climate change, local resource, and the slow demographic ... · the slow demographic transition in...

Climate change, local resource, and the slow

demographic transition in stagnant Africa

Thang Dao∗ and Matthias Kalkuhl†

November 22, 2018

Abstract

This paper considers the persistent eects of climate change on the speed of demographic transi-

tion, and hence the size of population in Africa. These eects are transmitted through interactions

between the education gender gap within families, fertility, and the local environment, through which

the demographic transition is delayed. Environmental conditions aect intra-household labor alloca-

tion due to impacts on local resources and infrastructure which has gender-biased impacts on labor

productivity. Examples include the collection of essential resources, e.g. clean water and re-wood,

by women for their families' daily lives. This persistent eect of climate change may oset (partially)

the role of technological progress and infrastructure inprovement in saving the time of women for their

housework duties. Hence, the gender inequality in education/income is upheld, delaying declines in

fertility and creating population momentum. The bigger population, in turn, degrades local resource

and the environment through expanded production. The interplay between local resource, gender

inequality and population, under the persistent eect of climate change, may thus generate a slow

demographic transition and the stagnation of Africa.

1 Introduction

The demographic transitions are dierential across countries and regions. While developed

countries have completed their demographic transitions, the least developed ones in Africa have

been experiencing persistently high fertility rates and slow demographic transitions which create

a population momentum for several decades to come. Population growth is documented as one of

key driving forces of global emission (IPCC 2014) and one of the main causes of natural resource

depletion and environmental degradation (Ehrlich and Holden 1971, Meadow et al. 1972, Cleaver

and Schreiber 1994, Kates 1996, Foster et al. 2002). The observation of dierential fertility rates

across regions suggest us several related research questions need to be answered. Why have the

least developed regions, which are most vulnerable to climate change, experienced persistent

high fertility rate and slow demographic transition? Does climate change have any eect on

∗Mercator Research Institute on Global Commons and Climate Change, Torgauer Str. 12 - 15, 10829 Berlin, Germany. Email:[email protected].†Mercator Research Institute on Global Commons and Climate Change, Torgauer Str. 12 - 15, 10829 Berlin, Germany; and

University of Potsdam. Email: [email protected].

1

1 INTRODUCTION

fertility decision? Answering these questions and understanding the mechanisms in explaining

the slow demographic transition in least developed regions such as Africa, in our point of view,

would be very important for combating global climate change through policies enabling to control

population growth in relation to environmental protection.

While the eects of population growth on climate change are included in many numerical

works (O'Neill et al. 2010, Wheeler and Hammer 2010, Lutz and Striessnig 2015, Scovronick

et al. 2017, among others), the investigations on the eects of climate change and natural

resource depletion on population growth are surprisingly rare in the related economic literature.

There is a small empirical literature studying the eects of environmental and local resource

conditions on fertility at micro level in developing countries (e.g., Grace et al. 2016, Aggarwal et

al. 2001, Filmer and Pritchett 2002, Nankhuni and Findeis 2004, Biddlecom et al. 2005). These

studies show that the environmental conditions aect households' fertility decisions through

the channels of health and the intra-household division of labor. In addition, historians like

Diamond (2005) emphasize how overpopulation and excessive environmental degradation, e.g.

due to deforestation, had been responsible for the collapses of some past societies such as the

Easter Island and the Maya of Central America. While precise explanations for the collapse of

the population of the Easter Island remain controversial, environmental feedback mechanisms

feature a prominent role in many research (Brander and Tayler 1998, Reuveny and Decker 2000,

de la Croix and Dottori 2008). That is to say, fundamentally theoretical research on the interplay

between population and cenvironmental change should be developed in the context of climate

change in the 21st century in least develop regions. The paper at hand attempts to develop such

an analytical framework for this task.

The modern demographic transition theories focus on non-environmental factors in explaning

the demographic transitions which mostly experienced in developed countries. The rst promi-

nent theory, which is linked to the rise in income per capita, was proposed by Becker (1960).

Becker argues that the rise in per capita income leads to a decline in fertility because of the

associated increase in the opportunity cost of raising children. In a rened model, Becker and

Lewis (1973) suggest that the income elasticity with respect to investments in child quality is

high, while the income elasticity for child quantity is low and, therefore, fertility declines when

income increases. The second theory links fertility to the decline in infant and child mortal-

ity. Developed by demographers, this theory incorporates the empirical observation that in

many countries a decline in child mortality preceded a decline in fertility (Montgomery and

Cohen 1998). O'Hara (1975) argues that the decline in infant and child mortality reduces gross

fertility, but not necessarily net fertility (i.e. the demand for surviving children). Moreover,

O'Hara points out that demographic transition only occurs if parents substitute child quality for

child quantity. The third theory is the old-age security hypothesis (Neher 1971, Caldwell 1976,

Boldrin and Jones 2002). This theory suggests that when capital markets or public safety nets

are absent, children are treated as an asset that permits parents to transfer income to old age.

2

1 INTRODUCTION

The development of capital markets reduces the demand for children, which, in turn, leads to

demographic transition. The forth theory, proposed by Galor and Weil (2000) and Galor and

Moav (2002), highlights the rise in the demand for human capital. They argue that technological

change raises the demand for human capital and the returns to educational investments. This, in

turn, changes the tradeo between quantity and quality of children, thus generates demographic

transition. The fth theory is related to the decline in the gender gap with respect to income

and employment during the development process. The decline in the gender gap is reected by

the rise in women's relative income and the associated increase in women's bargaining power.

This theory, developed by Galor and Weil (1996), Doepke and Tertilt (2009), Diebolt and Per-

rin (2013), and recently Dao et al. (2018), proposes that the rise in women's relative income

increases the opportunity cost of child-rearing and, thus, reduces fertility.

While the modern demographic transition theories neglect environmental and resource fac-

tors, the natural environment and resource plays a dominant role in early works on population

dynamics. Particularly, in the inuential treatise An Essay on the Principle of Population,

Thomas R. Malthus (1798) argued that as population growth is limited by natural resources,

the population will eventually converge to a stable level. Despite Malthus's warning, two last

centuries experienced high rates of population growth along with climate change and environ-

mental degradation. The fertility rates in the most vulnerable regions to climate change have

been being persistently high in spite of the climate damage on production and local resource.

This stylized fact suggests the existence of an alternative pattern of demographic transition for

these least developed regions in which climate change may play a crucial role. Unied growth

theory (Galor and Weil 1996, Galor and Weil 2000, Galor and Moav 2002, and Galor 2011)

generates a mechanism that economies escape Malthusian stagnation to enter the modern sus-

tained growth regime along with the demographic transition. This theory, however, so far does

not aim at explaining the current stagnation in Africa in relation to climate change. Our paper

contributes to the literature by incorporating climate and local resource factors into a unied

growth framework to study the persistent eect of climate change on fertility and the feedback

loop between population growth, availability of local resources, gender inequality in education

investment, and technological progress. It provides us an understanding the stagnation of Africa

in the context of climate change.

The rest of paper is organized as follows. The next section introduces the model and derives

the competitive equilibrium dynamics. Section 3 denes equilibria and the dynamic system of the

economy. Section 4 analyses the evolution of population and local resource, and the convergence

to conditional steady states. Section 5 analyzes the impact of extreme climate change and the

mechanism which leads the environmental disaster to occur. The global evolution of the economy

is discussed in section 6. Section 7 concludes the paper and discuss about further works for this

research agenda.

3

2 THE MODEL

2 The model

By local resource we mean the entire geographical environment and natural resource of the

economy which support daily lives of people such as water resource, re-wood, quality of the

air, etc. We consider an economy consisting Lt identical households at any period t ∈ N. For

simplicity, we assume that each household consists of 2 persons, as a couple, with gender male

and female. Each member of a household lives for two periods childhood and adulthood

and is endowed with one unit of time when adult. Adult members of the household (i) allocate

their total time between supplying labor, doing housework such as collecting essential resources

for daily life (e.g. re-wood and clean water), and child-rearing, (ii) decide how much to invest

in their ospring's human capital, and (iii) consume the remainder of their income. As in Becker

(1985, p.52), we assume that within a household the child-rearing and housework is taken care of

by only the woman which, in accordance to cross-cultural evidence. The time for doing housework

in any period t, φt = φ(Rt, at) ∈ [0, 1], depends on the availability of local resource Rt and the

level of technology at of the economy. In particular, the following assumption describes the

properties of function φ(R, a).

Assumption 1. φi(R, a) < 0 with i ∈ R, a, φRR(R, a) > 0, and limR→+∞

lima→+∞

φ(R, a) = 0.

Assumption 1 implies that a fact that the availability of local resource, such as water and re-

wood, saves the time for women in least developed regions in collecting them for their family's

daily lives. In addition, the level of technology of the economy not only improves the productivity

of production but also saves the women's time for housework. Indeed, the level of technology

of the economy could be reected by the development of the infrastructure such as the diusion

of electricity and pipe clean water system which strongly aect the women's time allocation

for housework. Greenwood et al. (2005) argue that technological progress plays a crucial role

in household's allocation of time. Indeed, the appearance and the generalization of appliances

such as washing machines, vacuum cleaners, refrigerators (mid 20th century) as well as frozen

foods and ready meals (mid-late 20th century) has freed a considerable amount of time from

housework that can be used otherwise.

Regarding child-rearing, each household's ospring consisting of 2-people, multi-gender

households as well, for the sake of simplicity requires to devote an amount of time ρt > 0,

while the human capital of each member of the child household is assumed to be a power

θ ∈ [12, 1) of the parent household's educational investment in each of them, et for a male child

and et for a female child respectively.1 We assume that the time for raising children depends

positively on the population density Lt/X, i.e. ρt = ρ(Lt/X), where X is the size of land, which

is xed, of the economy. For simplicity, we normalize X = 1, hence ρt = ρ(Lt). We introduce

the following assumption on functional form of ρ(L).

Assumption 2. ρ′(L) > 0 and ρ′′(L) < 0 for all L ≥ 0.

1The condition θ ≥ 12implies plausibly that the returns to education do not decrease so quickly.

4

2.1 Households 2 THE MODEL

This assumption implies that the cost in time of raising children is increasing concave in

population density.2

2.1 Households

Adult households at t have preferences, and make choices, over their consumption, the number

of children, and the potential income that the latter could earn when adults as a function of

the human capital they are endowed with by their parents' educational investment. Specically,

households maximize their utility (1) below under their budget constraint (2) and time constraint

of the women (3),

maxct,nt>0et,et≥0

(1− γ) ln ct + γ ln(ntwt+1[eθt + (1− φt+1)eθt ]

)(1)

subject to

ct + nt(et + et) ≤ wt[eθt−1 + (1− φt − ρtnt)eθt−1

](2)

φt + ρtnt ≤ 1 (3)

given the wage rate wt+1 per ecient unit of labor at t+1; the fractions of household time φt, φt+1

needed for housework at t, t+1;3 and past choices of educational investments et−1 and et−1 made

by the household's parents where, at each period t, ct is adult household consumption, nt is the

number of (household-)children per household.4 The parameter γ ∈ (0, 1) captures the weight

of the household's ospring in its utility.

Since our paper focuses on the current economies of Africa whose fertility rates have been

fallen slowly i.e. these economies had experienced beyond the peaks of fertility rates along

with the increasing participation of female labor supply. From this fact, we introduce the

following assumption guaranteeing that the time constraint of women is not binding.

Assumption 3. φ(0, 0) ≤ 1−[

γ(1−θ)1−γ(1−θ)

]1−θ.

The following lemma characterizes the optimal choice of a representative household.

Lemma 1. Under assumption 3, the optimal choice of a representative household, which is the

interior solution to (1) subject to (2) amd (3), is

ct = (1− γ)wt[eθt−1 + (1− φt)eθt−1] (4)

2This idea is introduced in Goodsell (1937) and Thompson (1938), recently cited by De la Croix and Gosseries (2012) and Daoand Dávila (2013) to take into account that when households have small dwellings, child production is more costly and householdshave fewer children. More recently, De la Croix and Gobbi (2017) provide empirical evidence to conrm this idea.

3We assume that, when assessing the children's potential income, the household perfectly foresees φt+1, which is a consequenceof its own choices due to the technological progress induced by its educational investments but not what is the result of itschildren's choices (hence the absence of ρt+1nt+1 in the household's objective).

4We assume that the gender birth ratio (male over female) is 1 : 1 which is closed to the natural gender ratio 1.05 : 1.

5

2.2 Production, technology, and locally natural resources 2 THE MODEL

et =θ

1− θρt

1 + (1− φt+1)1

1−θwte

θt−1 (5)

et =θ

1− θρt

1 + (1− φt+1)1θ−1

wteθt−1 (6)

nt =γ(1− θ)

ρt

[eθt−1

eθt−1

+ 1− φt]

(7)

Proof. The proof is fairly straightforward and available upon request.

From (4) to (7), we have: (i) consumption is proportional to the household's potential income

i.e. net of housework costs; (ii) educational investments are proportional to the woman's

potential income; and (iii) fertility is increasing in the human capital gender gap resulting from

dierentiated educational investments between the man and the woman of the household.

2.2 Production, technology, and locally natural resources

We assume that the nal good production in each period t is linear in the eective units of labor

supplied in-elastically by the household adults

Yt = atHt = atLtht

where at is the level of technology; Ht = Ltht is aggregate human capital in nal good production,

and ht = eθt−1 + (1−φt− ρtnt)eθt is the average human capital that the representative household

devotes to the production.5 The marginal return to human capital is

wt = at

The technology evolves according to

at+1 = (1 + gt)at

where gt = g(eθt−1+eθt−1

at

)is the rate of technological progress between t and t + 1. We assume

that

g (x) , g′(x)

> 0 if x > x

= 0 if x ∈ [0, x]

5For sake of simplicity, we abstract from physical capital. The physical capital, indeed, can be introduced without changing thequalitative results of the model if we consider a small and open economy with perfect physical capital mobility. In this case, we denethe constant returns scale technology-augmented labor production function as Yt = F (Kt, atHt), and we extent the households tolive for three periods with saving motives.

6

2.3 Some static analyses 2 THE MODEL

That is to say, the rate of technological progress is non-decreasing in average human capital

and non-increasing in level of technology; and the higher level of technology at requires a su-

ciently high average human capital ht in production to guarantee a strictly positive technological

progress.

The locally natural resource or locally environmental quality index, Rt+1, evolves according

to the following dynamics equation

Rt+1 = Rt + Ω(Rt, Et)− ξtYt

where ξtYt is the damage on locally natural resources coming from the production activity Yt,

ξt > 0 measures the dirtiness of production in period t. We assume that ξt = ξatht

, which

captures the idea that higher level of technology and average human capital go hand in hand

with cleaner production. Without loss of generality, we normalize ξ = 1. The function Ω(Rt, Et)

is the regeneration capacity of local resource which depending on the quality of local resource

itself, Rt, and on the exogenous global climate qualiy index Et. An increase in Et has an impact

on the regeneration capacity of locally natural environment and resources.6

Assumption 4. The function Ω(R,E) is continuous in its variables and satises the following

properties

(i) ΩE(R,E) < 0, Ω(0, E) = Ω(R(E), E) = 0 for all E ≥ 0

(ii) ΩR(R,E) > (<)(=) 0 if R < (>)(=) R(E) ∈ (0, R(E))

(iii) ΩR(R,E) > −1 and Ω(R,+∞) = 0 for all R.

2.3 Some static analyses

In this section, we carry out some important static analyses which help us understand the eects

of technological progress and availability of local resources on education gender gap and fertility.

Denition 1. (Education gender gap) The education gender gap in any t is the relative gap in

education investments between the man and the woman in a representative household. That is,

ζt =et−1

et−1

.

By denition 1, from equations (5) and (6), with one period lagged, we have

6We follow the laws of thermodynamics as mentioned in Georgescu-Rogen (1971) in order to set up the general form of functionΩ(R,E). Indeed, the state of the environment is constrained by biophysical principles, specically through two processes: (i) theentropic process and (ii) the preservation process. The economy is basically a closed system with respect to material. Accordingto the law of material or energy conservation, material is neither lost nor created in any transformation process. The entropicprocess transforms the availability of material or energy in a closed system in the sense that the available energy is continuouslytransformed into unavailable energy until it disappears completely. Fortunately, the economy is an open system with respect toenergy. The preservation process refers to the constant receiving of solar radiation, which provides energy to compensate for theentropic process, thus making resources renewable. That is to say, while natural and human transformation processes destroy theavailability of material, new energy inows provide energy to recollect material and energy and to oset the destruction. Thisexplains the equilibrium in our ecology systems and the renewable nature of natural resources (Smulders 1995a,b).

7

2.3 Some static analyses 2 THE MODEL

Figure 1: Regeneration capacity of the local resource

ζt = [1− φ(Rt, at)]1θ−1 ≡ ζ(Rt, at) (8)

and hence, the fertility rate is determined by

nt =γ(1− θ)ρ(Lt)

[1− φ(Rt, at)]

1θ−1 + 1− φ(Rt, at)

≡ n(Lt, Rt, at) (9)

Proposition 1 states some important eects of local resources and technology on gender equal-

ity in education and fertility.

Proposition 1. The following properties hold:

(i) ζ(Rt, at) > 1 ∀(Rt, at) ≥ (0, 0), and ζi(Rt, at) < 0 with i ∈ R, a;(ii) nj(Lt, Rt, at) < 0 with j ∈ L,R, a.

Proof. The proofs for these statement are fairly straightforward, obtained from equations (8),

(9), and properties of functions ρ(L) and φ(R, a).

The statement (i) in Proposition 1 tells us that men always receive more education investment

than women. That is because of the asymmetric time allocation across gender within households

as reected in equation (8). Since the time for housework is unpaid and taken by women

while the potential income of child-household also constitutes the utility for their parents, then

a parent-household chooses optimally to invest more education for their sons than for their

daughters. The relative education gender gap ζt, however, decreases in both the availability of

local resource Rt and level of technology at. That is because both the increase in availability of

local resource and technological progress save women's time for housework. When the women's

time for housework, φ(Rt, at), is expected to be reduced, a parent-household will have incentive

invest more education for their daughters since the initial imbalance makes the marginal return

8

3 EQUILIBRIA AND DYNAMICS

to female education higher while the total education investment for one couple of children (one

son and one daughter), derived from equations (5) and (6) with one period lagged, is

et−1 + et−1 =θ

1− θρt−1wt−1e

θt−2

which is given, independent on at and Rt, and determined by past (predetermined) variables.

Therefore, in order to increase education investment for the daughters then a parent-household

has to reduce education investment for the sons. By which, the relative education gender gap is

reduced.

The statement (ii) tells that the fertility rate nt decreases in the size of population Lt, the

availability of local resources Rt, and the level of technology at. That is rather intuitive. Bigger

size of population Lt, i.e. higher population density, increases the cost in time raising children

then the demand of children declines. Higher availability of local resources Rt and/or higher

level of technology at save the women's time for housework then women are endowed more

education investment which increase opportunity cost of childrearing, making the substitution

eect dominates income eect. Hence, the fertility rate is reduced.

3 Equilibria and dynamics

The competitive equilibria of the economy are characterized by: (i) the household's utility

maximization under its constraints, (ii) the aggregate output equating the total return to human

capital, (iii) the dynamics of population size, (iv) the dynamics of technology, and (v) the

dynamics of locally natural environment and resource. That is to say a competitive equilibrium

(ct, et, et, nt, at+1, Lt+1, Rt+1, wt) is determined by the following system of equations

ct = (1− γ)wt(eθt−1 + [1− φ(Rt, at)]e

θt−1) (10)

et =θ

1− θρ(Lt)

1 + [1− φ(Rt+1, at+1)]1

1−θwte

θt−1 (11)

et =θ

1− θρ(Lt)

1 + [1− φ(Rt+1, at+1)]1θ−1

wteθt−1 (12)

nt =γ(1− θ)ρ(Lt)

[1− φ(Rt, at)]

θθ−1 + 1− φ(Rt, at)

(13)

at+1 =

[1 + g

(eθt−1 + eθt−1

at

)]at (14)

Lt+1 = ntLt (15)

9

4 EVOLUTION OF LOCAL RESOURCE AND POPULATION

Rt+1 = Rt + Ω(Rt, Et)− ξtYt (16)

wt = at (17)

given an exogenous global climate change index Et ≥ 0.

The competitive equilibria are fully characterized by the following reduced system describing

the equilibrium dynamics of the educational investments for male and female children et and et,

the level of technology at+1, the population size Lt+1, as well as the locally natural environment

and resources Rt+1.

et =θ

1− θρ(Lt)

1 + [1− φ(Rt+1, at+1)]1

1−θate

θt−1

et =θ

1− θρ(Lt)

1 + [1− φ(Rt+1, at+1)]1θ−1

ateθt−1

at+1 =

[1 + g


at

)]at

Lt+1 =γ(1− θ)ρ(Lt)

[1− φ(Rt, at)]

θθ−1 + 1− φ(Rt, at)

Lt

Rt+1 = Rt + Ω(Rt, Et)− Lt

given initial conditions e−1, e−1, a0, L0, R0> 0, and a sequence of exogenous climate change

indexes

(Et)+∞t=0

.

4 Evolution of local resource and population

In this section, we study the evolution of the economy conditionally on a given level of technology

a and a given climate change index E.

4.1 Conditional steady states

Lt+1 =γ(1− θ)ρ(Lt)

[1− φ(Rt, a)]

θθ−1 + 1− φ(Rt, a)

Lt (18)

Rt+1 = Rt + Ω(Rt, E)− Lt (19)

Given any level of technology a and global climate change index E, we dene the loci LL(a)

and RR(E) as follows

10

4.1 Conditional steady states 4 EVOLUTION OF LOCAL RESOURCE AND POPULATION

LL(a) ≡

(Lt, Rt) ∈ <2+ : Lt+1 = Lt

i.e. Lt = ρ−1

(γ(1− θ)

[[1− φ(Rt, a)]

θθ−1 + 1− φ(Rt, a)

])≡ ψ(Rt; a) (20)

and

RR(E) ≡

(Lt, Rt) ∈ <2+ : Rt+1 = Rt

i.e. Lt = Ω(Rt;E) (21)

Lemma 2. ψa(R; a) < 0, and ψR(R; a) < 0, ψRR(R; a) > 0.

Proof. By applying the implicit function theorem for equation (20) with respects to Lt, Rt, a,

and note that θ ∈ [12, 1), we have

ψi(R; a) =γ

ρ′(L)

(θ[1− φ(R, a)]

1θ−1 − 1 + θ

)φi(R, a) < 0, i ∈ R, a

and

ψRR(R; a) =γ

ρ′(L)

θ

1− θ[1− φ]

2−θθ−1φ2

R +(θ[1− φ]

1θ−1 − 1 + θ

)φRR

− ρ′′(L)

ρ′(L)ψ2R > 0

The properties of the function ψ(R; a), as stated in lemma 2, and those of the function

Ω(R;E) are very important in determining the number of conditional steady states as pointed

out in proposition 2. In addition, as will be apparent in the next subsections, they allow us to

study the impacts of technological progress and climate change on the size of population and

the availability of local resource at the conditional steady states.

Lemma 3. For the dynamic system characterized by equations (18) and (19), then

(i) Lt+1 − Lt

> 0 if Lt < ψ(Rt; a)

= 0 if Lt = ψ(Rt; a)

< 0 if Lt > ψ(Rt; a)

and (ii) Rt+1 −Rt

> 0 if Lt < Ω(Rt;E)

= 0 if Lt = Ω(Rt;E)

< 0 if Lt > Ω(Rt;E)

Proof. The proofs are fairly straightforward.

11


Given a level of technology a and a global climate change index E, the conditional steady

states of the economy are determined by the intersections between the loci LL(a) and RR(E).

Analytically, they are the solutions R to the following equation

Ω(R;E) = ψ(R; a) (22)

The following proposition states about the existence of conditional steady states and their

properties.

Proposition 2. Given a level of technology a and a global environmental quality index E, then

(i) There exists at most two distinct conditional steady states;

(ii) If there exists steady state(s), there is at least a steady state at which it holds R > R(E);

(iii) If there exists two distinct conditional steady states characterized by R > R, then:

(iii.a) the one with higher availability of local resource R is a locally stable node;

(iii.b) if ψR(R; a) ≥ −1, the steady state with R∗ = R is a saddle node.

Proof. (i) From equation (22), the strict concavity of Ω(R;E) in R and the strict convexity of

ψ(R; a) in R guarantees that there exists at most two distinct condition steady states.

(ii) We consider rst the case that there is only one conditional steady state. Since ψ(0; a) >

Ω(0;E) = 0, ψ(R(E); a) > Ω(R(E);E) = 0, and the strict concavity of Ω(R;E) in R and

the strict convexity ofψ(R; a) in R, this case occurs when the curves Ω(R;E) and ψ(R; a) are

tangent to each other at such the steady state (R, L). That is to say, at steady state it holds

ΩR(R;E) = ψR(R; a). From lemma 2 we have ΩR(R;E) < 0, and hence from assumption 4 and

equation (21) we have R > R(E).

In case there are two distinct conditional steady states characterized by 0 < R < R < R(E),

let us suppose a negation that R ≤ R(E). Hence, Ω(R;E) > Ω(R;E) while ψ(R; a) < ψ(R; a).

In addition, we have Ω(R;E) = ψ(R; a). Therefore, Ω(R;E) > ψ(R; a) which contradicts the

property that Ω(R;E) = ψ(R; a). That is to say, R > R(E).

(iii) In order to examine the local stability of steady states, we linearize the dynamic system,

which is characterized by two equations (18) and (19), around the steady states. The associated

Jacobian matrix appears as

J =

1− η∗L η∗ψR(R∗; a)

−1 1 + ΩR(R∗;E)

where η∗ = L∗ρ′(L∗)

ρ(L∗)is the elasticity of cost raising children with respect to the population density

evaluated at a steady state (R∗, L∗). By assumption 2 we have η∗ ∈ (0, 1).

The trace and determinant of the Jacobian matrix J are respectively

Tr(J) = 1− η∗ + 1 + ΩR(R∗;E) > 0

12


det(J) = (1− η∗) [1 + ΩR(R∗;E)] + η∗LψR(R∗; a)

Since ψR(R∗; a) < 0, it is straightforward that Tr(J)2 > 4 det(J). That is to say the charac-

teristic polynomial

C(λ) ≡ λ2 − Tr(J)λ+ det(J) = 0

has two real distinct eigenvalues λ1 < λ2, which satisfy following properties

λ1 + λ2 = Tr(J) and λ1λ2 = det(J)

It is fairly straightforward to prove that ψR(R; a) > ΩR(R;E) and ψR(R; a) < ΩR(R;E). So,

we have

C(1) = η∗L [ψR(R∗; a)− ΩR(R∗;E)]

> 0 if R∗ = R

< 0 if R∗ = R(23)

(iii.a) For the steady state with R∗ = R : We have −1 < ΩR(R;E) < ψR(R; a) < 0, then

λ1 + λ2 = Tr(J) ∈ (0, 2) and λ1λ2 = det(J) ∈ (−1, 1). We now that −1 < λ1 < λ2 < 1. Indeed,

since λ1 +λ2 < 2 then λ1 < 1. Let us now suppose a negation that λ1 ≤ −1. In this case λ2 > 1

because λ1 +λ2 > 0. Hence, λ1λ2 < −1 which contradicts the property det(J) > −1. Therefore,

−1 < λ1 < 1. Since, at R∗ = R, C(1) > 0, then λ2 < 1. In summary, it holds −1 < λ1 < λ2 < 1.

That is to say, the steady state with R∗ = R is a stable node.

(iii.b) For the steady state with R∗ = R : From (23) we have C(1) < 0 then λ1 < 1 < λ2.

Since ψR(R; a) ≥ −1 then det(J) > −1, therefore

C(−1) = 4 + 2ΩR(R;E)− 2η −[ΩR(R;E)− ψR(R; a)

]η > 2 + ΩR(R;E) + ψR(R; a) > 0

where ηL ∈ (0, 1) is the elasticity of cost raising children with respect to the population density

evaluated at the steady state (R, L). Hence, λ1 ∈ (−1, 1) while λ2 > 1. That is to say, the

steady state (R, L) is a stable node.

Proposition 2 characterizes the number of conditional steady states and their properties, par-

ticularly their stability. In the case that two distinct conditional steady states prevails, the

condition ψR(R; a) ≥ −1 is just a sucient condition under which the steady state (R, L) is

a stable node.7 We will focus on this most interesting case (the case of two distinct condi-

7Indeed, we have room to introduce the following condition on parameters and functional forms under which ψR(R; a) ≥ −1always holds. It is

γ(θ[1− φ(0, 0)]

1θ−1 − 1 + θ

)φR(0, 0) + ρ′(sup Ω(R;E)) ≥ 0

13


tional steady states). The diagram in gure 2 depicts the existence of and the convergences to

conditional steady states. In this case, there is a unique saddle path (stable arm) leading to

the conditional steady state (R, L). This saddle path can be represented by a monotonically

increasing function L(R;E, a), and there exists an availability level of resource, which depends

on level of technology and global environmental index, R(a,E) ∈ [0, R).

Figure 2: Dynamics and convergence to conditional steady states

Denition 2. (Environmental disaster) The economy is called falling in an environmental

disaster in some period t if Rt ≤ 0.

Proposition 3. If, in some period t, the state of economy is characterized by (Rt, Lt) such that

Lt > L(Rt; a,E) or Rt < R(a,E)

then the environmental disaster will occur.

Proposition 3 characterizes dangerous area which locates in the left side of the saddle path,

as depicted graphically by the diagram in gure 2. Proposition 3 tells us that, if the economy

falls into this dangerous area then in the long run the environmental disaster will occur. This

result implies an important implication that, as will be more apparent in the later analyses,

under a suciently strong environmental shock putting the economy into the dagerous area,

without suciently technological progress, then, in the long run, the economy will fall in an

environmental disaster.

14

4.2 Impact of technological progress 4 EVOLUTION OF LOCAL RESOURCE AND POPULATION

4.2 Impact of technological progress

In this subsection, we consider the impact of technological progress on the availability of local

resources and the size of population in the long run. Let us focus on the case that the economy

converges to a conditional stable steady state. We know from lemma 2 that ψa(Rt; a) < 0.

That is to say, the increase in the level of technology a makes the locus LL(a) move downward

as depicted in gure 3. The downward shift of LL locus from LL(a) to LL(a′) makes the

economy tend to converge to a conditional steady state with higher availability of local resource,

R(a′, E) > R(a,E), and smaller size of population, L(a′, E) < L(a,E).


The economic intuitions for the convergence to the conditional steady state with higher avail-

ability of local resource and smaller size of population, under the impact of increasing technology,

is following. As analyzed in section 2.3, the technological progress reduces fertility rate because

it makes the substitution eect dominate the income eect through increasing education invest-

ment for daughters relatively more compared to the sons'. The lower fertility rate leads to the

smaller size of population, which will be associated with better human capital and higher female

labor force participation, in the future. The smaller size of population, in turn, as reected in

equation (19), damages less the local resource, making the availability of local resource to be

higher. The higher availability of local resource, in turn, saves more time of women in doing

housework for their families' daily lives, and hence, reinforcing the convergence to the conditional

steady state (R(a′, E), L(a′, E)).

15

4.3 Impact of climate change 4 EVOLUTION OF LOCAL RESOURCE AND POPULATION

4.3 Impact of climate change

We study in this subsection the impact of climate change on the availability of local resource

and the size of population in the long run. As in the previous subsection, we also focus on the

case that the economy converges to a conditional stable steady state. We have from assumption

4 that ΩE(R,E) < 0. That is to say, the increase in climate change makes the RR locus shifts

downward from RR(E) to RR(E ′) as illustrated in gure 4. This downward shift of the locus RR

make the conditional stable steady state move from (R(a,E), L(a,E)) to (R(a,E ′), L(a,E ′))

with R(a,E ′) < R(a,E) and L(a,E) > L(a,E ′).

The mechanism associated with the persistent impact of climate change is as following. The

increase in climate change reduces the regeneration capacity of the local resource, thus it makes

the local resource scarer, requiring women to spend more time in collecting essential resources

for their families' daily lives. Through this channel, the education gender gap is enlarged be-

cause households tend to reduce education investment for their daughters and they instead invest

more in education for their sons. By which, the fertility will be increased, creating bigger size

of population in the future associated with low average human capital. The bigger population

and lower average human capital, in turn, damage local resource through (more dirty) produc-

tion activity. The feedback loop between local resource, population, and gender inequality in

edcuation, under the persistent eect of increasing climate change, lead to the convergence to

the conditional stable steady state (R(a,E ′), L(a,E ′)).


16

5 CLIMATE CHANGE AND DISASTER

Comparing to the eects of technological progress, the increasing climate change has opposite

eects on the long run local resources and population through the opposite mechanism linking

women's time in doing housework, education investment across genders, and fertility.

5 Climate change and disaster

The analyses and phase diagrams in the previous section suggest us that when the climate change

is too extreme such that the RR(E) locus always lies below the LL(a) locus i.e. RR(E) ∩LL(a) = Ø as depicted in gure 5, then the environmental disaster will occur. In this section,

we will characterize the precise conditions on climate change and level of technology under which

the interactions between population and and local resource will lead to environmental disaster

regardless the starting state (R0, L0) ∈ <2++ of the economy.

Figure 5: Extreme climate change

In order to characterize the condition when climate change always leads to an environmental

disaster, we study the economy under the following assumption

Assumption 5. LL(0) ∩RR(E) 6= Ø for some E < +∞.

In order to facilitate the analyses, in subsection 5.1, we consider the conditional convergence

to an environmental disaster in the sense that we x the level of technology a when we study

the dynamics. In subsection 5.2 we instead study the unconditional convergence by allowing the

level of technology to change over time.

17

5.1 Conditional convergence to an environmental disaster 5 CLIMATE CHANGE AND DISASTER

5.1 Conditional convergence to an environmental disaster

Proposition 4 provides us the properties of conditional dynamics of population Lt and local

resource Rt which, under a certain conditions, generate a monotonic congerence to an environ-

mental disaster.

Proposition 4. If LL(a)∩RR(E) = Ø and in some T , the state of the economy (RT , LT ) ∈ <2++

is characterized by

LT ∈ (Ω(RT , E), ψ(RT , a)) (24)

then the economy converges monotonically in an environmental disaster, in which for all t ≥ T

until the environmental disaster occurs, the following properties holds

Rt+1 < Rt and Lt < Lt+1 ∈ (Ω(Rt+1, E), ψ(Rt+1, a)) .

Proof. We have for

RT+1 = RT + Ω(RT , E)− LT < RT (25)

and, as in (18),

LT+1 = γ(1− θ)

[1− φ(RT , a)]θθ−1 + 1− φ(RT , a)

LTρ(LT )

= ρ(ψ(RT , a))LTρ(LT )

because, as dened in (20), ψ(RT , a) = ρ−1(γ(1− θ)

[1− φ(RT , a)]

θθ−1 + 1− φ(RT , a)

).

Hence, by assumption 2, equation (25), and ψR(R, a) < 0, we obtain

LT = ρ(LT )LTρ(LT )

< LT+1 < ρ(ψ(RT , a))ψ(RT , a)

ρ(ψ(RT , a))= ψ(RT , a) < ψ(RT+1, a) (26)

We prove next that LT+1 > Ω(RT+1, E). Indeed, it is trivially true for RT , RT+1 ∈ (0, R(E)].

For the case RT , RT+1 ∈ (R(E), R(E)), since ΩR(R,E) < 0 for RT+1 ∈ (R(E), R(E)) and

LT+1 > LT , we have

Ω(RT+1, E) = Ω(RT + Ω(RT )− LT , E) < Ω(RT + Ω(RT )− LT+1, E) (27)

Let us dene the following function

G(LT+1, RT ) = LT+1 − Ω(RT + Ω(RT )− LT+1, E) (28)

We have

GL(LT+1, RT ) = 1 + ΩR(RT + Ω(RT )− LT+1, E) > 0

18


since , by assumption 4, ΩR(RT , E) > −1.

Note from (26) that LT+1 > LT > Ω(RT , E), hence we have

G(LT+1, RT ) > G(Ω(RT , E), RT ) = 0 (29)

Combining (27), (28), (29), (26), and (25), we have

LT < LT+1 ∈ (Ω(RT+1, E), ψ(RT+1, a)) and RT+1 < RT .

Thus, by inducing for all t > T , we obtain the results stated in proposition 4.

Proposition 4 implies that, in the absence of technological progress8, when the economy falls

into a certain area characterized by the condition (24), the size of population will increase

monotonically while the availability of local resource will decreases monotonically, making the

environmental disaster to occur in a nite time. The economic intuition for these dynamics is

rather obvious. Lt ∈ (Ω(Rt, E), ψ(Rt, a)) implies two aspects of the population size Lt in relation

to the availability of local resource Rt. First, Lt > Ω(Rt, E) means that the size of population

Lt, and hence labor force, is relatively high compared to the regeneration capacity Ω(Rt, E)

of contemporary local resource. Hence, the environmental damage from aggregate production

exceeds the regeneration of local resource, making the availability of local resource decline. The

decline in the availability of local resource, as pointed out in an earlier section, enhances the

fertility rate through enlarging gender gap in education and hence gender gap in income

because it requires women to spend more time for their housework duties, making their parents

invest less in education for them. Second, Lt < ψ(Rt, a) indicates that the size of population Lt,

and hence population density, is not suciently high to make the cost in time of raising children

physically expensive enough such that the demand for children of the representative household

is nt ≤ 1. Thus, the size of population becomes bigger over time. The mechanism of interactions

between population and local resource creates a monotonic convergence to an environmental

disaster.

The directions of motion of the population size and the availability of local resource as

pointed out in lemma 3 tell us that when the economy starts from a state such that L0 /∈(Ω(R0, E), ψ(R0, a)) then it will enter the area LT ∈ (Ω(RT , E), ψ(RT , a)) in some nite time

T . And eventually, the economy will converge monotonically in an environmental disaster.

The next proposition characterizes the level of climate change under which the environmental

disaster will occur.

Proposition 5. Given any level of technology a unchanged (more strictly, non-increasing) over

time, there exists a unique threshold E∗(a) > 0 such that for all E > E∗(a) then the environ-

8or if technology progresses too slowly

19


Figure 6: Extreme climate change

mental disaster will occur, regardless starting state of the economy (R0, L0) ∈ <2++. Moreover,

E ′∗(a) > 0.

Proof. We need to prove that for all given a, there exists a unique pair (R∗, E∗) ∈ <2++ such that

the following system of equations (30) and (31) hold

Ω(R∗, E∗)− ψ(R∗, a) = 0 (30)

ΩR(R∗, E∗)− ψR(R∗, a) = 0 (31)

This system of equations tells us that the loci LL(a) and RR(E) are tangent to each other

at (R∗, E∗). Thus, by the strict concavity of Ω(R,E) in R and strict convexity of ψ(R, a) in R

when R ∈ (0, R(E)), the system above implies that

LL(a) ∩RR(E∗) = (R∗, ψ(R∗, a))

By assumptions 4 and 5, there always exists E∗ > 0 such that the equation (30) has at least

a solution R∗. By applying the implicit function theorem for the equation (30) with respect to

E∗ and a, we have E∗ to be a function of a,

E∗ = E∗(a) with E ′∗(a) =ψa(R∗, a)

ΩE(R∗, E∗)> 0

20

5.2 Unconditional convergence to an environmental disaster 5 CLIMATE CHANGE AND DISASTER

We now complete the proof by showing that, for all a, there exists a unique R∗ making

equation (31) holds. Indeed, let us dene

4(R; a) = ΩR(R,E∗(a))− ψR(R, a)

We have, for all R ∈ (0, R(E∗(a))),

4R(R; a) = ΩRR(R,E∗(a))− ψRR(R, a) < 0; 4(R(E∗(a)); a) > 0 and 4(R(E∗(a)); a) < 0.

Thus, there exists a unique R∗ ∈ (R(E∗(a)), R(E∗(a))) to be the solution to 4(R; a) = 0.

Hence, there exists a unique pair (R∗, E∗) ∈ (0, R(E∗)) × <++ solving the system of equations

(30) and (31).

That is to say, for a given a there exists a unique E∗ = E∗(a) such that the loci LL(a) and

RR(E) are tangent to each other. For any E > E∗(a), then LL(0) ∩ RR(E) = Ø. Hence, the

evolutionary processes of Rt and Lt, as depicted by the directions of motions in gure 5, will

lead the economy to fall in the environmental disaster.

Proposition 5 provides an important condition under which an economy will fall in an envi-

ronmental disaster. It implies that if the level of climate change is too strong relatively compared

to the level of technology of the economy i.e. when it holds E > E∗(a) then in the long run

the availability of local resource is zero, R = 0, i.e. the environmental disaster will occur, even

the economy start from an initial state characterized by small size of population L0 and high

availability of local resource R0. In the absence of technological progress, the interplay between

population and local resource, under the persistent eect of climate change, make the size of

population increase along with the decrease in the availability of local resource until the latter

reaches zero.

5.2 Unconditional convergence to an environmental disaster

In this section, we relax the conditions that level of technology and global environmental index

are xed. We allow these variables change over time. Let dene AT =

(aτ )+∞τ=T

and ET =

(Eτ )+∞τ=T

to be respectively the set of levels of technology and the set of global climate change

indexes from some period T ≥ 0 onwards.

Proposition 6. If LL(supAT )∩RR(inf ET ) = Ø and in some period T , the state of the economy

(RT , LT ) ∈ <2++ is characterized by

LT ∈ (Ω(RT , inf ET ), ψ(RT , supAT ))

21

6 GLOBAL DYNAMICS AND DISCUSSION

then the economy converges monotonically in an environmental disaster, in which for all t ≥ T

until the environmental disaster occurs, the following properties holds

Rt+1 < Rt and Lt < Lt+1 ∈ (Ω(Rt+1, inf Et+1), ψ(Rt+1, supAt+1)) .

Proof. The proof for this proposition is similar to that for proposition 4

The following theorem extends the result stated in proposition 5.

Proposition 7. If in some period T ,

inf ET > E∗(supAT ) or equivalently supAT < E−1∗ (inf ET ) ,

then environmental disaster occurs in some nite time t ≥ T .

Proof. Under the condition stated in proposition 7, in any period t ≥ T , it holds LL(at) ∩RR(Et) = Ø i.e. there is no any conditional steady state. Hence, from the directions of

motions ofRt and Lt as pointed out in lemma 3, the economy eventually falls in the environmental

disaster.

6 Global dynamics and discussion

The global dynamics of the economy is governed by the following system of equations

et =θ

1− θρ(Lt)

1 + [1− φ(Rt+1, at+1)]1

1−θate

θt−1 (32)

et =θ

1− θρ(Lt)

1 + [1− φ(Rt+1, at+1)]1θ−1

ateθt−1 (33)

at+1 =

[1 + g


at

)]at (34)

Lt+1 =γ(1− θ)ρ(Lt)

[1− φ(Rt, at)]

θθ−1 + 1− φ(Rt, at)

Lt (35)

Rt+1 = Rt + Ω(Rt, Et)− Lt (36)

given initial conditions e−1 > 0, e−1 > 0, a0 > 0, L0 > 0, R0 > 0, and given an exogenous

sequence of global carbon concentration index (Et)+∞t=0 .

In addition to the previous section, we study in this section the impact of climate change

on technological progress. We learn from previous section that, the level of technology has

a negative impact on fertility through narrowing the gender in education investment. Hence,

22

6 GLOBAL DYNAMICS AND DISCUSSION

we show in this section that by aecting on technological progress, climate change has other

reinforcing eects on gender inequality in education and fertility in addition to the eects we

studied in the previous section. The following proposition states the impact of climate change

on technological progress

Proposition 8. An increasing in global climate change index Et in period t slows down the rate

of technological progress gt+1 between periods t+ 1 and t+ 2.

Proof. In eect, we come back with the education gender gap as dened in denition 1,

ζt+1 =etet

= [1− φ(Rt+1, at+1)]1θ−1 ≡ ζ(Rt+1, at+1)

where in any period t+ 1, the variables at+1 and Rt+1 are predetermined by equations (34) and

(36) respectively.

We have

∂ζ(Rt+1, at+1)

∂Et=

1

1− θ[1− φ(Rt+1, at+1)]

2−θθ−1φR(Rt+1, at+1)ΩE(Rt, Et) > 0

which implies that the climate change enlarges the gender inequality in education investment,

while the total education investment for a couple of children is

et + et =θ

1− θρ(Lt)ate

θt−1

which decreases in the climate change index Et.

The decrease in total education investment for a couple of children, the reallocation in edu-

cation investment is more biased towards the son, and the concavity of human capital formation

function together induce the decline in the rate of technological progress.

The theoretical results so far may suggest us a mechanism to explain the slow demographic

transition in least developed regions which are most vulnerable to climate change, accompanied

with gender inequality in education. The climate change persistently plays a role to maintain

high rates of fertility in these regions through several channels. First, it damages the local

resources which are essential for life, and thus requiring women to spend more time in collecting

resource for their families. When households anticipate that their daughters will supply less the

labor supply due to spending more time for doing housework, they will invest less in education

for the daughters and invest more education for their sons instead. Through this channel,

the gender inequality in education, which is biased to the sons, is maintained. Hence, the

high fertility rate is also upheld because the opportunity cost of child-rearing become relatively

cheap, making income eect to dominate substitution eect. Second, climate change directly

decreases the return to human capital, reducing the income of households, and thus indirectly

makes households invest less in education for their ospring. That is to say that climate change

23

7 CONCLUSION AND FURTHER WORKS

slows down the rate of technological progress because it decreases the average human capital of

the economy. The low level of technology, in turn, inhibits the development of infrastructures,

such as electricity and pipe clean water systems, as well as inhibits the diusion of saving-time

home appliances which all save the time for women in doing housework. Therefore, through

this channel, climate change also upholds gender inequality in education and thus maintain high

fertility rate.

7 Conclusion and further works

This paper advances a theory integrating two emerging and important topics in the literature,

demography and climate change in order to explain the slow demographic transition in least

developed regions of Africa. It provide a novel mechanism to explain the slow demographic

transition and stagnation of least developed regions. It also characterize the conditions on the

size of population and availability of local resources as well as the extreme of climate change

through which environmental disaster will occur in nite time.

The global dynamics of the model needs numerical exercises to illustrate the mechanism which

leads to slow demographic transition along with the slow technological progress and the persistent

gender inequality in education, as well as the mechanism which leads to environmental disaster.

Several hypotheses, further research questions, and policy implications should be addressed from

the theoretical results of the model. They are, but not limited in this list: (i) The damage of

climate change on local resource increases gender inequality in education/income (and hence

fertility) through its asymmetric eects on the time allocations between genders; (ii) Is only

public investment in education enough for triggering development in least developed regions?;

(iii) Using carbon tax revenue to invest in essential infrastructure (eg. clean pipe water system

and electricity) may trigger development in least developed regions. These further works are on

going in our research agenda.

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