Using Chebyshev polynomials 

Get Complete Project Material File(s) Now! »

Irreversibility and uncertainty

Irreversibility of investment amounts to saying that undertaking investment projects results in some unrecoverable initial cost —the so-called sunk cost. Uncertainty on future benefits and costs of investments is also important. In fact, it was the combined effect of irreversibility and uncertainty that led Weisbrod [1964] to introduce the con-cept of option value. He argues that if a decision has irreversible consequences, then the flexibility —the “option”— to choose the timing of that decision should be included in a cost-benefit analysis. The concept of option value has two different interpretations. The first views option value as a risk premium paid by risk averse consumers to reduce the impact of uncertainty in the supply of an environmental good [Cicchetti and Freeman, 1971, Schmalensee, 1972]. The second interpretation stresses the intertemporal aspect of irreversibility and the arrival of new information over time [Arrow and Fisher, 1974, Henry, 1974]. This second interpretation is the one that has been used more exten-sively in the literature on real options [Myers, 1977]. Examples are the entry, exit, and temporary shutdowns and re-startups on investment and output decisions; the impli-cations of construction time —and the option to abandon construction— for the value of a project; and the determinants of a firm’s choice of capacity [McDonald and Siegel, 1986, Brennan and Schwartz, 1985, Pindyck, 1988].
To gain more intuition about irreversibility, uncertainty, and about what the option value means, let us consider the model proposed by Olsen and Stensland [1988]. Their model consists of an industry based on non-renewable resources (e.g. off-shore oil extraction) having to decide whether or not shutting down its activity. First at all, the shut down decision can be thought to be irreversible. In Olsen and Stensland’s model irreversibility means that firms cannot restart operations after shutting down (probably because of prohibitive high costs). Additionally, Olsen and Stensland assume that economic conditions —particularly prices and production quantities— are uncertain. Uncertainty jointly with irreversibility imply that firm’s decision involves the exercising, or “killing” of an option —the option to optimally shut down at any time in the future. In such a framework, there is a value of waiting —the option value— since the firm has always the possibility to postpone the shut down in order to learn more about the present and future payoffs. As a result, the usual rule of “shutting down activity when the marginal production costs saved equal the marginal revenue foregone” is not longer valid. In order to shut down, the marginal production costs saved must exceed the marginal revenue foregone by an amount equal to the value of keeping the option alive [Pindyck, 1988, Dixit and Pindyck, 1994]. Notice that the opportunity of shutting down (or the opportunity to invest, or any other opportunity under uncertainty and irreversibility) is analogous to a option on a common stock. In this case we have a (put) option giving us the right (the exercise price of the option), but not the obligation (because of the opportunity to wait) to shut down (sell) activities (the underlying security), the value of which fluctuates stochastically [Pindyck, 1991]. Of course, in the case of complete irreversibility the salvage value of the firm is zero [Dixit and Pindyck, 1994, Dangl and Wirl, 2004].
Uncertainty and irreversibility has also being considered in some other environmen-tal problems. For instance, Brennan and Schwartz [1985] analyze the operation of a mine, which can be temporarily closed; Paddock et al. [1988] propose a model to value offshore petroleum leases as a function of the market price of oil; and Clarke and Reed [1990] study the preservation of natural wilderness reserves. A standard result is that, in the presence of environmental irreversibility, a standard cost-benefit analysis is bi-ased against conservation [Arrow and Fisher, 1974, Henry, 1974]. Freixas and Laffont [1984] generalize this result, and Conrad [1980] links option value to the expected value of information.
In some more recent papers, Pindyck [2000, 2002, 2007] explains the role of uncer-tainty and irreversibility in environmental policy. In the case of global warming-oriented policies for instance, uncertainty appears as we do not know how much average temper-atures will rise with or without reduced CO2 emissions, nor do we know the economic impact of higher temperatures. Second, irreversibility appears in, for instance, policies aimed at reducing environmental degradation. There are some sunk costs taking the form of discrete investments (e.g. coal-burning utilities might be forced to install scrub-bers, or firms might have to scrap existing machines and invest in more fuel-efficient ones), or they can take the form of expenditure flows. These sunk costs create an opportunity cost of adopting a policy now, rather than waiting for more information about ecological impacts and their economic consequences. There is also some irre-versibility in environmental damage. For example, atmospheric accumulations of GHG are long lasting; even if we were to drastically reduce GHG emissions, atmospheric concentration levels would take many years to fall. This means that adopting a policy now rather than waiting has a sunk benefit, that is a negative opportunity cost. Then traditional cost-benefit analysis will be biased against policy adoption.
What about our particular cases of energy efficiency and on the substitution of resources? Consider the case of resources substitution. Imagine a firm trying to decide whether or not stop using non-renewable (fossil fuels) in favor of renewable inputs in energy production. Uncertainty can appear in a lot of ways, but it seems particularly relevant to focus on the uncertainty over the availability of resources [Pindyck, 1980, Epaulard and Pommeret, 2003, Smith and Son, 2005]. Specifically, we can have an idea of their current stock, but we do not know too much about their future availability. And even if we knew how much of the resources are expected to be available in the future, we would not know the resulting effect on firm’s decisions. For instance the more renewable resources are expected to be available in the future, the more the firms are tempted to adopt them sooner to produce energy. Moreover, the adoption of renewables imposes sunk costs on society. It is in particular the case if an offshore oil platform is being transformed into an offshore wind farm. Once this decision is made, almost all of the previously installed capital needs to be dismantled, and very few parts of the old facility can be re-utilized. This process is clearly very expensive in terms of time and money. As before, there is an option value in the sense that investors can wait for more information before getting rid of the offshore oil platforms.
Consider now the specific case of a homeowner investing in new “equipment” in order to reduce his energy bill. We can safely assume that this investment is irre-versible, as uninstalling the new equipment can be unreasonably costly or because the old equipment has been scrapped. Some uncertainty can also be considered. For in-stance, the evolution of the household’s wealth is not fully know, particularly if we assume that wealth depends on risky assets. Under this circumstances, household’s de-cision to invest can be considered to have an option value, as she can always postpone the investment in order to learn whether future wealth is increasing or not. If there is no arbitrage between household’s consumption and adoption —i.e. in a partial equi-librium setting, in order to invest in this energy saving technology, the marginal value of the new equipment must exceed the purchase and installation cost, by an amount equal to the value of keeping the investment option alive. In the case of an arbitrage between consumption and adoption —i.e. in a general equilibrium setting, the optimal adoption timing is not only sensitive to uncertainty, but also sensitive to the degree of agents’ risk aversion (Hugonnier et al. [2008]; see also Pommeret and Schubert [2009] for an example in environmental economics).

On numerically solving Bellman equations

In this dissertation we face ourselves with functional equations that are complex and cannot be fully solved using standard analytical techniques. In particular, analytical solutions for the models in chapters 3 and 5 are only available in the case of a zero discount rate. There is a huge debate in environmental economics about the way future utilities should be discounted, especially when today’s decisions have very long-term consequences (see, for instance, Portney and Weyant [1999] for a survey). As we are not particularly inclined in favour of a zero or a positive discount rate, a methodology to solve for all cases becomes necessary. Additionally, the model in chapter 4 can never be fully solved analytically. At least for the models we are presenting here, the problem is the high non-linearity of the value functions and the fact that they have to satisfy some boundary conditions. Then, we need to rely on numerical methods.
As suggested by Judd [1992, 1998], we use the approximation properties of Cheby-shev polynomials to compute stable non-diverging solutions of the Hamilton–Jacoby– Bellman equations. Specifically, we transform the value functions and the given condi-tions into expressions with unknown Chebyshev coefficients. By using this representa-tion, our original problem of solving partial differential equations reduces to a problem of solving simple systems of non-linear equations. The model of chapter 3 is solved by using an algorithm based on Newton’s method [Miranda and Fackler, 2004, Dangl and Wirl, 2004]; while the models in chapters 4 and 5 are solved by using the methodology presented in chapter 2.5 In particular, we transform the value function and the given conditions into matrix equations with unknown Chebyshev coefficients. By doing like this, our problem reduces to one of solving a simple system of algebraic equations. This methodology is a secondary by-product of this dissertation, but it constitutes by itself an original contribution to the numerical methods literature.

READ  Nonprofit Organizations and Entrepreneurship

Energy efficiency

Home renovations are generally asserted to be a highly effective means for households to lower expenditures on energy —and to indirectly reduce GHG— through improving efficiency, and they become therefore a key target for environmental policies. Indeed, cost-benefit analyses point to the economic viability of these systems even if the comfort co-benefits such as improvements in indoor air quality and protection against noise are not taken into account [Jakob, 2006, Ott et al., 2006]. However, actual investment in these systems is still relatively rare [Banfi et al., 2008]. In chapter 3 we aim to explain the home renovation decision of households in a theoretical model. Specifically, we try to explain the slow diffusion of energy efficient investments —the so-called energy paradox or energy-efficiency gap [Jaffe and Stavins, 1994a].
The existing literature that explains the energy paradox (Hassett and Metcalf [1995], for instance) considers partial equilibrium settings, and therefore ignores the interac-tion between optimal consumption and optimal adoption, as well as the notions of risk aversion, or intertemporal substitutions. In those settings consumers behave like firms when deciding energy-saving technology adoption. To challenge these results, we reconsider the joint effect of irreversibility and uncertainty on the energy-efficiency investment decision in a general equilibrium framework [Hugonnier et al., 2008, Pom-meret and Schubert, 2009]. In particular, we wonder whether the explanation of the energy paradox based on the existence of an option value still valid in a more realistic and general model.
To tackle this issue we consider the specific case of a homeowner who may invest in new insulation, or double glazing in order to reduce her energy bill. We assume that this investment is irreversible, as uninstalling the new “equipment” can be unreasonably costly or because the old equipment has been scrapped. We also assume that the benefits of such energy-saving technologies as well as the financial returns on household’s savings are assumed to be stochastic. Our results imply that the threshold triggering adoption depends not only on technological parameters but on preference parameters as well. Additionally, we show that while uncertainty on energy-saving technologies efficiency hardly affects adoption timing, uncertainty on financial returns fosters it. We also find that the existence of an option value does not rule out the energy paradox.

On the substitution of resources I

Substitution of fossil fuels provides permanent GHG emission reductions. Many economies in the world have therefore incentives to move towards renewable energy. In chapter 4 we consider a model of technology switching in which energy can be produced from two available inputs: non-renewable resources (fossil fuels) and renewable resources. We are particularly interested in tackling the following issue: At what point should society stop using non-renewable resources to produce energy and start producing from renewables?
We propose a model in which the available resources (fossil fuels, and renewable resources such as water, wind, solar, biomass) are perfect substitutes in energy pro-duction, and their stocks are stochastic. We assume that firms start producing energy using only fossil fuels, but the possibility to carry out an irreversible investment to switch to the other input is always open. Due to our particular assumptions, we obtain a value function before the switch that is S-shaped. This is completely new in the literature of technology switching, where the resulting value function is mostly concave [Dixit and Pindyck, 1994]. Another novelty of our model is that firms do not switch immediately in the case of switching cost being equal to zero (Pommeret and Schubert [2009]; see also chapter 3). This results from the higher profits the firms get from using nonrenewable resources, particularly if they are abundant.
We find that uncertainty plays a clear role in the decision to switch. The more the uncertainty about the availability of the non-renewable resources, the sooner the firms switch to the renewable resources; and the more the uncertainty about the availability of renewable resources, the later the switching time. The optimal switching time is also sensitive to energy demand, costs, and the relative productivity of resources parameters.

On the substitution of resources II

We already know that one of the policies commonly undertaken by many countries in order to reduce GHG emissions is to substitute dirty energy sources, such as coal, oil and gas, with a cleaner and renewable energy source, such as solar and wind energy. However, it seems that fossil fuels will continue to be an important part of the energy mix around the world even by 2050, a year in which a reduction of about 80 percent of total 1990-emissions is expected. Our theory is that as long as renewable energies are not very advanced and widespread, (i) industry will still need a percentage of energy that derives from dirty resources, and (ii) the provision of clean energy itself will require dirty resource at least as materials to build the plants (think of solar panels for instance).
In chapter 5 we model an economy having access to two different energy sources. The first one comes from a natural polluting resource, such as fossil fuels. The second one comes from a backstop natural resource, such as solar radiation. In particular, we consider the case of solar radiation being converted into energy by means of solar panels. There are two productive sectors in the economy. The first one is dedicated to manufacturing the backstop resources. At any time, this sector requires both fossil fuels and the energy provided by the backstop already available. We therefore account for the need of fossil fuels to provide clean energy. The second sector is devoted to production of the consumption good. Initially it uses energy coming exclusively from fossil fuels. However, it has always the possibility of switching to a new technology in which energy comes from both types of resources. Under this setting we seek to appraise what happens for the energy adoption decision if we account for the need of dirty resources even in an economy using clean energy.
In modeling this switching decision, we account for the uncertainty over the future costs and benefits. In particular, we assume that the accumulation of the backstop, and the increase in pollution stock —which in our case is equal to the resource extraction— are stochastic. We also introduce the irreversibilities associated with environmental policy. Specifically, adoption of the cleaner technology imposes sunk cost on the con-sumption sector. Our results imply that, under uncertainty and irreversibility, the incentives to switch to the cleaner technology depend on the relative importance of fossil fuels in the production of consumption goods after the switch. We also find that technological improvement in the solar panels sector is of some importance in order to switch to cleaner technologies. If the technological change implies that the backstop can be produced with relatively less of the fossil fuels, the adoption occurs sooner.

Table of contents :

I Introduction (French version) 
Le contexte
Irreversibilité et incertitude
Sur la résolution numérique des équations de Bellman
Efficacité énergétique
Sur la substitution des ressources I
Sur la substitution des ressources II
II English version 
1 Introduction
1.1 Context
1.2 Irreversibility and uncertainty
1.3 On numerically solving Bellman equations
1.4 Energy efficiency
1.5 On the substitution of resources I
1.6 On the substitution of resources II
2 Using Chebyshev polynomials
2.1 Introduction
2.2 The optimal stopping problem
2.2.1 Description of the model
2.2.2 Solving the model analytically
2.3 The matrix model
2.3.1 A numerical approximation of the value function
2.3.2 Collocating the collocation points
2.3.3 Value matching condition and Chebyshev coefficients
2.3.4 Smooth pasting condition and the optimal stopping time
2.3.5 Results
2.4 Introduction to non-linear optimal stopping problems
2.4.1 Linearizing the non-linear equation
2.4.2 Using the smooth pasting condition
2.4.3 Results
2.5 Conclusion
2.A An illustrative example with N=2
3 Energy-saving Technology 
3.1 Introduction
3.2 The Model
3.3 The Optimal Path after Adoption
3.4 The Optimal Adoption Timing with no Discounting
3.4.1 The Marginal Value of Wealth
3.4.2 Boundary Conditions
3.4.3 Comparative Statics
3.5 The Optimal Adoption Timing with Discounting
3.5.1 An Approximate Value Function Before Adoption
3.5.2 Choosing the Coefficients
3.5.3 Results
3.6 Conclusion
3.A Solving the Optimal Program after Adoption
3.B Solving the Optimal Program Before Adoption
3.C Comparative Statics
4 Switching from Non-Renewable to Renewable Resources 
4.1 Introduction
4.2 The model
4.3 Three steps to solve for the optimal switching time
4.3.1 Step 1: Production with non-renewable inputs
4.3.2 Step 2: The optimal path after the switch
4.3.3 Step 3: The opportunity to switch and the optimal switching time
4.4 The natural resource’s renewal function and the steady state behaviour of the resource after the switch
4.4.1 The natural resource’s renewal function
4.4.2 The steady state behaviour of the resource after the switch
4.5 Results
4.6 Conclusion
4.A Numerical approximations and more on sensitivity analysis
4.B Solving the Hamilton-Jacobi-Bellman equation
4.B.1 The non-regulated process case
4.B.2 The regulated process case
5 Switching to clean(er) technologies 
5.1 Introduction
5.2 The model
5.3 The optimal path after the switch, and the case without an option to switch
5.3.1 The after the switch case
5.3.2 The no option to switch case
5.4 The optimal switching time, the undiscounted case
5.5 Comparative Statics
5.5.1 No technological improvement in the solar panel process
5.5.2 Technological improvement in the solar panels sector
5.6 The optimal switching time, the discounted case
5.6.1 A numerical approximation of the value function
5.6.2 Results
5.7 Conclusions
5.A Proof of Proposition 5.1
5.B Proof of Proposition 5.2
5.C Proof of Propositions 5.3 and 5.4
5.C.1 Proof of Proposition 5.3
5.C.2 Proof of Proposition 5.4
6 Conclusion 


Related Posts