3.1 Positive feedback between the two endogenous variables

We observe from (17) and (18) the similarities as well as differences for the two systems: ψ = θ _b or ψ = ϕ _b. In sections 3.2 and 3.3, we will consider their differences by studying each of them individually.Footnote ¹⁷ Before that, we first focus on the common elements of these two systems of equations, which are given by the terms $\partial \tilde{S}( R^\ast ; \;\theta _b) /\partial R$ in (13) and $\partial \tilde{R}( S^\ast ; \;\theta _b, \;\phi _b) /\partial S$ in (14). These common elements, which concern the interaction between optimal schooling years and retirement age, exhibit interesting properties, as given in the following proposition. In all the propositions in this paper, we assume that the second-order conditions (A.7) to (A.9) hold for a meaningful maximization problem. (We have verified that they are satisfied computationally in our analysis in section 4.)

Proposition 1. For the life-cycle model given by (1) to (4),

(a) $\partial \tilde{S}( R^\ast ; \;\theta _b) /\partial R > 0$: anticipating that an exogenous shock will shift up (respectively down) the retirement age function, the agent changes the schooling years in the same direction; and

(b) $\partial \tilde{R}( S^\ast ; \;\theta _b, \;\phi _b) /\partial S > 0$: a rise (respectively fall) in schooling years leads to a subsequent change of retirement age in the same direction.

Proof. See Appendix A.

The intuition of Proposition 1 is as follows. We observe from the first-order condition (7) that the retirement age affects the marginal benefit, but not the marginal cost, of increasing schooling years. When the retirement age rises (say, in response to an exogenous shock), it shifts up the marginal benefit schedule because of a longer duration to reap the benefit of education. With an unchanged marginal cost schedule, the increase in retirement age induces the optimal schooling years to move in the same direction, as given in Proposition 1(a).

According to the first-order condition (8), a change in schooling years has two effects on the marginal benefit of continuing working: a term related to human capital function and another related to the normalized consumption level. We show from (6) and (7) that at the optimal schooling years, the effect on normalized consumption level becomes zero.Footnote ¹⁸ As a result, only one effect is related to the rate of return of accumulating human capital, as given in (14). Since the rate of return is positive in the relevant region, agents with more schooling will retire later, as given in Proposition 1(b).

According to Proposition 1, $\partial \tilde{S}( R^\ast ; \;\theta _b) /\partial R > 0$, and $\partial \tilde{R}( S^\ast ; \;\theta _b, \;\phi _b) /\partial S > 0.$ The changes in the two endogenous variables S* and R* (caused by a particular exogenous shock, e.g.) reinforce each other.Footnote ¹⁹ Positive feedback exists, and this contrasts with the other possibility of negative feedback in which the two derivatives are of opposite signs.Footnote ²⁰

In the process of showing the presence of positive feedback in the above system, our analysis also contributes to the literature about the Ben-Porath mechanism [such as Ben-Porath (Reference Ben-Porath1967); and Hazan (Reference Hazan2009)]. We show in Proposition 1(a) that, in anticipating, for example, an increase in retirement age in the future because of an exogenous shock, the individual chooses longer schooling years to receive the higher benefit of human capital accumulation. Proposition 1(a) extends Ben-Porath's (Reference Ben-Porath1967) result, which is proved assuming the retirement age is fixed, to an environment in which both schooling years and retirement age are choice variables.Footnote ²¹

3.2 Combining endogenous feedback and exogenous productivity increase

We first examine how an increase in the productivity parameter (ϕ _b), other things being equal, affects optimal schooling years and retirement age. The analysis is simpler in this case because productivity increase has no direct effect on schooling years $( \partial \tilde{S}( R^\ast ; \;\theta _b) /\partial \phi _b = 0)$. The results are summarized in the following proposition:

Proposition 2. Consider the life-cycle model given by (1) to (4). If

(20)$$0 < \sigma < 1, \;$$

then ∂S*/∂ϕ _b < 0, and ∂R*/∂ϕ _b < 0.

Proof. See Appendix A.

Proposition 2 states that an increase in the productivity parameter (ϕ _b) causes schooling years and retirement age to decrease, leading to a positive co-movement of these two variables in this model. The intuition of Proposition 2 is as follows. As observed in (17) and (18), which are obtained from solving (11) and (12) simultaneously, the impact of an exogenous shock (ϕ _b) on S* and R* can be expressed as the sum of two terms: the exogenous shock component ($\partial \tilde{S}( R^\ast ; \;\theta _b) /\partial \psi$ in (17) or $\partial \tilde{R}( S^\ast ; \;\theta _b, \;\phi _b) /\partial \psi$ in (18)) and the endogenous feedback component ($( \partial \tilde{S}( R^\ast ; \;\theta _b) /\partial R) \,( \partial \tilde{R}( S^\ast ; \;\theta _b, \;\phi _b) /\partial \psi )$ in (17) or $( \partial \tilde{R}( S^\ast ; \;\theta _b, \;\phi _b) /\partial S) \,( \partial \tilde{S}( R^\ast ; \;\theta _b) /\partial \psi )$ in (18)). Proposition 1, which concerns the feedback (or interaction) term, shows that both $\partial \tilde{S}( R^\ast ; \;\theta _b) /\partial R$ and $\partial \tilde{R}( S^\ast ; \;\phi _b, \;\theta _b) /\partial S$ are positive.Footnote ²² We call this feature the positive endogenous effect. Together with second-order condition (A.9), we see from (17) and (18), with ψ = ϕ _b, that when $\partial \tilde{S}( R^\ast ; \;\theta _b) /\partial \phi _b = 0$ (the direct effect of the productivity shock on schooling years is zero), each of the two total effects (∂S*/∂ϕ _b or ∂R*/∂ϕ _b) is positively related to $\partial \tilde{R}( S^\ast ; \;\theta _b, \;\phi _b) /\partial \phi _b$ (the direct effect of the productivity shock on retirement age). When 0 < σ < 1 according to (20), the income effect dominates the substitution effect, leading to negative exogenous effect (i.e., $\partial \tilde{R}( S^\ast ; \;\theta _b, \;\phi _b) /\partial \phi _b < 0$). Combining the negative exogenous effect and the positive endogenous effect leads to the negative total effect for the productivity shock in Proposition 2. Since the total effects for the productivity shock on S* and R* are negative, a positive co-movement of these two variables occurs after a productivity shock.

For the sake of completeness, we summarize in the following proposition the remaining two cases about the value of σ. The proof, which is only slightly different from that of Proposition 2, is omitted.Footnote ²³

Proposition 3. (a) When σ = 1, an increase in productivity level has no effect on both S* and R*.

(b) When σ > 1, an increase in productivity level leads to increases in both S* and R*.

3.3 Combining endogenous feedback and exogenous mortality decline

We now study how a change in the mortality parameter (θ _b), other things being equal, affects optimal schooling years and retirement age.

Solving (11) and (12) simultaneously gives (17) and (18), with ψ = θ _b. As in section 3.2, it is helpful to decompose the total effect of a mortality decline on schooling years or retirement age into the exogenous (shock) and endogenous (feedback) components. According to Proposition 1, the effects caused by the feedback terms are positive. On the other hand, the exogenous components are given by the two direct effects caused by the mortality decline: $\partial \tilde{S}( R^\ast ; \;\theta _b) /\partial \theta _b$ and $\partial \tilde{R}( S^\ast ; \;\theta _b, \;\phi _b) /\partial \theta _b$. We can easily see from (17) and (18) that the total effect on either schooling years or retirement age (∂S*/∂θ _b or ∂R*/∂θ _b) is a linear combination of these two direct effects.

We first examine these two direct effects separately, and then combine them to obtain the total effect.

3.3.1 The direct effect of a mortality decline on school years

Based on (A.14) as well as (A.7) and (A.18) in Appendix A, a positive value of $\partial \tilde{S}( R^\ast ; \;\theta _b) /\partial \theta _b$ is equivalent to

(21)$$\displaystyle{{\int _{S^\ast }^{R^\ast } \exp ( {-rx} ) l( {x; \;\theta_b} ) \left[{\int_{S^\ast }^x \left({-\displaystyle{{\partial \mu ( {t; \;\theta_b} ) } \over {\partial \theta_b}}} \right){\rm d}t} \right]{\rm d}x} \over {\int _{S^\ast }^{R^\ast } \exp ( {-rx} ) l( {x; \;\theta_b} ) {\rm d}x}} > 0.$$

According to (7), a mortality decline affects optimal schooling years (S*) directly through a higher future income stream (by increasing the survival probabilities from S* to R*) in the marginal benefit schedule and a foregone current income (by increasing the survival probability at age S*) in the marginal cost schedule. The mortality rate at the current age only affects future survival probabilities but not past survival probabilities; see also (1) and (A.17). Since the survival probabilities of age R* and above do not appear in (7), the effect of a change in θ _b on μ(.; θ _b) for ages after R* is irrelevant. Moreover, the analysis in (A.18) shows that the effects of a change in θ _b on μ(t; θ _b) for t ≤ S* on the marginal benefit and marginal cost schedules exactly cancel out. Thus, the direct effect of a change in θ _b on optimal schooling years depends only on its impact on μ(t; θ _b) for t ∈ [S*, R*].Footnote ²⁴ Equation (21) has a useful interpretation that the linear combination of the effects of a change in θ _b on the survival probabilities from age S* to age R*, which only appear in the marginal benefit schedule, is positive.

A more intuitive interpretation can further be obtained in the special case that the mortality decline process causes decreases in the mortality rates of the working years from S* to R*. In this case,

(22)$$\displaystyle{{-\partial \mu ( {t; \;\theta_b} ) } \over {\partial \theta _b}} > 0, \;\forall t\in [ {S^\ast , \;R^\ast } ] .$$

Since (22) is a sufficient condition for (21), we can easily see that (21) is satisfied when a change in θ _b decreases mortality rates during working years. The observed mortality decline in the twentieth century usually reduces mortality rates for most (but not necessarily all) ages. Thus, −∂μ(t; θ _b)/∂θ _b may be negative for some t, and (22) may not hold. However, the above arguments suggest that, based on the linear-combination interpretation described in the previous paragraph, (21) likely holds for many empirically relevant mortality decline processes.Footnote ²⁵

3.3.2 The direct effect of a mortality decline on retirement age

We observe from (8) that a mortality decline affects the normalized consumption level on the marginal benefit schedule and the disutility of labor term on the marginal cost schedule. According to (A.19) in Appendix A, the effect of a mortality decline on the consumption level can be decomposed into two effects, which are called the lifetime human wealth effect and the years-to-consume effect, following (23) of D'Albis et al. (Reference D'Albis, Lau and Sánchez-Romero2012).Footnote ²⁶ Combining (A.15) and (A.19), we show that the sign of $\partial \tilde{R}( S^\ast ; \;\theta _b, \;\phi _b) /\partial \theta _b$ is the same as that of

(23)$$\eqalign{& \displaystyle{1 \over \sigma }\displaystyle{{\int _0^T \exp \{ {-[ {( {1-\sigma } ) r + \sigma \rho } ] x} \} l( {x; \;\theta_b} ) \left[{\int_0^x \left({-\displaystyle{{\partial \mu ( {t; \;\theta_b} ) } \over {\partial \theta_b}}} \right){\rm d}t} \right]{\rm d}x} \over {\int _0^T \exp \{ {-[ {( {1-\sigma } ) r + \sigma \rho } ] x} \} l( {x; \;\theta_b} ) {\rm d}x}} + \displaystyle{{\left({-\displaystyle{{\partial \nu ( {R^\ast ; \;\theta_b} ) } \over {\partial \theta_b}}} \right)} \over {\nu ( {R^\ast ; \;\theta_b} ) }} \cr & \cr & \quad -\displaystyle{1 \over \sigma }\displaystyle{{\int _{S^\ast }^{R^\ast } \exp ( {-rx} ) l( {x; \;\theta_b} ) \left[{\int_0^x \left({-\displaystyle{{\partial \mu ( {t; \;\theta_b} ) } \over {\partial \theta_b}}} \right){\rm d}t} \right]{\rm d}x} \over {\int _{S^\ast }^{R^\ast } \exp ( {-rx} ) l( {x; \;\theta_b} ) {\rm d}x}}.} $$

Summing up the above analysis, there are three components in the direct effect of a mortality decline on retirement age: the lifetime human wealth effect (by shifting down the marginal benefit schedule), the years-to-consume effect (by shifting up the marginal benefit schedule), and the compression of morbidity effect (by shifting down the marginal cost schedule). If the sum of the years-to-consume and compression of morbidity effects is at least as large as the lifetime human wealth effect, then (23) is non-negative, and $\partial \tilde{R}( S^\ast ; \;\theta _b, \;\phi _b) /\partial \theta _b \ge 0$. In the main model considered by D'Albis et al. (Reference D'Albis, Lau and Sánchez-Romero2012), in which only the lifetime human wealth and years-to-consume effects exist, they argue that when a mortality decline concentrates on old ages, the lifetime human wealth effect is absent, and the years-to-consume effect is present, resulting in a delay in retirement. On the other hand, if a mortality decline concentrates on younger ages, then the lifetime human wealth effect may dominate, leading to earlier retirement. Thus, a mortality decline that simultaneously affects mortality rates at different ages will generally have an ambiguous effect on retirement age. In the life-cycle model described in section 2, with schooling years being endogenously determined and with an additional compression of morbidity effect, the various effects examined in D'Albis et al. (Reference D'Albis, Lau and Sánchez-Romero2012) are also relevant, leading to the same conclusion that the direct effect of a general mortality decline process on retirement age is usually ambiguous.

4.1 Specifications of the baseline model

As far as possible, the specifications in our baseline model are those commonly used in the literature. We use the Gompertz-Makeham specification for the survival function, as in Heijdra and Romp (Reference Heijdra and Romp2009) and Bloom et al. (Reference Bloom, Canning and Moore2014). The Gompertz-Makeham survival function, which involves three parameters, is given as

(24)$$l( x; \;\theta _b) = l^{GM}( x; \;\theta _b) = \exp \left\{{-\mu_{b, 0}x-\displaystyle{{\mu_{b, 1}} \over {\mu_{b, 2}}}[ {\exp ( \mu_{b, 2}x) -1} ] } \right\}, \;$$

where μ _b,i (i = 0, 1, 2) is related to the mortality parameter θ _b defined before, as follows:

(25)$$\mu _{b, i} = \mu _i( \theta _b) .$$

The corresponding age-specific mortality rate function is given by $\mu ^{GM}( x; \;\theta _b) = {-}{1 \over {l^{GM}( x; \;\theta _b) }}{{\partial l^{GM}( x; \;\theta _b) } \over {\partial x}} = \mu _{b, 0} + \mu _{b, 1}\exp ( \mu _{b, 2}x)$.Footnote ²⁹ Note that the coefficients are cohort-specific.

Following the specification of the disutility of labor function in Bloom et al. (Reference Bloom, Canning and Moore2014), the disutility of labor or study function of cohort b individuals is assumed to be proportional to that cohort's age-specific mortality rate function:

(26)$$\nu ( x; \;\theta _b) = \delta \mu ^{GM}( x; \;\theta _b) = \delta [ {\mu_{b, 0} + \mu_{b, 1}\exp ( \mu_{b, 2}x) } ] , \;$$

where δ > 0.

For the human capital function, we assume

(27)$$h( S) = \exp ( {\gamma S^\lambda } ) , \;$$

where γ > 0 and 0 < λ ≤ 1. This functional form is consistent with those in the literature, such as Cervellati and Sunde (Reference Cervellati and Sunde2013) and Cai and Lau (Reference Cai and Lau2017). According to this specification, the rate of return to schooling is

(28)$$\displaystyle{{{h}^{\prime}( S) } \over {h( S) }} = \gamma \lambda S^{\lambda -1}, \;$$

which is a decreasing function of S when λ is strictly less than 1.

Finally, we assume a constant growth rate in productivity, as in Restuccia and Vandenbroucke (Reference Restuccia and Vandenbroucke2013) and Bloom et al. (Reference Bloom, Canning and Moore2014). Normalizing the productivity index for the 1900 cohort as 1, this index for cohort b is given by

(29)$$\phi _b = \exp [ {g( b-1900) } ] , \;$$

where g (g > 0) is the growth rate of the productivity level.

4.2 Calibration

The values of the parameters in the baseline model are chosen to match those in the literature as far as possible. (We will also use other parameter values in the sensitivity analysis.) In particular, we choose ρ = 0.03, r = 0.03, and g = 0.0127, following Bloom et al. (Reference Bloom, Canning and Moore2014). We set σ, the intertemporal elasticity of substitution, to be 0.6. Following Boucekkine et al. (Reference Boucekkine, de la Croix and Licandro2003), we assume that N = 10.Footnote ³⁰ Finally, we assume that T = 100, which corresponds to a maximum biological age (T + N) of 110.

To estimate the parameters of the survival functions, we minimize the sum of squared residuals between the survival probability based on the Gompertz-Makeham specification (24) and the data of US men from the Berkeley Mortality Database. For each cohort, we first transform the data to obtain the survival probabilities conditional on reaching age N and then choose the three survival parameters (μ _b,0, μ _b,1, and μ _b,2) to minimize

(30)$$SSR^l( {\mu_{b, 0}, \;\mu_{b, 1}, \;\mu_{b, 2}} ) = \mathop \sum \limits_{x = 1}^T \left[{\exp \left\{{-\mu_{b, 0}x-\displaystyle{{\mu_{b, 1}} \over {\mu_{b, 2}}}[ {\exp ( \mu_{b, 2}x) -1} ] } \right\}-\displaystyle{{l_{N + x}^{data} } \over {l_N^{data} }}} \right]^2, \;$$

where $l_x^{data}$ is the survival probability data up to age x. Figure 2 shows the comparison of the survival probability data and that of the estimated Gompertz-Makeham model, for the beginning and ending years (1900 and 2000) as well as the mid-point (1950) of the US data in the Berkeley Mortality Database. We observe that the fit is good in every case. The fits in other years, which are not shown, are also good.

Figure 2. Survival probability: data and estimated Gompertz-Makeham survival functions.

For the estimation of parameters γ and λ in the human capital function h(S), we also use the non-linear least-squares method. Specifically, we choose γ and λ to minimize

(31)$$SSR^S( \gamma , \;\lambda ) = \mathop \sum \limits_{x = 0}^{15} \{ {S_{1900 + 5x}^\ast ( \gamma , \;\lambda ) -[ {S_{1900 + 5x}^{data} -( N-6) } ] } \} ^2, \;$$

where $S_b^\ast ( \gamma , \;\lambda )$ denotes the optimal schooling years of cohort b individuals calculated from the model, and $S_b^{data}$ is from the schooling years data set (for US men) used in Goldin and Katz (Reference Goldin and Katz2008), which starts in 1876 and ends in 1975.Footnote ³¹ There are only 16 observations, for the data in the 5-year interval, covered in both the Berkeley Mortality Database and Goldin and Katz (Reference Goldin and Katz2008). According to the estimation result, $\hat{\gamma } = 0.0482, \;\hat{\lambda } = 0.948$, and the root mean squared error (RMSE) is 0.302, which is reasonably small. The estimated schooling data fits the actual data reasonably well, as seen in Figure 3.

Figure 3. Schooling years.

Finally, we calibrate parameter δ of the disutility function such that the optimal retirement age for the 1900 cohort is 65.Footnote ³² The parameter values of the baseline case are summarized in Table 1.

Table 1. Parameters of the baseline model

4.3 Effects of mortality decline and productivity increase

In the quantitative analysis, we focus on 21 cohorts of US men, starting from 1900 and increasing every 5 years up to 2000. For each cohort, we use the specifications and parameters discussed above, together with the survival probability data of the corresponding cohort. The results of the baseline case are shown in Figure 4, with the optimal retirement ages given in the upper panel and the optimal schooling years given in the lower panel.

Figure 4. Baseline results.

The optimal retirement age rises in early cohorts (from an imposed value of 65 in 1900) until reaching the peak of 69.0 for the 1950 cohort and falls gradually afterward to 67.9 in 2000. We observe from Figure 4 that the rate of increase in retirement age from 1900 to 1950 is larger in magnitude than the rate of decrease from 1950 to 2000. The path of optimal schooling years, on the other hand, shows a rather different profile compared with that of the retirement age. The optimal schooling years series starts from 8.22 years for the 1900 cohort, increases all the way up to 13.71 years for the 1990 cohort, and then decreases slightly to 13.67 years for the 2000 cohort. We also observe from Figure 4 that the rate of increase in schooling years was quite high in the earlier decades, but the increase slowed down in the middle of the century.

To understand the trend of the optimal retirement age and schooling years paths, we perform decomposition exercises by isolating the effects of the two factors individually. Specifically, when analyzing the effect of mortality decline only, we shut down the productivity increase channel by using g = 0 in (29). On the other hand, when analyzing the effect of productivity increase only, we use μ _b,j = μ _1900,j (j = 0, 1, 2) in (24) and (26). In other words, we switch off the mortality decline channel by assuming that the survival functions in later decades are the same as the 1900 version. Note that we keep other parameters as those in Table 1 in these decomposition exercises.

We first look at the effect on optimal retirement age in the upper panel of Figure 4. When the productivity level is held constant, the retirement age increases monotonically from 65 to 78.1 in response to reductions in mortality rates. We also observe that the retirement age increases at a decreasing rate, implying that the mortality effect becomes weaker over time. This reflects the well-known fact that the rate of mortality decline in developed countries slowed down in the second half of the twentieth century. As observed in Figure 2, the shifting out of the survival functions from 1950 to 2000 is smaller than that from 1900 to 1950. On the other hand, the optimal retirement age decreases from 65 to 53.0 if only a productivity increase (and thus increased lifetime wealth) occurs. Note that the change in retirement age in response to a productivity increase is almost linear, unlike the effect of a mortality decline.

We observe similar effects on optimal schooling years. When only a mortality decline occurs, optimal schooling years increase monotonically from 8.22 to 18.45. On the other hand, when only a productivity increase occurs, the corresponding figure decreases from 8.22 years to 5.29. Again, we observe that the schooling years path is concave when only a mortality decline occurs, and is rather linear when only a productivity increase occurs.

The two results on retirement age (the positive effect of mortality decline and the negative effect of productivity increase) are consistent with those in Bloom et al. (Reference Bloom, Canning and Moore2014). In this sense, their qualitative results, which are developed for a model with exogenous retirement age, are also relevant when the retirement age is endogenously determined. On the other hand, we find that introducing the choice of schooling years brings new qualitative and quantitative results when we consider the effects of simultaneous changes in mortality and productivity. Bloom et al. (Reference Bloom, Canning and Moore2014) show that the magnitude of the decrease in optimal retirement age in response to the productivity increase is about twice that of the increase in retirement age in response to the mortality decline. As a result, the optimal retirement age decreases from 1901 to 1951 as well as from 1951 to 1996 [Bloom et al. (Reference Bloom, Canning and Moore2014), Table 3]. Our analysis, however, shows that the optimal retirement age increases in the first half of the century and then decreases in the second half. The intuition of this difference can be traced to the concave shape of the path of schooling years, which is caused by the substantial decline of mortality in the first half of the century and the decreasing rate of return of human capital formation. Bloom et al. (Reference Bloom, Canning and Moore2014) do not consider schooling choice, and thus, any indirect effect through schooling years on the retirement age is not captured.Footnote ³³

Regarding the impact on schooling years, the positive effect of mortality decline is consistent with that of Restuccia and Vandenbroucke (Reference Restuccia and Vandenbroucke2013). On the other hand, the negative effect of productivity increase on schooling years is different from the positive effect in Restuccia and Vandenbroucke (Reference Restuccia and Vandenbroucke2013). We will re-examine this issue in section 5.

4.4 Sensitivity analysis

We then perform the sensitivity analysis regarding the parameters listed in Table 1. The results of the sensitivity analysis are reported in a Supplementary Appendix available upon request.

We summarize the results briefly. The computational results are robust with respect to the interest rate (r), the subjective discount rate (ρ), the age when individuals begin making economic decisions (N), and the maximum age in the model (T), at least when they are within some relevant ranges. They are less robust with respect to the growth rate of productivity (g) and the intertemporal elasticity of substitution (σ) individually, but we also find that simultaneous changes in g and σ can lead to retirement age and schooling years paths close to the baseline case.

For each of the analyzed cases, we also perform decomposition exercises by focusing on one exogenous change (mortality decline or productivity increase) only. We find that in each case, the optimal retirement age and schooling years always change monotonically, and these two variables always move in the same direction.

5.1 Positive feedback

Similar to the analysis in section 3, we first obtain the relations between the two endogenous variables. We show that

(35)$$\displaystyle{{\partial \tilde{S}( R^\ast ; \;\phi _b) } \over {\partial R}} = \displaystyle{{\phi _b{h}^{\prime}( S^\ast ) \exp ( {-}rR^\ast ) l( R^\ast ) + \displaystyle{1 \over \sigma }\eta ( S^\ast , \;R^\ast ; \;\phi _b) \left[{\displaystyle{{\exp ( {-}rR^\ast ) l( R^\ast ) } \over {\int_{S^\ast }^{R^\ast } \exp ( {-}rx) l( x) {\rm d}x}}} \right]} \over {\Delta _S}}, \;$$

and that

(36)$$\displaystyle{{\partial \tilde{R}( S^\ast ; \;\phi _b) } \over {\partial S}} = \displaystyle{{\displaystyle{{{h}^{\prime}( S^\ast ) } \over {h( S^\ast ) }}-\displaystyle{1 \over \sigma }\displaystyle{1 \over {c^n( 0, \;S^\ast , \;R^\ast ) }}\displaystyle{{\partial c^n( 0, \;S^\ast , \;R^\ast ) } \over {\partial S}}} \over {r-\rho + \displaystyle{1 \over \sigma }\displaystyle{{\exp ( {-}rR^\ast ) l( R^\ast ) } \over {\int _{S^\ast }^{R^\ast } \exp ( {-}rx) l( x) {\rm d}x}} + \displaystyle{1 \over {\nu ( R^\ast ) }}\displaystyle{{\partial \nu ( R^\ast ) } \over {\partial x}}}}, \;$$

where

(37)$$\eqalign{& \Delta _S = \phi _b\exp ( {-}rS^\ast ) l( S^\ast ) h( S^\ast ) \left[{\displaystyle{{2{h}^{\prime}( S^\ast ) } \over {h( S^\ast ) }}-\displaystyle{{{h}^{\prime \prime}( S^\ast ) } \over {{h}^{\prime}( S^\ast ) }}-\mu ( S^\ast ) -r} \right]\cr & \quad + \eta ( S^\ast , \;R^\ast ; \;\phi _b) \left[{\displaystyle{{{h}^{\prime \prime}( S^\ast ) } \over {{h}^{\prime}( S^\ast ) }} + \displaystyle{1 \over \sigma }\displaystyle{{\eta ( S^\ast , \;R^\ast ; \;\phi_b) } \over {\phi_bh( S^\ast ) \int_{S^\ast }^{R^\ast } \exp ( {-}rx) l( x) {\rm d}x}} + \mu ( S^\ast ) + \rho } \right]> 0.} $$

The intuition of various terms in (35) and (36) is as follows. A change in retirement age (R) affects the first-order condition (33) of schooling years through both the productivity-enhancing and direct utility benefit terms. First, it affects the duration that the individual can reap the benefit of schooling. This is represented by the first term on the numerator of (35). We label this the Ben-Porath effect. Second, it affects the direct utility term (through the marginal utility of consumption), as given in the second term of the numerator of (35). We label it the direct utility of schooling effect. Note that when ζ = 0, this effect disappears. On the other hand, a change in schooling years (S) affects the first-order condition (8) of retirement age in two ways. First, it affects the wage rate through the term h ^′(S*)/h(S*) related to the human capital formation. This is given in the first term on the numerator of (36). We label it the return to schooling effect. Second, a change in schooling years affects the individual's lifetime wealth, as captured by the level of normalized consumption at age 0. This is given in the second term on the numerator of (36). Moreover, we show that $( {-}1/c^n( 0, \;S^\ast , \;R^\ast ) ) \,\,( \partial c^n( 0, \;S^\ast , \,\;R^\ast ) /\partial S) \, = \, \eta ( S^\ast , \;R^\ast ; \,\;\phi _b ) / [ \phi _bh( S^\ast ) \int _{S^\ast }^{R^\ast } \,\exp ( {-}rx) l( x) {\rm d}x ]$ and that it is positive when ζ is positive. We label it the consumption level effect. On the other hand, for the model in section 2, (1/c ⁿ(0, S*, R*)) (∂c ⁿ(0, S*, R*)/∂S) = 0 because of (7).Footnote ³⁷ As a result, the consumption level effect disappears in that model.

We conclude from (35) and (36) that $\partial \tilde{S}( R^\ast ; \;\phi _b) /\partial R > 0$ and that $\partial \tilde{R}( S^\ast ; \;\phi _b) /\partial S > 0$. Thus, positive feedback between them continues to hold in the presence of the direct utility benefit of schooling.

5.2 Positive feedback (but possibly negative co-movement) between schooling and retirement choices

To examine the total effect of productivity increase on schooling years or retirement age, we first consider the two direct effects. For this model, we show that $\partial \tilde{R}( S^\ast ; \;\phi _b) /\partial \phi _b$ is the same as (A.16) of the model in section 2, but

(38)$$\displaystyle{{\partial \tilde{S}( R^\ast ; \;\phi _b) } \over {\partial \phi _b}} = \displaystyle{{-\left({1-\displaystyle{1 \over \sigma }} \right)\displaystyle{1 \over {\phi _b}}\eta ( S^\ast , \;R^\ast ; \;\phi _b) } \over {\Delta _S}}.$$

If 0 < σ < 1 according to (20), then one direct effect is positive $( \partial \tilde{S}( R^\ast ; \;\phi _b) /\partial \phi _b > 0)$ and the other negative $( \partial \tilde{R}( S^\ast ; \;\phi _b) /\partial \phi _b < 0)$. In contrast to the model in section 2, the two direct effects are opposite in sign in the extended model. Combining this feature with positive feedback between the two endogenous variables (positive endogenous effect), the total effects of the productivity increase on schooling years and retirement age may be of the same sign or of different signs, as given in the following proposition:

Proposition 6. Consider the life-cycle model with the direct utility benefit of schooling and an exogenous process of productivity increase, as given by (1), (3), (4), and (32), where θ _b is cohort-invariant. If (20) holds, then

(a) ∂S*/∂ϕ _b < 0 and ∂R*/∂ϕ _b < 0 if

(39)$$0 < \zeta < \left[{\displaystyle{{\exp [ {-\rho ( R^\ast{-}S^\ast ) } ] \displaystyle{{l( R^\ast ) } \over {l( S^\ast ) }}\displaystyle{{{h}^{\prime}( S^\ast ) } \over {h( S^\ast ) }}} \over {r-\rho + \displaystyle{1 \over {\nu ( R^\ast ) }}\displaystyle{{\partial \nu ( R^\ast ) } \over {\partial x}}}}} \right]\nu ( R^\ast ) ; \;$$

(b) ∂S*/∂ϕ _b > 0 and ∂R*/∂ϕ _b < 0 if

$$\left[{\displaystyle{{\exp [ {-\rho ( R^\ast{-}S^\ast ) } ] \displaystyle{{l( R^\ast ) } \over {l( S^\ast ) }}\displaystyle{{{h}^{\prime}( S^\ast ) } \over {h( S^\ast ) }}} \over {r-\rho + \displaystyle{1 \over {\nu ( R^\ast ) }}\displaystyle{{\partial \nu ( R^\ast ) } \over {\partial x}}}}} \right]\nu ( R^\ast ) < \zeta $$

(40)$$< \left[{\displaystyle{{\exp [ {( r-\rho ) ( R^\ast{-}S^\ast ) } ] \left[{\displaystyle{{2{h}^{\prime}( S^\ast ) } \over {h( S^\ast ) }}-\displaystyle{{{h}^{\prime \prime}( S^\ast ) } \over {{h}^{\prime}( S^\ast ) }}-\mu ( S^\ast ) -r} \right]} \over {\displaystyle{{{h}^{\prime}( S^\ast ) } \over {h( S^\ast ) }}-\displaystyle{{{h}^{\prime \prime}( S^\ast ) } \over {{h}^{\prime}( S^\ast ) }}-\mu ( S^\ast ) -\rho }}} \right]\nu ( R^\ast ) ; \;$$

and

(c) ∂S*/∂ϕ _b > 0 and ∂R*/∂ϕ _b > 0 if

(41)$$\zeta > \left[{\displaystyle{{\exp [ {( r-\rho ) ( R^\ast{-}S^\ast ) } ] \left[{\displaystyle{{2{h}^{\prime}( S^\ast ) } \over {h( S^\ast ) }}-\displaystyle{{{h}^{\prime \prime}( S^\ast ) } \over {{h}^{\prime}( S^\ast ) }}-\mu ( S^\ast ) -r} \right]} \over {\displaystyle{{{h}^{\prime}( S^\ast ) } \over {h( S^\ast ) }}-\displaystyle{{{h}^{\prime \prime}( S^\ast ) } \over {{h}^{\prime}( S^\ast ) }}-\mu ( S^\ast ) -\rho }}} \right]\nu ( R^\ast ) .$$

Proposition 6 specifies the conditions that lead to the three possible outcomes. These outcomes are represented graphically in Figure 5.Footnote ³⁸ The interpretations of the conditions are as follows. When the extent of the utility benefit of schooling (ζ) is small and (39) holds, then the direct effect of productivity increase on schooling years is not strong enough. Thus, the indirect effect dominates and the total effect (∂S*/∂ϕ _b) is still negative. Similarly, ∂R*/∂ϕ _b is negative. A productivity increase causes a positive co-movement of S* and R*. The model in section 2, with ζ = 0, could be treated as a limiting case of case (a). When ζ is large enough, the direct effect dominates and the total effect ∂S*/∂ϕ _b turns positive. Furthermore, we find that when ζ is larger than the first threshold in (40) but not the second one, we have the results ∂S*/∂ϕ _b > 0 and ∂R*/∂ϕ _b < 0. (The two thresholds correspond to the upward-sloping dotted lines in Figure 5.) In contrast to case (a), there is a negative co-movement of S* and R* in case (b). When ζ is substantially large to pass the higher threshold, we have ∂S*/∂ϕ _b > 0 and ∂R*/∂ϕ _b > 0, as in case (c).Footnote ³⁹ The co-movement of S* and R* becomes positive again.

Figure 5. The impact of productivity increase for the extended model.

The results in Proposition 6 imply that the extended model is able to explain the negative co-movement of schooling years and retirement age, provided that the flow utility of schooling is in the intermediate range given by (40). Moreover, they have relevance for the results in Restuccia and Vandenbroucke (Reference Restuccia and Vandenbroucke2013). In their paper, retirement age is assumed to be exogenous. We can see from (17) and (38) that we only need ζ > 0 to guarantee that $\partial \tilde{S}( R^\ast ; \;\phi _b) /\partial \phi _b > 0$ in that environment. On the other hand, when retirement age is also a choice variable, we need parameter ζ to be not only positive, but also larger than the lower threshold in (40) to explain the positive total effect of productivity increase on schooling years (∂S*/∂ϕ _b > 0).