تنزیل شبه هذلولی و بازنشستگی

Some people have self-control problems regularly. This paper adds endogenous retirement to Laibson’s quasi-hyperbolic discounting savings model [Quarterly Journal of Economics 112 (1997) 443–477]. Earlier selves think that the deciding self tends to retire too early and may save less to induce later retirement. Still earlier selves may think the pre-retirement self does this too much, saving more to induce early retirement. The consumption pattern may be different from that with exponential discounting. Other observational non-equivalence includes the impact of changing mandatory retirement rules or work incentives on savings and a possibly negative marginal propensity to consume out of increased future earnings. Naive agents are briefly considered.

If you are one of the vast majority of people who think they are saving too little of their income.1 The natural conclusion is that you have self-control problems. If, in addition, you argued to yourself that saving more today would only lead to spending more tomorrow, and thus there is no point in saving for retirement, at least there is a small consolation: you are a sophisticated decision-maker with self-control problems. And self-control problems can extend beyond savings decisions. A thirty-something Italian, one of us met in Prague, had decided that it wasn’t worth looking for a job anymore, because even if he got himself to do it and found one, he would quit shortly thereafter, anyway. It is exactly these kinds of agents our paper is concerned with: people who have self-control problems but realize this and behave according to it. A very clean way to model such actors is through the introduction of quasi-hyperbolic discounting.2 This form of discounting sets up a conflict between the preferences of different intertemporal selves. With assumptions of no commitment and that the agent takes into account her self-control problem, savings decisions can then be modeled as an equilibrium in a sequential game played by the different selves. This modeling paradigm avoids the common connection made between preference changes and cognitive failures,3 and is therefore closer to standard economic analysis. The agent in the model understands perfectly the consequences of her actions, and acts optimally within the constraints imposed by her discount function, which the psychological evidence seems to support at least some of the time,4 and the absence of easily available commitment. Laibson (1997a) analyzed actors of the above kind in detail. His key result is that sophisticated actors with a quasi-hyperbolic discount structure undersave; that is, all intertemporal selves could be made better off if all of them saved a little bit more. Since each self consumes too much from earlier selves’ point of view, each of them would agree to increase savings a little bit in exchange for later selves doing the same. We adapt Laibson’s basic setup for the analysis of the effect of endogenous retirement decisions on savings behavior. The addition is simply that in each of the models there is a single period (period 0) in which the agent can choose whether to work or retire. Working costs the agent some utility, but she is compensated for it with extra wealth. We assume that commitment is not possible: agents cannot precommit to a decision concerning retirement, nor to any consumption level. The paper characterizes the savings and retirement outcomes with these preferences as a function of lifetime income and of the additional earnings if retirement is delayed. There are three types of individual outcomes. Saving and early retirement could be the same as in the situation where work in period 0 were not an option. Similarly, saving and delayed retirement could be the same as in the situation where retirement in period 0 were not an option. Interestingly, the former early retirement outcome can be the equilibrium when the late retirement outcome would be the equilibrium if the prior self could commit the later self to work. We refer to the higher level of savings to accommodate such retirement as ‘resigned oversaving.’ A third and distinctive possibility is that savings could be just low enough to ‘force’ work, which we refer to as ‘strategic undersaving.’ This outcome does not match either of the outcomes where there is not a choice about retirement. It is not surprising that the removal of choice can change both the retirement age and savings, as is also true with time consistent preferences. What is different is that the savings to ‘force work’ can be lower than they would be if retirement were not an option, even though the retirement outcome is the same with and without a choice. With time-consistent preferences, removing an option that is not chosen cannot change behavior. While the paper contrasts outcomes with and without a choice about work without explicitly modeling a change in the underlying economic environment, the results can be interpreted as relating to policy changes. The simplest interpretation is the introduction of mandatory retirement, thereby replacing a choice whether to retire or not by definite retirement. With time consistent preferences, a worker retiring before the new mandatory retirement age would not change behavior because of the introduction of mandatory retirement. The same is true with the quasi-hyperbolic discounting that we model. The alternative of the disappearance of the opportunity to retire is more complicated to envision and more interesting. Consider a worker who could retire at the earliest age of eligibility for (illiquid) social security benefits, but chooses to work for one more period. Assume she is doing some saving in every period and so satisfies the first order conditions we analyze. If the earliest retirement age were increased, (with benefits unchanged at the later retirement age) a time consistent worker would not change behavior. However, a quasi-hyperbolic worker might respond by saving more while still retiring at the same age. In this case, greater savings did not happen when there was a choice because with greater savings, the later self would have chosen early retirement, while the earlier self preferred later retirement. Note that all the selves prefer the changed outcome when the early retirement option is removed. We defer a systematic analysis of the effects of social security to a later paper that recognizes liquidity issues. The paper also considers a setting where the earlier self can commit the later self to a given retirement age, although no commitment is possible on future savings. In part, this is simply a way to pick out the interesting examples of removing options. In part it can be interpreted in terms of a choice between two different employers with different defined benefit plans. Consider a worker choosing between two firms. As a function of the length of career each firm offers a lifetime compensation level. A time-consistent worker would plan savings and retirement based on the maximal level of lifetime income for each retirement age (the outer envelope). However, a quasi-hyperbolic worker would also pay attention to the incentives to work inherent in the lifetime earnings profile. Choosing between the firms might be equivalent to a commitment device on work if the firms differ in the payoff to the last period of work while not differing in lifetime compensation for the planned length of career. If one offers little, while the other offers such a large amount that work will be worthwhile at the optimal retirement age with optimal savings, a commitment to a firm is effectively a commitment on retirement age. We start with the simplest model that is relevant in (quasi-)hyperbolic discounting: a three-period model in which the middle period is the retirement decision period, period 0. The crucial intuition is that part of the payoff from self 0’s working accrues to self 1 through higher savings. But in a quasi-hyperbolic framework, self 0 cares less about self 1 relative to self 0 than self −1 does. So there will be circumstances where self −1 would want self 0 to work (for the benefit of self 1), but self 0 does not want to work. In order to avoid this outcome, self −1 might save less (than she would if she could commit self 0 to work) to ‘force’ self 0 to work. On the other hand, if self −1 would like self 0 to work, but it is too expensive to achieve that without commitment, she will save more (than if she could commit self 0 to work) to help finance self 0’s unavoidable retirement. Note the qualitative distinction between a change in self −1’s saving (compared to a setting with commitment) to induce a retirement decision and to accommodate one. Here, we can get lower saving to block the ‘threat’ of retirement and higher saving to accommodate it. Things get much more complicated when we allow for more periods before retirement. In a four-period model we show a possible conflict that a later self plans to retire too late, not too early, from earlier self’s point of view. This is because with quasi-hyperbolic discounting successive selves agree in what the later selves should do, but they don’t agree on how much it is worth to induce them to do it. And the earlier pre-retirement self will always prefer for the later pre-retirement self to save more than the later wants to save. Thus we can get higher saving to ‘encourage’ early retirement. The paper also considers how a retirement decision affects the ability to observationally distinguish quasi-hyperbolic and exponential discounting.5 The most radical difference from the predictions of consistent preference models emerges when we consider the effect of an increase in wage level in the endogenous retirement period. In a situation of strategic undersaving, the need for lower savings to induce work is relaxed through higher earnings, so the agent will save more, giving a negative marginal propensity to consume out of changes in future earnings. Also, we briefly discuss the potential outcomes under the assumption of naiveté, that each self falsely assumes that the others will comply with her plans. Since there is no game in this case, the analysis is considerably simpler. One interesting implication of naiveté is the possibility that the selves before the deciding self plan to retire late, but the deciding self chooses to retire early, leading to an update in lifetime wealth and thus a drop in the consumption path at retirement.

This paper makes an addition to the classic quasi-hyperbolic discounting savings model. Its technical contributions are minor—most of the analysis is possible with little more than the tools developed by David Laibson. However, the interaction of two decisions, with the one (savings) available as a tool to influence the other (retirement), changes the classic model in a few interesting ways. One is the possibility of additional undersaving with the eventual consequence of making the self with a choice poor enough so that she will want to work. This strategic undersaving occurs in addition to the undersaving that characterizes the equilibrium without a retirement decision. It therefore aggravates an already inefficient outcome, and is likely to be bad for all selves. The other, and perhaps more novel, effect is the possibility of higher saving than when commitment is possible. Higher saving can occur for two reasons: either because it is too costly in terms of discounted utility to make the deciding self work, and thus one would rather finance her retirement, or because self t≤−1 is too eager to work long and it is worth saving more to make her choose early retirement. Unlike undersaving, it is not in general bad for the individual—it can mitigate the overconsumption equilibrium of the classic model. In fact, higher saving seems never to be Pareto-worsening: the later selves, at least, should be happy about getting more savings. We also noted some effects of mandates that are not present with exponential discounting. It might be possible to find ‘natural experiments’ changing work and retirement opportunities. The theoretical model would benefit from two major extensions. One is the introduction of more periods when the agent can choose whether to work. We have solved a model of this sort without savings: in each period, the agent can decide whether or not to retire (the retirement decision being final,) and consumption just equals income or benefits. To make it an interesting problem, one has to assume, for example, a benefit profile that increases with the age of retirement. In equilibrium, the agent retires too early: the retirement date is Pareto-dominated by a later retirement date. No such results emerge in our models with savings, but they might if there are more periods of retirement decisions.22 Another useful extension would be the investigation of liquidity constraints in this context. They are clearly important in practice, and they change the nature of equilibria with quasi-hyperbolic discounting considerably. They would play an important role in the analysis of social security since the payment of benefits as an annuity can have independent effects from the mandate to save. A perplexing aspect of quasi-hyperbolic discounting models is a question that is very hard to answer: why don’t people take advantage of annuity-type commitment devices to overcome their undersaving problem? These financial tools are readily available but rarely used. Some modestly satisfactory reasons can be brought up. First, if there is a bequest motive, then, just like in many exponential discounting models, annuities look less attractive than without a bequest motive. Second, the annuities market is quite complicated, and there are good reasons for boundedly rational people not to enter markets they know little about. The latter seems to indicate that as people learn about annuities they may come into broader use. Even if that happens, the commitment is unlikely to be full, leaving at least some room for quasi-hyperbolic discounting effects. In the absence of annuities, there is of course a wide-spread institutional structure that serves as a commitment device for agents happy or unhappy about it: social security. We plan to study the implications of the joint mandates of savings and receipt of social security benefits as a real annuity in a later paper.