3.1. Production and the environment

The production of the composite good is carried out by a representative firm. As in Varvarigos (Reference Varvarigos2013), the output is produced according to constant returns to scale technology. The firm combines units of efficient work, $L_{t}h_{t}$ – where $h_{t}$ is the human capital per worker – and capital stock, $K_{t}$:

(1)\begin{equation} Y_{t} = AK_{t}^{\alpha }\left( L_{t}h_{t}\right) ^{1-\alpha }, \end{equation}

where $A>0$ is the technology level and $\alpha \in (0,1)$ is the share of capital in production.

During the production process, the firm emits pollution, which induces a negative impact on human capital accumulation. To correct this externality, the government implements a tax $\tau \in (0,1)$ on production. Tax revenue is used to fund pollution emissions reduction, $m_{t}$, with $m_{t} = \xi \tau Y_{t}$ and $\xi \in ]0,1]$ representing the efficiency of abatement expenditures. The firm's profit is:

(2)\begin{equation} \Pi _{t} = A(1-\tau )K_{t}^{\alpha }\left( L_{t}h_{t}\right) ^{1-\alpha}-w_{t}h_{t}L_{t}-R_{t}K_{t}. \end{equation}

For simplicity, capital is assumed to fully depreciate in one period. Denoting the wage for one unit of efficient labor by $w_{t}$ and the return factor of capital by $R_{t}\equiv 1 + r_{t}$, with $r_{t}$ being the interest rate, factor prices are:

(3)\begin{align} w_{t} & = A(1-\alpha )(1-\tau )K_{t}^{\alpha }\left( L_{t}h_{t}\right)^{-\alpha }, \end{align}

(4)\begin{align} R_{t} & = A\alpha (1-\tau )K_{t}^{\alpha -1}\left( L_{t}h_{t}\right)^{1-\alpha }. \end{align}

The production sector generates a pollution flow with respect to $\Omega \in ]0;1]$, the pollution intensity of production. The dynamics of the pollution stock, given in equation (5), also depend on $a\in ]0;1]$, the natural absorption rate of pollution.

(5)\begin{equation} Z_{t + 1} = \Omega Y_{t}-m_{t} + (1-a)Z_{t} = (\Omega -\xi \tau )AK_{t}^{\alpha} \left( L_{t}h_{t}\right) ^{1-\alpha } + (1-a)Z_{t}. \end{equation}

Any local pollution such as pesticides, metals or POPs in soil or freshwater could be considered in equation (5). Here, if $a = 1$, the model describes a pollution flow. If $a = 0$, all emissions of pollution remain in the environment and the only sustainable way of dealing with the pollution is to completely abate emissions. Abatement efforts could include water treatment, waste processing or changes in technology to reduce emissions.

3.2. Household behavior

In this economy, there are three generations: children, adults and retirees. At each period $t$, adults choose their number of children $n_{t}$, depending on the cost of fertility. As in de la Croix and Gosseries (Reference de la Croix and Gosseries2012), raising children requires a fraction $\sigma N_{t}^{\delta }n_{t}$ of income, where $0<\delta <1$ captures the overcrowding effect: the larger the population, the more costly fertility is.Footnote ⁶ This assumption is in line with empirical evidence described in works such as de la Croix and Gobbi (Reference de la Croix and Gobbi2017) or Sibly et al. (Reference Sibly, Hone, Clutton-Brock, Lutz and Qiang2002).

The emigration rate is denoted by $\rho \in \lbrack 0,1[$. Migration implies that only $(1-\rho )n_{t}N_{t}$ children stay in the domestic country as adults. The other $\rho n_{t}N_{t}$ children migrate to countries where wages are greater. The evolution of the size of the adult generation (or the labor force) is represented by:

(6)\begin{equation} N_{t + 1} = n_{t}N_{t}(1-\rho ). \end{equation}

Adults born in $t-1$ care about their adult and old age consumption levels, denoted by $c_t$ and $d_{t + 1}$, respectively. Agents’ preferences are represented by the following utility function, according to the psychological discount factor $\beta$:

(7)\begin{equation} U(c_{t},d_{t + 1}) = \ln (c_{t}) + \beta \ln (d_{t + 1}). \end{equation}

When they are children, individuals are reared by their parents and receive an education to acquire a human capital level $h_{t}$. Adults in the domestic country supply one inelastic unit of labor to earn a wage $w_{t}$ per unit of human capital $h_{t}$. A fraction $\gamma$ of their revenue is transferred to their parents. The rest of their income is devoted to consumption $c_{t}$, savings $s_{t}$ and children's education $n_{t}e_{t}$. Adults in foreign countries can claim higher wages, which are assumed to be proportional to the domestic wage, such that $w_{t}^{F}\equiv \varepsilon w_{t}$, where $\varepsilon >1$ denotes the net gain from migration. The parameter $\varepsilon$ can be interpreted as the increase in income that can be obtained in the receiving country over the income in the domestic area, from which the costs associated with migration are subtracted. Migrants also transfer a share $\gamma$ of their revenue to their parents. Thus, parents receive remittances from their children abroad and intergenerational transfers from children in the domestic area. The budget constraint in the first period is given by:

(8)\begin{equation} c_{t} + s_{t} + n_{t}e_{t} = w_{t}h_{t}(1-\gamma -\sigma n_{t}N_{t}^{\delta }).\end{equation}

Human capital per child $h_{t + 1}$ depends on education expenditures per child $e_{t}>0$ (null values would bring the stock of human capital to 0), the parents’ human capital $h_{t}$ and the pollution level, which affects the efficiency of human capital accumulation $\theta (Z_{t})$. This function is defined for positive or null real values of $Z_{t}$ and decreases with the pollution stock, such that $\theta ^{\prime }(Z_{t})<0$. Therefore, a polluted environment deteriorates the children's ability to accumulate human capital, regardless of whether it comes from their parents’ human capital or investments in education. The children's human capital $h_{t + 1}$ is written as:

(9)\begin{equation} h_{t + 1} = \theta (Z_{t})h_{t}^{1-\mu }e_{t}^{\mu }, \end{equation}

where $0<\mu <1$ represents the efficiency of education.

When they are old, agents only consume their savings remunerated at the return factor $R_{t + 1}$ and the intergenerational transfers sent by their children, wherever the children live. The budget constraint in the second period is written as:

(10)\begin{equation} d_{t + 1} = s_{t}R_{t + 1} + n_{t}\gamma (1-\rho )w_{t + 1}h_{t + 1} + n_{t}\gamma \rho\varepsilon w_{t + 1}h_{t + 1}. \end{equation}

Hence, individuals face a first tradeoff between adult versus old age consumption. Second, they choose between savings or children's transfers to fund their consumption when old. Finally, they decide whether to increase education or fertility to raise the amount of intergenerational transfers received. Here, $\gamma (1-\rho + \rho \varepsilon )w_{t + 1}h_{t + 1}$ is the share of children's income received by parents. In the rest of the paper, the share of the children's income transferred is denoted by $\Lambda _{h} = \gamma (1-\rho + \rho \varepsilon )$. It is increased by $\varepsilon ,\rho$ and $\gamma$, which are the net gain from migration, the emigration rate and the intergenerational transfer rate, respectively. The consumer program is summarized by:

\begin{align*} \underset{c_{t},s_{t},e_{t},n_{t}}{\max}U(c_{t},d_{t + 1}) & = \ln (c_{t}) + \beta \ln (d_{t + 1}) \\ \textrm{s.t.}\quad c_{t} + s_{t} + n_{t}e_{t} & = w_{t}h_{t}(1-\gamma -\sigma n_{t}N_{t}^{\delta }) \\ d_{t + 1} & = s_{t}R_{t + 1} + n_{t}\Lambda _{h}w_{t + 1}h_{t + 1} \\ h_{t + 1} & = \theta (Z_{t})h_{t}^{1-\mu }e_{t}^{\mu }. \end{align*}

To solve this model, constraints are substituted into the utility function, which leads to the first-order condition (FOC) of the household's problem. The FOC with respect to $s_{t}$ shows the consumption tradeoff over the life cycle (equation (11)). It depends on the psychological discount factor, $\beta$, and the return factor on savings, $R_{t + 1}$. The two other FOCs of the household's problem with respect to education and fertility – equations (12) and (13) – suggest that the remuneration from intergenerational transfers and savings should be equal in equilibrium.

(11)\begin{align} \frac{1}{c_{t}} & = \frac{\beta R_{t + 1}}{d_{t + 1}} \end{align}

(12)\begin{align} \frac{1}{c_{t}} & = \frac{\beta \mu \Lambda _{h}w_{t + 1}h_{t + 1}}{e_{t}d_{t + 1}} \end{align}

(13)\begin{align} \frac{1}{c_{t}} & = \frac{1}{\sigma N_{t}^{\delta }w_{t}h_{t} + e_{t}}\frac{\beta \Lambda _{h}w_{t + 1}h_{t + 1}}{d_{t + 1}}. \end{align}

Combining (12) and (13) yields a first no-arbitrage condition. It ensures that the household is indifferent between education and fertility by equating the opportunity costs of education and fertility. It directly determines the level of education expenditures:

(14)\begin{align} \frac{\mu \Lambda _{h}w_{t + 1}h_{t + 1}}{e_{t}} & = \frac{\Lambda_{h}w_{t + 1}h_{t + 1}}{\sigma N_{t}^{\delta }w_{t}h_{t} + e_{t}} \end{align}

(15)\begin{align} e_{t}^{\star } & = \frac{\mu \sigma N_{t}^{\delta }}{1-\mu }w_{t}h_{t}. \end{align}

Education choice depends solely on adult income, the cost of raising children and the efficiency of education. Moreover, the adult population increases the congestion effect, which results in higher costs of fertility. Thus, the larger the income and/or the adult population size, the higher the education expenditures are.

Another no-arbitrage condition is obtained thanks to equations (11) and (13). It equates the returns from savings $R_{t + 1}$ and future intergenerational transfers relative to the cost of children. Thus, the condition ensures that the household is indifferent between investments in children and savings. Moreover, when combined with the optimal education choice, this gives rise to a relationship essentially between equilibrium input prices at $t + 1$:

(16)\begin{align} R_{t + 1} & = \frac{\Lambda _{h}w_{t + 1}h_{t + 1}}{\sigma N_{t}^{\delta}w_{t}h_{t} + e_{t}} \end{align}

(17)\begin{align} R_{t + 1} & = \frac{(1-\mu )\Lambda _{h}}{\sigma }\frac{w_{t + 1}h_{t + 1}}{w_{t}h_{t}N_{t}^{\delta }}. \end{align}

Rewriting the adult's budget constraint according to the optimal choice of education summarizes the investments in children – i.e., fertility and human capital – in one term: $n_{t}({\sigma w_{t}h_{t}N_{t}^{\delta }}/{ (1-\mu )})$. This term represents a part of the adults’ investments in their old age consumption. In other words, it is the adult available income that is not consumed in the first period. It is denoted by $x_{t}$ in equation (19):

(18)\begin{align} w_{t}h_{t}(1-\gamma ) & = c_{t} + s_{t} + n_{t}\frac{\sigma w_{t}h_{t}N_{t}^{\delta }}{1-\mu } \end{align}

(19)\begin{align} x_{t} & = s_{t} + n_{t}\frac{\sigma w_{t}h_{t}N_{t}^{\delta }}{1-\mu }. \end{align}

The next step is to solve the intertemporal optimization problem of the household. To do so, expression (17) and the budget constraints are introduced in the FOC with respect to the savings (equation (11)) to obtain a new expression for $x_{t}\equiv ({\beta (1-\gamma )}/{(1 + \beta )}) w_{t}h_{t}$ (equation (20)). It also leads directly to an expression for adult consumption:

(20)\begin{align} \beta R_{t + 1}c_{t} = d_{t + 1} &\Leftrightarrow \frac{\beta (1-\gamma )}{1 + \beta }w_{t}h_{t} = s_{t} + n_{t} \frac{\sigma w_{t}h_{t}N_{t}^{\delta }}{1-\mu} \end{align}

(21)\begin{align} &\Leftrightarrow c_{t} = \frac{1-\gamma }{1 + \beta }w_{t}h_{t}. \end{align}

Equations (20) and (21) depict the tradeoff between adult consumption and investments in old age consumption, i.e., between $c_{t}$ and $x_{t}$. As expected, the discount factor, $\beta$, has a positive (negative) effect on $x_{t}$ ($c_{t}$). Second, both variables are negatively affected by the intergenerational transfer rate, $\gamma$, which generates a negative income effect. Nevertheless, $\gamma$ has an ambiguous effect on old age consumption, $d_{t + 1}$, and since higher transfers can be received when old, this lowers the burden on investing when an adult.

From the two expressions for $x_{t}$, it is possible to obtain the initial relationship between savings and investments in intergenerational transfers:

(22)\begin{equation} s_{t} = w_{t}h_{t}\left( \frac{\beta (1-\gamma )}{1 + \beta }-n_{t}\frac{\sigma N_{t}^{\delta }}{1-\mu }\right). \end{equation}

At this point, the tradeoff between fertility and savings is not solved. It is necessary to obtain an additional equation between savings and fertility. To do so, note that the representative household has perfect foresight regarding future returns from investment (de la Croix and Michel, Reference de la Croix and Michel2002). As a result, the optimal choices of the households simultaneously solve their intertemporal income optimization problem and the market clearing conditions (MCCs), given in the following equations:

(23)\begin{align} K_{t + 1} & = s_{t}N_{t} \end{align}

(24)\begin{align} L_{t + 1} & = N_{t + 1} = n_{t}N_{t}(1-\rho ) \end{align}

(25)\begin{align} h_{t + 1} & = \theta (Z_{t})e_{t}^{\mu }h_{t}^{1-\mu }. \end{align}

Combining the input prices, $w_{t + 1}$ and $R_{t + 1}$ (given by equations (3) and (4)) and the MCCs in equality (17) yields the following equation:

(26)\begin{equation} s_{t} = n_{t}\left[ \frac{1-\alpha }{\alpha }\frac{(1-\mu )\Lambda _{h}}{(1-\rho) \sigma w_{t}h_{t}N_{t}^{\delta }}\right] ^{-1}. \end{equation}

Finally, solving the system defined by equations (22) and (26) leads to the household's optimal choices:

(27)\begin{align} s_{t}^{\star} & = \frac{\beta \alpha (1-\rho )(1-\gamma )}{(1 + \beta )[\alpha (1-\rho ) + \Lambda _{h}(1-\alpha )]}w_{t}h_{t}, \end{align}

(28)\begin{align} n_{t}^{\star } & = \frac{\beta \Lambda _{h}(1-\gamma )(1-\alpha )(1-\mu )}{\sigma (1 + \beta )[\alpha (1-\rho ) + \Lambda _{h}(1-\alpha )]}N_{t}^{-\delta }. \end{align}

A comparative static analysis of household choices, given by equations (15), (27) and (28), is conducted with respect to migration-related parameters.Footnote ⁷ First, for the tradeoff between savings and investments in children to fund old age consumption – i.e., $s_{t}$ versus $n_{t}({\sigma w_{t}h_{t}N_{t}^{\delta }}/{(1-\mu )})$ – savings are negatively correlated with $\Lambda _{h}$, the share of children's income received by parents. Thus, increases in the net gain from migration, $\varepsilon$, the emigration rate $\rho$ and/or the intergenerational transfer, $\gamma$, increase children's investments at the expense of savings. For $\gamma$, this is aggravated by the negative income effect that comes from the term $(1-\gamma )$, as described earlier.

Second, the tradeoff between fertility and education – i.e., $n_{t}$ and $e_{t}$ – depends on the incentives to invest in education and, on the other hand, on migration. As noted above, households choose to invest in education expenditures if the efficiency of education and the child-rearing costs, measured by $\mu$ and $\sigma N_{t}^{\delta }$, respectively, are higher. Through the overcrowding effect, migration may change the latter term. Specifically, $n_{t}$ is positively correlated with $\rho$ and $\varepsilon$, while the impact of $\gamma$ depends on the interaction between the share of the children's income received $\Lambda _{h}$ and the negative income effect induced by $(1-\gamma )$. This interaction is captured by the condition below:

\[ \frac{\partial n_{t}}{\partial \gamma }>0\Leftrightarrow \frac{\gamma }{1-\gamma }<\frac{\alpha (1-\rho )}{\Lambda _{h}(1-\alpha )}. \]

If migration leads to a strong increase in population size, fertility decreases to the benefit of education expenditures because of the overcrowding effect. Therefore, migration can affect education through its impact on fertility. To clarify this mechanism, it is necessary to first study the population dynamics and labor market.

Finally, as households do not take into account the environmental externality, their optimal choices cannot reflect the dynamics of human capital. Therefore, agents will invest in education, fertility or capital in the same way as in a situation without the externality.Footnote ⁸

4.1. Case 1: equilibrium with the total abatement of pollution emissions

This case emerges only if the efficiency of the abatement effort is high enough and/or if the pollution intensity of production is low enough to have $\xi \tau = \Omega$. With a long-term zero pollution stock, $\bar {\theta } = \theta (0)$ denotes the efficiency of human capital accumulation. In that case, the ratio of capital to efficient units of labor $k_{t}$ can be defined as:

(30)\begin{equation} k_{t + 1}\equiv \frac{K_{t + 1}}{N_{t + 1}h_{t + 1}} = \left(\frac{A\sigma (1-\tau)}{1-\mu }\right)^{1-\mu} \frac{\alpha }{\bar{\theta}\Lambda _{h}[\mu(1-\alpha )]^{\mu }}k_{t}^{\alpha (1-\mu )}N_{t}^{\delta (1-\mu )}. \end{equation}

The growth factors of the stocks of physical capital, human capital per capita and the adult population are denoted by $g_{t}^{K}$, $g_{t}^{h}$ and $g_{t}^{N}$, respectively.

(31)\begin{align} g_{t}^{K} = \frac{K_{t + 1}}{K_{t}} & = \Psi \alpha A(1-\tau )(1-\rho)k_{t}^{\alpha -1} \end{align}

(32)\begin{align} g_{t}^{h} = \frac{h_{t + 1}}{h_{t}} & = \bar{\theta} \left[\frac{\mu A\sigma(1-\alpha )(1-\tau )}{1-\mu }\right] ^{\mu }k_{t}^{\alpha \mu }N_{t}^{\mu\delta } \end{align}

(33)\begin{align} g_{t}^{N} = \frac{N_{t + 1}}{N_{t}} & = \Psi \Lambda _{h}(1-\rho )\frac{(1-\mu )}{\sigma }N_{t}^{-\delta }. \end{align}

Equation (30) depicts the labor force growth rate, which depends solely on the economy's structural parameters and on population size. Consequently, population dynamics affect both human and physical capital stocks; however, the reverse is not true. The growth rate of the population is directly given by $n_{t}^{\ast }(1-\rho )$, and in the long run $n^{\ast } = 1/(1-\rho )$. It is possible to decrease the adult generation size until $N^{\star }$ is reached. This is the case if the initial population size is larger than $N^{\star }$, the steady-state value of the labor force:

(34)\begin{equation} N^{\star } = \left[ \Psi (1-\rho )\Lambda _{h}\frac{(1-\mu )}{\sigma }\right]^{{1}/{\delta }}. \end{equation}

All the parameters that have a positive effect on $n_{t}$, except for $\rho$, have a positive effect on $N^{\star }$. The ambiguous effect of the emigration rate is due to its positive (negative) impact on the number of children (adults). It depends on the following condition:

(35)\begin{equation} \frac{\partial N^{\star }}{\partial \rho }>0\Leftrightarrow \frac{1-\rho}{1-\rho + \rho \varepsilon }>\sqrt{\frac{\gamma (1-\alpha )}{\alpha(\varepsilon -1)}}. \end{equation}

Proposition 2: Due to the constant returns to scale for human and physical capital, the economy reaches a balanced growth path (BGP), where the system satisfies Proposition 1, and the stock of physical and efficient units of labor grows at the same constant rate $g_{\textrm {BGP}} = g^{K} = g^{h}$. Therefore, the long-term ratio of capital per unit of efficient labor is constant: $k_{t}\equiv {K_{t}}/{L_{t}h_{t}} = k_{\textrm {BGP}}$. The values of $k$ and $g$ in the unique locally stable equilibrium are:

(36)\begin{align} k_{\textrm{BGP}} & = \left[\frac{\alpha \lbrack \Psi A(1-\rho )(1-\tau )]^{1-\mu }}{\bar{\theta} [\mu \Lambda _{h}(1-\alpha )]^{\mu }}\right]^{{1}/{(1-\alpha(1-\mu))}} \end{align}

(37)\begin{align} g_{\textrm{BGP}} & = \left[ [\bar{\theta}[\mu \Lambda _{h}(1-\alpha )]^{\mu}]^{1-\alpha } [\Psi \alpha ^{\alpha }A(1-\rho )(1-\tau )]^{\mu }\right] ^{{1}/{(1-\alpha (1-\mu ))}}. \end{align}

The positive effects of $A$ and $\sigma$ on the long-term ratio of capital per unit of efficient labor, $k_{\textrm {BGP}}$, result from the increase in production and from the decrease in the number of children due to the extra cost – i.e., the decrease in the size of the next generation. The negative impact of the other parameters on this ratio is due to the increase in the number of units of efficient labor in the economy – with respect to $\varepsilon$, $\bar {\theta }$, $\gamma$ and $\rho$. Finally, while the effect of pollution is completely canceled out, the economic cost of this operation leads to a reduction in the capital stock per efficient unit of labor with respect to $\tau$.

Proposition 4: On the BGP, economic growth, $g_{\textrm {BGP}}$, is positively impacted by the technology factor, $A$, the psychological discount factor, $\beta$, the efficiency of human capital accumulation, $\bar {\theta }$, and the net gain from migration, $\varepsilon$. The tax rate has a negative effect on economic growth, while the effects of the intergenerational transfer rate, $\gamma$, and the emigration rate, $\rho$, depend on the conditions below:

(38)\begin{align} \frac{\partial g_{\textrm{BGP}}}{\partial \rho }>0 &\Leftrightarrow \frac{1-\rho }{1-\rho + \rho \varepsilon }>\frac{[\varepsilon -(1-\alpha )(\varepsilon-1)(1-\rho )]}{\alpha (\varepsilon -1)(1-\rho )} \end{align}

(39)\begin{align} \frac{\partial g_{\textrm{BGP}}}{\partial \gamma }>0 &\Leftrightarrow \frac{1-\alpha}{\alpha }\frac{1-(1-\alpha )(1-\gamma )}{(1-\alpha )(1-\rho )-\gamma }>\frac{(1-\rho )}{\Lambda _{h}}. \end{align}

First, note that the long-term growth factor directly gives the growth in production per worker and thus in production per capita because of the constant population size. In that case, the growth factor of the production per capita can be directly translated into the growth in utility, which depends strongly on consumption (see online appendix B.3 for details).

On the one hand, some effects are easy to describe. For $k_{\textrm {BGP}}$, the tax rate represents the cost for the economy to cope with environmental degradation. Therefore, the higher pollution emissions are, the higher the tax rate and the lower the growth rate. Increases in the technological factor, $A$, and in the efficiency of human capital accumulation, $\bar {\theta }$, lead to a more efficient economy. Increases in the psychological discount factor, $\beta$, result in higher investments in the future and subsequently in an increase in economic growth. Moreover, an increase in the net gain from migration, $\varepsilon$, enhances the income of the old age generation and per capita production growth.

On the other hand, the effects of the emigration rate, $\rho$, and of the intergenerational transfer rate, $\gamma$, are quite intricate. Conditions are obtained for the sign of the derivatives of the growth factor on the BGP with respect to the emigration rate and the intergenerational transfer (see online appendix B.2 for details); however, they are difficult to interpret. This is explained by the opposite effects that are observed on household choices and on aggregate variables.

First, the emigration rate, $\rho$, creates an incentive to have more children through the increase in the gains from migration. However, because there are more adults who leave the territory in the next period, it can lead to a decrease in the number of units of efficient labor. As the population size increases, education expenditures can increase because of the congestion effect, which is reinforced by the substitution effect between investments in children and savings that occurs when the emigration rate increases. In that context, higher migration can lead to an increase in human capital and thus to an increase in income, which is directly given by $w_th_t$. Therefore, the effect of the emigration rate on the capital stock is threefold. By increasing the number of children, there is a rise in child-rearing expenditures and thus a decrease in savings, which is the substitution effect, on the intensive margin. In addition, the extensive margin also affects the capital stock. Migration can lead to a decrease (an increase) in the adult population size, which induces a reduction (a rise) in the capital stock because of the smaller (larger) number of contributors. Finally, there is a migration effect on education expenditures and thus on human capital dynamics and income. In a wealthier economy, even if the share of the income devoted to savings is reduced, the capital stock might be larger, therefore the relationship between migration and economic growth depends on the combination of those three effects.

Second, the intergenerational transfer rate, $\gamma$, reduces fertility because of the negative income effect described above, but only when $\gamma$ is very high. For low values, $\gamma$ should lead to an increase in investment in children. Therefore, while the mechanisms are different, their impacts through the net gains from migration are the same as those of $\rho$. If small, it leads to a rise in the units of efficient labor, and if high, savings are low, and the population size and human capital may decline.

In conclusion, three main intuitions can be derived from this first case. If the environmental externality can be completely canceled out, the economy can reach a BGP where there is green growth. The higher the externality, the lower economic growth will be. Second, there is a strong tradeoff between intergenerational transfers and savings because migration enhances the net gain from the children's transfers. In that context, positive impacts from migration on the capital stock and the economy are possible, but only if there is a gain in human capital and in the labor force. Consequently, and this is the last intuition, it is possible to have an emigration rate that has a negative impact on the economic growth of these countries because of the combined effects on capital stock and on units of efficient labor stock. However, due to the complexity of the conditions obtained for the emigration rate and the intergenerational transfers, it is difficult to capture the respective magnitudes of these different effects. This could be clarified by a numerical analysis of two Caribbean islands: Barbados and Jamaica.

Ait Benhamou and Cassin (Reference Ait Benhamou and Cassin2020) conduct similar analyses on five Caribbean islands using a model without environmental degradation or endogenous fertility. Their parameters calibrated to study migration can be used directly in the present work, except for the fertility cost, $\sigma$, and the congestion parameter $\delta$ (see online appendix C for the detailed calibration method and the parameters values). Barbados and Jamaica have been selected for this analysis because they present very different features. Indeed, Jamaica relies heavily on migration ($\rho = 0.49$) because both the gain from migration, $\varepsilon$, and the intergenerational transfer rates are high ($\varepsilon = 6.58$ and $\gamma = 0.2$). In comparison, those values are lower in Barbados, $\rho = 0.37$, $\varepsilon = 1.91$ and $\gamma = 0.12$. Moreover, for this illustration, the environmental parameters are set to completely nullify the externality caused by production, $\Omega$, the natural absorption rate, $a$, the efficiency of the environmental policy, $\xi$, and the tax rate, $\tau$, and are thus set to 0.2, 0.5, 0.8 and 0.25, respectively.

Figure 4 displays economic growth according to the emigration rate and the share of income transferred to the parents for Barbados and Jamaica.Footnote ⁹ The relationship between economic growth and migration or remittances is described by inverted U-shaped curves. This result is in line with the literature, which holds that migration should not be overly large to prevent the depletion of productive capital stocks. In Jamaica, high migration levels lead to a strong substitution effect, and both observed parameters $\gamma$ and $\rho$ (represented by circles) are higher than their optimal level (depicted by the triangles). Barbados is not strongly impacted by migration – the curve on the left is relatively flat – because the net gain from migration is small, and thus even with a relatively high level of migration, the loss in terms of economic growth is small. This means that the negative outcomes from the migration rate, in particular, the reduction in savings, are higher if the country relies heavily on migration through higher remittances or higher gains from migration. In contrast, if the development stage of the sending country is close to those of the receiving economies, migration may not lead to a brain gain, but it does not have a negative impact on economic growth.

Figure 4. BGP values of the growth factor ($g_{\textrm {BGP}}$) according to parameters $\rho$ and $\gamma$.

4.2. Case 2: equilibrium with partial abatement of pollution

The next step of the analysis is to compute the steady state of the economy if pollution is not completely abated, thus if $Z^\star >0$.

Proposition 5: A steady state (SS) is an equilibrium satisfying Proposition 1 and where $N_{t} = N^{\star }$, $K_{t} = K^{\star }$, $h_{t} = h^{\star }$ and $Z_{t} = Z^{\star }$ are constant:

(40)\begin{align} N^{\star } & = \left[ \frac{(1-\mu )(1-\rho )\Lambda _{h}}{\sigma }\Psi\right] ^{{1}/{\delta }} \end{align}

(41)\begin{align} K^{\star } & = \alpha (1-\alpha )(1-\tau )\Psi (1-\rho )\frac{a\theta^{-1}(\chi )}{\Omega -\xi \tau } \nonumber\\ h^{\star } & = \frac{a\theta ^{-1}(\chi )}{(\Omega -\xi \tau )A}\left[ \alpha A(1-\tau )\right]^{-{\alpha }/{(1-\alpha)}} \left[ \frac{\sigma }{\Lambda_{h}(1-\mu )}\right] ^{{1}/{\delta }} \nonumber\\ &\quad \times (\Psi (1-\rho ))^{-{(1-\alpha(1-\delta ))}/{\delta (1-\alpha )}} \end{align}

(42)\begin{align} Z^{\star } & = \theta ^{-1}(\chi ), \end{align}

where $\chi = [\mu \Lambda _{h}(1-\alpha )]^{-\mu }[\Psi A\alpha ^{\alpha }(1-\tau )(1-\rho )]^{-{\mu }/{(1-\alpha )}}$ is the efficiency of human capital accumulation in the steady state and $\theta ^{-1}(.)$ is the inverse function of $\theta (Z_{t})$.

The model is a four-dimensional problem, and it is quite difficult to study the stability of the equilibrium. Instead, the rest of this section offers some insights into the mechanisms that might affect the dynamics of the model. In the previous case, when the population size converges to its steady-state value, human and physical capital grow at the same rate (see online appendix B.1). Here, the pollution stock prevents human capital from increasing in the long run. Therefore, the focus of this discussion is on the determinants of the pollution dynamics: the damage function, $\theta (Z_{t})$, the natural absorption rate, $a$, and the pollution intensity, $\Omega$.

The numerical analysis is conducted with Barbados’ parameters; however, the robustness test results for Jamaica are displayed in online appendix D.2 (see table A2). Two functions for $\theta (Z_{t})$ that respect the conditions given in the model (see section 3) are tested. This means that those functions are defined for positive or null values of $Z_{t}$ and their first derivatives with respect to $Z_{t}$ are negative. The two functions tested are $\theta _{1}(Z_{t}) = {\bar {\theta }}/{(1 + Z_{t})}$ and $\theta _{2}(Z_{t}) = {\bar {\theta }}/{(1 + Z_{t}^{2})}$.

In figure 5, the dynamics of production ($Y_{t}$), the capital stock ($K_{t}$), the pollution stock ($Z_{t}$) and human capital ($h_{t}$) are displayed. The plain line represents the benchmark economy with $\theta _{1}(Z_{t})$, while the dashed line displays the results for $\theta _{2}(Z_{t})$. The first specification results in a stable equilibrium with damped oscillations, while the second exhibits regular oscillations around the steady state. It appears that the second function is concave for some values of $Z_{t}$, which leads to an unstable equilibrium (see online appendix D.2 for details). A sufficient assumption to ensure the stability of the equilibrium should be that $\theta ^{\prime }(Z_{t})<0$ and $\theta ^{\prime \prime }(Z_{t})\geq 0$. Moreover, it is clear that the initial pollution stock might affect the stability of the equilibrium. In the rest of the article, the results are computed with the function $\theta _{1}(Z_{t})$ and, for simplicity, the subscript is removed.

Figure 5. The effect of $\theta (Z_{t})$ on the stability of the steady state.

Second, supplementary analyses of the stability have been conducted for pollution intensity, $\Omega ,$ and the natural absorption of pollution, $a$ (see online appendix D.2 for the details). Those parameters seems to impact the amplitude of the oscillations, the time necessary to reach the equilibrium, and steady-state values of production, human capital and physical capital stock. However, they do not impact the steady-state values of the pollution stock nor the stability of the equilibrium. In the presence of pollution emissions, human capital cannot increase without leading to a rise in pollution. Due to the externality, this leads to an abrupt decrease in human capital and the capital stock (because of income loss). With the loss of productive capital, production is lessened, which thus reduces the pollution stock. When pollution is low, another cycle begins, with human capital accumulation, growing production and physical capital. However, increases – for all variables – are slower because future human capital depends on past values of human capital. Consequently, the economy reaches a steady state with damped oscillations. In the steady state, the increase in human capital that would normally occur is exactly compensated by the reduction in abilities due to pollution. This cyclical convergence has also been found by Varvarigos (Reference Varvarigos2013) with similar mechanisms. The level where this equilibrium exists depends on $\chi$, the efficiency of human capital accumulation on the steady state. Before the steady state is reached, a high $\Omega$, or a low $a$, accelerates the accumulation of the pollution stock. This results in a larger cycle amplitude. However, the steady-state pollution stock is reached when the marginal increase in human capital is exactly compensated by the marginal loss due to the reduction in cognitive skills. Consequently, the long-term pollution stock depends only on parameters in $\chi$.