2.1 Endogenous fertility and population dynamics

Within the human population, there is a large number of families indexed by $ i\in \{1,\ldots, m\}$ . Each family $i$ has an adult population $N_{i,t}$ at time $t$ . Therefore, the total adult population size at time $t$ is

(1) \begin{equation} N_{t}=\sum _{i=1}^{m}N_{i,t}\text{.} \end{equation}

Each family $i$ is endowed with an exogenous brain size denoted as $b_{i}\in \lbrack b^{\min },b^{\max }]$ , where $b_{i}$ is heterogeneous across families and follows a general distribution within the lower bound $b^{\min }$ and upper bound $b^{\max }$ on brain size.

Given the metabolic costs of the brain, a family with a larger brain size faces a higher subsistence requirement on per capita consumption denoted as $ \kappa _{i}=\kappa (b_{i})$ , which is assumed to be an increasing function in brain size $b_{i}$ . We consider overlapping generations of agents, and each agent lives for two periods. Each adult agent of family $i$ has the following utility function $u_{i,t}$ at time $t$ :

(2) \begin{equation} u_{i,t}=(1-\gamma )\ln (c_{i,t}-\kappa _{i})+\gamma \ln n_{i,t+1}\text{,} \end{equation}

where $\gamma \in (0,1)$ is the degree of preference on fertility relative to consumption $c_{i,t}$ . $n_{i,t+1}$ is the agent’s number of children, who then become adults at time $t+1$ . Raising children is costly, and the level of consumption net of the fertility cost is

(3) \begin{equation} c_{i,t}=y_{i,t}-\rho n_{i,t+1}\text{,} \end{equation}

where the parameter $\rho \gt 0$ determines the cost of fertility and $y_{i,t}$ is the per capita output of food production in family $i$ .

The utility-maximizing level of consumption is

(4) \begin{equation} c_{i,t}=(1-\gamma )y_{i,t}+\gamma \kappa _{i}\text{,} \end{equation}

and the utility-maximizing level of fertility is

(5) \begin{equation} n_{i,t+1}=\frac{\gamma }{\rho }(y_{i,t}-\kappa _{i})\text{,} \end{equation}

where fertility cost $\rho$ is identical across families for simplicity. Equations ([4]) and ([5]) show that a family with a larger brain size $b_{i}$ allocates a larger amount of food output to consumption (due to the higher subsistence requirement $\kappa (b_{i})$ ) at the expense of fertility. Therefore, if a larger brain size does not carry a cognitive advantage, then families with larger brains would have an evolutionary disadvantage.

Each adult agent in family $i$ has $n_{i,t+1}$ children, and the number of adult agents in family $i$ at time $t$ is $N_{i,t}$ . Therefore, the law of motion for the adult population size in family $i$ is

(6) \begin{equation} N_{i,t+1}=n_{i,t+1}N_{i,t}=\frac{\gamma }{\rho }(y_{i,t}-\kappa _{i})N_{i,t} \text{,} \end{equation}

which is decreasing in the subsistence requirement $\kappa _{i}$ . The growth rate of $N_{i,t}$ at time $t$ is

(7) \begin{equation} \frac{\Delta N_{i,t}}{N_{i,t}}\equiv \frac{N_{i,t+1}-N_{i,t}}{N_{i,t}}=\frac{ \gamma }{\rho }(y_{i,t}-\kappa _{i})-1\text{,} \end{equation}

and the growth rate of total adult population $N_{t}$ at time $t$ is

(8) \begin{equation} \frac{\Delta N_{t}}{N_{t}}=\sum _{i=1}^{m}s_{i,t}\frac{\Delta N_{i,t}}{N_{i,t} }=\frac{\gamma }{\rho }\sum _{i=1}^{m}s_{i,t}(y_{i,t}-\kappa _{i})-1\text{,} \end{equation}

where $s_{i,t}\equiv N_{i,t}/N_{t}$ and $\Delta N_{t}/N_{t}$ will be simply referred to as the population growth rate.

2.2 Hunting-gathering

To capture the cognitive advantage of a larger human brain, we assume that the level of hunting-gathering productivity denoted as $\theta _{i}=\theta (b_{i})$ in each family $i$ is also increasing in its brain size $b_{i}$ . The food production function of family $i$ is

(9) \begin{equation} Y_{i,t}=\theta _{i}(lN_{i,t})^{\alpha }(Z_{i,t})^{1-\alpha }\text{,} \end{equation}

where $lN_{i,t}$ and $Z_{i,t}$ are, respectively, the amount of labor and land devoted to hunting-gathering by family $i$ . Individual labor supply $l\gt 0$ is exogenous, and the parameter $\alpha \in (0,1)$ measures labor intensity of the hunting-gathering process.

For simplicity, the amount of land occupied by family $i$ for hunting-gathering is assumed to be proportional to its population share $ s_{i,t}$ :Footnote ⁹

(10) \begin{equation} Z_{i,t}=s_{i,t}Z=\frac{N_{i,t}}{N_{t}}Z\text{.} \end{equation}

Substituting ([10]) into ([9]) yields the level of food output per capita in family $i$ as

(11) \begin{equation} y_{i,t}\equiv \frac{Y_{i,t}}{N_{i,t}}=\frac{\theta _{i}(lN_{i,t})^{\alpha }(Z_{i,t})^{1-\alpha }}{N_{i,t}}=\theta _{i}l^{\alpha }\left ( \frac{Z}{N_{t}} \right ) ^{1-\alpha }\text{,} \end{equation}

which is increasing in the family’s brain size $b_{i}$ via its hunting-gathering productivity $\theta (b_{i})$ .

3.1 Expanding brain size in human evolution

If the positive effect of brain size $b_{i}$ on fertility always dominates its negative effect (i.e.,

(13) \begin{equation} \frac{\partial g_{i,t}}{\partial b_{i}}\gt 0\Leftrightarrow l^{\alpha }\left ( \frac{Z}{N_{t}}\right ) ^{1-\alpha }\frac{\partial \theta _{i}}{\partial b_{i} }\gt \frac{\partial \kappa _{i}}{\partial b_{i}} \end{equation}

for all $b_{i}\in \lbrack b^{\min },b^{\max }]$ and $N_{t}\in \lbrack N_{0},N^{\ast }]$ where $N^{\ast }$ is the steady-state level of total population),Footnote ¹¹ then the population growth rate of family $ i$ is increasing in its brain size $b_{i}$ . In this case, the growth rate of the population share of family $i$ is also rising in its brain size:Footnote ¹²

\begin{equation*} \Delta s_{i,t}\approx \frac {\Delta N_{i,t}}{N_{t}}=s_{i,t}g_{i,t}\Rightarrow \frac {\partial \Delta s_{i,t}/s_{i,t}}{\partial b_{i}}\approx \frac {\partial g_{i,t}}{\partial b_{i}}\text {.} \end{equation*}

As a result, families with the largest brain size would have an evolutionary advantage and eventually dominate the population. In the long run, $ s_{i,t}(b_{i}=b^{\max })\rightarrow 1$ because given ([13]), its population growth rate $g_{i,t}(b_{i}=b^{\max })$ in ([12]) is the highest among all families such that $s_{j,t}(b_{j}\lt b^{\max })\rightarrow 0$ . In this case, total population $N_{t}$ converges to the steady-state population level of the families with the largest brain size $b^{\max }$ :

(14) \begin{equation} \lim _{t\rightarrow \infty }N_{t}\rightarrow N^{\ast }=\left [ \frac{\gamma \theta (b^{\max })l^{\alpha }}{\rho +\gamma \kappa (b^{\max })}\right ] ^{1/(1-\alpha )}Z\text{,} \end{equation}

which is derived from ([12]) by setting $b_{i}=b^{\max }$ .

As their population share $s_{i,t}(b_{i}=b^{\max })$ rises over time due to their higher population growth rate, the average brain size $b_{t}\equiv \sum _{i=1}^{m}s_{i,t}b_{i}$ of human population also increases over time, which is consistent with the rising trend in human brain size that started over 2 million years ago. Eventually, the average brain size converges to the upper bound (i.e., $b_{t}\rightarrow b^{\max }$ as $s_{i,t}(b_{i}=b^{ \max })\rightarrow 1$ ). As the average brain size increases over time, the average level of hunting-gathering productivity $\theta _{t}\equiv \sum _{i=1}^{m}s_{i,t}\theta _{i}$ also rises over time and converges to $ \theta (b^{\max })$ .Footnote ¹³

When does this expanding brain size in human evolution occur? To explore this question, we substitute $N^{\ast }$ from ([4]) into $N_{t}$ in ([13]) to derive

(15) \begin{equation} \frac{\rho +\gamma \kappa (b^{\max })}{\gamma \theta (b^{\max })}\theta ^{\prime }(b^{\max })\gt \kappa ^{\prime }(b^{\max })\text{.} \end{equation}

Recall from ([13]) that the positive effect of $b_{i}$ is decreasing in $N_{t}$ . So, if the inequality in ([13]) holds for $N^{\ast }$ , it would also hold for $N_{t}\in \lbrack N_{0},N^{\ast }]$ . In the following section, we consider a parametric example to demonstrate when ([15]) holds.

3.1.1 A parametric example

Suppose $\theta (b_{i})$ and $\kappa (b_{i})$ take the following functional form: $\theta (b_{i})=1+\overline{\theta }b_{i}$ and $\kappa (b_{i})= \overline{\kappa }b_{i}$ , where $\overline{\theta }\gt 0$ and $\overline{\kappa }\gt 0$ are productivity and cost parameters, respectively. To be more precise, as the productivity parameter $\overline{\theta }$ increases, hunting-gathering productivity $\theta (b_{i})$ rises, and this positive marginal effect of $\overline{\theta }$ on hunting-gathering productivity $ \theta (b_{i})$ is increasing in brain size $b_{i}$ .Footnote ¹⁴ Similarly, as brain size $b_{i}$ increases, hunting-gathering productivity $\theta (b_{i})$ also rises, and this positive marginal effect of $b_{i}$ on hunting-gathering productivity $ \theta (b_{i})$ is also increasing in the parameter $\overline{\theta }$ , which therefore captures the marginal effect of brain size on hunting-gathering productivity. In other words, an increase in the parameter $\overline{\theta }$ represents an improvement in hunting-gathering productivity that is complementary to brain size (i.e., a family $i$ with a larger brain size $b_{i}$ is better able to benefit from the increase in $ \overline{\theta }$ ). For example, the discovery of stone tools (an exogenous increase in $\overline{\theta }$ in our model) is only useful for a hominin species that is intelligent enough to make use of the tools.

Given $\theta (b_{i})=1+\overline{\theta }b_{i}$ and $\kappa (b_{i})= \overline{\kappa }b_{i}$ , ([15]) simplifies to

(16) \begin{equation} \overline{\theta }\gt \frac{\gamma }{\rho }\overline{\kappa }\text{.} \end{equation}

In this case, in order for human population to evolve toward a larger average brain size, the productivity parameter $\overline{\theta }$ needs to be sufficiently large relative to the product of the cost parameter $ \overline{\kappa }$ and the ratio $\gamma /\rho$ .Footnote ¹⁵ The comparison between hunting-gathering productivity and metabolic cost is intuitive, whereas the ratio $\gamma /\rho$ also plays a role because fertility and population size are increasing in fertility preference $\gamma$ and decreasing in fertility cost $\rho$ . As ([13]) shows, a larger population reduces the marginal benefit of brain size on hunting-gathering productivity due to the decreasing returns to scale in food production. An implication of ([16]) is that a high level of hunting-gathering productivity $\overline{\theta }$ that is complementary to brain size (e.g., the discovery of using fire in hunting animals and cooking food or the development of “increasingly sophisticated blades, handaxes, and flint and limestone tools”) helps to trigger the emergence of an expanding brain size in human evolution;Footnote ¹⁶ see for example, Gowlett (Reference Gowlett2016) who provides evidence that the discovery of fire “has fuelled the increase in brain size through the Pleistocene.”

3.2 Optimal brain size in human evolution

We now explore the more interesting case in which the average brain size $ b_{t}$ evolves toward an interior steady state $b^{\ast }\in (b^{\min },b^{\max })$ . If there exists a certain level of population threshold $ \widetilde{N}$ under which

(17) \begin{equation} l^{\alpha }\left ( \frac{Z}{\widetilde{N}}\right ) ^{1-\alpha }\frac{\partial \theta (b^{\ast })}{\partial b_{i}}=\frac{\partial \kappa (b^{\ast })}{ \partial b_{i}}\text{,} \end{equation}

then there exists an interior optimal brain size $b^{\ast }\in (b^{\min },b^{\max })$ from an evolutionary viewpoint. To determine this optimal brain size, we also need the steady-state population level:

(18) \begin{equation} N_{i}^{\ast }(b_{i}=b^{\ast })=\left [ \frac{\gamma \theta (b^{\ast })l^{\alpha }}{\rho +\gamma \kappa (b^{\ast })}\right ] ^{1/(1-\alpha )}Z= \widetilde{N}\text{,} \end{equation}

where $\widetilde{N}$ and $b^{\ast }$ are determined jointly by ([17]) and ([18]). In other words, families with the optimal brain size $b^{\ast }$ have the highest steady-state population growth rate and dominate the population in the long run (i.e., $s_{i,t}(b_{i}=b^{\ast })\rightarrow 1$ because its steady-state population growth rate $g_{i}^{\ast }(b_{i}=b^{\ast })=0$ is the highest among all families such that $g_{j}^{\ast }(b_{j}\neq b^{\ast })\lt 0$ ).Footnote ¹⁷

Combining ([17]) and ([18]) yields the following condition that determines $ b^{\ast }$ :

(19) \begin{equation} \frac{\theta ^{\prime }(b^{\ast })}{\kappa ^{\prime }(b^{\ast })}=\frac{ \gamma \theta (b^{\ast })}{\rho +\gamma \kappa (b^{\ast })}\text{.} \end{equation}

The left-hand side of ([19]) is weakly decreasing in $b^{\ast }$ because $ \theta ^{\prime \prime }\leq 0$ and $\kappa ^{\prime \prime }\geq 0$ . As for the right-hand side of ([19]) defined as $\Phi (b^{\ast })\equiv \gamma \theta (b^{\ast })/[\rho +\gamma \kappa (b^{\ast })]$ , it can be shown that ([19]) implies $\Phi ^{\prime }(b^{\ast })=0$ . Therefore, we need to assume that at least one of $\theta ^{\prime \prime }\leq 0$ and $\kappa ^{\prime \prime }\geq 0$ is a strict inequality to ensure the existence of a solution $ b^{\ast }$ . If $b^{\ast }$ falls within the range of brain size $b_{i}\in \lbrack b^{\min },b^{\max }]$ , then families with the optimal brain size $ b^{\ast }$ will dominate the population in the long run and have a steady-state level of population that is equal to the threshold $\widetilde{N}$ in ([18]). In this case, given an initial value $b_{0}$ , the average brain size $b_{t}\equiv \sum _{i=1}^{m}s_{i,t}b_{i}$ rises toward $b^{\ast }$ as $ s_{i,t}(b_{i}=b^{\ast })\rightarrow 1$ .Footnote ¹⁸

3.2.1 Another parametric example

Suppose $\theta (b_{i})$ and $\kappa (b_{i})$ now take the following functional form: $\theta (b_{i})=1+\overline{\theta }b_{i}$ and $\kappa (b_{i})=\overline{\kappa }b_{i}^{2}$ , where $\overline{\theta }\gt 0$ and $ \overline{\kappa }\gt 0$ are parameters. In this case, the optimal brain size $ b^{\ast }$ from ([19]) is determined by the following quadratic equation:

(20) \begin{equation} (b^{\ast })^{2}+\frac{2b^{\ast }}{\overline{\theta }}=\frac{\rho }{\gamma \overline{\kappa }}\text{,} \end{equation}

in which the solution $b^{\ast }\gt 0$ is increasing in hunting-gathering productivity $\overline{\theta }$ that is complementary to brain size. If $ \overline{\theta }$ becomes sufficiently large, it is possible for $b^{\ast }$ to even exceed the upper bound $b^{\max }$ , in which case the average brain size $b_{t}$ converges to $b^{\max }$ as in Section 3.1. Solving ([20]) yields

(21) \begin{equation} b^{\ast }=-\frac{1}{\overline{\theta }}+\sqrt{\frac{1}{\overline{\theta }^{2} }+\frac{\rho }{\gamma \overline{\kappa }}}\in (b^{\min },b^{\max })\text{,} \end{equation}

which depends positively on the parameter $\overline{\theta }$ and the composite parameter $\rho /(\gamma \overline{\kappa })$ . Proposition2 summarizes our result in this section.

Proposition 2. Suppose $\theta (b_{i})=1+\overline{\theta }b_{i}$ and $\kappa (b_{i})= \overline{\kappa }b_{i}^{2}$ . Then, $b^{\ast }$ in ([ 21 ]) is the evolutionarily optimal human brain size, which is increasing in hunting-gathering productivity $\overline{\theta }$ that is complementary to brain size.

3.2.2 Quantitative analysis

In this section, we compute the value of the optimal brain size $b^{\ast }$ in ([21]) and explore its quantitative implications. The introduction features a discussion on the range of brain size of different human species. Table 1 presents a summary based on the midpoint of the range of brain size of each of these species. It also uses the brain size of the first human species, Homo habilis, as the benchmark and presents the relative brain size of the subsequent human species.

Table 1. Human brain size

Suppose the brain size of each human species was optimal (given the level of hunting-gathering productivity $\overline{\theta }$ ) at its time. Then, we compute the change in $\overline{\theta }$ required for the optimal brain size to increase from $b^{\ast }=1$ for Homo habilis to $ b^{\ast }=2.26$ for Homo sapiens. In addition, this simulation also involves the fertility cost parameter $\rho$ , the fertility preference parameter $\gamma$ and the subsistence consumption parameter $ \overline{\kappa }$ . Specifically, the simulation requires an assigned value of the composite parameter $\rho /(\gamma \overline{\kappa })$ . Given that we do not know its value, we consider a range to examine how the value of this composite parameter affects our calculation. Table 2 presents the results. As $\rho /(\gamma \overline{\kappa })\rightarrow 5$ , the percent increase in $\overline{\theta }$ (required for $b^{\ast }$ to increase from 1 to 2.26) approaches infinity. Therefore, we start with a value of 6 for $ \rho /(\gamma \overline{\kappa })$ as a lower bound. Then, we examine how large the percent increase in $\overline{\theta }$ would be under a range of values of $\rho /(\gamma \overline{\kappa })$ for illustration. It turns out that as $\rho /(\gamma \overline{\kappa })$ increases, the percent increase in $\overline{\theta }$ remains over 100%. Thus, we conclude that the dramatic increase in human brain size over the last 2.5 million years was driven by a rapid improvement in hunting-gathering productivity $\overline{ \theta }$ that is complementary to brain size.

Table 2. Changes in productivity $\overline{\theta }$

We now consider hominin brain evolution over the last 10 million years. According to DeSilva et al. (Reference DeSilva, Traniello, Claxton and Fannin2021), the cranial capacity in fossil apes (Miocene hominids) from 10 million years ago is about 250 $cm^{3}$ , which is about 40% of the brain size of Homo habilis. If we consider a value of 10 for $\rho /(\gamma \overline{\kappa })$ as a benchmark, then the implied value of $\overline{\theta }$ for $b^{\ast }=0.40$ is 0.08. Suppose $\overline{\theta }$ increases linearly from 0.08 to 0.22 between 10 million years ago and 2.5 million years ago and from 0.22 to 0.92 between 2.5 millions years ago and the dawn of the Agricultural Revolution about 10,000 years ago. Then, we simulate the path of the optimal brain size $b^{\ast }$ in ([21]) as $\overline{\theta }$ gradually increases from 10 million years ago to recent times in Figure 1, which closely replicates the trend of hominin brain size in DeSilva et al. (Reference DeSilva, Traniello, Claxton and Fannin2021).Footnote ¹⁹ Specifically, the growth trend of hominin brain size was steadily slow from 10 million years ago to 2.5 million years ago and then accelerated about 2.5 million years ago before decelerating until recent times. This pattern is robust to other values of $\rho /(\gamma \overline{\kappa })$ with the recent deceleration becoming less rapid at larger values of $\rho /(\gamma \overline{\kappa })$ .Footnote ²⁰

Figure 1. Simulated trend of hominin brain size.

4.1 Scale effect

In this section, we introduce a scale effect on hunting-gathering productivity to our baseline model.Footnote ²² For simplicity, we assume that subsistence requirement $\kappa _{i}=\kappa$ is homogeneous across families. In this case, our baseline model predicts that families with the largest brain size $b^{\max }$ would dominate the population by having the highest hunting-gathering productivity $\theta (b^{\max })$ and the highest population growth rate. Suppose we now modify the hunting-gathering productivity function as $\theta (s_{i,t}b_{i})$ to be increasing in $s_{i,t}b_{i}$ , where $s_{i,t}\equiv N_{i,t}/N_{t}$ is the population share of family $i$ and captures a scale effect of relative population size on hunting-gathering productivity. For example, a larger family group may be able to occupy a more fertile area of land for hunting-gathering purposes. In this case, the population growth rate of family $i$ in ([12]) becomes

(22) \begin{equation} g_{i,t}=\frac{\gamma }{\rho }\left [ \theta (s_{i,t}b_{i})l^{\alpha }\left ( \frac{Z}{N_{t}}\right ) ^{1-\alpha }-\kappa \right ] -1\text{,} \end{equation}

which shows that the relative population size $s_{i,t}$ now plays a role. The family that dominates in the long run may no longer be the one with the largest brain size $b^{\max }$ . Instead, the initial population share $ s_{i,0}$ also matters. Specifically, the family with the largest $ s_{i,0}b_{i}$ dominates the population in the long run as its population share $s_{i,t}$ rises over time and eventually converges to unity (i.e., $ s_{i,t}\rightarrow 1$ ). In other words, a human species with a relatively large brain size (e.g., the Neanderthals) may not survive in the long run because its initial population size is relatively small.

4.2 Body mass

Our model assumes that a larger brain size carries a benefit to productivity but requires higher subsistence consumption. Considering the evolution of human body mass, Lagerlof (Reference Lagerlof2007) and Dalgaard and Strulik (Reference Dalgaard and Strulik2015) also assume that a higher body mass has a positive effect on productivity but requires higher subsistence consumption. Therefore, one natural question is how the evolution of human body mass differs from the evolution of human brain size.

Empirically, Lagerlof (Reference Lagerlof2007) documents that the average human body mass increased from 61.8 kg roughly 1.8 million years ago to 76.0 kg about 36-75 thousand years ago.Footnote ²³ This 23% increase in human body mass is significant but not as much as the increase in human brain size, which more than doubled in the last 2 million years. Therefore, the increase in human brain size is not a mere reflection of a higher human body mass. Rather, human brain size has increased relative to human body mass.Footnote ²⁴

Theoretically, if we modify the hunting-gathering productivity function as $ \theta _{i}=\theta (B_{i})$ and the subsistence requirement function as $ \kappa _{i}=\kappa (B_{i})$ , which are both increasing in the body mass $ B_{i}$ of family $i\in \{1,\ldots, m\}$ . Then, this alternative model could generate the same implication that the average human body mass rises over time. However, given that the increase in human brain size is quantitatively more prominent in the data than the increase in human body mass, it is arguably more important to apply an evolutionary growth model (like the one in this study) to the evolution of human brain size than that of human body mass.Footnote ²⁵