2.1. Preferences and choices

Young adults, who make all allocative decisions, have preferences over lotteries involving current consumption, consumption in old age and, if they are altruistic, the level of (net) full income accruing to each of the children in their care upon reaching adulthood in $t+1$ , denoted by $(1 - \rho )\bar{y}_{t+1} \equiv (1 - \rho )\bar{Y}_{t+1}/N^2_{t+1}$ . The latter is the central measure of the size of the children’s opportunity set on reaching young adulthood and hence exercises a heavy influence on the level of well-being they can attain. When choosing an allocation $(c^2_t, e_t, S_t)$ , young adults must forecast mortality and destruction rates in the coming period, higher levels of which weaken the effect of current investments in human and physical capital on $\bar{y}_{t+1}$ . If these forecasts are unerring, those who survive through old age will obtain $c^3_{t+1}$ , as given by (1), which the law of large numbers renders virtually non-stochastic. The stochastic element in the lotteries in question therefore arises only from the individual risks of failing to reach old age and, where altruism toward the children is concerned, that the latter will suffer the misfortune to die prematurely in young adulthood. If, in contrast, the outbreaks of war and disease in the future are viewed as stochastic events, there will be systemic risks, which are analyzed in Sections 5–8.

4.1. Backwardness

We seek to establish conditions that yield $e_t^0=0\ \forall t$ . Along such a path,

\begin{equation*} \bar {y}_t\left(e_t^0 = 0\right) = y_t\!\left(e_t^0 = 0\right) = F\!\left (1 - q^2_t + n_t\gamma, \dfrac {\sigma _t s_{t-1}}{n_{t-1}}\right ) \; \forall t, \end{equation*}

since $\lambda _t = z_th(0)\lambda _{t-1} + 1 = 1\ \forall t$ . As defined above, the path is stationary, so the index $t$ can be dropped. From the f.o.c. (see Appendix A), we have

\begin{equation*} u^{\prime}\!\left(c^2\right) = \sigma \!\left(1-q^2+\beta n\right)F_2\!\left (1 - q^2 + n \gamma, \dfrac {\sigma s}{n}\right ) \Omega (e^0 _t = 0), \end{equation*}

where $\nu_t\equiv b\frac{(1-q^2_{t+1})n_t}{(1-q^2_t)}$ and

\begin{equation*} n_t\Omega _t \equiv \frac {\rho \delta n_t}{1 - q^2_t} \cdot u^{\prime}\!\left (\frac {n_t \rho \bar {y}_{t+1}}{\left(1 - q^2_t\right)\left(1 - q^3_{t+1}\right)}\right ) + (1 - \rho )\nu _tv^{\prime}[(1 - \rho )\bar {y}_{t+1}]. \end{equation*}

Since $l_t = 1-q^2 +\gamma\! n$ , the budget constraint (7) specializes to

\begin{equation*} \left(1-q^2+\beta n\right)c^2 + s = (1- \rho )F\!\left(1-q^2 +\gamma\! n, \sigma s/ n\right), \end{equation*}

and (6) to

\begin{equation*} c^3 = \dfrac {n \rho }{\left(1 - q^2\right)(1-q^3)} \cdot F\!\left(1-q^2 +\gamma\! n, \sigma s/n\right). \end{equation*}

Substituting for $c^2$ and $c^3$ in the first-order conditions, we obtain an equation in $s$ , which features the constellation $(n,q^2,q^3, \sigma )$ , the preference parameters $(b, \delta )$ and the generational parameters $(\rho, \beta, \gamma )$ . The smallest positive value of $s$ that satisfies this equation is denoted by $s^b = s^b\left(n,q^2,q^3, \sigma, \ldots \right)$ .

The final step is to examine the counterpart of (8) when $e^0_t=0\ \forall t$ . Rearranging terms, we obtain

(9) \begin{align} (w + \gamma ) \sigma F_2 \!\left(1-q^2 + \gamma\! n, \sigma s^b/n\right) \geq \left(1 - q^2\right)zh^{\prime}(0). \end{align}

A marginal investment in a child’s education will yield $zh^{\prime}(0)$ units of human capital in the next period, with the fraction $1 - q^2$ of all children later surviving early adulthood, and so contributing to output. The cost of this investment involves the sum of the opportunity and direct costs of education at the margin, measured in units of human capital. When $\lambda _t = 1$ , this combined direct cost is $(\gamma + w)$ for each child. The latter is surely less than the basic endowment of unity, for a child is much less productive than an uneducated adult and $w$ is the teacher-pupil ratio, with some allowance for an administrative overhead.

The alternative is to invest in physical capital. The marginal product thereof, $F_2$ , is a pure number, since capital is made of the same stuff as output. When adjusted by the survival rate $\sigma$ , it measures the marginal yield of investing in physical capital, the proportional claim on future full income being $\rho$ for both forms of investment. Hence, $\sigma F_2$ is the opportunity cost of a marginal investment in education.

The existence of a poverty trap in this model depends heavily on the steepness of $h$ at $e = 0$ , as seen from the right-hand side of (9). Since we assume that $h^{\prime}(0)$ is finite, it is possible that the marginal pay-off to investment in physical capital is larger than that of investment in human capital, even in the absence of schooling. If this be the case, parents would invest only in physical capital, and to such degree as merely to maintain the (normalized) stock of capital, keeping the economy in a perpetual poverty trap.

It is arguable that $h^{\prime}(0)$ is at most $h(1)$ (footnote 4). Since $s^b \lt (1 - \rho )F\big(1-q^2 + \gamma\! n, \sigma s^b/n\big)$ , condition (9) is not an exacting requirement, even though $\gamma + w \lt 1$ . If (9) indeed holds as a strict inequality, then by continuity, it will be preserved when there are sufficiently small changes in $\mathbf{q}$ and $\sigma$ . This establishes:

4.2. Steady-state growth

The (asymptotic) rate of growth of $\lambda _t$ and $s_t$ at any fixed $e$ , denoted by $g(e)$ , is given by $g(e) = zh(e) - 1$ . A growth path with $e_t = e$ is feasible if and only if $zh(e) \gt 1$ . Each such path is effectively defined by the value of $e$ and the initial values of $s$ and $\lambda$ , where $c^2_t, c^3_t, s_t, y_t$ and $\bar{y}_t$ also grow at the asymptotic rate $g(e)$ and the potential contribution of child labor can be neglected for all sufficiently large $t$ .

Since the first derivatives of $F$ are homogeneous of degree zero,

(10) \begin{align} u^{\prime}\!\left(c^2_t\right)/ \Omega _t = \sigma \!\left(1-q^2+\beta n\right) \cdot F_2\!\left [\left(1 - q^2\right)\zeta (e), \sigma/n\right ], \end{align}

where $\mathbf{q}_t$ and $F_2$ are constant, and $\forall e_t \in [0,1]$

(11) \begin{equation} \Omega _t = \frac{\rho \delta }{1 - q^2} \cdot u^{\prime}\!\left [\frac{n \rho \bar{y}_{t+1}}{\left(1 - q^2\right)\left(1 - q^3\right)}\right ] + (1 - \rho )b v^{\prime}\!\left[(1 - \rho )\bar{y}_{t+1}\right]. \end{equation}

It is seen that $\Omega _t$ cannot be changing at the same rate as $u^{\prime}\!\left(c^2_t\right)$ unless either $u$ and $v$ are identical, or there is no altruism ( $b = 0$ ). In the latter case, $u^{\prime}\!\left(c^3_{t+1}\right)/u^{\prime}\!\left(c^2_t\right)$ must also be constant. That is to say, steady-state growth is possible only with some restrictions on preferences beyond those underpinning (4). The requirement that $u^{\prime}\!\left(c^3_{t+1}\right)/u^{\prime}\!\left(c^2_t\right)$ be constant motivates a standard assumption:

Independently of restrictions on preferences, the f.o.c. yield, along such a path,

(12) \begin{align} \frac{F_1\!\left [\left(1 - q^2\right)\zeta (e), \sigma/n\right ]}{F_2\!\left [\left(1 - q^2\right)\zeta (e), \sigma/n\right ]} \cdot \left[\left(1-q^2\right)zh^{\prime}(e)\right] \ge \sigma wF_1\left [\left(1 - q^2 - wne\right)\zeta (e), \sigma/n\right ], \; e \le 1. \end{align}

The ratio $F_1/F_2$ is the (constant) $|MRTS|$ Footnote ⁵ in producing full income, $F_1$ on the r.h.s. is the marginal product of human capital, and both are evaluated at $\zeta (e)$ and $\sigma/n$ .

4.2.1. No altruism

Since $v$ plays no role, (10) may be written as

(13) \begin{equation} F_2 \big [\left(1 - q^2\right)\zeta (e), \sigma/n \big ] = \dfrac{\left(1 - q^2\right)\left[(1+g(e))\left(c^3_t/c^2_t\right)\right]^{\xi }}{\delta \rho \sigma (1-q^2 + \beta n)}. \end{equation}

Lemma 1. Let $e$ vary parametrically to yield steady-state growth paths. Then, $\zeta$ is increasing in $e$ for all $F$ that are:

1. (i) sufficiently close to Cobb-Douglas in form, provided $\xi \le 1$ ; or

2. (ii) members of the CES family whose absolute value of the elasticity of substitution, $|1/(\epsilon - 1)|$ , is at most 1, provided $\xi + \epsilon \le 1$ .

Proof. See Appendix A.

Remark: The condition $\xi \le 1$ in part (i) can be weakened to include values exceeding, but sufficiently close to, 1. Regarding part (ii), if $\epsilon = -1$ , the result holds for all $\xi \le 2$ .

Under what conditions are steady-state paths possible? Let $e^p$ denote the smallest value of $e$ satisfying $zh(e) = 1$ , where $e^p \gt 0$ in virtue of $h(0) = 0$ . If $e^p\ge 1$ , there exists no steady-state growth path.

Suppose, therefore, that $e^p \lt 1$ . Under the conditions of Lemma 1, $\zeta (1) \gt \zeta (e) \; \forall e \lt 1$ . Hence, if the optimality condition (12) is violated at $e = 1$ , then, by definition, progress is ruled out. If, with progress ruled out, the l.h.s. of (12) exceeds the r.h.s. at $e^p$ , then by continuity, there exists at least one value of $e \in (e^p,1)$ such that (12) holds as an equality. This establishes:

Proposition 2. In the absence of altruism, there are three possibilities when $e$ is parametric.

1. (i) If $e^p \ge 1$ , there exists no steady-state growth path.

If $e^p \lt 1$ and $F$ satisfies the conditions in Lemma 1 , then:

1. (ii) if ( 12 ) is violated at $e = 1$ and the l.h.s. exceeds the r.h.s. at $e^p$ , there exists at least one steady-state growth path such that $e \in (e^p, 1)$ ;

2. (iii) if ( 12 ) holds at $e = 1$ , the progressive state is a possible path, and if it holds as a strict inequality, then that path is also locally stable.

The direct costs of education, which arise from the teacher-pupil ratio $w$ , exert a strong influence on which of these possibilities holds. If $w$ is sufficiently close to zero, it follows from (12) that progress is the only possible outcome, which accords with intuition. In fact, the educational system is a fairly heavy user of its own output, so that the other outcomes cannot be ruled out.

In view of the role played by condition (12), we have established

Corollary 1. The parametric growth paths defined by parts (ii) and (iii) will be sustained by families’ optimal choices.

4.2.2. Altruism

If $v$ differs from $u$ in the degree of concavity, it follows at once from (11) that $\Omega _t$ cannot be changing at a constant rate along a steady-state growth path.

Assumption 2. Let $v$ also be iso-elastic: $v\!\left[(1 - \rho )\bar{y}_{t+1}\right] = \left[(1 - \rho )\bar{y}_{t+1}\right]^{1 - \eta }{/}(1 - \eta )$ .

With $v$ in play in this form, (10) becomes:

\begin{align*} &F_2\!\left [\left(1-q^2\right)\zeta (e), \sigma/n\right ]=\\[10pt]& \quad \frac{\left [(1+g(e))c_{t}^3/c_t^2\right ]^\xi }{\sigma \!\left(1-q^2+\beta n\right)}\left (\frac{\rho \delta }{1-q^2}+(1-\rho )^{1-\eta }b\!\left [c_t^3(1+g(e))\right ]^{-\eta +\xi }\left (\frac{\rho n}{(1-q^3)\left(1-q^2\right)}\right )^\eta \right )^{-1},&& \end{align*}

so that Lemma 1 continues to hold.

This equation may also be expressed as

(14) \begin{align} \nonumber u^{\prime}\!\left(c^2_t\right)/ \Omega _t = & \left [\frac{\rho \delta }{1 - q^2} \cdot \left (\frac{n \rho }{\left(1 - q^2\right)(1 - q^3)}\right )^{ - \xi } + (1 - \rho )^{1 - \eta }b \cdot \bar{y}_{t+1}^{(\xi - \eta )}\right ]^{-1} \left (\frac{\bar{y}_{t+1}}{c^2_t}\right )^{\xi }\\ = & \; \sigma \!\left(1-q^2+\beta n\right) \cdot F_2\!\left [\left(1 - q^2\right)\zeta (e), \sigma/ n\right ]. \end{align}

Given that the economy is on a steady-state growth path, the r.h.s. is constant, attaining its upper limit when $e = 1$ , since $\zeta$ is increasing in $e$ . The expression in brackets is constant if, and only if, $\xi = \eta$ ; so that we distinguish among three cases. First, it is seen that if $v$ is less concave than $u$ , that is, $\eta \lt \xi$ , then the terms involving $\bar{y}_{t+1}$ will grow without bound, which implies from (34) that $e^0_t = 1$ , and hence that the path is locally stable. Second, in the borderline case $\xi = \eta$ , $u^{\prime}\!\left(c_t^2\right)/\Omega _t$ is indeed constant. Altruism introduces the additional, constant term $(1 - \rho )^{1 - \eta }b$ into the said expression, thus inducing an increase in $\zeta (e)$ if $e \lt 1$ . Attaining the state of progress is then more likely than in the absence of altruism. Third, if $\eta \gt \xi$ , the said expression approaches a limit as $t$ becomes very large. In effect, altruism ceases and the results of Section 4.2.1 apply. This argument establishes:

Proposition 3. With altruism and iso-elastic preferences, and given $zh(1) \gt 1$ , the possible steady-state growth paths are as follows.

1. (i) If $u$ and $v$ differ, with $\eta \lt \xi$ , then progress is the sole steady-state path that can be supported by families’ optimizing decisions. It is also locally stable.

2. (ii) If $\eta = \xi$ , there may exist steady-state growth paths with incompletely educated populations, but the state of progress is also possible as a limiting case.

3. (iii) If $\eta \gt \xi$ , Proposition 2 applies.

What is the intuition for these findings? A necessary and sufficient condition for $\Omega _t$ to fall at the same rate as $u^{\prime}\!\left(c^2_t\right)$ in the presence of altruism is $\xi = \eta$ . Relaxing that restriction, consider the path $e_t = 1$ , along which $\Omega _t$ can fall at a rate less than $g(1)$ without violating the conditions for optimality. Investing yields old-age provision and fosters the children’s well-being in adulthood. Since $\bar{y}_{t+1}$ is the argument of both $u\!\left(c^3_{t+1}\right)$ and $v$ , it suffices that $v$ be less strongly concave than $u$ in order to maintain steady-state growth. If parents are perfectly selfish, the concavity of $u$ rules alone, so that $e^0_t \lt 1$ is possible.

If $u$ is less strongly concave than $v$ , then along any steady growth path, $v^{\prime}$ is falling faster than $u^{\prime}(c^2_{t})$ and $u^{\prime}\!\left(c^3_{t+1}\right)$ . The relative contribution of altruism goes asymptotically to zero as $t$ becomes arbitrarily large, and with it, $e^0_t$ may slip below 1.

4.2.3. A special case: Log-land

In light of the foregoing results and its ubiquity in applications, the logarithmic case warrants further examination.

Proposition 4. Suppose $u = \ln c, v = \ln\! [(1- \rho )\bar{y}_{t+1}]$ . If

(15) \begin{equation} \dfrac{n}{\left(1 - q^2\right)\left[\delta \!\left(1 - q^3\right) + bn\right](1 - \rho )} \ge \dfrac{ zh^{\prime}(0)}{\gamma + w} \end{equation}

and, when $F$ is Cobb-Douglas $\left(y_t = A\cdot l_t^{1-\alpha }k_t^\alpha \right)$ ,

\begin{equation*} zh^{\prime}(1) \ge \alpha \sigma ^{\alpha } n^{1 - \alpha } w A \cdot \frac {[\zeta (1)]^{1 - \alpha }}{(1 - q^2 - wn)^{\alpha }}, \end{equation*}

then the economy possesses at least three steady-state equilibria: backwardness, progress, and one or more (stationary) states with constant values of $e_t^0 \in (0,1)$ . Backwardness and progress are locally stable. Among the third type, that having the smallest value of $e_t$ is unstable.

Proof. See Appendix A. Condition (15) is independent of $F$ . The l.h.s. depends only on fertility and mortality rates, and the social norm and preference parameters $\rho, b$ and $\delta$ ; the r.h.s. depends only on those representing the costs of education and the associated marginal yield of human capital at $e_t = 0$ .

5.1. The occurrence of war

The first step is to examine how $I_t$ and $q^2_t$ affect the set of feasible current choices, which are independent of $\pi _{t+1}$ . In state $I_t$ , the budget set is denoted by

\begin{equation*} S(I_t) = \left\{c^2_t, e_t, s_t\,:\, (19), c^2_t \ge 0, e_t \in [0,1], s_t \ge 0 \right\}, \; I_t = \{0,1\}. \end{equation*}

It is seen from condition (39) in Appendix B that if the ratio of survival rates, $\sigma _t(I_t)/\!\left(1 - q_t^2(I_t)\right)$ , is independent of the current state, then the outer frontier of the feasible set is affected only by the mortality rate $q^2_t(I_t)$ . If the said ratio is indeed independent of $I_t$ , an increase in $q^2_t$ , whether associated with war or not, also makes $c^2_t$ cheaper relative to $s_t$ .

The extreme allocations of $S(I_t = 0)$ ’s outer frontier, which involve positive outlays on at most two of savings, education, and consumption, are depicted as the points A, B, C, and D, respectively, in Figure 2. The corresponding allocations in the state of war $(I_t = 1)$ are A $^{\prime}$ , B $^{\prime}$ , C $^{\prime}$ , and D $^{\prime}$ . The allocations on the whole of the frontier are examined in Appendix B. To summarize: A sufficient condition for $\left(dc^2_t/ dq^2_t(I_t)\right)_{ \, e_t = s_t = 0} \lt 0 \; \forall \lambda _t$ is $\beta \gt \gamma$ , which is not a very strong requirement. If the ratio of survival rates, $\sigma _t(I_t)/\big(1 - q_t^2(I_t)\big)$ , is fixed for each current state $I_t$ and $\beta \gt \gamma$ , the outer frontier of the feasible set $S(I_t)$ will contract inwards everywhere as the mortality rate $q^2_t(I_t)$ rises. If the said ratio is the same for both states, the contraction from $S(0)$ to $S(1)$ represents the effects of an outbreak of war.

Figure 2. Feasible sets of consumption and investment.

To complete the argument, consider the case where $e_t = 1$ is infeasible for sufficiently large values of $q^2_t(I_t)$ . Suppose that when $q^2_t(I_t) = 0$ , the maximal values of $c^2_t$ and $s_t$ , respectively, are both positive, as depicted in the figure when ABCD (or A $^{\prime}$ B $^{\prime}$ C $^{\prime}$ D $^{\prime}$ when $I_t = 1$ ) corresponds to a zero level of such premature mortality. As $q^2_t(I_t)$ progressively increases, BD will shift toward G until the allocations B and D coincide at G $(e_t = 1, c^2_t = s_t = 0)$ . Further increases in $q^2_t(I_t)$ will reduce the maximal feasible level of $e_t$ below one, with the associated allocation moving downwards along the $e_t$ -axis toward the origin O. Since AC also shifts progressively inwards toward O, the outer frontier of $S(I_t)$ contracts everywhere as $q^2_t(I_t)$ increases.

The contraction of the feasible set established above points to unambiguous income effects, for horizontal lines on the frontier move inwards in parallel and $c^2_t$ and $\bar{y}_{t+1}$ are normal goods, with $\bar{y}_{t+1}$ increasing in $e_t$ and $s_t$ . Changes in survival rates also imply changes in marginal rates of transformation. As noted above, given the current state $I_t$ , an increase in $q^2_t(I_t)$ makes $c^2_t$ cheaper relative to $s_t$ , as does an outbreak of war if this event leaves the ratio of survival rates unchanged. Turning to the marginal rate of transformation between $s_t$ and $e_t$ , we obtain from (18), on fixing $c_t$ ,

\begin{equation*} \text{MRT}_{se}(I_t) = - \bigg (n_t(w\lambda _t + \gamma ) F_1 \bigg [(1 - q_t^2(I_t) - wn_t e_t)\lambda _t + n_t\gamma (1 - e_t), \,\sigma _t(I_t)\cdot \dfrac {s_{t-1}}{n_{t-1}} \bigg ] \bigg )^{-1}. \end{equation*}

For any given $I_t$ and $e_t$ , an increase in $q^2_t(I_t)$ will increase $F_1$ and so reduce $|\text{MRT}_{se}(I_t)|$ : $s_t$ will become cheaper relative to $e_t$ , as intuition suggests.

An outbreak of war has ambiguous effects on $\text{MRT}_{se}(I_t)$ . Since $F_1$ is homogeneous of degree zero, $\text{MRT}_{se}(I_t)$ can be expressed in the form

\begin{equation*} \text{MRT}_{se}(I_t) = - \bigg ( n_t(w\lambda _t + \gamma ) F_1 \bigg [\lambda _t + \dfrac {n_t[\gamma - (w\lambda _t + \gamma )e_t]}{1 - q_t^2(I_t)}, \, \dfrac {\sigma _t(I_t)}{1 - q_t^2(I_t)} \cdot \dfrac {s_{t-1}}{n_{t-1}} \bigg ] \bigg )^{-1}. \end{equation*}

Suppose, as before, that the ratio of survival rates is independent of $I_t$ , but with $q^2_t(1) \gt q^2_t(0)$ . It is seen that the associated increase in mortality reduces, leaves unchanged, or increases, the (normalized) input of human capital according to $e_t \; ^{\le }_{\gt } \; \gamma/(\gamma + w\lambda _t)$ . For sufficiently large values of $\lambda _t$ , the expression on the r.h.s. will be very small, so that war in the current period will make $s_t$ cheaper relative to $e_t$ for all values of $e_t$ . The converse holds when $\lambda _t$ is close to one; for $\gamma/(\gamma + w\lambda _t)$ is then close to one, and the $n_t/\big(1 - q_t^2(I_t)\big)$ children cared for by each surviving young adult constitute a potentially large pool of labor, relatively speaking. The outbreak of war reduces the opportunity cost of their labor and so makes investment in their education more attractive relative to investment in physical capital. We summarize these findings as

It seems rather unlikely that the associated changes in the marginal rates of transformation will offset the reduction in investment arising from the adverse income effect.

5.2. The probability of war

Intuition suggests that an increase in the prior probability of war in the future will depress investment in the present. It will now be demonstrated that this is indeed so in our framework provided an additional condition holds.

The feasible set in period $t$ , as defined by (18), is independent of $\pi _{t+1}$ , so that changes in the latter will affect decisions only through $V(I_t)$ . Inspection of (16) reveals that the weight on the altruism term $v$ is increasing in $\pi _{t+1}$ , since $q^2_{t+1}(1) \gt q^2_{t+1}(0)$ . Where the terms involving old age are concerned, the probability of surviving into full old age is increasing in the probability of peace, $\pi _{t+1}$ . Weighing against this, the pay-off received by each of the survivors depends on the number of claimants as well as the size of the common pot. It is seen from (17) that for any given $(e_t, s_t, I_t), \; c^3_{t+1}(I_t; 0) \; ^{\ge }_{\lt } \; c^3_{t+1}(I_t; 1)$ according as

\begin{equation*} \frac {F\!\left[{\left(1-q^2_{t+1}(0)\right)\lambda _{t+1}(e_t) + n_{t+1}\gamma }, \sigma _{t+1}(0) s_t/n_t\right]}{F\!\left[{\left(1-q^2_{t+1}(1)\right)\lambda _{t+1}(e_t) + n_{t+1}\gamma }, \sigma _{t+1}(1) s_t/n_t\right]} \; ^{\ge }_{\lt } \; \frac {1-q_{t+1}^3(0)}{1-q^3_{t+1}(1)} \, .\end{equation*}

The numerator on the l.h.s. is the level of full income in period $t+1$ when peace prevails, and the denominator is the corresponding level when war does so. The r.h.s. is the corresponding ratio of survival rates into old age. Both ratios exceed 1, but it is very likely that the former ratio is the larger; for war is likely to take a proportionally heavier toll on young adults, and it will surely destroy some of the capital stock. It is plausible, therefore, that

(19) \begin{equation} \frac{F\!\left[{\left(1-q^2_{t+1}(0)\right)\lambda _{t+1}(e_t) + n_{t+1}\gamma }, \sigma _{t+1}(0) s_t/n_t\right]}{F\!\left[{\left(1-q^2_{t+1}(1)\right)\lambda _{t+1}(e_t) + n_{t+1}\gamma }, \sigma _{t+1}(1) s_t/n_t\right]} \; \ge \frac{1-q_{t+1}^3(0)}{1-q^3_{t+1}(1)} \, \end{equation}

holds. This suffices to ensure that the second term on the r.h.s. of (16), which may be expressed as $E_{I_{t+1}}u\!\left[c^3_{t+1}(I_t; I_{t+1})\right]$ , is increasing in $\pi _{t+1}(I_t = 0,1)$ , and so establishes

The argument in Section 5.1 indicates how the balance between $e_t$ and $s_t$ is affected by the probability of war in the following period depends in a complicated way on the differences in survival rates between the two states. The weight on the altruism term $v$ in $V_t$ is decreasing in the future mortality rate among young adults, and the stronger the degree of altruism, as represented by $b$ , the larger will be the absolute size of the reduction in the said weight. Yet $u$ and $v$ share the argument $\bar{y}_{t+1}$ in the same way, so that even the strong prospect of a rather destructive war is unlikely to induce much substitution between the two forms of investment beyond any differential effect on survival rates.

5.3. Pestilence

An unexpected outbreak of pestilence, such as the Black Death, is an asymmetric shock of a different kind, carrying off much of the population, but leaving the current capital stock untouched. This will be a windfall for the survivors, but it will profit them little if physical and human capital are poor substitutes in production—indeed, not at all if they are strict complements. If, in contrast, they are perfect substitutes, then the windfall will yield a correspondingly large income effect, which may be sufficiently strong to propel an economy out of backwardness onto a growth path, even with perpetual, but not unduly destructive warfare.Footnote ⁷

A belief that adult mortality will fall in the future will make current investment, especially in schooling, more attractive, thus promoting growth. An outbreak of heavy hostilities or disease that shocks the population into pessimism will therefore have ambiguous effects on accumulation.

7.1. Backwardness and progress: constant destruction rates

The following conditions must be satisfied for both states to be equilibria:

1. (i) condition (9) must hold as a strict inequality for backwardness $\left(e^0_t = 0 \; \forall t\right)$ to be a locally stable equilibrium;

2. (ii) $zh(1)\gt 1$ , so that unbounded growth results when $e_t=1 \, \forall t$ ; and

3. (iii) condition (12) yields $e^0_t = 1$ along the steady-state path $e_t = 1 \; \forall t$ .

Let $u(c_t)=\ln c_t$ and $v[(1 - \rho )\bar{y}_{t+1})] =\ln\! [(1 - \rho )\bar{y}_{t+1})]$ .

Condition (i). A sufficient condition for (9) to hold is (15). It is seen that there exists a measurable subset of the parameters involved such that (9) will indeed hold.

Condition (ii). Let $h(e) = d_1\cdot e - d_2\cdot e^{d_3}$ , so that $h(1) = d_1 - d_2$ . To illustrate, if $h = e$ , then $h^{\prime}(e) = h(1) = 1$ , and $z \gt 1$ yields $g(1) \gt 0$ .

Condition (iii). Let $F$ be Cobb-Douglas: $y_t = A l_t^{1-\alpha }k_t^\alpha$ . It is proved in Appendix C that progress ( $e^0_t = 1 \; \forall t, g(1) \gt 0$ ) will be an equilibrium path if, and only if,

(20) \begin{align} & \frac{h^{\prime}(1)}{h(1)} \geq \frac{nw}{(1-q^2-wn)} \cdot \frac{1 + \alpha \!\left[\delta \!\left(1 - q^3\right) + bn\right]}{\left[\delta \!\left(1 - q^3\right) + bn\right]} \cdot \frac{1}{1 - \rho \!\left [\left(1 - q^2\right)/(1 - q^2 - wn)\right ]^{1-\alpha }}, \end{align}

where $h^{\prime}(1)/h(1)$ will be close to 1 if $h$ is weakly concave.

7.2. Transitions to permanent peace or war

We augment the analysis of the stochastic setting of Section 6 by examining the transition to peace or war when these are ultimately established as permanent states. Changing expectations play a central role along these paths.

Suppose war breaks out in period $t$ . Agents form some sharp prior, $1 - \pi _{t+1} \gt 0$ , that war will also occur in the next period, and make their investment decisions in $t$ accordingly. Given the resulting normalized endowments and the prior $\pi _{t+1}$ , households choose $(e^0_t, s^0_t)$ . If war occurs again in period $t+1$ , the resulting normalized endowments, suppressing the time subscripts for $n_t$ , $\sigma _t$ and $q^2_t$ , will be

\begin{equation*}\bar {l}_{t+1} = \left(1 - q^2(1)\right)\lambda _{t+1}\left(e^0_t\right) + n\gamma \; \text {and}\; k_{t+1} = \sigma (1)s^0_{t}/\!\left[\left(1 - q^2(1)\right)n\right]. \end{equation*}

Two wars in a row may not be out of the ordinary, so it is quite possible that $\pi$ is not revised. Indeed, in an environment wherein the discrete variate $I_t$ is i.i.d. and memories are sufficiently long, $\pi _{t+1}$ will be fixed, a state of affairs analyzed in Section 7.3.

Now suppose that, for some reason or other, peace is confidently expected in period $t+2$ and $\left(e^0_{t+1}, s^0_{t+1}\right)$ are chosen accordingly. Suppose peace also actually rules, so that the resulting normalized endowments in period $t+2$ are

(21) \begin{align} \bar{l}_{t+2} = \left(1 - q^2(0)\right)\lambda _{t+2}\left(e^0_{t+1}\right) + n\gamma \; \text{and}\; k_{t+2} = \sigma (0)s^0_{t+1}/\!\left[\left(1 - q^2(0)\right)n\right]. \end{align}

There is the alternative, happier possibility that peace rules in period $t+1$ . In that event, the normalized endowments will be

\begin{equation*}\bar {l}_{t+1} = \left(1 - q^2(0)\right)\lambda _{t+1}\left(e^0_{t}\right) + n\gamma \;\text {and} \; k_{t+1} = \sigma (0)s^0_{t}/\!\left[\left(1 - q^2(0)\right)n\right] \end{equation*}

and the calculations for period $t+2$ then proceed as before.

Turning from the transition to the long run, suppose peace reigns thereafter and is confidently expected to do so. Given the values of the state variables in period $t$ , $\left(\lambda _t\left(e^0_{t-1}\right), s^0_{t-1}\right)$ , it can be checked whether the economy will recover from one or two wars, or fail to do so, whereby the starting endowments at $t+2$ after two wars are given by (21). The same holds, mutatis mutandis, for a transition to the regime of perpetual war.

A particular limitation of the two-period phase $t$ and $t+1$ during which war can occur is that its influence on decisions ex ante is confined to period $t$ . Two consecutive adverse shocks are possible, but the (exogenous) change in beliefs to the certainty of peace from $t+2$ onwards makes ultimate recovery more likely. Suppose the said phase is extended to three periods, thus yielding an ex ante influence on decisions in both periods $t$ and $t+1$ . The possible sequences are

\begin{equation*} \{1,1,1\}, \{1,1,0\}, \{1,0,1\}, \{1,0,0\}, \{0,1,1\}, \{0,1,0\}, \{0,0,1\}, \{0,0,0\}. \end{equation*}

We concentrate on the grimmest outcome, namely, three consecutive periods of war.

The constellation of parameter values must be extended to cover war and peace. Let the values in Table 1 hold in the state of peace, so there is a poverty trap, even with unbroken peace, {0,0,0}, as the actual outcome in the first three periods. The associated long-run value of $\zeta _{t+1}$ along the path of progress, $\zeta _{t+1}(I_t = 0;\, e_t = 1)$ , is $49.26$ , as yielded by (42), with $I_t = 0 \; \forall t$ . Let the prior probability that war occurs in periods 1 and 2, $1 - \pi _{t+1} \; ( t =0,1)$ , be 0.5, and let the mortality rates in that state be $q^2_t(1) = 0.25, q^3_t(1) = 0.35$ , with $\sigma _t(1) = 0.4$ .

If the state variables $\lambda _t$ and $k_t$ are very large, extremely heavy losses will have to occur in order to reduce the normalized endowments to levels such that even $e_t = 1$ will not be optimal, let alone set in train a certain collapse into backwardness. Suppose, therefore, that the initial values of human and physical capital, $\lambda _0$ and $k_0$ , which are inherited from period $t = -1$ , are sufficiently small for a sequence of shocks as severe as {1,1,1} to rule out any path to progress. Recalling the results in Section 6, the stationary (critical) values of $\lambda ^*$ are now in play. Under perpetual peace, expected as well as realized, $\lambda ^*(0) = 3.1988$ . With $\pi _{t+1} = 0.5$ , the critical value of $\lambda _0$ when the realized sequence is indeed $\{0,0,0\}$ is $3.3856$ ; but when the outcome is three periods of war, $\{1,1,1\}$ , the said value is $3.9976$ . To complete the initial conditions, let $\zeta _0 = 20$ .

The trajectories of $\lambda _t$ and $\zeta _t$ for each of the values $\lambda _0 = 3, 3.6, 4, 4.2$ and $10$ are depicted in Figure 3. With the attendant heavy destruction of physical capital, three periods of war generate an immediate and sharp upward spike in $\zeta _t$ , regardless of the ultimate outcome. If $\lambda _0$ is close to the critical value $3.9976$ , the trajectories of $\zeta _t$ have more than one local extremum. The trajectory for $\lambda _0 = 3.6$ follows a spike at $t = 2$ by first undershooting, and then converging from below to the value under permanent backwardness. That for $\lambda _0 = 4$ also attains a local maximum at $t = 3$ , before ultimately and slowly converging from below to the value under progress. That for $\lambda _0 = 4.2$ possesses three extrema and converges from above. These oscillations indicate complicated and long-drawn-out transitional dynamics near critical values of the boundary conditions. These dynamics are less apparent in the trajectories of $\lambda _t$ . The trajectory for $\lambda _0 = 4$ recovers only extremely slowly, despite the favorable change in beliefs, from the three episodes of war. In contrast, the recovery of the trajectory for the slightly more favorable $\lambda _0 = 4.2$ is complete by $t= 4$ , and rapid growth sets in from $t = 6$ onwards, indicating the sensitivity of the long-run path to small changes in $\lambda _0$ in the range between $3.6$ and $4.2$ . As for $\lambda _0 = 10$ , its trajectory is little affected at any point.

Figure 3. Trajectories of $\zeta _t$ and $\ln \lambda _t$ : three consecutive periods of war before peace reigns.

7.3. A stationary stochastic environment

We examine the varying fortunes of these five initial configurations as events subsequently unfold in a different setting, namely, one wherein $\pi _t = 0.5$ in all periods, with the associated unchanging, rational belief that $\pi _{t+1} = 0.5$ . For each value of $\lambda _0$ , a draw is made from the binomial distribution $Pr(I_{t} = 0) = 0.5$ and the resulting values of all relevant outcomes are calculated for period $t = 0$ and the start of $t = 1$ . At the latter point in time, a new draw is made, and so on and so forth, up to the start of period $t = n$ , thus yielding the realized path $\{\lambda _t\}_{t=0}^{t=n}$ . Since the process is i.i.d., there are $2^{k}$ distinct, equi-probable sequences $\{I_t\}_{t=0}^{t=k-1}$ up to the start of period $k$ , yielding a distribution of the $n$ -vector variate $\{\lambda _t(I_t)\}_{t=1}^{t=n}$ , whose $k$ -th element is $\lambda \!\left(\{I_t\}_{t=0}^{t=k-1}, \lambda _0, \pi _{t+1} = 0.5\right), \; \forall k \ge 1$ . More than one of these sequences may have landed up in backwardness by the start of period $k$ , in which event, the number of distinct values of $\lambda _k$ would be correspondingly smaller than $2^{k}$ . At all events, the distributions’ upper and lower limits are generated by the sequences $\{I_t = 0\}_{t=0}^{t=n-1}$ and $\{I_t = 1\}_{t=0}^{t=n-1}$ , respectively.

The resulting distributions of $\lambda _t$ are depicted in Figure 4. The courses of the upper and lower limits are shown in the left-hand panel; the frequency distributions of $\lambda _{14}$ are shown in the right-hand panel. Recall from Section 7.2 that when $\pi _{t+1} = 0.5 \; (t = 0, 1)$ , the initial realized sequence is $\{0,0,0\}$ and there is also peace forever after, the critical value of $\lambda _0$ is $3.3856$ . It follows that when $\pi _{t+1} = 0.5 \; \forall t$ , all sequences that ever involve $\lambda _t \le 3.3856$ for some $t$ are doomed to backwardness. This sufficient condition is not, however, necessary for that outcome. When $\lambda _0 = 3$ , all sequences will end up in that state by $t = 8$ , though $\zeta _t$ will continue to fluctuate thereafter with the states of war and peace. The distribution of $\zeta _{14}$ comprises four, separate dense clusters, the lowest with values of about 19, the highest with values of about 27.

Figure 4. The distributions of trajectories of $\ln \lambda _t$ in a stationary stochastic environment.

For values of $\lambda _0$ lying in the range 3.6 to 4.2, Ecclesiastes, 9:11 sums up the ensuing trajectories aptly: “[ $\ldots$ ] fortune and chance happeneth to them all.”—albeit the chances of ultimately attaining progress are slim when $\lambda _0$ is even a little less than $4$ . When $\lambda _0 = 3.6$ , the sequence $\{I_t = 1\}_{t=0}^{t=5}$ yields backwardness for good from period $t = 6$ onwards. At that point in time, about 70% of all the possible 128 sequences yield $\lambda _6 \lt 3.3856$ , rising to 94% of all possible 16,384 sequences at $t=14$ . When $\lambda _0 = 4$ , the sequence $\{I_t = 1\}_{t=0}^{t=6}$ yields backwardness for good from period $t = 7$ onwards, and 38% of all possible 16,384 sequences yield $\lambda _{14} \lt 3.3856$ . When $\lambda _0 = 4.2$ , that fraction is still 12.5%; but about 7 out of 9 attain $\lambda _{14} \gt 4.004$ , the critical value under perpetual war, and hence the assurance of ultimately enjoying progress indefinitely.

Far removed from such starting values, $\lambda _0 = 10$ provides a springboard for robust, but unsteady, growth. Progress is maintained whatever events come to pass $\left(e^0 _t = 1 \; \forall t\right)$ ; but since mortality and destruction rates fluctuate, so do the growth rates of output and investment in physical capital, and hence the level of $\zeta _t$ . The distribution of the latter in period $t = 14$ comprises six separate, dense clusters, the lowest with values of about 53, the highest with values of about 87. In the worst possible outcome, war reigns perpetually, and if expectations are revised accordingly, the associated path is classed among those in Section 4.2. The same holds, mutatis mutandis, for perpetual peace (see Figure 3 from $t = 3$ onwards.)

8.1. Calibration: 1910–1990

The data sources and the series constructed from them are discussed in detail in Bell et al. (Reference Bell, Bruhns, Gersbach and Völker2004). The GDP series was taken from the Penn World Tables (hereinafter, PWT), with starting year 1950 (see Table 2). In view of the fairly strong annual fluctuations in GDP, 5-year average levels were employed. The first population census was conducted in 1948, followed by others in 1962, 1969, and then decennially. After some minor smoothing and interpolation, the census data yield fertility and mortality rates, as reported in Table 2. The series for $e_t$ was constructed from the census tables dealing with years of completed education by age cohort. For the present calibration, a series for the physical capital stock, $K_t$ , is also needed. This, too, is taken from the PWT.Footnote ⁹

Table 2. Calibration for Kenya, 1920–1990

Exogenous values: Bell et al. (Reference Bell, Bruhns, Gersbach and Völker2004, Reference Bell, Bruhns and Gersbach2006a), and text.

8.1.1. First stage

This stage involves only the technologies for producing output and human capital. It holds for all values of $e_t$ , however chosen. The input of human capital employed in producing the aggregate good is

(22) \begin{equation} L_t = \left(0.9N^1_t + 0.5N^2_t\right)(1 - e_t)\gamma + \left (0.5N^2_t \lambda _t + \sum _{a = 3}^{5} N^a_t\lambda _{t + 10(2-a)}\right ) - w \lambda _{t-10}\left(0.9N^1_t + 0.5N^2_t\right)e_t, \end{equation}

where, in connection with this step only, $N^a_t$ denotes the number of people in the age cohort $a$ ( $a = 0$ , ages 0–4; $a = 1$ , ages 5–14; $a = 2$ , ages 25–34; $\ldots$ ; $a = 7$ , ages 75+). The resulting level of output of the composite good is

(23) \begin{equation} Y_t = A_t L_t^{2/3}K_t^{1/3}, \end{equation}

whereby it should be remarked that Kenya is an open economy exporting primary products, so that fluctuations in world markets surely affect the TFP parameter $A_t$ .

Turning to the educational technology, there is some evidence that the school system became overburdened from about 1970 [Bell et al. (Reference Bell, Bruhns and Gersbach2006a)], so that $z_t$ , too, may have varied. We choose the simple form $h(e_t) = e_t$ and allow $z_t$ , as well as $A_t$ , to shift over time. Thus,

(24) \begin{equation} \lambda _{t+10} = z_t e_t \lambda _{t-10} + 1. \end{equation}

Given $\lambda _{1900} = \lambda _{1910} = \lambda _{1920} = 1$ [Bell et al. (Reference Bell, Bruhns and Gersbach2006a)] and the values of $e_t$ for 1920–1980, (24) yields the values of $\lambda _{1930}, \lambda _{1940}, \ldots, \lambda _{1990}$ given any values of $z_{1920}, z_{1930}, \ldots, z_{1980}$ .

GDP comprises the output of the composite good and investment in education. Valuing teachers’ input of human capital at the latter’s marginal product, measured in units of the composite good, we have

(25) \begin{equation} GDP_t = A_t L_t^{2/3}K_t^{1/3} + (2/3)A_t \left(K_t/L_t\right)^{1/3} \left[w \lambda _{t-10}\left(0.9N^1_t + 0.5N^2_t\right)e_t\right].\end{equation}

Table 2 reports two solutions. The large values of $z_t$ up to 1970 reflect the fact that the education of the population began earlier, so that, by definition, Kenya had not remained mired in the state of “backwardness”. In both solutions, the second break occurs in 1980, with $z_t$ taking the values 1.44 and 1.75 from then onwards, values implying long-run annual growth rates of 1.84 and 2.84%, respectively.Footnote ¹⁰

O’Connell and Ndulu (Reference O’Connell and Ndulu2000), using entirely different methods and sources in their investigation of the experience of growth in sub-Saharan Africa, also arrive at an estimated long-run annual growth rate for Kenya of 2%. Virtually identical is Sachs and Warner’s (Reference Sachs and Warner1997) estimate of 1.9%. That these estimates lie comfortably in the interval $[1.84, 2.84]$ provides independent support for the values derived here, where solution 2 should be viewed as a somewhat optimistic springboard for developments after 1990.

8.1.2. Second stage

Associated with each of solutions 1 and 2 in the first stage are the parameters $(b, w, \delta, \rho, \eta )$ . The method of estimating them rests on the household’s f.o.c., which are set out in Appendix A. Those w.r.t. $e_t$ and $s_t$ do not involve the current term $u^{\prime}\!\left(c_t^2\right)$ ; instead, there is a weighted pair of derivatives pertaining to the next period:

\begin{equation*}W_{t+1} \equiv \delta \rho u^{\prime}\!\left(c_{t+1}^3\right) + b\!\left(1-q_{t+1}^2\right)(1-\rho ) v^{\prime}[(1-\rho )\bar {y}_{t+1}], \end{equation*}

whereby $v = \left[(1 - \rho )\bar{y}_{t+1}\right]^{1 - \eta }/(1 - \eta )$ . Since $e_t \in (0,1)$ in the period of calibration, (30) holds as an equality, as does (29), $S_t$ being always positive.

In Log-land ( $\eta = 1$ ), $W_{t+1}$ specializes to

\begin{equation*} W_{t+1} = \left [ \delta \left(1 - q_{t+1}^3\right) \dfrac {N^3_{t+1}}{N^2_{t+1}} + b\!\left(1 - q^2_{t+1}\right) \right ]\frac {1}{\bar {y}_{t+1}}, \end{equation*}

which is independent of $\rho$ , and $V_t$ takes the special form

\begin{equation*} V_t = \ln \left(c_t^2\right) + \left [\delta \!\left(1-q^3_{t+1}\right) + \dfrac {b\!\left(1-q_{t+1}^2\right)}{\left(1-q_{t}^2\right)}n_t \right ] \ln \bar {y}_{t+1} + R, \end{equation*}

where $R$ denotes various terms involving parameters, exogenous variables, and logs thereof, terms that have no influence on the optimum. It is seen that the taste parameters $\delta$ and $b$ are essentially perfect substitutes; for given fertility and mortality rates, their values can be varied arbitrarily while preserving the value of the expression in brackets, subject only to the requirements $\delta \in (0,1)$ and $b \ge 0$ . We therefore choose the “standard” value $\delta = 0.85$ .

It remains to determine the Lagrange multiplier $\mu _t$ associated with the budget constraint—see (27). Noting that we have data for $y_t$ and $s_t$ for the period 1950–1990, and so can derive $\bar{y}_t$ from the parameters estimated in the first stage, the assumption that $u = \ln c^2_t$ and (28) yield $\mu _t = 1/\!\left[\left(1-q_t^2+\beta n_t\right)c^2_t\right]$ . The normalized form of the family’s budget identity is $\left(1-q_t^2+\beta n_t\right)c^2_t + s_t + \rho \bar{y}_t = y_t$ . Substituting for $c^2_t$ , we obtain $\mu _t = 1/(y_t - s_t - \rho \bar{y}_t)$ , which can then be employed in (30). Rearranging, we obtain

(26) \begin{equation} \dfrac{\partial \Phi _t}{\partial e_t } = W \cdot \dfrac{n_t}{\big(1-q_t^2\big)} \dfrac{\partial \bar{y}_{t+1}}{\partial \lambda _{t+1}}\dfrac{\partial \lambda _{t+1}}{\partial e_t} + \dfrac{\bar{y}_{t+1}}{\left(y_t - s_t - \rho \bar{y}_t\right)}\dfrac{\partial y_t}{\partial e_t} = 0. \end{equation}

In Log-land, the second term marks the sole appearance of $\rho$ in the system to be estimated. Observe also that the social norm parameter $\beta$ appears nowhere in the above scheme of equations, and so cannot be estimated.

In 1980, Kenya’s people could have had no inkling of the AIDS epidemic that was to befall them later in that decade. Their decisions about education in 1980 were therefore based on expectations about future mortality derived from the experience of falling rates over the previous three or four decades. For this reason, we apply the above procedure using only the 4 years 1950, 1960, 1970, and 1980, and the counterfactual values of $q^2_t$ and $q^3_t$ for 1990 estimated by Bell et al. (Reference Bell, Bruhns and Gersbach2006a), as reported in Table 3.

In Log-land, there are just the three parameters $b, w$ , and $\rho$ , whereby $w$ exerts a weak influence on stage 1, which is taken into account. Proceeding from solution 1 in stage 1, the pair of years (1960, 1980) in stage 2 yields one solution wherein $\rho$ is fairly close to $\alpha$ , there is modest altruism ( $b = 0.092$ ) and $w = 0.0710$ ; this solution is reported in the final panel of Table 2. All the other second-stage solutions from this pair of years involve implausibly small values of $\rho$ or $b \lt 0$ . Of all the other pairs of years, only (1950, 1980) bears examination, and its solutions are marked by $b\lt 0$ . Proceeding from solution 2 in stage 1, the pair of years (1950, 1980) yields the sub-constellation $\rho = 0.342, b = 0.024$ and $w = 0.120$ in stage 2, as reported in Table 2. All other sub-constellations arising from this pair involve either $b \lt 0$ or implausible values of $\rho$ . Other pairs of years fail in one way or another, often yielding implausibly large values of $\rho$ or $w$ .

There remain the variants wherein $\eta$ is free. It turns out that this additional flexibility is not needed for the purposes of this particular calibration (see Appendix D).

8.2. Transitions to an end of the epidemic

Proceeding as in Section 7.2, we posit a definite end of the epidemic after three 20-year periods, that is, in 2050. We also assume that Kenya will be spared war or a serious insurgency. The full-scale outbreak is revealed in 1990. All agents then form some prior that it will continue in the period starting in 2010, perhaps somewhat attenuated, or mercifully end altogether. In the latter event, the falling course of mortality over the period 1950–1980 would be restored. The agents make their investment decisions accordingly. By assumption, the worst happens again in 2010, and a prior is formed; and once more in 2030, but then with the certainty of full, permanent relief in 2050. The associated mortality rates for each of these two possible states are set out in Table 3. Given what must have been great uncertainty in the minds of Kenya’s people in the early 1990s about the course of the epidemic and the likelihood of effective and available antiretroviral therapies, we set the prior probability of a continuation in 2010 and 2030 at 0.5.

Table 3. Premature adult mortality rates and fertility: with and without AIDS

Source: Bell et al. (Reference Bell, Bruhns and Gersbach2006a).

Solutions 1 and 2 yield the trajectories of $\zeta _{t}$ and $\ln \lambda _t$ over the period 1990–2070 depicted in Figures 5 and 6, respectively. Although $z_{t}h(1) \gt 1$ from 1990 onwards in both solutions, thus satisfying the central necessary condition for long-run growth, $z_{t}h(1) \gt 1$ does not suffice to bring about that state: the initial conditions must also be sufficiently favorable. Both solutions involve an initial setback: $\lambda _{2010} \lt \lambda _{1990}$ . In 2030, moreover, recovery of the initial level is still incomplete in solution 1 and scarcely fulsome in solution 2. More encouraging is the attainment of full education for that cohort in both solutions, with $\lambda _t$ growing at the steady rate $z_{t}h(1) - 1$ thereafter. Despite the severity of the outbreak early on, a fall into backwardness is not threatened.

Figure 5. Trajectories of $\zeta _t$ and $\ln \lambda _t$ for Solution 1: three realized waves of AIDS.

Figure 6. Trajectories of $\zeta _t$ and $\ln \lambda _t$ for Solution 2: three realized waves of AIDS.

Figure 7. Kenya: trajectories of $\ln \lambda _t$ and the distribution of $\ln \lambda _{2070}$ , solution 1.

It must be emphasized that these trajectories stem from forecasts of mortality that rest on the available evidence and demographic modeling in the period 2002–2004, together with the admittedly speculative prior $\pi _{t+1} = 0.5$ . In fact, Kenya’s epidemic has indeed continued into the present, with an estimated prevalence rate of 4.9% in 2018 [Kenya (2020)]. Yet changes in behavior and the extensive provision of antiretroviral treatment have held mortality rates in check, at levels well below those in Table 3. As for an early setback, GDP per capita actually fell by about 6% between 1990 and 2001, and gross primary school enrollments also declined in that decade [Kimalu et al. (Reference Kimalu, Nafula, Manda, Bedi, Mwabu and Kimenyi2001)]. The sharp setback in Figures 5 and 6 is admittedly a rather strong amplification of actual early outcomes, but it captures their essence.

8.3. A permanently stochastic environment

To complete this exploration of Kenya’s development in the time of AIDS up to 2070, we proceed as in Section 7.3, treating the stochastic environment as the values taken by the mortality rates described in Table 3. As before, it is assumed that, whereas $\mathbf{q}_{1990}$ was revealed to the population at that time as a fact, their beliefs about $\mathbf{q}_{2010}$ put the respective probabilities of occurrence at one half—a Solomonic judgment in such circumstances. The same is assumed for each of the succeeding periods 2010, 2030, and 2050. As in Section 8.2, war is ruled out, leaving pestilence as the only hazard. We provide the respective trajectories in Figures 7 and 8.

When governed by solution 1, Kenya’s immediate prospects were rather poor, even with the counterfactual, no-AIDS mortality profile as the actual outcome in 2010. Yet in all of the $16({=} 2^4)$ binary sequences, all cohorts of children enjoy a full education from 2030 onwards. In eight of those sequences, $\lambda _{2070}$ is at least 3.4 times greater than its value in 1990 (see Figure 7). In the two worst, it is just 2.4 times greater—a decidedly modest achievement after 80 years of development. In solution 2, the corresponding ratios are 4.8 and 2.9, respectively (see Figure 8).

In fact, the widespread provision of antiretroviral therapies and changes in behavior have resulted in a course of the epidemic less dire than that forecast in Table 3. If supported by improvements in the conduct of economic policy, and with more sanguine beliefs, the long-term outlook will be a deal better than that portrayed in Figure 8.

Figure 8. Kenya: trajectories of $\ln \lambda _t$ and the distribution of $\ln \lambda _{2070}$ , solution 2.