2. Data and method

We use the extensive register data from Statistics Denmark that tracks each individual in the Danish population through a Central Person Register number. We consider all individuals above age 49 in the time period 1982–2019. The individual-specific data contain information on age, gender, year, time of death, and marital and cohabiting status. We define seven different marital states: married, widowed, divorced, never married, cohabiting-widowed, cohabiting-divorced, cohabiting-never married. Individuals are tracked over time as their marital status changes. This is one of the many advantages of using the Danish register data: it provides detailed information throughout an individual's lifespan, and not just, e.g., at the time of death. Thus, we allow individuals to move between the different states until two years prior to death. This assumption avoids flows toward the single groups in the year of death. The importance of tracking individuals over time and accounting for transitions between marital states rather than focusing on one stage in life (death) is discussed in Robards et al. (Reference Robards, Evandrou, Falkingham and Vlachantoni2012). Brown and Wright (Reference Brown and Wright2017) also highlight that increases in divorce rates at older ages make the static marital status as a proxy for longevity risk less appropriate.

A cohabiting couple is defined using the following definition by Statistics Denmark: (1) they live at the same address, (2) they are of opposite sex, (3) the age difference is less than 15 years, (4) they are not closely related, (5) no other adults (above age 18) live at the address. If two individuals who live together have shared children living at the same address, these individuals are also characterized as a cohabiting couple (regardless of the criteria listed above).

Table 1 shows the yearly average, minimum, and maximum counts of exposures and deaths for the period 1982–2019 for the entire Danish population above age 49 for females and males. Here, exposures refer to the number of people at risk of dying in a particular marital-gender subgroup. Each year, we consider on average 1,245,213 females with 27,120 deaths and 945,323 males with 26,297 deaths. The largest group consists of married individuals for both females and males. The fact that females are on average living longer than males causes the number in the widowed female group to be higher: it is almost as large as the married female group. Moreover, this effect is also driven by the age difference for married couples in which females are typically the youngest.

Table 1. Yearly average, minimum, and maximum exposure and death counts for the period 1982–2019 for the total Danish population above age 49

In order to visualize demographic changes over time, Figure 1 plots the percentage share of married, cohabiting, and single individuals of the entire Danish population above age 49. Since the percentage share of cohabiting individuals is very low compared to married and single individuals, we have plotted the married (blue) and single (green) series on the left axis, while the cohabiting series (red) is plotted on the right axis. The figure shows that there is a small but consistent movement from the married to the cohabitation state (especially after 2008), while the single group is relatively stable over time (percentage share around 35%).

Figure 1. Percentage share of married, cohabiting, and single individuals in the total Danish population above age 49. Source: Statistics Denmark–StatBank.dk/FAM100N.

We group all individuals into 5-year age intervals to avoid few deaths at older ages and top-coded the data at age 95. Thus, we define the age intervals $a_{x}\in \left\{50\hbox {-}54,\; 55\hbox {-}59,\; \ldots ,\; 95 + \right\}$ where x refers to a specific age at the lower bound of an age interval. For each age interval, a _x, and year, t ∈ {1982, 1983, … , 2019}, we calculate marital-specific death rates:

(1)$$m^{i}\left(t,\; a_{x}\right) = {d^{i}\left(t,\; a_{x}\right)\over E^{i}\left(t,\; a_{x}\right)},\; $$

for i = {married, widowed, divorced, never married, cohabiting-widowed, cohabiting-divorced, cohabiting-never married}. The number of deaths in group i is denoted by d ⁱ(t, a _x) and E ⁱ(t, a _x) denotes the exposure for group i. Note that, when disaggregating individuals into seven marital groups for each gender, the cohabitation classifications display very few deaths for some years. Consequently, the corresponding death rates and life expectancies become noisy. In the Internet Appendix IA.2, we show life expectancies based on death rates that are smoothed over the year dimension using P-splines [Eilers and Marx (Reference Eilers and Marx1996)].Footnote ¹

The purpose of this paper is to investigate the effect of marital status on the remaining lifetime (i.e., life expectancy) of an individual at a given age, x. Thus, we calculate period life expectancies for each of the seven marital groups using standard life table techniques [see Preston et al. (Reference Preston, Heuveline and Guillot2001)]. The period life expectancy at age x is a recursive function, f, of the age interval death rates in (1) at all age intervals above age x:

(2)$$e^{i}\left(t,\; x\right) = f\left(m^{i}\left(t,\; a_{x}\right),\; m^{i}\left(t,\; a_{x + h}\right),\; m^{i}\left(t,\; a_{x + 2h}\right)...,\; m^{i}\left(t,\; a_{\bar{x}}\right)\right).$$

Note that here h = 5 is the width (in years) of the age intervals and $\bar {x} = 95$ is the lower bound of the last age interval. We assume that deaths occur half way through the age interval. For more details, we refer to Preston et al. (Reference Preston, Heuveline and Guillot2001), pages 38–51. Life expectancy calculations are based on artificial cohorts, since we compute period life expectancies (i.e., using all age-specific death rates in a given year). Thus, if an individual changes marital group from one year to the next, he/she will contribute to the life expectancy calculation of two different martial groups in two different years. In this paper, we calculate period life expectancies at ages 50 and 70.Footnote ²

Note that we calculate period life expectancies as opposed to cohort life expectancies. As discussed in Pitacco et al. (Reference Pitacco, Denuit, Haberman and Olivieri2009), period life expectancies are based on a set of age-specific death rates for a given year, and therefore do not allow for future changes in mortality. On the contrary, cohort life expectancies allow for changes in mortality in later years, since these are based on a set of age-specific death rates of a given cohort. However, this advantage comes at the cost of estimation and projection errors: in order to get cohort life expectancies in later periods, one needs to rely on projections of future death rates for a given cohort [see Pitacco et al. (Reference Pitacco, Denuit, Haberman and Olivieri2009), for details]. Using mortality data for the US and Sweden, Goldstein and Wachter (Reference Goldstein and Wachter2006) show that cohort life expectancies are consistently higher compared to period life expectancies. Using Danish data, Alvarez et al. (Reference Alvarez, Kallestrup-Lamb and Kjærgaard2021) look at the indexation of the retirement age based on life expectancies. They confirm the result that cohort life expectancies are higher compared to period life expectancies, but argue that the size of the gap between the two methods is not overwhelming.

Figure 2 plots gender-specific life expectancies at ages 50 and 70 for married, singles, and cohabitors. The figure confirms results previously found in the literature: married and cohabiting individuals live longer lives compared to singles with a gap in life expectancies of 2.5–7.5 years in 2019. This difference is more pronounced for men than women: the difference in gap size between men and women is up to 3 years in 2019. In this paper, we elaborate on these findings by investigating how different types of living arrangements affect the life expectancy patterns, and whether these effects are statistically significant.

Figure 2. Life expectancies at age 50 and 70 for married, singles, and cohabitors. (a) Female LE at age 50. (b) Male LE at age 50. (c) Female LE at age 70. (d) Male LE at age 70. Note: Life expectancies are based on 5-year age interval death rates.

2.1 Welch's t-test

We use Welch's t-test [Welch (Reference Welch1947)] to formally test whether the differences in life expectancy among various living arrangements are significant. In general, Welch's t-test tests whether two groups have equal means while allowing for unequal sample sizes and variances. Alternatively, we could have performed a classical ANOVA test [Fisher (Reference Fisher1925)] of equality in means between groups. However, ANOVA assumes that the sample sizes and variances in all groups are equal. The constant variance assumption is unlikely to be fulfilled in our case, since we expect the variances of life expectancies to differ across the different living arrangements. In particular, we will see in the figures below that the cohabitation groups have higher standard deviations. A similar argument holds for the sample size.

Booth et al. (Reference Booth, Hyndman, Tickle and De Jong2006) perform single and multiple t-tests (ANOVA) to determine whether several distinct methods for forecasting life expectancies are equal to the actual, realized life expectancies. They show that a direct translation of significant differences in mortality rates to significant differences in expected lifetime do not exist. This substantiates our choice of testing for significant differences in life expectancies.

The test statistic between the life expectancies of group i and j is defined as

(3)$$t_{x, i, j} = {\bar{e}^{i}\left(x\right)- \bar{e}^{\,j}\left(x\right)\over \sqrt{s^2_{\bar{e}^{i}\left(x\right)} + s^2_{\bar{e}^{\,j}\left(x\right)}}},\; $$

where $\bar {e}^{i}( x)$ is the life expectancy sample mean over years and $s^2_{\bar {e}^{i}( x) }$ is the sample standard error. The latter is defined by the sample standard deviation, $s_{e^{i}( x) }$, divided by the square root of the sample size, N _i: $s_{\bar {e}^{i}( x) } = s_{e^{i}( x) }/\sqrt {N_i}$. Here N _i = 2019 − 1982 + 1 = 38 for all groups i. We perform two-tailed tests for all combinations of married and cohabiting groups (as first term i) versus all single groups (as second term j)Footnote ³ for different ages using actual death rates for life expectancy calculations.

4. Case study: long-term care

Most developed countries are experiencing population aging [United Nations (2019)], as life expectancy is increasing. The risk of dying during post-retirement ages has trended downwards [Rau et al. (Reference Rau, Soroko, Jasilionis and Vaupel2008); Zuo et al. (Reference Zuo, Jiang, Guo, Feldman and Tuljapurkar2018)], such that individuals from more recent cohorts spend a longer time in retirement compared to previous ones [Sanderson and Scherbov (Reference Sanderson and Scherbov2010, Reference Sanderson and Scherbov2017)]. In Denmark, like most other European countries, the old-age dependency ratioFootnote ⁵ is currently at around 30% and forecasted to increase further to 50% in 2070.Footnote ⁶ In line with this development, the costs of national healthcare expenditures are continuously rising [Martin et al. (Reference Martin, Hartman, Lassman and Catlin2021)]. In 2011, 11% of Danish GDP was spent on the healthcare system while 22% of the total healthcare expenditures was spent on people in the last 3 years of life [see French et al. (Reference French, McCauley, Aragon, Bakx, Chalkley, Chen, Christensen, Chuang, Côté-Sergent, De Nardi, Fan, Échevin, Geoffard, Gastaldi-Ménager, Gørtz, Ibuka, Jones, Kallestrup-Lamb, Karlsson, Klein, de Lagasnerie, Michaud, O'Donnell, Rice, Skinner, van Doorslaer, Ziebarth and Kelly2017)]. Thus, there is a growing concern over the impact on future sustainability of healthcare systems.

Although the population aged 65 and above consume healthcare resources at a much higher rate, these older individuals also hold a potential to limit the expenditure levels as married spouses are more likely to serve as informal caregivers [see Wachterman and Sommers (Reference Wachterman and Sommers2006)]. However, an increasing number of individuals are misclassified as singles even though they are in fact living with a spouse, see Figure 1. Scarce information about unmarried cohabiting couples has prevented researchers from investigating the support function of cohabitation. Moreover, given the protective effect of both marriage and cohabitation in terms of life expectancy shown above, there are reasons to suspect that healthcare utilization and thus healthcare expenditures too will differ by marital status. As the first in the literature, we show how end-of-life long-term care (LTC) expenditures vary across different types of singles including cohabitation. In particular, we estimate the mean annual LTC spending that occurs in the final year of life. These costs typically increase substantially close to death as the majority of older people experience declining health and progressive disability at the end of their lives, see Lunney et al. (Reference Lunney, Lynn, Foley, Lipson and Guralnik2003).

Note that sharing of resources among spouses (like health insurance benefits or finances) is not a concern in our study as the Danish healthcare system is characterized as a Beveridgian model. Thus, taxes finance healthcare costs, and citizenship enables free and equal access to universal health insuranceFootnote ⁷. Moreover, this limits the effect of bequest motives on the distribution of LTC utilization and expenditures between marital groups. In other countries, such as the US, bequests play a significant role in the demand for long-term care [see Groneck (Reference Groneck2017)].

This study uses unique register data on long-term care supplied by Statistics Denmark for the years 2011 and 2012. The registers contain individual-level information for the entire Danish population allowing us to link anonymized data at the individual level on demographics such as age, gender, and marital status to information on various forms of long-term care. On home care and home nurses, we have access to number of hours supplied on a daily basis from the municipality. These services are delivered either in people's own homes or in residential housing for the elderly. We group these two categories into one, labeled home help. Moreover, we have access to a register that identifies whether individuals were residents in nursing homes or residential homes for the elderly during the year. We label this category retirement homes.

The registers do not contain specific information on expenditures. Instead we assess total expenditures on LTC by combining information from OECD Health Data (categories HC3.1 and HC3.2) with expenditure information from the Danish Ministry of Health (2009, 2010) and the Danish Economic Council (2009) as in French et al. (Reference French, McCauley, Aragon, Bakx, Chalkley, Chen, Christensen, Chuang, Côté-Sergent, De Nardi, Fan, Échevin, Geoffard, Gastaldi-Ménager, Gørtz, Ibuka, Jones, Kallestrup-Lamb, Karlsson, Klein, de Lagasnerie, Michaud, O'Donnell, Rice, Skinner, van Doorslaer, Ziebarth and Kelly2017). Thus, we allocated these imputed costs to individuals based on the micro data on nursing home incidence, and hours of use of home care and home nurses, respectively.Footnote ⁸ All numbers are inflated to 2014 price levels. We consider our imputation of LTC at the individual level to be the best possible given the information available. To determine health expenditures as individuals approach the time of death, we acknowledge that medical spending in a given year of death does not represent a full year of expenses.Footnote ⁹ Therefore, we exploit the daily nature of the register data to track the 2012 death cohort over their last 12 months prior to death (covering the years 2011 and 2012) and calculate their LTC costs. We observe 47,794 decedents above the age of 49 in 2012.

During the last couple of decades, it has been a deliberate policy of the Danish government to keep elderly individuals in their own homes for as long as possible [Rostgaard et al. (Reference Rostgaard, Glendinning, Gori, Kröger, Osterle, Szebehely, Thoebald, Timonen and Vabo2011)] and when necessary provide support from home careers, home nurses, etc. Nursing homes are more costly and reserved for the very fragile and ill elderly [Nielsen et al. (Reference Nielsen, Bekker-Jeppesen, Almer and Andreasen2016)]. Therefore, only a small fraction of a cohort will be residents in such a home. Figure 5 shows the average cost spent on home help or retirement homes out of all individuals who passed away in 2012. Thus, within each marital group, the total costs on home help and retirement homes are both divided by the total number of individuals dying in that group. By conditioning on those who died we are able to infer which martial group had the highest level of consumption of a given expenditure type. Note that the figure does not display the average cost of e.g. placing an individual in a retirement home. This would be uninformative for our comparison as the average costs conditional on residing in a retirement home are unlikely to differ significantly between marital states.

Figure 5. Average 1-year medical spendings on home help and retirement homes per individual for all individuals dying in 2012. Note: Numbers are reported in US Dollars and 2014 price levels. The average age in each group is from left to right for females 73, 76, 68, 64, 86, 75, 77 and for males 76, 79, 70, 64, 84, 70, 69.

The average 1-year medical spendings on LTC per individual for the 2012 death cohort are displayed in Figure 5. We find that cohabiting individuals have lower LTC cost in the last 12 months of their lives compared to non-cohabiting singles. The gray bar represents expenditures to home help and the blue bar to retirement homes. Thus, singles that are cohabiting display a spending pattern that closely mimic the behavior of married individuals. Never married individuals that are cohabiting show the lowest spendings across all types of marital groups whereas widowed individuals display the highest level of spendings. Among the married and cohabiting individuals, cohabiting-widows stand out as the home help expenditures are much higher compared to married and cohabiting-divorcees and never married. This indicates that cohabiting-widows do not to the same extent receive informal care from their cohabiting male spouses.

Singles spend on average two and a half times as much on long-term care compared to married and cohabiting individuals. In particular, we find that singles on average spend almost twice as much on home help and four times as much on retirement homes. It is clear from Figure 5 that women to a larger degree serve as informal caregivers as married and cohabiting males have lower average costs. Finally, we see that single women have higher average LTC costs compared to men reflecting the higher life expectancies of females. We apply the Welch's t-test to determine whether expenditures are in fact significantly different. We confirm that this indeed is the case and the results can be found in Appendix A, Tables A.1, A.2, and A.3.

The demand and need for care will increase dramatically over the next decades as a result of changing population demographics [European Commission (2020)]. This will likely change the availability of informal caregivers in the future. Understanding variations in long-term care expenditures between and within different types of singles are important when evaluating the economics of end-of-life care. Our results strongly suggest that not only married individuals but also singles that are cohabiting have the potential to limit the expenditure levels due to the level of informal care they provide.