated by a continuum of risk-averse agents with unit mass, who work, consume and retire. Every agent has endowment a0
0 at time 0. There are two kinds of shocks to agents, namely
the labor productivity shocks and the retirement shock, both are idiosyncratic and private information. The pension designer collects contributions and provides a lump sum of pension once agents retire.
3.1
Technology and Setting
Since labor choice is discrete, which is either to work or to retire, one’s income is just his labor productivity. We extend to continuum labor in the later section. If an agent remains working, his ‡ow of income at time t is d t , where labor productivity Brownian motion of drift We denote Bt =
t
> 0 and volatility
t
is a
> 0 with probability measure P ( ).
t = as the associated standard Brownian motion, which is
interpreted as the labor productivity shock. The maximal length of working life is the time until the retirement shock hits. The date of the retirement shock follows an independent Poisson process with arrival rate . There is a saving technology with a rate of return r, which is exclusive to the pension designer. An event consists of the entire history of labor productivities. In particular, an event ! in the event space
represents a duple consisting of a continuous path B
with a Werner measure P ( ) of path
and a path taking value of zero or one which
indicates whether retirement shock has hit by or before, with Poisson measure P of arrival rate .
3.2
Actions: Retirement, Contribution, Consumption and Quit
Retirement is irreversible and triggered either involuntarily by the Poisson retirement shock, or voluntarily by agent’s choice. Each agent can retire any time before they are
6
hit by the retirement shock. Once he is hit by the retirement shock, he is no longer able to work and must retire. The pension designer cannot distinguish whether it is the retirement shock or the agent’s choice that leads to retirement. Let TR (!) :
! R+
denotes the stopping time of early retirement, which happens before the retirement shock. Once the agent retires, his continuation value then can be speci…ed by a value function U (a) : R+ ! R+ which is twice di¤erentiable, increasing, and strictly concave. We also assume Ua (a)
dU (a) =da is bounded and in particular Ua (a) is
strictly bounded below from zero. Without loss of generality we assume U (0) = 0. The shape of the value function after retirement, U , captures the utility of leisure after retirement, and the institution of pension bene…t. The details after retirement are beyond our focus, which is pension contributions, and thus we assume it can be conveniently summarized by a pension function U . A technical convenience is that we deport the stochastic transversality condition to U , so we do not need to worry about the possibility of Ponzi game. Let dYt denote the contribution normalized to the volatility of labor productivity , and thus dYt is the actual contribution at time t, which is given by agent’s budget: dYt = d
t
(1)
ct dt;
where ct is agent’s consumption ‡ow at time t. If ct is observed, and thus
t,
then
the …rst best can be achieved and the problem becomes trivial. So we assume hidden consumption in our model. Agents stop contributing after they retire or quit the mechanism. Agents can always quit the mechanism any time and leads to an outside option with continuation value Vmin . Quit is irreversible. Let TL (!) :
! R+ denote
the stopping time when the agent quits the de…ned-contribution mechanism.
3.3
Mechanism: De…ned-Contribution Pension
The benevolent pension designer collects contribution during agent’s working life, and o¤ers pension whenever agents retire. The level of pension depends on the public
7
history of contribution Y T until retirement and the retirement time T as A Y T ; T : C[0; 1)
[0; 1) ! R+ .
In particular, the pension designer only observes history of pension contribution Y fYt ; 0
t < 1g, but not its decomposition which involves agent’s consumption and
actual labor productivity, so the mechanism cannot be speci…ed on these unobserved variables. From the perspective of the pension designer, the DC mechanism A Y T ; T
is
Y constructed to be progressively measurable on …ltration fFtY g1 t=0 , where Ft is the
augmented …ltration2 generated by Y and P0 (Y ) is the probability measure of con-
tribution path Y . In particular, P0 (Y ) is the probability measure of a standard Brownian motion (associated with the reported labor productivity b, which can be
di¤erent from the true labor productivity ) with drift
ct =
at time t. The
…ltration FtY is interpreted as the public information of contributions until time t.
3.4
Preferences and Incentives
Incentives of agents are their continuation utility. Given a de…ned-contribution mechanism A Y T ; T , agent’s continuation value vt , given the private information Ft and true probability space ( ; fFt g1 t=0 ; P ( ) vt =
max
C 0;b;Y;TR ;TL
E
8 Z < :
TR ^TL
e
(r+ )(s t)
P ) is:
s
[u (Cs ) + U (A (Y ; s))] ds
t
+e
(r+ )(TR ^TL t)
1TR
TL U
A Y
TR
; TR
;
(2)
subject to the evolution of observed contribution dYt = d 2
+ 1TR >TL Vmin
Ft
9 =
t
Ct dt = dbt
ct dt:
The augmented …ltration FtY generated by Y is the product -…eld of the …ltration generated by Y t and P0 -null sets. The reason to consider augmented …ltration generated by Y rather than …ltration generated by Y is that the latter is not right-continuous, hence usual conditions fail to be held. Usual conditions are needed for Martingale Representation Theorem, the existence RCLL modi…cation and the existence of the weak solution to backward sotchastic di¤erential equation, which are applied in this paper.
8
,
Since the retirement shock and the labor productivity shocks are independent, without loss of generality we can focus on the true probability space of contribution C[0; 1); fFtY g1 t=0 ; P (Y ) . The true probability measure of contribution P (Y ) is the
probability measure of a Brownian motion with drift
Ct = at time t.
It is obvious in (2) that participation to the mechanism at time t is implicit, which is given by the fact that: vt
maxfVmin ; U A Y t ; t g;
(3)
otherwise, the typical agent would quit immediately (TL = t) if vt < Vmin , or would retire immediately (TR = t) if vt < U (A (Y t ; t)). Verbally, for agents who are still choosing actions under a mechanism, it must be that they have not retired nor quited the mechanism before. We assume u (c) : R+ ! R+ is twice di¤erentiable, increasing, and strictly con-
cave. We also assume both u (c) and uc (c) are Lipschitz continuous and satisfy the linear growth condition. Without loss of generality we assume u (0) = 0. Given a DC mechanism A Y T ; T , we denote the set D of the maximizers to (2)
as :
D (A)
f(c; Y; TR ; TL ) : (c; ; Y; TR ; TL ) solves (2) at t = 0 given A Y T ; T g. (4)
Consider the pension designer recommends c (Y ), TR (Y ) and TL (Y ) according to the observed history of contribution Y . We say that a de…ned-contribution mechanism A Y T ; T is incentive compatible is if there is process Y such that (c (Y ) ; Y; TR (Y ) ; TL (Y )) 2 D (A). Also, we say that TR and c can be supported by an incentive-compatible mech-
anism A (Y; T ) if there there exists Y and TL such that (c; Y; TR ; TL ) 2 D (A).
The di¢ culty in solving (2) is its lack of recursive structure. We cannot apply
standard HJB approach to solve vt . In particular, the choice of contribution dYt at time t will a¤ect all sequential ‡ows of value through A (Y s ; s), for s
t. This is
essentially the feature of DC plans where the level of pension depends on the path of contributions. The following proposition establishes the evolution of vt under some incentive9
compatible mechanisms: Proposition 1 Under the maintained assumptions, given v0 > Vmin , any stopping time TR of fFtY g1 t=0 and any non-negative, progressively measurable and bounded
process c can be supported by an incentive-compatible mechanism A Y T ; T in C[0; 1); fFtY g1 t=0 ; P (Y ) Furthermore, vt is the unique strong solution to dvt =
(
u (ct ) U (At )] dt + uc (ct ) dBt , if ct > 0 U (At )] dt + qt dBt , if ct = 0
[(r + ) vt [(r + ) vt
(5)
where q is an arbitrary positive, bounded, progressively measurable process and A is an arbitrary progressively measurable process such that qt
uc (0) ;
At 2
0; U
1
(vt ) :
In particular, the supporting Y is given by dYt =
ct
dt + dBt :
The supporting TL is given by: TL = inf ft : vt = Vmin g :
(6)
The supporting incentive-compatible mechanism is: T
A Y ;T =
(
U 1 (vTR ) if T = TR AT , otherwise,
(7)
Proof. See Appendix 2. The proof of proposition 1 involves a change to the "risk-neutral space" C[0; 1); fFtY g1 t=0 ; PY (Y ) .
Under such a probability space, Y becomes a standard Brownian motion without drift.
The risk-neutral space serves as a bridge connecting the true space C[0; 1); fFtY g1 t=0 ; P (Y ) and the reported space C[0; 1); fFtY g1 t=0 ; P0 (Y ) , which are the same if appropriate
incentive constraints are satis…ed. We leave all the details and technical discussion to 10
Appendix 1. The proof of proposition 1 also establish a version of taxation principle (for example see Albanesi and Sleet (2006)), which shows that any direct mechanism b can be supported by some DC mechanism (the allocation depends instead on B) A Y T;T :
At time 0, the mechanism promises a lifetime utility v0 . If there were none of any
‡ow of utility from consumption nor retirement, the drift of promised continuation utility has to be (r + ) vt in order to keep the level of promised continuation utility at vt .
So the drift of promised continuation utility has to be reduced by u (ct ) +
U (A (Y t ; t)). That is why the drift of promised continuation utility is (r + ) vt u (ct )
U (A (Y t ; t)).
The pension designer provides incentives to encourage high contribution by manipulating the sensitivity of path of vt to dYt , which is given by qt
uc (ct ). The
sensitivity qt is set such that agents have no incentive to contribute less or more than the recommended contribution dYt = (
ct ) dt + dBt , which can also be re-
covered from the recommended consumption ct . Suppose the agent contributes less, then his promised continuation utility decreases by qt . Since this also implies that the agent consumes more than the recommended consumption ct , then his ‡ow of utility increases by uc (ct ). So the agent has no net gain or loss under qt = uc (ct ) if he contributes other than the recommended level. Hence the recommended contribution is incentive compatible. An exception happens when the pension designer recommends ct = 0. Since consumption reaches its lower bound, the pension designer only needs to provide one-side incentive such that the agent do not …nd desirable to consume more. Hence the pension designer can set the sensitivity qt at any level higher than uc (0). Then, suppose the agent contributes less, then his promised continuation utility decreases by more than uc (0), but his ‡ow of utility increases by uc (0) only. So the agent has net loss if he consumes more than the recommended level. The pension designer provides incentive to avoid the moral hazard of early retirement by providing the level of pension less than the promised continuation utility. This is captured by the incentive constraint At
U
1
(vt ). The agents who retires
immediately can collect pension At and leads to value U (At ). The pension designer 11
promises the agent higher continuation utility vt than the retirement value U (At ) if he chooses to stay in the mechanism rather than retire early. At the recommended stopping time of early retirement TR , the pension designer provides the pension ATR = U
1
(vTR ) high enough such that the retirement value is
given by U (ATR ) = vTR . The agent becomes indi¤erent to retire early to leave in the mechanism. So TR is incentive compatible. At the recommended stopping time to quit the mechanism TL , the pension designer promises the continuation utility vTL = Vmin low enough such than the agent becomes indi¤erent to take the outside option. So TL is incentive compatible.
4
The Problem of Pension Designer
By duality and assuming that the law of the large number applies, the objective of the benevolent pension designer can be stated as follows. First, given a initial promised lifetime utility V0 , where V0 > Vmin , the pension designer solves A (Y; T ) that maximizes the continuation revenue G such that recommendation (c; Y; TR ; TL ) are incentive compatible:
G (V0 )
max
c;Y;TR ;TL ;A
EP0
8 Z < :
TR ^TL
e
(r+ )t
t
[dYt
A (Y ; t) dt]
0
e
(r+ )TR
1TR
TL A
Y
TR
; TR
v0 = V0 ; (c; Y; TR ; TL ) 2 D (A) ; dYt =
F0Y
9 = ;
s.t.
(8)
ct dt + dBt ; Y0 = 0;
The expectation is taken under the reported probability space C[0; 1); fFtY g1 t=0 ; P0 (Y ) .
The reported probability space becomes the true space C[0; 1); fFtY g1 t=0 ; P (Y ) as recommendations are incentive compatible. The continuation revenue function G (V0 )
represents the most e¢ cient DC mechanism to deliver lifetime utility V0 . Second, the benevolent pension designer solves a static problem which maximizes V0 subject to a0
G (V0 ), that is the optimal-incentive DC mechanism which is feasible. Note
G satis…es a boundary condition G (Vmin ) = 0. Agents quit immediately if the pension designer promises continuation utility Vmin . The continuation revenue of the DC 12
mechanism after the agent quits becomes zero. Given proposition 1, the revenue given by (8) can be written as:
G (V0 )
dvt =
(
max E
c;TR ;TL ; q uc (0);A
[(r + ) vt [(r + ) vt
8 Z < :
TR ^TL
e
(r+ )t
ct
At dt
0
e
(r+ )TR
1TR
TL U
1
(vTR )
F0Y
9 = ;
s.t.
(9)
u (ct ) U (At )] dt + uc (ct ) dBt , if ct > 0 ; given v0 = V0 U (At )] dt + qt dBt , if ct = 0 (10) At 2 0; U
1
(vt )
TL = inf ft : vt = Vmin g :
(11)
Hence the revenue G (v) of the optimal mechanism solves the following HJB with v as the state variable: 8 > > > > < (r + ) G (v) = max > > > > :
max
c 0;A2[0;U
1 (v)]
max
q uc (0);A2[0;U
(
1 (v)]
c + Gv (v) [(r + ) v +[ (
u (c)
uc (c)]2 Gvv 2
(v) A + Gv (v) [(r + ) v
+(
q)2 Gvv 2
(v)
U (A)]
A
U (A)] )
) 9 > > > > =
(12)
subject to the boundary condition at v = Vmin : (13)
G (Vmin ) = 0; and the smooth-pasting condition at some v = vTR : G (vTR ) =
U
1
(vTR ) ; Gv (vTR ) =
1 : Ua (U 1 (vTR ))
(14)
Let vTR denote the level of promised continuation value when the pension designer recommends agents to retire early, and ATR =
G (vTR ) denote the corresponding
level of pension. Then the optimal feasible mechanism promises to deliver maximal v0 such that G (v0 ) + a0
0.
Having introduced the HJB equation, we are ready to solve for the optimal DC mechanism, which is stated in the following proposition: 13
> > > > ;
;
Proposition 2 Under the maintained assumptions, (a) There are unique continuous di¤erentiable solution G (v) and unique vTR that solve the second-order ODE (12) such that (13) and (14) are satis…ed. (b) G (v) is concave in v and G (v)
1
U
(v) for all v
Vmin , where the equality
holds at v = vTR (c) If Gv (vt )
0, then ct = 0 and qt = uc (0). If Gv (vt ) < 0, then ct solves Gv (vt ) =
1 + uc (ct )
2
(15)
ucc (ct ) Gvv (vt ) :
(d) TR and TL are given by: TR = infft : vt = vTR g; TL = infft : vt = Vmin g: (e) If Gv (vt )
(16)
0, then At = 0. If Gv (vt ) < 0, then At is increasing in vt and
given by At = min U
1
(vt ) ; Ua 1
Gv (vt )
1
:
(17)
Proof. See Appendix 2. Based on the observed contributions Y t , the continuation utility vt promised by the optimal mechanism is : dvt =
(r + ) vt u (ct ) + uc (ct ) dYt
U (At )
uc (ct )
ct
dt
given v0 = V0 ;
(18)
where ct solves (15) and At solves (17). Note uc (ct ) > 0, so higher contribution dYt implies a higher change in th continuation utility promised to the agent, and vice versa. The pension designer ful…lls the promised continuation utility by recommending consumption ct and, more importantly, by providing pensions At whenever agents are hit by the retirement shock.
4.1
Three Phases of the Optimal Mechanism
To understand the optimal mechanism, we divide the life of a working agent into three phases, according to the level of pension At he is entitled to collect under the 14
optimal mechanism. Figure 1 illustrates the three phases with a numerical example.
4.1.1
The Punishment Phase
There is a region of vt such that Gv (vt )
0. We call that region a punishment phase.
Since G (v) is strictly concave (by proposition 2(b)), a punishment phase occurs when vt is low and in particular around v = Vmin . Integrating (5) from time 0 to t, the value of vt is low when low labor productivities are realized for su¢ cient length of long. This also implies a history of low contributions. To punish the agent with a history of low contributions, the pension designer promises low continuation utility, recommends low consumption (ct = 0 by proposition 2(c)) and pays no pension (At = 0 by proposition 2(c)) when the agent is hit by the retirement shock. If the agent keeps contributing low for su¢ cient length of time during a punishment phase such that vt = Vmin , the pension designer recommends the agent leaving the mechanism to take the outside option. This is the harshest punishment the pension designer can give. In order to leave a punishment phase, agents have to provide high contributions for su¢ cient length of time until Gv (vt ) < 0. 4.1.2
The Accumulation Phase
Suppose the level of promised continuation utility is in a modest region where Gv (vt ) < 0 and Ua 1
Gv (vt )
1
< U
1
(vt ). We call that region an accumulation phase.
During an accumulation phase, the level of pension starts accumulating with contributions.
The pension designer recommends positive consumption (ct > 0 by
proposition 2(c)) and pays pension (At > 0 by proposition 2(c)) according to At = Ua 1
Gv (vt )
1
. An agent with higher vt can collect higher pension At (by proposi-
tion 2(e)) when he is hit by the retirement shock. This encourages agents to provide high contributions to the mechanism. If the agent contributes low for su¢ cient length of time during an accumulation phase such that Gv (vt )
0, then he will enter a pun-
ishment phase. This happens when low labor productivity are realized for su¢ cient length of time. So an accumulation phase that features a large region of vt implies that 15
Figure 1: Upper panel: the plots of G (v) and U 1 (v), under a range of v. Lower panel: the plot of A (v). Parameters: u (c) = 1 e c , U (a) = 2:5a0:4 , = = 1, r = 0:05, = 1=40, and Vmin = 2:5 (0:01)0:4 (ie the outside option is the utility of the pension which is equal to one percentage of average labor productivity). 16
the mechanism provides a large bu¤er for low labor productivities again a punishment phase. 4.1.3
The Retirement Phase
If the agent contributes high for su¢ cient length of time during an accumulation phase such that Gv (vt ) < 0 and Ua 1
Gv (vt )
1
U
1
(vt ), then he will enter
another phase which is referred as a retirement phase. This is the region of high continuation utility, which happens a history of high contributions for su¢ cient length of time. During a retirement phase, the incentive constraint of early retirement, ie U (At )
vt , is binding. If the pension designer ignores the incentive constraint of
early retirement, then the working agent will …nd desirable to mimic being hit by the retirement shock and hence collect the pension immediately. During a retirement phase, the pension designer is bound to pay high level of pension. The level of incentive compatible pension is binding at At = U
1
(vt ) such that the agent is
indi¤erent between early retirement and working. Since G (v) is strictly concave (by proposition 2(b)), a retirement phase occurs around v = vTR . If the agent keeps contributing high for su¢ cient length of time during a retirement phase such that vt raises to reach vTR , the pension designer recommends the agent early retirement to collect the pension At = U
1
(vTR ). By the smooth-pasting condition, this is the
highest level of pension the pension designer gives, in other words the highest reward it can give to encourage a history of high contributions. If the agent contributes low for su¢ cient length of time during a retirement phase such that Ua 1 U
1
Gv (vt )
1
<
(vt ), then he will go back to an accumulation phase.
It is optimal to recommend agents to retire early when promised continuation value vt is su¢ ciently high to hits vTR .
Due to the income e¤ect, maintaining promised
continuation value higher than vTR is too costly to the pension designer. Given vt is promised, the marginal bene…t of promised continuation utility is Gv (vt ). If the agent is recommended to retire early, the pension designer has to provide U
1
(vt ) units of
pension in order to keep the promise. So the marginal cost for the pension of early retirement at vt is
1=Ua (U
1
(vt )). At the optimal stopping time to recommend
early retirement, the marginal revenue equal to the marginal cost, which is given by 17
the smooth-pasting condition (14). It is never optimal to recommend early retirement if smooth-pasting condition never satis…ed. Note that "inverse-Euler equation" in Rogerson (1985) and Sannikov (2008) does not necessarily hold under the optimal mechanism. This is because of hidden consumption. Inverse- Euler equation states
1=uc (ct ) is a martingale. Applying Ito
lemma on Gv (vt ), then the drift of Gv (vt ) is [1 + Gv (vt ) Ua (At )]
dAt : dvt
So under a punishment phase (dAt =dvt = 0) or an accumulation phase (1 =
Gv (vt ) Ua (At )),
the above is Gv (vt ) is martingale. Under a retirement phase, we have 1+Gv (vt ) Ua (At ) < 0 and dAt =dvt > 0, hence Gv (vt ) is supermartingale. Recall under the optimal mechanism, from the …rst order condition of c (15), we have 1 = uc (ct )
Gv (vt ) +
2
ucc (ct ) Gvv (vt ) :
Compared to Sannikov (2008), there is an extra term term,
2
ucc (ct ) Gvv (vt ). Without this
1=uc (ct ) is equal Gv (vt ), which is a martingale under a punishment phase or
an accumulation phase. The left hand side is the marginal cost of delivering a util at time t. The right hand side is the decomposition of marginal bene…t. The …rst term is the marginal bene…t from decreasing the promised continuation utility. For additional util delivered at time t, the promised continuation utility is ful…lled and decreases by a unit. The second term is the marginal bene…t of providing incentives for contribution with respect to additional util. As discussed before, in order to make the increase of consumption, hence the increase of util, incentive-compatible, the pension designer also has to reduce volatility of promised continuation utility qt = uc (ct ). We interpret Gvv (vt ) as the marginal bene…t of variance of vt , in term of revenue. Since the variance of vt is ( uc (ct ))2 , so to increase addition util at time t, the pension designer needs to implement additional @ ( uc (ct ))2 =@u (ct ) =
2
ucc (ct )
variance of vt . Since Gv (vt ) is supermartingale under retirement phase (Gv (vt ) < 0) and Gv (vt ) is martingale under the other two phases, it is expected that an agent most likely spend most of his life under a retirement phase. 18
4.2
Optimal Transfers and Contribution Deductions
Having characterized the optimal mechanism, then it is important to see how insurance and incentive are actually worked out. Interpret
Gt =
G (vt ) as the balance
in the "pension account", which is the level of pension the agent can collect at the recommended stopping time of voluntary retirement TR , ie ATR =
GTR . Also, the
agent is recommended to quit the mechanism when the pension balance becomes zero. Substituting Ito lemma into (12), the evolution of the pension balance is d ( Gt ) =
(r + ) Gt
At + (1 + uc (ct ) Gv (vt ))
ct
dt
uc (ct ) Gv (vt ) dYt ; which can be expressed in a distortion form: d ( Gt ) = ( rGt + bt ) dt + (1 where bt and
t
t)
are given by bt dt = t
( Gt
At ) dt +
(21)
= 1 + uc (ct ) Gv (vt ) ;
deduction rate. Contributions are subsidized when implies
t
t
(20)
t Et dYt ;
We call bt the (lump-sum) transfer to pension balance and deducted when
(19)
dYt ;
> 0. Laissez faire implies
t
= 0 and
t
t
the contribution
< 0, and contributions are Gt = At . Perfect insurance
= 1. Since agents are subject to income shock, so do their pension contribu-
tions, hence a perfect insurance implies the pension balance is completely insensitive to agent’s current contribution. The distortion form helps to understand how a DC mechanism works. The accumulation of pension balance is distorted by transfer bt and contribution deduction rate
t
in order to provide incentive and insurance of pension accumulation against
shocks. In particular, the pension designer can always divert and pool share
t
of
the agent’s contribution and redistribute back to agents as transfer. This is captured by the part
t Et dYt
constituting to the transfer bt dt. Another part 19
( Gt
At ) dt
captures the gain (loss if negative) of pension balance after that some agents are hit by the retirement shock and take away At from their pension balances. So the difference ( Gt
At ) will be gained by the agents remained in the mechanism. Given
the Poisson process, there are dt share of agent hit by the retirement shock. So the gain for the agent remained in the mechanism is
( Gt
At ) dt.
One might doubt the use of contribution deductions and transfers, as they seem to cancel each other. In fact, they have di¤erent e¤ects on providing insurance and incentive. Recall that
t
= 1 re‡ects perfect insurance, and
t
= 0 re‡ects laissez faire.
Allocations of laissez faire are always incentive compatible but there is no insurance on pension. So a value of
t
which is closer to one than zero actually re‡ects that
more weigh is put on insurance rather than incentive. On the other hand, a transfer bt re‡ects a subsidy to the pension balance. Roughly speaking, a higher bt can be attained through either higher
t
or lower pension At . The
former re‡ects that a subsidy to pension balance takes the form of pension insurance. The later re‡ects a trade-o¤ of welfare between working agents and retiring agent. A lower value of At re‡ects that the subsidy to pension balance actually comes from redistributing resources from retiring agents to working agents. Also, the choice of At is subject to incentive constraint of early retirement, that is At
U
1
(vt ), so a
low value of transfer bt may induce agent retiring early. The following proposition characterizes the distortions: Proposition 3 Under the maintained assumptions, Gv (vt )
0, then
t
1 and is decreasing in vt :
t
2 (0; 1) if Gv (vt ) < 0. If
Proof. See Appendix 2. Under the optimal mechanism, by Proposition 3, the contribution deduction rate t
is positive and less than unity during an accumulation phase and retirement phase
(Gv (vt ) < 0). The contribution deduction rate
t
is greater than unity during a
punishment phase. This implies that it is always optimal to deduct contributions rather than to subsidize so. Recall that the …rst order condition of ct implies 1 = uc (ct )
Gv (vt ) + 20
2
ucc (ct ) Gvv (vt ) :
The term
2
ucc (ct ) Gvv (vt ) captures marginal bene…t of providing incentives for con-
tribution with respect to additional util. If it were zero, ie, providing incentive is costless, then we would have
t
= 0, that is, contributions are not deducted. Hence
the positive contribution deduction rates re‡ect the fact that it is costly to maintain agent’s incentive . The contribution deduction rate is positive and less than unity during an accumulation phase and an retirement phase because the pension designer provides insurance on pension accumulation. When
t
2 (0; 1), the pension balance is not sensitive to
the ‡uctuating contribution. The higher contribution deduction rate insurance since the redistribution part
t Et dYt
t
implies higher
of transfer bt is pooled amongst agents
to share the shock. The contribution deduction rate is greater than unity during a punishment phase (Gv (vt )
0) because the pension designer provides punishment to discourage low
contribution. When
t
> 1, then contribution deduction is so high that further
contributions actually reduce the pension balance. This is used as a punishment device to agents, along with recommending ct = 0. Agents’ pensions only start accumulating after that their pension balances reach some level with
t
< 1.
On the other hand, the transfer bt can be positive or negative. The part ( Gt
At )
of bt is strictly negative and decreasing in vt during a punishment phase. Recall that the part ( Gt
At ) is the deduction on pension balance once the agent retires due
to the retirement shock. The value of Gt + At is interpreted as the penalty (reward if positive) for retiring the pension balance. This penalty reallocates resource from retiring agents to working agents. So during a punishment phase, the optimal mechanism features Gt >
At , hence there is a penalty for accidental retirement. The
magnitude of the penalty is increasing in vt . On the other hand, since G (v) touches U
1
(v) at v = vTR from above, and at v = vTR we have ATR = U
vt is around the neighborhood of vTR , we have Gt > retirement phase, where At = U
1
1
(vTR ), so when
At . This is the region of
(vt ). So during a retirement phase, there is also a
penalty for accidental retirement but the magnitude of the penalty is decreasing in vt . The penalty is zero at the recommended stopping time for early retirement TR .
21
5
Comparison between the Optimal Mechanism and Laissez Faire
To see the source of welfare improvement by the optimal mechanism, it would be helpful to compare the allocation under optimal DC mechanism with laissez faire. Suppose instead of the optimal mechanism, agents have access to saving through pension designer. Imagine that agents can open and contribute to a pension account at pension designer without any distortion. Given the level of balance a, the evolution of the agent’s continuation utility V (a) solves the HJB equation: 2
(r + ) V (a) = max cLF
u c
LF
0
+ U (a) + Va (a) ra +
c
LF
+
2
Vaa (a) ; (22)
with the boundary condition (23)
V (0) = Vmin ; and there exists aTR that satis…es the smooth-pasting condition V (aTR ) = U (aTR ) ; Va (aTR ) = Ua (aTR ) :
(24)
That is, the agent retires early when his balance reaches aTR , the counterpart of ATR under the optimal mechanism. Follow similar steps in the proof of proposition 2(a), it is straight-forward to show V (a) exists and is unique. We compare the allocations of the two environments based on the same continuation utility, that is V (at ) = vt . In other words, the level of balance under laissez faire supports the same continuation utility as the one under the optimal mechanism. This is essentially the comparison of two solutions V
1
(v) and
G (v). The following
proposition characterizes the main di¤erences: Proposition 4 Under the maintained assumptions, (a) aTR < ATR (b)
G (v)
vt = Vmin (c)
V
1
(v), for all v 2 [Vmin ; U (aTR )], and the equality holds only when
uc (ct ) Gv (vt ) < 22
< ct (d) Given V (at ) = vt and Gv (vt ) < 0, cLF t Proof. See Appendix 2. Proposition 4(b) is intuitive, as the pension designer can always replicate the allocation of laissez faire by doing nothing, so the optimal mechanism must take strictly less resource to deliver the same level of continuation utility. According to proposition 4(a), when agents retire voluntarily, pension is higher under the optimal mechanism than under laissez faire. Because contributions are deducted, agents contribute less under the optimal mechanism than laissez faire. This is re‡ected by proposition 4(d), which concludes agents consume more under the optimal mechanism than laissez faire. But the insurance provided by the optimal mechanism induces agents to stay longer in the mechanism. This postpones agents’ voluntary retirement so agents contribute for a longer length of time. In sum agents contribute less but retire later under the optimal mechanism. It turns out the the e¤ect of longer working life dominates thus agents can retire with higher pension under the optimal mechanism. A source of e¢ ciency gain comes from insuring pension accumulation, which re‡ects by Proposition 4(c). In particular, recall in (19) that there is less volatility of pension balance under the optimal mechanism. So there is less motivation to save for self-insurance, so consumption increases. This explains why consumption is higher under the optimal mechanism, as stated by proposition 4(d). A trade-o¤ of providing insurance of pension accumulation is that it results in lower drift of pension balance than the one under laissez faire. Given the same pension balance, on average pension is accumulate slower under the optimal mechanism. It is because of the higher consumption under the optimal mechanism. In sum, pension under optimal mechanism features lower drift and lower volatility than laissez faire. One may note that, under the optimal mechanism, there could be higher or lower volatility of promised continuation utility than laissez faire. Recall under the optimal mechanism, the evolution of promised continuation utility is given by dvt = [(r + ) vt
u (ct )
23
U (At )] dt + qt dBt ,
and under laissez faire, the evolution of promised continuation utility is u cLF
dVt = (r + ) Vt
U (a) dt + dBt ;
So the volatility of promised continuation utility is lower under the optimal mechanism than laissez faire if and only qt < 1. By proposition 3, under an accumulation phase or retirement phase, we have qt <
1 : Gv (vt )
Since we know G (v) is concave from proposition 2(b), hence the right hand side Gv (vt )
1
is decreasing. Suppose there is v 0 such that Gv (v 0 ) = 1, then for any
promised continuation utility vt higher than v 0 , we have less volatility of promised continuation utility under the optimal mechanism than laissez faire. This must be true for the agent who is under retirement phase. In other words, for agents who are about to retire voluntarily, they enjoy lower ‡uctuation in promised continuation utility until the retirement.
6
Comparative Statics of
and
To understand how shocks to life-cycle a¤ect the level of optimal pension with asymmetric information, we perform the comparative statics analysis of section. Recall that and
1
a lower
and
in this
captures the likelihood of being hit by the retirement shock,
is the expected remaining time before being hit by the retirement shock. So captures an economy with more healthy agent, or the fact that agents can
work longer. On the other hand, So a higher
captures the volatility of labor productivity shock.
re‡ects a mor risky environment.
The following proposition characterizes the comparative statics of
and :
Proposition 5 Under the maintained assumptions, let X i , for i = 1; 2 denote the variable under an economy with
=
i
, where
then (a) vT1R < vT2R and A1TR < A2TR (b) G1 (v) < G2 (v) for all v 2 (Vmin ; vT1R ] 24
1
>
2
and other parameters equal,
Let X i , for i = 3; 4 denote the variable under an economy with 1
>
2
=
i 2
, where
and other parameters equal, then
(c) vT3R < vT4R and A3TR < A4TR (d) G3 (v) < G4 (v) for all v 2 (Vmin ; vT3R ] Proof. See Appendix 2. In general, proposition 5 states that the optimal mechanism can provide more pension for voluntary retirement when either
is lower or
is lower. The optimal
mechanism also becomes more e¢ cient to deliver promised continuation utility when either
is lower or
is lower. Consider
of early retirement At
U
1
becomes lower. The incentive constraint
(vt ) does not change. However there are less agents
being hit by the retirement shock. So the pension designer prepares less resource for the pension for accidental retirement. This also means there are more agents who remains in the mechanism to contribute for the pension accumulation. This implies the pension designer has more resource to provide for the pension for voluntary retirement. On the other hand, consider
becomes lower. Recall that the marginal cost of
delivering additional util at time t is, given the …rst order condition of ct ,: 1 = uc (ct ) So when
Gv (vt ) +
2
ucc (ct ) Gvv (vt )
becomes lower it is likely that the marginal cost of delivering additional
util is also lower. It is because agents become more di¢ cult to misreport the labor productivity shock. An agent can always underreport the labor productivity shock in order to have higher consumption. Suppose the agent underreport such as to have one more unit of consumption. Then he has to underreport
1
unit of labor productivity
shock in order to match the observed contribution. So when
becomes lower, the
agent has to underreport more. In the continuous time, as high degree is negligible, we can think of the pension designer maintain a linear punishment for every unit of labor productivity shock misreported, so the total punishment becomes higher under lower .
25
7
Extensions
We want to show that the result that it is optimal to deduct contributions from the pension balance is robust to environments with di¤erent labor choice and di¤erent information structure. We …rst consider that agents can choose a continuum labor e¤ort, rather than zero or one. Then we consider that the pension design can observe agents’income process on top of the history of contributions.
7.1
Endogenous Labor
Suppose agents can also choose the level of (private) labor lt
0, and the contribution
process becomes dYt =
lt
ct dt + dBt :
(25)
Let u (c; l) denote the ‡ow of agent’s utility. Assume the regularity condition for maximization: ul (c; l) < 0; ull (c; l) < 0; ucl (c; l) < 0; uc (c; l) > 0; ucc (c; l) < 0; ucc (c; l) ull (c; l) > ucl (c; l)2 : De…ne C (q) and L (q) which solve the system of equations: q = ul (C; L) ;
q = uc (C; L) ; where uc (c; l) and tions imply
(26)
ul (C; L) = have …nite upper bounds. Then the above assumpCq (q) < 0 and Lq (q) > 0:
Denote the maximum of two upper bound as q. Then it is straight-forward to show that the evolution of promised continuation utility under some incentive compatible mechanism is dvt = [(r + ) vt
u (C (qt ) ; L (qt ))
26
U (At )] dt + qt dBt :
(27)
The revenue of the mechanism is given by the HJB: (r + ) G (v) =
max
q ;q;A2[0;U
1 (v)]
(
2
L (q) C (q) A + ( 2q) Gvv (v) +Gv (v) [(r + ) v u (C (q) ; L (q))
U (A)]
) (28)
subject to the boundary condition and the corresponding smooth-pasting condition. It is straight-forward to extend the proof of proposition 2(a) and (b) to show G (v) exists, is unique and strictly concave. Then, by Ito lemma and the above, the evolution of pension balance is d ( Gt ) =
(r + ) Gt + L (qt )
= ( rGt + bt ) dt + (1
C (qt ) t)
( Gt
qt Gv (vt ) dBt
dYt ;
where the transfer bt and contribution deduction rate bt =
At dt
At ) +
t
are given by
t Et dYt ;
= 1 + qt Gv :
t
The distortion form is the same the exogenous labor case. The …rst order condition of q, if q is not binding at q = q, implies 2
1 + qt Gv (vt ) =
qt Gvv (vt ) Lq (qt ) Cq (qt )
0:
If q is binding at q = q then we have 2
1 + qt Gv (vt ) >
qt Gvv (vt ) Lq (qt ) Cq (qt )
0:
So we always have t
2 (0; 1) ; if Gv (vt ) < 0;
t
> 1, if Gv (vt )
27
0:
(29)
hence contribution deduction rate is always positive. So with endogenous labor, contributions to the optimal mechanism are also always deducted, as in the exogenous labor case.
7.2
Observable Income
Suppose the pension designer can also observe agent’s income (normalized with respect to ): (30)
dIt = lt dt + dBt ; as well as the contribution: dYt =
lt
(31)
ct dt + dBt :
Combining both information, the pension designer knows about agents’consumption ct . Thus the pension designer can set ct as a control variable. The problem becomes similar to principal-agent problem as in Sannikov (2008), except that agents are under the risk of retirement shock. The pension mechanism depends on agents’income and retirement time as: A IT ; T : Then it is straight-forward to that show the evolution of promised continuation utility under some incentive compatible mechanism is dvt = [(r + ) vt where L0 (q; c) solves
u (ct ; L0 (qt ; ct ))
(32)
U (At )] dt + qt dBt ;
q = ul (c; L0 ). Note that L0q (q; c) > 0. The revenue of the
mechanism is given by the HJB: (r + ) G (v) =
max
c 0;q2[0;q]; A2[0;U 1 (v)]
(
2
L0 (q; c) c A + ( 2q) Gvv (v) +Gv (v) [(r + ) v u (c; L0 (q; c))
U (A)]
)
(33)
subject to the boundary condition and the corresponding smooth-pasting condition. It is straight-forward to extend the proof of proposition 2(a) and (b) to show G (v)
28
exists, is unique and strictly concave. Then, by Ito lemma and the above, the evolution of pension balance in the distortion form is d ( Gt ) = ( rGt + bt ) dt + (1
t)
dYt ;
where the transfer and contribution deduction rate are given by bt = t
( Gt
At ) + + t Et dYt ;
= 1 + qt Gv :
The …rst order condition of q, if q is not binding at q = q, implies 2
qt Gvv (vt ) L0q (qt ; ct )
1 + qt Gv (vt ) =
0
Note that if q is binding at q = q then we have 2
qt Gvv (vt ) L0q (qt ; ct )
1 + qt Gv (vt ) >
0:
So we always have t
2 (0; 1) ; if Gv (vt ) < 0;
t
> 1, if Gv (vt )
(34)
0:
hence contribution deduction rate is always positive. So, with observable income, contributions to the optimal mechanism are also always deducted, as in the exogenous labor case.
8
Conclusion
The optimal de…ned-contribution pension mechanism features a punishment phase, an accumulation phase and a retirement phase. During a punishment phase, there are low consumptions and no pension for accidental retirement. To move from a punishment phase to an accumulation phase, agents have to provide high contributions for 29
su¢ cient length of time. During an accumulation phase, agent’s pensions is strictly increasing in the promised continuation utility, which in turn roughly depends on the length of high contributions. If the agent contributes high for su¢ cient length of time, he will move from an accumulation phase to a retirement phase. During a retirement phase, the incentive constraint of early retirement is binding, hence the pension designer is bound to pay high level of pension. If the agent keeps contributing high for su¢ cient length of time during a retirement phase, the pension designer recommends the agent early retirement to collect the highest level of pension it would give. If the agent contributes low for su¢ cient length of time during a retirement phase, then he will go back to an accumulation phase. In sum, we study voluntary de…ned-contribution pension plans with asymmetric information. This leads to a dynamic incentive problem of early retirement and low contributions over time. The optimal incentive compatible mechanism can be implemented through transfers and contribution deductions from agents’ pensions over time. We …nd that the level of pension is higher under the optimal mechanism than under laissez faire. Working agents enjoy higher consumption, contribute less, and retire later. The level of pension for the voluntary retirement decreases when the probability of the retirement shock increases, when the volatility of the labor productivity shock increases, and when the …rst best contribution is maintained. We …nd that the result that it is optimal to deduct contributions is robust to environments with continuum labor choice and with observable income.
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9
Appendix 1
9.1
Risk-Neutral Measure
Since the participation constraint and incentive constraint of the extensive margin of labor productivity (able workers not to retire earlier than "natural") are simply given by vt
maxfVmin ; U (A (Y t ; t))g, we …rst suppose it is satis…ed (and veri…ed later)
and focus on the incentive constraint of the intensive margin. Considera "risk-neutral world" C[0; 1); fFtY g1 t=0 ; PY (Y ) , where Y becomes a
standard Brownian motion under some probability space C[0; 1); fFtY g1 t=0 ; PY (Y ) .
The pension designer can always recommend agents consumption c to be progressive b can be constructed from an observed contribution measurable3 . A Brownian report B Y under any arbitrary progressive measurable recommendation c, not necessarily
incentive-compatible, as
bt = Yt B
which is a Brownian motion with drift
Z
t
cs
ds;
0
ct = under the risk-neutral space C[0; 1); fFtY g1 t=0 ; PY (Y )
Let Ct denote agent’s actual consumption solving (2), hence the true labor productivity shock dBt is given by
Bt = Yt
Z
t
Cs
ds:
0
So agents report truthfully if and only if the recommended consumption is followed, ie, c = C. In other words, if the recommended consumption c is incentive compatible, ie (c (Y ) ; Y; TR (Y ) ; TL (Y )) 2 D (A), then agents report truthfully.
Agent’s reporting strategy is the probability measure P0 such that the report
b is a standard Brownian motion in C[0; 1); fFtY g1 B t=0 ; P0 (Y ) . Such P0 can be
constructed by applying Girsanov theorem. First, …xed a stopping time T , de…ne a 3
Such progressive measurable c always exists. Since c is F Y -measuarble and adapted, c has a progressive measurable modi…cation (Proposition 1.1.12 KS). Y is a modi…cation of X if for every t 0 we have Pr [Xt = Yt ] = 1. X is progressive measuraeble (with respect to th …ltration fFtY g) if for every t 0 and Borel set A 2 B (R), the set f(s; !) ; 0 s t; ! 2 ; Xs (!) 2 Ag belongs to the product -…eld B (R) FtY .
33
continuous local martingale in C[0; 1); fFtY g1 t=0 ; PY (Y ) : t (c
t
;T)
exp
Z
t^T
cs
dYs
0
with
0
= 1. Indeed
t
Z
t^T
2
cs
!
(35)
ds ;
0
is martingale if c satis…es Novikov condition (Corollary 3.5.13
KS): EPY for all t.
1 2
exp
1 2
Z
t^T
(cs )2 ds
0
< 1;
(36)
A su¢ cient condition is that c is …nite a.s. We assume recommended
consumption c is constructed such as to satisfy Novikov condition throughout this paper. We aslo assume the parameters such that T is …nite a.s., so together with (36), these imply
t
is a uniformly integrable martigale. Then by Girsanov theorem
(Theorem 3.5.1 KS), there is a unique probability measure P0 such that the probability n o b t 2 F Y , where 0 t < T , is of a report B b t = EP 1 bt t (ct ; T ) ; for all t P0 B Y B
T,
(37)
b becomes a standard Brownian motion in C[0; T ); fF Y gT ; P0 (Y ) . and the report B t t=0
In particular, the Radon-Nikodym derivative connects the constructed measure P0 with risk neutral measure PY under C[0; 1); fFtY g1 t=0 ; PY (Y ) : dP0 =
t
ct dPY :
From the point of view of pension designer, consider the agent’s problem (2) which is formulated under C[0; T ); fFtY gTt=0 ; P0 (Y ) , which is equivalent to the following problem under risk-neutral space C[0; 1); fFtY g1 t=0 ; PY (Y ) for all t
vt =
max
b R ;TL c 0;B;T
EPY
8 Z > > > <
TR ^TL ^T
e
(r+ )(s t)
s
T:
t
[u (cs ) + U (A (Y ; t))] ds
t
+e > > > : +e
(r+ )(TR ^T t) (r+ )(TL
TR ^T ; TR ^ T TL TR ^T U A Y t) 1TR ^T >TL TL Vmin
1TR ^T
34
FtY
9 > > > = > > > ;
(38)
;
s.t. the density
t
evolves according to d
t
t
=
(
( )
ct ) dYt :
If the pension designer set recommendation as the maximizer c; TR and TL solving b = B and P0 = P as the solution. (38), and set T = TR , then we have c = C, hence B The problem stated in (38) becomes (2) when P0 = P. So the maximizer (c; TR ; TL ) to (38) with respect to Y constitutes the set of incentive compatible action D (A). 9.1.1
Stochastic Pontryagin Principle and Incentive Compatibility
Applying stochastic Pontryagin principle detailed in Yong and Zhou (1999), we can de…ne a system of co-state equations which is su¢ cient to support the maximum. With slight abuse of notations, we denote v 0 where q
fqt ; fFtY g1 t=0 ; 0
fvt0 ; fFtY g1 t=0 ; 0
t < 1g and q,
t < 1g, as the co-states of drift and volatility to
t
in the agent’s problem (38) under the risk-neutral space C[0; 1); fFtY g1 t=0 ; PY (Y ) .
Co-states can be interpreted as the supporting prices for agent’s optimal choices. The current value Hamiltonian is H ( t ; qt )
max ct
t
U A Y t; t
u (ct )
+ qt
t
ct
:
(39)
The co-state of drift vt0 is the solution to the following backward stochastic di¤erential equation (BSDE) under the risk-neutral space C[0; 1); fFtY g1 t=0 ; PY (Y ) : (
dvt0 = rvt0
@Ht @ t TR
vT0 R = U A Y
dt + qt dYt ; ; TR
; vT0 L = Vmin .
(40)
Given (40), the stopping time to retire and to quit can be determined by v 0 : TR = inf t : vt0 = U A Y t ; t
35
; TL = inf ft : vt0 = Vmin g .
(41)
The …rst order conditions for ct is ct = C (qt ) ; where C (x) is de…ned as C (x)
(
uc 1 (min (x; uc (0))) , if x 1, if x < 0
0
(42)
The following lemma is needed to prove proposition 1. Lemma 1 (a) Given any incentive-compatible mechanism A Y T ; T . There exists a process q, such that (C (q) ; Y; TR ; TL ) 2 D (A) under C[0; 1); fFtY g1 t=0 ; P (Y ) , where TR = inf ft : vt0 = U (A (Y t ; t))g, TL = inf ft : vt0 = Vmin g and v 0 is given by (
U (A (Y t ; t))] dt + qt dBt ; dvt0 = [(r + ) vt0 u (C (qt )) vT0 R = U A Y TR ; TR ; vT0 L = Vmin
(43)
(b) If q is bounded and progressively measurable, then v 0 given by (43) is the promised continuation utility v. The volatility of promised continuation utility vt is qt . Incentive compatibility of TR implies vt0
U (A (Y t ; t)) :
Proof. Lemma 1(a). To recover D (A) under original space C[0; 1); fFtY g1 t=0 ; P (Y ) , we can apply Girsanov theorem once more on C[0; 1); fFtY g1 t=0 ; PY (Y ) to construct b in C[0; 1); fFtY g1 the reported Brownian shock B t=0 ; P0 (Y ) : C (qt )
bt = dYt dB
dt:
(44)
After substituting (44) into (40) under measure P0 , we have (43). Then we have (C (q) ; Y; TR (Y ) ; TL (Y )) 2 D (A) under the space C[0; 1); fFtY g1 t=0 ; P0 (Y ) , where b = B and P0 = Y is given by (44). Since (C (q) ; Y; TR (Y ) ; TL (Y )) 2 D (A), we have B
P, so (C (q) ; Y; TR (Y ) ; TL (Y )) 2 D (A) under the original space C[0; 1); fFtY g1 t=0 ; P (Y ) as well.
Lemma 1(b). For any bounded qt , all the relevant functions in (43) are Lipschitz continuous and satisfy linear growth condition, so the strong solution vt0 exists. Note
36
that (43) implies (r+ )t
dvt0 (r + ) e (r+ )t vt0 dt = + e (r+ )t U (A (Y t ; t)) dt
e
)
e
+
RT t
, vt0 = E
e
(r+ )t
U (qt ) dt + e
(r+ )t
qt dBt ;
R T (r+ )s e (r+ )T vT0 e U (qs ) ds t R T (r+ )s = for any T: s U (A (Y ; s)) ds + t e qs dBs
(r+ )t 0 vt + (r+ )s
e 8 Z < :
TR ^TL
e
(r+ )(s t)
s
[U (qs ) + U (A (Y ; s))] ds
t
+e
(r+ )(TR ^TL t)
1TR
TL U
A Y
TR
; TR
+ 1TR >TL Vmin
FtY
9 = ;
= vt ;
where the second line follows integration by part over any T , and the last line takes T to in…nity.
10 10.1
Appendix 2 Proof of Proposition 1
Proof. Proposition 1. Note U (0) < Vmin , so vt must pass through Vmin …rst then U (0). Fix an arbitrary positive, bounded, progressively measurable process q and an arbitrary progressively measurable process A as in proposition 1. Construct a forward stochastic di¤erential equation as dvt00 = [(r + ) vt00
u (C (qt ))
bt ; given v000 = v0 : U (At )] dt + qt dB
(45)
By the construction in proposition 1, the pension designer recommends the agent to quit when vt hits Vmin , so the above implies the pension designer never recommends agent to retire early with zero pension. On the other hand, by the construction in proposition 1, the pension designer recommends the agent to quit at t = TR and provides pension ATR = U
1
(vTR ). These imply vT00R = U (ATR ) ; vT00L = Vmin :
It is obvious that the constructed process v 00 from the forward stochastic di¤erential equation (45) is a solution to the backward stochastic di¤erential equation (40). That 37
is v 00 = v 0 . The supporting Y is C (qt )
dYt =
dt + dBt ; Y0 = 0:
By the Stochastic Pontryagin Principle, thus (C (q (Y )) ; Y; TR (Y ) ; TL (Y )) 2 D (A) is
incentive compatible. By lemma 1(a) and (b), v 00 is the continuation utility promised by the mechanism A. Note by lemma 1(b) that, a necessary condition of an incentive compatible mechanism is that At 2 [0; U
1
(vt )]. Also, note that by the de…nition of C (q), we have
ct = C (qt ). So we have ct = 0 if and only if qt
uc (0). Hence the evolution of vt
given by (43) in lemma 1(a) is dvt = where qt
(
[(r + ) vt [(r + ) vt
u (ct ) U (At )] dt + uc (ct ) dBt , if ct > 0 U (At )] dt + qt dBt , if ct = 0
uc (0) and At 2 [0; U
1
(46)
(vt )]. It is straight-forward to show that the strong
solution to the above exists and is unique (Theorem 5.2.9 and Theorem 5.2.5 KS).
10.2
Proof of Proposition 2
10.2.1
Proof of Proposition 2(a) and (b)
Instead of the original pension designer’s problem, consider solution G0 to the following second order ODE: (r + ) G0 (v) =
max1
q;A2[0;U =
(v)]
(
C (q) + G0v (v) [(r + ) v
u (C (q))
( q)2 0 U (A)] + Gvv (v) 2 (47)
subject to the initial condition condition at v = Vmin : G0 (Vmin ) = 0; G0v (Vmin ) = : where
(48)
is some constant. Hence solution G0 ignores the smooth-pasting condition
(14). Lemma 2 Given
0, the solution G0 (v; ) to the second order ODE (47) satisfy38
)
A
ing (48) exists and unique. Proof. First we note can rewrite (47) as G0vv (v) =
min
q 0;A2[0;U =
(r + ) G0 (v)
+ C (q) + A
1 (v)]
G0v (v) [(r + ) v ( q)2 =2
u (C (q))
U (A)]
:
(49)
To see, let q and A denote the minimizer, then we have , G0vv (v)
A+G0v (v) [(r + ) v
C (q)
U (q)
U (A)]+
( q)2 0 Gvv (v) ; 8q 6= q ; A 2 = 0; U 2
where the equality holds when q = q and A = A , which coincides with the HJB (12). Note that G0vv (v) de…ned above is Lipschitz continuous. Then apply PicardLindelof theorem and we have that the solution G0 (v; ) and G0v (v; ) exist given G0 (Vmin ; ) = 0 and G0v (v; ) = . Lemma 3 Given
1
>
2,
the solution G0 (v;
i ),
once at v = Vmin .
i 2 f1; 2 only intersect each other
Proof. Suppose not. Then there is v 0 > Vmin such that G0v (v 0 ; G0 (v;
1)
G0vv (v 0 ;
> G0 (v; 1)
= >
2)
and G0v (v; min
q 0;A2[0;U =
min
q 0;A2[0;U =
= G0vv (v 0 ; So by continuity of G0 (v; G0v
(v
0
";
2 ),
1 (v 0 )]
1 (v 0 )]
(
(
1)
> G0v (v;
1 ( q)2 =2 1 ( q)2 =2
( (
1)
1)
= G0v (v 0 ; 2), and
for all v 2 (Vmin ; v 0 ). Then we have
(r + ) G0 (v 0 ; 1 ) G0v (v 0 ; 1 ) [(r + ) v 0
+ C (q) + A u (C (q)) U (A)]
(r + ) G0 (v 0 ; 2 ) G0v (v 0 ; 2 ) [(r + ) v 0
+ C (q) + A u (C (q)) U (A)]
2) :
1)
and G0 (v;
2 ),
there is " > 0 such that G0v (v 0 0
which is contradiction. Thus G (v;
1)
0
and G (v;
2)
";
1)
<
have no inter-
section other than v = Vmin . Lemma 4 Given , the solution G0 (v; ) is concave. Proof. Note that G0 (v; ) is bounded. Suppose G0vv (v 0 ; ) > 0 at some v 0 , then we have q = 1 (recall C (1) = 0)as the maximizer to (47), so G0 (v 0 ; ) is unbounded, which is contradiction.
39
)) ))
;
1
(v)
Proof. Proposition 2(a) and 2(b). De…ne a relaxed smooth-pasting condition for a function G0 (v; ) as that, there exists vTR such that G0 (vTR ; ) = We want to show there exists v 0 > Vmin . Consider 0 00
0
(50)
(vTR ) :
such that there exists vTR such that (50) holds.
Suppose not, then we have G0 (v; ) >
Fixed any
1
U
U v0
=
U 1
1
(v) for all v and all
. Fixed any
(v 0 ) < 0: Vmin
. By lemma 4, we know G0 (v;
00
00
) is concave, so G0v (v; 0
Then integrating both side from v = Vmin to v = v 0 , we have G0 (v 0 ;
)
0
)
U
1
.
(v 0 ),
which is contradiction. Then we want to show that any G0 (v; ) satisfying (50) at some vTR (maybe multiple), then there exists some vTR such that G0v (vTR ; ) G0 (vTR ; ) =
1
U
(vTR ). That is, G0 (v; ) crosses
not. Then for all vTR such that G0 (vTR ; ) = Ua (U
1
U
U 1
1
Ua (U
1
1
(vTR ))
and
(v) from above. Suppose
(vTR ), we have G0v (vTR ; ) >
1
(vTR )) . But since any intersection of two continuous functions must have
alternating order of slopes, so that implies there is one intersection, hence unique vTR . Hence we have G0 (v; ) < that G0 (Vmin ; ) = 0 >
U
U 1
1
(v) for all v < vTR , which contradicts to the fact
(Vmin ).
Since the proof of Picard-Lindelof theorem implies the set of solution G0 (v; ) with di¤erent
is continuous in the initial values (omit to proof here), then by continuity
and we increase
from some
which leads to the corresponding G0 (v; ) satis…es
(50) and there exists some vTR such that G0v (vTR ; ) G0 (v; 1) = 1, then by the continuity of G0 (v; ) in that G0v (vTR ;
0
)=
Ua (U
1
(vTR ))
1
Then
0
Ua (U
1
1
(vTR )) . Since
, there must exists
0
such
is the maximal G0 (v; ) over the set
of solution G0 (v; ) to (47) given initial conditions V (0) = Vmin that satis…es (13) and (50). So that G0 (v; ) is solution G (v) of (12) satisfying (13) and (14). Proof. Proposition 2(c) to 2(e)To show 2(c), note the …rst order condition of q is: qt = Cq (qt )
Gv (vt ) qt + 1 = uc (ct ) ; 2 G (v ) vv t 40
(51)
rearranging terms then we have (15) To show 2(d), note that the smooth-pasting condition (14) and proposition 2 imply the stopping times TR and TL are given by (16). To show 2(e), …rst consider Gv (vt ) < 0, then the …rst order condition of A in (47) is 0; where the equality holds if and only if the incentive constraint
Gv (v) Ua (A) 1 A
U
1
1
(v) is not binding. This implies A = min U
which is (17). Note that both U
1
(v) and Ua 1
Gv (v)
1
(v) ; Ua 1
Gv (v)
1
;
are strictly increasing in
v, so does the minimum of both. On the other hand, consider Gv (vt )
0 then the
left hand side of the above is negative, hence A will take the lower bound, which is zero.
10.3
Proof of Proposition 3
Proof. Proposition 3. Note that from the …rst order condition of q in (47), we have qt = since Cq (qt ) < 0 and Gvv (vt ) t
Cq (qt ) t = uc (ct ) > 0: 2 G (v ) vv t
0 by proposition 2(b), then we have
= 1 + uc (ct ) Gv (vt ). Suppose Gv (vt ) < 0, then we have
0, then we have
t
1. In particulat, we have ct = 0 and
Since G (v) is concave, so
10.4
t
t
t
> 0. Recall that
< 1. Suppose Gv (vt ) t
= 1 + uc (0) Gv (vt ).
is increasing in vt .
Proof of Proposition 4
We need the following lemma to prove proposition 4, which allows us to compare G (v) and
1
V
(v).
Lemma 5 G0 (v) 0
rG (v) =
C
V
(q)+G0v
1
(v) solves the HJB:
(v) [(r + ) v
u (C (q))
where q = Va (a).
41
( q)2 0 U ( G (v))]+ Gvv (v) ; (52) 2 0
Proof. Denote v = V ( G0 (v)), and a =
G0 (v), then the left hand side of (52) is
ra, and the right hand side can be rewritten as "
=
=
1 Va (a) [ Va (a) 2
C (q)]
[(r + ) v
u (C (q))
U ( G0 (v))] +
0
(r + n) v + U ( G (v)) 1 6 6 + maxc u (c) + Va (a) [ra + ra + 6 Va (a) 4 2 2 q V (a) + Vaa2(a) Vra (a) 2 aa " # 2 2 q ra + Vaa (a) 1 = ra: 2 Va (a)
c] +
2
2
Vaa (a)
Proof. Proposition 4. To show 4(b), suppose not, and Gv (Vmin )
3
Vaa (a) 2
q Va (a)
2
#
;
o 7 7 7; 5
G0v (Vmin ). Then
there is v 0 > Vmin such that there exists " > 0 where G (v 0 ) = G0 (v 0 ), G (v) < G0 (v) and Gv (v) > G0v (v) for all v 2 (v 0 Gvv (v 0 ) =
min1
q;A2[0;U
"; v 0 ). Recall we can express Gvv (v 0 ) as
(r + ) G (v 0 ) (v 0 )]
1 < ( Va (G0 (v 0 )))2 =2
(
+ C (q) + A
(r + ) G0 (v 0 ) G0v (v 0 ) [(r + ) v 0
Gv (v 0 ) [(r + ) v 0 ( q)2 =2
+ C (Va (G0 (v 0 ))) u (C (Va (G0 (v 0 ))))
u (C (q))
G0 (v 0 ) U ( G0 (v 0 ))]
= G0vv (v 0 ) ; where the last line follows lemma 5. So by continuity of Gv and G0v , there is "0 > 0 such that Gv (v 0
"0 ) > G0v (v 0
Gv (Vmin ) > G0v (Vmin ), or Gv (Vmin )
"0 ), which is contradiction. So either we have G0v (Vmin ) and there is no intersection other
than v = Vmin . Then we want to show the case Gv (Vmin )
G0v (Vmin ) is impossible. Suppose that,
since there is no intersection between G (v) and G0 (v), so we have G (v) < G0 (v) for all v > V min. But that implies G is not optimal as it is dominated by G0 for all v: the pension designer can always replicate laissez faire allocation which is incentive compatible. So we must have Gv (Vmin ) > G0v (Vmin ). Finally, we want to show there is no intersection given Gv (Vmin ) > G0v (Vmin ). Suppose not, then there is v 0 > Vmin such that G (v 0 ) = G0 (v 0 ), and Gv (v 0 ) < G0v (v 0 ), 42
U (A)] )
;
as the order of slopes must be alternating for each intersection. By repeating the above proof but substitute Vmin by v 0 , then we show such v 0 does not exist. So there must be no intersection. To show 4(a), note that at vt = V (aTR ), agents retire early under laissez faire. This implies we have
V
U 1
1
(V (aTR )) =
aTR =
V
1
(V (aTR )). Since by the above lemma,
(V (aTR )) < G (V (aTR )), so agents under optimal mechanism do not
retire at vt = V (aTR ), hence vTR > V (aTR ) and ATR = U
1
U
1
(V (aTR )) <
U
1
(vTR ), so
(vTR ) > aTR .
To show 4(c), we compare agent’s …rst order conditions under laissez faire and optimal mechanism respectively. = Va ( G (vt )) = uc cLF t
Under laissez faire, the …rst order condition is
1=Gv (vt ); under the optimal mechanism, the …rst order
condition is uc (ct ) = qt . Since from Proposition 3, we know qt <
1=Gv (vt ), so by
the concavity of u we establish ct > cLF t .
10.5
Proof of Proposition 5
We skip the proof of proposition 5(c) and (d) since it is similar to the proof of 5(a) and (b). We need the following lemma to show proposition 5(a) and (b) Lemma 6 Suppose there is v 0 such that G1 (v 0 ) = G2 (v 0 ) and G1v (v 0 ) > G2v (v 0 ). Then we have G1 (v) > G2 (v) for all v > v 0 . Proof. Suppose not, then there is v 00 > v 0 such that G1v (v 00 ) = G2v (v 00 ), and for all v 2 (v 0 ; v 00 ) we have G1v (v) > G2v (v) and G1 (v) > G2 (v). Note that G2 (vt ) G2v (vt ) vt + A2t + G2v (vt ) U (A2t ) is concave so G2 (vt )
0: To see so, note that by proposition 2(b) G2 (v)
G1v (vt ) vt
0. On the other hand, suppose G1v (vt )
0,
then by proposition 2(e) we have A2t = 0 and A2t + G1v (vt ) U (A2t ) = 0. Suppose G1v (vt ) < 0, then we have @ [A2t + G1v (vt ) U (A2t )] =@A2t we have A2t +G1v (vt ) U (A2t ) A2t + G1v (vt ) U (A2t )
0. So concluding two case
0 for all vt . Hence we have the sum G2 (vt ) G1v (vt ) vt +
0.
Then rewrite G1vv (v 00 ) as G1vv (v 00 ) =
min
q 0;A1 2[0;U =
1 (v 0 )]
(
1 ( q)2 =2
(
(r + 1 ) G1 (v 00 ) + C (q) + 1 A1 1 G1v (v 00 ) [(r + ) v 00 u (C (q)) U (A1 )] 43
))
;
>
min
q 0;A2 2[0;U =
1 (v 0 )]
(
1 ( q)2 =2
(
(r+) G2 (v 00 ) + 2 [G2 (v 00 )
+ C (q) G2v (v 00 ) [rv 00 u (C (q))] G1v (v 00 ) v 00 + A2 + G1v (v 00 ) U (A2 )]
= G2vv (v 00 ) ; where the second last line exploits the result G2 (vt ) G2v (vt ) vt +A2t +G2v (vt ) U (A2t ) 0. So by continuity of G1 (v) and G2 (v), there is " > 0 such that G1v (v 00 G2v (v 00
") <
"), which is contradiction. Thus G1 (v) and G2 (v) have no intersection for
all v > v 0 . Hence G1 (v) > G2 (v) for all v > v 0 . Proof. Proposition 5(a) and (b). Suppose G1v (Vmin ) > G2v (Vmin ). Then by lemma 6 we have G1 (v) > G2 (v) for all v > Vmin . But by proposition 2(b) we have G2 (v)
U
1
(v), hence combining two inequality we have G1 (v) >
1
U
(v) so vT1R
does not exist and this leads to contradiction. So we must have G1v (Vmin ) < G2v (Vmin ) : Suppose G1 (v) < G2 (v) for all v > Vmin . Then at v = vT2R we have
U
1
vT2R =
G2 vT2R > G1 vT2R , which contradicts to proposition 2(b) by taking G = G1 . Hence G1 (v) and G2 (v) must intersect twice, one at v = Vmin . At another intersection v = v 0 6= Vmin , since G1v (Vmin ) < G2v (Vmin ) and intersections of two functions must
have alternating slopes, so we have G1v (v 0 ) > G2v (v 0 ) and G1 (v 0 ) = G2 (v 0 ). Apply lemma 6 again then we know there does not exist the third intersection. In sum we establish that G1 (v) < G2 (v) for all v 2 (Vmin ; v 0 ) and G1 (v) > G2 (v) for all v > v 0 . Suppose vT1R > v 0 . Then we have
U
1
vT1R
= G1 vT1R
> G2 vT1R , which
contradicts to proposition 2(b) by taking G = G2 . So we have vT1R Suppose vT2R < v 0 . Then we have
U
1
vT2R
= G2 vT2R
v0.
> G1 vT2R , which
contradicts to proposition 2(b) by taking G = G2 . So we have vT2R
v0.
Combining the above two case we prove vT1R < vT2R . We can never have vT1R = v 0 = vT2R , otherwise by the smoothi-pasting conditions we have G1v vT1R = G1v (v 0 ) = 1
U
(v 0 ) = G2v (v 0 ) = G2v vT2R , which contradicts to the fact that G1v (v 0 ) > G2v (v 0 ).
The smooth-pasting conditions also implies that A1TR = U
1
vT1R < U
1
vT2R =
A2TR . Finally, since vT1R 2 (Vmin ; v 0 ), then we have G1 (v) < G2 (v) for all v 2 (Vmin ; vT1R ].
44
))
