Mental Accounting, Discretionary Saving, and Public Pensions

Transcription

1 Mental Accounting, Discretionary Saving, and Public Pensions Tomasz Sulka The University of Edinburgh February 15, 2016 Abstract This paper develops a model of consumption-saving decisions which assumes the existence of cognitive costs of decision-making and selfcontrol problems. The model provides micro-foundations for the theory of mental accounting (endogenous account formation and nonfungibility of wealth). The arising of inertia is partially explained by self-control issues and a level of income. The model provides a theoretical support for automatic enrolment into pension schemes, which is currently being implemented in the UK. However, the impact of the reform on aggregate saving remains ambiguous. A stylised numerical exercise replicates empirical findings on the impact of automatic enrolment on savings behaviour and suggests that automatic enrolment with a default contribution rate equal to the minimum UK requirement maximises welfare. 1 Introduction Long-term planning is difficult. Moreover, even when an appropriate plan is devised, some individuals may find it hard to fulfil. These two observations are ignored by standard economic models, which assume perfect and costless optimisation, and no temptations. However, when considered in a context of retirement-related decision-making, they might be helpful in explaining some of the empirical phenomena that are inconsistent with classical models. The main purpose of this paper is to introduce a model that attempts to account for observed patterns in individual behaviour and to discuss its 1

2 policy implications. The simple model proposed in this paper makes two major assumptions. Firstly, it is assumed that realising a need to save for one s retirement and coming up with an appropriate plan requires effort and therefore is costly. Secondly, once a plan has been established, an individual may deviate from it because giving up current consumption for a future gain requires a certain degree of self-control. A rich body of empirical evidence suggests that log-term planning is indeed difficult. The nationally-representative Attitudes to Pensions survey is administered by the UK s Department for Work and Pensions. According to the 2012 edition of the survey, 19% of respondents had no private savings of any kind. Having no accumulated wealth was found to be positively correlated with reporting having difficulties with financial decisions. 71% of women (56% of men) agreed with the statement that sometimes pensions seem so complicated that I cannot really understand the best thing to do, while 28% (13%) said that dealing with pensions scares them. 39% of young people avoid thinking about retirement. These results are paired with poor knowledge of the state pension system (MacLeod et al. 2012). Furthermore, in their retirement-related decision-making people often rely on simplifying heuristics, which may lead to predictable behavioural biases, such as the powerful default option effect or the framing effects (e.g. Agnew 2006; Benartzi, Thaler 2001, 2004, 2007, 2013; Brown et al. 2011; Card, Ransom 2011; Choi et al. 2004; Duflo, Saez 2003; Feldman 2010; Goda et al. 2014; Knoll 2010; Madrian, Shea 2001). Finally, recall the popularity of costly voluntary services of advising on and managing one s pension savings. All these observations suggest that in fact many individuals find it challenging to decide whether or not, or how much, to save for their retirement. Even when the necessity to save has been acknowledged and the appropriate plan has been devised, an individual may fail to stick to it. This motivates the second assumption as saving means limiting one s current consumption voluntarily - a task that requires discipline and self-control. As many as 45% respondents of the Attitudes to Pensions survey strongly agreed with a statement that they should have started saving earlier (but they have not). Self-control issues are also reflected by the fact that people tend to rely on various commitment devices to control their spending (e.g. Antonides et al. 2011; Ashraf et al. 2006; Binswanger, Carman 2012; Graham, Isaac 2002; Laibson 1997; Loewenstein, Thaler 1989; Shefrin, Thaler 1988). This paper develops a simple, two-system model of consumption-saving decisions within a context of retirement saving. A dual structure of preferences is assumed in order to account for the self-control problems that may affect saving. A planner is the economic agent s far-sighted self that would like to maximise lifetime utility. However, the consumption-saving decisions 2

3 are made by a doer - the economic agent s impatient self that only cares about instantaneous utility. Self-control is exerted by a planner by using an internal commitment device in a form of savings targets. The savings targets specify a desired level of saving and have an impact on the doer s preferences by inducing a feeling of guilt when the desired target is missed. 1 It must be noted, however, that the existence of the savings targets is assumed primarily for modelling purposes and not in attempt to accurately simulate decision-making processes. Furthermore, coming up with an appropriate savings plan is a challenging task. For that reason the model assumes the existence of cognitive costs of setting up the savings targets. The introduction of cognitive costs of decision-making directly results in a possibility of inaction. Both inaction and self-control problems have been studied extensively, have received considerable empirical support, and have been a centrepiece of policy debate in the discussed context. However, I am not aware of other economic models that would formally account for the possibility of inaction using simple transaction costs. It is, arguably, a vital addition as the possibility of inaction is a key determinant of long-term decision-making. Moreover, the interaction between self-control and cognitive costs is important for determining behaviour. For instance, self-control problems partially explain the arising of inertia (i.e. an individual with severe self-control problems may be prevented from saving by even small cognitive costs). In relation to the literature on mental accounting (e.g. Antonides et al. 2011; Davies et al. 2009; Heath, Soll 1996; Levin 1998; Shafir, Thaler 2006; Shefrin, Thaler 1988; Thaler 1985, 1990, 1999), the theoretical contribution of this paper is to provide the important micro-foundations for non-fungibility of wealth and endogenous (mental) account formation. The discussed model is characterised by endogenous decisions on whether or not to set up a mental account and what the financial target for this account should be. Non-fungibility of wealth, which implies that different kinds of income are used in different ways, is generated by the model as discretionary saving and public pension wealth are shown to be only imperfect substitutes. This is due to the fact that contributions towards a public pension are compulsory, and do not require any planning or discipline. The model implies non-fungibility even when different sources of wealth are of the same liquidity. Additionally, when the accumulated savings are not perfectly liquid, the 1 Support for such a commitment mechanism is provided by the economic literature on budgetary rules (e.g. Antonides et al. 2011; Binswanger, Carman 2012; Heath, Soll 1996) and the psychological literature on the role of guilt in exerting self-control (e.g. Giner- Sorolla 2001; Han et al. 2014; Hofmann, Fisher 2012; Kotabe, Hofmann 2015; Strecher et al. 1995). 3

4 model allows to analyse the effectiveness of both internal and external commitment devices within a unified framework. From a perspective of a two-system literature (e.g. Benhabib, Bisin 2005; Brocas, Carillo 2008; Fudenberg, Levine 2006), this paper constitutes a particular application of this modelling technique to study aspects of public pension scheme design. What is more, the discussed model allows for the additional impact of external commitment devices and offers a specific channel through which otherwise abstract self-control may be exerted, namely the savings targets. 2 The predictions of the model explain behavioural phenomena captured by the Attitudes to Pensions survey, which are briefly recalled throughout the paper and provide additional support for the proposed framework. In particular, the model suggests why individuals who are not engaging in consumption smoothing (i.e. the hand-to-mouth consumers) are characterised by poor self-control, lack financial expertise, and come from low-income households. In order to discuss the policy implications of the model, two recent pension system reforms implemented in the UK are considered. The proposed framework provides a theoretical support for the introduction of automatic enrolment into workplace pension schemes. This is due to the fact that the individuals who are most likely to be affected by the reform (i.e. the handto-mouth consumers) are at the same time shown to be most likely to benefit from the scheme, even when it is actuarially unfair or inconsistent with individual time preferences. Alternatively, this result may be interpreted as a potential explanation for the default option effect (e.g. Benartzi, Thaler 2004; Choi et al. 2004; Madrian, Shea 2001). However, the impact of the scheme on aggregate saving is equivocal, as some individuals are forced to increase their savings, while others become discouraged and stop engaging in discretionary saving after its introduction. The impact of the second reform (the introduction of a single-tier pension, which is supposed to simplify the decision environment and incentivise discretionary saving) on propensity of low-income individuals to save also remains ambiguous. A stylised numerical application of the model employing the data on earnings in the UK economy is performed in order to present the theoretical results in a particular context. Three hypothetical reforms (introducing a compulsory, but unfair, pension scheme and introducing automatic enrol- 2 The comparison of the two-system models with alternative theories, such as the model of quasi-hyperbolic discounting (e.g. Laibson 1997; O Donoghue, Rabin 1999, 2001) or the temptation model (Gul, Pesendorfer 2001) is beyond the scope of this paper. However, the particular way in which one models self-control problems should not significantly affect the derived results. 4

5 ment with differing contribution rates) are considered. This simple exercise suggests that current non-savers could be made better off by an introduction of a compulsory, but strikingly unfair, public pension scheme. For instance, a hand-to-mouth consumer with median income would be made better off by a scheme that effectively deprives him of almost 10% of his lifetime earnings. Furthermore, an U-shaped relation emerges between aggregate saving and levels of default contribution rates which are tailored to individual time preferences. This result suggests why automatic enrolment paired with low contribution rates may reduce savings rates. Automatic enrolment with high contribution rates, on the other hand, increases aggregate saving. Finally, the exercise suggests that automatic enrolment with constant default contribution rates (expressed as percentages of agent s income), which may be inconsistent with individual time preferences, maximises welfare for a level of contributions corresponding to the minimum UK s requirement. The model also replicates a stylised fact that the introduction of automatic enrolment reduces variation in accumulated pension wealth. The remainder of this paper is organised as follows. The model is introduced and its behavioural implications are presented in Section 2. Section 3 focuses on policy implications of the proposed framework. In Section 4 a stylised numerical exercise is presented. Discussion on limitations of the model and potential extensions is provided in Section 5. Section 6 concludes. 2 The Model 2.1 The economic environment There are three periods indexed by t {1, 2, 3}. The economic agent s disposable income is deterministic and equal to w 1, w 2 (wages) and b (pension benefits) in periods 1, 2, and 3 respectively. Periods 1 and 2 correspond to working life of an agent, whereas period 3 is interpreted as retirement. Parameter b is interpreted as a level of social benefits that are granted to an individual upon reaching period 3. The timing of retirement is exogenous. An agent has no initial wealth. Following a standard approach in the behavioural literature, there is no uncertainty in the model in order to abstract from any effects of precautionary motives on saving and focus on saving decisions motivated solely by an intention to smooth one s consumption over the life cycle. 3 The analysis is limited to income profiles characterised by high earnings 3 For examples see: (Benhabib, Bisin 2005; Bowman et al. 1999; Diamond, Koszegi 2003; Fudenberg, Levine 2006; Laibson 1997; Shefrin, Thaler 1988). 5

6 in periods 1 and 2, and low earnings in period 3, which imply that an economic agent would achieve consumption smoothing by shifting his income from periods 1 and 2 towards period 3. This simplifying limitation allows to focus exclusively on retirement saving without considering potential issues of smoothing one s consumption between periods 1 and 2. One may argue that this assumption is more plausible in case of individuals who are approaching their retirement. 4 Savings earn interest of rate r per period and they are illiquid in the sense that when depleted in period 2 (i.e. before retirement) an agent obtains only fraction g of the nominal value of savings, g [0, 1]. Illiquidity of wealth serves as an external commitment device in the model. 5 Lastly, the model abstracts from possibility of borrowing. 2.2 Preferences The model assumes a dual structure of preferences. A doer (the economic agent s impatient, myopic self) would like to maximise his instantaneous utility, while a planner (the economic agent s far-sighted, smoothing-oriented self) acts so as to maximise the lifetime utility function, which is defined as a discounted sequence of doers instantaneous utilities. A doer has complete control over resources within a given period. A planner can exert some influence on a doer by using an internal commitment device in a form of savings targets. A savings target in period t, α t, is chosen exclusively by a planner, i.e. a doer treats the target as exogenously given. The actual level of savings chosen by a doer in period t is denoted by a t. Note that ideas of the theory of mental accounting are incorporated into the model by assuming the existence of (mental) savings targets - particular financial goals set by an economic agent for himself in order to control his spending. The income that is actually saved is deposited into an illiquid savings account. The costs of depleting the existing savings account, resulting 4 While this paper explicitly focuses on retirement-related decision-making, a similar approach could be employed to analyse behaviour of individuals who face periods of very high income followed by periods of low income (e.g. farmers, professional athletes). 5 In the present context one may also be interested in modelling endogenous investment decisions. Under the set-up considered, this could be accounted for as a choice between various savings accounts characterised by different levels of interest rates and liquidity. In practice, more illiquid savings devices are often characterised by higher interest earnings. As it is shown in Section 3 for the income profiles assumed, economic agents prefers lower levels of liquidity. Given this observation, there is no interesting trade-off to analyse, because all individuals irrespective of their characteristics would choose a savings account characterised by the lowest possible level of liquidity and enjoy high interest earnings. Therefore levels of r and g are assumed to be exogenously given. 6

7 from the assumption of illiquidity, should be interpreted in financial terms. In other words, the savings targets are mental, but the savings accounts themselves are not. Doer s instantaneous utility in period t can be written as: U t = log(c t ) Ψ(a t, α t ), where: Ψ(a t, α t ) = { ψ(α t a t ) 2 for a t α t 0 for a t > α t In order to assume away the impact of income and substitution effects, suppose that instantaneous utility of consumption has a logarithmic form. 6 C t denotes consumption (equivalently: consumption expenditures) in period t. Savings targets serve their role of an internal commitment device because a doer feels bad (i.e. obtains negative utility) if he misses the target. This can be interpreted as guilt that one feels when one fails to follow well-intended plans. Suppose that the magnitude of this painful sensation is increasing in the magnitude by which the target was missed. The self-control parameter ψ > 0 measures the intensity of the painful sensation. The above functional form is assumed for analytical convenience, but in principle any function that is increasing in the magnitude by which the target was missed could be used. 7 A doer maximises U t subject to the budget constraint: C t w t (a t (1 + r)a t 1 ), provided that no dissaving takes place in period t. Recall that dissaving can only occur in period 2. If a doer depletes the savings account in that period his budget constraint becomes C 2 w 2 + g[(1 + r)a 1 a 2 ]. A planner s goal, on the other hand, is to maximise lifetime utility. However, setting the savings targets is costly. Denote the cognitive costs of setting the savings targets by Φ. The planner s objective function is then: where U = U 1 + δu 2 + δ 2 U 3 Φ(α 1, α 2 ), 6 This type of a utility function is also reference-independent. However, in deterministic settings choices made under reference dependence coincide with choices made under reference independence (Koszegi, Rabin 2006). 7 For instance, the above function may enter the optimisation problem in a caseindependent form, which does not affect the solution. 7

8 φ if α 1 > 0 Φ(α 1, α 2 ) = δφ if α 1 = 0, α 2 > 0 0 if α 1 = α 2 = 0 and φ > 0 denotes the one-off cognitive cost of decision-making that is borne in period in which the target is set for the first time. 8 It is therefore implicitly assumed that the cognitive costs of decision-making are borne at the time when one starts saving for one s retirement. This is consistent with interpreting φ as either effort of financial planning or changing one s habits, or costs of acquiring relevant knowledge. The cognitive costs do not depend on specific values of the savings targets. In other words, the effort of designing an appropriate saving schedule is assumed to be independent of the resulting savings target itself. Parameter δ (0, 1) denotes the intertemporal discount factor. Note that while doers preferences are time-inconsistent (each doer is pathologically myopic), a planner is fully time-consistent. The description of the model is completed by outlining the timing within each period. Firstly, a planner makes his decision regarding the level of the savings target. Then income becomes available to a doer and he decides how much to spend and how much to save given the target. 2.3 Characterisation of behaviour Decision-making of a doer Focus on the period-by-period decision-making of doers for given savings targets α (all derivations underlying this subsection can be found in Appendix A). In period 1 a doer selects the actual level of savings a 1 so as to maximise U 1 subject to the budget constraint C 1 w 1 a 1. This yields the following unique solution: a 1 = w 1+α 1 (w 1 α 1 ) ψ, 2 provided that the above equation is strictly positive. In case there is no savings target (α 1 = 0), the borrowing constraint binds and no savings are made by a doer (a 1 = 0). The larger ψ, which measures the magnitude of 8 When a planner sets a savings target in period 1, he is free to choose a new savings target for period 2. Note that this assumption implies decoupling the planner s decision when to start saving from the time when the cognitive cost is borne. 8

9 disutility for deviating from the savings target, the closer the actual savings made by a doer to the target. As ψ increases, a 1 converges to α 1. It is thus clear why ψ can be interpreted as a self-control parameter. Economic agents characterised by greater ψ can exert more self-control in the sense that their actual saving levels are closer to the selected savings targets. What is more, even though the painful sensation resulting from missing the savings target is increasing in ψ for a fixed deviation from the target, it is no longer the case once the actual saving by a doer is adjusted upwards. Because the level of saving is converging to the savings target as the self-control parameter increases, the total painful sensation is in fact decreasing in ψ for a given target. The level of savings a 1 is also increasing in w 1 and α 1, but note that for given α 1 the returns on savings r do not enter the doer s maximisation problem. That is because a doer does not save in order to smooth the consumption over the life cycle, but merely to avoid the punishment for not meeting the already established financial target. In period 2 a doer may either save according to the same decision rule as in period 1, or decide to run down the accumulated savings from period 1. Recall that the analysis focuses on cases in which α 2 (1 + r)a 1, which means that some saving is assumed to be desired in period 2. 9 The fact that the accumulated savings are illiquid implies a discontinuity in the budget constraint. Given the discontinuity, a unique solution can be written in a case-dependent form: w 2 +(1+r)a 1 +α 2 (w 2 +(1+r)a 1 α 2 ) ψ for [α 2 2 (1 + r)a 1 ] 1 2ψw 2 g a 2 = (1 + r)a 1 for [α 2 (1 + r)a 1 ] ( w 2 g +(1+r)a 1 +α 2 ( w 2 g +(1+r)a 1 α 2 ) ψ for [α 2 2 (1 + r)a 1 ] g 2ψw 2 1 2ψw 2, 2ψw 2 ) In the first case above a doer is saving according to a decision rule analogical to the decision rule for period 1. In the second case a doer carries over the accumulated savings from period 1 to period 3 (i.e. he neither saves, nor dissaves). In the third case dissaving occurs. Which of these takes place is determined by the value of the required savings for period 2, [α 2 (1 + r)a 1 ]. The above conditions state that saving takes place (or: no dissaving occurs) provided that the required saving is high enough. The better self-control (higher ψ), the less restrictive the above conditions (lower levels of required 9 Whether or not this condition holds at the optimum depends not only on the shape of an income profile, but also on values of r and δ. Income profiles and parameters for which saving is optimal in both periods 1 and 2 are presented in Section 4. 9

10 savings are necessary in order to prevent a doer from invading the savings account). Similarly, the lower g, the less restrictive the above conditions. In other words, the more illiquid the savings, the less likely it is that they will be depleted before retirement. What is more, when the savings account is invaded the level of withdrawal is increasing in g. This justifies referring to illiquidity of savings as the external commitment device. 10 Independent of the scenario, the level of savings left for period 3 is weakly increasing in w 2, α 2 and ψ. As in period 1, a 2 converges to α 2 as ψ increases. In period 3 there is no optimisation problem to solve. A retired doer simply consumes whatever wealth is available to him and obtains utility of: U 3 = log(b + (1 + r)a 2 ), because there is no financial punishment for running down the savings account in period Decision-making of a planner Given doers decision rules, a planner has three options to choose from. He can either set the savings targets in both periods 1 and 2, or set the savings target for period 2 only, or set no savings targets at all. Which action leads to the highest level of lifetime utility depends on how large the benefits from consumption smoothing are relative to the cognitive costs. Firstly, consider how levels of lifetime utility change with parameters ψ and φ at the optimum (derivations of Remarks 1-3 can be found in Appendix B). Remark 1 (impact of self-control and cognitive costs on utility) At the optimum, the lifetime utility obtained by an agent is: 1. weakly increasing in ψ, 2. weakly decreasing in φ. The above result formalises the statement that economic agents with better self-control are able to smooth their consumption to a greater extent without inducing overly intense painful sensations. That is because the total painful sensation related to inducing a given level of saving is decreasing 10 Note that there is no discontinuity in either the budget constraint or the doer s decision rule for g = 1. Only for g < 1 one obtains the above interval for values of required savings for which a doer neither invades nor contributes towards the savings account. 10

11 in ψ. The impact of cognitive costs on welfare is straightforward. They either directly decrease the lifetime utility of individuals who save, or prevent remaining agents from engaging in potentially beneficial consumption smoothing. Remark 2 (possibility of inertia) There exist non-negative A(ψ), B(ψ) and C(ψ), where A and B are increasing in ψ and A B, A C for any ψ, such that: 1. No saving takes place at all (i.e. no savings targets are set by a planner) if φ A and φ 1 δ B; 2. saving takes place only in period 2 if φ < 1 B and φ(1 δ) C; δ 3. saving takes place in both periods 1 and 2 if φ < A and φ(1 δ) < C. The above observation simply states that an individual will not save, and just consume his current income in every period, if the cognitive costs of decision-making are higher than the utility gains from consumption smoothing. As implied by Remark 1, those gains are lower for individuals with low degrees of self-control and therefore the agents that are most likely to not save are the agents characterised by either high φ or low ψ. For any given level of cognitive costs, the poorer self-control, the less likely it is that an agent will smooth his consumption over the life-cycle. Furthermore, with high cognitive costs small self-control problems might suffice to prevent an economic agent from saving. Cognitive costs and poor self-control can thus be seen as substitutes when interpreted as factors that prevent an individual from saving for his retirement. One can also observe that the utility gains from consumption smoothing are lower for economic agents characterised by low incomes. 11 In the light of the presented model inertia can be partially explained by the lack of self-control. To my best knowledge, no alternative formal models link the possibility of inaction to the level of self-control or the level of income. For various levels of the self-control parameter ψ, Figure 1 shows the values of the cognitive cost φ for which a planner is indifferent between saving in both periods and simply consuming his current income across the life cycle. Note that this marginal level of the cognitive cost (which is equal to benefits 11 That is income profiles which are characterised by low levels of income keeping the variability of income constant. For any ɛ > 1 benefits from consumption smoothing are greater for an income profile (ɛw 1, ɛw 2, ɛb) than (w 1, w 2, b). 11

12 from consumption smoothing) is increasing in ψ. This is a direct implication of Remark Figure 1: Marginal φ w 1 = 21962, w 2 = 31193, b = 8048, (1 + r) = (1.04) 20, δ = (0.97) 20 In theory, the hand-to-mouth consumers do not engage in consumption smoothing for at least one of the three reasons. They either lack necessary self-control, or they have limited cognitive capacity (resulting in high costs of decision-making), or they are characterised by a low income profile. In reality, however, these three types are unlikely to be disjoint. Firstly, there exist some evidence that cognitive abilities are correlated with self-control (e.g. Benjamin et al. 2013; Camerer 2013), which would suggest that parameters ψ and φ are negatively correlated. Secondly, well-known stylised facts from labour economics indicate that incomes of highly educated workers are significantly higher (e.g. Borjas 2010; Cahuc, Zylberberg 2004). To the extent to which high education is an indicator of high cognitive abilities, these regularities might be interpreted as suggesting a negative correlation between φ and a level of income. Alternatively, such a correlation could arise 12 Parameter values used to generate all the figures correspond to values employed in Section 4. 12

13 due to the fact that individuals from low-income households often lack access to suitable financial institutions and might be characterised by higher levels of a stress-related cognitive load (Mullainathan, Shafir 2013). Even though no formal assumptions are made regarding the correlation between ψ, φ, and the level of income, one can imagine those variables as being interdependent. Recall that according to the Attitudes to Pensions survey, 19% of respondents could be classified as hand-to-mouth consumers. Compared to the rest of the sample, these individuals were significantly more likely to come from a low-income household and exhibit symptoms of poor financial self-control and high cognitive costs of decision-making. 13 These results are consistent with the predictions of the model. Remark 2 highlights that the condition for not saving is more likely to be satisfied for low values of ψ, high values of φ, and a low income profile. The economic agent whose behaviour is described by the standard theories is nested within the model as an agent with zero cognitive costs and perfect self-control (ψ + ). Call this individual a classical agent. In case of a classical agent, doers are completely obedient (i.e. a = α) and therefore the savings targets devised by a planner are implemented without any disutility for missing the targets. The next remark states how the optimal choices of an agent with self-control issues (i.e. ψ < + ) or non-zero cognitive costs compare with the choices of a classical agent. Remark 3 (upper bound for optimal saving and possibility of overshooting ) Let α C 1 denote the optimal level of savings (and the savings target) of the classical agent in period 1. For the optimal choices of an otherwise identical agent with ψ < + or φ > 0 it must be the case that a 1 α C 1, while it is possible that either α 1 α C 1, or α 1 > α C 1. Moreover, a 1 is increasing in ψ. Similar remarks apply to behaviour in period 2. The above states that the actual savings made by any agent cannot exceed the savings of the classical agent at the optimum. In other words, α C 1 constitutes the upper bound for savings in period 1, because agents with self-control problems cannot achieve the same level of consumption smoothing as the classical agent without imposing an excessively painful sensation. 13 More specifically, 29% of those individuals (compared to 59% in the rest of the sample) said that they were keeping up with their bills and credit commitments without any difficulties; 35% (58%) reported that they were putting some money aside for emergency situations; 41% (12%) said that they would have no idea about what they needed to do when making important financial decisions, such as taking out a mortgage, loan, or pension ; 37% (16 %) reported their financial knowledge to be poor; 84% (51 %) stated that they had no idea what their retirement income would be (MacLeod et al. 2012). 13

14 They consider the trade-off between consumption smoothing and disutility for missing the high savings targets and therefore they find it optimal to save less. Similar remarks apply when one compares two agents with imperfect self-control - an individual characterised by higher ψ optimally saves more. Furthermore, high cognitive costs may prevent an agent from saving altogether. However, under specific conditions it may still happen that the savings target is set above α1 C. This situation is referred to as overshooting because a planner deliberately sets too high a target in order to elicit the desired level of saving. Finally, note that the optimal level of saving is (weakly) decreasing in both self-control issues and high cognitive costs. Although both factors influence the decision-making via separate channels, they have the same qualitative impact on the optimal saving. The presence of cognitive costs and self-control problems results in public pension wealth and discretionary saving being only imperfect substitutes. The following proposition formalises the notion of non-fungibility of these two kinds of wealth. What is important, in the setting of this model the two kinds of income are non-fungible even without assuming any differences in levels of liquidity. Proposition 1 is the only derived result which uses a reduced, two-period version of the model. That is because its only purpose is to provide an illustration of the non-fungibility result, which is more clear under the two-period version. Proposition 1 (non-fungibility of discretionary saving and public pension wealth) Focusing on decision-making in period 2, denote the optimal level of discretionary saving by a 2. If an agent is characterised by either positive cognitive costs or imperfect self-control, discretionary saving a and public pension wealth are no longer perfect substitutes, i.e. 2 is not b constant. More precisely: 1. For φ = 0 and ψ +, a 2 b is constant (a 2 is linear in b). 2. For φ > 0 and ψ +, a 2 is discontinuous, but piecewise constant b (a 2 is discontinuous, but piecewise linear in b). 3. For φ = 0 and ψ < +, non-linear in b). a 2 b is non-constant (a 2 is continuous and 4. For φ > 0 and ψ < +, a 2 is discontinuous and non-constant (a b 2 is discontinuous and non-linear in b). 14

15 For derivation see Appendix C. Due to the cognitive costs, the optimal level of discretionary saving is changing discontinuously with b. As the public pension system becomes more generous, the benefits from consumption smoothing become limited. Eventually, above a certain threshold for b the utility gains from discretionary saving become smaller than the cognitive costs of decision-making, preventing an economic agent from saving privately. At this threshold a 2 is discontinuous in b. The introduction of self-control issues results in the optimal level of discretionary saving being non-linear in the level of pension benefits (below the threshold). That is due to the fact that when choosing optimal a 2 a planner takes into account the painful sensation resulting from setting the savings targets and not only the consumptionsmoothing aspect of saving. Note that both self-control issues and cognitive costs would imply non-fungibility in isolation, but, as in case of Remark 3, the impact of ψ and φ is exerted via separate channels. Figure 2 shows optimal levels of saving a 2 as a function of pension benefits b. There is a clear discontinuity in a 2, the location of which depends on the self-control parameter. Economic agents with low degrees of self-control give up on discretionary saving sooner than agents with better self-control. Figure 2: Discretionary saving in period 2 w 2 = 31193, b = 8048, (1 + r) = (1.04) 20, δ = (0.97) 20, φ =

16 3 Policy implications This section presents the policy implications of the existence of cognitive costs and self-control problems. Firstly, the model predicts preference for illiquidity of pension wealth. Subsequently, two major reforms of the UK pension system are discussed in more detail. The model provides a theoretical justification for the introduction of automatic enrolment into workplace pension schemes, because the individuals who are most likely to be affected by the reform (the hand-to-mouth consumers) are at the same time most likely to benefit from it. However, its impact on aggregate saving is ambiguous and depends not only on individual s characteristics (levels of self-control and cognitive costs), but also on a degree of default coverage. Automatic enrolment paired with low contribution rates is likely to reduce aggregate saving, for instance. The second considered reform is the conversion of public pension benefits into a single-tier scheme, which is supposed to simplify the decision environment and incentivise discretionary saving. In case of members of low-income households (the presumed hand-to-mouth consumers), the reform additionally results in a top-up of pension benefits and therefore its effect on propensity to save is ambiguous. Consistently with other theories assuming the existence of self-control problems, the model predicts preference for illiquidity of pension wealth, denoted by parameter g. This preference might be used to rationalise the observed policies of restricting access to one s pension wealth, employed for example in the UK (legally binding pension ages) or in the US (financial penalties for depleting a 401(k) account). However, even though all economic agents prefer their retirement wealth to be characterised by lower levels of liquidity, only those with poor self-control are effectively willing to pay for an access to an illiquid savings account. Such regularity can only be derived within a model that incorporates both internal and external commitment devices. Refer to Appendix D for a formal derivation and a more detailed discussion. The first discussed reform of the UK s pension system is the introduction of automatic enrolment into workplace pension schemes by the Pensions Act As stated in the official report by the Department for Work and Pensions, automatic enrolment is designed to harness the natural tendency towards inertia that people display in pension behaviour, by making people opt out, rather than opt in to a workplace pension (MacLeod et al. 2012). Importantly, the reform also specifies minimum contribution rates. To be eligible for automatic enrolment one needs to be over 22 years old, not currently saving into a qualifying scheme, below the State Pension Age, and 16

17 have earnings above the specified, but not restrictive, threshold. 14 What is interesting, even though employees can opt out of their workplace pension schemes, they are automatically enrolled again after a period of 3 years. Similar reforms were also considered in the US and New Zealand (Iwry 2006). The following discussion focuses on welfare and saving implications of the introduction of automatic enrolment, for simplicity modelled as the introduction of a compulsory scheme. 15 It must be noted that pension schemes are introduced into the model solely as policy devices which help individuals smooth their consumption across the life cycle by collecting contributions in periods 1 and 2, and providing pension benefits in period 3. Specifically, any redistributive properties of the schemes, which are often of great importance, are disregarded in the analysis. Consider an agent with an income profile (w 1, w 2, b) who saves in both periods in absence of the scheme. Let (Ĉ1, Ĉ2, Ĉ3) denote the levels of consumption that would maximise consumption utility subject to the budget constraints: (Ĉ1, Ĉ2, Ĉ3) = argmax {log(c 1 ) + δlog(c 2 ) + δ 2 log(c 3 )}, s.t. C 1 + C 2 (1+r) + C 3 w (1+r) w 2 (1+r) + b ; C (1+r) 2 1 w 1 ; C 2 w 2 + (1 + r)(w 1 C 1 ). Definition 1 (compulsory pension scheme) A compulsory pension scheme effectively confiscates all the agent s resources and pays out zĉ1, zĉ2, zĉ3 in periods 1, 2 and 3, z (0, 1]. This is achieved by collecting a contribution of τ 1 = w 1 zĉ1 in period 1, τ 2 = w 2 zĉ2 in period 2, and providing a total benefit of zĉ3 in period 3. For z < 1 the scheme is said to be actuarially unfair because a fraction of wealth is lost when transferred across time by the scheme, which implies that zĉ3 b < (1 + r) 2 τ 1 + (1 + r)τ 2. This can be interpreted as representing either administrative costs of operating the scheme, or the fact that public pension schemes may offer lower returns to contributions, or also the possibility that the scheme is not perfectly fitted to individual s time preferences. That is because by changing z continuously between 0 and 1 one can obtain all levels of lifetime utility attainable by an agent with a given income profile. 14 For example, the threshold was set equal to 8105 in the tax year. 15 Given well-documented and long-lasting effects of changing the default option (e.g. Benartzi, Thaler 2004; Benartzi, Thaler 2007; Madrian, Shea 2001), it may be argued that this is not too radical a simplification. Moreover, such an effect can be micro-founded by assuming that there exist mental or time costs of opting out. 17

18 Proposition 2 (actuarially unfair pension scheme) The above actuarially unfair, but compulsory, scheme makes the agent strictly better off provided that: φ + E(ψ) > (1 + δ + δ 2 )log( 1 z ), where E is decreasing in ψ, lim ψ E(ψ) = 0. The derivation can be found in Appendix E. The fact that the additional pension scheme is compulsory (or: that an agent is automatically enrolled into it) has two effects on utility. Firstly, an individual does not need to bear the cognitive costs of decision-making. And secondly, he does not need to exert any self-control in order to control his spending. In case of an actuarially fair pension scheme (z = 1), the impact on individual s welfare is unambiguous - for economic agents with any degree of self-control issues or cognitive costs the scheme strictly improves welfare. In case of an actuarially unfair pension scheme the present value of contributions exceeds the present value of benefits, i.e. a certain amount of agent s income is lost when transferred across the life cycle. However, the scheme might still be welfareimproving because it relieves him from bearing the cognitive costs or feeling the painful sensation. If the sum of these two effects dominates the lost income effect, an agent is strictly better off. For a given value of parameter z the scheme is more likely to be welfare-improving for agents characterised by either high φ or low ψ. As in case of previous results, the impacts of self-control problems and high cognitive costs on welfare implications are of the same qualitative nature. In contrast to previous results, however, the impacts of the two factors are disjoint, i.e. the interactions between ψ and φ do not affect the above welfare criterion. 16 Figure 3 shows various levels of the variable z for which an individual with a given level of self-control and cognitive costs is indifferent between saving on his own and participating in an actuarially unfair pension scheme. For higher values of φ economic agents are willing to accept greater income losses related to participating in a scheme (lower z). Individuals with better self-control are willing to accept smaller losses. 16 Recall that Proposition 2 presents the welfare condition for an economic agent who either saves privately or participates in a compulsory pension scheme. When one additionally allows for non-saving, the left-hand side of the above condition becomes case-dependent, but nonetheless it remains (weakly) increasing in φ and (weakly) decreasing in ψ. This is illustrated numerically in Section 4. 18

19 Figure 3: Marginal z w 1 = 21962, w 2 = 31193, b = 8048, (1 + r) = (1.04) 20, δ = (0.97) 20 The theory developed above allows to make some further remarks. The hand-to-mouth consumers (characterised by low income, poor self-control, and high cognitive costs) are most likely to be affected by the introduction of automatic enrolment due to the design of the reform. What is crucial, this group of economic agents is also most likely to benefit from the existence of a compulsory scheme, even when it is unfair. Proposition 2 provides a theoretical justification for automatic enrolment, because it suggests that people who are expected to be influenced by the reform are at the same time most likely to benefit from it. Under an alternative interpretation, Proposition 2 suggests why changing the default option may have such large effects on propensity of individuals to participate in pension schemes. The hand-to-mouth consumers, who are typically affected by changing the default option, are most likely to find unfair schemes welfare-improving. In other words, individuals who had not engaged in discretionary saving are most likely to remain opted-in after the default option has been changed, even when the considered scheme is actuarially unfair or not perfectly consistent with their time preferences. This is also reflected in the Attitudes to Pensions survey. Among those 19

20 eligible for automatic enrolment, as many as 64% had never heard about the reform. However, 68% agreed that it was a good idea, while 70% reported that they were likely to stay in the scheme once enrolled (MacLeod et al. 2012). Apart from welfare implications of automatic enrolment, one may also be interested in examining its potential impact on aggregate saving, defined as a sum of discretionary saving and default (compulsory) contributions. In the standard models with no cognitive costs or self-control problems, the two sources of pension wealth are perfectly substitutable provided that the compulsory pension scheme is actuarially fair and does not force agents to save too much. In this case the aggregate saving is invariant to the level of contributions. However, this is not true in the setting of the discussed model. The introduction of automatic enrolment might either increase or decrease aggregate saving. Proposition 3 ( discouraged saving and forced saving ) Consider two agents characterised by the same income profile (w 1, w 2, b). Agent 1 has no access to a compulsory pension scheme and if he decides to save, he allocates a 1 into an illiquid savings account in period 1. Agent 2 has access to an actuarially fair compulsory pension scheme that collects a contribution of τ 1 (0, a 1) in period 1 and automatically transfers it into the savings account. Denote the optimal level of discretionary saving by agent 2, in case he decides to save, by a 1. The agents are otherwise identical. Then there exist increasing F (ψ), G(ψ), where F > G for any ψ, such that: 1. For φ F > G neither of the agents engages in discretionary saving, in which case the aggregate saving of agent 2 is greater, i.e. τ 1 > 0 ( forced saving ). 2. For F > φ G agent 1 saves on his own while agent 2 does not, which implies that the aggregate saving of agent 1 is greater, i.e. a 1 > τ 1 ( discouraged saving ). 3. For F > G > φ both agents perform discretionary saving, in which case their levels of aggregate saving are equal, i.e. a 1 = τ 1 + a 1. Analogical remarks apply to savings be- Moreover, G is decreasing in τ 1. haviour in period 2. For derivation see Appendix F. In presence of cognitive costs of decisionmaking and self-control problems the introduction of automatic enrolment 20

21 might indeed result in some individuals being compelled to save. This outcome is more likely for economic agents characterised by poor self-control and high cognitive costs, the likely hand-to-mouth consumers. On the other hand, the aggregate saving of economic agents characterised by high ψ and low φ (the classical agents) remains unaffected by the change. There also exists an in-between group of individuals for whom G > φ F. After the introduction of automatic enrolment these agents do not engage in discretionary saving, even though they would save in absence of the scheme. That is due to the fact that after the scheme has been introduced, the potential benefits from consumption smoothing are diminished. If they become smaller than the cognitive costs an individual becomes discouraged and decides not to save privately. Note that the propensity to stop engaging in discretionary saving depends on the interaction between the level of cognitive costs and self-control. For instance, individuals with poor self-control need to have lower cognitive costs in order to become discouraged (for any given level of τ 1 ). In sum, the effect on aggregate saving is equivocal, it may be either increased, decreased, or unaffected as a result of the reform, depending on individual characteristics. Additionally, note that the impact of the level of default coverage on aggregate saving is also ambiguous. On the one hand, for higher values of τ 1 an agent is compelled to save more by the scheme. On the other hand, he is less likely to engage in discretionary saving. This mechanism is in line with empirical studies which report that the impact of introducing automatic enrolment on aggregate saving is ambiguous and crucially depends on the default contribution rates (Choi et al. 2004; Madrian, Shea 2001). Turn to the second major reform of the UK pension system, the introduction of the single-tier pension system by the Pensions Act Currently, pension benefits consist of two main elements: flat-rate Basic State Pension and earnings-related State Second Pension. Replacing the two schemes with a single flat-rate benefit, which will take place in April 2016, is believed by the policy-makers to simplify the system and, as a result, provide a clear incentive to save. As stated in the official document by the Department for Work and Pensions: Pensioner incomes represent a very complex aggregation of state and private payments, making it difficult for anyone to predict what income they will receive in retirement. The implementation of the single-tier pension will significantly simplify the pension system, helping people to understand what they will get from the State when they retire. 17 The Attitudes to Pensions survey indeed documents much confusion about 17 Quoted from: The single-tier pension: a simple foundation for saving (2013). 21