1. An Updated Analysis of Optimal Carbon Pricing
with Flow and Stock Externalities
Jeff Mollins
Memorial University of Newfoundland

2. Mollins 1
Table of Contents
Abstract.............................................................................................................................................2
Sections
1. Introduction ............................................................................................................................2
2. Theoretical Model ...................................................................................................................3
3. Functional Forms.....................................................................................................................7
4. Parameter Values...................................................................................................................10
5. Results..................................................................................................................................14
6. Conclusion............................................................................................................................16
Appendix .........................................................................................................................................18
References .......................................................................................................................................20
This essay is submitted in partial fulfillment of
the requirements for the Bachelor of Arts with
first degree honours

3. Mollins 2
Abstract
The growing literature on environmental and natural resource pricing points to the
urgency of optimal policy measures. Using fossil fuel data, Farzin (1996) showed that the
traditional Pigouvian taxation was insufficient in a dynamic framework. Instead, carbon and
depletion taxes should rise over time after consideration of stock externalities and the
corresponding threshold effect of atmospheric carbon. This essay will use Farzin’s model to
assess the changes in tax policy since the time of original publication using his intertemporal,
integrated model with updated data. The effects of carbon sinks and flow externalities are also
added to the model to align it more closely with real-world occurances. Using more recent
studies on the effects and future of climate change, the results call for immediate and stringent
policy. Sensitivity analysis shows high levels of variation in policy and optimal delay times
based on the marginal cost of abatement. Additionally, since 1996 optimal taxes have increased
dramatically, and higher levels of abatement are needed to decrease net emissions, likely due to a
lower long-term price elasticity of demand for fossil fuels.
1. Introduction
The issue of environmental and natural resource pricing is one of increasing importance
and controversy. Of course, the phrase ‘optimal pricing’ implies policy instruments such as
taxation to reconcile the divorce of private prices and the social costs of resource extraction and
consumption. There are other policy instruments for emission control and overexploitation of a
natural or environmental resource use including transferable permits, subsidies for emissions
reduction, performance standards, and severance taxes. To be sure, there is rarely a ‘perfect’
choice of policy for any governing body because the very act of making a choice necessitates a
trade-off in this case: no particular instrument represents the best solution for all of the concerns
involved in policy-making. It is often the case that there are multiple market failures, and hybrid
solutions are typically appropriate, as noted by Goulder & Parry (2008). Nonetheless, the
benefits for taxation are relatively abundant, such as high incentive to innovate, lower
administrative costs (as compared to, say, emission standards), and operational flexibility for
firms (OECD, 2011). This paper addresses some of the main concerns with fossil fuel taxation
using updated figures in the model by Farzin (1996)1. Namely, these concerns are the dynamic
considerations as opposed to static calculations, inclusion of both flow and stock externalities,
and the incorporation of environmental as well as resource depletion externalities. Since Farzin’s
paper, these concepts have been adapted and expanded on in multiple studies (Newell & Pizer,
1 Henceforth referred to simply as ‘Farzin’s paper’.

4. Mollins 3
2003; Dangl & Wirl, 2007; Strand, 2010; Ploegf & Withagen, 2014). The proverbial threshold
effect explicitly designed in the model has not become irrelevant but rather a more essential
component of emission control policy (Marchal, et al., 2011). As such, while the model assumes
that climate stability cannot be achieved, as opposed to the “450 scenario” outlined in the World
Energy Outlook (2011), the concept of delaying the critical level of atmospheric stock pollution
has very practical applications. The aim of this paper is threefold: first it resolves the theoretical
model in detail and reinterprets some of the major components to correspond with more recent
literature. Secondly, it extends the model slightly by introducing carbon sink effects as well as
more realistic assumptions about emission flow externalities. Thirdly, it will use updated figures
and information in the calibration of optimal taxation, emissions, and abatement.
This paper is organized in the following way: Section 2 will outline and explain the
theoretical model and the changes made from Farzin’s paper. Section 3 will describe the linear
forms that the model will adopt. Section 4 will evaluate the parameters using current figures and
statistics. Next, Section 5 will calculate and analyse the optimal values and taxation. Lastly,
Section 6 will provide concluding remarks and observations about the paper and future research.
2. Theoretical Model
Using Farzin’s paper as a guide and reference the theoretical model is described, while all
equations are solved in detail in the appendix.
The inverse demand function for fossil fuels is described as ( )p x , such that the economy,
at time t, exploits the resource at the rate of ( )x t . There are two types of intertemporal
externalities that plague the process across time, the first being in regards to the exploitation
costs. These are private expenditures and will rise the larger the amount to be extracted, but they
also rise as the stock of the resource is depleted due to the deposits being harder to reach. To use
an analogy, the low hanging fruit get picked first after which the more difficult to reach fruit are
sought, which is concordant with the Herfindahl rule that states low-cost reserves will be
exploited before high-cost reserves (Herfindahl, 1967). The current extraction rate is given as
( )x t , while the cumulative exploitation is such that X(t)=
0
( )
t
x d  . The total cost of fossil fuel
exploitation is given as:
( ( ), ( ))tC C x t X t (1)
Farzin notes that the marginal current exploitation cost could be larger at higher exploitation
rates ( 0xxC  ) and, holding the exploitation rate constant, for higher depletion rates (

5. Mollins 4
0xX XxC C  ). In our view, the latter assumption is sound but the former may be inaccurate.
Higher exploitation rates can imply larger marginal costs in the short run due to the reality of
decreasing marginal productivity. However, the very purpose of this paper is to analyse pricing
dynamically, as such, over time firms typically capitalize on economies of scale, the learning-by-
doing process, as well as a heavier investment in research and development activities. This is
outlined by the “learning curve effect” in the literature (Siegal, 1985) and the “tug-of-war” of
depletion costs and technology innovation (Lindholt, 2013). Nevertheless, in the model
simulation, technology is held constant and linear specifications are assumed so this deviation of
argument does not have a significant impact on the final results. Another important feature in the
model is that it is assumed that there is an infinite supply of fossil fuels for extraction at
increasing costs as represented by depletion costs (CX>0).
The second intertemporal externality relates to the environmental pollution created by
consumption. Once again, there is both flow and stock externalities. The environment is
damaged by the current use of the resource and the subsequent pollution, as well as the
accumulated stock pollution rising over time. Following Farzin, we denote the stock of the
externality as S(t) and the flow as s(t), then the overall cost of environmental damage at time t is
given by:
( ( ), ( )),tD D S t s t (2)
In the case of fossil fuels, and for the purpose of this paper, pollution will be exclusively
examined in terms of atmospheric levels. Additionally, it is assumed that the flow externality is
fixed in tandem to the resource extraction: ( ) ( )s t x t , where α is the proportion of the emission
that remains in the atmosphere.
There also exists a cost of abating pollution, equal to a marginal cost of abatement
multiplied by n(t) units to be abated:
( ( ))tE E n t (3)
The marginal abatement cost is unambiguously positive here ( 0nE  ). The question as to
whether or not the marginal costs rise over time is subject to some debate. On the one hand,
much like marginal extraction costs, innovation can often lead to a downward shift of the
marginal abatement curve and the more to be abated the more time and technology will be
dedicated to this end. On the other hand, as Farzin notes (p. 35), it is empirically supported that
marginal costs rise with the amount to be abated.
All the same, the net emission signals the accumulation of carbon stock pollutant2:
2 Where a dot above a variably denotes a time derivative.

6. Mollins 5
( ) ( )S s t n t

  (4)
Alternatively,
( ) ( ) ( )S z t x t n t

   (5)
It is assumed that the net flow of emission, z(t), is positive (z(t)≥0). However, unlike Farzin’s
work, the inclusion of the α term implies that there is re-absorption of a percentage of the carbon
emitted via anthropogenic carbon sinks such as oceans and trees. This is a slightly more
optimistic stance than Farzin, and closer to reality than α=1, which means that every unit of
carbon emitted by fossil fuel consumption is retained in the atmosphere, which is not accurate
according to the literature (Gruber & Sarmiento, 2002). Unfortunately, the inclusion of α could
imply the violation of the rule that ( ) 0z t  , because we are only taking a percentage of x(t) but it
is still possible that x(t)=n(t), such that αx(t)< n(t). In the real world, carbon sinks will continue
to absorb carbon from the atmosphere even at x(t)= n(t) = 0, which alleviates the contradiction.3
Equation (2) can therefore be rewritten as:
( ( ), ( )),tD D z t s t (6)
Thus, the flow of net social benefits from fossil fuel use is4
0
( , , , ) ( ) ( , ) ( , ) ( )
x
W x n X S p s ds C x X D z S E n    (7)
To find the optimal values of resource use, taxation, and abatement one must maximize the
discounted present value flow of net social benefits:
{ 0, 0}
0
max ( )t
x n
e W t dt


 
  (8)
s.t. ,X x

 (8a)
S z x n

   (8b)
0(0) 0X X  (8c)
0(0) 0S S  (8d)
3 ( ) 0z t  will still beused as a condition for simplecalculationslater in this paper.
4 The time index of variables will notbe included when appropriatefor aesthetic and flowpurposes.

7. Mollins 6
Where ρ>0 is the constant social discount rate.
The current value Hamiltonian is thus
     *, , * ( ), ( )H W x n X S t tx t z t    (9)
Where λ(t) and μ(t) are the shadow externality costs of cumulative resource extraction and
cumulative environmental pollution, respectively.5 It shall be assumed here that there is an
interior solution, despite using a linear model in the simulation. The first-order necessary
conditions for the problem are:
( ) 0x z
H
p x C D
x
 

     

(9a)
0z n
H
D E
n


   

(9b)
xC  

   (9c)
SD  

   (9d)
lim ( ) ( ) 0t
t
e t X t


 (9e)
lim ( ) ( ) 0t
t
e t S t


 (9f)
Condition (9a) can be interpreted as a pricing rule for fossil fuels, such that the shadow
price must include both flow and stock costs of resource and environmental exploitation.
Using the above conditions one can solve for the optimal paths of the shadow costs:
( )
( ) ( ) ( )t
X
t
t e C d 
  

 
  (10)
( )
( ) ( ) ( )t
S
t
t e D d 
  

 
  (11)
The resource scarcity rent, λ(t), is described as the present value discounted sum of
cumulative marginal depletion costs. This is intuitively appealing as scarcity rent is defined as
5 Unlikehis earlier work (Farzin,1992),Farzin subtracts the shadowstock externality costs.This can be a sourceof
confusion,as itcan be easily interpreted that subtractinga costis synonymous with addinga benefit. This paper
will followthe logic thatbecause λ and μ arecosts,they are positiveand should therefore be subtracted in the
Hamiltonian (i.e.λ(t)> 0, μ(t)> 0).

8. Mollins 7
the wedge between price and marginal extraction costs; the extra cost in future extraction costs
due to the depletion from the current period's extraction makes sense as the compensation
between the resource price and marginal cost.6 The time path of the scarcity rent is a question of
considerable debate. Of course, the Hotelling rule states that resource prices should be growing
at the rate of interest, but in his seminal work Hotelling did not consider extraction costs that are
stock dependant, as fossil fuels unambiguously are (see: Livernois & Martin, 2001; Khanna,
2001). Devarajan and Fisher (1981) showed that scarcity rent would grow at the rate of interest
minus the percentage increase in extraction cost. This is in line with equation (10): there is
discounting involved, which implies that it will grow at the rate of interest, ρ, times the change in
marginal depletion costs while the extra flow costs from past extraction is reflected in the higher
marginal extraction cost (Cx).7
Equation (11) tells a similar story for the environmental scarcity rent, μ(t). It is equal to
the discounted sum of cumulative marginal pollution damages. It is μ(t) that will be used to
identify the proper carbon tax, much like λ is the wedge between price and cost, μ can be
understood as the wedge between social and private costs, or the user cost.
This is the theoretical basis of the model; the next section will now define the forms of
the aforementioned functions.
3. Functional Forms
For the purpose of simulations, the functions in this paper shall take the following linear
forms as described by Farzin8:
Resource demand:
( )P x a bx  (12)
Resource exploitation cost:
( , )C x X cx X  (13)
Environmental damage:
6 It is assumed that the fossil fuel industry is,for the purpose of this paper, perfectly competitive.
7 As briefly discussed,the change in depletion costs aredependent on many things, such as technology, new
depositdiscoveries,and the amount extracted in the current period. In the model simulation section of this paper,
the Devarajan and Fisher concept will beroughly followed: constantdepletion costs,but stock effects that subtract
from the growth of prices and scarcity rent. Sensitivity analysiswill beperformed in lieu of variablecosts.
8 All parameters aregreater than zero unless otherwiseindicated

9. Mollins 8
ˆ0, S<S
( , ) ,
ˆ0, S S
{D z S dz S 

 
 
(14)
Marginal abatement costs:
nE En (15)
Where
ˆS is the level of pollution stock that causes significant economic damages and is
the so-called critical pollution level.
Equation (14) suggests that marginal stock damages are not actually felt until the critical
stock level, or the carbon stock at which there is no chance of cleaning it back up. The question
is thus whether to delay abatement and avoid the costs associated with it, or to abate now and
delay the stock damages while incurring the costs of cleaning up in the present. In this sense, it is
only considered possible to optimally delay the time in which the stock pollutant reaches its
critical level.
If T>0 is the optimal delay time where ˆ( )S T S , then with the linear specifications listed
above optimization problem (9) becomes:
2
0 0
1max ( ) ( ) ( ) ( , ) ( )
2{ }
x
t
e a bx dx cx X d x n t T S En dt
   


        (16)
where ( , )t T is an auxiliary function used in order to realise equation (14), such that,
0, 0<t<T
( , )
1, t
{t T
T
 

(17)
The current value Hamiltonian is the same as before with the first-order conditions taking the
form in (9a)-(9f). However, using the auxiliary function, (9d) becomes:
( , )t T  

  (18)
Which implies,
0 , 0<t<T 

  (19a)
, T<t<  

   (19b)
Using these last two equations, it can be seen that

10. Mollins 9
( )
, 0<t T
( )
, T t<
{
T t
e
t

 

 
 


 
(19)
Additionally, using equations (10) and (9c), it follows that
( ) , 0 t<t     (20)
The shadow environmental scarcity rent thus rises at the social discount rate up until time T, at
which it reaches the level of   and stays there. Of course, the assumption in equation (14) that
there are no stock damages before the critical level is unrealistic; the detrimental effects of
carbon stock do not just occur suddenly one day, but is rather a gradual process. To deal with this
inaccuracy, this paper will attempt to estimate δ so that it reflects the damages of the
accumulated carbon at time T, which will be much higher than any other point (t< T). Therefore,
equation (21) can be interpreted as the discounted damage incurred at the critical level of stock
pollution. Farzin (p. 21) interprets equation (20) as preparatory scarcity rent in anticipation of the
stock damages felt entirely after time T. Using the above explanation, equation (20) can be re-
interpreted as environmental scarcity rent (which can be understood as the optimal dynamic
carbon tax) reflecting stock damages that supposedly grows at the social discount rate up to time
T when marginal stock damages reaches its highest point yet (δ). At this point, damages are
irreversible and the delayed benefit of abatement (lower future externality costs) indicated by the
environmental scarcity rent remains constant at   .
To solve for the optimal path of fossil fuel extraction and marginal emissions, equations
(20) and (21) are substituted into (9a):
( )
[ ]/ , 0<t T
*( )
[ ( ) / ]/ , T t<
{
T t
a c d e b
x t
a c d b

   
  
 
    

     
(21)
Additionally, using μ from (20) and (9b), the optimal abatement path for fossil fuels is obtained:
( )
[d+ e ]/ , 0<t T
*( )
[d+ ]/E, T t<
{
T t
E
n t

 
 
 


 
(22)
Equation (23) suggests that resource use should decrease over time from its initial point until
time T when it remains at a constant level afterwards. The optimal abatement path indicates that
abatement should rise over time until time T when it too becomes constant. It is worth noting that
the assumption that the stock of pollution is irreversible is not entirely accurate in our view, even
after the critical level S(T)= ˆS (Izrael, et al., 2007). The commonly accepted temperature level at
which there will be significant and irreversible impacts is 2 Celsius, although even this level is
estimated subject to a probability, albeit a high one (e.g. Parry, et.al. 2007). All the same, the
conclusion is inescapable that pollution reduction must begin immediately, and must grow at

11. Mollins 10
least up until the critical level of carbon stock. Static Pigouvian taxation is therefore non-optimal
with a growing stock of pollution and dynamic considerations.
The major question of the magnitude of the optimal delay time remains. To solve for T
one must take (8b) to show:
0
0
( ) [ ( ) ( )]
t
S t x n d S      (23)
Upon substitution of equation (22) and (23) into (24), this becomes
2
20
ˆ( ) [( ) ( )] ( )( )(1 )T
b S S a c d b E T b E e     


         (24)
Upon examination of (25) it can be seen that changes in some of the parameters could
potentially greatly affect the optimal delay time. Sensitivity analysis will be performed on some
of the parameters to simulate both variation across time and uncertainty in valuation. The
following section will now deal with the calibration of parameter values.
4. Parameter Values
The difficulty of evaluating many of the parameters in the model comes from the fact that they
are often unobservable, so averaging and second best approaches are used in many cases. Also,
measurement values are in 2011 U.S. dollars due to the high accessibility of data from this year9.
Carbon concentration. Most estimates of carbon concentration in the atmosphere are expressed
in parts per million (ppm) of CO2.One must distinguish between carbon concentrations and that
of carbon dioxide. The atomic weight of carbon is 12 atomic mass units, which means carbon
dioxide has a weight of 44.10 This implies that one tonne of carbon is equal to 3.67 tonnes of
CO2. It is also common practice to examine current and future atmospheric carbon concentration
as compared to pre-industrial levels, where it is assumed to be mostly devoid of anthropogenic
additions (Cao & Caldeira, 2010). For the sake of consistency, the initial atmospheric
concentration of carbon S0 and the critical level ˆS will be measured in regards to the aggregate
tonnes of carbon accumulated. Accordingly, there is currently a concentration of 400 ppm of
CO2 in the atmosphere. To convert it to tonnes of carbon, the U.S. Department of Energy (2012)
calculates that 1 ppm of atmospheric CO2 is equal to about 2.13 billion tonnes of carbon (GtC),
based on the weight of the Earth’s atmosphere. Hence, S0 is found to be about 850 GtC. There is
now a general consensus that the raising of global temperature by 2 °C will lead to the critical
9 This paper also uses themetric system, while Farzin uses the Imperial system.
10 CO2 has two units of oxygen, with a weight of 16 atomic mass units,and one unit of carbon:12+2*16=44.

12. Mollins 11
level, which is associated with a concentration of about 450 ppm of CO2 (Marchal, et al., 2011).
It can thus be determined that ˆS is equal to approximately 960 GtC.
Parameters a and b. To find these parameters, the elasticity of demand for fossil fuels must be
first identified. In the literature, there is significant discrepancy among findings for the demand
elasticity among different countries and areas as well as for the different types of fossil fuels. It
should be emphasized that for the calculations in this model, it is the long-term price elasticity of
demand that should be used. This distinction is necessary because the long-term elasticity
considers potential changes in technology, while the short-term elasticity captures adjustment
costs. Therefore, by using the long-term price elasticity of demand, one can capture to some
degree exogenous changes in technology over time, which is otherwise not included in the
model. The estimations of the long-run elasticity are often hard to obtain, because of the
uncertainty inherent in long-run predictions. Estimates range for oil elasticity from about 0.08
(IMF, 2011), to about 0.4 (Baumeister & Peersman, 2013; Javan & Zahran, 2015), but rarely
went much higher than this for recent data11. Natural gas price elasticities range in a similar way,
typically from 0.1 to about 0.4 (Dagher, 2012). The elasticity estimates for coal were less
abundant, but it is estimated that these were approximately the same as natural gas and oil (IEA,
Fuel Competition in Power Generation and Elasticities of Substitution, 2012). Based on this
literature, we consider a value of 0.3 as an appropriate figure. Solving for and a and b using
equation (11) yields:
0 (1 1 )a p   (25)
And,
0 0( )(1 )b p x  (26)
Where ε is the price elasticity of demand and 0p and 0x are point observations of price and
elasticity. The amount demanded is about 4.2 billion tonnes of oil, 113 trillion cubic feet of
natural gas (2.825 billion tonnes of oil equivalent, or b.t.o.e.), and 7.6 billion tonnes of coal (5.25
b.t.o.e.), which corresponds to about 8.4 billion tonnes of carbon (GtC).12 The quantity
demanded is assumed to be constant over the whole time period examined. The price of fossil
fuels is slightly more difficult to evaluate at a single point. The recent trends in pricing,
particularly with oil, have provided much uncertainty about future values. The prices in 2011
were quite high, and correspond to a global average of around $550 per tC (IEA, 2011).
However, recent events may make this average somewhat obsolete as prices have dropped by
60% from the 2011 levels, and it is possible that they may not rebound fully as distortions
11 These studies also often noted that for data before the 1990s,priceelasticities weremuch higher and were
more consistentwith Farzin’s approximation of 0.75.
12 Sources: (World Coal Association,2015),(EIA, 2012).The computed figures are in linewith those by the OECD
(Marchal,et al.,2011).

13. Mollins 12
become “baked” into prices (IEA, 2015). Therefore, lower prices are also to be considered for
each of the fossil fuels, implying an average of $425 per tC for medium-range prices and $266
per tC for low-range13. Using this information and formulae, a and b are allowed to vary together
in sensitivity analysis using high, medium, and low prices, such that: a = [2376, 1843, 1154], and
b = [218, 169,106].
Parameter E. The marginal abatement costs are typically measured by one of two
approaches: bottom-up and top-down. The former14 stems from very specific processes, often at
an individual level, and focuses on existing inefficiencies in energy use (switching to fluorescent
lighting, recycling, using a hybrid car). The latter analyzes costs from a macroeconomic
perspective: it evaluates target abatement across the entire economy compared to business as
usual baselines (BAU) (Cline, 2011). The fact that this paper endeavors to calculate optimal
pricing of fossil fuel use for optimal tax purposes, a top-down model seems most appropriate.
Even after restricting the scope of consideration to this model, there still exists significant
variation in the specifications to be used. For example, some models have different stabilization
targets (how much emission to be reduced), backstop technology, baseline emissions, carbon
capture, among other variables.15 The model that was originally used in Farzin’s paper cannot
simply be recycled due to outdated data. Using the findings of the RICE model by W. Nordhaus
(2010) and applied by W. Cline (2011), it is estimated that by 2050 to abate the 75% of BAU
emissions suggested by the Copenhagen Accord (UNFCCC, 2010), total cost to abate would be
1.15 percent of global GDP in 2050. This corresponds to $3800 (U.S. 2011 dollars), and
abatement of (14.5)(.75)=10.9 GtC, which implies a marginal cost of $348.6 per tC. Using the
linear formula in (14), En=348.6, where n is the amount to be abated, E=348.6/10.9=32. To use
Farzin’s expression, this is equivalent to E=463.7R, where R is the percentage to be abated. This
is most definitely on the high end of marginal abatement cost estimates, as noted by (Baker,
Clarke, & Shittu, 2008). As such, E is allowed to vary downward to Farzin’s original estimate
which will also compensate for some learning-by-doing and technology improvements, such that
E= [22, 27, 32].
Parameter c and d. The parameter c can be interpreted as the average global worldwide lifting
costs (production costs) for fossil fuels. The most readily available estimates are specifically for
oil and gas, and are roughly $11 per boe (EIA, 2011), (Aguilera, Eggert, Lagos C.C., & Tilton,
2009). If the carbon content of fossil fuels is averaged out to roughly 20 KgC/MMBtu, then
c=$99 per tC. Parameter d is slightly harder to estimate because the flow externalities are
unobservable, and more importantly, vary greatly depending on the area and technology in a
particular case. For example, car emissions would cause much less damage to people’s health in
a spread out rural setting compared to a busy city. Assuming, as Farzin did, that these costs are
13 The medium range prices areused for baselinecalculations.
14 Sometimes called theengineering approach.
15 However, only some of these modifications havebeen found to causestatistically differentmarginal costs (Kuik,
Brander, & Tol, 2009)

14. Mollins 13
equal to one dollar per boe is not realistic. He likely did this because of mathematical
convenience, as was also the argument for setting α =1, but also because of the literature at the
time, which has been significantly developed since then. In fact, one study found the monetary
value of health impacts to be significantly larger than the retail price of the fossil fuels and
potentially cost thousands of dollars per tC (Machol & Rizk, 2013). In a tamer version of this
methodology, external costs from electricity generation via healthcare and materials damage has
been shown to be about $1.27-$4.35 per GJ for certain technologies (El-Kordy, Badr, Abed, &
Ibrahim, 2002),16 which is approximately $10 per boe or $90 per tC. For the purpose of
comparability, the model will also be calculated assuming d=$1 per boe.
Parameter α. As mentioned before, the natural carbon sinks absorb a certain amount of carbon
based on the quantity of emissions. A few studies (Gruber & Sarmiento, 2002; Sabine, et al.,
2004) have found that these sinks could subtract over a third of total yearly emissions from
atmospheric carbon concentrations. However, other human activities like deforestation can
reverse some of these natural processes and release the carbon back into the atmosphere.
Therefore, α is conservatively estimated to be 0.7.
Parameters γ and δ. The marginal depletion costs, γ, are defined as the costs imposed on future
extraction due to an extra unit of current extraction and thus lower resource stock. To appraise
this value, resource scarcity rent is approximated by the finding and development (F&D) costs of
fossil fuels. This makes economic sense because resource rent can be understood to define the
opportunity costs associated with the exploitation of a particular deposit, where the next best
alternative in this case would be to find and operate a different deposit. The F&D costs for low-
cost regions (OPEC mostly) are valued at $5-12 per boe (IEA, 2014), other higher cost regions
will obviously bring up the average, but global estimates are typically in the mid-teens (EIA,
2011). As such, if λ is equal to $15 per boe or about $135 per tC, γ can be calculated in the
following manner using (19): γ = ρλ =.02(135) = 2.7 per tC. This is more than double Farzin’s γ,
but is reasonable considering the extraction rate of fossil fuels and the low-hanging fruit
behaviour mentioned above.
Recall that in the definition for δ, it was stated that it was an approximation of the stock
damages felt at time T, to simulate the gradual effects of stock damages rising at the social
discount rate. Therefore, δ must be measured at the optimal delay time by using formula (8b) and
substituting in equations (20) and (21) to attain ( ) / ( )a c b E d          , which is
the upper-bound limit for   . Using the base values for each of the parameters, it can be
computed that 0 1.9  . It will thus be assumed, the same as Farzin, that δ = 1.5.
16 This is region specific,as isthe Machol and Rizk study, so should be taken as very rough estimates.

15. Mollins 14
5. Results
Table 1. Optimal valuation of parameters, sensitivity analysis.
T* μ(0) μ(T) x(0) x(T) n(0) n(T) z(0) z(T)
price
high 49.00 28.15 75.00 9.32 9.17 3.69 5.16 2.83 1.26
medium 55.00 24.97 75.00 8.87 8.67 3.59 5.16 2.62 0.91
low 80.00 15.14 75.00 7.74 7.34 3.29 5.16 2.13 -0.02
T* μ(0) μ(T) x(0) x(T) n(0) n(T) z(0) z(T)
E
32 55.00 24.97 75.00 8.88 8.67 3.59 5.16 2.62 0.91
27 77.00 16.08 75.00 8.91 8.67 3.93 6.11 2.31 -0.04
22 124.00 6.28 75.00 8.95 8.67 4.38 7.50 1.89 -1.43
T* μ(0) μ(T) x(0) x(T) n(0) n(T) z(0) z(T)
ρ
0.02 55.00 24.97 75.00 8.88 8.67 3.59 5.16 2.62 0.91
0.03 37.00 23.86 50.00 9.18 9.04 3.33 4.38 3.10 1.95
0.05 30.00 16.46 30.00 9.43 9.33 3.02 3.75 3.58 2.78
T* μ(0) μ(T) x(0) x(T) n(0) n(T) z(0) z(T)
d = 9 25.00 45.49 75.00 9.27 9.15 1.70 2.63 4.78 3.78
α = 1 26.00 44.59 75.00 8.71 8.54 4.21 5.16 4.51 3.38
The benchmark case exhibits an optimal delay time of 55 years. This is just under half of
what Farzin predicted in 1996 reflecting the higher level of difficulty to control emissions the
longer appropriate policy implementation is delayed. The higher costs of extraction, the larger
(and growing) emission rates and stock of carbon, as well as the updated and lower ‘critical
level’ of atmospheric carbon, all contribute to more expensive emissions control policy.
However, this is somewhat mitigated by the higher flow costs of carbon used in this model. A
higher value for the d parameter raises the social cost of carbon and thus increases the damages
foregone (or benefits received) from abating emissions. This is also true for α, such that when
anthropogenic carbon sinks are introduced, it decreases the net emissions each year, which
pushes back the time at which the critical level of stock pollution is reached.
Another observation from Table 1 is that there is relatively large sensitivity to the delay
time from the parameters as compared to Farzin’s earlier numbers. Likely, a major reason that
this higher variation exists is because of the lower critical level of stock pollution and a higher

16. Mollins 15
current level. This indicates that there is much less of an ‘allowance’ to work with, and a change
in values will create bigger impacts in a shorter-term situation. It is important to note that this
linear model does not explicitly account for technological change, but the increasing costs
associated with more carbon to be abated was simulated in the original calculation of parameter
E. This is the reason abatement cost sensitivity were only calculated at lower levels than the
benchmark. Therefore, the high level of sensitivity for this parameter can be ascribed to an
intense impact of technology on policy decisions, and are increasingly influential the lower the
value assigned to parameter E. This finding provides further evidence that a tax policy would be
the most effective in this case because, as mentioned above, among the possible policy
instruments taxation theoretically ranks among the highest for incentive to innovate; a claim that
has been empirically demonstrated for pollution taxes (OECD, 2010). Additional research and
development incentives such as research grants and subsidies would be advantageous, and can be
done despite rising costs because of the increasing marginal benefits derived. Furthermore, at
lower levels of E the net emissions at time T become negative. As mentioned before, this violates
the assumption that z(t) ≥0, but because natural sinks continue to absorb carbon even if x(t)= n(t)
= 0, this still corresponds to real world outcomes.
Table 1 implies the following optimal emissions path for the benchmark values:
0.02
8.98 0.10 , 0 t 60
*( )
8.67, t 60
{
t
e
x t
  


(27)
And the optimal abatement path such that,
0.02
2.8 0.71 , 0 t 60
*( )
5.16, t 60
{
t
e
n t
  


(28)
An interesting finding here is that while Farzin contributes the decrease in net emissions to 66%
from more abatement and 34% from lower consumption from the present to time T, the above
results show about 88% of the change in net emissions coming from higher abatement. This is a
result of a higher dependency for fossil fuels in this essay’s model, meaning abatement
technology is cheaper to implement as compared to switching away from oil, gas, and coal.
The optimal tax is calculated by adding a constant depletion charge of λ= $135/ tC, a
flow emissions charge of d= $90/ tC, and a stock pollution tax that changes over time in the
following manner:
0.02
24.47 , 0 t 60
*( )
75, t 60
{
t
e
t
 


(29)
Equation (30) says that the carbon stock tax should start at $24.47/ tC and rise at a
constant rate for 55 years at which point it will remain at $75/ tC. The results in Table 1 showed

17. Mollins 16
that, in particular, changes in the price do not significantly impact the carbon stock tax. A
possible explanation for this is that in the model quantity demanded did not change, regardless of
the price. Of course, this is unrealistic, and in a more sophisticated simulation demand alterations
could possibly lead to a necessary change in taxation to compensate for a lower or higher price
of fossil fuels.
The taxation that has been discussed here is on the higher end of the typical estimates for
carbon taxes. British Columbia currently is the only province in Canada with a carbon tax, and it
is equal to $30 per tonne of CO2 or $110/ tC. The tax shown in this paper starts at a total of $249/
tC, however, this taxation includes the resource depletion tax, which is often not considered in
carbon taxing. Therefore, the social cost of carbon was really calculated at about $160/ tC. 1 Tol
(2010) completed a literature review and showed that the 95th percentile for the social cost of
carbon was as high as $676/ tC, so whether one includes the depletion tax or not, these values are
certainly not unrealistic. As one can observe from the differences between the calculations seen
here and Farzin’s total of $64/ tC, the cost will continue to rise as policies fall short of social
optimality, and pollution stocks rise. It should be noted that in this case these taxes should be
implemented separately, rather than as a single tax. Although this causes higher administrative
costs, the incentivisation to adapt to the policies will be most potent when directed at a single
externality (Goulder & Parry, 2008). Of course, separating flow and stock externality taxes is
likely unfeasible, but depletion and carbon taxes ought to be distinct.
6. Conclusion
This paper used the model outlined by Farzin (1996) to show how optimal taxation has
changed 20 years later. The model was altered slightly to reflect more realistic assumptions
about the natural carbon cycle and the current flow damages of carbon. While these changes did
increase the optimal delay time of reaching the critical stock level of pollution, the tax still
increased by just under a factor of 4. These numbers reflect the increasing urgency of the
adaptation of socially optimal policy measures. Of course, the values presented in this simple
model cannot accurately express every real world variable and nuance perfectly. Technology
advancements, population increase, and hybrid policy measures are just a few elements that
could sophisticate the model greatly. As previously mentioned, the actual values assigned to each
parameter, especially that of ˆS , are subject to a certain probability and the timing and extent of
economic damage suffered is fairly unpredictable. In other words, uncertainty is a huge
complication for the issue of optimal policy, and as Christiansen and Smith (2013) opined any
significant shift in one element could mean an entirely new policy environment. The increasing
nature of the tax as suggested by the research is also a matter of concern for policy makers.
However, this particular issue can be overcome through stepwise and predictable increases as
well as redistribution of the tax base on equity grounds, although this latter strategy is practically

18. Mollins 17
quite difficult (OECD, 2011). Finally, each of these parameters can vary quite substantially
depending on the nation in question, so these numbers should be taken as an average across the
globe. It will ultimately be up to sovereign nations to make the distinction of how these averages
apply to them, and it will be the responsibility of international agencies to advise each nation on
the subject.

19. Mollins 18
Appendix
Equations (9c) and (9d) are solved in the same manner, and thus this appendix will only include
a proof of the former. This process will be repeated for other similar equations.
We know that the equation of motion for the current-value Hamiltonian takes the form when
subtracting the shadow costs:
H
X
 

 

(A.1.1)
Which implies,
XC 

  (A.1.2)
And can be re-arranged to,
XC  

   (A.1.3)
The proof of (10) and (11) is as follows:
( ) ( )t
Xt e A C e d 
 
   (A.2.1)
Where CX is a function of time t, and A is a constant. Rearranging and setting t = T,
( )T T
Xe T A C e dT 
 
   (A.2.2)
We know from (9e) that the LHS is equal to zero, because it is not optimal to let X(t) rise
consistently with time, which means that:
T
XA C e dT
  (A.2.3)
Substituting this into (A.2.1) yields
( ) ( )t T t
X Xt e C e dT C e dt  
  
   (A.2.4)
Or,

20. Mollins 19
( )
( ) t
X
t
t C e d 
 

 
  (A.2.5)
Equation (20) and (21) are solved by using the general solution for a differential equation, as
follows:
( ) ( )t t
t e A e dt 
  
   (A.3.1)
Rearranging and setting t= T,
e ( )T T
T A e dT 
  
   (A.3.2)
From equation (9f) it is known that the LHS is equal to zero, and following the procedure in
(A.2),
( ) t
t
t e e d 
  


  (A.3.3)
Solving the integral yields
( )
( ) ( ) |t
t
t e   
 
  (A.3.4)
Where the right hand side is simply equal to 

. The above methodology is used to solve both
(21) and for (20) when T t  . In the case that 0 t T  , it is known that ( )T   ; upon
using the general solution for this particular problem and then solving for A gives:
T
A e 


 (A.3.5)
Finally, substituting (A.3.5) back into the general solution gives equation (20) for 0 t T  , i.e.
( )
( ) T t
t e  
 
 (A.3.6)

21. Mollins 20
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