Silviase/jgov_v1.4_full_ann huggingface dataset:
### 2.4 R&D sector

Next, we consider the technology involved in developing a new intermediate good. Before entering the intermediate goods market, an entrepreneur must undertake R&D activity to develop a new blueprint. R&D activities require labor input and the R&D sector is perfectly competitive. We assume the following R&D technologies:

$$ A_{t+1}-A_{t}=\theta_{t}L_{A,t}, $$

where $ \theta_{t} $ and $ L_{A_{t}} $ represent the productivity of R&D and the amount of labor devoted to R&D. $ A_{t+1}-A_{t} $ measures new intermediate goods. Following Romer (1990), Grossman and Helpman (1991), and Jones (1995), the productivity of R&D depends on current knowledge that is a result of previous R&D activities. We assume the following productivity of R&D:

$$ \theta_{t}=\bar{\theta}A_{t}^{\phi}. $$

Thus, $ A_{t} $ also represents the stock of knowledge. As discussed by Jones (1995), we consider the parameter range where $ 0 < \phi < 1 $ . If an entrepreneur can invent a new blueprint, he/she acquires $ v_{t} $ , where $ v_{t} $ denotes the market values of new blueprints. The profit of entrepreneurs is given by $ \pi_{t}^{A} = v_{t}(A_{t+1} - A_{t}) - w_{t}L_{A,t} = (v_{t}\theta_{t} - w_{t})L_{A,t} $ . Perfect competition prevails in R&D activities. Consequently, when R&D is undertaken, the following equality holds:

$$ v_{t}{=}\frac{w_{t}}{\bar{\theta}A_{t}^{\phi}}. $$

On the other hand, when $ v_{t}\theta_{t}
or for the purchase of existing stocks. Hence, the asset market clearing condition is given by

$$ w_{t}L_{A,t}+A_{t}v_{t}=s_{t}N_{t}. $$

Finally, we consider the equilibrium condition of the final goods market. The final goods are used for consumption and for raising children. The final goods clearing condition is as follows: $ Y_{t}=c_{t}^{y}N_{t}+c_{t}^{o}pN_{t-1}+\delta n_{t}N_{t} $ . Note that because the survival probability is p, the population size of old individuals at period t becomes $ pN_{t-1} $ .

### 3. Equilibrium

### 3.1 Dynamics

We now show equilibrium paths in this economy. Substituting (12) into (7) yields

$$ Y_{t}{=}\Big(\frac{\alpha^{2}}{w_{t}}\Big)^{\frac{\alpha}{1-\alpha}}A_{t}L_{Y,t}. $$

Using (8) and (19), we obtain

$$ w_{t}=\hat{\alpha}A_{t}^{1-\alpha}, $$

where $ \hat{\alpha}\equiv\alpha^{2\alpha}(1-\alpha)^{1-\alpha} $ . This implies that an increase in $ A_{t} $ raises the wage rate $ w_{t} $ . From (6) and (20), we obtain

$$ n_{t}{=}\frac{\gamma\hat{\alpha}A_{t}^{1-\alpha}}{(1+p\beta+\gamma)(\rho\hat{\alpha}A_{t}^{1-\alpha}+\delta)}. $$

(5), (16), and (18) yield

$$ L_{A,t}{=}\frac{s_{t}N_{t}}{w_{t}}{-}\frac{A_{t}v_{t}}{w_{t}}{=}\frac{p\beta N_{t}}{1+p\beta{+}\gamma}{-}\frac{1}{\bar{\theta}}A_{t}^{1{-}\phi}. $$

The investment in R&D is obtained from a difference between total savings and the purchase of existing stocks. Using (14), (15), and (22), we obtain

$$ A_{t+1}{=}\frac{p\beta\bar{\theta}}{1+p\beta{+}\gamma}A_{t}^{\phi}N_{t}. $$

Equations (1), (21), and (23) characterize the dynamic system with respect to $ A_{t} $ and $ N_{t} $ . Note that both $ A_{t} $ and $ N_{t} $ in period t are predetermined variables.

### 3.2 Initial conditions for sustainable growth

First, we examine whether $ N_{t+1} \geq N_t $ or $ N_{t+1} < N_t $ at each point on the $ (A_t, N_t) $ plane.

(1) implies that $ N_{t+1} \geq N_{t} $ holds when $ n_{t} \geq 1 $ . From (20) and Lemma 1, we obtain
$$ n_{t}\gtrless1\Longleftrightarrow A_{t}\gtrless\left\{\frac{\delta\left(1+p\beta+\gamma\right)}{\hat{\alpha}[\gamma-(1+p\beta+\gamma)\rho]}\right\}^{\frac{1}{1-\alpha}}\equiv\overline{A}. $$

Therefore, we have $ N_{t+1}>N_{t} $ when $ A_{t}>\bar{A} $ and $ N_{t+1} A_t $ or $ A_{t+1} = A_t $ at each point on the $ (A_t, N_t) $ plane. Because there is no product obsolescence, $ A_{t+1} < A_t $ does not hold in this study. Setting $ A_{t+1} = A_t $ in (23) leads to

$$ N_{t}{=}\frac{1+p\beta+\gamma}{p\beta\bar{\theta}}A_{t}^{1-\phi}{\equiv}\Phi(A_{t}). $$

Hence, we have $ A_{t+1} > A_t $ above this locus and $ A_{t+1} = A_t $ below this locus (hereafter, the no R&D region). Using these results, we can present a phase diagram as in Figure 1.

As shown in Figure 1, the point $ (\bar{A}, \bar{N}) $ is saddle-point stable, where $ \bar{N} \equiv \Phi(\bar{A})^{3} $ . Note that $ A_{t} $ and $ N_{t} $ are predetermined variables at time t. Hence, the initial state of the economy is given by a point $ (A_{0}, N_{0}) $ on the $ (A_{t}, N_{t}) $ plane. For an economy where the initial state $ (A_{0}, N_{0}) $ is in the upper-right side of the saddle-point stable arm, $ A_{t} $ increases and $ N_{t} $ decreases at first. After exceeding the vertical line $ \bar{A} $ , both $ A_{t} $ and $ N_{t} $ increase forever, that is, economic growth is sustainable. In contrast, for an economy where the initial state $ (A_{0}, N_{0}) $ is in the lower-left side of the saddle-point stable arm, the economy reaches the no R&D region in a finite time. After that time, $ A_{t} $ remains constant and $ N_{t} $ decreases at a constant rate. That is, the population decreases and R&D activities are not conducted. This path falls into the trap in the long run.

These results show that the saddle-point stable arm in Figure 1 represents the threshold of $ A_{t} $ for each level of $ N_{t} $ to sustain economic growth. In summary, we can

Figure 1: Phase diagram of the dynamics of $ A_{t} $ and $ N_{t} $ .
Population Dynamics, Longer Life Expectancy, and Child-Rearing Policies in an R&D-based Growth Model with Overlapping Generations

state the following proposition:

Proposition 1. There is a threshold curve to sustain economic growth. The threshold curve is downward sloping on the $ (A_{t}, N_{t}) $ plane.

From (20), the wage rate, $ w_{t} $ , is determined by the stock of knowledge $ A_{t} $ . Thus, a sufficiently small $ A_{t} $ implies that $ w_{t} $ is also low. From Lemma 1, the fertility rate $ n_{t} $ tends to be lower than 1. This induces the economy to fall into the trap. However, if the population size, $ N_{t} $ , is sufficiently large, the total savings become large. From (22), this expands investment in R&D, and hence, the growth rate of $ A_{t} $ increases. The economy can exceed the threshold value $ \bar{A} $ and attain sustainable growth.

### 3.3 Effects of a rise in life expectancy

We examine the effects of a rise in life expectancy (that is, an increase in p) on the sustainability of economic growth. Differentiating $ \bar{A} $ with respect to p yields

$$ \frac{\partial\overline{A}}{\partial p}=\frac{1}{1-\alpha}\Big\{\frac{\delta(1+p\beta+\gamma)}{\widehat{\alpha}[\gamma-(1+p\beta+\gamma)\rho]}\Big\}^{\frac{\alpha}{1-\alpha}}\frac{\delta\gamma\beta}{\widehat{\alpha}[\gamma-(1+p\beta+\gamma)\rho]^{2}}>0. $$

Moreover, from (24), we obtain $ \partial\Phi(A_{t})/\partial p<0 $ . A rise in p shifts the $ A_{t}=\bar{A} $ locus rightward and the $ N_{t}=\Phi(A_{t}) $ locus downward (see Figure 2). These results are summarized in the following proposition:

Proposition 2. An increase in life expectancy p enlarges the region where R&D is conducted, whereas it shrinks the region where population increases.

Figure 2 : Effects of an increase in p.
From (5) and (6), a higher p implies a higher per capita saving $ s_{t} $ and a lower fertility rate $ n_{t} $ . Because individuals survive with a higher probability, they decrease the number of children and increase the savings for consumption in their old age. From (22), we see that increasing per capita saving accelerates the investment in R&D. From these results, the R&D region expands and the region where population increases shrinks. The sustainability of economic growth depends on these opposite effects. We will see the effects of p on the sustainability of economic growth by numerical examples in Section 6.

### 4. Child-rearing policies

As discussed in the previous section, an increase in life expectancy negatively impacts the fertility rate. Therefore, we analyze the effects of child rearing policies aimed at increasing the fertility rate $ ^{4} $ . First, we introduce a child allowance. Suppose the government subsidizes a fraction of $ z \in [0, 1] $ of the rearing cost paid in the final goods. Second, we introduce a childcare service as follows. We assume that the rearing cost spent by young individuals' time $ \rho $ depends on public goods $ B_{t} $ . One unit of a public good requires one unit of labor input. $ \rho $ is a function of per capita $ B_{t} $ ; that is, $ \rho(b_{t}) $ , where $ b_{t} \equiv B_{t}/N_{t} $ . In addition, $ \rho(b_{t}) $ is assumed to satisfy the following properties: $ \rho'(b) < 0 $ and $ \rho''(b) > 0 $ . The government expenditure is financed by a lump-sum tax $ \tau_{t} $ on each young individual. In each period, the government runs a balanced budget. Thus, the government's budget constraint in period t can be expressed as

$$ \tau_{t}=z\delta n_{t}+w_{t}b_{t}. $$

For simplicity, we assume that the government keeps $ b_{t} $ constant (that is, $ b_{t}=b $ ) and that $ \tau_{t} $ is determined to satisfy the government budget constraint.

Under these policies, we rewrite the budget constraint of each young individual as follows: $ c_{t}^{y} + s_{t} + (1 - z)\delta n_{t} = [1 - \rho(b_{t})n_{t}]w_{t} - \tau_{t} $ . Individuals' optimal decisions (5) and (6) become

$$ s_{t}{=}\frac{p\beta(w_{t}{-}\tau_{t})}{1{+}p\beta{+}\gamma}, $$

$$ n_{t}{=}\frac{\gamma(w_{t}{-}\tau_{t})}{(1{+}p\beta{+}\gamma)[\rho(b)w_{t}{+}(1{-}z)\delta]}. $$
Population Dynamics, Longer Life Expectancy, and Child-Rearing Policies in an R&D-based Growth Model with Overlapping Generations

Using (20), (25), (26), and (27), we obtain

$$ s_{t}{=}\frac{(1{-}b)p\beta[(1{-}z)\delta{+}\rho(b)\hat{\alpha}A_{t}^{1-\alpha}]}{(1{+}p\beta{+}\gamma)\rho(b)\hat{\alpha}A_{t}^{1-\alpha}{+}[(1{-}z)(1{+}p\beta){+}\gamma]\delta}. $$

$$ n_{t}{=}\frac{(1{-}b)\gamma\bar{\alpha}A_{t}^{1-\alpha}}{(1{+}p\beta{+}\gamma)\rho(b)\bar{\alpha}A_{t}^{1-\alpha}{+}[(1{-}z)(1{+}p\beta){+}\gamma]\delta}. $$

(16), (18), and (28) yield

$$ L_{A,t}\mathrm{=}\frac{(1-b)p\beta[(1-z)\delta+p(b)\hat{\alpha}A_{t}^{1-\alpha}]N_{t}}{(1+p\beta+\gamma)\rho(b)\hat{\alpha}A_{t}^{1-\alpha}+[(1-z)(1+p\beta)+\gamma]\delta}-\frac{1}{\bar{\theta}}A_{t}^{1-\phi}. $$

As mentioned above, the investment in R&D is obtained from the difference between total savings and the purchase of existing stocks even under child rearing policies. From (14), (15), and (30), we obtain

$$ A_{t+1}{=}\frac{(1{-}b)p\beta[(1{-}z)\delta{+}\rho(b)\hat{\alpha}A_{t}^{1{-}\alpha}]\bar{\theta}A_{t}^{\phi}N_{t}}{(1{+}p\beta{+}\gamma)\rho(b)\hat{\alpha}A_{t}^{1{-}\alpha}{+}[(1{-}z)(1{+}p\beta){+}\gamma]\theta}. $$

Equations (1), (29), and (31) characterize the dynamic system with respect to $ A_{t} $ and $ N_{t} $ .

We investigate the phase diagram under child-rearing policies. Using (29), we obtain

$$ n_{t}\gtrless1\Leftrightarrow A_{t}\gtrless\Big\{\frac{\delta[(1-z)(1+p\beta)+\gamma]}{\widehat{\alpha}[(1-b)\gamma-(1+p\beta+\gamma)\rho(b)]}\Big\}^{\frac{1}{1-\alpha}}\equiv\widehat{A}. $$

Therefore, we have $ N_{t+1} \gtrless N_{t} $ when $ A_{t} \gtrless \widehat{A} $ . To ensure the existence of $ \widehat{A} $ , we impose the following assumption (see Appendix B for details) $ ^{5} $ :

Assumption 2. $ \gamma>(1+p\beta+\gamma)\rho(0) $ and $ 0\leq b<\hat{b} $ , where $ \hat{b} $ is defined as $ (1-\hat{b})\gamma-(1+p\beta+\gamma)\rho(\hat{b})=0 $ .

Next, we examine whether $ A_{t+1} > A_t $ or $ A_{t+1} = A_t $ at each point on the $ (A_t, N_t) $ plane. Setting $ A_{t+1} = A_t $ in (31) leads to

$$ N_{t}{=}\frac{(1+p\beta+\gamma)\rho(b)\hat{\alpha}A_{t}^{1-\alpha}{+[(1-z)(1+p\beta)+\gamma]\delta}}{p\beta\bar{\theta}(1-b)[(1-z)\delta+\rho(b)\hat{\alpha}A_{t}^{1-\alpha}]}A_{t}^{1-\phi}{\equiv}\Gamma(A_{t}). $$

We have $ A_{t+1} > A_t $ when $ N_t > \Gamma(A_t) $ and $ A_{t+1} = A_t $ when $ N_t \leq \Gamma(A_t) $ . We show in Appendix C that $ N_t = \Gamma(A_t) $ locus is increasing in $ A_t $ on the $ (A_t, N_t) $ plane under the following assumption:
### Assumption 3. $ z \leq \min\{\hat{z}, 1\} $ , where $ \hat{z} $ is defined in Appendix C

Under these two assumptions, we can conclude that the phase diagram of child-rearing policies is similar to that of the laissez-faire economy.

Effects of child allowance

We begin with the effects of changes in z on the sustainability of economic growth. From (32), we obtain $ \partial\widehat{A}/\partial z<0 $ , and thus, a rise in z shifts the $ A_{t}=\widehat{A} $ locus leftward. Differentiating $ \Gamma(A_{t}) $ with respect to z, we obtain

$$ \frac{\partial\Gamma(A_{t})}{\partial z}=\frac{\gamma\delta A_{t}^{1-\phi}}{p\beta\bar{\theta}(1-b)}\frac{\delta+\rho(b)\bar{\alpha}A_{t}^{1-\alpha}}{\left[(1-z)\delta+\rho(b)\bar{\alpha}A_{t}^{1-\alpha}\right]^{2}}>0. $$

Therefore, when the government increases z, the locus of $ N_{t}=\Gamma(A_{t}) $ shifts upward. We can depict these shifts in Figure 3 and state the following proposition:

Proposition 3. A rise in the child allowance z reduces the region where R&D is conducted, whereas it enlarges the region where population increases.

The effects of child allowance is opposite to those of the increase in life expectancy. Subsidizing the rearing cost positively impacts the fertility rate. However, from $ (25) $ and $ (26) $ , a higher z leads to a higher tax burden on young individuals. This negative effect reduces per capita savings, and decreases investment in R&D. From these results, when the government raises z, the R&D region shrinks and the region where population increases expands. We also consider the effects of z on the sustainability of economic growth through numerical examples in Section 6.

Figure 3 : Effects of an increase in z or b when $ 0 \leq b < \tilde{b} $
Population Dynamics, Longer Life Expectancy, and Child-Rearing Policies in an R&D-based Growth Model with Overlapping Generations

Effects of child care service

Next, we examine the effects of changes in b. Differentiating $ \widehat{A} $ with respect to b yields

$$ \frac{\partial\widehat{A}}{\partial b}=\frac{[\gamma+(1+p\beta+\gamma)\rho^{\prime}(b)]\widehat{A}}{(1-\alpha)[(1-b)\gamma-(1+p\beta+\gamma)\rho(b)]}. $$

From the existence condition of $ \widehat{A} $ , we obtain $ (1-b)\gamma-(1+p\beta+\gamma)\rho(b)>0 $ if $ 0\leq b<\widehat{b} $ . Thus, the sign of $ \partial\widehat{A}/\partial b $ is determined by the sign of the term $ \gamma+(1+p\beta+\gamma)\rho'(b) $ . Here, we impose the following assumption:

Assumption 4. $ \gamma+(1-p\beta+\gamma)\rho^{\prime}(0)<0. $

Under Assumption 4, we can show in Appendix D that $ \tilde{b}<\tilde{b} $ and $ \partial\widehat{A}/\partial b\lessgtr0 $ if $ b\lessgtr\tilde{b} $ , where $ \tilde{b} $ is defined as $ \gamma+(1+p\beta+r)\rho'(\tilde{b})=0 $ . We then differentiate $ \Gamma(A_{t}) $ with respect to b as follows:

$$ \begin{align*}\frac{\partial\Gamma(A_{t})}{\partial b}=&\frac{A_{t}^{1-\phi}}{p\beta\theta}\Biggl\{\frac{[(1-z)(1+p\beta)+\gamma]\delta+(1+p\beta+\gamma)\rho(b)\hat{\alpha}A_{t}^{1-\alpha}}{(1-b)^{2}[(1-z)\delta+\rho(b)\hat{\alpha}A_{t}^{1-\alpha}]}\\&-\frac{z\gamma\delta\rho^{\prime}(b)\hat{\alpha}A_{t}^{1-\alpha}}{(1-b)[(1-z)\delta+\rho(b)\hat{\alpha}A_{t}^{1-\alpha}]^{2}}\Biggr\}>0.\end{align*} $$

We summarize these results as follows. If $ 0 \leq b < \tilde{b} $ , an increase in b shifts the $ A_{t} = \tilde{A} $ locus leftward and the $ N_{t} = \Gamma(A_{t}) $ locus upward (see Figure 3). On the other hand, if $ \tilde{b} < b < \tilde{b} $ , a rise in b shifts the $ A_{t} = \tilde{A} $ locus rightward and the $ N_{t} = \Gamma(A_{t}) $ locus upward (see Figure 4). Thus, we have the following proposition:

Proposition 4. When $ 0 \leq b < \tilde{b} $ , an increase in childcare service b decreases the region

Figure 4 : Effects of an increase in b when $ \tilde{b} \leq b < \hat{b} $
where R&D is conducted, whereas it expands the region where population increases. When $ \tilde{b} 0(dn_{t}/db < 0) $ holds $ ^{6} $ . As well as the effect of z, a rise in b decreases per capita savings because of the tax burden effect. Therefore, if $ 0 \leq b < \tilde{b} $ , the effect of b is similar to that of z. When the government increases b, the R&D region shrinks and the region where population increases expands. In contrast, if $ \tilde{b} < b < \tilde{b} $ , the tax burden effect is very large. An increase in b decreases both regions where R&D is conducted and where population increases. In this case, expanding childcare service is not appropriate for the sustainability of economic growth. We will see the effects of b on the sustainability of economic growth through the numerical examples presented in Section 6.

### 5. Growth rates

Thus far, we have focused on the sustainability of economic growth. We must consider the growth rate of per capita GDP. As in Strulik et al. (2013) and Hashimoto and Tabata (2016), we define per capita GDP as $ y_{t} \equiv Y_{t}/N_{t} $ . Substituting (20) into (19), we obtain $ y_{t} = [\alpha^{2}/(1-\alpha)]^{\alpha} A_{t}^{1-\alpha} l_{Y,t} $ , where $ l_{Y,t} \equiv L_{Y,t}/N_{t} $ . The growth rate of per capita GDP is as follows:

$$ 1+g_{y,t}\equiv\frac{y_{t+1}}{y_{t}}=\left(\frac{A_{t+1}}{A_{t}}\right)^{1-\alpha}\frac{l_{Y,t+1}}{l_{Y,t}} $$
The growth rate of per capita GDP is determined by the growth of $ A_{t} $ and by the changes in $ l_{Y,t} $ . Let us define the growth rate of $ A_{t} $ as $ 1 + g_{A,t} \equiv A_{t+1}/A_{t} $ . If $ L_{A,t} > 0 $ , we can calculate $ g_{A,t} $ using (31). We then consider the labor market clearing condition to derive $ l_{Y,t} $ . Using (12) and (20), the labor input into the production of intermediate goods becomes $ A_{t}x_{t} = \alpha^{2}L_{Y,t}/(1-\alpha) $ . From (30), when R&D is conducted, $ L_{A,t} > 0 $ holds. Under child-rearing policies, (17) can be expressed as $ L_{Y,t} + A_{t}x_{t} + L_{A,t} = [1 - b - \rho(b)n_{t}]N_{t} $ . Hence, these results imply

$$ l_{Y,t}{=}\frac{1{-}\alpha}{1{-}\alpha{+}\alpha^{2}}[1{-}b{-}\rho(b)n_{t}{-}l_{A,t}]{~i f~}L_{A,t}{>}0, $$

where $ l_{A,t} \equiv L_{A,t}/N_{t} $ . Using (29), (30), (31), (34), and (35), we can calculate the growth rate of per capita GDP $ g_{y,t} $ . Furthermore, we can show $ g_{y,t} = 0 $ if $ L_{A,t} = 0^{7} $ .

If the economy is on a sustainable growth path, $ A_{t} $ and $ N_{t} $ grow forever. We define the balanced growth path (BGP) as the state of the economy at which the growth rates and the employment share of each sector become constant. Using (29), $ n_{t} $ along the BGP is given by

$$ \lim_{A_{t}\to\infty}n_{t}=\frac{(1-b)\gamma}{(1+p\beta+\gamma)\rho(b)}\equiv n^{*}. $$

In this study, asterisks represent variables along the BGP. As shown in Appendix E, we can derive the growth rates along the BGP as follows:

$$ \begin{aligned}&1+g_{A}^{*}=(n^{*})^{\frac{1}{1-\phi}},\\&1+g_{y}^{*}=(n^{*})^{\frac{1-\alpha}{1-\phi}}.\end{aligned} $$

As in Jones (1995), the growth rates along the BGP are determined by the fertility rate. We then consider the effects of changes in p, z, and b on $ n^{*} $ . From (36), a rise in p reduces $ n^{*} $ and z does not affect $ n^{*} $ . As shown in Appendix F, we obtain

$$ \frac{\partial n^{*}}{\partial b}\gtrless0\Leftrightarrow b\gtrless\bar{b}, $$

where $ \bar{b} $ is defined as $ \rho(\bar{b}) + (1 - \bar{b})\rho'(\bar{b}) = 0 $ . Therefore, there exists $ \bar{b} $ that maximizes the long-run growth rate. However, does the economy achieve sustainable growth under this value? How long does it take to attain this maximal rate? We will see these answers in the following numerical analyses.

At the end of this section, we mention why z has no effect on $ n^{*} $ . As discussed above, when the stock of knowledge $ A_{t} $ is larger, the wage rate $ w_{t} $ is higher. If the economy
achieves sustainable growth, the wage rate continues to rise. Thus, the rearing cost paid in final goods, $ (1-z)\delta n_{t} $ , becomes negligibly low and the individual's fertility decision depends on the rearing time cost in the long run $ ^{8} $ .

### 6. Numerical results

We analyze the model numerically because investigating the aforementioned effects of changes in p, z, and b on the sustainability of economic growth and the growth rates along the transitional dynamics is too complicated. First, we consider the effects of an increase in life expectancy as discussed in Subsection 3.3. In the dynamic system characterized by equations (1), (21), and (23), we adopt the following parameters: $ \beta=1 $ , $ \gamma=0.3 $ , $ \alpha=0.8 $ , $ \delta=0.1 $ , $ \phi=0.7 $ , $ \bar{\theta}=1 $ , and $ \rho=0.1 $ . Under $ A_{0}=20 $ , we calculate the initial value of population size that converges to the stable saddle-point $ (\bar{A}, \bar{N}) $ . Let us define $ N_{sad} $ as this initial value. The solid and dashed lines in Figure 5 present $ N_{sad} $ and $ \bar{N} $ for each value of $ p \in [0.4, 0.8] $ . From Figure 5 and Proposition 2, when p increases, the point $ (\bar{A}, \bar{N}) $ shifts up and rightward and $ N_{sad} $ rises. This shows that the saddle-point stable arm shifts up and rightward, that is, the region of sustainable growth shrinks. As mentioned in Subsection 3.3, a0n increase in p raises the R&D region and reduces the region where population increases. As shown in Figure 5, the latter exceeds the former effect.

Next, we set $ A_{0}=20 $ and $ N_{0}=40 $ and vary the value of p from 0.4 to 0.8 at intervals

Figure 5 : Effects of changes in p on the sustainability of economic growth. The solid line is $ N_{sad} $ . The dashed line is $ \bar{N} $ .
of 0.1. Figure 6 presents the evolution of $ n_{t} $ , $ g_{A,t} $ , and $ g_{y,t} $ for each value of p. To see the short-run effects, we provide the short-run values of $ n_{t} $ , $ g_{A,t} $ , and $ g_{y,t} $ in Table 1. Initially, population size $ N_{t} $ decreases, whereas the stock of knowledge $ A_{t} $ grows because $ n_{t}<1 $ and $ g_{A,t}>0 $ from Table 1. When p=0.4, p=0.5, and p=0.6, the economy attains sustainable growth because $ n_{t}>1 $ holds in a finite time. However, when p=0.7 and p=0.8, the economy falls into the trap (that is, $ n_{t}<1 $ and $ g_{A,t}=0 $ hold in a finite time). This result is consistent with that of Figure 5. As mentioned in Subsection 3.3, an increase in p raises per capita saving $ s_{t} $ and decreases the fertility rate $ n_{t} $ . Table 1 shows that investment in R&D initially expands but starts to decrease after a few periods. From (22), as the population size falls, the total savings decrease, and this reduces investment in R&D. Table 1 also shows that the initial increase in R&D

Table 1: The short-run values of $ n_{t} $ , $ g_{A,t} $ , and $ g_{y,t} $ in Figure 6.

n_{t}
t	p=0.4	p=0.5	p=0.6	p=0.7	p=0.8
0	0.847	0.800	0.758	0.720	0.686
1	0.965	0.925	0.887	0.850	0.815
2	1.033	0.993	0.953	0.915	0.878
3	1.076	1.033	0.990	0.947	0.906
4	1.108	1.061	1.011	0.963	0.917
5	1.136	1.082	1.026	0.971	0.918
6	1.163	1.101	1.036	0.973	0.918

g_{A,t}
t	p=0.4	p=0.5	p=0.6	p=0.7	p=0.8
0	2.831	3.523	4.142	4.699	5.203
1	1.169	1.301	1.385	1.435	1.461
2	0.660	0.659	0.629	0.585	0.532
3	0.472	0.415	0.342	0.262	0.183
4	0.410	0.318	0.216	0.115	0.019
5	0.409	0.287	0.160	0.040	0.000
6	0.444	0.290	0.138	0.000	0.000

g_{y,t}
t	p=0.4	p=0.5	p=0.6	p=0.7	p=0.8
0	0.223	0.260	0.293	0.324	0.355
1	0.145	0.165	0.185	0.206	0.227
2	0.099	0.106	0.112	0.119	0.128
3	0.076	0.072	0.068	0.065	0.064
4	0.067	0.056	0.044	0.033	0.008
5	0.067	0.049	0.031	0.015	0.000
6	0.073	0.049	0.025	0.000	0.000
Figure 6 : Numerical example (life expectancy rising).

$$ \rightarrow p=0.4\quad\rightarrow p=0.5\quad\rightarrow p=0.6\quad\rightarrow p=0.7\quad\rightarrow p=0.8 $$

investment is not large, and as a result, an economy of higher p can be trapped in the no R&D region. With regard to the growth rate of per capita GDP $ g_{y,t} $ Figure 6 shows that the path of $ g_{y,t} $ is similar to that of $ g_{A,t} $ . This implies that the effect of changes in $ l_{Y,t} $ on $ g_{y,t} $ is sufficiently small from (34). As in Section 5, Figure 6 shows that a higher p leads to lower fertility and growth rates in the long run.

## Child-rearing policies

To examine the effects of child-rearing policies, we specify the functional form of rearing time cost $ \rho(b) $ as follows: