Solving for Equilibria Numerically

The dynamic equilibrium conditions are described by differential equation (5.16) for the open-loop Nash equilibrium and (5.34) for the feedback Nash equilibrium. Because we have only one state variable, both problems collapse into a set of differential equations with respect to time, so we can use the same numerical method for both equilibria. We use a Runge-Kutta shooting algorithm, simulating the open-loop Nash and feedback equilibrium user cost of oil using (5.16) and (5.39) respectively, and using (5.5) to simulate the path of the oil stock. We do not know the initial value for the user cost of oil q, so to find it we introduce a long-run steady state condition, which must hold as time approaches infinity.

For the quadratic and exponential production functions we assume that the marginal productivity at zero is bounded at a low enough value such that some oil is left in situ. Thus the long-run steady state condition for the oil stock is:

(5.40) G(S) + D0(E0+ S0− S)/ρ = F0(0).

(For the isoelastic and non-HARA production case where the marginal productivity of oil is infinite, the final condition becomes more complex. We describe how to derive the final condition in section 5.7.3 below.) Since the long-run steady state described in (5.40) is only approached asymptotically and never actually reached, we approximate it by the conditions:

(5.41) R < ε, G(S) + D0(E0+ S0− s)/ρ < ε,

for an arbitrary small value of ε. We use a binary search algorithm to find the initial price q(0), which produces an oil extraction path that satisfies (5.40). We then use (5.16) and (5.39) to back out the open-loop Nash or feedback Nash equilibrium carbon

tax levels, respectively.

5.7.3

Solving for Equilibria with Infinite Marginal Production

at Zero

If the productivity of the marginal unit of oil is greater than the extraction cost of the marginal unit of oil, asymptotically oil is fully exhausted albeit that we never actually get to a point where R = 0, S = 0. Thus we must find a way to derive a different final condition. Let oil demand growth be g(t) = R(t)_R(t)˙ . Note that as the oil price increases, oil demand decreases so demand growth is negative. Now assume that as time goes to infinity, the demand growth rate converges to some finite value limt→∞g(t) = ¯g (while

we do not have a proof of this conjecture, we can show that it holds numerically). Let T be the point in time wheng(t) = 0 i.e. g(T ) = ¯˙ g. Then it must hold that for all t > T

(5.42) R(t) = eg(t−T )¯ _{R(T ).}

Substituting (5.39) in the differential equation for the oil stock (5.5) we get the condi- tion for the stock of oil at time T :9

(5.43) S(T ) = R(T )

Z ∞ t=T

eg(t−T )¯ _{dt =}R(T )

−¯g .

We can now use the same Runge-Kutta algorithm that we used for the quadratic and exponential production function. We find time T numerically by using the Runge- Kutta method to simulate the model until the growth rate of demand converges, and use (5.43) as the final condition to check whether the oil extraction path satisfies the equilibrium conditions.

Conclusion

Preventing global climate change is a difficult problem with many complicating factors. This dissertation addresses several issues related to the economics of climate change, presenting insights as to how uncertainty, renewable energy innovation, and fossil fuel scarcity impact our environmental policy decisions.

Chapter two investigates the topic of climate change skepticism, the fact that in spite of extensive scientific evidence, a number of people and policy makers believe that climate change is not caused by man-made emissions. In this chapter, I construct a model in which the level of policy makers’ skepticism is formally defined as the proba- bility that climate change is random rather than emission-driven. Another key feature of the study is that climate change damages are modeled as a potential catastrophe whose probability increases with global average temperature. I then solve the model for a variety of skepticism levels, identifying the optimal emission levels for fully-aware, fully-skeptical and moderately-skeptical policy makers. The main finding is that the as- sumption that damages are catastrophic has a much bigger effect on optimal emissions than the level of skepticism. Even full skeptics should favor some emission reduction, though unfortunately insufficient to prevent significant climate change, while moderate skeptics (those that believe that there is a 50% chance that climate change is anthro- pogenic) should favor emission reduction which limit temperature increases to below 2 degrees.

The third chapter also studies uncertainty in climate change economics, and focuses on renewable energy innovation. In this study, I model the firm’s decision to invest in developing cheaper renewables given that the prices of the fossil fuels that compete with these renewables are highly volatile. I demonstrate that the fossil price volatility can have two opposite effects. On the one hand, investing in making renewables cheaper can serve as a hedge against unpredictable fossil fuel prices, such that the return on investment increases with uncertainty. On the other hand, in uncertain market conditions there is an incentive to delay investment until there is more information

(i.e., until fossil fuel prices rise or drop significantly) pushing investment in renewables down. I find that the second consideration significantly dominates the first, resulting in an overall negative effect of volatility on renewable energy innovation.

When considering optimal policy to phase in renewables instead of dirty fossil fuels, it is also important to consider the variety of fossil fuel types. The fourth chapter examines the optimal renewables subsidy when multiple types of fossil fuels are present in the market. The model considers three energy sources - dirty and abundant coal, relatively cleaner but scarce oil, and expensive renewables. Together with my co-author Jacob Jannsenn I use this model to calculate the optimal subsidy that a government should implement in case the first-best tool (a carbon tax) is not available. The main result is that the government should wait until all oil is depleted, and a fair amount of coal has been utilized to subsidize renewables. Our results confirm other findings in the literature that subsidies are largely ineffective in battling climate change.

Unfortunately, fossil fuels, instead of being distributed uniformly, are heavily con- centrated in a small set of nations, giving these nations significant monopoly power in the fossil fuel market. It is important to consider this phenomenon when thinking about optimal climate change policy, as the regions where these fossil fuel stocks are concentrated are not always concerned with global well-being. The fifth chapter studies the case when the fossil fuel reserves are concentrated in one region (a monopolistic exporter) while another region (an environmentalist importer) is concerned with cli- mate change. Together with my co-authors, Professors Rick van der Ploeg and Cees Withagen, I construct a framework where Oilrabia (the exporter) and Industria (the importer) play a game, in which Oilrabia sets the price of oil while Industria sets the carbon tax. The key focus of this study is to investigate how the nature of oil demand in Industria affects its efforts to battle climate change and steal Oilrabia’s monopoly rents. To achieve this goal we solve the model for a variety of demand functions. It is demonstrated that for the HARA class demand functions, the oil exporter is the conservationist’s best friend, with the competition between the exporter and the im- porter increasing oil prices, and hence reducing emissions. However, a key new result is that there are demand functions for which the monopolist actually has an incentive to increase initial extraction causing more climate change to occur.

In addition to the theoretical findings each study contains useful quantitative in- sights for environmental policy. The second chapter presents specific emission paths for a variety of skepticism levels, which can serve as a tool of persuasion to moder- ately skeptical policy-makers and voters for supporting climate change mitigation. The third chapter suggests that in order for significant cost reduction in renewables to be achieved, the government may need to increase renewable R&D subsidies in times of volatile fossil fuel prices. On the other hand, the fourth chapter argues against subsi-

gas, pushing for a carbon tax when at all possible and only for a limited use of subsidies to phase out dirty coal. The fifth chapter’s results serve as a cautionary tale against making strong assumptions about oil demand in the importer country in environmen- tal policy calculations. Hence, its main lesson for policy makers is to consider the full demand structure when calculating the optimal carbon tax, rather than assuming a single elasticity parameter or a simplistic linear demand function.

Although I try to address some of the complexities of the climate change problem in my models, every one of them can be explored further leaving plenty of room for further research. The second chapter focuses on a fairly extreme case of catastrophic climate change damages. While some analysis is performed to test the robustness of the results to the extreme specification of catastrophic damages, fully solving a model with less extreme climate change damages would lead to even more understanding about when skepticism can be damaging. A full robustness check testing how using different functional forms (i.e. CES production function, CARA utility) affects the incentives and ability of the moderate skeptic to prevent climate change would yield a more complete picture of the problem. Likely these exercises will confirm some of the findings, although for many variations the differences between the fully skeptic policy maker and the moderately skeptic one would be less stark. Another area of further study is to introduce learning into the model, similar to many other studies on climate change (Kelly and Kolstad, 1999; Leach, 2007; van Wijnbergen and Willems, 2015). Extending the model in this fashion would further determine whether skepticism can lead to significant climate change damages, or whether learning can mitigate the initial skepticism level (an unlikely result given the existing literature on learning in climate economics). For the remaining three studies, the next step would be to extend the models to general equilibrium. Adding capital accumulation and an endogenous inter- est rate would not only produce more convincing results but also lead to a variety of new insights. In the third chapter, capital accumulation would provide an additional channel to hedge against uncertainty, decreasing the attractiveness of renewable energy investment. For the fourth chapter the ability to accumulate capital, would lead to an optimal renewable subsidy that is variable, rather than flat. Additionally for the third chapter, a natural extension would be to add knowledge spillovers, common in the innovation literature and even explored in some renewable energy innovation studies (for example Fischer et al. (2013)). Furthermore, as climate change is a global prob- lem involving multiple sovereign nations, introducing multiple agents into the models studying skepticism and renewable energy innovation is another promising direction for providing relevant insights. A model with multiple nations with various levels of skep- ticism is likely to demonstrate the difficulty of achieving an international agreement

on climate change, a particularly relevant result for climate negotiation.

Lastly, though I have briefly discussed policy implications throughout my disserta- tion, steps can be taken to make the analysis even more policy-relevant. Most of the studies above do not explicitly model policy decisions. Hence, formally adding policy decision variables, and finding the optimal levels of government intervention, should confirm the initial insights presented in the studies. This would be particularly useful for studying renewable energy innovation, as a variety of policy measures (from inno- vation subsidies to renewable subsidies or a carbon tax) can be examined. In this vein, an interesting extension for the second chapter on skepticism is to determine whether policy makers of different levels of skepticism prefer a carbon tax or a carbon permit scheme. Another step that would increase the policy relevance of the results in this dissertation is robust empirical analysis. While there is some calibration done in all the models, performing an empirical study to find evidence of the effects predicted by theory in data would significantly strengthen the policy-oriented conclusions. For example, Kellogg (2014) empirically calculates the effect of fossil fuel price volatility on drilling investment, finding evidence of an investment reduction in line with real option theory. A similar empirical study finding evidence of that effect on renewable energy R&D, would provide a compelling argument for subsidizing innovation when fossil fuel prices are volatile. This work is left for future research.

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