Business, Sauder School of
Strategy and Business Economics, Division of
DSpace
UBCV
Siami, Navid
Doctor of Philosophy - PhD
University of British Columbia
This dissertation is a collection of three essays that studies decision-making under uncertainty in a crisis context. The first essay examines the rational behavior of decision-making authority in providing water to a municipality facing drought. The forward-looking decision-making authority optimally chooses the size and time of building a desalination plant. The essay formulates the question as a dynamic programming problem with uncertainty in rainfall. By solving the problem numerically, the essay studies the behavior of the model through simulation. The correlation patterns produced by the simulations from the model are consistent with the correlation patterns observed in real-world data. The second essay examines the interactions of three different green innovations to combat climate change. It revisits Jevons’ paradox, which states that if demand is sufficiently elastic, an improvement in the fuel efficiency (FE) increases the flow of fuel consumption and, in the modern context of climate change, the flow of carbon emissions. An improvement in fuel efficiency also increases the stock of total carbon emissions. However, with carbon capture and storage (CCS), the effect on the total carbon emissions stock depends on the time pattern of emissions. Strong enough innovation in CCS reverses the paradoxical impact of FE. A similar (but reversed) logic holds for innovation in clean energy (CE) backstops. CE innovations reduce the stock of total carbon emissions. However, the improving CCS technology reduces the benefits of clean energy innovation, and strong enough, CCS reverses their beneficial impact. The third essay examines the interaction of re-election pressure and asymmetric information in politicians’ decision-making. Politicians and voters update their beliefs in a Bayesian way. Politicians have asymmetric information and seek re-election. The closer the election, the more the incumbent politicians refrain from implementing the welfare-maximizing policy and deviate towards their voting base’s bias. The theoretical model’s implications are tested with a dataset on gubernatorial decisions during the national Covid-19 crisis. A difference-in-difference empirical strategy shows that the governors who had an upcoming election in 2020 were biased towards their base. The Democrat and Republican governors who did not face a forthcoming election behaved statistically similar to each other.
Three Essays on Crisis Managementunder UncertaintyDrought, Climate Change, and PandemicbyNavid SiamiBSc, Sharif University of Technology, 2012MSc, The University of British Columbia, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Business Administration - Strategy and Business Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2021© Navid Siami 2021The following individuals certify that they have read, and recommend to the Facultyof Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:Three essays on crisis management under uncertainty: drought, climate change, and pandemicsubmitted by Navid Siami in partial fulfillment of the requirements forthe degree of Doctor Of Philosphyin Business Administration - Strategy and Business EconomicsExamining CommitteeJames Brander, Business Administration - Strategy and Business Economic, UBCCo-supervisorRalph Winter, Business Administration - Strategy and Business Economic, UBCCo-supervisorWerner Antweiler, Business Administration - Strategy and Business Economic, UBCUniversity ExaminerBrian Copeland, Economics, UBCUniversity ExaminerPaul Schrimpf, Economics, UBCSupervisory Committee MemberAdditional Supervisory Committee Members:Alvaro Parrra, Business Administration - Strategy and Business Economic, UBCSupervisory Committee MemberiiAbstractThis dissertation is a collection of three essays that studies decision-making under uncer-tainty in a crisis context.The first essay examines the rational behavior of decision-making authority in providingwater to a municipality facing drought. The forward-looking decision-making authorityoptimally chooses the size and time of building a desalination plant. The essay formulatesthe question as a dynamic programming problem with uncertainty in rainfall. By solvingthe problem numerically, the essay studies the behavior of the model through simulation.The correlation patterns produced by the simulations from the model are consistent withthe correlation patterns observed in real-world data.The second essay examines the interactions of three different green innovations to com-bat climate change. It revisits Jevons’ paradox, which states that if demand is sufficientlyelastic, an improvement in the fuel efficiency (FE) increases the flow of fuel consumptionand, in the modern context of climate change, the flow of carbon emissions. An improve-ment in fuel efficiency also increases the stock of total carbon emissions. However, withcarbon capture and storage (CCS), the effect on the total carbon emissions stock dependson the time pattern of emissions. Strong enough innovation in CCS reverses the para-doxical impact of FE. A similar (but reversed) logic holds for innovation in clean energy(CE) backstops. CE innovations reduce the stock of total carbon emissions. However,the improving CCS technology reduces the benefits of clean energy innovation, and strongiiiAbstractenough, CCS reverses their beneficial impact.The third essay examines the interaction of re-election pressure and asymmetric in-formation in politicians’ decision-making. Politicians and voters update their beliefs in aBayesian way. Politicians have asymmetric information and seek re-election. The closerthe election, the more the incumbent politicians refrain from implementing the welfare-maximizing policy and deviate towards their voting base’s bias. The theoretical model’simplications are tested with a dataset on gubernatorial decisions during the national Covid-19 crisis. A difference-in-difference empirical strategy shows that the governors who had anupcoming election in 2020 were biased towards their base. The Democrat and Republicangovernors who did not face a forthcoming election behaved statistically similar to eachother.ivLay SummaryThis thesis aims to provide a framework to improve our understanding of crisis managementand decision-making under uncertainty. To achieve this goal, I studied three different crisesand relevant decisions around those crises.The first essay concerns local droughts and the decision of cities to install some waterdesalination capacity to combat drought. In the second essay, I studied the role of innova-tion in the fight against the global warming crisis. The last essay concerns the 2020 covidpandemic and the decisions of politicians to close down economic activity.The contribution was to model different aspects of crisis decision-making and showthe implications of uncertainty in different situations. It is possible to use some of theframeworks developed in this paper to study the behavior of decision-makers and measurethe quality of their decisions.vPrefaceChapters two and four of this dissertation are my original independent work. Chapterthree is joint work with one of my advisers, Professor Ralph A. Winter of the UBC SauderSchool of Business. In this chapter, we collaborated on developing the propositions andtheir proofs. Part of this chapter is published as Siami, Winter (2021) in the journalEconomics Letters under the title of “Jevons’ paradox revisited: Implications for climatechange.”viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Drought . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13viiTable of Contents2.4.1 Variables and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 142.4.2 Loss Function and Constraints . . . . . . . . . . . . . . . . . . . . . 162.4.3 Bellman Equation and Solution Concept . . . . . . . . . . . . . . . 182.5 Testing the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Climate Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3.1 The Green Innovation Portfolio and Damage Function . . . . . . . . 503.3.2 Vanilla Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4 Clean Energy Innovation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.1 Unexpected CE innovation on its own is a boon . . . . . . . . . . . 563.4.2 CE-CCS interaction: The devil is in the details . . . . . . . . . . . 573.4.3 CE innovation threat and a green paradox . . . . . . . . . . . . . . 593.5 Fuel Efficiency Innovation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.5.1 Paradoxical effect of an unexpected FE innovation . . . . . . . . . . 623.5.2 CCS innovation saves the day: Inelastic demand . . . . . . . . . . . 633.5.3 CCS-FE interaction: General demand . . . . . . . . . . . . . . . . . 643.5.4 Last piece of the puzzle: the threat of FE innovation . . . . . . . . 653.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 Pandemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75viiiTable of Contents4.3 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3.2 Solution for a unified public . . . . . . . . . . . . . . . . . . . . . . 814.3.3 Solution when the voters are sorted . . . . . . . . . . . . . . . . . . 834.3.4 Endogenizing the politician’s information gathering . . . . . . . . . 894.4 Empirics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4.1 Empirical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.4.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.4.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110AppendicesA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119A.1 Optimal Utilization Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119A.2 Mistakes in Probability Estimation . . . . . . . . . . . . . . . . . . . . . . 122A.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126A.3.1 Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127A.3.2 Estimation of demand elasticity . . . . . . . . . . . . . . . . . . . . 129A.3.3 Estimation of utility function . . . . . . . . . . . . . . . . . . . . . . 131B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137B.1 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137ixTable of ContentsB.2 Extending the CCS Technology . . . . . . . . . . . . . . . . . . . . . . . . 141B.2.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141B.2.2 Clean-energy innovation . . . . . . . . . . . . . . . . . . . . . . . . 143B.2.3 Fuel-efficiency innovation . . . . . . . . . . . . . . . . . . . . . . . . 145C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146C.1 On the Microfoundations of the Loss Fucntion . . . . . . . . . . . . . . . . 146xList of Tables2.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4 Simulated Data, Random Effects . . . . . . . . . . . . . . . . . . . . . . . . 382.5 Actual Data, Random Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 Party Affiliation and Election . . . . . . . . . . . . . . . . . . . . . . . . . . 984.2 Closing Days—Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . 994.3 Re-opening Days—Summary Statistics . . . . . . . . . . . . . . . . . . . . . 994.4 Main Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.6 Robustness to Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.7 Robustness to Measures of Severity . . . . . . . . . . . . . . . . . . . . . . . 104A.1 Variables Affecting Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128A.2 Demand Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130A.3 Demand Elasticity—IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131A.4 Fixed Effects—Capacity Built Is the D.V. . . . . . . . . . . . . . . . . . . . 132A.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136xiList of Figures2.1 Capacity building, bad rainfall . . . . . . . . . . . . . . . . . . . . . . . . . 262.2 Capacity building, good rainfall . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 Next period water reserves, bad rainfall, low fixed cost . . . . . . . . . . . . 282.4 Next period water reserves, good rainfall, low fixed cost . . . . . . . . . . . 292.5 Next period reserves, good rainfall, high fixed costs . . . . . . . . . . . . . . 302.6 Slice of future water reserves vs. current capacity . . . . . . . . . . . . . . . 312.7 Next period reserves, bad rainfall . . . . . . . . . . . . . . . . . . . . . . . . 322.8 Continuation value, bad rainfall . . . . . . . . . . . . . . . . . . . . . . . . . 332.9 Continuation value, good rainfall . . . . . . . . . . . . . . . . . . . . . . . . 344.1 Daily β3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102A.1 Realized capacity over time . . . . . . . . . . . . . . . . . . . . . . . . . . . 120A.2 Usage over time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121A.3 Average capacity over time . . . . . . . . . . . . . . . . . . . . . . . . . . . 122A.4 Average capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123A.5 Average utilization rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.6 Effects of mistakes on capacity . . . . . . . . . . . . . . . . . . . . . . . . . 125A.7 Effects of mistakes on utility . . . . . . . . . . . . . . . . . . . . . . . . . . . 126A.8 Relative rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133xiiAcknowledgementsFirst and foremost, I want to thank my advisors at Sauder School of Business, ProfessorsJames Brander and Ralph Winter, for their continuous support of my PhD studies. Theyspent generous amounts of their time to help me understand the nuances of thinking like aneconomist. I owe a lot to Professor Brander for teaching me how to write like an economist,and specifically helping me to organize my thoughts with his feedback on the second andfourth chapters. I owe a lot to Professor Winter for teaching me how to create precisemodels. Also, he was my coauthor on the third chapter of the thesis and has taught methe rigor of conducting high-quality research. He was also very generous with his academicresources and acquired the data used in the second chapter of my thesis.After that, I want to thank to my advisor at Vancouver School of Economics, ProfessorPaul Schrimpf, for his help. Whenever I was stuck on a minute detail, he was more thangenerous with his time and patience. He helped me think through mathematical proofsand computational technics, especially in the second chapter of my thesis.I need to thank Professors Alvaro Parra, Werner Antweiler, Tom Ross, Sanghoon Lee,and Limin Fang, who were supportive all the time and were generous with their feedbacks.I have learned a lot from them by discussing my problem in their offices, in division’sannual parties, and over a cup of coffee.Also, I want to thank all the faculty and students at Sauder School of Business forbeing supportive and providing suggestions during and after seminars.xiiiAcknowledgementsI want to specifically thank Julian Vahl, a fellow PhD student at Sauder for being aneverlasting source of interesting ideas and rigouros thoughts. I came up with the idea ofmy forth chapter during a walk with him on the Spanish Banks in Vancouver. He alsoproofread many of my works and made my words much more clear.All the shortcomings in this thesis are mine.xivDedicationI want to dedicate my work to the memory of Herbert A. Simon. the genius who insistedthat our rationality is bounded, yet we need to think about this limited rationality in arational way, even though it may be limited.xvChapter 1IntroductionThis thesis is a collection of three essays that study decision-making under uncertainty inthe context of a crisis. The first essay studies the dynamic decisions of municipal author-ities to provide desalination plants to combat drought. In this essay, I try to answer thisquestion, “What is the optimal time to build a desalination plant, and what is its optimalcapacity when there is uncertainty in rainfall?” and test whether the behavior of munici-palities matches the optimal predicted behavior. The second essay studies interactions ofdifferent exogenous green innovations and how they affect the long-term cumulative green-house gasses in the biosphere and combat climate change. In this essay, my coauthor and Itry to answer this question,“In the absence of the first-best policy of emission pricing, howdoes green innovation affect the total cumulative emission?” and shed light on the optimalsecond-best policies. The third essay studies the decisions of gubernatorial authorities tolimit economic activities to combat the pandemic. In this essay, I address this questions,“How does the ballot box pressure affect the decisions of the politicians to limit economicactivity” and find out “how the interaction of uncertainty and ballot box pressure candeviate politicians from implementing the welfare-maximizing policy.”The first essay examines the rational and forward-looking behavior of benevolent decision-making authority. The decision maker solves a dynamic programming problem with un-certainty in rainfall to optimally choose the size and time of building a desalination plant.We can observe correlation patterns between the model parameters and the time and size1Chapter 1. Introductionof building a plant through simulation of multiple scenarios. I test these patterns againstthe real-world data and show that they are consistent.The second essay examines the effect of different green innovations and how they inter-act as a portfolio. Without correct carbon pricing, innovations are the second-best policiesto fight the adverse effects of climate change. Economists have been aware of unintendedconsequences and paradoxical effects of green innovations, at least since Jevons. Jevonsparadox states that if demand is sufficiently elastic, an improvement in the fuel efficiency(FE) of internal combustion engines increases fuel consumption flow and, in the moderncontext of climate change, raises the flow of carbon emissions. This paper revisits Jevons’paradox and show that an improvement in fuel efficiency increases the stock of total car-bon emissions. However, if carbon capture and storage (CCS) is operating, the effect onthe total carbon emissions stock depends on the time pattern of emissions. With strongenough innovation in CCS, the paradoxical impact of ICE efficiency is reversed. A similar(but reversed) logic holds for innovation in clean energy backstops. Clean energy innova-tions reduce the stock of total carbon emissions on their own. However, an improving CCStechnology reduces the benefits of clean energy innovation, and when strong enough, CCScan reverse their beneficial impact.The third essay examines the interaction of re-election pressure and asymmetric infor-mation in decision-making under uncertainty by politicians. Politicians and voters formtheir beliefs and update their information in a Bayesian way. Politicians have asymmet-ric information and seek re-election. This paper shows that given exogenous informationstructure, election timing interacts with the decision quality of the politicians. The closerthe election, the more the incumbent politicians refrain from implementing the welfare-maximizing policy and deviate toward their voting base’s bias. Moreover, endogenizingthe politicians’ information gathering leads to a higher level of information asymmetry2Chapter 1. Introductionbetween the politicians and voters. The theoretical model’s implications are tested witha dataset on gubernatorial decisions during the national Covid-19 crisis. A difference-in-differences empirical strategy shows that the governors who had an upcoming election in2020 were biased towards their base. The Democrat and Republican governors who didnot face a forthcoming election behaved similar to each other statistically.Economists use uncertainty to describe a situation where an agent, a player, or adecision-maker has imperfect information relevant to their decision-making objective. Thesource of uncertainty may be natural (as in the realization of next month’s rainfall orrealization of global warming), or it could be about other agents’ beliefs and actions. Thisthesis studies both of these situations.Another distinguishing dimension between different uncertainty classes is whether theuncertainty is quantifiable (risk) or non-quantifiable (Knightian). On one end of this spec-trum lies the perfectly quantifiable uncertainty, where all possible outcomes and the prob-ability of each realization are perfectly known beforehand (like rolling a dice). On theother end of this spectrum is the idea of a fundamental degree of ignorance, a limit toknowledge, and an essential unpredictability of future events. I used the term Uncertaintyin this thesis to refer to a quantifiable degree of imperfect knowledge.In the first essay, the primary source of uncertainty is the unpredictability of rainfall,and I assume that the decision-maker knows the distribution of rainfall. In the secondessay, uncertainty comes from the fact that innovation is not predictable. Although Imodel innovation with a Poisson process of a known magnitude that reduces the priceof an input, the results are more general and do not depend on this quantification forexpositional purposes. In the final essay, the source of uncertainty is how events unfold,and people’s beliefs evolve about specified actions. The underlying model is a Poissonprocess of known characteristics, but the results are more general like the second essay.3Chapter 1. IntroductionThe results will hold if people’s beliefs converge to the truth over time.In this essay, the crisis is used to describe a situation where the public and decision-makers believe that inaction (or status quo) leads to severe difficulty for the public witha relatively high probability. These challenging situations give unique opportunities tostudy the public decision-making process. The decision-makers are under pressure to makedecisions without having all the necessary information. Since the stakes are perceived tobe high, we expect decision-makers to pay much attention to these decisions. Moreover, acrisis occurs in a time of heightened uncertainty, making it much more essential to studycrisis management under uncertainty.In the first essay, a city faces a drought crisis, and the planners have to decide whetheror not to build a desalination plant to combat that risk. The second paper concerns theclimate change crisis and how innovation affects the long-run climate outcome. The finalessay concerns the Covid crisis and the political decisions to contain the virus spread.All three chapters tackle the same fundamental question of crisis management under un-certainty, though the methodologies vary widely. Although the science of decision-makingis at least as old as the ancient Greeks, the application of various mathematical and com-putational tools to everyday decision-making is relatively young, and the field suffers froma lack of consensus among its practitioners. One group of practitioners, mainly compu-tational economists and rational expectation theorists, emphasize the role of probabilisticrational dynamic models to explain the optimal behavior of decision makers directly. Withthe recent progress in computation power and the computational tools developed around it,more scientists are attracted to this camp of thinking about decision-making under uncer-tainty. The first essay of this thesis is primarily in line with the concerns and standards ofthis field—the first model detailed aspects of the decision-making problem at hand. Thentry I solve the model that resembles reality as much as the computational power allows4Chapter 1. Introductionand compare the model’s predictions with what is observed in the real world to see if thereis a good match between the models and reality.Another group of practitioners, mainly empirical economists and applied theorists, be-lieve that many real-world limitations play an essential role in decision-making. Therefore,an indirect approach to the problem is necessary. For example, in many cases, even thoughthe first best solution is clear, it is impossible to be implemented due to complex politicaland psychological biases. The second essay is in line with this style of thinking. In thisessay, we assume the first-best policy (carbon pricing) cannot be implemented withoutanalyzing the underlying political reasons. Therefore, instead of studying the dynamics ofthe interaction between the environment, economy, and decision makers, we focus on onemeasure of environmental variables (cumulative emissions in the biosphere) and study theeffect of one set of economic variables (an innovation portfolio). There is ample scientificevidence that cumulative emissions are a good predictor of the long-term effect of climatechange, and in this paper, our focus is on long-term decision-making under uncertainty.The third group of practitioners, primarily political economists and industrial orga-nization theorists, focuses on the incentives of the decision maker and aims to model aspecific behavior of a decision maker, rather than to assume that the decision maker isbenevolent or it actually solves the problem we expect a benevolent decision maker tosolve. The third essay is in line with this kind of thinking. Politicians, who are supposedto act quickly in an uncertain environment of international crisis (e.g., pandemic), look attheir electoral incentives and makes decisions that maximize their chance of re-election,rather than maximizing social welfare.These three essays tackle the problem of decision-making by finding an optimal solutionin a simplified world. For the sake of completeness, I must add that in contrast to theseways of thinking, there is also a fourth group of practitioners, primarily behavioral and5Chapter 1. Introductionexperimental economists, who put an emphasis on the limitations of human rationality.They study psychological biases, heuristics, and actual behaviors of decision makers tomake a satisfactory decision in a more realistic and complex world. In this thesis, I do notstudy this important approach to decision-making.6Chapter 2Markovian rainfall andDesalination Demand2.1 IntroductionWater shortages have caused significant problems throughout history. At present, manycities struggle with water supply problems, especially in areas where droughts are common.Desalination, the technology that turns saltwater into freshwater, has become an importanttool for ensuring adequate water supplies. As of 2019, more than 300 million people reliedon desalination for some or all of their daily water needs.However, desalination plants are costly to build and costly to operate. Therefore,building or expanding desalination capacity is an important and difficult decision for manypolitical jurisdictions, especially because criticism of such decisions is commonplace. Some-times current desalination capacity is insufficient to meet current municipal needs, and yetsometimes, costly excess capacity is left unused. One source of difficulty in making decisionsis the underlying randomness of rainfall patterns. Merely observing ex post insufficient orexcess desalination capacity does not necessarily mean that one has made poor decisions.Studying decision-making with rainfall uncertainty is the focus of this paper.The main research question is whether the observed decisions are consistent with arational expectation model of rainfall. First, I develop a model for how a rational decision72.1. Introductionmaker provides desalination capacity optimally. Then I ask whether the actual records ofdesalination capacity construction are consistent with the model.The decision to build some capacity depends on how this capacity will be used. There-fore, both aspects of building and operating the desalination plant are explicitly presentin the dynamic model. The model is a dynamic programming problem with uncertaintyin rainfall with numerous state and control variables. I solve the model numerically forgiven parameter values to obtain the capacity trajectory. Simulating the model for differ-ent parameter values and different realizations of the uncertainty provides the comparativedynamics needed to characterize the model numerically.Then I compare the dynamics obtained from simulation with the dynamics observed inreal-world desalination construction decisions. The time-series data consist of all the munic-ipal desalination capacity built worldwide, the historical rainfall, temperature, population,and average GDP for more than 60 years. This comparison implicitly tests whether actualdesalination decisions can reasonably match the outcome of a rational decision-makingprocess.The correlation dynamics produced by the model are consistent with the actual dataon construction decisions. For example, one interesting property of the model is that thecities with higher variation in their rainfall patterns would, other things held equal, buildmore desalination capacity. A similar correlation is present in the data. Desalination worksas an insurance instrument against probable unfavorable rainfall outcomes.This analysis is consistent with the hypothesis that decision makers are, on average,approximating optimal decision rules. However, one advantage of explicit modeling ofrational decision-making is that it can be expanded and used as a decision-making tool.The model is flexible enough to incorporate the data on the actual usage of the plant.Section 2.2 contains a brief review of the related literature; section 2.3 summarizes the82.2. Literature Reviewdata, section 2.4 studies the theoretical model. Section 2.5 tests the theoretical model, andsection 2.6 concludes.2.2 Literature ReviewThis paper combines two aspects of inventory management and investment under uncer-tainty in the context of water provision with desalination. These aspects are unified ina single dynamic programming equation that takes short-run and long-run trade-offs intoaccount. Therefore, the current paper is related to a broader literature in inventory man-agement (e.g., Gustafson (1958), Samuelson (1971) , Scheinkman, Schechtman (1983), andAsche et al. (2015)) and investment under uncertainty which was pioneered by Lucas,Prescott (1971), Kydland, Prescott (1977), and Kydland et al. (1982).The first application of dynamic programming to water management dates back tothe works of Burt (1964), who studied the optimal allocation of groundwater. Tsur et al.(1991) improved on Burt’s work and showed the value of groundwater as an insuranceoption (or, as they call it, a buffer) to surface water supplies. Truong (2012) studied theeffects of water storage capacity on the agricultural sector.Xie, Zilberman (2018) showedthe substitutability and complementarity patterns of water storage capacity and water useefficiency. This paper is the first to apply dynamic programming to water managementwith the option to build a desalination plant.The literature on desalination plants has recognized the importance of desalinationplants as an insurance strategy to mitigate the risk of drought (e.g., Schoengold, Zilberman(2007), Bensoussan, Farhi (2010), Clarke (2014)). Scholars have also studied factors thataffect desalination demand (Ghaffour (2009), Ghaffour et al. (2013) , and Amy et al.(2017) and how to prepare for future capacity building (?. But a quantitative analysis ofdesalination as an insurance policy is missing from the literature.92.2. Literature ReviewThis paper contributes to this literature by highlighting the role of uncertainties in-troduced by rainfall patterns and the trade-offs that municipalities face in their optimalchoice. Here, I develop a framework for studying the optimal timing of construction andoptimal size of desalination plants.Many economists used dynamic programming problems with continuous decision vari-ables and inequality constraints to model optimal timing problems (e.g.,Deaton (1991)).The standard way to deal with inequality constraints is to discretize the choice space torule out any violation of the inequality constraint by design. This discretization increasesthe dimension of the state space, so it suffers from the curse of dimensionality.Rendahl (2015) proposed the method of time iteration. This is a faster method basedon the corresponding problem’s Euler equation. This method is faster than traditionalvalue function iteration methods when we need to discretize a continuous state space. Inthis paper, I implement the time-iteration algorithm to solve the dynamic programmingmodel.It is necessary to clarify the relationship between this work and relatively well-developedliterature on dynamic choice models because this literature empirically studies dynamicdecision making under uncertainty. The famous bus engine replacement model by Rust(1987) is suitable for comparing this literature and the current work. Rust estimated adynamic discrete choice model by solving a fixed-point problem for every iteration of theestimation procedure because, for every iteration, a fixed-point problem has to be solved;his method suffers from the curse of dimensionality (meaning that the state space growsexponentially as the number of state variables increase).Hotz, Miller (1993), Ai, Chen (2003), and Su, Judd (2012) allow the econometriciansto sidestep one aspect of the curse of dimensionality. They directly estimated the valuefunctions and inferred the model’s parameters from the estimated value functions without102.3. Datathe need to calculate a fixed-point problem on each iteration.The current work contributes to understanding dynamic models in two ways. First,rather than estimating the parameters (or a family of non-parametric functions) of anunderlying dynamic model, in this paper, I test whether or not the dynamics producedby the model match with the real-world data. Second, these models are data intensive.Multiple observations of each decision maker are necessary to estimate the underlyingmodel. With the proposed simulation approach in this paper, there is not need to estimateeach agent’s behavior explicitly. Instead, I compare the collective patterns produced bythe model’s simulation and what is observed in the real-world data.2.3 DataThe information on desalination plants is from the DesalData.com. This database hascomprehensive information on all desalination plants built all around the globe. The ge-olocation of each plant, the building date, and information such as capacity, technology,and construction prices are variables of interest. There were 19,232 plants in the data asof March of 2017. I selected 3,262 plants in municipalities with a population of more than300,000 in January 2017. This criterion narrows down the focus of this paper to municipaldemand for desalination plants.The data on the cities’ population are from the population division of the UnitedNations. This dataset consists of information on the population of cities with a currentpopulation larger than 300,000. The geolocation of these cities is used to match eachdesalination plant with the corresponding city. A match happens if a desalination plant iswithin 100km of the city’s centroid. If a desalination plant is within the boundaries of twocities, the closest city is chosen.112.3. DataTable 2.1: Descriptive StatisticsVariable Min. Median Mean Max.Capacity 1 1000 14388 880000Nominal Prices 3.500e+04 1.170e+06 3.164e+07 1.800e+09Population(/1000) 0.0 668.5 2064.6 35861.0Average Rainfall 4.39 309.15 620.07 3452.37Average Temperature 3.118 23.194 21.330 29.168GDP per Capita 168 16094 20333 88565-The data on rainfall are from the U.S. Department of Commerce—National Oceanicand Atmospheric Administration website. The data used in this paper are monthly averagerainfall from 1900 onward on a 1◦by 1◦grid. Each city is matched to its closet grid. Themonthly rainfall levels are summed up to provide the annual rainfall for each city. Long-term average rainfall (mean rainfall from 1900 to the present), the long-term standarddeviation of rainfall, and each year’s rainfall average are calculated and used in the analysis.The data on temperature are from CRUTEM (Berkeley). The resolution of the datais similar to the rainfall data, and all the relevant variables are similarly calculated. TheGDP per capita and populations are from World Development Indicators of the WorldBank. I calculated the year-to-year GDP per capita from this source.122.4. Demand2.4 DemandThe theoretical framework for modeling desalination is a time-discrete dynamic program-ming problem. A benevolent central planning authority wants to provide water to a city.The planning authority faces a trade-off between two types of losses, population dissatis-faction, and monetary costs.If it does not provide enough water, the population gets dissatisfied, and that incursa loss. However, there are long-term and short-term costs associated with building andrunning the plant, respectively. In the long-term, the planning authority chooses how muchdesalination capacity to build. In the short run, it has to decide how much of the installedcapacity to use.There are three main mechanisms that derive the investment behavior and usage rateof the model: the trade-offs between utilizing the plant and not providing enough water,the trade-off between building the plant (and incurring a cost) now and running the riskof not having enough water, and the interaction of these trade-offs.The first trade-off is easy to understand if we assume the city’s optimal water need isconstant and it has built all of its required desalination plants or cannot build any. In bothsituations, the planner acts similar to an agent that wants to smoothen its consumptionover time. In the first case, if the planner’s constraints bind, they will provide enoughwater such that the marginal utility of providing water in each period equals the marginalcost of providing desalinated water. In the second case, the planner acts like a consumerwho maximizes its consumption utility intertemporally when its income is uncertain.The second trade-off is easy to understand if we assume that the planner has to eitherprovide a fixed amount of water to the municipality (if it has the water) and zero otherwise.This planner will not build a plant if there is enough water to provide next year no matterwhat the realization of rainfall is and would always build a plant if it does not have enough132.4. Demandwater to provide again no matter what the realization of rainfall. The planner is indifferentbetween building and not building if the expected value of not providing water equals thecost of building the plant. So there is a threshold amount of water in the reservoirs abovewhich the planner would not build and below which would always build.The interaction effect comes into play while cities have not yet transitioned to theirsteady-state equilibrium level of desalination capacity. Before transitioning to the equilib-rium level of capacity, if the planner not only takes next-period water needs into accountbut also considers that if in this year they provide more water to the city or desalinate justenough water to meet this year’s needs, then they increase the chance of having to build aplant in the following year. Therefore, before reaching the equilibrium levels, the plannerdesalinates more water and provides less water than if it had an equilibrium capacity level.This effect can only be captured in a dynamic programming framework.2.4.1 Variables and DynamicsThree state variables describe the dynamic programming problem:1. Kt desalination capacity at the beginning of period t;2. Wt water reserves available at the beginning of period t;3. Rt rainfall in period t.All these states are observable at the beginning of the period. After observing these statevariables, the planner decides on three variables:1. wft amount of fresh water to provide;2. wdt amount of desalinated water to provide;3. xt amount of capacity to build.142.4. DemandDefine S = (Kt,Wt, Rt) as the vector of state variables and d = (wft , wdt , xt) as the vectorof decision variables.The state variables evolve according to three dynamic equations Dt. The evolution ofcapacity is deterministic; capacity in next period is the capacity from the beginning of thisperiod plus the capacity built in this period. The plants are assumed to have an infinitelife-time.Kt+1 = Kt + xt (2.1)The evolution of water reserves is deterministic conditional on current period rainfall.Water reserves available during the period t (Wt+1) comprise the sum of current rainfallRt, and water remained from the last period minus the freshwater provided.Wt+1 = Wt +Rt − wftThis equation does not take water evaporation into account; therefore, the amount ofreserves might diverge. Taking water evaporation into account, I modify the equation inthe following way, where (θ) is the fraction of water that evaporates from the reserves:Wt+1 = (1− θ)Wt +Rt − wft . (2.2)Rainfall follows a Markov distribution. For simplicity, we assume that rainfall can onlyhave two states (h,l) with corresponding rainfall levels Rh, Rl. We denote the probabilityof remaining in state i by pi. Finally, the transition matrix is given by:Θ = ph 1− ph1− pl pl . (2.3)152.4. DemandThe first element in the first row of Θ is the probability that the state remains in highrainfall (ph), and the second element is the probability of moving from high to low rainfall(1− ph). In the second row, the first element is the probability of moving from low to highrainfall (1 − pl). The second element is the probability of remaining in the low rainfallstate.The variable Rt is known at time t. Its future value Rt+1 is a probabilistic functionof current value. Variation in rainfall is the only source of exogenous uncertainty in thismodel.Equations 2.1, 2.2, and the Markov processes described in 2.3 constitute the dynamicequations. These equations fully describes the transition of the state St+1 = Dt(St, dt).2.4.2 Loss Function and ConstraintsIn each period, the planning authority incures a per-period loss of lt. Its goal is to minimizethe stream of discounted (β) expected losses subject to constraints (Gt ≥ 0) and dynamics(St+1 = D(St, dt)).L = mind0,d1,...∞∑t=0βt(Et[lt]) s.t. Gt ≥ 0, St+1 = D(St, dt) (2.4)In equation 2.4 Et is the expectation conditional on the informations available at timet.This per-period loss does not directly depend on time. It depends on time only throughthe state of the system (St) and the planner’s decision (dt) at time t. Therefore, it is possibleto drop the t subscript on l and write is as a function of St, dt:lt ≡ l(St, dt). (2.5)162.4. DemandThe loss in each period consists of three terms.l(St, dt) = (αP − (wft + wdt ))2︸ ︷︷ ︸1+ (FI(xt ≥ 0) + κxt)︸ ︷︷ ︸2+ cwdt︸︷︷︸3(2.6)1. the loss of the planning authority from water underprovision;2. the cost of building new desalination capacity;3. the cost of using current capacity.The first term in equation 2.6, P , represents population of the city. On average, eachcity targets α units of water per capita annually. The planner provides wft + wdt liters ofwater to the city dwellers. Any deviation from the target level gives the planner a quadraticloss. This damage is meant to be a convex function of the amount of under-provision. Asecond-degree polynomial simplifies the computation while preserves the convexity.The second term in equation 2.6 concerns the cost of building new capacity. It costs κto build one unit of capacity. The planner decides to build xt units of capacity. F modelsthe economies of scale, and I is the indicator function. If the planner chooses to build somecapacity, they must pay a fixed cost to build the plant.And finally, in the last term, it costs c to use one unit of an already built capacity.The last term is a linear function of the amount of provided desalinated water, which liesbetween zero and current desalination capacity.Note that the loss function defined as a convex utility function incorporates many as-pects of our perception when we think of water shortage. If available water is slightly belowthe abundance level (desired level), then society adopts simple measures to compensate forthis: water efficiency and conservation technologies would be adopted (e.g., lower pressuretap and showers, reuse water for specific purposes), certain activities would be limited (no172.4. Demandextravagance large fountain in the middle of a park), or water prices may increase. All ofthese incur some minor costs or cause a minor inconvenience.At the mid-size level of water shortage, the government may start to import water, banor severely limit certain water-consuming activities, or even enact some rationing measures.AT higher levels of water shortage, the government may limit the water supply to the bareminimum. Cities may start to leave cities, and people who remain may suffer diseases oreven premature death due to water shortage. A convex utility function captures all of theseeffects in a relatively simple formulation.The set of constraints (Gt ≥ 0) limits the choice set of the planner.1. xt ≥ 0 : the planner can only build new capacity and cannot reduce capacity.2. Kt ≥ wdt ≥ 0 : The desalinated water cannot exceed current capacity, and it cannotbe negative.3. Wt(1 − θ) + Rt ≥ wft ≥ 0 : The fresh water cannot exceed the water available, andit cannot be negative.2.4.3 Bellman Equation and Solution ConceptTo solve the stochastic dynamic programming problem, I first rewrite the loss functionas a Bellman equation, then simplify it analytically, and finally use known quantitativemethods to numerically solve it.Let’s start from the loss minimization problem of equation 2.4. Multiply both sides of2.4 by a minus sign and define V ∗(S0) ≡ −L. This converts the problem to a maximizationproblem shown in equation 2.7.V ∗(S0) = maxd0,d1,...(∞∑t=0βtEt[−lt]), s.t. Gt ≥ 0, St+1 = D(St, dt) (2.7)182.4. DemandV ∗(S), which we call ex ante value function (similar to ?) plays an import role insetting up the Bellman equation. This function tells us intuitively the lifetime value ofbeing in a state S. Bellman’s principle of optimality allows us to translate the infinitehorizon programming problem to a corresponding functional equation for the ex ante valuefunction of ?.Bellman’s principle of optimality states that the solutions of equation 2.7 is equivalent tothe solutions of the functional equation 2.8, if some minimal assumptions are met (theorem9.4 of ?). Equation 2.8 is called the Bellman equationV (S0) = maxd0(− l(S0, d0) + βE(V (S1))s.t. G(S0, d0) = 0, S1 = D(S0, d0) (2.8)In our case the decision space is Euclidean (d ∈ R3) and −l ≤ 0. Therefore, theconditions of theorem 9.4 of SL are satisfied. Hence, the solutions of equations 2.8 and 2.7are equivalent. Intuitively, equation 2.8 tells us that one obtains the ex ante value of beingin a state (before realization of uncertainties) by making the optimal decision to balanceout the current utility and future expected continuation value.Abstractly speaking, the Bellman equation maps from functional space (space of boundedfunctions from R3 to R) to itself with the sup norm. Equation 2.8 is concisely written asV = T (V ), where T is the right-hand side of equation 2.8 for any S0 including the con-straints.First, I need to show that this equation is a contraction mapping, meaning that thesup norm distance between two output functions of the functional will be smaller than thedistance between any two input functions. This means that for any two functions V1 and192.4. DemandV2 there exists some 0 ≤ k < 1 such that||T (V1)− (V2)||∞ ≤ k||V1 − V2||∞ (2.9)I take advantage of the Blackwell sufficient condition to prove that the current Bellmanequation is a contraction mapping. Blackwell Sufficient Condition states that (Theorem3.3 ?):Let X ⊆ Rl and B(X) be the space of bounded function f : X → R, with a supnorm.T is a contraction mapping with modulus β if:a. [Monotonicity] If f, g ∈ B(X) and ∀x ∈ X : f(x) ≤ g(x), Then ∀x ∈ X : (Tf)(x) ≤(Tg)(x)b. [Discounting] ∃β ∈ (0, 1) such that ∀f ∈ B(X), a ≥ 0, x ∈ X : [T (f + a)](x) ≤T (f)(x) + βaIt suffices to show that these monotonicity and discounting hold for the current Bellmanequation and that the Bellman equation is bounded on B(X). Let’s clarify the correspon-dence between the variables in Blackwell sufficient condition and our problem:X ⊆ R3, is the space of state triplet S := (Kt,Wt, Rt) ∈ XB(X) : the space of bounded functions on X → R, V (Kt,Wt, Rt) ∈ B(X)T := maxD0[u(D0, S0) + βpV (S1(D0) + β(1− p)V (S2(D0)]Proof.202.4. Demand[Monotonicity] :V1(S) ≤ V2(S), ∀S ∈ B(S) (∗)T (V1(S0)) = maxD0[u(S0, D0) + βpV1(S1(D0)) + β(1− p)V1(S2(D0))]DefineDˆ0 := arg maxD0[u(S0, D0) + βpV1(S1(D0)) + β(1− p)V1(S2(D0))]T (V1(S0)) = u(S0, Dˆ0) + βpV1(S1(Dˆ0)) + β(1− p)V1(S2(Dˆ0))from (*) :T (V1(S0)) ≤ u(S0, Dˆ0) + βpV2(S1(Dˆ0)) + β(1− p)V2(S2(Dˆ0))Then using the definition of max, the RHS of last equation is smaller than the RHS ofthe following:T (V1(S0)) ≤ maxD0[u(S0, D0) + βpV2(S1(D0)) + β(1− p)V2(S2(D0))]Therefeore,T (V1(S0) ≤ T (V2(S0))Monotonicity is satisfied.[Discounting]212.4. DemandT (V + a) = maxD0[u(D0, S0) + βp(V (S1(D0) + a) + β(1− p)(V (S2(D0) + a)]T (V + a) = maxD0[u(D0, S0) + βp(V (S1(D0)) + β(1− p)(V (S2(D0)) + βa]T (V + a) ≤ maxD0[u(D0, S0) + βp(V (S1(D0)) + β(1− p)(V (S2(D0))] + maxD0(βa)T (V + a) ≤ maxD0[u(D0, S0) + βp(V (S1(D0)) + β(1− p)(V (S2(D0))] + βaDiscounting is satisfied.[bounds]V is bounded from above by 0 and it is bounded from below bymin(u)1− βSince it is established that T is a contraction mapping, the Banach fixed point theorem(Theorem 3.2 in ?) guarantees that T admits a unique fixed point and that value functioniteration converges to that fixed point.The number of states and decisions are large (9 dimensions, 3 state variables, and 3 de-cision variables of period 0 and 3 state variables of period 1), so the curse of dimensionalitymakes the problem hard to solve numerically. We take three steps to make the problemmanageable.In the first step, note that the variable wdt only appears in the per-period loss function(l) and inequality constraints (G ≥ 0). It does not appear in the dynamic equation (St+1 =Dt). This allows us to take the derivative of equation 2.4 with respect to wd and obtain222.4. Demandthe first-order condition.We start from the Bellman equation 2.8 and use the Dynamic equation to replaceS1 = D(S0, d0). We can specify the exact dependence of S1 on S0 and d0 . This gives theequation:V (S0) = maxd0(− l(S0, d0) + βE(V (K0 + x0, (1− θ)W0 +R0 − wf0 , R1)))s.t. G0 > 0.To get the F.O.C. for wd0 , we must write down the Lagrangian with the inequalityconstraints.L = −l(S0, d0) + βE(V (K0 + x0, (1− θ)W0 +R0 − wf0 , R1)) + λG0Then the F.O.C. would be given by the derivative of L with respect to wd0 .−∂l(S0, d0)∂wd0+ λ∂G0∂wd0= 0Where the inequalities do not bind, we get λ = 0 and the following solution holds:wd0 = P − wf0 −c2.If the inequalities bind, either wd0 = 0 or wd0 = K0. Therefore, the final solution wouldbe given by equation 2.10. This provides wd as a function of only one state variable (Kt)and one decision variable (wf ).wd0 = max(0,min(Kt, P − wf0 −c2)) (2.10)Therefore, by plugging the value of wdt (2.10) into the Bellman equation 2.8, we can232.4. Demandreduce the dimension of the decision space from 3 to 2 (total dim: 9 → 8).The next step is to get rid of the dynamic equation as a constraint. We can directlyplug in the dynamic constraints (equations 2.1 and 2.2) to rewrite the Bellman equationas a function of state variables of this period and state variables of the next period. In thisway, we eliminate one constraint and reduce the dimension of the problem from 8 to 6.Although the problem is simplified significantly, the curse of dimensionality still pre-vents convergence in a reasonable time. For the contraction mapping to converge in areasonable amount of time, the quality of the first guess becomes crucial: the better theinitial guess, the lower the number of iterations.To develop a reasonable initial guess, I develop an intuitive heuristic specific to thecurrent problem that significantly helps find the starting point for the optimization. Theidea behind this heuristic is intuitive and straightforward. It suggests that instead of havinga planner that makes two decisions simultaneously, have two decision makers that take twoindependent decisions, taking the decision of the other decision maker as given.This intuition behind this model is similar to the solution concept of a Cournot game.The first player takes the behavior of the second player as given and plays its optimalstrategy. The second player does so by taking the behavior of the first player as given. Byreiterating this process, the solution converges to the answer.In the actual problem, the planning authority takes two related decisions. It decideson how much capacity to build in the next period and how much water to keep in thereservoirs.The heuristic suggests that first, I fix the capacity and allow the first planner to findthe optimal amount of water remaining in the reservoirs. This heuristic is a 2-dimensionaldynamic programming problem, and it can be solved quickly, using conventional methods.I used time-iteration, which is usually faster than value function iteration in 2-dimensional242.4. Demandproblems.After that, I let the second planner choose the optimal capacity to be built while takingthe solution of the first planner as given. This problem is also 2-dimensional and easy tosolve.I am then to repeat these two stages by feeding the answer of each of them to the otherone. After a few iterations, the result of this procedure serves as a starting point for thevalue-function iteration procedure. This procedure makes the algorithm converge faster ina reasonable number of steps by providing a suitable initial guess.Figures 2.1-2.9 depict the behavior of some variables of the model as a function ofcurrent capacity and water reserves. These graphs are obtained for some specific parametersof the model, but they capture the important qualitative behavior of the model.Figure 2.1 shows capacity building as a function of current capacity and water reserves.The population water target (P ) in this model is 50, and the minimum rainfall is 40.The maximum amount of desalination must be smaller than 10, and we can see that it isbetween 8 and 9.252.4. DemandFigure 2.1: Capacity building as a function of current capacity and water reservesduring years with unfavorable rainfallFigure 2.2 shows that there would be no capacity building in the good rainfall years.Because in good years the rainfall is 65, which is much larger than the population watertarget (P = 50), it is not efficient to build capacity in good years. But this is not a generalresult, and for some parameters some capacity would be built even in the good years.262.4. DemandFigure 2.2: No capacity would be built in years with favorable rainfallThe first two figures showed capacity building, and the next two figures show the amountof next period water reserves for good and bad rainfall years. Generally, the amount ofreserves will be higher in good years.272.4. DemandFigure 2.3: With small fixed costs the reserves are independent from currentcapacity. The plot is for years with bad rainfall.For the parameters of this simulation, the amount of reserves are independent from thecurrent capacity. This is not a general feature of the model. For high enough fixed costs,the reserves will depend on current capacity.282.4. DemandFigure 2.4: With small fixed costs the reserves are independent from currentcapacity. The plot is for years with good rainfall.The fixed cost (F ) and variable costs (κ) of building a plan in last two figures wererelatively low. The next few figures show the behavior of the solution under these param-eters.292.4. DemandFigure 2.5: For high enough fixed costs the reserves will depend on currentcapacityFigure 2.5 shows the 3-dimenstional plot of next period reserves for the case with highcosts of building a desalination plant for good rainfall draws.302.4. DemandFigure 2.6: Future water reserves vs. current capacity: for current capacityabove 8.5 the decision maker would keep smaller reserves for the next periodbecause there would be enough capacity even in a low rainfall realization.Figure 2.6 shows a slice of the 3-dimenstional plot in Figure 2.5. For a fixed level ofcurrent water reserves, future water reserves are plotted against current capacity. If thecurrent capacity is high enough (above 8.5), the decision maker would keep smaller reservesfor the next period because there would be enough capacity in the next period even in alow rainfall realization.312.4. DemandFigure 2.7: Next period reserves in bad years are qualitatively similar to theones in good yearsFigure 2.7 shows next period reserves for parameters similar to Figure 2.5 but for abad rainfall realization. The results are qualitatively similar.322.4. DemandFigure 2.8: In a year with bad rainfall if water reserves are low, not havingenough desalination capacity is very costlyFigures 2.8 and 2.9 show the continuation value function for both good and bad rainfalldraws. The continuation value function is continuous. It is lower when the rainfall is badbecasue the municipality has to use desalination capacity or provide a less than idealamount of water. For the same reasons, the function is smaller when there is less waterreserves or less installed capacity.332.5. Testing the modelFigure 2.9: In a year with good rainfall, when water reserves are low, not havingenough desalination capacity is not as costly as good rainfall years, but itincreases the probability of building a desalination plant soon.2.5 Testing the modelA theoretical model is a simplified version of reality. A model captures some essence ofreality and neglects the rest. At this stage, I am primarily concerned about the qualitativeimplications of the model. Many simplifications have been made in the literature. Forexample, the dynamic programming model has only two possible states, yet I classified anyyear with below (above) median rainfall as bad (good). In reality, the decision maker mightconsider more than two states, or the criteria for good vs. bad year might be different.There are other potential issues. It may be that transition probabilities from one stateto another are not always Markovian, or the loss function of a municipality is differentfrom my specifications.The task is further complicated because, in our dynamic model, many variables are in342.5. Testing the modelprinciple observable to the decision maker, but the econometrician only has noisy proxiesof the data. Table 2.2 shows what data the decision maker observes versus what data isavailable to the econometrician.If the model is rich enough, and the econometrician simulates the model for manydifferent parameters, it is possible to identify some correlations between the variables ofthe model’s solution. These correlations should qualitatively match the correlations thatwe observe in the data. Note that the data is a noisy proxy of what the decision makercould observe.What I do is similar to what empirical researchers do. They explain how a modelbehaves. Iin many cases, they do so verbally or symbolically. They make a quantitativeprediction (e.g., signs of some parameters) in a regression model based on those explana-tions. Then by running a regression, they test the proposed hypothesis. Instead of verballyexplaining the model, I propose and numerically solve a dynamic programming problemfor multiple random parameters. Then I simulate numerous draws and develop a predic-tion for the signs of many observable variables in a separate regression with the real-worlddata. Then I run the regression with the data and test those hypotheses. Specifically, Isolve the dynamic model for fifty different random parameters drawn from jointly normaldistributions.352.5. Testing the model[H]Table 2.2: Simulation ParametersVariable Model ProxyP population observedR volume of water made average monthlyavailable by rainfall rainfall in mmθ annual evaporation average monthlyrate temperatureσR difference in average sd of annualannual rainfall rainfallα water target per -population unitc yearly cost of -using capacityκ marginal cost of partially observedbuilding more capacityF fixed cost of building partially observedµ conversion rate of GDP per capitawaterunderprovision-The parameters of the distribution are listed in Table 2.3. Some of these variablesin the model have a corresponding variable in the real-world data. The variables in thereal-world data are proxies for the variables in the theoretical model.362.5. Testing the modelTable 2.3: Simulation ParametersVariable Proxy DistributionP Population N(50,5)µ GDP per Capita N(1.5,5)θ average Temperature N(0.2,0.02)R average Rainfall N(50,10)σR sd Rainfall N(5,3)α - 1γ - 0.1f - 1F - 0-Each solution of the model for these fifty different parameters represents one city. Eachcity goes through one hundred years of simulation stream. But this may introduce a spuri-ous correlation because some cities may build desalination capacity early on, not becausethe parameters dictate that they are more likely to build early but because they randomlygot bad initial rainfall draws. If each city gets multiple rainfall realization streams, we cancapture a city’s average behavior. Therefore, each city is simulated for 200 realizations.Eventually, there would be fifty distinct city types, 200 rainfall realizations streams, anda hundred years of simulation for each city-realization pair. Both the simulated data andthe real data are panels, and each observation corresponds to one year for one city.The simulated data contains information on the simulated endogenous variable (whenand what capacity is being built) and respective parameters and timing of rainfall realiza-372.5. Testing the modeltions. The goal is to compare the sign of coefficients in a regression with the simulateddata and the real-world data. If the model is a good approximation of reality, we expectthe regression results for both tables to have similar signs.The empirical model that I use to compare the simulation data and real-world data isshown in equation 2.11. In the following model, Ui is a city-specific random effect, and itis the error term distributed with 0 mean.dit = β0 + β1Pi + β2Rit + β3R¯i + β4θ¯i + β5µi + β6σi + Ui + it (2.11)The regression Table 2.4 shows the coefficients of a random effect model 2.11 appliedto the simulated data.Table 2.4: Simulated Data, Random EffectsCapacity Coeff. std. error t-Value(Intercept) 0.47 0.077 6.08 ***population 0.0121 0.0008 15.2 ***relativerain -0.59 0.03 -17.0 ***ave Rain -0.0128 0.0004 -31.0 ***ave Temp 1.07 0.24 4.46 ***GDP per capita 0.018 0.009 1.96 .sd Rain 0.01595 0.0017 9.3 ***Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’0.1-Table 2.5 applies the same empirical model of equation 2.11 to the real data.382.5. Testing the modelTable 2.5: Actual Data, Random EffectsCapacity Coeff. std. error t-Value(Intercept) -2.55e+03 2.84e+02 -9.0 ***Population 1.45e-01 3.8e-02 3.9 ***relative rain -5.39e+02 1.47e+02 -3.7 ***ave Rain -5.6e-01 1.7e-01 -3.4 ***ave Temp 9.1e+01 1.5e+01 6.2 ***GDP per capita 7.60e-02 4.3e-03 17.7 ***sd Rain 7.00e+03 9.0+02 7.8 ***Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1-We should compare the results in Tables (2.4 and 2.5) qualitatively only, meaning thatwhat matters are the sign and significance, but the values do not matter. Comparing thetwo tables tells us that all the six variables have similar signs and are statistically significant.This match indicates that the proposed dynamic demand model captures essential featuresof reality, and its predictions agree with real-world data.One alternative hypothesis would be that factors other than temperature and rainfallaffect the timing of the desalination plant. For example, cities could decide how muchdesalination they need in the long run and randomly build the plant. It is also possiblethat the institutional hurdles determining the available budget for desalination play a moredecisive role in building a plant. This hypothesis means that building patterns would beindependent of rainfall. In this hypothesis, there would be no correlation between rainfallpatterns and building desalination. Another important feature that does not seem intuitiveis that this model and real-world data predict that cities with higher s.d. in rainfall would392.5. Testing the modelbuild more desalination capacity. It could well be that cities rely on desalination for theirentire water needs. Still, it seems that desalination acts more as an insurance policy thanthe primary water provision source.The reason behind the dramatic increase of desalinatoin capacity in the previous yearsis a combination of multiple factors. The model predicts that each city has an equilibriumlevel of desalination based on its geographic characteristics, demographic characteristics,price of desalination construction, and price of desalination operation. The significantfactors that affect demand for desalination have been widely studied. Some of the moreimportant ones that explain the major adoption of desalination plants are: an increase inpopulation, an increase in marginal willingness to pay for water in super arid areas (thinkof population booms in prosperous cities in the middle east and Mediterranean), a sharpdrop in desalination construction cost, an improvement in desalination efficiency.It is theoretically possible for an increase in extreme weather events and a drop inaverage rainfall levels to cause an increase in the adoption of desalination, but it has not yetbeen documented to this day. This model can allow for all of the aforementioned changesand give a specific and quantitative prediction on the equilibrium level of desalinationcapacity for each city in the long run.I estimated another version of this model regarding the relative value of constructinga plant in good vs. bad times. I moved this estimation, along with some other analysesregarding optimal utilization rate, the possibility of mistakes on rainfall probabilities, andestimating the price elasticity of demand for construction of desalination plants, to Ap-pendix A. These results are not directly contributing to the central intellectual contributionof this paper but are part of my explorations about decision-making regarding desalinationplants.402.6. Conclusion2.6 ConclusionThis paper first investigates the question of optimal timing and optimal size of buildingdesalination plants, given the uncertainties inherent to rainfall patterns using seventy-six-years-long panel data on desalination plant construction worldwide. Secondly, this paperaims to explicitly define the criteria of rationality in a specific problem and test whetherdecision makers act consistently with these criteria in this limited domain.This paper adopts a dynamic programming framework to answer the first question. Arational decision maker maximizes the expected utility of building the plant. The decision-making authority considers decisions on both the plant construction and plant use. It findsthe optimal time to build a plant and its size. All three aspects of optimal use, optimaltiming, and optimal size of the plants are critical issues that experts face in the real world.By calibrating the elements of this model to dimensions of cities in need of building adesalination plant, this model can serve as a tool to provide a baseline for the needs of acity and a tool to measure the quality of decisions made by decision-making authorities.To compare the behavior of the real-world decision makers with the optimal behaviorprovided by this model, I compare the correlations predicted by simulations from the modeland the correlations observed in the real-world data. All of the compared variables hadthe correct signs and were significant in the correct directions.In this paper, the micro-level behavior of the decision maker was observed. Admittedly,this is not the only micro-level rational model to write for decision-making in this context.This paper is evidence that the predictions of a dynamic model are consistent with real-world behavior on average. To develop more robust tests, we should develop further modelsand gather more data on the usage of desalination plants and local characteristics.Another limitation of this paper is that I implicitly assumed that the managementof desalination plants is relatively similar in different cities. Any difference observe in412.6. Conclusionconstruction patterns is due to observables or year-city-specific shocks. With more dataon desalination plants’ annual usage rate, it would interesting to see if there are inherentdifferences in the management of these plants.This model can be used to make some some predictions about desalination adoptiondue to climate change. In this model, the rainfall patterns follow a Markov process in itssteady state. If we model climate change as a one time shift in the probabilities of differentrainfall patterns, which increases the standard deviation of rainfall and decreases the meanvalue of rainfall in arid areas, then this model predicts that:1. Cities in arid areas would build more desalination per capita to compensate for lowerrainfall levels2. Cities would generally build more desalination to use desalination as an insurancepolicy against extreme events3. It can explain how does the timing of these transitions work: cities would waituntil the available freshwater in the reservoir to fall below a threshold, and then they startbuilding desalination capacityAnother aspect worthy of exploration is relaxing or improving the rationality assump-tion of decision makers. In this paper I assumed that decision makers believe that theirneed for water follows a Markov process that is a function of observables and shocks. Inprinciple, the municipalities can overreact or underreact to observables, or they might havesuperior information on predicting water needs that the econometrician does not observe.It is interesting to know how decision-making authorities come up with decisions consistentwith a relatively complex dynamic programming problem.42Chapter 3Can Innovation Help Us in theBattle against the ClimateChange?3.1 IntroductionClimate scientists, economists, and politicians run out of superlatives while describing thedisastrous long-term effects of climate change. They agree on the enormity of climatechange’s threat. But when it comes to action, the agreement seems to fade away. Whileclimate scientists emphasize the long-term catastrophic effects of climate change, politicianstend to think in the short-term. Economists are somewhere in between, balancing the short-to mid-term costs of mitigating climate change and its mid-to-long- term consequences ofaction (or lack of it). There is also a sea of disagreement about the best instruments tofight climate change. Some climate scientists, overwhelmed by the sheer magnitude ofthis disaster, prefer an outright ban of any form of greenhouse gas emission. At the sametime, politicians tend to prefer innovation policies to combat climate change. Economists,again, take the middle ground by accepting the merits of both emission restrictions andinnovation policy. Their studies show a preference for carbon pricing, either through taxesor cap and trade systems.433.1. IntroductionSome OECD countries like Canada, Japan, Chile, and some European countries imple-mented some forms of carbon pricing. But according to the World Bank 1, as of 2021, lessthan 22 percent of the total carbon emissions would be priced. This does not take intoaccount the fact that the prices in many jurisdictions vary, and they may be far below themarginal damage of carbon.In the United States, carbon pricing proved to be politically impossible at the federallevel, as evident from the latest environmental policy of the Biden administration. In 2022,the Biden administration would increase the budget to fight climate change from $22 billionto more than $36 billion without any indication of a carbon tax. The main bulk of thisfunding would be spent on research, innovation, and infrastructure.The emphasis on innovation without a clear taxation policy raises a fundamental ques-tion that we plan to address in this chapter: Is innovation even helpful in the battleagainst climate change? In some cases, it does, but a closer look at the portfolio ofinnovations and their interactions raises multiple paradoxes that rule out a simple answerto this question.In this paper, we set out the economic logic of the impact of innovation on climatechange. We primarily focus on the portfolio of the most promising green technologicalinnovations that have been pursued in the past decades: clean energy innovation, innovationin fuel efficiency of internal combustion engines, and innovation in carbon capture andstorage. We ask when innovations help the battle against climate change and how theseinnovations interact with each other.Our focus is to study the long-term effect of a green innovation portfolio on climatechange. While the economic literature neglected these long-term effects and focused onthe short-to-medium range, we know from climate studies that long-term (beyond fifty1https://carbonpricingdashboard.worldbank.org/443.1. Introductionyears) climate outcomes are much more predictable than short-to-medium-range climateoutcomes. Many climate studies emphasize the long-term disastrous effects of carbonemission and climate change. These studies show that the long-term climate outcome isindependent of the emission patterns. A simple measure of cumulative carbon emissions isa sufficient statistic for long-term climate outcomes. Therefore, we study the effect of thegreen innovation portfolio on one measure of impact: cumulative carbon emission.While trying to answer when innovation helps us against climate change, we revisitsome of the well-known strong green paradoxes and introduce some new ones. A stronggreen paradox arises when a policy or innovation intended to improve climate outcomesend up causing greater cumulative environmental damage. In the rest of this introduction,we set out the green paradoxes that we visit in this paper.The history of strong green paradoxes dates back to the late nineteenth century. Jevons(1865) first observed an improvement in energy efficiency involves the so-called reboundeffect: the demand for output will increase as fuel costs fall, and in terms of the overallimpact, this increase in demand may increase the overall flow of fuel consumption andcarbon emissions.Our first contribution is the introduction of a new paradox: an improvement in fuelefficiency will always increase the total stock of fossil fuel extracted. Fossil fuel depositsvary in the cost of extraction. In a simple setting, fossil fuel deposits are extracted in orderof extraction cost until they reach a point where the marginal cost of extraction, say c1,justifies using a clean-energy backstop, which is the alternative to an internal combustionengine. If the fuel efficiency (FE) of internal combustion engines increases, with a drop inthe fuel requirement per unit output, fossil fuels will continue to be used until the marginalcost of extraction at a higher level, say c2, is reached. All fuel with extraction costs in(c1, c2) will be extracted under the new technology instead of being left in the ground, thus453.1. Introductionadding to total carbon emissions. Innovation in fuel efficiency on its own leads to a stronggreen paradox.The logic applies not only to internal combustion engines but also to fossil fuel pro-duction of energy. With an increase in fossil fuel energy production efficiency (as ith theintroduction of combined-cycle natural gas generation in the 1960s), fossil fuel will be usedto a greater extent for energy production. The switch to a clean alternative such as solaror wind power would take place at a higher cost of fossil fuel extraction.Our next contribution involves innovation in carbon capture and storage (CCS). Aninnovation in CCS (given that it is implemented) always improves long-term climate out-comes. Moreover, this kind of innovation alleviates and may even reverse the paradoxicalresult of a FE innovation. If energy demand is inelastic enough, a substantial innovation inFE pushes more emissions into the future; these emissions will occur when CCS technologyis more advanced so that the emissions net of CCS are reduced.Then we turn our attention to innovation in clean energy (CE). It is well-known in theliterature that innovation in clean energy (e.g., wind energy, solar energy) or even the threatof it reduces the expected future rents of fossil fuel owners, hence reducing the current priceof fossil fuel. If demand is elastic enough, current emission will increase, meaning that CEinnovation causes a weak green paradox: a green-intentioned policy (innovation) leads togreater short-term environmental damage.We establish that the effect of CE innovation on its own is always positive for the long-term cumulative emissions. CE acts as a backstop (or alternative) to fossil fuel deposits.With cheaper green backstops, the deposits that would otherwise be extracted are left inthe ground. Hence, CE innovation will not lead to a strong green paradox on its own.We show, as our final contribution, that for an elastic demand, the CE innovation isless beneficial if CCS improves over time: the greater the share of emissions that occurs463.1. Introductionin the near future (when CCS technology is less advanced than in the distant future) thegreater the emissions net of carbon captured and sequestered. CE innovation acceleratesthe fossil fuels extraction flow and, therefore, the flow of emissions. For strong enoughinnovation in CCS, the CE innovation becomes harmful, and the interaction of these twoinnovations yields a strong green paradox.The starting point of our analysis is the Hotelling model of an exhaustible fossil fuelresource with variable costs. Throughout the paper, we represent innovations in FE and CEas Poisson processes on a one-time improvement of known magnitude, following establishedliterature. We represent CCS innovation, in contrast, by an exogenous and increasingfraction a(t) of emissions that are captured at zero cost over time. This representationreflects an underlying assumption that the government (regulator), not firms, bears thecost of the CCS. In reality, this cost is mainly the cost of storage and the cost of thenetwork to transport CO2 to storage.In the appendix B, we extend the model to allow for the endogenous choice of a by theregulatory agency by assuming a CCS technology represented by a cost function c(a; θt),in which θt is a technological parameter representing (possibly) improving technology overtime. In the extension, the regulator decides how intensively to use existing CCS technologyat any time given the cost function and the state of technology in the dimensions of cleanenergy or fuel efficiency. The use of CCS changes with the discovery of a clean energysubstitute or an increase in fuel efficiency. We show that our central results are preservedin this extension.In the next section, we give a brief survey of the literature. Then we introduce thetheoretical setup and discuss our assumptions in section 3. In sections 4 and 5, we studythe effects of FE innovation and CE innovation, respectively. In section 6, we conclude bydiscussing the limitations of our results and the policy implications of our paper.473.2. Literature Review3.2 Literature ReviewFossil fuel innovation affects both the flow pattern of emissions and the total stock ofemissions. The impact on the flow of emissions involves the rebound effect studied byFullerton, Ta (2019), Gillingham et al. (2016) and possibly Jevons’ paradox (1865).2 Anincrease in the efficiency of car engines, for example, will increase the current flow of demandfor gasoline if the elasticity of the demand for car travel is sufficiently high because thecost per mile of travel falls with the innovation; the extra miles driven may offset the lowerfuel needed per mile. The rebound effect is well known. But a paradoxical stock effect ofan innovation in fuel efficiency has, to our knowledge, gone unnoticed.The paper contributes to the green paradox literature. In his seminal work, (Sinn(2012)) coined the term “the Green Paradox”. According to Ploeg van der, Withagen(2012), the green paradox is the idea that well-intended policies to improve the climateoutcome lead to outcomes that are below the social optimal and counter-productive. Thisis because such policies reduce the expected rent collectible by fossil fuel producers, andthey choose to produce more fuels and sell them at a lower price. More specifically, thispaper contributes to the literature on innovation and green paradox, which focuses on theunintended consequence of innovation policies to improve the climate outcome.Werf, Maria (2012) review more than 20 papers on the climate change policy andthe green paradox. Several papers in the literature have investigated the interaction ofinnovation and climate change (Hoel (2009), Ploeg van der, Withagen (2012), Winter(2014), Acemoglu et al. (2016)). The models developed in most of these papers containassumptions that imply that in the long-term the environment will converge to a steady-state independent of any temporary effects of policy.2Jevons’ paradox is a rebound effect from an increase in the efficiency of internal combustion enginesthat is so strong that it more than offsets the direct, negative effect of the increased efficiency on fueldemand.483.2. Literature ReviewThis independence is explicit in Hoel (2009) and Ploeg van der, Withagen (2012). ButAcemoglu et al. (2016) operate at a higher level of abstraction: environmental qualityis represented at a state variable St, which either reaches 0 (“environmental disaster”)during any period of carbon emissions and remains at that level, or converges to completeenvironmental recovery in the long-term. The assumption of a categorical connectionbetween short run policy and the long-term environment (“disaster” or perfection) rulesout consideration of the effects we examine in this paper.Winter (2014) pairs an economic model of innovation in clean energy with climate dy-namics that incorporate feedback effects. Dynamical systems with feedback effects yieldtipping-point dynamics, which link short-term to long-term effects of innovation. In thispaper we explore a different link between innovation and long-term climate outcome, mo-tivated by the scientific evidence cited above, that cumulative carbon emissions are a goodindicator of the long-term climate impact of the emissions. Gans (2012) examines the in-centives for innovation in clean technologies induced by policy. Gans considers essentiallythe same portfolio of technologies as this paper, and establishes a parallel result: onlyfor technologies that directly abate carbon pollution is there an unambiguously positiveimpact on innovation. In our paper, only for these technologies is exogenous innovationunambiguously beneficial. Gans investigates incentives for innovation. We investigate amore basic issue, the impact of exogenous innovation.Our paper is related to the literature on second-best climate policies and their in-teractions.Fischer, Newell (2008), Grimaud, Lafforgue (2008), and Hart (2019) assessedthe interaction of carbon tax and clean energy subsidies. They showed that they bothhave substantial welfare effects in the optimal general equilibrium and quantified differentdeviations from the first best.Lafforgue et al. (2008), Hoel, Jensen (2012), and Kalkuhl et al. (2015) examined the493.3. Setupoptimal pattern of CCS with constant technology when there is clean energy innovation.However, they did not consider improving CCS technology and infrastructure. With animproving CCS, the percentage of emissions captured may increase over time.Because the theoretical starting point of our paper is the Hotelling model of resourceextraction, it is worth discussing the relevant literature on Hotelling model and empiricalevidences for it. It has been empirically established that in the short-run, aggregate oilproduction is inelastic in price (Griffin (1985) ; Dahl et al. (1991); Ramcharran (2002);Gu¨ntner (2014).As Cairns (2014) points out, Hotelling style models will break-down in the short-runbecause production cannot be re-arranged at will. But aggregate oil production, in the longrun, is determined by both how much firms produce from existing wells (intensive margin)and how many new drilling projects do firms initiate (extensive margin). A recent study(Anderson et al. (2018)) shows that an extension of the Hotelling model (that takes drillinginto account) explains why the Hotelling model can only hold in the long run. Because ourscope of analysis is long-run, the Hotelling model deems appropriate.3.3 Setup3.3.1 The Green Innovation Portfolio and Damage FunctionWe set out three classes of clean-tech innovation, clean energy (CE), fuel efficiency (FE),and carbon capture and storage (CCS). We study both the ex post realization and the exante threat of FE and CE innovations on their own and their interaction with a steadyimprovement in CCS technology.Clean energy (CE) alternatives consist mainly of solar energy and wind energy. Thecost of generating electricity from solar power depends upon geography, but the cost of503.3. Setupsolar panels per kw generated has dropped by 2 orders of magnitude since the 1970’s.Solar power bids for generating electricity have in places reached $ 50 dollars per mega-watt hour, substantially less than the cost of coal,3 but the cost of storingand transmissionof this energy has to this point ruled out significant global market share to date for eithersolar or wind energy.Fuel efficiency (FE) innovation in the of internal combustion engines in particular andany form of fossil fuel production of energy in general is our second source of innovation.This innovation is illustrated by the increases in average automobile fuel efficiency of about90 percent since 1975. 4 and in efficiency gains of generating electricity from fossil fuel,such as the development of combined-cycle fossil fuel plants in the 1960’s.We represent innovation in a clean energy alternative more simply by the exogenousdiscovery of a single new technology. Following much of the economic literature on innova-tion, we assume a Poisson process for the discovery of the new innovation, with probabilityof discovery ρdt in any instant dt at any date for which the innovation has not been discov-ered. We adopt the same process in examining the impact of fuel efficiency innovation. Forboth clean-energy and fuel-efficiency innovation, we distinguish between the ex post effectof a realized discovery on long-term climate change and the ex ante effect of the threat ofinnovation, as this threat alters the equilibrium in the existing market for fossil fuel as anexhaustible resource.Carbon capture and storage (CCS), or sequestration consists of capturing thecarbon dioxide produced with combustion and moving the gas to a storage facility such asan underground geological site.5 Innovation in this area is promising but highly uncertain:IPCC(2005) Metz et al. (2005) predicts that the economic potential of CCS will be between3Jeremy Berke (2018)4https://www.epa.gov/fuel-economy-trends/highlights-co2-and-fuel-economy-trends#Highlight25We set aside innovation (prior to T , the maximum date of significant carbon emissions) direct carboncapture and sequestration from air as well as other green-tech innovation such as large scale batteries.513.3. Setup10% and 55% of the total carbon mitigation effort until year 2100. We represent CCS byan increase over time in the fraction a(t) of emissions captured. CCS is exogenous, costlessand implemented by regulation. We represent CCS simply by an increase over time in thefraction a(t) of emissions captured.We consider the ex post and ex ante impact of innovation of various types on thecumulative carbon emissions over a finite period [0, T ] over which carbon emissions arepositive. (T is the termination date of fossil fuel extraction with no prospect of innovation.)Our focus on total emissions is motivated by scientific evidence that cumulative emissionsare a close proxy for the long-term impact of emissions on climate. A recent editorialin Nature Allen et al. (2009) discussed evidence supporting the “insight that cumulativeCO2 emissions determine the long-term climate outcome, essentially independent of theevolution of CO2 releases over time.”6It is helpful to elaborate on the physics of why the pattern of emissions of a givenamount of cumulative emissions might be irrelevant, or nearly irrelevant, within the rangeof parameters contained in a realistic climate model. Why might an earlier path of agiven amount of carbon emissions, for example, have only a small impact on the state ofthe environment at T? Such a shift would (1) have a negative (i.e., beneficial) impacton the atmospheric concentration of carbon at T because the emitted carbon would haveon average a longer time to settle back to the earth’s surface. But this beneficial impactwould be offset or mitigated by: (2) a significant portion of carbon emitted remainingin the atmosphere for hundreds or thousands of years; (3) the feedback effects of highertemperature (realized earlier) on the flow of carbon into the atmosphere, even if not tothe extent of triggering runaway global warming; (4) the earlier settling of carbon into the6Nature Geoscience Vol.2, December 2009. This editorial refers to Meinshausen et al (2009), whichreports that for the wide set of emission scenarios modeled, cumulative carbon emissions up to 2050 is arobust indicator of the probability that twenty-first century warming will not exceed 2 degrees C relativeto pre-industrial temperatures.523.3. Setupocean in reducing the capacity of the ocean to absorb additional carbon and heat; and,(5) the carbon and heat settling earlier into the ocean has itself a longer time to damagethe ocean through increased acidity, higher temperatures and decreased oxygen. From theNature(2009) editorial cited before, in terms of the net long-term environmental impact,these effects evidently balance out. In seeking to understand the long-term environmentalimpact of carbon emissions, it is therefore, reasonable to focus as a first approximation ona simple parameter, cumulative emissions.Government policy to reduce carbon emissions falls into two general classes: carbonpricing (taxes or cap-and-trade) or other means of incentivizing reductions in emissions,and policies to encourage innovation that would reduce carbon emissions. Although carbonpricing is the preferred policy, for political reasons, incentivizing innovation is more widelyimplemented. With optimal carbon pricing, all policies are beneficial, and there wouldbe no green paradox. But the current levels of carbon prices are well below the optimalcarbon price.Formulating an optimal policy on carbon pricing and stimulating endogenous innova-tion requires an understanding of the impact of exogenous innovation. If a technologicaldiscovery or the mere possibility of a discovery makes climate change worse, for example,this should be known. Our focus is limited to the consideration of the long-term impactof innovation. While the long-term environmental impact is only one component of thetotal welfare impact of innovation, it is a critical component and often under-emphasizedin economic models. By limiting our focus to long-term impact of exogenous innovation,we can set out a potentially complex economic logic in simple terms.By long-term damage, we mean any damage that may occur after extracting all fossilfuels. Because many geophysical interactions are very slow, it may take centuries afterraised carbon levels that these adverse effects come into effect. This damage includes533.3. Setupan increasing extinction probability of various life forms on earth, including humans. Ifthe extinction probability spikes decades after extracting all fossil fuels, then the totaldamage is a function of cumulative carbon emission after the exhaustion of all fossil fuels.Our analysis explores the possibility of strong green paradoxes in green innovations withdamage function dependencies as described.3.3.2 Vanilla ModelOur starting point is the standard Hotelling model of a competitive market for an ex-haustible resource, fossil fuels. The distribution of deposits at various extraction costs isG(c), with density g(c) and support [c, c]. We assume a stationary demand for an outputsuch as miles driven, Q(P ), which can be produced by either an internal combustion engineor a clean energy backstop such as electric cars. The backstop supply of output costs y0dollars per unit. The ICE requires inputs of cost z other than fuel. The fuel requirementis λ0 per unit output.The markets for both machines and fuel are competitive, with prices over time given byP (t) and p(t). We adopt a regularity condition on output demand, Q(P ), that the elasticityof demand be non-decreasing in price. The variables q(t) and x(t) represent both the flowand cumulative fuel extracted and (by choice of units and in the absence of sequestration)the flow and cumulative amounts of carbon emissions. Finally, there is a constant interestrate, r.The equilibrium of this well-known model is studied in the literature(Heal (1976),Stiglitz (1976)). The resource is extracted in order of lowest extraction-cost deposits untila date T at which production of energy switches entirely to the backstop. The dynamicsof p(t), q(t), c(t), and x(t) (price and quantity of fuel, cost of extraction, and cumulative543.3. Setupemissions) are determined by the following conditions:q(t) = λ0Q(z + λ0p(t)). (3.1)The date of extraction of a deposit of cost c, t˜(c), maximizes the present value of rentsfor the owner of the deposit:t˜ = argmaxt[p(t)− c]e−rt. (3.2)The function c(t) is the inverse of t˜(c).From the first-order conditions for the maximization problem (3.2) we get the followingdifferential equation:p˙(t)p(t)− c(t) = r. (3.3)with a terminal condition given by:p(T ) = c(T ) = (y0 − z)/λ0 (3.4)Here, T is the date of the switch to the backstop. At T , the costs of the two technologiesare equal: z + λ0c(T ) = y0.The variable x(t) is the cumulative extraction of fossil fuel (first carbon reservoir).Without sequestration (CCS), this variable also represents the cumulative addition of car-bon to the atmosphere(second carbon reservoir). The total extraction of fossil fuel is givenby:x(T ) = G((y0 − z)/λ0). (3.5)CCS is represented by the fraction of emission that is captured from emission a(t).553.4. Clean Energy InnovationWith CCS, the dynamics of carbon in two reservoirs changes: q(t) still measures the rateof carbon removed from the earth, but now the rate of carbon added to the atmosphere isnow [1− a(t)]q(t).7Each technology is represented by one parameter in our model. The parameter yrepresents the per-unit cost of CE technology. Lower levels of y0 means cheaper (superior)technology. The FE technology is represented by λ0. A smaller λ means more efficient(superior) technology. And finally, a represents CCS technology. Larger a means more(superior) sequestration technology.3.4 Clean Energy InnovationInitially, the per-unit cost of CE backstop is y0. But there is the possibility of a singleinnovation at any time that would lower the costs to y1 < y0. The probability of discoveryof the new CE technology follows a Poisson process: if at date t the new backstop has notbeen discovered, then with probability ρdt the new technology will be discovered in theinterval [t, t+ dt] for dt vanishingly small.3.4.1 Unexpected CE innovation on its own is a boonIn asking about the impact of innovation on climate change, we distinguish between the expost effect of realized innovation, and the ex ante effect that of the threat of innovation. Toisolate the ex post effect, suppose that innovation has zero (or vanishingly small) probabilitybut is possible, and consider the impact of a realized discovery of the new technology.Before discovery, the equilibrium follows the same path as if innovation were impossible.7A third reservoir, the cumulative carbon stored, is potentially important in reality since the under-ground space required to store CO2 is several times the space taken up by the fossil fuel prior to extractionand combustion. Storage space may be limited. We set aside this complication in our simple model, how-ever, assuming that the cost of CCS does not include the shadow price of a limited amount of storagespace.563.4. Clean Energy InnovationAfter a discovery, the equilibrium is described by the set of equations above, with y0replaced by y1. The time of switch to backstop is given by t˜((y1 − z)/λ0). If discoveryoccurs at a date at or beyond t˜((y1 − z)/λ0), then fossil fuel extraction stops immediatelyand production switches to the new backstop technology.The following proposition is immediate.Proposition 1 The ex post effect of the discovery of a new clean-energy technology at adate t ∈ [0, t˜((y1 − z)/λ0)] is a reduction in cumulative emissions, E, by the followingamount:(y0−z)/λ0∫(y1−z)/λ0g(c)dc = G(y0 − zλ0)−G(y1 − zλ0).The effect of a discovery at date t ∈ [t˜(b1),T] is a reduction in cumulative emissionsby:G(t˜((y0 − z)/λ0))−G(c(t)).An unexpected CE innovation always reduces the long-term cumulative carbon and itis beneficial for the climate in the long-term.3.4.2 CE-CCS interaction: The devil is in the detailsTo study the interaction of CE and CCS, first we need to define innovation in CCS technol-ogy. Innovation in CCS is taken to be a monotonic increase over time in a(t), starting ata(0) = 0. “Stronger” innovation in CCS is the partial order given by: a1(t) is stronger thana2(t) if a′1(t) > a′2(t) for all t. (A statement S is true with “sufficiently strong” innovationin CCS means that there exists a pattern of innovation a∗(t) such that for all a(t) strongerthan a∗(t) the statement S is true.)573.4. Clean Energy InnovationCCS does not affect the patterns of extraction, but it reduces the amount of carbonemitted to the atmosphere. Therefore, CCS innovation is beneficial on it its own. Withno innovation in clean energy, the impact on total net emissions of an innovation in CCSfrom a1(t) to a2(t), given the equilibrium flow of fossil fuel q(t) is∫ T0 [a2(t)− a1(t)]q(t)dt.This generalizes easily to the case of clean-energy innovation at any date:Proposition 2 Given the discovery of clean energy technology at any date t, or no inno-vation in clean energy, an increase in CCS efficiency lowers cumulative carbon emissions.To apply the clean-energy model in the presence of CCS innovation in the next section,we need to characterize the impact of innovation not just on cumulative extraction but alsoon the entire path of extraction. To this end, let p(s) ≡ p(x−1(s)) be the equilibrium pricepath as a function of the amount s extracted to date in the basic model; with innovationintroduced, p0(s) represents the price path when the backstop technology fixed at y0 andp1(s; τ) is the price path for t > τ when the new technology is discovered at date τ. Fort > τ , the current extraction cost is c1(t; τ), the inverse of this function (in t) is t˜1(c; τ)and total extraction to date is x1(t; τ).Innovation accelerates extraction:Lemma 1 With discovery of the new backstop y1at date τ , we have: p1(s; τ) < p0(s), forall s > x0(τ); for all t > τ , x1(t; τ) > x0(t); and for all c > c(τ), t˜1(c; τ) < t˜0(c).Proof: in the Appendix.The interaction of CCS innovation and clean-energy innovation is central. The acceler-ation of emissions that results from clean energy innovation brings the emissions from thefossil fuel available at each extraction cost c forward in time to a date when CCS is lessdeveloped.583.4. Clean Energy InnovationProposition 3 In the presence of CCS innovation a(t), the impact of clean energy dis-covery at a date τ ∈ [0, t˜0((y0 − z)/λ0)] is a change in cumulative emissions given by(y1−z)/λ0∫c(τ)[a(t˜0(c)− a(t˜1(c; τ)]dG(c)−(y0−z)/λ0∫(y1−z)/λ1[1− a(t˜0(c))]dG(c) (3.6)For sufficiently strong innovation in CCS, the first term on the right-hand side of (3.6)dominates, so that total cumulative emissions increase with the clean energy discovery.Proof. In (3.6), the first term represents the increase in emissions net of CCS of thoseunits of carbon that are emitted with or without the discovery of the new clean-energyalternative. This term is positive (by Lemma 1) because all deposits of extraction costbetween (y1 − z)/λ1 and (y0 − z)/λ0 are extracted more quickly and therefore, subjectto the lower value of a(t). The second term represents the emissions released under theoriginal clean energy technology but not under the new technology.Choosing a∗(t) greater than 1− ε over, for arbitrarily small ε, makes the second termarbitrarily small for all innovation patterns stronger than a∗(t). Thus, for sufficiently stronginnovation in CCS, the discovery of clean energy leaves the environment worse off in thelong-term.3.4.3 CE innovation threat and a green paradoxWe have, to this point, examined the impact of a realized discovery on the long-term climateoutcome. To isolate the ex ante effect of a threat on innovation, we compare two pathsfor fossil fuel prices: the price path that the fossil fuel market would follow if innovationis impossible, and the price path that the market follows if innovation is possible (withprobability ρdt in a small interval dt) but never realized. This comparison captures theimpact of the threat of innovation on prices and consequently on the extraction path and593.4. Clean Energy Innovationcumulative extraction.We start with the model of clean-energy innovation. The examination of the ex anteimpact of clean energy innovation requires ex ante price paths for both the innovation-is-impossible case and the innovation-is -possible case. The former price path is given by(3.3). The innovation-is-possible price path we derive from a recursive model with statevariables (x, δ), where x is the cumulative extraction to date and δ takes on the value 0 ifthe new technology has not been discovered to date and the value 1 if it has. We againlet p0(t) denote the price path prior to innovation and p1(t; τ) denote the price at date tfollowing discovery of the technology at date τ ≤ t. Thus, p1(t; t) is the equilibrium pricein the market established immediately upon discovery at date t. The differential equationfor the ex ante price path is determined by the condition that the owner of a deposit ofextraction cost c be indifferent, as dt approaches 0, between extracting at date t˜(c) andextracting at date t˜(c) + dt. This condition is:p0(t)− c(t) = limdt→0e−rdt[(1− ρdt)p0(t+ dt) + ρdt · p1(t; t)− c(t)]. (3.7)Taking this limit yields the following lemma, proved in the Appendix B:Lemma 2 With clean-energy innovation under the parameter ρ, the price path prior todiscovery satisfies:p˙0(t) = r{[p0(t)− c(t)] + ρ[p0(t)− p1(t, t)]}. (3.8)We denote the variables before and after discovery with the subscripts 0 and 1. Theendogenous variables without subscripts refer to the innovation-is-impossible case. Equa-tion (3.8) differs from the price innovation-impossible path, (3.3), only in the inclusion ofthe last term on the right-hand side. This term reflects the additional opportunity cost forwhich the holder of reserves of cost c(t) must be compensated to be indifferent between603.5. Fuel Efficiency Innovationholding the reserves for an additional instant and selling them at t: the threat of a capitalloss with a price drop to p1(1, 1).Proposition 7 below characterizes the impact of the threat of unrealized innovation onthe price path: the price starts lower as a result of the threat and moves more steeply,under (3.8) than under (3.3). This means that the date at which reserves of cost c areextracted, t˜(c), is reduced by the threat of innovation, for all c up to the terminal cost, b.The earlier date means that for each c, emissions at c are subject to CCS at any earlierdate. If there is innovation in CCS, then this earlier date leads to greater net emissions.The proposition is proved in the appendix.Proposition 4 (a) In the absence of CCS innovation, the ex ante impact of clean-energyinnovation on cumulative emissions is zero.(b) With CCS innovation , a˙(t) > 0, the ex ante impact of clean energy innovation is toincrease emissions by:∫ (y0−z)/λ00[a(t˜(c))− a(t˜0(c))g(c)dc > 0The threat of innovation does not affect the gross cumulative carbon emissions (i.e.,emissions prior to the application of CCS) but increases net emissions because each unitof fossil fuel is subject to the less well-developed CCS technology at an earlier date.3.5 Fuel Efficiency InnovationIn this section, we turn off innovation in CE technology and fix the backstop cost at y.Our focus in this section is the effect of innovation in FE technology. First, we study the613.5. Fuel Efficiency Innovationeffect of a realized but unexpected innovation in FE. Then we study the interaction withCCS in two scenarios: first, an inelastic demand. This example helps solidify the intuition.And later, we allow for a general demand function, which brings the Jevons paradox intoplay. Finally, we study the effect of the threat of innovation in FE.3.5.1 Paradoxical effect of an unexpected FE innovationSimilar to CE innovation, we consider a single innovation in fossil fuel efficiency thatwould lower the fuel requirements per unit of machine output from λ0 to λ1. Only thedate of discovery is uncertain, and this date follows a Poisson process with ρdt being theprobability of an exogenous innovation in a small interval dt if the innovation has not yetbeen discovered.8 In the absence of CCS innovation, the cumulative amount of carbonemission after innovation is given by changing λ0 to λ1 in equation 3.5.x(T ) = G((y − z)/λ1)The following proposition summarizes the FE innovation paradox.Proposition 5 Suppose that with an initial technology described by fuel requirements perunit of λ0, the terminal date for fossil fuel extraction in this model is T . An innovationin [0, T ] that reduces the fuel requirements to λ1 < λ0 raises the cumulative fuel extractionand carbon emissions by:G((y − z)/λ1)−G((y − z)/λo).Fuel of extraction cost in the interval [(y − z)/λ0, (y − z)/λ1 becomes economic only8We assume that innovation requires that the market for fossil fuels has a positive quantity; that is, weignore the possibility that new innovation occurs after the fossil fuels market is closed that would renderfossil fuels once again economic.623.5. Fuel Efficiency Innovationwith the discovery of the new fossil fuel technology. The new technology allows each unit offuel to produce more machine output, offsetting the higher cost. The proposition is easilygeneralized to the case where the backstop technology is improving over time(i.e. where yis a decreasing function of t). 93.5.2 CCS innovation saves the day: Inelastic demandThe impact of fuel efficiency innovation if CCS innovation is in place depends on theelasticity of demand for machine output. We start with the assumption that this demandis perfectly inelastic, at a fixed quantity h. In interacting fuel efficiency innovation andCCS, we have in mind ICE’s in industrial production as well as electricity production,rather than transportation.With the original technology, the date of extraction of a fuel reserve of cost c is given byt˜0(c) = G(c)/λ0h for values of c up to the maximum cost of reserves extracted, (y− z)/λ0,because fossil fuel is extracted at the rate λ0h. With discovery of a new fuel-efficiencytechnology at date τ , the rate of emissions drops to λ1h. That is, the emissions rate isidentical up to the reserves of cost c0(τ), but then decreases to λ1h up to the (higher)terminal cost (y − z)/λ1. Thus for c > c0(τ),t˜1(c; τ) =G(c0(τ))λ0h+G(c)−G(c0(τ))λ1h>G(c)λ0h= t˜0(c). (3.9)Given CCS innovation, a′(t) > 0. Hence, from (3.9), a(t˜1(c; τ) > a(t˜0(c)).Proposition 6 With the discovery of a new fossil fuel technology at date τ ∈ [0, (y−z)/λ],9And we conjecture that endogenizing innovation in the clean backstop magnifies the detrimentaleffect of an increase in fuel efficiency. Successful innovation in clean energy would be rewarded with delayedpresence in the market if the innovator were competing against a higher FE. This would decrease theincentives to innovation and the reduced innovation would in turn allow the fossil fuel power generation tosurvive until the fossil fuel extraction cost reached a higher level.633.5. Fuel Efficiency Innovationgiven CCS innovation, cumulative emissions fall by∫ (y−z)/λ0c0(τ)[a(t˜1(c; τ))− a(t˜0(c))]dG(c)−∫ (y−z)/λ1(y−z)/λ0[1− a(t˜1(c; τ))]dG(c) (3.10)For sufficiently strong innovation in CCS, the first term dominates, and the discoverydecreases cumulative emissions.There is no change in emissions up to the discovery date τ(i.e., no change in emissionsfrom reserves of cost c ∈ [c, c(τ)]). Beyond this cost interval, the first term of (3.10)represents the reduction in cumulative emissions net of CCS from those deposits that areused under either technology. This is positive because the lower rate of emissions followingfuel-efficiency innovation means that t˜1(c; τ) > t˜(c) in this interval. The emission of carbonfrom the deposits of cost c, which had resulted in net emissions of 1− a(t˜0(c)) prior to thediscovery of the new fossil fuel technology, result in smaller net emissions of 1− a(t˜1(c; τ))after the discovery. The second term of (3.10) represents the emissions from deposits thatare rendered economic only by the more efficient new fuel technology. For sufficiently strongCCS innovation, this second term is smaller than the first term, proving the proposition.3.5.3 CCS-FE interaction: General demandWe have to this point discussed two paradoxes. FE innovation on its own can damagethe environment in the long-term. CE innovation with sufficiently strong CSS innovationhas the same effect. If demand is sufficiently elastic, then in extending the model of FEinnovation the logic brings into play a third paradox, Jevons paradox. The increase infuel efficiency reduces the cost and thus the competitive price for machine output. If thedemand is elastic, this price reduction raises the quantity demanded of machine output,643.5. Fuel Efficiency Innovationoffsets the lower rate of fuel required per mile, and increases the demand for fuel.10Jevons’ paradox, or the rebound effect, like our fuel-efficiency paradox, is about theimpact of a decrease in the fuel requirements of internal combustion engines and electricgeneration from fossil fuels. Jevons’ paradox describes the possible short run effect. It doesnot enter the economic logic of long-term impact when CCS innovation is zero, because itis irrelevant for long-term gross output. But whether Jevons’ paradox holds matters whenwe have CCS innovation in the portfolio because this affects the time path of emissions.In our model the derived demand for fuel is given by q(p;λ) = λQ(z + λp), where Q isthe demand for machine output. Jevons’ paradox, dq/dλ < 0, holds if the elasticity of Qexceeds 1/λ. It is easily verified that ∂2q/∂λ∂p < 0, so that if Jevons paradox holds at aparticular value of p, then it holds at all higher p. Because in equilibrium p is increasingover time, Jevons paradox holds for all time if it holds for p(0) and holds at no time if itdoes not hold at p(T ). Proposition 5 follows directly.Proposition 7 Suppose that the elasticity of demand of Q(P ) is increasing in price. Thena sufficient condition for fuel efficiency innovation to increase cumulative emissions with(or without) CCS innovation is that the elasticity of demand for machine output exceed1/λ at the price P = z + λp(0). A sufficient condition for fuel efficiency innovation todecrease cumulative emissions with sufficiently strong CCS innovation is that the elasticityof demand for machine output be less than 1/λ at the price P = z + λp(T ).3.5.4 Last piece of the puzzle: the threat of FE innovationIn the fuel efficiency model, existing technology in the production of machine output usesλ0 units of fossil fuel per unit output and requires z dollars per unit output in additionalinput expenditures. A backstop technology is available at cost per unit y. Innovation10Jevons (1965) argued that the increased efficiency of internal combustion engines might actuallyincrease the demand for fuel. The (misplaced) concern at the time was that reserves of coal would run out.653.5. Fuel Efficiency Innovationoccurs with probability ρdt in any instant dt and consists of discovery of a new technologyλ1 < λ0.The ex ante impact of fuel efficiency innovation refers to the impact on cumulativecarbon emissions of the threat, ρ, of innovation - when the innovation is never actuallyrealized. The ex ante impact operates through the impact of ρ on the price. In theclean-energy innovation model, we compare the ex ante price path with the innovation-is-impossible price path. If the ex ante price path is lower, then the fixed amount of cumulativeemissions, G((y − x)/λ), is released at an earlier time with predictable consequences if wehave CCS innovation in the existing portfolio of innovation.The dynamics of the endogenous variables (using the same notation as in the ex anteclean-energy model) p0(t), q0(t), c0(t), x0(t); p1(t; τ), q1(t; τ), c(t; τ) are governed by the fol-lowing conditions.Using a logic parallel to the ex ante case for clean energy, one can show that the exante price path follows equation (3.8). The terminal condition is p0(T ) = c(T ) = (y−x)/λ.(That is, if the new technology is not discovered, fossil fuel extraction continues until thealternative technology is equally costly.) The cumulative gross emissions, G((y − x)/λ,are unaffected by the threat of innovation. The ex ante price given the state x is lessthan the innovation-impossible price path if and only if the price falls with discovery(i.e.,if p1(t; t) < p0(t), equivalently, if p1(c; τ) < p0(c)).Unlike the case in the clean-energy innovation model, this latter inequality holds, if atall, only in early years. There are two forces at work. First, the new technology reducesfuture consumption conditional upon innovation, assuming demand elasticity below thecritical point for Jevons’ paradox to kick in. This increases the length of time that the fuelwill last, reducing its scarcity value. To take an extreme example, suppose that the currentcost of extraction is zero, but fossil fuel will run out (or become prohibitively expensive663.5. Fuel Efficiency Innovationto extract) in one year. Then the current price in the market for fossil fuels is boundedbelow by [(y − z)/λ0]/(1 + r) because holders of reserves have the option to wait one yearto sell at (y − z)/λ0. This fossil fuel price may be very high, if y is high. But supposethat the innovation is so miraculous that if it occurs it reduces the requirements of fossilfuel per mile from λ0 to λ˙1 = 0.001 · λ0 with the effect that fossil fuel will be available atzero extraction cost for about 1,000 years. The price of fossil fuel will fall to about zero.The threat of the capital loss from innovation therefore, reduces the fossil fuel price in thisexample, accelerating extraction and emissions. In the presence of CCS innovation, thisbrings emissions forward to a time of low CCS effectiveness. Net emissions rise.The second force underlying the ex ante impact of fuel efficiency innovation worksin the opposite direction. Consider an innovation at the exact date, T, of at which themarket would switch to the alternative technology downstream. (T = G−1((y−z)/λ0) TheHotelling rent, p(t)− c(t), jumps from zero to a positive value. (The price after innovationfollows an innovation-impossible path, and Hotelling rent is always positive before the enddate.) From continuity of the price paths, there is an interval [t, T ] over which the impactof an innovation on price is always positive.11 The ex ante impact of innovation over thisinterval is therefore, to raise price, due to the promise of the capital gain. This lowersthe output for sufficiently elastic demand, deferring emissions to the future. With CCSinnovation in place, net cumulative emissions are reduced. In short, the threat of fuelefficiency innovation always helps net emissions late in the game.The following proposition formalizes this discussion:Proposition 8(a) In the absence of CCS innovation, the ex ante impact of fuel-efficiency innovation oncumulative emissions is zero.11This interval may include t = 0.673.6. Conclusion(b) With CCS innovation but perfectly inelastic demand for machine output, the ex anteimpact is again zero.(c) With CCS innovation and d′(p) < 0, the ex ante impact of fuel efficiency innovation,beyond some tˆ < T, is to reduce cumulative net emissions.The proof is in the appendix B.3.6 ConclusionTo formulate optimal climate change policy, we must first understand the impact of exoge-nous innovation. We show in this paper that the impact of the main types of innovationdepends on the portfolio of existing innovation. In two cases, clean-tech innovation canhave a perverse long-term impact both in terms of realized innovation and, under furtherconditions, the ex ante threat of innovation.The models in the paper relied on two technical assumptions (Hotelling Model for pricepath of fossil fuels and innovation modeled as an unexpected one-time reduction of someinput), both of which can be relaxed in an extension of this work. The first assumption isstrong in that it requires the price of fossil fuels to increase with the annual interest rate.Our results will still hold if fuel producers collect rent on top of the extraction cost andthat rent decreases if renewable prices drop.The second assumption (of a one-time Poisson process) can be relaxed to allow for mul-tiple Poisson processes. We showed that both an unexpected innovation and anticipation ofinnovation lead to qualitatively similar results. If the results hold for one Poisson process,it will hold for multiple Poisson processes as well. Therefore, if one can model gradualinnovation as the sum of multiple incoming Poisson processes (with different probabilitiesfor the size of each), our results would still hold.683.6. ConclusionWe emphasize that our results about perverse impacts of innovation do not make thisan “anti-innovation” paper. The paradoxical results all pertain to a world with no carbontaxes. With optimal Pigouvian carbon taxes, all emission externalities would be inter-nalized and innovation, by expanding the social planner’s choice set, could only increasewelfare. The main policy implication of this paper is to underscore the importance of car-bon pricing. Without carbon taxes, one cannot even be sure that innovation has positivevalue.Policy discussions have not appreciated the limitations of innovation as a sole instru-ment for tackling climate change. One example is the Green New Deal, a pair of resolutionssubmitted to US Congress in 2019, sponsored by Rep. Alexandria Ocasio-Cortez (D-NY)and Sen. Ed Markey (D-MA). These resolutions discuss proposed goals of reducing green-house gas emissions, among other objectives. The goal of reducing emissions is laudatory,but the Green New Deal resolutions contain no mention of carbon taxes whatsoever. In-stead, the proposed methods for achieving the goals include “making public investments inthe research and development of new clean and renewable energy technologies and indus-tries”. 12 Our message is that carbon pricing is essential even to ensure that the researchand development of clean energy technologies has a positive value, let alone be the centraltool for reducing greenhouse gas emissions.The results of our analysis are one piece of a large issue, the optimal policy towardsclimate change. This policy would consider not only the long-term impact of innovation butalso the full welfare, taking into account short- and medium-term effects. A fuller analysiswould also consider the sensitivity of innovation size and likelihood to government subsidiesof innovation. In our model, the potential perverse impact of clean-energy innovation ofinnovation (when the rate of improvement in CCS is substantial) would disappear if the12p.12 of the House resolution693.6. Conclusioninnovation were sufficiently large. A large innovation in clean energy would eliminate allemissions immediately. This suggests a non-concavity in optimal innovation policy. Smallsubsidies to innovation may have a negative value in the absence of carbon pricing, whereasthe marginal value of innovation subsidies may be invariably positive at a sufficiently highsubsidy level. The policy implication of this non-concavity; in an optimal world, innovationsubsidies cannot be half-hearted.70Chapter 4Asymmetric Information,Reelection Pressure and PoliticalDecision Making underUncertainty4.1 IntroductionA foundational principle of a well-functioning democratic government is that elected leadershave strong incentives to act in the public or voters’ interest. We expect a politician whoacts against this interest to lose electoral support and be replaced in a future election.However, some considerations may undercut this principle. One possibility is thatvoters may be less informed about major issues than are elected officials. A politicianseeking re-election may prefer to implement policies consistent with the beliefs of poorlyinformed voters rather than enact better (welfare-superior) but less popular policies.In a nutshell, the public relies on the politician to acquire relevant information andmake informed decisions. The politician is better informed, but the relatively ill-informedelectorate chooses the politician’s re-election fate. The information asymmetry createstension between decisions in the voters’ interest and what merely looks good to the voters.714.1. IntroductionThis tension manifests itself in the decisions that the politician makes and the politician’sinformation-gathering efforts.This paper studies different aspects of this tension, theoretically and empirically. Iexamine the interaction of the next election’s proximity with politicians’ distortion towardspolicies that appear to be in the public’s interest - but are not. We expect the asymmetryof information between politicians and the public to diminish over time. Therefore, thepoliticians who have an imminent upcoming election will be more likely to act toward theill-informed voters’ beliefs. On the contrary, the politicians facing distant elections canact based on voters’ interests (and ignore voters’ misguided beliefs), knowing that theywill update their belief until the election. I also examine how a lack of incentive for thepolitician to act in the public’s interest leads to the politician’s sub-optimal informationacquisition.In the paper’s theory section, an incumbent politician has to decide on independentissues in an uncertain environment (e.g., response to a crisis). Each particular issue consistsof two opposing extremes (i.e., act strongly vs. ignore the crisis). The politicians can chooseanywhere between the two extremes.The politicians’ objective is to get re-elected (or their party to stay in power). They actto maximize their re-election probability at the election date in expectation. The votersare not active players. They judge politicians’ actions based on the information availableto them. The incumbent politicians’ chance of re-election declines if their decision deviatesfrom the public’s belief about the optimal action. The decline is proportional to the sizeof the deviation and the quality of public information.The politician receives an unbiased signal of the best policy before making their deci-sion. Before making any decision, the public receives a noisier version of the same signal.The politician forms an expectation of the public’s belief on the election day and takes ac-724.1. Introductiontion. The public receives many more independent signals until the election day and keepsupdating its belief. The public can have a diffuse or non-diffuse prior, and their signalsmay be more or less informative than the politician’s signal.I consider two possibilities for the public’s belief and solve the model for those cases:unified and polarized. A unified public means that voters agree about the probabilitiesof the optimal action, whereas a polarized public means that voters sort themselves intotwo opposing groups. The group members agree within itself but disagree between the twogroups.In either case, with an innocuous assumption about information asymmetry, I show thatthe politician deviates from implementing the best action conditional on their informationset. Moreover, the closer the election, the more the politician deviates from implementingthe welfare optimizing policy, holding everything else constant. The required assumptionis that the politician has superior information at the time of decision-making about thebest action compared to the public. Some (but not all) of this superior information fadesaway over time as the public receives more informative signals.A polarized public implies that any politician must consider two opposing beliefs andits effect on their popularity when making a decision. Polarized elections are decided byvoter turnout, and turnouts are proportional to candidates’ popularity. Following thislogic, I assume that the politicians care about their popularity among their supporters,hoping to excite them to turn out, more than their popularity among their opposition.This assumption allows me to pin down that the direction of the politicians’ deviationfrom the optimal decision. Their bias is towards their voting base.As the last theoretical contribution, I endogenize the quality of the politician’s signal.Exogenous signal quality is unrealistic because the politician can exert time and resourcesto obtain a higher quality signal. We should investigate if endogenizing the signal quality734.1. Introductionaffects the result. I show that previous results still hold. Moreover, I show that the signal’squality is directly proportional to voters’ signal quality and is inversely proportional tothe election timing. A politician with a later election spends more resources to acquire ahigher quality signal.In the empirical section of the paper, I use a difference-in-differences strategy to testthe following prediction: two politicians who are identical except for their next election’stiming behave differently during the crisis. In early 2020, the United States was hit by theCovid-19 pandemic. In this setting, US governors are responsible for deciding on multiplepolicies to limit business activities and protect citizens’ health in their state. About halfof these governors were Republican, and the other half were Democrats. One-fifth of thesegovernors have an upcoming election in 2020.The voting base of Democrats assess the threat of the disease higher than the Re-publicans and required a more robust response from the government. The voting base ofRepublicans, however, emphasizes the individual liberties of citizens and oppose decisivegovernment intervention. The situation is highly uncertain since the threat of the dis-ease’s infectivity and lethality is unknown, and there are doubts about the effectiveness ofdifferent policies.Eventually, the data shows that the Democrat governors who have an upcoming electionin 2020 are the most likely to close economic activity. The Democrats and Republicanswho do not have an upcoming election behaved statistically similar to each other. Finally,the Republicans who have an upcoming election in 2020 are the least likely to close anyeconomic activity.The empirical data are consistent with the hypothesis that the governors with an up-coming election are biased towards their voters. In contrast, the ones without an upcomingelection implement their best expectation. It is worth noting that this empirical strategy744.2. Literature Reviewdoes not require knowing what the optimal (i.e., correct and informed) action is. It onlyrelies on the observed decisions made by politicians.In the next section, I review the relevant literature. In section 3, I proceed to explainthe model and derive the theoretical results. In section 4, I explain the data, set up themodel’s empirical implications, and present the results. And I conclude in section 5.4.2 Literature ReviewRetrospective voting is the idea that voters hold the politicians (or parties) accountablefor their actions during their tenure. Key (1966) formalizes this idea in his seminal work,the Responsible Electorate. Retrospective voting readily invites the idea of the principal-agent problem faced by the public. Barro (1973) and Ferejohn (1986) model the moralhazard problem in the accountability of the politician to allocate economic resources effi-ciently. Reed (1994) addresses the adverse selection problem by introducing heterogeneouspoliticians.Nordhaus (1975) introduces the idea of political business cycles. If the electorate votesretrospectively, the incumbent chooses the economic policy that maximizes its probabilityof re-election, which is different from the optimal policy. The incumbent increases its re-election chance by “manipulating” the economy and increasing welfare in the election year(Nordhaus (1975), Suzuki (1992)). The incumbent may also “surf” the economy (i.e., tocall elections more (Palmer, Whitten (2000), Roper, Andrews (2003)) or less Smith (2003)opportunistically in favorable economic times). Kayser (2005) studies the case where theincumbent both surfs and manipulates the economy for re-election.In recent years, researchers started to focus more on studying retrospective voting onissues other than national macroeconomic policies (e.g. school performanceBerry, Howell (2007), handling disasters Healy, Malhotra (2010), and relative754.2. Literature Reviewunemployment in a city Hopkins, Pettingill (2018)). This literature concludes that if apolicy influences a tangible outcome and is within the politician’s sphere of control, thenthe incumbent’s poor decisions harm their chances in the next election. Moreover, thesepapers suggest that the incumbent’s punishment is mediated by media attention level onan issue.A laboratory experiment by Woon (2012) confirms these conclusions. Woon asserts that“retrospective voting is a simple heuristic that voters use to cope with a cognitively difficultinference and decision problem and, in addition, suggest that voters have a preference foraccountability.”Healy, Malhotra (2013), raise a few open research questions. I partially address two ofthem in this paper. They uncovered that no one has studied the effect of retrospective vot-ing on policy outcomes. They also state that no one studied the interaction of polarizationand political sorting with retrospective voting.This brings us to the literature on political polarization, specifically the effects of polar-ization on political decision-making. The consensus among political scientists is that theAmerican political elites have become more polarized over the past few decades Fiorina,Abrams (2008). Most of the literature show polarization by focusing on congress roll callsand deviation from the party ranks McCarty et al. (2006). In addition to that, a historicalanalysis of politicians’ speech by Gentzkow et al. (2019) shows polarization has significantlyincreased among the political elite since the 1990s.But there is an ongoing debate on the degree to which the American public is polarized.Some academics Abramowitz, Saunders (2008) assert that polarization has grown amongthe American public, and people hold more extreme and opposing beliefs. Others pointout Fiorina et al. (2008) that only party sorting has increased among the public, andeven though opinions are not more polarized, voter-party identification is much stronger.764.2. Literature ReviewPolitical parties play a significant role in this polarization by increasing control over theirmembers and setting a polarized agenda Canen et al. (2020). Whether the public hasbecome more polarized (holds more extreme views) or it becomes more sorted (more peopleself identify with political parties and its politics), it caused “the nation as a whole to holdmore aligned political identities, which has strengthened partisan identity” Mason (2015).Grumbach (2018) analyzes sixteen issues of gubernatorial policies across US statesto study the interaction of decision making and polarization. He shows that party controlpredicts socioeconomic decisions in polarized topics like health care but not in nonpolarizedareas like criminal justice. Moreover, the policy disparity between parties on polarizedissues has increased from 1970 up to 2014.Gentzkow, Shapiro (2010) show that the media find it economically optimal to de-viate from reporting an unbiased version of the news. They found evidence that profit-maximizing American newspapers respond to their readership biases by introducing a slanttowards their base. In a similar vein, I show here that incumbent politicians find it optimalto introduce a slant towards their base in implementing policies.In this paper, I make use of the findings of the literature on public learning and opinionformation. Acemoglu et al. (2006) show that it is theoretically possible for the public to beBaysian and holds polarizing opinions . Others showed that pre-existing polarized opinionsmight lead to a different assessment of similar information (Gerber, Green (1999), Dixit,Weibull (2007)).This paper can be classified as part of the emerging literature on responses to Covid-19.I contribute to this literature by studying how the governor’s party and the election timingaffect the governments’ decision for implementing social distancing policies. Chernozhukovet al. (2021) study the effect of government policies in the contagion of Covid 19. In arecent study, Allcott et al. (2020) show that Democrats and Republicans perceive Covid-19774.3. Theoretical Modeldifferently. Democrats see Covid 19 as more dangerous and expect more stringent behaviorfrom the government. In the absence of the aforementioned political considerations, Alvarezet al. (2021) theoretically study the optimal policies to respond to Covid-19 . To my bestknowledge, this is the first paper that discusses the political challenges in finding andimplementing the optimal policy.4.3 Theoretical ModelThis section formalizes a game between politicians and voters. Incumbent politicians (andpolitical parties) make decisions during their tenure. The politician wants to win re-election, whereas the voters want the politician to make the welfare-maximizing decision.We need to formally distinguish among welfare maximizing, the politicians’ informationabout it, and what voters believe. These three do not always fully agree. There is aninformation asymmetry between politicians and voters.This information asymmetry interacts with election timing through an informationchannel. Voters learn new information after the politician makes a decision. The politiciansform expectations about what voters will learn until the election and bases the decision onthat expectation. If the election is in the distant future and enough information becomesavailable to the voters at the election day, the politicians tend to act based on their superiorinformation. Nevertheless, if the election is in the near future and politicians do not expectthe voters to update their information, they tend to act more similarly to the voters’ priors.The model allows for a polarized electorate, and it also paves the way for endogenizingthe information gathering efforts of the politician. The model yields a clear difference indifferences style prediction that can be tested with real-world data on politicians’ decisions.784.3. Theoretical Model4.3.1 SetupTo formally model the interaction between the politician and voters, assume that naturesets the welfare-maximizing action and sends a signal to the politician and the public. Thepolitician may be a Democrat or Republican. The public may be unified and receive asignal or be polarized, and each group receives a signal separately.Every player updates their belief according to their signal. The politician takes action.Then the public receives new signals. Finally, an election takes place. For some politicians,this election is early, and for the others, it is later.For every politician, the model has the following timing:• Nature sets the welfare-maximizing action for an issue.• Nature sends a signal to the politician, and the politician updates their belief with adiffuse prior belief.• Nature sends a noisier version of the same signal to the public, and the public updatesits belief with its prior belief.• The politician chooses their action.• The public receives N independent signals from nature and updates their beliefs untilthe election day.• Election takes place (it may be early or late).The real line represents the issue. A point represents the politician’s decision on thereal line s ∈ IR. I assume that there exists a unique decision point on the issue line thatmaximizes social welfare. Still, neither the politician nor the public knows the optimaldecision point with certainty. They both have a belief about the optimal decision.794.3. Theoretical ModelA belief is a normal distribution ∼ N (sx, σx) over the issue. This belief represents thetrue probability that a belief holder assigns every decision to be the welfare-maximizingdecision. The higher the standard deviation (σx), the lower the certainty of the beliefholder.Take the welfare-maximizing action to be so. The politician has a diffuse prior. Theythen receive a signal sg about the optimal action. The signal is an unbiased randomvariable with known uncertainty σg. Therefore, the signal has the distribution N (so, σg).The realization of this signal is sg. Following the Bayes rule, their belief on the best policyafter receiving the signal will be N (sg, σg).Similarly, the public receives a signal sg + g. g is a mean-zero normally distributedrandom variable. The public combines the signal using its initial prior and form its belief.Therefore, the posterior belief will be normally distributed as N (sp, σp). It is possibleto write sp and σp as a function of the prior and the signal, which will be done later.Nevertheless, this suffices for the moment.We can think of the public’s belief as the wide-held opinion of the voters. Ideology,past shared experience, and the media slant all influence public opinion. Therefore, itconsists of prior information, potential bias, and a noisier version of the politician’s signal.The politician knows about this belief; hence, it has no extra information for a politician’sinference of so. They can safely ignore this belief for inferring so.After the politician and the public both receive the signal, the politician chooses theiraction. However, the public’s belief is dynamic. For simplicity, assume that the publicreceives exactly N signals from nature up to the election day. Each signal is consistentand distributed as N (so, σy). Label every realization of these signals as sˆip. Therefore, the804.3. Theoretical Modelpublic opinion on the election day would be normal with the following distribution:sp(T ) ∼ N (spσ2y +∑Ni=1 sˆipσ2pσ2y +Nσ2p,σ2pσ2yσ2y +Nσ2p) (4.1)The public wants the politician to implement the welfare-maximizing policy, but itdoes not know about the information set available to the politician. The politician maybe incompetent and have high uncertainty. They may have some ulterior motives andnot implement the best policy, or they get unlucky, and the signal’s realization is far off.Irrespective of the reason, the politician will lose popularity if they deviate from the meanof public belief.4.3.2 Solution for a unified publicThe politician wants to maximize their re-election probability. In Appendix C, I show howto derive a loss-function approach for such a politician. With a unified public, the followingloss-function (re-election probability) captures the politician’s objective:P = P0 −(s∗ − sp(T )σp(T ))2. (4.2)The politician does not have to act (s∗) based on what they believe to be the best policy(sg). As new information emerges, the public’s belief about the optimality of politician’saction changes. The politician wants to maximize their expected re-election probability.This probability is negatively proportional to the squared difference of what they did andwhat the public holds to be true on average at the election day (T). The coefficient ofproportionality is the inverse of the public’s certainty about the best action.If issues are independent of each other, they enter the probability function in a separableadditive way. As a result, the politician can make decisions on each issue separately.814.3. Theoretical ModelTherefore, as long as each issue is independent of all others, the model can be generalizedto multiple issues.The politician acts to maximize their expected re-election probability conditional onthe information set they have at the decision-making time:s∗ = argmaxsP0 − E[(s− sp(T )σp(T ))2|I]. (4.3)Proposition 9 With a unified public, the politician’s optimal action is a linear combina-tion of the politician’s belief and public’s belief.s∗ =1σ2y +Nσ2p(spσ2y +Nsgσ2p). (4.4)Proof. From equation 4.1, the politician knows the voters’ uncertainty on the electionday:σp(T ) =σ2pσ2yσ2y +Nσ2p.Therefore, the following first-order condition is obtained:s∗ = E[sp(T )|I]. (4.5)Using the linearity of the expectation operator and equation 4.1 we get:s∗ =spσ2y +∑Ni=1 E[sˆip|I]σ2pσ2y +Nσ2p. (4.6)The politician’s expectation of the optimal action given their information (E[sˆip|I]) is sg,hence equation 4.4 holds.The welfare-maximizing action from the societal perspective, conditional on the infor-mation available at the decision-making time, is sg. This information is available to the824.3. Theoretical Modelpolitician, yet proposition 9 tells us that they refrain from implementing the optimal ac-tion. The amount of deviation decreases with voters’ informedness. This channel is the“action channel” of political inefficiency.Corollary 1 The politician’s deviation from the welfare-maximizing action reduces withvoters information.∂(|s∗ − sg|)∂σ2y> 0,∂(|s∗ − sg|)∂N< 0. (4.7)Proof. The deviation from the welfare-maximizing action is given by:|s∗ − sg| =σ2yσ2y +Nσ2p|sp − sg|. (4.8)Higher N means the voters receive more signal and lower σy means that the votersreceive better quality signals. Both increase voters’ information.∂(|s∗ − sg|)∂σ2y=Nσ2p(σ2y +Nσ2p)2|sp − sg| > 0.and∂(|s∗ − sg|)∂N=−σ2p(σ2y +Nσ2p)2|sp − sg| < 0.Note that if the public will be adequately informed by the election day, meaning thateither σy → 0 or N →∞, the politician will act based on their best knowledge.4.3.3 Solution when the voters are sortedIf there is more than one group in public, a single number cannot summarize the publicbelief, and the politician pays attention to multiple opposing opinions. The case where834.3. Theoretical Modelvoters sort themselves into two groups is most relevant in a polarized society. In theAmerican society, Democrats hold more left-leaning ideologies and pay attention to similarmedia. It is similarly true for the Republicans and those right-leaning ideologies.The loss function introduced in equation 4.2 should be modified to incorporate bothopposing views. The problem is set for an incumbent Democrat, but the logic is similar ifthe incumbent is Republican. If the incumbent’s base judges the politician’s action poorly,they might get demoralized, refrain from voting, or even switch to the opposition. Yet ifthe opposition’s base judges the politician’s action poorly, they might get mobilized andturn out more than usual for the opposition. Hence, each action affects the two groupsseparately to make different decisions. (For more discussion, see Appendix C) The followingexpression gives the incumbent Democrat’s re-election probability:P = P0 − α(s∗ − sd(T )σd(T ))2 − (s∗ − sr(T )σr(T ))2. (4.9)The subscript r denotes Republican, and d denotes Democrat. The parameter α de-notes the relative strength of the politician’s decision effect on different voting groups’turnout. With costly voting and a large enough ideological distance between two parties,the elections are decided by turnouts and less so by voters, switching parties. In such anenvironment, it is plausible that a Democrat cares more about exciting their base, ratherthan influencing Republican voters’ opinion. Therefore, for a Democrat politician α > 1.The politician chooses the action that maximizes the expected weighted re-electionprobability among both groups conditional on their information set:s∗ = argmaxx{P0 − E[α(s∗ − sd(T )σd(T ))2 − (s∗ − sr(T )σr(T ))2|I]}. (4.10)Proposition 10 With a polarized public, the politician’s optimal action is a linear combi-844.3. Theoretical Modelnation of the politician’s belief and both groups’ beliefs.s∗ =1σ2d + ασ2r(ασ2rσ2y +Nσ2d(sdσ2y +Nsgσ2d) +σ2dσ2y +Nσ2r(srσ2y +Nsgσ2r )) (4.11)Proof. Because the politician knows σD(T ) and σR(T ) at the time of decision-making,we can bring those out of expectations. The first-order condition for this maximizationproblem is the following:s∗ =1σ2d + ασ2r(ασ2rE[sd(T )|I] + σ2dE[sr(T )|I])Using the equation 4.1 for both Democrats and Republicans and linearity of the expectationoperator we get:s∗ =1σ2d + ασ2r(ασ2rsdσ2y +Nσ2dE[sˆid|I]σ2y +Nσ2d+ σ2dsrσ2y +Nσ2rE[sˆir|I]σ2y +Nσ2r)The fact that E[sˆid(te)|I] = E[sˆir(te)|I] = sg gives equation 4.11.From equation 4.11, we can derive the comparative statics for how partisanship interactswith election timing. Equation 4.11 gives the incumbent Democrat’s action (s∗d). We canobtain a similar expression for an incumbent Republican (s∗r) by symmetry (d→ r, r → d).Then, by calculating s∗d − s∗r we can compare the level of partisanship for earlier vs. laterelection timing (small vs. larger N).From this point onward, I set σd = σr := σp, mainly because there is no evidencethat Democrats and Republicans’ uncertainty levels are different, and this assumptionmarginally simplifies the results. Moreover, by abstracting away from the effects caused by854.3. Theoretical Modelthe voting groups’ uncertainty levels, we can focus on the effects caused by their belief’sdifference, which is empirically documented.The next corollary that readily follows from proposition 10 is the foundation for themain empirical tests. This corollary gives a testable difference-in-differences predictionfor the behavior of politicians. Specifically, the equation∂|s∗d − s∗r |∂N< 0 predicts that thedifference between Democrats and Republicans who face an imminent election is largerthan the similar difference if the election is in the distant future.Corollary 2 The difference in politicians’ actions is proportional to the difference in theirbases’ opinion.s∗d − s∗r =α− 1(1 + α)(1 + σ2pNσ2y)(sd − sr) (4.12)The difference size shrinks with better-informed voters.∂|s∗d − s∗r |∂N< 0,∂|s∗d − s∗r |∂σ2y> 0 (4.13)Proof. Replace both σd and σr with σp in equation 4.11 to get s∗d for the Democratincumbent:s∗d =1σ2p + ασ2p(ασ2pσ2y +Nσ2p(sdσ2y +Nsgσ2p) +σ2pσ2y +Nσ2p(srσ2y +Nsgσ2p)).This yields:s∗d =σ2y(αsd + sr) +Nσ2psg(1 + α)(1 + α)(σ2y +Nσ2p).Using the symmetry d → r, r → d, we can similarly obtain for the Republican incum-bent:864.3. Theoretical Models∗r =σ2y(αsr + sd) +Nσ2psg(1 + α)(1 + α)(σ2y +Nσ2p).Subtracting s∗r from s∗d gives:s∗d − s∗r =α− 1(1 + α)(1 + σ2pNσ2y)(sd − sr).By taking the derivative of |s∗d − s∗r | with respect to N :∂|s∗d − s∗r |∂N= −σ2pσ2yα− 1(1 + α)(1 + σ2pNσ2y)2|sd − sr| < 0., and σ2y :∂|s∗d − s∗r |∂σ2y=σ2pNσ4yα− 1(1 + α)(1 + σ2pNσ2y)2|sd − sr| > 0.Informative signals reduce polarization. The more efficient the voters become in gettinginformed, the less polarized the politicians would act. We expected this result becausevoters’ initial bias is the source of polarization, and with better information, this initialbias fades away.It is evident from equation 4.13 that as elections are getting closer, the politicians startto act more partisan. So in the cross-section, we expect politicians who have an upcomingelection act more partisan than politicians who would not face an upcoming election.In addition to the comparative statics, it is insightful to examine expression 4.11 forpolitician’s action s∗ in the limiting cases. The next corollary concerns two limiting casesthat the new incoming signals are highly informative (large N and small σy) and not874.3. Theoretical Modelinformative at all (N = 0 and large σy). Define the variable Iy ≡σ2yN, as the measure ofnew signals’ informativeness.Corollary 3 If signals are highly informative, Iy → 0, both Democrat and Republicanincumbents act based on their best information sg.limIy→0s∗d = limIy→0s∗r = sg (4.14)If signals are not informative, Iy → ∞, both Democrat and Republican incumbents ignoretheir information (sg) and act closer to their base’s beliefs, respectively.limIy→∞s∗d =11 + α(sr + αsd) (4.15)limIy→∞s∗r =11 + α(sd + αsr) (4.16)Proof. The result follows from taking the limit of equation 4.11 for Democrats and thecorresponding equation for Republicans.Highly informative signals fully break inefficiencies. Both politicians implement theirexpectation of the optimal policy, and they behave similarly. However, when the newsignals are not informative, politicians ignore facts and implement a policy biased towardstheir base.Polarization stems from two different sources, both of which are necessary for a disparityin politicians’ behaviors. The first one is a divergence in initial opinions (bias) of differentgroups sd 6= sr. However, as it is evident is equation 4.12, it is not enough that these twosocial groups have different opinions for politicians to act differently. It is also necessarythat the relative influence that the politician’s actions have over their political base to be884.3. Theoretical Modellarger than their influence on the opposition α > 1.4.3.4 Endogenizing the politician’s information gatheringThe politician’s action in both unified and polarized cases crucially depends on the ratioof the public’s uncertainty (σp) and the signal’s quality (σy). In the previous sections, Ihave assumed that nature gives this information structure. This assumption is not real-istic because, in many situations, a substantial amount of public learning depends on thepolitician’s information gatherings.The public’s information has no predictive value conditional on the politician’s infor-mation, and it consists of the politician’s signal plus noise. We can interpret the noise asa measure of the politician’s transparency and the public’s political awareness. The priorbelief of the public is normally distributed with a standard deviation of σ0, and the noise’sstandard deviation is σn then:σp =(σ2g + σ2n)0.5σ0(σ2g + σ2n)0.5 + σ0. (4.17)The politician chooses the quality of their signal (σg). They can increase their effort toget a better signal(lower σg) of the optimal action. The cost of getting better informationis proportional to the inverse of the signal’s variance:θσ2gBecause the intuition does not change with heterogeneous voters, I set the case up witha unified public for the sake of expositional simplicity. As in equation 4.2, the politiciancares about their popularity in public:P = P0 −(s∗ − sp(T )σp(T ))2.The timing of events is similar to before, except for the first step. After nature chooses894.3. Theoretical Modelthe optimal action, the politician decides on the quality of the signal. They face a tradeoffbetween being more informed and paying an extra cost of information. Their utility isgiven by:U = P − θσ2g. (4.18)The politician does not know what their realization or the public’s realizations wouldbe. However, she knows the public’s ability to accurately determine the optimal decision bythe election day (σy, σp). Therefore, they maximize their (unconditional) expected utility.σ∗g = argmaxσgE[P0 −(s∗ − sp(T )σp(T ))2 − θσ2g] (4.19)The politician optimizes the equation with backward induction. First, they solve howthey would act given a signal quality (as in proposition 1) and then chooses the optimalsignal quality that maximizes their utility.Proposition 11 The signal quality must satisfy the following first-order condition:2θσ3g=∂∂σg(Nσ2p(σ2y +Nσ2g)σ2y(σ2y +Nσ2p)). (4.20)Proof. Use equations 4.1 and 4.4 to replace the values of s∗, sp(T ), σp(T ), and write downthe first-order condition of equation 4.20:2θσ3g=∂∂σg(σ2y +Nσ2pσ2yσ2pE[(spσ2y +Nsgσ2pσ2y +Nσ2p− spσ2y +∑Ni=1 sˆipσ2pσ2y +Nσ2p)2]),which will be further simplified to:2θσ3g=∂∂σg(σ4pσ2yσ2p(σ2y +Nσ2p)E[(Nsg −N∑i=1sˆip)2]).904.3. Theoretical ModelAfter adding Ns0 and subtracting it from the term inside the expectation, and notingthat sg and sˆip are independent random variables with mean s0, we get:2θσ3g=∂∂σg(σ2pσ2y(σ2y +Nσ2p)[E[N2(sg − s0)2] + E[( N∑i=1(sˆip − s0))2]]),which yields the result.By plugging the definition of σp (equation 4.17) into equation 4.20, we can numericallysolve for the optimal quality of the signal. However, it is possible to study two limitingbehaviors relevant to the intuition analytically. One is when the public’s initial signal isvery noisy, so the politician’s signal quality does not affect the public. The other is whenthe public has a diffuse prior and maximally learns from the politician’s signal.In the next corollary, we consider the case that the public receives a very noisy signal(σn → ∞). Label this signal quality as σˆg. Recall that we have shown new signals’informativeness with Iy ≡σ2yNCorollary 4 If the public receives a very noisy signal (σn →∞), the optimal signal qualityincreases with voters’ signal informative quality.σˆ2g := limσN→∞σ∗g =√Iyθ(1 +σ2yNσ20)(4.21)Proof. First, calculate the limit for σo from equation 4.17:limσN→∞σp = σ0Then use the first-order condition in proposition 3 and replace σp with σ0 to get:2θσ3g=Nσ20σ2y(σ2y +Nσ20)∂∂σg(σ2y +Nσ2g) =2N2σ20σgσ2y(σ2y +Nσ20).914.3. Theoretical ModelBy moving the terms around, we get:σˆg4 =θσ2y(σ2y +Nσ20)N2σ20.The square root of it gives equation 4.21.Since the initial signal that the public receives is not informative, the public’s belief’suncertainty on the election day is proportional to the information quality (Iy) and theirinitial uncertainty. Since this initial information is a form of bias, the more assured of theirinitial biased opinion, the less the politician can dissuade them, and their informationwill not be useful. Smaller σ0 decreases politician’s incentive to obtain higher qualityinformation.Next, consider the case that the public has a diffuse prior (σ0 →∞). Label this signalquality as σ˜g.Corollary 5 If the public has a diffuse prior (σ0 →∞), the optimal signal quality increaseswith voters’ signal informative quality.σ˜2g = limσ0→∞σ∗g =√Iyθ1√1− Nσ2nσ2y(σ2y +Nσ˜2p)2(4.22)Proof. The proof is similar to the proof of corollary 3.1.In the first scenario, the noise shields the politician’s information from the voters;therefore, they can use all of that information to their benefit. In the second scenario, someof the information would leak to voters, and they would use that to judge the politician.Controlling for the signal quality, the politician seeks a lower quality signal if the voterscan learn from the politician’s information.The next proposition concerns the general relationship between voters’ signal’s infor-924.3. Theoretical Modelmative quality and the politician’s signal quality.Proposition 12 The optimal signal quality increases with voters’ signal’s informativequality.∂σ∗g∂σy> 0,∂σ∗g∂N< 0 (4.23)Moreover, if voters are uninformed (Iy →∞), the politician obtains no signal.limIy→∞σ∗g =∞ (4.24)Proof. For the first part of the proposition, I use the envelope theorem on equation 4.19and use the results from equation 4.20 to write the following equation for the derivativewith respect to σy:∂σ∗g∂σy= − ∂∂σy(Nσ2p(σ2y +Nσ2g)σ2y(σ2y +Nσ2p))Taking the partial derivative gives:∂σ∗g∂σy= −Nσ2p2σy(σ4y +Nσ2yσ2p)− (σ2y +Nσ2g)(4σ3y + 2Nσyσ2p)(σ4y +Nσ2yσ2p)2,which gives:∂σ∗g∂σy= Nσ2p2σ5y + (Nσ2g)(4σ3y + 2Nσyσ2p)(σ4y +Nσ2yσ2p)2> 0,and similarly for the derivative with respect to N :∂σ∗g∂N= − ∂∂N(Nσ2p(σ2y +Nσ2g)σ2y(σ2y +Nσ2p))< 0.For the second part of the proposition, assume the contrary and call the finite value of thelimit S:limIy→∞σ∗g = S.934.4. EmpiricsThen take the limit of the first-order condition given by proposition 3 (equation 4.20):limIy→∞2θσ3g= limIy→∞∂∂σg(Nσ2p(σ2y +Nσ2g)σ2y(σ2y +Nσ2p)).We can change the order of derivative and limit, and use the fact that the limit of σp andσg are finite to get:limIy→∞2θσ3g=∂∂σglimIy→∞(Nσ2pσ2y)= limIy→∞(σ2pIy).The left-hand side is finite by assumption, but the right-hand side grows unboundedly.In the first two propositions, we explored the “action channel” of political inefficiency.The politician has the relevant information, yet she refrained from implementing the effi-cient option. Nevertheless, proposition 4 tells us that if voters cannot collect information byelection time, the politician will not have any incentives to obtain any information. Propo-sition 4 is the “information channel” that prevents the political system from implementingan efficient solution.4.4 EmpiricsThe empirical section is an indirect test of the theory developed in the previous section.Specifically, I test the implication of proposition 2, corollary 2.1, which states that thedifference between Democrats and Republicans shrinks the further their election since thepoliticians expect the voters to be more informed regarding later elections than earlier ones.The context of this section is gubernatorial decisions regarding the economy shutdownduring the Covid-19 crisis.944.4. Empirics4.4.1 Empirical SetupAlternative models that explain the difference between Democrat and Republican politi-cians’ actions implicitly or explicitly assume preference for actions based on the politicians’types. The politician either acts in their own best judgment and preference or their votingbase’s best interest and preference. In either case, this should not have any interactionwith the election timing. On the contrary, this paper, based on information asymmetry,predicts that the incumbent’s bias will be intensified when the election is closer or thesignals are noisier.The optimal action is not directly observable. Hence, it is not possible to directlymeasure politicians’ bias from optimal behavior. If we suppose Democrats and Republicanshave different optimal actions, it seems impossible to distinguish whether this action is inline with the corresponding optimal action, or it is a biased behavior to get re-elected. Ageneral disparity of behaviors is consistent with rational decision makers with heterogeneouspreferences and strategic bias towards the base.The Covid-19 crisis allowed us to test the idea that elections interact with politicaldecision making. In 2020, the Covid-19 virus hit all US states. The governments at thestate level were partially responsible for responding to the crisis. They enacted policiesto protect civilians’ life. This paper focuses on a subset of these policies that entailed atrade-off between economic objectives and health objectives.The subset consists of governors’ policies on whether to close specific businesses ornot. Some governors closed all but essential businesses early on, while others allowed allbusinesses to remain open. These decisions embody a clear trade-off between potentialhealth risk and imminent economic costs. Early closure of businesses guarantees that thespread of the virus slows down, but it initially hurts the local economy. The optimaldecision on business closure depends on both the perceived severity and infectivity of the954.4. Empiricsvirus and the magnitude of economic costs.The literature shows that in response to Covid-19, Democrats and Republicans’ votingbase had different opinions about how severe the condition is and what the proper responseof the government should be. Democrats believed the virus to be more dangerous comparedto what Republicans believed. Democrats also expected the government to react quicklyand more intensely. They required the government to limit a broader range of economicactivity to prevent the spread of the virus early on. On the other hand, Republicans down-played the government’s role and put more emphasis on individual freedom and economiccosts of such policies.Out of fifty US states, about half of the governors were Republicans (26), and the otherhalf were Democrats. Eleven of these governors had an upcoming election in 2020. Thisvariation allows us to test whether the next election’s timing impacted governors’ decisions.Regardless of the election timing, we expect that both parties’ governors acted morein line with their bases and differently from each other. We expect a Democrat to limiteconomic activity more. Taking election timing into account, the model predicts that aRepublican governor with an upcoming election will be further away from other Democratson the issue line than a Republican without an upcoming election. The model predictsthat a Republican with an upcoming election is the least likely to close any business. Incontrast, a Democrat with an upcoming election is most likely to close all businesses.I test the implications of the theoretical model by estimating the following differencein differences linear probability regression model:yi = β0 + β1IR + β2I>2020 + β3IR × I>2020 + β4Si + i. (4.25)The dependent variable represents decisions made by governors. If a governor closes aspecific business, then the corresponding y equals one; else, it is zero. The first independent964.4. Empiricsvariable (IR) is an indicator function that equals one if the governor is a Republican andzero otherwise. The second independent variable (I>2020) is an indicator function for facingan election later than 2020 (distant elections). If a politician had an upcoming electionin 2020, this variable is zero. The third independent variable is the interaction of being aRepublican and facing an election later than 2020. And finally, Si is a variable that linearlycontrols the severity of the disease in the state during the decision-making period.The difference between Democrats and Republicans who face an election in 2020 isgiven by:E[yi|D, 2020]− E[yi|R, 2020] = β0 − (β0 + β1) = −β1, (4.26)and the difference between Democrats and Republicans who face an election later than2020 is given by:E[yi|D,> 2020]− E[yi|R,> 2020] = β0 + β2 − (β0 + β1 + β2 + β3) = −β1 − β3. (4.27)Using equation 4.26 and 4.27 we can calculate the difference of these two differences:(E[yi|D, 2020]− E[yi|R, 2020])− (E[yi|D,> 2020]− E[yi|R,> 2020]) = β3. (4.28)The central coefficient of interest is β3, which captures the change in the differencebetween Republicans and Democrats as the next election’s time increases. The theorypredicts this coefficient to be positive. A competing theory that only relies on preferencespredicts β1 + β3 to be positive and β3 to be zero.974.4. Empirics4.4.2 DataThe dependent variable consists of decisions of governors to close down economic activity inthe middle of the first phase of the pandemic. Each governor had to make different decisionsas to whether to close down an economic activity or not. There are two independentvariables that are relevant to the main result: Whether a governor’s term is finishing in2020 and whether the governor is a Democrat. There is also a series of control variablesthat capture the level of virus severity in each state. The control variables used were thecumulative number of deaths and the cumulative number of positive cases per capita ineach state.The data regarding the governors’ political parties and their next election are scrappedfrom Wikipedia. Republicans govern twenty-six and Democrats govern twenty-four states.Eleven of these states have an election forthcoming in 2020, and thirty-nine do not havean election in the coming year.Table 4.1: Party Affiliation and ElectionParty Election 2020 Election Later Start < 2016 TotalRepulican 7 19 7 26Democrat 4 20 7 24The summary statistic for governors-The data regarding the Covid-19 policies were collected by Raifman et al. (2020). Thisdataset contains multiple variables on when each state put some policies into place andremoved them later. Specifically, it contains data related to economics vs. health trade-off.The decisions are on whether the restaurants, the theaters, the gyms, and non-essentialbusinesses are closed on a particular day.984.4. EmpiricsTable 4.2: Closing Days—Summary StatisticsVariable count min 0.25 % 0.5% 0.75% maxClosed Theatres 50 76 78 82 85 neverClosed Dine-in Restaurants 50 75 77 78 80 neverClosed Gyms 50 76 78 81 85 neverClosed Non-essential Businesses 50 79 84 87 92 neverThe summary statistic closing day, starting from the beginning of 2020-Table 4.2 summarizes the distribution of closing decisions for the relevant variables.Similarly, Table 4.3 summarizes the distribution of re-opening decisions for those variables.From these two variables, it is possible to reconstruct whether a particular business withina state was closed on any given day.Table 4.3: Re-opening Days—Summary StatisticsVariable count min 0.25 % 0.5% 0.75% maxOpened Theatres 50 118 143 163 237 (not opened)Opened Dine-in Restaurants 50 115 128 139 153 (not opened)Opened Gyms 50 115 136 153 171 (not opened)Opened Non-essential Businesses 50 111 122 129 138 (not opened)The summary statistic for the day of openning, starting from the beginning of 2020-All of the decisions are included as separate decisions. For each day in the data, thedimension of y (200) would be the number of governors (50) times the number of decisions(4).994.4. Empirics4.4.3 Empirical ResultsThe first column of Table 4.4 shows the result of the regression equation 4.25 for the mainspecification. Columns 2 and 3 shows the regression with probit and logit specifications. Inall of these specifications, the interaction coefficient is statistically significant. The ballotbox’s pressure makes politicians 30.5 percent more likely to act partisan and please theirbase without acting in their interest.Table 4.4: Main SpecificationsClose down an activity:(Linear) (Probit) (Logit)Const 1.003∗∗∗ 7.267∗∗∗ 17.766∗∗∗(0.002) (0.758) (1.451)Rep -0.249∗∗ -5.277∗∗∗ -14.242∗∗∗(0.126) (0.422) (0.927)” > 2020” -0.071∗∗ -4.731∗∗∗ -13.275∗∗∗(0.034) (0.292) (0.773)Rep×” > 2020” 0.305∗∗ 6.115∗∗∗ 16.040∗∗∗(0.131) (0.623) (1.437)Severity 0.016∗ 5.174∗ 9.426∗(0.009) (2.697) (5.105)Observations 192 192 192R2 0.103 0.2443 0.2383Adjusted R2 0.084Residual Std. Error 0.241(df = 187) 1.000(df = 187) 1.000(df = 187)F Statistic 2.344∗ (df = 4.0; 187.0) (df = 4; 187.0) (df = 4; 187.0)Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01Most of this effect comes from Democrats, and Republicans’ different behavior whohave an election in 2020 (−β1 > 0). The behavior difference of politicians from differentparties who did not have an election in 2020 is not statistically significant at the 95 per-cent level. Moreover, the sign of this difference is in the opposite direction of the base’s1004.4. Empiricsopinion (−β1− β3 = −0.0558 < 0). Republicans who did not face an election in 2020 weremore likely to limit economic activity and act not in line with their base’s belief. Table 4.5summarizes these results.Table 4.5: SummaryClose down an activity:(Linear) (Probit) (Logit)Difference election later (−β1 − β3) -0.0558* -0.8384** 1.798**(0.0363) (0.465) (1.091)Difference election 2020 (−β1) 0.2489** -5.277*** -14.24***(0.1263) (0.422) (0.927)Difference in differences (β3) 0.305** 6.115*** 16.04***(0.131) (0.623) (1.437)Marginal effect 0.305** 0.390* 0.416***(0.131) (0.250) (0.170)Partisanship increases for decisions made cloeser to electionsTo compare the results between linear and non-linear (Probit, Logit) model we need tolook at the marginal effect which is given by equation 4.28. These results are statisticallyand economically significant. The logit model is statistically significant at 0.01 level, whilethe linear is at 0.05 level, and the probit model is at 0.10 level. Logit seems to be a betterfit for the data, but in all specifications, the results are statistically different from zero.These models predict that election pressure significantly increases the difference be-tween Republicans and Democrats. Democrats and Republicans, under re-election pres-sure, act between %30 to %40 more polarized than their counterparts who are not underreelection pressure.The decisions of each governor are correlated. It is necessary to make sure the correla-tion between different governors decisions does not artificially inflate the significance of theresults. I have done so by clustering the standard errors around each state. The reported1014.4. Empiricserror bars in Table 4.4 are robust to clustering around states.Figure 4.1: Daily β3Democrats and Republicans with an election in 2020 behave differently as soon as 10 daysafter implementing the first policyIn the main specification, the decision y is whether a business is closed as of April the20th. On this day, states have implemented the first phase of limitations, and no state hasre-opened any business yet. Figure 4.1 shows the evolution of this interaction term fromthe time that first business is closed on mid-March until the time that states re-opened thebusinesses by early May.A non-linear control for severity led to excluding from the regression the two outlierstates of South Dakota and Wyoming, which had the smallest number of deaths per capitain this early period. Both results are robust to dropping/non-dropping multiple states percapita. Table 4.6 summarizes the regression results controlling for the different exclusionof states with low severity. Only the linear probability model is reported; nevertheless, inother specifications (probit and logit), the results’ statistical significance does not change.1024.4. EmpiricsTable 4.6: Robustness to OutliersClose down an activity:(No exclusion) (Drop 3 states) (Drop 7 states)Const 1.004∗∗∗ 1.003∗∗∗ 1.003∗∗∗(0.003) (0.002) (0.002)Rep -0.249∗∗ -0.249∗∗ -0.299∗(0.126) (0.126) (0.168)” > 2020” -0.072∗∗ -0.071∗∗ -0.060∗(0.034) (0.034) (0.034)Rep ×” > 2020” 0.242∗ 0.305∗∗ 0.344∗∗(0.141) (0.131) (0.172)Severity 0.019∗ 0.016∗ 0.014(0.011) (0.009) (0.009)Observations 200 188 172R2 0.060 0.101 0.132Adjusted R2 0.041 0.082 0.111Residual Std. Error (df) 0.281(195) 0.244(183) 0.231(167)F Statistic (df=4) 2.634∗∗ 2.340∗ 1.830Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01The measure of severity was the number of deaths per capita from the beginning ofthe epidemic until April 20th. In Table 4.7, the number of infected per capita is used as ameasure of severity. Another specification includes a quadratic control of severity.1034.5. ConclusionTable 4.7: Robustness to Measures of SeverityClose down an activity:(Cases as Severity Measure) (Quadratic Mesaure)Const 1.007∗∗∗ 1.029∗∗∗(0.004) (0.016)Rep -0.249∗∗ -0.245∗(0.126) (0.126)> 2020 -0.078∗∗ -0.088∗∗(0.036) (0.039)Rep× > 2020 0.308∗∗ 0.315∗∗(0.131) (0.132)Severity 0.022∗ 0.133∗∗(0.012) (0.064)Severity2 -0.019∗∗(0.009)Observations 192 192R2 0.106 0.114Adjusted R2 0.086 0.090Residual Std. Error 0.241(df = 187) 0.240(df = 186)F Statistic 2.374∗ (df = 4.0; 187.0) 1.948∗ (df = 5.0; 186.0)Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01The results of Table 4.5 is robust to all of these specifications. As is shown in Tables 4.6and 4.7, the results are robust to including all states, different functional forms (polynomial)for severity, and different measures of severity.4.5 ConclusionThis paper shows how higher election pressure can lead politicians to act more partisan.The theory suggests that voters’ different priors and politicians expecting them not to learnthe full truth until the next upcoming election is a potential cause of partisanship. With alonger election horizon, parties act more similarly and, based on the theory, more optimal.1044.5. ConclusionThe theory sheds light on why more polarized democracies might be poorly equippedwith instruments to respond effectively to a crisis. I discuss how the politician’s decisionto acquire information interacts with the social signals’ informativeness and election dates.When the election is close, the politician spends fewer resources on information acquisition.On top of that, the closer the election, the less they weigh their decision towards thatinformation set, controlling for their information quality.In a more general setting, the interaction between a politician and the voting base issimilar to a principal-agent “P-A” problem. The principals (here, voters) choose an agentto make decisions in an uncertain environment for their benefit. Information asymmetryposes a challenge for the principal to monitor the agent’s behavior efficiently and for theagent to signal the quality of their decisions.Similar to conventional P-A problems, the politician is better informed. They makemany decisions over their tenure, and voters later evaluate their performance on electionday. The information asymmetry causes an opinion divide between the politician and thevoters at the time of decision making, as does the fact that these opinions evolve until theelection day as more information immerges.In this paper, voters are Bayesian learners, and they perfectly update their beliefs givena new signal. This assumption does not hold in reality (people are far from perfect Bayesianupdaters). But the results from this model can be extended as long as some learninghappens and the voters’ beliefs get closer to the truth about the welfare-maximizing policy.A further avenue for research is to design mechanisms to increase politicians’ incentivesto acquire high-quality information and stick to the best expectation. I did not explicitlyformulate the individual rationality and incentive compatibility of the politician, but a fullanalysis must consider it.This paper provides evidence that the pressure of an upcoming oversight induces the1054.5. Conclusiondecision maker to deviate from implementing the optimal policy towards the principal’sbias. Although the example is tailored for a political representative, it can be generalized toother managerial settings. For example, a manager who faces an uncertain decision beforea significant event (e.g., merger, critical board meeting) deviates from implementing theoptimal action. Can we find similar evidence to show that the manager is more likely todeviate towards the principal’s bias if they expect it to affect their evaluation?If this effect can be found in other settings, we should examine other possible mech-anisms that would create similar empirical results. In this paper, I have explained therational behavior of an agent who forms an expectation of voters’ information set at theelection date. The mechanism at play may be much more complicated than what wasdiscussed. More research is needed to pin down how this shift occurs. Is it a deliberatecalculation of the agent to implement a policy that maximizes their popularity, or is itcoming from a psychological bias?Maybe the psychological pressure of an upcoming election makes it easier for the politi-cian to think of what voters want, which shifts the politicians towards their base’s favoritepolicies. Finding out the mechanism behind this bias is an avenue for further research inmanagerial decision making under uncertainty.106Chapter 5ConclusionThis thesis consists of three chapters that study crisis management under uncertaintyfrom the lens of exactly solving simplified models. Each chapter focuses on one aspect ofsimplification and studies a real-world problem.Chapter two studies a municipal decision-making authority to provide a public good(water security) to its citizens. In this chapter, I abstract away from complexities arisingfrom political economies of providing a public good and focus on the trade-off that abenevolent decision maker faces. The decision maker uses all of the information availableto them. They solve a dynamic problem that balances the risk of not having enough waterand the cost of building and operating an expensive water provision facility. The modelfirst explains what a benevolent decision maker would do and then tests the correlationpatterns produced by the real-world data against the expected correlation produced by thebenevolent planner.In chapter three, the decision-making’s scope is global (climate change), and the decisionmaker aims to understand the implications of innovation policies for long-term cumulativeatmospheric CO2. In this chapter, we abstract away from the optimal dynamic policy. Weassume that the decision maker cannot implement the first-best policy (carbon pricing)due to political reasons. Then we ask a theoretical question about whether the effect ofan exogenous innovation in clean energy or fuel efficiency is always beneficial for long-term cumulative CO2. We introduce new occurrences of the strong green paradox in fuel107Chapter 5. Conclusionefficiency innovation and the interaction of innovation in carbon capture and storage andclean energy innovation.Chapter four concerns decision-making at the gubernatorial level to combat the Covid-19 pandemic. In contrast to the first two chapters, the decision maker is not benevolentand maximizes their own (party) utility. The decision-making authority faces a trade-offbetween the policy that they find beneficial for society and the policy that merely looksgood to the public. There is inherent uncertainty in the unfolding of future events and theinformation that the public learns in the future. I develop a theoretical model and predictthe behavior of a utility-maximizing politician and test the results against the observedbehavior of US governors during the Covid crisis. I show that their behavior is consistentwith the predicted behavior in that the politician who faces an imminent election caresmore and acts more similarly to what their base deems appropriate.Decision-making researchers have a plethora of tools at their disposal to examine howauthorities manage crises in uncertain situations. But we are yet to find a framework thatunites different aspects of decision-making. There are numerous questions regarding thescope of the problem, decision makers’ incentives and goals, limitations they face, and eventhe analytical framework to study decision-making under uncertainty. With the adventof novel computation techniques (e.g., reinforcement learning) and their wide adoption inmultiple governing bodies, we will need more convergence in decision-making science.We may need to re-evaluate what we call a “rational” decision. 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Water storage capacity versus water use efficiency: Substi-tutes or complements? // Journal of the Association of Environmental and ResourceEconomists. 1 2018. 5. 265–299.118Appendix AThis appendix consists of three sections inspired by the model developed in the secondchapter. First, I present the pattern of usage and define the concept of optimal utilizationrate. Second, I discuss how does a mistake in the wrong estimation of rainfall probabilitiesaffects decision-making. Finally, I estimate the relative value of being in a good year usinga simpler version of the dynamic model.A.1 Optimal Utilization RateThe dynamic model is solved numerically, and we are interested in further studying themodel and understanding some implications of it. I achieve this by fixing most parametersof the model and change only one or two parameters and simulate the behavior of suchcity.The first thing that I look into is how the capacity of a typical city evolves and howthis capacity is used. For a city that faces no economy of scale (F = 0), the accumulatedcapacity looks similar to Figure A.1. Initially the city gets a few good draws and thereis no need to build any capacity. Then after some bad draws, the city starts to build adesalination plants, and finally some more bad draws and city builds the optimal level ofdesalination that they maintain forever.119A.1. Optimal Utilization RateFigure A.1: Realized capacity over timeConditional on building some capacity the city uses some of the capacity in some yearsand lets it remain idle in others. The blue line in Figure A.2 shows the relative amount ofused capacity for each year. For this specific city, the city wont use the capacity in goodyears (because it gets more waters than it needs) but as you can see in average (the yellowline) only 22 percent of the capacity is used. This usage is spread out over time. Varianceof usage is correlated with variance of rainfall.This pattern of sporadic use of desalination capacity is typical behaviour of desalinationusage in many cities. Any city that builds desalination for combating bad draws of rainfalland not for base water needs should see a behavior of this sort. If rainfall patterns accrossdifferent cities are not heavily correlated, rainfall patterns of this sort suggests a need formobile desalination plants. Imagine a large desalination plant installed on a vessel thatcan move around the world and provides water on demand. This vessel would have amuch higher utilization rate and further studies into this area might prove this to be costeffective.120A.1. Optimal Utilization RateThe realization of built capacity in Figure A.1 depends on the specific random rainfallrealization. In order to average across these rainfall realizations I simulated different rainfallpatterns for each city 1000 times for 300 years. Then each point is the average built capacityfor each year. In Figure A.3 each line represents the average capacity built for each city.Figure A.2: Usage over timeEach line in this figure represents different cities that are faced with changing rainfallvariations between good and bad years (δ = r2 − r1). After twenty-five turns (years)in simulation most cities converge to the optimal desalination capacity. As expected anincrease in rainfall variation leads to an increase in the optimal capacity level.121A.2. Mistakes in Probability EstimationFigure A.3: Average capacity over timeA.2 Mistakes in Probability EstimationMunicipalities have to decide on their model of weather dynamics. In the basic setting,the municipality needs to have an estimation of two probabilities (p11, p22). If there is nopredictive power in the data, the planner should set p11 = p22 = 0.5. But the data showsdifferent levels of predictivity for different cities (p11 6= 0.5, p22 6= 0.5). The question wewill ask in this section is how a mistake in estimating these probabilities would affect theoutcome of this model.The estimation mistake can go both ways. The “availability heuristic” is a psychologicalbias. It means that there is a “tendency to overestimate the likelihood of events with greateravailability in memory, which can be influenced by how recent the memories are or howunusual or emotionally charged they may be.” This bias may lead to an overestimation ofthe probability of remaining in bad draws. Another psychological bias is Gambler’s fallacy.It means a “tendency to think that future probabilities are altered by past events when inreality they are unchanged.” This bias may lead to an underestimation of the probabilityof remaining in bad draws.122A.2. Mistakes in Probability EstimationSuppose a city’s population in aggregate shows tendencies toward any of these biases.In that case, a decision maker’s choice might be affected because the decision maker caresboth about the outcome of the decision and the public’s judgment.As in a previous section, I simulated 1,000 cities for 300 years for each remainingprobability p11 = p22. Figure A.4 shows the average capacity built over time. The finalcapacities of cities are almost equal, but the rate at which they are built is different. FigureA.5 shows the average utilization rate as a function of the probability of remaining in a(good or bad) state. As probabilities of remaining in a state increase, more capacity isused on average.Figure A.4: Average capacity over time for different probabilities p11 = p22123A.2. Mistakes in Probability EstimationFigure A.5: Average utilization rate vs. probability of remaining in a statep11 = p22Last plots were about how the optimal behaviour changes with probabilities if there isno mistake in the estimation of probabilities. Now I simulate the same 1000 cities for 300years and look at how much the utility changes if a city makes its decisions based on amistaken model rainfall probabilities.The y-axix of Figure A.6 is the weighted sum (β) of the mean (over different cities)of capacities built with the mistaken beliefs minus the mean of capacities built with thecorrect belief.y =300∑t=0βt110001000∑i=1(dˆti − dti,p)If y > 0, the optimal thing to do is to build the capacities earlier, and y < 0 meansthe opposite. We can see that for making a small mistake the effect is negligible, butfor larger mistakes, underestimation leads to building later and overestimation leads to124A.2. Mistakes in Probability Estimationbuilding earlier.Figure A.6: Effect of mistakes on capacity, actual transition probability is p11 =p22 = 0.55The y-axix of Figure A.7 is the weighted sum (β) of the mean (over different cities) ofutilities with the mistaken beliefs minus the mean of utilies with the correct belief.y =300∑t=0βt110001000∑i=1(Uˆ ti − U ti,p)The y-axis shows the utility lost due to mismanagement. For the worst mistake (be-lieving probability is 0.75 when it is really 0.55) the effect is still very small ( 0.008) wherelifetime utility is around −45. This shows that for the current variables, the mistake inestimating the correct form of the weather dynamics is negligible.125A.3. EstimationFigure A.7: Effect of mistakes on utility, actual transition probability is p11 =p22 = 0.55A.3 EstimationIt is not possible to fully estimate all the parameters of the dynamic demand model but I canmake some progresses by making some simplifying assumptions. It is helpful to estimatethe long-term price elasticity of demand for desalination capacity and a structural modelthat specifies the relative value of plants in years with favorable and unfavourable rainfalldraws.Estimation of the price elasticity requires some supply shifters that do not affect demandseparately, and I will discuss them in the next section. With the current data, one canestimate the long-term price elasticity with some simplifying assumptions.I would continue to estimate a simple structural model which abstracts away fromquestions about plant size and quantifies the value of building a plant in years with a bad126A.3. Estimationdraw of rainfall relative to the similar value in a year with good draw.A.3.1 SupplyI choose a model-free approach to understand supply. Table A.1 shows what variables affectthe volume unit price of constructing a desalination plant using a linear regression modelwith fixed effects on each subregion and a dummy for technology type. The coefficient forEast Asia/ Pacific is set to zero, so all the subregion fixed effects are measured with respectto that.The coefficient of year variable captures the annual decreasing of prices. This is partiallycaused by process innovation and learning by doing that consistently reduces the cost ofconstruction of desalination plants. The coefficient on time variable (year) captures thiseffect. Each year the price of building a new plant drops on average by about 4.0 percent.The technology fixed effect captures the other aspect of innovation. Major desalinationplants are designed based on one of three technologies: multi-effect distillation (MED),multi-stage flash MSF, and reverse osmosis (RO). In Table A.1 the coefficient of MEDis set to zero, and other technologies are measured with respect to this technology. RO(the most recent technology) is a revolutionary technology and brought down prices by 61percent in comparison with with MED or MSF.The negative coefficient on log(Capacity) captures economies of scale. Building a planttwice the size of a normal plant would reduce the price by 3.6 percent.127A.3. EstimationTable A.1: Variables Affecting Pricelog(Price per Volume) Coeff. std. error t-Value(Intercept) 8.49 0.13 63.0 ***Eastern Europe / Central Asia -0.54 0.28 -1.912 .Latin America / Caribbean -0.22 0.15 -1.46Middle East / North Africa -0.624 0.064 -9.7 ***North America -0.929 0.071 -13.0 ***Southern Asia -0.44 0.15 -2.91 **Sub-Saharan Africa -0.26 0.23 -1.17Western Europe -0.517 0.079 -6.60 ***MSF (Multi-stage Flash) 0.15 0.11 1.39RO (Reverse Osmosis) -0.611 0.089 -6.9 ***log(Capacity) -0.0365 0.0096 -3.8 ***year -0.0396 0.0020 -20.4 ***Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1-In further sections, I treat innovation as an exogenous supply shifter. Two cities withsimilar observables (population, city size, and climate) but facing different supply due toavailability of more modern (hence cheaper) technology would identify the elasticity ofdemand. Similarly two cities with the same observables except population are helpful inidentification, because one city is able to build a larger plant and face a smaller price percapita.The results in Table A.1 guide us in finding supply shifter and from there instrumentalvariables that help identify the demand elasticity.128A.3. EstimationA.3.2 Estimation of demand elasticityThe demand model should be convincing that in the long-term cities move to the equlibriumcapacity that is determined by supply and demand fundamentals. A city is in equilibriumif it has passed bad rainfall years and has built its equilibrium capacity.Demand fundamentals include some observables like GDP, population, and climate pat-terns (e.g., standard deviation of rainfall) and some unobservables like the cost of waterunderprovision, average water usage, and local geographic characteristics. Supply funda-mentals include the cost of technology and local economic characteristics (e.g., wages, landvalue, local infrastructure, and environmental characteristics).To estimate the long-term elasticty of demand, I make the following simplifying as-sumptions:1. The desalination capacity is in equilibrium over a five-year period.2. The log of equilibrium demand capacity is a linear function of log of prices.3. Conditional on GDP per capita, population of cities, and local climate patterns, citiesare similar up to a random constant term and random shocks.4. Average annual prices of desalination construction that city “i” faces are independentof city “j”’s demand for desalination.From these four assumptions, I can proceed to the following specification:Cit = AitpβitRα1it Gα2it Pα3it Tα4it . (A.1)where Ait = Uieit and Ui is coming from a random distribution and it is the errorterm.129A.3. EstimationIn equation A.1, C represtents cumulative capacity available for each city, p the pre-dicted price that a city face in building new capacity, R rainfall, G the GDP, P populationof the city, T temperature. All the explanatory variables are averaged over five-year peri-ods.For each city-year observation, the value of p is predicted using a model with year andcountry fixed effects. p is the price that cities face if they choose to build a desalinationplant (though in many years they have not).Two major worries are that the city specific term A might be correlated with pricesand the elasticity β is not measured correctly. To address this issue, I take advantage ofan instrument for price based on assumption 4.Because the average annual price of desalination in cities other than “i” is excludedfrom demand of city “i”, the average price of desalination in cities other than “i” in year“t” p−it can be used as an IV for pitTable A.2: Demand Elasticitylog(Price per Volume) Coeff. std. error t-Value(Intercept) -304 30 -7.86 ***log(price.per.cap) -0.74 0.12 -6.00 ***log(rainfall) 0.054 0.099 0.54log(GDPperCap) 0.728 0.074 9.77 ***log(temp + 273) 54 6.7 8.08 ***log(city.pop) 0.376 0.09 4.2 ***Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’1-Table A.2 summarizes the result of regression A.1. The price elasticy of demand is130A.3. Estimation−0.74. As you may have noticed, after controlling for temperature, rainfall is no longersignificant. This shows that temperature is sufficient statistics for long-term climate pat-terns, and five years is a suitable time scale for the cities to reach equilibrium.Table A.3: Demand Elasticity—IV(Intercept) -284 39 -7.2 ***log(price.per.cap) -0.96 0.13 -7.2 ***log(rainfall) 0.072 0.099 0.72log(GDPperCap) 0.635 0.078 8.2 ***log(temp + 273) 51.0 6.8 7.5 ***log(city.pop) 0.300 0.090 3.2 **Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05‘.’ 0.1 ‘ ’ 1-Table A.3 replicates table A.2 with price of desalination in other countries as an in-strument. The results are qualitatively similar, and price elasticity in this case is closer to−1.A.3.3 Estimation of utility functionIn the last section we studied demand from a long-term aspect, and it became clear thatthe fluctuations in rainfall does not affect demand after controlling for average temperatureover a five-year period.This is in stark contrast to the effect of rainfall in the short run. Both the predictionof the theoretical dynamic programming model and the results of regression in Table 2.5suggest that rainfall affect the timing of building the plants.Further evidence supports that lower rainfall in short-term is correlated to an increased131A.3. Estimationdesalination demand. Assuming that city characteristics does not change much from thisyear to the next, we can take advantage of fixed effect models and show that how a relativelybad draw of rainfall is correlated with the city’s higher demand for desalination.Table A.4: Fixed Effects—Capacity Built Is the D.V.Capacity Coeff. std. error t-Valuelag(relativerain, -1) -110 142 -0.77lag(relativerain, 0) 582 142 -4.1 ***lag(relativerain, 1) -347 143 -2.4 *lag(relativerain, 2) -296 143 -2.1 *lag(relativerain, 3) -48 143 -0.33Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’0.1 ‘ ’ 1-The dependent variable in Table A.4 is Capacity. The independent variables, relativerain (rainfall in a given year divided by mean of the city rainfall) and its lead and lags, aredefined by dividing corresponding year’s rainfall by average rainfalls. The correlation isvery strong for current rainfall and significant for the last two previous rainfall. But it soonstarts to fade away for years further in time and the year after the plant is awarded. Thisevidence supports the causal story that a bad draw of rainfall trigerrs building the plant.Municipalities choose to build a desalination plant when the supply of water is stressed.Figure A.8 shows the same effect in a plot from a new perspective. The y-axis representsthe mean of relative rainfall for all cities in year t (x-axix) after a plant is built.132A.3. EstimationFigure A.8: relative rainfall around the date of constructionThis evidence motivates me to estimate the relative value of a desalination plant. In thissection I eventually estimate the value of building a plant in favorable condition comparedto an unfavorable one.I estimate a simplified version of the dynamic model. The water levels in each city canbe in two states. In good g state, the water reserve is high enough to provide water. Inbad b state, the rainfall is not enough to provide the required water.The city can always choose to build a desalination plant, but it takes time for it tobe built. Therefore, cities face a tradeoff between building now and paying the costs orpostponing the construction and risking incurring the cost of not having enough water.Eventually, I want to estimate (u1(ρ) − u0(ρ)), the utility difference of building andnot building as a function of relative rainfall (ρ =RR¯), and Pgg, Pbb, the probabilities of133A.3. Estimationremaining in good or bad states. This allows me to quantify the effect of short term rainfallfluctuations on demand.The estimation framework is a discrete choice model. The planner can choose whetherto build a plant. There is only one state variable. An unobserved variable that capturesif the city is in water shortage. This unobserved state variable in principle depends onmany characteristic which includes: current desalination capacity, temperature, other in-frastructes and population. The unobserved variable represents all of these charecteristicsin a concise manner.With probability that is to be estimated any given city is in state g or similarly in stateb. With probabilities Pgg, Pbb any given city will remain in its current state correspondinglyand with probabilities 1− Pgg, 1− Pbb the city will move to the other state.The static value of not building a plant is normalized to 0. The static value of buildinga plant for city i at time t is given by:uit = −αρit + νit + it.αρit represents the utility of relatively more rainfall for city i. νit represents the hiddenMarkov state. it is the idiosyncratic preference shock. I assume that the shock is distibutedi.i.d. type-1 extreme value.Following Bajari et. al. ? I use four sets of moment conditions, and in total I get 30moment equalities. Two sets of moments are related to conditional correlation of choicesover time.In city i, at time t, the city chooses j. j ∈ 0, 1 is the decision of whether to build adesalination plant or not.The first set of moment conditions is related to conditional correlation of choices one134A.3. Estimationperiod apart:M jh1 = E[Zt(dij,t+1dij,t −K∑k,p=1Πkpσkij(t+ 1)σpih(t))], (A.2)where Zt is an instrumental variable, Πkp represent the stationary distribution ofPr(νit = zk, νit = zp), dij,t is a dummy that is 1 if city i made decision j at time t.And σkij is the unobserved choice probability of city i choosing decision j in state k.Similarly, the second set of moment conditions is related to conditional correlation ofchoices two periods apart:M jh2 = E[Zt(dij,t+2dij,t −K∑k,p,r=1ΠkrΠrpσkij(t+ 1)σpih(t))]. (A.3)The two remaining sets of the moment conditions are related to the Bellman equation.The first one is the Bellman equation for not building choice:M3 = E[Zt(Vt) + βK∑k,p=1Πk,plog(σki0(t+ 1))− βVt+1], (A.4)where Vt is the conditional value function and β is the discount factor. Similarly, thesecond one is the Bellman equation for building a plant:M4 = E[Zt(Vt) + βK∑k,p=1Πk,plog(σki0)− βVt+1 − αρit +K∑k,p=1Πkplog(σkij(t)σki0(t))]. (A.5)These moment equalities and a set of moment inequalities which states that all proba-bilities have to be between zero and one identifies the model.135A.3. EstimationTable A.5: ResultsParameter ValuePbb 0.43Pgg 0.97α 0.86Table A.5 shows the results of the estimation procedure. Three estimated parameterssummarizes the data with respect to the effect of rainfall on the utility of building adesalination plants.Each city which is in good state remains in a good state with high probability (0.97).So, there is a small chance that this city goes to a bad state. Then almost 57 percent ofcities move out of a bad state because they either build a plant or the whether conditionchanges. Then the remaining 43 percent remains in a bad state.The parameter of interest α is positive. This shows that building a desalination plantis more valuable to cities in when they get relatively low level of rainfall compared to whenthey rainfall levels are high.136Appendix BIn the first section of this appendix, we offer proofs of lemmas and propositions that weremoved from the body of chapter three for ease of exposition. Moreover, in that chapter,we treated the CSS innovation as exogenous. The second section of this appendix providesan extension of the model that endogenizes CCS innovation and proves that the resultshold under endogenous CCS innovation.B.1 ProofsLemma 1. With discovery of the new backstop y1 at date τ , we have: p1(s; τ) < p0(s),for all s > x0(τ); for all t > τ , x1(t; τ) > x0(t); and for all c > c(τ), t˜1(c; τ) < t˜0(c).Proof. At any extraction amount s, equation (3.3) implies that dsdt · dPds = r[p(s) − c(s)].From dsdt = x˙(t) = q(p(t)), this yields:dPds= r[p(s)− c(s)]/q(p(s) (B.1)where c(s) is the marginal extraction cost at s, i.e. c(·) ≡ G−1(·).This is a differentialequation describing the evolution of p in s. Note that at s∗given by c(s∗) = b1, p1(s∗; τ) =b1 < p0(s∗), since p0(s∗) covers not only the extraction cost but also the (positive) Hotellingrent at any s satisfying c(s) < b0. Thus at s∗,the price path is lower after innovation.Suppose that for some sˆ, the paths cross:p1(sˆ; τ) = p0(sˆ)137B.1. ProofsThen because both price paths extending from sˆ satisfy (B.1), the price paths mustcoincide. But this contradicts p1(s∗; τ) < p(s∗). Therefore, p1(s; τ) < p0(s) for all s.It follows that s˙, given by q(p) is greater at any s post innovation. Therefore, the pre-innovation path and the innovation paths in t start with the same value x(τ), but at anyhigher value of s, x˙is greater post-innovation. It follows that x1(t; τ) > x0(t).Lemma 2: With clean-energy innovation under the parameter ρ, the price path priorto discovery satisfiesp˙0(t) = r{[p0(t)− c(t)] + ρ[p0(t)− p1(t, t)]}.Proof. We aim to show that equation 3.7 in the textp0(t)− c(t) = limdt→0e−rdt[(1− ρdt)p0(t+ dt) + ρdt · p1(t; t)− c(t)] (3.7)yields the differential equationp˙0(t) = r{[p0(t)− c(t)] + ρ[p0(t)− p1(t, t)]} (3.8)We expand the limiting expression on the RHS of (3.7) via Taylor series expansion. Leto(dt) denote all terms involving dt of order higher than 1. Equation (3.7) becomes:p0(t)− c(t) = limdt→0[1− rdt+ o(dt)] · {(1− ρdt) [p˙0(t) + p˙0(t)dt+ o(dt)] + ρp1(t; t)dt− c(t)}138B.1. Proofs= limdt→0(1− rdt− ρdt)p0(t)(1 + p˙0(t)p0(t)dt) + ρp1(t; t)dt− (1− rdt)c(t) + o(dt)= limdt→0[p0(t)− c(t)] + p0(t)[ p˙0(t)p0(t)dt− rdt− ρdt] + ρp1(t; t)dt+ rc(t)dt+ o(dt),from which we get0 = limdt→0p0(t)[p˙0(t)p0(t)dt− rdt− ρdt] + ρp1(t; t)dt+ rc(t)dt+ o(dt).Dividing both sides by the non-zero term dt yields:0 = limdt→0p0(t)[p˙0(t)p0(t)− r − ρ] + ρp1(t; t) + rc(t) + o(dt)dt.The last term disappears in the limit. We can re-arrange the equation to getp˙0(t) = (r + ρ)(p0(t))− ρp1(t; t)− rc(t),which is equivalent to equation 3.8.Proposition 4: (a) In the absence of CCS innovation, the ex ante impact of clean-energy innovation on cumulative emissions is zero.(b) With CCS innovation , a˙(t) > 0, the ex ante impact of clean energy innovation is toincrease emissions by:∫ (y0−z)/λ00[a(t˜(c))− a(t˜0(c)]g(c)dc > 0.Proof. To prove (a), define b ≡ (y0 − z)/λ0 note that the both the innovation-possible139B.1. Proofsprice path and the innovation-impossible price path terminate at p = b, with cumulativeextraction (and emissions) given by G(b). If a(t) is constant at a, the net emissions undereither price path are aG(b). The cumulative emissions are therefore, unaffected by the(unrealized) threat of innovation.To prove (b), note that it is easily demonstrated first that p1(t, t) < p0(t), i.e. thecurrent price falls upon discovery of the clean energy technology at lower cost. For anyprice and cumulative emissions paths, p(t) and x(t), define P (x) = p(x−1(x)). LetP0(x) andP (x) describe the prices under the innovation-possible and innovation-impossible modelsas functions of cumulative extraction to date. We first show that P0(x) < P (x) for all xless than the final value, x = G(b).Note that P0(x) = P (x) = b at x = G(b). At this end-point, we have:∂P∂x=dp/dtdx/dt=p˙q(p).Similarly,∂P0dx=p˙0q(p0).The denominators of the right-hand sides of these equations are the same because at theend-point, p = p0 = b.And p˙ < p˙0 from comparing equations 3.3 and 3.8. Therefore, ∂P/∂x < ∂P0/∂x at theend point, p = b. Therefore, within a neighborhood of the endpoint, x = G(b), we haveP0(x) < P (x) Suppose that at some values x < G(b), P0(x) ≥ P (x). Let xˆ be the largestsuch value. It follows the differentiability of the price functions that P0(xˆ) = P (xˆ) andthat evaluated at xˆ, ∂P/∂x ≥ ∂P0/∂x. (The two price functions either cross or are tangentat xˆ.)But at xˆ, the two functions have the values of p. ∂P/∂x = p˙/q(p) (following our earlier140B.2. Extending the CCS Technologyderivation) and similarly ∂P0/∂x = p˙0/q(p0). Comparing equations 3.3 and 3.8 again, wehave ∂P/∂x < ∂P0/∂x at xˆ, which is a contradiction. Hence, P0(x) <P (x) for x < G(b).From the assumption of downward-sloping demand, P0(x) < P (x) implies that Q0(x) >Q(x), where these are the flows of demands. For each of these functions, Q(x) = dx/dtevaluated at the inverse of x(t). Thus we have two variables, x0(t) and x(t), for whichx0(0) = x(0) = 0 and x˙ < x˙0, evaluated not at the same time, but at the same stock y.That is, ∀y, x˙(x−1imp(y)) < x˙0(x−1o (y)). This implies that x grows more slowly than x0 andtherefore,that ∀t > 0, x(t) < x0(t). Because c(t) = G−1(x(t)), this in turn means thatc(t) < c0(t) and that t˜(c) < t˜0(c).Part (b) of the proposition then follows.B.2 Extending the CCS TechnologyB.2.1 FrameworkTo this point, we have represented CCS as a function a(t) representing the fraction ofcarbon emissions that is captured with the technology existing at time t. This allows asimple framework to outline the impact of exogenous innovation, but is based on a narrowassumption on technology (an assumption of zero marginal cost of CCS up to a(t) at each t.The question arises as to whether our results obtain with a more general CCS technology inwhich the extent of CCS at any time is variable and chosen endogenously by the planner.13We address this question for the analysis of the ex post impact of innovation, for bothclean energy and fuel-efficiency innovation. We represent the cost of CCS of a fraction a ofemissions q as a function q · c(a; θ), where θ is the current state of technology. We assume13We will continue to assume that the costs of CCS (mainly the costs of storage and the transportationnetwork) are borne by the government, not by suppliers. The dynamic regulatory game in which CCSstandards are set by the regulator but costs borned privately is important but beyond the scope of thispaper.141B.2. Extending the CCS Technologythat ca > 0,caa > 0, cθ < 0 and cθa < 0. Thus c is increasing and convex in a and bettertechnology reduces the marginal cost of a. The state of technology at date t is θt, and byinnovation or improvement over time in CCS technology we mean that θt is increasing overtime.Because emissions are a pure externality, the equilibrium extraction rate and emissions,q(t) is uninfluenced by changes in a. The regulator therefore, takes the path q(t) asexogenous in deciding on the path a(t). The regulator’s objective in setting a(t) at any timeis to minimize costs, comprised of the cost of CCS and the long-run cost of emissions. Welet T be a date beyond which all emissions have terminated under any scenarios involvinginnovation at any date and let the welfare cost (damages) of adding to cumulative emissions(i.e. the stock of emissions at date T ), as of date t, be e−r(T−t)D(E). As of date 0, giventhe exogenous gross emissions path q(t), the regulator’s objective function ismina(t) e−rTD T∫0[1− a(t)]q(t)dt+ T∫0e−rtq(t)c (a(t); θt) dt (B.2)and the regulator’s objective function at subsequent dates is the obvious extension. LettingE be the cumulative net emissions,∫ T0 [1−a(t)]q(t)dt, the first-order condition for the point-wise choice of a(t) yieldsc′(a(t)) = e−r(T−t)D′(E) (B.3)The left-hand side of (B.3) represents the marginal cost of reducing emissions by one unit;14the right-hand side represents the marginal benefit as of date t of reducing emissions byone unit.14Since emissions are [1−a(t)]q(t), the cost of reducing emissions by q(t) units is c′(a(t))q(t);the marginalcost of reducing emissions by only 1 unit are a fraction 1/q(t) of this, which equals c′(a(t)).142B.2. Extending the CCS TechnologyB.2.2 Clean-energy innovationFor the clean-energy innovation model of the previous section, let q0(t) represent the exante path of emissions, i.e. the path followed when innovation is impossible and let q1(t; τ)represent the path when innovation occurs at date τ . These paths are exogenous in theregulator’s problem (B.2). We let a0(t) be the solution to the problem given q0(t). Becauseinnovation occurs with probability zero in the ex post model, in solving for a0(t) theregulator need not contemplate the possibility of innovation. Once innovation occurs atdate τ , we now let the regulator’s CCS strategy respond to innovation: the solution to(B.2) given q1(t; τ) is denoted by a1(t; τ).Suppose that the improvement in technology θt were so strong, with the consequenceof a rapidly increasing a0(t) that∫ T0a0(t)[q1(t; τ)− q0(t)]dt > 0That is, if the regulator did not respond to innovation by readjusting the optimal CCSstrategy, then innovation would have the paradoxical effect of increasing total emissions.This is essentially the case analyzed in section 2.3, with a(t) = a0(t) being taken as givenrather than responsive to innovation. The question for us now is whether endogenizingthe regulator’s optimal response to innovation will negate our prediction of a paradoxicalincrease in total emissions if CCS technology is improving sufficiently rapidly. The followingproposition shows that the optimal adaptation of CCS by the regulator mitigates, but doesnot eliminate, a paradoxical increase in emissions with innovation.Let the cumulative net emissions under the no-innovation path be E0 =∫ T0 a0(t)q0(t)dtand under the innovation-at-τ path be E1 =∫ T0 a1(t; τ)q1(t; τ)dt . (Both definitions areconditioned on CCS strategy being optimal, i.e. solving (B.2) with the corresponding143B.2. Extending the CCS Technologyq(t).) Now let Eˆ1 be the cumulative net emissions under the path q1(t, τ) and the (sub-optimal) non-innovative CCS, a0(t); Eˆ1 is what the cumulative net emissions would beif the regulator did not adjust CCS at all in response to the change in net emissionswith innovation. Given q1(t; τ) and q0(t), suppose that E0 < Eˆ1. That is, suppose thatemissions would increase with innovation at τ if the CCS were taken as fixed at a0(t). ThenE0 < E1 < Eˆ1. The proposition states that any (paradoxical) increase in emissions withclean-energy innovation at τ, is preserved when the optimal CCS strategy is endogenous.The regulator mitigates but does not eliminate the increase in emissions. The proof ofthe proposition is simple, and sketched here. If a0(t) were unchanged with innovation,with the result that total net emissions equaledEˆ1, then the marginal cost of decreasingemissions (the left-hand side of (B.3) would remain unchanged. But because D(·) is convex,the increase in emissions from E0 to Eˆ1 would increase the marginal benefit of decreasingemissions. The first-order condition (B.3) would be violated. Because of the convexity ofc(·) and D(·), i.e. the second-order conditions for (B.2), an increase in a(t) is optimal. Thisimplies E1 < Eˆ1, the second inequality in the proposition. Next, suppose arguendo thatE1 ≤ E0. This would require an increase in a(t), with innovation, for at least some t ≥ τ .But this increase would raise the marginal cost of reducing emissions, the left hand sideof (B.3) by the convexity of c(·). But since E1 ≤ E0 by hypothesis, the marginal benefit(the right hand side of (B.3)) would not have risen with innovation. Again the first-ordercondition (B.3) would be violated. An increase in emissions would be optimal. This provesE0 < E1. In short, the optimal response to any innovation that increased emissions wouldbe to increase CCS but not to the degree that the increase in emissions were eliminated.For an innovation in clean energy to yield an increase in total net emissions under optimalregulatory choice of CCS, it is sufficient that the innovation yield an increase in emissionswith CCS policy remaining at a0(t). This corollary means that our basic model, in144B.2. Extending the CCS Technologywhich a(t) is exogenous, captures the conditions under which innovation may will increaseemissions, providing the exogenous a(t) is interpreted as the CCS that would be optimalin the absence of innovation.B.2.3 Fuel-efficiency innovationThe identical analysis analysis and proposition to the case of optimal regulatory choiceof CCS in response to a discovery of new fuel-efficiency technology. Let the definitionsof q1(t; τ), E1and Eˆ1 now correspond to the event of a fossil-fuel efficiency innovation atτ . Given q1(t; τ) and q0(t), suppose that E0 < Eˆ1. That is, suppose that emissions wouldincrease with innovation at τ if the CCS were taken as fixed at a0(t). Then E0 < E1 < Eˆ1.145Appendix CIn the fourth chapter, the politician decided based on minimizing a loss function. Thisappendix explores the microfoundations of this loss function. It shows that the loss functionis equivalent to an enhanced version of the well-known spatial voting model.C.1 On the Microfoundations of the Loss FucntionSince the Hotelling-Downs model of spatial voting was developed, researchers have pro-posed numerous variants of the model to explain voters’ behavior. The main theoreticalresult from this literature is that when two politicians compete, their platforms on relevantissues are attracted to the median voter. Short-term frictions, uncertainty about the me-dian voter’s belief, and primary elections prevent the politicians from becoming identical;nevertheless, political parties should become more similar over time and have to fight overwinning the median voter.However, in the real world, over the last four decades, not only have the parties failed toconverge, but also literature showed an increased polarization among the political parties.Moreover, many pundits believe that elections are decided by which party can excite itsbase more strongly, rather than by convincing the median voter to switch to its side.Finally, the spatial voting models do not have any prediction about turn-outs.The facts mentioned above makes it is necessary to enhance the contemporary spa-tial voting models to explain voters’ behavior. The politician uses this enhanced modelto predict voters’ behavior. Exploiting the new model, I show that the politician’s opti-mal behavior (to the first non-zero approximation) is given by a simple form, which was146C.1. On the Microfoundations of the Loss Fucntionintroduced in equations 4.2 and 4.9.As in the discrete choice literature, voter’s i utility for the incumbent I is given by uiI ,and similarly for the opposition O, the preference is uiE . The voter faces a cost ci. Thiscost is comprised of the actual cost and opportunity cost of going to the polling booth andcasting a vote and all other emotional and intellectual costs of voting. The voter casts hisvote for the incumbent (opposition) if uiI − uiO > ci, (uiO − uiI > ci), and abstains fromvoting otherwise.There are two eqaully large types of voters (D or R). All utility and cost terms can bewritten as an average plus a mean zero normal shock. For a Democrat voter, we can write:uiI = u¯ID + iIuiO = u¯OD + iOci = c¯+ ic(C.1)And similarly for a Republican voter, we have:uiI = u¯IR + iIuiO = u¯OR + iOci = c¯+ ic(C.2)The incumbent’s vote share is given by the following equation:Pr(u¯ID − u¯OD − c¯ > iO + iC − iI) + Pr(u¯IR − u¯OR − c¯ > iO + iC − iI) =2− Φ( u¯ID − u¯OD − c¯σ)− Φ( u¯IR − u¯OR − c¯σ)(C.3)147C.1. On the Microfoundations of the Loss Fucntionwhere Φ is the cumulative distribution function of the normal distribution, and σ isthe standard deviation of iO + iC − iI .The incumbent wins the election if their vote share is higher than their opponents’.Therefore, the incumbent wishes to maximize the difference between their shares and theiropponents’. their actions can only affect u¯ID and u¯IRmax[Φ(u¯OD − u¯ID − c¯σ) + Φ(u¯OR − u¯IR − c¯σ)−Φ(u¯ID − u¯OD − c¯σ)− Φ( u¯IR − u¯OR − c¯σ)] (C.4)This brings us to the following first-order condition:−u¯′IL(φ(u¯OD − u¯ID − c¯σ) + φ(u¯ID − u¯OD − c¯σ))−u¯′IR(φ(u¯OR − u¯IR − c¯σ) + φ(u¯IR − u¯OR − c¯σ)) = 0(C.5)Equation C.5 has an straightforward interpretation. Every decision that the incumbentmakes has four effects. It causes some of their base to switch between voting for theincumbent to 1. the opposition or 2. abstention or it causes some of their opposition baseto switch between voting for the opposition to 3. their or 4. abstention.To obtain the loss function in the theoretical section of the article, I have to specifywhat u¯IL and u¯IC are.u¯ID = u¯0ID − θID(s− sD(T )σD(T ))2u¯IR = u¯0IR − θIR(s− sR(T )σR(T ))2 (C.6)148C.1. On the Microfoundations of the Loss FucntionIn equation C.6, θID is the attention that average Democrats pay to the incumbent’sdecisions and θIR is the attention of average Republicans to the incumbent’s decision. Fromequations C.5,and C.6, the value α in 4.9 is reconstructed:α =θIDθIRφ(u¯OD − u¯ID − c¯σ) + φ(u¯ID − u¯OD − c¯σ)φ(u¯OR − u¯IR − c¯σ) + φ(u¯IR − u¯OR − c¯σ)(C.7)If the preferences are symmetric (u¯ID = u¯OR, u¯IR = u¯OD), then alpha is simplified to:α =θIDθIR(C.8)Therefore, α > 1 if the incumbent’s decision affects the Democrat base’s utility morethan it affects the Republican’s base utility, which is a reasonable assumption for a Demo-crat incumbent.149
Thesis/Dissertation
eng
Business Administration - Strategy and Business Economics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Brander, James A.
Winter, Ralph Albert
Three essays on crisis management under uncertainty : drought, climate change, and pandemic
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