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Updated, 7 March 2004
ECON/SØK500 Natural resource economics, Spring 2003
Olav Bjerkholt:
Lecture notes on the Theory of Nonrenewable Resources
5. Taxation of nonrenewable resources
The taxation of nonrenewable resources in general and of oil in particular has generated a
literature of its own. Oil is a particularly profitable nonrenewable resource and is heavily
taxed in many countries (although several producing countries sell oil domestically at low
prices).
The purpose of taxation of nonrenewable resources differs from taxation of other
industries as it can be said to be directed towards collecting a share of the resource rent
(“oil rent”) for the public. Usually it is the case in producing countries that the public
owns the (underground) resource while various operators, some of which may be publicly
owned, run the resource-extracting firms and are exposed to taxation.
The taxation instruments, i.e. the different kinds of taxes applied, may not be special to
the nonrenewable resource. Often several kinds of taxation are applied at the same time.
The efficiency of different kinds of taxation can be looked at from two perspectives. One
is the fiscal efficiency, that is how successful or useful taxation is for collecting the
resource rent. The other is the (in-)efficiency in the sense of creating distortions in the
market, usually measured as deviations from a non-tax situation.
Taxation is one side of the fiscal aspect of nonrenewable resources, the other side being
the use of the resource rent or the share of it that has been gathered through taxation and
ownership. For many oil countries the fiscal management of the oil rent has major
importance for national economic development as well as for the distribution of the
benefits from oil extraction.
The analysis of taxation issues and particular efficiency aspects can be quite complicated
as taxation aimed at capturing the resource rent often comes on top of corporate income
taxation and other kinds of taxation as well. The tax rules will often pay attention to
incentives for exploration and also for secondary recovery.
We will discuss taxation only in a very stylized way within the simple models we have
looked at so far, but try to make this treatment fairly general. The focus will be on how
the taxation rules affect the price and extraction profiles, relative to the competitive
solution, through the incentives of the operators.
We will not deal with many other aspects such that e.g. also non-oil producing
governments can behave strategically with regard to taxation of nonrenewable resources,
trying to capture part of the oil rent through import tariffs or tax on oil products. Neither
will we deal with the fact that different jurisdictions (oil regions) have different tax rules,
allowing international operators some leeway for strategic games. In general, studying
distortion effects of taxation ought to take into consideration the already existing
distortions, hence should be conducted in a second-best framework.
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Perman et al. (2003) has just a short and limited section (15.7) on taxation. It shows two
important results: (1) a tax on the resource rent, pt  b , is neutral in its effect on the
optimal extraction path, (2) a revenue tax, i.e. a tax on pt Rt , is distorting and changes the
optimal extraction pattern to that of production with higher unit cost. Section 15.7 should
be studied in addition to the discussion of these lecture notes, although the content
overlaps. (Be aware of confusions in the terminology, Perman et al. uses “royalty tax” for
“resource rent tax”, while “royalty tax” below means something else.)
The method we apply is to derive the impact of different kinds of taxes through
comparison with the competitive solution, supported by intuitive reasoning. We do this in
two stylized settings. The first is the simple Hotelling model with constant unit costs. In
the second setting we allow cost to be a function of the extraction rate. This will look
more complicated, but is not really difficult. In both of these settings the shadow price of
the remaining resource will be positive and thus the entire stock will be extracted. This
simplifies the reasoning.
More comprehensive theoretical and empirical treatments would also take into
consideration that unit costs depend upon the remaining stock (thus taxation will easily
affect the total amount of the stock extracted) and deal with investments and capital stock
explicitly. We leave these issues aside in this context.
For the analysis below we introduce the following types of taxes written as function
σ(.) of one or more of the following variables: extraction (Rt), price (pt), resource stock
(St), time (t):
σ(R, t) = s(t)Rt
royalty/severance tax
σ(R,p,t)= s(t)ptRt
ad valorem royalty/severance/revenue tax
σ (t)
franchise tax/license fee/fixed property tax
= f(t)
σ(S,t) = g(t)St
property tax
σ(R,p,t)= u (t )[ pt  b ]Rt
resource rent/net price/profit tax
We first look at the simple Hotelling model with a time independent linear demand
schedule with a choke point and constant unit costs and mark the competitive non-tax
solution with ^ as the reference case. The solution of this model can be set out as
follows:
(5.1)
pˆ t     Rˆt
(5.2)
qˆt  pˆ t  b
(5.3)
qˆt  rqˆt
Tˆ
(5.4)
S0   Rˆt dt
0
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(5.5)
pˆ Tˆ  
Let us then consider royalty taxation with rate s(t). We then must have that the arbitrage
principle instead of (5.3) is
(5.6)
d [qt  s (t )]
 r[qt  s (t )]
dt
From (5.6) it follows immediately that if s(t) increases with rate r then all the equations
(5.1)-(5.5) are fulfilled also under the tax regime and the solution is thus the same as the
reference case, i.e. the tax is neutral. Hence, no distortion under royalty taxation if the
royalty rate increases with rate r. By varying the initial rate s(0) any part of the resource
rent can be collected, as long as s(0)  pˆ t  b .
If on the other hand s(t) is constant over time, it follows that q must increase at a rate less
than r, causing a similar effect as a lower discount rate, namely postponing of extraction
through a higher initial price and a correspondingly longer extraction period.
Let us then consider a tax on the resource rent. Also in this case only (5.3) gets a different
specification. The arbitrage principle in this case will be
(5.7)
d [qt (1  u (t ))]
 rqt (1  u (t ))
dt
From (5.7) we can see that u(t) constant implies that q grows at rate r, hence no
distortion. On the other hand it also follows easily that du/dt>0 implies that q grows at a
rate higher than r, and vice versa when du/dt<0, with extraction shifted
forwards/backwards in time, respectively.
All the effects we have looked are supported by intuitive reasoning. E.g. a constant
royalty tax is reduced in real value by postponing production, while an increasing
resource rent tax rate can be counteracted by moving production forward in time.
We leave the study of the other tax alternatives under constant unit costs as exercises for
those who are interested.
Let us then introduce a slightly more ambitious analysis by allowing unit costs to depend
upon the extraction rate, replacing b with b( Rt ) . If we go back to the maximization
problem under these assumptions, writing the tax function in an all-encompassing form
as σ(R,p,S,t) we have the Hamiltonian in current value as
(5.8)
H C  pt Rt  b( Rt )   ( Rt , pt , St , t )  t Rt
By the maximum principle we have the first-order condition
(5.9)
pt  bR ( Rt )   R  t
with the resource rent changing over time according to
(5.10)
t  r t   S
The transversality condition – H C  0 at T – implies that
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(5.11)
pT 
b( RT ) 

 T
RT
RT
which combined with first order condition at T gives that
(5.12)
bR ( RT )   R 
b( RT ) 

RT
RT
Let us now do a little trick and consider (5.8)-(5.12) as the equations holding for a
resource producing firm. Assuming all firms identical and reconsider the equations as
those holding for the resource producing industry. We then have to endogenize the price
via the demand equation and thus replace pt in (5.9) with pt = P(Rt).
What we then do is to differentiate (5.9) totally wrt. t, using (5.10) and (5.9) to eliminate
 t and  t , respectively. After rearranging we arrive at
(5.13)
Rt ( pR  bRR   RR   Rp 
dP
)  r[ p  bR   R ]   S   Rt   RS R
dR
This is the differential equation for Rt, useful for studying the effects of alternative taxes.
In the non-tax case the equation simplifies to
(5.14)
Rˆ  ( pˆ R  bRR )  r[ pˆ  bR ]
We see immediately that R decreases over time. When cost is a function of the extraction
rate, the ultimate extraction rate is not zero, but the extraction rate that minimizes average
cost, as indeed follows from (5.11).
Consider now the franchise tax, i.e. σ = f(t). If we first reason intuitively about this tax
which is paid by a fixed amount, possibly varying over time, for the entire period
extraction takes place, we might expect that the imposition of the tax would give an
incentive to shorten the extraction period, which indeed turns out to be the case. How can
we see this? (5.13) reduces under franchise taxation to
(5.15)
R  ( pR  bRR )  r[ p  bR ]
The differential equation is identical to that of the non-tax case! Does this mean that the
extraction paths are equal? Not necessarily, it implies only that if the two extraction paths
had the same extraction rate at the same point of time they had to coincide. We see,
however, from the transversality condition (5.12) that the marginal cost at the terminal
point T is higher than that of the non-tax case at Tˆ , and thus the terminal extraction level
RT must be higher than RˆTˆ . As total extraction are the same with and without taxation the
only possible profile for extraction under franchise taxation is with a higher initial
extraction, hence lower initial price, and a shorter extraction period. Drawn in a diagram
the higher rate of extraction under franchise taxation until T must exactly match the nontax extraction from T to Tˆ .
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Finally, let us check what we find for the royalty tax, i.e. σ = s(t)R, in this setting. We
see now from the terminal condition (5.12) that we must have RT  RˆTˆ , (but not
necessarily T  Tˆ ).
The differential equation (5.13) now becomes
R  ( pR  bRR )  r[ p  bR  s (t )]  s
(5.16)
If the royalty rate is constant, we see from (5.16) that if at some t Rt  Rˆt , then Rt must
cut Rˆ ˆ from below. Hence, the only possible solution is that R starts out lower (price
t
T
higher) than Rˆt , cuts Rˆ r from below and continues beyond Tˆ to T ending with extraction
rate R equal to Rˆ ˆ .
T
T
We have still only dealt with the simplest cases of taxation, but we end the discussion of
taxation issues here.