# Optimal Extraction Paths of Coal

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### Chapter 1: Introduction

### 1.1. Motivation

According to the World Energy Outlook (WEO 2007)[1], global carbon dioxide (CO2) emissions will increase by 1,8 % per year from 2005 to 2030, and 2 % per year for the period 2030-2050.[2] From 12.446 Mt of CO2 equivalent in 2002, emissions will reach 15.833 Mt in 2030 for OECD countries - an average increase of 1,1 % per year. CO2 is the most important anthropogenic greenhouse gas (GHG), which is contributing to global warming. The primary source of the increased atmospheric concentration of CO2 since the pre-industrial period results from fossil fuel use, with land-use change providing another significant but smaller contribution.[3] Continued greenhouse gas emissions at or above current rates would cause further warming and induce many changes in the global climate system during the 21st century.[4]

According to the Nuclear Energy Agency and the International Energy Agency the power generation sector will contribute to almost half the increase in global emissions between 2002 and 2030, and will remain the single biggest CO2-emitting sector in 2030. In OECD countries, its emissions will rise from 4.793 Mt of CO2 in 2002 to 6.191 Mt of CO2 in 2030, but the share will remain constant.[5]

Today, power generation emits 65 % of industrial emissions of CO2 in OECD countries and is likely to become instrumental in countries' strategies to reduce greenhouse gas emissions.[6] One of such instruments is the Kyoto Protocol.

Under the United Nations Framework Convention on Climate Change (UNFCCC), more than 180 countries have recognised the need to stabilise the concentration of GHG in the atmosphere, which are causing climate change. The Kyoto Protocol to the UNFCCC, was adopted at the third session of the Conference of Parties in 1997 in Kyoto, Japan. It entered into force on 16 February 2005 with 184 Parties of the Convention who have ratified to date.[7]

The major feature of the Kyoto Protocol is that it sets binding targets for 37 industrialized countries (including Germany) and the European Community for reducing GHG emissions. These amount to an average of five percent of the 1990 levels over the five-year period 2008-2012.[8]

The Kyoto Protocol includes specific "flexible mechanisms" such as Emissions Trading, the Clean Development Mechanism (CDM) and Joint Implementation (JI) for the countries to be able to reach their mandatory emission limits.

Emissions trading, as set out in Article 17 of the Kyoto Protocol, allows countries that have emission units to spare - emissions permitted to them but not "used" - to sell this excess capacity to countries that exceed their targets. Thus, a new commodity was created in the form of emission reduction or removal assets. Since CO2 is the principal greenhouse gas, people speak simply of trading in carbon. Carbon is now tracked and traded like any other commodity. This is known as the "carbon market".[9] In European countries the emissions trading system is the European Union Emissions Trading Scheme (EU ETS), the largest system nowadays.

The CDM, defined in Article 12 of the Protocol, allows a country with an emission reduction or emission limitation commitment under the Kyoto Protocol (Annex B Party) to implement emission reduction projects in developing countries. Such projects can earn saleable certified emission reduction credits, each equivalent to one ton of CO2, which can be counted towards meeting the Kyoto targets.

A CDM project activity might involve, for example, a rural electrification project using solar panels or the installation of more energy-efficient boilers.[10]

The JI mechanism, defined in Article 6 of the Kyoto Protocol, allows a country with an emission reduction or limitation commitment under the Kyoto Protocol (Annex B Party) to earn emission reduction units from an emission-reduction or emission removal project in another Annex B Party, each equivalent to one ton of CO2, which can be counted towards meeting its Kyoto target.

JI offers Parties a flexible and cost-efficient means of fulfilling a part of their Kyoto commitments, while the host Party benefits from foreign investment and technology transfer.[11]

Germany is one of the world's largest energy consumers and ranks third in total CO2 emissions within the G-7, after the USA and Japan.[12] Annually, Germany produces around 850 millions tons of CO2 equivalent gases, which is approximately 2,8 % of all world's CO2 emissions.[13] On 31 May 2002, the Kyoto Protocol was ratified by Germany. After entering it into force Germany has played an active role in the European and world carbon markets.

Electricity production in Germany is largely based on burning exhaustible resources, causing high CO2 emissions. That makes the issue of CO2 trade crucial for German power plants and the economy in whole.

In 2008, the total amount of gross electricity supplied in Germany was around 639,1 TWh[14], that is slightly higher in comparison to the previous year. Nevertheless, during last years there is a tendency of increase in electricity supply (See Table 1).

The electricity supply in Germany is based on several technologies and fuels. The distribution of net electricity supply in last years in Germany is shown in Table 1. Electricity production in 2008, as in previous years, was based mainly on coal-fired (hard coal and lignite) steam turbine (43,6 %) and nuclear (22,3 %) power plants.[15]

### Energy source

Since the share of the coal based power plants in Germany is large and the amount of electricity produced is still growing, the impact of the CO2 emissions trade on the economy of these plants is very significant.

According to data provided by the Nuclear Energy Agency and the International Energy Agency, the price for coal is rising during the economic lifetime of the coal-firing plants.[16] This rise partly can be caused by additional CO2 costs.

The largest impact of the emissions trading on the electricity generation cost is felt by the lignite-fired power plants followed by the hard coal-fired power plants, since lignite while burning is producing more emissions than hard coal.[17] With an assumed emission price of 20 €/tCO2 the power generation costs of the lignite-fired power plant would increase by 63 % from 25,4 €/MWh to 41,4 €/MWh, whereas the generation costs with hard coal-fired would rise by 48 % from 30,2 €/MWh to 44,8 €/MWh.[18]

The competitiveness of the coal-fired plants is also influenced by including the CO2 prices into the costs. 1 represents marginal cost curve based on the total installed capacity and facilities' operating costs for Europe.[19] As can be seen, the addition of CO2 price to the production costs can make coal power plants less competitive. The sequence of most of electricity plants stays the same after addition of 20 €/tCO2 to the costs, though coal based power plants move to the side of less competitive plants.

These facts and evident changes raise many questions such as following: how long will electricity from fossil fuels stay competitive, how the extraction of fossil fuels is influenced by CO2 prices.

### 1.2. Problem definition

From all of the above it can clearly be seen that the CO2 price is influencing the value of coal and its extraction path.[20] Questions this thesis is dealing with are how the extraction path is affected by the CO2 price, and what the optimal path of using coal is. For many companies, i.e. in coal mining and coal utilizing, this question is essential, since they already face significant changes in profitability. The thesis is aimed at describing the optimal extraction path of exhaustible resource (coal) without and then with CO2 considerations. That will allow to compare and to see the changes in paths. Coal-related industries will be discussed here, but similarly the approaches can be used for other exhaustible fossil fuels.

Since coal is an exhaustible resource, for describing its optimal extraction path we will use the exhaustible resource economic theory, to be more precise, Hotelling's theory, which determines the optimal extraction path of exhaustible resource. Hotelling's rule is one of the required conditions of optimality of the extraction path. The optimal extraction path means that the miner is maximising his profit if he follows this path.

Besides that, we widen the scope of the work and change the condition of maximising the profit and look at the case when a miner aims to prolong the life-time of the mine as much as possible. We will also consider different markets types: competitive and monopoly. For modelling all the scenarios in the mentioned conditions, a single mine which is situated in Germany will be used, and we will assume that all coal is burned at the power plant for production of electricity which belongs to the same company as the mine.

### 1.3. Relevance

We aim to determine how the EU ETS is influencing the extraction path of the coal and its value. This question is very important for the mine owner, as it allows him to choose the right strategy for production and exploitation, depending on the new market conditions with costs for CO2. That is essential for the economic survival of the miner. And for us, the task is therefore to determine the influence of CO2 price on the extraction path of a coal mine. First, we will construct the model without consideration of CO2 price in two different market conditions, and afterwards we include CO2 price considerations. As mentioned before, we will discuss the case when a miner wants to maximize the life-time of the mine. The reasons for that might be to save jobs or governmental directives. This case also will be studied in different markets.

### 1.4. Goals

The goal of the work is to construct simplified models, on the base of Hotelling's rule theory, which will determine the optimal extraction paths of coal and extraction paths leading to maximization of life-time, for one single mine situated in Germany in different market conditions without and with CO2 price consideration. Afterwards, on the base of models including into them numerical data, we aim to show the scale of the CO2 price affecting the extraction path.

### 1.5. Structure

The current chapter, chapter one, gives an introduction into the topic, determines the goals of the paper, explains the motivation of the research done in the work, supports it with topical data.

The second chapter contains the theoretical base for the further research. It describes Hotelling's rule extraction of exhaustible resources, discusses the crucial points of the theory, and gives the basic model of optimal extraction of exhaustible resource.

In the third chapter, models of optimal extraction of coal in different conditions are developed. At the beginning, the models represent the optimal extraction path of competitive market and then monopoly market. Next, cases are discussed in which the company is maximising the life-time of the mine also in two market types. Afterwards, the CO2 price is integrated into the models, and the change in extraction paths is described. At the end, two numerical examples are given, and calculated to find two optimal extraction paths without CO2 and then with it.

The last chapter, chapter four, gives the summary of the whole master thesis and its results.

### Chapter 2: The theory of exhaustible resources

2.1. Overview

This chapter is dedicated to Hotelling's theory itself, since we use it to determine extraction paths of coal. It contains the theoretical background for further models construction, and allows to understand the theory deeper. Next, Hotelling's rule is discussed. Afterwards, we discuss different parameters which can influence the rule, since these considerations are necessary for construction of the models and making appropriate assumptions for them. At the end of this chapter the basic model of optimal extraction of exhaustible resource is given. On the basis of this model, in the following chapter, we will build models with considerations of different market conditions and CO2 price.

The main questions of the economics of exhaustible resources are: what is the optimal rate of exploration of the resource by company, the price path of the exhaustible resource and how does it change through time? These are the questions which we are interested in. And since coal is exhaustible resource, this theory is applicable to our case.

Exhaustible resources are those that are available in fixed quantities. They don't exhibit significant growth or renewal over the time. Coal is exhaustible resource; its amount in deposits is fixed and doesn't grow over time. Pindyck distinguishes between exhaustible and non-renewable resources[21] by noting that, while the latter do not exhibit growth or regeneration, new reserves can be acquired through exploratory effort and discovery.[22] Since the first one is more wide spread, in this work the term exhaustible resources will be used for indication of this type of resources.

In 1914 L. C. Gray dealt with questions of natural resource economics. He examined the supply behaviour over time of an individual extractor who anticipates a sequence of real prices and attempts to maximize discounted profits.[23] Harold Hotelling extended Gray's theory by predicting the sequence of market prices that Gray took as given in his work “The Economics of Exhaustible Resources” in 1931, which then became a seminal paper on the economics of exhaustible resources.[24]

### 2.1.1. Hotelling's rule

Hotelling's rule, as described in his paper entitled “The Economics of Exhaustible Resources”, is an economic theory, pointing out how the prices should behave under a specified (and very restrictive) set of conditions.[25]

It states that competitive mine owners, maximizing the present value of their initial reserves, should extract a quantity such that price of the exhaustible resource rise at the rate of interest.[26] In other words, if we assume that P0 is the initial price of the resource, Pt is the price of resource at some point of time, i is interest rate, then:[27]

(1)Hotelling's rule is based on the following assumptions:[28]

§ the mine owner's objective is to maximize the present value of his current and future profits. This requires that extraction takes place along an efficient path in a competitive industry equilibrium, which implies that all mines are identical in terms of costs and that they are all price takers in a perfect and instantaneous market of information.

§ the mine is perfectly competitive and has no control over the price it receives for its production.

§ mine production is not constrained by existing capacity; it may produce as much or as little as it likes at any time during the life of the mine.

§ the ore deposit has a capitalized value. That is, a copper or gold deposit in the ground is a capital asset to its owner (and society) in the same way as any other production facility. Furthermore, he assumed that the richest and most accessible deposits would be mined first, and that increasing scarcity (after exhaustion of the best mines) would confer capitalized value on inferior deposits, which could then be mined.

§ the resource stock is homogenous and consequently there is no uncertainty about the size, grade and tonnage of the ore deposit. Current and future prices and extraction costs are known. This implies that an ore body has uniform quality or grade throughout and that there is no change in grade of the ore as mining proceeds. Miners and grade control officers, who endeavour to supply the mill only with ore above a certain grade, recognize this fifth assumption to be major departure from reality. The topic of uncertain reserves is discussed in more details in section 2.1.5 of the thesis.

§ The sixth assumption is that the costs of mining or extraction do not change as the orebody is depleted. Again, this assumption does not recognize that all mines face increasing costs as the ores are depleted. Underground mining costs increase as the mining face becomes longer and deeper and moves further away from the shaft system, while in open pit operations haul roads become longer and pits become progressively larger and deeper. A rider to Hotelling's assumption that the marginal unit (standard mining unit) is accessible at the same constant cost, is the assumption that the marginal cost of extraction in this particular case is zero. In addition, it implies that the market price and the rate of extraction are connected by a stable, downward sloping demand curve for the resource.[29] In this constrained model the size of the remaining stock declines without ever being augmented by exploration discoveries. To the topic of cost of extraction is also dedicated the section 2.1.4 of the thesis.

§ The final assumption is that there is no technological improvement during the life of the mine and that no new additions to the resource stock are contributed by exploration. Sections 2.1.7 and 2.1.8. are discussing technological progress and “backstop” resources, which are also connected to technological progress.

Hotelling's model predicts a general rise in commodity prices over time. The model has been used by numerous authors as a useful reference point in discussions on the various dimensions of mineral supply and availability. Among the factors that the model helps introduce are that:[30]

§ Prices are a useful indicator of scarcity, if markets are functioning well (section 2.1.3 is discussing the question of resource scarcity)

§ The effects of exploration and technological innovation significantly and importantly influence mineral availability over time

§ Market structure matters (competition versus monopoly)

§ Mineral resources are not homogeneous

§ Backstop technologies limit the degree to which prices can increase

§ Substitution is an important response to increased scarcity

§ Changes in demand influence price and availability.

In other words, the model provides a vehicle for introducing the various dimensions of mineral supply and scarcity.[31]

But since Hotelling's rule uses a number of assumptions, it might not coincide with reality completely. The next part discusses the empirical validation of Hotelling's rule.

### 2.1.2. Empirical validation of Hotelling's rule

All the assumptions of the model mentioned before diminish the potential value of the application of the model for the miner in the real world. In an attempt to validate Hotelling's rule, much research effort has been directed to empirical testing of that theory. But unfortunately, till now there is no consensus of opinion coming from empirical analysis.[32]

One way of testing Hotelling's rule seems to be clear: collect time-series data on the price of a resource, and see if the proportionate growth rate of the price is equal to r. This was done by Barnett and Morse. They found that resource prices - including iron, copper, silver and timber - fell over time, which was a most disconcerting result for proponents of the standard theory.[33] Other research came up with absolutely different results which could not assess whether the theory is right or wrong.

But the problem is far more difficult than this to settle, and a direct examination of resource prices is not a reasonable way to proceed. The variable Pt in Hotelling's rule is the net price (or rent, or royalty) of the resource, not its market price. Roughly speaking, these are related as follows:

pt= Pt +b (2)

where pt is the gross (or market) price of the extracted resource, Pt is net price of the resource (unextracted), and b - the marginal extraction cost. According to the equation (2), if the marginal cost of extraction is falling, pt might be falling even though Pt is rising. So, evidence of falling market prices cannot, in itself, be regarded as invalidating the Hotelling principle.[34]

This suggests that the right data to use is the resource net price, but this is an unobservable variable as well as i. So it's possible to construct a proxy for it, by subtracting marginal costs from the gross market price to arrive at the net price. This difficult approach was pursued by a number of researchers. Slade made one the earliest studies of this type. She concluded that some resources have U-shaped quadratic price paths, having fallen in the past due to changes in demand or costs of extraction, but later rising.[35] The other study of this type is by Stollery's, which generally supported the Hotelling hypothesis with an example of the nickel market by calculating the resource rent per ton of nickel.[36] Thirdly, Halvorsen and Smith tested the theory and concluded, that “using data for the Canadian metal mining industry, the empirical implications of the theory of exhaustible resources are strongly rejected”.[37]

If it can be shown that prices for exhaustible resource did not rise at the rate i, it does not necessarily mean that Hotelling's rule is not right. There are several circumstances where the resource prices may fall over time even where Hotelling's rule is being followed. For example, a sequence of new mineral discoveries could lead to a downward-sloping path of the resource's net price. Pindyck first demonstrated that in his seminal paper. If the resource extraction takes place in non-competitive markets, the net price will also rise less quickly than the discount rate. And in the presence of technical progress continually reducing extraction costs, the market price may fall over time, thereby contradicting a simple Hotelling rule.[38]

Named before facts show numerous contradictions which researchers face while dealing with Hotelling's rule. But inspite of all these problems, the theory remains appealing. In their conclusion, Devarajan and Fisher state that Hotelling's article is “the sole source of work in a vigorously growing branch of economics”.[39] Solow stated that, “Good theory is usually trying to tell you something, even if it is not the literal truth”.[40] So although the economics of exhaustible resources does not cover the real world of mining and mineral extraction to any large extent, it is still worthwhile to re-examine the theory. Also, many studies relaxed the assumptions of Hotelling, which introduced flexibility and widened the scope of the model applications.[41]

Next some of the most important factors influencing the Hotelling model will be discussed.

As can be clearly seen from formula 1, the main variable is the price of the resource. On what does it depend? Which parameters function is it? As in the thesis will be considered a single mine case, in the discussion we take into consideration mainly single mine factors, which are:

§ scarcity rent ( see section 2.1.3)

§ cost of extraction (see section 2.1.4)

§ uncertain reserves - the amount of the resource left in the mine, discovery of new reserves (see section 2.1.5)

§ demand in the market (see section 2.1.6)

§ technological progress (see section 2.1.7)

§ “backstop” technologies (see section 2.1.8)

§ market structure: competitive (see section 3.3.1) or monopoly (see section 3.3.2)

Now we have a closer look at these parameters, since further description of the scenarios in different markets might require taking some of the facts into consideration.

### 2.1.3. Resource Scarcity

Hotelling's rule is determining the price of exhaustible resource and the extraction path of it. This price, along with other costs, covers resource scarcity, and a large part of the Hotelling's theory is dedicated to resource scarcity. Since it may influence the price of the resource and the extraction path, we discuss it more in details.

Worries about resource scarcity can be traced back to medieval times in Britain, and have surfaced periodically ever since. The scarcity of land was central to the theories of Malthus and other classical economists.

What do we mean by resource scarcity? One use of the term - to be called absolute scarcity - holds that all resources are scarce, as the availability of resources is fixed and finite at any point in time, while the wants which resource use can satisfy are not limited.[42]

But this is not the usual meaning of the term in general discussions about natural resource scarcity. In these cases, scarcity tends to be used to indicate that the natural resource is becoming harder to obtain, and requires more of other resources to obtain it. The relevant costs to include in measures of scarcity are both private and external costs. It is important to recognize that, if private extraction costs are not rising over time, social costs may rise if negative externalities such as environmental degradation or depletion of common property resources are increasing as a consequence of extraction of the natural resource. Thus, a rising opportunity cost of obtaining the resource is an indicator of scarcity - let us call this use of the term relative scarcity.[43]

There are several indicators that one might use to assess the degree of scarcity of particular natural resources, and natural resources in general including physical indicators (such as reserve quantities or reserve-to-consumption ratios), marginal resource extraction cost, marginal exploration and discovery costs, market prices, and resource rents.

Scarcity is concerned with the real opportunity cost of acquiring additional quantities of the resource. This suggests that the marginal extraction cost of obtaining the resource from existing reserves would be an appropriate indicator of scarcity. Unfortunately, no clear inference about scarcity can be drawn from extraction cost data alone. Barnett and Morse, studying marginal resource extraction costs, found no evidence of increasing scarcity, except for forestry.[44]

The most commonly used scarcity indicator is time-series data on real (that is, inflation-adjusted) market prices. It is here that the affinity between tests of scarcity and tests of the Hotelling principle is most apparent. Market price data are readily available, easy to use and, like all asset prices, are forward-looking, to some extent at least. Use of price data has three main problems. First, prices are often distorted as a consequence of taxes, subsidies, exchange controls and other governmental interventions. Reliable measures need to be corrected for such distortions. Secondly, the real price index tends to be very sensitive to the choice of deflator. Should nominal prices be deflated by a retail or wholesale price index (and for which basket of goods), by the GDP deflator, or by some input price index such as manufacturing wages?[45]

The third major problem with resource price data is that market prices do not in general measure the right thing. An ideal price measure would reflect the net price of the resource. Hotelling's rule shows that it rises through time as the resource becomes progressively scarcer. But net resource prices are not directly observed variables, and so it is rather difficult to use them as a basis for empirical analysis.[46]

Stern distinguishes two major concepts of scarcity: exchange scarcity and use scarcity. Rents and prices measure the private exchange scarcity of stocks and commodities, respectively, for those wishing to purchase them. They are not necessarily good measures of scarcity for society as a whole or for resource owners. Though originally intended as an indicator of the classical natural or real price, unit cost can be reinterpreted as an indicator of use scarcity. Unit cost or related measures are possible indicators of use scarcity but are not perfect either as a social scarcity indicator - they do not reflect downstream technical improvements in resource use, availability of substitutes, or, as in the case of price, the impact of environmental damage associated with resource extraction and use on welfare. All individual indicators of scarcity have limitations. There is no “correct” way to measure resource scarcity.[47]

### 2.1.4. Cost of extraction

The cost of extraction of an exhaustible resource is discussed in this section, since these costs, similarly to resource scarcity, are also included in the price of resource. Any changes in them can affect the resource price and the extraction path of it, and further we need to make appropriate assumptions.

A number of researchers have attempted to provide deterministic explanations for deviations from the Hotelling price path based on the properties of the extraction cost function [Solow and Wan (1976), Hanson (1980), and Roumasset, Isaak, and Fesharaki (1983)]. They argue that, holding technology and knowledge of the stock of the resource constant, the most easily accessible sources of the resource will be exploited first. This suggests that extraction costs should rise over time, and this will affect the resource price path [Dasgupta and Heal (1974, 1979)]. However, extraction costs alone-unless changed unexpectedly-do not explain why prices have not risen.[48]

### 2.1.5. Uncertain Reserves

The change in reserves may influence the resource scarcity value, the price of the resource and demand in the market, any of these changes affects the Hotelling's rule. We discuss reserves change to have better understanding of it, as then we need to make an assumption about it to construct the model.

Changes in extraction and exploration technology all affect the size of the stock of proven, or extractible, reserves. This uncertainty about the reserve base contrasts with another underlying assumption in the Hotelling model. Constant real appreciation in exhaustible resource prices is derived in this model because the reserve stock is known with certainty (as are the demand function and extraction costs). In practice, however, reserves are not known with certainty and have increased dramatically over time, often in large, discrete leaps.[49]

The effect of uncertain reserves on the optimal depletion path has been examined in a number of studies. An unanticipated shock to reserves can cause a shift among optimal paths. A sudden, unanticipated increase in proven reserves causes the price trajectory to fall to assure full resource exhaustion. Observed prices in these models fall sharply when the discovery is made.[50]

In addition to unanticipated shocks to the reserve base, a number of these models address the impact of endogenous exploration behaviour on the resource price path. As shown by Arrow and Chang, exploration tends to accelerate as the stock of known reserves declines and the price of the resource rises. With major new discoveries, exploration tends to slow until scarcity again becomes important.[51] The implied price path, therefore, is one that rises and falls, with little apparent trend.

As pointed out by Pindyck, uncertainty about the stock of reserves is consistent with observed price behavior, although such uncertainty does not fully explain that behaviour.[52] Clearly, reserve shocks have played an important role in preventing the Limits to Growth scenario from occurring by consistently raising the size of the resource stock. The timing of reserve discoveries and shifts in price trajectories, however, do not coincide precisely as the theory would predict. Announcements of large new deposits have sometimes caused prices to move, but often there is little immediate response.[53]

In any case, the frequency with which shocks to the reserve base have occurred - either because of luck or because of the endogenous response of enhanced exploration activity - raises an important issue regarding the degree to which these resources really are exhaustible. The steady rise in reserves, despite growing demand, which depict a steady upward trend in consumption), may argue for decreasing scarcity value of the resource over time.[54]

D.B. Reynolds (1999) shows 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.[55]

### 2.1.6. Demand

Similarly to previously discussed factors, any change in demand can cause fluctuations in price of the resource. Besides that, demand is closely related to the market conditions factor, and they both play roles in Hotelling's theory. Now we discuss demand, to become more familiar with it to be able to manipulate with it while constructing models. Market conditions will be discussed in the following chapter during the model construction.

Technical change affecting the demand for a resource also may be an important factor in the observed failure of the Hotelling model. A key assumption of the model is that demand for the resource is known and predictable. In reality, however, dramatic changes in use patterns, the availability of alternatives, and variations in resource use intensity have caused frequent shifts in the demand for the resources. For example, the discovery of semiconductors and silicon chips significantly reduced the demand for copper wiring. Increased energy efficiency in automobiles, including substitution of aluminium and plastic for steel, had a direct impact on iron and petroleum demand.[56]

These technological shocks result, in part, from a direct response to perceived shortages-reflected in rising prices and from spin-off discoveries in other applications. In the short-run, most resource demand is highly inelastic. Over the longer-term, however, substitutes tend to develop that allow much greater substitutability. Often, the emergence of the substitutes leads to relatively sudden shifts in demand when the product appears, typically exceeding expectations of resource producers. When these shifts occur, the expected consumption path is altered, and the optimal depletion path changes.[57]

Such changes in demand can lead to consistent errors in the estimation of demand. Adjustments by producers to those errors then can affect the observed price path for resources.

Relatively simple models of resource depletion have been developed for the case where alternative technologies exist. In the simplest form, a "backstop" technology is assumed to exist in perfectly elastic supply at some price.[58]

A more complicated version of the process considers the optimal depletion problem when the alternative technology is endogenously determined. The miner must then choose a price and production schedule that maximizes profits taking into account the effect that the price level will have on encouraging alternatives. This approach, however, continues to predict monotonically rising resource prices, which is not always true, as explained before.[59]

### 2.1.7. Technological Progress and Stock Effects

The topic of technological progress in the context of extraction of exhaustible resources is closely connected to demand, cost of extraction, reserves uncertainty and “backstop resources”. Certainly, this parameter should be considered in models construction.

C.-Y. Cynthia Lin and G. Wagner (2007) claim that “stock effects increase extraction costs and are consistent with rising resource prices, technological progress lowers extraction costs and causes prices to decline. Slade accounts for both of these effects in her theoretical model explaining U-shaped price trends”.[60]

C.-Y. Cynthia Lin and G. Wagner argue, that “most resource prices have not increased between 1970 and 2004, largely due to technological innovation in both the supply side and the demand side was able to offset stock effects”.[61]

In comparison to technological progress, very similar effects on the extraction of exhaustible resources have the “backstop” resources.

### 2.1.8. “Backstop” Resources

As already mentioned before, “backstop” resources can affect the price and extraction path of the resource, and in whole can make the resource absolutely non competitive, therefore, we discuss at this parameter.

Suppose there is some other resource which is a perfect substitute for the exhaustible resource in question. Suppose also that this alternative, or “backstop” resource can be supplied at some high cost but in fairly large quantities so that it is inexhaustible for all practical purposes. Since the backstop has a virtually unlimited supply, its price will be just sufficient to cover its marginal extraction cost.[62]

Implicitly, backstop technologies are assumed to be renewable. Ethanol fuel from renewable corn and sugar is frequently seen as a backstop for petroleum. However, a backstop technology can itself be exhaustible. For example, coal-based electric transportation would itself be an exhaustible backstop for finite petroleum resources.[63]

In the presence of a backstop, there is a ceiling on the net price of the exhaustible resource. In theory, as soon as the price of the exhaustible resource just exceeds the price of the backstop, the former will be priced out of the market and the demand would be entirely satisfied by the latter resource. The net price of the exhaustible resource will rise at the interest rate till it is completely exhausted. Exactly at that instant the net price would be equal to the price of the backstop and production would shift from the exhaustible to the backstop resource.[64]

After the discussion of the main factors playing important roles in the optimal extraction of exhaustible resources, we describe the basic model of optimal extraction of exhaustible resource which afterwards is a base for construction of models for more complicated scenarios.

### 2.2. Basic model of optimal extraction of exhaustible resource

In the thesis, with the help of Hotelling's rule, optimal extraction models for different scenarios are constructed and then CO2 price considerations are included in these models. Afterwards, in the numerical example, we want to show the scale of affect of the CO2 price. To proceed it, we describe models in mathematical way.

For construction of models in different market conditions in the following parts of the work, we describe the basic model of optimal extraction of an exhaustible resource[65], which leads to Hotelling's rule.

For now assume that the extraction costs are not playing any role and are neglected, and P is defined as the net price of the exhaustible resource, that is the price after deduction of the cost of extraction. Let P(R) denote the inverse demand function for the resource, indicating that the resource net price is a function of the quantity extracted, R. The social utility, N(R),[66] from consuming a quantity R of the resource may be defined as

(3)

which is illustrated by the shaded area in 2, where the demand curve is non-linear.

By differentiating the total utility with respect to R, the rate of the resource extraction and use is obtained

(4)

which states that the marginal social utility of resource use equals the net price of the resource.

Assume that the intertemporal social welfare function is utilitarian. Future utility is discounted at the instantaneous social utility interest rate i. Then the value of social welfare, W, over an interval of time from period 0 to period T can be expressed as

(5)

The problem is to make social-welfare-maximizing choices of a quantity of the resource extracted in each period (Rt, for t=0 to t=T), and the optimal point in time (T) when the resource will be completely extracted, which is subject to a constraint

(6)

where S0 is the total initial stock of the exhaustible resource.[67]

Then the remaining stock of the natural resource at time t, St, is defined as

(7)

then by differentiation with respect to time we obtain ,where

=dS/dt (8)

the rate of change of the remaining resource stock with respect to time.

So the dynamic optimization problem involves the choice of a path of resource extraction Rt over the interval t=0 to t=T that satisfies the resource stock constraint and which maximises social welfare, W. To denote it mathematically:

(9)

subject to .[68]

Common sense suggests one condition that must be satisfied if W is to be maximized. Rt must be chosen such that the discounted marginal utility is equal at each moment in time, that is,

(10)

If the discounted marginal utilities from resource extraction were not equal in every period, then total welfare W could be increased by shifting some extraction from a period with a relatively low discounted marginal utility to a period with relatively high discounted marginal utility. Rearranging the path of extraction in this way would raise W. It must, therefore, be the case that welfare can only be maximised when discounted marginal utilities are equal, means constant, too. This requirement is equivalent to the requirement that the discounted net price is constant as well. Mathematically

(11)

Rearranging this condition leads to

(1)

Equation (1) is the Hotelling efficiency rule. It states that the net price or royalty Pt of a exhaustible resource should rise at a rate equal to the social utility interest rate, i, if the social value of the resource is to be maximised.[69]

Now we go deeper into details of the model and determine more parameters of the model, since we will need them to construct models for different market conditions and include CO2 price considerations.

To make it possible to determine the optimal value of T, the optimal rate of extraction at each point in time, the optimal initial value of the resource net price and other parameters, one additional piece of information is necessary: to determine the form of the resource demand function. Suppose that the resource demand function can be written as

(12)

which is illustrated in 2. As mentioned before, the relationship in this between P and R is non-linear, and it exhibits zero demand at some finite price level (P(RT) = K). K is the so-called choke price for this resource, meaning that the demand for the resource is driven to zero or is “choked off” at this price. At the choke price people using the services of this resource would switch demand to some alternative, substitute, exhaustible resource, or to an alternative final product not using that resource as an input. a is a constant positive number in this model which depends on the elasticity of demand.[70]

The knowledge of a particular resource demand function, Hotelling's efficiency rule, an initial value for the resource stock and a final value for the resource stock allows determining the optimal value of T, optimal rate of extraction at each point in time, the optimal initial value of the resource net price. The further solution will depend on the resource demand function and in case of different function the result may be different.

An optimal solution must have the property that the stock goes to zero at exactly the time that demand goes to zero. In order for demand to be zero at time T, the net price must reach the choke price at time T, that means PT = K. This together with equation (10) implies

(13)

From equations (1) and (12) we get:

(14)

After substituting K we obtain

(15)

Now it is possible to find an expression for the rate at which the resource should be extracted along the optimal path. To find the optimal time period, T, over which extraction should take place, recall that the fixed stock constraint is

(16)

and so by substitution for Rt from equation (15) we obtain

(17)

(18)

l

or

(19)

Then we determine the initial royalty level, P0, from equation 12:

(20)

To obtain an expression for the resource royalty at time t, substitute equation 1 into the expression just derived for the initial royalty level to obtain the required condition:[71]

(21)

The optimal initial extraction level is, from the equation 14,

(22)

This model describes the basic model of optimal extraction of exhaustible resource. In the next chapter, more detailed assumptions are made, and the basic model is used to construct models for different scenarios.

### Chapter 3: Model Construction

### 3.1. Assumptions

For constructing models on efficient extraction of coal in different markets conditions pursuing different aims and finding out the impact of the CO2 price, we need to simplify real conditions.

In our model we look at the single mine, that has a fixed known amount of the resource in the deposit (S0). The mine is situated in Germany. As is usual, in Germany, the extracted coal is used to produce electricity. It is assumed that all extracted coal from the mine is burned at the coal-fired power plant. Besides that, assume that the owner of the mine and of the plant is one company. The operator of the electricity plant is obliged to participate in The EU ETS.

According to the Kyoto Protocol, the Commission has no view on what the price of allowances for CO2 in the EU ETS should be. The price is a function of supply and demand as in any other free market. Market intermediaries quote prices for allowances offered or bid for. The Commission does not intervene in the allowance market. In the case of distortions, the competition law is ruling the market.[72]

As follows from the EU ETS, the CO2 price is an exogenous variable, the company cannot influence it. For the beginning of the model construction it is assumed that this price is constant and equal to C. This is not describing the reality, where the price is volatile, but for the beginning it makes the basic model construction easier and more transparent. Later, the model with CO2 price change will be discussed.

Since Hotelling's rule constitutes the base for our model, it is necessary to mention the interest rate which plays a major role in Hotelling's theory. There are two types of interest rates: social and private. In the modelling we use one term of interest rate, and assume that it is the social discount rate which in the modelling is equal to the private interest rate.

In the previous chapter in the basic model description we used the social welfare and social utility terms. Since we are discussing a single mine case now, in this chapter terms utility and welfare will be used to indicate private utility and welfare.

Besides that, demand determines the extraction rate. Assume that demand is a non-linear inverse function determined by equation (12).

Some of the above mentioned factors influence the optimal extraction condition of Hotelling and are needed for the modelling. For building the model, we assume that the extraction costs (marginal cost of extraction) stay constant over time, since the examined time horizon is short (till 2013). For the same reason, there is no backstop technology in the model, and the technology used stays the same over time, and there are no new deposits found during this period.

### 3.2. CO2 Price Modelling

Since it's assumed that all coal extracted is burned at the plant owned by one company, we consider that these two processes are one. Then, a CO2 price which has to be paid for CO2 emissions is directly linked to the whole process. Depending on the amount extracted and then burned, the plant owner must bear the costs of the CO2 emissions. In a certain sense, this is as if he was paying a tax. The more coal we burn, the more CO2 we produce for which we have to pay a “tax” based on the CO2 price. There are two parameters which determine the “tax”: CO2 price by itself and the amount of coal extracted and burned. To start the analysis we assume that the CO2 price is constant and equal to C. As mentioned before, this allows to construct a simple model. Since the sum we pay for CO2 emissions depends on the amount extracted and then burned, it can be approached as specific tax or tax per unit.[73]

Let Pt be the competitive price of a unit of an unextracted resource (i.e. royalty price) and let pt denote the consumer price of a unit of extracted resource (gross price). For simplicity of exposition, suppose that the average extraction cost is a constant b. Let i denote the competitive rate of interest. Suppose that the “tax”, fixed CO2 price, is a specific tax Ct on the unit of carbon dioxide emitted.[74] Then mathematically we define:

(23)

The effect of the “tax” is formally identical to the effect of a once for all increase in average extraction cost. In its impact it is analogous to the impact of a constant specific sales tax. The “tax” certainly introduces a distortion, and in particular, results in higher initial consumer prices and consequently greater conservation. It is also clear that the effect of the CO2 price is to reduce the value of the deposit, since p0<p*0, where p0 is the initial price with the tax in force, and p*0 is the initial royalty price of the resource in the absence of the tax.[75]

Using the basic model of optimal exhaustible resource extraction and the approach of integrating the CO2 price into the model we construct other models of extraction of exhaustible resource in different conditions.

In the next part, the basic model in different market conditions and pursuing different aims is described. Afterwards, the approach described in this part will be integrated into models to see the influence of CO2 prices on the extraction path.

### 3.3. Basic model without CO2 price considerations

3.3.1. Basic model in a competitive market

Using the assumptions mentioned before, the basic model for optimal extraction of an exhaustible resource is constructed. In this part, two different conditions in a competitive market are analysed: when the company is maximizing its profit, and when the company tries to prolong the life-time of the mine.

The perfectly competitive market means that each company and consumer is so small that its input and output decisions have no significant impact on the prices in the market.[76] Hence, the company only can decide on the choice of company.

Suppose there are several competitive companies in the market, one of which is our. Use the subscript j to denote out company. Since all companies are in a competitive market, they have to accept the fixed price at any moment in time, and market royalty will be identical over companies. Given market royalty Pt, each company chooses an amount to extract, Rt, to maximize its profits.[77]

Mathematically, the company's objective is to maximize

(24)

where U is the company's profit and i is the market interest rate. There is a constraint on the stock, which means that the company cannot extract more than the fixed initial stock S0 over the whole time horizon. The profit maximizing extraction path is obtained when the company selects an extraction amount Rt at each time, t=0 to t=T, so that its discounted marginal profit will be the same at any moment in time t, that is,

(25)

for t=0 to t=T, where MU is the company's marginal profit function. If discounted marginal profits were not the same over time, total profits could be increased by switching extraction between time periods so that more was extracted when discounted profits were high and less when they were low. The result that the discounted marginal profit is the same at any moment in time implies that[78]

or (26)

This is Hotelling's efficiency rule, which is the required condition for profit maximisation in the competitive market, so that the market net price of the resource must grow over time at the rate i.[79]

The results of solving equations for royalty, extraction, and depletion time are shown in the Table 2.[80]

Initial point (t=0)

Interim point (t=t)

Final point (t=T)

Royalty, P

Extraction, R

Depletion time

### Table 2 - Royalty, extraction rate and depletion time in competitive market. Source: Perman et al. (2003), p. 516.

As mentioned before, Pt stands for the net price (royalty) of the exhaustible resource with fixed stock S0, Rt represents the total extraction of the resource at time t, i indicates the interest rate, T denotes the exhaustion time of the natural resource, K and a are fixed parameters.[81]

3 portrays the solution of the optimal extraction path of an exhaustible resource in a competitive market.

3 - Graphical representation of solutions to the optimal resource depletion model in competitive market. Source: Perman et al. (2003), page 517.

In the upper right quadrant, the net price is shown rising exponentially at the interest rate, i, thereby satisfying Hotelling's rule. The upper left quadrant shows the resource demand curve with a choke price K. The lower left quadrant gives the optimal extraction path of the exhaustible resource, which, in case of a competitive market, is represented by a linearly declining function of time.

At initial moment of time the net price of the resource is at P0 and then it grows until it reaches the choke price K at time T. At this moment, demand for the resource goes to zero, and the accumulated extraction of the resource (the shaded area beneath the extraction path) is exactly equal to the total initial resource stock, S0. The lower right quadrant maps the time axes by 45° line.

Next, the model for maximizing profit in monopoly conditions is described, and both models are compared in order to see the differences between optimal extraction paths in two different market conditions.

### 3.3.2. Basic model in a monopoly market

In this part, the company is seen as a single extractor of the exhaustible resource, and, depending on its output choices, it can influence the market prices.

Since the company controls all deposits of the exhaustible resource, it chooses the extraction path that maximizes its discounted profit over time. Thus, it selects the net price Pt (or royalty) and chooses the output Rt so as to maximize

(27)

subject to

(28)

where Ut = P(Rt)Rt.[82]

For the same reason as in the case of perfect competition, the profit-maximizing solution is obtained by choosing a path for R such that the discounted marginal profit will be the same at any time

(29)

that is

(30)

### Chapter 4: Conclusion

The new economic environment in the energy market, caused by emissions trade, raises one of the main questions of how the CO2 price will influence the extraction path of exhaustible resources. This thesis was dedicated to this question.

For answering it, Hotelling's theory of optimal extraction of exhaustible resources, which determines how the net price for the resource should grow, was used. The theory uses a number of strict assumptions. On the base of this theory and its basic models other models were constructed.

This thesis described optimal extraction paths for coal, but approaches used here and main conclusions can be used for determining extraction paths of many other exhaustible resources.

In whole, the thesis studied eight different scenarios. In all of them, the case of a single mine situated in Germany was used. At the beginning, the case where company's target is to maximise its profit in two different markets, in perfect competition and then in monopoly was discussed. Afterwards, were developed two models were developed to determine extraction paths of the company, when its target is to maximise the life-time of the mine, also in two market conditions.

Next, into developed four models were included considerations of the CO2 price. All models used a number of assumptions, such as the extraction costs and interest rate are constant over time; the demand is determined by non-linear inverse curve; there is no backstop technology. Also at the beginning the CO2 price is assumed to be constant; then this assumption is changed to the one which says that the price is growing linearly.

The models, determining extraction paths for a company which is maximising the profit, showed that in a competitive market the resource will be extracted faster than in a monopoly market. This is caused by the fact that in a monopoly market the company puts a higher price for the resource at the beginning. That reduces the demand for it and results in a lower extraction rate. But in both markets the overall tendency is similar: the price is rising and the extraction rate is decreasing over time.

In the case where the company is maximising the life-time of the mine, we came to the conclusion that in both market types the extraction rate will stay the same over time, and this rate has to satisfy the condition that the company covers the constant extraction costs and opportunity costs. But in monopoly market the owner will put a higher price and that results in lower demand, and consequently has a lower rate of extraction. That means that in a monopoly market, time till complete exhaustion of the exhaustible resource will be longer.

Lastly, all models were examined with the CO2 price, which was approached in the model as a specific tax, since the company has to pay for each unit of produced CO2. The inclusion of the CO2 price into models gave the following results. In models where the company is maximising its profit, the CO2 price has a similar effect as an increase in extraction costs. That means that since the initial price will be higher the demand will decrease and the extraction rate too, but then net price of the resource won't grow as steep as before. Time till complete depletion is longer. That is true for both markets, and taking into consideration the fact that monopoly market conditions have the resource saving effect in comparison to competitive market, the inclusion of CO2 prices results in an even higher saving rate of the resource in monopoly market.

The discussion of the life-time maximising case with consideration of the CO2 price resulted in the conclusion that costs for emissions in both market conditions will raise the prices for the resource and decrease the extraction rate. That will lead to longer time period in which the resource is extracted.

The thesis is not aimed at describing CO2 price changes, but since the assumption that CO2 price is constant does not depict the reality, we briefly discussed the situation when the price for CO2 is changing, and growing linearly. And as the CO2 price in comparison to the price of the resource is sufficiently large, as in cases before, it will reduce extraction at all times.

At the end, using the models developed in the thesis, two numerical examples were calculated. They allow us clearly to see the scale of the CO2 price impact on the extraction paths of coal. The examples were given for the conditions when the company is maximising its profit. These numerical examples showed that change in price of the resource caused by the CO2 price is significant for the considered assumptions and is reducing the demand. Another conclusion which follows from calculations is that the initial price and amount extracted, which were assumed for the calculation, were not optimal for the assumed market conditions and demand function. In case of perfect competition market the price is too high and the extraction rate is low. And in monopoly market on the contrary the price should be much higher than the assumed and the output lower.

In conclusion, the CO2 price is significantly increasing prices of exhaustible resources which are sources of CO2 emissions. That leads to a decrease in demand and lower extraction rates. If the prices for CO2 rise too high, the price for the resource could reach its choke price and then the resource won't be extracted any more. That means that mining and power producing companies for their survival should somehow mitigate the consequences of the introduction of carbon dioxide market. Some of the measures could become carbon capture storages (CCS), which will delete the costs for emissions for companies at all, but add costs for the storage maintenance, or to use the resource for several purposes at once. For example to use coal not only for electricity production, but also heat or gasoline. That will allow to distribute the costs for emissions between different products, then the impact on the price of one product won't be such significant.

The topic of the CO2 price impact is very urgent nowadays, and this thesis was motivated by it. In the discussion, many questions were raised, which can become the reason for other research works, especially the topic of the mitigation of the economic impacts from emissions trading for power-producing companies.

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