How Much Retirement Income is Needed to Maintain the Living - - PDF document

How Much Retirement Income is Needed to Maintain the Living Standard?

Julian Schmied∗Christian Dudel†

Abstract In light of growing old-age dependency, public pensions are decreasing in many developed

• countries. As a consequence, private savings and occupational pension schemes have become

more important for old age income. But how much income from these sources is needed to compensate for the reduction in public pensions is unclear, as there is no definite standard for how much retirement income individuals actually need. We contribute by estimating constant living standard net replacement rates (CLS-NRRs) based on expenditure data. CLS-NRRs measure how high the ratio of retirement income to pre-retirement income needs to be to keep the living standard unchanged by retirement. We apply modern nonparametric and traditional parametric methods. Using the Income and Expenditure Survey of Germany we show, that these rates mostly exceed 90 %. This is substantially more than a commonly quoted figure in the pension literature.

Keywords: replacement rates; pension adequacy; partial identification; semiparametric estimation; nonparametric estimation; JEL Classification: C14 H55 J14

∗Max-Planck-Institute for Demographic Research †Max-Planck-Institute for Demographic Research

1 Introduction

The proportion of individuals aged 65 and older has been increasing in the majority of developed countries, going along with an increase of the retired population. For the coming decades this process of population aging is expected to continue. This puts social security systems under pressure, and public pension schemes in particular. As a reaction, public pensions have been reduced in many countries, leading to a decrease in pension replacement rates, i.e., retired persons receive a smaller share of their previously earned labor income. For instance, while up to the early 90s the German statutory pension scheme provided average net replacement rates of about 70% (Boersch-Supan and Schnabel 1998), this number has decreased to 53% in 2014 (OECD 2015). Private and occupational schemes have become more important due to this decline in replace- ment rates provided by public pensions. How much individuals need to save in addition to public pensions to compensate for the decrease in replacement rates is unclear, though. The question is how high the replacement rate needs to be. While several heuristics can be found in the literature, usually suggesting replacement rate of around 70% (e.g., Haveman et al. 2007, Benartzi 2012), they lack an empirical basis. Empirical approaches are less common and lead to conflicting results. One example are approaches based

• n economic life cycle models, which yield a broad range of replacement rates. For instance,

Hamermesh (1984) and Mitchell and Moore (1998) report findings for the US between 80% and 90%, while one can derive a replacement rate of 66% from the model of Scholz et al. (2006). This paper contributes to the literature by proposing, applying, and comparing three approaches for the estimation of “optimal” replacement rates. We define “optimal” replacement rates as what we will call “constant living standard net replacement rates” (CLS-NRRs). CLS-NRRs capture how much retirement income is needed to achieve the same living standard as during working life. As such they can be used as a guideline for individual savings and as a reference point for assessing pension adequacy. The three approaches we use vary considerably in complexity and the underlying assumptions. For all of them the basic idea is similar to the estimation of equivalence scales (Dudel et al. 2016). Essentially, given data on one or several welfare indicators, the income levels of retired and pre-retired individuals are sought which give the same values for the welfare indicators. Apart from this basic idea, approaches differ considerably. The first approach is in the spirit of Engel (1857) and fully parametric, the second approach was originally proposed by Pendakur (1999) and is semiparametric, and the third one is nonparametric and based on Dudel (2015). The parametric approach has the virtue that it is easily understood and implemented, but it rests

• n strong assumptions. The nonparametric approach, on the other hand, is more complex and

computationally intensive, but avoids restrictive assumptions of the parametric approach. The semiparametric approach falls in between. The results are based on the German Income and Expenditure Survey (“Einkommens- und Verbrauchsstrichprobe”; EVS). The remainder of this paper is structured as follows. The expenditure based approaches for the estimation of CLS-NRRs are discussed in section 2. Section 3 describes the data. In section 4 we present preliminary results while section 5 concludes. 1

2 Methodology

2.1 Problem description

The problem we tackle in this paper can be approached from two perspectives. The first perspective follows economic theory. Let q(p, y, x) be a demand function depending on prices p, income y, and socioeconomic characteristics x, where we assume that x includes a dummy variable d capturing whether an individual is retired or not. u(q, x) is the utility derived from demand and v(y, p, x) is the indirect utility function. Using these functions we define an income function y(u, p, x) = minz[z|v(z, p, x) = u] which returns the smallest possible income which allows a utility level of u given p and x. Now let x1 and x0 be two vectors which only differ with respect to d which equals 1 for x1 and equals 0 for x0. Constant living standard net replacement rates (CLS-NRRs) now can be defined as φ(u, p) = y(u, p, x1) y(u, p, x0). (1) This means that we are interested in the ratio of incomes before and after retirement which give the same level of welfare, keeping everything else constant. This problem is essentially the same as estimating equivalence scales (Dudel et al. 2016). Throughout we assume that prices are fixed and can safely be ignored, because of which it suffices to write φ(u) = y(u, x1)/y(u, x0). Note that φ(u) still depends on the level of utility. Equation (1) states the problem from an economic perspective. The second perspective is the estimation problem which follows from this and which can be phrased in terms of the counterfactual approach (Rubin 2005). Let y1

i (u) be the income individual i needs to attain some utility level u

when being retired and y0

i (u) is the income level when not being retired. Moreover, let ud i be the

utility level the individual achieves depending on whether she is retired or not as captured by d. CLS-NRRs can be defined as y1

i (u0 i )

y0

i (u0 i ).

(2) The problem is that we can either observe y0 or y1 and not both. Moreover, y1

i (u1 i ) will be observed

and not y1

i (u0 i ).

In the potential outcomes literature, usually three basic assumptions are invoked to overcome problems like the above. First, the unconfoundedness assumption requires that the pair y1, y0 is independent of d conditional on some other covariates x. This requires that selection into retirement can be controlled for. Second, the probability of d = 1 needs to be positive for all values

• f x (overlap condition). Third, the stable unit treatment value assumption states essentially that
• bservations are independent, i.e., individual i does not influence individual j.

Given these three assumptions, usually expectations of quantities of interest are identified. Unfortunately, this is not the case for (2). More specifically, taking expectations we have E y1(u0) y0(u0)

• = E
• y1(u0)
• E
• y0(u0)

− 1 E

• y0(u0)

Cov y1(u0) y0(u0), y0(u0)

• .

(3) The first ratio on the right hand side is identified given above assumptions, but the second term is not. This follows from the fact that the joint distribution of y0 and y1 is not identified (Dudel 2015). The second term essentially captures how much CLS-NRRs depend on the pre-retirement

• income. The three approaches we use to estimate CLS-NRRs are all derived from the equivalence

scale literature and differ in their assumptions about the second term in (3), among other things. 2

2.2 Parametric approach

The parametric approach we employ dates back to Engel (1857) and assumes that CLS-NRRs do not depend on the baseline income level, i.e., that the second term on the right hand side of equation (3) equals zero. This implies that φ(u) = φ, meaning that CLS-NRRs do not depend on utility. In the equivalence scale literature this assumption is often called “independence of base” (Lewbel 1989). The approach works as follows. Assume that the share of income spend for the expenditure for food is a valid indicator of the utility level and denote it with wi. In what follows, we will use the expenditure share for food, but the approaches presented here are not restricted to it and

• ther welfare indicators are possible. The food expenditure share is regressed on income y, the

retirement indicator d, and other covariates x, wi = α + β log(yi) + λxi + γdi + ui. (4) Estimates of the coefficients are plugged into (4), which is equated for both d = 1 and d = 0, i.e. ˆ α + ˆ β log(y0) + ˆ λx = ˆ α + ˆ β log(y1) + ˆ γ + ˆ λx. (5) Solving for y1

y0 yields

φ = y1 y0 = exp

• − ˆ

γ ˆ β

• .

(6) As already noted above this approach assumes that CLS-NRRs do not depend on reference income. However, during working life well paid persons might accumulate wealth over time, which they can benefit from to maintain their welfare status after they stop working Smith (2003). Consequently, they have a lower CLS-NRR than low-income workers. Moreover, this approach requires linearity of the effect of log income. This might hold for some commodity groups but clearly not for others (e.g., Banks et al. 1997). Using another specification, e.g., adding a quadratic income term, is in principle possible, but complicates the solution considerably.

2.3 Semiparametric approach

Blundell et al. (1998) used a semiparametric approach to estimate equivalence scales (see also Pendakur 1999, Wilke 2006, Stengos et al. 2006), which avoids the linearity assumption of the parametric approach. To explain their approach for the case of CLS-NRR, let wd(p, y) be the food expenditure share given income y, prices p, and retirement status d. The semiparametric approach then assumes that w0(p, log(y)) = w1(p, log(y) + φ) + µ(p), (7)

• r assuming fixed prices,

w0(log(y)) = w1(log(y) + φ) + µ. (8) The idea is that the food expenditure share as a function of (log) income looks similar before and after retirement, except that after retirement it is shifted by φ and an additional term µ. µ(p) reflects price elasticity, but assuming fixed prices µ is hard to interpret. To derive φ and µ, nonparametric estimates of w0 and w1 as functions of log(y) are calculated. Given estimates ˆ w0 and ˆ w1, values of φ and µ which minimize S(φ, µ) are found using grid search, where S(φ, µ) = 1 n

n

• 1
• ˆ

w1(log[yi] + φ) − µ − ˆ w0(log[yi]) 2 . (9) 3

Just like the parametric approach, the semiparametric method assumes that CLS-NRRs do not depend on reference income, but it avoids strong assumptions on functional form. This benefit comes at the cost of loosing the possibility to easily control for additional covariates x. A potential remedy is the combination of the semiparametric approach with a matching approach, which allows to remove differences with respect to x between retirees and non-retirees (Dudel et al. 2017). In a first step we apply one-to-one matching using the Gower distance (Dettmann et al. 2011), which is appropriate for both quantitative and qualitative variables, and in a second step the procedure

• utlined above is applied.

2.4 Partial identified nonparametric approach

The parametric and the semiparametric approach both assume that the second term in equation (3) is zero and CLS-NRRs do not depend on reference income y0. Without this assumption, (2) is not identified, as only the marginal distributions of y1 and y0 are known and not their joint

• distribution. Following Frechét (1951) knowledge of the marginal distributions of y1 and y0 is

enough to calculate bounds on the joint distribution. From these, bounds on (2) can be derived, and more specifically bounds on Cov (y1/y0, y1) (Dudel 2015). If the bounds for Cov (y1/y0, y1) are below or above 0, then the assumption that CLS-NRRs do not depend on income can be rejected. The upper bound is determined by βL = ∫ 1 F−1

1 (t)

F−1

0 (t)dt

(10) where F−1

d (t) is the quantile function of the conditional marginal distribution of y1 and y0, respec-

• tively. The lower bound can be found by replacing F−1

0 (t) with F−1 0 (1 − t).

In contrast to previous approaches this identification strategy does not rely on the assumption that CLS-NRRs are independent from individual welfare level. Further, it does not rely on any specific shape for individual welfare functions. Covariates can also easily be included (for details see Dudel 2015). While this approach is thus light regarding assumptions, no point estimate can be achieved and interval estimates might not be informative if they are wide.

3 Data

To estimate CLS-NRR for the case of Germany we used the most recent German Income and Expenditure Survey of 2013, ("Einkommen- und Verbraucherstrichprobe"; EVS), provided by the German federal statistical office. The EVS is a quinquennial cross-sectional survey and includes detailed information on household consumption and income. We reduced the sample to single households to avoid inter-dependencies related to the income

• f the spouse. A person is considered as retired if one self-reports one’s employment status as

a pensioner and is 60-64 years old (n=503). The reference group consists of non-retired persons within the same age range (n=735). We took into account control variable such as gender, season, population density of the residual area, age, education and whether or not the person lives in the former GDR. Finally, we included a variable that captures whether a person lives in his own apartment or is a tenure. Prior to the analysis, we excluded persons on welfare benefits and observations with a monthly income above 7000. On the other hand, we kept observations with very low income in the sample 4

Table 1: Summary statistics

After matching Pre-retirees Retirees Pre-retirees Retirees n 738 511 508 508 Age, mean 61.6 62.5 61.9 62.5 Age, min 60 60 60 60 Age, max 64 64 64 64 Monthly net income (in 1000 EURO), mean 2.3 1.6 2.4 1.8 Monthly net income (in 1000 EURO), min 0.6 0.6 0.6 0.6 Monthly net income (in 1000 EURO), max 11 10.8 11.3 10.8 Food expenses (in PERCENT), mean 10.7 13.9 10 12.3 Food expenses (in PERCENT), min 0.7 0.9 0.7 0.8 Food expenses (in PERCENT), max 42.9 40.2 42.9 39.9 Share of males (in PERCENT) 30.1 29.7 29.9 29.3 Share of former GDR (in PERCENT) 21.4 26.4 25.4 26.4 Share of homeowner (in PERCENT) 44.2 39.3 39.2 39 Share of highly educated (in PERCENT) 45.0 35.0 40.4 34.8 Share of low educated (in PERCENT) 19.6 27 21.9 27 Share of people from rural areas (in PERCENT) 11.2 14.3 12.2 13.8 * Including persons with "Fachabitur" or "Abitur", ** Including persons with no degree or a degree from "Hauptschule" Source: EVS 2013, author’s calculations.

to cover self-employed persons (e.g., artists) who might earn a small amount in one quarter but a lot in the subsequent quarter (the EVS reporting period was reduced from 1 year to 3 month). To measure the welfare level we follow the approach of Engel (1857) and evaluate expenditures for non-durable food bought in stores as a share of total expenditures. As a function of individuals’ (log) net income, this yields the Engel curves. Other commodity groups, such as clothes are also possible to examine (Rothbarth 1943). However, those are not as regularly consumed as food and given the short reporting period of 3 months, the category suffers from a large number of zero expenditures which potentially underestimates the true consumption of a person. Table 1 describes control and explanatory variables of our sample across retirees and pre- retirees: The small share of male observations in this age group is consistent with high mortality rates of men in the 40s and 50s, which has lead to a high share of (single) widows in 2013. Also, we observe some structural difference between the two groups. For instance, the retired group has a smaller share of highly educated persons. The matching step described in Section 2.3 reduces this heterogeneity (See Table 1 Column 4-5).

4 Results

In Table 2 we present, by method, our results for the constant living standard net replacement rates. The bottom of the table can be interpreted as: The retirement income needs to be 86−111% percent

• f ones ultimate working life paycheck, if one intends to maintain the standard of living. While

based on very mild assumptions these bounds are rather wide. However, they are a good starting point to get an idea what results are possible. In theory, it is expected that the replacement rate is below 100% since retirees face lower living 5

Table 2: Constand living standard net replacement rates by method

Method CLS-NRR Parametric 106 % Semiparametric 92 % Semiparametric (matched) 95 % Partial identified nonparametric bound 86 - 111 % Source: EVS 2013, author’s calculations.

costs: Work related expenses such as commuting and clothes disappear and one has not to save anymore for old age (Munnell and Soto 2005). However, the parametric approach yielded a CLS-NRR of 106%. While the result is not entirely unlikely it remains surprisingly high. We explain this by the methods’ underlying assumptions, in particular the assumption of linear Engel curves. As Banks et al. (1997) demonstrated, this does not hold or hold only approximately for most commodity groups. And also in this case, linearity can be approximately asserted for middle incomes only (as shown in Figure 1). Allowing non-linear Engel curves, a more plausible CLS-NRR of 92% was identified by the semiparametric approach. After homogenizing the reference and the non-reference group with a matching technique, and thus take into account socio-demographic characteristics, the rate increased to 95%. Table 3: Constant living standarad net replacement rates by subgroups

High Education Low Education Former GDR West Germany CLS-NRR 87 % 100 % 85 % 96% n 367 742 295 962 Source: EVS 2013, author’s calculations.

Even though we control for socio-demographic factors, there might still be unobserved hetero-

• geneity. Therefore, we show some within group estimates in Table 3. It is hard to predict which

group will have the higher rate, though. For instance, highly educated people might eat more out

• f the house and thus have lower food costs. On the other hand, they are more interested in health

and spend proportionally more on high quality food. In Table 3 we show that the CLS-NRR for low educated person is indeed higher. It is unknown though whether this comes from changing expenditure structures during the retirement process or the high correlation between income and education. Germany was separated until 1989 and while many aspects have converged, pension entitlements are still slightly different in East Germany. Hence, we calculated CLS-NRR for the eastern region

• separately. The results suggests, that in West Germany one needs a larger fraction of one’s income

to maintain the living standard.

5 Discussion and concluding remarks

In this paper we identified constant living standard net replacement rates with expenditure data. Our results are comparable to studies where the living standard was addressed by subjective measures (Dudel et al. 2016, Binswanger and Schunk 2012). With rates mostly exceeding 90%, however, 6

Figure 1: Linear vs. non-linear fit of Engel curves

• ●
• 7.5

8.0 8.5 9.0 9.5 10.0 10.5 0.0 0.1 0.2 0.3 0.4 log(Net income) Food share linear fit non−linear fit

Note: Share of food consumed at home; Nonparametric Engel curves are calculated by Kernel estimation (Nadaraya-Watson Estimator). Source: Author’s calculations, Data: EVS 2013

we found substantially higher rates than life cycle models suggest (e.g., Scholz et al. 2006). Most importantly, our results call into question the optimal replacement rates of 70% which financial advisor usually propose as a rule of thumb (e.g., Benartzi 2012). Further, by comparing our “optimal” replacement rate and the 53% replacement rate provided by the German statutory pension system OECD (2015), a large saving gap emerges. While replacement rates are predicted to further decrease as a consequence of demographic aging (Werding 2016), individuals are increasingly held responsible to provide themselves for a decent old age living. Since the poor tend use a larger fraction of their income for consumption, this certainly raises questions on old age income inequality. We believe that, although this approach has its limitations (e.g., results can not be generalized to couple households), its methods are data driven and do not rely on strong assumptions with respect to the individuals’ rational behavior. Moreover, data requirements are low, given that household expenditure data is broadly available in many countries. 7

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