Economic grwoth, demographic and renewable resources: A dynamical … · 2003-03-07 · capital. The human population dynamics are in uenced by the per capita intake of goods from

Economic growth, demographics, and renewable resources: A

dynamical systems approach

John M. Anderies

Mailing address: CSIRO Sustainable EcosystemsGPO Box 284Canberra, ACT 2601e-mail: [email protected]

DRAFT VERSION

Running Title: Demographics and economic growth

June 14, 2001

1

Growth and Demographics — DRAFT 2

Abstract

A two sector growth model including human demographics and a renewable resource base is

developed and analyzed. The influence of economic structure, investment, and the interplay

between demographics and growth on the evolution of the system is studied. The model is very

robust over a wide parameter range and exhibits two basic classes of behavior: convergence

to a state with a stable population and constant consumption over time or “overshoot and

collapse” where periods of welfare increase are followed by periods of welfare decline. It

is shown via a combination of numerical bifurcation techniques and simple scaling arguments

that the key determinant of model behavior is the relationship between per capita consumption,

fertility and mortality, and the relative time scales implicit in the economic, demographic, and

resource systems. Also, technological change that increases productivity favors “overshoot and

collapse” type dynamics while technological change that reduces the environmental impact of

production has no effect on the dynamics of the model and thus cannot prevent “overshoot

and collapse” type behavior.

Key Words: investment, development, sustainability, demographic transition, system dy-

namics.


Introduction

There has been a long association between the formal neoclassical theory of economic growth

and the study of environmental policy and sustainability. The most recent wave of work in

this area probably began with the well known work of Solow (1974) who used growth models

to study intergenerational equity with exhaustible resources, and papers by Stiglitz (1974),

and Dasgupta and Heal (1974) which address the optimal depletion of exhaustible resources.

This basic work has been extended in many directions including studying the feasibility of

maintaining consumption (Cass and Mitra, 1991) and optimal investment rules for doing so

(given constant consumption over time as a goal) (Hartwick, 1977).

These models have led to research on how they might be used to define “sustainability”, a

term for which there are many definitions (Pezzey, 1992). Toman et al. (1994) provide a neat

summary of this area of research and Pezzey (1997) has explored many subtleties concerning

using such models to define sustainability.

Growth models have been used more broadly (than the optimal consumption of a depletable

resource base) by incorporating such elements as renewable resources, assimilative capacity,

pollution, and defensive expenditures. Beltratti (1997) gives an excellent summary of this

area of research and its implications for policy. The main thrust of the work mentioned so far

is to better understand under what conditions (optimal) sustained economic development is

possible in the face of some type of physical environmental constraint.

Another line of work stemming from formal growth theory relates to social, rather than

physical, limits to growth outlined in the classic book by Hirsh (1977). The point here is that

the interaction of social and economic growth processes determines whether the latter can

deliver increased benefits to society. The crux of the argument is that a significant proportion

of consumption is what Hirsch terms “positional” in that its main function is to maintain

or increase ones position in the income hierarchy. Several papers have followed that address

“status” effects and the empirical evidence that growth is not increasing utility -i.e. people

are not getting any happier as economies grow (Keely, 2000).

My modest aim here is to bring together some general ideas from each of these areas

to study the influence of social processes on growth paths in an economy with a renewable

resource constraint. The social process I focus on is the demographic transition whereby

changing economic structure resulting from growth influences demographic patterns. By using

a combination of analytical and numerical bifurcation techniques, the influence of the interac-

tion of economic and demographic processes on growth paths is explored in several different

economic contexts.

The model

The model is a representative agent, two sector, three factor growth model with a renewable

resource base. The two sectors are agriculture and manufacturing (hereafter labeled as sectors


1 and 2, respectively), and the three factors are labor, human-made capital, and natural

capital. The human population dynamics are influenced by the per capita intake of goods

from each of these sectors.

Firms

I represent agricultural production with the Cobb-Douglass technology:

Y1 = E1(t)kαrr Lα11 K1−α1

1 (1)

where Y1 is output, kr, K1 and L1 represent natural capital, human-made capital and labor

in sector 1, respectively, and E1 is a time dependent measure of efficiency. The technology is

constant returns to scale in labor and human-made capital, but exhibits increasing returns in

natural capital. The presence of of kr in equation 1 is meant to capture the dependence of

agriculture on ecological attributes such as soil structure, fertility, and hydrological processes.

Thus kr should be interpreted not as a stock measure, but as a quality measure. Doubling

labor and capital on doubly fertile land can more than double output.

Similarly, manufacturing output, Y2 is given by

Y2 = E2(t)Lα22 K1−α2

2 , (2)

but in this case there is no dependence of E2 on natural capital. I do not include this depen-

dence for two reasons: exhaustible resources have been studied in great detail in the context

of economic growth (Solow, 1974; Hartwick, 1977; Dasgupta and Heal, 1974), and it does not

add significantly to the ideas developed here (Anderies, 1998b). Exogenous (labor augmenting)

technical progress can be incorporated by making Ei functions of time. Endogenous growth

can be modeled by setting αi = 0.

Consumers

In the economic growth literature, it has become customary to represent household behavior

as an intertemporal optimization problem, i.e. the standard Ramsey model (Barro and Sali-i

Martin, 1995). However, I have chosen to assume a constant savings rate as in the standard

Solow-Swan model. I do so for two reasons. First, including two sectors and the interaction of

demographic processes and economic growth requires that the model be more complex than

the usual one sector growth model. A constant savings rate makes the analysis more clear.

Second, although savings rates vary cross-sectionally, at the aggregate level they are constant

over time.1 Thus, individual agents are motivated to save by variety of forces i.e. consumption1See Frank (1985) for an interesting interpretation why this is so. He argues that for people lower in the

income hierarchy, consumption may have great importance for its demonstration effect. As people rise up theincome distribution, demonstration becomes less important, and savings rates go up. In this case, savings rateswould be determined by the shape of the income distribution, not the overall magnitude of the distribution.Thus as aggregate income increases, if the distribution remains the same, the aggregate savings rate will besame.


smoothing, life cycle considerations, etc. which translates into a constant savings rate at the

aggregate level. In this context, a constant savings rate is a reasonable representation of reality.

Thus identical consumers solve the problem:

max U(q1, q2) = (q1)c1(q2)1−c1 (3)

subject to: P1q1 + P2q2 ≤ (1− s)M (4)

where U is utility, q1 and q2 are the per capita consumption rates of agricultural and manufac-

turing goods, P1 and P2, are their respective prices, M is per capita income, s is the savings

rate, and c1 is the preference for agricultural goods. The familiar solution is (see appendix for

details):

q1 =c1M(1− s)

P1and q2 =

(1− c1)M(1− s)P2

. (5)

General Equilibrium

Let the human population level at time t be denoted by h(t). Each of the h identical consumers

owns the same quantity of capital stock. Let labor and human-made capital factor prices be

given by w and r respectively. There are five markets in the economy, two factor, two output

markets, and an investment market. Equilibrium occurs when output and factor prices reach

equilibrium. Because technology is constant returns to scale and there are no adjustment

costs, firms are indifferent to scale. Thus firms will make use of all investment in new capital,

i.e. investment levels are completely determined by sY where Y is total output and thus the

investment market is always in equilibrium. I assume that the manufacturing sector supplies

both investment and consumer goods at the same price. The main elements of the computation

of the general equilibrium are highlighted below. For full details see the appendix.

The total demand for agricultural and manufactured goods is then

Y D1 = hq1 (6)

Y D2 = hq2 +

sM

P2(7)

where the superscript indicates “demanded”. Because of our assumption of constant returns

to scale, the zero profit condition specifies the supply output levels:

P1YS

1 = wL1 + rK1 (8a)

P2YS

2 = wL2 + rK2 (8b)

where the superscript indicates “supplied”. Again because of constant returns to scale, there

is nothing in (6), (7), (8a), or (8b) to determine the scale of production. Equations (6) and

(7) determine the proportions of total income spent in each sector and profit maximization

by firms determines the capital labor ratios in each sector. To set the scale of production, I

assume full factor utilization, i.e.:

L1 + L2 = h (9a)

K1 +K2 = kh. (9b)


Given r and w, equilibrium in output markets is computed by setting (6) equal to (8a)

and (7) equal to (8b). The economy comes to equilibrium when the rental rates of labor and

capital adjust to reflect the actual relative scarcity of labor and human-made capital given by

h and kh and factor markets clear. In equilibrium we have

L1 = βh (10a)

L2 = (1− β)h (10b)

K1 = γkh (10c)

K2 = (1− γ)kh (10d)

where β and γ are the proportions of labor and capital devoted to agriculture given by

β =((

1c1(1− s)

− 1)α2

α1+ 1)−1

(11)

γ =((

1c1(1− s)

− 1)

1− α2

1− α1+ 1)−1

, (12)

respectively. A glance at equation 11 reveals that the fraction of labor directed to agriculture

is an increasing function of preferences for agricultural goods (c1) and the productivity of

labor in agriculture (α1) and a decreasing function of productivity of labor in manufacturing

(α2). The same statement holds for γ with the word labor replaced with human-made capital.

Finally, combining equations (1), (2), and (10) yields

Y1 = E1βα1γ1−α1kαrr hα1k1−α1

h (13)

Y2 = E2(1− β)α2(1− γ)1−α1hα1k1−α1h (14)

Investment, I, is computed by substituting the expression for q2 in expression (5) into (8b)

and noting that investment is a proportion of total manufacturing good output (see appendix).

This yields

I =sY2

1− c1(1− s). (15)

Dynamic stock equations

In general terms, the three stocks in the model h, kh, and kr evolve according to the differential

equations

h = G(h, kh, kr)h (16a)

kh = I − δkh (16b)

kr = F (h, kh, kr)−D(h, kh, kr) (16c)

where G is the per capita growth rate of the human population and δ is the depreciation rate

of human-made capital. The function F represents the regeneration rate of natural capital

while D represents the degradation of natural capital caused by economic activity.


Demographics

Over short time scales implicitly assumed in standard growth models, exogenously determined

exponential human population growth is a reasonable approximation. Over longer time scales,

this assumption is unreasonable. Demographics are linked to economic and natural resource

systems and thus should be endogenized. There have been several growth models that have

endogenized fertility with varying objectives, but often in an attempt to better understand

the mechanisms that underly relationships between economic growth, fertility, and mortality

observed empirically, e.g. (Raut, 1990; Barro and Sali-i Martin, 1995; Galor and Weil, 2000).

My purpose, rather, is to understand how these relationships might, via the demographic

transition, affect the nature of growth paths in an economy that relies on renewable resources.

In his recent work on human demography, Cohen (1995) notes that the “idealized historical

pattern” of the demographic transition occurs in four stages: 1) population has both high birth

and death rates that are nearly equal leading to slow population growth. 2) Death rate falls,

birth rate remains high leading to rapid population growth (mortality transition). 3) Birth

rate falls (fertility transition). 4) Birth and death rates are both low and nearly equal and the

population stabilizes at a higher level than at stage one.

As Cohen notes, there is often confusion surrounding the term “demographic transition”

as to whether it is interpreted as the historical process just discussed, or the hypothetical

mechanism by which the historical process occurs. Of course, for a model, we must build in

the hypothetical mechanism. To do so, I make only the most basic assumptions: the negative

income-child quantity relationship observed in developing economies is taken for granted and

mortality is assumed to be negatively correlated with nutrition. I doubt anyone would argue

with such basic assumptions. The difficulty is building them into the model in a reasonable

way.

To develop the relationships between birth and death rates, I assume that per capita man-

ufactured goods output is strongly correlated with overall economic development, e.g. with

more developed infrastructure, access to health care and education, etc. Increased manufac-

tured goods output reduces death rates through improved health care (“mortality transition”).

It reduces birthrates when economic development and associated social change puts downward

pressure on birth rates due to, for example, the increased marginal cost of children and chang-

ing preferences (“fertility transition”). Note that in this context, birth rates are associated

with fertility decisions as opposed to a physical measure of birth rates. Increased per capita

output of agricultural goods reduces the death rate through improved nutrition. It puts up-

ward pressure on birth rates through improved overall physical health which may, for example,

cause earlier onset of and higher fertility in females. It may also have had an influence on fer-

tility decisions in some societies - i.e. the better off one is, the more children one might decide

to have.

To capture these four elements, i.e. the effects of manufactured and agricultural goods on


birth and death rates we write

G(h, kh, kr) = G(q1(h, kh, kr), qm(h, kh)) = b(q1, qm)− d(q1, qm) (17)

where b(·) and d(·) have the obvious interpretations, and qm is total per capita manufacturing

goods output as opposed to q2 which represents per capita manufacturing goods consumption.

Here, I assume that

b(q1, qm) = b0(1− e−b1q1)e−b2qm . (18)

The term b0(1 − e−b1q1) represents increases in birth rates up to a maximum of b0 as q1 (i.e.

nutrition) increases. The parameter b1 measures the sensitivity of birth rates to such increases.

The term e−b2qm represents downward pressure on birth rates as qm increases (fertility transi-

tion). Again, b2 measures the sensitivity of birth rates to changes in qm. Similarly, I assume

d(q1, qm) = d0e−q1(d1+d2qm). (19)

Improved nutrition reduces death rates (i.e. through improved immunity, etc.) via the term

q1d1 while improved infrastructure reduces death rates via the term q1d2qm (mortality tran-

sition) where as above, d1 and d2 are sensitivity parameters. Note that death rates can only

be reduced by improved infrastructure when q1 > 0, i.e. people still starve when they have no

food regardless of the level of development of the society in which they live. The parameter

d0 measures the maximum death rate with no nutritional intake.

This simple demographic model captures the four basic linkages between population dy-

namics and the structure of the economy. The parameters bi, and di, i = 1, 2, 3 measure the

sensitivity of human population dynamics to economic structure.

The renewable resource system

The basic assumption in the model is that natural capital provides inputs for agricultural

production. Key inputs include clean irrigation water, soil nutrients, and soil structure, all of

which are provided by interacting biological processes. Tree and plant communities dramati-

cally influence hydrological cycles that affect groundwater and soil salinity, a major problem

in many areas (Lefkoff and Gorelick, 1990; Quiggin, 1988). Agricultural practices necessarily

disturb these communities and reduce the services they contribute to both surface and ground

water quality maintenance with, in some cases, drastic consequences. Further, soil is much

more than its material matrix. Complex communities of organisms live within the soil, cycling

nutrients and maintaining soil structure. Agricultural practices can disturb these communities

thus reducing their ability to perform these tasks. To maintain productivity, the goods and

services supplied by these forms of natural capital must be replaced by substitutes such as

fertilizer and complex water management systems generated using human-made capital.

Thus, natural capital is set of interlinked animal and plant populations and metabolic

processes. When disturbed, these processes can exhibit highly nonlinear behavior causing

the natural capital base to change rapidly and unpredictably (Carpenter et al., 1999). This


nonlinear behavior is the major difference between this model and depletable resource models

mentioned in the introduction. In the latter, a removal of x units of resource results in an x

unit reduction in the capital base regardless of its state (provided that it is larger than x). In

contrast, in a natural system, a removal of x trees may have a very small or very large effect

depending on the the state of the system (Carpenter et al., 1999).

It is impossible to accurately model the ecological processes described above. For our

purposes, all that need be captured is the fact that on time scales relevant to economic systems

the populations described above, if undisturbed, will grow up to maximum levels limited by the

physical environment. A simple way to capture this behavior is with a logistic growth function;

a very common way to model density dependent regeneration of a bioresource (Clark, 1990;

Anderies, 1998a; Brander and Taylor, 1998). Natural capital regenerates itself logistically and

is degraded by agricultural production, thus

F (h, kh, kr) = nrkr(1− kr) (20a)

D(h, kh, kr) = ηY1. (20b)

Note that I have not included a term for manufacturing output, Y2, which certainly can

degrade natural capital. Inclusion of such a term would needlessly complicate the analysis

of the model. It would have a minor effect on some results which I will address later. By

choice of scaling, kr lies in the interval [0, 1], nr is the aggregate intrinsic regeneration rate

of the renewable resource base, and η is a parameter that measures the impact of farming on

the natural capital base. This is the same basic set up used in (Brander and Taylor, 1998;

Anderies, 1998a).

Analysis of the Model

The model is analyzed in three sections. First, for a given economic structure, I explore

the influence of investment levels and the sensitivity of the birth rate to increases in per

capita manufacturing output on the dynamics of the model. Next, the influence of different

assumptions about economic structure on these results is examined. Finally, the possible

influence of technological change on the model dynamics is addressed. Throughout the analysis

of the model we will be interested in how changing certain parameters changes the location

and stability properties of the steady state and what different qualitative behaviors of growth

paths are possible.

There are two main classes of development paths possible in the model: Trajectories will

either approach an equilibrium either monotonically, or after a series of damped oscillations,

or not at all. If trajectories do not approach an equilibrium, they will approach a limit cycle

and undergo persistent oscillations. Figure 1 illustrates these possibilities. In figure 1 (A), any


reasonable initial condition with high biophysical capital and low population will evolve to a

stable steady state (perhaps through a series of damped oscillations). In figure 1 (B), on the

other hand, no reasonable initial condition with high biophysical capital and low population

will evolve to a steady state, and will instead converge to a limit cycle. The existence of the

limit cycle acts as a barrier between any initial condition and a long term sustainable growth

path. From a development perspective, oscillations are undesirable because they represent

periods of increasing per capita consumption followed by periods of decreasing per capita

consumption. The intergenerational equity problems associated with such trajectories are

obvious.

Limit Cycle

Stable FixedPoint

Initial Point

Natural Capital

Hum

an P

opul

atio

n

Hum

an P

opul

atio

n

Natural Capital

PointUnstable Fixed

Initial Point

(A) (B)

Figure 1: Two main model structures: (a) attainable steady state, (b) unattainable steadystate and limit cycle.

We wish to explore how changing model parameters effect the qualitative behavior of the

model. Unfortunately, even with simple models, such analysis is not easy. The two behaviors

(A) and (B) are separated by a Hopf bifurcation which occurs when, as a parameter is varied,

the steady state changes from being locally stable to unstable, and a periodic orbit develops

around the steady state. For simple systems, standard analytic methods of dynamical systems

theory can be applied reasonably easily (Kuznetsov, 1995). For more complex models, such

analysis becomes increasingly difficult and may be impractical. The main tool I employ here is a

numerical technique known as pseudo arclength continuation available in the software package

Auto (Doedel, 1981). By combining known analytical results with powerful computational

algorithms, models that are analytically tractable, but for which the analysis is impractical or

cumbersome can be efficiently, thoroughly, and accurately analyzed.

The analysis amounts to starting at a known fixed point of the system and tracking its

stability as a parameter is varied in very small steps. By locating points where the stabil-

ity of the fixed point changes, we can detect local bifurcations and use these to divide the

parameter space as mentioned above. This powerful tool for analyzing dynamical systems is

freely available. Interested readers should visit http://www.iam.ubc.ca/guides/xppaut/ for

more details and download information. For more details on the application of the method for


ecological models see (van Coller, 1997).

Critical points

There are sixteen parameters in the model. Fortunately, many of them merely scale the

variables and thus can be fixed by choice of units. An analysis of the critical points of the model

and their stability helps bound a reasonable range of values for the parameters. The model

exhibits three critical points (see appendix for details): (h, kh, kr) = (0, 0, 0), (h, kh, kr) =

(0, 0, 1), and (h, kh, kr) = (h∗, k∗h, k∗r) such that h∗ > 0, k∗h > 0 and 0 < k∗r < 1. The point

(0, 0, 0) is unstable, meaning that if not exploited, natural capital will increase to its maximum.

The stability of the point (0, 0, 1) depends on parameter choices. Stability of this point means

that the ecological economic system represented by this parameter set is not “viable”. That

is, the resource base is not sufficiently productive to support a human population with a given

technology. If this point is unstable, the system is “viable”. Given a viable parameter set,

it follows that there exists and interior equilibrium point (see appendix for details). Table 1

summarizes a set of viable parameters used in the model analysis.

Parameter Definition Value

Economic parameters

α1 Labor productivity in resource sector variesα2 Labor productivity in manufacturing sector variesαr resource productivity 0.75c1 resource good preference 0.3δ depreciation rate of capital 0.05s savings rate varies

Ei(t) Efficiency factor in sector i varies

Ecological parameters

η effect of resource good production on resource base 0.1nr intrinsic rate of increase of resource base 0.1

Demographic parameters

b0 maximum birth rate 0.1d0 maximum death rate 0.2b1 sensitivity of birth rate to resource good intake 1b2 sensitivity of birth rate to manufactured good intake variesd1 sensitivity of death rate to resource good intake 5d2 sensitivity of death rate to manufactured good intake varies

Table 1: Parameter values used in the model for the numerical bifurcation analysis.

Of interest is the stability of the interior equilibrium point as a function of model parame-


ters. A change in its stability marks the transition from the qualitative behavior in graph (A),

to the qualitative behavior in graph (B) in figure 1. The results of the analysis will be presented

in terms of the boundary between these two regions of qualitatively different behaviors.

The savings rate and the demographic transition

I begin the analysis by exploring the model behavior as a function of s (which should be

interpreted as gross fixed capital expenditure) and b2 which controls the “fertility transition”.

For this baseline case, I assume no technical progress and set Ei = 1 for i = 1, 2. The

parameters are set as shown in table 1 and α1 = 0.7 and α2 = 0.3. Finally, d2 is set to zero

for the moment.

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

savings rate(A)

popu

lati

onde

nsit

y

natural capital(B)

>

>

<

Figure 2: Graph (A) shows the bifurcation diagram for the model with the basic parameterset from table 1 as s is varied. Graph (B) shows the trajectories in phase space that eventuallyconverge to the long run configurations (i.e. stable equilibrium or limit cycle) shown in (A)for a low savings rate (s = 0.09, dashed line) and a high savings rate (s = 0.23, solid line).The horizontal lines show the correspondence between (A) and (B). For s = 0.09, the modelconverges and for s = 0.23, the model overshoots and collapses.

Figure 2 summarizes the long term behavior of the model as a function of s with b2 = d2 =

0. Figure 2 (A) is a bifurcation diagram that plots the long run equilibrium population for the

model for different levels of investment. The solid line for lower values of s indicates that these

long run equilibria are stable. The dashed portion of the curve for higher values s indicates

that the equilibria are unstable and the system will never approach them. Rather, for value

of s above the Hopf bifurcation point near where the curve of heavy circles intersects the line,

the system will approach a stable limit cycle, whose amplitude is shown by the heavy solid

circles emanating from the curve of fixed points. Figure 2 (B) shows the transient behavior

and long run equilibria in phase space corresponding to points in Figure 2 (A) as indicated by

the horizontal lines which connect points in the bifurcation diagram with their counterparts

in phase space.

As is clear from the diagram, increased savings rates destabilize the model. Investment

tends to increase productive capacity which allows the population, in the short term, to over-


exploit resource base (Clark et al., 1979). A high level of investment enables the population

to grow beyond the long run carrying capacity of the resource base. The population must

subsequently (perhaps painfully) adjust downwards. This “overshoot and collapse” like be-

havior is generated by different time scales in the nonlinear model. If the time scale on which

productive capacity grows is faster (slower) than that on which the resource base regenerates

itself, the more (less) prone the system will be to overshoot and collapse. This is highlighted

in figure 3 which shows population and per capita output as functions of time for s = 0.09 (the

approximate savings rate that maximizes the long run sustainable population) and s = 0.23

corresponding to the two cases shown in figure 2.

0

0.2

0.4

0.6

0.8

0 100 200 300 400 5000

0.5

1

1.5

2

2.5

0 50 100 150 200 250 300 350 400 450 500

time (years)(A)

popu

lati

onde

nsit

y

time (years)(B)

per-

capi

taou

tput

Figure 3: Graph (A) shows the population density over time for s = 0.09 (solid line) ands = 0.23 (dashed line). Graph (B) shows the corresponding per capita output of manufactured(solid) and agricultural (dashed) for s = 0.1 (light) and s = 0.23 (bold).

Notice in figure 2 (A) that the time scale for population expansion with s = 0.23 is about

2.5 times faster than with s = 0.09. This is enabled by the higher per-capita manufacturing

output generated by higher investment. In this case per capita agricultural output can’t be

maintained as shown in figure 2 (B) (bold curves). Although peak agricultural output when

s = 0.09 can’t be maintained either, the downward adjustment is very small (light curves in

figure 2 (B)). If s is smaller still, there is no downward adjustment in agricultural output and

the the system approaches equilibrium monotonically. This suggests that, in the case with

constant technology, to avoid overshoot and collapse behavior, the rate of economic growth

should be calibrated to time scales intrinsic to the natural resource system.

The importance of relative time scales in determining model dynamics can be made explicit

by rescaling time. Define τ ≡ nrt. The new variable, τ , measures time relative to the natural

time scale implicit in nr. The model can then be rewritten with respect to this new timescale

using the chain rule, i.e.dh

dτ=dh

dt

dt

dτ=dh

dt

1nr. (21)


Performing this operation on each of the equations in the model yields

dh

dτ= (b(q1, qm)− d(q1, qm))h (22)

dkhdτ

=sY2

1− c1(1− s)− δkh (23)

dkrdτ

= kr(1− kr)− ηY1 (24)

where ˆ indicates division by nr, i.e. s =s

nr. The topological properties of this model are

equivalent to the original model. For a given set of parameters, this model will exhibit a Hopf

bifurcation when s exceeds a certain level. Here the interpretation is more revealing – as s

increases, the relative time scale in the economic system becomes faster than in the natural

system. For the parameter set used in the example, a Hopf bifurcation would occur when

s = snr≈ 0.2

0.1 = 2, i.e. when the economic time scale is roughly twice the natural time scale.

Thus, what is important is not the absolute savings rate, but the savings rate relative to the

intrinsic replacement rate of the resource base.

Of course, the results just presented are strongly influenced by the underlying economic

and demographic structure. Depending on these, for example, society might enjoy more rapid

growth without the overshoot and collapse problem discussed above. Growth paths are gov-

erned by four major drivers: the investment rate, the relationship between the birth and death

rates and the consumption of manufactured goods (measured by the parameters b2 and d2),

the relative productivity of labor versus capital within and across sectors (measured by the

parameters α1 and α2), and technological progress. I will explore each of these drivers in turn.

In the analysis thus far, b2 and d2 have been set to zero. Equilibrium occurs when birth

and death rates are equal, regardless of their magnitude. Thus a state with high birth and

high death rates (less desirable) or low birth and death rates (more desirable) can be stable

and the transition between these states is the crux of the demographic transition. With no

fertility transition, the model is stabilized by resource degradation, i.e. the “positive check” in

the Malthusian model. Figure 4 which shows birth and death rates for the first 200 years that

correspond to the trajectories shown in 3 illustrates this point. The main effect of the rate of

investment is not in determining the long run birth and death rates, but the magnitude of the

difference between them as the resource base is degraded. The heavy lines in figure 4 show the

birth (dashed) and death (solid) for s = 0.23 while the light lines show the same for s = 0.09.

In the first case, large increases in productive capacity cause a large difference in birth and

death rates leading to rapid population growth (around a maximum of 4% per year). This leads

to resource degradation and eventually to very high death rates peaking around year 170 at

8% annually. In the latter case, the difference between birth and death rates is relatively small

resulting in a growth rate of less than 1.7 % per annum. This results in a slower degradation

of the resource base as the birth and death rates approach their equilibrium value of around

3.1%. In both cases, the equilibrium “throughput” is high. A more desirable situation would

be one with low “throughput”, achieved by an increase in b2, i.e. the “preventive check” in


0

0.02

0.04

0.06

0.08

0 50 100 150 200

time (years)

birt

han

dde

ath

rate

s

Figure 4: The solid(dashed) lines show death(birth) rates over time for s = 0.09 (light lines)and s = 0.23 (heavy lines).

the Malthusian model.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

0 50 100 150 200 250 300 350 400 450

0

0.2

0.4

0.6

0.8

0 50 100 150 200 250 300 350 400 450

0

0.2

0.4

0.6

0.8

0 50 100 150 200 250 300 350 400 450 savings rate

b 2

O

C

Figure 5: Depicted in the graph on the right is the separation of s − b2 parameter spaceinto two regions. Points on the curve represent combinations of s and b2 for which a Hopfbifurcation occurs, i.e. a qualitative change in model behavior occurs at these points. Thethree graphs on the right shows human population density as a function of time for parametervalues located by the dotted lines. For all s - b2 parameter combinations in region “C”, themodel converges to long run equilibrium. For all s - b2 parameter combinations in region “O”,the model oscillates indefinitely.


Figure 5 summarizes the effect the fertility transition has on growth paths. The graph

on the right shows the two regions in s − b2 parameter space that exhibit qualitatively dif-

ferent behavior. The curve that separates the two regions is generated by locating the Hopf

bifurcation point for each combination of s − b2. The three graphs on the left show the pop-

ulation trajectories over time corresponding to the points in parameter space to which they

are connected with a dotted line. The cases shown in the top two graphs both eventually

converge to a long run equilibrium as do all of the cases with parameter combinations in the

region marked “C” (convergence). The bottom graph never converges; it oscillates indefinitely

as do all cases for parameter combinations in the shaded region marked “O” (overshoot and

collapse). Notice that even though the model can experience some overshoot and collapse as

it converges for parameter combinations in region “C”, they are much less severe than the

overshoot and collapse for parameter combinations in region “O”. It is interesting to note that

recent archeological work suggests that human societies that have exhibited “O” type behavior

are very common (Tainter, 1988; Redman, 1999).

The interpretation is straightforward: the higher the value of b2 (the strength of the fertility

transition) the larger the savings rate s can be before the model moves into the region exhibiting

overshoot and collapse dynamics. In fact, if b2 > 1 the system does not undergo a Hopf

bifurcation. The reason that the curve separating the two regions bends over as s increases

beyond 0.45 is that with extremely high savings rates the population can’t afford to feed itself

and thus grows very slowly (not a very realistic scenario). Thus increasing b2 allows faster

growth without overshoot and enables the system to reach stage four of the demographic

transition with low birth and death rates. For example, with b2 = 1, and s = 0.2, the

equilibrium birth (and death rate) is approximately 1.1% versus 3.1% with b2 = 0.

The parameter b2 could have many different physical interpretations - i.e., how would two

economic systems with different values of b2 all else being equal differ? I offer one possibility.

Different distributions of income across a particular economy can give rise to the same average

per capita income. The homotheticity of the Cobb-Douglas utility function makes income

distribution irrelevant when computing aggregate demand (demand is a linear function of

income), but the same may not be true of birth rates. Suppose, as with preferences, each

agent has the same response for birth rate to consumption. If this function is nonlinear, then

the aggregate birth rate will depend on income distribution.

For example, if economic development is not even, some individuals might enjoy certain

benefits that reduce mortality without experiencing other aspects of the development process

that might suppress birth rates. In this case the response of the birth rate to consumption levels

would be weak (modeled by a low value of b2). If, on the other hand, economic growth is more

even and income is distributed evenly, birth rates would fall off more quickly as consumption

increased because more individuals in the population would reduce births for the same level of

per capita intake (high b2). Whatever the physical interpretation of b2, the key point is that

it is largely socially determined.


Figure 6 illustrates the effect the other demographic parameter, d2 has on the model. Recall

that d2 measures the sensitivity of death rates to per capita manufactured goods output.

With both b2 and d2 non-zero, economic growth sets up a tug-of-war between the fertility

transition (stage 3 of the demographic transition), and the mortality transition (stage 2 of

the demographic transition). Figure 6 (A) shows how the bifurcation boundary shifts upward

and to the left as d2 is increased from 0 to 1. When d2 is zero, the “O” region is indicated

by dark grey shading. When d2 is 1, the “O” region is enlarged by the area shown with light

grey shading. The larger the effect growth has on reducing death rates the greater must be its

effect on reducing birth rates to avoid overshoot and collapse dynamics. This common-sense

result highlights the double-edged-sword nature of economic development. Finally figure 6 (A)

shows the explicit dependence of the minimum strength of the fertility transition, b2, required

to avoid overshoot and collapse dynamics on the strength of the mortality transition, d2. The

dotted (solid) line shows this relationship for s = 0.15 (s = 0.2). The larger savings rate

increases the size of the overshoot and collapse region from the dark grey shaded area to the

light grey plus the dark grey shaded areas.

0

0.5

1

1.5

2

2.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

O

C

O

C

s

(A)

b 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5 3

d2

b 2

(B)

Figure 6: In graph (A), The dashed line is a portion of the curve shown in figure 5 ford2 = 0. The grey region below the dashed line marked “O” has the same interpretation asin figure 5. The solid curve is the same for d2 = 0. The movement of the curve up andto the left expands the size of the overshoot and collapse region by an amount shown bythe light gray shading. The ligh Graph (B) the tradeoff between b2 and d2 (the influence ofmanufactured good consumption on birth and death rates, respectively). Points on the linesrepresent parameter combinations for which a Hopf bifurcation occurs for s = 0.15 (dot-dashedline) and s = 0.2 (solid line). These lines divide parameter space into regions with qualitativelydifferent behavior.


Economic structure

Thus far, the analysis has summarized the model behavior as a function of demographic

parameters. Another critical factor in the model is economic structure which has to key

features: the overall labor (or capital) intensity and the relative labor (or capital) intensity in

each of the sectors. In the analysis thus far, α1 = 0.3, and α2 = 0.7. From 11 it is easy to see

that the labor fraction in the ith sector decreases with decreasing αi. Increasing α2 relative

to α1 increases the proportion of available labor devoted to manufacturing. The parameter

choice used in the analysis thus far (α1 = 0.3, and α2 = 0.7) is then roughly consistent with a

modern society where relatively less labor is employed in agriculture than in manufacturing.

The question of interest here is the relationship between the fertility transition, i.e. the

value of b2 and the relative and overall labor (or capital) intensity of the economy. Figure 7

summarizes this relationship. The savings rate is set at 0.2 and all other parameters are

as shown in table 1 with Ei both still set at 1, and d2 = 0. The heavy curve shows the

boundary between overshoot and collapse and convergence type dynamics in b2–α1 parameter

space with α2 set at 0.7. The lighter curve shows the same for α2 set at 0.5. Again, for

parameter combinations in region “C” or “O” the model converges or overshoots and collapses,

respectively.

C

O0

1

2

3

4

5

6

0 0.2 0.4 0.6

labor productivity in agriculture, α1

Sens

itiv

ity

ofbi

rth

rate

tom

anuf

actu

red

good

outp

ut,b

2

Figure 7: Division of b2 – α1 parameter space into regions of qualitatively different behavior.In region “C” the model converges. In region “O” the model overshoots and collapses. Theheavy (light) line shows the division between the two regions when α2 = 0.7 (α2 = 0.5).

For a given value of b2, as the labor intensity in agriculture decreases the system moves

from the “C” region to the “O” region. The lower b2, the higher the value of α1 below

which the system will exhibit overshoot and collapse behavior. Thus, the less labor intense

agricultural production is, the stronger must be the effect of increased per capita manufactured

good output on decreasing birth rates. Thus in societies where the bulk of labor is engaged


in manufacturing, the more important social processes governing fertility become if overshoot

and collapse is to be avoided.

The lighter curve shows the what happens when the economy is more capital intense

overall.. In this case, the manufacturing sector is more capital intense than in the first case.

Two points are worth noting. First, when b2 is zero, the labor intensity below which the system

becomes unstable is higher (approximately 0.46 when α2 = 0.5 versus 0.35 when α2 = 0.7).

Second, for larger values of b2, the difference between the boundaries becomes less marked.

This is due to the fact that when the capital intensity increases in the manufacturing sector,

per capita output of of manufactured goods (q2) increases which increases the strength of the

downward pressure on birth rates for a given value of b2.

In summary, the structure of the economy has a strong influence on the dynamics of the

model. The more labor intense the agricultural sector the less likely the model will exhibit

overshoot and collapse behavior. The less labor intense the agricultural sector, the higher the

value of b2 must be to avoid overshoot and collapse. For example when α1 = 0.2, b2 must be

above 1.7 to avoid overshoot and collapse. When α1 = 0.3, b2 need only be above 0.7 to avoid

overshoot and collapse. Thus, the more society devotes labor to manufacturing and relies on

capital to produce food, the more important social processes such as the “fertility transition”

become in avoiding degradation of the natural resource base.

Technological change

The above analysis highlighted the important role that different (static) technological assump-

tions play in the model. As the ecological economic system evolves over time technology will

change. How would dynamically evolving technology effect the results presented thus far?

Unfortunately, this is an extremely difficult question because of the fundamentally speculative

nature of technological change. In the case of endogenous growth theory where technological

change has been a focus, the question is how innovation might cause growth through, for ex-

ample, expanding product variety or rising product quality (Grossman and Helpman, 1991),

investment in human capital (Lucas, 1988), and knowledge spillovers (Romer, 1986). These

models treat knowledge as an aggregate which works to improve aggregate productivity. In

this case, no assumptions are required about in exactly what areas the knowledge and pro-

ductivity improvements are taking place and assuming that these processes are taking place

somewhere most of the time is plausible.

As highlighted in the previous section, in our model it is much more important in which

sector knowledge and productivity improvements come. For example, overall productivity

growth in an economy might be the result of small segment of the economy (e.g. computer

hardware and the U.S. economy in the decade from 1990-2000 (Gordon, 1999)). It is irrelevant

that the productivity gains are exceedingly narrow, they still generate growth. On the other

hand, it is difficult to say how these productivity gains might affect our ability to rehabilitate

soils degraded by salinization. One would have to assume similar gains in productivity in the


biotechnology sector. The point is, when addressing specific problems that have explicit spatial

and temporal components, one must assume the timely arrival of the appropriate knowledge.

Such assumptions are far more speculative than those of the much more general nature made

in aggregate growth models. As such, the meaning of technological change in a model such as

ours is much more dubious.

Technology can affect several things in the model: productivity in the manufacturing and

agricultural sector and the “environmental friendliness” of agricultural production. Includ-

ing exogenous technical change amounts to making E1 and E2 increasing functions of time.

Including technological change that reduces the environmental impact of agriculture can be

modeled by making η a decreasing function of time.

To explore the role of technological progress via increases in E1 and E2 we fix the demo-

graphic parameters and treat Ei as parameters. By varying them we can get an idea of how

the qualitative behavior of the model might change as Ei change over time. These quantities

affect the model dynamics in three basic ways: E1 and E2 affect human population dynamics

by increasing qi, E1 increases the per capita impact on the environment via equation (20),

and E2 increases the rate of growth of the capital stock (and thus the long run capital-labor

ratio). Figure 8 summarizes the effects of change Ei on the model. In order to compare with

previous results, the parameters are set as shown in table 1 with α1 = 0.3, α2 = 0.7, s = 0.1,

and b2 = d2 = 1.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

E1

E2

O

C

Figure 8: Division of E2 − E1 parameter space into convergence (”C”) and overshoot andcollapse (”O”) regions. The arrows indicate possible paths E2 and E1 might take as theyevolve over time.

The interpretation of the shading and labels is the same as with previous figures. The solid

arrows indicate trajectories that Ei(t) might take over time. Notice that for this parameter

combination with Ei both fixed at 1 as previously, the model will converge to a long run

equilibrium and will not overshoot and collapse. As is clear from the figure, technological


advance that increases productivity is destabilizing. The more even the advance of E1(t) and

E2(t) (i.e. the 45o line) the sooner the model becomes unstable. The more uneven the advance

of E1(t) and E2(t), the longer it takes for the model to become unstable.

The destabilization occurs through the increasing impact of agricultural production on the

natural resource base given by

ηY1 = ηE1βα1γ1−α1kαrr hα1k1−α1

h , (25)

directly through E1 and indirectly through E2 increasing kh. To prevent the inevitable desta-

bilization of the model, this impact must eventually stop increasing. This can be achieved

several ways. The most direct way is, of course, for technological progress to cease before E1

and E2 move into region “O”. Another way is through technological progress that causes η(t)

to decrease over time at a rate fast enough compensate for increases in Ei(t). Thus as pro-

duction becomes more efficient, its impact on the environment must decrease over time. The

model highlights the fact that outside a simple one sector growth model the role technological

progress may play in achieving sustainable outcomes is not obvious. A balancing act between

different types of progress is required.

Finally, consider a case where the main effect of technological progress is to reduce the

impact of production on the resource base (i.e. Ei are constant and η(t) is a decreasing

function over time). Such technological progress has no effect on the basic topology of the

model - it does nothing more than to scale the human and human-made capital populations.

To see this make the following change of variables:

h = ηh (26)

kh = ηkh. (27)

The new variables h and kh measure the capital and human populations in units of η, i.e. in

terms of their impact on the environment. Then

Y1(h, kh, kr) = E1βα1γ1−α1kαrr

(h

η

)α1(khη

)1−α1

=Y1(h, kh, kr)

η(28)

Y2(h, kh) = E2(1− β)α2(1− γ)1−α2

(h

η

)α2(khη

)1−α1

=Y2(h, kh)

η. (29)

Next note that sincekhh

=kh

h,

q1(h, kh, kr) = q1(h, kh, kr) (30)

q2(h, kh) = q2(h, kh), (31)

(32)


Using these expressions we can rewrite the model in terms of h, and kh:

˙h = ηh = η(b(q1, qm)− d(q1, qm))

h

η= (b(q1, qm)− d(q1, qm))h (33)

˙kh = ηkh = η

sY2(h,kh)η

1− c1(1− s)− δ kh

η

=sY2(h, kh)

1− c1(1− s)− δkh (34)

kr = nrkr(1− kr)− ηY1(h, kh, kr)

η= nrkr(1− kr)− Y1(h, kh, kr). (35)

Note that equations (33) and (34) are identical to the original model- all that has changed are

the variable names. Equation (35) appears different but, since equation 28 holds, is not. The

qualitative behavior of this model is identical to the original model. the rescaled variables can

be converted back into the original variables using the simple expressions

h(t) =h(t)η

and kh(t) =kh(t)η

. (36)

Notice that the smaller η (the more environmentally friendly production) is the larger h(t)

and kh(t). Thus if the parameter set is in region “O” in figure 5, the model will overshoot

and collapse no matter what η is. The only difference between different values of η will be

the overall scale the economic system attains before it collapses. Thus technological progress

towards cleaner technology can do nothing to prevent overshoot and collapse. Only changes

in social parameters (e.g. increasing b2 or decreasing s) can prevent overshoot and collapse.

Concluding remarks

In this paper I developed and analyzed a simple growth model which includes the dynamic

interaction of an economic system based on representative agents with identical preferences

with a simple demographic model and a renewable resource base exhibiting logistic growth.

We certainly have no hope of representing an actual ecological economic system with a model

made up of such simple ingredients. However, three points concerning the interaction of

demographics and growth that emerge from the model analysis are worth making.

First, somewhat analogous to growth models with depletable resources which suggest that

there is a minimum investment level required to maintain consumption, the model developed

here suggests that there is a maximum allowable investment level set by renewable natural

capital constraints. The overshoot and collapse dynamic is generated by differing time scales

in the economic, demographic, and natural systems. The larger the economic and demographic

time scales are relative to the time scale on which natural capital regenerates itself, the more

prone to overshoot and collapse the system is.

Second, the relationship between model stability and investment rates depends on economic

and social structure as parametrized in the model. The basic result is summarized in figure 6

(A): the stronger the impact of economic growth on reducing fertility (higher b2), the less prone

to overshoot and collapse the model will be. However, economic growth sets up a tug-of-war


between the mortality and fertility transitions. As shown in figure 6 (B), the more effective

growth is in reducing death rates, the stronger it must be in reducing birth rates. Increases in

the productivity of labor tends to destabilize the system as shown in figure 7, increasing the

importance of b1 and the social mechanisms that would tend to increase it.

Third, technological change does not play as important role in avoiding overshoot and

collapse as do demographic processes. Productivity increasing technological change (increasing

Ei(nn) favors overshoot and collapse type dynamics (figure 8). The destabilizing effect of such

chance can be mediated by technological change that reduces the environmental impact of

economic processes (decreasing η). There is a tug-of-war between increases in productivity

and environmental friendliness. Increasing environmental friendliness has no influence on

whether the model overshoots and collapses or attains a long run sustainable equilibrium

(equations (26) through (35). At best, the right combination of technological progress which

reduces the environmental impact of production faster than it increases factor productivity

can buy time in the model system to allow the critical social processes governing the increase

of b2 to progress.

The model is very robust over a wide parameter range and the different behaviors exhibited

share a common thread: the key aspect of those ecological economic systems capable of a

smooth transition from a natural capital to a human-made capital intensive state is not so

strongly related to the nature of technology and productive capacity per worker (as in the

neoclassical models) as to the evolution of sufficient social capital to manage that capacity.

The greater this productive capacity, the more prone to overshoot and collapse the model

becomes, and the more critical becomes the role of social capital in avoiding this possibility.

Put another way, technical innovation gives rise to the need for social and cultural innovation.

Many past human ecological systems that have managed resources well have done it through

ritual or religious representation of resource management (Berkes et al., 1995), i.e. with rich

endowments of social and cultural capital. It is unlikely that smart elders at some point de-

cided to invest heavily in social capital and invent intricate and brilliant resource management

systems which they then codified in ritual. It is more probable that effective resource man-

agement and the social institutions through which it was implemented, such as complex ritual

and religious practices, evolved through selection. The time scale needed for the latter method

to develop resource management strategies makes the more active approach more attractive

to say the least. Unfortunately, because of strong potential profit asymmetries, growth in

social capital may be more difficult to achieve than for human-made capital at a time when

the former may be critical.


1 Appendix

Consumer optimization

To derive the demand functions in (5) from the constrained optimization problem given in (4)

we define the Lagrangian:

L ≡ U(q1, q2)− λ[P1q1 + P2q2 −M(1− s)]. (37)

The first order conditions are

∂L∂q1

= c1qc1−11 q1−c1

2 − λP1 =c1U

q1− λP1 = 0 (38)

∂L∂q2

= (1− c1)qc11 q−c12 − λP2 =

(1− c1)Uq2

− λP2 = 0 (39)

Adding (38) and (39) yields

U

λ= P1q1 + P2q2 = M(1− s) (40)

Solving (38) and (39) for q1 and q2, respectively, and substituting the right hand side of (40)

for Uλ yields (5).

General equilibrium

Assume firms rent capital from individuals. Savings is used to purchase investment goods from

the manufacturing sector which are added to the capital stock. The price of investment goods

is the same as for consumer goods. Perfectly competitive firms solve:

max πi(Li,Ki) = PiYi − wLi − rKi

Li,Ki(41)

where πi(Li,Ki) is profit in sector i = 1, 2. The first order conditions are

∂π

∂Li=αiPiYiLi

− w = 0 (42)

∂π

∂Ki=

(1− αi)PiYiKi

− w = 0 (43)

Adding (42) and (43) for i = 1, 2 yields the supply equations 8 (a) and (b).

Output markets clear when

P1YD

1 = P1YS

1 ⇐⇒ c1hM(1− s) = wL1 + rK1 (44)

P2YD

2 = P2YS

2 ⇐⇒ (1− c1)hM(1− s) + sM = wL2 + rK2 (45)

Since technology is constant returns to scale, there is nothing in the equations so far to de-

termine the scale of economic output. I thus assume full capital utilization as given by the

conditions in (9a) and (9b).


Factor markets clear when labor and capital rental rates, w and r adjust to reflect the

relative scarcity of total available labor, h, and capital, kh. Equations (42), (43), (44), (9a)

and (9b) can be used to calculate K1 as follows: Note that

hM = wh+ rkh (46)

by definition. Given r and w, (42) and (43) imply that firms will choose capital-labor ratios

that satisfyKi

Li=w(1− αi)

rαi. (47)

This combined with (46) allows (44) to be written as

c1(1− s)[rK1

1− α1+r(kh −K1)

1− α2

]=rα1K1

1− α1+ rK1 (48)

which when solved for K1 yields (10c) with γ given by (12). The expression for K2 in (10d)

follows immediately from (9b).

Given this capital allocation, the labor demand in each sector dictated by 47 is

L1 =rγkhα1

w(1− α1)L1 =

r(1− γ)khα2

w(1− α2)(49)

To close the system, choose r as the numeraire and set it equal to one. It is now easy to

see that labor demand is determined by total available capital stock, and wage rates. If total

labor demand is higher (lower) than the total available, h, there will be upward (downward)

pressure on the wage rate. The wage rate will adjust until (9a) is satisfied, factor markets

clear, and the whole system converges to the general equilibrium. Substituting the expressions

for labor demand in (49) into (9a) and solving for the equilibrium wage rate yields

w =khh

[γα1

(1− α1)+

(1− γ)α2

(1− α2)

](50)

Substituting this expression for the wage rate into the labor demand equations in (49) yields

the equilibrium labor allocations given in (10a) and (10b) with β given by (11).

1.1 Investment

Note that as with most models of growth and the environment, total investment is completely

supply side driven. This is due to the fact that there are no adjustment costs in the model. Re-

call from (45) that the total demand for manufactured goods is comprised of total consumption

goods, hq2 and investment goods,shM

P2. At equilibrium,

P2Y2 = h(1− c1)M(1− s) + hsM (51)

which when rearranged giveshM

P2=

Y2

1− c1(1− s). (52)

Thus investment is given by

I =shM

P2=

sY2

1− c1(1− s). (53)


Equilibria and local stability analysis

The model,

h = (b(q1, qm)− d(q1, qm))h (54)

kh =sY2

1− c1(1− s)− δkh (55)

kr = nrkr(1− kr)− ηY1 (56)

has three equilibria:

(h, kh, kr) = (0, 0, 0) (57)

(h, kh, kr) = (0, 0, 1) (58)

(h, kh, kr) = (h∗, k∗h, k∗r), h

∗ > 0, k∗h > 0, k∗r ∈ (0, 1). (59)

The first two equilibria follow immediately from the model equations. We will establish con-

ditions for the existence of the third (non-trivial) equilibrium below.

Standard methods for checking the stability of the first two points are not applicable

because the Jacobian does not exist when h and kh are zero. It is easy to show that the point

(0, 0, 0) is unstable by considering a perturbation of the form (0, 0, ε). For such a perturbation,

both h and kh are zero while kr > 0. Thus the perturbation will tend to grow, and the system

will move away from (0, 0, 0). This implies (0, 0, 0) is unstable.

To check the stability of the second point, consider a perturbation of the form (ε1, ε2, 0).

Define

A ≡[sE2(1− β)α2(1− γ)1−α2

δ(1− c1(1− s))

] 1α2

. (60)

Equation 55 implieskhh≤A⇐⇒ kh ≥ 0. (61)

Thus perturbations such that ε1 >ε2A

will tend to grow in the kh dimension untilkhh→ A.

Similarly, perturbations such that ε1 <ε2A

will tend to decay in the kh dimension untilkhh→ A.

The question of the stability of this equilibrium (and the existence of the third equilibrium)

then hinges on whether this perturbation will grow in the h dimension as well, or will decay

back to zero.

Fix all the parameters in the model except b0. Rewriting equations (13) and (14) in per-

capita terms yields

q1(x, kr) = E1βα1γ1−α1kαrr x1−α1 (62)

q2(x) = E2(1− β)α2(1− γ)1−α1x1−α1 (63)

(64)

where x is the capital labor ratio,khh

. Choose b0 such that

b0 >d0e−q1(A,1)(d1+d2q2(A))

e−b2q2(A)(1− e−b1q1(A,1))(65)


For parameter sets satisfying (65), at the equilibrium capital labor ratio A and with kr = 1,

h > 0 so the population will grow. Increasing the population will tend to decrease x below

A, all else being equal. When this occurs, equation (61) implies that kh will grow. Thus a

perturbation of the form above will tend to grow in both h and kh dimensions away from

(0,0,1) - i.e. (0,0,1) is unstable. If, on the other hand, the inequality in condition (65) is

reversed, at the equilibrium capital labor ratio A and with kr = 1, h < 0 so the population

will decay. As h decreases, x will tend to increase above A, all else being equal. When this

occurs, equation (61) implies that kh will decay. Thus a perturbation of the form above will

tend to decay in both h and kh dimensions back to (0,0,1) - i.e. (0,0,1) is stable.

Condition (65) describes a parameter set for a “viable” system. Parameter combinations

that satisfy (65) yield a model system in which the resource base can support a human and

human-made capital population. Otherwise, the resource base, given technology and prefer-

ences, cannot meet the needs of the human population. Our final task is, given a “viable”

system, to establish the existence and uniqueness of the equilibrium point (h∗, k∗h, k∗r).

If this equilibrium exists,k∗hh∗

= A, and (54) implies that

ξ(k∗r) ≡ b0e−b2q2(A)(1− e−b1q1(A,k∗r ))− d0e−q1(A,k∗r )(d1+d2q2(A)) = 0. (66)

Note that if (65) is satisfied, ξ(1) > 0 and it is easy to check that ξ(0) < 0. The continuity

of ξ and the Intermediate Value Theorem imply that ∃ k∗r ∈ (0, 1) such that ξ(k∗r) = 0. Next

note that

ξ′(k∗r) = b0e−b2q2(A)e−b1q1(A,k∗r )b1

∂q1

∂kr+ d0e

−q1(A,k∗r )(d1+d2q2(A))(d1 + d2q2(A))∂q1

∂kr> 0. (67)

The Mean Value Theorem along with (67) implies that ∃ at most one k∗r such that ξ(k∗r) = 0.

This combined with the existence statement above implies that there exists a unique k∗r such

that ξ(k∗r) = 0. Then

h∗ =k∗rnr(1− k∗r)

ηE1βα1γ1−α1(k∗r)αrA1−α1and (68)

k∗h = Ah∗. (69)

Thus, for parameter sets satisfying (65), we have established the existence and uniqueness

of an equilibrium point (h∗, k∗h, k∗r) with h∗ > 0, k∗h > 0, k∗r ∈ (0, 1). In general, there is no

closed form solution for equation (66). Thus the analysis of the equilibrium point (h∗, k∗h, k∗r)

is carried out numerically as outlined in the text.


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