Strategic resource extraction, learning-by-doing and the incidence of environmental taxes Andrew Leach a,∗ and Ujjayant Chakravorty b a School of Business, University of Alberta (CIRANO, CABREE) 3-40K Business Building, University of Alberta, Edmonton, Alberta, T6G 2R6, Canada b Department of Economics College of Business, University of Central Florida PO Box 161400 Orlando FL 32816-1400 USA October, 2007 Abstract In this paper, we examine the implications of induced innovation through learning-by-doing when energy is supplied by both traditional resource firms and firms exploiting an emerging technology which is an experience good. The resource owners face competing incentives to extract rents from the resource and to prevent expansion of the new technology. We examine how carbon taxes and subsidy policies for research and development are likely to affect behaviour in this context, and how these effects change with increased learning-by-doing rates. We show that rather than harnessing the power of endogenous technological change to reduce emissions, carbon taxation may have less effect on emissions the greater is the potential for technological change. Key words: Resource Extraction; Climate Change; Induced Innovation; Learning-by-doing. JEL classification : TBD ∗ Corresponding author. Email address: [email protected]. Preliminary version. Please contact the corresponding author for an updated version. 23 October 2007
Transcript
Strategic resource extraction, learning-by-doing and
the incidence of environmental taxes
Andrew Leach a,∗ and Ujjayant Chakravorty b
aSchool of Business, University of Alberta (CIRANO, CABREE)3-40K Business Building, University of Alberta, Edmonton, Alberta, T6G 2R6, Canada
bDepartment of Economics College of Business, University of Central FloridaPO Box 161400 Orlando FL 32816-1400 USA
October, 2007
Abstract
In this paper, we examine the implications of induced innovation through learning-by-doing whenenergy is supplied by both traditional resource firms and firms exploiting an emerging technologywhich is an experience good. The resource owners face competing incentives to extract rents fromthe resource and to prevent expansion of the new technology. We examine how carbon taxes andsubsidy policies for research and development are likely to affect behaviour in this context,and how these effects change with increased learning-by-doing rates. We show that rather thanharnessing the power of endogenous technological change to reduce emissions, carbon taxationmay have less effect on emissions the greater is the potential for technological change.
Preliminary version. Please contact the corresponding author for anupdated version.
23 October 2007
1 Introduction
The threat of anthropogenic climate change has led to a myriad of policy responses. A
common thread through many of these policies is the premise that the combination of
higher traditional energy prices and available substitutes which may be experience goods
will play a key role in eventual emissions reductions even where these are currently more
expensive. In this paper, we argue that innovation may not be a magic bullet for climate
change mitigation. We show that where resource extraction is separate from the production
of energy from new, emissions-free technology, strategic incentives to maintain market
share may lead to higher extraction of the resource relative to a case with no competition,
and that these effects are enhanced if learning-by-doing can eventually reduce the cost of
the substitute technology.
The notion of strategic over-extraction to avoid competition is not new, and has been
explored for the management of both renewable and non-renewable resources. Crabbe and
Long (1993) examines the interaction of owners and poachers in the fishery, where resource
owners react to the potential entry of poachers by catching more fish. In their model,
poachers are less efficient, but still induce a quantity response from the resource owner.
Our model is similar, but learning-by-doing in the substitute implies that in our case the
entrant’s cost disadvantage is affected by the actions of the incumbent firm. Further, as in
Crabbe and Long the entrants are price-takers, but in our case they internalize the dynamic
incentives provided through learning-by-doing, and so will produce below marginal cost in
the present to increase their future profits. Mason and Polasky (1994,2002) each examine
the impact of future competition on the management of a common property resource.
Mason and Polasky (2002) shows that the actions of a monopolist under the threat of
entry may lead to extinction of the harvested resource, while competition from the outset
would not. We question whether an analogous result may occur where the enhanced
threat of competition from a substitute which is an experience good may lead to earlier
exhaustion of the finite resource.
Our results highlight the tradeoff between potential welfare losses due to externalities and
those due to market power. In our case, the maximization of rents from the resource stock
through conservation may be discouraged since these rents would be available for eventual
capture by emerging technologies. Conversely, the potential supply from the emerging
2
technology will limit the exercise of market power. In equilibrium, dynamic incentives on
the part of the resource-owner firm to reduce future competition and on the part of the
new technology firm to increase experience may lead to either over-exploitation of the
resource or over-use of the alternative technology for strategic gain.
Harris and Vickers (1995) examine a related problem for non-renewable resources. In
their case, a resource owner must take into account that the likelihood of their consumers
developing an alternative source of energy increases in current period rent extraction. The
Harris and Vickers problem is related to similar work by Dasgupta, Gilbert and Stiglitz
(1982) since in each case, the Hotelling rule is modified to characterize optimal resource
management when the timing of a future substitute is endogenous. Similarly, in Cairns and
Long (1991), excessive rent-seeking by resource owners leads to diminished future rents
as a result of regulation. In each case above, there exists uncertainty as to the timing of
the reduction in available rents, which occurs either as a result of entry or regulation.
Our model assumes perfect foresight so some of the findings of the above work do not
relate.We study a case where the resource owner can actively influence the future costs
of the emerging technology. In this sense, the problems are similar, since in both cases
actions today influence the cost of available substitutes at some time in the future.
These developments all suggest that the pricing of resources by their owners will not
be independent of the state of current substitute technology, the states of current and
future climate policies, or the potential of future cost reductions in substitutes. However,
literature on climate change policy has largely ignored these links. In papers such as Popp
(2004, 2006), resource prices are fixed in both policy and no-policy scenarios.
Our paper is unique in that it combines policy, learning-by-doing, and strategic extraction.
We show that enhanced learning potential is akin to an enhanced threat of entry, and
can have similar results to those shown in previous work - the higher the quality of
the substitute or the faster the rate of learning, the higher the resource extraction rate.
Interestingly, this result only holds to a point. For cases where learning is very fast or
the substitute is initially of low cost, the ability to prevent its expansion is lost, and so
incentives to “make hay while the sun shines” dominate, and lead to lower extraction and
higher rents in earlier periods.
In our environment, the effects of climate policy are altered in interesting ways relative
3
to an environment without strategic behaviour and learning. Most significantly, a carbon
tax has sharply differing effects on energy prices depending on the underlying economic
environment. The carbon tax has three counteracting effects. First, it raises the cost of
the traditional resource, which we should expect to lead to increases in energy prices.
In so doing, however, it makes the emerging technology relatively more competitive and
increases the long-term value of the alternative technology, and so leads to more production
at prices below marginal cost which lowers overall energy prices. Finally, the tipping of the
competitive balance in favour of the emerging technologies means that the resource-owner
may see fewer gains to conservation and thus increase production to extract more rents
today. Depending on which effect dominates, energy prices will be affected differently.
Traditionally, we have thought of induced innovation occurring as a result of carbon
taxation; we highlight the reverse result. In our case, carbon taxation may have less effect
on emissions the greater is the potential for induced innovation.
The paper proceeds as follows. In Section 2, we develop the general model of the economy.
We characterize the dynamic, competitive equilibrium implied by this model and describe
the algorithm used to solve it in Section 3. In Section 4, we characterize the competi-
tive equilibrium with and without policy intervention to illustrate the role of strategic
incentives. Section 5 tests the sensitivity of results to modeling assumptions. Section 6
concludes.
2 The Model
The model characterizes a partial-equilibrium economy which uses energy for goods pro-
duction at decreasing returns to scale. Energy may be derived from two sources: fossil fuels
or alternative energy. Emissions from the use of fossil fuels contribute to climate change,
which in turn reduces future productivity. 1 Alternative energy is an experience good, and
so both its total and marginal production costs decline with cumulative production. For
the main results reported below, we restrict attention to the case perfect substitutabil-
1 We opt for a model structure which is simplified relative to that proposed in most IntegratedAssessment Models (IAM’s). These simplifications are not crucial here since our purpose is notto provide predictions of the magnitude of climate change or to compute optimal climate changepolicy, but rather to provide meaningful comparative analysis for policy analysis.
4
ity between energy sources, however this assumption is relaxed as part of the sensitivity
analysis. Denote the quantity of energy supplied by qt, and fossil and alternative energy
supplies by ft and at, and let qt = ft +at. The industrial organization of the energy sector
is that of an oligopoly with n price setting resource extraction firms and a price-taking
fringe firm exploiting the alternative energy technology. Below, we explore each sector of
the model before describing and solving for the competitive equilibrium.
2.1 Energy Demand
Energy is paid its marginal revenue product in production. The productivity of the econ-
omy is negatively affected by the accumulation of atmospheric pollutants produced in
conjunction with fossil-fuel-based energy. Denote by Z the concentration of the stock
pollutant. Denote the total revenue product of energy consumption by P (q, Z), and the
marginal revenue product of energy by p(q, Z) = Pq(q, Z) > 0. Further, P (q, Z) satisfies
An initial, known stock of fossil fuels, X0 is equally owned and extracted by n ≥ 1
symmetric firms. Denote by Xi,t the resource stock owned by an individual firm and
denote that firm’s extraction rate by fi,t, so that:
Xi,t+1 = Xi,t − fi,t, (1)
and
Xt+1 = Xt − nfi,t = Xt − ft. (2)
The resource stock is subject to extraction and delivery costs which are constant in both
in the intensity of extraction in each period and cumulative extraction. Let these costs be
given by marginal condition cX(ft) = cX .
2 This structure is analogous to that used in the integrated assessment literature, which tra-ditionally assumes that the total factor productivity of the economy is negatively affected byclimate change resulting from the accumulation of greenhouse gases in the atmosphere.
5
2.3 Emissions and Damages due to Climate Change
Emissions from the production of energy from fossil fuels contribute to a stock pollutant.
We assume that fossil fuels are measured in emissions units, such that emissions in each
period are equal to the volume of extracted resources ft.3 Emissions augment an existing
atmospheric stock, Z, which decays exogenously at rate δz. Therefore, Zt+1 = ft + (1 −
δz)Zt. The accumulation of the stock pollutant, Z, negatively affects the marginal and
total productivity of energy as discussed above.
2.4 Alternative Energy Supply
An emissions-free substitute for fossil fuels exists for energy production. Further, alterna-
tive energy is an experience good, such that future costs of production are decreasing in
current production levels.
Let the marginal cost of production of alternative energy be increasing in production in
any period (a), and decreasing in the level of accumulated experience (A). In particular,
let c(a,A) = c0A−η + csa, η > 0. By construction, cA(A, a) < 0, cAA(A, a) > 0. Experience
thus always reduces both the marginal and total costs of production, but at exhibits
decreasing returns to scale. A doubling of experience induces a reduction of a factor of
1 − 2−η in the cost-intercept-shift term A−η, which reduces both the marginal and total
costs of producing emissions-free energy.
Given that there exists a positive level of initial experience, A(0) = A0 > 0, experience
evolves according to a learning-by-doing law of motion as:
At+1 = At + a. (3)
3 Competitive Equilibrium
A competitive equilibrium is defined in which firms each choose supply to maximize the net
present value of profits under perfect foresight, with the following industrial organization
3 We do not allow for technological change in the transformation of resources to energy. Werethis to be the case, we would have to separately specify emissions rates per unit extraction.
6
assumptions. First, we assume that a single alternative energy firm exists and acts as
both an emissions- and price-taker. The firm does not internalize the effect of an increase
in alternative energy supply today on current energy prices or on future levels of GHG
accumulation. However, the firm does internalize the future value of present experience.
We assume that n ≥ 1 resource-extraction firm(s) pay(s) the constant extraction cost
and internalize(s) the effect of carbon emissions on future resource rents and act(s) as the
price-setting Stackelberg-leader(s) in the energy market. The resource extraction decisions
of the firm(s) will therefore include all relevant consequences of resource extraction, taking
the actions of other extraction firms as given.
The state space is the same for both firms; the profits of each firm will be affected by
current resource stocks (X), current experience levels (A), and current atmospheric ac-
cumulation of carbon (Z). In the characterization of the competitive economy, we also
introduce carbon taxes and alternative energy subsidies, denoted by τc and τa respectively.
With minor modifications, the indirect profit functions of each firm represent contraction
mappings over this state and policy space. 4 The simultaneous solution to the dynamic
programs developed below constitutes a subgame perfect equilibrium of the economy.
3.1 Optimal behaviour: Alternative Energy Supply Firm
The alternative energy supply firm takes fossil fuel supply as given and maximizes profits
from alternative energy supply over time. Let f(X,K,Z) be the total supply of fossil fuels
at any point in the state space. Let τatrepresent a subsidy to alternative energy supply.
The alternative energy firm solves the following dynamic program:
V1(X,A,Z) = maxa
p(
f(X,A,Z) + a, Z)
a −∫ a
0c (A, a) da + τaa (4)
+βV1(X′, A′, Z ′) + γaa
4 The model as written is not stationary in accumulated experience, A. In solving the dynamicprogram, we redefine the the model in terms of the cost-intercept-shift term, A−η, which willalways be bounded from below by zero and from above by its initial value, A
−η0 .
7
subject to
X ′ = X − f(X,A,Z) (5)
Z ′ = f(X,A,Z) + (1 − δZ)Z (6)
A′ = a + A (7)
and γa again denotes the shadow value for the non-negativity constraint on a. The first
order condition for the optimal choice of a given state (X,A,Z) is:
p(
f(X,A,Z), a, Z)
− c (A, a) + τa + β∂V1(X
′, A′, Z ′)
∂a+ γa = 0 (8)
while the accompanying complimentary slackness condition is:
γa ≥ 0 and γaa = 0. (9)
The future value derived from experience, ∂V1(X′,A′,Z′)∂a
, will be critical in the results shown
below. Intuitively, this term should be decreasing in resource stocks, decreasing in ac-
cumulated experience, and increasing under both carbon taxes and alternative energy
subsidies.
The solution to this dynamic program yields a policy function, a(X,A,Z|f) which de-
scribes its behaviour at each point in the state space, given a level of fossil fuel extraction.
3.2 Optimal Behaviour: Resource Extraction Firms
Each resource extraction firm takes account of the alternative energy firm’s policy function
for energy supply and takes as given the extraction decisions of the other (n− 1) resource
firms. Let fi represent the individual firm’s decision variable and let (n − 1)fj represent
the supply of the other (n−1) resource owners (where n¿1), and recall that a(X,A,Z|fi +
(n − 1)fj) represents the supply of alternative energy at a given point in the state space
(the solution to the dynamic program described in Section 3.1). Recall that τf represents
an emissions tax. Each resource-extraction firm solves the following dynamic program:
V2(X,A,Z) = maxfi
p(
fi + (n − 1)fj + a(X,A,Z|fi + (n − 1)fj), Z)
fi − cXfi
−τffi + βV2(X′, A′, Z ′) − vf
(X
n− fi
)
+ γffi (10)
8
subject to
X ′ = X − (n − 1)fj − fi (11)
Z ′ = (n − 1)fj + fi + (1 − δZ)Z (12)
A′ = A + a(X,A,Z|fi + (n − 1)fj) (13)
In the equations above, vf and γf denote the shadow values for the individual extraction
constraint,(
Xn− fi
)
,and non-negativity constraint on fi respectively.
In equilibrium, the assumption of symmetric, resource-extracting firms implies that fi +
(n − 1)fj = nfi = f which we substitute where appropriate to derive the following first
order condition for optimality:
p(
f + a(X,A,Z|f), Z)
+ fip′(
f + a(X,A,Z|f), Z)
(
1 +∂a(X,A,Z|f)
∂fi
)
− cX − τf
−β∂V2(X
′, A′, Z ′)
∂X ′+ β
∂V2(X′, A′, Z ′)
∂A′
∂A′
∂a(X,A,Z|nf)
∂a(X,A,Z|nf)
∂fi
+β∂V2(X
′, A′, Z ′)
∂Z ′− vf + γf = 0 (14)
while the accompanying complimentary slackness conditions for the stock and non-negativity
constraints are:
vf ≥ 0 and vf
(X
n− fi
)
= 0, (15)
γf ≥ 0 and γff = 0 (16)
Each resource extraction firm internalizes the elasticity of alternative energy supply with
respect to its own extraction and the implied future reduction in rents resulting from learn-
ing. To do so, it must take into account the change in alternative energy supplied today
that will result from a change in resource extraction, given by ∂a(X,A,Z|f)∂fi
. This substitution
effect lessens the ability of the firm to exercise market power. Where ∂a(X,A,Z|f)∂fi
= −1,
any reduction in resource extraction today will be completely compensated for by alter-
native energy supplies, and (14) will mimic price-taking behaviour, even when n = 1.
Conversely, when there is no substitution at the margin, a single resource-extracting firm
is a standard monopolist. It must also internalize a second effect which is captured by∂V2(X′,A′,Z′)
∂A′
∂A′
∂a(X,A,Z|f)∂a(X,A,Z|f)
∂fi, through which the cost of inducing increased alternative
9
energy supplies today is a reduction in its ability to extract rents in the future by allowing
learning-by-doing. Learning plays a key role here since this term will be different from
zero only when there is a positive learning-by-doing rate. The faster the learning rate,
the larger will be the magnitude of this future cost of allowing alternative energy supply
today.
Each of these incentives will be weaker as n grows. First, for larger n, the total net present
value of rents is smaller for each firm, and each firm has less ability to influence the market
quantity and price. This reduces the first strategic incentive to reduce extraction. Second,
as the potential rents earned are lowered, so are the losses in net present value of rents from
allowing the alternative energy firm to learn. This reduces the magnitude of the second
strategic incentive. In fact, as n grows, deterrence of alternative energy supply becomes
a public good: each firm gains from maintaining a high-cost substitute, but in a Nash
equilibrium, it will be optimal for each firm to shirk and profit from others’ deterrence
activities. As such, equilibrium deterrence efforts will be decreasing in n. 5
3.3 Numerical Solution and Simulation
The simultaneous solutions to the dynamic programs defined in (4) and (10) will allow us
to define the subgame perfect equilibrium for the competitive economy.
Due to the structure employed, there will be no closed-form solution, so the value func-
tions will be approximated numerically. In particular, the model is solved using an al-
gorithm, analogous to those used in Kelly and Kolstad (1999) and Leach (2007), which
characterizes the solution to each value function as the fixed point of a neural network
approximation derived by iterating over each firm’s value (profit) function over a finite
set of grid points. 6 A complete description of the algorithm used in the present analysis
is presented in Appendix A.
5 The same effect would occur if we treated resources as a commons with a fixed extractioncost. In this case, as n tends to infinity each firm is a price-taker and no rents are earned in anyperiod (except the period of exhaustion) and so all strategic motives disappear.6 For a detailed discussion of neural networks, the interested reader is referred to Hassoun(1995).
10
3.4 Calibration
While the intent of the present study is not to predict the magnitude of energy supply or
climate change but rather to provide informative comparative dynamics, we use parame-
ters and functional forms which are comparable to those used in other models of climate
change in the literature.
We adopt a total revenue product function of Ω(Z)qθ, with θ ∈ (0, 1) and Ω(Z) =Ω
(1+ξ1Z+ξ2Z2),ξ1 = 2 × 10−5, ξ2 = 1.6 × 10−7. This function is calibrated to approximately
match the marginal revenue product of energy and the marginal damage of emissions in
the global economy in the first period of the Nordhaus and Boyer (2000) DICE-99 model.
The energy share of production, θ = .05 is within the range of parameter values used in
the literature. 7
The cost of fossil fuel extraction is calibrated to be equal to the constant component of the
Nordhaus and Boyer (2000) function, $113 per ton. We do not adjust for a fixed markup
as Nordhaus and Boyer do since our firms are price-setting and choose the rent in each
period to maximize their gain from resource extraction.
The delivered cost of fossil fuels serves as an anchor for the cost of alternative energy. We
assume that the lower bound of the cost of alternative power is less than the extraction
cost of fossil fuels, and that the slope of the marginal cost of alternative energy is such that
θΩ(a∗)θ−1 = ca(∞, a∗) < cx; i.e. when the cost of alternative energy is at its lower bound,
the equilibrium energy price will be below the extraction cost of fossil fuel resources.
4 Results and Discussion
Below, we first characterize optimal reaction functions at given points in the state space
(the simultaneous solution to the dynamic programs at differing states of the world),
as well as presenting simulations of the economy from particular starting values. In all
reaction functions which follow below, the resource stock is fixed at X = 200 which
7 We specify climate damages as arising directly as a result of accumulated carbon, which allowsus to solve the model without accounting for a temperature state. We adopt a simple specificationwhere accumulated carbon stocks contribute quadratically to reducing total factor productivity.
11
01
23
45
Cho
sen
Res
ourc
e E
xtra
ctio
n
0 .05 .1 .15Intercept of MC of Alternative Energy
No learning (0%) Slow Learning (10%)Fast Learning (25%) Very Fast Learning (50%)
Fig. 1. Optimal extraction decisions of the resource-owning firm conditional on the technologystate of its competitor for each of three learning scenarios.
represents 25 years of equilibrium demand were price to be set to extraction cost, and so
scarcity is not an immediate issue. Further, carbon accumulation is set at Z = 0. These
values are also used as the initial conditions for the simulations, along with an initial
alternative technology with a y-intercept of its marginal cost function equal to extraction
cost. To emphasize strategic responses, we assume monopoly (n = 1) in the resource
sector.
4.1 Learning and Emissions in the Strategic Equilibrium
In this section, we characterize the effects of learning-by-doing in a substitute technology
on the optimal decisions of the firm(s) extracting finite resources and the firms exploiting
the new technology for alternative energy production. We show how learning-by-doing af-
fects incentives to use resource extraction for maintaining market share and for protecting
future rents.
Figure 1 shows the optimal extraction decisions of the resource firm in the absence of
any policies as a function of the accumulated experience in alternative energy production
for each of three learning scenarios. Relative to the case with no learning, extraction is
lower for all learning rates considered, however this relationship is not strictly monotonic
in learning rates. An examination of the underlying equilibrium strategies leading to this
12
105
110
115
120
Mar
gina
l Rev
enue
of E
nerg
y ($
/tCE
)
0 .05 .1 .15Intercept of MC of Alternative Energy
No learning (0%) Slow Learning (10%)Fast Learning (25%) Very Fast Learning (50%)Extraction Cost ($113/ton)
Reaction functions by learning rate
Fig. 2. Resource firm reaction functions for each learning rate as a function of current alternativetechnology cost. Extraction cost is $113 per ton.
outcome reveals interesting tradeoffs.
Figure 2 shows the reaction function for the resource-owner. Where there is no learning,
the optimal decision is to set marginal revenue equal to the extraction cost plus a resource
rent. Intuitively, the rent charged is lower the cheaper is the alternative energy technol-
ogy. However, where learning is possible, the firm deviates from a traditional strategy and
internalizes the benefit of preventing learning which lowers the true cost of resource ex-
traction. This effect is strongest for the 10% and 25% learning rates as a result of the fact
that, with still higher rates of learning, there are fewer future rents for the resource-owner
to lose, and so comparatively less incentive to protect them by over-extraction today. For
slower learning rates, a strategy to exclude future competition makes sense, but as learn-
ing becomes faster, the ability to keep these technologies out of the market dissipates. As
such, the strategy of the resource extraction firm is to extract more rents today if learning
is non-existent or very fast, and to over-extract in the intermediate case of slow learning.
In all cases with learning, a more expensive current alternative technology implies greater
incentives to decrease rents today in favour of reducing future competition. This is as a
result of the fact that exponential learning rates imply that more expensive technologies
experience faster cost-reduction, and therefore pose a greater threat to rents relative to
the current situation.
13
020
4060
Sha
dow
Val
ue o
f Alte
rnat
ive
Ene
rgy
Sup
ply
($/tC
E)
0 .05 .1 .15Intercept of MC of Alternative Energy
No learning (0%) Slow Learning (10%)Fast Learning (25%) Very Fast Learning (50%)
Fig. 3. Amount by which the alternative energy firm is willing to produce above marginal cost,as a function of the current cost of alternative energy.
For the alternative energy firm, we might expect faster learning to always imply a higher
future benefit to current production and thus a greater willingness to produce at a price
below marginal cost today. In fact, this is not the case in the strategic equilibrium. Figure
3 shows that in equilibrium, these incentives are strongest in the 25% learning case unless
the technology is currently very low-cost. In relative terms, the 50% learning rate is
sufficiently fast that the technology becomes viable without as much extra effort on the
part of the alternative energy firm. So, while the total value to the firm is higher, its
willingness to over-produce is lower. This occurs in conjunction with the lower resource
equilibrium rent extraction in the 25% learning rate case shown above.
With the 10 and 25% rates, both firms have strategic incentives to over-produce to increase
future rents. The upshot of this is the observed faster resource extraction with the higher
learning rates when the resource firm strategic incentives dominate the alternative energy
sector over-production incentives. Also, since the alternative energy firm is willing to
produce below marginal cost, and the resource extraction firm is willing to accept lower
marginal revenues, we will see lower equilibrium energy prices than in the no learning
scenario, ceteris paribus. While intuition suggests that the extraction of rents by resource
owners will drive the development of substitute technologies, the above suggests that the
inverse effect is also important: that the existence of a substitute could lead to fewer rents,
No learning (0%) Slow Learning (10%)Fast Learning (25%) Very Fast Learning (50%)
Fig. 4. Markup applied by a monopolist to the marginal extraction cost, as a function of thecurrent cost of alternative energy.
more extraction, and lower energy prices.
A key motivation for this work is the assumption that resource prices will follow a
Hotelling-style rent curve in previous papers such as Nordhaus (2002) and Popp (2004,
2006). In such an environment, the impact of introducing learning-by-doing or other forms
of endogenous technological change will be overstated since these resource pricing increases
will render these technologies viable over time and reduce carbon emissions abatement
costs. Above, we have shown that firm behaviour in both resource and alternative sectors
will be altered significantly by learning rates in the absence of policies. In Figure 4, we
show the monopolist’s pricing curve for resources as a function of remaining stocks for
each learning rate, holding fixed the alternative sector marginal cost. This figure empha-
sizes that, rather than modeling endogenous technological change as occurring as a result
of resource prices which increase over time, it is imperative to internalize the reality that
optimal resource prices will be lower, and may increase less rapidly, the greater is the
potential for endogenous technological change.
4.2 Policies in a strategic environment
When policies are introduced in a strategic environment, the present and future playing
fields are altered for both parties. An emissions tax increases the marginal cost to the
15
resource firm, and erodes its ability to earn rent from the resource. This is magnified in the
presence of the substitute energy source. First, there is less to gain by maintaining market
share since the available rents are lower. Second, the competing technology is relatively
more competitive, which also decreases expected future rents. From the alternative energy
firm’s point of view, a carbon tax renders the technology it exploits more cost-competitive,
and the future gains to learning-by-doing today are increased since it can gain more
market share in a shorter time. A subsidy to alternative energy has many of the same
effects, altering the competitive balance in favour of the emerging technology. Below,
we demonstrate the consequences of introducing a $10/ton carbon tax and a $10/ton
equivalent alternative energy subsidy each imposed across all possible states of the world.
4.2.1 Carbon Taxes
When we set carbon taxes, we generally expect that three things will occur. First, we
expect that energy prices will rise. Second, we expect that emissions will fall. Third, we
expect that alternative energy supply will increase and, where possible, that this will
drive increased experience accumulation and eventual cost reductions in the alternative
sector. While these expectations are realized in most cases in our model, we show that
learning rates and present states of alternative technologies have substantial influence on
the effects of emissions taxes.
Tax incidence on energy prices
In a standard model, the incidence of a carbon tax on the energy price will be determined
by the relative elasticities of the marginal revenue product and marginal cost functions.
This is also true here, although the determination of the relative elasticities is not obvious.
In our case, the carbon tax also affects the degree to which strategic incentives lead firms
to deviate from a standard marginal revenue equals marginal cost rule. As a result, the
effects of carbon taxes are not always as expected. Consider Figure ?? which shows the
change in energy prices resulting from a $10 per ton carbon tax for each of the three
learning scenarios considered. At first glance, this figure shows no obvious pattern, as it
is the result of countervailing underlying incentives. In order to understand these results,
we examine the incentives which determine the equilibrium prices.
16
34
56
Cha
nge
in E
nerg
y P
rices
($/
tCE
)
0 .05 .1 .15Intercept of MC of Alternative Energy
No learning (0%) Slow Learning (10%)Fast Learning (25%) Very Fast Learning (50%)
Fig. 5. Change in equilibrium energy prices after the imposition of a $10 per ton carbon tax asa function of the cost of the initial unit of alternative energy.
−2
−1
01
2
Cha
nge
in M
argi
nal F
utur
e V
alue
of F
ossi
l Fue
ls (
$/tC
E)
0 .05 .1 .15Intercept of MC of Alternative Energy
No learning (0%) Slow Learning (10%)Fast Learning (25%) Very Fast Learning (50%)
Fig. 6. Change in future values of present alternative energy production after the imposition ofa $10 per ton carbon tax as a function of the cost of the initial unit of alternative energy.
The resource firm sets its marginal revenue of extraction equal to the marginal extraction
cost plus the carbon tax and the cost of resource use today in terms of lost future stock,
net of the benefit of resource use today in terms of preventing learning-by-doing in the
alternative sector. The future value of in situ resources is captured in the resource rent,
and the changes to the resource rent induced by the imposition of the carbon tax are
17
shown in Figure 6. Without learning, the marginal value of resources is reduced. This is a
continuation of the intuitive effect of lower resource rents where alternative technologies
are more competitive. The carbon tax renders competitive alternative technologies that
were previously too costly, and as a result the resource rents are reduced over these states.
Conversely, for the learning scenarios, more costly technologies now represent a greater
threat to future rents, and strategic attempts to exclude alternative energy production
may be less effective. The large increase in rents charged in the 10% learning rate case is
evidence of this effect.−
4−
20
24
Mar
gina
l Fut
ure
Val
ueof
Alte
rnat
ive
Ene
rgy
Pro
duct
ion
($/tC
E)
0 .05 .1 .15Intercept of MC of Alternative Energy
No learning (0%) Slow Learning (10%)Fast Learning (25%) Very Fast Learning (50%)
Fig. 7. Change in future values of present alternative energy production after the imposition ofa $10 per ton carbon tax as a function of the cost of the initial unit of alternative energy.
There is an offsetting story to be told on the alternative energy side of the market. Figure
7 shows that the future values of alternative energy supply today may either increase or
decrease after the introduction of the carbon tax. Recall that these values are interpreted
as the amount which the firm is willing to accept a marginal cost above price in order
to benefit from learning-by-doing. Where no learning occurs, there is no future value of
today’s production, and that does not change after a carbon tax. In the 25% learning case,
the firm will be more aggressive in its production decisions for technologies at or below
the extraction cost of carbon, and will be less aggressive for more costly technologies. For
the 10% learning case, the opposite is true. For more costly technologies, the resource firm
was shown above to see less interest in preserving market share. As a result, there are
18
−10
0−
80−
60−
40−
200
Em
issi
ons
Red
uctio
ns (
%)
0 .05 .1 .15Intercept of MC of Alternative Energy
No learning (0%) Slow Learning (10%)Fast Learning (25%) Very Fast Learning (50%)
The effect of carbon taxes on emissions
Fig. 8. Percentage reduction in emissions after the imposition of a $50 per ton carbon tax as afunction of the cost of alternative energy.
further gains to be made from over-production today, so we see an aggressive response.
However, while there are also more gains to learning introduced by the carbon tax in the
50% learning case, we see the firm being less aggressive. Since the resource firm passes on
higher prices to market and does not alter its rent-seeking, the alternative energy firm’s
job is done in part by the carbon tax, and it will be able to increase production without
pricing as far below marginal cost. As a result of this type of strategic evoloution, the
effect of the carbon tax may be lower for higher learning rates.
Tax effects on emissions and experience
The goal of setting a carbon tax is to reduce emissions, and Figure 8 presents our findings
on this front. In all cases, the imposition of a carbon tax reduces emissions, however
the effects vary significantly in both the state of experience and the learning rate. As a
consequence of assumed perfect substitutability, a carbon tax will eliminate all emissions
for cases where the equilibrium price of alternative energy alone would be less than the
cost of resource extraction and the tax. As the cost of the alternative energy substitute
increases, the emissions reduction achieved by a carbon tax decreases. In addition, these
results show that the emissions reduction effects of a carbon tax should not be expected
to be monotonic in the learning rate.
19
Regardless of the strategic response, carbon taxes always provide an increased incentive
for experience accumulation and will increase alternative energy supply. However, on this
measure, it is important to note that with any learning and any degree of market power,
these incentives will not be as strong as that which would be attributed in a model without
strategic considerations.
4.2.2 Alternative Energy Subsidies
Alternative energy subsidies are often intended to provide a “foot-in-the-door” for new
energy sources with fewer or no harmful emissions. In our context, however, we will show
that this may have perverse consequences. A subsidy to alternative energy supply when
alternative energy is an experience good will erode the future value of resource stocks,
and eliminate a key motive for conservation by the resource owner.
To illustrate the effects of introducing an alternative energy subsidy, we examine a case
where, regardless of their current cost, carbon-free substitutes are offered a subsidy of $10
per ton of carbon equivalent energy supplied. Many of the effects discussed here will be
the inverse of those demonstrated for the carbon tax, but the imposition of the subsidy
on the price-takng rather than the price-setting sector is an important distinction. We
provide a more concise discussion of the results in this section, highlighting the differences
between the subsidy and the tax.
In Figure 9, we see the effect of alternative energy subsidies on emissions, while Figure
10 shows the effects of the same subsidies on alternative energy supplies. In fact, rather
than providing a “foot-in-the-door” for technologies far from being viable, the subsidy has
large impacts in states where the alternative energy technology already has a relatively
low cost, while it has less effect in states where alternative energy costs and thus learning
potentials are high.
Figure 9 reveals the potential for perverse consequences of subsidies to alternative energy
subsidies. Where alternative energy technologies are very costly, there is little incentive for
the resource firm to react. When these technologies are provided a subsidy, they become
cost-competitive and it will be optimal for the resource firm to increase extraction to
maintain a competitive advantage in the face of the newly lower-cost competition. So,
20
−10
0−
80−
60−
40−
200
Em
issi
ons
Red
uctio
ns (
%)
0 .05 .1 .15Intercept of MC of Alternative Energy
No learning (0%) Slow Learning (10%)Fast Learning (25%) Very Fast Learning (50%)
The effect of alternative energy subsidies on carbon emissions
Fig. 9. Change in carbon emissions after the imposition of a $25 per ton of carbon equivalentsubsidy.
1020
3040
50In
crea
se in
Alte
rnat
ive
Ene
rgy
Sup
ply
(%)
0 .05 .1 .15Intercept of MC of Alternative Energy
No learning (0%) Slow Learning (10%)Fast Learning (25%) Very Fast Learning (50%)
The effect of subsidies on alternative energy supply
Fig. 10. Increase in alternative energy production after the imposition of a $25 per ton of carbonequivalent subsidy.
while it is true that the policies lead to increases in alternative energy supply, and therefore
learning, they may also lead to perverse incentives for increased emissions if the technology
being subsidized is currently cost-prohibitive.
In contrast to the carbon tax, which disadvantages the price setter and gives an advantage
21
−2
−1
01
Incr
ease
in M
argi
nal F
utur
e V
alue
o
f Alte
rnat
ive
Ene
rgy
Sup
ply
($/tC
E)
0 .05 .1 .15Intercept of MC of Alternative Energy
No learning (0%) Slow Learning (10%)Fast Learning (25%) Very Fast Learning (50%)
The effect of subsidies on alternative energy rents
Fig. 11. Increase in marginal future value of present alternative energy production after theimposition of a $25 per ton of carbon equivalent subsidy.
to the fringe, the subsidy seeks the same results by giving an advantage to the competitive
fringe. If market power is very strong, and rents are sufficiently large at the margin, the
price-setter will be able to mitigate the subsidy by increasing extraction and incurring a
loss of rents to maintain energy prices. From the point of view of the alternative energy
firm, if the energy price decreases by close to the amount of the subsidy, the changes
brought by the subsidy are minimal.
Subsidies do not necessarily increase the future value of alternative energy production
today. In some sense, the subsidy replaces some of the original incentives which lead
the alternative energy firm to supply energy at marginal costs greater than price. This
effect, shown in Figure 11, is again non-monotonic in the learning rate. In the 50% case,
increasing the subsidy rate has a negative on future rents in almost all cases. With this
learning rate, cost-reduction occurs quickly even without over-production and the subsidy
leads to greater quantities of production at market prices. As a result, there is less incentive
to over-produce to capture future rents. For other learning rates, the incentives to over
produce are accelerated for a set of increasingly expensive technology states, and reduced
for others relative to the no-subsidy case. What is important here is that to 10 to 20% of
the subsidy value may be captured by the firm through a reduction in previous willingness
to produce above marginal cost.
22
Overall, we see that that it is important to consider the impact of emissions policy, learn-
ing, and competition on the resource firm’s decision making with respect to extraction.
The traditional Hotelling rent curve is certainly not invariant to introduced emissions
control policy or to the nature of the substitute advantaged by the policy.
5 Sensitivity Analysis
The results reported above are likely to be affected by three key assumptions. First, the
assumption that a single firm owns extraction rights to all resources and can extract
monopoly rents may overstate the role for strategic responses. Second, the assumption
that all returns to learning-by-doing are captured by producing firm may over-state the
willingness to produce below marginal cost. Finally, the assumption of perfect substi-
tutability of emerging technologies for traditional energy sources may again over-state
the value of alternative energy production experience and the potential loss of extractive
resource rents. Below, each of these assumptions are removed, and key results compared
to those shown earlier. While some results change in magnitude, the general message of
the analysis is not significantly altered.
5.1 Market Power in the Resource Sector
Many of the results above depend on the fact that the resource firm can exercise market
power to extract rents and to strategically prevent expansion of the alternative technology.
In this section, we impose a Cournot duopoly structure in the resource sector, thereby
reducing the available rent for each firm.
Under a duopoly, the basic result that extraction will be an important strategic tool to
reduce future competition still holds. However, the non-monotonicity results relating to the
25% learning rate are lost or more muted. This is not unexpected, since the introduction
of a competing duopolist reduces the rewards of limiting future competition which was
a more important motive in the 25% case. As we move toward perfect competition in
the resource sector, we would expect to see the incentives on the alternative side take
over, and the results would then be dominated by the alternative production increasing
monotonically in the learning rate.
23
5.2 Myopic Behaviour in the Alternative Energy Sector
Here, we remove the assumption that alternative energy firms are able to appropriate all
of the future value from cost reductions accruing as a result of current production. The
numerical solution above implies that the alternative-energy firm is able to extract all of
the rents accruing as a result of future cost reduction due to current production decisions.
Any weakening of this condition will reduce the incentive to over-produce relative to the
quantity at which the energy price equals the current marginal cost. We explore a case
where all future claims on today’s experience are eliminated, and the industry acts as
a myopic competitive fringe. This effectively mimics the case where patent protection is
non-existent, and experience in alternative energy production is a public good. It is also
comparable to the cases originally explored in Crabbe and Long where a competitive fringe
exists but does not optimize in a dynamic sense.
By construction, we will find a decrease in the alternative energy supplied, ceteris paribus,
for all scenarios with positive learning rates. We are interested more in the degree to which
our results are sensitive to the assumption of dynamic optimization under an analog of
complete patent protection. We find the non-monotonicity results hold when this assump-
tion is removed. As a result of the parameterization of the model, earning any rents at all
will result in some production from the alternative energy sector. With the 50% learning
rate, this implies that the alternative source will soon dominate the market, and there is
little margin for strategic interference. When we allow for strategic actions by the monop-
olist, and ignore the strategic motives for the alternative sector, the resource owner does
not need to over-extract to the same degree, and so extraction is lower in all cases.
5.3 Perfect Substitutability of Resources and Alternative En-ergy Technology
To be completed.
24
6 Conclusion
This paper examines the role of learning-by-doing and environmental regulation on the
extraction decisions of a strategic nonrenewable resource owner. In general, policy design
has tended to focus on the roles of carbon taxes and scarcity in increasing energy prices
and focussing on the role of higher prices in making alternative technologies viable. We
show a reverse role. The higher the quality of the substitute, or the faster is learning, the
higher will be the no-policy rate of resource extraction, which implies lower energy prices.
In such a context, carbon taxes can have an important role in accelerating development of
alternative technologies, however they may not always lead large energy price increases.
The latter has important consequences for climate policy evaluation, which has concen-
trated on the economy-wide effects induced by higher energy prices following from carbon
taxes.
References
[1] R. D. Cairns and N. V. Long. Rent seeking with uncertain opposition. European Economic
Review, pages 1223–1235, 1991.
[2] P. Crabbe and N. V. Long. Entry Deterrence and Overexploitation of the Fishery. Journal
of Economic Dynamics and Control, 17:679–704, 1993.
[3] P. Dasgupta, R.J.Gilbert, and J. Stiglitz. Invention and Innovation Under AlternativeMarket Structures: The Case of Natural Resources. The Review of Economic Studies,49(4):567–582, 1982.
[4] C. Harris and J. Vickers. Innovation and Natural Resources: A Dynamic Game withUncertainty. Rand Journal of Economics, 26(3):418–430, 1995.
[5] M. H. Hassoun. Fundamentals of Artificial Neural Networks. The MIT Press, Cambridge,Massachusetts, USA, 1995.
[6] K. L. Judd. Numerical Methods in Economics. Massachusetts Institute of Technology,Cambridge, Mass., USA, 1998.
[7] D. L. Kelly and C. D. Kolstad. Bayesian learning, growth and pollution. Journal of Economic
Dynamics and Control, 23:491–518, 1999.
[8] D. L. Kelly and C. D. Kolstad. Solving growth models with an environmental sector. Journal
of Computational Economics, 18:217–235, 2001.
25
[9] A. Leach. The climate change learning curve. Journal of Economic Dynamics and Control,31:1728–1752, 2007.
[10] C. F. Mason and S. Polasky. Entry deterrence in the commons. International Economic
Review, 35(2):507–525, 1994.
[11] C. F. Mason and S. Polasky. Strategic Preemption in a Common Property Resource: AContinuous Time Approach. Environmental and Resource Economics, 23:255–278, 2002.
[12] W. D. Nordhaus and J. Boyer. Warming the World. MIT Press, Cambridge, Mass., 2000.
26
Table 1Parameter values and state variable initial conditions used for simula-tions
Parameter Description Calibrated Value
Energy Demand
Ω0 Total factor productivity 18.1936
θ Energy share of production .05
Carbon Sector Cost Function
cx Constant extraction costs .113
Alternative Sector Cost Function
ζ1 Marginal cost intercept: alternative sector .1
ζ2 Determines learning rates (reported) a 0, 0.25, 0.5
ζ3 Slope of marginal alternative energy cost .00144702
Climate
ξ Emissions intensity of fossil fuels 1
(1 − δz) Atmospheric retention of carbon .986
D1 Linear Damage Coefficient .01
D2 Quadratic Damage Coefficient .04
State Variable Initial Conditions for Projections
X0 Initial resource stock 400
Z0 Initial carbon concentration, all cases 100
a This parameter value is set such that the reduction in marginal cost fora doubling of accumulated experience corresponds to the learning ratespecified.
27
A Approximate Numerical Solution
In order to solve the value functions, we use a technique described in Kelly and Kolstad
(1999, 2001) and Leach (2007), which characterizes the fixed point of the value function
using an iterative algorithm combined with a neural network approximation of the value
function over a finite set of grid points.
The neural network approximation is defined as follows. 8 Define by L = 20 the number
of nodes in the hidden layer, and let n = 3 represent the number of state variables in the
model, such that the state space is Rn. Denote by x ∈ R
n+1 a set of real-valued signals to
the network, with the first element (x0 = 1) being a bias signal (analogous to a constant),
and the remaining elements being the state vector for a particular point in the state space.
Let χ1 be a (n + 1) × L matrix of inner weights, and let z(x, χ1 be a (L + 1) × 1 vector,
with the first element (z0 = 1) being a bias signal. The additional elements (z1..zL) are
the output values from the hidden layer of the network, zl = tanh(χ′1lx)∀l = 1..L, where
χ1l represents a column of χ1. The L + 1 elements of z(x, χ1) are then aggregated using
outer weights χ2, a (L + 1) × 1 vector. We can thus express the approximation as:
Φ(x|χ) = χ′2 (z(x, χ1)) . (A.1)
Using Φk(si|χk) to denote the approximation to the value (revenue) function of firm k
defined over R4, we use the following iterative algorithm to solve the simultaneous solution
to the value functions in the competitive equilibrium. 9
Denote the set of choices by aj(si)Ni=1 and f j(si)
Ni=1 for alternative energy supply and
resource extraction respectively, for iteration j.
Algorithm 1 Algorithm Preliminaries: Choose a convergence criterion ǫ, number of
neural network nodes L, and starting values for the weights χ1,k and χ2,k in Φk(x|χk) for
k = a, f. Define ranges for each of the state variables to be covered by a grid, and a
number of points N to make up the grid. Draw N points from a 3-dimensional
low-discrepancy sequence and transform these points to meet the desired bounds of the
state space. 10 We set ǫ = 10−3, L=20, and N=1000. We set state space bounds so that
8 For a detailed discussion of neural networks, the interested reader is again referred to Hassoun(1995).9 The approximation at state space point si is defined over R
4 - the 3-dimensional state space,plus a constant. We use the notation si without the addition of the constant for simplicity.10 We use a Halton sequence to draw a set of grid points which are uniformly distributed within
28
0 < X < 1000, −100 < Z < 500, and 0 < A−ζ2 < .15.
Step 1: Solve the maximization problems given in (4) and (10), taking other firm’s
choices as given. For iteration j, at each each state si, aj(si) solves:
p(
f j−1(si), aj(si), Z
)
− c(
A, aj(si)))
+ τa + β∂Φa((si
′|xi, f j−1(si), aj(si))|χa))
∂a+ γa = 0,
(A.2)
while f j(si) solves:
p(
f j(si) + aj−1(si), Z)
+ f j(si)p′(
f j(si) + aj−1(si), Z)
(
1 +∂aj(si)
∂f
)
− cX − τf
−β∂Φf ((si
′|si, f j(si), aj−1(si))|χf ))
∂X ′+ β
∂Φf ((si′|si, f j(si), aj−1(si))|χf ))
∂A′
∂A′
∂aj(si)
∂aj(si)
∂f
+βφ∂Φf ((si
′|si, f j(si), aj−1(si))|χf ))
∂Z ′− vf + γf = 0. (A.3)
The displacement term, ∂aj(si)∂f
, is calculated using a one-sided, numerical derivative.
Step 2: Denote by Πk(aj(si), f
j(si), si) the profit of firm k at state si given choices.
Update the value for each firm k at each point on the grid as:
Vj+1k (si) = Πk(a
j(si), fj(si), si) +
1
1 − βΦk((si
′|si, f j(si), aj(si))|χk)) (A.4)
Step 3: Use updated values V j+1k (si)
Ni=1 to solve for new weights χ1,k and χ2,k that
minimize ‖V j+1k (s) − Φk(s|χk))‖.
Step 4: Return to Step 1 unless∑N
i=1
(
Vj+1k (si) − V
jk (si)
)2
< ǫ for each firm.
As specified in the text, we make a change to the model so that it can be solved using the
algorithm detailed above. Since the model is not stationary in accumulated experience, A,
we redefine the the model in terms of the cost-intercept-shift term, A−η, which will always
be bounded from below by zero and from above by its initial value, A−η0 . The other state
variables, X and Z are implicitly bounded as Z can never exceed Z(0)+X(0) and cannot
be negative, and X can never exceed its initial value and is bounded by zero from below.
We use negative values of Z on the state variable grid to allow the approximation to pick
up curvature close to Z=0. The model remains stationary for negative values of Z, since
Z1 = δZZ0 + E > Z0 < 0 ∀ ≥ 0.
the state space. For details on low-discrepancy sequences, see Judd (1998).
