Optimizing the Life-Cycle Path of Pension Premium Payments and the Pension Ambition in the Netherlands

Abstract

We determine the optimal life-cycle path of pension premiums during the working ages and pension benefits during retirement, assuming that households can choose these freely. We calibrate the model on Dutch data. The optimization takes into account the fact that incomes generally rise during the working ages and that children are generally present in young households. Both features lead to an upward sloping path of optimal pension premium rates over the working life, while under the current pension system this is rather flat. The welfare gains from implementing the optimal pension system depend on the specification of the model, but can be sizable. A potentially lower return on pension assets in the future implies a further delay of optimal pension premiums during the working ages and a lower optimal pension benefit after retirement. Differentiating the path of pension premiums and benefits according to the level of educational attainment of the main income earner brings only small additional welfare gains compared to the situation in which pension premium rates are based on the average income profile in the Dutch economy.

Appendix

1.1 Summary Statistics

See Tables 9 and 10.

Table 9 Household income summary statistics

Table 10 Household composition summary statistics

1.2 Optimal Pension Premium Rates with Progressive Income Tax Rates

In this appendix we check the robustness of the results to the assumption that income tax rates are flat. For simplicity, the baseline scenario makes the assumption of flat tax rates. This assumption may not be innocuous, because a progressive income tax rate will flatten the life-cycle profile of net household income. Consequently, the scope for age dependent pension premium rates declines.

Throughout the paper we use the policy variables (for example the value of the pension franchise and the pension cap) from the year 2010, the middle of the period for which the income data is available (2006–2013). However, in the case of the tax code, we choose to use the average tax rates corresponding to the year 2020. We make this choice because income tax rates have become substantially more progressive since 2010. The relationship between the gross income and the average tax rate we assume is similar to the one estimated in Vrouwerff (2021) (see Fig. 14, left panel). Because information about the average tax rates after the AOW age is missing in Vrouwerff (2021), the income tax is kept flat at 22%, the same as in the baseline scenario. The average tax rates corresponding to the life-cycle income profile of the average Dutch household are shown in Fig. 14, right panel.

Fig. 14

Progressive average income tax rates

Using the progressive tax rate depicted in Fig. 14, we compute the optimal consumption and savings schedule. The results are presented in Fig. 15. Comparing these with the results from the baseline scenario (Fig. 4), we see that the life-cycle profile of the savings rates is slightly less upward sloping. Wealth accumulation and pension benefits are also smaller in the case of both the earner and the family model. The differences are however small. The welfare gains from implementing the optimal pension system decline to 2.5% from 3.1% in the case of the family model and 1.3% from 2.6% in the case of the earner model. If we impose borrowing constraints, welfare gains drop to 2.4% and 1%, respectively.

Fig. 15

The current pension system (solid line), the family model (dashed line) and earner model (circles): optimal wealth, consumption, savings ratio and pension benefits

1.3 Derivation of the Optimization Problem

Appendix 1.3 contains a step by step derivation of the equations presented in the paper. We start directly with the family model and mention what changes must be implemented in the derivations in order to obtain the earner model.

The household chooses each period how much to consume ($c_t$) and implicitly how much wealth to accumulate ($a_t$) by maximizing the present value of the utility of standardized consumption:

$$\begin{aligned} \sum _{t=1}^{t=T} \beta ^{t-1}\psi _t n_t u\left( \frac{c_t}{eq_t}\right) \end{aligned}$$

(22)

where $\beta$ is the time preference parameter, $\psi _t$ is the probability to survive until time t, $n_t$ is the number of members in a household, $\sigma$ is the inverse of the elasticity of intertemporal subsitution and $eq_t$ equals the equivalence factor at time t. In the earner model, the number of household members ($n_t$) is equal to 2 at every age because the consumption of children is not valued in the utility function. Consequently, this variable drops from the subsequent equations.

The decision of the household is subject at each age t to a budget constraint:

$$\begin{aligned} y_t + Ra_{t-1} + tr_t = c_t + tax_t + a_t \end{aligned}$$

(23)

where $y_t$ is the income of the household comprised of labor income until the age when the pay-as-you-go benefit is received ($t_{aow}$) and the pay-as-you-go pension benefit (AOW) afterwards, R is the rate of return, $a_t$ is the wealth, the per capital transfers $tr_t$ come from the fact that a fraction of the population alive at time $t-1$ dies and the bequests are divided between the members still alive at time t.

The size of the bequests left by the people who die is given by:

$$\begin{aligned} B_t = R a_{t-1} l_{t-1} \Big ( 1 - \frac{\psi _t}{\psi _{t-1}} \Big ) \end{aligned}$$

(24)

where $R a_{t-1}$ is the per capita wealth accumulated by people alive at time $t-1$, $l_{t-1}$ is the number of people alive at time $t-1$, $\frac{\psi _t}{\psi _{t-1}}$ is the probability to survive between period $t-1$ and t and $\Big ( 1- \frac{\psi _t}{\psi _{t-1}} \Big )$ is the probability to die between period $t-1$ and t. We divide total bequests among the people that are still alive at time t:

$$\begin{aligned} \frac{B_t}{l_t} = R a_{t-1} \frac{l_{t-1}}{l_t} \Big ( 1- \frac{\psi _t}{\psi _{t-1}} \Big ) \end{aligned}$$

(25)

Next we take into account that the ratio of people alive at time t and people alive at time $t-1$ is equal to the probability to survive between periods $t-1$ and t:

$$\begin{aligned} \frac{l_t}{l_{t-1}} = \frac{\psi _t}{\psi _{t-1}} \end{aligned}$$

(26)

We substitute relation 26 into 25 and obtain the per capita transfer coming from bequests:

$$\begin{aligned} \frac{B_t}{l_t}= & {} R a_{t-1} \frac{\psi _{t-1}}{\psi _t} \Big ( 1- \frac{\psi _t}{\psi _{t-1}} \Big )\end{aligned}$$

(27)

$$\begin{aligned} tr_t= & {} \frac{B_t}{l_t} = R a_{t-1} \Big ( \frac{\psi _{t-1}}{\psi _t} -1 \Big ) \end{aligned}$$

(28)

Finally, we substitute 28 in the period by period per capita budget constraint from relation 23 and obtain:

$$\begin{aligned} y_t \psi _t + R a_{t-1} \psi _{t-1} = c_t \psi _t + tax_t \psi _t + a_t \psi _t \end{aligned}$$

(29)

We use the individual budget constraints to obtain the lifetime budget constraint:

$$\begin{aligned} \begin{aligned} \sum _{t=1}^{T} \frac{\psi _t (c_{t}+tax_t)}{R^{t-1}} = \sum _{t=1}^{T} \frac{ \psi _{t} y_{t}}{R^{t-1}} \end{aligned} \end{aligned}$$

(30)

The consumption can be described by:

$$\begin{aligned} c_t = (y_t-p_t)(1-\omega _t) \end{aligned}$$

(31)

where $p_t$ equals the amount of pension money (either being saved or dis-saved) and $\omega _t$ equals the percentage of tax on income net of pension premiums paid at age t: 36% until the age of 65 and 22% afterwards. For simplicity, average and marginal tax rates in both phases are set equal.¹⁷ The tax paid is equal to $tax_t = (y_t-p_t)\omega _t$, so expenditures (consumption and tax) can be rewritten as¹⁸:

$$\begin{aligned} c_t + tax_t = (y_t-p_t)(1-\omega _t) + (y_t-p_t)\omega _t = \frac{c_t}{1-\omega _t} \end{aligned}$$

(32)

Therefore, Eq. 30 can be rewritten as:

$$\begin{aligned} \sum _{t=1}^{T} \frac{\psi _t c_{t}}{R^{t-1}(1-\omega _t)} = \sum _{t=1}^{T} \frac{ \psi _{t} y_{t}}{R^{t-1}} \end{aligned}$$

(33)

The problem that the household solves becomes:

$$\begin{aligned} \begin{aligned} \max _{\{c_t\}} \sum _{t=1}^{t=T} \beta ^{t-1}\psi _t n_t \frac{(\frac{c_t}{eq_t})^{1-\sigma }}{1-\sigma } + \lambda \left( \sum _{t=1}^{T} \frac{ \psi _{t} y_{t}}{R^{t-1}}- \sum _{t=1}^{T} \frac{\psi _t c_{t}}{R^{t-1}(1-\omega _t)} \right) \end{aligned} \end{aligned}$$

(34)

The first order conditions of the above maximization problem are:

$$\begin{aligned} \beta ^{t-1} n_t \left( \frac{1}{eq_t} \right) ^{1- \sigma } c_t^{-\sigma } = \frac{\lambda }{R^{t-1} (1-\omega _t)} \end{aligned}$$

(35)

It follows that the relationship between first period consumption ($c_1$) and the consumption in period t ($c_t$) is given by:

$$\begin{aligned} c_t = c_1 (\beta R )^{\frac{t-1}{\sigma }} \left( \frac{n_t}{n_1} \right) ^{\frac{1}{\sigma }} \left( \frac{eq_1}{eq_t}\right) ^{\frac{1 - \sigma }{\sigma }}\left( \frac{1-\omega _t}{1-\omega _1} \right) ^{\frac{1}{\sigma }} , \forall t=2,\ldots ,T \end{aligned}$$

(36)

Next, we substitute Eq. 36 in the life-time budget constraint of the household from Eq. 33:

$$\begin{aligned} \begin{aligned} \sum _{t=1}^T \frac{\psi _t c_1 (\beta R)^{\frac{t-1}{\sigma }} \left( \frac{n_t}{n_1} \right) ^{\frac{1}{\sigma }} \left( \frac{eq_1}{eq_t}\right) ^{\frac{1 - \sigma }{\sigma }} \left( \frac{1-\omega _t}{1-\omega _1} \right) ^{\frac{1}{\sigma }} }{R^{t-1} (1-\omega _t)} =\sum _{t=1}^{T} \frac{ \psi _{t} y_{t}}{R^{t-1}} \end{aligned} \end{aligned}$$

(37)

Now we can derive a closed-form solution for the first period consumption of the household:

$$\begin{aligned} c_1 = \frac{\sum _{t=1}^{T} \frac{ \psi _{t} y_{t}}{R^{t-1}} }{\sum _{t=1}^T \frac{\psi _t (\beta R)^{\frac{t-1}{\sigma }} \left( \frac{n_t}{n_1} \right) ^{\frac{1}{\sigma }} \left( \frac{eq_1}{eq_t}\right) ^{\frac{1 - \sigma }{\sigma }} \left( \frac{1-\omega _t}{1-\omega _1} \right) ^{\frac{1}{\sigma }} }{R^{t-1} (1-\omega _t)}}. \end{aligned}$$

(38)