descriptive test of the hotelling model with data from the rare
TRANSCRIPT
Descriptive Test of the Hotelling Model with Data from the
Rare Earths Market
Esther Divovich
Advisor Richard Walker
Northwestern University
This Version: June 6, 2011
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Acknowledgements
First and foremost, I would like to thank my thesis advisor, Professor Richard Walker, for his
accessibility, patience, and perceptive input through the various twist and turns of performing
independent research. His continued guidance not only made this thesis a possibility, but taught
me a great deal along the way. Likewise, Kathleen Murphy, the Social Science Data Services
librarian, was an invaluable resource for obtaining the necessary data sets for my analysis.
Lastly, I would like to thank all of the teachers who have given me the tools and the passion for
learning. Within this thesis, are the imprints every single teacher I’ve ever had the privilege of
learning from, ranging from my first-grade math teacher, to my highschool English teacher, and
most importantly, to all of my MMSS and Economics professors at Northwestern.
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Section 1: Introduction
The dilemma of the optimal extraction of nonrenewable resources has faced leaders,
organizations, and policy-makers consistently in the 20th
century: how much of a nonrenewable
resource to extract now, and how much to save for the future?
Harold Hotelling answered this question in 1931 with his model of the optimal extraction
of nonrenewable resources. He claimed that exhaustible resources were extracted and valued
differently than renewable resources. While real prices of sustainable commodities, such as corn
or wheat, should stay constant year to year, the real prices of nonrenewable resources rise at the
rate of interest. Since then, Hotelling’s “r-percent” rule has been expanded upon to become its
own subfield in economics.
The next immediate question to ask is: how well does Hotelling’s model hold when tested
against the data? The Hotelling model was tested extensively during the 1970’s as a result of the
oil price shocks from the OPEC embargo. After the 1970’s, however, when commodity prices
stabilized, mainstream economics has moved away from the theoretical study of Hotelling’s
model. The recent increase in real commodity prices by 286% from 1998 – 2008 alludes to the
re-emergence of this branch of economic literature.
This paper undertakes an empirical test of the Hotelling model for the rare earths market
from 1955-2009. The rare earth market is of particular interest because of the rapid growth in
technology, which has caused a surge in demand for the rare earths. In addition, while the
Hotelling model has been tested with data from nearly every major nonrenewable resource, it has
never been tested on rare earths market data.
The next section, Section 2, will deal with the theory. Section 2.1 will introduce dynamic
programming, the key theoretical fundament of Hotelling’s model. Section 2.2 will derive
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Hotelling’s basic findings using dynamic programming, and Section 2.3 will outline the
theoretical extensions to the basic model that have been developed since 1931.
Section 3 will present an empirical test of Hotelling’s model with market data on the rare
earths from 1955-2009. A review of the empirical body of literature in Section 3.1 will
differentiate between various formal and descriptive tests of the Hotelling model. Section 3.2
will outline the data. Section 3.3 will describe the methodology of the test we perform, which is a
descriptive rather than a formal test due to the unavailability of firm cost data. Section 3.4 will
present the results, which show mixed support for the Hotelling model in the rare earths data,
although the Hotelling model cannot be formally accepted or rejected due to the descriptive
nature of the test.
Section 2: Theory
2.1 Dynamic Programming
The key question that the Hotelling model seeks to answer is how a producer of a
nonrenewable resource maximizes his intertemporal profit stream over continuous time. Basic
Lagrangian optimization can no longer handle such a multi-faceted problem.1 We will abstract
from the specific application of the Hotelling model for the time being, and introduce the integral
tool of dynamic programming.2
The basic problem of dynamic programming is to maximize
(1)
subject to
(2)
1 A.K. Dixit
2 Obstfeld
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The objective function, (1), is the discounted utility function from time 0 to . The
continuously compounded discount rate is denoted, r. The constraint, , is a function of both
the flow (control) variable, q(t), and the state variable, S(t). It is a key point to distinguish
between the flow variable and the state variable. The state variable is a fixed known entity at a
certain point in time. The flow variable, however, is best known as a rate where units denote the
change in a state variable “per time.” For example, output is a state variable, whereas GDP is a
flow variable (since it is computed annually).
In equation (2), S(0) is the initial quantity of the state variable at time zero. The key to
solving this problem, as we will see later, is that the maximized value of (1) depends solely on
S(0), the predetermined initial value of the state variable. The dot over S(t) denotes a time
derivative, such that
.
The flow variable is also known as the control variable because it is the variable under
the control of the entity maximizing the objective function, (1). This is why we will eventually
differentiate (1) with respect to q(t) in order to obtain the optimal path. The optimal path, ,
is the solution the problem. In real-life applications, it can represent a household’s decision of
how much to consume, or in the case of Hotelling’s model, how much a firm will choose to
extract of an exhaustible resource.
The first step to solving this problem involves assuming that there exists a maximized
value function, , which is the lifetime utility a producer gains from following the optimal
extraction path. The value function is the answer to the question: if the producer picks the
optimal extraction path, , what is the maximum level of utility he will gain? Thus, with the
defined terms and
, the maximized value function can be written as,
(3)
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where the maximized value of (1) depends solely on S(0). Hence, this problem is stationary – it
does not change with the passage of time. This implies that at any time onward to T, a new
decision maker would make the same optimal choice path because S(0) would be the same for
both decision makers. Another common term for stationarity is “dynamic consistency” because
dynamically (over time) the solution is consistent. As a result, (3) can be rewritten as
+ for all T (4)
Since the value function is the solution to our optimization problem, we know that
(5)
By the assumption of dynamic consistency, the maximized value function holds for all
so we can take
, which results in,
(6)
where the constraint is ]. This is equivalent to
0 = (7)
Let us define the first two terms of the maximized function as the Hamiltonian equation,
H = (8)
Maximizing the Hamiltonian with respect to the control variable (extraction) will give us the
static efficiency condition for solving the producer’s intertemporal problem,
: (9)
The static efficiency condition says that at each (static) point in time, the marginal benefit
is directly offset by the marginal cost, , which is the potential value at a
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future point in time. This marginal cost is the user cost (shadow cost) of the
constraint .
Since q(S) and it is the solution to the Hamiltonian, then it automatically satisfies the
original equation (7). If we substitute back into (7), we can further optimize to solve for the
dynamic efficiency condition,
0 = (10)
By envelope theorem, we know that
(11)
where
(12)
Substituting (12) into (11) gives us the optimized function
(13)
Let us denote the user cost,
Also, by definition, the term =
Thus, (13) can be rewritten as
(14)
which gives us the dynamic efficiency condition,
(15)
or
(16)
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The tool of dynamic programming has given us two necessary conditions, static and
dynamic, in order to solve continuous time maximization. In the next section, we will apply this
critical tool to the problem of solving a producer’s optimal extraction path of a non-renewable
resource.
2.2 Hotelling’s Model
Hotelling developed the basic model for the optimal extraction rate of a nonrenewable
resource by a firm extracting from a known resource stock. The basic model assumes a perfectly
competitive market, although the general structure can accommodate the case of a monopoly as
well.
A producer attempts to maximize his discounted profit stream, which is modeled by the
objective function, J,
(17)
where r is the discount rate the firm faces and his profit is defined by the benefit gained from
selling the resource minus the cost. The functions depend on the control and state variables, q(t)
and S(t). The control variable, q(t), is the rate of extraction the producer chooses at each point in
time. The state variable, S(t), is the amount of the stock remaining at time t.
However, the producer is also subject to the following constraints,
(18)
(19)
(20)
(21)
With the tool of dynamic programming, we are able to define the Hamiltonian equation,
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H = (22)
where is the shadow price on the resource constraint.
Maximizing the Hamiltonian with respect to the control variable (extraction) will give us
the static efficiency condition for solving the producer’s intertemporal problem,
: (23)
In the Hotelling model, the static efficiency condition says that at each (static) point in
time, the producer’s marginal utility, , from extracting and selling the nonrenewable
resource is directly offset by the producer’s potential utility from extracting at a future point in
time. Also from dynamic programming we know that the dynamic efficiency condition is
(24)
or
(25)
Equation (26) is the famous “r-percent rule”, that the marginal value of the nonrenewable
resource will increase at the rate of interest.
2.3 Literature Review of the Theory: Extensions to Hotelling’s Model
In the dynamic world of extracting nonrenewable resources, the basic assumptions on the
cost function may not be sufficient to account for real life complications. Such complications
include exploration, uncertainty about the size of reserves, durability effects (recycling and
stockpiling), imperfect competition, taxation effects, stock effect, and technical changes.3 For the
sake of this review, however, we will focus on three major extensions to the Hotelling model:
depletion (stock effect), exploration, and technical advances.
3 Slade and Thille
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In order to model the role of depletion, the intuitive logic is that extraction costs depend
not only on the current extraction, q(t), but also on the remaining stock of reserves, S(t). This is
due to the fact that the cheapest, most readily available resources are extracted first, followed
sequentially by less economic reserves deeper in the ground that are more difficult to extract. For
minerals, this is known as declining ore quality. While the static efficiency condition remains
unchanged, the dynamic efficiency condition becomes,
(26)
Another real-life complication of the Hotelling model that alters the cost function is
exploration. Exploration is an added cost, but the benefits of discoveries lower cost because they
increase the stock of cumulative discoveries, D. The rate of change of discoveries is
where e is the exploratory effort at cost . The constraint on the objective function with
exploration becomes,
(27)
and the new static efficiency condition is,
(28)
The role of technical advancement has also been modeled as a realistic extension to the
general Hotelling model. Technical innovation lowers marginal cost such that the new cost
function becomes C(q,S,T). The static efficiency condition is,
(29)
where is the rate at which marginal costs fall due to changes in technology.
All three of these theoretical extensions predict that the marginal cost function is U-
shaped. Initially, the marginal cost of production is extremely high until efficient technology is
established. Over time, this drop in cost is offset by the diminishing rate of technological
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advances, at which point the marginal cost of production continues to rise as the resource grows
in scarcity. For the case of depletion, stock effect is initially minimal. A U-shaped cost function
is formed over time when ore quality decreases and extraction is more costly. Similarly,
discoveries of reserves are far more common in the first stages of production of a nonrenewable
resource. In the later stages, exploration becomes more expensive as the rate of discoveries
diminishes.
The theory presented in this section will serve as the foundation for the next section,
where we will test the theory of the Hotelling model empirically.
Section 3: Empirical Test of the Hotelling Model on Rare Earths Market Data, 1955 – 2009
3.1 Literature Review of Empirical Tests
Empirical tests of the Hotelling model can be divided into two broad categories:
descriptive and formal tests. Formal tests derive an appropriate cost function (such as the ones
outlined in the previous section) and test the Hotelling model with the appropriate data.
Specifically, cost data must be available to perform such a test. However, due to the proprietary
nature of cost data and the difficulty in obtaining it, formal tests are limited. These tests lead to
more conclusive results but also restrict the cost function to one specific form. Descriptive tests,
however, do not commit to a specific functional form. Rather, descriptive tests study the role of
market prices in the market equilibrium. They use market data to assess which of these models
are consistent.
The first to empirically assess the validity of the Hotelling model were Barnett and Morse
(1963), who used a descriptive approach. They concluded that the observed fall in mineral-
commodity prices over time indicated that scarcity was not a concern. Further descriptive tests
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were then undertaken by Heal and Barrow (1980), who compared metal price movements to
interest rates, but found that changes in interest rates, not interest rate levels, predicted prices.
Other descriptive studies found evidence that price paths are U-shaped, as was predicted
by the various theoretical extensions included in Section 2.3. One example of such a study was
performed by Slade (1982), who found that quadratic trends revealed upturns of real prices in
mineral commodities in the 1970’s, which she attributes to technical change.
Formal tests have had mixed results. Stollery (1983) obtains cost data from INCO, the
Canadian firm that is a monopolistic force in the world nickel industry. He concludes that the
Hotelling model is congruent with the data and a discount rate of 15%. In contrast to this
conclusion, Farrow (1985), rejects the hypothesis that proprietary data from a mining firm was
consistent with the theoretical Hotelling model. Although other structural have been performed,
such as those of Halvorsen and Smith (1984, 1991) and Young (1992), among others, evidence
in support of the Hotelling model from both formal and descriptive tests has been mixed and
varies based on the type of nonrenewable resource data tested.
3.2 Rare Earths Market Data, 1955-2009
As seen in the previous section, the Hotelling model has been tested both descriptively
and formally with market data from nonrenewable resources such as oil, copper, tin, and
numerous others. However, a test of the Hotelling model has never been performed with data
from the rare earths market. This section will present the data used, both historical and
quantitative, in order to perform such a test. Section 3.2.1 will give a general background of the
rare earths and their uses. Section 3.2.2 will describe the empirical data set utilized in testing the
Hotelling model. Section 3.2.3 will summarize the historical market trends taking place in the
rare earths market from 1955-2009.
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3.2.1 General Background
The rare earths elements are the relatively abundant group of 17 chemical elements in the
periodic table, specifically the 15 lanthanides along with scandium and yttrium. Although the
actual elements are abundant in the earths crust, the reason these resources are considered “rare”
is the scarcity of their concentrated, economically exploitable reserves, which are known as rare
earth minerals.
The principal rare earth minerals are bastnäsite, monazite, loparite, and the lateritic ion-
adsorption clays. The extraction of these minerals is the first step to a ten-day process in which
the actual rare earth elements are separated from one another. Rare earth producers require
significant capital and technological expertise, as well as licensing for mines which can be very
costly due the environmental impacts on the surrounding area.
The uses of the rare earths are extremely diverse, ranging from everyday devices
(computer memory, DVD's, rechargeable batteries, cell phones, magnets, fluorescent lighting) to
critical defense applications and medical devices. Recently, demand for rare earths has
dramatically increased due the heightened production of high-tech products and various political
factors. This makes the optimal extraction of rare earths an important area of study.
3.2.2 Empirical Data
Data on rare earth prices and production is available from the U.S. Geological Survey
(USGS) website which provides Historical Statistics for Mineral and Material Commodities in
the United States. Price data is reported in real terms as well as nominal. Data on interest rates
was found on the Federal Reserve website. The rate used in analysis was the 3-month Treasury
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bill secondary market rate, on a discount basis, from 1954-2009. Also, data on the annual CPI
(Consumer Price Index) was obtained from the Bureau of Labor Statistics website from 1955-
2009.
3.2.3 Historical Research on the Rare Earths Market
Historical research on the rare earths market was conducted with data from the Bureau of
Mines Minerals Yearbook, which outlined in detail the major events in the rare earths market
during the years in question. Knowledge in the historical events of the market will be utilized
later to see if the empirical test performed is congruent with reality. In specific, the roles of
depletion, exploration, and technology (which were the three theoretical extensions reviewed in
the literature review of the theory) will be summarized.
Technology – Technological advancement in the rare earths industry was most pronounced in the
late 1950’s and early 1960’s when efficient separation techniques for the rare earth elements
were developed. The concentrated, pure form of the rare earth elements are extremely difficult to
separate from the minerals and from one another due to their small ionic size. Thus,
technological advances are central in the production of rare earths, which require complex
separating technology such as ion exchange, fractional crystallization, and liquid-liquid
extraction.
In the late 1950’s, intensive research was published by the Bureau of Mines describing
solvent extraction, ion exchange, and other methods of rare earth extraction. At this point, the
price of rare earths metals began to drop due to “advances in extraction and separation
technology, rapid expansion of procession facilities, and competition among major producers”.
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By the early 1960's, the methods of solvent extraction and ion exchange were well-
researched; however these findings were still being applied to the industrial setting, specifically
for the separation of bastnäsite. For example, in 1962, the mill at Mountain Pass, California was
able to expand production of high-grade bastnäsite concentrate through improved efficiency.
In 1961, techniques using liquid-liquid extraction were presented at the Second Rare-
Earth Conference in the US. Likewise, Russian scientists used rapid liquid-liquid extraction on a
semi-industrial scale. Most technological advances continued until the mid-1960's, at which
point separation techniques were well-established and enabled research to focus on the properties
of rare earths and their numerous applications.
Some research did occur in the 1970's, mostly focusing on the separation of Yttrium and
overall cost efficiency. However technological discovery was pretty minimal until 1988 when
China was able to improve the recovery rate of rare earths such that the operating cost of
recovery was lowered by over 80%.4 Afterward, research efforts temporarily heightened in the
late 1980’s due to China’s entry in the world market, but quickly subsided. Since then,
technological advances regarding the uses of rare earths have dominated over improvements in
separation technology.
Exploration – In the rare earths market, exploration for the minerals of bastnäsite, monazite,
xenotime, and rare-earths bearing clay was rarely conducted specifically for the rare earths.
Rather, most mineral sand deposits were found in search for ilmenite and other titanium
minerals. Also, because many of the rare earths are naturally radioactive, many discoveries were
made during exploration for uranium and thorium. Once located however, there are substantial
4 Rare Earths Minerals and Metals, Mineral Yearbook 1988.
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costs in determining if the ore is economically extractable. This often involves constructing a
pilot plant in order to understand if the grade and purity of the minerals are economical.
Exploration can be divided into the search for heavy minerals and light minerals. Heavy
minerals such as monazite and xenotime survive continued erosion, generally through river and
sea currents, and form economic heavy-mineral deposits. Light minerals, such as bastnäsite are
formed when molten magma intrudes the earths crust at points of weakness and crystallizes. If
this occurs at or near the surface at a sufficient grade it may be economic. Exploration techniques
employed to locate the rare earths, as well as other minerals, include surface and airborne
reconnaissance with magnetometric and radiometric equipment.
From the years of 1955-2009, there were significant discoveries that decreased the
marginal cost of rare earth production (due to increased stock). The first large discovery took
place in 1961 in Argentina, followed in 1965 by a very large rare-earth monazite discovery in
Malawi as well as in South-West Africa in 1969. In the 1970’s, discoveries were made in
Western Australia, Burundi, India, and Brazil.
In 1980, one of the largest discoveries of rare earths to date took place in the Bayan Obo
mining region of Mainland China, which increased world reserves nearly two-fold. This was
followed by numerous discoveries in Canada: in 1982, a substantial discovery of yttrium-
beryllium-zirconium deposit was found near Strange Lake in Quebec, and in 1984, yttrium-
beryllium deposits were also discovered at Thor Lake, in the Northwest Territories of Canada.
Discoveries in the late 1980’s were made in Germany, Venezuela, Zaire, Gabon, Sri Lanka, and
Western Australia. Specifically in Australia, the discovery Mount Weld, one of the richest grade
rare earth deposits in the world, contained reserves of 15.4 million tons grading at 11.2% REO.
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World reserves were dramatically revised upward in 1991 from 62 million tons of
contained REO to 85 million tons of contained REO due to additional data on the
Commonwealth of Independent States, which reported 21.4 million tons. China’s share of world
reserves, which was thought to be 78%, dropped to 50%. In 1992, another revision of C.I.S. data
caused world reserves to be revised upward to 100 million tons of contained REO, such that
China’s reserves dropped to 43%.
Since then, reserves have stayed fairly constant although discoveries were made in
northern Mozambique, New South Whales (Australia), and India.
Stock Effect – Stock effect (decreasing ore quality over time) has not historically been a large
issue for rare earths producers. In some cases, however, it has been apparent. Beginning in the
1970’s, Australia’s production of monazite decreased by 12.6% because of leaner monazite
contents of the ore. Similarly, India’s production decreased in 1979-1980 due to lower
concentrations in mineral sands, requiring them to install new technology to provide higher feed
grades. With the rise of China as the main producer of rare earths, low-grade ores may become a
problem for Chinese producers in the future.
3.3 Methodology
The data presented on the rare earths market from 1955 – 2009 will be used to
empirically test the validity of the general Hotelling model. Due to the proprietary nature of cost
data for the rare earths market, cost data was unavailable. As a result, the Hotelling model cannot
be formally accepted or rejected because a formal model on the cost function cannot be
performed. However, the descriptive test described below will still be able to provide interesting
results.
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In order to test the Hotelling model without the availability of cost data for rare earths
market, we will first make the assumption that the theoretical Hotelling model holds. Another
way to re-write the general equation of the Hotelling model is,
(30)
In the above equation, we have data on nominal prices, , as well as data on nominal
interest rates, , from 1955 – 2009. Data is nominal due to the underlying theory of rational
expectations. The expected rise in price is equivalent to the expected rise in interest rate.
We will use this available data to empirically fit the “best-case” marginal cost function,
. This approach was used by Ellis and Halvorsen (2002), who estimate the functional form of
market power in a monopoly industry. This analysis was done as a component of their empirical
test on the Hotelling model. In specific, they restrict their function to a polynomial function and
use the assumption that the Hotelling model holds in order to estimate the appropriate
coefficients on the polynomial. After we obtain the estimated time path of marginal cost, we will
compare it with the historical data of what actually happened.
In addition to the assumption that the Hotelling model holds, however, this test also
makes an assumption on the nature of the rare earths industry. More specifically, we are
assuming that the rare earths market is perfectly competitive. This enables us to use price data,
, as a proxy for the benefit of extracting a nonrenewable resources where the benefit is
defined as the area under the demand curve. Without this assumption we would have to replace
price data, , with another proxy for benefit. For example, if the market were a monopolistic,
we would have to obtain data on marginal revenue, , which would be extremely difficult.
This assumption of a perfectly competitive industry is reasonable for the rare earths
market. From 1955-1990, the rare earths market has been extremely competitive and divided
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amongst numerous producing countries. The years where perfect competition may be an issue
occurred from 1990-2009, when China began its ascent as a major producer of rare earths.
China’s production grew dramatically from producing 30% of world rare earths in 1990, to
currently producing nearly 95%.
At first glance, China’s monopoly power would contradict our assumption of a perfectly
competitive industry. However, due to the decentralization of production (historically it has been
controlled by the individual localities rather than the central government) rare-earth producers
have struggled to maintain profitability. Competition between local governments has resulted in
overproduction and has deflated prices. In addition, illegal smuggling of rare-earths has also
decreased prices, limiting the profit of individual firms.5 Since 2007, the Chinese government
has been struggling to centralize production in order to preserve their resources and improve
profitability, but the given the scope of the data set from 1955-2009 and the lack of tangible
results thus far, this fact has little effect on the assumption that the rare earths market resembles a
perfectly competitive industry.
Now that the major assumptions have been made, we can define the marginal cost
function of rare earth production, , as a function of time. We will perform two separate tests.
The first test will include a wide variety of functional forms including a constant, a linear time
argument, quadratic time argument, a cubed time argument, a square root time argument, and a
logarithmic time argument. The “best-case” result would mirror the time path of marginal cost
from 1955-2009. In order to account for the nominal terms, and , we deflate each variable
by the .
Test 1:
(31)
5 USGS, “China’s Rare-Earth Industry”
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The second test enables us to model environmental cost, which is not a cost included in
the marginal cost function. This is because environmental costs began at a specific point in time,
marking a structural shift rather than a continuous process. In 1988, the radioactive byproduct of
rare earth production, thorium, became far more costly to process, ship, and dispose of. Costs
have only increased since then and are continuing to rise. Thus, a dummy variable for the years
after 1988, as well as an interaction term with this dummy variable, were included in the second
test to proxy for the structural shift in environmental costs.
Test 2:
(32)
By substituting and (the derivative of with respect to time) of each test into
(30) and rearranging terms (see Appendix I for algebraic calculations), we can simplify the
equation into the empirically testable forms shown in (33) and (34). This involves generating the
variables and , such that
running the OLS regressions below (with the constant withheld) will yield the appropriate
coefficients for the time path of marginal cost,
Test 1: (33)
Test 2:
(34)
3.4 Results
Before analyzing the results, it is important to once again highlight that the descriptive
test performed does not have the power to formally accept or reject the Hotelling model for the
rare earths market. Nonetheless, analyzing the "best-case" marginal time paths in conjunction
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with the historical research of the rare earth market from 1955-2009 will deliver meaningful
conclusions.
The “best-case” marginal cost function for each test had a positive initial marginal cost in
the constant term, the most statistically significant coefficients, and the highest . The
functional form for Test 1 that performed best with respect to the data is reported below, along
with Graph 1.1 and Graph 1.2,
(Test 1)
Graph 1.1: Best-Fit Functional Form, 1955-2009
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Best-Fit Functional Form, 1955-2009
MC(t)=381278+41623t-253050(t)^.5
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Graph 1.2: Best-Fit Functional Form, 1960-1970
As is seen by the results of Test 1, the "best-case" marginal cost function is negative in
the mid-1960's when marginal cost is at its minimum. This is inconsistent with the definition of
marginal cost, which is a powerful result because this marginal cost function is the “best-case”
scenario. Given we are assuming the Hotelling model holds, such an unrealistic result can be
seen weakly as a contradiction of the Hotelling model.
However, the results for Test II did not have these contradictory results. The "best-fit"
functional form of Test 2, along with Graph 2.1, Graph 2.2, Graph 2.3 are reported below,
(Test 2)
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Best-Fit Functional Form, 1960-1970
MC(t)=381278+41623t-253050(t)^.5
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Graph 2.1: Best-Fit Functional Form with Structural Break, 1955-2009
Graph 2.2: Best-Fit Functional Form with Structural Break, 1987-1989
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82
19
83
19
84
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85
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86
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87
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88
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89
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90
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91
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92
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93
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00
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01
20
02
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03
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04
20
05
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06
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07
20
08
20
09
Time in Years (1955-2009)
Best-Fit Functional Form with Structural Break, 1955-2009
MC(t)=40912.3+10.8t^3-20762.2 ln t+19899post1988-425post1988t
320000
340000
360000
380000
400000
1987 1988 1989
Time in Years (1987-1989)
Best-Fit Functional Form with Structural Break, 1987-1989
MC(t)
Divovich 24
Graph 2.3: Best-Fit Functional Form with Structural Break, 1960-1970
The structural break in 1988 is hardly apparent in Graph 2.1. A closer look at the
structural break is seen in Graph 2.2, which shows that during 1988 marginal cost increased if
the Hotelling model holds true. Given the increase in environmental costs, this seems reasonable.
Most importantly, allowing for this structural shift kept the marginal cost function positive, as
seen in Graph 2.3.
Both Test I and Test II predict U-shaped marginal cost functions, which are consistent
with the theoretical and empirical literature to date. Likewise, both cost functions are at a
minimum around year 1964 (for Test 1 the minimum is at the start of 1964, while for Test 2 the
minimum is between 1963 and 1964).
Is this compatible with what actually took place in the rare earths market? As was
described earlier, technological advances were at their peak in the late 1950's and early 1960's
when the major separating technologies were being developed. By 1964, such technologies were
well-developed and were being successfully applied on an industrial scale. After the mid-1960's,
0
5000
10000
15000
20000
25000
1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970
Time (1960-1970)
Best-Fit Functional Form with Structural Break, 1960-1970
MC(t)=40912.3+10.8t^3-20762.2 ln t+19899post1988-425post1988t
Divovich 25
however, technological developments that decreased the marginal cost of rare earth production
were scarce. While discoveries of rare earths have been fairly constant over time and stock effect
has been minimal, technological change has been, arguably, the most important factor embedded
in the marginal cost of production.
Section 4: Conclusion
With the key theoretical tool of dynamic programming, along with data on the rare earths
market from 1955-2009, we were able to derive the basic Hotelling model and perform a
descriptive test on the rare earths market. We were unable to formally prove or disprove the
Hotelling model due to lack of cost data, however, we were able to derive the “best-case” time
path of marginal cost from 1955 – 2009. Results were in line with the current theoretical and
empirical literature that predicts U-shaped cost curves. In addition, the minimum point of the
cost curve, in year 1964, was aligned with the technological advances in the rare earths market,
lending credibility to the Hotelling model for this market. However, without allowing a structural
shift for increased environmental costs, the Hotelling model was not consistent with the data.
Future tests should seek cost data in order to generate a formal test of the Hotelling model in the
rare earths market. The optimal extraction of this nonrenewable resource is an economically and
politically viable area of interest for future research.
Divovich 26
References
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Divovich 27
Appendix 1
Below is the algebraic restructuring which enables us to test the Hotelling model
empirically and find the coefficients of marginal cost with respect to time,