EXHAUSTIBLE RESOURCES, TECHNOLOGY TRANSITION, AND
ENDOGENOUS FERTILITY IN AN OVERLAPPING-GENERATIONS MODEL
The paper presents a synthesis of the economics of exhaustible resources and that of endogenous fertility
in an overlapping-generations framework. The consumption good is produced from energy – provided by a
backstop or from a stock of fossil fuels – and labor. The paper studies how the backstop is substituted for
the fossil fuel technology and how the economy is sustained by renewable energy in the long run. The
paper also studies the impact that an exhaustible resource has on the population.
Keywords: Exhaustible Resources, Endogenous Fertility, Overlapping Generations.
JEL Classification: J13, Q30
1
1. INTRODUCTION
Malthusian stagnation – an explosive population and increasing scarcity of natural
resources – has been the motivation behind numerous simulation studies that culminated
in the publication by The Club of Rome a popular book entitled “The Limits to Growth,”
authored by Meadows et. al. (1972), a team of leading MIT engineers. At first, academic
economists objected to the trend projections used to simulate the future, arguing that
human beings act rationally in the face of resource scarcity, and will solve the
overpopulation problem by making the number of offspring part of the economic decision
and the resource scarcity problem by finding a substitute. Not long after the publication
of this unpleasant prophecy, the oil crisis in the beginning of the 70’s rang the alarm bell,
calling for further inquiries on a real problem that cannot be dismissed cavalierly.
While the role of exhaustible resources in economic growth has almost been thoroughly
explored (see the well-known Review of Economic Studies 1974 Symposium), the
question of endogenous population and fertility decision has only attracted attention –
once again – more recently (see Becker and Barro (1988), Barro and Becker (1989),
Becker, Murphy, and Tamura (1990), Galor and Weil (1999, 2000), among others). To
our knowledge, a synthesis of these two strands of literature has not been undertaken,
despite the fact that such a synthesis certainly contributes to our understanding of the real
world. This is the task that we propose to accomplish in this paper.
In the first strand of literature, most of economic models considered are of the RamseyKoopmans type in which the size of the population – assumed to be exogenously given –
has seldom been a concern. The basic questions addressed by this strand of the literature
concern the pattern of resource depletion over time and the implications of resource
exhaustibility in the context of economic growth. Some fine analyses of these questions
can be found in Dasgupta and Heal (1974, 1979). Along the equilibrium path, the
resource price rises at the rate of return for holding assets – the so-called Hotelling rule –
and this would warrant allocation efficiency, with the resource being depleted
asymptotically. As to the question of whether economic growth is sustainable in the face
of increasing resource scarcity, Stiglitz (1974) introduced exogenous neutral technical
progress, and found that growth is possible with a stationary depletion policy determined
2
by the saving rate given in the economy, even if the resource input is essential, For the
case of Cobb-Douglas production in which the resource factor is essential, but not
important, a non-degenerate steady state is also possible. In general, without exogenous
technical progress, and when the resource is essential in the production process, the
economy would sink in the long run to the trivial steady state in which both the resource
stock and the capital stock vanish. This unpleasant outcome could only be avoided when
the exhaustible resource can be substituted for by a reproducible capital. The existence
and characterization of the optimal solution have also been fully analyzed – in the setting
of an optimal growth model – by Mitra (1980) and more recently by Cass and Mitra
(1991) for a wide range of technological possibilities, allowing for unbounded
consumption in the long run; see Krautkraemer (1998) for a comprehensive survey of this
strand of the literature.
With respect to the second strand of the literature on economic growth in which the
size of the population is endogenous, only reproducible capital has been considered as a
factor of production besides labor. In the class of models that follow the Ramsey-Solow
tradition in which all economic decisions are conferred to a single infinitely lived agent (a
central planner, or the head of a dynasty), the population size tends to a stationary level;
see, for example Razin and U Ben-Zion (1975) or Nerlove, Razin, and Sadka (1986).
When capital is human – as in Barro and Becker, op. cit. – rather than physical, the
appropriate model is of the Uzawa-Lucas variety; see Lucas (1988). Using also the
dynastic-utility formulation, these authors showed that the economy exhibits exponential
growth at a rate equal to a positive endogenous fertility rate. Works in this direction have
been carefully surveyed in Tamura (2000). In the overlapping-generations framework,
Samuelson (1975) investigated the optimal size of the population in the long run, and
pointed out that the incentive to maintain an increasing fertility rate would ultimately lead
to an inefficient allocation outcome. Erhlich and Lui (1991) further discussed this
question, and a concise literature survey was provided in Erhlich and Lui (1997).
Surprisingly, there are not many studies bringing together natural resources and
population in a comprehensive synthetic model of economic growth. Exceptions are
Nerlove, Razin, and Sadka (1987) and Eckstein, Stern, and Wolpin (1988). These authors
focused on indestructible land as a production factor. Nerlove et al. relied upon the
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dynastic-utility approach to show the efficiency of the market outcome with endogenous
population, while Eckstein et al. used the overlapping-generations framework, and
demonstrated that as long as the fertility decision is taken into account, the population
growth will not be excessive; the market outcome will be efficient; and the economy will
reach a stationary equilibrium in which the long run consumption is above the Malthusian
subsistence level. The value of land depends on the time path of land per capita, and since
land is fixed in quantity, the problem of over-accumulation of capital is simply ruled out.
To our knowledge, Nerlove (1993) is the only study that pieces together the use of a
renewable resource and the fertility decision. Studies linking exhaustible resources with
fertility decisions are simply non-existent; on this point, see Nerlove and Rault (1997),
and Robinson and Srinivasan (1997).
In this paper, we make use of the overlapping-generations model à la AllaisSamuelson-Diamond tradition to formulate and analyze the relationships among
exhaustible resources, technology transition,1 and endogenous fertility. The reason why
we formulate our problem in the context of an overlapping-generations model is that an
overlapping-generations model is the appropriate framework to analyze fertility decision.
In our model, children are treated as a consumption good, and the number of offspring in
each period is the result of the lifetime utility maximization of the young generation of
that period. Our formulation thus stands in contrast with the dynastic formulations of
Becker and Barro (1988) and Barro and Becker (1989) in which the utility of a parent
depends on her own consumption, the number of offspring she raises, and the total
utilities of all her offspring, and in which the decisions on fertility are made at the
beginning of time by the head of the dynasty. Unlike the dynastic-utility formulation,
which assumes perfect foresight on the part of the head of the dynasty and to whom the
task of inter-temporal resource planning is assigned, the overlapping-generations model
provides a decentralized setting. Our model is thus somewhat more market oriented than
the central-planning view; the latter requires perfect foresight on the part of the head of
the dynasty and a binding contract across generations. In addition to the fact that under
1
The transition from the fossil fuel technology to a backstop has been analyzed under the partial
equilibrium framework by Dasgupta and Stiglitz (1981), Hung and Quyen (1993, 1994), among others.
Tahvonen and Salo (2001) examined the substitution between fossil fuels and renewable energy in the
Ramsey model, but the renewable energy is not the backstop technology we consider in this paper.
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the overlapping-generations framework the society is composed of mortal individuals
who can trade through their lifetime – and with resource assets which may act as stores of
values for saving purpose – we think that our modeling strategy is more appropriate. At
least, it does not go against the strong empirical evidence that does not support the
hypothesis that the members of extended families are altruistically linked in the way
presumed by the dynastic type model; see, for example Atonji et al. (1992).
Three classes of economic agents co-exist in each period: a young generation, an old
generation, and competitive firms producing a consumption good. The consumption good
is produced from two inputs – a composite energy input and labor – according to a
standard neo-classical production function. The composite energy input is generated by
combining two sources of energy – energy from fossil fuels and renewable energy.
Renewable energy is produced by a backstop from capital, say solar collectors. Although
energy is an essential input in the production of the consumption good, neither the
backstop nor fossil fuels is essential in generating the composite energy input. The
composite energy input used in the production of the consumption good can be generated
by (i) burning only fossil fuels, (ii) using only the backstop, or (iii) combining fossil fuels
and renewable energy.
The consumption good can also be used as investment goods to augment the stock of
backstop capital. While oil can be extracted at negligible cost, its ultimate stock is
limited. The backstop, on the other hand, can provide a perpetual flow of energy.
However, energy produced by the backstop requires capital, and as the stock of fossil
fuels dwindles, it is imperative that backstop capital be accumulated to avoid a drastic cut
in consumption. Thus the transition from oil to backstop becomes an interesting question.
In our model, economic agents interact on four markets – oil, capital, labor, and the
consumption good. The two real assets in the economy – capital and oil – represent the
only possible forms of saving, and in any period they are owned by the old generation of
that period. An individual works when she is young, and retires when she is old. A young
individual has one unit of time that she supplies in-elastically on the labor market. She
has to allocate her wages among current consumption, raising children, and saving for
old-age consumption. The lifetime utility of a young individual depends on her current
consumption, her old-age consumption, and the number of offspring she raises, and these
5
variables are assumed to be separable in the lifetime utility function. Furthermore, future
utilities of consumption are discounted, and the single-period sub-utility function of
consumption is assumed to be strictly concave, strictly increasing, and satisfy the familiar
Inada conditions. As for the sub-utility function of offspring, it is assumed to be strictly
concave and strictly increasing as the number of offspring rises from 0 to a saturation
level. However, unlike the single-period sub-utility function of consumption, the
marginal utility of offspring is assumed to be finite when the number of offspring is 0. It
is this assumption that gives our model a Malthusian flavor: when wages are low, the
birthrate will be low.
The introduction of endogenous fertility into an overlapping-generations model of
exhaustible resources and economic growth changes the nature of the problem of
resource depletion in two fundamental ways. First, it is not the resource scarcity in
absolute terms, but the resource scarcity per capita that matters. Although the absolute
size of the remaining resource stock necessarily declines through time due to extraction,
the resource endowment per worker might rise or fall through time, depending on the
birthrates chosen by the successive young generations, and this leads to a more general
Hotelling rule which is related to the birthrate (the biological interest rate). Second, the
temporary equilibria can no longer be computed recursively, and the existence of a
competitive equilibrium becomes problematic, especially because the market size
becomes endogenous due to the fertility decision of economic agents. Furthermore, a
birthrate that is close to 0 in one period leads to an abundance of resource or capital
endowment per worker in the next period, and this means that many limiting arguments
must be deployed to prevent the population from collapsing in finite time.
The mechanism that operates in our model can be described as follows. In any period, if
the resource scarcity per worker – the endowments of oil and capital per worker – are
high, the energy input per worker will be high, leading to a high wage rate. The high
wage rate in turn induces a high level of current consumption, a high level of future
consumption, and more importantly, a number of offspring close to the saturation level –
assuming that the cost in terms of real resources of raising a child is constant. The
endowments per worker in the next period, although still high, will be lower than that of
the current period. On the other hand, when the resources per worker are scarce, wages
6
will be low, with the ensuing consequences that current consumption, future
consumption, and fertility will all be low. In particular, if wages are extremely low, most
of the wages will be devoted to consumption, and fertility will be extremely low. This
last result follows from the assumption that the Inada conditions are imposed on the
single-period sub-utility function of consumption, but not on the sub-utility function of
offspring. Thus when the wage rate descends to a critical level, fertility will decline to 0,
leading to an abundance of resources per worker in the next period. An abundance of
resources per worker in the next period, as already argued, leads to a high fertility rate in
that period, and allows the population to bounce back.
The findings of our paper can be described as follows. When fossil fuels are abundant
at the beginning, a competitive equilibrium in our model consists of three phases: the preinnovation phase, the transition phase, and the post-innovation phase.
During the pre-innovation phase, the energy inputs used in the production of the
consumption good come solely from oil. Initially, the price of oil is low and the birthrates
are high. Along the equilibrium path, the oil endowment per worker falls rapidly due to
the high birthrates. The price of oil rises steadily as fossil fuels become less and less
abundant, and the rising oil prices provide the signal for the competitive firms to begin
the substitution of renewable energy for fossil fuels.
If the population is stable or growing in the long run, oil alone cannot sustain the
economy indefinitely, and the backstop must be brought into use at some time to meet
part of the energy needs of the economy. The time interval that encompasses the
introduction of the backstop and the end of extraction activities constitutes the transition
phase during which the technology substitution – backstop for fossil fuels – takes place.
During this phase, the price of oil continues to rise and the birthrate continues to fall. The
rising oil prices induce the competitive firms to substitute renewable energy for fossil
fuels. The transition phase ends when the price of oil relative to the price of renewable
energy reaches its choke price level.
The post-innovation phase begins after all extraction activities have been terminated,
either due to oil exhaustion or because the competitive equilibrium in question involves
incomplete oil depletion. We demonstrate that the economy might have multiple
7
equilibria – one under which the stock of fossil fuels is completely depleted and one
under which depletion is incomplete.
Under an equilibrium with complete depletion, the exhaustible only has a level, but not
a growth effect on the population. A large stock of fossil fuels induces high birthrates at
the beginning. However, after the resource stock has been depleted, the evolution of the
population is completely determined by the capital labor ratio. In the long run, the
resource stock has only an impact on the absolute size of the population, not on its
growth rate. The behavior of the economy during the post-innovation phase exhibits
complex dynamics. Depending on the functional forms and values of the parameters that
characterize preferences and technologies, the convergence to the steady might be
monotone or in damped oscillation. There might even be cycles. The study of this
complex dynamics lies outside the scope of our work for the time.
Incomplete oil depletion merits some elaboration. Because the stock of an exhaustible
resource necessarily declines with use over time, it is natural to presume that it is
exhausted – in finite time or asymptotically. In conventional models of resource
extraction – à la Hotelling or in the tradition of optimal growth – the resource stock is
always completely depleted. In the model of optimal growth, the pattern of resource
extraction is determined by a central planner, and leaving part of the resource stock in
situ unexploited is clearly not optimal. The Hotelling model of resource extraction is a
partial equilibrium model, and the perfectly competitive equilibrium is socially optimal,
which means that the resource stock is also completely depleted. Furthermore, resource
exhaustion will occur in finite time if its choke price is finite, and asymptotically if its
choke price is infinite. However, in an overlapping-generations model, there is no a priori
reason to believe that the resource stock will be completely depleted – either in finite
time or in infinite time – when the resource is not essential and when consumers choose
their own number of offspring as well as the assets in which to invest. It is thus
conceivable that successive young generations might put part of their savings into oil, and
thus part of the resource stock might be left in situ unexploited as a store of value to be
transferred from one generation to the next.
In order for incomplete depletion to occur, two conditions must be met at all times.
First, the price of oil relative to the price of renewable energy after all extraction
8
activities have been terminated must be greater than or equal to the choke price level to
discourage the competitive firms from generating part of the energy inputs used in the
production of the consumption good by burning fossil fuels. Second, to induce the
successive young generations to continue buying the part of the resource stock left in situ,
the price of oil must obey Hotelling rule and rise at the rate of return to capital
investment. The rise in time of the resource price implies a rise in the value – expressed
in the numéraire – of the part of the resource stock left in situ, and this can only occur if
the population is rising so that more and more young individuals each puts more or less
the same amount of savings (in terms of the numéraire) into oil.
We demonstrate that in the long run under a competitive equilibrium with incomplete
depletion, the capital labor ratio, the birthrate, and the value of the oil investment – in
terms of the numéraire, not in physical units – of a young individual are all unique.
Furthermore, the population must be either stable or growing in the long run. In the long
run, if the population is stable, then the price of oil will also be stable. On the other hand,
if the steady-state birthrate exceeds unity, then in steady state the resource price will rise
geometrically through time at a rate equal to the rate of return on capital investment, and
in steady state this rate is equal to the birthrate. This surprising feature – an oil bubble so
to speak – is first encountered here. The oil bubble is driven and sustained by population
growth.
A competitive equilibrium with incomplete oil exhaustion is obviously not efficient.
Although oil has an intrinsic value as an input in the production of the consumption good,
the part of the oil stock left in situ unexploited serves as a store of value, a means through
which successive young generations transfer their incomes made during their working
days to days of retirement. In this manner, the part of the oil stock left in situ unexploited
serves a basic function of money. In contrast with paper money, which has no intrinsic
value, and might have a zero price in equilibrium, oil left under the ground unexploited
always has a positive value, which, according to our version of Hotelling rule in general
equilibrium setting, must appreciate at the rate of return to capital investment, which, in
steady state, is equal to the birthrate.
We demonstrate that fhe model might have multiple equilibria. That is, from the same
initial conditions the economy might evolve along two distinct paths of economic
9
development – one under which the resource stock is completely depleted and one under
which depletion is incomplete. In steady state, the path with incomplete depletion has a
lower capital labor ratio and a lower birthrate than the path with complete information.
Thus unlike the path with complete depletion, which has a level, but not a growth effect
on the population, the path with incomplete depletion has a negative growth effect on the
population in the long run.
The paper is organized as follows. In Section 2, the overlapping-generations model is
presented. Section 3 contains a discussion of the competitive equilibrium for an economy
with reproducible capital. The statement of the existence of a competitive equilibrium for
an economy endowed with a stock of fossil fuels and some properties of such an
equilibrium are given in Section 4. In Section 5, we discuss the possibilities of complete
and incomplete depletion. In Section 6, we bring together all the disparate elements into a
synthetic characterization of the competitive equilibria that emerge from our model.
Section 7 contains some concluding remarks. The paper has a technical addendum in
which the proof of the existence of a competitive equilibrium for an economy endowed
with a stock of fossil fuels and the proof of the existence of a confining interval for the
price of energy (Lemma 1) are given.
2. THE MODEL
2.1. The Technologies
The perfectly competitive firms produce a consumption good from energy and labor
according to a standard neoclassical production function, say Y  F ( E , L) , where
Y denotes the output; E a composite energy input; and L the labor input. In what
follows, we let e  E / L denote the composite energy input per worker, and
f (e)  F (e,1) denote the output of the consumption good produced by a worker.
In our economy, the composite energy input – or simply energy input – is produced by
combining two different sources of energy: energy generated by burning fossil fuels and
renewable energy, produced by a backstop technology. While oil can be extracted at
negligible cost, its ultimate stock is limited. The backstop, on the other hand, can provide
a perpetual flow of energy. However, the production of renewable energy requires
investments in backstop capital. Our model allows for the possibility of substitution
10
between energy generated by burning fossil fuels and renewable energy. More
specifically, we assume that in each period the composite energy input E used in the
production of the consumption good depends on both the stock of backstop capital K
and the amount of oil extracted Q , and write the output of the composite energy input
as E  E (Q, K ) . We assume that E (Q, K ) is defined for all Q  0, K  0, and has the
following properties. First, it is linear homogenous, and it is continuously differentiable
in the region Q  0, K  0. Second, E (0,0)  0; E (Q ,0)  0 if Q  0; and E (0, K )  0 if
K  0. Third, the marginal product of each source of energy in the production of the
composite energy input is (i) positive, but diminishing, (ii) non-decreasing when more of
the other source of energy is used; and (iii) bounded above. Fourth, the marginal rate of
technical substitution – renewable energy for fossil fuels – along an energy isoquant is
greater than or equal to 1. The following figure depicts an isoquant of the composite
energy production function. Observe that a composite energy isoquant meets both axes.
Furthermore,   , is the marginal rate of technical substitution when only fossil fuels is
used is greater than or equal to 1, and   , the marginal rate of technical substitution when
only the backstop is used, is finite.
K
slope
E Q, K
constant
slope
O
Q
An isoquant of the composite energy production technology
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Using the assumption that E is linear homogenous, we can write E (0, K )  KE (0,1) .
Without any loss in generality, we shall set E (0,1)  1 , i.e., the composite energy
produced from one unit of backstop capital is equal to 1. Our formulation of the
generation of the composite energy input thus allows for the possible substitution of one
source of energy for another, and that oil is not an essential input in the generation of the
composite energy input.2
If Qt is the amount of oil extracted in period t , and K t is the stock of backstop capital,
also in that period, then the total composite energy produced in period t is E (Qt , Kt ).
Furthermore, if Lt is the labor input used in period t , then the output of the consumption
good in that period is Yt  F ( E (Qt , K t ), Lt ) . We assume that the consumption good can
also be used as investment goods to augment the stock of backstop capital. As time goes
on, and the stock of fossil fuels dwindles, it is imperative that investments in the backstop
be made to prevent a drastic reduction in consumption. The accumulation of backstop
capital only influences the output of the consumption good indirectly through the amount
of renewable energy delivered by the backstop to the economy. In what follows, we shall
assume that backstop capital depreciates at rate  , 0    1 . Because there is only one
kind of capital, namely backstop capital, in the model, we shall also refer to backstop
capital simply as capital and its rental rate as the price of renewable energy.
2.2. Economic Agents
In the economy, three classes of economic agents coexist in each period: a young
generation, an old generation, and competitive firms producing the consumption good.
These economic agents interact on four markets – oil, capital, labor, and the consumption
good. The competitive firms buy oil on the oil market and rent capital on the capital
market, and use these factors to generate the composite energy input, which is then
combined with labor to produce the consumption good. An individual works when she is
2
If oil is an essential input in the production of the composite energy input, then it must be used in each
period to prevent the composite energy input and a fortiori the output of the consumption good from falling
to 0. Furthermore, because the stock of fossil fuels necessarily declines with use over time, it will approach
0 asymptotically in the long run. This scenario has been analyzed exhaustively in the literature, and is
neither interesting nor sustainable.
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young. She allocates her wages among current consumption, raising children, and saving
for her old-age consumption. The two real assets in the economy are oil and capital,
which represent the only possible forms of saving.
At the beginning of each period t , t  0,1,..., the state of the economy is represented by
a list ( X t , K t , N t0 , N t1 ) , where X t , Kt , N t0 , and N t1 represent, respectively, the remaining
oil stock, the stock of capital, the number of young individuals, and the number of old
individuals. The initial state of the economy, i.e., ( X 0 , K 0 , N 00 , N 01 ), is assumed to be
known. The consumption good in each period is taken as the numéraire, and for each
t  0,1,..., let t ,  t , and  t denote, respectively, the price of oil, the rental rate of capital,
and the wage rate – all in period t.
2.2.1. Producers of the Consumption Good
In each period t , the competitive firms in the consumption good sector buy oil and rent
backstop capital to generate the composite energy input needed, and then combine the
composite energy input generated with labor hired on the labor to produce the
consumption good. The competitive firms solve the following profit maximization
problem:
max (Q ,K ,L,Y ) [Y  tQ   t K  t L]
subject to the technological constraint Y  F ( E (Q, K ), L), where Q , K , L , and Y
represent, respectively, the oil input, the capital input, the labor input, and the output of
the consumption good. Let (Qt , Kt , Lt , Yt ) be a solution of the preceding profit
maximization problem. Note that (i) Qt  0, Kt  0 if t /  t    , (ii) Qt  0, Kt  0 if
   t /  t    . When    t /  t , we have Qt  0, Kt  0.
A variable that plays a fundamental role in out analysis is the cost of producing one unit
of composite energy that we also refer to as the price of energy. If t and  t are,
respectively, the price of oil and the rental rate of capital that prevail in period t , then the
price of energy in that period is given by
p(t ,  t )  min ( Q , K ) t Q   t K subject to E (Q , K )  1 .
13
The first-order condition that characterizes the optimal level of energy input per worker
in that period is then given by f ' (e)  p(t ,  t )  0 .
2.2.2. The Old Generation
The real assets in each period are owned by the old generation of that period. An old
individual in period t owns X t / N t1 units of oil and K t / N t1 units of capital. Her
consumption is thus given by ct1  [t X t  (1     t ) K t ] / N t1 .
2.2.3. The Young Generation
A young individual owns nothing except for 1 unit of labor that she supplies inelastically on the labor market. She has to allocate her labor income among current
consumption, raising children, and saving for old-age consumption. A lifetime plan for a


young individual of period t is a list ct0 , ct11 , bt , xt 1 , kt 1 , where ct0 , ct11 , bt , xt 1 , and
kt 1 denote, respectively, her current consumption, her old-age consumption, the number
of offspring she raises – at the constant cost h in terms of real resources per child – the
amount of oil she buys as investment, and the amount of capital she buys – also for
investment purposes. The lifetime utility associated with such a lifetime plan is assumed
to be given by
(1)
u(ct0 )  u(ct11 )  v(bt ),
where u (c ) is the single-period sub-utility function associated with consumption, and
v (b ) is the sub-utility function of offspring. Also,  , 0    1, is a parameter
representing the factor she uses to discount future utility. It should be emphasized that for
parents children have an intrinsic value, and the number of offspring is here considered as
a consumption good from their viewpoint.
We shall suppose that the sub-utility function of consumption u (c ) is defined for all
c  0.
It is continuously differentiable, strictly concave, and strictly increasing.
Furthermore, it satisfies the following Inada conditions: imc 0u ' (c)   and
imc u' (c)  0. As for the sub-utility function of offspring, v (b ) is defined for all
b  0. It is continuously differentiable and concave. Furthermore, there exists a saturation
number of offspring b max  1 such that v ' (b)  0, 0  b  bmax , and v' (b)  0, b  bmax .
14
The problem of a young individual in period t is to find a feasible lifetime plan that
maximizes (1) subject to the following inter-temporal budget constraint
(2)
ct0  ct11 / rt 1  hbt  t  0,
where we have let
(3)
rt 1  max t 1 / t , 1    t 1
denote the interest rates she earns on her savings. The budget constraint (2) together with
the following first-order conditions characterize the optimal lifetime plan of a young
individual of period t.
(4)
u' ( ct0 )    0,
(5)
u' (ct11 )   / rt 1  0,
(6)
v' (bt )  h  0,
with equality holding in (6) if bt  0. Note that in these first-order conditions  is the
multiplier associated with the budget constraint (2).
The saving of the individual is st  t  ct0  hbt , and the division of saving between oil
and capital depends on their relative rates of return. If the rate of return on oil investment
is higher than that of capital investment, then all the savings of the individual are put into
oil, and we have xt 1  st / t , kt 1  0. On the other hand, if the rate oil return to oil
investment is lower than that of capital investments, then all the savings will be put into
capital, and we have xt 1  0 , kt 1  st . When t 1 / t  1    t 1, the individual is
indifferent between oil and capital. In this case, kt 1 can assume any value between 0 and
st , with xt 1  ( st  kt 1 ) / t . When it is necessary to make explicit the dependence of the
optimal lifetime plan on labor income and the rate of return on savings, we shall write
ct0  c 0 (t , rt 1 ), ct11  c1 (t , rt 1 ), bt  b(t , rt 1 ), st  s(t , rt 1 ).
To obtain sharper results, we shall assume that current consumption, old-age
consumption, and offspring are gross substitutes. It follows from this assumption that the
current consumption of a young individual and the number of offspring she raises will
decline when the discounted price of future consumption declines. Hence saving is an
increasing function of the rate of interest. On the other hand, for a young individual,
15
current consumption, old-age consumption, and offspring are all normal goods. Thus we
expect ct0 , ct11 , and bt to rise with t . Because ct11  rt 1 st , saving also rises with t .
When the birthrate is positive, it follows from (4), (5), and (6) that
u' (ct0 )  rt 1u' (ct11 )  v ' (bt ) / h  v' (0) / h.
Thus, when bt  0, the labor income of a young individual of period t must satisfy the
following condition:
t  ct0  ct11 / rt 1  hbt  [u' ]1 (v' (0) / h) 


1
u'1  v' (0) .
rt 1
 hrt 1 
Now let
 ( rt 1 )  [u' ]1 (v' (0) / h) 


1
u'1  v' (0) .
rt 1
 hrt 1 
As defined,  ( rt 1 ) represents the critical wage rate at or below which an individual of
period t will choose not to raise any children when rt 1 is the rate of return she earns on
her savings, Now if  ( rt 1 ) is the wage rate that prevails in period t , then using the
assumption that current consumption, old-age consumption, and children are gross
substitutes, we can assert that a young individual of period t still chooses not to raise any
children if rt 1 rises. Hence  ( rt 1 ) is strictly increasing in rt 1.
In any period t , a young individual can earn at least the rate of return
1    t 1  1   by investing her savings in capital. Hence the theoretically lowest rate
of return on savings is 1   , and the theoretically lowest wage rate at or below which a
young individual will not choose to raise children no matter what the rate of return she
earns on her saving is
(7)
 min ( )  [u' ]1 (v' (0) / h) 
1
u'1  v' (0) .
1
 h (1   ) 
Let emin ( ) denote the value of e defined implicitly by  min ( )  f (e)  ef ' (e). As
defined, emin ( ) is the critical energy input per worker at or below which a worker will
choose not to raise children. The price of energy that is associated with the critical energy
16
input per worker is then given by p max ( )  f ' (emin ( )). In what follows, we refer to
p max ( ) as the critical price of energy.
When the labor income descends to the critical level  min ( ) , the birthrate approaches
0, but the saving for old-age consumption – although low – is still bounded below and
away from 0, allowing for a high saving per offspring s(  t , rt 1 ) /b(  t , rt 1 ). The high
saving per child means a high energy endowment per worker in the next period, with the
ensuing consequence of a high wage rate in that period. A high wage rate in the next
period in turn induces a high birthrate in that period, and this gives the population a
chance to bounce back. The saving offspring ratio also tends to infinity when labor
income tends to infinity. The reason is that the birthrate, although rises with income,
remains bounded above by the saturation level bmax , while saving increases without
bound. Thus when the wage rate is high, the cost of raising children becomes a negligible
fraction of labor income, and most of the labor income is spent on current consumption
and on investments to provide for old-age consumption.
2.3. Definition of Competitive Equilibrium
Let P  (t ,  t , t )t0 be a price system. An allocation induced by P is a list of infinite
sequences
A  c01 , (ct0 , ct11 , bt , xt 1 , kt 1 )t0 , (Qt , Kt , Lt , Yt )t0 , ( X t , Kt , N t0 , N t1 ) 
with the following properties: Under the price system P ,
(i) c01  [0 X 0  (1    0 ) K 0 ] / N 01 ;
(ii) (ct0 , ct11 , bt , xt 1 , kt 1 ) is the optimal lifetime plan for a young individual of period t ;
(iii) (Qt , Kt , Lt , Yt ) is an optimal production plan of the competitive firms in the
consumption good sector in period t ;
(iv) N t11  N t0 ;
(v) N t01  N t0bt ;
(vi) X t 1  N t0 xt 1.
The pair P, A  is said to constitute a competitive equilibrium if the following marketclearing conditions are satisfied for each t  0,1,...,
17
(vii) X t 1  Qt  X t ,
(viii) K t  N t01kt ,
(ix) Lt  N t0 ,
(x) N t1ct1  N t0 (ct0  hbt  kt 1 )  Yt  (1   ) K t .
3. COMPETITIVE EQUILIBRIUM FOR AN ECONOMY WITH REPRODUCIBLE
CAPITAL
In this section, we consider an economy that has no oil or that has exhausted its stock of
fossil fuels. For such an economy, renewable energy is the only source of energy used in
the production of the consumption good. This corresponds to the post-innovation phase
whose beginning, for expositional purpose, is taken to be period 0. Thus the economy


begins in state K 0 , N 00 , N 01 in period 0,
with K0  0, N 00  0, and N 01  0. The total
energy input produced by the backstop in period 0 is E (0, K0 )  K0 , and the energy input
per worker in that period is thus e0   0, , where we have let  0  K0 / N 00 denote the
capital labor ratio. To prevent the population from becoming extinct at the end of period
1, we shall assume that  0  e min ( ).
Because one unit of capital produces one unit of energy, the price of energy is equal to
the rental rate of capital. Thus, in period 0, the equilibrium rental rate of capital and the
equilibrium wage rate are given by  0  f ' ( 0 ) and 0  f ( 0 )   0 f ' ( 0 ), respectively.
The consumption of an old individual in period 0 is c01  (1     0 ) K 0 / N 01 . As for a
young individual of period 0, if 1 is the rental rate of capital that she expects to prevail
in period 1, then her current consumption, her old-age consumption, the number of
offspring she raises, and her saving under the form of capital are given, respectively, by
c 0 (0 , 1    1 ), c1 (0 , 1    1 ), b(0 , 1    1 ), and k (0 , 1    1 ). The capital
labor ratio in period 1 that is generated by this optimal lifetime plan is
 (0 , 1    1 )  k (0 , 1    1 ) / b(0 , 1    1 ) ,
and the rental rate of capital in period 1 that is generated by this capital labor ratio is
f '  (0 ,1    1 ) .
18
The curve 1  f '  (0 ,1    1 )  is continuous. Because a young individual must
save for her old-age consumption, and because her labor income  0 exceeds the critical
level  min ( ), , the number of offspring she raises is positive even if the rental rate of
capital in period 1 is equal to 0. Hence the capital labor ratio that is generated by her
utility-maximizing
behavior
must
be
positive,
but
finite,
and
this
means
f '  (0 ,1    1 )   0. Now as 1 rises from 0 to p max ( ), the capital labor ratio
generated by her utility-maximizing behavior also rises because of the assumption that
current consumption, old-age consumption, and offspring are gross substitutes. Hence the
curve 1  f '  (0 ,1    1 )  is downward sloping.
Now as 1 rises steadily, if b(0 ,1    1 )  0 for some value 1  ~1, then
 (0 ,1    1 )   when 1  ~1 , which means that f '  (0 ,1    1 )   0 as
1  ~1 , and the curve 1  f '  (0 ,1    1 ) must have crossed the 45-degree line
before 1 reaches ~1. On the other hand, if b(0 , 1 )  0 for all 1  0, then the curve
1  f '  (0 ,1    1 ) must also cross the 45-degree line at a single point. In either
case, the curve 1  f '  (0 ,1    1 )  crosses the 45-degree line at a single point,
which represents the equilibrium rental rate of capital in period 1. Note that at this
equilibrium the birthrate is positive. The equilibrium rental rate of capital in period 1
satisfies the following condition:
(8)
f '  (0 , 1    1 )   1.
The condition (8) defines implicitly the equilibrium rental rate of capital in period 1,
given 0 , the wage rate that prevails in period 0. Because there is a one-to-one
relationship between the wage rate and the marginal product of energy, the equilibrium
rental rate of capital in period 1 can be expressed as a function of the rental rate of capital
that prevails in period 0, and we shall represent this relationship as 1  ( 0 ). In general,
 : t 1  ( t ), t  0, expresses the transition of the equilibrium rental rate of capital
from one period to the next.
In period 1, the equilibrium rental rate of capital is 1  ( 0 ) . The equilibrium capital
labor ratio and the equilibrium wage rate in period 1 are then given, respectively, by
19
1  [ f ' ]1 ( 1 ) and 1  f (1 )  1 f ' (1 ). Let ( c00 , c11 , b0 , k1 ) denote the optimal lifetime
plan for a young individual of period 0. The state of the economy at the beginning of

 

period 1 is K1 , N10 , N11  N 00k1 , N 00b0 , N 00 .
The procedure used to obtain 1, 1 , ( c00 , c11 , b0 , k1 ), and K1 , N10 , N11  can be repeated
ad infinitum to obtain a price system P  t , t t0 and an allocation induced by P, say
A  c01 , (ct0 , ct11, bt , kt 1 )t0 , ( Kt , Lt ,Yt )t0 , ( Kt , Nt0 , N 1y ,


where ct0 , ct11 , bt , kt 1 is the optimal lifetime plan of a young individual of period t and
and Kt , Lt , Y  is an optimal production plan in period t under the price system P. Also,
N t1  N t01 and N t01  N t0bt . The pair P, A , thus constructed, constitutes the unique
competitive equilibrium for an economy without fossil fuels or an economy that has
exhausted its stock of fossil fuels.
PROPOSITION 1: For an economy that has no oil resources, but that is sustained by a
source of renewable energy – provided by a backstop technology – there exists a unique
competitive equilibrium. Furthermore, such an economy also has a steady state.
PROOF: The first statement of Proposition 1, which bears resemblance to Theorem 13.1,
found in Azariadis (2000), has already been proved. To prove the second statement, note
that when   p max ( ), the wage rate corresponding to  tends to the critical level
 min ( ), and this means that the capital labor ratio in the following period tending to
infinity, which in turn implies that the rental rate of capital in the following period will
tend to 0; that is, im  p max ( ) (  )  0. When   0 , the capital labor ratio and the wage
rate associated with  both tend to infinity. Because the current wage rate tends to
infinity, the current consumption and the future consumption of a young individual both
tend to infinity – even when the rental rate of capital in the next period is 0. Also, the
number of offspring raised by a young individual will rise to the saturation level b max .
Hence the capital labor ratio generated by the maximizing behavior of a young individual
of period 0 will tend to infinity, which implies that the rental rate capital in period 1 will
tend to 0; that is, im 0(  )  0. Finally, note that as   0, the output per worker is
20
less than the capital labor ratio (the energy input per worker) due to the Inada condition
ime f ' (e)  0. Furthermore, the saving per worker is strictly less than her labor income,
which in turn is strictly less than the amount of the consumption good she produces. This
last result together with the result that the number of offspring she raises tends to its
saturation level imply that the capital labor ratio in the next period is strictly less than the
current capital labor ratio when  is small; that is, ' (  )  1 for all  in a small right
neighborhood of 0. We can now conclude that the curve  :   (  ), 0    p max ( ),
must cross the 45-degree line at least once, and the rental rate of capital at such a crossing
represents the rental rate of capital in a steady state.
■
The shape of the curve  :   (  ), 0    p max ( ), – first rising from 0, then
returning to 0 – suggests a possible rich dynamics. Depending on the preferences, the
technology, and the values of their parameters, the economy might converge to a steady
state in a monotone manner or in damped oscillation. There might even exist cycles.
4. COMPETITIVE EQUILIBRIUM FOR AN ECONOMY ENDOWED WITH A
STOCK OF FOSSIL FUELS
The presence of an exhaustible resource whose stock declines through time makes the
problem non-stationary, and the existence of a competitive equilibrium becomes
problematic. Furthermore, the equilibrium can no longer be computed recursively, as is
the case of an economy without a stock of fossil fuels that is analyzed in Section 3. The
existence of a competitive equilibrium for an economy that is endowed with a stock of
fossil fuels becomes problematic. Our existence proof proceeds in three stages. First, we
show that when the economy is truncated at the end of a finite number of periods, the
economy thus obtained has a competitive equilibrium. The definition of the competitive
equilibrium for the truncated economy is exactly the same as that for the infinite time
horizon economy given earlier, except that a young individual in the last period has no
future to plan for and thus will consume all the wages she earns. Next, we deploy a series
of technical arguments to establish some upper and lower bounds on the prices, the
birthrates, and the energy endowments per worker in each period that apply uniformly
21
across truncated economies. In the third stage, we use these bounds to show that a
sequence of competitive equilibria – one competitive equilibrium for each truncated
economy – has a subsequence that converges in the product topology of a denumerable
family of finite-dimensional Euclidean spaces, and the limit of the subsequence is a
competitive equilibrium of the infinite time horizon economy. The technical arguments of
the existence proof are provided in the technical addendum of the paper. The following
proposition asserts the existence of a competitive equilibrium for an economy endowed
with a stock of fossil fuels and gives some properties of the competitive equilibrium. In
this proposition,
et  E (Qt , K t ) / N t0 is the energy input per worker in period t , and
pt  f ' (et ) is the price of energy also in period t .
PROPOSITION 2: An economy that is endowed with a stock of fossil fuels and possibly a
stock of capital has a competitive equilibrium, say P, A , with
P  (t ,  t , t )t 0 ,
A  c01 , (ct0 , ct11 , bt , xt 1 , k t 1 ) t0 , (Qt , St , Lt ,Yt ) t0 , ( X t , K t , N t0 , N t1 ) t0 ,
which has the following properties:
(i) The birthrate is positive in each period.
(ii) The price of energy is positive but strictly lower than the critical level  max ( ) :
0  pt  f ' et   p max ( ) ,
(t  0,1,...) .
(iii) The price of oil relative to the rental rate of capital is bounded below by   , i.e.,
t /  t    ,
(t  0,1,...) .
To see why the birthrate is positive in each period, note that if bt  0 for some t , then
there will be no workers in period t  1. If there are no workers in period t  1, then a
young individual of period t will not invest in oil because there will be no worker to use
the energy generated from the fossil fuels to produce the consumption good. Investing in
oil thus serves no purpose, and thus a young individual of period t will only invest in
capital because she can at least consume the depreciated capital. The capital accumulated
by the young generation of period t will be used by the competitive firms in period t  1
22
to produce the composite energy inputs needed in the production of the consumption
good. Their demand for labor will then be positive due to the Inada condition
imL0F ( E , L) / L  , and in equilibrium must be met by a positive labor force. Thus
there will be an excess demand for labor in period t  1 if the birthrate in period t is 0,
and such a situation cannot arise in equilibrium. Using the result that the birthrate is
positive for every period, we can assert that the wage rate is higher than the critical level
 min ( ) for all the periods, from which follows (ii) of Proposition 2.
As for (iii) of Proposition 2, note that if  t  0, then backstop capital will be used to
produce renewable energy in period t , and we must have t /  t    , On the other hand,
if  t  0, then all the energy needs of the economy in period t must be met by burning
fossil fuels. In this case, we must have t /  t    , Furthermore, if t /  t    , then  t
can be lowered until the strict inequality turns into an equality without disturbing the
initial equilibrium. Indeed, when the inequality is strict, no capital will be used to
generate part of the energy input used in the production of the consumption good in
period t . Furthermore, if capital is not used in generating the energy input used by the
competitive firms to produce the consumption good in period t , a young individual of
period t  1 does not invest in capital. Lowering the rental rate of capital in period t until
the strict inequality turns into an equality will not induce the competitive firms to use
capital in generating part of its composite energy input needed in the production of the
consumption good. Also, lowering the rental rate of capital in period t lowers the rate of
return to capital investment for a young individual of period t  1 , and thus will not
induce her to change her investment behavior, either.
The following lemma goes further than (ii) of Proposition 2 in characterizing the
evolution of the price of energy.
LEMMA 1: For each value  ,0    1, of the rate of capital depreciation, there exist two
critical values for the price of energy – a lower critical value p  ( ) and an upper
critical value p  ( ) – with 0  p  ( )  p  ( )  p max ( ), which have the following
properties.
23
For any competitive equilibrium – of a truncated economy or of the infinite time
horizon economy – if p t and pt 1 are the prices of energy in two successive periods, then
the following conditions must hold:
(i) pt 1  p  ( );
(ii) if pt  p  ( ), then pt  pt 1 ;
(iii) if p  ( )  pt  p  ( ), then p  ( )  pt 1  p  ( ).
Furthermore,
(iv) p  ( ) is bounded below and away from 0 when  is small.
Lemma 1 is proved in the technical addendum. Although the proof of Lemma 1 involves
numerous limiting arguments, the economic intuition behind this lemma is quite simple.
When the price of energy is close to the critical level p max ( ) , the number of offspring
raised by a young individual will be very low, leading to a very high energy endowment
per worker and thus a very low price of energy in the following period. That is, a very
high energy price in the current period will be pushed back dramatically in the next
period by the fertility decision of the young generation of the current period. On the other
hand, a low price of energy means a high wage rate, which will induce a young individual
to choose a high level of current consumption, a high level of saving, and a birthrate that
is close to the saturation level. A high birthrate means an energy endowment per worker
in the next period that is lower than its current level, with the ensuing consequence of a
higher energy price in the next period. Lemma 1 asserts that except possibly for a few
initial periods the price of energy will be evolving inside a confining interval whose
endpoints do not take on the extreme values of 0 and p max ( ).
5. ENDOGENOUS FERTILITY, COMPLETE, AND INCOMPLETE RESOURCE
DEPLETION
Whenever extraction activities are carried out, the price of oil relative to the price of
renewable energy must be below its choke price level to induce the competitive firms into
using fossil fuels to generate part of the energy input needed in the production of the
consumption good. Furthermore, during the time interval that extraction activities are
24
being carried out, it is necessary for the successive young generations to put part of their
savings in oil, and this – according to Hotelling rule – implies that the rate of return to oil
investment must be at least equal to 1     t , the net rate of return earned by capital
investment. In normal circumstances and after a reasonable long time interval, we expect
that 1     t ,  1, i.e., the net rate of return to capital investment exceeds unity, or
equivalently the marginal product of capital is greater than the rate of capital
depreciation. When this condition is met, the price of oil must rise through time
geometrically at a rate that is greater than or equal to 1    t ,  1. Because the price of
energy is bounded above by p max ( ), the price of renewable energy must be bounded
above when the price of oil rises indefinitely, and this means that the price of oil relative
to the price of renewable energy will reach its choke price level in finite time. Two
conflicting forces thus operate at the same as long as extraction activities are being
carried out. The price of oil must not be too high to discourage the competitive firms
from using this exhaustible resource to generate part of the energy input used in the
production of the consumption good. The price of oil must also rise through time to
induce the successive young generations into holding oil in their investment portfolios.
Thus we should expect extraction activities to be terminated in finite time.
Now the first-order condition that characterizes the demand for capital in period t by
the competitive firms is f ' ( E ( qt ,  t ))E ( qt ,  t ) /  t  t  0, with equality holding if
 t  0. Using the fact that E ( qt ,  t ) /  t  E (0,  t ) /  t  1, we can then assert that
f ' ( E ( qt ,  t ))  t  0. The condition 1    t ,  1 will be met if the technology used in
the production of the consumption good is highly productive so that the marginal product
of energy is greater than the rate of capital depreciation. Another way that this condition
can be met is when the rate of capital depreciation is low. Indeed, according to Lemma 1,
for any competitive equilibrium – of a truncated economy or of the economy with infinite
time horizon – the price of energy will enter a confining interval after some initial
periods. That is, after a finite number of periods the price of energy will be bounded
below and away from 0, and a fortiori both the price of oil and the rental rate of capital
will also be bounded below and away from 0. Thus, if the rate of capital depreciation is
low, then after some initial periods the net rate of return to capital investment will be
25
bounded below and away from 1, and this implies that as long as the successive young
generations still invest in oil, the price of this asset must rise geometrically at a rate above
unity. The following lemma states formally the results just discussed.
LEMMA 2: If the rate of capital depreciation is not too high, then extraction activities are
terminated in finite time.
Lemma 2 asserts that when the rate of capital depreciation is low, extraction activities
are terminated in finite time. However, it is silent on the question of whether the resource
stock is completely depleted when extraction activities are terminated. The following
proposition gives a sufficient condition for the existence of a competitive equilibrium
under which the stock of fossil fuels is exhausted in finite time.
PROPOSITION 3: If the rate of capital depreciation is not too high, then there exists a
competitive equilibrium under which the stock of fossil fuels is exhausted in finite time.
PROOF: We claim that if the rate of capital depreciation is low enough, then there exists
a positive integer, say T , such that when the time horizon of a truncated economy is
greater than or equal to T , the stock of fossil fuels will be depleted in or before period
T . Indeed, if the claim is not true, then for each integer n  1 , there exists a time horizon
T ( n )  n, and a competitive equilibrium E T ( n ) for he truncated economy with time
horizon T (n ), such that the remaining oil stock at the beginning of period T (n ) under
this competitive equilibrium is positive. Because oil exhaustion always occurs under a
competitive equilibrium for a truncated economy, the remaining stock of fossil fuels will
be depleted in period T (n ) under the competitive equilibrium E T ( n ) . Furthermore,
because all the young generations before period T (n ) invest in oil, the price of oil will be
arbitrarily large when n is large. On the other hand, because the price of energy enters its
confining interval after some initial periods, a large value for the price of oil necessarily
means a rental rate of capital that is bounded above. Thus in the last period of the
competitive equilibrium E T ( n ) the price of oil relative to the price of renewable energy
must exceed its choke price level, i.e., oil will not be used to generate part of the energy
26
input used in the production of the consumption good T (n ). We thus arrive at two
contradictory results if the claim is not true.
Apply the diagonal trick to the sequence E T , T  1,2,..., we obtain in the limit a
competitive equilibrium of the infinite time horizon economy. Under this competitive
equilibrium the stock of fossil fuels is exhausted in finite time.
■
The next question to ask is whether there exists an equilibrium under which part of the
resource stock is left in situ unexploited at the time extraction activities are terminated.
To find an answer to this question, consider a competitive equilibrium under which the
oil stock is partially depleted, and that T is the last period it is exploited. Under such a
scenario, we have 0  QT  X T , and Qt  0, t  T . Because all the young generations of
period T and after put their savings in both oil and capital, the rates of return of these two
assets must be the same, i.e.,
(9)
t 1 / t  1     t 1 ,
(t  T ).
Furthermore, because after time T the energy input into the production of the
consumption good consists only of renewable energy, the price of oil relative to the price
of renewable energy must be greater than or equal to its choke price level, i.e.,
(10)
t /  t    ,
(t  T ).
Also, after paying – out of her wages – for current consumption, raising children, and
capital investment, a young individual of period t  T must still have some funds left for
oil investment, and the value – in terms of the numéraire – of the oil investment of such
an individual is given by
(11)
t X T / N t0  t  ct0  hbt  bt t 1.
Forwarding (11) by one period, then dividing the result by (11), we obtain
(12)
t 1  ct01  hbt 1   t  2bt 1 t 1 N t0 1     t 1


t  ct0  hbt   t 1bt
t N t01
bt
Note that the first equality in (12) has been obtained with the help of (11) and the second
equality with the help of (9). If a steady state in which the oil stock is partially depleted,
then in the limit, (12) becomes
(13)
[1     ] / b  1,
27
where we have let   lim t   t ,   lim t  t and b  lim t  bt . Equation (13) asserts
that when depletion is incomplete, the steady-state rate of return to capital investment is
equal to the biological rate of interest. Also, in the limit, the right-hand side of (11)
becomes
(14)
f (k )   f ' ( )  c 0  hb  b   0,
which is the value – expressed in the numéraire – of the oil investment made by a young
individual in steady state. In (14), c 0  lim t   ct0 denotes the current consumption of a
young individual in steady state.
PROPOSITION 4: Consider a competitive equilibrium under which extraction activities are
terminated in finite time and at the time extraction activities are terminated part of the
resource stock is left in situ unexploited. We have the following results:
(i) In steady state, the capital labor ratio, the birthrate, and the value of the oil
investment – expressed in terms of the numéraire – of a young individual are all unique;
(ii) Furthermore, b  lim t   bt .  1 , i.e., the population is stable or growing in the long
run.
(iii) Also, in steady state the price of oil grows geometrically at a rate equal to the
long-run birthrate.
PROOF: (i) According to (13), the steady-state capital labor ratio  is the value of  that
satisfies the following condition:
(15)
1    f ' ( )  b( f ( )  f ' ( ),1    f ' ( )) .
The left-hand side of (15) is strictly positive and strictly decreasing to 1   as the capital
labor ratio rises without bound from emin ( ). As for the right-hand side of (15), as 
rises from the critical energy input level emin ( ), the wage rate rises with  , but the rate
of interest moves in the opposite direction. Hence the right-hand side of (15) is strictly
increasing from 0 and approaches the saturation number of offspring when  rises from
emin ( ) to infinity. Thus there exists a unique value of   emin ( ) that satisfied (15).
The uniqueness of b and the uniqueness of the value of the oil investment made by a
young individual in steady state follow from the uniqueness of  .
28
(ii) Next, we show that b  1. Indeed, if b  1 , then 1      1 according to (13), and
it then follows from (9) that the price of oil will tend to 0 in the long run. On the other
hand, we know that the price of energy is bounded below and away from 0. Hence when
the price of oil tends to 0, the rental rate of capital must remain bounded below and away
from 0, and this means that the price of oil relative to the price of renewable energy will
tend to 0 in the long run. This last result implies that in the long run only fossil fuels are
used to generate the energy input used in the production of the consumption good,
contradicting the premise of the reductio ad absurdum argument.
(iii) Finally, note that according to (13), we have 1      b . Furthermore, according
to (9), the price of oil rises geometrically at rate 1     in steady state. Hence in steady
state the price of oil grows geometrically at a rate equal to the long-run birthrate.
■
The necessary conditions that characterize the steady state of a competitive equilibrium
with incomplete depletion are also sufficient for the existence of such an equilibrium, as
shown by the following proposition. The proof is constructive.
PROPOSITION 5: Let  be the unique value of  that satisfies (15),   f ' ( ),
  f ( )   f ' ( ), and b  b( ,1     ). Suppose that
(16)
b  1,
and
(17)
  c0 ( ,1     )  (h   )b  0 .
Next, let  0   ,  0   , 0   , b0  b , and 0 be any value satisfying 0 /  0    .
Define
(18)
0  0  c0 (0 ,1     0 )  (h   0 )b0  / 0 .
Consider an economy that begins in period 0 with 0 as the initial oil endowment per
worker and  0 as the initial capital labor ratio. Let P  t , t ,t t0 be the price system


defined by t ,  t , t   0b0t ,  0 , 0 . Then P  t , t ,t t0 is the prevailing price system
of a competitive equilibrium under which part of the stock of fossil fuels is left in situ
unexploited.
29
PROOF: Note that (16) ensures that the population is stable or growing in the long run.
Also, (17) asserts that in steady state what is left of the labor income of a young
individual – after the costs of current consumption, raising children, and buying capital
have been accounted for – is still positive so that some oil can be purchased.
It is straightforward to verify that when the price system P  t , t ,t t0 prevails, a
young individual of each period t  0,1,..., will have (i) the same labor income; (ii) the
same current and old-age consumption; (iii) the same number of children, (iv) the same
capital investment per child, and (v) the same amount of real resources for oil investment.
Under this price system, fossil fuels are never used, and the capital labor ratio is  . With
the price system thus constructed, the production plan of the competitive firms, and the
lifetime plans that are induced by this price system constitute a competitive equilibrium.
The competitive equilibrium constructed in this manner is a steady state for the economy
under which the part of the oil stock left in situ will never be exploited.
■
Proposition 5 asserts the uniqueness of the value expressed in the numéraire of the oil
investment, not the physical amount of the oil investment. Because 0 is only constrained
to satisfy the inequality 0 /  0    , there are infinitely many values for the initial price
of oil to choose from, and according to (18), a high chosen value for 0 corresponds to a
low value for 0 . Thus, the steady-state oil endowment per worker is indeterminate under
the competitive equilibrium constructed in Proposition 5. A resolution of this
indeterminacy can only be obtained by appealing to the initial state of the economy.
When the conditions of Proposition 5 are satisfied, there exists a competitive
equilibrium with incomplete depletion. If, in addition, the rate of capital depreciation is
not too high, then for the same initial conditions stated in this proposition, we can invoke
Proposition 3 to assert the existence a competitive equilibrium under which the stock of
fossil fuels is completely depleted in finite time. Hence our model might have multiple
equilibria. The following proposition gives a comparison of these two equilibria.
30
PROPOSITION 6: Suppose that the economy has two equilibria – one with complete and one
with incomplete depletion. Also, suppose that the steady state under the competitive
equilibrium with complete depletion is unique. We have the following results:
(i) The steady-state capital labor ratio and the steady-state birthrate is lower under the
equilibrium with incomplete depletion than under that with complete depletion.
(ii) Under the equilibrium with complete depletion, the resource stock has a level effect,
but not a growth effect, on the population. A large resource stock induces large
population in absolute terms, but has no impact on the long-run birthrate. The resource
stock has no impact on the long-run capital labor ratio, either.
(iii) Under the equilibrium with incomplete depletion, the resource stock has a negative
growth effect on the population. The hoarding of the exhaustible resource leaves a young
individual with less funds for capital investment and raising children, and the final
results – relative to the equilibrium with complete depletion – are a lower capital labor
ratio and a lower birthrate in the long run.,
PROOF: (i) Let  ( )  f ( )  f ' ( ) and
 ( )  ( )  c0 ( ),1    f ' ( )  b( ),1    f ' ( )   h .
We have  (e min ( ))   min ( )  c 0  min ( ),1    p max ( )   0. Furthermore, using the
Inada condition on the production function of the consumption good we can show that
 ( )  0 when  is large. Hence the curve  :    ( ),   e min ( ), must cross the
horizontal axis at least once, and the value of  at a crossing represents a steady state for
the economy after its stock of fossil fuels has been completely depleted. Because
 ( )  0 at the steady-state capital labor ratio under incomplete depletion, the steadystate capital labor ratio must be lower under incomplete depletion than under complete
depletion under the hypothesis that the steady state of the latter equilibrium is unique. A
lower steady-state capital labor ratio implies a lower steady-state birthrate.
(ii) This statement follows immediately from the fact that a large resource stock induces
high birthrates in the beginning. However, after the stock has been exhausted, the
economy evolves as if it has no oil and is endowed only with a stock of backstop capital.
(iii) In the case of the equilibrium with incomplete depletion, the resource stock induces
a lower steady-state capital labor ratio and a lower steady-state birthrate.
■
31
6. A CHARACTERIZATION OF THE COMPETITIVE EQUILIBRIUM
To concentrate on the influence of the exhaustible resource on fertility decisions and on
the process of technology substitution, we shall suppose that in period 0 the economy is
endowed with a large stock of fossil fuels and that backstop capital is yet to be
accumulated. Also, we shall not consider the case the population might become
asymptotically extinct in time, and presume that under the competitive equilibrium the
population is stable or growing in the long run.
When the initial stock of fossil fuels is large, the competitive equilibrium consists of
three phases: the pre-innovation phase, the transition phase, and the post-innovation
phase. To alleviate some of the technical arguments concerning the limiting behavior of
the economy when the energy endowments per worker are extremely high or close to the
critical level emin , we shall assume that for any positive value of e, we have
o
o
o
f (e)  ef ' (e)  b ef ' (e), where b is a constant satisfying 1  b  b max . That is, labor
income is not too high relative to the earning of the composite energy factor.
6.1. The Pre-innovation Phase
First, we claim that when the initial oil endowment per worker is large, the price of oil
in period 0 must be low. Indeed, if the price of oil remains outside a right neighborhood
of 0, then the oil input per worker will be bounded above, and consequently, the wage
rate will be bounded above, which in turn means that the demand for oil investment by a
young individual of period 0 will be bounded above. Thus there will be an excess supply
of oil in period 0, a situation that cannot arise in equilibrium.
Because of the low price of oil in period 0, the wage rate will be high in this period.
The high wage rate induces a high level of current consumption, a number of offspring
close to the saturation level b max , and a high level of old-age consumption. The high oldage consumption necessarily requires a high level of energy endowments per worker in
the following period; that is, either 1 or 1 will be high.
Second, we claim that E (1, 1 )  E ( q0 ,0), i.e., the composite energy input obtained
from the backstop and the energy generated by burning all the fossil fuels in period 1 is
32
strictly lower than the composite energy input generated in period 0. To see why, note
that the saving per offspring of a young individual of period 0 is
(19)


s0
 t 1
 01  1  f ' ( q0 E (1,0)) E (1,0) t 1 
.
b0
f
'
(
q
E
(
1
,
0
))
E
(
1
,
0
)
0


Because saving is necessarily less than labor income, we must have
o
(20)
s0 0 b

 E ( q0 ,0) f ' ( E ( q0 ,0))  E ( q0 ,0) f ' ( E ( q0 ,0))  q0 E (1,0) f ' ( q0 E (1,0)).
b0 b0 b0
o
In (20) the second inequality is due to assumption f (e)  ef ' (e)  b ef ' (e), and the third
inequality is due to fact that b0 is close to the saturation number of offspring and the
o
assumption b  bmax . It follows from (19) and (20) that
(21)
1 
1
f ' ( q0 E (1,0)) E (1,0)
 q0 .
Because price of oil in period 0 is low, the marginal product of oil in the production of
the consumption good is low, and the Inada condition on the production function of the
consumption good implies that f ' ( q0 E (1,0)) E (1,0)  1, from which we obtain
(22)
1  1 / f ' ( q0 E (1,0)) E (1,0).
Hence
(23)


1
E 1 , 1   E 1  1 ,0  E  1 
,0   E ( q0 ,0).
f ' ( q0 E (1,0)) E (1,0) 

In (23) the first inequality is due to assumption that along an energy isoquant the
marginal rate of technical substitution of renewable energy for oil is greater than or equal
to unity; the second inequality due to (22); and the third inequality due to (21). The
second claim is proved. It follows from the second claim that the price of energy in
period 1 is higher than that in period 0.
Third, we claim that the price of oil in period 1 is higher than that in period 0. Indeed,
we have
(24)
1  f ' ( E ( q1, 1 )) D1E ( q1, 1 )  f ' ( E ( q1, 1 )) D1E ( q1,0)  f ' ( E ( q0 ,0)) E (1,0)  0 .
Note that in (24) the second inequality is due to the assumption that in the generation of
the composite energy input a rise in the input of renewable energy raises the marginal
33
product of fossil fuels. Also, the third inequality is due to (23), the fact that q1  1, and
the diminishing returns of the composite energy input in the production of the
consumption good.
Fourth, we claim that a young individual of period 0 will not invest in backstop capital.
Indeed, if 1  0, then the rate of return to capital investment must be at least equal to the
rate of return to oil investment, i.e.,
(25)
1    1 
1
 1.
0
Furthermore, if 1  0, then renewable energy constitutes part of the energy input used in
the production of the consumption good in period 1, and we must also have
(26)
1
  .
1
There are two possibilities to consider: (i) 1 remains bounded when 0 grows
indefinitely large, and (ii) 1 is large when 0 becomes indefinitely large. Under
possibility (i), 1 must be large, and the price of oil in period 1 is low. Using this result
and (26), we can assert that the rental rate of capital in period 1 will be low. Under
possibility (ii), 1 is large, which in turn implies that the rental rate of capital in period 1
will be low. In either case, we arrive at the result that 1    1  1, contradicting (25).
With 1  0, we must have 1  q0  0 .
Because a finite stock of fossil fuels alone cannot sustain a stable or growing
population in the long run, the backstop must be brought into use in finite time. Let
the first time the backstop is brought into use. The time interval
be
constitutes
the pre-innovation phase of the competitive equilibrium. During this phase, only fossil
fuels provide the energy input used in the production of the consumption good, and the
following conditions are satisfied:
(27)
t
 1    t ,
t 1
(t  1,..., T1  1),
(28)
t
  ,
t
(t  0,..., T1  1).
Using (28), we can rewrite (27) as
34
(29)
t 
  1   t 1
,
   t 1
(t  1,..., T1  1),
Now the curve     (1   ) /(    ), 0      is above the forty-five degree line
for     , and this implies that the price of oil is rising during the pre-innovation phase
if f ' ( E ( 0 ,0))    , i.e., if the rate of capital depreciation is small.
During the pre-innovation phase of the competitive equilibrium, the oil endowment per
worker falls rapidly. There are two reasons behind this fast decline. First, the price of oil
must be low to induce the competitive firms into using more of this source of energy.
Second, the low price of oil also means a high wage rate, which in turn induces a high
birthrate. The combined effect of these two factors results in less and less oil available for
each worker in the successive periods. One implication of the fast decline in the oil
endowment per worker is a slow-down in the population growth. As the price of oil rises,
the wage rate declines, and this in turn induces a fall in the birthrate, every other thing
equal. According to (29), during the pre-innovation phase the rate of return that a young
individual earns on her savings is bounded below by   1    /(   t 1 ), and this lower
bound is rising through time as long as the oil endowment is still large or if the rate of
capital depreciation is low.
6.2. The Transition Phase
When the backstop is first brought into use, the price of oil relative to the rental rate of
capital will rise above its lower bound level   . It is conceivable that T1 /  T1 might rise
above the choke price level   , and induces an abrupt substitution of the backstop for the
fossil fuel technology. Intuitively, if the choke price level is high, i.e., if it is difficult to
substitute renewable energy for fossil fuels completely, then the substitution will be
carried out gradually, and the two technologies will co-exist for sometime before the
backstop completely takes over the burden of meeting the energy needs of the economy,
say in period
. The time interval
constitutes the transition phase.
Before the backstop is brought into use, the price of oil relative to the price of
renewable energy is equal to   . In particular, T1 1 / T1 1    . If fossil fuels do not
35
constitute part of the energy input used in the production of the consumption good in
period T1 , then we must have T1 / T1    , and it follows from this inequality that
T
  T
  T

 max
.
T 1 T 1
p E (1,0)
1
1
1
1
1
Under such a scenario,
(30)
T



 (1     T )   T  max
 1  (1   ).
T 1
 p E (1,0) 
1
1
1
1
Observe that if   is large,   / p max E (1,0) will be large. Furthermore, we can also
expect that  T1 is bounded below and away from 0. Thus, the differential rate of return to
oil investment over capital investment, as represented by (30), will be positive, and this
contradicts the premise that the technology substitution is abrupt.
In a period, say t, during the transition phase, we must have
(31)
t / t 1  1    t ,
and
(32)
   t /  t    .
Letting  t  t / t , we can rewrite (31) as
(33)
t 
 t 1   t 1
,
 t  t 1
As in the pre-innovation phase, if the rate of capital depletion is low, we will have
t  t 1. Furthermore, if the choke price of oil relative to renewable energy is not too
high, then we can expect that t / t 1   t 1    /( t  t 1 ) will also be rising during the
transition phase. Again as our discussion of the pre-innovation phase, the rise in energy
prices means a fall in the wage rate. Also, a rise in the rate of return to saving will induce
a young individual to save more at the expense of children. Thus, the birthrate continues
to fall in the second phase.
6.3. The Post-innovation Phase
The post-innovation phase begins after all extraction activities are terminated, either
due to oil exhaustion or incomplete depletion.
36
If the stock oil fossil fuels has been completely depleted when the third phase begins,
then the evolution of the economy from this time on is completely determined by the
backstop technology and the preferences, as described in Section 3. Depending on the
values of the parameters, the economy might converge to a steady state in a monotone
manner, in damped oscillation, or it might converge to a stable cycle. The existence of an
exhaustible resource has only a level effect, not a growth effect, on the population. At the
beginning, the abundance of fossil fuels makes it possible for the population to grow
rapidly and for capital to accumulate in a less painful manner. One can visualize through
time the process of transforming oil into the consumption good and into backstop capital.
In the long run oil does not influence the birthrate; its only impact is to allow for a
population with a larger absolute size.
In the case the competitive equilibrium involves incomplete depletion, the steady-state
capital labor ratio and the steady-state birthrate are both unique. When the rate of capital
depreciation is low, then for the same initial condition the economy also has a
competitive equilibrium under which depletion is complete. If we assume that this
equilibrium has a unique steady state – for the sake of comparison – then in steady state
the capital labor ratio and the birthrate are both lower under incomplete than under
complete oil exhaustion. The resource stock has a negative growth effect on the
population under this scenario.
6.4. Numerical Example
This numerical example aims at highlighting the characterization of competitive
equilibria that arise in our model. Suppose that backstop capital and fossil fuels are
perfect substitute in generating the composite energy input, and that preferences are
represented by the following lifetime utility function: Logc0  Logc1  v(b), with
v(b)  Log (b  1), 0  b  bˆ, where  is a positive parameter and b̂ is a constant greater
than 1 but less than the saturation number of offspring b max . As specified, the singleperiod sub-utility function is logarithmic and the sub-utility function of offspring is also
logarithmic in the relevant range [0, bˆ]. Saving and the number of offspring depend only
on labor income, not on the rate of return to saving, and the optimal lifetime plan for a
young individual of period t is given by
37
ct0 
h  t
 (h  t )
  h(1   )
 (h  t )
, ct11 
rt 1 , bt  t
, st 
.
1   
1   
h(1     )
1   
As for the output of the consumption good produced by a worker, we assume that it is
given by f (e)  e , 0    1. The following values for the parameters are assumed:
  0.10,   0.77,   0.73, h  0.13, and   0.15. Also, the initial oil endowment per
worker and the initial capital labor ratio are assumed to be given by 0  2.011 and
 0  0, respectively. For this numerical example, we are able to find two competitive
equilibria, one under which the oil stock is exhausted in finite time, and one under which
the oil stock is only partially depleted.
Table I presents the equilibrium under which the oil stock is depleted in finite time.
Under this equilibrium, the oil stock is completely exhausted at the end of period 5. After
that the economy is completely sustained by the backstop. The two technologies co-exist
during four periods, and the economy enters a steady state – without any oil left – in
period 6. The steady state capital labor ratio is 0.263, and the steady state birthrate is
1.068.
TABLE I
COMPETITIVE EQUILIBRIUM WITH COMPLETE OIL EXHAUSTION
(  0.10,   0.77,   0.73, h  0.13,   0.15, 0  2.011,  0  0)
Period
bt
t
t
qt
t
t
t
0
2.011
0
0.480
0.194
0.194 0.836 1.179
1
1.418
0
0.453
0.204
0.204 0.831 1.168
2
0.826
0.084 0.337
0.218
0.218 0.825 1.155
3
0.424
0.161 0.222
0.237
0.237 0.818 1.138
4
0.177
0.214 0.127
0.264
0.264 0.808 1.116
5
0.044
0.246 0.044
0.305
0.305 0.795 1.087
6
0
0.261
0
0.335
0.335 0.787 1.068
7
0
0.263
0
0.332
0.332 0.788 1.068
8
0
0.263
0
0.332
0.332 0.788 1.068
…
…
…
…
…
t
0
0.263
0
0.332
…
…
…
0.332 0.788 1.068
38
The second equilibrium we find is presented in Table II.
TABLE II
COMPETITIVE EQUILIBRIUM WITH INCOMPLETE OIL EXHAUSTION
(  0.10,   0.77,   0.73, h  0.13,   0.15, 0  2.011,  0  0)
Period
t
t
qt
t
0
2.011
0
0.480
0.194
0.194 0.836 1.179
1
1.898
0
0.453
0.204
0.204 0.831 1.168
2
1.253
0
0.421
0.218
0.218 0.825 1.155
3
0.721
0.097 0.287
0.237
0.237 0.818 1.138
4
0.381
0.165 0.175
0.264
0.264 0.808 1.116
5
0.185
0.209 0.081
0.305
0.305 0.795 1.087
6
0.095
0.232
0
0.372
0.372 0.778 1.047
…
…
…
…
…
t
0.095
b6t  6
0.232
0
0.372b6t  6
t
…
t
…
bt
…
0.372 0.778 1.047
As can be seen from Table II, for the first three periods, all the energy requirements of
the economy are met by drawing down the oil resources. Capital begins to be
accumulated at the end of the third period, and energy produced by the backstop
technology provides part of the energy inputs in the fourth period. The two technologies
– fossil fuels and the backstop – are both exploited during three periods, with the
backstop gradually replacing oil. From period 6 on, the oil stock is not exploited
anymore. The amount of oil that remains at the beginning of period 6 is left in situ,
unexploited, and all the energy requirements of the economy are met by the backstop
technology. The economy enters a steady state at the beginning of period 6. In steady, the
capital labor ratio is 0.232, and the population as well as the aggregate capital stock
39
grows geometrically at the rate of 1.047, which is also the rate of return to capital
investment. Note that the rate of capital depreciation has been deliberately chosen to be
low,   0.15, so that an equilibrium under which the oil stock is depleted in finite time
exists, as asserted by Proposition 4. Also, note that the capital share in national income
(   0.1 0) and the cost of raising a child ( h  0.13) are chosen sufficiently low to ensure
that an equilibrium with incomplete oil exhaustion exists. Finally, observe that under both
equilibria, the convergence to steady state is monotone, due to logarithmic preferences,
and that in steady state, the capital labor ratio as well as the birthrate are both lower under
the equilibrium with incomplete exhaustion.
7. CONCLUSION
Our model can be extended to include human capital. In the extended model, the cost of
raising children and the sub-utility function of offspring will depend on the number and
quality of offspring, which jointly determine the effective labor supply in the following
period. Furthermore, for a young individual, the number of offspring is an inferior good,
while the quality of a child is a superior good, so that the quantity-quality trade-off
should favor quantity over quality at low labor income, but quality over quantity at high
labor income. For such an extended model, the discovery of a large stock of an
exhaustible resource leads to rise in the effective wage rate, which can set in motion the
demographic transition – according to Lucas’ the theory of economic development (see
Lucas (2003) – by making the quantity-quality trade-off easier.
This paper does not deal with some important normative issues. It is well known that
inefficiency of various kinds usually arise in overlapping-generations models. In addition
to the eventual under-accumulation of capital, we now face the possibility of an oil
bubble, the over-accumulation of a resource asset. Besides the prescription of affecting
the pattern of capital accumulation through taxes on bequests, gifts, etc. in order to
restore economic efficiency, the obvious policy implication of our work is how to induce
complete resource depletion by some public intervention which should, at some point in
time, discourage unproductive resource hoarding under the form of a stock in situ as a
resource bubble. Also, public intervention may alter the resource ownership over time:
instead of grandfathering the entire resource stock to the fist generation. An authority
40
argues that it is a common property and proposes to distribute the right to use the stock to
different future generations; see Gerlag and Keyser (2001), or Valente (2006) for
analyses in the absence of population consideration. Besides efficiency, there is also the
inter-generational fairness issue. Leaving the exploitation of a stock of an exhaustible
resource to current selfish agents with finite lifetimes, who find resource depletion
profitable, is not necessarily compatible with the well-being of future generations; see,
for instance, Mourmouras (1993). If some light could be shed with their simple
overlapping-generations setting, the analytical task will be much more involved with our
model where population is endogenous. Because of the general equilibrium repercussion
of any policy on the rest of the economy, this is rather cumbersome, and would lead us
too far afield.
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