Intertemporal Production Decisions for Depletable
Resources
Qian Shujing
International College. Ningbo University
Ningbo, Zhejiang Province, P.R.China
qianshujing@nbu.edu.cn
Abstract—A depletable resource in the ground is like money in
the bank and it tends to be overused in the present unless there
are institutions created that provide resource users with a way to
benefit from conservation. If complete property rights exist,
profit-maximizing extraction firms take user cost as well as
extraction cost into consideration when they make intertemporal
production decisions. They will conserve any resources that
would be more profitably extracted in the future. And if
resources produced by a monopolist, he will charge a high price
and depletes the resources more slowly.
Keywords-depletable resources; interest rate;user cost; incomplete
property right; monopolist
I.
INTRODUCTION
When production today affects sale or costs in the future,
we faces intertemporal production decisions. One of the good
examples is the production of a depletable resources.
Resources such as oil, natural gas, coal, copper, iron, lead,
zinc are all depletable because there are actual fixed supply in
the earth and are renewed so slowly from a human time
perspective, say oil and coal will take millions of years to
form out of decaying plants and animals. When the owner of
an oil well pumps oil today, less oil is available for future
production. As decision maker choose how much to produce
today, they have to take the characteristic of depletable
resources into account. Such decisions involve comparison
between costs and benefits today with costs and benefits in the
future.
Once decision maker can weigh the costs and benefits of
present use against the costs and benefits of future use,
decision maker can choose the optimal amount of production
now. The key and popular tool to use is present value. We
introduce it first and see how it works. Then we will exploit
our understanding by discussion production decision making
in following situation:
(1) Incomplete property right exist
(2) Decision maker is a monopolist rather than a
competitive industry.
want to get $100 in 5 years, how much money do we need to
save today? The answer is $78.35 because if $78.35 is invested
at an interest rate of 5% per year, it will grow into $100 in 5
5
years ( 78.35  (1  5%)  100 ). Say, $78.35 today can be
thought of as being equivalent to $100 in 5 years because it is
possible to transform $78.35 today into $100 in 5 years by
simply investing it at the market rate of interest.
B. The Production Decision of an Individual Resource
Producer
Suppose you have an oil well. The well contains 1000
barrels of oil and the marginal and average cost are constant at
$10 per barrel. Should you produce all the oil today to should
you save it for future use?
Use the general rule of MR=MC, you might think the
optimal output is when market price of oil is $10 per barrel.
However, this choice ignores the opportunity cost of using up
the oil today so that it is not available for future. Every bit of
resource that is extracted and sold today comes at the cost of
not being able to extract it and sell it in the future, we call this
cost as the user cost of extraction. As shown in Figure 1, the
company’s marginal production cost or extraction cost is $10.
the user cost is the difference between price and marginal
extraction cost. It rises over time because as the resource
remaining in the ground becomes scarcer, the opportunity cost
of depleting another unit becomes higher.
Figure 1. Extraction Cost and User Cost of a Depletable Resources
II.
PRESENT USE VERSUS FUTURE USE
A. Intuition of calculating present value
Intuition of calculating present value is simple. Suppose
that the current market rate of interest is 5% per year. If we
Therefore production decision rule is simply: Keep all your
oil if you expect its price less its extraction cost to rise faster
than the rate of interest (Pindcky et al. 2008). Extract and sell
all of it if you expect price less extraction cost to rise at less
than the rate of interest. If the expected price less extraction
cost to rise at exactly the rate of interest, then it would be
indifferent between extracting the oil now and leaving it in the
ground for future use. Letting Pt be the price of oil this year,
Pt+1 the price next year, and c the cost of extraction, we can
write this production rule as follows:
If
( P t 1 c)  (1  R)( Pt  c) , keep the oil in the ground.
If
( P t 1 c)  (1  R)( Pt  c) , sell all the oil now.
If
( P t 1 c)  (1  R)( Pt  c) , makes no difference.
Given the expectation about the growth rate of oil prices,
we can use this rule to determine production. But how are
interest rates determined?
C. Determines of Interest Rate
We have seen how market interest rates are used to help
make intermporal production decisions. The interest rate is the
price that borrowers pay lenders to use their funds. Like any
market price, interest rates are determined by supply and
demand for loanable funds.
As shown in Figure 2, the supply of loanable funds comes
from households who wish to save part of their incomes in
order to consume more in future. The demand for loanable
funds has two components. First, some households want to
consume more than their current incomes. Because their
incomes are low now but are expected to grow, or because
they want to make a large purchase that must be paid by future
income. These households are willing to pay interest in return
for not having to wait to consume in the future. It is the curve
labeled DH. The second source of demand for loanable funds
is firms that want to make capital investments. Often firms
borrow to invest because the flow of profit from an investment
comes in the future. It is labeled DF. Changes in demand or
supply cause changes in interest rates.
III.
OPTIMAL EXTRACTION LEVEL IN COMPETITIVE
INDUSTRY
A. Perfect Competition leads to Conserve Resources
An exhaustible resource in the ground is like money in the
bank and must earn a comparable return. Therefore, the goal of
profit-maximizing extraction firms is not to simply pump as
fast as possible. Instead, they are interested in extracting
resources at whatever rate will maximize their streams of profit
over time. Suppose there were no OPEC cartel and there were
many competitive oil producers. If any producers want to
maximize their profit, they will follow the rules we stated
above. This means that price less marginal cost must rises at
exactly the rate of interest. If price less cost were to rise faster
than the rate of interest, no one would sell any oil. Then this
would drive up the current price. If price less cost were to rise
at a rate less than the rate of interest, everyone would try to sell
all of their oil immediately. Consequently, this would drive the
current price down.
See figure 3, if a firm only takes account of current
extraction costs, shown as EC, it will produce Q0 units of
output currently. If a firm also takes account of user costs,
shown as UC, it will produce only Q1 units of output. At this
production level, price for all selling units are exceeds the sum
of extraction costs and user cost.
Figure 3 Choosing the Optimal Extraction Level
See figure 4, with the opportunity cost of depleting another
unit becomes higher, shown as increasing user cost from UC0
to UC1, the firm reduces current production from Q0 to Q1 in
order to be able to extract more in the future and take
advantage of the increase in future profitability.
Figure 2 Supply and Demand for Loanable Funds
Government policy can determine the interest rate too. For
example, when government runs a large deficit, it will have to
borrow to balance the deficit, shifting the total demand for
loanable fund to the right; government can create money,
shifting the supply of loanable funds to the right. Change in
demand and supply cause changes in interest rate. Although
there are several interest rates, rather than a single market
interest rate, it stills a workable tool to be used in assessing
present value.
Figure 4 An Increase in Expected Future Profits Leads to Less Current
Extraction
This incentive structure is very useful to society because
it means that our limited supplies of nonrenewable resources
will be conserved for future extraction and use if extraction
firms expect that demand and hence profits in the future will
be higher than they are today.
B. Icomplete Property Right Lead to Excessive Present Use
We just demonstrated that profit-maximizing extraction
companies are happy to decrease current extraction if they can
benefit financially from doing so. This pleasant result breaks
down completely if weak of uncertain property rights do not
allow extraction companies to profit from conserving resources
for future use (McConnell et al, 2009). As a result, the firm will
take into account only current extraction costs.
One of good example is conflict diamonds. In war zones in
African, diamonds are mined in order to provide the hard
currently to finance military activities. However, most of these
civil war are very unpredictable, so that control of the mines is
slipping from one army to another depending on the tide of war.
Because nobody can be sure of controlling a mine for more
than a few months, extraction rates are always extremely high.
This behavior is very wasteful because once the war finally
ends and money is needed to rebuild the country, whichever
side wins will find precious few diamonds left to help pay for
the reconstruction. Unfortunately, incomplete property rights
lead to extraction at far too rapid a pace.
It is no surprise that if resource users have no way of
benefiting from the conservation of a resource, they will use
too much of it in the present and not save enough of it for
future use, even if future use would be more beneficial than
present use.
IV.
OPTIMAL EXTRACTION LEVEL IN MONOPOLIST
INDUSTRY
What if the resource is produced by a monopolist rather
than by a competitive industry? Should price less marginal cost
still rise at the rate of interest?
Suppose a monopolist is deciding between keeping an
incremental unit of a resource in the ground, or producing and
selling it. The value of that unit is the marginal revenue less the
marginal cost. The unit should be left in the ground if its value
is expected to rise faster than the rate of interest; it should be
produced and sold if its value is expected to rise at less than the
rate of interest. Since the monopolist controls total output, it
will produce at optimal level:
( MRt 1  c)  (1  R)( MRt  c)
This rule also holds for a competitive firm. For a
competitive firm, however, marginal revenue equals the market
price. For a monopolist who faces a downward-sloping demand
curve, price is greater than marginal revenue. See Figure 3.
Figure 3 Monopolist’s Marginal Revenue and Price
Therefore, if marginal revenue less marginal cost rises at
the rate of interest, price less marginal cost will rise at less than
the rate of interest. Showed as following:
( Pt 1  c)  ( MRt 1  c)  (1  R)( MRt  c)
Till now we draw an interesting result that a monopolist is
more conservationist than a competitive industry. In exercising
monopoly power, the monopolist charges a higher price and
depletes the resource more slowly.
V.
CONCLUSIONS
In economic theory, a depletable resource in the ground is
like money in the bank. If complete property rights exist,
profit-maximizing extraction firms take user cost as well as
extraction cost into consideration when they make
intertemporal production decisions. They will conserve any
resources that would be more profitably extracted in the future.
And if resources produced by a monopolist, he will charge a
high price and depletes the resources more slowly.
But in reality, resource depletion has not been very important as
a determinant of resource prices over the past few decades. See Table
1, the user cost of crude oil and natural gas is a substantial component
of price. For other resources, user cost almost had nothing with price
fluctuations. Much more important have been market structure and
changes in market demand. Mainly because oil and natural gas are
more depletable than others. For example, known and potentially
discoverable inground reserves of oil and natural gas are equal to
only 50 to 100 years of current consumption. While coal and iron
have a proven and potential reserve base equal to several hundred or
even thousands of years of current consumption.
Resource
Crude Oil
Natural Gas
Uranium
Copper
Bauxite
Nickel
Iron Ore
Gold
User Cost/ Competitive Price
0.4 to 0.5
0.4 to 0.5
0.1 to 0.2
0.2 to 0.3
0.05 to 0.2
0.1 to 0.3
0.1 to 0.2
0.05 to 0.1
Table 1. User Cost as a Fraction of Competitive Price
We should keep in mind that the role of depletion should
not be ignored. Over the long term, it will be the ultimate
determinant of resource prices.
REFERENCES
[1]
[2]
C.R.McConnell, S.L.Brue, and S.M.Flynn, Economics. McGraw-Hill,
2009.
R.S. Pindyck, Micoroeconomics, Peasron, 2008