by *Douglas
B. Reynolds*

Abstract

Natural Resource prices and costs of extraction have declined simultaneously
with increasing quantities of extraction for a long time. In a Hotelling sense
this indicates decreasing scarcity since low cost resources normally would be
used first and quantities of extraction normally would decrease over time. The
main reason for the trend being opposite to Hotelling characteristics is usually
thought to be due to technological innovation. However, an alternative reason
for decreasing costs and prices and increasing quantities of extraction may
be due to Georgescu-Roegan’s (1972) concept of "Bonanza" where there is
only the appearance of decreasing scarcity. Norgaard’s (1990) "Mayflower"
problem can be used to model an alternative neo-classical approach to resource
extraction and scarcity. In this paper, a model of resource exploration is developed
where the explorer does not know total reserves of the resource base as he searches
for and extracts the natural resource. The explorer never entirely knows how
big the resource base is but does gain information about the potential location
of new reserves as discovery proceeds. That reduces exploration costs. The lower
exploration costs can cause the price to fall over time, until eventually scarcity
of the resource causes the price to rise. The true scarcity is only revealed
towards the end of exhaustion. The model shows that it is possible to have several
years of increasing production simultaneous with lower prices and costs until
a sudden, intense price rise occurs with a huge cut in production. When technology
is able to cut costs and increase the reserve base, the decline in prices and
costs and the increase in production can last longer. However even with better
technology, it is still possible for a sharp increase in price as long as demand
is growing faster than technological innovation. The problem is that the true
size of the resource base is never known. Society does not know if technology
is actually overcoming scarcity or not until demand for a resource outstrips
supplies.** It is even possible for a price shock of incredible magnitude to surprise
an economy within one or two years after a hundred years of declining prices
and increasing production.**

KEYWORDS: Resources, Scarcity, Technology, Exploration, Information, Hubbert, Norgaard’s Mayflower problem

**The Mineral Economy:** **How Prices and Costs Can Falsely Signal
Decreasing Scarcity**

**Introduction**

According to Hotelling (1931) there are 3 things, we can say about resource extraction for a finite resource. 1) Firms will always try to extract low cost resources before high cost resources causing resource extraction costs to increase over time. 2) In general, greater scarcity caused of in-situ reserves increases the value of a resource causing the price to increase over time. 3) Since the price of a resource increases over time, then demand and production should decrease over time. Barnett and Morse (1964), and more lately Scott and Pearse (1992) and Simon (1996), show that all of these theoretical trends have not occurred. Costs of extraction and prices of natural resources have mostly decreased over time, not just in recent years but in many cases for decades. Quantities extracted and produced of most natural resources have also increased over the same long time frame. The only logical explanation is that technology and substitutes are in fact more powerful than scarcity.

In this paper is shown an alternative model for scarcity based on Norgaard’s (1990) Mayflower problem that shows how it is possible for there to appear to be long trends of decreasing scarcity only to have a sudden and dramatic increase in scarcity. The model here shows that long run resource exploration cost, and resource price declines simultaneous with increasing resource extraction, do not necessarily signify decreasing scarcity. The paper shows that the true power of technology in over coming scarcity may be difficult to determine. It is also shown how price and cost measures of scarcity may fail even when prices and costs have declined for many years.

Hall and Hall (1984) define scarcity in terms of the cost of extracting a resource, which is based on Hotelling’s original concepts. Malthusian scarcity is where in-situ stocks are finite and Ricardian scarcity is where stocks are not finite, but costs of extracting them increase as you extract more. Concentrating on Malthusian scarcity, Hall and Hall define Malthusian Stock Scarcity (MSS) where resource stocks are finite but are of uniform quality so that the cost of producing them stays relatively constant with constant technology. However instead of looking at the cost of extraction to determine scarcity, this paper will look purely at the costs of exploration. Extraction costs will be zero. The assumption of zero extraction costs can be modified later, nevertheless this assumption changes the emphasis of determining scarcity away from the costs of producing a resource and toward the cost of finding a resource.

Norgaard looks at the difficulty of exploring for resources in the "Mayflower" problem where information about the location of low cost natural resources is incomplete. When the Mayflower pilgrims came to America, they settled at the first most convenient location possible in New England even though the soil there was not as good as the soil in say the Midwest. If the Mayflower pilgrims had used the available resources optimally in a Hotelling sense, they would have taken the Mayflower to the Gulf of Mexico and up the Mississippi River all the way to Iowa in order to start their farms there. Or at least within a generation or two, the new settlers would have moved all their settlements to the Midwest. At least part of the reason that the Mayflower pilgrims and subsequent generations did not go to the Midwest quickly was because they did not know that there was good soil there. Their actions imply that finding information about resource location is an unavoidable cost of the supply of natural resources.

**Monte Carlo Modeling**

In order to understand the economics of mineral exploration and the Mayflower problem, consider a Monte Carlo simulation. Menard and Sharman (1975) show in a Monte Carlo model that if oil explorers in the US drilled random oil wells throughout the US starting in 1859, then the super giant East Texas oil field would have been discovered at the very least by 1902, given the historical rate of exploration. East Texas was actually discovered in 1930. Other giant and super giant fields also undoubtedly would have been found earlier by merely drilling exploration oil wells randomly. The problem with the analysis is that Menard and Sharman know something that the early oil explorers did not know, namely that the East Texas oil field exists. In other words, the Monte Carlo modelers have perfect information that twenty-two giant oil fields in the US exist before they run their model, but the early oil explorers did not have this information and did not know if even one giant oil field existed until the first one was found. Therefore, consider an alternative set up.

A Monte Carlo model is rather like playing roulette where there are black and red slots on the roulette wheel for the ball to fall into. Black is oil, red is dry. The probability of getting black on a bet that costs the amount of money for drilling an oil well depends on how many black slots there are as a proportion of all the slots. If you were to play the gamble, the expected results are exactly equal to the probability of getting black. However, different roulette wheels like different countries and different regions have different probabilities of success. For example compare finding oil in the Persian Gulf to that of Switzerland. In other words, there are different roulette tables with different probabilities.

Assume then that there is a large room full of roulette tables, and that gamblers will come into the room to play the tables. Based on their expected probability of winnings, they will be more willing to bet on the tables with the higher probability of black since those tables will have a higher expected gain. Now assume one more thing. All the roulette tables are covered and mixed before the gamblers enter the room. In that case, the gamblers do not know which tables have a high proportion of black slots and which have a high proportion of red slots. They do not even know if any tables have any black slots. The gamblers have no idea what their chances of winning are or even if there is anything worth betting on or not.

As the gamblers come into the room with the covered roulette tables, they must try to get information about what the tables are like. What strategy will they use? Undoubtedly, the gamblers will wait to see what other gamblers do. If one gambler does make a bet and does get a black slot, chances are good that at least that particular roulette table has more black slots. The gamblers race to that table and start betting. However as the bets are made, every time a slot is hit, it is removed and not replaced. Eventually the black slots are used up and the gamblers will have to leave old tables and start taking their chances on newer tables.

The gamblers do not randomly play on all the roulette tables, as implied by a pure Monte Carlo model. For all they know, all the tables could have only red. There is no way of knowing a priori what the probability is that any black slots exist. In this situation, the gamblers have in their mind a subjective probability of winning at any given table. However, their subjective probability is different than the actual probability of the table. When the gambler goes to a table where previous gamblers won black, he has information to judge the probability that he also will win based on past experience. That is his own subjective probability. It is based on available information. A gamblers subjective probability will be higher at a table with a past history of winning bets than with an unknown table where there is no history. The subjective probability of getting black at a new table may be lower or even much lower than the actual probability of getting black at that table. Therefore there is a difference between a gamblers or a mineral explorers subjective probability of finding a mineral and the actual probability of finding a mineral, should he drill a well.

This is the Mayflower problem. At first there is very little information and few bets are made, i.e. explorations for minerals. Then as more information is found, more betting occurs because the information helps increase the probability of success until eventually scarcity brings to an end any chance of finding new resources. The mineral economy below is derived from this Mayflower problem. By using this model, we get a very different perspective on how a resource dependent economy will act over time and on the interpretation of the effect of better technology on scarcity.

**The Mineral Economy Model**

Two economic models describe a resource-based economy in its purest form: the "corn" economy and the "hardtack" economy explained below. For a complete analysis of these models see Page (1979). We can call a third type of resource model the "mineral" economy. The mineral economy model is similar to a hardtack economy model except that the mineral – the hardtack – is hidden. The hidden hardtack must be explored for and extracted before it can be used. To explain the models, consider Robinson Crusoe stranded on a deserted island. In the corn economy scenario, he has some corn with him. He can choose to use varying amounts of the corn to either plant or eat depending on what his intertemporal preference is. If he eats more corn now, he has less corn to plant for future output. On the other hand if he plants more corn, he will have more future output but less to eat right now. In an infinite horizon world, such as a tribe of constant population on the island, a corn economy can last indefinitely.

In the second scenario, Crusoe is faced with a much more sobering possibility of only having a finite amount of hardtack. Again, depending on his personal preference, his years of life and the amount of hardtack he has, he faces several scenarios for exhausting that hardtack. He can choose to eat more now or save his stock and eat more later, but his reserves are limited. He may even not survive long if his supply is too small. If we extend Crusoe's life indefinitely by replacing him with a tribe on the island with a constant population and still a finite amount of hardtack, then that society faces the inevitable dilemma of sooner or later running out of hardtack and dying off. Better technology, though, could over come such a grave scenario.

However, neither of these scenarios for Robinson Crusoe parallels the real world, where our economies are to a large degree based on underground non-renewable non-recyclable resources, such as oil, coal and natural gas. In the real world, unlike a hardtack economy, these natural resources are not stored in such a manner so that we can quantify them costlessly, the way we can quantify hardtack stores costlessly. Rather, many natural resources are hidden from the economy until they are found. The hidden resources create an unavoidable cost for finding the minerals. That is an exploration cost. Even if extracting the mineral is costless, there is still an exploration cost implied by the search process that influences the rate of extraction. Using the characteristics of the Mayflower problem, we can create a model for the cost of exploration over time and then use that model to analyze scarcity and how powerful technology really is for overcoming scarcity.

Consider a situation where Robinson Crusoe lands on a deserted island with no food. Assume that previous shipwrecks with no survivors have left hardtack on the island. If Crusoe is able to find these old shipwrecks and the hardtack on them, he can survive by eating the lost hardtack. However, let us further assume that these previous shipwrecks happened many years before Crusoe's shipwreck, such that all the ships have been buried deep in the sand and are not visible from the surface. Crusoe can survive by eating this leftover hardtack, which is his "mineral" resource, if Crusoe can find it. This situation is the mineral economy. In the mineral economy assume there is Malthusian Stock Scarcity where the resources are of constant quality and even free to extract. The only problem is knowing how much resources there are and finding them.

When Crusoe first lands on the island, he must hurry to find food where he can. Because no previous shipwrecks can be seen from above ground, Crusoe does not know if any hard tack even exists underground and if it does exist, how much there is. Nevertheless with no other option, Crusoe takes a chance and digs for hidden hardtack. Eventually he gets a break and finds one shipwreck and manages to find the left over hardtack of that ship. Such a find will give Crusoe new information about where other shipwrecks may be because often times ships will wreck at the same location. This is similar to all the gamblers with the covered roulette wheels all going to the one table where there was a black slot found and betting on it. Therefore, with this new information about one ship's location, Crusoe will be better able to find other ships and his cost of searching will decrease. The more hardtack that Crusoe finds, the more information he has, and the easier it is to find even more hardtack.

Eventually though, because the supply of hardtack is finite, depletion sets in and Crusoe finds less and less hardtack. Therefore, even though Crusoe has more and more information about where the hardtack is, the depletion of the resource base creates higher costs of exploration, since it is hard to find the last few remaining stocks the rate of discovery decreases. Crusoe's supply of hardtack left to be found dwindles until he can no longer survive.

Two long run forces, then, drive Crusoe’s mineral economy. First, there is the information effect where the more information Crusoe has about the location of the hardtack, the cheaper it is to find and the more he is able to find. Second, there is the depletion effect where the less hardtack that remains to be found, the costlier it is to find and the less he can find. These forces create a pattern of first an increasing and then a decreasing rate of discovery. Uhler (1976) calls the two opposite forces of information and depletion the information and depletion effects.

**An Exploration Information Cost Model**

Assume that Crusoe’s technology for finding and using the hardtack stays
constant and actual extraction is costless. Changes to these assumptions are
possible after the basic model is developed. Assume every time Crusoe explores
a location for hardtack, he must eat h_{c} of hardtack in order to give
himself the physical energy to look. This is his cost of exploration effort.
If Crusoe searches in 20 locations, he must eat 20 times h_{c} hardtack.
Furthermore, assume for simplicity that when hardtack is discovered in any one
location, that it is always the same amount of hardtack, call it h_{d}.
While most minerals occur in large and small loads or pools, rather than constant
loads, it is assumed that a large load is the same as a lot of small ones that
are near each other. Therefore, hardtack that is left in the ground from previous
shipwrecks is always packed in small tins or boxes that are the same h_{d} size. The tins are separate and must be discovered separately. Smith and Paddock
(1984) suggest based on empirical evidence that minerals such as oil occur in
large and small pools or loads and that the largest pools or loads of minerals
are usually found first and then subsequently smaller pools and loads are found
within one region. However, their empirical evidence is not general enough,
since in many cases the largest loads of minerals are not the very first reserves
found. For example, the East Texas oil field, Prudhoe Bay, and the great Ghwar
oil field in Saudi Arabia were not the first oil fields found even though these
are some of the largest fields on Earth. In general, small loads of minerals
are found first, then large ones and then small ones again. For Crusoe, it is
assumed that each tin is separate and equal, however large loads of minerals
can be simulated by having lots of tins being close together. When one is found
others nearby are easy to find. Hence:

h_{c} = Pounds of hardtack consumed per exploration location that
is searched, i.e. effort cost.

h_{d} = Pounds of hardtack found per exploration location when the
exploration is

successful. h_{d} is the pounds of hardtack found in each tin. All
tins are

assumed constant for all successful exploration locations.

h_{d} > h_{c}

In order to discover Crusoe's marginal cost of extraction for any given rate
of extraction, it is necessary to know the probability of success for each individual
search and how that probability changes as the rate of extraction changes. Crusoe’s
exploration of any given location is not always successful. There is a probability
of success or failure. The probability for Crusoe's success rate is called the
probability weight function, P_{r}, which is modeled formally below.
P_{r} is a function of ultimately recoverable reserves, the current
rate of exploration, and information, which is dependent on cumulative discovery.
P_{r} models Crusoe's subjective probability of finding more hardtack
rather like a probability coin. Every time Crusoe expends effort to search for
hardtack at a specific location, his success of finding more hardtack is like
flipping a coin. However, the probability of success is not necessarily 50/50.
Instead, the probability weight of the coin can be anywhere from 0/100 to 100/0
for success, depending on Crusoe's subjective probability. That subjective probability
depends on information such as how successful previous explorations have been
and exploration progress. The subjective marginal benefit of exploration for
Crusoe is then h_{d× } P_{r} or alternatively
the marginal cost is h_{c}/ P_{r}.

In the Monte Carlo game with the gamblers, there was the actual probability
of taking a gamble on the roulette wheel, which the gamblers did not know, and
their own subjective probability of gambling at that table. Like the gamblers
in an unknown Monte Carlo simulation, Crusoe too has no idea if there is any
hard tack at all under ground or, if there is some, where it is. Nevertheless,
Crusoe will use whatever information he can garner from past experience to choose
the best location possible. His subjective probability, P_{r}, will
change as information changes.

There are two distinct aspects of how information affects Crusoe’s exploration process. When Crusoe begins his search for hardtack at the beginning of the day, he will have in his mind all possible information about where hardtack may or may not be. He will begin his search at the location where his subjective probability of success is the highest. That is, Crusoe will explore for hardtack first in the most probable locations. He will continue searching at second and third best locations. Therefore, as the day passes, Crusoe will have a lower and lower probability of success.

During the day, however, Crusoe may also gain new information, which could help his search. For example, Crusoe may discover hardtack and start to see a pattern about where hardtack usually occurs underground. This may increase his probability of success. Therefore, there are two distinct forces that affect the probability of success. One force is where the old information becomes less and less useful in helping Crusoe make successful new discoveries. The other force is where new information discovered during the course of exploration helps Crusoe raise his probability of success. In order to understand these two forces better let us separate them and model them differently. For simplicity, assume that Crusoe can only use one set of information during a given day and that any new information he finds he cannot assimilate until the next day. This will separate the effects of information into a short run (within a day) and a long run (longer than one day) time frame. In other words, Crusoe will use information rather like capital. "Information capital" is Crusoe's accumulated information, i.e. his stock of information. Information capital is like a stock of capital machinery, it cannot change in the short run. Only in the long run can new information be added to the current stock of information, causing the information capital to increase.

Short Run Characteristics of Probability

In the short run, Crusoe has a finite stock of information, i.e. information
capital that cannot change, to use for looking for more hardtack. The information
capital is a complex data configuration. New information and data cannot be
easily integrated into the existing information capital just like a new machinery
cannot be easily and quickly integrated into an existing stock of capital. Therefore,
all new information cannot be assessed until the next day. However, Crusoe can
use the existing information capital to look for hardtack over an infinite combination
of places during the current day. He will look for hardtack first in the places
where his subjective probability of successfully finding hardtack is the highest
and then in second and third best locations. As he looks, though, the probability
of success declines. The reason for the decline is because the information capital
is subject to the law of diminishing returns which causes the value of that
information stock to decrease. In other words, the law of diminishing returns
degrades the value of Crusoe's information capital as he uses it so that it
is less and less useful the more it is used during the day. The marginal productivity
of the information capital declines as effort and therefore output increases.
This is similar to how the marginal productivity of capital itself declines
as output increases. The effect of the degrading value of old information is
to cause a decreasing probability of success. A convenient formula for P_{r} to model this short run value of the information capital is the following.

P_{r} = exp[- Q/L ]

Q = Q(S)

where

dQ/dS > 0

and

Q = the daily quantity discovered or current rate of discovery

S = Effort in terms of numbers of searches per day

L = a long run trend parameter

The short run probability will act as a decline curve as the rate of discovery
increases. The rate of effort, S, is an independent variable chosen by Crusoe,
which will determine the rate of discovery. If effort increases, discovery increases
and probability falls. At the beginning of the day it may take two exploration
tries to find hardtack, but as the day wears on, it may take Crusoe ten tries.
Therefore, the probability of finding that last tin of hardtack decreases from
50% to 10%, and hence the marginal cost rises. The lower probability of success
is due to the diminishing returns on the current information capital which increases
marginal cost. Crusoe will tend to look for hardtack during the day until his
cost of exploration is close to or equal to his benefit of successful discovery,
h_{d}. Setting the probability as a function of the rate of discovery
rather than effort makes the model easier to work with. The more one discovers
the resource during the day, the more one uses the current information capital.
The more the current information capital is used, the less valuable it is. The
less valuable the information capital is, the lower the probability of success.
Therefore marginal cost rises.

Long Run Characteristics of Probability

Recall in the Monte Carlo situation above that initial gambles, like initial efforts to find a resource, are very unsuccessful due to a lack of knowledge and therefore a great deal of effort is used to find only a small amount of resource. Eventually, however, with greater information, it takes fewer efforts to find a given amount of resource. Therefore, in the long run, new information will increase the information capital and increase the probability of success, such that the probability of finding the mineral will increase. Concurrently, however, there is depletion. In addition to long run information gains, the depletion effect will cause less of a resource to be found. As a resource depletes to nothing, the probability of finding more of that resource must also decline. Therefore, in the long run, two effects on probability are modeled: the information effect and the depletion effect. See Uhler.

With greater information capital Crusoe can find more hardtack for a given level of effort. Greater cumulative discovery implies greater information capital, which will cause a higher probability of success. This is the information effect. However, with greater depletion, Crusoe will find less hardtack with a given level of effort. Greater cumulative discovery equates to less remaining resources in the ground available to find, which will cause a lower probability of success. This is the depletion effect. The information effect will dominate initially, and the depletion effect will dominate subsequently. The probability weight function as a whole shifts first outward and upward and then inward and downward with cumulative discovery. The result is a small-large-small sequence of the discovery and extraction of hardtack in the long run. These long run effects on probability are modeled using cumulative discovery.

Greater cumulative discovery of hardtack implies that there have been greater exploration efforts, and such efforts must increase the amount of information. Therefore, cumulative discovery is used as the long run information capital indicator. It is also possible to use cumulative exploration as an information indicator, but if unsuccessful search locations actually do provide information, then they will lead to more discovery. Thus cumulative discovery is just as good an indicator of information capital as cumulative exploration efforts. The benefit of using cumulative discovery rather than cumulative effort is that one variable can be used for both the information and depletion effects more simply.

Greater information capital shifts the probability weight function higher.
That shift is an information effect shift, where greater cumulative discovery __implicitly__ signals greater information capital. As the stock of undiscovered
reserves depletes to zero, though, the probability of finding new hardtack must
eventually become zero, because there is no more hardtack left to find. The
depletion effect then is where less hardtack reserves make it more difficult
to find new hardtack causing the probability weight function to shift down.
The depletion effect depends __explicitly__ on greater cumulative discovery,
because higher cumulative discovery equates to a lower stock of remaining undiscovered
reserves.

In order to model the information and depletion effect, assume the long run
trend parameter, L, shifts the probability function P_{r} in the manner
explained above. A likely equation for the trend is M. King Hubbert's life cycle
hypothesis (Hubbert 1962). Hubbert’s life cycle hypothesis which was used
to explain the long run trend of oil production in the U.S. ideally models a
mineral economy faced with the Mayflower problem. Hubbert’s model has
a small-large-small sequence of discovery and production of a mineral that would
be expected in the mineral economy. For an economic analysis of Hubbert see
Cleveland (1991), Cleveland and Kaufmann (1991), McCabe (1998), and Edwards
(1997). Hubbert’s original model using L as the trend for current discovery
is a time dependent logistics curve as follows:

L = a× U× {1 + exp[-a(t-t_{o})]}^{-2} × exp[-a(t-t_{o})] (1)

where

a = an amplitude parameter. This determines the height

and width of the Hubbert curve

t_{o} = The time at which the peak in production

occurs.

t = Time.

U = Ultimately Recoverable Reserves

L = The trend of the rate of discovery

However, it is possible to subsume the time element of Hubbert’s logistics curve in order to make it a pure cumulative discovery model that can be used to measure the combined effects of information and depletion. Hubbert’s cumulative discovery model is as follows,

CQD = U/{1 + exp[-a(t-t_{o})]}

Which can be rewritten as

(t-t_{o}) = {Ln[U/CQD – 1]}/ -a (2)

where

CQD = Cumulative Discovery

Substituting equation 2 into equation 1 gives

L = a× CQD - a× CQD^{2}/U
(3)

which can be called the Hubbert trend.

J.J. Wiorkowski [1981] developed a range of logistics curves similar to Hubbert's life cycle curve, using what is called the Richards function (F.J. Richards [1959]). From Wiorkowski and Hubbert, a generalized trend in terms of cumulative discovery can be used to describe L over time. The generalized Hubbert trend is as follows.

L = a×CQD×U^{ (m -1)}/m - a×CQD^{m} ×CQD/
U×m

and, when m = 0,

L = a×CQD× lnU/ U - a×lnCQD×CQD/ U

where

m = shape parameter, -1 < m < ¥

Equation 3 was Hubbert’s original trend where m = 1.

The Hubbert trend, L, drives the probability weight function in the prescribed manner hypothesized by the mineral economy. It models an early low probability of success that increases due to the information effect. It then models a subsequent decreasing probability of success due to the depletion effect. In the short run, L does not change so that as Q increases there is diminishing returns to finite information capital causing a decline curve for the probability weight function. The long run trend will cause the short run probability function curve to shift out initially and then inward in later periods.

**A Mineral Market Simulation**

Since Crusoe does not know how much hardtack is under the ground, it could
be one million times his most opulent lifetime consumption or only a weeks worth
of food, he cannot perform an intertemporal maximization. If Crusoe’s
discount rate is zero, he simply lives at a subsistence level. However, for
any other discount rate above zero, he just does not have enough information
to maximize utility. Nor can a tribe of constant population. As long as a tribe
has a rate of discount greater than zero, then they cannot use the Hotelling
principle, since they do not know if the resource base will last a thousand
generations or less than one. Therefore, a simple Crusoe model cannot be solved.
However, if we assume that Crusoe or a tribe believes that the resource is so
vast that it verges on being infinite, i.e. that the probability of discovery
always goes up in the long run, then there would be no Hotelling rent. The resource
is seemingly renewable. Crusoe will each day explore up to the point where marginal
cost equals h_{d}. Consider this possibility more generally.

Assume instead of Crusoe by himself, that there is a large economy and that there is a mineral industry for one particular essential natural resource input into that economy. The market for the mineral is within a larger market place. The marginal cost of finding the mineral for all firms in the industry is:

MC = E_{c}/h_{d}×P_{r}

where

E_{c} = cost of effort in the industry

MC = aggregated marginal costs of the industry

P_{r} = exp(-Q/L)

P_{r} is determined by the industry wide rate of discovery. When the
industry extraction rate, Q, increases, P_{r} decreases due to industry
wide diminishing returns to information capital. Note, the industry is made
up of many firms and all firms use each others information, therefore, this
is an efficient market where all information is available to the other players,
similar to the Monte Carlo gambling simulation above. Costs are minimized, profits
maximized. h_{d} is the marginal unit of extraction, therefore let h_{d} = 1 for convenience. The cost of actual extraction is assumed zero. That assumption
can be changed to add an extraction cost that also decreases with time, but
the general conclusion will be the same. Therefore in this simulation it is
assumed that the aggregated marginal cost includes both extraction and exploration
costs. In that case:

MC = E_{c× }exp(Q/L) (4)

This gives the following characteristics:

dMC/dCQD < 0 when the information effect dominates

and

dMC/dCQD > 0 when the depletion effect dominates

In addition technological effects for supply can be added to equation 3 and 4. Let:

E_{c} = E_{co× }exp(-T_{c× }t)

where

E_{co} = Initial cost of effort in the industry

T_{c} = Cost technology factor (the percent reduction in costs per
year)

Also

U = U_{o× }exp(T_{u× }t)

where

U_{o} = Initial Ultimately Recoverable Reserves

T_{u} = Ultimately Recoverable Reserve technology factor (the percent

increase in ultimately recoverable reserves per year due to technology)

Consider a simple demand curve for the resource that reflects a growing economy.

Q = G × (P)^{-1/3} (5)

and

G = G_{o× }exp(g× t)

Where

G = economic production of the entire economy

G_{o} = initial economic production of the economy

g = economic growth factor for the economy

P = price for a given quantity demanded

According to the Hotelling rule, rent will increase over time. However, if uncertainty is large enough this may not happen. See Watkins (1992). For example, if resource prices have been falling for a very long time, and they look to continue to fall for a long time, then producers will set P = MC. There will be no Hotelling rent. Producers believe the resource is infinite due to new technology. When technology is thought to be the cause of a long price decline, then it is easy to conclude that technology will continue to improve indefinitely and cause prices to fall indefinitely. In that case, a resource producer maximizes present value not by withholding exploration for, or extraction of, resources, but by producing more now before prices go even lower. The producer maximizes present value simply by producing as much as profitable in the current year irrespective of rent. Producers would generally set P = MC.

Consider a model simulation using the following values for equations 3, 4, and 5:

U_{o } = 100,000

a = 0.15

T_{u} = 1% per year, i.e. slower than economic growth

G_{0} = 500

g = 3% per year, i.e. faster than the rate of technological growth of the resource

E_{c0} = 1

E_{c13} = 1, E_{c} is held constant in the initial 13 periods
to give the system

stability

T_{c} = 5% per year after the first thirteen years.

In the simulation, the technology to increase ultimately recoverable reserves
is improving at a slower pace than economic growth. However, the effect of technology
on the resource base to increase its recoverable reserves is unknown to the
market and remains unknown until late in the exploration process. Therefore
the market sets P = MC. In each period, Q and P are solved using equations 3,
4 and 5, then the effects of technology are added. Since equation 4 and 5 are
indeterminate for Q, Q_{t-1} is used to solve for Q_{t}. The
difference between Q_{t} and Q_{t-1} is small enough so that
the substitution creates no problem.

The results of the simulation can be seen in figure 1 and 2. In figure
1, the supply Hubbert trend (L), the quantity produced (Q) (which is the
equilibrium of supply and demand), and price (P) are shown as a function of
cumulative production (CQD). Actual production is less than the supply trend
potential, i.e. the Hubbert trend, for most of the cycle. It is only after the
supply Hubbert trend reaches its point of decrease that production finally gets
close to and goes above the Hubbert trend. The Hubbert supply trend is only
the *trend* of supply, it does not determine absolute supply. The actual
supply can be above or below the Hubbert trend although supply becomes increasingly
inelastic above the Hubbert trend.** Scarcity is revealed only when production
goes well above the Hubbert trend. At that point, the price finally starts increasing
and production finally starts decreasing. Before that point, it is not possible
to know how forceful technology is for increasing potential reserves. Suddenly,
scarcity is revealed but unlike the Hotelling model, there is no forewarning.**

In figure 2 is shown the price trend as a function of time. Notice that the price,
as well as the costs that determine price, which is not shown, decline for many
years before the price finally increases. Furthermore, during those price decreases,
quantities extracted of the resource are constantly increasing. In fact it is
possible to model a situation where price decreases and supply quantity increases
continue for hundreds or even thousands of years before the price finally rises
and quantities finally decline.* However, the change from one trend to the other
can happen within a few years even though the trends before the change over
lasted for hundreds of years. In other words, the information effect can dominate
the depletion effect for a very long time before depletion finally and dramatically
dominates. If you add to the information effect technology, then it takes even
longer before depletion dominates. Nevertheless, the power of technology to
overcome scarcity is not known until it is revealed late in the cycle.*

The problem with using price, cost or quantities produced as a measure of
scarcity is that up until the point where demand crosses the Hubbert trend,
it is unclear how powerful technology is for decreasing scarcity. Only when
prices start actually rising due to the depletion effect can society know how
big the Hubbert cycle is and how much technology has stretched the resource
base. Before that point in time, the information effect obscures the size of
the resource base and the true power of technology for reducing scarcity.** An
additional problem is that since price declines for a long period of time before
the price shock, it is unclear what substitutes for the resource are available
and at what price. There has been no gradual increasing price and therefore
no signal to inventors of when a backstop will be needed, and no signal to consumers
of when the backstop will be needed to be used. The market is not properly prepared
for the change over.**

However, once the price reaches its lowest level in figure 2, producers of the resource will realize that the resource is a lot more
scarce then they had previously thought. They will suddenly understand that
the price will start increasing and they will cut production in order to maximize
intertemporal profit via the Hotelling principle. This cut in production will
cause the price shock to be much more sever. In addition, Reynolds (1997) shows
that the backstop price of a resource can unexpectedly rise due to the loss
of entropy subsidies. **It is conceivable that after one hundred years of price
decline and production increase, that a resource can have a ten or one hundred
fold price increase within a year or two, with a corresponding decline in production.
Given that short run elasticities of demand and substitution are much more inelastic,
then long run elasticities, this can create quite a problem for an economy to
say the least***.*

**Conclusion**

In the Hotelling model, there is perfect information about the backstop price,
the quantity of the resource, the demand for the resource, and the cost of extracting
the resource. With perfect knowledge, rent of a resource increases at the interest
rate up to the backstop price. Uncertainty theory can be added to the Hotelling
model to try to find out how trends of prices and quantities change under various
assumptions when the resource base is unknown. However if there is enough uncertainty
it is almost impossible to use the Hotelling principle. In this model is shown
how the uncertainty of the resource base can obscure the actual trend of scarcity
and the true power of technology and create a price shock. Such a price shock
can occur after a very long price and cost decrease simultaneous to a long production
increase. This model shows theoretically that empirical data on cost and price
declines as well as data on increases in quantities of extraction do not necessarily
imply decreasing scarcity. Indeed, such data may not even be useful for warning
about impending price shocks. **What is even more foreboding is that once producers
do know that scarcity is much greater than previously thought, the price shock
will become all the more severe. Resource producers will cut back production
due to increasing prices causing prices to increase even further. The so called
back stop price may be much higher than expected especially since the loss of
entropy subsidies can make the backstop price increase substantially.**

Acknowledgments: Special thanks to many anonymous reviewers, who gave very helpful comments. Also thanks to Darwin C. Hall who saw the importance of this paper.

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Captions to illustrations

Figure 1. Price, Production, and Hubbert Supply Trend as a Function of Cumulative Production

Figure 2. Price and Production as a Function of Time