The effect of co-payments in Long Term Care on the distribution of income and risk. First Draft.

Transcription

1 The effect of co-payments in Long Term Care on the distribution of income and risk. First Draft. Bram Wouterse 1,2,5, Arjen Hussem 1,3,5, and Albert Wong 4,5 1 CPB Netherlands Bureau for Economic Policy Analysis, The Hague, The Netherlands 2 Erasmus School of Health Policy & Management, Erasmus University Rotterdam, Rotterdam, The Netherlands 3 PGGM, Zeist, The Netherlands 4 National Institute for Public Health and the Environment, Bilthoven, The Netherlands 5 Network for Studies on Pensions, Ageing and Retirement, Tilburg, The Netherlands January 10, 2018 Abstract Population aging leads to concerns about the financial sustainability of collective long term care insurance systems. One way to keep public spending in check is by increasing the role of co-payments. An interesting feature of the copayments that have been introduced in the Netherlands is that they are incomeand wealth dependent. This dependency allows the fine-tuning of effects across income groups, but can also distort consumption decisions of the elderly. Modeling long term care expenditures over the lifecycle is challenging because of their very uneven distribution, with a small proportion of elderly experiencing very high costs. We use a very flexible semi-parametric nearest-neighbor approach to estimate lifecycle paths of long term care spending. We apply this approach to an extensive administrative data set for the entire Dutch elderly population. The estimated paths are then used as inputs in a stochastic lifecycle decision model for retirees. We analyze the effects of different co-payment schemes on the distribution of consumption and risk across income groups. We also analyze the effects of the schemes on the optimal annuitization share of pension wealth. We find that, compared to a uniform co-payment independent of income and wealth, income- 1

2 and especially wealth-dependent co-payments lead to much lower welfare costs for group with low financial means, at the same time, the welfare costs of the groups with the lowest means increase only slightly. Including bequest motives and health-state dependent utility in the model leads to lower estimated welfare losses due to co-payments (compared to full insurance). Co-payments that only depend on pension income, and not on financial wealth, distort the annuitization decision as they decrease the optimal share of pension wealth that should be annuitized. 1 Introduction The aging of the population, and the resulting increase in the number of elderly with disabilities, has put the provision and financing of long term care (LTC) at the forefront of the policy debate. While some countries are only now setting up forms of public LTC insurance, countries such as the Netherlands, with a strong tradition of extensive public LTC insurance, are seeking to increase private financing to keep public spending in check (Colombo and Mercier, 2012). One way to do so, is by increasing the level of co-payments. An interesting feature of the Dutch system is that the co-payments are income- and wealth-dependent. This dependency allows the fine-tuning of the financial impact of co-payments across income- and wealth groups, but it also distorts saving and annuitization decisions of individuals. Co-payments affect the distribution of spendable income across pensioners. Elderly with low incomes use more LTC, on average, than those with high incomes (Bakx et al., 2016). Public insurance of LTC thus redistributes income from high to low income groups, while increasing the role of co-payments limits this redistribution. Just like, for instance, income inequality (Aaberge and Mogstad, 2015), these redistributions are ideally measured over the whole lifecycle: the lifetime distribution of LTC costs is much more equal than that in one particular year. This means that a cross-sectional analysis might overestimate the distributional effects (Hussem et al., 2016). Further, an analysis of the effect of co-payments across income groups should not only include the average (ex-post) redistribution of payments and benefits, but also the effects on risk across income groups. Even when, let s say, the average effect of co-payments on consumption is low, there might still be a chance that an older individual is confronted with very high costs. As shown by McClellan and Skinner (2006), in their analysis of lifetime redistributions across income groups in Medicare, insurance against risk is indeed an important part of the value of public health insurance, especially for low income groups. The insurance value is specifically important in LTC, where buying insurance on the private market is difficult or not possible at all (Brown and Finkelstein, 2007). Co-payments also affect saving behavior. The introduction of high co-payments means that the elderly are confronted with the risk of potentially very substantial costs during their remaining life, for which they need to hold liquid wealth. Indeed, studies 2

3 for the U.S. find that precautionary savings for LTC costs can, partially, explain why the elderly hold relatively large amounts of financial wealth, even at high ages, and why they choose not to fully annuitize this wealth (De Nardi et al., 2010; Ameriks et al., 2011; Peijnenburg et al., 2017). Income- and wealth-dependent co-payments make the effect on saving behavior more ambiguous: the elderly still want to hold liquid assets to pay for LTC costs and smooth consumption, while at the same time the co-payments are an implicit tax on wealth, reducing the desire to hold wealth at high ages. When the implicit tax on (pension) income differs from that on wealth, this also affects the annuitization decision. An important empirical challenge in assessing the effects of co-payments in LTC is the modeling of the lifecycle distribution of LTC costs. This is hard to do parametrically: a large part of the population does not have any LTC costs at all, while a small group of individuals experiences very high costs persisting over many years. Existing approaches use autoregressive models (De Nardi et al., 2010; French and Jones, 2004) or Markov models (Ameriks et al., 2011) to estimate time dynamics in LTC costs. However, these models require a variety of assumptions that most often cannot be justified on the basis of the data alone (Wong et al., 2016). A recent study by Hurd et al. (2017) for the U.S. shows the relevance of a non-parametric approach. They estimate lifetime LTC use and out-of-pocket spending using the HRS. They compare their nonparametric method, based on matching, to a more standard parametric Markov model. They find that the risk (the chances of extreme use or costs) as estimated non parametrically is substantially greater than the risk as estimated by the parametric model. The use of non-parametric approaches to model lifecycle distributions is not only relevant for LTC costs, but for instance also for the complex dynamics of income over the lifecycle (De Nardi et al., 2016). In this paper, we use a semi parametric nearest neighbor approach developed by Wong et al. (2016) and Hussem et al. (2016) to estimate the lifetime distribution of LTC costs. The main advantages of this approach are its flexibility and the ability to use it on short periods of panel data. We apply the nearest neighbor algorithm to estimate 20,000 synthetic lifecycle paths using a rich administrative data that includes information on LTC spending, household status, income, and wealth for the entire Dutch elderly population. The estimated paths serve as inputs in a stochastic lifecycle decision model for retirees. This model determines optimal consumption and saving behavior of elderly for different levels of initial wealth and pensions, taking into account their financial risk under different co-payment regimes for LTC. We use a simulation based algorithm developed by Koijen et al. (2010) to solve the model. The use of health dynamics in a lifecycle model adds additional challenges: the health model should be sophisticated enough to capture the complex dynamics in health, while at the same time be parsimonious enough so that its use in a structural life-cycle model is computationally manageable (De Nardi et al., 2017). The combination of the synthetic lifecycle paths with the approach of Koijen et al. (2010) fulfills both these requirements,. We analyze the effects of different forms of income- and wealth-dependent co- 3

4 payments that are implemented or considered by policy makers in the Netherlands. Our model allows us to analyze effects over the entire lifetime after the pension age and to incorporate the value of insurance. We focus on the effects on (average) consumption and welfare (certainty equivalent consumption) across income- and wealth-groups. We also assess the effects of the co-payment schemes on the optimal annuitization share of pension wealth. Dutch elderly s financial wealth consists largely of annuitized pension wealth. Therefore, the question whether a larger role for co-payments in LTC requires a more flexible pension payout system is of specific policy relevance. 2 The Dutch Long Term Care system Unlike many other developed countries, the Netherlands has a long tradition of providing collective LTC through social insurance. It also has one of the most extensive collective LTC arrangements in the world (Colombo and Mercier, 2012). Almost all health spending is financed through collective social insurance or taxes. Since the LTC reform of 2015 (Maarse and Jeurissen, 2016), LTC is financed through two programs. Relatively light forms of care, such as personal assistance, are financed through the Social Support Act (WMO). The provision of this type of care is a responsibility of municipalities. They get a financial contribution out of the general means of the national government, depending on the composition of their population. Intensive forms of care for patients who are in permanent need of care (often in an institutional setting) are financed through a social insurance called the Long Term Care Act (WLZ). This type of care is paid for by a social insurance. The premium is included in the income tax and is a fixed percentage of taxable income in the first and second income brackets. In this paper, we use data from before the reform. Before 2015, both home care and institutional care was financed through a collective insurance called the AWBZ. The premium for the AWBZ was collected through the income tax and is a percentage of taxable income (including pension income) in the first and second income brackets. A small part of the costs were covered by out-of-pocket payments. In the current system, out-of-pocket payments play a larger role. They depend on the spending power, which is a measure of the income and wealth of the individual, and differ according to the type of care and living situation. They can vary for home care from 247 euros per year plus 15 percent of spending power minus a euros threshold to the maximum of the actual costs. For institutional care these payments can be 12.5 percent of spending power for people with a partner living at home to 75 percent of net income with a maximum of 26,983 euros per year (all amounts 2014). The spending power is defined as gross income plus 12 percent of financial wealth. For wealth a threshold is used of 21,139 euros which can go up to 49,123 for people with low incomes. 4

5 3 Long term care spending over the lifecycle 3.1 Source data We use administrative data on LTC use from the Dutch Central Administrative Office (CAK). These data cover the period The data include information on all formal LTC use in the Netherlands. The data contain information on the type of care (institutional care, nursing home care, personal home care, and support) and the amount of care used (in days for institutional care, and in hours for home care). We derive costs of LTC from use in hours/days in the CAK database and the tariffs provided by the Dutch Health Authority (NZA) for extramural care and derived from the CAK and CVZ annual reports for intramural care. The LTC data is combined using a unique identification number with other datasets in the special data environment of Statistics Netherlands. The linkage to the Dutch Municipal Register provides basic information on everyone enlisted in a Dutch municipality. From this register we obtain date of death, age, sex and marital status. Additionally, we link the data to the Death Causes Registry for the entire Dutch population, and administrative data and survey data on all income sources from Statistics Netherlands (Regionaal InkomensonderzoekRIO), and marital status from Statistics Netherlands (Gehuwdheidsbestand, VRLHUWELIJKS-GESCHIEDENISBUS). 3.2 The nearest neighbor algorithm As inputs for our model we use synthetic lifecycle paths estimated by Hussem et al. (2016). These lifecycle paths of LTC costs have been estimated using a nearest neighbor resampling method. Although there are many specific implementations, the idea behind nearest neighbor matching (NNM) is that we want to match an observation from one group (for instance a treatment group) to the most similar observation from another group (for instance the control group). NNM uses a distance metric to determine, based on the covariate values, which observation form the other group is the nearest. Some of the first implementations of NNM in a time series or panel context are by Farmer and Sidorowich (1987) and Hsieh (1991). The LTC paths we use have been estimated using the approach developed by Wong et al. (2016) who have implemented a nearest neighbor resampling method to estimate lifecycle paths of curative care costs. The basic idea of the NNM algorithm is that we want to simulate N individual lifecycle realizations of LTC spending. Each simulated lifecycle will consist of an age series Z i = {Za=0, i Za=1, i..., Za=A i i }. Za i is a vector containing LTC spending and income as the other variable of interest of individual i at age a. a = 0 denotes the starting age and A i is the age of death. To correct for wage and LTC inflation in the period and the difference in scales we use the observed percentiles for income and LTC costs respectively for the matching process. Our data is a relatively short panel containing observed values of the variables of interest Y j a,t for individuals j = 1,.., J 5

6 over time periods t = 1,.., T. The algorithm works as follows. Suppose we already have a simulated lifecycle path for an individual up to age A: Z i = {Z0, i Z1, i.., ZA i }. To extend this lifecycle path to age A + 1 we consider all individuals in our data who have age A + 1 in period T. We pick the individual who s life history over the last p age years Y j = {Y j A p+1,t p,..., Y j A,T 1 } is most similar to {Zi A p+1,..., Zi A }. Note that, because we want to extend the lifecycle by one period, and the time length of the panel is T, we can use a maximum age lag p of T 1 years. When we have picked an individual j, we use YA+1,T J as our simulated realization of Zi A+1. Then, to obtain a realization for age A + 2 we can repeat the procedure using all individuals in the data with age A + 2 at time T, matching on the life history over ages A p +2 to A +p+1. This procedure is repeated until i is matched to an individual who dies in period T. The time periods t = 1,.., T and the number of lags p will generally depend on the data at hand. When the available panel data is long enough, the number of lags can be determined by comparing model performance across different choices of p. See Wong et al. (2016) for examples. To initialize the algorithm, we use all individuals with age a = 0 at time T. For these individuals we have data on Y over T 1 ages before the starting age a = 0. 1 We include the information on the last p 1 ages in the simulated lifecycle path, so we start with Z i = {Z p+1, i, Z0}. i To match a simulated lifecycle path to an observation from the data we use k- nearest neighbor matching. We measure the distance between two p-long blocks z and y using a distance measure d(z, y). We use the Euclidean measure. This measure is defined as d(z, y) = (y z) T (y z), (1) Out of the k-nearest neighbors, one neighbor is randomly drawn. We use the simulated paths as described in Hussem et al. (2016) with k = 2. They used data for three years ( ), and therefore have two lags (p = 2). The data is stratified by sex, age, and household status (e.g. men are only matched to other men) and matched on income and LTC expenditures. They simulated 10,000 paths for women and 10,000 for men. These paths start at birth, but we start the simulation exercise at the pension age which we assume to be 70 years and thus select those individuals who are alive at age 70. A more detailed description of the matching procedure can be found in Hussem et al. (2016). In line with literature and to simplify the individual s utility function we focus on singles (people living on their own). Therefore the expenditures of the partner are not taken into account and everyone is assumed to be single. As over 75% of lifetime LTC expenditures is institutional care and this type of care is distributed very unevenly we do not take home care (extramural care) into account. 1 Obviously no information is available when the simulation starts at the age of 0. 6

7 3.3 Including information on wealth The lifecycle paths that we currently use do not include information on wealth. Therefore, we include financial wealth (excluding the value of the own house) ex-post (similarly to Hussem et al. (2017)). We use the relationship between income and financial wealth for all Dutch single 70 year olds in We group these individuals in income (inq) and wealth (wq) quintiles. This gives us 25 combinations of income and wealth groups. For each of these combinations we calculate the average financial wealth. Each of these groups is given a weight w inq,wq based on the relative size of that combination of wealth- and income group in the 70-year old population. We also group each individual in the lifecycle paths to an income quintile inq, based on his or her income at age 70. Then we can assign initial wealth at 70 using the average wealth by income- and wealth groups we have estimated for the whole Dutch population of 70 year olds. We use each individual in the lifecycle paths five times: each time we assign to the individual an initial wealth level that is equal to average wealth in one of the wealth quintiles. We weight each replicated observation by the weight of that particular combination of income and wealth, w inq,wq, in the total Dutch population 3.4 Estimation results As described in Hussem et al. (2016) the resulting lifecycles can be interpreted as a cohort of newborns in 2006 and how they would accumulate income and LTC costs during an entire lifetime. The assumptions are that the levels of both transition probabilities of income and costs remain constant at those observed in the calendar year 2006 and the state of the long-term care system in 2006 stays the same. The method replicates the true transitional probabilities (e.g. death rate or probability of becoming a LTC-user), and the distribution of income and LTC from the source data conditional on the variables considered. As the population composition (e.g. by age and household type) in the synthetic lifecycle paths differs from the source data the total annualized amounts in the lifecycle paths are not equal to those amounts in the source data. The lifecycle paths are representative for a stationary population based on the behavior and institutions of the period. Future changes in mortality and LTC use are therefore not taken into account. Figure 1 shows that 1.5% of the population has LTC expenditures at the age of 70. Of this population 86% has died at the age of 95 and 4% has LTC expenditures. At the age of 90 almost 10% of the original population has long term care costs. Almost half of these individuals has costs higher than 25,000 euros per year. Table 1 shows the average annual and lifetime LTC costs in the lifecycle paths used. As many people might need LTC in some point of their life, the standard deviation of the amounts are much lower over lifetime than per year. The unevenness of the lifetime distribution is also shown in Figure 2. 60% of the individuals have no LTC costs at all, 2 Data obtained from CBS 7

8 Dead Costs > 25k euro Costs > 10k and < 25k Costs < 10k euro No costs Age Figure 1: Population composition and distribution of annual LTC costs by age. for 93% it remains under 200,000 euros over the lifetime. For only a few it can go up to 1 million euros. Figure 3 shows lifetime LTC costs, lifetime income and initial wealth across spending power quintiles. Spending power is defined as the total, discounted, lifetime consumption (without LTC costs) 3. The figure shows that higher spending power quintiles have, naturally, both more remaining lifetime income and higher initial financial wealth. Lower spending power quintiles have higher LTC costs, which reflects their lower average health. 4 As financial assets are not included in the matching procedure to construct the lifecycles, the differences in costs between spending power quintiles 3 This is defined as the certainty equivalent consumption in the baseline (without any co-payment). See Section 4.1 for the definition of certainty equivalent consumption. 4 As the Dutch system is based on social insurance with low co-payment rates, differences in costs reflect differences in need for LTC and not differences in willingness or ability to pay for LTC. 8

9 lifetime costs (x 1000 euro) Figure 2: Distribution of lifetime LTC costs. Table 1: LTC costs in the lifecycle paths, from age 70 onwards average std/average per year 2, lifetime 42, is purely driven by the relationship between income and LTC costs. 9

10 x 1000 euro LTC costs Income Assets Spending Power Quintile Figure 3: The relationship between LTC, financial wealth, and income (present value) at age 70, per quintile of spending power. 10

11 4 A model of lifecycle consumption after retirement 4.1 The model The estimated lifecycle paths provide a semi-parametric distribution function of LTC costs and mortality. We implement a standard lifecycle model with rational and forward looking individuals to model consumption and saving behavior given this distribution. Mortality risk and the development of LTC spending over life are based on the lifecycle paths. Consumption and saving behavior, conditional on initial wealth, are determined by the lifecycle model. To be able to use the semi-parametric lifecycle paths in the optimization problem, we implement a simulation-based maximization algorithm developed by Koijen et al. (2010). The basic model We model the consumption and savings decisions of individuals after retirement. An individual starts at the pension age, t = 0, with initial wealth W 0. He uses this wealth to finance consumption over the remaining time periods t 1,..., T. The individual faces uncertainty about the duration of remaining life and the amount of LTC copayments. We assume that the individual only derives utility out of consumption (later we will introduce a bequest motive). The individual wants to maximize his expected utility over remaining lifetime. With a time-separable utility function the individual s maximization problem then is: [ T ( )] t E(V 0 ) = E β t u(c t ) p s, (2) t=0 withp s the probability of surviving period s, and β the discount factor. Each period, the individual has to choose the amount of his wealth W t he wants to consume now (c t ), and the amount he wants to save for later (m t ). The individual is also faced with co-payments for LTC costs h t which function as exogenous shocks in wealth. He faces the following annual budget constraint: s=0 c t + m t + h t = W t. (3) We impose the borrowing constraint W t 0.The timing is such that first h t has to be paid, and then the individual decides how to divide his remaining wealth between c t and m t. We treat the level of private LTC spending, h t, as given: the individual does not weight utility gained from h t against utility from c t, but instead h t is an exogenous shock in W t. The utility function is defined as a standard CRRA function: u(c t ) = c1 γ t 1 γ. (4) 11

12 This implies that individuals want to smooth consumption evenly over the lifecycle. Wealth growths with the risk free interest rate r 1, so that Extensions W t+1 = m t r. (5) We extend the model in four ways. First, we allow for (partial) annuitization. Individuals can choose to annuitize a share α of their initial wealth at the age of retirement. In that case, they will receive an actuarially fair constant annual income y α : y α = The budget constraint then becomes T t=0 αw 0 [ r t s=t s=0 p ]. (6) s c t + m t + h t = W t + y α. (7) The individual s maximization problem now not only involves choosing consumption in each period, but also deciding on the share of initial wealth he wants to annuitize. Second, we allow the level of co-payments to depend on wealth and pension income. Let hc t be the total LTC spending an individual needs in period t. This spending is exogenous. Private LTC spending, h t, is not necessarily equal to hc t, but depends on the co-payment rules set by the government. We use the following general co-payment rule: h t = min[τhc t, ν y y α + ν w W t δ, µ]. (8) This general rule allows us to emulate the Dutch co-payment system, but also to include other variants, such as a nominal co-payment independent of spending power. The government sets the parameters τ, ν y, ν w, δ, and µ. The parameter τ determines what share of total health care spending has to be paid by the individual himself. The parameters ν y and ν w determine the maximum share of income and wealth that have to be spent on co-payments. The parameter δ is a sort of deductible: a fixed amount of income that is exempted from the co-payments. There government can also set a maximum µ on annual co-payments. The way the government sets the co-payment rules affects the optimization problem of the individuals. When ν y > 0 or ν w > 0, co-payments are no longer fully exogenous since they depend on the annual savings and the annuitization share of initial wealth chosen by the individual. As a comparison to the variants with co-payments, we also include a variant where health care spending is financed out of a, income-dependent, premium π. In that case individuals will pay a contribution to the health care system, regardless of their own use, but depending on their pension income: πy α. 12

13 Third, we include a bequest motive. We assume that the individual derives utility from the level of wealth W death he leaves at time of death. We use the same bequest function as De Nardi et al. (2010): g(w death ) = θ (W death + ξ) 1 γ, (9) 1 γ where θ determines the strength of the bequest motive and ξ the curvature of the bequest function. Fourth, we allow for health state-dependent utility. The utility an individual derives from non-health care consumption could depend on his health status (disability). Finkelstein et al. (2013) find that an increase in the number of chronic diseases has a significant negative impact in the marginal utility of consumption. A priori, however, the effect of poor health could go both ways: individuals might derive less utility from things like eating out or recreation, but at the same demand for things like domestic help, wheelchairs, and stairlifts might increase (Meyer and Mok, 2009). Indeed, as pointed out by Peijnenburg et al. (2017), there is no consensus in the empirical literature on the size and even the sign of the effect. In our case, a negative effect on the marginal utility of consumption might be more likely, as institutional care in the Netherlands is relatively comprehensive and encompasses most additional consumption needs (housing, cleaning) related to disability. To include state-dependent utility, we use the following commonly used adaptation of the utility function in Equation (4) (Palumbo, 1999; De Nardi et al., 2010; Peijnenburg et al., 2017): u(c t ) = (1 κhe t ) c1 γ t 1 γ. (10) The variable he = I(hc t > 0) is a dummy indicator for poor health, which we define as an individual having any institutional LTC use in period t. The parameter κ determines the relative change in the marginal utility of consumption in poor health (he t = 1) compared to good health (he t = 0). When κ < 0, marginal utility is lower in poor health. When κ = 0, marginal utility is equal in both health states. Outcome The main outcome measure we will use to present welfare effects of different financing schemes across groups is certainty equivalent consumption: ( ) CEC = u 1 E(V 0 ) ( T t=0 βt t s=0 p ) (11) s More specifically, we will the change in CEC for a financing variant v compared to a baseline financing variant v = 1. in the baseline financing variant, all health care spending is financed out of premiums. For group g and alternative v we will show CEC g,v CEC g,1 CEC g,1, (12) 13

14 with CEC g,1 is the CEC for group g under the baseline variant. 4.2 Numerical approach The basic model The individual s maximization problem can be solved using a dynamic programming approach. In this approach, the lifecycle optimization problem is divided into smaller yearly optimization problems. The algorithm starts at the last time period T, and is then solved backwards recursively using Bellman equations. In each period the optimization problem can be written as max [E(U t ) = u(c t ) + E t [V t+1 (m t )]. (13) We solve this problem using the approach developed by Koijen et al. (2010). This approach has been applied to LTC financing for the U.S. by Peijnenburg et al. (2017). The approach combines the method of endogenous gridpoints (Carroll, 2006) with a simulation based approximation of the expected values (Brandt et al., 2005). A simulation based approach is well suited to use in combination with the lifecycle paths. Other approaches generally approximate the stochastic processes by a limited number of discrete states. Instead, the method of Koijen et al. (2010) allows us to directly use the lifecycle paths as inputs. Often, the maximization problem in Equation (13) is solved for a finite number of possible values of wealth (on a grid) at the beginning of a period W t. The solution for other values of W t is then obtained by (linear) interpolation between the gridpoints. Instead, in the endogenous gridpoints method, a grid is used for m t the amount of wealth at the end of period t after consumption and health spending. Optimal consumption c t in period t is then determined given the amount of wealth that is left at the end of t. This method avoids the need for numerical optimization to determine c t. An endogenous gridpoint for W t is determined afterwards by summing up m t, optimal consumption c t, and LTC spending h t. To see how the algorithm works, let s start in the final period T. If an individual is still alive at period T, he consumes all his remaining wealth. So optimal consumption is given by: c T = W T h T, (14) and u T = u(c T ). For period T 1, we define a fixed grid with j = 1,.., J gridpoints m j,t 1 for wealth after consumption and LTC spending. Because the wealth level after consumption in T 1 is already known, the corresponding level of consumption c T 1 is given by the first order condition: c j,t 1 = (E(βc γ T r m j,t 1 )) 1 γ. (15) This is the standard Euler condition implying that individuals want to smooth consumption evenly over remaining lifetime. 14

15 To determine E(βc γ T r m j,t 1 ) we use a simulation approach. The idea is similar to a Monte-Carlo approach: the lifecycle-paths give us a large number of random draws from the stochastic process determining mortality and LTC spending. The expected value can then be estimated by averaging over these draws. Note that for each individual (path) the realized consumption in T conditional on m T 1 is given by the fact that (if still alive) the individual will consume all the wealth he has left: (W T m j,t 1 ) = m j,t 1 and (c T m j,t 1) = m j,t 1 h T. To determine the expected value we regress these realizations of consumption at T on (a polynomial expansion) of the state variables (background characteristics and LTC spending) at time T 1. This gives E(βc γ T r m j,t 1 ) θf(x T 1 ), (16) with x t 1 a vector with the state variables in period t 1 and f() a polynomial expansion of some order. We estimate this equation using a GLM with log-link, to ensure that the estimated values are strictly positive. The expected values are then obtained by using the predictions from the regression model (conditional on the state variables), and this also provides the optimal level of consumption in period T 1 given m T 1. We have to perform this procedure for each gridpoint, and thus have to run a regression for each gridpoint. 5 Now that we have the optimal consumption levels c j,t 1 for each fixed gridpoint for wealth m j,t 1 at the end of period T 1, we can create a grid with endogenous gridpoints for wealth W j,t 1 at the beginning of period T 1. These are given by W j,t 1 = c j,t 1 + h T 1 + m j,t 1. (17) The level of initial wealth at the beginning of T 1 is determined by the level of wealth that is saved at T 2. So we now have the set-up for the iterative algorithm. Because the endogenous gridpoints W j,t 1 are not necessarily the same as the gridpoint we use for m T 2, we use linear interpolation to obtain the levels of optimal consumption in T 1 belonging to the gridpoints m j,t 2 for wealth saved at the end of period T 2 for each individual (path). This then, allows us to estimate expected optimal consumption at T 1 using the same regression as in Equation (16). This gives optimal consumption in T 2. And this in turn determines the endogenous gridpoints for W T 2. We can iteratively perform this algorithm for periods down to t = 1. In the end, we have the optimal consumption at each period for the endogenous gridpoints W 1,t,..W J,T. We have a series of (different) endogenous gridpoints and optimal consumption for each individual (path) i. Now that we have the consumption rules, we can use these to simulate consumption and saving behavior of the individuals in the lifecycle sample. We do this by assigning 5 As pointed out by Koijen et al. (2010) the endogenous grid method facilitates the use of this regression based approach. The optimal consumption can be derived analytically from the Euler equation (15) once the conditional expectation is known, so we do not need to determine this numerically. Else we would have to run for each gridpoint a (non-linear) regression at each iteration of the numerical optimization process, instead of just once for each gridpoint. 15

16 an amount of initial wealth at the start of the first period to each individual. We can then simulate forward. Extensions The inclusion of a bequest motive and state-dependent utility in the optimization procedure is relatively straightforward. The same is true for including a fixed pension income. To estimate optimal annuitization shares we run the algorithm over a grid of values for α between 0 and 1. The value of α determines which share of each individual s actual initial wealth (taken from the lifecycle data) is annuitized. Comparison of average certainty equivalent consumption, for individuals with the same initial characteristics, over the values of α then gives the optimal annuitization share. The policy variants require another adaptation to the numerical approach. Wealthdependent co-payments put an implicit tax on savings. Individuals have to include this tax when making decisions on current consumption. Specifically we adapt the Euler equation (15) by including the expected marginal implicit tax on wealth. Income- and wealth dependent co-payments also affect the optimal annuitization share of pension wealth, but this does not require any additional change in the numerical approach. 4.3 Policy variants The effects of co-payments on welfare In the Dutch system, co-payments depend on both income and wealth. We emulate the Dutch co-payment scheme in 2015 using the formula in Equation (8). In this scheme, 75 % of income and 9 % of financial wealth 6 is included in the co-payment. There is a deductible of 4,500 euros and a maximum co-payment of 27,000 euros. To study the effect of introducing or abolishing the income and/or wealth dependency of the co-payments we define the following policy variants. The parameters of the other variants are set in such a way that they finance an equal amount of aggregated LTC costs as the variant based on the current system. We focus only on the financing of the share of health care costs financed out of co-payments in the current system (which in our model is equal to 27 percent of total LTC costs). This means that in the variants with premiums instead of co-payments, these premiums only have to raise the amount of LTC costs currently financed out of co-payments. Variant 1: Uniform (additional) premium π as a fixed percentage of income for the 70+ population, no co-payments. τ = 0, ν y = 0, ν w = 0, δ = 0, and µ = % of financial wealth is added to the income definition used to calculate the co-payment. As 75 % of this income definition is included, this means that 0.12*0.75 = 9 % of financial wealth is included. 16

17 Variant 2: Actuarially fair (additional) premium π g, group specific fixed percentage of income for the 70+ population, no co-payments. τ = 0, ν y = 0, ν w = 0, δ = 0, and µ = 0. Variant 3: Income dependent co-payment of 89% of income. τ = 1, ν y =.89, ν w = 0, δ = 4500, and µ = 27, 000. Variant 4: Income and wealth dependent co-payment of 75% of (income + 12% wealth). τ = 1, ν y = 0.75, ν w = 0.09,, δ = 4500, and µ = 27, 000. Variant 5: Fixed co-payment of 35% of LTC costs τ = 0.35, ν y = 1, ν w = 1,, δ = 0, and µ = 27, 000. In all cases, co-payments do no exceed the actual LTC costs. The consumption floor is set at 7,000 euros. We assign a fixed income stream (y) to each individual path based on income at the starting age (70). To assign (not annuitized) initial wealth to individuals we use the approach described in Section 3.3. The other parameters are set as described in Table 2. We do not include bequest motives or health-state dependent marginal utility in the main analysis. In a set of sensitivity analyses we do include these. Table 2: Values of parameters in the main specification r β θ 0 ξ 0 κ 0 The effect of co-payments on annuitization There is an ongoing (policy) discussion on the role of pensions in case of an increase in private LTC spending. In the Netherlands, for instance, there are debates about both the co-payment scheme in LTC as well as more flexible pension pay out systems. The reason why these two discussions are related is that co-payments in LTC makes individuals want to hold more of their pension wealth in the form of liquid assets as a precaution. As discussed by Peijnenburg et al. (2017), the optimal annuitization share will depend on the timing of LTC costs: when LTC costs occur relatively early in retired life, full annuitization is not optimal, while when LTC costs occur at the end of life (near) full annuitization might still be optimal. 17

18 We analyze the effect of co-payments on annuitization by estimating the optimal annuitization share of pension wealth in each policy variant. We do this for each spending power group separately. As the starting point we take the current value of pension wealth, which is fully annuitized in the Netherlands. As in the previous paragraph, we take the income at age 70 as a fixed annual income stream y over the rest of life. The second pillar annual pension payout y p is then equal to y minus the state pension. We calculate pension wealth W p 0 by taking the net present value of the second pillar pension at the age of retirement. We then run the algorithm over a range of values for the annuitization share α. In each case, the lump sum payout is equal to (1 α)w p 0, and the annual pension payout is equal to y p α calculated as in Equation (6). The optimal annuitization share is calculated by taking the value of α that gives the highest certainty equivalent consumption which is determined for each spending power group. 5 Results 5.1 Income and wealth dependent co-payments Main results Table 3 shows the average consumption and certainty equivalent consumption in each payment variant (v). Results are reported for the total population and by spending power quintile (spq). For the total population, average consumption is higher in the case of full insurance (v = 1, 2) than in the case of co-payments (v = 3, 4, 5). In the case of a uniform premium, for instance, average consumption is 27,967 euros, while in the case of a uniform co-payment it is 26,753 euros. This lower consumption is due to additional savings: co-payments induce individuals to hold more financial wealth as a precaution. As some of this wealth is not spent before death, average lifetime consumption decreases. This effect is strongest in the case of uniform co-payments that are independent of income and wealth. The drop in CEC when going from full insurance to co-payments is even larger than the drop in average consumption. The CEC for the total population goes from 27,747 euros in the case of a uniform premium to 25,999 in the case of a uniform copayment. This additional drop is due to increasing uncertainty. Co-payments lead to uncertain LTC payments, which in some cases can be very substantial. As individuals are risk-averse, this uncertainty leads to a welfare loss. Table 3 also shows considerable differences in consumption across spending power groups. The lowest spending power quintile has an average consumption (in variant 1) of 11,540 euros, while the highest spending power quintile has 54,941 euros. We further analyze the results by subgroup by looking at Figures 4 and 5. Figure 4 shows how much of lifetime income (pension income and initial wealth, starting at 70) each spending power groups uses on average for consumption, bequests and HC payments ( health care, both premiums and co-payments for LTC). First, HC payments form a larger part of lifetime income for low spending power groups 18

19 Table 3: Consumption and certainty equivalent consumption per policy variant, and spending power group (spq). Main specification. total spq1 spq2 spq3 spq4 spq5 Consumption 1 Uniform premium 27,967 11,540 16,626 21,654 35,076 54,941 2 Actuarially fair premium 27,994 10,809 16,404 21,717 35,294 55,748 3 Inc. dep. co-pay 27,330 11,298 16,064 20,950 34,399 53,941 4 Inc. & wealth dep. co-pay 27,523 11,432 16,378 21,335 34,409 54,061 5 Uniform co-pay 26,735 10,872 15,069 20,322 33,665 53,748 CEC 1 Uniform premium 27,747 11,504 16,590 21,593 34,836 54,211 2 Actuarially fair premium 27,773 10,774 16,368 21,655 35,059 55,009 3 Inc. dep. co-pay 26,569 10,971 15,511 20,147 33,488 52,729 4 Inc. & wealth dep. co-pay 26,782 11,172 15,960 20,723 33,407 52,648 5 Uniform co-pay 25,999 10,568 14,411 19,345 32,769 52,901 than for higher groups. Second, the percentage of income used for LTC decreases with spending power, even in variants 3 and 4 that have wealth and/or income dependent co-payments. Second, co-payments indeed induce additional savings (the part of lifetime consumption that is left as bequest is higher), but mostly for the higher spending power groups. Third, uniform co-payments sometimes lead to such substantial spending shocks that the income and wealth of members of the lowest spending power quintile, and to a lesser extent the second, drop below the consumption floor. This leads to a transfer, which is paid out of total co-payment revenues. Figure 5 zooms in on average consumption and CEC by spending power group. The other variants are compared to the base case (variant 1) where co-payments are zero and LTC costs are paid out of a uniform fixed premium. The table shows the loss of CEC in variant v compared to variant 1: in loss in average consumption, CEC v CEC 1 CEC 1 C v C 1 C 1 C v C 1 C 1 CEC v CEC 1 CEC 1. This loss is decomposed (in red), and loss due to an increase in risk: (in green). The figure shows that the design of the co-payment matters: all variants raise the same revenue, but the distribution of costs across groups differs greatly. Fixed copayments, independent of income and wealth, lead to the highest loss in CEC for the lower spending power quintiles (mainly 2 and 3). Income, and especially income- and wealth dependent co-payments lead to a much lower loss in CEC for these groups, while the loss for the highest spending power group increases only slightly. Including risk in the assessment of different variants also matters. For the co-payment variants, welfare loss due to increased risk form a substantial part of total welfare loss. Only for the highest spending power group, risk is a relatively small part of total welfare loss. The only variant where risk does not play a role is the actuarially fair premiums, 19

20 as that variant does not induce any uncertainty in payments. Lower spending power groups, especially the first quintile, are worse off in this variant compared to the base case with a uniform premium: as LTC use is higher for low income groups, an uniform premium redistributes costs from low to high spending power groups, where an actuarially fair premium does not. Strikingly, the actuarially fair premium leads to the lowest welfare for the lowest spending power group of all variants, except the uniform co-payment. An interesting finding is that, in case of uniform co-payments, spending quintiles 2 and 3 have a higher welfare loss than quintile 1. The income of members of quintile 1 is relatively close to the consumption floor, so that the net payments (co-payments minus the income transfer due to an income drop below the consumption floor) are relatively limited. Groups 2 and 3 hardly benefit from the safety net and do have to pay the high uniform co-payments. 20

21 % of lifetime income Consumption Transfer Bequest HC payment 1 Premium (%Inc) 2 Act fair prem 3 Inc dep. 4 Inc & wealth dep. 5 Fixed copay Figure 4: Share of lifetime resources spent on consumption, bequests, and LTC. By spending power group (spq) and policy variant. 21

22 spq1 spq2 spq3 spq4 spq5 % loss compared to baseline Average Uncertainty 2 Act fair prem 3 Inc dep. 4 Inc & wealth dep. 5 Fixed copay Figure 5: How large is the loss in CEC compared to the baseline policy with a uniform premium and no co-payments ((CEC v CEC 1 )/CEC 1 ). Per policy variant (v) and spending power group (spq). 22

23 Sensitivity analysis Table 4 shows (CEC v CEC 1 )/CEC 1 by spending power group for the main specification and for two sensitivity analyses. In the first sensitivity analysis, we include a bequest motive. We do this by setting θ = 2.3 in Equation (9). We recalibrate the co-payment parameters so that all variants again raise the same revenue as the variant based on the current system (v = 3). The presence of a bequest motive generally decreases the utility loss form introducing co-payments (compared to full insurance): as financial wealth left at death now still has utility, through the bequest, the precautionary savings induced by co-payments are less costly in terms of utility than in the case without a bequest motive. Table 4: The loss in CEC compared to the baseline policy (v 1 ) with a uniform premium and no copayments. By spending power group (spq) and policy variant. For different parameter settings. total spq1 spq2 spq3 spq4 spq5 Main specification 2 Actuarially fair premium Inc. dep. co-pay Inc. & wealth dep. co-pay Uniform co-pay Bequest motive (θ = 2.3) 2 Actuarially fair premium Inc. dep. co-pay Inc. & wealth dep. co-pay Uniform co-pay State-dependent utility (κ = 0.2) 2 Actuarially fair premium Inc. dep. co-pay Inc. & wealth dep. co-pay Uniform co-pay As a second sensitivity analysis, we assess the influence of health state-dependent utility by setting κ in Equation (10) to 0.2. This means that the marginal utility of consumption is 20 percent lower in poor health compared to that in good health. De Nardi et al. (2010) choose a similar value for κ and it seems to be at the more extreme side of the range of values found by Finkelstein et al. (2013). The results can again be found in Table 4. The loss in CEC in the variants with co-payments (v = 3, 4, 5) compared to the variant without co-payments (v = 1) is smaller than in the baseline specification: the marginal utility is now lower in poor health, which means that it is no longer optimal to fully smooth consumption between health states. Replacing a full insurance (v = 1, 2) with a co-payment system allows individuals to shift consumption towards years in good health. However, even with state-dependent utility, co-payments lead to a loss in CEC compared to a full insurance, for all spending power groups: the utility 23

24 loss due to risk is larger than the gain from shifting consumption to good health years. 5.2 The effect of co-payments on annuitization Main results Table 5 shows the results for the case where individuals can freely choose which part of their pension wealth is annuitized at the age of retirement. The table shows the annuitization shares (α) that provide the highest CEC for a spending power group and a particular co-payment variant. The first result that stands out is that, even with co-payments, full annuitization is optimal for most groups. This might be due to the fact that a large part of LTC spending happens very late in retired life (Hussem et al., 2016). Another explanation might be the distribution of income and wealth across the spending power groups: the pension wealth for low spending power groups might be too low to both provide a sufficiently high income stream and sufficiently high levels of liquid wealth, to protect against substantial LTC co-payments, at the same time. High spending power groups, on the other hand, already have substantial amounts of financial wealth which provide sufficient protection against LTC co-payments. For them, using a part of pension wealth as additional precautionary savings is unnecessary. A second result that stands out is that, when optimal annuitization shares do differ from 1, the results are in line with what we would expect: co-payments that do depend on pension income, but not on financial wealth, lower the optimal annuitization share. For both the second and third spending power group, the optimal share is 0.95 in case of an income-dependent co-payment and 1 in all other variants. The only exception is spending power group 1, where the optimal annuitization share is lowest in the variant with income and wealth dependent co-payments. Table 5: The effects on welfare when pension wealth can be freely annuitized. CEC and optimal annuitization share (α). By spending power group (spq) and policy variant. spq1 spq2 spq3 spq4 spq5 1 Uniform premium CEC 11,101 16,411 21,316 33,285 50,585 α Actuarially fair premium CEC 10,258 16,167 21,339 33,580 50,819 α Inc. dep. co-pay CEC 10,629 15,368 19,858 31,710 49,546 α Inc. & wealth co-pay CEC 10, ,238 α Uniform co-pay CEC 10,329 14,239 18,960 31,109 48,439 α