The endogenization of the choice of participation in the active population,Footnote 5 Lt, as a share of the total population, Nt, is conducted by introducing the relation between the active population and the entire population, Lt/Nt, in the utility function. This is the inverse of the dependency ratio in the form \( 1+{D}_t=\frac{N_t}{L_t} \), or \( {D}_t=\frac{N_t-{L}_t}{L_t} \), which is the ratio of all dependent persons, not only the old, per worker. It has a Frisch parameter of labour supply, ϑ, which captures the effect of a change in the share of the active population. Being involved in the active population or dedicating a high share of the population to the active population, has a negative effect on utility. On the other hand, being inactive, or involving few labour resources in the active population, will lead to low output per capita, which will result in low consumption and, hence, low utility. Hence, there must be an optimal labour supply. Consumers are assumed to maximize their utility function, which is assumed to be
$$ {U}_t={\sum}_{t=0}^{\infty }{\beta}^t{N}_t\ \left(\frac{c_t^{1-\sigma }}{\left(1-\sigma \right)}-\xi \frac{{\left(\frac{L_t}{N_t}\right)}^{1+\vartheta }}{1+\vartheta}\right) $$
The expression above shows the utility function of the entire population. Here 0 < β < 1 is the subjective discount factor; σ > 0 is the intertemporal elasticity of substitution for consumption with σ ≠ 1 in order to avoid division by zero; ϑ > 0 is the Frisch elasticity parameter for labour supply; ξ > 0 is a parameter, which measures the disutility of participation in the active population relative to the consumption part of utility; ct is individual consumption. Lt is the size of the population active in work or leisure as in the training model of Wallenius (2011). Nt is the size of the entire population normalized to unity in Wallenius (2011) and Malik (2013) both using the same utility function.Footnote 6 The economy consists, by assumption, of output-producing firms and labour- and capital-supplying consumers. Output is formed by a Cobb-Douglas production function and is determined by physical capital, Kt, and efficient labour. Efficient labour, (1 − et)htLt, is the product of individual human capital, ht, and the part of the active population, which is not in education, (1 − et)Lt. The households decide between spending their time in production (1 − et) for immediate output generation and education, et, to increase their productivity for later production. A human capital externality is added as, \( {\overline{h}}_t^{\epsilon } \), modelled after Lucas (1988), to include the influence of the average skill level on the economy. This forms the production function
$$ {Y}_t=A{\left({K}_t\right)}^{1-\alpha }{\left(\left(1-{e}_t\right){h}_t{L}_t\right)}^{\alpha }{{\overline{h}}_t}^{\epsilon } $$
(1)
The demand for physical and human capital is determined in a firm, which maximizes profits:
$$ \underset{\left(1-{e}_t\right),{K}_t}{\max}\uppi =\mathrm{A}{\left({K}_t\ \right)}^{1-\alpha }{\left(\left(1-{e}_t\right){h}_t{L}_t\right)}^{\alpha }{{\overline{h}}_t}^{\epsilon }-{\omega}_t\left(1-{e}_t\right){h}_t{L}_t-{r}_{kt}{K}_t $$
$$ {\omega}_t=\frac{\alpha {Y}_t}{\left(1-{e}_t\right){h}_t{L}_t} $$
(2)
$$ {r}_{kt}=\left(1-\alpha \right)\frac{Y_t}{K_t} $$
(3)
Equations (2) and (3) represent first-order conditions, equating marginal productivity of labour and capital to wages for efficient labour and rental rates. Consumers face the following budget constraint:
$$ {N}_t{c}_t+{K}_{t+1}-\left(1-{\delta}_k\right){K}_t={\omega}_t\left(1-{e}_t\right){h}_t{L}_t+{r}_{kt}{K}_t+{B}_{t+1}-\left(1+r\left(\frac{B_t}{Y_t}\right)\right){B}_t $$
(4)
The right-hand side of Eq. (4) represents the total income which is the income from labour, ωt(1 − et)htLt, the income from capital rent, rktKt, and the foreign debt from outside the economy’s borders minus the interest and re-payments, \( {B}_{t+1}-\left(1+r\left(\frac{B_t}{Y_t}\right)\right){B}_t \). \( r\left(\frac{B_t}{Y_t}\right) \) is the debt-dependent interest rate. The left-hand side of Eq. (4) represents the spending on consumption, Ntct, and capital investment, Kt + 1 − (1 − δk)Kt. Output is the numéraire, and capital, consumption, wages and debt are measured in the same unit. For the budget constraint to hold, both sides must be equal at all times. We do not model a government or a pension system separately. The firm has zero profits by Euler’s theorem because of constant returns to scale (Hellwig and Irmen 2001). The household budget and that of the country are balanced through the inclusion of debt. Consumption is not differentiated in regard to age or (not) working. This is implicit in the assumption of equal consumption of all for a given point in time.
Human capital formation is determined by the time share spent in education, et, with diminishing or constant returns to scale. Equation (5) shows how human capital is formed with the productivity parameter, γ ≤ 1, the knowledge efficiency coefficient, F, and depreciation of human capital, δh.Footnote 7
$$ {h}_{t+1}=F\ {e}_t^{\gamma }{h}_t+\left(1-{\delta}_h\right){h}_t $$
(5)
Uzawa (1965) and Lucas (1988) use a zero depreciation rate. But for the treatment of ageing the increase in the loss of qualifications of workers can be captured by an increase in this depreciation rate and therefore it must be included and correspondingly the calibration is different than without it. We assume that h-terms on the right-hand side have exponent unity in line with recent evidence favouring fully over semi-endogenous growth theory (Ha and Howitt 2007; Madsen 2008; Ziesemer 2020). Consumers maximize their utility subject to their budget constraint, Eq. (4), and the human capital formation function, Eq. (5), given the parameters and initial values of ht, and Kt-Bt. The maximization program for the consumers is:
$$ \underset{c_t,{e}_t,{L}_t,{B}_{t+1},{K}_{t+1},{h}_{t+1}}{\max }{\sum}_{t=0}^{\infty }{\upbeta}^t\left({N}_t\left(\ \frac{c_t^{1-\sigma }}{\left(1-\sigma \right)}-\xi \frac{{\left(\frac{L_t}{N_t}\right)}^{1+\vartheta }}{1+\vartheta}\right)-{\boldsymbol{\mu}}_{\boldsymbol{t}}\kern0.5em \left[{N}_t{c}_t+{K}_{t+1}-\left(1-{\delta}_k\right){K}_t-{\omega}_t\left(1-{e}_t\right){h}_t{L}_t-{r}_{kt}{K}_t-{B}_{t+1}+\left(1+r\left(\frac{B_t}{Y_t}\right)\right){B}_t\right]-{\boldsymbol{\mu}}_{\boldsymbol{ht}}\ \left[{h}_{t+1}-F\ {e}_t^{\gamma }\ {h}_t-\left(1-{\delta}_h\right){h}_t\right]\right) $$
The first-order conditions for an interior solutionFootnote 8 are as follows:
$$ {\mathrm{c}}_t:{c}_t^{-\sigma }={\mu}_t $$
(6)
$$ {\mathrm{e}}_t:\kern2em {\mu}_t{\omega}_t{L}_t={\mu}_{ht} F\gamma {e}_t^{\gamma -1} $$
(7)
$$ {\mathrm{L}}_t:\xi {N}_t^{-\vartheta }{L}_t^{\vartheta }={\mu}_t{\omega}_t\left(1-{e}_t\right){h}_t $$
(8)
$$ {\mathrm{B}}_{t+1}:{\mu}_t={\beta \mu}_{t+1}\left(1+r\left(\frac{B_{t+1}}{Y_{t+1}}\right)\right)+\beta {\mu}_{t+1}\frac{B_{t+1}}{Y_{t+1}}{r}^{\prime}\left(\frac{B_{t+1}}{Y_{t+1}}\right) $$
(9)
$$ {\mathrm{K}}_{t+1}:{\mu}_t=\beta {\mu}_{t+1}\left(1-{\delta}_k+{r}_{kt+1}\right) $$
(10)
$$ {\mathrm{h}}_{t+1}:{\mu}_{ht}=\beta \left[{\mu}_{t+1}{\omega}_{t+1}\left(1-{e}_{t+1}\right){L}_{t+1}+{\mu}_{ht+1}F{e}_{t+1}^{\gamma }+\left(1-{\delta}_h\right){\mu}_{ht+1}\right] $$
(11)
The following transversality conditions hold by assumption:
-
I.
$$ \underset{t\to \infty }{\lim }{\beta}^t{\mu}_t{K}_t=0 $$
-
II.
$$ \underset{t\to \infty }{\lim }{\beta}^t{\mu}_{ht}{h}_t=0 $$
The system of Eqs. (1)–(11) determine the eleven endogenous variables, Yt, Kt, Lt, ht, et, ωt, rkt, ct, Bt, μt, and μht. The rates of return to physical capital, bonds, human capital and, future labour input can be derived. They are displayed in Eqs. (12a-e).
$$ \frac{1}{\beta }{\left(\frac{c_{t+1}}{c_t}\right)}^{\sigma }=\frac{\mu_t}{\beta\ {\mu}_{t+1}}={R}_{Bt+1}={R}_{Kt+1}={R}_{Ht+1}={R}_{Lt+1} $$
(12a)
$$ {R}_{Bt+1}=1+r\left(\frac{B_{t+1}}{Y_{t+1}}\right)\left(1+{\eta}_{rb}\right) $$
(12b)
$$ {R}_{Kt+1}=1-{\delta}_k+\left(1-\alpha \right)\frac{Y_{t+1}}{K_{t+1}} $$
(12c)
$$ {R}_{Ht+1}=\left(1+{g}_{\omega}\right)\left(1+{g}_L\right) F\gamma {e}_t^{\gamma -1}\left[1-{e}_{t+1}+\frac{1}{\gamma }{e}_{t+1}+\frac{1-{\delta}_h}{F\gamma {e}_{t+1}^{\gamma -1}}\right] $$
(12d)
$$ {R}_{Lt+1}=\frac{{\left(1+{g}_N\right)}^{\vartheta}\left(1+{g}_{\omega}\right)\left(1+{g}_{1-e}\right)\left(1+{g}_h\right)}{\beta {\left(1+{g}_L\right)}^{\vartheta }} $$
(12e)
The first part of (12a) is derived from (6), where the other equivalencies on the right hand side follow from (12b-e), obtained by rearranging Eqs. (8), (9), (10) and (11) to solve for \( \frac{\mu_t}{\beta\ {\mu}_{t+1}} \). Equation (12b) follows straight forwardly from (9) with \( {\eta}_{rb}=\frac{B_{t+1}}{Y_{t+1}}\frac{r^{\prime}\left(\frac{B_{t+1}}{Y_{t+1}}\right)}{r\left(\frac{B_{t+1}}{Y_{t+1}}\right)} \); (12c) is derived from (10) where rkt + 1 is replaced by the expression in (3); Eq. (12d) is derived from (11) where μht is replaced by the relation in (7). The relation (12e) for the rate of return of the active population size is derived from (8).
If \( \frac{B_t}{Y_t} \), \( r\left(\frac{B_t}{Y_t}\right) \) and its elasticity are assumed to be constant this leads to a constant growth rate of ct through (12a, b):
$$ \mathbf{1}+{\boldsymbol{g}}_{\boldsymbol{c}}={\left[\beta \left(1+r\left(\frac{B_t}{Y_t}\right)\left(1+{\eta}_{rb}\right)\right)\right]}^{\frac{1}{\sigma }} $$
(12a’)
From this follow constant RKt + 1 and RHt + 1 and RLt + 1 in (12c), (12d) and (12e) respectively. Constancy of \( \frac{Y_{t+1}}{K_{t+1}} \) follows directly from constant RKt + 1 in (12c), which can be expressed as: \( \frac{Y_{t+1}}{K_{t+1}}=A{\left(\frac{K_{t+1}}{\left(1-{e}_{t+1}\right){h}_{t+1}^{1+\frac{\epsilon }{\alpha }}{L}_{t+1}}\right)}^{-\alpha } \). This implies equality of the growth rates of the numerator and the denominator:
$$ 1+{g}_Y=1+{g}_K=\left(1+{g}_{1-e}\right){\left(1+{g}_h\right)}^{1+\frac{\epsilon }{\alpha }}\left(1+{g}_L\right) $$
(12c)
Equation (12c)I shows that the growth rates of output and capital depend on the change of time spent in production, the growth rate of human capital, and the growth rate of labour supply.
The growth rate of wages is crucial to determine the rates of return to labour and human capital. Wages are determined from Eq. (2):
$$ 1+{g}_{\omega }=\frac{1+{g}_Y}{\left(1+{g}_{1-e}\right)\left(1+{g}_h\right)\left(1+{g}_L\right)} $$
(2)
Inserting (12c)I into (2)I leads to:
$$ 1+{g}_{\omega }={\left(1+{g}_h\right)}^{\frac{\epsilon }{\alpha }} $$
((2)II)
Together with the human capital formation function (5), this shows that gω is constant if et is constant (i.e. if ge = 0). Equation (2)II shows that the development of wages per labour efficiency unit depends solely on that of human capital. Wage growth per labour efficiency unit is only positive if the externalities are positive.
The growth rate of the active population, (1 + gL), can be derived from Eq. (12e). With the expression in Eq. (12b), RLt + 1 can be replaced by \( 1+r\left(\frac{B_t}{Y_t}\right)\left(1+{\eta}_{rb}\right) \). Plugging Eq. (2)II into Eq. (12e), and thus eliminating (1 + gω) leads to
$$ \left(1+{g}_L\right)={\left(1+{g}_N\right)\left(\frac{{\left(1+{g}_h\right)}^{1+\frac{\epsilon }{\alpha }}\ \left(1+{g}_{1-e}\right)}{\beta \left(1+r\left(\frac{B_t}{Y_t}\right)\left(1+{\eta}_{rb}\right)\ \right)}\right)}^{\frac{1}{\vartheta }} $$
((12e)I)
From (12c)I and (12e)I we find the growth of the GDP per capita of the population.
$$ \frac{1+{g}_Y}{1+{g}_N}={\left[{\left(1+{g}_h\right)}^{1+\frac{\epsilon }{\alpha }}\right]}^{1+\frac{1}{\vartheta }}{\left(\frac{1\ }{\beta \left(1+r\left(\frac{B_t}{Y_t}\right)\left(1+{\eta}_{rb}\right)\ \right)}\right)}^{\frac{1}{\vartheta }} $$
(12f)
For given education time e, human capital growth rate gh and interest rate r or debt b, lower population growth translates one-to-one into lower labour growth. For a constant interest rate and population growth rate, the steady state growth rate of labour depends on the endogenous development of individual human capital, and through this on time spent in education or production. Because the time spent in production, (1 − et), can by assumption not exceed 1, its growth rate cannot be stable at any other value than zero. Equation (12e)I shows that if the choice of the size of labour is endogenized through negative utility associated with labour supply, the steady state growth rate of labour depends on time spent in education. If time spent in education and the interest rate are constant, so is (1 + gL). For positive values of the Frisch parameter, the fraction in parenthesis will determine whether we have gL < gN. If the denominator including the discount factor and the interest rate is larger (smaller) than the numerator, this will (not) be the case.
In order to find the optimal time spent in education, we have to derive the dynamics of et. We do this by relating et and et + 1 to only exogenous variables. The main equation used is Eq. (12d). To show how the rate of return of human capital relates to time spent in education only, (1 + gω) can be replaced in Eq. (12d) using Eq. (2)II and Eq. (5):
$$ {R}_{Ht+1}={\left(F\ {e}_t^{\gamma }+\left(1-{\delta}_h\right)\right)}^{\frac{\epsilon }{\alpha }}\left(1+{g}_L\right)\ F\gamma {e}_t^{\gamma -1}\left[\left(1-{e}_{t+1}\right)+\frac{1}{\gamma }{e}_{t+1}+\frac{\left(1-{\delta}_h\right)}{F\gamma {e}_{t+1}^{\gamma -1}}\right] $$
((12d)I)
Next, the endogenous variables gL and gh are replaced. With Eq. (12e)I replacing the growth rate of the labour force, (1 + gL), Eq. (12d)I becomes
$$ {R}_{Ht+1}\kern0.5em ={\left(1+{g}_h\right)}^{\frac{\epsilon }{\alpha }}\ {\left(1+{g}_N\right)\left(\frac{{\left(1+{g}_h\right)}^{1+\frac{\epsilon }{\alpha }}\ \left(1+{g}_{1-e}\right)}{\beta \left(1+r\left(\frac{B_t}{Y_t}\right)\left(1+{\eta}_{rb}\right)\ \right)}\right)}^{\frac{1}{\vartheta }} F\gamma {e}_t^{\gamma -1}\left[\left(1-{e}_{t+1}\right)+\frac{1}{\gamma }{e}_{t+1}+\frac{\left(1-{\delta}_h\right)}{F\gamma {e}_{t+1}^{\gamma -1}}\right] $$
Replacing \( \left(1+{g}_h\right)=F\ {e}_t^{\gamma }+\left(1-{\delta}_h\right) \) and \( \left(1+{g}_{1-e}\right)=\frac{1-{e}_{t+1}}{1-{e}_t} \) yields:
$$ {R}_{Ht+1}\kern0.5em ={\left(F\ {e}_t^{\gamma }+\left(1-{\delta}_h\right)\right)}^{\frac{\epsilon \left(1+\vartheta \right)+\alpha }{\vartheta \alpha}}\left(1+{g}_N\right){\left(\frac{1-{e}_{t+1}}{1-{e}_t}\right)}^{\frac{1}{\vartheta }}{\left(\frac{1}{\beta \left(1+r\left(\frac{B_t}{Y_t}\right)\left(1+{\eta}_{rb}\right)\right)}\right)}^{\frac{1}{\vartheta }} F\gamma \left[{e}_t^{\gamma -1}-{e}_t^{\gamma}\frac{e_{t+1}}{e_t}+{e}_t^{\gamma}\frac{1}{\gamma}\frac{e_{t+1}}{e_t}+{\left(\frac{e_t}{e_{t+1}}\right)}^{\gamma -1}\frac{\left(1-{\delta}_h\right)}{F\gamma}\right] $$
((12d)II)
This relates et and et + 1 to the interest rate and exogenous variables and parameters. \( \frac{e_{t+1}}{e_t} \) can then be replaced by 1 + ge. Since \( 1+{g}_e=\frac{e_{t+1}}{e_t}\leftrightarrow {e}_{t+1}=\left(1+{g}_e\right){e}_t \), the expression \( {\left(\frac{1-{e}_{t+1}}{1-{e}_t}\right)}^{\frac{1}{\vartheta }} \) thus becomes \( {\left(\frac{1-\left(1+{g}_e\right){e}_t}{1-{e}_t}\right)}^{\frac{1}{\vartheta }} \). This leads to:
$$ {R}_{Ht+1}\kern0.5em ={\left(F\ {e}_t^{\gamma }+\left(1-{\delta}_h\right)\right)}^{\frac{\epsilon \left(1+\vartheta \right)+\alpha }{\vartheta \alpha}}\left(1+{g}_N\right){\left(\frac{1-\left(1+{g}_e\right){e}_t}{1-{e}_t}\right)}^{\frac{1}{\vartheta }}{\left(\frac{1}{\beta \left(1+r\left(\frac{B_t}{Y_t}\right)\left(1+{\eta}_{rb}\right)\right)}\right)}^{\frac{1}{\vartheta }} F\gamma \left[{e}_t^{\gamma -1}-{e}_t^{\gamma}\left(1+{g}_e\right)+{e}_t^{\gamma}\frac{1}{\gamma}\left(1+{g}_e\right)+{\left(\frac{1}{1+{g}_e}\right)}^{\gamma -1}\frac{\left(1-{\delta}_h\right)}{F\gamma}\right] $$
((12d)III)
Because of the relations in Eqs. (12b) and (12a), RHt + 1 equals \( 1+r\left(\frac{B_t}{Y_t}\right)\left(1+{\eta}_{rb}\right) \); if the interest rate is constant, the both sides of Eq. (12d)III is constant as well. This is the dynamic equation that determines how et develops over time, depending on the interest rate, the population growth rate and several parameters. With help of this equation, we can analyze the stability of et. Unfortunately, the above expression cannot be solved for et, or ge analytically. If we could solve for the growth rate of e, the function probably would be non-linear in e, 1 + r and 1 + gN. The next section takes an empirical approach to this relation.