Download Sustainability and substitution of exhaustible natural resources

Sustainability and substitution of exhaustible
natural resources
How structural change affects long-term R&D-investments
Lucas Bretschger* and Sjak Smulders**
June 2010
Abstract
We study long-run growth in a multi-sector economy with non-renewable
resource use and endogenous innovations. Unlike recent capital resource
models, we find that poor input substitution need not be detrimental for
sustainable growth; on the contrary, combined with resource depletion it
fosters structural change, which helps to sustain research investments. We
derive the properties of the transition path, show which sectors survive in
the long run, and discuss whether the economy approximates a steady state
with or without a scale effect. The results continue to hold when some
sectors exhibit perfect competition.
Keywords :
Growth, non-renewable resources, substitution, investment
incentives, endogenous technological change, sustainability
JEL-Classification : Q20, Q30, O41, O33
* Corresponding author: CER-ETH Center of Economic Research at ETH Zurich,
ETH-Zentrum, ZUE F7, CH-8092 Zurich, [email protected].
** Department of Economics, Tilburg University, P.O.Box 90153, 5000 LE Tilburg,
The Netherlands, [email protected].
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1
Introduction
The recent surge in oil prices and the debate on climate policies have reawakened
interest in whether and how natural resource scarcity impairs development prospects
for future generations. The critical question is to which degree capital and future
technologies can substitute for decreasing input of natural resources like fossil fuels.
In this process, input substitution and innovations are complemented by sectoral
change, which is increasingly emerging as a crucial mechanism to obtain
sustainability. The International Energy Agency states that, in the last decades,
reductions in energy intensity were due to a “combination of structural changes and
efficiency improvements” (IEA 2008, p.15), while the Stern Review concludes that
“structural changes in economies will have a significant impact on their emissions”
(Stern 2007, p.180).
The present paper analyses the impact of natural resource depletion and the
effects of input substitution on sectoral change and long-term growth. Our main
finding is that poor substitution between natural resources and man-made inputs is
not detrimental for long-run growth, once structural change and endogenous
innovations are taken into account. In our multi-sector setting, poor input substitution
may promote sectoral change and enhance investment activities. The main
mechanism underlying this result is sectoral reallocation of labour. Given the
depletion of the resource, sectors with relatively good substitution possibilities are
less vulnerable to rising resource prices and increase employment at the cost of the
other sectors. When the relative size of the innovative sectors increases, innovation
incentives remain intact in the long run and sustainability can be obtained. Thus,
unlike in the existing literature, long-run growth can be sustained under free market
conditions even when input substitution is poor. However, without structural change,
poor input substitution leads to a non-sustainable future, despite the assumption of
scale-dependent learning in the economy.
We also show how intersectoral factor reallocation can neutralise the scale
effect often present in endogenous growth models. This happens despite the
assumption that learning spillovers are proportional to aggregate investments. We
thus provide a new approach to eliminate scale effects in growth models. The relative
size of the different substitution elasticities governs the process. In addition, we
explicitly derive how many and which sectors survive in the long run, which depends
on initial conditions. The number of surviving sectors determines whether the
economy approximates a steady state with or without a scale effect.
The paper builds on the seminal contributions on growth and natural resource
use, see Solow (1974a) and Dasgupta and Heal (1974) and (1979). The problem of poor
input substitution in a one-sector model is treated in Solow (1974b, p.11) and briefly
discussed by Stiglitz (1974), but only in combination with exogenous technological
change. The paper relates to the endogenous growth literature by adopting the
modelling of endogenous technological change from Romer (1990) and Grossman and
Helpman (1991). The combination of endogenous knowledge accumulation and nonrenewable resource depletion is also analysed in Barbier (1999), Scholz and Ziemes
(1999), Schou (2000), Groth and Schou (2002), and Grimaud and Rougé (2003). But
with the exception of Lopez et al. (2007), these models focus on one-sector economies
or unitary elasticities, thereby abstracting from sectoral shifts due to resource
2
depletion. Elasticities of substitution below unity are analysed in some models of
renewable resources and endogenous growth, see Bovenberg and Smulders (1995),
Bretschger (1998), and Peretto (2009), but not in the context of non-renewable
resources. The problems regarding the “scale effects on growth” have been discussed
extensively in Jones (1995, 1999). The impact of structural change on growth is treated
in Kongsamut, Rebelo, and Xie (2001). Ngai and Pissarides (2007) study the
implications of different sectoral productivity growth rates for structural change. In a
setting with resource depletion in small open economies, Lopez et al. (2007) state that
shrinking resource-dependent sectors and expanding knowledge-intensive sectors are
consistent with well accepted stylized facts. We build on these observations and
present a model where productivity growth and all the prices are fully endogenous.
In order not to predetermine our major sustainability results by optimistic
model assumptions, we aim at using a precautionary framework, especially with
regard to the core elements of the model. This is why we extensively explore the
consequences of sub-unitary elasticities of input substitution. Moreover, to stay on the
conservative side, we do not allow for research which is directly aimed at improving
the productivity of certain inputs as in Acemoglu (2002). The application of this
directed technical change on natural resource use is included in Smulders and de
Nooij (2003), di Maria and Valente (2008), and Pittel and Bretschger (2010). Here we
look at resource-using technical change and assume that innovations are equally used
in all the sectors, like computers or mobile phones. Learning by doing is assumed to
support sectoral productivity in a Hicks-neutral way, which appears to be realistic.
We show what happens when the sectors are heterogeneous, each with different
opportunities for substitution, innovation, and learning spillovers. This allows us to
analyse how resource depletion drives sectoral shifts and, consequently, has an
impact on the incentives for innovation. We do not build on the accumulation of
physical capital in order to do justice to material balance principles. Finally, we
extend our framework by including the possibility that certain sectors exhibit perfect
competition.
The remainder of the paper is organised as follows. In section 2, the theoretical
multi-sector model of the economy is presented. Section 3 shows how the model can
be solved. Section 4 provides results for long-run growth for different types of
parameter and substitution conditions. Section 5 considers transitional dynamics.
Section 6 generalises the results by introducing sectors with perfect competition and
section 7 concludes.
2
The model
2.1 Production
The economy consists of J different sectors (j = 1, 2,…, J) producing sectoral output Yj
with intermediate input Kj and exhaustible natural resource input Rj, according to the
following CES production function:
3
σ j −1
σj
⎡
Y j = Aj ⋅ ⎢v j K j
+ (1 − v j ) R j
⎢
⎣
v
1− v
Y j = Aj K j j R j j ,
σj
σ j −1 σ −1
j
σj
⎤
⎥
⎥
⎦
,
σ j ≠1
(1)
σ j =1
where 0 < v j < 1 and σ j > 0 are time-invariant parameters; Aj is the total-factorproductivity index, and σj is the constant elasticity of substitution between natural
resource input and produced intermediate input. Time indices are omitted whenever
there is no ambiguity.
Kj is an aggregate of heterogeneous intermediate varieties, Kji. At each moment
in time, a mass of N (t) varieties is available in each sector; the constant elasticity of
substitution between the varieties equals 1/(1 − β j ) > 1 (with 0 < β j ≤ 1 ). Aggregate
intermediate input in sector j is I j (t ) = ∫ K ji (t )di . In a symmetrical equilibrium, the
quantity of variety i in sector j, denoted by K ji , is equal for all varieties i, i.e. we have
K ji = I j / N , ∀i, and:
Kj =
(∫
N
0
βj
K ji di
)
1
βj
=N
ψj
⋅Ij
(2)
where ψ j = (1 − β j ) / β j measures the gains from diversification. According to (1) and
(2), holding I j constant, output increases with the number of intermediates, N . The
ψ
term N j captures the gains from specialization in the use of intermediates, as
introduced by Ethier (1982) and common in the endogenous growth literature
(Romer, 1990 and Grossman and Helpman, 1991). Note that the multiplication of I j
ψ
with the productivity parameter N j stems from the symmetry in the CES-function in
(2) and not from a Cobb Douglas assumption. In addition, we assume that, through
learning by doing, gains from specialisation spill over to total factor productivity in
sector j in the following way:
Aj = N
δj
(3)
where δ j measures the strength of spillovers. Accordingly, when specialization
increases, production possibilities shift in two distinct directions: total factor
productivity improves, and intermediate inputs become more productive relative to
resource inputs.
Perfect competition prevails in the market for Yj-goods; resources are mobile
between the sectors. Producers take prices of sectoral output, resource inputs and
intermediate goods (denoted by pYj , pR and pKj , respectively) as given. They
N
maximise total profits pYjY j − pR RY − ∫ pKj K ji di , subject to (1) and (2). Relative demand
0
for intermediates and resources then becomes:
⎛ vj
=⎜
R j ⎜⎝ 1 − v j
Ij
σj
⎞ ⎛ pKj ⎞
⎟⎟ ⎜
⎟
⎠ ⎝ pR ⎠
−σ j
N
− (1−σ j )ψ j
.
(4)
4
Equation (4) reveals that if (1 − σ j )ψ j < 0, technological change (i.e. an increase in N) is
resource-saving; if (1 − σ j )ψ j > 0, it is resource-using; if σ j = 1 , technical progress is
Hicks-neutral. To remain on the conservative side with respect to technological
opportunities we assume poor substitution, i.e. 0 < σ j ≤ 1 , so that – as ψ j > 0 – we rule
out resource-saving technical change.
The N intermediate goods are produced by monopolists. One unit of
production requires one unit of labour L, so that the unit cost is w. The monopolists
sell to all J sectors. In any sector j, the profit-maximising monopolistic supplier faces a
price elasticity of demand equal to −1/(1 − β j ) . As in the standard Dixit-Stiglitz
approach, this follows from the Yj-producers demand for Kj. It follows that the
optimal price to be charged in sector j equals:
pKj = w / β j .
(5)
Due to sector-specific price setting, the profit margins intermediate goods producers
realise generally differ across sectors; they are zero in sectors with β j = 1 . Total profits
for the supplier of a single intermediate goods π are, from (2) and (5):
⎛
J
⎝
j =1
⎞1
.
N
⎠
π = ⎜ ∑ wψ j I j ⎟
(6)
Profits are used to cover the upfront investment cost in the production of K-goods,
which consist of the cost of acquiring the blueprint for the intermediate good. Each
blueprint contains the know-how for the production of one intermediate K.
2.2 Innovation
Innovation expands intermediate goods variety, N, as in Grossman and Helpman
(1991, Chapter 3). We define the innovation rate as:
g = N / N
(7)
The R&D sector produces a flow N of blueprints for new varieties. It is assumed that
an increase in variety increases the stock of public knowledge on which R&D builds
so that research costs decline with N. In particular, per blueprint, a/N units of labour
are required. Research efforts and associated knowledge spillovers provide the
driving force for long-run growth.
There is free entry into R&D. Whenever the cost to develop a new blueprint,
w ⋅ (a / N ) , is lower than the market value of a blueprint, denoted by pN, entry will
compete away the rent. Hence we have:
aw / N ≥ pN
with equality (inequality) if g > 0 (g = 0).
(8)
5
The market value of a blueprint follows from the condition that investors earn the
market interest rate r when investing in blueprints, thus earning profits π and capital
gains p N :
π + p N = r ⋅ pN
(9)
We combine (6)-(9) to get:
J
Ij
j =1
a
g > 0 ⇒ r − wˆ = ∑ψ j
−g
(10)
Equation (10) characterizes the return to innovation in sector j, which increases in the
size of j, the mark-up over marginal cost in j, and the productivity in the research lab;
it decreases with the innovation growth rate.
2.3 Factor markets
The total stock of the non-renewable resource at time τ is denoted by S (τ ) . It is
depleted according to:
J
S = −∑ R j ,
S (0) given,
S (τ ) ≥ 0 ,
(11)
j =1
which reflects that any flow of resource use depletes the total resource stock
proportionally, that the resource stock is predetermined, and that the stock can never
become negative.
The labour market is in equilibrium that is the fixed supply L equals labour
demand in intermediate goods production and in research:
J
L = ∑ I j + ag
(12)
j =1
2.4 Households
The representative household inelastically supplies L units of labour; it owns the
resource stock with value pRS and equity in intermediate goods firms with value pNN.
The household consumption choice is determined in two steps. First, the household
decides at each point in time how to allocate total consumption expenditure across
different goods. The consumption index is given by Cobb-Douglas preferences over
the sectoral consumption goods, i.e.:
J
C = ∏ Y j j , with
j =1
φ
J
∑φ
j =1
j
= 1,
so that the first order conditions for maximum C with given prices read:
(13)
6
( pYj ⋅ Y j ) / ( pYj ' ⋅ Y j ' ) = φ j / φ j '
(∀j , j ') .
(14)
According to (14), specification (13) implies constant expenditure shares φ j for
sectoral output. In the second step, consumers choose the intertemporal profile of
total expenditure by maximising utility over an infinite horizon subject to their
intertemporal budget constraint. We assume a logarithmic utility function and
constant discount rate ρ so that the household born at time t maximises:
∞
U (t ) = ∫ e − ρ (τ −t ) ln C (τ )dτ ,
(15)
V = rV + pR R + wL − pC C ,
(16)
t
subject to (11) and
with V = pN N . The current-value Hamiltonian of this optimisation problem reads:
H = ln C + μ1 [ rV + pR R + wL − pC C ] − μ2 R
(17)
where μ1 , μ2 denote the costate variables. Necessary conditions for an interior solution
are given by the following first order and transversality conditions:
1/ C = pC μ1
μ1 pR = μ2
μ1 = ρμ1 − r μ1
μ 2 = ρμ 2
(18)
(19)
(20)
(21)
lim[ μ1 (τ )V (τ )]e − ρ (τ −t ) = 0
(22)
lim[ μ 2 (τ ) S (τ )]e − ρ (τ −t ) = 0
(23)
τ →∞
τ →∞
The transversality conditions (22) and (23) require that total firm and resource wealth
each approaches a value of zero in the long run. Differentiating (18) logarithmically
with respect to time and using (20) yields the Keynes-Ramsey rule:
pˆ C + Cˆ = r − ρ ,
(24)
where hats denote growth rates and pC is the consumer price index. The equation
states that the growth rate of consumer expenditures is equal to the difference
between the nominal interest rate r and the discount rate ρ . Differentiating (19)
logarithmically with respect to time and using (20) and (21) gives the Hotelling rule:
pˆ R = r
(25)
7
which guarantees that households are exactly indifferent between selling resources
(and investing the profit with interest rate r ) and preserving the stock of resources
(and earning capital gains because of increases in the resource price).
3
Solving the model
To characterise the dynamics of the system we introduce the intermediate goods
value shares in the different sectors, v j , and the proportional extraction rates uj and a
variable y, defined by:
p I
v j ≡ Kj j
(26)
p jY j
u j ≡ Rj / S
(27)
y ≡ pC C / w
.
(28)
Using (4), (5), (25) and (26) as well as (24) and (28) we derive the following differential
equations, see the appendix:
(
vˆ j = −(1 − v j )(1 − σ j ) r − wˆ + ψ j g
(
)
(29)
)
Iˆ j = ⎡⎣1 − (1 − v j )(1 − σ j ) ⎤⎦ r − wˆ +ψ j g −ψ j g − ρ
(30)
Recall that Ij is the amount of labour employed to produce intermediates for sector j.
Hence, (30) characterizes the dynamics of the employment share of sector j , the
conventional measure for structural change. Furthermore, using (20) and expressing
the transversality condition (22) in percentage changes yields:1
limτ →∞ r (τ ) − wˆ (τ ) ≥ 0
for limτ →∞ g (τ ) > 0 .
(31)
By combining (29) and (31) it becomes clear that intermediate goods shares decrease
(i.e. vˆ j < 0 ) when input substitution is poor ( σ j < 1 ) and stay constant ( vˆ j = 0 ) with
unitary elasticities ( σ j = 1 ) which is the well-known Cobb-Douglas case. Both variants
will be considered in the following. We now state:
Lemma 1
The dynamics of the system are described by the equations (32)-(35):
τ
τ
− ∫ [ r ( s )] ds
t
1 From (20) we have μ (τ ) = μ (t )e ∫t [ ρ − r ( s )]ds . Inserting in (22) yields lim
= 0 with
τ →∞ V (τ )e
1
1
V = pN N and pN according to (8), giving Vˆ = wˆ and (31). Combining (21) and (23) in the same way
yields limτ →∞ S (τ ) = 0 .
8
yˆ =
y
B2 − ( ρ + g ) ,
a
(32)
y
L⎫
⎧
vˆ j = −(1 − v j )(1 − σ j ) ⎨[ B1 (1 −ψ j ) + B2 ] − (1 −ψ j ) ⎬ ,
a
a⎭
⎩
g=
L y
− B1 ,
a a
(33)
(34)
uˆ j = (1 − σ j ) yˆ + vˆ j + (1 − σ j )ψ j g − σ j ρ + u j ,
J
J
j =1
j =1
(35)
where B1 = ∑ β jφ j v j , B2 = ∑ (1 − β j )φ j v j .
Proof in appendix. □
The steady state of the system is defined by v j = u j = 0 . Note that also I j = 0 in the
long run. We use the fact that the system is decomposable: (32)-(34) are sufficient to
find equilibrium innovation (and consumption) growth. This will be the focus in the
following. After finding equilibrium values for y, v j , and g from (32)-(34) and
furthermore employing (30), the equilibrium depletion rate in (35) can be derived
separately.
4
Long-run growth
4.1 Cobb-Douglas case
With unitary elasticities between man-made inputs and natural resources, i.e. σ j = 1
(∀j ) , we arrive at:
Proposition 1
Provided that σ j = 1 (∀j ) there are no transitional dynamics and the
innovation rate equals:
L
⎧
⎫
g = max ⎨ (1 − θ ) − θρ , 0 ⎬
a
⎩
⎭
J
J
j =1
j =1
where θ ≡ ∑ β jφ jν j / ∑ φ jν j .
(36)
9
Proof
From (29) we observe that vˆ j = 0 ; the v j s are predetermined (see 1) and we
can use the system consisting of (32) and (34), which is unstable in y, that is it
immediately jumps to a state where yˆ = 0 . Solving (32) and (34) for g yields (36). □
The rate of innovation is stimulated by a higher supply of labour L, a lower unit input
coefficient in research a, higher mark-up rates 1/β (affecting θ), and a lower discount
rate ρ . This corresponds to the findings in research-driven growth models. The new
result here is that the sector specific combinations of mark-ups 1/βj, expenditure
shares φ j , and production elasticities v j determine the weights θ and 1 – θ valuing
the relative impact of efficient labour L/a and the discount rate. Naturally, a high
mark-up in a specific sector is the more favourable for innovation the larger are the
intermediate goods and expenditure shares of that sector. With σ j = 1 (∀j ) there is no
sectoral change in the economy. In the symmetric case with identical mark-ups in all
the sectors, i.e. β1 = β 2 = ... = β , we obtain for the innovation rate the expression
g = (1 − β )( L / a) − βρ ≥ 0 which is identical to the result in the one-sector model
without non-renewable resource inputs of Grossman and Helpman (1991, p. 61).
4.2 Poor input substitution
In the more general case of poor input substitution, σ j < 1 (∀j ) , we have sectoral
change during transition to the long-run steady state. Specifically, labour is moving
between the different sectors as well as between intermediates’ production and the
research sector. The nature of the steady state then depends on how many and which
sectors survive the transition process. A sector is said to survive if there is constant
positive labour input into a sector in the steady state; a vanishing sector has an
asymptotically zero employment level in the steady state.
We first study a long-run equilibrium where two sectors survive in the long
run, which corresponds to one of the two feasible states (as analysed below in Section
5). The two unequal sectors are labelled j″ and j’, where we assume
σ j′′ ≠ σ j ' and ψ j′′ ≠ ψ j ' .
(37)
The other sectors experience a constant outflow of labour but still produce
infinitesimal output in the long run due to (14). We now state:
Proposition 3
With two sectors surviving in the long run, we obtain that: (i) the
innovation rate is positive when the sector with better input substitution (higher σ) has
sufficiently stronger gains from diversification (viz. higher (1 – σ)ψ), (ii) innovation activities
increase with the discount rate, (iii) growth does not vary with the size of the labour force.
Proof Let sector j′ and j″ be the only sectors with non-vanishing employment in the
steady state, so that Iˆ j′ = Iˆ j ′′ = 0 . From (29) we find v j′ → 0, v j′′ → 0 . Substituting these
10
results into (30), once for j=j′ and once j=j″, we find two equations in two unknowns, g
and r − wˆ . Solving for g, using the non-negativity constraint, we find:
⎧⎪
⎫⎪
(σ j′′ − σ j ' ) ρ
, 0 ⎬ ≡ g j′j′′ .
g = max ⎨
⎪⎩ σ j ' (1 − σ j′′ )ψ j′′ − σ j′′ (1 − σ j ' )ψ j ' ⎪⎭
Hence, g > 0 requires (1 − σ j′ )ψ j′ > (1 − σ j′′ )ψ j′′ if σ j′ > σ j′′ .
(38)
□
The main result is that innovation can be sustained even with poor input
substitution in all sectors. According to (38), a positive innovation rate requires that
the sector with highest substitution elasticity σ is sufficiently responsive to technical
change. The economic intuition behind this result is that two opposing – but
inseparable – forces from depletion and technological change determine labour
allocation. On the one hand, we have a “differential substitution effect”: as the
resource stock is depleted, labour tends to move to the sector with the better
substitution possibilities. On the other hand, there is an “innovation effect”:
innovation causes resource-using technological change (since ψ > 0) and increases the
demand for resources in the sector with high (1 − σ )ψ , as is shown in (4), which makes
it harder for this sector to employ labour. In the steady state, the two effects exactly
offset each other and both sectors keep employing labour so that innovation remains
economically attractive. Notably, the larger is heterogeneity with respect to input
substitution the higher becomes the innovation rate.
The innovation rate increases with the discount rate. The reason is that
resource conservation and innovation investments do not necessarily move in the
same direction. An increase in the discount rate makes investors less patient so that
the resource stock is depleted faster. This implies that the differential substitution
effect becomes stronger and there is room for a stronger innovation effect to
counteract. Although discounting reduces investment in the resource by speeding up
depletion, it makes investment in innovation more attractive if depletion expands the
sector with the better substitution possibilities. That entry cost in R&D depend
negatively on the discount rate resembles the findings of earlier non-scale growth
models, see Peretto and Smulders (2002).
Interestingly, an increase in the labour force has no effect on the innovation
rate, as it would make the innovation effect dominate the depletion effect (which is
fixed at ρ) but this is self-defeating. Thus when two sectors survive we get an
equilibrium without the scale effect, i.e. g becomes independent of L, which has been
emphasised as an important feature in growth models e.g. by Jones (1995, 1999).
In the same line we now analyse the case with only a single sector surviving in
the long run, which is fundamentally different from the symmetric case above. It will
turn out to be equally important in the next section. We can show:
11
Proposition 4
With poor input substitution and a single sector surviving in the long run,
sustainable development is feasible when the discount rate is not too high; the growth rate
exhibits a scale effect.
Proof
Let sector s be the only sector with non-vanishing employment in the steady
state, so that I j = 0, j ≠ s , and Iˆs = 0 . From (29) we get vs = 0 . Now combining these
results with (10), (12) and (30), we find:
⎧σ ψ L / a − ρ ⎫
g = max ⎨ s s
, 0⎬ ≡ g s . □
⎩ σ s +ψ s
⎭
(39)
Contrary to the two-sector case, the growth rate in (39) depends on the size of the
labour force and the discount rate has a negative impact on the innovation rate.
Similar to the two-sector case, poor input substitution can be compatible with
sustained innovation activities.
4.3 Consumption growth
To study how resource dependence affects growth of consumption rather than
innovation, we need to calculate output growth in all sectors. Besides innovation, only
the depletion of resource inputs drives the growth process, since aggregate labour
input is constant. For the Cobb-Douglas case we arrive at:
Proposition 5
Assuming unitary elasticities in production steady-state consumption
growth in the long run is given by:
J
Cˆ = ∑ φ j ⎡⎣(δ j + ψ j v j ) g − (1 − v j ) ρ ⎤⎦ .
(40)
j =1
Proof
In order to satisfy the transversality condition (23), both the stock of
resources and sectoral resource use must decline at the rate − Sˆ = − Rˆ j = ρ .
Differentiating consumption (13) and production functions (1) and (2) with respect to
time and combining we obtain (40). □
Consumption grows at a positive rate only if innovation (at rate g, see first term at
right-hand side) is sufficiently large to offset the decline in resource inputs (at the rate
ρ, see second term). Consumption growth is bigger, the larger are productivity
spillovers (δ). A lower discount rate (ρ) reduces resource depletion and implies a
smaller drag on growth from the scarcity of non-renewable resources.
For the case of poor input substitution we state:
12
Proposition 6
Assuming poor input substitution in production steady-state consumption
growth in the long run is given by:
⎛ J
⎞
Cˆ = ⎜ ∑ φ jδ j ⎟ g − ρ
⎜ j =1
⎟
⎝
⎠
(41)
where g is determined according to the results in sections 4.2 and 4.3.
Proof
We use v j instead of v j and take v j (∞) = 0 in the long run to simplify (40)
and to obtain (41). □
Whenever endogenous knowledge accumulation affects the productivity of Yproduction ( δ > 0 ), innovation implicitly raises the productivity of resources which
makes long-run consumption growth technically feasible. However, in the market
equilibrium, consumption grows only if incentives to innovate are large enough
relative to the incentive to deplete the resource stock. This requires:
(i)
sufficiently large spillovers δ ,
(ii)
a sufficiently high innovation rate g, and
(iii) a sufficiently low discount rate ρ .
Under these conditions, the discount rate affects consumption growth in an
ambiguous way: in the scale equilibrium, an increase in the discount rate lowers both
innovation and consumption growth; however, for the non-scale equilibrium, an
increase in the discount rate raises both innovation and consumption growth. Because
the discount rate has an impact on the nature of the long-term equilibrium, we obtain
an inverted-V shaped relationship between the discount rate and long-run
consumption growth.
5 Sector Structure Convergence
We now study which sectors survive if the economy starts from an initial situation
with many sectors. Specifically, we state:
Proposition 7
If input substitution is poor in all sectors and sectors are heterogeneous, i.e.
σ j ≠ σ j ' and/or ψ j ≠ ψ j ' for all j ≠ j ′ , the economy convergences either to two sectors or to
a single sector with constant positive employment in the long run.
Proof
From (29), we find v j → 0 in the long run. From (30) with v j = 0 we get:
>
> 1−σ j
ρ
Iˆ j 0 ⇔ (r − wˆ )
ψ jg −
<
< σj
σj
.
(42)
Recall that Iˆ j < 0 means a vanishing sector, while Iˆ j > 0 cannot occur in a steady state
since it violates the labour constraint. Hence, if sector j survives, we must have Iˆ j = 0 .
13
Let us first hypothetically assume that no sector survives. Then (12) implies g = L/a, so
that by (10) and (12) r − wˆ = − L / a < 0. Since this violates the transversality condition
(31), an equilibrium without any surviving sector is not feasible. Now suppose three
or more sectors survive. Then, there are three or more conditions like (42) with
equality signs in only two unknowns, g and r − wˆ ; hence this is also impossible.2 But if
two sectors survive, say j′ and j″, (42) holds with equalities for j′ and j″. This gives
two equations in two unknowns and the solution for g given by (38). This can be an
equilibrium only if all other sectors are vanishing, which requires:
Iˆ j < 0 ⇔ (1/ σ j′ − 1)ψ j′ g j′j′′ − ρ / σ j′ = (r − wˆ ) < (1/ σ j − 1)ψ j g j′j′′ − ρ / σ j , ∀j ≠ j ′, j ≠ j ′′ .
Finally, if one sector, say s, survives, (42) holds with equalities for j = s and together
with (10) and (12) we can solve for the growth rate as given in (39). This can be an
equilibrium only if all other sectors are vanishing, which requires:
Iˆ j < 0 ⇔ (1/ σ s − 1)ψ s g s − ρ / σ s = r − wˆ < (1/ σ j − 1)ψ j g s − ρ / σ j for all j ≠ s . □
We illustrate the implications for the long-run growth rate in Figures 1 and 2.
The upper part of Figure 1 shows the long-run Iˆ j = 0 -locus, as defined by (42), in the
g , r − wˆ plane for one particular sector j. If the long-run equilibrium is a pair (g, r − wˆ )
located above the locus, this sector would expand, Iˆ j > 0 , which cannot be
maintained indefinitely. A pair (g, r − wˆ ) below the locus implies that the sector
vanishes. If the sector survives in the long run, the equilibrium innovation and
interest rate must be on the line. In the lower panel we depict equation (39) to indicate
the amount of labour needed to sustain a long-run growth rate with sector j only.
In Figure 2 we depict the same loci, but now for three sectors. For a given
growth rate g, one or two sectors can survive in the steady state, provided the
associated pair (g, r − wˆ ) does not imply that another sector expands (which would in
the end violate the labour constraint). Since points above any Iˆ j = 0 locus implies that
the corresponding sector would expand indefinitely, the equilibrium pair (g, r − wˆ )
must be on the lowest Iˆ j = 0 locus in the upper panel (corresponding to the surviving
sector), and hence below the loci of the other sectors. Since the loci may intersect, it is
possible that this lowest point is on two loci, in which case two sectors survive. Once
we have identified the surviving sectors in the upper panel, we can plot the associated
labour requirements from (39); see the upward sloping segments in the lower panel.
The vertical segments correspond to equilibria with two sectors: the growth rate is
then independent of the labour supply as shown in proposition 3.
The lower panel of Figure 2 thus visualizes how the endogenous innovation
rate depends on labour supply. The labour endowment (divided by the given a)
determines which sectors are able to survive. For example, with a small labour
endowment L / a < l1* , the market for innovation is small which prevents too fast
resource-using progress. Sector 1 has superior substitution possibilities (σ1 is high)
and by the “differential substitution effect” it drives all other sectors out of the market
2 A pair (g, r − wˆ ) is only consistent with Iˆ = 0 in three or more sectors under specific knife-edge conj
ditions, which means that we would need an exact relationship between exogenous parameters while
any arbitrarily small deviation from it causes the transition to the results of the main text.
14
when over time the resource becomes scarcer. However, when the labour endowment
is larger, resource-using technical progress becomes more profitable. Since sector 1 is
more affected by technical progress (its value for (1 – σ)ψ is bigger), faster technical
change makes sector 1 relatively more dependent on resources. This makes it harder
for sector 1 to compete for scarce resource with sector 2. At labour endowment
L / a = l1* sector 1 can no longer drive sector 2 out of the market and both survive. With
further increases in the scale of the economy, sector 1 loses competitiveness. When the
scale is larger than l2* , only sector 2 survives.3
It thus emerges that
(i)
the two cases analysed in subsection 4.3 are the only relevant long-run
equilibria;
(ii)
sectors survive according to the relative strength of the innovation effect
and the differential substitution effect;
(iii) the labour endowment drives the innovation effect and thus governs the
selection of surviving sectors.
6 Sectors with perfect competition
We now show that our results continue to hold when some sectors in the economy
have perfect competition and do thus not contribute to the compensation of
innovation efforts; this makes the results even more general. Specifically, we consider
dynamics in a two-sector framework with a “traditional” sector T with perfect
competition and a “high-tech” sector H according to the sectors considered above. It
is useful to start again with the Cobb Douglas case (unitary input elasticities) as a
reference point. We find:
Proposition 8
Assuming the economy consists of two sectors with unitary substitution
elasticities, labelled H (“high-tech”) and T (“traditional”), with H having monopolistic and T
perfect competition, i.e. β H = β and , βT = 1 , the innovation rate becomes:
⎧ (1 − β ) L / a − βρ − ρ (vT / vH )φT / φH ⎫
g = max ⎨
,0 ⎬ .
1 + (vT / vH )φT / φH
⎩
⎭
Proof
(43)
Use (32) and (34) for yˆ = 0 and predetermined v j and insert the conditions
β H = β and βT = 1 to obtain (43). □
The innovation rate decreases with (vT / vH )φH / φT , which captures three
effects. First, since innovation takes place in the H-sector only, a lower expenditure
share on H-goods (lower φH ) reduces innovation. Second, since innovation is
3 It can be shown that the effect of different substitution possibilities on structural change continues to
hold for the case of good input substitution but this is not the focus of the paper.
15
embodied in intermediate goods in the H-sector, a smaller role for intermediates, as
measured by a smaller intermediate goods share vH , decreases the market for
innovations, which makes research less profitable. Finally, innovation is low when the
share of non-renewable resources in the T-sector is low (high vT ). If the T-sector relies
heavily on resources rather than labour input, less labour is allocated to this sector,
and more becomes available for the research sector.
We next turn to the case in which substitution is more difficult in the
traditional sector T than in the high-tech sector H, which – as above – has a unitary
elasticity of substitution. We then arrive at the next proposition:
Proposition 9
If 1 = σ H > σ T > 0 the rate of innovation is non-decreasing over time; its
long-run value is determined by g = max{0, (1 − β )( L / a) − βρ} . An increase in the resource
stock reduces innovation.
Proof
Because of relatively poor substitution in the T-sector, T-output becomes
relatively more expensive and labour moves out of this sector when the resource
stock gets depleted (see (29) with r − wˆ > 0 and vT gradually declining to zero). As a
result, the T-sector vanishes; only conditions in the H-sector determine innovation in
the long run.4 □
Indeed, the steady-state innovation rate is the same as the one derived in (36)
for a single sector. Another implication is the “resource curse”-effect, stated in the
second part of the proposition. A high resource stock benefits mainly the T-sector if
this sector has the lowest substitution possibilities. This sector expands at the cost of
the innovating sector in response to a higher resource stock, and the smaller size of
the innovating sector makes innovation less profitable.
For the case of a unitary elasticity in the T-sector and poor substitution in the
H-sector we state:
Proposition 10
If 1 = σ T > σ H > 0 the innovation rate is non-increasing over time,
becomes zero in finite time and then remains zero. In an equilibrium with innovation, an
increase in the resource (knowledge) stock increases (reduces) innovation.
Proof
Relatively poor substitution in the high-tech sector has opposite effects
compared to poor substitution in the traditional sector: now resource depletion makes
the innovating sector relatively more expensive and shifts labour to the traditional
sector. As a result, research incentives fade away with the depletion of the resource
stock and consumption steadily declines in the steady state. Furthermore, a higher
resource stock now expands the H-sector and this may spur innovation in the short
run. With resource-using technological change, a higher knowledge stock increases the
demand for scarce resources in the H-sector, which reduces its size and thus
innovation incentives.
Finally, assuming poor substitution in both sectors H and T we state:
4 Phase diagrams for the different cases in section 6 are available from the authors upon request.
16
Proposition 11
With poor input substitution both in the high-tech and the traditional
sector, the steady state innovation rate is given by:
g =0
if 0 < σ H ≤ σ T < 1 or
if 0 < σ T < σ H < 1 and L < L ,
(44a)
g=
(1 − β )σ H L / a − βρ
≡ g scale
(1 − σ H ) βψ + σ H
if 0 < σ T < σ H < 1 and L ≤ L ≤ L ,
(44b)
g=
ρ (σ H − σ T )
≡ g nonscale
ψ (1 − σ H )σ T
if 0 < σ T < σ H < 1 and L > L ,
(44c)
⎛ β (1 − σ H )ψ + σ H − σ T ⎞ a ρ
⎛ β ⎞ aρ
where L ≡ ⎜
, L ≡⎜
, βH = β , ψ H =ψ .
⎟
⎟
⎝ 1− β ⎠ σ H
⎝ (1 − σ H )ψ (1 − β ) ⎠ σ T
Proof
We know that the value shares in both sectors continue to fall so that they
must eventually approach zero in the steady state. There are three possible steady
states: (i) an interior steady state with g, IT and I H strictly positive and constant, (ii)
a corner solution with IT = 0 zero, and (iii) a corner solution with g = 0. Applying the
expressions in (38) and (39) to the case of a single innovative sector yields the
expressions in (44). □
A necessary condition for innovation to remain active in the long run is that
substitution possibilities in the high-tech sector H are better than in the traditional
sector T. As the resource stock is depleted, the sector with poorest substitution
possibilities is hurt most, i.e. demand shifts away from it because the sector faces the
steepest increase in costs. Hence, if the H-sector suffers from poorest substitution, it
shrinks over time, which sooner or later makes innovation unprofitable. In contrast, if
the T-sector has poorest substitution possibilities labour moves towards the
knowledge-using sector which increases the incentives to innovate. If, in addition, the
labour force is large relative to the discount rate ( L > L ), the long-run size of the Hsector is large enough to sustain incentives to invest in new firms. If the labour
supply is still relatively small ( L < L < L ), the growing scarcity of resources drives
labour out of the T-sector. An exogenous increase in the labour force eventually ends
up in the Y-sector and in innovation. Again, this implies a scale effect: growth is
increasing in the size of the economy as measured by the labour force. However, this
scale effect only applies for small L.
7.
Conclusions
In this paper we have used a multi-sector framework in which differences in sectoral
substitution opportunities cause labour reallocation when the resource stock is
depleted. Endogenous innovation generates technological change and, as a by-
17
product, public knowledge, on which further innovation and production can build.
Combined with a sufficiently low discount rate, knowledge spillovers would be
sufficient to keep growth going in a model without natural resources (like the
standard endogenous growth model) or with natural resources and good input
substitution (our Cobb Douglas case). However, we have shown that with poor input
substitution, knowledge spillovers can only sustain growth if substitution in the more
innovative sectors is larger than in the less innovative sectors, including sectors
without any innovation opportunities. In this case, the increasing scarcity-price of
resources makes the sector with lower innovation opportunities relatively expensive,
shifting consumer demand towards the innovating sector and increasing the
incentives for innovation.
The model suggests that, in order to obtain sustainability, we should rely on
sufficient structural change rather than on (unrealistic) high substitution elasticities or
the ability to direct innovations exclusively at improving the productivity of natural
resources. Provided that the highly innovating sectors are more flexible with regard
to input substitution than less innovative sectors long run growth is likely to be
positive according to our findings. The detailed results also have interesting new
implications. With relatively poor substitution in sectors without innovation
opportunities, long-run consumption growth may be higher with poorer substitution
and, during transition, resource abundance may reduce innovation incentives.
Furthermore, the size of the elasticities of substitution, rather than resource and
labour endowments, bound the rate of growth. As a result, depending on certain
threshold levels, the scale of the economy has no effect on long-run growth.
We have made some simplifying assumptions that may be relaxed in future
research. First, the model could be extended with separate R&D activities for capital
augmenting and energy augmenting innovations. Differences in substitution across
sectors would still determine sectoral allocation in response to resource depletion and
thus market size for innovation, but more complex patterns of innovation over time
could arise. Second, we have abstracted from resource extraction costs and pollution
from resource use, which may be taxed by the government.5 These features may
change the price profile of the resource but they hit both consumer sectors in the same
way. As the effects of price changes in the different sectors work in opposite
directions, the quality of our results is not expected to change substantially when
enlarging the general model set-up in this way. Third, as the paper focuses on market
solutions, the issue of optimal policies has not been discussed. Resource use produces
no negative externalities in this model, only R&D generates positive spillovers which
lead, as in the original “Romer-type” approach to R&D, to positive subsidies for
innovations in the social optimum.
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20
Figures
r − wˆ
I j > 0
ρ /σ j
) ψ j (1 − σ j ) / σ
j
I j < 0
g
L/a
) (σ j + ψ j ) / σ jψ j
ρ / (σ j +ψ j )
g
Fig. 1: Single sector
r − wˆ
1 survives
2 survives
3 survives
1
2
3
g
L/a
l4*
l3*
l2*
l1*
g1,2
g
g2,3
Fig. 2: Multiple sectors.
21
Appendix
To find (29) we multiply (4) by pKj / pR to get:
⎛ vj
=⎜
1 − v j ⎜⎝ 1 − v j
vj
σj
1−σ j
⎞ ⎛ pKj ⎞
⎟⎟ ⎜
⎟
⎠ ⎝ pR ⎠
N
− (1−σ j )ψ j
.
(A.1)
Log-differentiating (A.1) and inserting pˆ Kj = wˆ from (5) and pˆ R = r from (25) yields
vˆ j /(1 − v j ) = (1 − σ j )( wˆ − r ) − (1 − σ j )ψ g
(A.2)
which - by rearranging - gives (29).
In order to find the dynamics as given in Lemma 1 we insert (5) into (26) and observe
that p jY j = φ j pCY from (13) and (14) to get:
wI j = β j v jφ j pC C
(A.3)
which is inserted in profits (6) to have:
π = B2
pC C
N
(A.4)
with B2 as defined in the main text. From (26), (5) and (13) we equally get:
I j = β j v jφ j y
(A.5)
with y given by (28). Log differentiating (A.5) and using (A.2) and (24) gives (30). We
solve (24) for the interest rate r ; dividing (9) by pN yields another expression for r, so
that with (8) and (A.4) we derive:
B p C
r = ρ + Cˆ + pˆ C = 2 C + wˆ − g
aw
(A.6)
By noting that yˆ = pˆ C + Cˆ − wˆ (see 28) we get (32). Writing (12) as
L J
g = −∑Ij /a
a j =1
,
(A.7)
using (A.5), and the definition of B1 from the main text yields (34). Inserting (34) in
(32) yields
22
y
L
.
(A.8)
( B2 + B1 ) − − ρ
a
a
Now replacing r − wˆ by ρ + ŷ in (29) and inserting (A.8) and (34) yields (33). Finally,
we build growth rates of (4) and use (5) and (25) to write:
yˆ =
Iˆ j − (uˆ j − u j ) = σ j ( r − wˆ ) − (1 − σ j )ψ j g
which is solved for uˆ j . Using (26), r − wˆ = ρ + ŷ and (A.5) then delivers (35).
(A.9)