Confuser Cost
Abstract
Inconsistent and interchangeable usage of marginal user cost (MUC), net price,
resource rent, shadow price and royalty promotes confusion and policy errors regarding
resource valuation and extraction incentives. We provide distinct concepts that facilitate
more transparent articulation of first order conditions for both renewable and nonrenewable resources, regardless of whether an internal optimum obtains. In particular, we
show that the Pearce formulation of the first-order condition requires the MUC being
defined as the costate variable of the current value Hamiltonian, not the present-value
definition often provided. While resource economics originated as a branch of capital
theory, user cost in the theory of optimal investment has since come to mean the implicit
rental price, not the asset value foregone. We clarify these meanings in a sustainable
growth context by extending Jorgenson’s canonical model of capital accumulation.
Key words: marginal user cost, renewable resources, royalty, shadow price, user cost of
capital
1
1. Introduction
Certain fundamental concepts have been used so frequently in the economics
literature that after a time there is general acceptance that the concepts are universally
understood and no longer need clarification or explicit definitions. Examination of
current usage, however, sometimes reveals that a concept is not universally understood,
or that its underlying meaning has become confused. For example, Morey (1984)
suggested that “consumer surplus” was used to cover such a multitude of concepts that it
had become "confuser surplus". In what follows, we suggest that a similar problem has
occurred with respect to “marginal user cost” since it was first coined by Keynes in 1936.
Marginal user cost (MUC)is a central concept in resource economics. In order to
allocate resources efficiently over time, the opportunity cost of having a smaller resource
stock in the future must be taken into account. Despite the importance of this concept,
there are a number of ambiguities and inconsistencies in its usage.
In resource economics, the words “marginal user cost,” “net price,” “resource
rent,” “royalty,” and “shadow price” are commonly used interchangeably. For example,
Hartwick and Olewiler (1997, p.271) write:
“Rent in this case is referred to by various authors as user cost, royalty,
dynamic rent, or Hotelling rent. Five names for the same thing!”
A similar treatment can be found in Neher (1990, p.289):
“It [royalty] is being generated by recognition of the ultimate scarcity of the
resource and so can be thought of as a scarcity value enjoyed by the
resource owner. Sometimes it is referred to as user cost ….”
2
However, some economists recognize the difference and point out that these terms must
be used with care. Pearce and Turner (1989, p.273) summarize the confusion as follows:
“In the literature R is known as the royalty …, the resource rent (or rental)
…, the depletion premium, and the marginal user cost. The different terms
are unfortunate since they add to the potential for confusion. The reader
simply has to watch out for the context in which they are used.”
Additional confusion and inconsistency arise in optimal control settings, where
marginal user cost is interpreted as the costate variable in both the present and current
value Hamiltonians. Moreover, the evolution of “user cost” in resource economics and
capital theory has led to different meanings, even though they both are descendants of
Keynes’s General Theory.
In what follows, we provide definitions of marginal user cost, net price and
royalty that are distinct, are consistent with their historical meanings, and that support a
useful interpretation of an extended Hotelling condition for renewable as well as nonrenewable resources. We illustrate how a failure to associate these terms with distinct
meanings can and has led to misleading policy recommendations. In the last section, we
discuss how user cost has come to have different meanings in capital theory, thereby
compounding the confusion.1
2. Definitions and Relationships
The objective of this section is to provide a taxonomy that will both distinguish
the terms and illuminate their relationships.
2.1. Net price and royalty
3
The net price of a resource is the difference between market price and its
extraction cost (p-c). It is also referred to as “resource rent” or “in situ price”.
The term "royalty" originated as the amount charged for resource extraction on
crown lands. However, many resource economists use royalty to mean net price, e.g.
Fisher (1981, p.14), Neher (1990, p.289), Perman et al. (2003, p.512), and (Pearce and
Turner 1989, p. 272).2However, “royalty” continues to be used in policy circles to mean
the payment by the lessee to the owner of the property, typically as a percentage of the
value of the minerals that are mined (e.g., WEAL 2005, Alberta Department of Energy,
2007; Heaps and Helliwell, 1985). Accordingly, we use “royalty” in its original and
policy sense as the payment to the owner for the right to use the resource (see e.g.,
Gaudet et al., 1995).3
2.2. Marginal user cost
Marginal user cost, first introduced by Keynes (1936), is refined by Clark (2005,
p.106) as “... the loss in value when a capital asset is reduced by one marginal unit.”
Even though primarily concerned with capital theory, Keynes (1936) also understood the
importance of the concept for resource economics:
“… if a ton of copper is used up today it cannot be used tomorrow, and
the value which the copper would have for the purposes of tomorrow must
clearly be reckoned as a part of the marginal cost” (p. 73).
Similarly, Toman (1986, p.343) describes MUC as
“… the absolute size of the incremental reduction in the present value of
maximum future profit caused by an incremental increase in the current
extraction rate.”
4
These definitions roughly accord with the common intuition that optimal extraction in
each period requires that the net price on the last extracted unit equals its MUC (e.g.
Scott, 1953; Howe, 1979; Warford 1997; Easter and Liu 2005).
For the case where resource-use imparts a positive or negative externality, the
equation has been expanded to require that the marginal benefit of resource extraction
equals the marginal opportunity cost, which is the sum of the marginal extraction cost
(c), the marginal user cost (MUC), and the marginal externality cost (MEC). MUC is the
loss of asset value from harvesting the marginal unit now. For the case of fund pollution,
MEC (also known as the marginal damage cost) is the external cost of the
contemporaneous marginal damage inflicted by the emission of pollutants. For stock
pollution, say the accumulation of carbon dioxide in the atmosphere, MEC is the shadow
price of carbon (Nordhaus, 1991) in the optimal solution. Assuming that marginal benefit
is given by the resource price, p, we have the "Pearce rule" for efficient resource
extraction (Pearce and Markandya, 1989; Pearce, Markandya and Barbier, 1989; Pearce
and Turner, 1989) requiring that p = c + MUC + MEC. In other words, the net price of
resource extraction, p - c must be equal to the sum of MUC and MEC. This equation, if it
can be rigorously derived, has enormous value as a unifying equation of resource and
environmental economics and a prescriptive rule for corrective taxation, resource pricing,
and green accounting.
For present purposes, however, we abstract from externalities and focus on the
derivation and associated interpretation of the reduced optimality condition, p - c =
MUC.In a highly regarded reference, Clark (2005) defines MUC as the costate variable in
5
the present-value Hamiltonian and represents the intertemporal allocation problem for a
resource manager,
∞
max Vt =
xt
∫e
− rt
[ p − c( S t )]xt dt
… (1)
t =0
s.t. S& t = g ( S t ) − x t ,
as the maximized value of the present value Hamiltonian,
H = e − rt [ p − c ( S t )] x t + α t [ g ( S t ) − x t ] ,
… (2)
wherep is resource price, c is the unit extraction cost, x is resource extraction, S is
resource stock, g is the natural growth function of a renewable resource, r is discount
rate, and α is the costate variable.4 The corresponding first-order condition for the
problem is:
… (3)5
e − rt [ p − c ( S t )] = α t
Clark also notes that the costate variable, α, from the above is the marginal user cost, i.e.
the “the loss in value when a capital asset is reduced by one marginal unit” (p.106).
Equation 3 is simple and serves to unify renewable and non-renewable resource
economics inasmuch as the same equation is valid for both cases.6 However, condition
(3) requires that marginal user cost equal the discounted net price. Thus, defining MUC
as the costate variable of the present value Hamiltonian is inconsistent with the condition,
p - c = MUC.
This confusion can be avoided by defining marginal user cost as the costate
variable of the current value Hamiltonian.7 In this way, the loss in value is reckoned at
the time the resource is extracted, not at the outset of the allocation problem. Considering
the standard renewable resource optimization problem in its more general form,8 a social
6
planner chooses the resource extraction path to maximize the present value of net benefit
over time, i.e.:
∞
max Vt =
xt
∫e
− rt
t =0
⎡ xt
⎤
⎢ ∫ p ( z ) dz − c( S t ) xt ⎥ dt
⎣⎢ 0
⎦⎥
… (4)
s.t. S& t = g ( S t ) − x t
The corresponding current-value Hamiltonian can be written as:
x
~ t
H = ∫ p( z )dz − c( S t ) Rt + λt [ g ( S t ) − xt ]
… (5)
0
Where λt is the costate variable for the resource stock (or the current shadow price of the
resource). Mathematically, λt can be interpreted as the marginal value of the state variable
at time t in terms of value at t (Kamien and Schwartz 1991, p.165), or
∂Vt * 9
. The
∂S t
corresponding optimal condition for an interior solution can be written as:
p ( x t ) − c ( S t ) = λt
… (6)
i.e. net price = MUC. Since the present value and current value costate variables are equal
at time t0, the present value costate variable is equal to the MUC only when considering
the extraction occurring at time t0. This has presumably added to the confusion.
2.3. Summary
In summary, net price, royalty and marginal user cost are distinct concepts. To
allocate a resource optimally, it must be used such that its net marginal benefit (or net
price) equals marginal user cost. Royalty refers to the actual amount charged to a lessee
for the right to use a resource.
7
3. The Shadow Price of Extraction
Net price, royalty and MUC are distinct concepts. Regarding these concepts as
interchangeable can lead not only to semantic confusion but to policy errors as well. For
example, economists sometimes suggest charging concessionaires of public leasing
contracts an extraction fee equal to net price. In the forestry context, for example, Repetto
(1988) and Gillis (1988) recommend setting the stumpage price equal to the resource
rent, p - c. Similarly, in environmental accounting, the shadow price of a resource is often
estimated as net price (e.g. United Nations et al. 2003; Hecht 2005). As noted above,
however, net price is only equal to the first-best shadow price if the resource is being
extracted according to the optimal program (for an interior solution). In the common case
wherein the resource is being overused, the observed net price is less than the shadow
price. Accordingly, the net price underestimates the marginal value of the resource for
both environmental accounting and for charging resource concessionaires.
Clarification of these terms and understanding the difference between the
observed net price and the shadow price of the resource are also important in formulating
policies regarding ecosystem services. Pricing of ecosystem services has been
increasingly recognized as a means to promote efficiency in resource use (e.g., Barbier,
2007; Heal et al., 2005).
Understanding the theoretical foundations of ecosystem
valuation is an important part of the decision regarding whether a particular measurement
approach will provide a good approximation. Failing to recognize the meaning of
different terms can create confusion and obstruct proper ecosystem pricing and valuation.
Dasgupta (2007) acknowledges that net price can severely underestimate the value of a
resource but did not suggest a solution for the problem, noting instead that the suggested
8
practice may be defendable on pragmatic grounds. Understanding the difference between
net price and shadow price is presumably a prerequisite for exercising judgment about
what proxy measurements are appropriate in which circumstances.
For example, if the resource is open-accessed, resource extraction occurs where
price is equal to the unit extraction cost (xOA in figure 1). Charging the observed net price
(equal to zero) would not change anything and could not improve efficiency. In the
example of a public forest, the first-best policy is to charge concessionaires a per unit
extraction fee (royalty or stumpage fee) equal to the marginal user cost evaluated at the
optimum x*, equivalently, the net price at x* as shown (not the observed net price).10
Note that in the optimal solution, the necessary condition depicted in figure 1, i.e., pt= ct
+ MUCt, holds for all time periods. (In the standard autonomous case and starting with an
initial stock of the resource greater than its steady state value, the inverse demand
function is stationary; the total marginal cost, c + MUC, increases over time, implying a
monotonic increase in the shadow price up to its steady state value.)
p(x)
c + MUC
MUCOA
c + tax (royalty)
c (extraction cost)
x*
xOA
extraction, x
Figure 1. Charging net price is only efficient at x*
9
As shown in figure 1, the optimal policy for resource extraction on public lands is
to charge an extraction fee equal to the marginal user cost or net price evaluated at the
optimum. For a privately-owned resource in the absence of externalities, the first-best
policy is not to charge for extraction; the private owner already internalizes the
opportunity costs of harvest. However, if resource extraction generates an externality,
extraction fees for both lessees and private owners must be adjusted accordingly.
A caveat is in order regarding boundary conditions. If the optimal solution is not
an internal one, then the condition, net price = MUC, need not apply. For example, in the
case of a small, open economy, the resource price is exogenous and optimal extraction
may follow a most rapid approach path to the steady state, constrained only by a
maximum harvest level.11 Along the optimal path, price is constant and extraction cost is
increasing. As a result, net price is decreasing. However, as a resource is depleted, its
MUC (or shadow price) is increasing. This can be explained using the definition and
analysis in section 2. From (4), the optimal condition for an interior solution requires that
net price equal MUC. However, for a boundary solution, net price is different from MUC.
Where resource extraction is constrained by the maximum harvest level, however, the
constraint prevents extraction from increasing the MUC until it equals the net price, i.e.
MUC is less than net price at the constrained optimum. In this case, charging the
observed (decreasing) net price confiscates more than the necessary MUC required to
induce efficient extraction. Similarly, if the resource stock cannot be depleted below
some critical threshold because of irreversibilities or precautionary considerations,
optimality is characterized by net price equaling MUC plus a term involving the binding
constraint.
10
4. Marginal user cost vs. User cost of capital
Another source of confusion arises from the inconsistent use of the term “user
cost” in resource economics and capital theory. It is natural to believe that “user cost” and
“marginal user cost” refer to the same user cost but only one of them is a marginal
concept. We show in this section, however, that both of them are marginal concepts but
that the two implicit meanings of user cost are different.
The term user cost of capital is correctly credited to Keynes. This concept was
clarified by Scott (1953) who formalized the unified capital theoretic foundations of
investment theory and resource economics. Scott explains that:
“User cost, as distinguished from opportunity cost, is a term invented by
Keynes. The term is currently applied to the opportunity cost of putting
goods and resources to a certain use, future alternative revenues being
given equal weight with present alternative uses” (p.369).
Clark and Munro (1975) further developed the capital theoretic foundations of
resource economics along the lines suggested by Scott (1953 and 1955), noting that the
marginal user cost of resource extraction is the change in optimized present value
evaluated at the time of resource extraction, previously referred to as the “imputed
price,” or “imputed demand price of capital”.12
In modern investment theory, the term “user cost of capital”, commonly attributed
to Jorgenson (1963), refers to the implicit rental price of capital services (harkening back
to Walras’s, 1874, price of capital services). For example, Diewert (2005) explains the
user cost of capital as the “net cost of using the new asset for period t,” and calls it the
“end of period vintage rental price” of capital.13
11
In order to clarify the distinct meanings of user cost in both resource economics
and the theory of investment demand, we consider the neoclassical model of sustainable
growth with a natural resource (e.g. Krautkraemer, 1985; Toman, Pezzey, and
Krautkraemer 1995).14 A social planner maximizes social welfare given by (7), where
U(C) is utility of the representative consumer, UC>0, UCC<0, and C is consumption.
Given that the rate of time preference is ρ, the optimization problem can be written as:
∞
max Wt =
Ct
s.t.
∫e
− ρt
U (Ct )dt
… (7)
t =0
K& t = F ( K t , xt ) − δK t − c( S t ) xt − Ct
S& = g ( S ) − x
t
t
t
where Kt is the man-made capital at time t;
δ is the depreciation rate of capital.15
From the above optimization problem, the corresponding current value
Hamiltonian is:
H = U (C t ) + μ t [ F ( K t , xt ) − δK t − θ ( S t ) xt − C t ] + λt [G ( S t ) − xt ]
… (8)
Where μt and λt are the current shadow prices of capital stock and resource stock
respectively.
The first-order conditions for an internal solution require equalities between
marginal benefit and marginal cost of investment and resource use and equations of
motion for capital and the resource stock. In problems such as (7), the shadow price or
costate variable of capital ( μ ) is usually interpreted as the marginal benefit of adding a
unit of capital. If we think instead of subtracting a unit of capital (through use and
depreciation), we can understand the Keynesian MUC terminology.
12
Manipulating the first-order conditions, one can derive the efficiency conditions:
Fx − θ =
1
{F&x + ( Fx − θ ) g ' ( S ) − θ ' ( S ) g ' ( S )}
( FK − δ )
FK = δ + ρ + η (C )
C&
C
… (9)
… (10)
whereη (C ) , the consumption elasticity of marginal utility, is −
U CC C
. Equation (9) is a
UC
growth-theoretic form of the Hotelling condition, extended to allow for renewables. For
an internal optimum, extraction must equate net marginal benefit to MUC. For a nonrenewable resource, G ( X ) = 0 , the condition simplifies to: Fx − θ =
F&x 16
.
FK − δ
Equation (10) is the familiar Ramsey condition. The right-hand side of equation
(14) is the “implicit rental of one unit of capital service per period of time”. The optimal
condition for the investment requires that the capital must be accumulated such that its
marginal value is equal to its implicit rental price.17 Jorgenson (1963, p.249) calls the
R.H.S. of equation (13) the “user cost of capital.”18
Ironically, “marginal user cost” as referred to in resource economics is consistent
with the Keynesian definition, even though Keynes was defining the term primarily for
its use in capital theory. Marginal user cost in this sense represents the cost of
depleting/accumulating an incremental unit of capital/resource. At the optimum, this
marginal user cost must be equal to the net marginal benefit of that incremental change.
However, since Jorgenson’s (1963) canonical article, the term “user cost of capital” has
come to mean the implicit rental price of capital services (e.g. Diewert, 2005; Diewert
and Schreyer, 2008). User cost of capital, in this context, is the implicit rental price,
13
which must be equal to the marginal product of capital, not the net marginal benefit of an
incremental investment.
In the typical situation where the initial values of capital and resources are lower
and higher than their steady-state values respectively, the first order conditions for our
sustainable growth problem require accumulating capital such that the marginal product
of capital equals the user cost of capital (implicit rental price) and depleting the resource
each period until net marginal benefit of the resource equals its marginal user cost
(foregone asset value).
5. Concluding remarks
Inconsistent and interchangeable usage of marginal user cost (MUC), net price,
resource rent, shadow price and royalty promotes confusion and policy errors regarding
resource valuation and extraction incentives. We provide distinct concepts that facilitate
more transparent articulation of first order conditions for both renewable and nonrenewable resources. In particular, we show that the Pearce formulation of the first-order
condition requires that MUC be defined as the costate variable of the current value
Hamiltonian, not the present-value definition often provided. Failing to distinguish
between differences in these concepts can contribute to policy and accounting errors.
Along the optimal path, the first order condition requires that the net price of the
resource is equal to its MUC. The equality between these two terms holds only for the
interior optimal solution. For example, if a resource is not optimally extracted or the
extraction rate is limited by a maximum constraint, net price will not be equal to the
marginal user cost. To incentivize efficient resource extraction on public lands, for
example, resource concessionaires should be charged royalties equal to their
14
corresponding marginal user costs. If the resource is being excessively depleted, charging
net price will not restore efficiency. Similarly, marginal user cost should be the basis of
resource valuation, both for project evaluation and the construction of "green" income
accounts. Net price should be used as a proxy only when the internal optimum obtains.
The fact that user costs in resource economics and post-Keynesian investment
theory are different concepts adds to the confusion. Even though Keynes defined
marginal user cost primarily in the context of capital accumulation, his definition
survived in resource economics but not in capital theory. Following Jorgenson’s (1963)
canonical article on capital accumulation, the “user cost of capital" has come to mean the
implicit rental price of capital services. To clarify, we consider a neoclassical growth
model with both capital and a natural resource in the production function. Optimal
growth in the model requires that a Ramsey and an extended Hotelling condition are
satisfied. The Ramsey condition requires that the marginal product of capital equals the
Jorgensonian user cost of capital, i.e. its implicit rental price. The extended Hotelling
condition requires that the net marginal benefit of resource extraction equals the marginal
user cost of the resource, i.e. the loss of present value of the resource stock at the time the
resource is extracted.
There is a growing recognition of the need for systems analysis for interacting
resources, prioritizing conservation spending, and sustainable development generally
(e.g. Barbier, 1990; Turner, 2002; Taylor and Brock, 2006; Dale and Polasky 2007). The
meanings and distinctions as used above should in no way imply that the full marginal
cost of resource use should ignore these considerations. Rather, the framework needs to
be extended to include irreversibilities, externalities, and interactions between resource
15
stocks. Hopefully, the distinctions introduced here will facilitate the derivation of policy
prescriptions from the optimality and equilibrium conditions of such problems.
Notes:
1
In Morey’s “Confuser Surplus,” the confusion arises because of one term having multiple,
inconsistent meanings. In the present case, the confusion results from multiple terms being
sometimes used to mean the same thing and other times to mean different things.
2
Pearce and Turner note that while "royalty" "derives from the sovereign's rights to property in
the ground," it is now used to mean "the price in the ground [p-c(X)]."
3
This meaning is also consistent with royalty as payment to the owner of intellectual property
(e.g., WTO, 2006; NAFTA, 1992, Article 415).
4
Some notations of variable are different from those used in Clark (2005).
5
This condition is same as equations 4.19, and 4.32 in Clark (2005).
6
Non-renewable resource extraction can be modeled as a special case wherein g(S) = 0. While
equation 3 remains unchanged, the expression for the costate variable is simplified.
7
This is consistent with the treatment in Clark and Munro (1975), who equate marginal user cost
to ψe rt , where ψ is the costate variable in the present value Hamiltonian. Conrad and Clark
(1987, p. 92) use the same definition as ours, in the context of an open-access resource regulation
problem, although they use the term user cost in place of marginal user cost.
8
Following Clark (2005), equation (4) is written from the resource manager’s perspective, with
exogenous prices. To solve for competitive prices endogenously, one need only combine these
conditions with a specification for demand.
9
The symbol (*) is used to designate values of variables along the optimal trajectory.
16
10
Analogously, a corrective tax to internalize a static externality should be set equal to the
marginal externality cost evaluated at the optimal, not the current, level of emissions (Baumol
and Oates,1971).
11
The small, open economy assumption for fixed resource prices follows Taylor (2008).
12
We would like to thank Edward Barbier for making us aware of Scott’s (1953 and 1955)
clarification of Keynesian marginal user cost and elucidation of its role in resource economics.
13
Diewert’s “stock value of the asset” defined in the same paper as the “discounted future service
flows that the asset is expected to yield in the future period” is closely related to “marginal user
cost” as used in resource economics (see also Diewert, 2005).
14
We normalize labor (L=1) and abstract from the population growth and technological progress.
15
Other notations are defined in the previous section.
16
This is comparable to the Hotelling partial equilibrium condition: ( p − c ) = p& / r , and is
consistent with the optimal condition, net price equals MUC.
17
As in the neoclassical growth context, consumption is the numeraire and capital is reckoned in
the same units, i.e. its price is one.
18
To interpret Ramsey condition using Keynes’s marginal user cost, we can rearrange (10) using
&
− U CC C
where the
the first-order condition for optimal investment, μ = U C . This yields, U =
C
FK − δ − ρ
left hand side is the marginal net benefit of capital depletion in terms of the incremental utility
gain afforded. The right hand side is the marginal user cost of capital – the foregone utility
present value from depleting a unit of capital today.
17
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