2. The model

Our economy is populated by N _j,m,t agents where m denotes their type, j their age, and t the calendar year. Agents live for j = 1, 2, 3, …, J periods, π _j,t denotes their survival probability, with π _J,t = 0 at the maximum age. The probability of survival is homogeneous within a birth cohort, that is, common across m types of agents. Entry at j = 1, which corresponds to age 21 in reality, allows us to abstract from the education choice. Denote $N_{j, t} = \sum _mN_{j, m, t}$.

In the model, we compare the initial (ISS) and the final steady state (FSS). The economy is fully annuitized, that is, both private voluntary assets and incentivized OAS yield a rate of return including unintentional bequests (see also Bruce and Turnovsky, Reference Bruce and Turnovsky2013). Thus, we ignore the utility of life span uncertainty insurance, which typically differentiates private voluntary savings from public social security. Note that with this solution, there are no longer bequests to enter on the income side of the budget constraint.

We denote the behavioral types of agents by m. The first type is the fully rational agent typically featured in economic models. However, growing literature demonstrates that a small fraction of households behave in a manner consistent with complete rationality. For example, the consumption of retirees is substantially lower than that of the working-age population, whereas the difference is so large that it cannot be explained away by the decline in work-related consumption expenditures (Haider and Stephens, Reference Haider and Stephens2007; Battistin et al., Reference Battistin, Brugiavini, Rettore and Weber2009). To account for this empirical regularity, our model also features agents whose behavior is characterized by incomplete rationality.

2.1 Preferences

Agents of all types derive utility from consumption c _j,m,t and leisure (1 − l _j,m,t), where l _j,m,t denotes endogenous and perfectly elastic labor supply. We assume the instantaneous utility function of

(1)$$u( {c_{\,j, m, t}, \;l_{\,j, m, t}} ) = \log [ {} c_{\,j, m, t}^\phi ( {1-l_{\,j, m, t}} ) ^{1-\phi }] , \;$$

where ϕ denotes the weight of leisure in utility, identical across and within cohorts. Note that with log utility substitution and income effect cancel out, which is a conservative assumption from the perspective of this study.

Denote gross consumption by ${\cal C}_{j, m, t} = ( {1 + \tau_t^c } ) c_{j, m, t}$, where c _j,m,t is the experienced consumption and $\tau _t^c$ is the consumption taxation. Analogously, denote labor income earned by agents with $j < \bar{J}$ by ${\cal I}_{j, m, t} = ( {1-\tau^l-\tau } ) w_{j, t}l_{j, m, t}$ with w _j,t denoting age-specific wages. Labor income tax is denoted by τ ^l, and accordingly, τ denotes the contribution rate to mandatory and universal social security. Net pension benefits are denoted by ${\cal I}_{j, m, t} = ( {1-\tau^l} ) b_{j, m, t}$ for $j \ge \bar{J}$. The net capital income of an agent is denoted by ${\cal K}_{j, m, t} = \{ {[ {1 + r_t( {1-\tau^k} ) } ] ( {N_{j-1, t}/N_{j, t}} ) -1} \} a_{j-1, m, t}$, with r _t denoting the interest rate in the economy and a _j,m,t denoting the stock of assets. The annuitization of the economy implies that agents receive the survivor premium on assets.

Given that the agents can borrow during the working period against the OAS income (in particular, the financially illiterate agents would find it optimal), we impose a no-borrowing constraint, that is, $\forall _{j, m, t}\;a_{j, m, t} \ge 0$. Absent bequest motive, the terminal condition is given by a _J,m,t = 0. For brevity, it is convenient to define a sequence of lifetime consumption starting from age j in period t that has a length J − j as $\tilde{c}_{j, m, t} = \{ {c_{j, m, t}, \;c_{j + 1, m, t + 1}, \;\ldots , \;c_{J, m, t + J-j}} \}$ and analogously for sequence of lifetime labor supply $\tilde{l}_{j, m, t}$ and assets $\tilde{a}_{j, m, t}$.

2.2 Behavioral heterogeneity

Fully rational agents find optimum consumption and leisure path by solving the following problem:

(2)$$\eqalign{U_{\,j, m, t} = & \max {\{ {\tilde{c}}_{\,j, m, t}, {\tilde{l}}_{\,j, m, t}, {\tilde{a}}_{\,j, m, t}\} }u( c_{\,j, m, t}, \;l_{\,j, m, t}) \cr & \quad + \sum\limits_{s = 1}^{J-j} {\delta ^s} \displaystyle{{\pi _{\,j + s, t + s}} \over {\pi _{\,j, t}}}u( c_{\,j + s, m, t + s}, \;l_{\,j + s, m, t + s}) } $$

(3)$${\rm subject\ to}\colon \;a_{\,j, m, t}-a_{\,j-1, m, t-1} = {\cal I}_{\,j, m, t} + {\cal K}_{\,j, m, t} + \Upsilon -{\cal C}_{\,j, m, t} + {\cal T}_t.$$

where index m = FR and δ signifies time discounting; $\Upsilon$ denotes a lump-sum tax in the economy. We define intra-cohort lump-sum transfers ${\cal T}_t$, see equation (11) below. It is useful to represent the intertemporal behavior of the agents as marginal rate of substitution (MRS), which takes the following form for the fully rational agents.

(4)$$\eqalign{{\rm MRS\ at}\;age = & \; j\colon \displaystyle{{u_c( c_{\,j, m, t}, \;l_{\,j, m, t}) } \over {u_c( c_{\,j + 1, m, t + 1}, \;l_{\,j + 1, m + 1, t + 1}) }}\cdot \displaystyle{1 \over \delta }\cdot \displaystyle{{\pi _{\,j, t}} \over {\pi _{\,j + 1, t + 1}}} \cr = & \; \mu _{\,j + 1, t + 1} + ( 1-\tau ^k) {\bar{r}}_{\,j + 1, t + 1}, \;} $$

with $\bar{r}_{j, t} = \mu _{j, t}\cdot r_t$, where r _t follows from financial markets equilibrium, and μ _j,t = N _j−1,t−1/N _j,t is the survivor premium. The Euler condition generally holds only when the non-negative constraint on assets is non-binding. Otherwise, the LHS of the Euler equation for subsequent ages is greater than the RHS.

Time-inconsistent agents solve the following problem:

(5)$$\eqalign{U_{\,j, m, t} = & \max {\{ {\tilde{c}}_{\,j, m, t}, {\tilde{l}}_{\,j, m, t}, {\tilde{a}}_{\,j, m, t}\} }u( c_{\,j, m, t}, \;l_{\,j, m, t}) \cr & \quad + \beta \sum\limits_{s = 1}^{J-j} {\delta ^s} \displaystyle{{\pi _{\,j + s, t + s}} \over {\pi _{\,j, t}}}u( c_{\,j + s, m, t + s}, \;l_{\,j + s, m, t + s}) } $$

(6)$${\rm subject\ to}\colon \;a_{\,j, m, t}-a_{\,j-1, m, t-1} = {\cal I}_{\,j, m, t} + {\cal K}_{\,j, m, t} + \Upsilon -{\cal C}_{\,j, m, t} + {\cal T}_t.$$

where index m = TI. We denote additional discounting applied to all future periods by β. It is the so-called quasi-hyperbolic or naive time inconsistency. The MRS implied for the time-inconsistent agents is given by

(7)$$\eqalign{{\rm MRS\ at}\;age = & \; j\colon \displaystyle{{u_c( c_{\,j, m, t}, \;l_{\,j, m, t}) } \over {u_c( c_{\,j + 1, m, t + 1}, \;l_{\,j + 1, m + 1, t + 1}) }}\cdot \displaystyle{1 \over {\beta \delta }}\cdot \displaystyle{{\pi _{\,j, t}} \over {\pi _{\,j + 1, t + 1}}} \cr = & \; \mu _{\,j + 1, t + 1} + ( 1-\tau ^k) {\bar{r}}_{\,j + 1, t + 1}.} $$

For fully rational agents, MRS evaluated at j + s horizons is consistent with equation (4). It is not the case for time-inconsistent agents, for whom equation (7) holds between j and j + 1 at age j. The optimum s periods ahead, that is between j and j + s when evaluated at j is given by:

$$\displaystyle{{u_c( c_{\,j, m, t}, \;l_{\,j, m, t}) } \over {u_c( c_{\,j + s, m, t + s}, \;l_{\,j + 1, m + 1, t + 1}) }}\cdot \displaystyle{1 \over { \beta \delta ^s}}\displaystyle{{\pi _{\,j, t}} \over {\pi _{\,j + s, t}}} = \prod\limits_{i = 1}^s {( \mu _{\,j + s, t} + ( } 1-\tau ^k) \bar{r}_{\,j + s, t}) .$$

Time inconsistent-agents progressing in age systemically under-save. Note that in a limit, time-inconsistent agents behave like hand-to-mouth agents: with the infinite preference for the present period (β → 0), these agents will consume entire income instantaneously. Time-inconsistency implies that an agent sets an optimal plan but cannot implement it.

Financially illiterate agents solve the following problem:

(8)$$\eqalign{U_{\,j, m, t} = & \max {\{ {\tilde{c}}_{\,j, m, t}, {\tilde{l}}_{\,j, m, t}, {\tilde{a}}_{\,j, m, t}\} }u( c_{\,j, m, t}, \;l_{\,j, m, t}) \cr & \quad + \sum\limits_{s = 1}^{J-j} {\delta ^s} \displaystyle{{\pi _{\,j + s, t + s}} \over {\pi _{\,j, t}}}u( c_{\,j + s, m, t + s}, \;l_{\,j + s, m, t + s}) } $$

(9)$${\rm subject\ to}\colon \;a_{\,j, m, t}-a_{\,j-1, m, t-1} = {\cal I}_{\,j, m, t} + {\cal K}_{\,j, m, t} + \Upsilon -{\cal C}_{\,j, m, t}.$$

where index m = FI. Hence, they are able to put funds aside but cannot receive capital income gains.

The implied MRS of the financially illiterate agents is given by

(10)$${\rm MRS\ at}\;age = j\colon \displaystyle{{u_c( c_{\,j, m, t}, \;l_{\,j, m, t}) } \over {u_c( c_{\,j + 1, m, t + 1}, \;l_{\,j + 1, m + 1, t + 1}) }}\cdot \displaystyle{1 \over \delta }\cdot \displaystyle{{\pi _{\,j, t}} \over {\pi _{\,j + 1, t + 1}}} = 1$$

Given that the financially illiterate agents obtain no interest, they have no return on income withheld, that is, postponing consumption.

Asset market clearing. The interest rate on assets accumulated by financially illiterate agents is transferred to financially literate agents. It is implemented through lump-sum instantaneous transfers. Interest accrued by assets stored by financially illiterate agents is transferred to financially literate agents as

(11)$${\cal T}_t = \displaystyle{{\sum\nolimits_{m = FI} {\sum\nolimits_{\,j = 1}^J {N_{\,j, m, t}} } {\cal K}_{\,j, m, t}} \over {\sum\nolimits_{m\ne FI} {\sum\nolimits_{\,j = 1}^J {N_{\,j, m, t}} } }}.$$

Since ${\cal T}_t$ enters the budget constraint of financially literate agents as a lump-sum transfer, it does not affect their inter-temporal choices. Section 2.5 defines the market clearing interest rate.

When discussing the results, we systematically report the results for fully rational agents, that is, agents who display no time-inconsistency and are financially literate. We then report agents with varying degrees of time-inconsistency among both financially literate and financially illiterate households. We describe in section 5 the assumed population structure.

2.3 Social security

There is a universal pay-as-you-go defined benefit social security with an exogenous contribution rate τ and exogenous replacement ρ at retirement ($j = \bar{J}$). The contemporaneous benefits are financed from contemporaneous contributions. The value of the old age pension benefit for a cohort retiring in period t is given by:

(12)$$\forall _m\colon \;b_{\bar{J}, m, t} = \rho \cdot \sum\limits_{\,j = 1}^{\bar{J}-1} {w_{\,j, t}} l_{\,j, m, t}\quad {\rm and}\quad \forall _{\,j > \bar{J}}\colon \;b_{\,j, m, t} = ( 1 + g_t) b_{\,j-1, m, t}$$

where g _t denotes the economy payroll growth rate (in the steady state with constant population growth, it is equal to the exogenous technological progress), and w _j,t = ω _j × w _t denotes the age-specific productivity profile. Note that the agents, when solving their problem, include the future pension benefit in their labor derivative (Bütler, Reference Bütler2002). The German pension system is of a defined benefit character, but citizens record points in the pension system, which reflect the multiplicative of their productivity relative to the contemporaneous average in the economy, thus yielding strong labor supply incentives. Adopting the implicit taxation approach in the intra-temporal choice allows to reflect this feature.Footnote ⁸ In the PAYG DB system, as characterizes Germany, social security budget constraint is given by

(13)$$B_t = \mathop \sum \limits_m \mathop \sum \limits_J^{\,j = \bar{J}} N_{\,j, m, t}b_{\,j, m, t} = \tau _tw_tL_t + {\cal S}_t, \;$$

where we denote the social security deficit by ${\cal S}_t$ (it is negative if social security runs a surplus). The government clears the balance of the pension system. The universal, mandatory pension system does not change between our scenarios, although we study the change in the pension system parameters.

2.4 Government

There are four types of taxes: labor income, capital income, consumption, and lump sum tax. Per capita public spending G _t is stationary in both the initial and the final steady states, that is, it grows at γ _t. The government spends G _t, services public debt r _tD _t and balances social security following equation (13), financed through tax revenues and issuance of new public debt.

(14)$${\cal R}_t = \tau ^lw_tL_t + \tau ^lB_t + \tau ^kr_tA_t + \tau _t^c C_t + \Upsilon _t\sum\limits_{\,j = 1}^J {\sum\limits_m {N_{\,j, m, t}} } $$

(15)$${\cal R}_t = G_t + {\cal S}_t + r_tD_t, \;$$

where C _t, B _t, and A _t denote aggregate consumption, pensions benefits, and assets, respectively, whereas w _tL _t denotes labor revenue in the economy; we denote by ${\cal R}_t$ total tax revenue of the government. We set the debt in the initial steady state at par with the data. The final steady state debt to GDP ratio is the same; otherwise, it would generate welfare effects on its own.

In the initial steady state, we calibrate the tax rates to match the tax revenue shares in GDP, following Mendoza et al. (Reference Mendoza, Razin and Tesar1994). In the initial steady state, we close the government budget constraint portrayed in equations (14) and (15) with a lump-sum tax ($\Upsilon$). When the economy changes between the initial steady state and one of the studied final steady states, the consumption taxes adjust accordingly. Note that G _t and $\Upsilon _t\sum _j\sum _mN_{j, m, t}$ adjust due to population change. Note also that in our simulations, the contribution rate τ or replacement rate ρ adjusts to maintain ${\cal S}_t = 0$ in the final steady state despite longevity. As Section 4 describes, in the initial steady state, the social security is also balanced, in line with the data.

2.5 Production

The economy produces a composite consumption good. A representative firm employs labor and capital. Without loss of generality, we assume a Cobb-Douglas production function: $Y_t = K_t^\alpha ( {z_tL_t} ) ^{1-\alpha }$, with K _t denoting capital and L _t denoting labor aggregates. Capital depreciates at the rate of d. In line with empirical evidence, the economy features labor augmenting exogenous technological progress, γ _t = z _t+1/z _t. Given this parametrization, a representative firm optimizes labor and capital demand, thus yielding the price of capital and wage:

(16)$$r_t = \alpha K_t^{\alpha -1} ( {z_tL_t} ) ^{1-\alpha }-d\quad {\rm and}\quad w_t = ( {1-\alpha } ) K_t^\alpha z_t^{1-\alpha } L_t^{-\alpha } .$$

3. Policy reforms

Our model features the common initial steady state across all policy experiments. In all scenarios, longevity is higher in the final steady state than in the initial one, in line with the demographic projections. In the baseline scenario, no new instruments are introduced in the final steady state. In the reform scenarios, we offer several variants of the OAS schemes.

In the baseline scenario, we obtain changes in macroeconomic aggregates purely due to changes in life expectancy. Notably, longevity implies that the pension system stops being fiscally neutral: either the contribution rate has to increase or the replacement rate has to decline. The size of either of these adjustments is obtained in the model.

1. (a) We raise contributions to keep social security in balance. The required adjustment in the contribution rate informs on the size of a potential corresponding capital-based OAS scheme with fiscal incentives. In other words, we use the magnitude of this rise to set the size of OAS schemes.

2. (b) We reduce the replacement rate (pension benefits) to keep social security in balance. The comparison of the scenarios of increased contribution rates versus reduced replacement rates informs on the preferences of agents towards laissez faire, that is, the size of the pension system. We also keep the scenario of pension decline as our baseline for studying the effects of OAS programs.

Thus, our baseline is consistent with the following policy choice: reduce pensions to maintain fiscal stability in light of longevity and let laissez-faire private voluntary savings compensate for longer life at retirement with lower pensions. Given that we adjust the size of the OAS instruments to be fiscally equivalent, in practice, the exempt may signify full or partial tax exemption or even negative taxation (i.e., generally preferential tax treatment or a subsidy). In other words, the size of the exempt adjusts endogenously.

The key reforms in this study concern the introduction of incentivized OAS schemes. Recall that unincentivized, private voluntary savings for old-age consumption are subject to full taxation at contribution and accumulation phases but are exempt from any dedicated taxation at the disbursement stage. The government-subsidized old-age voluntary pension schemes are essentially tax incentives, that is, partial or full tax exemption. Following the OECD taxonomy, we introduce incentives to voluntary pension savings at the contribution, accumulation, and disbursement stages. Government-incentivized OAS instruments are typically capped in size (OECD, 2018, i.e., the tax-exempt eligibility is limited to a certain level of contributions or a certain level of assets). We use the implied rise in contribution rates from the baseline policy experiments to set this cap. Using OECD taxonomy, our OAS policy experiments are:

1. 1. typical OAS schemes, denoted as E-E-T (contributions and accumulation are exempt, pension benefits are taxed);

2. 2. T-T-E (taxed-taxed-exempt: contribution and accumulation stages are taxed, but disbursement is exempt from taxation) with proportional subsidy;

3. 3. T-T-E with a lump-sum subsidy.

Since we introduce various schemes, keeping them comparable in terms of fiscal size is imperative. For the OAS schemes to be comparable with the changes in the pension system parameters, we evaluate the indispensable rise in the contribution rate in the public pension system to keep the pension system balanced in light of longevity. We obtain the (cap of) contribution rate to the OAS by comparing the contribution rate with and without longevity. We thus set the rate of contribution to the E-E-T scheme (a cap), and to maintain internal consistency, we fully exempt these funds from labor taxation, social security contributions (in the contribution stage), and capital income taxation (in the accumulation stage). To keep the T-T-E scheme comparable in size, we measure fiscal expenditure on the tax exempts in the E-E-T scheme as the share of GDP and then assign a subsidy to retirees at disbursement to maintain the same level of fiscal expenditure as with the E-E-T scheme.

Thus, the tax incentives are equivalent, that is, we obtain the size of exempt such that in each scenario, the same amount of government funds (in % of GDP) is allocated to the households. The participation decision is voluntary: the households use up the entire cap but are free to choose the age at which they contribute to OAS. The specific formulae for each instrument are relegated to Appendix A. Note that OAS schemes are not a part of social security. OAS schemes induce adjustments in labor supply and wages, which affect the balance of the pension system, but OAS instruments do not enter directly into the pension system balance depicted in equation (13).

Introducing OAS schemes does affect the government budget constraint described by equation (14). First, not entire earned income (labor income in E-E-T and pension income in T-T-E schemes) is subject to labor taxation due to the OAS incentives. For the same reason, not entire capital income is subject to taxation (in the E-E-T scheme). It is reflected in the following fiscal closure. Let $\Xi _t = G_t-\Upsilon _t\sum _j\sum _mN_{j, m, t}$, then:

(20)$$t = FSS\colon \tau _t^c = \displaystyle{{\Xi _t + r_tD_t-\tau ^kr_t{\tilde{A}}_t-\tau ^lw_t{\tilde{L}}_t-\tau ^lB_t-{\rm {\mathbb I}}\tau ^l\sum\nolimits_{\,j = \bar{J}}^J {\sum\nolimits_m {{\rm taxable\ OAS\ benefits}} } } \over {C_t}}.$$

In this notation, ${\rm I}$ captures the fact that in some final steady states, the OAS schemes are implemented. Consequently, $r_t\tilde{A}_t$ denotes taxable capital income, which need not be the same as r _tA _t in the presence of OAS schemes, because with E-E-T scheme capital proceeds in the OAS scheme are exempt from taxation. Likewise, participants in either of the three OAS schemes do not pay labor income tax on contributions to the OAS, hence $w_t\tilde{L}_t < w_tL_t$ if at least one agent endogenously chooses to participate in E-E-T for at least one period. Finally, the rules for taxation of OAS benefits differ between the three studied schemes and are described in more detail in section 3.

Introducing OAS schemes changes the equilibrium definition, as per Section 2.2. Namely, the consumer choice variables include participation in OAS: the maximization problem is subject to altered budget constraints, given the relevant definition of ${\cal I}^r$ instead of ${\cal I}$ in case of participation.

The reforms are summarized in Table 1. First, we study an E-E-T scheme. For fully rational households, an E-E-T scheme implies that the interest rate in the OAS is higher than on voluntary saving (assets accumulated in the scheme are exempt from capital income taxation) and labor tax is lower (contributions to the E-E-T scheme are savings with a high return rather than taxes). Consequently, households may save less in voluntary assets (the OAS brings a higher rate of return) and effectively work less (the same number of hours worked brings a higher return over a lifetime). The implications are the same for households with access to financial markets (financially literate) but with time-inconsistent preferences. Finally, for financially illiterate households, OAS creates access to financial markets, thus dramatically raising their rate of return on savings. This group of households will most likely increase savings, but they also obtain a source of additional income, which reduces their incentives to work.

Table 1.

Policy reforms – summary

Notes: all instruments are fiscally equivalent in that the amount of tax redemptions or subsidies is the same as a share of GDP. The instruments also have the same contribution (see section 6). Participation is endogenous (voluntary) in all instruments.

In both T-T-E instruments, at the disbursement stage, households not merely receive their savings exempt from taxes but also an additional subsidy. We study two different kinds of subsidies. In the proportional T-T-E scheme, retirees receive a subsidy proportional to their pension benefit. In this setup, the individuals internalize the size of the future subsidy in their intra-temporal choices through the implicit link between labor supply and future pensions. In the T-T-E flat scheme, retirees receive a lump sum transfer every period during their retirement. This scheme is equivalent in fiscal terms, but since the transfer is a lump sum (the same for all participants), it does not enter the labor supply decisions during the agents' working periods.

For a fully rational agent, the proportional T-T-E instrument implies the same adjustments as the E-E-T instrument, so long as the two types of instruments are of similar magnitudes. However, a T-T-E flat scheme breaks the direct proportionality between labor supply and lifetime asset smoothing because the subsidy amount does not depend on the hours worked.

For financially literate agents with time-inconsistent preferences, T-T-E instruments provide lower incentives to save early in life than the E-E-T instrument. Moreover, since gains are only observed in the future, time-inconsistent agents assert that the incentives in T-T-E instruments as smaller than in the E-E-T instrument.

For financially illiterate agents, T-T-E instruments, in parallel to E-E-T, provide access to the market interest rate, thus fostering incentives to save at all. Since proceeds from capital accumulated in OAS are taxed, these incentives are lower than in the E-E-T scheme. Lump sum transfer in T-T-E flat instrument reinforces this effect of dampening the incentives.

4. Calibration

The economy is calibrated to replicate the features of the contemporaneous German economy. Demographics and technological progress are entirely exogenous in our model. We use the demographic forecast from Eurostat for 2080 to obtain survival rates for the final steady state. We assume the contemporaneous survival rates to reflect the initial steady state.

Following European Commission (2018), we set the rate of exogenous technological progress at 1% per annum in the initial steady state. We assume the capital income share in the economy to be α = 33%. Hours worked are calibrated using OECD Employment Outlook data. Our calibration features age-specific productivity profile ω _j. We follow Fehr et al. (Reference Fehr, Kallweit and Kindermann2015) in setting the age profiles.

We use national accounts data from Eurostat to obtain the target aggregates for the investment rate, government spending, and public debt as a share of GDP. We use averages for 1995–2017 as targets. We assume the interest rate of 6%, which is a conventional assumption for Germany.Footnote ¹² This interest rate, together with the investment rate, defines the depreciation rate in this economy.

We use OECD Tax Database data to obtain tax revenues as a share of GDP for consumption, labor, and capital taxes in the initial steady state. We follow Mendoza et al. (Reference Mendoza, Razin and Tesar1994) to obtain effective tax rates from these aggregates. We use averages for 1995–2017 as targets.

We use the share of pension expenditure in GDP to obtain the replacement rate. The Aging Working Group report for Germany (European Commission, 2018) provides data on the share of pension benefits in GDP.Footnote ¹³ We assume the social security to be balanced in the initial steady state, thus directly implying a contribution rate.Footnote ¹⁴ Relying on OECD (2018), we set the retirement eligibility age at $\bar{J} = 65$.

Table 2 reports the calibration of the macroeconomic aggregates in the model economy in the initial steady state. Note that our model economy is fully annuitized, whereas the aggregate German economy is not.

Table 2.

Calibration of the macroeconomic parameters

Note: We denote subcohort by m. AWG: Aging Working Group. National accounts data from Eurostat. Employment data from OECD Employment Outlook. Tax data from OECD Tax Database. We obtain the values of the parameters by calibrating the economy, which is not fully annuitized. This calibration refers to a capital share in the economy α = 33%. We report the alternative calibration of the initial steady-state with a capital share in the economy α = 45% in Table F.1 in the Appendix. The interest rate reported in Table 2 is r _t as per equation (16). In the model with full annuitization on top of the interest rate, r _t, financially literate agents get annuity premium, μ _j,t. This renders the net interest rate faced by financially literate agents age-specific.

5. Behavioral (intra-cohort) heterogeneity

Observational data, especially if only cross-sectional, give limited opportunities to identify the scope of behavioral biases of incompletely rational agents. It is due to two major reasons. First, having no instantaneous savings is consistent with a wide variety of behavioral biases and life situations: hand-to-mouth behavior, a high degree of time inconsistency, and financial illiteracy – they all yield virtually no asset accumulation in the early years of the career.

Similarly, lack of instantaneous savings is also consistent with repaying credit, for example, mortgage, if individuals pursue investment in real estate as an old-age saving strategy. Second, observing instantaneous savings may be consistent with precautionary motives and savings for indivisible consumption goods and services (e.g., relatively more expensive durable goods or more expensive services such as children's education or vacation travel). It is particularly troublesome in cross-sectional data, where individual asset and consumption decisions are observed over time.

Given these constraints, empirical evidence on the behavioral structure of populations worldwide is scarce. We thus model our economy with the interest of insights into the varying degree of time inconsistency and its potential interaction with financial literacy. Accordingly, equinumerous subgroups of agents exhibit a different degree of time inconsistency. These subgroups are again split into financially literate and financially illiterate groups.

More specifically, our population consists of agents with varying degrees of time inconsistency as follows: equinumerous eleven subcohorts with β ∈ {0.5, 0.55, …, 1}. Within each such subcohort, 50% of agents are characterized by a lack of financial literacy, and 50% of agents are financially literate. Overall, that implies twenty-two subcohorts, each of the same share in a cohort of a given age, reflecting various behavioral patterns. This behavioral structure does not intend to reflect the prevalence of incomplete rationality in the population. Instead, it helps us to study in detail specific types of individuals.

In Figure 1, we portray the heterogeneity of asset accumulation lifetime profiles across the types of agents. Notably, agents lacking financial literacy are characterized by meager savings even if they do not suffer from time inconsistency. Agents with no financial literacy also postpone the period in life in which they begin to accumulate assets. A higher degree of time inconsistency further delays the moment of accumulating assets. Financially illiterate agents also de-accumulate assets faster than fully rational or financially literate agents with comparable levels of time inconsistency. Two mechanisms are at play, both working in the same direction. First, since these agents earn no interest on private voluntary savings even after retirement, their assets do not increase in value for lack of accrual. Second, postponing consumption comes at a high price: their MRS remains bound by 1, as revealed in equation (10). Agents with time inconsistency also de-accumulate assets faster than fully rational agents.

Figure 1.

Assets: private voluntary savings in initial steady state – calibration for α = 33%.

Note: figure portrays baseline profiles of asset accumulation across various types of agents. Dashed lines are used for financially literate agents. Solid lines are used for financially illiterate agents. Darker shades of gray signify β parameter closer to 1 (i.e., lower extent of time inconsistency). Thick black lines denote agents with β = 1 for reference. Fully rational agents are financially literate with no time inconsistency (β = 1, thick dashed lines).

In Figure 2, we portray the proxy for saving regret across the diverse group of incompletely rational agents. The fully rational agents are represented in this figure as financially literate with no time inconsistency (the bottom right corner of the graph). Intuitively, fully rational agents display no saving regret. Overall, the greater the departure from the profile of a fully rational agent, the greater the share of permanent consumption that an agent would give up to obtain the assets of a fully rational agent at retirement. This share reaches approximately 30–40% (depending on the metric) for the financially illiterate agents and as much as 45–60% for the financially literate agents with a high degree of time inconsistency (β ≈ 0.5).

Figure 2.

Saving regret, welfare in the spirit of Imrohoroğlu et al. (Reference Imrohoroğlu, Imrohoroğlu and Joines2003).

Note: saving regret computed following the procedure described in Appendix B.4. Saving regret is expressed in terms of consumption equivalent (% of lifetime consumption). The baseline scenario assumes pension decline to maintain pension system balance. Welfare in the spirit of Imrohoroğlu et al. (Reference Imrohoroğlu, Imrohoroğlu and Joines2003), see Appendix B.

Among financially literate agents, the ‘saving regret’ proxy is the highest for the highest degree of time inconsistency, which stems from the fact that these households deviate from the fully rational optimization the most. For financially illiterate agents, regardless of the degree of time inconsistency, the regret remains high, reflecting a high share of lifetime income lost due to the inability to accrue interest. The fact that welfare loss from not behaving ‘rationally’ about asset accumulation is greater for financially literate agents with extreme time inconsistency than for financially illiterate ones reflects the fact that financial literacy is modeled through the budget constraint, and those agents receive no interest on their assets past retirement (welfare gain from having their asset gap bridged is lower). The differentiated level and patterns of saving regret corroborate the case for studying financial literacy and time inconsistency as separate phenomena.