Optimal extraction of the Dutch natural gas reserves : the development of an optimal extraction model with historical prices

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University of Amsterdam Faculty of Economics and Business The Amsterdam School of Economics

Optimal Extraction of the Dutch Natural Gas

Reserves; The Development of an Optimal

Extraction Model With Historical Prices

A thesis submitted to The Amsterdam School of Economics in partial fulfillment of the requirements

for the degree

MASTER OF SCIENCE in

ECONOMICS

by

R.W.B. VAN DER WEIJST Amsterdam, The Netherlands

April 2016

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Preface

The state of the economy caught my eye when I first started following the news at a young age. This interest led to my decision to gain deeper knowledge in this field by undertaking a master’s in economics at the University of Amsterdam. My passion for mathematics and interest in economics have been combined in this thesis, which was written to fulfill the requirements of this program. It gave me the unique chance to conduct research in the field of optimal re-source extraction. I would like to take this opportunity to thank my supervisor, Dr. E.W.M.T. Westerhout, for his feedback on my draft. I would also like to thank Dr. W.E. Romp for our conversations and for the suggestions that he offered that helped me to choose this topic. Fur-thermore, I would like to thank all of my family and friends for their support.

R.W.B. van der Weijst 17 April 2016

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List of Symbols

α Factor in the used adjusted discounted cost function

AC Adjusted discounted cost function

C Discounted cost function

∆ Time step (equal to 1 year in the numerical part of this paper)

f Discounted profit function

γ Time distribution in the underlying continuous functions

Oe Function of the observed extraction rate in the past, Oe: t → R Os Function of the observed amount of stock left in the past Os: t → R ω Underlying continuous function of the extraction rate

p Discounted average profit per unit gas extracted Π Total discounted government income

q Extraction rate

Q Total amount of natural gas extracted

ri The long-term interest rates in year i as explained in the data section s Stock

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Abstract

Hundreds of gas fields have been discovered in the Dutch subsurface, including the Groningen gas field, the largest gas field in Europe. The income for the Dutch State obtained from the extraction of natural gas is a substantial part of the total income of the government. The natural gas fields have contributed to between 2% to over 18% of the total Dutch State income since 1973. This income depends on several factors, such as the price of natural gas, production costs, taxing policy, and also the extraction policy. The extraction policy determines the extraction rate of natural gas and has been called sub-optimal by various researchers and institutions due to the irrational combination of limited market development and resource overhang, i.e. the abundant stock of natural gas that was discovered in 1959.

This paper aims to uncover what the optimal natural gas extraction rate in the Netherlands would have been after 1973, such that state income from gas reserves would have been maximized. Furthermore, this paper investigates how this optimal extraction rate compares to rates obtained by strategies that use known historical profits per cubic meter of gas. An optimal extraction model based on the theory of Harold Hotelling is developed to answer the research question. The model uses the observed income of the Dutch State per cubic meter of natural gas to determine what the optimal extraction rate would have been. The income for the Dutch State is numerically optimized using this developed model. Certain extraction strategies are tested to identify what the total government income would have been if they had been implemented. A numerical solution for the extraction rate optimization problem has been determined based on the historical income of the Dutch State per cubic meter of natural gas. The solution illustrates a significant difference between the observed extraction rate and the optimal extraction rate. The tested optimal extraction strategies suggest that certain strategies would have enlarged the total income yielded from natural gas for the Dutch State, especially under special circumstances. This could be of importance in determining the future extraction policy of the Netherlands.

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Introduction

The Netherlands has extracted large quantities of natural gas since the discovery of the Gronin-gen gas field, the largest gas field in Europe, in 1959. The gas reserves have influenced the GDP of the Netherlands and have impacted its fiscal policy. The share of state income yielded from the natural gas fields has varied between 2% to over 18% since 1973. Between 2003 and 2014, the percentage of total state income yielded from the fields was between 5% and 10%. The value of these gas reserves is immense, and extraction rate policy determines how much money these reserves yield.

The extraction rate of the Dutch gas reserves has been called sub-optimal by various researchers and institutions because the government has aimed to restrict the supply in the past. Research has been conducted on the expenditure of income from natural gas and on the future optimal extraction policy. However, no research has been conducted on what the optimal extraction rate would have been and what the impact of such a rate would have been on the Dutch state’s income.

In the literature, there are various optimal extraction models that forecast the optimal extraction rate of natural resources. There are models for different markets, such as monopolies, oligopolies, and perfect competition. Further, advanced models have been developed that include the number of wells, technology developments, and variable production costs. All of these optimal extraction models focus on the future and deal with unknown prices. The outcomes of these models can be implemented in practice to determine production levels. This paper aims to combine the theory of non-renewable resource extraction optimization with information on historical natural gas prices and knowledge related to the historical gas market. The model developed in this paper contrasts with the models in the literature in that it uses historical prices rather than unknown future prices. Further, the objective of this research is not to identify an optimal extraction rate for the government to implement now. Rather, this paper aims to answer the question: What would have been the optimal extraction rate of natural gas in the Netherlands after 1973 such that State income from the gas reserves would have been maximized? Furthermore, this paper investigates how this optimal extraction rate compares to rates obtained by strategies that use known historical profits per cubic meter of gas.

The model used to determine the optimal extraction rate in this paper is based on the theory of Harold Hotelling (Hotelling, 1931). The model uses the observed income for the Dutch state per cubic meter of natural gas to determine what the optimal extraction rate would have been. In this paper, the assumptions about the demand for natural gas are based on historical information related to the natural gas market. The income for the Dutch state was numerically optimized using the developed model.

Policy makers can not predict future natural gas prices exactly. Therefore, drawing a conclusion about the observed policy based on the optimal rate, given observed prices, is not possible.

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Therefore, this paper tests a selection of strategies that could have been implemented by policy makers, e.g. extraction strategies not based on knowledge of future factors such as future prices. The extraction rates obtained using these strategies were determined and compared with the optimal extraction rate.

Chapter 1 of this paper focuses on the role of the Dutch state in the extraction of natural gas and the historical gas market. Chapter 2.1 outlines the current models in the literature. In Section 2.2, an optimal extraction model for Dutch natural gas resources is developed. Section 2.3 describes extraction rate strategies and Section 2.4 presents the data. Chapter 3 presents the numerical results and discusses the implications of the results. It contains solutions regarding the question of what the optimal extraction rate would have been based on different assumptions. The results related to the strategies discussed in Chapter 2 are compared with the optimal extraction rate, to answer the second part of the research question. The last chapter contains the conclusion and discusses the limitations of the study. It also offers suggestions for further research.

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1

Natural Gas Market and Political Policy

1.1 Dutch Gas Reserves

In Slochteren, the Netherlands, the Groningen natural gas field was discovered by the ’Neder-landse Aardolie Maatschappij B.V.’ (NAM) on July 22, 1959. The Groningen gas field is the largest gas field in Western Europe. The gas in the Groningen field is very concentrated com-pared with gas from other fields with the same amount of gas. It is also a special gas field because production costs associated with it are very low in comparison with those of other gas fields (Correlj´e, 1998). The total amount of gas in the Dutch subsurface and the Groningen gas field in particular cannot be precisely determined; estimates have varied over time. At the time of discovery, it was expected that the field contained at least 150 billion Sm3 and at most 400 billion Sm3 (“Alles over de gasbel”, 2009). In 1967, this estimate increased to 2,000 billion Sm3 and in 2010, it increased again to 2,750 Sm3.

Besides the Groningen gas field, the Netherlands has many other small on and offshore natural gas fields. Exploitation of these fields is relatively difficult and expensive in comparison with that of the Groningen gas field. Further, the composition of natural gas from different gas fields can differ, as can the pressure. Facilities that use gas, such as boiler or electricity plants, are often only able to deal with one kind of gas composition. In the Netherlands, the equipment in such plants is made for gas with a composition and pressure similar to that of the Groningen gas field. Therefore, some smaller fields with natural gas of different compositions are sold to other parties, so that the facilities do not need to be changed, e.g. gas from the Annerveen field is exclusively sold to Italy (Correl´e, 1998). Natural gas with different compositions is also sometimes blended with Groningen gas so that it can be used by the same equipment. The differences between the Groningen gas field and the small gas fields have led to the government putting a special policy in place for the smaller fields, the so-called Small Fields Policy. Nitrogen plants can convert gas with a caloric value different from that of the gas from the Groningen field to make it suitable for use in the Netherlands.

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1.2 Organizational Structure of the Dutch Gas System

1.2.1 Public-Private Partnership

The Dutch government has chosen to regulate the gas market in order to optimize profits from the gas fields for the Dutch state. Specifically, it has set up a public-private partnership. Her-man de Jong, director of the N.W. Posthumus Institute, stated that the Dutch public-private partnership, as described in the memo of De Pous (De Pous, 1962), was a false front to disguise the government stake. He states that Shell and Esso did not want the so-called ’sheikh effect’, which means that the government has a large stake in the extraction and sale of natural gas. If the private companies allowed the Dutch government to have an important stake, their own stakes would be negatively affected in oil-producing countries in the Middle East (“Alles over de gasbel”, 2009). However, other options were seen as less valuable for the Dutch state since this option would allow the price of gas to be higher than in a competitive market. Furthermore, this partnership ensured that private companies would not receive high profits.

The N.V. Nederlandse Gasunie (Gasunie) was founded in 1963 and developed the infrastructure to transport natural gas to Dutch households. Initially, there was an active campaign to stimu-late the demand for natural gas and to connect households to the natural gas system for cooking and heating. Later, this plan shifted to limiting market development, especially abroad, to guarantee supply to Dutch users. This limited market development was viewed as economically sub-optimal because of the simultaneous existence of a resource overhang (Odell, 1995, 1997). The Gasunie was responsible for coordinating the commercialization of the gas reserves. The Gasunie was given the right of ’first refusal’ and was therefore able to purchase all gas produced in the Dutch onshore and offshore fields (Correlj´e, 2000). The organization had to make sure that the sales and the exploration were in harmony, as stated in the Nota De Pous. Control over supply was seen as the responsibility of the state. This was not only to limit market development in the long run, but also to avoid disruptions of the energy market (see Correlj´e, 1998).

After the discovery of the Groningen gas field, there was a monopoly on the gas supply. However, this monopoly slowly disappeared, as elaborated in the section on the historical development of the gas market.

1.2.2 Small Fields Policy

The production costs associated with the small fields are much higher than the costs associated with the Groningen gas field. Extracting gas from the small fields was often barely profitable. Therefore, the government had to decide what to do with the small fields. One possibility was to not start extracting from these fields until after the Groningen gas field was totally spent. In the meantime, the technology could develop and make the extraction of gas from these fields more profitable. The low profitability in combination with the different composition of the gas from the small fields led to the Dutch State implementing the Small Fields Policy in 1974 (GasTerra,n.d.-c). The government decided to begin extracting gas from the small fields in order to blend this gas with the Groningen gas. The essential part of this policy is that it guarantees a buyer (Gasunie) for extracted gas from these fields. Since the implementation of this policy, gas from the Groningen gas field has been extracted in a flexible manner to deal with demand fluctuations and so that gas from the small fields can be completely extracted (MEZ, 2008).

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In the winter, demand for natural gas is higher, especially when temperatures are very low. Nowadays, demand can be so high that extraction may not meet demand. In summer, production is higher than demand to compensate for the shortage in winter. The extra gas is then stored in fields in Langelo and Grijpskerk, and when demand is high in winter, it is sold. Due to the Small Fields Policy, gas from small fields can continue to be extracted in summer when demand is lower. Extraction of gas from the Groningen field is reduced during the summer (Correlj´e, 1998).

1.2.3 Alternative Policies

The Dutch government had to decide how to convert the gas resources into cash and they decided to go for a public-private partnership. Total government control would have been another possibility. The advantage would have been that no profit could flow to parties other than the state. However, it is not known whether this approach would have led to as efficient and profitable exploration of the reserves as an approach geared toward driving profit for private firms. Researchers also claim that this option would have negatively affected the stake of Dutch oil-producing companies in the Middle East (“Alles over de gasbel”, 2009). Another alternative would have been to leave the exploration of the reserves to the free market. However, high initial investment costs for setting up the gas system and building a distribution network could have slowed down the development of the gas market.

Besides using alternative policies for extraction and distribution, the commercialization of the gas reserves could have been organized differently as well. The objectives of the Gasunie could have been different. One objective could have been guaranteeing supply for the future. The organization could have changed the restriction on gas supply to maintain sufficient proven reserves to satisfy Dutch consumption needs for the upcoming years.

The possibility of an alternative gas market where the production rate depends on the price of natural gas instead of on the memos predetermined by the Dutch government is crucial to the model in this thesis. The long-term exporting contracts made a stable production rate possible. The short-term contracts could have been signed when gas prices were relatively high and more gas could have been imported when the gas price was low. The possibility of this more ’flexible’ gas market, where it is easy to import and export gas, depends heavily on the international gas market. The historical development of this market has changed a great deal over time and is described in Section 1.5.

1.3 Income for the Dutch State

The Dutch state generates income from the natural gas fields via three different channels. It receives money from the dividends of companies in which it has a stake, corporate tax, and concessions. The last channel is growing rapidly; in 2000, the income from concessions was 1.9 billion euro and in 2012, it was 10.4 billion euro (Centraal Bureau voor de Statistiek [CBS], 2013). In 2012, about 15.8% of income from natural gas fields for the Dutch state came from dividends, about 71.9% came from concessions, and about 12.4% came from corporate tax (CBS, 2013). The composition of the income of the state has changed over time as the structure of the sector has changed; new fields have been exploited and the production and profitability of gas fields have changed (see Figure 1.1, which contains the composition of income between 1996 and 2010 [Rossum & Swertz, 2010]).

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Figure 1.1: State Income of Natural Gas by Income Type

Not only has the decomposition of income changed, but the total amount of income for the Dutch state from natural gas extraction has significantly changed over time as well, as illustrated in Figure 1.2. Figure 1.2 also illustrates the Dutch state’s income from natural gas as a percentage of total income, which has varied significantly as well.

There is a special law for the Groningen gas field, the MOR, Meeropbrengst regeling Groningen. This law ensures that 85-95% of the revenue surplus goes to the State instead of 70% (Van Der Hoeven, 2008). It was implemented so that the Dutch State would profit from the extraordinarily low production costs associated with this field.

1.4 Pricing Policy

The Groningen gas field was discovered under the terms of unified concession and was only operated by the NAM with a state interest. After the discovery of the Groningen gas field, the price had to be determined. Because of its near-perfect monopoly, the Dutch State could determine the pricing policy. However, demand for natural gas would depend on the exogenous prices of alternatives, e.g. coal and oil. Esso suggested linking the price of natural gas to substitutes, such as gas oil and fuel oil (Heren, 1999). The Dutch government decided to go for this plan. As a consequence, the price of Groningen gas exceeded production costs. This price determination also ensured that the price was not much lower than the price of alternatives. Further, it ensured that consumers did not have to pay more for gas than for substitute fuels. Odell (1975) states that Europe was used to high-priced energy in the 1960s due to the protected

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Figure 1.2: State Income of Natural Gas

coal market and due to the oil product market being not fully competitive. Both factors made high profits possible. This pricing policy could be maintained as long as the regional monopoly on the production of natural gas was maintained. There were no alternative sources of natural gas within a wide area in Europe. Odell (1969) argues that the lack of alternative sources combined with the unified concession created a near-perfect monopoly on the gas market. The Gasunie was officially made responsible for price determination. Via this system, the Dutch State has no direct influence on pricing policy. However, in practice, the government is always consulted on strategic issues. It initiates discussions if an issue is in the national interest, even though the Ministry of Economic Affairs (EZ) officially confined its responsibility for approving the decisions for pricing (Correlj´e & Odell, 2000).

Over time, the price of gas has become more and more determined by the price developments on the virtual marketplaces, such as the National Balancing Point (NBP) and the Title Transfer Facility (TTF). On virtual marketplaces, ownership of natural gas can change several times before it is used. However, prices for exported Dutch gas are still determined mainly by long-term contracts (GasTerra, n.d.-b).

1.5 Historical Development of the Gas Market

1.5.1 The Gas Market after the Discovery of the Groningen Gas Field New sources of natural gas in Eurasia were exploited soon after the monopoly was exercised. In the United Kingdom, offshore gas fields were found and exploited, and natural gas from the

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Union of Soviet Socialist Republics became available in parts of Europe via pipelines. Liquid natural gas from Africa became available as well. Figure 4.1 in the appendix shows the estimated demand for natural gas in the 1970s, as estimated in 1969 by the NAM. The prices in Guilder cents per m3 _{are provided in the circles. The size of each circle represents the demand. The} demand for the largest circles was 3.5 billion m3 (Northern Germany), 4 billion Sm3 (United Kingdom), and 5 billion m3 (Ruhr region). The demand from Central Germany and Northern Italy was 2.5 billion Sm3 and the demand from Belgium was 3.0 billion m3_{. Demand from} Northern France was 2 billion m3 and demand for all of the smallest circles was estimated to be 1.5 billion m3.

In his memo, De Pous mentioned that due to new technology, such as nuclear power, it was questionable whether the relative value of natural gas is preserved on a long-term basis. There-fore, he concluded that the extraction of gas from the Groningen gas field had to begin as soon as possible (“Het gasdebat van 1967”, 2015a). Indeed, extraction began soon after the discovery. However, the uncertainty about the relative value in the long term did not lead to an extrac-tion of force. Instead, the actual policy was to limit market development, as Odell (1995) has suggested.

Many selling covenants to export the gas were made at the end of the 1960s. Those contracts often had a duration of 10, 20, or 25 years. Most of the contracts had a clause which allowed the price to be adjusted periodically. Prices were linked to oil prices, but the contracts also often contained a clause that determined a maximum price for the gas. Dutch buyers of gas often dealt with a minimum price, but this was not the case for all export contracts. One exception to this link between oil and gas prices was a contract formed in 1971 with Italy. This contract had a predetermined price that was remarkably lower than prices for imported gas from other countries. Shell’s president, Wagner, later explained that this was due to pressure from the Minister of Foreign Affairs Luns. NATO did not want Italy to be too dependent on gas from the USSR (Correlj´e, 1998).

October 6, 1973 was an important day in the history of the energy markets. Egyptian and Syrian troops attacked Israel in an attempt to regain the land occupied by Israel since 1967. On October 24, 1973, the Organization of the Petroleum Exporting Countries (OPEC) decided to boycott some Western countries to punish them for their pro-Israel stance during the October War. The U.S.A., the Netherlands, the United Kingdom, Japan, and Canada suffered a total oil embargo (Hellema, Wiebes & Witte, 2004). Prices of oil rose worldwide and in Western Europe, restrictions on the use of oil products were put in place e.g. car-free-Sundays (Kasteleijn, 2013). The fear about the stability of the oil supply tightened the oil and gas price links. This enabled the Dutch government to boost revenues from Dutch natural gas (Odell, 1979), as competitive forces were confined by this crisis (Correlj´e & Odell, 2000). This also changed the view on natural gas. Natural gas became seen as scarce, and the government implemented measures to stimulate the discovery and extraction of gas from fields than the Groningen gas field. The Dutch state decided to shorten the duration of new contracts. The government also decided to change over from contracts that had a clause for a maximum price of gas to contracts with closer links between the oil and gas price in order to profit more when oil prices rise (Correlj´e, 1998).

The gas market became more international in the 1970s as new players arose and suppliers started to act oligopolistically (Correlj´e & Odell, 2000). As gas was seen as a scarce commodity, Dutch and European governments began to take action (Odell, 1973). The gas-producing countries for the European market decided to keep the demand for gas low. At the same time, the supply was kept low on purpose in the Netherlands although demand, especially from abroad, was way

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higher (Odell, 1984). Gas exports were restricted to maintain sufficient proven reserves to satisfy Dutch consumption needs for the upcoming years. Therefore, the Dutch extraction policy was deliberately sub-optimal in economic terms (Correlj´e, 1998). Table 1.3 depicts the natural gas demand in Western Europe; this demand was partly influenced by the Dutch policy to keep demand down.

Figure 1.3: Natural Gas Demand in Western Europe

Gas suppliers, such as the Soviet Union, were expanding their markets in Western Europe in the seventies. The USSR increased exports, for example, to the Ruhr area. Norway’s exports also increased, as the country invested more in the development of their gas fields and also in gas delivery systems responsible for shipping gas to the European mainland. Algeria expanded their markets with a pipeline to Italy and with the export of LNG, for example to Belgium, which was completely dependent on Dutch gas beforehand. Also, the UK decided to connect its then isolated gas system to the European mainland (Odell,1988). The share of Dutch gas that went to the market on the mainland of Western Europe declined from 54% in 1971 to 36.4% in 1981. However, Dutch exports rose from 25.2 billion Sm3 to 43 billion Sm3 in those same years. The market share kept on decreasing, although the production quantity did not fluctuate much after 1981 (Correlj´e & Odell, 2000). This evolution changed the Netherlands’ position in the gas market. The Netherlands slowly became a price taker on the gas market and its near-perfect monopoly disappeared.

After the 1970s, demand for oil went down on the mainland of Western Europe, but demand for gas went up from 106 109 Sm3 in 1971 to 356 109 Sm3 in 1998. The role of the Netherlands in the continental Western European gas market declined further from a 36.1% share in 1981 to an 18.6% share in 1991. After a small increase in market share some years later, it dropped again to 16.2% in 1998.

The lower Dutch gas market share combined with the Small Fields Policy led to the Groningen gas field losing its function as a ’catalyst’ of the gas system. The Dutch gas sector was no longer the regional initiator; instead, it became a player that responded to the market (Correlj´e & Odell, 2000).

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changes in the import and export of gas over time. It summarizes the information on exports from the previous tables and suggests that use in the Netherlands is quite constant. Exports have changed over time, reaching a minimum at the end of the 1980s. On the other hand, the quantity of gas imports has been growing over the last 20 years.

Figure 1.4: Import and Export of Dutch Natural Gas

1.5.2 New Gas Law and Liberalization

In the 1990s, the government was planning to change the gas market. This was due to pressure from the European Union to liberalize the gas market. However, Odell states that the desire to change the gas market was mainly due to the government recognizing that extraction was economically sub-optimal and the irrational combination of limited market development and the simultaneous existence of a resource overhang in the past (Odell, 1995, 1997).

The right of first refusal of all Dutch gas by the Gasunie was withdrawn by the Dutch state in 1994 at the time that the European Hydrocarbons Directive was accepted (European Commu-nity, 1994). After that, a memo from the Ministry of Economic Affairs, the Derde Energienota (Third Energy Paper) (MEZ, 1995), along with the New Gas Law (MEZ, 1998), changed the Dutch gas system. New suppliers and traders of gas were allowed to negotiate access to transport and distribution channels due to this new law.

The requirement to maintain sufficient proven reserves was now considered excessive (Odell, 1992, 1997). Mr. Verberg, president of the Gasunie until 2004, admitted in the newspaper ’Financieele Dagblad’ in 1998 that the restriction on production was illogical (Correlj´e, 2000). The New Gas Law changed this objective formally and made the Gasunie no longer responsible for guaranteeing supply in the long term. It also changed the objective that Dutch Gas should be mainly reserved for Dutch consumers because of its scarcity (Odell, 1987; Witteveen-Hevinga,

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1998; Algemene Rekenkamer, 1999)

1.5.3 Current Situation

In 2015, Minister of Economic Affairs H. Kamp decided to lower the production of natural gas (Bereentsen & Koot, 2015). This decision was due to earthquakes that arose in Groningen. According to this decision, wells in Loppersum in particular will extract less or no gas. In the short term, lower production will decrease income for the Dutch State, as estimated by the Dutch government. This decision was made just after a significant drop in the price of oil and gas and will lower the income from gas for the Dutch government even more. According to the CEO of Shell, Van Beurden, we can expect oil and gas prices to stay low for many years. Also, several investment banks, such as Goldman Sachs, have lowered expectations of future oil and gas prices up to 2020 (Van Dijk, 2015). Lower levels of production are making the Netherlands more dependent on foreign gas. Therefore, the Dutch Minister of Economic Affairs is searching for opportunities to import more gas from Russia. He has also decided to invest in the production of nitrogen, which can be used to change the caloric value of foreign natural gas to make it suitable for Dutch households (Weissink, 2015). However, this will increase the Netherlands’ dependence on Russian gas. Therefore, H. Kamp is looking for opportunities to extract more gas from new fields in the North Sea so that the Netherlands can become more independent from imports from Russia (“Kamp wil meer aardgas”, 2015b). These political decisions suggest that the Dutch government is not just optimizing the value of gas in economic terms; political factors also play an important role in the natural gas policy of the Netherlands.

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2

Optimal Extraction Model

2.1 Models in the Literature

There are different types of models in the literature on optimal extraction of natural resources. The objective of the models is to maximize the sum of the discounted profits over the determined period. In these models, extraction is seen as multiple real options. Each option can be exercised once and entails fully extracting the resource. Optimal extraction models consist of an objective and constraints. These constraints include the restriction on total extraction; no more than the total initial stock can be extracted over time. Models in the literature take different factors into account, such as technological progress, taxes, price expectations, environment externalities, national security externalities, and the role of the interest rates.

In general, two types of models in the literature can be distinguished: models with prices determined endogenously and models with prices determined exogenously. These models have been developed to determine the optimal extraction rate now and in the future. It is important to note that the (future) prices are unknown in these models. In the models with prices determined exogenously, the price of the natural resource is a function that is independent of the extraction rate and the stock. These models may use either stochastic functions or deterministic functions. Only deterministic functions are applied in the model used in this thesis for simplicity’s sake. The models with endogenously determined prices are used for players that can influence the price, like players with large oil fields in the Middle East, or for scarce non-renewable resources with limited suppliers and no substitutes, e.g. diamonds. Models with exogenously determined prices are developed for price takers.

In the optimal extraction models in the literature, a distinction between continuous time models and discrete time models can also be found. The discrete time models can be derived from the continuous models for a better understanding of those models. The discrete models are especially of interest for this article, since the data related to the extraction rate, interest rate, Dutch gas reserves, and income are all known per year.

The basis of these models is to maximize income, which consists of price multiplied by the quan-tity extracted minus the total costs. The cost functions for the models of extracting exhaustible resources vary. Certain models have fixed costs per unit extracted, others have a cost function that depends on stock and extraction rate. Certain models have advanced cost functions and take many factors into account, for example, the costs that arise from more wells needing to be built when production increases.

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Mathematician, statistician, and economist Harold Hotelling lays the foundations of optimal resource extraction models in his 1931 paper. His paper was ’neglected’ due to its mathematical difficulty until the energy crises of the 1970s arose. This limited the accessibility of his theories to only a few economists at that time (Devarajan & Fisher, 1981). Hotelling’s model does not necessary take many factors into account and can be adjusted based on the information that is known. The model developed in this thesis is an adjustment of his model that optimizes incomes over a time period. Therefore, this model is explained in more detail in the coming pages. His optimal extraction theory is based on a pricing theory that he describes in his article. The Hotelling rule describes the price of exhaustible resources. The rule for a competitive market is given and is sufficient for explaining his models.

Theorem 2.1.1 (Hotelling Rule). The price of a non-renewable resource grows at a rate equal

to the interest rate. This theory holds in a competitive resource industry equilibrium and also holds for optimal extraction. In formula, form this is:

Pt= P0ert (2.1)

with Pt being the net (nominal) price received after paying the cost of extraction and placing upon the market at time t, P0 being the initial (nominal) net price received after paying the cost

of extraction and placing upon the market, and r being the nominal interest rate. Therefore, ert is the present value of a unit of profit to be obtained

If this rule did not hold and prices increased more over time, then it would be more valuable to extract nothing now and wait to extract everything in the future. If the price was lower, it would be optimal to extract everything directly. Extracting all reserves at once would mean an infinite extraction rate in a continuous time model. It should be noted that the pricing theory on which his models are based has been replaced with known past prices in the developed model in this thesis. However, it is important to mention to make sure his assumptions are not violated since this rule is crucial in his models.

Based on Hotelling’s rule, a competitive resource owner would extract natural resources at a socially optimal rate and the undiscounted value would grow at the same rate as the interest rate (Hotelling, 1931). Hotelling states that in the case of negligible (or non-existing) extraction costs, the extraction rate is equal for all periods. Also, the present value of a unit extracted is the same in all periods, and there is no incentive to shift extraction amongst periods. If the extraction costs are not negligible and increase less than the interest rate, and demand is also stable, then output declines monotonically in this model.

Hotelling has described models with different markets, such as monopoly markets and competi-tive markets. He has developed models with an increase in extraction costs for lower stock, e.g. increasing production costs when the cumulative production is higher. He has also described models with demand influenced by the cumulative production, for example those developed for diamond production. Hotelling has also elaborated models with fixed investment costs.

In the case of a monopoly, the monopolist raises the price higher than it would be raised in a competitive industry and restricts output. The Hotelling rule does not apply in this case; instead, the marginal revenue grows at the same rate as the interest rate. It is then undecided if the price rises more or less rapidly compared to the price in a competitive industry. However, Hotelling thinks the rise would be less rapid due to observations he made in practice.

Besides the cases of a monopoly and perfect competition, Harold Hotelling also considered the case in which there are a few competing sellers. In this case, a Cournot-Nash equilibrium can be used to come up with a solution for optimal extraction. Here, each resource owner chooses

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the extraction rate to optimize his profit, while taking the extraction rates of others as given (Salant, 1976).

The Hotelling model with exogenously determined gas prices is the base for the developed model. A major difference between his model and the model developed for this thesis lies in these prices. In Hotelling’s model, the exogenous prices are based on predictions, whereas in the developed model, they are based on observation.

In the models that Hotelling describes, cumulative production can influence the extraction rate due to changes in demand and changes in extraction costs. Demand can go down if the cumu-lative production is high, as in the case of the diamonds example. This demand trend is not so relevant for oil and gas. However, the costs of extraction can go up due to the higher produc-tion costs related to oil and gas fields that are more depleted. In gas fields, pressure decreases through extraction and it becomes harder to extract gas over time as the fields become emptier. In such a case, it can become necessary to inject substances, like nitrogen, into these fields to increase pressure and be able to extract the gas that is left.

Uncertainty is an important factor for optimal extraction models. Hotelling did not address the issue of uncertainty, although he states that uncertainty of the stock size could have an impact. Hotelling mentions that the uncertainty of demand, in addition to that of supply, has an impact on optimal extraction models. After the 1970s, researchers developed models that take this kind of uncertainty into account. However, uncertainties relating to demand and supply do not apply to an extraction model with known prices. It is clear that uncertainty about the impact of supply on price and demand does influence optimal extraction.

The main aim of an optimal extraction model based on the theory of Hotelling is to maximize total profit:

Π =

Z ∞

0

f(q, s, t) dt (2.2)

by choosing q(t) or s(t), which are subject to:

             ∂s ∂t = −q(t) ≤ 0 s(0) = s0 q(t) ≥ 0 s(t) ≥ 0

Here, f : R3 → R is the discounted profit function. Thus, f can be considered f (q, s, t) =

R(q, s, t) − C(q, s, t), where R: R3_{→ R is the revenue function and C: R}3 _{→ R is the discounted} cost function. The discounted profit depends on both the discounted cost function and the dis-counted price function. Since this is a model in which disdis-counted prices and costs only depend on the extraction rate, the cumulative extraction, and time, f is a function of q, s, and t. Here,

q(t) is the extraction rate at time t and s(t) is the stock that is not extracted at time t. Choos-ing q(t) or choosChoos-ing s(t) are equivalent in the case of there beChoos-ing no new discoveries of stock because of the relationship: ∂s_∂t = −q(t). This relationship can be expressed as q = ρ(s) = −∂s_∂t and s = ρ−1_{(q). Therefore, the discounted profit function f can also be written in the form}

f : R2→ R with f (q, t) or f (s, t).

The real profit that has been observed in the past is discounted in this model using the real interest rates. The constraints are that initial stock s0 is given, such that the extraction rate is equal to the negative change of stock, the quantity extracted at any time is positive, and the stock cannot become negative.

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An optimal feedback solution can then be derived analytically in the continuous case. Assume ∃ ¯q >0, such that ∂f_∂q >0 and ∂_∂q2f2 <0 ∀ q, 0 < q < ¯q

Then, the elasticity of marginal discounted profit is defined by:

σ(q) ≡ − ∂2 f ∂q2q ∂f ∂q ∀ q < ¯q (2.3)

Using the assumption that σ(q) is integrable ∀ q < ¯q, the next integral can be defined

θ(q) ≡

Z q

0

σ(x) dx (2.4)

Note that the following equation hold lim

q→¯qθ(q) = ∞ (2.5)

and that this implies

lim

s→∞q(s) = ¯q (2.6)

In feedback form, q is a function of the state of resource s, so q = ρ(s). The optimal policy in feedback form is then represented by:

q= ρ(s) = θ−1(δs) ∀ s (2.7)

with δ being the discount rate.

The proof that this is the optimal solution is provided by Rouillon (2013). More advanced mathematical derivations written in Hotelling’s paper are used in the section. This section develops the optimal gas extraction model for the Dutch gas fields.

2.2 Optimal Extraction Model of the Dutch Natural Gas

Re-sources

2.2.1 Model Preconditions

The objective of the optimal extraction model developed in this paper is to maximize total government income over the total extraction period. In this model, the discounted profit depends on the extraction rate that is chosen. Therefore, extraction rates in this model vary from the observed extraction rates of the past. There are certain restrictions to the choice of extraction rates that ensure a realistic model regarding the historical gas market. Other assumptions also need to be made to estimate the production costs and expected income per cubic meter of extracted gas.

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Discounted Profit Function

The discounted profit function had to be determined in order to develop a model in the spirit of the theory of Hotelling. However, the discounted price of natural gas and discounted production costs are not explicitly known. The income of natural gas for the Dutch State is based on taxes, concessions, and dividends. It is not a simple sum of price time quantity minus total costs. The different contracts with parties described in Chapter 1 make it impossible to explicitly state the price of gas. The price of gas has changed over time as the near-perfect monopoly has slowly disappeared.

However, the profit of the Dutch state per year and the amount of gas extracted is explicitly known. This information contains the profit per cubic meter of gas extracted. After discounting, this is the f function in Hotelling’s model: f (Oe, Os, t) = R(Oe, s, t) − C(Oe, Os, t), where Oe is

the function of the observed extraction rate, Oe: t → R, and Os is the function of the observed

amount of stock left in the past, Os: t → R. The revenue function R is the contribution of the

revenue of gas sales per year to the income of the government. The discounted cost function C is the contribution of total extraction costs to the income of the Dutch State, e.g. how these extracting costs negatively influence the natural gas income of the Dutch state. This formula does not require knowing the actual production costs for firms and the actual prices in the optimization model.

In the model, the pricing of Dutch gas is assumed to be exogenous, e.g. the production rate does not influence the discounted price of natural gas. The pricing policy was determined by the different contracts that were signed, and the objective of this article is not to question the pricing policy. The contracts were signed with different prices and, as stated by Odell and explained in the first chapter, the demand at these price levels was higher than the production. Therefore, maintaining the price and selling more natural gas would have been possible. A near-perfect monopoly usually leads to endogenous pricing. However, the decision to let the price depend on other factors, such as the oil price, made the pricing exogenous. This pricing policy is assumed in this paper and different pricing policies are not investigated in this study. When the Dutch ceased to dominate the gas industry, it became even more obvious that the price is exogenous. Over the last few decades, the Netherlands has become a smaller gas player. The globalization of the natural gas market also led to the Dutch loss of pricing power. Therefore, the assumption of exogenous prices also holds for the entire period of 1973-2014.

Demand and Supply

A change of extraction rates at any point in time changes supply. The impact on demand is not explicitly known. The impact of a change of supply on the national income had to be known or estimated to be able to develop an optimal extraction model. As mentioned in Chapter 1, the Dutch state kept supply down on purpose, although demand, especially from abroad, was much higher in the 1970s (Odell, 1984, 1995, 1997). Odell states that this difference between demand and supply grew during the first oil crisis in 1973. Gas exports were restricted to maintain sufficient proven reserves to satisfy Dutch consumption needs for the upcoming years. Odell argues that demand was higher at the existing price level, which can be interpreted as an argument for assuming that an increase in supply would have led to an equal increase in sales of natural gas. This assumption can also be used as an argument that an increase in supply would not have decreased the price since 1973. These conclusions are reasonable in an exogenous pricing model up to a limited level of higher production than observed in the past.

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If the production of natural gas is much higher in the model than observed in the past, it is more likely that supply would have been higher than demand or that prices would have to drop, which is not possible in an exogenous model. Therefore, a maximum production rate is assumed in the model. The model also assumes that an increase in supply would have led to an equal increase in sales of natural gas with no changes in the price since 1973.

Assumptions also had to be made to deal with the case of a lower extraction rate than observed in the past. As concluded in Chapter 1, the gas market became more international in the 1970s as new suppliers arose. The market also started to act oligopolistically, due to the supply of gas from Algeria (LNG), Norway, and Russia (Correlj´e & Odell, 2000). The rise of new supplier countries can explain the situation where the extraction rate in the model is lower than that observed in the past. This gas could be bought to provide the promised amount of natural gas. However, this assumption can only be made as long as those other countries could meet the demand. It is unrealistic to assume that the 42% market share of Dutch gas in 1976 on the mainland of Western Europe was produced by these other countries. Rather, it is assumed that the lower production of Dutch natural gas could be partly compensated by the higher production of other suppliers. Therefore, it is assumed that such a compensation could made up to a certain barrier, namely a minimum production of the Dutch gas fields.

The minimum production between 1973 and 2014 was 6.7 · 1010_Sm3 _{and was reached in 1988.} Maximum production in this time period was 10.1·1010Sm3 and was reached in 1976. Therefore, it was assumed that the minimum amount of gas that had to be extracted in this entire period was 6 · 1010_Sm3 _{and the maximum was 11 · 10}10_Sm3 _{(about 10% below the minimum observed} production rate and 10% above the maximum observed rate). In the results section, calculations are made with these values as the minimum and maximum extraction rates per year. Within this minimum and maximum range, the price of natural gas is considered exogenously determined, e.g. independent of the stock and extraction rate. This price is the observed discounted price from the past. If the costs are considered both constant per cubic meter of gas and the same as observed in the past and the discounted price is constant per cubic meter, then the average discounted profit per cubic meter of gas can be used to determine the total discounted profit, depending on the extraction rate.

Optimization Horizon

The optimization horizon could influence the outcome of the optimization. The data in these periods have an impact on the outcome of the model and the assumptions that are made in the model are not evenly realistic in all periods. The Netherlands was the main supplier of gas for the Western European mainland in 1971, with a market share of 54%. This number declined to 42% in 1976, 36.4% in 1981, and 26.8% in 1986. The fact that the Netherlands accounted for the main market share of natural gas can raise the question of whether it was possible to import more gas from Algeria, Russia, Norway, and the UK during those years, as is assumed in the model, especially considering that gas exports are often limited because of the necessity of having a gas pipe for non-liquified gas. The gas market became more flexible over time and importing and exporting opportunities increased; both of these facts make the model more realistic. Therefore, this paper only determines the impact of optimization for a shorter and later period. The period of 1974-2014 was shortened by 25% and by 50% to observe the impact of different periods on the outcome of the model. The scenarios of optimization beginning in 1984 and 1994 have therefore been investigated, as has the scenario of optimization beginning in 1974. These different periods illustrate the effect of the optimization.

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Discounting

The real discounted income was used in order to be able to compare the income of all of the years. In the entire mathematical model, all prices, costs, and income are real and discounted. To determine real income (Ir(t)) from the nominal income (In(t)) of year t, the following formula

is used:

Ir(t) = In(t)

Gt · 100 (2.8)

Here, Gt is the GDP deflator and 2014 is the base year. The choice of base year was made

independent of the optimal extraction rate. Only the total government income differs in real terms. The GDP deflator is precisely defined in the data section. The total government income, which is maximized by the model, was then defined as the sum of the real incomes of all the years. The real prices were discounted using the real interest rates. This was done in the same way as discounting using the GDP deflator.

Id(t) = Ir(t) T −1

Y

t=1

(1 + rt) (2.9)

Here rtis the interest rate of year t and T the last year of the optimization period (2014).

Discovery rates

Some optimal extraction models depend on discovery rates. They assume a certain discovery rate of new gas fields. The discoveries of new reserves, e.g. changes in estimates of the gas fields, in the past are known. In the optimization model in this article, discovery rates are therefore not taken into account. The total reserves at the beginning of the optimization period were set equal to the total reserves known today minus the amount extracted at the beginning of the optimization period. This is the total amount of reserves that is left today plus the amount of gas extracted since the beginning of the optimization period.

Variety of Gas Fields

The costs for exploration and extraction are different for the different fields. The Groningen gas field, with its very low costs, was used to balance demand and supply. The Small Field Policy, implemented by the government, ensured that the small onshore and offshore fields, which had higher production costs, were exploited at the same time as the Groningen gas field. This choice was made to optimize total production over a long-term horizon. In the literature, there is no evidence that this approach is not optimal. This policy also ensures that the high production necessary in winter can be attained. Therefore, in the model developed in this paper, there is no division between the order of extracting gas from the different fields. Gas form all fields is extracted simultaneously and the Small Fields Policy is therefore seen as granted. Therefore, the model does not have to consider differences in costs and income related to extraction from the different fields.

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Total Quantity Extracted

It could have been optimal to have extracted less (or more) gas, such that more (less) reserves would have been left today, especially if the discounted price in the future would have been higher (lower) than in the past. Estimates of the value of the gas reserves that are left are based on many assumptions, such as market development and future prices. Therefore, the total extracted quantity in the optimization scenario is considered as equal to the observed total extracted quantity. This means that the total quantity extracted in the past must be equal to the total quantity extracted in the optimal extraction scenario.

R&D

New technology for extracting natural gas is developed over time. Due to soap (and also nitrogen) injection, it is possible to extract gas from an almost empty field (“Putten ontwateren”, n.d.). Extracting became easier and cheaper over time because of the development of technology. Experience, for example of the NAM, also made it easier to extract gas from the Groningen gas field and therefore more profitable. The impact of these developments was not known at the beginning of the exploration of the Dutch gas fields. The reduction in costs due to these developments and the resulting higher profit for the Dutch state, for example due to higher tax income, have to be taken into account in the model. The higher discounted profit per cubic meter of natural gas extracted due to technological developments for companies leads to lower production costs. This in turn leads to a higher discounted profit per cubic meter of gas for the Dutch state, for example via taxes. Therefore, the technological developments were directly included in the model via the discounted cost function. These developments are then reflected in the discounted profit function that uses the average discounted profit per cubic meter of gas for the Dutch state and that depends on this discounted cost function.

2.2.2 Mathematical Optimal Extraction Model

A mathematical model was developed to maximize the income of the government (Π) from natural gas extraction. The model is based on the ideas of Hotelling and the stated assumptions. The total discounted profit is the sum of the discounted profits f over all of the years. Here, f is the discounted profit function per year. The model optimizes the total discounted profit for the Dutch state:

Πtotal= T

X

t=1

ft (2.10)

where the year of the first extraction is indicated by t and the last year of extraction is indicated by T . In the case of searching for the optimal path, T is a time in the past.

Since f : R2→ R is defined by f (q, t) = R(q, t) − C(q, t), this leads to: Πtotal=

T

X

t=1

(R(q, t) − C(q, t)) (2.11)

The revenue function R is the contribution of the revenue of gas sales per year to the income of the government from natural gas. The discounted cost function C is the contribution of the total

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extraction costs to the income of the Dutch state, e.g. how these extracting costs negatively influence the natural gas income of the Dutch state.

The discounted price P is seen as a deterministic value for every year and is exogenously deter-mined so the discounted price does not depend on the extraction rate. This yields:

Πtotal = T

X

t=1

(Pt· q − C(q, t)) (2.12)

Now, we define the adjusted discounted cost function, which depends on the time t, the quantity extracted q, and on the the observed extraction rate, Oe: t → R.

ACt(q) = C(q, t) − C(Oe, t) (2.13)

Note that

ACt(Oe) = C(Oe, t) − C(Oe, t) = 0 (2.14)

The total discounted profit per cubic meter for the Dutch state is known per year for an extrac-tion rate of Oe(t) (function of the observed extraction rate in year t) per year, which is

Vt= Pt· Oe(t) − C(Oe, t) (2.15)

The average discounted profit per cubic meter, then, is:

vt=

Pt· Oe− C(Oe, t)

Oe = Pt− K(Oe, t) (2.16)

Where K(Oe, t) = C(O_Oee,t)

The optimization problem therefore reads Πtotal= T X t=1 (Pt· q − C(q, t)) = T X t=1 (Pt· q − C(q, t) + C(Oe, t) − C(Oe, t)) = (2.17) T X t=1 (vt· q − C(q, t) + C(Oe, t)) = T X t=1 (vt· q − AC(q, t)) (2.18)

Since the discounted values are used for the optimization, all values for profit and costs are discounted.

This optimization is subject to constraints. Extraction can only be positive so that

qt≥ 0 ∀ t ∈ {t|t ∈ N, 1 ≤ t ≤ T } (2.19)

The total natural gas reserves cannot be negative. Since stis a decreasing function for increasing tit is sufficient to have the constraint:

sT ≥ 0 (2.20)

Here, st is the stock at time t. Note that the following holds:

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With the assumption that the total amount of natural gas extracted in the period t until T is equal to what has actually been extracted in this period. This constraint is:

T

X

t=1

qt= Q (2.22)

with Q being the actual extracted amount of natural gas stock in this period.

The assumption of a minimum (∧) and maximum (∨) extraction rate can be expressed as: ∧ ≤ qi ≤ ∨ ∀ i ∈ {i|i ∈ N, 1 ≤ i ≤ T } (2.23)

Altogether, this leads to the optimization model, which can be summarized as:

Πtotal= T

X

t=1

(vt· qt− AC(qt, t)) (2.24)

by choosing q(t), subject to:

         q(t) ≥ 0 ∧ ≤ qi≤ ∨ ∀ i ∈ {i|i ∈ N, 1 ≤ i ≤ T } T P t=1 qt= Q

2.2.3 Discounted Cost Function

The adjusted discounted cost function AC(q, s) has to be determined. There are conditions that must hold for this function. Hotelling assumes in some of his elaborations that the costs are independent of the extraction rate and of the stock. In this case, the adjusted discounted cost function is equal to zero, but in other cases it is not.

James L. Sweeney and A.V. Kneese (1992) state that the cost function in a discrete time frame-work has to be derivable from the integral of cost in an underlying continuous time represen-tation. In order to avoid violation of assumptions that make the cost function realistic, the discrete discounted cost function is derived from the continuous function. After the develop-ment of this discrete discounted cost function, the adjusted discounted cost function is derived from the derived function. In this framework, the discrete quantity extracted should also be derivable from the integral of extraction:

qt=

Z t+∆

t

ω(γ) dγ (2.25)

Here, ∆ is the time step, this is taken equal to 1 year in this research, ω is the underlying continuous function of the extraction rate and γ is the time distribution in the underlying continuous functions. The cost function per year can then be expressed as:

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Ct(q, s) = Z t+∆ t c(ω(γ), s(γ)) dγ (2.26) Where s(γ) = st− Z γ t ω(γ) dγ (2.27)

Here, c is the underlying continuous cost function. This leads to the total discounted cost:

T

X

t=1

Ct(qt, st−1) (2.28)

Several restrictions exist, since the discrete time representation has an underlying continuous time representation. Based on Equation 2.26 it follows that the first partial derivative of the costs with respect to initial stock is given by:

∂Ct ∂st = Z t+∆ t ∂c ∂sdγ = ∂c ∂s∆ (2.29) Here, ∂c

∂S is evaluated at some point between t and t + ∆. The first partial derivative of the

costs with respect to initial stock is approximately proportional to the time step ∆. Based on Equation 2.29, it can also be observed that the sign of ∂Ct

∂St₋1 is equal to the sign of

∂c ∂s

Using Equation 2.26, one can determine the marginal extraction costs:

∂Ct ∂qt = ∂f ∂ω − Z t+∆ t ∂c ∂sdγ (2.30)

The derivation of this formula can be found in the appendix and is based on Sweeney (1992). With ∂c

∂ω evaluated at γ = t, Equation 2.29 and Equation 2.30 combined give: ∂Ct ∂qt + ∂Ct ∂st = ∂c ∂ω >0 (2.31)

Now, taking the derivative of this equation with respect to qt yields: ∂2Ct ∂q_t2 + ∂2Ct ∂st∂qt = ∂ 2_c ∂ω2 ∂ω(t) ∂qt (2.32)

This equation demonstrates that the derivative of marginal cost with respect to the extraction rate is the sum of two effects. It also illustrates that increasing the total extraction in the interval increases the instantaneous extraction rate at all times and also at t. An increase in the instantaneous extraction rate increases marginal cost. This means that the right hand side must be positive. Increasing the extraction rate at the beginning of the period reduces the reserves during the time that is left in the period. The reduction of stock then increases marginal cost even more if ∂2

Ct

∂st∂qt is negative.

If marginal costs are more sensitive to the extraction rate than to the stock level, as in most of the literature, then the following equation must hold:

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∂2Ct ∂q2_t >

∂2Ct ∂qtst+1

(2.33) A commonly used assumption is that the cost function is a weakly convex function in its argu-ments. Assuming this type of function can be explained by the fact that for higher production rates, the costs of extracting are higher per cubic meter due to loss of pressure. Then, this leads indirectly to an decrease in income for the Dutch state due to lower tax income from the firms that extract natural gas.

Definition 2.2.1 (Weak convexity). For any open interval (a, b) ⊂ R, define a weakly convex

function m : (a, b) → R as one for which

m(qx0+ (1 − q)x1) ≤ qm(x0) + (1 − q)m(x1) (2.34) ∀ x0, x1∈ (a, b) and ∀ q ∈ [0, 1] ∩ Q.

The Hessian matrix of a function C is positive semi-definite at every point if and only if the function C is weakly convex. The Hessian matrix is defined as follows:

H(x) ≡             ∂2C ∂x2₁ ∂2C ∂x1∂x2 . . . ∂ 2_C ∂x1∂xn ∂2C ∂x2∂x1 ∂2C ∂x2₂ . . . ∂2C ∂x2∂xn .. . ... . .. ... ∂2C ∂xn∂x1 ∂2C ∂xn∂x2 . . . ∂ 2_C ∂x2 n             . (2.35)

The principal minor determinants of a matrix are all positive or zero if and only if the matrix is positive semi-definite. The principal minor determinants of the Hessian matrix for the cost function are: ∂2Ct ∂q2 t ∂2Ct ∂S2 t−1 ! − ∂ 2_C t ∂S_t−1∂qt !2 (2.36) ∂2Ct ∂S2 t−1 (2.37) ∂2Ct ∂q2 t (2.38) Therefore, assuming that the function C is weakly convex, these principal minor determinants are non-negative. The discounted cost function that meets all of these requirements and that is found in the literature, with s and q separable, appears in the following form:

α₁(q − a)2+ β1(s − b)2+ α2· q + β2· s (2.39) Here, a, b, α1, α2, β1, and β2are constants. Note that the costs are also discounted so as to align with the discounted profit and discounted price functions. These discounted cost functions are

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based on models that do not use the average discounted profit, but the discounted price instead. Here, the linear terms are taken into account when the average discounted profit is used because the average profit is basically the price minus the total costs divided by the quantity. The dependence on s represents the cost dependence on the amount of stock. Since this model optimizes over the long term, higher costs need to be weathered anyway at some point in time because the pressure in gas fields decreases. Also, technology can mitigate higher costs due to lower stock over time. Therefore, this was included in the discounted cost function that is used for the calculations. This discounted cost function is then provided by:

α· q2 (2.40)

This leads directly to the following adjusted discounted cost function:

AC(q) = α · q2− α · Oe2 (2.41)

The results for different values of α, where α is chosen close to values that can be found in the literature, are located in Chapter 3.

2.2.4 Conditions for an Optimal Solution

To ensure that there was an optimal solution for the optimization problem, the theorem of Karush, Kuhn, and Tucker was used.

Theorem 2.2.1 (Karush-Kuhn-Tucker). Suppose that x ∈ RN is a maximum (or minimum) point of f (x) in the region:

R ≡ {x ∈ RN|φj(x) ≤ 0, j = 1, ..., m}

where f, φ1, ..., φm : RN → R are continuously differentiable functions. If the constraints φj satisfy some regularity conditions in the x then there exist λ ∈ Rm _{such that (x, λ) is a solution} of the system:              ∇f (x) −P_j_{= 1}m_λ j∇φj(x) = 0 φj(x) ≤ 0, j = 1, ..., m λj ≥ 0, j = 1, ..., m λjφj(x) = 0, j − 1, ..., m

(Kuhn & Tucker, 1950)

The discrete backward-looking model, described in this chapter appears in this form; therefore, if one follows this theorem then one will find an optimal solution.

2.3 Extraction Rate Strategies

A comparison between policy decisions and the optimal extraction rate as determined using the developed model cannot be made since future prices were not known when policy decisions were made. Optimal extraction strategies can be used to determine what extraction rate to use. These strategies can be applied without knowing the future price and could have been

Optimal extraction of the Dutch natural gas reserves : the development of an optimal extraction model with historical prices