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Growth on a Finite Planet: Resources, Technology andPopulation in the Long Run
Pietro F. Peretto Duke UniversitySimone Valente ETH Zürich
February 2, 2010
Abstract
We characterize the interactions between technological change, natural resource scarcityand population dynamics in a Schumpeterian model with endogenous fertility choices. Weshow that there exists a pseudo-Malthusian equilibrium in which population is constantand income grows at a constant rate: the equilibrium population level is determined byresource scarcity but is independent of technology. The stability properties are drivenby (i) the income reaction to increased resource scarcity and (ii) the fertility response toincome dynamics, and ultimately depend on whether labor and resources are complementsor substitutes in production. Under substitutability, the pseudo-Malthusian equilibrium isa global attractor. Under complementarity, population follows diverging paths of explosionor implosion.
Keywords Endogenous Innovation, Resource Scarcity, Population Growth, Fertility Choices
JEL codes E10, L16, O31, O40
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1 Introduction
Achieving sustained per capita income levels is a major aim of modern societies. The funda-mental concerns of economists and policymakers are generally related to three potential sourcesof intergenerational con�ict. The �rst is natural resource scarcity: the production process ofpost-industrial economies still relies on a �nite supply of minerals and fossil fuels, and thenon-renewable nature of these primary inputs implies that technical change will have a crucialrole in guaranteeing sustained output in the long run. The second concern refers to the processof environmental degradation which stems from the production process, and negatively a¤ectswelfare through pollution-related tradeo¤s and global environmental externalities. The thirdconcern is related to population growth, which is expected to exacerbate the two phenomenaby exerting further pressure on natural resource scarcity and environmental quality.
The modern growth literature analyzes the �rst two issues to a good extent, but generallyneglects the role of population. Resource scarcity, in particular, has been studied in generalequilibrium models featuring endogenous progress (e.g. Barbier, 1999; Scholz and Ziemes,1999) and directed technical change (Di Maria and Valente, 2008): these contributions em-phasize the importance of resource-augmenting technological progress but typically excludeinteractions with fertility choices by assuming exogenous population dynamics.1 In this paper,we take a �rst step in this direction and investigate the links between technical change, re-source scarcity and income levels when population size is endogenously determined by fertilitychoices.
The �rst key question is how technology responds to demographic forces. We tackle thispoint from the perspective of modern Schumpeterian theory, postulating that �nal productionconsists of a variety of di¤erentiated consumption goods and that each production unit istechnologically dynamic: �rms and entrepreneurs undertake R&D to learn how to use factorsof production more e¢ ciently and to design new products. This innovation process interactswith fertility rates and resource scarcity because each �rm�s output is produced by means oftwo factors of production, labor and a natural resource. Resource availability in each pointin time is determined by the exogenous natural endowment. Labor availability over timeis connected to the fertility choices of households � that we assume to be endogenous anddetermined by private utility maximization.
In our model, the process of product proliferation fragments the aggregate market intosubmarkets whose size does not increase with the size of the endowments: the endogenousprocess of entry, or exit, in the manufacturing business sterilizes the scale e¤ect in the longrun. This means that the e¤ect of endowments on growth is only temporary. In this con-text, we derive two main results. First, there exists a pseudo-Malthusian equilibrium in whichpopulation converges to a constant value while income per capita grows at a constant rate.Second, whether the economy converges to this equilibrium depends on the production elastic-ity of substitution between labor and resource. Under substitutability, the pseudo-Malthusianequilibrium is a global attractor and indeed determines the population level in the long run.Under complementarity, instead, the population level follows diverging paths implying eitherextinction or explosion.
1Similarly, the parallel literature on environmental degradation analyzes pollution externalities in the pres-ence of endogenous technical change (e.g. Bovenberg and Smulders, 1995) and studies the consequences ofinduced innovations for optimal policies (Grimaud and Rougè, 2005) in models where there is no feedbacke¤ect on population size.
2
We label our benchmark equilibrium as pseudo-Malthusian because, in this steady state,the key feedback that stabilizes population is resource scarcity. However, it is not a purelyMalthusian equilibrium because the associated level of population is not constrained by tech-nology parameters: if the economy converges to this steady state, population in the long runis determined by resource scarcity but is independent of technology.
Our second result �the stability properties of the pseudo-Malthusian equilibrium �hingeson two mechanisms that determine (i) the reaction of incomes to increased resource scarcityand (ii) the reaction of fertility rates to income levels. The �rst mechanism works as follows.Since population dynamics are determined by fertility choices, the endowment ratio �i.e. theratio between labor and resource � is endogenous. The response of the resource price to anincrease in the endowment ratio, in turn, determines the income earned by the owners of theresource �the households �and their expenditure on �nal consumption goods. This mechanismlinks resource scarcity to the size of the market for �nal goods: resource abundance a¤ects�rms�incentives to innovate and thereby the equilibrium market size, which is determined byR&D activity. The direction of these e¤ects crucially depends on whether labor and resourceare complements or substitutes in production. If the economy exhibits substitution, resourcedemand is elastic and a reduction in the relative endowment of the resource generates only amild increase in the equilibrium price: the quantity e¤ect dominates, resource income declinesand so does spending on consumption goods, which causes a temporary growth slowdown. Ifthe economy exhibits complementarity, instead, resource demand is inelastic and a reductionin the relative endowment of the resource induces a strong increase in the price: resourceincome increases and we have a temporary growth acceleration.
The second mechanism driving our results is the response of population through fertilitychoices. The bene�ts and costs of having children are represented by private utility and thetime cost for rearing each child, respectively. In equilibrium, the net fertility rate is positivelyrelated to disposable income. Consequently, the response of the fertility rate to increasedresource scarcity is directly linked to the income response described above. The crucial point isthat the fertility response becomes a contrasting or reinforcing element of population dynamicsdepending, again, on the elasticity of substitution between labor and resource:
� Under complementarity, increasing resource scarcity generates increasing incomes dueto very large increases in the resource price. In this case, the pseudo-Malthusian equi-librium acts as a threshold. If the initial population level is above that associated tothe pseudo-Malthusian steady state, the net fertility rate is initially positive; growingpopulation then implies increased resource scarcity, which generates higher income andthereby supports further expenditure and fertility. If the initial population level is belowthe threshold, instead, the net fertility rate is initially negative and population declineimplies decreased resource scarcity, lower income and expenditure, and thereby lowerfertility. In both cases, the absence of contrasting forces generates diverging paths ofpopulation size: explosion in the former case, extinction in the latter.
� Under substitutability, increasing resource scarcity generates decreasing incomes becausethe quantity e¤ect dominates. In this case, the pseudo-Malthusian equilibrium is aglobal attractor. If the initial population level is above the threshold, the net fertilityrate is positive but population growth implies increased resource scarcity that generateslower income, lower expenditure and thereby less incentive to raise children: the systemconverges to the constant level of population associated with the pseudo-Malthusian
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equilibrium. Symmetrically, if the initial population level is below the threshold, the netfertility rate is negative but population decilne implies decreased resource scarcity, higherincome and expenditure and thereby higher fertility rates until the pseudo-Malthusianequilibrium is achieved from below.
In general, our analysis unveils a theory of interactions between resources and populationthat di¤ers in several respects from the existing literature. Balanced growth models typicallyprovide theories of population growth as they hinge on equilibria with constant exponentialgrowth of the population mass �e.g. Barro and Sala-i-Martin (2004) �whereas our modelsuggests a theory of the population level. Clearly, long-run equilibria featuring exponentialpopulation growth are at odds with the real issue posed in resource economics that planetEarth has a �nite carrying capacity of people. Our result of pseudo-Malthusian steady stateimplies that the economy may achieve equilibria with constant population where populationsize (i) is determined by resource scarcity, (ii) is independent of technology, and (iii) is stilldi¤erent from the physical upper-bound imposed by the natural carrying capacity of thehabitat. Still, the carrying-capacity argument remains relevant in the context of our modelsince the pseudo-Malthusian steady state may not be the global attractor of the system.
2 The model
2.1 Overview
A representative household supplies labor services in a competitive market. It also borrowsand lends in a competitive market for �nancial assets. The household values variety andbuys as many di¤erentiated consumption goods as possible. Manufacturing �rms hire laborto produce di¤erentiated consumption goods, undertake R&D, or, in the case of entrants, setup operations. Production of consumption goods also requires a natural resource (e.g., land),which is supplied by the households. The economy starts out with a given range of goods, eachsupplied by one �rm. Entrepreneurs compare the present value of pro�ts from introducing anew good to the entry cost. They only target new product lines because entering an existingproduct line in Bertrand competition with the existing supplier leads to losses. Once in themarket, �rms establish in-house R&D facilities to produce cost-reducing innovations. As each�rm invests in R&D, it contributes to the pool of public knowledge and reduces the cost offuture R&D. This allows the economy to grow at a constant rate in steady state.
2.2 Households
The representative household maximizes
U(t) =
Z 1
te��(s�t) log u(s)ds; � > 0 (1)
where � is the discount rate. The household�s instantaneous preferences are over a continuumof di¤erentiated goods, the mass of children and the mass of adults:
log u = log
"Z N
0
�Xi
L
� ��1�
di
# ���1
+ (�+ 1) log (BL) ; � > 1; � > 0 (2)
4
where � is the elasticity of product substitution, Xi is the household�s purchase of each di¤er-entiated good, N is the mass of goods (the mass of �rms), B is the mass of children and L isthe mass of adults. The household likes children and adults equally with elasticity �+ 1.
For simplicity, and to bring to the forefront the novel features of our model, we work witha time cost of rearing children of per child.2 Letting b � B=L denote the fertility rate, thehousehold�s �ow budget constraint is
_A = rA+ wL (1� b) + p� Y; (3)
where A is assets holding, r is the rate of return on assets, w is the wage rate, L is laborsupply since there is no preference for leisure and children do not work, and Y is consumptionexpenditure. In addition to asset and labor income, the household receives rents from own-ership of the endowment, , of a non-exhaustible natural resource (e.g., land) whose marketprice is p. The household takes these terms as given. The law of motion of (adult) populationis
_L = L (b� d) ; (4)
where d is the (exogenous) death rate.This is a well known problem. Letting l denote the shadow value of L, the fertility decision
follows from the �rst-order conditions for L and b:
�
Ll+w (1� b)
lY+_L
L= ��
_l
l; (5)
�+ 1
b� wL
Y+ Ll = 0: (6)
The �rst-order conditions for A and Y yield
r = rA � �+_Y
Y: (7)
Taking as given this time-path of expenditure, the household then chooses Xi for i 2 [0; N ] tomaximize (2) subject to Y =
R N0 PiXidi. This yields the demand schedule for product i,
Xi = YP��iR N
0 P 1��i di: (8)
With a continuum of goods, �rms are atomistic and take the denominator of (8) as given;therefore, monopolistic competition prevails and �rms face isoelastic demand curves.
2All of our qualitative results go through if the cost per child is proportional (or even exactly equal) toparental consumption. In this case, we would write the budget constraint
_A = rA+ wL+ p� Y (1 + b)
and solve the model following the same procedure that we use here. The only di¤erence would be that therelevant resource constraint incroporating the cost of fertilty would be the equilibrium condition for the goodsmarket instead of that for the labor market.
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2.3 Manufacturing: Production and Innovation
The typical �rm produces with the technology
Xi = Z�i � F (LXi � �;Ri) ; 0 < � < 1; � > 0 (9)
where Xi is output, LXi is production employment, � is a �xed labor cost, Ri is resource use,Zi is the �rm�s stock of �rm-speci�c knowledge and F (�) is a standard neoclassical productionfunction homogeneous of degree one in its arguments. (9) gives rise to total cost
w�+ C(w; p)Z��i Xi; (10)
where C (�) is a standard unit-cost function homogeneous of degree one in its arguments.The �rm accumulates knowledge according to the R&D technology
_Zi = �KLZi ; � > 0 (11)
where _Zi measures the �ow of �rm-speci�c knowledge generated by an R&D project employingLZi units of labor for an interval of time dt and �K is the productivity of labor in R&D asdetermined by the exogenous parameter � and by the stock of public knowledge, K.
Public knowledge accumulates as a result of spillovers. When one �rm generates a new ideato improve the production process, it also generates general-purpose knowledge which is notexcludable and that other �rms can exploit in their own research e¤orts. Firms appropriatethe economic returns from �rm-speci�c knowledge but cannot prevent others from using thegeneral-purpose knowledge that spills over into the public domain. Formally, an R&D projectthat produces _Zi units of proprietary knowledge also generates _Zi units of public knowledge.The productivity of research is determined by some combination of all the di¤erent sources ofknowledge. A simple way of capturing this notion is to write
K =
Z N
0
1
NZidi;
which says that the technological frontier is determined by the average knowledge of all �rms.3
3 Equilibrium of the Market Economy
This section constructs the symmetric equilibrium of the manufacturing sector. It then charac-terizes the equilibrium of the primary sector. Finally, it imposes general equilibrium conditionsto determine the aggregate dynamics of the economy. The wage rate is the numeraire, i.e.,w � 1.
3.1 Partial Equilibrium of the Manufacturing Sector
The typical manufacturing �rm is subject to a death shock. Accordingly, it maximizes thepresent discounted value of net cash �ow,
Vi (t) =
Z 1
te�
R st [r(v)+�]dv�i(s)ds; � > 0
3For a detailed discussion of the microfoundations of a spillovers function of this class, see Peretto andSmulders (2002).
6
where e��t is the instantaneous probability of death. Using the cost function (10), instanta-neous pro�ts are
�Xi = [Pi � C(1; p)Z��i ]Xi � �� LZi ;
where LZi is R&D expenditure. Vi is the value of the �rm, the price of the ownership shareof an equity holder. The �rm maximizes Vi subject to the R&D technology (11), the demandschedule (8), Zi(t) > 0 (the initial knowledge stock is given), Zj(t0) for t0 � t and j 6= i(the �rm takes as given the rivals�innovation paths), and _Zj(t
0) � 0 for t0 � t (innovation isirreversible). The solution of this problem yields the (maximized) value of the �rm given thetime path of the number of �rms.
To characterize entry, I assume that upon payment of a sunk cost �PiXi, an entrepreneurcan create a new �rm that starts out its activity with productivity equal to the industryaverage.4 Once in the market, the new �rm implements price and R&D strategies that solvea problem identical to the one outlined above. Hence, a free entry equilibrium requires Vi =�PiXi.
The appendix shows that the equilibrium thus de�ned is symmetric and is characterizedby the factor demands:
LX = Y�� 1�
(1� S (p)) + �N ; (12)
R = Y�� 1�
S (p)
p; (13)
where
S (p) � pRi
C(w; p)Z��i Xi
=@ logC(w; p)
@ log p
is the resource share in the �rm�s total veriable cost.Associated to these factor demands are the returns to cost reduction and entry, respec-
tively:
r = rZ � �
�Y �(�� 1)
�N� LZ
N
�� �; (14)
r = rN �1
�
�1
�� N
Y
��+
LZN
��+ Y � N � �: (15)
The dividend price ratio in (15) depends on the gross pro�t margin 1� . Anticipating one of
the properties of the equilibria that we study below, note that in steady state the capital gaincomponent of this rate of return, Y � N , is zero. Hence, the feasibility condition 1
� > (r + �)�must hold. This simply says that the �rm expects to be able to repay the entry cost becauseit more than covers �xed operating and R&D costs.
3.2 General equilibrium
Equilibrium of the resource market requires R = . One can thus think of (13) as the equationthat determines the price of the resource, and therefore resource income for the household,given the level of economic activity as measured by expenditure on consumption goods Y .
4See Etro (2004) and, in particular, Peretto and Connolly (2007) for a more detailed discussion of themicrofoundations of this assumption.
7
The remainder of the model consists of the household�s budget constraint (3), the labordemand (12), and the returns to saving, cost reduction and entry in (7), (14) and (15). Thehousehold�s budget constraint becomes the labor market clearing condition (see the appendixfor the derivation):
L (1� b) = LN + LX + LZ ,
where LN is aggregate employment in entrepreneurial activity and LX + LZ is aggregateemployment in production and R&D operations of existing �rms. Assets market equilibriumrequires equalization of all rates of return (no-arbitrage), r = rA = rZ = rN , and that the valueof the household�s portfolio equal the value of the securities issued by �rms, A = NV = �Y .
A useful feature of the model is that we can combine the �rst-order conditions for Land b with the equilibrium conditions for the labor and assets markets to obtain a set ofequations that describe how the resource price a¤ects household income and thus expenditureon consumption goods and the fertility decision. Let h � lL so that we can rewrite (5) and(6) as:
�+1� by
= �h� _h; (16)
�+ 1
b�
y+ h = 0: (17)
Substituting A = �Y into (3) and using the rate of return to saving (7), we obtain, afterrearranging terms,
Y =L (1� b) + p
1� �� : (18)
Notice how all dynamic terms dropped out. This expression thus says that the ratio ofhousehold consumption expenditure to total household income is constant. This propertydramatically simpli�es the model�s dynamics because it allows us to owrk with a constantsaving ratio � as in the Solow model of capital accumulation, with the di¤erence that therethe saving ratio is exogenous while here it is endogenous.
Recall that we are focussing on a closed economy where the �xed domestic resource supply implies that the price of the resource re�ects scarcity. Then, letting y � Y
L denote expen-diture per capita and ! �
L denote the endowment ratio, we can use (13) and the conditionR = to write
p! = y�� 1�
S (p) : (19)
Substituting this expression into (18) to eliminate p!, we have that at any point in time giventhe fertility choice b we can describe the determination of the equilibrium values of y and p asthe intersection of the schedules:
y =1� b+ p!1� �� ; (20)
y =1� b
1� ��� ��1� S (p)
: (21)
The term 1� b is labor income per capita. The �rst schedule says how agents determine ex-penditure given resource income, the second how the expenditure decision determines resourceincome through the resource market�s determination of the resource price.
We can now use (17), (20) and (21) to construct decisions dated t as functions of the statevariables h and !. The following property of the demand function (13) is useful.
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Lemma 1 Let
� (p) � �@ logR (p)@ log p
= 1� @ logS (p)
@ log p= 1� @S (p)
@p
p
S (p):
Then, di¤erentiating (21) we have
d log y (p)
dp=
��1�
dS(p)dp
1� ��� ��1� S (p)
= ! [1� � (p)] :
Proof. See the Appendix.
� (p) is the elasticity of the demand for the resource R with respect to its price p, holdingconstant expenditure per capita y. It thus captures the partial equilibrium e¤ects of pricechanges in the resource market for given market size and regulates the shape of the incomerelation (21). Lemma 1 thus says that the e¤ect of changes in the resource price on expendituredepends on the degree of substitution between labor and the resource in the manufacturingtechnology. This degree of substitution is captured by the price elasticity of the demand forthe resource. The following proposition states the results formally, Figure 1 illustrates themechanism.
Proposition 2 Hold b constant. There are three cases.
1. Complementarity. This occurs when 1 > � (p) and the income relation (21) is amonotonically increasing function of p with domain p 2 [0;1) and codomain
y 2"
1� b1� ��� ��1
� S (0);
1� b1� ��� ��1
� S (1)
!:
Then there exists a unique equilibrium (p (b; !) ; y (b; !)) with the property:
dp (b; !)
db< 0;
dp (b; !)
d!< 0;
dy (b; !)
db< 0;
dy (b; !)
d!< 0:
2. Cobb-Douglas. This occurs when S is an exogenous constant, 1 = � (p) and the incomerelation (21) is the �at line
y =1� b
1� ��� ��1� S
:
Then there exists a unique equilibrium (pCD (b; !) ; yCD (b; !)) with the property:
dpCD (b; !)
db< 0;
dpCD (b; !)
d!< 0;
dyCD (b; !)
db< 0;
dyCD (b; !)
d!= 0:
9
3. Substitution. This occurs when 1 < � (p) and the income relation (21) is a monoton-ically decreasing function of p with domain p 2 [0;1) and codomain
y 2
1� b1� ��� ��1
� S (1);
1� b1� ��� ��1
� S (0)
#:
Then there exists a unique equilibrium (p (b; !) ; y (b; !)) with the property:
dp (b; !)
db< 0;
dp (b; !)
d!< 0;
dy (b; !)
db< 0;
dy (b; !)
d!> 0:
Proof. Consider �rst (19) and (20) and rewrite them as:
y =p
S (p)
�
�� 1!;
y =1� b+ p!1� �� :
The �rst schedule describes how expenditure drives demand for the factors of production andthereby determines resource income; the second how resource income determines expenditureon consumption goods. Note that
@�
pS(p)
�@p
=1
S (p)
�1� p
S (p)
@S (p)
@p
�=� (p)
S (p)> 0.
hence, changes in b yield shifts of the �rst schedule and movements along the second schedule.It then follows that y and p are decreasing in b for all ! and regardless of whether themanufacturing technology F (�) exhibits substitution or complementarity. Also:
b! 0)(p (0; !) = arg solve
np
S(p)���1! =
1+p!1���
oy (0; !) = 1+p(0;!)�!
1��� � y (!);
b! 1
)
8><>:p�1 ; !
�= arg solve
np
S(p)���1! =
p!1���
oy�1 ; !
�=
p�1 ;!��!
1��� � y (!):
Refer now to Figure 1. In the case of complementarity, depicted in the upper panel, we havethe following pattern. For ! ! 0 the expenditure line (20) is almost, but not quite, �atand intersects the income relation (21) for p ! 1 and y ! 1� b
1���� ��1�S(1) . As ! grows, the
expenditure line rotates counterclockwise and the intersection shifts left, tracing the incomerelation. We thus obtain that both p and y fall. As ! ! 1, the expenditure line becomesvertical and the intersection occurs at p! 0 and y ! 1� b
1���� ��1�S(0)
. In the case of substitution
in the lower panel, we have a similar pattern with the di¤erence that the income relation hasnegative slope so that y increases as the expenditure line rotates counterclockwise. Speci�cally,for ! ! 0 the expenditure line is almost, but not quite, �at and intersects the income relation
10
for p!1 and y ! 1� b1���� ��1
�S(1) . As ! grows, the expenditure line rotates counterclockwise
and the intersection shifts left yielding that p falls while y rises. As ! !1, the expenditureline becomes vertical and the intersection occurs at p! 0 and y ! 1� b
1���� ��1�S(0)
.
The e¤ect of the endowment ratio ! on the resource price is negative in all cases, while thee¤ect on expenditure changes sign according to the substitution possibilities between laborand the natural resource. The main message of this analysis thus is that resource abundanceraises expenditure, and thereby results into a larger market for manufacturing goods, whenthe economy exhibits substitution between labor and the natural resource. Conversely, whenthe economy exhibits complementarity resource abundance results into a smaller market formanufacturing goods.
We can now use this result to construct y and b as functions of the state variables h and!.
Proposition 3 There are three cases.
1. Complementarity. There exists a unique equilibrium (b (h; !) ; y (h; !)) with the prop-erty:
db (h; !)
dh> 0;
db (b; !)
d!< 0;
dy (h; !)
dh< 0;
dy (b; !)
d!< 0:
2. Cobb-Douglas. There exists a unique equilibrium (bCD (h; !) ; yCD (h; !)) with theproperty:
dbCD (h; !)
dh< 0;
dbCD (b; !)
d!= 0;
dyCD (h; !)
dh< 0;
dyCD (b; !)
d!= 0:
3. Substitution. There exists a unique equilibrium (b (h; !) ; y (h; !)) with the property:
db (h; !)
dh> 0;
db (b; !)
d!> 0;
dy (h; !)
dh< 0;
dy (b; !)
d!> 0:
Proof. Figure 2 plots the equilibrium as the interesection of the function y (b; !) fromProposition 2 with the fertility schedule
y =
h+ �+1b
obtained from (17). In all cases, as h rises the fertility schedule shifts down and yields amovement along the y (b; !) schedule. Accordingly, y falls and b rises. In particular, as h! 0,b ! 0 and y ! y (!), while as h ! 1, b ! 1= and y ! y (!). The di¤erence among
11
the three cases is the response to changes in !. In the case of complementarity, the y (b; !)schedule shifts down and yields a fall in both y and b. Also, as ! ! 0 we have
b (h; 0) = arg solve
(1� b
1� ��� ��1� S (1)
=
h+ �+1b
)
and as ! !1 we have
b (h;1) = arg solve(
1� b1� ��� ��1
� S (0)=
h+ �+1b
);
where S (0) < S (1). In the case of substitution, the y (b; !) schedule shifts up and yields arise in both y and b. The two limits above still apply with the di¤erence that S (0) > S (1).The Cobb-Douglas case is special in that the y (b; !) schedule does not move with ! and thusyields that y and b do not depend on !.
We can now do dynamics of the demography block of the model. Useful to characterizesteady state �rst.
3.3 The Steady State
We impose b = d in the fertility schedule (17) and the _h = 0 locus to obtain:
h =
y� �+ 1
d;
h =1
�
��+
1� dy
�:
We then solve these equations for
yss = (�+ d)� 1�+ ��+1d
:
This is a quasi-Malthusian equilibrium in that expenditure per capita depends only on pref-erences and demographic parameters. Neither the endowment of the natural resource nortechnology play a role.
Next, we impose b = d in the expenditure schedule (20) and the resource income schedule(21) to obtain
yss =1� d+ p!1� �� ;
yss =1� d
1� ��� ��1� S (p)
:
We then solve the resource income schedule (the second equation) for
pss = S�1��1� ��� 1� d
yss
��
�� 1
�;
12
and use this result to solve the expenditure schedule (the �rst equation) for
!ss =(1� ��) yss + d� 1
pss;
which yields
Lss = � pss
(1� ��) yss + d� 1 :
This is too has the Malthusian �avor that population is proportional to the resource endow-ment. What�s missing is technology. The point is that while the model determines the constantlevel of the population in the long-run, it does not detemine constant income per capita. Infact, this steady state exhibits constant endogenous growth.
3.4 Dynamics: Demography
We now use (16) and (17) to write the system:
_h = �h� �� 1� b (h; !)y (h; !)
;
_! = ! (d� b (h; !)) :
In general, _h � 0 forh � 1
�
��+
1� b (h; !)y (h; !)
�and _! � 0 for
d � b (h; !) :
Notice that ! = 0 is a steady state locus as well. To construct the steady-state loci in eachcase observe �rst that equation (20) allows us to write the _h = 0 locus as
h =1
�
��+
1� b (h; !)y (h; !)
�=1
�
��+ 1� ��� �� 1
�S (p)
�and the fertility schedule (17) as
h =
1� b
�1� ��� �� 1
�S (p)
�� �+ 1
b:
The advantage of these relations is that they contain the common term � (p) � 1 � �� ���1� S (p).
3.4.1 Complementarity
Proposition 2 says that, given b, as ! ! 0 ) p ! 1 ) S ! S (1) and as ! ! 1 )p ! 0 ) S ! S (0). In this case, we have S (1) > S (0) which implies � (0) > � (1). Wethus have that _h � 0 for h � h (!) j _h=0 where the locus on the right-hand side starts outat 1
� [�+ � (1)], is increasing in ! and ends at1� [�+ � (0)]. Next, we use the properties of
b (h; !) from Proposition 3 to argue that _! � 0 for h � h (!) j _!=0 where the locus on theright-hand side starts out at
1� d� (1)��+1d , is increasing in ! and ends at
1� d� (0)�
�+1d .
13
To compress notation, let
� ���+ 1
d+�
�
�� (1� d) (�+ d)� 1 =
1� bssyss
> 0:
This term is high when the preference for children and siblings, �, is strong and when thetime cost of reproduction per child, , is low. We obtain the following pattern.
1. � (0) > � (1) > �. Then h (0) j _!=0> h (0) j _h=0 and h (1) j _!=0> h (1) j _h=0, whichmeans that the _! = 0 locus is everywhere above the _h = 0 locus. The unique equilibriumtrajectory is a jump on the saddle path that converges asymptotically to the line h =1� [�+ � (0)]. Along this path the economy exhibits population shrinking.
2. � (0) > � > � (1). Then h (0) j _!=0< h (0) j _h=0 and h (1) j _!=0> h (1) j _h=0, whichmeans that the _! = 0 locus intersects the _h = 0 locus from below. The unique equilib-
rium trajectory is a jump on the saddle path that converges to�0; 1� [�+ � (1)]
�, if the
economy is initally to the left of (!ss; hss), or a jump on the saddle path that convergesasymptotically to the line h = 1
� [�+ � (0)], if it is initially to the right of (!ss; hss).
3. � > � (0) > � (1). Then h (0) j _!=0< h (0) j _h=0 and h (1) j _!=0< h (1) j _h=0, whichmeans that the _! = 0 locus is everywhere below the _h = 0 locus. The unique equilibrium
trajectory is a jump on the saddle path that converges to�0; 1� [�+ � (1)]
�where the
economy exhibits a constant exponential rate of population growth.
3.4.2 A very special case: the Cobb-Douglas
We have _h � 0 forh � 1
�
��+
1� bCD (h)yCD (h)
�;
where hCD (!) j _h=0 is independent of !. Similarly, _! � 0 for h � h (!) j _!=0 where the locuson the right-hand side is independent of !. In the phase diagram, therefore, the saddle pathis the _h = 0 locus itself. The equilibrium then features:
hssCD =1
�
��+ 1� ��� �� 1
�S
�;
bssCD = arg solve
(1� bCD
1� ��� ��1� S
=
1�
��+ 1� ��� ��1
� S�+ �+1
b
);
yssCD =1� bssCD
1� ��� ��1� S
;�_p
p
�ssCD
= ��_!
!
�ssCD
= d� bssCD:
The fertility rate is constant because resource income is proportional to expenditure and weassumed a pure time cost of reproduction. Only by remote chance bssCD = d. Consequently,the model predicts either constant population growth or constant population shrinking. The
14
literature typically imposes bssCD > d either by choosing d suitably low (e.g., Connolly andPeretto 2003) or by assuming that agents care about net fertility so that the Inada conditionimplies that agents choose b�d > 0 (e.g., Barro and Sala-i-Martin 2004). The important thingis that the Cobb-Douglas case does not produce a theory of the population level because itsuppresses the feedback from resource abundance to resource income by forcing the endogenousprice adjustment to cancel out with the change in the quantity.
3.4.3 Substitution
Proposition 2 says that, given b, as ! ! 0 ) p ! 1 ) S ! S (1) and as ! ! 1 )p ! 0 ) S ! S (0). In this case, we have S (1) < S (0) which implies � (0) < � (1). Wethus have that _h � 0 for h � h (!) j _h=0 where the locus on the right-hand side starts outat 1� [�+ � (1)], is decreasing in ! and ends at
1� [�+ � (0)]. Next, we use the properties of
b (h; !) from Proposition 3 to argue that _! � 0 for h � h (!) j _!=0 where the locus on theright-hand side starts out at
1� d� (1)��+1d , is decreasing in ! and ends at
1� d� (0)�
�+1d .
We obtain the following pattern.
1. � (1) > � (0) > �. Then h (0) j _!=0> h (0) j _h=0 and h (1) j _!=0> h (1) j _h=0, whichmeans that the _! = 0 locus is everywhere above the _h = 0 locus. The unique equilibriumtrajectory is a jump on the saddle path that converges asymptotically to the line h =1� [�+ � (0)]. Along this path the economy exhibits population shrinking.
2. � (1) > � > � (0). Then h (0) j _!=0> h (0) j _h=0 and h (1) j _!=0< h (1) j _h=0, whichmeans that the _! = 0 locus intersects the _h = 0 locus from above. There is a uniqueequilibrium trajectory: the economy jumps on the saddle path that converges to thesteady state (!ss; hss).
3. � > � (1) > � (0). Then h (0) j _!=0< h (0) j _h=0 and h (1) j _!=0< h (1) j _h=0, whichmeans that the _! = 0 locus is everywhere below the _h = 0 locus. The unique equilibrium
trajectory is a jump on the saddle path that converges to�0; 1� [�+ � (1)]
�where the
economy exhibits a constant exponential rate of population growth.
3.4.4 Remarks
A few remarks before going further.
� We rule out paths that hit horizontal axis because they either imply h < 0 or violatethe _h equation when h stops. We rule out paths that converge to the horizontal axisbecause they violate... what?
� Then, the population implosion path with _! > 0 must be asymptotic to the _h = 0 locusas it becomes �at.
� In each case then there are well-de�ned global dynamics.
� It should be easy to augment the model with something like subsistence consumptionto obtain that paths with _! > 0 in fact do not yield ! ! 1 but stop at some trueMalthusian steady state. Probably not worth doing explicitly.
15
� Interesting that under complementarity the quasi-Malthusian steady state is the dividingpoint between the path to population implosion and the path to population explosion.Under substitution, instead, it is the global attractor of the system. The intuition is thatunder complementarity the demand for land is inelastic and the associated price changevery large. That is, as ! ! 0 and land becomes very scarce, its price rises very fast,so fast in fact that resource income rises and thereby supports further expenditure andfertility. Similarly, as ! !1 and land becomes very abundant, its price falls so fast thatresource income falls and thereby sti�es expenditure and fertility. Under substitution, incontrast, resource income moves in the same direction of ! and thus sti�es expenditureand fertility when ! falls and supports them when ! rises.
� It is worth re�ecting on the fact that the asymptotic state with constant exponential pop-ulation growth is the global attractor of the system when preferences for people/childrenare su¢ ciently high and/or the reproduction cost su¢ ciently low. This leads us to ask:Since paths with constant exponential population growth in the presence of a �niteresource (land) endowment are silly, how do we get rid of them?
� The old-fashioned trick in the literature is to postulate a purely exogenous �xed (i.e.,it does not change with the mass of children) cost of reproduction per child in unitsof the �nal good instead of in units of time. This approach, proposed in Lucas (2002),introduces a congestion e¤ect of sort and gets the job done. (Interestingly, Barro andSala-i-Martin 2004 work with a �xed cost of reproduction per child that is beyond thecontrol of the family but that, because is proportional to capital per capita, does notcreate the congestion e¤ect that in Lucas yields a model of the population level but,rather, a model of the population growth rate.)
� It is a bit hard to interpret this assumption, however. A �xed cost of reproduction perchild entails a limit on how fast population can grow, not how much it can grow. Thatis, the assumption only says that population dynamics must obey a speed limit and doesnot say that there is an upper bound on the population level. Consequently, it dodgesthe real issue posed in resource/environmental economics (and in the public debate) thatplanet hearth has a �nite carrying capacity of people. (The reason why it works in theOLG model of Lucas is that the household faces a total cost of reproduction in unitsof goods that is the �xed cost per child times the mass of children. But the mass ofchildren is the fertility rate (a number between 0 and 1 which under the Cobb-Douglasassumption is independent of the land/population ratio) times the population size, whichyields that at any point in time the household bears a �xed cost of existence that is linearin population size. Now, this holds also in our model with time cost of reproduction,so what drives the di¤erence? When we add people to the family in the Lucas model,the cost of existence rises linearly while output rises less than linearly. Consequently,since the cost of existence is in units of output, population must stabilize at some levelwhere the marginal person produces extra output just su¢ cient to support himself. Ifthe cost of reproduction is in units of time, in contrast, the marginal person brings inthe wage and adds to the family existence cost in proportion to it. It is then possibleto choose parameters such that the net contribution of children wL (1� b) is strictlypositive. More importantly, this net contribution is independent of the land/populationratio in the Cobb-Douglas case since both wage and resource income are proportional
16
to the value of production so that [wL (1� b) + p] =Y is a constant. In a Lucas-likespeci�cation this object would be [wL� bL � PY + p] =Y . If we set PY � 1 and realizethat the Cobb-Douglas yields that [wL+ p] =Y is a constant, we see that the wholeratio must hit zero as L grows.
3.5 Getting rid of in�nite population in the presence of �nite resources: apositive check
The easiest way is to postulate that the mortality rate is endogenous with the property d0 (!) <0, that is, the death rate rises as scarcity increases, i.e., as ! falls. In fact, this allows us tocapture mathematically the very notion of a population time bomb by postulating that thereexists some ~! such that for ! � ~! we have d (!) > 1= so that _! = ! (d (!)� b (h; !)) > 0 forall h since b 2 [0; 1= ]. This modi�es all phase diagrams radically because it implies that the_! = 0 locus no longer has an intercept for ! = 0 but, rather, shoots up to in�nity as ! ! ~!.
The crucial consequence is that the point�0; 1� [�+ � (1)]
�is no longer reachable � as the
very notion of a population time bomb implies! However, instead of experiencing a disaster,the economy settles smoothly into the quasi-Malthusian steady state. Speci�cally:
1. Complementarity.
(a) Case 1 remains similar since the _! = 0 locus is again everywhere above the _h = 0locus.
(b) In case 2 there are now two steady states. The one to the left, which is new, issaddle-path stable, the one to the right, which is the old one, is unstable.
(c) In case 3 the fact that mortality is increasing in scarcity creates a new saddle-path stable quasi-Malthusian steady state that becomes the global attractor of thesystem.
2. Substitution.
(a) Case 1 remains similar since the _! = 0 locus is again everywhere above the _h = 0locus.
(b) Case 2 also remains similar since the _! = 0 locus again interesects the _h = 0 locusonly once.
(c) Case 3 changes in that the the _! = 0 locus now intersects the the _h = 0 locus andthus a new saddle-path stable steady state comes into existence.
This approach has the property, possibly questionable, that by construction the stable quasi-Malthusian steady state features a rising mortality rate as the economy approaches from theright with increasing scarcity. This is a positive check.
3.6 Getting rid of in�nite population in the presence of �nite resources: apreventive check
We postulate that each individual has a �xed requirement of land �. The insight is that theprice of land p shoots up to in�nity as ! ! � and, since the model features a feedback fromp to b, we get a preventive check.
17
The household budget now reads
_A = rA+ wL (1� b) + p (� �L)� Y;
where � �L is the net supply of land. The FOC for b and L now read:
�+1� b� p�
y= �h� _h;
�+ 1
b�
y+ h = 0:
Proceeding as before, we obtain that at any point in time given the fertility choice b we candescribe the determination of the equilibrium values of y and p as the intersection of theschedules:
y =1� b+ p (! � �)
1� �� ;
y =1� b
1� ��� ��1� S (p)
:
Hence, this part of the analysis is essentially identical if we de�ne s � !��. We get functionsb (h; s), y (h; s) and p (h; s) that we can use to characterize dynamics in (s; h) space using thetwo equations:
_h = �h� �� 1� b (h; s)� p (h; s)�y (h; s)
;
_s = (s+ �) (d� b (h; s)) :
The key is that the second equation no longer admits s = 0 as a steady-state locus where_s = 0. The economics is that as s! 0, i.e., as ! ! �, net land supply vanishes and the priceof land p shoots up to in�nity. This e¤ect, present also in the basic case, now has the morerealistic property that it drives the shadow value of family size h for which _h = 0 down to �1.Accordingly, in all cases the _h = 0 locus is now increasing and if it intersects the _s = 0 locus itdoes it from below thereby creating a saddle-path stable quasi-Malthusian steady state thatis the global attractor of the system.
This approach has the property, quite plausible and desirable in our view, that the preven-tive check disciplining fertility behavior in the presence of �nite land comes from the escalatingprice of land itself.
Let�s review what drives the di¤erence between the basic case and what done here. Inthe basic case the price feedback is present but works against a preventive check because theescalating land price produces a positive income e¤ect that induces fertility to rise. The pointis that the household � who takes all prices as given � gets the "wrong" signal since it hasno reason to internalize the fact that land scarcity drives down the marginal product of labor.In the setup of this exercise, in contrast, the household perceives the downside of land scarcitydirectly through the escalating land price that makes the �xed consumption of land per personexceedingly expensive.
18
4 Dynamics: Industry
Returns to innovation are increasing in x � Y=N , which is a measure of �rm size. Taking intoaccount the non-negativity constraint on R&D, we solve (11) and (14) for
Z = �LZN=
8<: x��(��1)� � r � � x > r+���(��1)
�
0 x � r+���(��1)
�
; (22)
Take (15) and (7) and write
N =1
�
�1
�� 1
x
��+
LZN
��� �� �: (23)
Using (22) yields
N =
8><>:1��(��1)
�� � (�+ �)� �� r+��
�1x x > r+�
��(��1)�
1�� � (�+ �)�
��1x x � r+�
��(��1)�
:
In symmetric equilibrium (2), (8) and the fact that manufacturing �rms set prices at amarkup �
��1 over marginal cost (see the Appendix) yield
log u = log
��� 1�
y
cZ�N
1��1
�;
wherec � C (1; p)
One can reinterpret the utility function (2) as a production function for a �nal homogenousgood assembled from intermediate goods, so that u is a measure of output, and de�ne aggregateTFP for this economy as
T � Z�N1��1 : (24)
Taking logs and time derivatives this yields
T (t) = �Z (t) +1
�� 1N (t) ;
where Z (t) is given by (22) and N (t) by (23).Now note that
_x
x=
_Y
Y�
_N
N
=1� � (�� 1)
��� (r + �)�
�� r+��
�
1
x:
This reduces to_x (t) = �'1 (t) � x (t) + '2 (t) ; (25)
19
where
'1 (t) �1� � (�� 1)
��� r (t)� �;
'2 (t) ��� r(t)+�
�
�:
This is a linear di¤erential equation with time-varying coe¢ cients that we can solve easily.Intuition: All we need to know to pin down industry dynamics, and the associated path ofTFP growth
T (t) = �
�x (t)
�� (�� 1)�
� r (t)� ��
(26)
+1
�� 1
"1� � (�� 1)
��� (�+ �)�
�� r(t)+��
�
1
x (t)
#;
is the path of the interest rate, which follows from the demography block according to
r = �+_Y
Y:
Time path of Y straightforward from phase diagram analysis of previous section.We have already stated that the feasibility condition 1
� > (r + �)� must hold. The quasi-Malthusian steady state features both vertical and horizontal R&D if �� > �+ � and
(�+ �)� +� (�� 1)
�<1
�< (�+ �)� +
��
�+ �
� (�� 1)�
:
It then yields
xss =Y ss
N ss=
�� �+��
1��(��1)� � (�+ �)�
; (27)
which substituted into (26) yields
T ss = Zss =��� (�+ �)
1��(��1)� � (�+ �)�
� (�� 1)�
� (�+ �) : (28)
This steady-state growth rate is independent of the endowments L and because there is noscale e¤ect.
5 Sustainability
OK, we have a model that predicts �nite population on a �nite planet. Now we can tacklesustainability. Log-di¤erentiating utility yields
u = y � c+ T :
In the quasi-Malthusian steady state, y = c = 0. Consequently, uss = T ss = Zss > 0.What if the economy is on the population explosion path? We have
u = y � S (p) � p+ T :
20
In the Cobb-Douglas case y = 0 so that growth is sustainable if T > Sp. Log-di¤erentiating
yCD =p
S
�
�� 1!;
where y is independent of !, yields p = �! so that, intuitively, sustainability requires theclassic condition T > SL.
In general, log-di¤erentiatingy =
p
S
�
�� 1!
yields
p = (y � !) 1
� (p)
so that the sustainability condition reads
u = [! (1� � (p))� S (p)] � p+ T :
Then, if there is complementarity 1 > � (p) and the bracket can be positive. (Intriguingly, thissays that there is a region de�ned by ! (1� � (p)) > S (p) where positive utility growth obtainseven if T = 0.) If there is substitution, instead, 1 < � (p) and the bracket is de�nitely negative.Bottom line: This is classic reasoning (Stiglitz 1974) with growth drag and technologicalchange both endogenous. Let�s focus on population growth and rewrite
u =! (1� � (p))� S (p)
� (p)� (y � !) + T :
Recall Lemma 1 and write
y =d log y (p)
dp
dp
dt= p! [1� � (p)] p = p!
1� � (p)� (p)
(y � !) :
Then
y ��1� p!1� � (p)
� (p)
�= �! � p!1� � (p)
� (p):
Consequently,
u =
!(1��(p))�S(p)�(p)
1� p! 1��(p)�(p)
� L+ T :
To dig deeper, consider the CES technology
Xi = Z�i [& (LXi � �)� + (1� &)R�i ]
1� ; � � 1:
The associated variable unit-cost function is
Ci = Z��i
h&�
1��1w
���1 + (1� &)�
1��1 p
���1i��1
�:
Then (recall that w � 1):
S (p) =1
1 +�
&1�&
� 11��
p�
1��
; � (p) = 1 +�
1� � (1� S (p)) :
21
As p!1 we get
S (1) =�1 if complementarity0 if substitution
) � (1) =�
1 if complementarity11�� if substitution
:
Now let
r0 � lim!!0
r = lim!!0
��+ Y
�= lim
!!0
��+ y + L
�= lim
!!0
0@�+ L �241 + p! 1��(p)�(p)
1� p! 1��(p)�(p)
351A= lim
!!0
0@�+ (b� d)241 + p! 1��(p)�(p)
1� p! 1��(p)�(p)
351A = �+ b0 � d;
whereb0 � lim
!!0b:
Then, we have that '1 and '2 in (25) are constant and
x (t)! x0 ��� r0+�
�1��(��1)
� � � (r0 + �):
Accordingly,
T (t)! T0 � �
24��� r0+�
�
���(��1)
�
1��(��1)� � � (r0 + �)
� r0 � �
35+ 1
�� 1 (r0 � �) :
Then,
u (t)! u0 ��� (r0 � �) + T0 if complementarity
T0 if substitution:
This is quite strong: Under substitution sustainability is guaranteed since land is not essential.Under complementarity we need to check that TFP growth is su¢ ciently strong. OK, let�s doit:
u0 = �
24��� r0+�
�
���(��1)
�
1��(��1)� � � (r0 + �)
� r0 � �
35� (1� ) (r0 � �) ;where 1
��1 � is social returns to variety and we use to disentangle the love-of-variety e¤ectfrom the markup e¤ect due to substitution across products. The advantage of this formulais that it contains only r0, the asymptotic interest rate, and thus yields the sustainabilitycondition in terms of the fundamentals. Assume < 1 for plausibility. u0 > 0 requires
�
24��� r0+�
�
���(��1)
�
1��(��1)� � � (r0 + �)
� r0 � �
35 > (1� ) (r0 � �) ;
22
where by construction r0 � �. Also note that the interior equilibrium with positive verticalR&D requires �� > r0 + � and
(r0 + �)� +� (�� 1)
�<1
�< (r0 + �)� +
��
r0 + �
� (�� 1)�
:
Then we can study graphically where the condition holds. Useful to rewrite condition as��� r0+�
�
���(��1)
�
1��(��1)� � � (r0 + �)
>
�1 +
1� �
�[(r0 + �)� (�+ �)] : (29)
Suppose �� < 1��(��1)�� . Then the LHS is decreasing in r0 + �. The RHS is increasing and is
0 at r0+ � = �+ �. Hence there is one intersection. Suppose �� > 1��(��1)�� . Then the LHS is
increasing in r0 + � and there are either no or two intersections.Recall that r0 = �+ b0 � d and note that in this case
b0 = arg solve
�1
�
��+
1
�� ��
�=
1� b
�1
�� ��
�� �+ 1
b
�:
What we get is two cases:
� (29) is always true because LHS > RHS for all r0. Then, whatever b0 is, it yields anr0 that is sustainable.
� (29) holds in a range r0 2 (�; r0) or r0 2 (�; r0) [�r0;
1��(��1)�� � �
�inside the feasible
set��; 1��(��1)�� � �
�. Then, b0 must be such that �+ b0 � d falls inside that range.
Either way, the result is that even under complementarity we �nd thick set of parameter valuesthat yield sustainability. Why? I think it is because R&D does not use land.
6 Conclusion
This paper investigates the links between technical change, resource scarcity and income levelswhen population size is endogenously determined by fertility choices. The crucial issue is tocharacterize the reaction of technology and income to resource scarcity and the subsequentresponse of demographic forces. Using a model of Schumpeterian growth, we obtain two mainresults. First, there exists a pseudo-Malthusian equilibrium in which population convergesto a constant value while income per capita grows at a constant rate. This equilibrium ispseudo-Malthusian because the key feedback that stabilizes population is resource scarcitybut the resulting population level is not constrained by technology.
The second result is that the stability properties of the equilibrium depend on the produc-tion elasticity of substitution between labor and resource: under substitutability, the pseudo-Malthusian equilibrium is a global attractor and indeed determines the population level inthe long run; under complementarity, instead, the population level follows diverging pathsimplying either extinction or explosion. We have shown that these result is determined bythe interplay of two mechanisms determining (i) the reaction of incomes to increased resourcescarcity and (ii) the reaction of fertility rates to income levels.
23
Under complementarity, the pseudo-Malthusian equilibrium acts as a threshold: if theinitial population level is above (below) that associated to the pseudo-Malthusian steady state,the net fertility rate is initially positive (negatice); growing (declining) population then impliesincreased (decreased) resource scarcity, which generates higher (lower) income and therebyhigher (lower) expenditure and higher (lower) fertility rates. The absence of contrasting forcesgenerates a diverging path where population size tends to explode (implode).
Under substitutability, the pseudo-Malthusian equilibrium is a global attractor: if the ini-tial population level is above (below) the threshold, the net fertility rate is positive (negative)but population growth (decline) implies increased (decreased) resource scarcity that generateslower (higher) income, lower (higher) expenditure and thereby less (greater) incentive to raisechildren: the system thus converges from above (below) to the constant level of populationassociated with the pseudo-Malthusian equilibrium.
To our knowlegde, the only contributions drawing explicit links between economic growth,resource use and population dynamics are two recent papers by Schäfer (2006) and Bretschger(2008). Schäfer (2006) analyzes a sophisticated OLG model in which production possibilitiesare constrained by resource scarcity, and fertility as well as the composition of the popula-tion in terms of skilled and unskilled households are endogenously determined. A numericalsimulation of the model emphasizes the role of education subsidies versus income transfers,and suggests that, due to the interactions between directed technical change and endogenouspopulation dynamics, policy turns out to be e¤ective with respect to the long-run growth ratedespite the absence of scale e¤ects. Bretschger (2008) considers a resource-constrained econ-omy with endogenous technical progress provided by knowledge accumulation, and assumesthat the population growth rate exhibits a precise response to technology improvements. Inthis framework, population growth and poor input substitution are not detrimental but, onthe contrary, even necessary for obtaining a sustainable consumption level.
With respect to these contributions, our analysis di¤ers in that we provide a full analyticalcharacterization of the interactions between resource scarcity and population growth underutility-maximizing fertility choices. Also, the nature of the pseudo-Malthusian equilibriumstudied here implies several di¤erences with respect to the existing literature on fertility.Balanced growth models typically provide theories of population growth as they hinge onequilibria with constant exponential growth of the population mass whereas our model suggestsa theory of the population level. In particular, our result of pseudo-Malthusian steady stateimplies that the economy may achieve equilibria with constant population where populationsize is determined by resource scarcity but is still below the physical upper-bound imposed bythe natural carrying capacity of the habitat. The carrying-capacity argument remains howeverrelevant because the pseudo-Malthusian steady state may not be the global attractor of thesystem.
A Appendix
The typical �rm�s behavior
To characterize the typical �rm�s behavior, consider the Current Value Hamiltonian
CV Hi = [Pi � C(1; p)Z��i ]Xi � �� LZi + zi�KLZi ;
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where the costate variable, zi, is the value of the marginal unit of knowledge. The �rm�sknowledge stock, Zi, is the state variable; R&D investment, LZi , and the product�s price, Pi,are the control variables. Firms take the public knowledge stock, K, as given.
Since the Hamiltonian is linear, one has three cases. The case 1 > zi�K implies thatthe value of the marginal unit of knowledge is lower than its cost. The �rm, then, does notinvest. The case 1 < zi�K implies that the value of the marginal unit of knowledge is higherthan its cost. Since the �rm demands an in�nite amount of labor to employ in R&D, thiscase violates the general equilibrium conditions and is ruled out. The �rst order conditionsfor the interior solution are given by equality between marginal revenue and marginal cost ofknowledge, 1 = zi�K, the constraint on the state variable, (11), the terminal condition,
lims!1
e�R st [r(v)+�]dvzi(s)Zi(s) = 0;
and a di¤erential equation in the costate variable,
r + � =_zizi+ �C(1; p)Z���1i
Xi
zi;
that de�nes the rate of return to R&D as the ratio between revenues from the knowledge stockand its shadow price plus (minus) the appreciation (depreciation) in the value of knowledge.The revenue from the marginal unit of knowledge is given by the cost reduction it yields timesthe scale of production to which it applies. The price strategy is
Pi = C(1; p)Z��i�
�� 1 : (A.1)
Peretto (1998, Proposition 1) shows that under the restriction 1 > � (�� 1) the �rm is alwaysat the interior solution, where 1 = zi�K holds, and equilibrium is symmetric.
The cost function (10) gives rise to the conditional factor demands:
LXi =@C (w; p)
@wZ��i Xi + �;
Ri =@C(w; p)
@pZ��i Xi:
Then, the price strategy (A.1), symmetry and aggregation across �rms yield (12) and (13).Also, in symmetric equilibrium K = Z = Zi yields _K=K = �LZ=N , where LZ is aggregate
R&D. Taking logs and time derivatives of 1 = zi�K and using the demand curve (8), theR&D technology (11) and the price strategy (A.1), one reduces the �rst-order conditions to(14).
Taking logs and time-derivatives of Vi yields
r + � =�XiVi
+_ViVi;
which is a perfect-foresight, no-arbitrage condition for the equilibrium of the capital market.It requires that the rate of return to �rm ownership equal the rate of return to a loan of size Vi.The rate of return to �rm ownership is the ratio between pro�ts and the �rm�s stock marketvalue plus the capital gain (loss) from the stock appreciation (depreciation).
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In symmetric equilibrium the demand curve (8) yields that the cost of entry is � YN . Thecorresponding demand for labor in entry is
LN =�_N + �N
��Y
N:
The case V > � YN yields an unbounded demand for labor in entry, LN = +1, and is ruledout since it violates the general equilibrium conditions. The case V < � YN yields LN = �1,which means that the non-negativity constraint on LN binds and _N = ��N , which impliesnegative net entry due to the death shock. Free-entry requires V = � YN . Using the pricestrategy (A.1), the rate of return to entry becomes (15).
The economy�s resources constraint
I now show that the household�s budget constraint reduces to the economy�s labor marketclearing condition. Starting from (3), recall that A = NV and (r + �)V = �X + _V . Substi-tuting into (3) yields
_NV = N�X + L+ p� Y:
Observing that N�X = NPX � LX � LZ � pR, NPX = Y , R = , and that the free entrycondition yields that total employment in entrepreneurial activity is LN = _NV , this becomes
L (1� b) = LN + LX + LZ :
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