Master thesis
University of Groningen
Faculty of Economics and Business
2018
Student number: S3398374
Valuing real option in gas-fired plant investment under EU climate policies
AbstractI compare the expansion behaviors of a gas-fired plant owner under two different situations. One is the normal situation, and another is with the consideration of the application of the ETS within European Union. I use a dynamic model to represent the uncertainty of demand size of electricity in the market and then investigate the research problem. The relationship between the demand size and the capacity of the plant which ensures the value of the plant not being hurt shows that the plant owner should expand more slowly when the ETS is applied to regulate the emission behavior of producer. A capacity cap is also used to investigate the expansion investment decision of the plant owner. And in this work, I use a numerical example to compare the results and present the expansion behaviors of the plant owner more clearly.
Keywords: Uncertainty of demand size, real option, expansion behaviors, Emission Trading System
1. Introduction
Nowadays, people gradually realize that the environmental problems are mainly caused by human beings. And among all the environmental issues that have already occurred, people care most about the global warming problem since it is said to influence all living creatures on the earth. Even though the impact or to be more serious the damage of global warming may only be seen in the long term, it is time for people to find effective solutions to tackle this problem. Instead of other possible reasons, similar to what Shaheen and Lipman (2007) mention in their work, it has been proved by many researchers that global warming is mainly caused by the emission of some greenhouse gas, mostly the carbon dioxide. The burning of conventional fuels, for example, oil, gas and some other types of fossil fuels are expected to be reasonable for the majority of carbon dioxide emission. And a large proportion of the consumption of fossil fuel is used for energy production, for example, electricity production. Nowadays, governments and some international organizations are starting to make great efforts to solve global warming. For example, the emission trading system has been applied within Europe Union to fight against global warming since 2005. And at the meanwhile, the Paris Agreement signed in 2016 within UNFCCC with a goal to limit the temperature rise within 2 degrees Celsius is taken to regulate the greenhouse gas emission behaviors of people as well.
global system and find benefits of implementing it. Despite advantages, there is also shortcoming for applying ETS. That is, they are not entirely costless. In other words, sometimes the environmental goal is only realized with the sacrifice of firms’ financial goals. For example, the application of the EU emission trading system brings some ‘side effects’ to the market apart from reducing carbon dioxide emission. This EU emission trading system (hereinafter referred to as ETS) scheme works under the principle of ‘cap and trade’. A total amount of carbon dioxide emission for all the participants in the system is set at the beginning of every year, and then the emission allowance is allocated to each participant for free or at a certain price. The emission allowance is tradable in the market. If one firm has run out of allowance, it has the right to purchase extra allowance at market price in the emission allowance trading market.1 And in contrary, one firm may earn extra money through selling part of
his emission rights to those who are in short of them.
Now, let us consider a gas-fired plant which is built in order to produce and provide electricity to families and factories. DCF analysis described in the book Risk and Return for Regulated Industries by Villadsen, Vilbert, Harris and Kolbe (2017) suggests that the value of this plant is sum of the present value of all the cash flow it can generate in the future. And the application of ETS is likely to increase the total cost of production and therefore is thought to influence the investment decisions of the plant owner. This paper is going to answer the question that how will the application of ETS influence the investment behavior of the plant owner.
This paper will be carried out under a fundamental assumption that the supply of electricity is monopolized, which means only this electricity producer masters the skill and technology to build gas-fired plants as well as to produce electricity, which means other people cannot mimic his business. Also, the producer is assumed to have enough capital to invest in such a gas-fired plant and have the right to expand the capacity of the plant if the electricity demand in the market increases a lot. In reality, the assumption that the plant owner is a monopoly is too limited, especially in Europe and North America where the electricity market is competitive. However, in some countries, for instance, China, the electricity supply is still monopolized by the government. It is true that some large firms may own a private electricity plant which allows them to serve their daily business. But electricity produced by them cannot be provided to others. Mostly, only the government can build gas-plants and supply electricity for family and business use. To simplify the problem, I use the assumption that the plant owner is a monopoly. And this assumption is expected to guarantee the rationality of the relationship
1 European Commission states in its official website that “…After each year a company must surrender enough allowances
between price and quantity which will be discussed in the next section.
Methodologically this paper is based on the real option theory as well as the theory of irreversible investment under uncertainty. John C. Hull describes real option theory in his book Options, Futures, and Other Derivatives. He states that there are many options included in most of investment projects and these options have value. For example, abandonment options, the option to stop business and expansion options, the option to expand capacity. In this paper I take the expansion option that a gas-fired plant owns as an example. If the electricity producer has the right to expand the plant capacity, he actually owns an option to increase the value of the plant. When the market demand size of electricity is small, the producer can only get profits from producing with current capacity. But his company, the plant, has great ‘growth opportunity’ if the demand size will increase. He can expand the capacity and then get profit through producing with the extra capacity. This expansion therefore is expected to add value to the plant and the market value of the plant is therefore the value of asset in place plus the value of growth opportunity. So the timing effect and uncertainty of future demand is expected to influence the value of the plant. In most of existing research papers, the uncertainty is resulted from a specific diffusion process. That is, the change of one variable is defined to follow the Geometric Brownian Motion.2 However, investment in projects like a gas-fired plant is somewhat
irreversible. The irreversibility of investment in projects has been discussed in many research works provide some reviews on it (see, e.g., Dixit and Pindyck, 1994; Murto and Nese, 2002). As a result, the decision maker must deal with the uncertainty very carefully.
In this work, I investigate a dynamic model to study the investment threshold of the plant with consideration of current EU climate policies (the ETS) and at the same time compare the plant owner’s investment behaviors. To be more concrete, in this work, I will investigate that if the demand size of the electricity in the market follows a Geometric Brownian Motion (hereinafter referred to as GBM) and therefore changes over time, how will the plant owner expand his plant under different limitations. Overall, this work can prove whether ETS is effective in reducing the carbon dioxide and can help a plant owner understand the difference of his expansion behaviors before and after the application of ETS.
There are many research papers that have discussed the similar topic with my work and they are providing perspectives on valuing real option in gas-fired or other type of electricity plants and on using Bellman equation to solve the problems. The following three research papers are the fundamental
2 dP=μPdt+σPdz is the expression of a standard Geometric Brownian Motion, showing how the variable P changes over
references of this work. Siddiqui and Maribu (2009) describes a detailed introduction and application of Bellman equation in solving the investment problems under uncertainty. In their work, they present the way how the Bellman equation is derived from the perpetual cash flow. Murto and Nese (2002) in their paper discuss the investment choices between a fossil fuel plant and a biomass plant when the cost of fossil fuel is under uncertainty. In their paper, the price of fossil fuel follows GBM and the owner of the plant can either invest in a fossil fuel plant or in a biomass plant. The biomass plant has a constant value while the value of the fossil fuel plant is uncertain because of the changing price of fossil fuel. They assume the plant receives fixed cash flow when the plant is operating and the cost of the production only consists of the cost for purchasing fossil fuel. Investment in a biomass plant is more recommended when the cost for fossil fuel is high while it is more valuable to invest in the fossil fuel plant with a low fossil fuel price. Murto and Nese (2002) in their paper solve the threshold price at which the value of the fossil fuel plant would equal to the value of a biomass plant by applying real option theory and using Bellman equation. Another research work by Pindyck(1998) studies the capacity choice of a monopoly firm. In his work, the demand size of the product in the market follows GBM. The relationship between the unit price of the product and the demand size as well as the production quantity is a linear function. And when the demand size increases a lot, the firm has the right to expand capacity. Except for the increased production cost, expansion of the firm requires extra investment cost, which is the installation cost of new capacity. Pindyck starts his work in determining the optimal initial capacity and then solves his problem by finding the threshold demand size of the expansion. In his paper, he also draws a conclusion that for some firms the value of future growth opportunities may take a large proportion in the total value of the firms. Apart from these three works, researches on real option are carried out intensively. Sarkis and Tamarkin (2008) analyze the real option problem for renewable energies under the condition GHG trading system is applied. Qin and Chu (2012) study the valuation of real option in wind power investment project and try to find the optimal time in making such an investment. Dobbs (2004) studies how the price cap regulation affects investor’s choice in deciding the capacity of a monopoly firm. Besides, Willems and Zwart (2017) make a further investigation on the price cap effect. They study the optimal regulation problem under asymmetric information.
investment threshold under these two situations if there exists an expansion cap, which works as a limitation for the expansion activity of the plant owner.
The result of this work indicates that compared with expansion behaviors under normal situation, the application of ETS requires the plant owner to expand more slowly related to the same evolvement trend of demand size. That is, if the demand size increases for the same amount, the plant owner can add more capacity to the plant to make the expansion valuable if he does not need to pay for the emission. And the application of the ETS also leads the price of the electricity at the moment of expansion to keep increasing while it stays constant in the normal case. Besides, if the capacity cap is adopted, with the appearance of the ETS, the plant owner has to wait longer for the demand size of electricity in the market to reach a higher level.
This research paper is organized as follows. Section 2 discusses the variables as well as the model of my work. In addition, I present the threshold of demand size under every situation in section 3. In section 4, I make some discussion of this work. I give the results of my research work in this section and at the same time give a numerical example to better present it. And finally, in section 5, I conclude this work and present the limitation of it.
2. Model
I consider in my work there is an electricity production investor who plans to build a gas-fired plant and produce electricity according to the market demand of it. Production and business are operated in continuous time. Unlike some other type of projects that can be de-invested, this plant owner who wants to invest in such a project should make decisions more carefully because of huge required capital and the irreversibility of the investment. The demand size 𝜃 (MW) in this research work reflects demand of electricity in the market and the producer can always observe it. The monopoly producer can determine how much to produce and certainly the production is limited by the capacity of the plant. That means, every moment the quantity of production cannot exceed the current capacity. The capacity of the gas-fired plant in this work stands for the maximum amount of electricity that the producer can produce with current available materials and human resources. Consider that the demand size evolves over time and is assumed to follow the specific Geometric Brownian Motion in the form of equation (1):
𝑑𝜃 = 𝜇𝜃𝑑𝑡 + 𝜎𝜃𝑑𝑧 (1)
defined as 𝑃 = 𝜃 ∗ 𝑄−𝛾 by the plant owner with 𝑄 is the production quantity and 𝛾 is the constant
elasticity with 0<γ<1.3 In order to make the problem less complicated, I regard the price P here as the
price which has included the tax effects as well as other small factors that have an impact on the price. So the revenue in the discrete time should be 𝑃𝑄, which is then 𝜃𝑄1−𝛾. The unit price of cost for
operating business is defined as 𝐶1 (EUR/MWh), and it is assumed to be constant over time. And compared with the electricity price P, 𝐶1 is always relatively small. This critical rule will be discussed
in the next section. Similarly, the unit cost here mainly represents the cost of gas consumption as well as the cost of operating the plant such as the management cost and has ignored some other minor factors. In a word, the cost for normal production has been adjusted to reflect the unit cost for per unit of electricity. So the cost for the plant at every moment is 𝐶1𝑄.4 To get the present value of the plant,
all the future cash flows that the producer will receive should be discounted. From the perspective of arbitrage pricing theory explained in Modern Portfolio Theory by Francis and Kim (2013), similar assets are expected to have similar cash flows, and therefore the cash flows of this gas-fired plant should be discounted by using the expected return rate of a similar tradable asset in the market. In this research paper, I use the term r to represent this discount rate. Except for the general production cost, to build or expand the plant requires an installation cost, which is I (EUR) for per unit capacity k (MWh). This installation cost per unit capacity keeps constant over time, and the effect of economics of scale is not included in the consideration.
The net cash flow which is also regarded as the profit of the plant at each moment is defined by 𝜋
(EUR), and 𝜋 is always calculated by revenues minus costs. Since the installation cost only happens
at the beginning of an investment and is not included in the following production activities, in the simplest situation where there is no cost for purchasing emission allowance, the profit is expressed as:
𝜋 = 𝜃𝑄1−𝛾− 𝐶
1𝑄. (2)
This research work emphasizes mainly on the impact of uncertainty of demand size on the expansion decisions. If the demand size of electricity in the market is always the same, it is easy to get the value of the gas-fired plant. And there is no necessary to expand capacity since the plant owner is only required to invest once for some capacity. But if the demand size of electricity develops as equation
3 The monopolist is able to decide the price for selling electricity and this relationship between price and quantity is
adopted by the plant owner in this paper. It can be somewhat explained by microeconomics and can be expressed either with a linear function or with a power function. In Pindyck(1988)’s research work about monopolist choices, he uses a linear function to show the relationship between price and quantity and investigate his research problem.
4 In reality, the amount of electricity production and the amount of gas consumption are not perfectly equal due to the
(1), the value of the plant is changing. And when the demand size reaches a certain high level, production with initial capacity will not be able to supply the market demand. The plant owner would like to expand the capacity of the plant. Then the application of ETS are expected to affect the value of the plant and the investment decision of the plant owner compared with that of the normal situation (without application of ETS).
No matter how the demand size of electricity indeed develops in the market, the plant owner has to decide and invest in a certain amount of initial capacity to carry out business. And this initial capacity should be determined carefully. To determine the initial optimal capacityK (MWh), I first investigate
the situation where the demand size of electricity in the market is constant. A constant demand size indicates that expansion is unnecessary and the value of the plant can be written as 𝜃𝐾−𝛾
𝑟 − 𝐶1𝐾
𝑟 − 𝐼𝐾.
By taking the first derivative of the function between the value of the plant and the capacity, we can know that the initial optimal capacity is going to satisfy 𝐾−𝛾 = (𝐼 ∗ 𝑟 + 𝐶
1)/((1 − 𝛾) ∗ 𝜃). However,
when considering that the process of demand size is stochastic and the plant owner can expand the capacity, the result of the initial optimal capacity is a little different. The stochastic situation will be discussed in the later part of this work.
As is mentioned above, every year companies are first allocated some emission rights (here regard that these rights are for free) and required to purchase more allowance for extra carbon dioxide emissions. To simplify the situation, in this work I assume that in the case the ETS is applied, the plant owner is firstly going to build the plant with a capacity equal to the initial optimal capacity in the normal situation. Therefore, it is not required to re-determine the initial optimal capacity again in this case. And electricity production within this amount only takes the general cost C1. In contrast, once the
production exceeds this level, the cost for emission rights is introduced into the production. As a result, the research problem is to compare the expansion behaviors of the plant owner while with or without the application of ETS.
In this work, the unit price for purchasing emission allowance is 𝐶2𝑄 (EUR/MWh).5 𝐶
2 is a constant.
And this unit price means the price changes with the electricity production and is indeed a function of the quantity of electricity production. The more electricity that the plant owner produces, the more
5 This is also a simplification of the research problem. In the emission trading market, the total number of emission rights
expensive that the emission rights will be. As a result, the extra cost of purchasing emission is 𝐶2𝑄 ∗(𝑄
-K) (EUR), where 𝑄 -K in this term refers to the quantity charged for emission cost. This extra cost
is only significant when Q is bigger than K. For Q smaller than K, this extra cost for emission is zero. The emission cost also becomes an additional production cost for the gas-fired plant to expand. Besides, similar to the normal production cost C1, 𝐶2𝑄 indeed represents the unit price of emission
right for producing per unit electricity.
In reality, if the plant owner always produces without adding new capacity, which means that he always has enough emission rights for production, he can even earn some additional money by selling his excess emission rights to others who need them. This additional revenue can make up for a small part of the plant value. But it should not change the expansion behaviors of the plant owner since it becomes zero at the moment when he decides to expand the capacity.
Finally, this research work is carried out under the consideration that the plant owner will always prefer to maximize the profit and hence maximize the value of the plant.
3. Investment threshold under different situations
Before implementing the investment, it is likely for the plant owner to predict the market demand size based on market information, government reports as well as other information and therefore build a specific amount capacity of the plant related to the demand size to carry out his business. I use the term
to represent the predicted demand size which is in line withK. And when production amount is higher than K, the emission cost is introduced. Later in this section, the relationship how is related with K will be stated.
Looking at equation (2) (actually just the effect of the cost factor in the profit equation), when the demand size is quite small, it is non-profitable for the plant owner to produce electricity with the full use of the initial optimal capacity. He would like to use only a part of the capacity to earn a positive profit. And since the demand size of electricity in this research work follows GBM, only when it becomes higher thanθ, which is also the predicted demand size in determining the initial optimal capacity, the plant owner will think of expanding the plant capacity. But expansion means making a further investment and certainly requires more capital. More concretely, the plant owner will need to pay the installation cost for the expanded capacity.
value of the plant, the plant owner will prefer not to expand no matter how high is the demand size of electricity in the market. Additionally, here in this work, I assume that the plant expansion can be realized instantly as long as the plant owner has decided and planned to expand capacity. This assumption means he can immediately build the additional capacity and use it for extra production. It is true that this assumption does not entirely stand in reality. In reality, the expansion may take a relatively long time, and it is possible that when the expanded plant is prepared to meet the extra need of electricity, the demand size has already fallen back to the original low level. So this assumption is used to guarantee the research significance of this work.
3.1 Investment threshold without limitation on carbon dioxide emission
I first investigate the investment threshold of expansion where there is no limitation of carbon dioxide emission. I define this investment situation as the investment under normal situation. That is, except for the installation cost and normal production cost described in this paper, there is no other kind of cost for expansion. The plant owner only cares about the relationship between the marginal value increased and the marginal cost of the expansion.
As is mentioned in the second section, the value of the gas-fired plant can be regarded as the same as the value of a certain tradable asset in the market. As a result, it is consisted of all the future profit flow that it can generate. In the following part of this work, I denote the term 𝑊(𝜃, 𝑄) to be the total value of the gas-fired plant. According to Siddiqui and Maribu (2009), a typical Bellman equation as equation (3) in the following can be used to show the relationship between the total value of the plant and its first as well as second derivatives.
𝑟𝑊(𝜃, 𝑄) = 𝜋(𝜃, 𝑄) + 𝜇𝜃𝜕𝑊 𝜕𝜃 + 1 2𝜎 2𝜃2 𝜕2𝑊 𝜕𝜃2 (3)
This equation shows that the most important determinant of the total value of this gas-fired plant is the profit in discrete time as well as the option to expand capacity under uncertainty. The profit term in equation (3) represents the cash flow in continuous time. Equation (3) also conveys that due to the uncertainty of demand size, the value of the plant is determined by the production quantity instead of the plant capacity.
𝑟𝑊(𝜃, 𝐾) = 𝜋(𝜃, 𝐾) + 𝜇𝜃𝜕𝑊 𝜕𝜃 + 1 2𝜎 2𝜃2 𝜕2𝑊 𝜕𝜃2. (4)
The total value of the gas-fired plant can be divided into two parts, one is the value of asset in place and another is the value of the ‘growth opportunities’. The value of asset in place is composed of the discounted future cash flow while the value of ‘growth opportunities’ refers to the option value. I denote by 𝑉(𝜃, 𝐾) and 𝐹(𝜃, 𝐾) the value of asset in place and the value of all options to expand respectively. Actually 𝑊(𝜃, 𝐾) here represents the maximum value of the plant at any level of capacity. As for the total value of plant, it is always driven by the actual production amount. And the true value may be smaller than the maximum value in reality. If the demand size of electricity drops after expansion, it might not be wise for the plant owner to produce with full capacity due to the effect of the cost variable, then the value of the plant is expected to be smaller. However, if the unit price of cost C1 is relatively small, the effect of the cost term for the final profit will be slight. That means, the
plant owner can approximately be regarded to use full capacity all the time. And there will not be big difference between the result of total value calculated by using equation (4) and the reality. The total value 𝑊(𝜃, 𝐾) can be driven by solving equation (4). The value of asset in place 𝑉(𝜃, 𝐾) while producing with full capacity is (𝜃 ∗ 𝐾−𝛾 (𝑟 − 𝜇)) − 𝐶
1∗ 𝐾/𝑟
⁄ . When plugging this 𝑉(𝜃, 𝐾) into equation (4), it is shown that 𝑉(𝜃, 𝐾) is a specific answer to it. In addition, the solution to a homogeneous Bellman equation can be derived by plugging in a power function. Then the solution of equation (5) 𝑟𝑊(𝜃, 𝐾) = 𝜇𝜃𝜕𝑊 𝜕𝜃 + 1 2𝜎 2𝜃2 𝜕2𝑊 𝜕𝜃2 (5) is proved to be 𝐹(𝜃, 𝐾) = 𝐴1(𝐾)𝜃𝛽1 + 𝐴 2(𝐾)𝜃𝛽2. (6) with 𝛽1 = 1 2− 𝜇 𝜎2+ √(𝜇−12𝜎2)2+2𝑟𝜎2 𝜎2 >1, (7) 𝛽2 = 1 2− 𝜇 𝜎2− √(𝜇−12𝜎2)2+2𝑟𝜎2 𝜎2 <0. (8)
𝑊(𝜃, 𝐾) =𝜃𝐾1−𝛾
𝑟−𝜇 − 𝐶1∗𝐾
𝑟 + 𝐴1(𝐾)𝜃
𝛽1. (9)
The expanding rule is that only when the marginal expansion cost equals to the marginal increase in the total value of this gas-fired plant, the plant owner will expand the current capacity. The additional expansion cost is the extra installation cost, I for per unit capacity k. As a result, the expansion rule suggests that:
𝜕𝑊
𝜕𝐾 = 𝐼 (10).
And to keep smooth transformation, the expansion should also obey:
𝜕2𝑊
𝜕𝐾𝜕𝜃= 0 (11).
Equation (10) and (11) together determine that the threshold of electricity demand size in the market for expansion activity under this situation is
𝜃1∗∗ 𝐾−𝛾= 𝐼+𝐶1𝑟
(1−𝛾𝑟−𝜇)(1−1
𝛽1)
. (12)
In fact, this equation (12) not only tells the relationship that the demand size and the capacity should satisfy when expansion activity happens. It also works in determining the optimal initial capacity. The rule that the plant owner prefers to maximize the value of the plant ensures the initial optimal capacity is always driven by taking the first derivative of equation (9). And the expansion conditions that equation (10) and (11) show indeed tell the minimum requirement to make unit capacity profitable, which is the emphasis of optimal capacity. As a result, equation (12) tells the optimal capacity related to the demand size of any level. After predicting the demand size in the market, the plant owner then can determine the initial optimal capacity of his plant K.
3.2 Investment threshold while taking the application of ETS into consideration
two-stage function:
𝜋(𝜃, 𝑄) = {𝜃𝑄
1−𝛾− 𝐶
1𝑄 + 𝐶2𝑄 ∗ (𝐾 − 𝑄), θ <θ
𝜃𝑄1−𝛾− 𝐶1𝑄 − 𝐶2𝑄 ∗ (𝑄 −𝐾), θ ≥θ. (13)
Besides, if the plant owner is required to purchase emission allowance as long as Q>0, the profit term at any level of demand size would always be 𝜃𝑄−𝛾− 𝐶
1𝑄 − 𝐶2𝑄2.
Again expansion happens only when the plant owner starts to use full capacity, which means >. If the demand size still keeps increasing, the cash flow at the expansion moment can be rewritten as π(𝜃, 𝐾) = 𝜃𝐾1−𝛾− 𝐶1𝐾 − 𝐶2𝐾(𝐾 −𝐾) . The cost for purchasing emission allowance is also a continuous expense. As a result, it is necessary to discount this cash outflow with the discount rate r as well. The total value of the plant with the value of option included is shown as equation (14).
𝑊(𝜃, 𝐾) =𝜃𝐾1−𝛾 𝑟−𝜇 − 𝐶1∗𝐾 𝑟 − 𝐶2𝐾(𝐾−𝐾) 𝑟 + 𝐵1(𝐾)𝜃 𝛽1 (14)
The expansion rules in the last section are still in effect here, and by reworking on equation (10) and (11), it leads the new investment threshold of demand size for expansion to be:
𝜃2∗∗ 𝐾−𝛾 = 𝐼+𝐶1𝑟+ 2𝐶2𝐾−𝐶2𝐾 𝑟 (𝑟−𝜇1−𝛾)(1−1 𝛽1) . (15)
Equation (12) and (15) stand when the demand size become higher than. These two equations convey the relationship between the demand size of electricity in the market and the capacity when the plant owner wants to expand the plant. The plant owner can always keep adding capacity to the plant as long as the capacity is in line with the demand size as equation (12) and (15) show. This expansion characteristic is also stated in Pindyck(1988)’s paper.
transition from not environmentally friendly energy to clean energy. As a result, policymakers do not hold an utterly supportive attitude on the unlimited expansion of a gas-fired plant. Furthermore, from the perspective of the market development, it might be harmful if there exists one specific firm that produces rather more carbon dioxide than any other firms. If this particular firm takes up most of the emission rights, it will hinder the business development of other firms. The truth in the market is that the quantity of carbon dioxide emission of every producer is somewhat scant when compared to the total number of emission rights.
In practice, some policies are formulated to regulate the entrepreneur infinite expansion behaviors. In some cases, expansion drives the price of product to rise, and a price cap is set by the government or the commercial association to deal with too high purchasing price and hence to protect customers. Apart from the price cap, methods like a quantity cap may also be used to regulate the expansion. A price cap or a quantity cap are both effective to limit the expansion and both can be used to study the expansion limitation effects on making investment decision. But in this work, I mainly investigate the impact of adopting a quantity cap. The reason is that no matter whether this quantity cap in reality indeed exists or not, there is an upper cap for the electricity production. Expansion is limited by many factors. The lack of sufficient investment capital, the land area available to build new plants or other limitations are all obstacles for the infinite expansion. And even the application of ETS can be a limitation for the plant owner to expand capacity. Because the total emission rights of carbon dioxide by countries within the European Union is fixed every year. As a consequence, no matter how high the demand size would be, theoretically the expansion activity cannot be carried out endlessly.
In the following part of this section, I investigate the effect of a quantity cap on the expansion decision of the plant owner when he should consider the cost of emission. The quantity cap in this work should represent the maximum capacity of the gas-fired plant. So it can also be regarded as the capacity cap or expansion cap. The capacity cap is defined as KC. In order to make sense, it should at least be larger
than the initial optimal capacity. That is KC>K. There is a corresponding investment threshold of
electricity demand size related to this cap in both situations.
The cap limits the behaviors of plant owner to add capacity more than KC to the plant. Because
investment which is made for capacity higher than KC is a kind of waste, the plant owner cannot receive
money from this specific capacity. Or in other words, any investment on capacity over KC becomes
non-profitable.
not preferable for any entrepreneur to expand only a small amount of capacity at a time. Because for every expansion activity, it takes a relatively long time and some fixed cost. The low efficiency of continuous expansion makes it a good choice for the plant owner to reduce expansion times. In some cases, he may choose to expand the gas-fired plant from initial capacity to the maximum capacity directly, which is also what I assume in the following part of this section to make the problem less complicated. And due to this potential expansion limitation, the plant owner is hence required to be more careful on dealing with the expansion threshold.
As a monopolist, the price of electricity is still decided by the plant owner through the equation 𝑃 = 𝜃 ∗ 𝑄−𝛾. I use the term PC to represent the corresponding price of electricity when capacity is KC. Since
now there is a capacity cap, the total value of the plant (the maximum value) after it has expanded to capacity KC can be easily determined. In the case that the demand size is high enough so that the plant
owner has implemented the expansion activity to capacity KC, there is going to be no expansion option
remained. The value of option finally becomes zero. The total value of the plant after expansion hence only comes from producing and selling electricity with initial and expanded capacity. As a consequence, unlike the equation (7) and (11) with the term of option value included, the option value is no longer included in the total value of the plant. Equation (13) is used to show the final value of the plant:
𝑊(𝜃, 𝐾𝐶) =𝜃𝐾𝐶1−𝛾 𝑟−𝜇 − 𝐶1∗𝐾𝐶 𝑟 − 𝐶2𝐾𝐶(𝐾𝐶−𝐾) 𝑟 − 𝐼(𝐾𝐶−𝐾) (16)
And again the value that equation (16) shows here is the maximum value of the gas-fired plant. In reality, the value of the plant is smaller due to the uncertainty of the demand size of electricity in the market. In most of the times, the plant owner does not make full use of the capacity of the plant. He only produces Q of electricity. That means, in most of the common time the total value of this plant after the expansion is 𝜃𝑄1−𝛾
𝑟−𝜇 − 𝐶1𝑄
𝑟 −
𝐶2𝑄(𝑄−𝐾)
𝑟 − 𝐼(𝐾𝐶−𝐾). And when the demand size drops a lot
and is belowθ, the cost for purchasing emission rights should be left out from the production cost. But as for the plant owner, he cares about the maximum value of the plant to make expansion investment. He is only willing to add capacity to the plant when the demand size of electricity in the market reaches the level related to the quantity cap because this threshold of demand size ensures that the maximum value of the plant can be realized. If the plant owner expands the production capacity when the demand size is below the threshold here, he takes the risk that his plant may never reach the maximum value.
The term 𝜃∗∗ is used to represent the threshold of demand size in this case. The profit term at which
value vanishes for small demand size. Therefore, the value of the gas-fired plant 𝑊(𝜃, 𝐾) is
𝜃∗𝐾1−𝛾 𝑟−𝜇 −
𝐶1𝐾
𝑟 + 𝐷1(𝐾)𝜃
𝛽1 when 𝜃 is below 𝜃∗∗ . When 𝜃 is above 𝜃∗∗ and the expansion is executed, the value of the plant is the maximum value. In order to find the threshold, the following equation (17) should be satisfied,
𝜃1∗∗∗𝐾1−𝛾 𝑟−𝜇 − 𝐶1𝐾 𝑟 + 𝐷1(𝐾)𝜃1 ∗∗𝛽1 =𝜃1∗∗𝐾𝐶1−𝛾 𝑟−𝜇 − 𝐶1∗𝐾𝐶 𝑟 − 𝐶2𝐾𝐶(𝐾𝐶−𝐾) 𝑟 − 𝐼(𝐾𝐶−𝐾). (17)
In addition, smooth pasting suggests to take the derivative of both sides of equation (17) with respect to . As a result: 𝐾1−𝛾 𝑟−𝜇 + 𝛽1𝐷1(𝐾)𝜃1 ∗∗𝛽1−1 = 𝐾𝐶1−𝛾 𝑟−𝜇 . (18)
Equation (17) together with (18) leads 𝜃1∗∗ to be:
𝜃1∗∗ = 𝛽1[ 𝐶1 𝑟(𝐾𝐶−𝐾)+ 𝐶2𝐾𝐶 𝑟 (𝐾𝐶−𝐾)+𝐼(𝐾𝐶−𝐾)] (𝛽1−1)(𝐾𝐶 1−𝛾 𝑟−𝜇 − 𝐾1−𝛾 𝑟−𝜇 ) . (19)
The result of equation (19) conveys the plant owner should expand the plant capacity to KC only when
the demand size of electricity in the market reaches this level or becomes higher than it. In addition, the value of the real option 𝐷1(𝐾)𝜃1∗∗𝛽1 in this case equals to (𝐶1
𝑟 (𝐾𝐶−𝐾) + 𝐶2𝐾𝐶
𝑟 (𝐾𝐶−𝐾) +
𝐼(𝐾𝐶−𝐾))/(𝛽1− 1).
Relatively, in the normal situation where there is no cost for extra carbon dioxide emission, the threshold of demand size to expand is quite similar to what equation (19) looks. When there is no cost for emission rights, the total value of this plant with no option left is 𝜃𝐾𝐶1−𝛾
𝑟−𝜇 − 𝐶1𝐾𝐶
𝑟 − 𝐼(𝐾𝐶−𝐾).
The corresponding threshold is hence expressed as equation (20). And it shows the only difference between these two thresholds coms from the components of the final maximum value of the gas-fired plant. 𝜃2∗∗ = 𝛽1[ 𝐶1 𝑟(𝐾𝐶−𝐾)+𝐼(𝐾𝐶−𝐾)] (𝛽1−1)(𝐾𝐶 1−𝛾 𝑟−𝜇 − 𝐾1−𝛾 𝑟−𝜇 ) (20)
Relatively, the value of expansion option 𝐷2(𝐾)𝜃𝛽1 here equals (𝐶1
𝑟 (𝐾𝐶−𝐾) + 𝐼(𝐾𝐶−𝐾))/
(𝛽1− 1).
true that with the quantity cap applied, there is a possibility that the price of electricity can be rather high if the demand size of electricity becomes dramatically higher than the expansion threshold. This outcome violates the principle of protecting customers. In general, except for the quantity cap, a combination of a price cap might do a better job in reducing the adverse effects of the infinite expansion. Therefore, policymakers may be willing to introduce an extra price cap.
4. A numerical example
In order to present the model and the result of it more clearly, I give a numerical example to help people know the differences in investor’s behavior under various situations. This example mainly serves as a tool to provide comparisons. The value of all parameters mentioned in the following part of this section is assumed and not fully reflects the actual world.
In most of the cases, as Koller (2015) writes in his book Corporate Valuation, CAPM or Fama French three-factor model is used to get the expected return of a specific asset and then serves as the discount rate. Besides, in some cases, the risk-free rate of return sometimes also works as a proxy for the expected return. The interest rate of German 10-years government bond is usually regarded as the risk-free rate. However, the interest rate of bond with this maturity has been pretty low for many years and so that of the German 30-years government bond. The average of German 10-years government bond during the past five years is only around 0.005.
It is hard to find a perfect related asset with the gas-fired plant, and the risk-free rate as 0,005 is extremely low. I consider the long-term yield of future cash flow of a perpetual model is probably higher than the current risk-free rates. As a result, the discount rate r of future cash flow in this work is hence just assumed to be 0.05. To make sense, the drift rate must be smaller than the discount rate. Therefore, the drift rate 𝜇 and the volatility 𝜎 of demand size are simply assumed to be 0.02 and 0.01 respectively. The value of volatility as 0.01 indeed is far from the reality. In most of the stock markets, the volatilities of stock returns are around 0.2 to 0.4. But to make the values of this numerical example easy to deal with, a relatively small volatility is used.
The elasticity of demand size is assumed to be 0.5 in this work. And the cost C1 for gas consumption
together with the normal business operation is assumed as 0.8. Relatively, the cost for emission rights
C2 is supposed to be 0.2. The installation cost of the capacity is made to be 20 EUR for every unit of
capacity that is used to produce 1 MW electricity.
With abovementioned information, equation (7) and (8) can be solved at the same time. That is, 𝛽1
and 𝛽2 are 2.5 and -401.5 respectively. After that, the exact relationships which equation (12) and (15) show between capacity and demand size can be found as well. Line graph 1 in the following illustrates these relationships. This graph only shows the result where the demand size starts from 3. Since expansion activities happen only when the demand size becomes higher than the initial predicted one. In this graph, the entire line without the application of ETS shows a steeper upward trend and is totally above the line with the application of ETS. In fact, if higher volatility according to the actual market conditions is adopted in this numerical example, the steeper upward trend can be found as well while the exact position of these two lines will be different.
The upward trend of these two lines in graph 1 indicates that under both situations when the demand size of electricity in the market goes up (higher than 2), the plant owner should gradually add more capacity to the plant related to its corresponding demand size. And according to graph 1, if expansion brings the additional cost for purchasing emission rights, every time regarding the same growth of demand size, the plant owner is required to add less capacity than that in the normal situation without the application of ETS to make the expansion activity worthwhile. When the demand size is the same under these two situations, the extra cost for emission rights makes the plant expanded smaller capacity. In other words, even the demand size evolves with the same trend under both situations, the plant owner should expand the plant more slowly in the situation where the ETS is applied.
Graph 1 Relationship between the demand size of electricity and plant capacity
Such expansion behavior is because expansion brings more expense when ETS is applied. And the additional cost brought to the investment makes expansion less profitable. As a consequence, if the plant is going to be expanded to the same capacity under these two situations, the plant owner has to wait for a longer for the demand size to reach a higher level in the situation with ETS applied to make up for the impact resulted from the extra cost.
The values of capacity and demand size in graph 1 also work in getting the electricity price at the 0 10 20 30 40 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 K
moment of expansion or making full use of the capacity. When there is no cost for purchasing emission rights, as equation (12) shows, the price of electricity at the moment of expansion is a constant. It stays 3.6 if the plant owner is using full capacity in this example. A stable purchasing price is conducive to the market stability. In contrary, when the ETS is adopted, the price of electricity at expansion moment keeps increasing, and in this example, it quickly becomes higher than 3.6. As is shown in equation (15), the unit price for purchasing emission rights is a positive linear function of the electricity production quantity, which finally results in the unit price at the time when the plant owner is using full capacity to be a positive linear function of the capacity. As a result, when demand size becomes higher than 2, the unit price of electricity in the situation without ETS applied remains the same with the change of demand size while the application of ETS makes it keep increasing with the increase of demand size. The comparison between the price of electricity in this example is shown in graph 2.
Graph 2 Electricity price related to its corresponding demand size at the moment of expansion
The uncertainty of demand size requires the plant owner to expand the plant based on the specific relationship like graph 1 shows. In fact, since the uncertainty of the demand size mainly comes from its volatility over time, the change of the volatility seems to have an impact on the change of the investment threshold. I use graph 3 to show a higher volatility (=0.08) of demand size affect this investment problem and the relationship between demand size and capacity of the plant. This graph tells that higher volatility can moderately speed the path of expansion no matter whether the ETS is applied or not. It means if the plant owner wants to expand the plant to certain capacity, higher volatility of demand size ensures him to wait for less time for the demand size to increase to its corresponding level. 3 5 7 9 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 P
Graph 3 The effect of the volatility on the relationship between capacity and the demand size of electricity
Next, I also use the numerical values above to investigate the capacity cap problem. The volatility is 0.01. Apart from the parameters above, I assume KC to be 5. And then I work out the threshold of
demand size when the expansion is carried only once from the initial capacity to the cap. By calculating equation (19) and (20), it is shown that if the ETS is used to regulate producer behavior, the plant owner should begin to expand the capacity of the plant when the demand size increases to 7.8. while it is only 5.0 if ETS is not applied. As a consequence, the application of ETS requires the expansion happen at a higher demand size.
Besides, the value of the expansion option 𝐷(𝐾)𝜃𝛽1 can be easily driven in this case. The option value 𝐷1(𝐾)𝜃𝛽1 with the ETS adopted is determined as 175.5 whereas in the normal situation 𝐷2(𝐾)𝜃𝛽1 is only 112.8. Apart from the option value, the total value of the gas-fired plant (the maximum value) in these two cases also shows some difference. If the demand size equals in the two situations and becomes higher than the expansion threshold, the total value of the plant after expansion with the application of ETS is smaller than that without application of ETS.
The higher value of demand size with the appearance of ETS here again indicates that the application of the ETS requires the plant owner to wait for a longer time for the demand size of electricity in the market to rise at a higher level so that the expansion activity can be worthwhile. And even though the total value of the plant may decrease, the option value of expansion instead increases a lot with the application of ETS. 𝐷1(𝐾)𝜃𝛽1 is much larger than 𝐷
2(𝐾)𝜃𝛽1. This result is due to the effect of timing
and the uncertainty of the demand size. In addition, the conclusion by Pindyck(1988) that the growth opportunity takes up a large portion of the total value of a firm is also proved in this work. In this work, the value of expansion option accounts for a large part of the total value of the gas-fired plant in both situations. 0 5 10 15 20 25 30 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 K
K(without ETS) K(higher volatility without ETS)
Ideally, a numerical example is expected to be related to the reality or at least reflect part of the truth. But overall, all the numerical values of parameters in this work is assumed and therefore are not so accurate or appropriate. The market information shows that a typical gas-fired plant is said to have a capacity around 400 MW. And such a typical plant can produce 400 MWh electricity in one hour for families and factories to use. In addition, the electricity price traded in the OTC markets within Europe is about 40 EUR/MWh. Hence, these numerical values assumed in this work may be not good proxies for the reality. In fact, I use more reasonable values to testify the conclusion driven by this numerical example. And I find the above conclusion still stands after I have changed the numerical values. As a whole, this numerical example can work to provide insights into how the expansion activity should be carried out by the plant owner.
5. Conclusion and limitation of this work
In conclusion, with other conditions to be the same, the application of ETS slows down the speed of expansion by the plant owner to ensure the profitability and requires him to wait for longer for the demand size of electricity to reach a higher level for expanding same amount of capacity. Additionally, with the demand size of electricity in the market increases, adoption of ETS will drive the electricity price at the moment of expansion to rise gradually and to be higher than that without the adoption of ETS.
Overall, this research work proves that the application of ETS plays a role in limiting the expansion behavior of the plant owner and in turns in dealing with the emission behaviors. The restriction on emission which brings additional cost makes the expansion becomes less encouraged. If the plant owner wants to keep the path with the increasing demand size and satisfy the market, it might be more profitable for him to turn to invest in more clean energy instead of expanding the gas-fired plant. Yu, He and Liu (2017) certify that ETS has a trend effect on renewable energy output. In other words, the application of ETS will be capable of helping realize the environmental goal.
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