3.1 Decision-making

Decisions on the four arrangements are made by the different actors by optimizing their remaining lifetime utility functions, where they take into account their wage income w, their already-standing other obligations OO_n like pension premiums agreed on in earlier contracts, mandatory pension premiums, and social security contributions, and their already secured other retirement benefits OB_n.

3.2 Decision-making by the individual worker

We start by looking for the individual savings pattern $\lpar {S_{20}^{{\rm \lpar IVS\rpar }} \comma \;S_{21}^{{\rm \lpar IVS\rpar }} \comma \;\ldots \comma \;S_{{\rm EW}}^{{\rm \lpar IVS\rpar }} } \rpar$, where we assume that individuals start working at the age of 20 and where we set EW = 64, and where we also assume that negative saving is impossible. If one buys a pension insurance contract at the age of 20 and a second contract at the age of 21, he will pay at the age of 21 a total premium amount of $S_{20}^{{\rm \lpar IVS\rpar }} + S_{21}^{{\rm \lpar IVS\rpar }}$. For simplicity, we assume that the individual assumes that his wage w and savings will not change over the coming years. Of course, this can be replaced by assuming a variable wage profile, but this will not substantially change the results of our paper. Moreover, for most individuals, the constant-wage-assumption seems to be a plausible behavioral assumption. On the other hand, each succeeding year the remaining lifetime utility function will change, at least due to the fact that the period till retirement is reduced by one year. The remaining lifetime utility functionFootnote ³ $\bar{U}_n$ of the decision-maker at age n looks like

(3.1)$$\bar{U}_n = W_n.U\lpar w-S_n^{{\rm \lpar IVS\rpar }} -{\rm O}{\rm O}_n\rpar + \lpar 1-W_n\rpar .U\lpar S_n^{{\rm \lpar IVS\rpar }} .G_n^{{\rm \lpar IVS\rpar }} \lpar i\rpar + {\rm O}{\rm B}_n\rpar \comma \;$$

where U(.) stands for the instantaneous utility function, with W _n( > 0) the weight attached to the remaining working life and 1 − W _n( > 0) the weight attached to the retirement period. When the individual grows older, the weight 1 − W _n on the retirement period will increase and the complementary weight W _n on the remaining working period will decrease.

The utility when working is $U\lpar w-S_n^{{\rm \lpar IVS\rpar }} -{\rm O}{\rm O}_n\rpar$, where the argument stands for the net consumption of workers and where $S_n^{{\rm \lpar IVS\rpar }}$ is the decision variable. The amount $S_n^{{\rm \lpar IVS\rpar }} .G_n^{{\rm \lpar IVS\rpar }}$ is the annual pension benefit derived from the new pension insurance. Clearly, the premium/benefit ratio $G_n^{{\rm \lpar IVS\rpar }} \lpar i\rpar$ is an increasing function of the prevailing interest rate i. The individual maximizes (3.1) with respect to $S_n^{{\rm \lpar IVS\rpar }}$. Since $G_n^{{\rm \lpar IVS\rpar }}$ increases with the interest rate, savings are a decreasing function in interest. The working individual has to take into account that there may be a mandatory occupational pension premium S ^(FF) and a mandatory social security contribution S ^(SS) to be paid as well, and perhaps premiums on voluntary pension insurance contracts closed in previous years, say $S_{20}^{{\rm \lpar IVS\rpar }} \comma \;\ldots \comma \;S_{n-1}^{{\rm \lpar IVS\rpar }}$. We call these amounts other obligations OO_n for short. Similarly, we define other benefits OB_n, consisting of occupational pension B ^(FF) = S ^(FF).G ^(FF), social security benefit B ^(SS) = S ^(SS).G ^(SS) and voluntary pensions $S_{20}^{{\rm \lpar IVS\rpar }} .G_{20}^{{\rm \lpar IVS\rpar }} \comma \;\ldots \comma \;S_{n-1}^{{\rm \lpar IVS\rpar }} .G_{n-1}^{{\rm \lpar IVS\rpar }}$ stemming from earlier pension contracts.

Here, we notice that we do not assume that an individual optimizes over a 45-period budget set, that is, he would apply dynamic programming at the age of 20 to plan his future consumption and savings at ages 10 or 30 years ahead. We stick to the relatively more realistic assumption that the individual will continue to save at the same rate for the years ahead. On the other hand, the individual has the possibility to adapt his/her savings pattern each working year, triggered by the fact that his utility function changes with age n as the retirement period draws closer.

If the instantaneous utility function is concave, i.e., the second-order derivative U′′ < 0, then it follows that the remaining lifetime utility function (3.1) is concave in $S_n^{{\rm \lpar IVS\rpar }}$ and consequently has a unique maximum. Since we exclude negative savings, the optimum may be a corner solution with $S_n^{{\rm \lpar IVS\rpar }} = 0$, in words, zero savings. There is still another instance where engaging in a voluntary pension insurance is not the first choice. If i < 0, a case which nowadays is not merely hypothetical in some countries, the individual would be better off hoarding the savings in cash rather than investing the savings in a voluntary pension contract at a negative interest rate. In that case, hoarding will yield $G_n^{{\rm \lpar HO\rpar }} = G_n^{{\rm \lpar IVS\rpar }} \lpar 0\rpar$ per dollar saved. In that case, the individual will maximize

(3.1a)$$\bar{U}_n = W_n.U\lpar w-S_n^{{\rm \lpar HO\rpar }} -{\rm O}{\rm O}_n\rpar + \lpar 1-W_n\rpar .U\lpar S_n^{{\rm \lpar HO\rpar }} .G_n^{{\rm \lpar HO\rpar }} + {\rm O}{\rm B}_n\rpar .$$

We see from (3.1) and (3.1a) that both an increase in OO_n and in OB_n will have a negative effect on the new savings.

The individual differentiates between four situations:

1. (a) If i > 0 and $\left[{{{{\rm d}{\bar{U}}_n} \over {{\rm d}S_n^{\lpar IVS\rpar } }}} \right]_{S_n^{{\rm \lpar IVS\rpar }} = 0} \ge 0$, then $S_n^{{\rm \lpar IVS\rpar }} \ge 0$ and zero hoardings

2. (b) If i > 0 and $\left[{{{{\rm d}{\bar{U}}_n} \over {{\rm d}S_n^{{\rm \lpar IVS\rpar }} }}} \right]_{S_n^{{\rm \lpar IVS\rpar }} = 0} \lt 0$, then $S_n^{{\rm \lpar IVS\rpar }} = 0$ and zero hoardings

3. (c) If i = 0 and $\left[{{{{\rm d}{\bar{U}}_n} \over {{\rm d}S_n^{{\rm \lpar HO\rpar }} }}} \right]_{S_n^{{\rm \lpar HO\rpar }} = 0} \ge 0$, then $S_n^{{\rm \lpar HO\rpar }} \ge 0$

4. (d) If i = 0 and $\left[{{{{\rm d}{\bar{U}}_n} \over {{\rm d}S_n^{{\rm \lpar HO\rpar }} }}} \right]_{S_n^{{\rm \lpar HO\rpar }} = 0} \lt 0$, then $S_n^{{\rm \lpar HO\rpar }} = 0$

Cases b. and d. may occur if the sum of pensions (FF and/or SS), built up thus far, is deemed already sufficient.

Finally, there is the border case i = 0, where the individual is indifferent between saving in pension contracts or hoarding in cash. These conditions have to be satisfied for each working cohort. For instance, if individuals start working at 20 and stop at 65, it implies 45 decisions to be made.

3.3 The median worker (MW)

Consider now the special position of the MW. As an individual, the MW may either save individually or hoard according to the behavioral rules just specified. However, as a MW, he is also the deciding cohort in the population of workers, that is, the age cohorts from 20 to 65. The decision here is whether there should be a mandatory funded occupational pension or not, and if so, what should be the size of that pension premium S ^(FF)? The pension-premium ratio for an individual pension starting at working age n is denoted by$G_n^{{\rm \lpar IVS\rpar }}$. If the MW opts for a fully-funded mandatory occupational pension, the corresponding pension-premium ratio is denoted by G ^(FF). If the mandatory pension covers all workers, it is identical to a pension contract starting at the beginning of the working period, i.e., at 20. Therefore, the corresponding pension/premium-ratio is then $G^{{\rm \lpar FF\rpar }} = G_{20}^{{\rm \lpar IVS\rpar }}$. It is now obvious that $G_n^{{\rm \lpar IVS\rpar }}$ is decreasing in n. More specifically, there holds $G_{{\rm MW}}^{{\rm \lpar IVS\rpar }} \lt G_{20}^{{\rm \lpar IVS\rpar }}$. It follows that the MW's first choice, if he is inclined to make additional pension savings, will be in favor of the mandatory occupational pension framework, since this presents better value for money than the individual contract would give. Hence, the MW will not go for an individual pension contract. However, it may be that the MW feels he already has enough pension contracts collected anyhow, and he will then abstain from the new mandatory contract as well. As a consequence, the negative decision would imply that there would be no mandatory occupational pension fund, because the MW is the one who decides on its existence. If he goes for the mandatory pension fund by choosing a (for him) optimum premium S ^(FF), then every worker, old and young, will have to participate in it, because it is a mandatory pension arrangement. If i > 0, the optimal premium is found by optimizing the remaining lifetime utility (3.1) with n = MW and $G_n^{{\rm \lpar IVS\rpar }}$ replaced by $G_{20}^{{\rm \lpar IVS\rpar }}$. If i < 0, the MW will hoard as well and optimize (3.1a).

3.4 The median voter (MV)

For the behavior of the MV, the situation is a bit different. If individual savings yield a better pension, that is, when $G_{{\rm MV}}^{{\rm \lpar IVS\rpar }} \gt G^{{\rm \lpar SS\rpar }}$, then the MV will opt for the individual arrangement, that is, either IVS or hoarding if i < 0, if he wants to create an additional pension. Then, there will not exist social security. If, on the contrary, $G_{{\rm MV}}^{{\rm \lpar IVS\rpar }} \lt G^{{\rm \lpar SS\rpar }}$, he will choose an additional social security pension if he wants to create additional pension. The optimal premium is found by optimizing the remaining lifetime utility (3.1) with n = MV and $G_n^{{\rm \lpar IVS\rpar }}$ replaced by $G_{}^{{\rm \lpar SS\rpar }}$. We notice that social security functions according to a PAYG-system. We have

$$S^{{\rm \lpar SS\rpar }}.P_{ {{\rm work}} \vert 20} = B^{{\rm \lpar SS\rpar }}.P_{ {{\rm ret}} \vert 20}.$$

Consequently G ^(SS) = (B ^(SS)/S ^(SS)) = (P _work|20/P _ret|20) is the support ratio, i.e., the inverse of the old-age dependency ratio.

3.5 The inner equilibrium

Since the joint optimization model just sketched consists of 2 × (65–20) + 2(2) = 94 interdependent first-order conditions in the 94 unknown S ^(IVS), S ^(FF), S ^(SS), and S ^(HO) corner solutions are possible, an analytical solution of this system is out of the question. Combining the conditions for S ^(IVS), S ^(FF), S ^(SS), and S ^(HO) the question is whether there is a numerical solution to the system

(3.2)$$\eqalign{S^{{\rm \lpar IVS\rpar }} = & f_{{\rm IVS}}\lpar S^{{\rm \lpar FF\rpar }}\comma \;S^{{\rm \lpar SS\rpar }}\comma \;S^{{\rm \lpar HO\rpar }}\vert {w\comma \;i} \rpar \cr S^{{\rm \lpar FF\rpar }} = & f_{{\rm FF}}\lpar S^{{\rm \lpar IVS\rpar }}\comma \;S^{{\rm \lpar SS\rpar }}\comma \;S^{{\rm \lpar HO\rpar }}\vert {w\comma \;i} \rpar \cr S^{{\rm \lpar SS\rpar }} = & f_{{\rm SS}}\lpar S^{{\rm \lpar IVS\rpar }}\comma \;S^{{\rm \lpar FF\rpar }}\comma \;S^{{\rm \lpar HO\rpar }}\vert {w\comma \;i} \rpar \cr S^{{\rm \lpar HO\rpar }} = & f_{{\rm HO}}\lpar S^{{\rm \lpar IVS\rpar }}\comma \;S^{{\rm \lpar FF\rpar }}\comma \;S^{{\rm \lpar SS\rpar }}\vert {w\comma \;i} \rpar \quad \forall S \ge 0} $$

for a given value interest rate i, where the f(.)s are short-hand notations for the optimization outcomes above. It appears that it is possible to find an equilibrium through iteration. In practice, as we will see later on, we always found a unique equilibrium. This equilibrium is found under the assumption that the interest rate i is exogenously fixed. We call this equilibrium solution the “inner” equilibrium. This assumption of an exogenously fixed interest rate is perhaps valid for a small open economy, but this assumption is not so realistic anymore, given the global mobility of nowadays. Dutch pension funds invest about 95% of their investments abroad. Rather, we have to see the whole world as a closed economy. Hence, we have to take into account that the equilibrium interest rate is endogenous as well. This “outer’ equilibrium is found by varying the interest rate over the corresponding inner equilibria until we find the rate for which capital demand equals capital supply. If this equilibrium entails a negative interest rate, we assume that the actual rate will fall to 0%, and that the remaining savings will be hoarded, since no one is interested in investing capital at a negative interest rate.Footnote ⁴

3.6 Capital, interest, and wages; the outer equilibrium

We have to endogenize the interest rate i and the corresponding wage rate w which had been taken to be exogenous up to this point. Thus, we introduce a capital market. We assume for simplicity that capital supply is only provided by individual savings and by the reserves of the occupational pension fund. Hence, we may write for the individual and collective accumulated savings per head of the population

(3.3)$$\eqalign{K^{{\rm \lpar IVS\rpar }} = & K^{{\rm \lpar IVS\rpar }}\lpar S^{{\rm \lpar IVS\rpar }}\semicolon \;i\comma \;D\rpar \cr K^{{\rm \lpar FF\rpar }} = & K^{{\rm \lpar FF\rpar }}\lpar S^{{\rm \lpar FF\rpar }}\semicolon \;i\comma \;D\rpar \comma \;} $$

where we summarize the demographic variables by D for the moment. This total capital supply per head of the population yields the capital per worker of

(3.4)$$k_{\sup } = \lpar K^{{\rm \lpar IVS\rpar }} + K^{{\rm \lpar FF\rpar }}\rpar /P_{ {{\rm work}} \vert {\rm SW}}. $$

We notice that this capital supply is a function of the interest rate i, since G ^(IVS) and G ^(FF) depend on i. Looking at (3.1) it can be seen that an interest increase leads to a decline in voluntary and mandatory savings. It follows that capital supply is decreasing in interest. We close the model by introducing a capital demand function per worker, denoted by k _dem(i), standing for the demand of an optimizing firm owner. The demand function k _dem(i) is monotonically decreasing in i as well. For a stable full-employment equilibrium, k _dem(i) is derived by maximizing the profit per worker. The marginal condition is

(3.5)$${\,f}^{\prime}\lpar k\rpar -\lpar i + \delta + \nu \rpar = 0\comma \;$$

where we assume a production f(k) per worker. Capital costs consist of three components, viz., interest, depreciation, and new investment to cope with population growth (or decline) to ensure a constant capital per head of the population.

An outer equilibrium is there where k _dem(i) = k _sup(i). In this general model, we cannot exclude that there will be more than one equilibrium. In our model to be specified hereafter, we find that both capital supply and demand fall with increasing interest, but that demand for zero interest is much higher than supply, while the demand curve is much more steeply falling than the supply curve with increasing interest.

Consequently, in the model specified hereafter, we found only one point of intersection where k _dem(i) = k _sup(i); the equalizing value of i is the equilibrium interest rate. An example is sketched in Figure 3 for a retirement age of 65, a birth rate of 0.10 (two children per couple) and an annual survival rate of 95% during retirement. This equilibrium is called the “outer” equilibrium.

Figure 3. Capital demand and supply curves.

It stands to reason that solving this for the equilibrium interest in the capital market requires an iterative solution. It follows that we have two sequential iteration processes: the inner loop converging to the inner equilibrium, which gives the total capital supply function per worker as a function of the interest rate i, and the outer loop where the interest i is varied until the equilibrium interest rate has been found at which supply and demand curves intersect each other. That final result is called the outer equilibrium.

In the next section, we will specify the model by choosing specific functions and parameter values. If the supply and demand curves intersect each other for a negative interest rate i, hoarding of part of the supply becomes relevant. The interest rate reaches its lower bound at i = 0 and the difference between supply and demand at i = 0 is hoarded. Whether the hoarding is done wholly by individuals or by the pension fund, or through a mixture of both, is not determined.

4.1 Demography

The population at time t is described by a vector $N_t^{\rm ^{\prime}} = \lpar N_{0\comma t}\comma \;\ldots \comma \;N_{100\comma t}\rpar$ where N _n,t stands for the number of people of age n at time t. The population develops according to the well-known Leslie (Reference Leslie1945) model [see also Lotka (Reference Lotka1907)] described by the matrix equation system

(4.1)$$\eqalign{& N_{0\comma t + 1} = \beta {\rm ^{\prime}}N_{-\comma t} \cr & N_{-\comma t + 1} = {\rm M}{\rm N}_{-\comma t}\comma \;} $$

where N _0,t stands for the number of newborns at time t, and $N_{-\comma t}^{\rm ^{\prime}} = \lpar N_{1\comma t}\comma \;\ldots \comma \;N_{100\comma t}\rpar$ stands for the vector of age cohorts from 1 to 100, where β stands for a vector of (age-specific) birth rates, and where M stands for a diagonal (100 × 100)-matrix of (age-specific) survival rates. The diagonal elements of M are also denoted as the vector μ. We assume that there is a fertility period of 10 years during which individuals may have children. This fertility period starts at the age of 25 and ends at 34. During that period, the annual birth rate is taken to be constant at β per individual. Since no difference is made between males and females, at β = 10% a couple is just reproducing [the expected number of children is 2(=2 × 10 × β)], if we exclude child mortality, as we do. Consequently, the population growth rate for β = 10% is ν = 0%. In order to investigate the effect of changes in the birth rate, we simulate the model for β = 7.5%, 10%, …, 30% where β = 7.5% stands for 1.5 children per couple and β = 30% stands for six children per couple.

It is well known from demographic theory that along the equilibrium path, the population is growing at a constant rate ν and has an age distribution p = (p ₀, …, p ₂₀, …, p ₆₅, …, p ₁₀₀). For ease of exposition, we shall assume no mortality, that is, μ = 1, before 65 and a constant annual survival rate μ < 1 from the age of 65 onwards. We will vary the annual survival rate from 0.93 to 0.98. We assume that individuals start working SW (StartWork) at the age of 20 and retire when they have reached the retirement age SP (StartPension). Their last working year is EW (EndWork = SP − 1).

Assuming that individuals younger than 20 do not work, from here on we denote the adult population share of the workers in the age interval [SW,  EW] by P _work and the share of the retired in the age interval [SP, 100] by P _ret, where we normalize such that P _work + P _ret = 1. We will vary SP from 63 to 72.

In order to gain insight into the effects of key demographic parameters, we present Table 1. We include the retirement age as a demographic variable, although strictly speaking it is not a demographic variable but is mostly determined by lawmakers. It is obvious that we can opt for a more sophisticated demographic model where the birth rate and survival patterns vary continuously with age; computationally, this is no problem. However, it implies that birth and survival patterns cannot be easily characterized by only one parameter each, which would obscure our analysis.

Table 1. Effect of demography on the demographic key variables

4.2 Pension systems

In modern societies, we mostly find a mixture of those systems simultaneously present, although the sizes of those systems differ between economies. The pension composition (PC) may be described by a vector

$$\lpar S_{{\rm SW}}^{{\rm \lpar IVS\rpar }} \comma \;\ldots \comma \;S_{{\rm EW}}^{{\rm \lpar IVS\rpar }} \comma \;B_{{\rm SW}}^{{\rm \lpar IVS\rpar }} \comma \;\ldots B_{{\rm EW}}^{{\rm \lpar IVS\rpar }} \comma \;S^{{\rm \lpar FF\rpar }}\comma \;B^{{\rm \lpar FF\rpar }}\comma \;S^{{\rm \lpar SS\rpar }}\comma \;B^{{\rm \lpar SS\rpar }}\comma \;S_{{\rm SW}}^{{\rm \lpar HO\rpar }} \comma \;\ldots \comma \;S_{{\rm EW}}^{{\rm \lpar HO\rpar }} \comma \;B_{{\rm SW}}^{{\rm \lpar HO\rpar }} \comma \;\ldots \comma \;B_{{\rm EW}}^{{\rm \lpar HO\rpar }} \rpar$$

or more briefly by (S ^(IVS), S ^(FF), S ^(SS), S ^(HO)). We assume that under voluntary individual saving the individual at age n (n = SW, …, EW) may buy a pension insurance contract according to which he agrees to pay a premium $S_n^{{\rm \lpar IVS\rpar }}$ for the rest of his working life in exchange for an annual pension of $B_n^{{\rm \lpar IVS\rpar }}$, starting at the retirement age. The link between premiums and benefits is given by the actuarial balance equation

(4.2)$$\eqalign{& S_n^{{\rm \lpar IVS\rpar }} .\left[{1 + \displaystyle{1 \over {1 + i}} + \cdots + {\left({\displaystyle{1 \over {1 + i}}} \right)}^{{\rm EW}-n}} \right] \cr & \quad= \,B_n^{{\rm \lpar IVS\rpar }} .\left[{{\left({\displaystyle{1 \over {1 + i}}} \right)}^{{\rm SP}-n}.\mu + \cdots + {\left({\displaystyle{1 \over {1 + i}}} \right)}^{{\rm EP}-n}.\mu^{{\rm EP}-{\rm EW}}} \right].} $$

We define the benefit-premium ratio $G_n^{{\rm \lpar IVS\rpar }}$ for the contract by $B_n^{{\rm \lpar IVS\rpar }} = G_n^{{\rm \lpar IVS\rpar }} .S_n^{{\rm \lpar IVS\rpar }}$. Benefits are proportional to the premium paid. The sum of those benefits for all voluntary pension contracts at the start of retirement, that is, the total individual pension, will be denoted by $\hat{B}_{{\rm EW}}^{{\rm \lpar IVS\rpar }} = \sum\nolimits_{n = {\rm SW}}^{{\rm EW}} {B_n^{{\rm \lpar IVS\rpar }} }$. For the hoarding benefits we get, similarly, $\hat{B}_{{\rm EW}}^{{\rm \lpar HO\rpar }} = \sum\nolimits_{n = {\rm SW}}^{{\rm EW}} {B_n^{{\rm \lpar HO\rpar }} }$. In a similar way, we denote the mandatory funded pension by its premium S ^(FF) and the corresponding benefit by B ^(FF). Since all age groups from SW = 20 onwards are obliged to participate in the mandatory system, this mandatory insurance is identical to the voluntary insurance in which we may participate at the age of 20. It follows that $G^{{\rm \lpar FF\rpar }} = G_{20}^{{\rm \lpar IVS\rpar }}$.

In Figure 4, we sketch the behavior of the G ^′s as functions of the interest rate i. The social security contribution is S ^(SS) and the corresponding benefit B ^(SS). There holds $G^{{\rm \lpar SS\rpar }} = {{B^{{\rm \lpar SS\rpar }}} \over {S^{{\rm \lpar SS\rpar }}}} = {{P_{{\rm work\vert 20}}} \over {P_{{\rm ret\vert 20}}}}$.

Figure 4. The behavior of $G_{{\rm MV}}^{{\rm \lpar IVS\rpar }} \comma \;G^{{\rm \lpar SS\rpar }}\comma \;G^{{\rm \lpar FF\rpar }}$as a function of the interest rate.

The MV may make a choice between IVS and social security. If $G^{{\rm \lpar SS\rpar }} \lt G_{{\rm MV}}^{{\rm \lpar IVS\rpar }}$, he will prefer to save individually instead of contributing to a social security system. This may be the case for high interest rates. In Figure 4, this occurs if the interest rate exceeds about 5.5%. If the MV prefers the individual pension, there will not be (a majority for) a social security system in the society.

In some societies, the institutional structure may be such that not all three systems are at work. For instance, in Chile there is no social security arrangement for old-age pensions on a PAYG-basis. In other countries, occupational pensions are mostly run on a pay-as-you-go basis. Hoarding in cash is a primitive last method of saving for old age. We refer to OECD (2017) for an international survey. In countries lacking a banking system, individual saving may be nearly impossible.

In this paper, we assume that all four pension schedules are accessible, even if some of those schedules are not actually used in the equilibrium. We assume that all voluntary and mandatory savings by individuals are eventually aimed at safeguarding an old-age pension. The retirement age SP is fixed here at 65. Later on we shall also vary the retirement age.

4.3 Capital supply

The resulting aggregate of individual saving reserves per working adult of age n ≤ EW, where EW is set at 64, is

(4.3a)$${\rm RES}_n^{{\rm (IVS)}} = \sum\limits_{\,j = {\rm SW}}^n {S_j^{{\rm (IVS)}} .\sum\limits_{m = j}^n {{(1 + i)}^{m-j}} } . $$

The individual reserves for a retiree at age n ≥ 56 are the present values of the future benefit flow

(4.3b)$${\rm RES}_n^{{\rm \lpar IVS\rpar }} = \hat{B}^{{\rm \lpar IVS\rpar }}.\sum\limits_{t = n}^{{\rm EP}-1} {\left({\displaystyle{\mu \over {1 + i}}} \right)} ^{t-n}n \ge {\rm SP} . $$

It follows that the average IVS reserve per head in the adult population is

(4.3c)$${\rm RE}{\rm S}^{{\rm \lpar IVS\rpar }} = \sum\limits_{n = {\rm SW}}^{{\rm EW}} {\,p_{n\vert {{\rm SW}} }} .{\rm RES}_n^{{\rm \lpar IVS\rpar }} + \sum\limits_{n = {\rm SP}}^{{\rm EP}} {\,p_{n\vert {{\rm SW}} }} .{\rm RES}_n^{{\rm \lpar IVS\rpar }} . $$

The per capita reserve in the mandatory FF-system is calculated likewise. It equals the individual pension contract for n = 20, where the premium S ^(FF) is determined by the MW. Hence, we get

(4.3d)$${\rm RE}{\rm S}^{{\rm \lpar FF\rpar }} = S^{{\rm \lpar FF\rpar }}\sum\limits_{\,j = {\rm SW}}^{{\rm EW}} {\,p_{\,j\vert {{\rm SW}} }.\sum\limits_{m = j}^{{\rm EW}} {{\lpar 1 + i\rpar }^{m-j}} } + S^{{\rm \lpar FF\rpar }}.G^{{\rm \lpar FF\rpar }}.\sum\limits_{\,j = {\rm SP}}^{{\rm EP}} {\,p_{\,j\vert {{\rm SW}} }} .\sum\limits_{t = j}^{{\rm EP}-1} {\left({\displaystyle{\mu \over {1 + i}}} \right)} ^{{\rm EP}-j-1}. $$

For other values of the retirement age SP the formulas have to be changed slightly because mortality may start before retirement when individuals are still at work.

The total capital supply per workplace is the sum of individual and collective savings. We have

(4.4)$$k_S\lpar i\rpar = \lpar {{\rm RE}{\rm S}^{{\rm \lpar IVS\rpar }}\lpar i\rpar + {\rm RE}{\rm S}^{{\rm \lpar FF\rpar }}\lpar i\rpar } \rpar /P_{{\rm work}\vert {{\rm SW}} }.$$

Since social security is run on a pay-as-you-go basis, it does not generate a reserve. The same holds for hoarding.

4.4 Parameter values

The choice of specific parameter values is a delicate one. There are many different estimates and they also vary between countries, between moments of estimation, and between the empirical estimation methods used. Since we are developing a general theory and our numerical simulations are only intended to get qualitative insights, we abstain from calibrating our parameter values in order to fit one specific country at a specific moment in time.

For the instantaneous utility function, we take the well-known Constant Relative Risk Aversion (CRRA) specification U(y) = y ^1−γ/(1 − γ), where we take γ = 3. In the literature, there are many estimates for γ, but they vary over a great range. See e.g., Gandelman and Hernandez-Murillo (Reference Gandelman and Hernandez-Murillo2015) and the recent survey by Outreville (Reference Outreville2015). See also Booij and Van Praag (Reference Booij and Van Praag2009). The value of 3 is somewhere in the middle of recent empirical estimates, but there is much uncertainty about it. The time weights are assumed to be

$$W_n = \displaystyle{{\sum\nolimits_{m = n}^{{\rm EW}} {\rho ^{m-n}} } \over {\sum\nolimits_{m = n}^{{\rm EP}} {\rho ^{m-n}} }}\comma \;$$

where the subjective time discount rate ρ is set equal to 0.89. There is a host of different estimates for ρ as well, but for macro-economic long-term decision settings, this value seems to be in the middle of the range [see Shane et al. (Reference Shane, Loewenstein and O'Donoghue2002)]. It appears that the outcomes of the model are very sensitive with respect to the value of ρ. We therefore tried several values.

For the production function, we take the traditional Cobb–Douglas function $Y = C.K^\alpha .L^{1-\alpha }$ where we use the traditional value α = 0.25. This value is debatable, too, since the capital elasticity varies a lot between industries and seems to increase over the years [see Piketty (Reference Piketty2014), Karabarbounis and Neiman (Reference Karabarbounis and Neiman2014), OECD (2015)]. Finally, we assume the depreciation rate to be δ = 10%. Also, here the value of the macro-depreciation rate is rather uncertain. We refer to Nadiri and Prucha (Reference Nadiri and Prucha1996) and a recent very down-to-earth but detailed catalogue of depreciation rates as prescribed by the New Zealand tax authorities [Taake (Reference Taake2017)].

6.1 The subjective time preference rate

We start by varying ρ from 0.88 up to 0.92, while setting β = 0.10, μ = 0.95, and SP = 65.

An increase in ρ implies that all parties put more weight on the retirement period. This will result in more capital and/or more social security. More savings will be reflected in more capital per worker. We see that if ρ increases from 0.88 to 0.92, capital per worker increases by 27% from 2.656 to 3.393.

This implies a capital elasticity with respect to ρ in the order of 5. It seems to imply, intuitively not implausibly, that the outcomes are rather sensitive with respect to ρ. We see that with an increase in ρ, the equilibrium interest rate falls from 2% to 0%.

In the fifth line of the table, the interest rate would become negative if we excluded the possibility of hoarding. In the last line, the outcomes are presented when hoarding is possible, i.e., when the interest rate is fixed at a lower limit of 0%. The amount hoarded in the last situation is about 10% according to the last column.Footnote ⁵ The net benefit-ratio is approximately 48.9/80.5, that is, around 60%. The total savings ratio is about 19.4%, of which 4.9% is spent on social security. Finally, we look at voluntary savings behavior over life. The individual with ρ = 0.88 starts at 20 with a tiny individual savings ratio of 0.7% which increases over life to 11.1% just before retirement. For an individual with a higher time preference of 0.92, the corresponding ratios are 4% and 10.8%, respectively.

6.2 Increasing longevity

We will now consider in Table 3 how the equilibrium changes if the survival rate μ is varied from 0.93 up to 0.98, keeping β = 0.10, SP = 65, and ρ = 0.89. Here, the interesting changes are seen in savings behavior. Individual savings dwindle when life expectation increases while the mandatory schedules gain in weight. The occupational pension premium increases from 2.5% to 3.5%, but the major change is in the role of social security. The social security premium rises from 0.6% to 9.1%, while the ratio of a fully-funded pension to the social security benefit 19.1/2.1 = 9.1 for μ = 0.93 (life expectancy 77) that ratio changes into 11.1/15.9 = 0.70 for μ = 0.98 (life expectancy 90).

Table 3. The effect of aging on the pension composition

Birth rate (β) = 0.10, age of retirement (SP) = 65, and time discount rate (ρ) = 0.89. For legend see Appendix.

Or in other words, individual pensions amount to 61% of total pension, the mandatory occupational pension to 35%, and social security to a meager 4% of total pension for μ = 0.93. For a rather old population, these fractions are 27%, 30%, and 43% respectively. Total pension as a fraction of gross wage falls from 55.1% to 36.9% and the net-benefit ratio falls rather dramatically from about 65% to 45%. The situation of workers does not change dramatically, but the situation for pensioners does deteriorate dramatically. If we may believe these figures, at least qualitatively, the future for an aging society appears bleak.

6.3 Increasing birth rate

The effect of varying the birth rate is not so straightforward. A consequence of a rising birth rate is a strongly growing labor force. The effect when capital is unchanged is that the capital per worker becomes scarcer. It follows that the interest rate will increase while the gross wage will fall. Indeed, we see that the interest rate increases from a moderate 1.4% to 15.2% when the number of children increases from 1.5 to a, for developed economies, unusual six children per couple. Actually, the interest rate rises much faster than the population growth rate. Gross wages fall from 1.002 to 0.715 and capital per job falls from 3.186 to 0.826. This capital thinning is due to the fact that the ratio of workers to retired increases (see Table 1) from 1.92 to 9.95 (Table 4).

Table 4. The effect of changes in the birth rate on the pension composition

Survival rate (μ) = 0.95, age of retirement (SP) = 65, and time discount (ρ) = 0.89. For legend see Appendix.

The tremendous increase in the interest rate to about 15% makes voluntary and mandatory saving very profitable with, as a result, very tiny savings, while social security vanishes. When the birth rate rises, the situation of the retired relative to that of the workers improves a great deal and to such an extent that retirees' pensions are much larger than net wages, which is indeed surprising. For Western countries where the birth rate hovers around 0.10 or below, we get rather low interest rates. Countries where the birth rate is still 0.20 or above are nowadays the less developed economies. In those countries, the whole pension system is clearly different from the one in our model as frequently there is not a well-developed IVS-, FF-, and/or SS-system. Moreover, the demography is rather different from ours with respect to the survival rate and the same probably holds for the subjective time discount rate ρ.

6.4 Increasing retirement age

Finally, let us consider the effect of the retirement age. We assume here that individuals of 72 are as efficient workers as those of 63, which is improbable in reality. We see a similar phenomenon as when the birth rate increases. Capital has to be spread over more workers with the effect that gross wage decreases and the interest rate increases. When the retirement age increases, there is a decline in the inequality between workers and the retired. If the retirement age increases to 71, we find that the retired become even better off than the workers due to the increase of the interest rate to 11.7% (Table 5).

Table 5. The effect of changes in the retirement age on the pension composition

Birth rate (β) = 0.10, survival rate (μ) = 0.95, and time discount (ρ) = 0.89. For legend see Appendix.

7.1 Realism

The dilemma behind economic modeling is that we have to choose between realism and transparency. If we opt for realism, we may end up with a jungle of details, actors, relationships, and variables. At the other extreme, we may have a simple elegant and transparent model, but it is so far simplified and stylized that it cannot be seen as a relevant description of reality. Moreover, by leaving out essential variables, we may find strongly biased effects of the remaining variables. We have looked for a compromise. Hence, some of our readers will object that our model is not realistic enough while others will complain that given the multiple simultaneous non-linear relationships in the model, we do not always get monotonic clear-cut effects which can be economically interpreted. However, the model in this paper can be easily extended to a more realistic demography, a heterogeneous labor force, a heterogeneous industrial sector, etc. It has to be seen as a first step. The main objective is to present a fresh way of thinking on the genesis of the pension composition as a mix of private savings (and incidentally hoarding) and the two main mandatory systems, which may be a stepping-stone to investigating the effects of changing demographics and retirement ages.

In this study, we assume a stable demography, that is, a fixed population growth rate (which is negative for β < 0.10) and a fixed age distribution. Clearly, this is unrealistic since the demographic parameters, i.e., birth and mortality rates, are never constant over time. However, since all demographic parameters change from one year to another, it is also not helpful to start from a specific population, say the American or the British, in a specific year, say 2016, and follow that population over a time period, when one wants to gain some insight into the general effects of demographic changes. The model which we have developed can be reformulated into a dynamic version, not starting from a stable equilibrium. But, even if we are not in an equilibrium situation and we assume birth and survival rates stay constant from now on, reaching the equilibrium path from the present disequilibrium would take many decades or even centuries. The attractiveness and the usefulness of studying an equilibrium model is that one can abstract from the specific peculiarities of different real situations, random shocks, intertemporal changes in values of model parameters, or in specifications of behavioral equations. We consider the stable equilibrium as the basic structure behind the reality. For country-specific studies that start from an actual demographic disequilibrium, we refer a.o. to Krueger and Ludwig (Reference Krueger and Ludwig2007), Börsch-Supan and Ludwig (Reference Börsch-Supan and Ludwig2010), Miles (Reference Miles1999).

7.2 What is novel?

One central, and to our knowledge novel point in our analysis is that we admit for the possibility of four simultaneously existing old-age support arrangements, viz., IVS, funded occupational pensions, social security on a pay-as-you-go-basis, and, as a residual component, the hoarding option. We call that mix the pension composition. That composition is not exogenously determined, but it is the joint result of the independent decisions of several parties, viz., all individual workers deciding on their individual savings, the MW (or trade union) as a representative of the body of workers deciding on the mandatory occupational pension system, and the MV as a representative of the electorate deciding on the existence and the size of social security [see also e.g., Galasso (Reference Galasso2008), Galasso and Profeta (Reference Galasso and Profeta2004), Bruce and Turnovsky (Reference Bruce and Turnovsky2013)]. All decision-makers act against the background of a specific demography and this demography determines their decisions in the last resort. And therefore, taking utility functions and the production function as given, the main macro-economic variables like wages, interest, and investments are in this model at the end determined by the demography as well. This stress on the different behavior of age cohorts and of two partly overlapping social classes, viz. active workers represented by the MW and the electorate as a whole, including the retired, represented by the MV, and the interpretation of the resulting pension composition as a Pareto equilibrium, seems to be novel as well.

We assume that in the economy the sources for capital investment are voluntary savings and mandatory savings for old age. In our time, the weight of institutional pension funds, pension insurance companies, and institutional savings funds is becoming overwhelming. We refer to Boeri et al. (Reference Boeri, Bovenberg, Coeuré and Roberts2006), Bijlsma et al. (Reference Bijlsma, Van Ewijk and Haaijen2014), Conference Board (2010), and Mitchell (Reference Mitchell2008). It would have been possible within this model to make an extension such that individuals could also save for private investment without the explicit goal of old-age provision, but this would not have changed the essential message of this paper. Moreover, our addition of a mandatory funded occupational pension, where the MW decides on the existence and the size of the pension, is also novel. With respect to individual savings, most authors assume that utility is a maximized subject to an intertemporal budget constraint. It is assumed that savings from one year may be used for consumption in the following year in order to smooth consumption, and that the citizen plans his/her savings and dis-savings for each future period over the remaining lifetime, given knowledge of his future annual incomes as well. This depicts a perfectly rational individual who has perfect knowledge of his future using the Euler conditions. But is such an assumption realistic when we face a future of about 45 years with 45 possible decision moments? Apart from the heroic assumption of the decision-making capacities of the individual, is it possible to know what the situation will be in the future many decades ahead? Instead, we make the rather naïve saving assumption that individuals expect their wages and their annual savings to remain constant over the years ahead. Each year to come, the individual will revise his savings decision based on the most recent situation. Although both assumptions do not seem perfectly realistic, we think that our assumption might be nearer to the truth in describing the savings behavior of ordinary humans than assuming an individual with perfect foresight on his lifetime 45-period budget equation. Our model can be generalized to encompass more general savings assumptions. For instance, we may assume that individuals decide each year on their savings for the current year only.

7.3 The main politically relevant results of our study are:

The finding that the room for governmental pension policy is rather restricted, because demography is the main determinant for the long-term equilibrium. It seems there are only a few possible political measures which all deal with the structure of old-age provisions: we may exclude one or more of the channels IVS, FF, or SS.Footnote ⁶ In this paper, we looked only at the triple combination IVS + FF + SS. Moreover, we may change the legal retirement age.

The demography appears to be a fundamental determinant of macro-economics, having effects on the wage rate, the interest rate, and capital per worker. This suggests that (the now frequently tabooized) population policy could (or even should) be a powerful instrument for reaching macro-economic targets (cf. Lee and Mason Reference Lee and Mason2010).

Aging of the population will result in a severe worsening of the net income of the retired.

Aging will also strongly increase the inequality between net wages and pensions to the disadvantage of the retired.

Increasing the retirement age would not have much effect on the financial situation of workers but it would improve the position of the retired.

Fertility increases will have a strong increasing effect on the interest rate.

Fertility increases may weaken social security and above a certain fertility rate social security may even vanish.

Fertility increases might strongly improve the situation of the retired.

Notwithstanding that this paper is based on a model which is oversimplified with respect to a number of issues, we believe that the way of thinking about the demographic problem in this paper sheds new light on one of the most threatening questions of our time: How do we provide for our old age, and what are the possibilities if we stick to the present institutional setup, where the pension composition is the joint result of the decisions of a number of parties?

Obviously, the model may be extended in many ways but already within the present setup its potential political relevance may be demonstrated by looking at a prediction for developed countries when we assume that they will stay at a low under-reproduction fertility level of 1.5 children per couple and a high survival rate of 0.98, that is a life expectancy of about 90 in our model, and that in the future 70 will be the new retirement age. Still, one step further would be to increase the maximum age in our model from 100 to 120 in order to reflect the increasing longevity in the present century. In Table 6, in the first line, we present the situation with a maximum age of 100 and in the second line the outcomes of the model when the maximum age is increased to 120.

Table 6. A look at a bleak future

For legend see Appendix.

When the maximum age is kept at 100, the rough prediction of our model would be a real interest rate of about 1.6%, an aggregate premium of about 16.4% of gross wage and a pension/net wage ratio of 39.8/83.6 = 48% (see Table 6). The capital per worker would be high at about 3.12. When we assume a maximum age of 120, which implies a lengthening of the potential retirement period from 30 to 50 years, the interest rate would increase to 2.7% and the aggregate premium to 21.8%. The pension/net wage ratio would again be about ½ while the gross wage would decrease by about 3.5%. Capital per job would decrease by about 10%. IVS would be nearly non-existent, while the social security premium would lean toward 18%.