Modeling Social Status and Fertility Decisions Under Differential Mortality

Abstract

Unlike most non-human social animals, the social status of humans does not consistently correlate with higher fertility and in many cases appears to suppress fertility. This discrepancy has been employed as an argument against the use of evolutionary biology to understand human behavior. However, some literature suggests that social status and its implications for survival during high-mortality events may imply that status-seeking at a cost to fertility may be an optimal strategy over the long term. Here, we propose a theoretical model, in which each generation trades-off between social status and fertility under different economic and environmental constraints. To our knowledge, the model we present here is the first to connect individual decisions of generations, strategies to maximize long-term biological fitness, and key environmental and economic conditions in a coherent stylized modeling framework. We use it, in particular, to explicate the conditions, under which the strategy of having a lower number of offspring with higher social status may result in higher biological fitness over the long term. Furthermore, we delineate sets of economic and environmental conditions, for which the dynasty shrinks and grows. As adaptation of individual preferences is costly, limited and may take generations, we argue that a sudden change in environmental or economic conditions may shift a dynasty from a growing to a declining trajectory, which may be irreversible. Also, we show that in some cases, a slight change in environmental conditions can lead to a regime switch of an optimal strategy maximizing biological fitness.

Appendix

Proof of Theorem 5.3

First, let us show that the optimal preference $\omega_{C}^{*} = 0$; for that let us assume the opposite. Let set $\left( {\hat{\omega }_{C} ,\hat{\omega }_{R} ,\hat{\omega }_{E} ,\hat{\omega }_{B} } \right)$, where $\hat{\omega }_{C} > 0$, be optimal. Then, as one can see from the formula for PGR in Lemma, set $\left( {\frac{{\hat{\omega }_{C} }}{2},\hat{\omega }_{R} ,\hat{\omega }_{E} ,\hat{\omega }_{B} + \frac{{\hat{\omega }_{C} }}{2}} \right)$ delivers even bigger value to PGR, as both denominators are greater in the latter case. This contradicts with the optimality of $\left( {\hat{\omega }_{C} ,\hat{\omega }_{R} ,\hat{\omega }_{E} ,\hat{\omega }_{B} } \right)$ and, therefore, $\omega_{C}^{*} = 0$ is optimal. Hereafter we assume, that $\omega_{C} = 0$, and, in turn, $\omega_{B} = 1 - \left( {{\omega }_{R} + {\omega }_{E} } \right)$.

Another observation is that ${{\text{PGR}}}\left( {{\omega }_{C} ,{\omega }_{R} ,{\omega }_{E} ,{\omega }_{B} } \right)$ is not defined for $\omega_{B} = 1$, or, given the previous assumption, for ${\omega }_{R} = {\omega }_{E} = 0$. So, below we suppose, that ${\omega }_{R} + {\omega }_{E} > 0$. Also, below, when we define optimal sets of preferences, we suppose that ${\omega }_{R}^{*} \ge 0$, ${\omega }_{E}^{*} \ge 0$, ${\omega }_{B}^{*} \ge 0$, and ${\omega }_{R}^{*} + {\omega }_{E}^{*} + {\omega }_{B}^{*} = 1$.

Let us rewrite the long-term population growth rate more explicitly than in Lemma using equalities (5.11), (5.12) as follows:

$$\begin{aligned} & {\text{PGR}}\\&\quad = \left\{ {\begin{array}{*{20}l} {\left( {\frac{{{\omega }_{E} + \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}{{{\omega }_{R} - \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}\frac{\kappa }{\lambda }} \right)^{\beta } \frac{{{\omega }_{R} - \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}{{1 - \omega_{B} }}\frac{{{s} + {r\lambda }\frac{{{\omega }_{E} + \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}{{{\omega }_{R} - \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}\frac{\kappa }{\lambda }}}{\kappa }} \hfill & {\text{if}\,\frac{{{\omega }_{E} + \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}{\lambda } \le \frac{{{\omega }_{R} - \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}{\kappa },} \hfill \\ {\frac{{{\omega }_{R} + {\omega }_{E} }}{{1 - \omega_{B} }}\frac{{{s} + {r\lambda }}}{\kappa + \lambda }} \hfill & {\text{if}\,\frac{{{\omega }_{E} + \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}{\lambda } > \frac{{{\omega }_{R} - \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}{\kappa }.} \hfill \\ \end{array} } \right. \end{aligned}$$

Substituting $\omega_{B} = 1 - \left( {{\omega }_{R} + {\omega }_{E} } \right)$, we simplify and obtain

$$\begin{aligned}&{\text{PGR}} = {\text{PGR}}\left( {\omega_{R} ,\omega_{E} } \right) \\&\quad= \left\{ {\begin{array}{*{20}l} {\left( {\frac{{{\omega }_{E} + \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}{{{\omega }_{R} - \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}\frac{\kappa }{\lambda }} \right)^{\beta } \frac{{\left[ {{\omega }_{R} - \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)} \right]\frac{s}{\kappa } + \left[ {{\omega }_{E} + \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)} \right]r}}{{{\omega }_{R} + {\omega }_{E} }}} \hfill & {\text{if}\,\frac{{{\omega }_{E} + \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}{\lambda } \le \frac{{{\omega }_{R} - \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}{\kappa },} \hfill \\ {\frac{{{s} + {r\lambda }}}{\kappa + \lambda }} \hfill & {\text{if}\,\frac{{{\omega }_{E} + \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}{\lambda } > \frac{{{\omega }_{R} - \beta \left( {{\omega }_{R} + {\omega }_{E} } \right)}}{\kappa }.} \hfill \\ \end{array} }\right.\end{aligned}$$

One can easily show that function ${\text{PGR}}\left( { \cdot , \cdot } \right)$ is continuous with respect to $\omega_{R}$, $\omega_{E}$ in domain $\omega_{R} \ge 0$, $\omega_{E} \ge 0$, $0 < \omega_{R} + \omega_{E} \le 1$ for any non-negative $\beta$, positive s, $\kappa$, $\lambda$, and an arbitrary r.

In what follows, we derive a maximizer of ${\text{PGR}}\left( {\omega_{R} ,\omega_{E} } \right)$ in the following three different cases: $\beta \ge \frac{\lambda }{\lambda + \kappa }$, $\beta = 0$, and $\beta \in \left( {0,\frac{\lambda }{\lambda + \kappa }} \right)$.

If $\beta \ge \frac{\lambda }{\lambda + \kappa }$ then, due to $\frac{\lambda }{\lambda + \kappa } \ge \frac{\lambda }{\lambda + \kappa }\frac{{\omega_{R} }}{{\omega_{R} + \omega_{E} }} \ge \frac{{\lambda \omega_{R} - \kappa \omega_{E} }}{{\left( {\lambda + \kappa } \right)\left( {\omega_{R} + \omega_{E} } \right)}}$, we obtain that $\beta \ge \frac{{\lambda \omega_{R} - \kappa \omega_{E} }}{{\left( {\lambda + \kappa } \right)\left( {\omega_{R} + \omega_{E} } \right)}}$, which is equivalent to $\frac{{\omega_{E} + \beta \left( {\omega_{R} + \omega_{E} } \right)}}{\lambda } \ge \frac{{\omega_{R} - \beta \left( {\omega_{R} + \omega_{E} } \right)}}{\kappa }$. In this case ${\text{PGR}}\left( {\omega_{R} ,\omega_{E} } \right) = {\text{PGR}}_{*} \equiv \frac{s + r\lambda }{\kappa + \lambda }$ is constant. Hence, any preferences $\omega_{R}^{*}$, $\omega_{E}^{*}$, $\omega_{B}^{*}$ are optimal and the optimal level of education is $\alpha_{*} = 1$. This proves case 4.

If $\beta = 0$, then the formula for ${\text{PGR}}\left( {\omega_{R} ,\omega_{E} } \right)$ has a simpler form which can be obtained by taking the corresponding limit as follows:

$${\text{PGR}}\left( {\omega_{R} ,\omega_{E} } \right) = \left\{ {\begin{array}{*{20}l} {\frac{{{\omega }_{R} \frac{s}{\kappa } + {\omega }_{E} {r}}}{{{\omega }_{R} + {\omega }_{E} }}} \hfill & {\text{if}\,\frac{{{\omega }_{E} }}{\lambda } \le \frac{{{\omega }_{R} }}{\kappa },} \hfill \\ {\frac{{{s} + {r\lambda }}}{\kappa + \lambda }} \hfill & {\text{if}\,\frac{{{\omega }_{E} }}{\lambda } > \frac{{{\omega }_{R} }}{\kappa }.} \hfill \\ \end{array} } \right.$$

Here, we consider three different subcases: $r = \frac{s}{\kappa }$, $r > \frac{s}{\kappa }$, $r < \frac{s}{\kappa }$.

If $r = \frac{s}{\kappa }$ then ${\text{PGR}}\left( {\omega_{R} ,\omega_{E} } \right) \equiv {{\text{PGR}}}_{*} = r$ is constant. Hence, any preferences ${\omega }_{R}^{*}$, ${\omega }_{E}^{*}$, ${\omega }_{B}^{*}$ are optimal and the optimal level of education in this case is $\alpha_{*} = \hbox{min} \left( {\frac{{\omega_{E}^{*} }}{{{\omega }_{R}^{*} }}\frac{\kappa }{\lambda },1} \right)$. This proves case 5.

Then, for technical reasons, we fix ${\omega }_{R} + {\omega }_{E} = {c} \in \left( {0,1} \right]$. Then

$${\text{PGR}}\left( {\omega_{R} } \right) = \left\{ {\begin{array}{*{20}l} {\frac{{{\omega }_{R} }}{c}\left( {\frac{s}{\kappa } - {r}} \right) + r} \hfill & {\text{if}\,\frac{\kappa c}{{{\lambda } + {\kappa }}} \le {\omega }_{R} \le c,} \hfill \\ {\frac{{{s} + {r\lambda }}}{\kappa + \lambda }} \hfill & {\text{if}\,0 \le {\omega }_{R} < \frac{\kappa c}{{{\lambda } + {\kappa }}}} \hfill \\ \end{array} } \right.$$

is a piecewise linear function of $\omega_{R}$.

If $r < \frac{s}{\kappa }$, then the maximum value of the long-term population growth rate is ${\text{PGR}}\left( {\omega_{R}^{*} } \right) = {{\text{PGR}}}_{*} = \frac{s}{\kappa }$, where ${\omega }_{R}^{*} = {c}$. Then ${\omega }_{E}^{*} = {c} - {\omega }_{R}^{*} = 0$. Thus, thanks to the independence of the maximum value of ${\text{PGR}}\left( \cdot \right)$ from the parameter $c \in \left( {0,1} \right]$, the optimal preferences are such that ${\omega }_{E}^{*} = 0$ and the optimal level of education is ${\alpha }_{*} = 0$. This proves case 1 (a) and case 1 (b) given $\beta = 0$.

If $r > \frac{s}{\kappa }$, then the maximum value of the long-term population growth rate is ${\text{PGR}}\left( {\omega_{R}^{*} } \right) = {{\text{PGR}}}_{*} = \frac{{{s} + {r\lambda }}}{\kappa + \lambda }$, where ${\omega }_{R}^{*} \in \left[ {0,\frac{\kappa c}{{{\lambda } + {\kappa }}}} \right]$. Then ${\omega }_{E}^{*} = {c} - {\omega }_{R}^{*} \in \left[ {\frac{\lambda c}{{\lambda + {\kappa }}},1} \right]$. By returning to the formula of ${\text{PGR}}\left( {\omega_{R} ,\omega_{E} } \right)$ and thanks to the independence of the maximum value of ${\text{PGR}}\left( \cdot \right)$ from parameter $c \in \left( {0,1} \right]$, we conclude that the optimal preferences are such that ${\omega }_{E}^{*} \ge \frac{\lambda }{\kappa }{\omega }_{R}^{*}$ and the optimal level of education is ${\alpha }_{*} = 1$. This proves case 3 (c).

Let us consider case $\beta \in \left( {0,\frac{\lambda }{\lambda + \kappa }} \right)$ and let $\omega_{R} > 0$. Then, we can rewrite the formula for the long-term population growth rate as follows:

$$\begin{aligned}&{\text{PGR}}\left( {\omega_{R} ,\omega_{E} } \right)\\&\quad = \left\{ {\begin{array}{*{20}l} {\left( {\frac{{\frac{{{\omega }_{E} }}{{{\omega }_{R} }} + \beta \left( {1 + \frac{{{\omega }_{E} }}{{{\omega }_{R} }}} \right)}}{{1 - \beta \left( {1 + \frac{{{\omega }_{E} }}{{{\omega }_{R} }}} \right)}}\frac{\kappa }{\lambda }} \right)^{\beta } \frac{{\left[ {1 - \beta \left( {1 + \frac{{{\omega }_{E} }}{{{\omega }_{R} }}} \right)} \right]\frac{s}{\kappa } + \left[ {\frac{{{\omega }_{E} }}{{{\omega }_{R} }} + \beta \left( {1 + \frac{{{\omega }_{E} }}{{{\omega }_{R} }}} \right)} \right]r}}{{\left( {1 + \frac{{{\omega }_{E} }}{{{\omega }_{R} }}} \right)}}} \hfill & {\text{if}\,\frac{{\frac{{{\omega }_{E} }}{{{\omega }_{R} }} + \beta \left( {1 + \frac{{{\omega }_{E} }}{{{\omega }_{R} }}} \right)}}{\lambda } \le \frac{{1 - \beta \left( {1 + \frac{{{\omega }_{E} }}{{{\omega }_{R} }}} \right)}}{\kappa },} \hfill \\ {\frac{{{s} + {r\lambda }}}{\kappa + \lambda }} \hfill & {\text{if}\,\frac{{\frac{{{\omega }_{E} }}{{{\omega }_{R} }} + \beta \left( {1 + \frac{{{\omega }_{E} }}{{{\omega }_{R} }}} \right)}}{\lambda } > \frac{{1 - \beta \left( {1 + \frac{{{\omega }_{E} }}{{{\omega }_{R} }}} \right)}}{\kappa }.} \hfill \\ \end{array} } \right.\end{aligned}$$

Hence, PGR appears to be dependent on $\frac{{\omega_{E} }}{{\omega_{R} }}$ only. Denoting $\omega_{ER} = \frac{{{\omega }_{E} }}{{{\omega }_{R} }}$, we have

$$\begin{aligned} &{\text{PGR}}\left( {\omega_{ER} } \right)\\ &\quad= \left\{ {\begin{array}{*{20}l} {\left( {\frac{{\omega_{ER} + \beta \left( {1 + \omega_{ER} } \right)}}{{1 - \beta \left( {1 + \omega_{ER} } \right)}}\frac{\kappa }{\lambda }} \right)^{\beta } \frac{{\left[ {1 - \beta \left( {1 + \omega_{ER} } \right)} \right]\frac{s}{\kappa } + \left[ {\omega_{ER} + \beta \left( {1 + \omega_{ER} } \right)} \right]r}}{{\left( {1 + \omega_{ER} } \right)}}} \hfill & {\text{if}\,0 \le \omega_{ER} \le \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}},} \hfill \\ {\frac{{{s} + {r\lambda }}}{\kappa + \lambda }} \hfill & {\text{if}\,\omega_{ER} > \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}.} \hfill \\ \end{array} } \right. \end{aligned}$$

Due to the continuity of function ${\text{PGR}}\left( { \cdot } \right)$ and being constant while $\omega_{{ER}} > \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}$, it is sufficient to consider the problem of its maximization only on segment $0 \le \omega_{{ER}} \le \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}$. The derivative becomes

$$\begin{aligned}&\frac{\partial }{{\partial \omega_{{ER}} }}{\text{PGR}}\left( {\omega_{{ER}} } \right) \\&\quad= \left( {\frac{{\omega_{{ER}} + \beta \left( {1 + \omega_{{ER}} } \right)}}{{1 - \beta \left( {1 + \omega_{{ER}} } \right)}}\frac{\kappa }{\lambda }} \right)^{\beta } \frac{{\beta \frac{s}{\kappa }\left( {\omega_{{ER}} } \right)^{2} + \left[ {\left( {1 + \beta } \right)r - \left( {1 - \beta } \right)\frac{s}{\kappa }} \right]\omega_{{ER}} + \beta r}}{{\left( {\omega_{{ER}} + \beta \left( {1 + \omega_{{ER}} } \right)} \right) \left( {1 - \beta \left( {1 + \omega_{{ER}} } \right)} \right)\left( {1 + \omega_{{ER}} } \right)^{2} }}. \end{aligned}$$

(5.13)

Here, we again will consider three different subcases ${r} \ge \frac{s}{\kappa }$, $r \le 0$ and ${r} \in \left( {0,\frac{s}{\kappa }} \right)$.

If ${r} \ge \frac{s}{\kappa }$, then $\left( {1 + \beta } \right)r - \left( {1 - \beta } \right)\frac{s}{\kappa } > 0$ and other multipliers and summands in formula (5.13) are also positive. Hence, $\frac{\partial }{{\partial \omega_{{ER}} }}{\text{PGR}}\left( {\omega_{{ER}} } \right) > 0$ and so ${\text{PGR}}\left( {\omega_{{ER}} } \right)$ increases monotonically in $\omega_{{ER}}$. Therefore, ${\text{PGR}}_{*} = {\text{PGR}}\left( {\omega_{{ER}}^{*} } \right) = \frac{{{s} + {r\lambda }}}{\kappa + \lambda }$, where $\omega_{{ER}}^{*} \ge \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}$; the optimal preferences are such that $\omega_{E}^{*} \ge \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}{\omega }_{R}^{*}$, and the optimal level of education is $\alpha_{*} = 1$. This proves case 3 (a).

Let us note, that stationary points of ${\text{PGR}}\left( {\omega_{ER} } \right)$ are given by equation:

$$\beta \frac{s}{\kappa }\left( {\omega_{{ER}} } \right)^{2} + \left[ {\left( {1 + \beta } \right)r - \left( {1 - \beta } \right)\frac{s}{\kappa }} \right]\omega_{{ER}} + \beta r = 0,$$

(5.14)

whose roots, if real, are

$$\omega_{{ER}}^{1,2} = \frac{1}{{2\beta \frac{s}{\kappa }}}\left( {\left[ {\left( {1 - \beta } \right)\frac{s}{\kappa } - \left( {1 + \beta } \right)r} \right] \pm \sqrt {\left( {\frac{s}{\kappa } - r} \right)\left[ {\left( {1 - \beta } \right)^{2} \frac{s}{\kappa } - \left( {1 + \beta } \right)^{2} r} \right]} } \right).$$

(5.15)

(subscript 1 corresponds to “−”, subscript 2 corresponds to “+“). As the left-hand side of (5.14) is quadratic with respect to $\omega_{{ER}}$, and other multipliers in (5.13) are positive, we conclude that ${\text{PGR}}\left( {\omega_{{ER}} } \right)$ increases for $\omega_{{ER}} \le {\omega }_{{ER}}^{1}$ and decreases for ${\omega }_{{ER}}^{1} \le \omega_{{ER}} \le {\omega }_{{ER}}^{2}$ and then it again increases for $\omega_{{ER}} \ge {\omega }_{{ER}}^{2}$.

If $r \le 0$, then ${\omega }_{{ER}}^{1} \le 0$ and ${\omega }_{{ER}}^{2} \ge 1$ and, hence, ${\text{PGR}}\left( {\omega_{{ER}} } \right)$ decreases on segment $\left[ {0,\frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}} \right]$. Therefore, ${\text{PGR}}_{*} = {\text{PGR}}\left( {\omega_{{ER}}^{*} } \right)$ = $\left( {\frac{\beta }{1 - \beta }\frac{\kappa }{\lambda }} \right)^{\beta } \left( {\left[ {1 - \beta } \right]\frac{s}{\kappa } + \beta r} \right)$, where $\omega_{{ER}}^{*} = 0$, with preferences such that ${\omega }_{E}^{*} = 0$ being optimal, and the optimal level of education is $\alpha_{*} = \frac{\beta }{1 - \beta }\frac{\kappa }{\lambda }$. This proves case 1 (a).

Finally, consider ${r} \in \left( {0,\frac{s}{\kappa }} \right)$. Consider the expression under the square root in (5.15): $\left( {1 - \beta } \right)^{2} \frac{s}{\kappa } - \left( {1 + \beta } \right)^{2} r$. Its roots are given by $\beta_{1,2} = \frac{{\left( {\sqrt {\frac{s}{\kappa }} \pm {\sqrt{r}} } \right)^{2} }}{{\frac{s}{\kappa } - {r}}}$. One can prove that $\beta_{1} = \frac{{\left( {\sqrt {\frac{s}{\kappa }} + {\sqrt{r}} } \right)^{2} }}{{\frac{s}{\kappa } - {r}}} > 1$ and ${\beta }_{2} = \frac{{\left( {\sqrt {\frac{s}{\kappa }} - {\sqrt{r}} } \right)^{2} }}{{\frac{s}{\kappa } - {r}}} < 1$.

If $\beta \in \left[ {\beta_{2} ,\frac{\lambda }{\lambda + \kappa }} \right)$, then the roots in (5.15) are either imaginary or real and coincide. Hence, due to monotone increasing of ${\text{PGR}}\left( \cdot \right)$ in this case, ${\text{PGR}}_{*} = {\text{PGR}}\left( {\omega_{{ER}}^{*} } \right) = \frac{{{s} + {r\lambda }}}{\kappa + \lambda }$, where $\omega_{{ER}}^{*} \ge \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}$; the optimal preferences are such that $\omega_{E}^{*} \ge \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}{\omega }_{R}^{*}$, and the optimal level of education is $\alpha_{*} = 1$. This proves case 3 (b) for $\beta \in \left[ {\beta_{2} ,\frac{\lambda }{\lambda + \kappa }} \right)$.

Now let us consider $\beta \in \left( {0,\beta_{2} } \right)$. In this case ${\bar{\omega }}_{{ER}}^{*} = {\omega }_{{ER}}^{1} > 1$ and function ${\text{PGR}}\left( \cdot \right)$ achieves its maximum on segment $\left[ {0,\frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}} \right]$ either on the border (at $\omega_{{ER}}^{*} \left( \beta \right) = 0$ or at $\omega_{{ER}}^{*} \left( \beta \right) = \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}$) or at the local maximum point ${\omega }_{{ER}}^{*} \left( {\beta } \right) = {\bar{\omega }}_{{ER}}^{*} \left( {\beta } \right)$ (if ${\bar{\omega }}_{{ER}}^{*} \left( {\beta } \right) \in \left[ {0,\frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}} \right]$). Note that ${\omega }_{{ER}}^{*} \left( \beta \right) \ne 0$ because ${\text{PGR}}\left( \cdot \right)$ increases for $0 \le {\omega }_{{ER}} \le {\bar{\omega }}_{{ER}}^{*}$. Let us write the values of ${\text{PGR}}\left( {\omega_{{ER}} } \right)$ at the other two points:

$${\text{PGR}}\left( {\frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}} \right) = \frac{{{s} + {r\lambda }}}{\kappa + \lambda } = d,$$

$$\begin{aligned}&{\text{PGR}}\left( {{\bar{\omega }}_{{ER}}^{*} \left( {\beta } \right)} \right) \\&\quad= \left( {\frac{{{\bar{\omega }}_{{ER}}^{*} + \beta \left( {1 + {\bar{\omega }}_{{ER}}^{*} } \right)}}{{1 - \beta \left( {1 + {\bar{\omega }}_{{ER}}^{*} } \right)}}\frac{\kappa }{\lambda }} \right)^{\beta } \frac{{\left[ {1 - \beta \left( {1 + {\bar{\omega }}_{{ER}}^{*} } \right)} \right]\frac{s}{\kappa } + \left[ {{\bar{\omega }}_{{ER}}^{*} + \beta \left( {1 + {\bar{\omega }}_{{ER}}^{*} } \right)} \right]r}}{{\left( {1 + {\bar{\omega }}_{{ER}}^{*} } \right)}}\\&\quad = {f}\left( {\beta } \right).\end{aligned}$$

Let us consider two subcases: $\lambda > \sqrt {\frac{s\kappa }{r}}$ and $\lambda \le \sqrt {\frac{s\kappa }{r}}$.

If $\lambda > \sqrt {\frac{s\kappa }{r}}$, one can prove that the following inequalities hold: $f\left( 0 \right) - d > 0$, $f\left( {\beta_{2} } \right) - d < 0$ and $f\left( \cdot \right)$ is a monotone decreasing function for $\beta \in \left( {0,\beta_{2} } \right)$. Then there exists a unique ${\bar{\beta }}_{*} \in \left( {0,\beta_{2} } \right)$ such that $f\left( {{\bar{\beta }}_{*} } \right) = d$.

Therefore, if $\beta \in \left( {0,{\bar{\beta }}_{*} } \right)$, then $f\left( {\beta } \right) > d$ and ${\text{PGR}}_{*} = {\text{PGR}}\left( {\omega_{{ER}}^{*} } \right) = {\text{PGR}}\left( {{\bar{\omega }}_{{ER}}^{*} } \right)$, where ${\omega }_{{ER}}^{*} = {\bar{\omega }}_{ER}^{*}$; the optimal preferences are such that ${\omega }_{E}^{*} = {\bar{\omega }}_{{ER}}^{*} {\omega }_{R}^{*} ,$ and the optimal level of education is $\alpha_{*} = \frac{{{\bar{\omega }}_{{ER}}^{*} + \beta \left( {1 + {\bar{\omega }}_{{ER}}^{*} } \right)}}{{1 - \beta \left( {1 + {\bar{\omega }}_{{ER}}^{*} } \right)}}\frac{\kappa }{\lambda }$ . This proves case 2 for $\lambda > \sqrt {\frac{s\kappa }{r}}$.

If $\beta \in \left[ {{\bar{\beta }}_{*} ,\beta_{2} } \right)$, then $f\left( {\beta } \right) < d$ and ${\text{PGR}}_{*} = {\text{PGR}}\left( {\omega_{{ER}}^{*} } \right) \equiv \frac{{{s} + {r\lambda }}}{\kappa + \lambda }$, where $\omega_{{ER}}^{*} \ge \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}$; the optimal preferences are such that $\omega_{E}^{*} \ge \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}{\omega }_{R}^{*}$ and the optimal level of education is $\alpha_{*} = 1$. This proves case 3 (b) for $\beta \in \left[ {{\bar{\beta }}_{*} ,\beta_{2} } \right)$ and $\lambda > \sqrt {\frac{s\kappa }{r}}$.

For the case $\lambda \le \sqrt {\frac{s\kappa }{r}}$ let us put ${\bar{\beta }}_{*} = \frac{{\lambda \left( {s - r\kappa } \right)}}{{r\lambda^{2} + \left( {s + r\kappa } \right)\lambda + s\kappa }} < \beta_{2}$. One can prove that $f\left( 0 \right) - d > 0$, $f\left( {{\bar{\beta }}_{*} } \right) = d$ and the function $f\left( \beta \right)$ is monotone decreasing for $\beta \in \left( {0,{\bar{\beta }}_{*} } \right)$.

Hence, if $\beta \in \left( {0,{\bar{\beta }}_{*} } \right)$, then ${\text{PGR}}_{*} = {\text{PGR}}\left( {\omega_{{ER}}^{*} } \right) = {\text{PGR}}\left( {{\bar{\omega }}_{{ER}}^{*} } \right)$ where ${\omega }_{{ER}}^{*} = {\bar{\omega }}_{ER}^{*}$; the optimal preferences are such that ${\omega }_{E}^{*} = {\bar{\omega }}_{{ER}}^{*} {\omega }_{R}^{*} ,$ and the optimal level of education is $\alpha_{*} = \frac{{{\bar{\omega }}_{{ER}}^{*} + \beta \left( {1 + {\bar{\omega }}_{{ER}}^{*} } \right)}}{{1 - \beta \left( {1 + {\bar{\omega }}_{{ER}}^{*} } \right)}}\frac{\kappa }{\lambda }$. This proves case 2 for $\lambda \le \sqrt {\frac{s\kappa }{r}}$.

If $\beta \in \left[ {{\bar{\beta }}_{*} ,\beta_{2} } \right)$, then ${\bar{\omega }}_{{ER}}^{*} \left( \beta \right) \ge {\bar{\omega }}_{{ER}}^{*} \left( {{\bar{\beta }}^{*} } \right) = \frac{\lambda r}{s} = \frac{{\left( {1 - {\bar{\beta }}^{*} } \right){\lambda } - {\bar{\beta }}^{*} {\kappa }}}{{\left( {1 + {\bar{\beta }}^{*} } \right){\kappa } + {\bar{\beta }}^{*} {\lambda }}} \ge \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}$ which means that local maximum point is outside the considered segment and, hence, ${\text{PGR}}_{*} = {\text{PGR}}\left( {\omega_{{ER}}^{*} } \right) \equiv \frac{{{s} + {r\lambda }}}{\kappa + \lambda }$, where $\omega_{{ER}}^{*} \ge \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}$; the optimal preferences are such that $\omega_{E}^{*} \ge \frac{{\left( {1 - {\beta }} \right){\lambda } - {\beta \kappa }}}{{\left( {1 + {\beta }} \right){\kappa } + {\beta \lambda }}}{\omega }_{R}^{*}$, and the optimal level of education is $\alpha_{*} = 1$. This proves case 3 (b) for $\beta \in \left[ {{\bar{\beta }}_{*} ,\beta_{2} } \right)$ and $\lambda \le \sqrt {\frac{s\kappa }{r}}$.

The proof is complete.

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