3.1.1 Equilibrium model

The economy is populated by individuals who live for two periods, and a family consists of a parent and his children. In the first period individuals receive the education provided by their parent. In the second period they form their own household, work, choose their fertility, and rear and educate their children.

Parents care about their own consumption, c, their expected number of surviving children, x(1 − δ), and the income, y, of their typical child. In addition, with mortality risk, parents are assumed to have a precautionary demand for children.Footnote ¹¹ A parent receives no utility from leisure, so a parent only works or spends time rearing and educating children. Thus the typical generation t parent i maximizes:Footnote ¹²

(5)$$\alpha {\rm ln}c_{it} + \lpar {1-\alpha} \rpar {\rm ln}x_{it} + \lpar {1-\alpha} \rpar {\rm ln}\lpar {1-\delta_{it}} \rpar -\displaystyle{{\delta _{it}} \over {2x_{it}\lpar {1-\delta_{it}} \rpar }} + \alpha \beta {\rm ln}y_{it + 1}$$

The typical t period parent's budget constraint is

(6)$$c_{it} = w_{it}h_{it}\lsqb {1-x_{it}\lpar {\theta + \tau_{it}} \rpar } \rsqb $$

where θ > 0 is the unavoidable time cost of child rearing, and τ _it is time spent teaching each child. Human capital next period depends on the investment time per child τ _it i.e.,

(7)$$h_{it + 1} = Ah_{it}\tau _{it}^\mu $$

where μ > 0 and A > 1 are efficiency parameters.Footnote ¹³ We assume that the cumulative mortality of young adults is a declining function of the average human capital of their generation:Footnote ¹⁴

(8)$$\delta _{it} = \min \lcub {\hat{\delta}, \Delta \exp \lpar {-\psi_1\bar{h}_{it + 1}^{\psi_2}} \rpar } \rcub $$

where $\bar{h}_{it + 1}$ is the average human capital of adult generation t + 1 and ψ ₁ and ψ ₂ are positive parameters. Hence whether a child born of a t generation parent, and hence a t + 1 generation adult, survives is an increasing function of the average human capital of their generation.Footnote ¹⁵

In the competitive equilibrium we have price taking behavior. Individuals are paid the marginal product of their human capital. The typical parent takes the wage per unit of human capital as given, and does not assume that his/her investment decisions have any effect on this wage, nor on the wage of any other worker. Furthermore the parent does not assume that their education provided to their children affects their child's wage.Footnote ¹⁶ Given that the production technology has constant returns to scale in the distribution of human capital in the economy, paying each worker their marginal product exactly exhausts output, which is evident from summing over all worker types in (2). Focusing only on the choice variables of a parent, the equilibrium problem for parent i in generation t can be written as:Footnote ¹⁷

(9)$$\mathop {\max} \limits_{\lcub {\tau_{it},x_{it}} \rcub } \left\{ {\alpha \ln \lpar {1-x_{it}\lsqb {\theta + \tau_{it}} \rsqb } \rpar + \lpar {1-\alpha} \rpar \ln x_{it}-\displaystyle{{\delta_{it}} \over {2x_{it}\lpar {1-\delta_{it}} \rpar }} + \alpha \beta \ln h_{it + 1}} \right\}$$

The first order conditions with respect to fertility and human capital investment produce the following expressions.

(10)$$\displaystyle{{\alpha \lpar {\theta + \tau_{it}} \rpar } \over {1-x_{it}\lpar {\theta + \tau_{it}} \rpar }} = \displaystyle{{1-\alpha} \over {x_{it}}} + \displaystyle{{\delta _{it}} \over {2x_{it}^2 \lpar {1-\delta_{it}} \rpar }}$$

(11)$$\displaystyle{{x_{it}\tau _{it}} \over {\lpar {1-x_{it}\lsqb {\theta + \tau_{it}} \rsqb } \rpar }} = \beta \mu $$

Observe that the first order condition for human capital investment in the equilibrium model is independent of mortality; the only way in which mortality risk affects human capital investment is indirectly through the fertility choice. Because of the non-convexity in the budget constraint induced by the quality–quantity interaction of (x _it, τ _it), we solve this problem using a grid on human capital investment rates given by:Footnote ¹⁸

(12)$$\tau _s = \left\{ {\displaystyle{{s\lpar {1-\theta} \rpar } \over k}} \right\}_{s = 1}^k $$

Then for each value of τ _s and imposing the homogeneity of the parents, we compute the mortality rate for these young adults using (8). We use (10) to solve for the fertility choice for each τ _s on the grid.Footnote ¹⁹ If we define m = δ/(2(1 − δ)), then it is easy to show that (∂x/∂m) > 0. So in a world with mortality risk, for any choice of human capital investment, fertility is higher than in a world with no mortality risk. After solving for the fertility choice as a function of the investment rate, we verify that (11) holds. We discard those solutions that do not satisfy this equilibrium first order condition.Footnote ²⁰ From this reduced set of investment and fertility candidate pairs, the utility maximizing choice is selected.

3.1.2 Equilibrium solution: zero mortality

Here we solve for the stationary zero mortality equilibrium solution. Using (10) it is easy to show that the fraction of resources spent on the next generation is constant and equal to 1 − α. Combining the results in (10) and (11), we find the equilibrium stationary fertility and investment time:

(13)$$x^{{\rm eq}} = \displaystyle{{1-\alpha -\alpha \beta \mu} \over \theta} $$

(14)$$\tau ^{{\rm eq}} = \displaystyle{{\alpha \beta \mu \theta} \over {1-\alpha -\alpha \beta \mu}} $$

In order to have a meaningful equilibrium we impose:

$$1-\alpha -\alpha \beta \mu \gt 0$$

Intuitively, an increase in β—i.e., if parents care more about their children's income—reduces fertility and increases the investment in children's human capital. Similarly, a higher θ reduces fertility and increases investment per child since the fixed cost of rearing a child is now higher. Finally, an increase in μ also increases investment in children's human capital, since the return to this investment is now higher.Footnote ²¹

3.1.3 ${\rm {\cal A}}$-Efficient model

In the ${\rm {\cal A}}$-efficient model, we consider the case in which a social planner internalizes the positive externality of population growth on TFP growth, as well as the positive externality of human capital investment on the next generation wages and next generation mortality risk. It is better to think of the planner's problem as actually an infinite sequence of separate planners.Footnote ²² In a given period, the planning problem maximizes the summation of the welfare of adults. The planner chooses the consumption for all adults in the coalition, their fertility, and their human capital investment in each child, but does not choose the number of grandchildren, nor the investment of human capital in the grandchildren. However, the planner does internalize the external effects of current fertility on the population next period and production, and also the effects of human capital of the next generation on the production coalition size and hence wages of that generation. The planner also considers the effect of human capital investment on the mortality risk facing the children, although the planner does not internalize the effect of the investment on the cost of human capital accumulation for the current parent's grandchildren.Footnote ²³ For convenience, and without loss of generality, we focus on the efficient solution in which all parents are treated equally. Thus in our analysis of the efficient solution, we are using the notion of ${\rm {\cal A}}$-efficiency of Golosov et al. (Reference Golosov, Jones and Tertilt2007): an allocation of (c _t, x _t, h _t+1) is ${\rm {\cal A}}$-efficient if there does not exist another allocation $\lpar {c_t^{\prime}, x_t^{\prime}, h_{t + 1}^{\prime}} \rpar$ that makes any subset of the adult population better off without making any adult worse off.Footnote ²⁴

Here we solve the planner's problem, and we only examine the case with identical individuals. We focus on the equal weight solution, and hence the representative parent. There are several regions that must be solved in order to characterize the efficient path. In the earliest days, human capital will be sufficiently low that no specialization occurs, and hence autarky is the method of production. Human capital accumulation may affect the mortality of the young adult, but not determine the scale of the production coalition. In the final stage, either the coordination technology will determine the size of the production coalition, or it will be determined by the fixed proportion of the total population, determined in (4). Since all individuals are identical, the earnings of a typical worker are given by:

$$y_{t + 1} = N_{t + 1}^{\omega -1} h_{t + 1}$$

$$N_{t + 1} = \min \lcub {\max \lcub {1,{\lsqb {\sigma {\bar{h}}_{t + 1}} \rsqb }^\lambda} \rcub,\max \lcub {1,\xi P_{t + 1}} \rcub } \rcub$$

The planner's problem can be written as:Footnote ²⁵

$$\eqalign{& \mathop {\max} \limits_{x_t,h_{t + 1}} \bigg\{ {\alpha {\rm ln}\lpar {1-x_t\lsqb {\theta + \tau_t} \rsqb } \rpar + \lpar {1-\alpha} \rpar {\rm ln}(x_t\lsqb {1-\delta \lpar {h_{t + 1})} \rsqb } \rpar } \cr & \left. {-\displaystyle{{\delta \lpar {h_{t + 1}} \rpar } \over {2x_t\lsqb {1-\delta \lpar {h_{t + 1}} \rpar } \rsqb }} + \alpha \beta \lsqb {\lpar {\omega -1} \rpar {\rm ln}N_{t + 1} + {\rm ln}h_{t + 1}} \rsqb } \bigg\}}$$

The first order conditions for fertility and human capital investment are:

(15)$$\displaystyle{{\alpha (\theta + \tau _t)} \over {1-x_t[\theta + \tau _t]}} = \displaystyle{{1-\alpha} \over {x_t}} + \displaystyle{{\delta (h_{t + 1})} \over {2x_t^2 [1-\delta (h_{t + 1})]}} + \displaystyle{{\alpha \beta (\omega -1)} \over {N_{t + 1}}}\displaystyle{{\partial N_{t + 1}} \over {\partial x_t}}$$

(16)$$\eqalign{\displaystyle{{\alpha x_t} \over {1-x_t[\theta + \tau _t]}} = & \displaystyle{{\alpha \beta \mu} \over {\tau _t}}-\displaystyle{1 \over {1-\delta (h_{t + 1})}}\displaystyle{{\partial \delta (h_{t + 1})} \over {\partial \tau _t}} \left\{ {1-\alpha + \displaystyle{1 \over {2x_t[1-\delta (h_{t + 1})]}}} \right\} \cr & + \displaystyle{{\alpha \beta (\omega -1)} \over {N_{t + 1}}}\displaystyle{{\partial N_{t + 1}} \over {\partial h_{t + 1}}}\displaystyle{{\partial h_{t + 1}} \over {\partial \tau _t}}} $$

$$\eqalign{\displaystyle{{\partial \delta (h_{t + 1})} \over {\partial \tau _t}} = & \left\{ {\matrix{ {0,} \hfill & {{\rm if}\,{\rm ln}\hat{\delta} -{\rm ln}\Delta -\psi_1h_{t + 1}^{\psi_2} \gt 0.} \hfill \cr {-\displaystyle{{\delta (h_{t + 1})\psi_1\psi_2\mu h_{t + 1}^{\psi_2}} \over {\tau_t}}} \hfill & {{\rm if}\,{\rm ln}\hat{\delta} -{\rm ln}\Delta -\psi_1h_{t + 1}^{\psi_2} \le 0.} \hfill \cr}} \right. \cr \displaystyle{{\partial N_{t + 1}} \over {\partial x_t}} = & \left\{ {\matrix{ {0,} \hfill & {{\rm if}\,N_{t + 1} = \max \{ 1,{[\sigma h_{t + 1}]}^\lambda \} \lt \xi P_{t + 1}} \hfill \cr {N_t(1-\delta (h_{t + 1}))} \hfill & {{\rm if}\,1 \lt N_{t + 1} = \xi P_{t + 1} \le {[\sigma h_{t + 1}]}^\lambda} \hfill \cr}} \right. \cr \displaystyle{{\partial N_{t + 1}} \over {\partial h_{t + 1}}}\displaystyle{{\partial h_{t + 1}} \over {\partial \tau _t}} = & \left\{ {\matrix{ {0,} \hfill & {{\rm if}\,N_{t + 1} = \max \{ 1,\xi P_{t + 1}\} \lt {[\sigma h_{t + 1}]}^\lambda} \hfill \cr {\displaystyle{{\lambda \mu \sigma^{\,\lambda} h_{t + 1}^\lambda} \over {\tau_t}}} \hfill & {{\rm if}\,1 \lt N_{t + 1} = {[\sigma h_{t + 1}]}^\lambda \le \xi P_{t + 1}} \hfill \cr}} \right.}$$

If human capital of the child does not affect his or her mortality rate, that is h _t+1 is below the critical threshold, and if ∂N _t+1/∂x _t = 0 and ∂N _t+1/∂h _t+1 = 0, then the two first order conditions of the social planner are identical to the first order conditions in the equilibrium solution. Thus for the earliest part of history, there is no difference in the path of human capital taken in the equilibrium solution and the efficient solution.

Now consider the range where children's human capital reduces their mortality, ∂δ/∂τ _t < 0. We further assume that the market size could be larger than autarky in the children's adulthood, i.e., N _t+1 >1. We use the same solution algorithm to solve for the ${\rm {\cal A}}$-efficient choice of fertility and human capital investment as we used in solving for the equilibrium choices. We use the same grid on possible investment rates, τ _t and the implied young adult mortality rate as in our equilibrium solution, see (12) and (8). For each value of τ _s, the first order condition for ${\rm {\cal A}}$-efficient fertility is given by (15). If the production coalition is determined by the evolution of population, then the first order condition for fertility can be written as:

$$\displaystyle{{\alpha \lpar {\theta + \tau_s} \rpar } \over {1-x_s\lsqb {\theta + \tau_s} \rsqb }} = \displaystyle{{1-\alpha + \alpha \beta \lpar {\omega -1} \rpar } \over {x_s}} + \displaystyle{{\delta \lpar {h_{st + 1}} \rpar } \over {2x_s^2 \lsqb {1-\delta \lpar {h_{st + 1}} \rpar } \rsqb }}$$

For any given investment rate τ _s, this is a quadratic function in x _s. Appendix B contains the exact form and the solution under this regime.

For each (τ _s, x _s) pair, we compute the utility of the representative parent, and the potential ${\rm {\cal A}}$-efficient solution is the maximizing pair. Unlike in the equilibrium solution, we do not constrain the choice of investment other than its feasibility. Implicitly the investment rate either satisfies (16) if interior, or is a corner solution. In either case, it is generally not equal to the equilibrium choice as (11) and (16) are not identical.

If the production coalition is determined by the coordination technology, i.e., 1 < N _t+1 = [σh _t+1]^λ < ξP _t+1, then the first order condition for fertility can be written as:

$$\displaystyle{{\alpha \lpar {\theta + \tau_s} \rpar } \over {1-x_s\lsqb {\theta + \tau_s} \rsqb }} = \displaystyle{{1-\alpha} \over {x_s}} + \displaystyle{{\delta \lpar {h_{st + 1}} \rpar } \over {2x_s^2 \lsqb {1-\delta \lpar {h_{st + 1}} \rpar } \rsqb }}$$

For this case, Appendix C presents the quadratic form that fertility solves for any choice of τ _s. The solution, under the assumption that the production coalition is determined by the coordination technology, is the (x _s,τ _s) that maximizes the utility of the representative parent.

We take the two candidate solutions from above, population is the determining coalition size, or if the coordination technology is the coalition size, and whichever regime maximizes the utility of the typical parent is the ${\rm {\cal A}}$-efficient solution.

For both the equilibrium solution and the ${\rm {\cal A}}$-efficient solution, we produce the time series on population, fertility, mortality, human capital, coalition size, and real per capita output. The evolution is given by:

$$h_{t + 1}^r = Ah_t^r \lpar {\tau_t^r} \rpar ^\mu$$

$$\delta \lpar {h_{t + 1}^r} \rpar = \min \lcub {\hat{\delta}, \Delta \exp \lpar {-\psi_1{\lcub {Ah_t^r {\lpar {\tau_t^r} \rpar }^\mu} \rcub }^{\psi_2}} \rpar } \rcub$$

$$P_{t + 1}^r = P_t^r \lsqb {1-\delta \lpar {h_{t + 1}^r} \rpar } \rsqb x_t^r$$

$$w_{t + 1} = \lpar {N_{t + 1}^r} \rpar ^{\omega -1}$$

where r refers to either equilibrium or ${\rm {\cal A}}$-efficient solutions.

3.1.4 ${\rm {\cal A}}$-Efficient model: zero mortality

In this section we characterize the zero mortality ${\rm {\cal A}}$-efficient solution. There are three possibilities, one of which we ignore for simple reasons. These solutions depend on the long run growth rate of the production coalition. One possibility is that the coalition grows at the stationary rate of fertility, x, and that the coordination costs are never binding: ξP _t+1 = ξP _tx = N _tx = N _t+1 <[σAh _tτ ^μ]^λ = [σh _t+1]^λ. A second possibility is that the coordination costs are always binding, and fertility is equal to the growth rate of the coalition arising from declining coordination costs: x = [Aτ ^μ]^λ, N _t+1 < ξP _tx. Finally, the third case is that the coordination costs determine the coalition size, and fertility exceeds the growth rate of the coalition: x > [Aτ ^μ]^λ, N _t+1 < ξP _tx. We disregard this third possibility as the fertility rate is below the rate of the equilibrium fertility in the scenario with zero mortality.Footnote ²⁶

In the first case, the ${\rm {\cal A}}$-efficient solution is characterized by the following first order conditions:

(17)$$\displaystyle{{\alpha \lpar {\theta + \tau} \rpar } \over {1-x\lpar {\theta + \tau} \rpar }} = \displaystyle{{1-\alpha} \over x} + \displaystyle{{\alpha \beta \lpar {\omega -1} \rpar } \over x}$$

(18)$$\displaystyle{{x\tau} \over {\lpar {1-x\lsqb {\theta + \tau} \rsqb } \rpar }} = \beta \mu $$

Solving these equations for the ${\rm {\cal A}}$-efficient solutions for fertility and investment produces:

(19)$$x^{{\rm eff}} = \displaystyle{{1-\alpha -\alpha \beta \mu + \alpha \beta \lpar {\omega -1} \rpar } \over {\theta \lsqb {1 + \alpha \beta \lpar {\omega -1} \rpar } \rsqb }}$$

(20)$$\tau ^{{\rm eff}} = \displaystyle{{\alpha \beta \mu \theta} \over {1-\alpha -\alpha \beta \mu + \alpha \beta \lpar {\omega -1} \rpar }}$$

It is clear that an increase in ω − 1 induces a higher fertility rate since the positive external effect of population on output per capita and wages is now higher. The corresponding level of investment per child decreases as ω − 1 increases. Utilizing (13) and (14), it is also easy to show that x ^eff > x ^eq and τ ^eff < τ ^eq.

In the second case, the solution requires that fertility be equal to the growth rate of the production coalition arising from the coordination costs. Thus we have:

$$x = \lsqb {A\tau^\mu} \rsqb ^\lambda$$

The ${\rm {\cal A}}$-efficient problem facing the planner becomes:

$$\mathop {\max} \limits_\tau \left\{ {\alpha \ln \lpar {1-{\lsqb {A\tau^\mu} \rsqb }^\lambda \lsqb {\theta + \tau} \rsqb } \rpar + \lpar {1-\alpha} \rpar \lambda \mu \ln \tau + \alpha \beta \lpar {\omega -1} \rpar \lambda \mu \ln \tau + \alpha \beta \mu \ln \tau}\, \right\}$$

The first order condition for the optimal human capital investment rate is:

(21)$$\displaystyle{{\lambda \mu A^\lambda \tau ^{\lambda \mu -1}\lpar {\theta + \tau} \rpar + {\lsqb {A\tau^\mu} \rsqb }^\lambda} \over {1-{\lsqb {A\tau^\mu} \rsqb }^\lambda \lpar {\theta + \tau} \rpar }} = \displaystyle{{\lpar {1-\alpha} \rpar \lambda \mu + \alpha \beta \lpar {\omega -1} \rpar \lambda \mu + \alpha \beta \mu} \over \tau} $$

We solve for both possible solutions (production coalition grows at the rate of fertility, and the production coalition is given by the coordination technology), and compare the utility attained by the representative household under each solution. The ${\rm {\cal A}}$-efficient solution arises from the maximum of these two possibilities. In our solutions, with small agglomeration returns to specialization, ω between 1.01 and 1.20, the first solution is optimal. In all other numerical solutions, the ${\rm {\cal A}}$-efficient solution corresponds to the second case, with human capital investment, τ, given by (21).

Table 1 contains the values of the parameters used for the numerical solutions. We used (13) and (14), and assumed stationary equilibrium fertility of x ^eq = 1.000000001 and τ ^eq = 0.375. The stationary equilibrium fertility is one of near stationary population. We chose this as the current rich countries are all below replacement fertility today, but we forecast that stationary population is more likely. Second, assuming that a period is 40 years, the equilibrium schooling is given by 40τ ^eq = 15, i.e., we assume that stationary equilibrium fertility is slightly less than college graduation. This is consistent with the 15 years of schooling of the 2010 generation in Denmark, Finland, Iceland, New Zealand, and the United States using the data from Tamura et al. (Reference Tamura, Dwyer, Devereux and Baier2019). In addition our choices of α, β, θ, and μ are designed to produce the following equilibrium balanced growth path outcomes: annual discount rate of β ^.025 = 0.9767, biological maximum fertility of (1/θ) = 11.4, and long run annualized per capita income growth of 1.7%, or a doubling of living standards every generation.

Table 1.

Parameter values

3.1.5 Comparing both setups under zero mortality

Here we present the stationary solutions for both the equilibrium solution and the ${\rm {\cal A}}$-efficient solution. While analytical solutions are available for the equilibrium solution and the first case in the ${\rm {\cal A}}$-efficient solution, the typical ${\rm {\cal A}}$-efficient solution is not analytical. Thus to facilitate the comparison between the equilibrium model and ${\rm {\cal A}}$-efficient model solutions, we present the stationary outcomes graphically. In the following figures, we present the stationary annualized income growth rates, fertility, and schooling as functions of the coordination cost parameter, λ, and the agglomeration returns to specialization, ω.

In Figure 2 we present two panels; the left side shows the stationary annualized income growth rate as a function of ω, and the right side shows the stationary annualized income growth rate as a function of λ. Observe that for fixed λ, the ${\rm {\cal A}}$-efficient income growth rate is non-monotonic in ω; it first falls until ω = 1.5, and then increases for higher values. Similarly, for fixed ω, the A-efficient income growth rate is monotonic in λ, but the relationship changes sign. For our ${\rm {\cal A}}$-efficient model solutions and ω < 3.5, ${\rm {\cal A}}$-efficient income growth is a decreasing function of λ. However, for higher values of ω, ${\rm {\cal A}}$-efficient income growth is monotonically increasing in λ. Notice that the equilibrium income growth rate exceeds the ${\rm {\cal A}}$-efficient income growth rate for all values of λ and ω < 2.75. For higher agglomeration returns to specialization, the ${\rm {\cal A}}$-efficient income growth rate exceeds the equilibrium income growth rate. As mentioned above, when the per capita income growth rate in the ${\rm {\cal A}}$-efficient allocation exceeds the per capita income growth rate in the equilibrium allocation, and all future generations have higher utility than their equilibrium counterparts.

Figure 2.

Stationary equilibrium and ${\rm {\cal A}}$-efficient annualized per capita income growth rates.

Figure 3 presents the stationary equilibrium fertility, and stationary ${\rm {\cal A}}$-efficient fertility for varying values for λ and ω. Stationary equilibrium fertility is independent of both ω and λ, because parents only get utility from children, and do not internalize the external benefit of population on earnings. For all values of these two parameters, stationary ${\rm {\cal A}}$-efficient fertility is higher than stationary equilibrium fertility. This is not a surprise, since at zero mortality, the equilibrium return to children comes only from the direct utility of having children, whereas for the ${\rm {\cal A}}$-efficient solution, it internalizes the externality on the next generation's earnings. Unlike the growth rate results, stationary ${\rm {\cal A}}$-efficient fertility is increasing in both λ and ω. In both cases, higher values of λ and ω increase the positive external benefit of population on earnings, and hence leads the ${\rm {\cal A}}$-efficient planner to increase stationary fertility.

Figure 3.

Stationary equilibrium and ${\rm {\cal A}}$-efficient fertility.

Figure 4 presents the stationary equilibrium schooling and stationary ${\rm {\cal A}}$-efficient schooling for varying values for λ and ω. As with fertility, the stationary equilibrium schooling is independent of both ω and λ. Unlike its fertility counterpart, stationary ${\rm {\cal A}}$-efficient schooling is not monotonically related to ω. For all values of λ, as the agglomeration return to specialization increases from slightly bigger than 1, to around 1.5, stationary ${\rm {\cal A}}$-efficient schooling declines. Nevertheless, once it reaches its minimum value, stationary ${\rm {\cal A}}$-efficient schooling is increasing in ω. For all values of λ between 1 and 25, stationary ${\rm {\cal A}}$-efficient schooling is increasing in ω for ω > 1.5. On the other hand, for all values of ω, stationary ${\rm {\cal A}}$-efficient schooling is monotonically decreasing in λ. The stationary ${\rm {\cal A}}$-efficient schooling comparative statics mirror the stationary ${\rm {\cal A}}$-efficient income growth comparative statics.

Figure 4.

Stationary equilibrium and ${\rm {\cal A}}$-efficient schooling.

3.1.6 Numerical solutions including non-zero mortality

In this section we present numerical solutions to compare and contrast the equilibrium time series with the ${\rm {\cal A}}$-efficient time series. One interesting feature displayed is the interplay between human capital growth and growth of the production coalition. If the long run coalition is bounded by the coordination cost, that is 1 < N _t = [σh _t]^λ < ξP _t, then even with high gains from specialization, a large ω, fertility in the ${\rm {\cal A}}$-efficient solution will not be given by (21), but rather x ^eff = [Aτ ^μ]^λ. If the long run coalition is bounded by the maximum fraction of the population, 1 < N _t = ξP _t < [σh _t]^λ, then fertility in the efficient solution will be given by (21).

We calibrate the equilibrium model to produce a demographic transition starting in 1800 and terminating in 2080.Footnote ²⁷ For all values of λ and ω, the equilibrium time series on human capital is identical, which implies that the mortality time series is also identical to all λ and ω. Thus the time series on mortality and fertility are independent of λ and ω.Footnote ²⁸ In all figures the equilibrium case assumes ω = 1.01 and λ = 2.5. By contrast, for all values of ω, the ${\rm {\cal A}}$-efficient solution has an earlier acceleration in growth, and a much earlier decline in mortality. The time series of mortality is presented in Figure 5. In the slowest case, ω = 1.01, the mortality decline begins in 5,680 BC, and is completed in 3,800 BC. However, for the next slowest case, both ω = 1.20 and ω = 1.50, the mortality decline begins in 16,560 BC and is completed in 15,360 BC. For ω = 2.00, the mortality decline begins in 17,160 BC and is completed in 16,160 BC. For ω = 6.00, the mortality decline begins in 17,760 BC and is complete in 17,360 BC. So as the returns to agglomeration from specialization increase from 1.01 to 6.00, the mortality decline begins 12,000 years earlier. It also becomes more rapid, dropping from 1880 years in length to 400 years in length. Higher ω increases the return to specialization and population. The ${\rm {\cal A}}$-efficient planner internalizes this by accelerating human capital accumulation to lower the coordinating costs of specialization. By accelerating human capital accumulation, the production coalition grows and additionally mortality can decline more rapidly. In all cases the ${\rm {\cal A}}$-efficient mortality declines occur and end before the equilibrium mortality decline. The onset is at worst only 7500 years earlier, and in the extreme case almost 20,000 years earlier.

Figure 5.

Equilibrium and ${\rm {\cal A}}$-efficient mortality.

Figure 6 shows the time series of total fertility rates. We produced total fertility rates by multiplying fertility in the model solutions by 2. The equilibrium time series path of total fertility reproduces the long classical-Malthusian world, interrupted by a brief and rapid fertility transition, and followed by a stationary total fertility rate of near zero population growth. This equilibrium fertility time series is identical across all values of ω. In all cases presented of ${\rm {\cal A}}$-efficient fertility, fertility actually is below the equilibrium fertility during the high-mortality regime, and even through the fertility transition in the equilibrium time series. With lower fertility comes the more rapid human capital accumulation as the planner internalizes the benefit of human capital accumulation in reducing young adult mortality. With the earlier onset of the ${\rm {\cal A}}$-efficient mortality decline, the ${\rm {\cal A}}$-efficient fertility declines to a new stationary value that also looks classical. About the time of the equilibrium fertility transition there is another large decline in the ${\rm {\cal A}}$-efficient fertility. This period corresponds to an ${\rm {\cal A}}$-efficient transition dynamic phase. Because world population is so much larger than the ${\rm {\cal A}}$-efficient production coalition, this period is marked by an extremely high rate of human capital investment and actual population decline. This period is also characterized by rapid growth in the size of the production coalition (much more rapid than stationary ${\rm {\cal A}}$-efficient fertility) and extremely high-transitional dynamic income growth rates. Finally, once human capital is sufficiently large that the production coalition is determined by ζP _t, then the stationary ${\rm {\cal A}}$-efficient fertility and human capital investment period are observed. In this phase, the ${\rm {\cal A}}$-efficient fertility exceeds the equilibrium fertility for all values of ω.

Figure 6.

Equilibrium and ${\rm {\cal A}}$-efficient total fertility rates.

Transition dynamics produce extremely high ${\rm {\cal A}}$-efficient per capita income growth rates. In order to keep things in scale, we constructed annualized growth rates over 1000 years.Footnote ²⁹ Thus each observation is given by:

(22)$$g_y = \displaystyle{{\ln y_{t + 25}-\ln y_t} \over {1000}},$$

where t represents adults in generation t, and each period is 40 years. Figure 7 shows the time series of the per capita income growth rates for the equilibrium solution and the ${\rm {\cal A}}$-efficient solutions. As mentioned above, during the transition dynamic phase of the ${\rm {\cal A}}$-efficient solution, human capital growth is above the stationary value and the growth rate of the production coalition is much greater than fertility. We plotted the growth rates on the log scale in order to show the differences in growth rates for the different values of ω more clearly.Footnote ³⁰

Figure 7.

Equilibrium and ${\rm {\cal A}}$-efficient growth rates of per capita income.

During the demographic transition, the ${\rm {\cal A}}$-efficient investment in human capital is much larger than the stationary value. Schooling is given by 40τ, and during the transition dynamic phase, the ${\rm {\cal A}}$-efficient schooling is at a boundary, that is to say, the planner chooses schooling just slightly below 40(1 − θ). In our solution this is roughly 33 years. This is presented in Figure 8.

Figure 8.

Equilibrium and efficient schooling.

Figure 9 shows the time series for log human capital. It is apparent that the demographic transition has accelerated human capital investment in the ${\rm {\cal A}}$-efficient solution before falling to a lower balanced growth path value.

Figure 9.

Equilibrium and ${\rm {\cal A}}$-efficient log human capital.

Figure 10 presents the time series of the log per capita output for the equilibrium solution and the ${\rm {\cal A}}$-efficient solutions. Reflecting the previous results on the growth rate of per capita output, the graph shows the acceleration in per capita income growth during the transition dynamics.

Figure 10.

Equilibrium and ${\rm {\cal A}}$-efficient log per capita output.

Finally, Figure 11 presents the time series of log population for the equilibrium solution and the equilibrium log production coalition, as well as the log ${\rm {\cal A}}$-efficient populations and log ${\rm {\cal A}}$-efficient production coalitions. The thick lines are the log population curves; one for the equilibrium case, and the others for different ω ${\rm {\cal A}}$-efficient time series. In all cases the thinner lines of the same color as the thick lines represent the corresponding log production coalition sizes. We observe that in the transition dynamic phase of the ${\rm {\cal A}}$-efficient solutions, log populations are declining, while the log production coalition sizes are growing more rapidly than stationary fertility.

Figure 11.

Equilibrium and ${\rm {\cal A}}$-efficient log population and log production coalition.