# Optimal time allocation in active retirement

## Abstract

We set up a lifecycle model of a retired scholar who chooses optimally the time devoted to different activities including physical activity, continued work and social engagement. While time spent in physical activity increases life expectancy, continued scientific publications increases the knowledge stock. We show the optimal trade off between these activities in retirement and its sensitivity with respect to alternative settings of the preference parameters.

## Keywords

Time allocation Active retirement Longevity Scientific production## 1 Motivation

With increasing survival to old age, the share of life time spent in retirement has increased over the last decades. While many studies investigate the transition from work to retirement, the time spent in retirement is less investigated. As recently argued in Sprod et al. (2017, p. 10), “The transition from a working lifestyle to one of retirement involves a reorganization of daily activities and the choice of activities has health consequences.” Hence, in light of productive and healthy aging it is important to gain a better understanding of how people spend their time in retirement.

In this paper we propose an optimal control model of productive and healthy aging in retirement. We focus our analysis on the active retirement phase of a university professor. We assume that our individual derives utility from scientific work (including research, attending conferences, etc.) and social engagement but suffers a utility loss from physical activity. These activities in turn determine the evolution of the knowledge and health stock. We assume that a higher knowledge stock left behind is positively valued by our individual and health is positively related to survival. Hence, an optimal tradeoff between time allocated to research and physical activity will exist. We investigate the optimal time allocation in retirement dependent on various specifications of preferences and technologies of knowledge and health accumulation.

## 2 The model

*x*with a fixed but guaranteed income until death. The consumption path and the wealth profile are assumed to have been optimally chosen during the working period. Hence, we focus on the optimal allocation of the available time in retirement among the three main activities: cognitive activity (\(\textsf {t}_c\)), physical activity (\(\textsf {t}_h\)), and social activity (\(\textsf {t}_s\)). Cognitive activity comprises the time devoted to producing papers, going to working meetings, attending conferences, workshops, and seminars. Physical activity accounts for the time spent on the gym, hiking in the mountains, etc; while the social activity represents the leisure time going to concerts, watching football, etc. The sum of all the available time at time

*t*is

*T*denotes the maximum number of available minutes per day.

^{1}We assume that the time spent on various activities are mutually exclusive; i.e. activities cannot be conducted in parallel to each other.

*D*) that in turn depends on the time spent on physical activity. To account for the evolution of health deficits we follow the work by Dalgaard and Strulik (2014). In particular, health deficits accumulate annually at a rate \(\mu (D-a)\) until the maximum number of health deficits \(\overline{D}\) is reached. The rate of accumulation of health deficits can be slowed down through physical activity (Dalgaard and Strulik 2017; Schuenemann et al. 2017; Strulik 2018) as follows

*K*. The stock of knowledge at age

*t*is defined as the total number of papers written until age

*t*,

*K*(

*t*). To produce papers, the scholar needs to devote time to cognitive activity. The number of papers published at each time is positively related to the exiting stock of knowledge of our scholar. Thus, the number of papers published at time

*t*is given by

^{2}

*t*is:

*U*. We use logarithmic utility functions on \(\textsf {t}_s\) and \(\textsf {t}_c\) and a convex function on the disutility of health activities in order to guarantee the existence and uniqueness of an optimal solution.

## 3 Optimal solution

*First-order conditions (FOCs)*

*Envelope conditions and transversality conditions*

^{3}

*Terminal age condition*

## 4 Calibration and simulation strategy

We use our model to study what kind of preferences allow scholars to enjoy a long retirement life, while continuing to accumulate scientific knowledge.

### 4.1 Calibration

*a*) at 0.04 and 0.02, respectively. The initial stock of health deficits \(D_x\) is set at 0.106, which corresponds to the estimated value of health deficits at age 78 reported by Mitnitski et al. (2002a), and the maximum number of health deficits \(\overline{D}\) at 0.22. We chose the value of \(\overline{D}\) that guarantees our scholar to reach 97.5 years without devoting time to physical activities. Then, we calibrate the parameters

*A*and \(\eta \) that account for the impact of physical activity on the accumulation of health deficits to 0.000363 and 0.7, respectively. The values of

*A*and \(\eta \) are set so as to obtain a difference between the highest longevity—720 min per day devoted to health activities—and the lowest longevity—0 min on health activities—of 7.35 years, which corresponds to the standard deviation of the life expectancy at age 78 for an Austrian male born in 1940 (Human Mortality Database 2018).

^{4}Given that we do not know the preference of our scholar towards the different activities, we examine alternative values for all the remaining parameters of the utility function. Specifically, we consider that the utility weight of cognitive activities \(\phi _c\) may range between 2 and 20, and the utility weight on social activities \(\phi _s\) between \(\phi _c\) and 20. We consider \(\phi _s\) to be always equal or greater than \(\phi _c\) in order to guarantee that the first-order condition for an optimal time devoted to cognitive activities is satisfied; i.e. \({\phi _s}/{\textsf {t}_s}-{\phi _c}/{\textsf {t}_c}>0\). The range of values for \(\phi _s\) and \(\phi _c\) are chosen in order to account for the impact of different weights of physical activity on the utility function. In particular, given that \(\phi _h\) and \(\omega \) are fixed, an increase in the values of \(\phi _c\) and \(\phi _s\) implies a decline in the relative importance of the disutility of physical activity on the instantaneous utility. The subjective discount factor is considered to take values between 0 and 0.02, which includes the average discount factor of 0.01, and the salvage parameter ranges between 0 and 5.

Parameter values

Parameter | Symbol | Value | Parameter | Symbol | Value |
---|---|---|---|---|---|

Time | Knowledge | ||||

Max. time per day (min) | \(\textsf {T}\) | 720 | Initial stock | \(K_x\) | 378 |

Current age |
| 78 | Returns to knowledge | ||

\(\theta \) | 0.0126 | ||||

Preferences | \(\gamma \) | 0.7 | |||

Fixed | \(\beta \) | 0.214819 | |||

Elasticity physical activity | \(\omega \) | 0.1 | Health deficits | ||

Weight physical act. | \(\phi _h\) | 0.015 | Natural force of aging | \(\mu \) | 0.043 |

Not fixed |
| 0.02 | |||

Weight cognitive activity | \(\phi _c\) | (2, 20) | Impact of physical activity |
| 0.000363 |

Weight social activity | \(\phi _s\) | (\(\phi _c\), 20) | \(\eta \) | 0.7 | |

Subj. discount factor | \(\rho \) | (0, 0.02) | Health deficits at | \(D_x\) | 0.106 |

Salvage parameter | \(\kappa \) | (0, 5) | Maximum deficits | \(\overline{D}\) | 0.22 |

### 4.2 Simulation strategy

*M*, that maps \(\psi \in \varPsi \subseteq \mathbb {R}_{+}^4\) into a set of output variables \(\pi \in \Pi \subseteq \mathbb {R}_{+}^2\):

*p*), which is calculated as \((K(\alpha )-378)/(\alpha -78)\), where 378 is the total number of papers published by our scholar until age 78.

Figure 2 shows the combination of possible average number of papers published per year (*p*) and longevity (\(\alpha \)), i.e. \(f(\alpha ,p)\), for all possible sets of preference weights \(\varPsi \). Each gray dot in Fig. 2 corresponds to a set of possible \(\psi \) values drawn from \(q(\psi )\). The shape of the joint density \(f(\alpha ,p)\) is due to the fact that not all combinations of longevity and number of papers per year are possible given the model *M*, the set of preference weights, and the boundary conditions. The marginal densities of *p* and \(\alpha \) are also shown at the top margin and at the right margin of Fig. 2, respectively.

According to (7) our scholar must devote time to each activity in order to attain a specific combination of average number of papers per year and longevity values. The average time devoted to each activity over the remaining lifespan, given the set of possible preference weights \(\varPsi \), is shown in Fig. 3. Blue dots depict the average time devoted to cognitive activities, green dots depict the average time devoted to social activities, and red dots correspond to the average time devoted to physical activities over the remaining life. Solid lines represent the mean of the average time devoted to each activity conditional on the maximum longevity (left panel) and average number of papers per year (right panel). From Fig. 3 we can observe that there is almost a one-to-one relationship between the physical activity and longevity (see left panel) and between cognitive activity and the average number of papers published per year (see right panel). The lack of a cloud of points around the mean of the average time devoted to physical activity implies that the maximum longevity is determined by \(\textsf {t}_h\) (see left panel), while the average number of papers per year is determined by the average time devoted to cognitive activity (see right panel). However, cognitive and social activities do not necessarily lead to a longer longevity, as the cloud of points in the left panel suggests, as well as social and physical activities are not directly translated, according to the cloud of points in the right panel, into a higher number of papers published per year. As a consequence, for instance, Fig. 3 shows that our scholar would need to devote around 5.8 h (\(=\) 350 min) per day on cognitive activity in order to produce over the remaining lifespan an average of five papers per year (see right panel), while whenever our scholar does not target a particular longevity the time spent on social activities and on physical activities is more flexible. By contrast, given that the total available time is assumed to be fixed, when our scholar seeks a specific average number of papers per year and a specific longevity, then the average time devoted to each activity is almost completely determined. Thus, we will use this information to study the optimal distribution of time across the different activities based on the outputs.

## 5 Results

Descriptive statistics of the preference parameters under three different scenarios

Symbol | Benchmark (A) | Case B | Case C | ||||
---|---|---|---|---|---|---|---|

Mean | SD | Mean | SD | Mean | SD | ||

Avg. number of papers per year |
| 5.02 | 0.14 | 4.99 | 0.14 | 4.06 | 0.14 |

Longevity | \(\alpha \) | 98.61 | 0.11 | 99.94 | 0.11 | 100.01 | 0.14 |

Weight cognitive activity | \(\phi _c\) | 5.078 | 1.389 | 16.702 | 2.489 | 14.663 | 2.971 |

Weight social activity | \(\phi _s\) | 12.662 | 3.920 | 18.537 | 1.446 | 18.233 | 1.507 |

Subjective discount factor | \(\rho \) | 0.016 | 0.004 | 0.003 | 0.002 | 0.005 | 0.003 |

Salvage parameter | \(\kappa \) | 2.512 | 1.345 | 3.182 | 0.707 | 0.727 | 0.646 |

Average distribution of time over the remaining life under three different scenarios

Symbol | Benchmark (A) | Case B | Case C | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Mean | SD | Mean | SD | Mean | SD | |||||

Avg. number of papers per year |
| 5.0 | 0.14 | 5.0 | 0.14 | 4.1 | 0.14 | |||

Longevity | \(\alpha \) | 98.6 | 0.11 | 99.9 | 0.11 | 100.0 | 0.14 | |||

Distribution of time over remaining life: | in % | in % | in % | |||||||

Cognitive activity in min/day | \(\textsf {t}_c\) | 350 | 49 | 11.5 | 348 | 48 | 12.1 | 270 | 37 | 11.1 |

Social activity in min/day | \(\textsf {t}_s\) | 322 | 45 | 13.5 | 219 | 30 | 14.2 | 291 | 40 | 16.9 |

Physical activity in min/day | \(\textsf {t}_h\) | 48 | 7 | 7.4 | 153 | 21 | 9.6 | 160 | 22 | 12.0 |

## 6 Extensions

In this section we propose three extensions to the optimal time allocation model (7). First, we assume the time devoted to physical activities does not generate disutility, i.e. \(\phi _h=0\). This model represents well the behavior of a “sportsman”. Second, we consider that the utility from the different activities declines as the stock of health deficits rises. We name this model as “health dependent utility” model. Third, we assume that our scholar’s productivity declines with increasing health deficits. This last model is named “cognitive decay” model.

### 6.1 Sportsman

Table 4 shows the descriptive statistics of the new preference weights characterizing cases A, B, and C in Fig. 5. The first thing to notice is that under the “sportsman” model only cases B and C, which both give an average longevity of 100 years, are possible. Second, comparing cases B and C we observe that publishing one less paper per year—from 5 papers to 4—is associated to scholars who get lower utility from cognitive activities, have a lower marginal value of knowledge, and discount the future pay-offs more.

Descriptive statistics of the preference parameters for cases A, B, and C

Symbol | Case A | Case B | Case C | ||||
---|---|---|---|---|---|---|---|

Mean | SD | Mean | SD | Mean | SD | ||

Avg. number of papers per year | | – | – | 4.99 | 0.14 | 4.04 | 0.14 |

Longevity | \(\alpha \) | – | – | 100.03 | 0.14 | 100.10 | 0.13 |

Weight cognitive activity | \(\phi _c\) | – | – | 13.471 | 4.432 | 8.457 | 3.297 |

Weight social activity | \(\phi _s\) | – | – | 16.177 | 3.320 | 16.079 | 2.976 |

Subjective discount factor | \(\rho \) | – | – | 0.013 | 0.004 | 0.016 | 0.003 |

Salvage parameter | \(\kappa \) | – | – | 3.687 | 0.794 | 2.434 | 0.882 |

### 6.2 Health dependent utility

^{5}

Descriptive statistics of the preference parameters for cases A, B, and C

Symbol | Case A | Case B | Case C | ||||
---|---|---|---|---|---|---|---|

Mean | SD | Mean | SD | Mean | SD | ||

Avg. number of papers per year |
| 5.01 | 0.15 | 4.99 | 0.14 | 4.07 | 0.13 |

Longevity | \(\alpha \) | 98.62 | 0.12 | 99.92 | 0.10 | 99.98 | 0.13 |

Weight cognitive activity | \(\phi _c\) | 5.342 | 1.422 | 17.092 | 2.228 | 15.245 | 2.662 |

Weight social activity | \(\phi _s\) | 11.937 | 4.006 | 18.720 | 1.261 | 18.517 | 1.280 |

Subjective discount factor | \(\rho \) | 0.016 | 0.004 | 0.003 | 0.002 | 0.004 | 0.003 |

Salvage parameter | \(\kappa \) | 1.820 | 1.125 | 2.492 | 0.584 | 0.492 | 0.422 |

### 6.3 Cognitive decay

*t*is given by

The average time devoted to each activity over the remaining life in the cognitive decay model is shown in Fig. 10. Comparing Figs. 3 and 10 we can observe that the average time spent on cognitive activities, conditional on the average number of papers per year, increases in the cognitive decay model relative to the benchmark model. However, there is no significant change between the benchmark model and the cognitive decay model on the time spent on the different activities in order to reach a specific longevity. Therefore, despite the fact that individuals devote the same average time across the different activities, in the cognitive decay model our scholar will reach the same longevity but lower average number of papers.

Descriptive statistics of the preference parameters for cases A, B, and C

Symbol | Case A | Case B | Case C | ||||
---|---|---|---|---|---|---|---|

Mean | SD | Mean | SD | Mean | SD | ||

Avg. number of papers per year |
| 5.00 | 0.14 | – | – | 3.98 | 0.14 |

Longevity | \(\alpha \) | 98.57 | 0.13 | – | – | 99.94 | 0.11 |

Weight cognitive activity | \(\phi _c\) | 7.992 | 2.966 | – | – | 16.885 | 2.233 |

Weight social activity | \(\phi _s\) | 11.376 | 2.696 | – | – | 18.513 | 1.471 |

Subjective discount factor | \(\rho \) | 0.014 | 0.004 | – | – | 0.003 | 0.002 |

Salvage parameter | \(\kappa \) | 3.532 | 1.028 | – | – | 2.800 | 0.862 |

## 7 Conclusions

As the life phase spent in retirement has increased over time, activity patterns in retirement have changed as well. Recent studies (e.g. Mergenthaler et al. 2018) have investigated the variation in these activity patterns identifying various types of retirees. However, so far, the role of an active retirement and its impact on the retiree’s health (and hence life expectancy) has not formally been investigated.

In this paper we built up a model of optimal time allocation in retirement allowing for feedback mechanisms of the choice of activity on the agent’s health. We calibrate our model to the life of a scientific scholar and assume that time in retirement can be allocated between cognitive activities, social activities and physical activity. While cognitive activity increases the stock of scientific papers produced, physical activity increases health and hence the life span of the retiree. We assume that our scholar derives utility out of cognitive and social activities and disutility from physical activity. Moreover, we introduce a salvage value for the papers written until the end of life.

Depending on the time preference of the retiree, the valuation of the salvage value of papers written, and the utility weight on the various activities we first study the optimal allocation of time in retirement and its impact on longevity and on the number of papers written. Then, we use the numerical results to investigate the behavioral characteristics that support specific outcomes of longevity and number of papers as well as the optimal allocation of time to reach these specific outcomes. Overall our results show a positive correlation between living longer and the joy of producing papers, being more patient, and the utility from leaving a large stock of knowledge to future generations.

To further study how physical activity impacts on longevity we also analyze three extensions of our benchmark model. In the first extension we have assumed that our scholar derives no disutility from physical activity. Such a set up allows for a pronounced shift in the longevity as the optimal time spent on physical activity increases, but at the expense of less papers being written. Since the health status will not only determine the longevity but may also affect cognitive and social outcomes, we have introduced two additional alternative specifications of our model framework. A model in which health is multiplicatively linked to the utility gained out of social and cognitive activities and another model in which the efficiency of cognitive activities to produce papers is reduced if health deteriorates. Our results indicate that these indirect effects of the health status on utility (case a) and on the efficiency of producing papers (case b) imply that in order to obtain the same value of the longevity as in the benchmark case the valuation of the stock of knowledge left to future generations needs to be reduced (case a) and alternatively increased (case b).

The results obtained in this paper assume that the time spent on each activity is mutually exclusive. Further possible extensions of the model set up can allow for the fact that either the time spent on physical activity can contribute to the time spent on social activities (e.g. exercising together with friends), or the time spent on physical activity can contribute to the time spent on cognitive activities (e.g. gaining additional innovative ideas during physical activity). Alternatively, we may also allow for interactions of the time spent on one activity to increase the efficiency of the time spent on other activities. Thus, our model should be considered a first attempt to gain more insight into the mutual feedback between an active retirement and longevity.

## Footnotes

- 1.
The value of

*T*denotes the total minutes per day net of sleep and other leisure activities. - 2.
- 3.
Feichtinger et al. (2019) assume that the salvage-value function of a scientist, who cares both about the stock of knowledge and about reputation, is \(\kappa _1K(\alpha )+\kappa _2 R(\alpha )\).

- 4.
Indeed, it can be shown that for \(\phi _c=\phi _s=1\) and if our scholar splits his time equally among the three different activities, the disutility from physical activity will be equal to the utility gained from social or cognitive activities; i.e. \(0.015\frac{(T/3)^{1+0.1}}{1+0.1}=\log (T/3)\).

- 5.
Given that \(D(\alpha )=\overline{D}\) and \(\mathcal {D}=D_x\) we have that \((\mathcal {D}/D(\alpha ))^\varepsilon =(0.106/0.22)^{0.5}\approx 0.75\), which is approximately 25% lower than the ratio \((\mathcal {D}/D(x))^\varepsilon \) at age

*x*.

## Notes

### Acknowledgements

Open access funding provided by TU Wien (TUW). We like to thank Mrs. Rivic for her research assistance and Gustav Feichtinger, Dieter Grass, Vladimir Veliov, Stefan Wrzaczek and participants at the symposium on occasion of the award of the Wissenschaftpreis der Österreichischen Forschungsgemeinschaft for useful comments and suggestions.

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