MANAGING BIOLOGICAL AND LINKED POLLUTANTS By Carson Jon Reeling A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Agricultural, Food and Resource Economics ŠDoctor of Philosophy 2015 ABSTRACT MANAGING BIOLOGICAL AND LINKED POLLUTANTS By Carson Jon Reeling This dissertation contains three chapters, each of which examine the management of environmental pollutants. The first chapter considers livestock producers™ incentives for self -protection against infectious disease, which can be viewed as a type of fibiological pollutant.fl Spillovers in disease protection create different ty pes of strategic interactions. Under certain conditions, multiple Nash equil ibria may arise with the possibility of coordination failure involving excessively low self -protection, in which case individuals™ expectations of others™ efforts d etermine which outcome arises. In prior studies, assumed technical relations between self -protection and infection probabilities drive the strategic interactions. We demonstrate that strategic relations can be endogenously determined and depend on the relative endogeneity of risk (RER), defined here as the degree to which individuals can take con trol of their own risks in a strategic setting. The potential for coordination failure may arise when RER is sufficiently small, whereas larger levels of RER may eliminate this possibility to ensure la rger levels of self -protection. We find that imposing a behaviorally -dependent indemnity may increase RER to eliminate the possibility of coordination failur e. We apply our analysis to the problem of livestock disease and illustrate the theory using a numerical example of the 2001 United Kingdom foot -and -mouth disease epidemic. The second chapter examines the problem of biological pollution from live animal movements. Prior literature asserts trade -related biological pollution externalities arise from the movement of contaminated goods . However, this literatu re ignores ( i) importers™ ability to reduce disease spillovers via private risk management choices and ( ii) the potential for strategic interactions to arise when an importer™s risk management measures simultaneously protect himself and others. This paper explores the design of efficient disease prevention policies when importers can mitigate disease risks to others. We demonstrate that the biological pollution externality extends beyond trade -related activities Šin contrast to prior work Šand derive efficient policy incentives to internalize the externality. We also find spillovers between importers may be chara cterized by strategic complementarities, leading to multiple Nash -equilibrium levels of risk -mitigating activities. Additional command -and -control policies may be needed alongside of incentives to achieve efficiency. The final chapter analyzes the managem ent of linked environmental pollutants. Advances in the understanding of pollutant generation, transport, and fate has increased researchers™ knowledge of the linkages between pollutants. A striking example involves reactive nitrogen (Nr), a family of poll utants which, once emitted into the environment, passes through multiple environmental media (e.g., air and water), causing economic damages in each. Traditional market -based policies manage different Nr species separately , and policy parameters, including pollutant caps, are often set independently (exogenously) of other market parameters . This paper examines the efficiency gains from an integrated market that allo ws trading across Nr species , both analytically and also using a numerical model of pollutant trading in the Susquehanna River Basin. When permit caps are exogenous, i t is demonstrate d that efficiency can be increased by integrating otherwise distinct markets for greenhouse gas and nutrient water pollutants . Specifically , integration allow s regula ted polluters to trade across pollutants at an optimally -chosen interpollutant exchange rate . Numerical results indicate the economic gains from integrating markets may be more than $200 million, depending on the level s of the permit caps . iv ACKNOWLEDGEMENTS I wish to extend my deepest appreciation to my advisor, Rick Horan, for providing patient guidance and expert training , and for funding my research over the past three years. Rick taught me the value of being patient, persistent, and persnickety, in addition to teaching me how to be an excellent economist. He is the consummate mentor, and I hope to have a sliver of his success in my own career. I als o wish to thank my doctoral committee , including Roy Black, Clo é Garnache, Chris Wolf, and Jinhua Zhao , for their comments and suggestions for improving this dissertation , as well as Scott Swinton for supporting my research during my first year at Michigan State. I am especially indebted to the Environmental Science and Policy Program at Michigan State for providing me with the Graduate Recruiting Fellowship. Finally, I am grateful for funding from the USDA National Institute of Food and Agriculture, Grant #2011-67023-30872, grant #1R01GM100471 -01 from the National Institute of General Medical Sciences (NIGMS) at the National Institutes of Health, and NSF grant #1414374 as part of the joint NSF -NIH -USDA Ecology and Evolution of Infectious Diseases program. T he contents of this dissertation are solely my responsibility and do not necessarily represent the official views of USDA or NIGMS. v TABLE OF CONTENTS LIST OF TABLES ......................................................................................................................... iv LIST OF FIGURES ....................................................................................................................... iv KEY TO ABBREVIATIONS ......................................................................................................... v CHAPTER 1 SELF -PROTECTION, STRATEGIC INTERACTIONS, AND THE RELATIVE ENDOGENEITY OF DISEASE RISKS ............................................................................ 1 1.1 Introduction ........................................................................................................................... 1 1.2 A Model of Infection Risks ................................................................................................... 4 1.3 Economic Model ................................................................................................................... 6 1.3.1 Incentives Arising from Strategic Interactions .............................................................. 7 1.4 Disease Prevention Policy and Strategic Interactio ns ......................................................... 13 1.5 Numerical Example: The 2001 UK Foot -and -Mouth Disease Epidemic ........................... 17 1.5.1 Model Specification ..................................................................................................... 18 1.5.2 Scenario (i): Baseline Case with No Disease Prevention Policy ................................. 23 1.5.3 Scenario (ii): Indemnities ............................................................................................. 25 1.6 Optimal Indemnity Design .................................................................................................. 28 1.7 Discussion and Conclusion ................................................................................................. 33 CHAPTER 2 ECONOMIC INCENTIVES FOR MANAGING BIOLOGICAL POLLUTION RISKS FROM TRADE ..................................................................................................... 35 2.1 Introducti on ......................................................................................................................... 35 2.2 Trade and Infection Risks Under Unilateral Spillovers ...................................................... 38 2.2.1 The Efficient and Privately -Optimal Outcomes under Unilateral Spillovers .............. 40 2.2.2 Incentives for Efficient Disease Control ...................................................................... 43 2.3 Trade and Infection Risks under Bilateral Sp illovers ......................................................... 47 2.3.1 The Efficient and Privately -Optimal Outcomes under Bilateral Externalities ............ 49 2.3.2 Incentives for Efficient Disease Control with Bilateral Externalities .......................... 50 2.3.3 Efficient Incentives in the Presence of Multiple Equilibria ......................................... 51 2.4 Discussion and Conclusion ................................................................................................. 53 CHAPTER 3 INTEGRATING POLLUTION MARKETS ACROSS ENVIRONMENTAL MEDIA: THE CASE OF REACTIVE NITROGEN ........................................................ 56 3.1 Introduction ......................................................................................................................... 56 3.2 A Model of Nr Pollution ..................................................................................................... 60 3.3 First -Best Nr Control .......................................................................................................... 62 3.4 Nr Abatement under Market Trading Schemes .................................................................. 63 3.4.1 Separate NO 3 and N 2O Markets when Caps are Chosen Efficiently ........................... 63 3.4.2 An Integrated Nr Market when Caps are Chosen Efficiently ...................................... 66 3.4.3 The Relative Efficiency of an Integrated Market when Caps are Exogenous to the Market ................................................................................................................................... 68 3.5 Numerical Model: Inter -pollutant Trading in the Susquehanna River Basin ..................... 71 3.5.1 Simulation Results ....................................................................................................... 73 vi 3.6 Discussion and Conclusion ................................................................................................. 77 APPENDICES .............................................................................................................................. 80 Appendix A Supplementary Information for Chapter 1 ........................................................... 81 Appendix B Supplementary Information for Chapter 2 ........................................................... 86 REFERENCES ............................................................................................................................. 91 iv LIST OF TABLES Table 1.1 Functional Forms and Parameters Used for the Numerical Example ........................... 22 Table 3.1 Simulation Parameters .................................................................................................. 74 Table 3.2 Results from Hypothetical Pollutant Trading Scenarios, SRB a .................................... 75 Table 3.3 Pollutant Caps under Each Market Setting ................................................................... 77 Table A.1 Description of Biosecurity Practices Used in Numerical Example ............................. 82 Table A.2 Sensitivity Analysis Results ......................................................................................... 85 Table B.1 Functional Forms and Parameter Values used in Illustrative Example ....................... 89 iv LIST OF FIGURES Figure 1.1 Best -response functions ................................................................................................. 9 Figure 1.2 The representative producer's best -response function with no policy ......................... 24 Figure 1.3 A beh aviorally -dependent indemnity .......................................................................... 26 Figure 1.4 a) A small and b) large behaviorally -independent indemnity ..................................... 27 Figure 1.5 A best -response function under an optimal indemnity ................................................ 32 Figure 2.1 Comparison of privately - and socially -optimal risk mitigation .................................. 41 Figure B.1 Comparison of privately - and socially -optimal risk mitigation when *0NDi ivv .......... 87 v KEY TO ABBREVIATIONS CO2 Carbon dioxide FMD Foot -and -mouth disease FOC First -order condition GHG Greenhouse gas LHS Left -hand side N2O Nitrous oxide NO3 Nitrate Nr Reactive nitrogen RER Relative endogeneity of risk RHS Right -hand side SNE Symmetric Nash equilibrium SRB Susquehanna River Basin 1 CHAPTER 1 SELF -PROTECTION, STRATEGI C INTERACTIONS, AND THE RELATIVE ENDOGENEITY OF DISEASE RISKS 1 1.1 Introduction A key behavior influencing the introduction and spread of infectious disease is the degree to which individuals self -protect from disease r isks (e.g., via immunization or biosecurity effort). Strategic interactions may arise in making self -protection choices , as individuals™ disease risks can depend on the self -protection behavior of others. Indeed, self -protection is an example of a private behavior that generates positive spillovers affecting the supply of a public good, i.e., infectious disease prevention (Buchanan and Kafoglis 1963; Olson and Zeckhauser 1970) . Different types of strategic relations may arise. Of particular consequence is when self -protection takes the form of a strategic complement, whereby an individual™s marginal incentives for self -protection increase with the self -protection of others. Relatively strong complementarities in the neighborhood of one equilibrium can lead to multiple Nash equilibria, with the potential for coordination failure in which all individuals adopt low levels of self -protection (Echenique 2004; Vives 2011) . Self -protection could also take the form of a strategic substitute, whereby an individual™s marginal incentives for self -protection decline with the self -protection of others. Coordination failure is not a concern for t his type of relationship (Hefti 2011). In prior studies of disease and strategic interactions, the type of strategic interaction is driven by assumed technical relations defining how one™s infection probabilit y is jointly affected 1 A version of this chapter was published in Reeling, C.J. and R.D. Horan. 2015. fiSelf -Protection, Strategic Interactions, and the Relative Endogeneity of Diseaes Risks.fl American Journal of Agricultural Economics 97(2): 422Œ468. 2 by the self -protection efforts of oneself and others . These tech nical relations generally depen d on the type of transmission. For instance, Barrett (2004) and Hennessy (2007b) find that self -protection to reduce spread of an existing livestock disease between herds is a strate gic substitute, while Hennessy (2008) finds that s elf -protection to prevent the introduction of a new livestock disease into a region is a strategic complement . Chen (2012) finds that social distancing is a strategic complement (substitute) when the rate of contact between individuals is decreasing (increasing) in the number of individuals. In all of this work, and in other work where strategic interactions exist but are not explicitly investigated (e.g., Fenichel et al. 2011) , individuals™ strategic behaviors are made in response to the spillovers created by particular technical relations that are taken as given rather than endogenously determined. This is true even in existing dynamic models in which strategic behaviors are both affected by and affect future risks (e.g., Fenichel et al. 2011; Chen 2012). Behavior in those models (i.e., social distancing or, equivalently, public avoidance, which could be viewed as a form of biosecurity) does not affect the nature of the technical relations or related spillovers that determine whether behaviors are strategic substitutes or complements, although behaviors can dynamically affect the magnitude of the spillovers (via effects on future states) to alter the strength of the strategic relations. We examine a more general case in which disease can be transmitt ed over two modes, introduction and spread, where the predominant technical relations (stemming from the type of risk exposure, introduction versus spread) and the associated spillover effects endogenously depend on self -protection efforts. Therefore, in c ontrast to prior work, the strategic relations are also endogenously determined. We show these relations depend on a new concept we refer to as the relative endogeneity of risk (RER), defined here as the degree to which an individual can take control of hi s or her own risks within a strategic setting where others™ actions also influence 3 one™s risks. When individuals™ self -protection gives them sufficient control over their own risks (so that an individual is not very reliant on spillovers from others™ self -protection), then self -protection is a weak strategic complement or strategic substitute and there will be at most one Nash equilibrium and no risk of coordination failure. Alternatively, when individuals™ self -protection does not yield sufficient control over their own risks (so that an individual is more reliant on spillovers from others™ self -protection), then self -protection is a strategic complement and coordination failure may become a concern. The policy implications arising from strategic interactio ns have largely gone unaddressed, although Chen (2012) does consider the potential consequences of coordination failure. We find that properly designed disease prevention and control policies can increas e RER to remov e the possibility of coordination failure while also increasing equilibrium levels of self -protection . In particular, we show such results can arise from indemnity payments that increase with the individual™s level of self -protection . The optimal design of these indemnities when there is a risk of coordination failure differs from traditional notions of optimal incentive design in strategic settings, however. Traditional approaches involve setting instrument parameters to ensure the first -best outcome is a N ash equilibrium (e.g., Acocella, Di Bartolomeo and Hughes -Hallett 2013) . This may be insufficient to guarantee the first -best outcome when there is a potential for coordination failure. We present an approach in which the first -best level of self -protectio n is a globally stable Nash equilibrium. In the sections that follow, we derive an analytical model of self -protection against disease risks and investigate how RER influences the strategic interactions among at -risk individuals. We then demonstrate how disease prevention policy can influence RER and, thus, 4 strategic relationships. A numerical model of the 2001 UK foot -and -mouth disease (FMD) outbreak demonstrates the theory , and is then used to explore optimal policy design . 1.2 A Model of Infection Risks We develop a simple model of endogenous infection risks to illustrate the incentives for self -protection. The model is framed in terms of livestock producers whose herds are at risk from the introduction and spread of either an existing or emerging infectious disease. However, the essential features of the model Šself -protection from infectious contacts along introduction and spread pathways, with strategic interactions arising along the spread pathway Šare also relevant to disease problems among humans and huma n-managed ecosystems. For simplicity, we adopt a one-period model of these processes. This simple framework captures the same sorts of risks and choices that would arise in more complex dynamic models, while allowing us to focus on str ategic elements of th e problem. Suppose a region contains a number of homogeneous livestock producers. Each producer™s herd faces disease risks along two path ways: introduction and spread. Consider the risks faced by a particular producer whose herd takes on the binary status of being either infected or uninfected. The probability that this producer™s herd becomes infected from sources outside the region (i.e., pathogen introduction) is PI .2 We assume )(iIP < 0 and )(IP [0,1 ] is the producer™s self -protection or biosecurity effort .3 The signs on the partial derivatives 2 Disease ecology models often model transmission either with the number of animals as being the primary unit of analysis (e.g., Heesterbeek and Roberts 1995) , or with the number of fipatchesfl or fisub -populationsfl being the primary unit (Hess 1996; Gog, Woodroffe and Swinton 2002; McCallum and Dobson 2002) . The latter case, often referred to as a metapopulation model, implicitly models population sizes within a patch as being fixed. Our approach, in which the farm is the primary unit of analysis, follows this latter approach. 3 All probabilities in the model are defined as conditional probabilities, i.e., conditional on the biosecurity decisions of the producer a nd his neighbors. Subscripts represent partial derivatives. 5 indicate that disease protection, defined as [1 Œ PI Introduction could occur via the importation of animals into the region. The producer™s herd is at risk of infection from direct or indirect contact from infected animals of neighboring pro ducers (i.e., pathogen spread). Herd contact may occur via people or wildlife acting as disease transmission vectors, via direc t contacts between animals in communal grazing areas or livestock exhibitions, or even via airborne droplets of water containing infectious agents. The probability that a producer™s herd becomes infected by spread depends not only on his own self -protectio f-protection of his neighbors. [0, 1] denote the biosecurity effort of the other producers in the region . a scalar for simplicity since all neighbors are homogeneous and make identical choices. 4 be thought of as either the effort level of another producer in a duopolistic setting, or as the average effort of other producers in the region. This follows the convention used by Vives (2005) for multiple, homogeneous firms. The prob ability that the representative producer becomes infected via spread, conditional on being uninfected, is written PS , with 0SiP and 0SiiP for i The function PS pread risks both directly (e.g., via biosecurity to prevent transmission across farms) and indirectly (e.g., by reducing the likelihood of other producers becoming infected). The probability that the producer becomes infected via either the introduction or spread pathway, assuming independence of the underlying random processes governing infection along 4 Heterogeneity in economic and/or technical relations, including heterogeneous disease risks over time and space, may arise in some settings to generate heterogeneous behaviors (Rahmandad and Ste rman 2008). In analyzing strategic economic behavior among many firms in non -disease settings, Vives (2005) demonstrates that the qualitative nature of strategic relationships and associated outcomes are preserved when heterogeneities are introduced withou t altering the underlying economic relations qualitatively (e.g., the heterogeneities do not affect whether a producer™s marginal incentives for an action are increasing or dec reasing in neighbors™ actions). Accordingly, while producer heterogeneities will certainly have quantitative impacts on economic and epidemiological outcomes, there may be only a limited impact on qualitative results. 6 these pathways, can be written 5 (1.1) ),()](1[)(),(SIIPPPP. 1.3 Economic Model Suppose the producer chooses his biosecurity effort to maximize expected profit, denoted E{}. The producer earns profits of H = RH Œ c when his herd is in the susceptible (i.e., non -infected or healthy) state and NH = RNH Œ c in the infected (or not healthy) state, where RH > RNH. Assume c The producer™s problem is (1.2) )(,)(,,1}{ max cPRcRPRPEHNHH where = (RH Œ RNH) > 0 represents losse s or damages due to infection. Problem ( 1.2) is solved as a Nash -Cournot game : the producer chooses his or her own biosecurity while taking others™ effort as given. Other producers behave analogously. Assuming an interior solution, t he first -order condition of problem ( 1.2) is 0}{E, which implies (1.3) )(),(cP. The left -hand side (LHS) represents the marginal benefits of biosecurity , which is the economic loss experienced in the infected sta te multiplied by the marginal impact of biosecurity on the probability of infection. At an interior optimum, this is equal to the marginal cost of biosecurity, denoted by t he right - to be uncommon because the marginal cost of taking some defensive action is likely to be quite 5 The probabilities of spread and introduction may be positively correlated in some settings. If so, then assuming ind ependence will bias the modeled probability of infection downward relative to the true level. 7 low rel ative to the marginal benefit. However, it may be possib le to have a corner solution explore this possibility below. Additional in sight arises from examin ing the marginal impact of biosecurity on the probability of infe ction , (1.4) ),()](1[),()()(),(SISIIPPPPPP. Increasing one™s biosecurity effort has three effects on the probability of infection, as indicated by the three right hand side (RHS) terms in equation ( 1.4). The first RHS term in ( 1.4), which is negative, pread risks constant. introduction to disease spread. Mathematically, the reduction in PI anied by an increase in 1 Œ PI , thereby placing more weight on the spread term PS 1.1). Intuitively, the risk of infection via spread becomes relatively more important as introduction risks are reduced and spread becomes the mo re likely pathway of infection. Hence, this term is positive so as to partially offse t the first RHS term. The final RHS term in ( 1.4), which is g introduction risks constant. 1.3.1 Incentives Arising from Strategic Interactions Condition ( 1.3) indicates that a producer™s marginal incentives for biosecurity depend on neighbors ™ biosecurity . The implicit solution to ( 1.3) is given by the best -response function , with }){/(}{EE.6 Note that E } < 0 if maximizes expected profits. 6 Define F(,) = E{}/ so that the producer™s best -response function () implicitly solves F(,) = 0. Totally differentiating this expression, () = Œ(F/)/( F/). 8 therefore depends on the sign of E }, which indicates how the producer ™s as neighbors increase . Hence, the slope of () reflects the strateg ic relation between producers. If E } > 0 expected profit is supermodular and biosecurity efforts among producers are global strategic complement s (Fudenberg and Tirole 1995) : a producer will incr ease (decrease) his efforts as his neighbors increase (decrease) theirs. S trategic complementarities result in multiple symmetric Nash equilibria (SNE) if the complementarities > 1 so that this SNE is unstable. Thi s case is presented in Figure 1 .1a. Three SNE occur where the producer™s best - 45° line. The SNE can be Pareto -ranked, with the high -effort equilibrium B being privately preferred (Vives 2005; Van Zandt and Vives 2007) . Equilibria A and B < 1 in the neighborhood of each point. 7 SNE C is T representing an fiexp ectational thresholdfl between A and C. Here, a producer™s Nash equilibrium biosecurity effort depends on his expectations about his neighbors™ effort. A common way of envisioning this within a static model is to think of a quasi -dynamic adjustment process, or tâtonnement process, in which is adjusted based on initial expectations about (Krugman 1991) . If the producer initially expects his neighbors T, then the system will be in the high -effort basin and proceed via the tâtonnement process to B. Alternatively, if he initially expects his neighbors T, then the system will be in the low -effort basin and proceed to A. This latter outcome represents coordination failure , a term that w e use as in Cooper et al. (1990) to define a situation where the 7 If the coordinates of A lay below the unit interval, then the origin would be a locally stable SNE. If the coordinates of B lie above the unit interval, then (1,1) would be a locally stable SNE. This is the case in our numerical example. 9 producers™ privately -preferred outcome B is not attained. Producers™ expectations therefore play a critical role in disease management outcomes when multiple SNE are present. Strategic comple mentarities can also coincide with a unique, globally stable SNE if the < 1 so that the SNE is stable (Vives 2011; Panagariya and Shibata 2000) . This case is presented in Figure 1.1b. Expec tations do not matter in this case, as the system will always converge to point A. If E }< 0 , then expected profit is submodular and biosecurity efforts are global strategic substitutes . In this case, a producer will decrease (increase) his effort as his neighbors increase (decrease) theirs . SNE | < 1 illustrated in Figure 1.1c, and so expectations do not matter. 8 Figure 1.1 Best -response functions Finally, it is possible that biosecurity may only exhibit strategic complementarity or substitution relations locally (i.e., E }< 0 or E }> 0 for sub - Multiple equilibria may arise in such cases as well, with the stabil ity properties described above 8 equilibrium (Hefti 2011) . Such an outcome is different from coordination failure because it involves some producers choosing high levels of bios ecurity effort. 10 being defined locally rather than globally. The strategic relationships depend on how neighbors™ biosecurity effort affect s the marginal technical effectiveness of a producer™s own biosecurity . If ŒP < 0, so that a neighbor ™s biosecurity reduces the marginal technical effectiveness of a producer™s own effort) , then E }< 0 and biosecurity is also a strategic substitute (Aronsson and Koskela 2011) . Conversely, if ŒP > 0, so that a neighbor™s biosecurity is a technical increases the marginal technical effectiveness of a producer™s own biosecurity ), then E }> 0 and biosecurity is also a strategic complement. Further insight is obtained by considering the strateg ic relationship from an alternative Šyet mathematically equivalent Šperspective that follows from Young™s theorem, E } = E}. P pathway: 0,)(1,SIPPP. This ing the probability of spread. Accordingly, we can say that the negative value of the above expression, E} = ŒP measures the extent to which the producer is reliant on neighbors for controlling his spread risks, with a larger valu e indicating greater reliance. risks: (1.5) ,)(1,)(,}{SISIPPPPPE The first RHS term in braces, which is positive, represents the degree to which the producer™s and hence the reliance on others to control tho se risks. The second RHS term in braces indicates 11 efforts in controlling spread risks. The overall sign of ( 1.5) is ambiguous. A positive value of (1.5) protection from spread risks. In other words, by increasing spread risks to his neighbors. This case arises when spillo ver effects , i.e., ),(SP > 0 (e.g., when jointly preventing wildlife that may act as a disease reservoir from entering common grazing areas ) or when Œ),(SP 0 but sufficiently small in magnitude (e.g., when cleaning share producer™s spread risks). A negative value of ( 1.5) (i.e., when ),(SP < 0 and sufficiently large in absolute magnitude ) for pro tection. In other words, by increasing risks away from his neighbors. Expression ( 1.5) can be rewritten as (1.6) PSIPPE1),()(}{, where P P /1/1SSPPP, and IIPP11 > 0. We use P to quantify the (scaled) relative endogeneity of risk (RER) of infection via spread, defined here as the degree to which an individual can take control of his or her own risks within a strategic setting where others™ actions also influence one™s risks. 9 The concept of RER extends more traditional notions of risk endogeneity (Shogren 1991) that focus on an individual™s ability to control the risk he or she faces, without regard for strategic effects. Specifically, P measures 9 We use the term firelativefl here in the same sense as Pratt™s (1964) coefficient of relative risk aversion, which is also an elasticity. 12 the extent to which the producer can take control of his own spread risks away from his neighbors, i.e., the degree to which spillover effects are endogenous to the producer. A positive value means that, by raising read risks away from his neighbors, whereas a negati ve value implies the opposite. The term P is then scaled by one™s self -protection impacts on exposure, > 0; the smaller is , the less exposure will there be to he relative amount of control over these risks is less of a concern. 10 By Young™s Theorem, we see that strategic relations and RER are fundamentally related: (1.7) self -protection is a strategic 1 RER iff complement substitute P. P is sufficiently large, then the producer can take control of his risks by substituting his own : biosecurity is a strategic substitute , the magnitude of which is increasing in P, other things equal . P is sufficiently small, then the producer enhances : biosecurity is a strategic complement , the magnitude of which is decreasing in P, other things equal . These re sults, along with our earlier results on the stability of SNE, suggest that a larger value of P is likely to reduce the risk of coordination failure. Finally, note that P only reflects a producer™s relative ability to control the probabilistic (not econo mic) component of spread risks, hence the superscript P The next section illustrates how the use of economic policy may modify expressions ( 1.6) and ( 1.7) to also reflect control over economic components of risk. 10 P could also be said to measure the relative degrees of control along the two pathways: spread and introduction. While this is also a valid perspective, we believe the focus on the relative degree to which one can endogenously control one™s own risk vis -à-vis the control exerted by neighbors is a more insightful and useful perspective. For instance, the endogenous risk perspective highlights the potential role of policy mechanisms that may offer more opportunities to control one™s economic risks. 13 1.4 Disease Prevention Policy and Strategic Interactions Up to this point, strategic interactions between producers have depended on technological relationships. This is a common finding in the literature (e.g., Barrett 2004; Hennessy 2007b, 2008; Chen 2012). However, little attention has been paid to the potential effects of disease prevention policy on these strategic interactions. In this section we consider the effect of disease prevention policy on producers™ marginal incentives for biosecurity and its effect on RER. For concreteness, assume the policy is an indemnity, which is the most commonly used instrument in practice (Hoag, Thilmany and Koontz 2006) . This is not an insurance program, as no premiums are paid. Rather, these are used as safety nets and to motivate producers to repor t disease outbreaks on their farm to regulatory authorities, limiting the potential for disease spread. Prior work has shown, however, that indemnities may reduce the incentives for biosecurity (e.g., Muhammad and Jones 2008) . We examine how a particular c lass of indemnity can enhance biosecurity incentives, thereby reducing adverse spillovers. Issues related to optimal policy design are explored following the numerical example below. Returning to our model from equations (1 .1) and ( 1.2), assume now that i nfected producers are compensated with an indemnity payment [0,1], denotes the share of damages for which the producer receives compensation. 11 The case of represents a behaviorally -dependent indemnity, so that compensation is greater when producers apply more biosecurity efforts. The case of = 0 (so that behaviorally -independent indemnity, so compensation does not depend on p roducers™ biosecurity efforts. The representative producer™s expected -profit maximization problem ( 1.2) is now 11 We relax the assumption that 14 )(,)(1}{ max cPREH, with first -order condition (1.8) )(),()(),()(1cPP. Consider first the special case of a behaviorally -independent indemnity ( = 0), which is the form indemnities typically take in practice (Hoag et al. 2006) . Here, the second LHS term vanishes from ( 1. to condition ( 1.3). Ot her things equal, this results in lower biosecurity effort, as is consistent with most prior work on indemnity payments (Bicknell, Wilen and Howitt 1999; Kuchler and Hamm 2000; Hennessy 2007a; Muhammad and Jones 2008) . Now consider the case of a behavioral ly-dependent indemnity ( LHS term in ( 1.8) is positive so as to increase the incentives for biosecurity. However, the overall incentives provided by the indemnity are ambiguous. Subtract condition ( 1.3) from ( 1.8) to obtain the diffe rence in the marginal benefits of biosecurity with and without the indemnity: (1.9) ),()(),()(1)(),(1)(PPPPPSIIS. P (1. infection, P). The second term in ( 1.9), which is positive, is the expected indemnity payment, holding the probability of infection fixed (i.e., only considering the probabilistic and economic compone nts of risk. The net effect is ambiguous in sign. Expression ( 1.9) is positive, so that indemnities increase one™s incentives for self - P 15 prior literature finds that in demnities decrease one™s incentives for self -protection (e.g., probability of infection may be smaller than its effect on increasing the payment amount. Gramig, Horan, and Wolf (2009) also find that biosecurity incentives can be increased by basing indemnities on observable disease outcomes that are correlated to biosecurity efforts (e.g. disease prevalence within a herd). However, they do not model strategic effects, which play an important role here. The effect of indemnities on the strategic interactions can be seen by differentiating ( 1.8) (1.10) ),()(),()(1}{PPE. Consider first the case of a behaviorally -independent indemnity ( = 0), in wh ich case the final RHS term vanishes. The indemnity does not impact the prevailing type of strategic relationship, as it does not affect the sign of E } relative to equation ( 1.5). However, relative to equation ( 1.5), the indemnity produces a scaling ef fect that reduces the magnitude of any complementarities or increases the magnitude of substitution. For instance, suppose there were strategic complementarities involving multiple equi libria prior to the indemnity. In reducing the magnitude of the complem eliminated. This means the behaviorally -independent indemnity may result in a unique SNE, numerica l example below illustrates this might be the case with a large indemnity. Multiple equilibria will persist if the reduction in complementarities is not too great, although the basin of attraction for the low -effort SNE may increase due to the reduced bios ecurity incentives under the indemnity. Hennessy (2007a) shows that the potential for coordination failure may increase 16 with a small indemnity, thereby increasing the importance of expectations. Next consider the case of a behaviorally -dependent indemnit y ( > 0) so that the final RHS term in ( 1.10) does not vanish . Now the sign and magnitude of E} depends on both a (technologically -defined) probabilistic relationship, ),(P, and an economic relationship, . Specifically, the term P larger indemnity that can be earned in indemnity. If they were strategic complements prior to the indemnity, the post -indemnity outcome is ambiguous and depends on the relative magnitude of the probabili stic and economic effects described above. At a minimum, the behaviorally -dependent indemnity will weaken the complementarities, and it could even change the sign of E} relative to the case of no indemnity or a behaviorally -independent indemnity. For i nstance, suppose there were strategic complementarities involving multiple equilibria prior to the indemnity. In reducing the eliminated. This means the behaviorally -depend ent indemnity may result in a unique SNE. But, in contrast to the case of the behaviorally -independent indemnity, the SNE may be at a large -dependent indemnity increases biosecurity incentives. Thus, a behaviorally -depende nt indemnity may eliminate both the possibility of coordination failure and the role of expectations. Such a case is explored in the numerical example below. The strategic impacts of a behaviorally -dependent indemnity a lso affect our measure of RER. Specif ically, we can derive the following condition: 17 (1.11) self -protection is a strategic , 1 RER iff complement substitute EP E E E = Œ Œ Œ )] > 0 when E reflects the producer™s ability to take control over his infection risks via the indemnity. The denominator E Condition ( 1.11) indicates that RER is unchanged for a behaviorally -independent E P as in condition ( 1.7). In the case of a behaviorally -dependent indemnity, however, condition ( 1.11) indicates that RER is increased by E. This new measure of RER quantifies the producer™s ab ility to take control over P) and the E). The more able a producer is to control his economic risks, the less likely are ther e to be strong strategic complementarities that generate the pote ntial for coordination failure. 1.5 Numerical Example: The 2001 UK Foot -and -Mouth Disease Epidemic We now explore our model through the use of an illustrative numerical example based on the 2001 UK FMD epidemic. The outbreak began in Northumberland County in northeast ern England in early February. The source of the outbreak was traced to a pig that had been fed infected m eat (Segarra and Rawson 2001) . The disease initially spread throughout Great Britain via animal movements, contaminated vehicles, and farm workers (Gibbens et al. 2001) . Nationwide movement restrictions were implemented in late February, at which point the virus continued to spread via airborne transmission (Donaldson and Alexander sen 2002) . At its greatest extent, the outbreak had spread throughout Great Britain a nd parts of France, Ireland, and the Netherlands. By the time the outbreak was over, more than six million cattle, sheep, pigs, 18 and other animals had been slaughtered in t he UK alone, with economic damages to producers, government, and the tourism sector totaling over £3 billion ($4.4 billion; Thompson et al. 2002) . We apply our model to the case of dairy farmers in Cumbria County, l ocated in northwestern England. Cumbria was the most heavily affected county during the 2001 outbreak, experiencing 44 percent of the total number of cases reported nationwide (Convery et al. 2005) . The next section des cribes the model specification. We then calculate Nash equilibria fo r two scenarios: (i) a baseline case in which no disease prevention policy exists and (ii) the case of behaviorally -depend ent and -independent policies. 1.5.1 Model Specification Consider a region of Cumbria County that is initially disease -free but at risk of i nfection from outside sources. We assume producers in this region are homogeneous in herd size, costs, and prices. The analysis is therefore based on the perspective of a representative producer. The representative producer™s herd is at risk from infection along two pathways: (i) primary infection via sources outside the region (i.e., pathogen introduction) and (ii) secondary infection via direct or indirect contact with neighboring producers who have had the pathogen introduced to their herds (i.e., pathog en spread). For simplicity, we adopt a static, one -period , non-spatial model of these processes, thereby representing behavior at the e arliest stages of an outbreak. This framework is consistent with prior work (Hennessy 2008) , except that we allow produce rs to choose biosecurity to protect their herds from both pathogen introduction and spread. 12 12 Hennessy (20 07b) also considers tertiary infections, i.e., when a producer infects another producer who infects another producer. This was the exclusive pathway through which neighbors™ biosecurity effort choices enter the representative producer™s profit -maximization problem. In contrast, we account for positive spillovers from neighbors™ efforts to prevent spread via reducing the effective number of contacts between herds. This means of controlling spread has not been examined previously in economic studies. Our assu mption that herds are not at risk from tertiary infections is a simplifying one, although it is unlikely to affect the general results of the model. 19 First con sider the introduction pathway. Let the probability that the pathogen is introduced into the representative producer™s herd be PI Œ is the probability probability per unit of biosecurity. Now consider spread from neighboring herds that have become infected. The pathogen is introduced into a n eighboring herd with probability PI Œ ). Spread to the representative producer™s herd may then occur v ia direct or indirect contact. Direct contact may occur in communal grazing areas or livestock exhibitions, whereas indirect contacts may invol ve disease transmission vectors such as people, wildlife, or airborne droplets of water containing infectious agents. Let N indirectly contact the representative producer™s herd in the absen ce of meas ures to avoid those contacts. Biosecurity effort by either the representative producer or his neighbors can reduce the effective number of contacts between herds, e.g., by limiting access to farm workers who travel between farms, by cleaning shar ed equipment, or by prohibiting the movement of anim als between neighboring farms. Suppose biosecurity reduces the producer™s effective number of contacts to N(1 Œ )(1 Œ biosecurity adopted. 13 This specification implies that neighbors™ biosecurity efforts are as effective in preventing contacts as the producer™s own efforts, e.g., because the biosecurity 13 Absent biosecurity, the representative producer can potentially make contact with each of his N neighbors . This represents a case of uniform mixing over the landscape. This is a somewhat extreme assumption, but it is commonly used in epidemiological modeling, particularly in metapopulation models (e.g., Watts et al. 2005) . It seems somewhat realistic here, in the absence of movement restrictions, since the disease is easily spread by movement of people and equipment across farms, which may be somewhat randomly distributed over the landscape. As biosecurity is adopted, more weight would have to be put on more localized (e.g., aerial) transmission. This is N. Epidemiological models often adopt a similar approac h to modeling spatial transmission by putting less weight on the fisusceptibility functionfl, which describes the susceptible portion of the population that is likely to come into contact with infected animals (Barlow 1995) . 20 technology used by each producer is the same. Given that the producer™s herd is u ninfected and contact does occur with an infected neighboring herd, let k Œ represent the probability that such contact spreads the pathogen to the producer™s herd. results in unit of biosecurity adopted. The probability the producer™s herd becomes infected from contact with a particular neighbor™s herd, conditional on that herd being infected, is then k)PI Note that k , as it is assumed that neighbors have no incentive to prevent disease from leaving their farm. Even so, neighbors™ biosecurity efforts produce positive spillovers in reducing spread, as their efforts reduce both the probability of pathogen introduction to the region and the number of contacts between herds, as described above. Given this specification, the probability that a producer™s herd becomes infected via spread from any of his neighbors Šconditional on the producer™s herd being uninfected Šcan be modeled as a Bernoulli process: 14 (1.12) PS Œ [1 Œ k PI ] . The expression [1 Œ k PI ] is the probability that a particular neighbor™s herd does not infect the producer™s herd, and so [1 Œ k PI represents the probability that the producer™s herd does not get infected by any of his neighbors™ herds. Thus, PS one infectious contact occurs via a neighbor™s herd. Note that PS N; as the number of producers in a region increases, the likelihood of an infectious contact with at least 14 Note that it is likely that the exponent -integer value . In the strictest sense, a Bernoulli process requires the number of trials to be in the set of nonnegative natural numbers, +. However, allowing non -integer values in equation ( 1.12) serves as a reasonable app roximation of the probability of infection and the differentiability allowed by this functional form allows for a greater level of analysis while preserving realistic assumptions about how biosecurity affects contact rates between herds. 21 one neighbor™s herd increases f or the representative producer. Likewise, PS Finally, the probability that a producer™s herd be comes infected via either pathway (introduction or spread) is found by substituting PI PS The producer™s problem ( 1.2) is parameterized using values taken or derived from economic and epidemiological studi es of the 2001 UK FMD outbreak. It is assumed that the parameters are homogeneous among all N producers. Allowing the parameters to vary among producers will affect the quantitative results of our example but will not qual itatively change how biosecurity affects infection risks and economic incentives. Thus, the insights into the strategic interactions provided by this numerical example will be unaffected. Parameter values are presented in Table 1 .1. The sources and derivat ion of each parameter are detailed in Appendix 22 Table 1.1 Functional Forms and Parameters Used for the Numerical Example Domain Form/Value in Numerical Example a Description Units Parameters [0, 1] 0.3125*# Probability of introducing infected animal, no biosecurity Unitless [0, 1] 1 % reduction in probability of introduction per unit of biosecurity Unitless [0, 1] 0.9*§ Probability of infection given contact occurs Unitless [0, 1] 0.15 % reduction in probability of infection from contact per unit of biosecurity Unitless 1,328* Cost parameter $ N 80ƒ Number of neighboring herds Herds [0, 1] 0.75*ƒ+ % reduction in contacts per unit of biosecurity Unitless = RH Œ RNH 26,404⁄ Cost of infection $ Variables [0, 1] ŒŒŒ The producer's biosecurity effort Units of effort [0, 1] ŒŒŒ Neighboring producers™ biosecurity effort Units of effort Functions PI PI [0, 1] PI Œ PI Œ Probability of introducing infected animal to farm Unitless PS [0, 1] PS Œ [1 Œ k PI Probability of infection from spread Unitless k [0, 1] k Œ Probability herd becomes infected given contact with infected herd Unitless [0, N] N Œ 1)(1 Œ Œ Effective number of neighboring farms Farms E{ E{ RH Œ P Œ c Expected profit $ c c 2 Cost of biosecurity effort $ a Note: * Chi et al. (2002) ; # Green et al. (2006) ; § Schley et al. (2009) ; ƒ Brennan et al. (2008) ; Defra (2011) ; ⁄ Thompson et al. (2002) ; + Ferguson et al. (2001) . Entries without superscripts are assumed values 23 A. Note that biosecurity is assumed to be only m arginally effective in reducing the infectivity of contact k This is because FMD is highly contagious and is capable of aerial transmission over relatively long distances (Mikkelsen et al. 2003; Ferguson et al. 2001) . Also because of a erial transmission, we assume biosecurity can only eliminate contact from neighbors located outside of a 2 km radius of the representative farm; within this radius, there is always positive probability of aerial transmission (Ferguson et al. 2001) . 1.5.2 Scenario (i): Baseline Case with No Disease Prevention Policy We first consider strategic interactions arising from the baseline case when no indemnity or other disease prevention policy is pr esent, i.e., the baseline case. We solve first -order condition ( 1.3) numerically using Mathematica 7.0 (Wolfram Research, Inc. 2008) to derive the producer™s best - ). This function is depicted by the solid, discontinuous curve in Figure 1.2. The only two equilibria present in the model are the two symmetri c Nash equilibria (SNE), A and B in Figure 1.2.15 The SNE can be Pareto -ranked, with the high -effort equilibrium B being privately preferred (Van Zandt and Vives 2007) .16 The SNE A and B < 1 at both points . The dashed vertical line running through point C represents a discontinuity in the response function that acts like an unstable threshold , dividing the graph into two basins of attraction: a low -effort basin with a stable SNE at A and a high -effort basin with a stable SNE at B.17 Expectations matter in Figure 15 We consider only pure -strategy Nash equilibria. Mixed strategies are often unstable (Harsanyi 1973; Echenique and Edlin 2004) , and can therefore be unreliable predictors of behavior. 16 Note that the high -effort equilibrium is also socially -preferred in this case since it represents the maximum l evel of biosecurity available. then it would not be Pareto optimal (Milgrom and Roberts 1990) . 17 The stability properties of the system follow from the fact that, according to the numerical results, E} < 0 for below the discontinuity and E} > 0 for above the discontinuity. Intuitively, when two stable equilibria are divided by a threshold, t he threshold must be unstable. It is therefore straightforward to hypothesize a tâtonnement process of adjustment to A for any expectation by the representative producer that places to the left of the 24 1. T, then the system proceeds via tâtonnement to B. Conversely, if producers expect their neighbors initially choose biosecuri T, then coordination failure occurs and the system proceeds via tâtonnement to A.18 Figure 1.2 The representative producer's best -response function with no policy The multiple, stable SNE in Figure 1.2 arise due to strong strategic complementarities within the neig hborhood of the discontinuity. Strategic complementarities in this region c an be seen in two ways. A and C. Second, the curve labeled RER = 1 in Figure 1.2 P = discontinuity or to B for any expectation to the right of the discontinuity . Note also that if a 45° line extending from the origin intersected the positively - > 1 at such a point. 18 We perform a sensitivity analysis for the baseline parameterization. For conciseness, we present the analysis in Appendix A. 25 1. P < 1, and thus by condition ( 1.7), biosecurity is a local strategic complement at A, as well as up to and beyo nd the expectational threshold. To the righ t of this P > 1, and thus biosecurity is a local strategic substitute at B. 1.5.3 Scenario (ii): Indemnit ies Consider next the case in which an infected producer receives an indemnity ets his losses due to disease. We begin by analyzing beh avior under a behaviorally -dependent indemnity, i.e., Assume for simplicity that receives compensation equal to 68 percent of his losses from infection if he fully self -protects. indifferent to fully self -protecting; a larger amount would actually reduce his biosecurity incentives (see below). Using the relation for -order condition (1.3) for the best - represented by the solid curve in Figure 1.3. The behaviorally -dependent indemnity drastically changes the producer™s best -response function relative to Figure 1. -effort basin Šand the low -effort equilibrium AŠis eliminated. The only SNE remaining in Figure 1.3 is the high -effort equilibrium B. The expectational threshold and the low -effort equilibrium have been eliminated in Figure 1.3 because the indemnity has i complementarities at lower effort levels. Indeed, curve RER = 1 now lies to the left of the RER = 1 curve from Figure 1.2, as the RER = 1 curve is now a contour line denoting the locus of points for whi P E = 1 (see condition ( 1.11)). Since the unique SNE is located to the right of 26 RER = 1 in Figure 1.3, biosecurity is a strategic substitute at equilibrium , and B is globally duces similar qualitative results, although the high -effort equilibrium B is reduced slightly. Intuitively, a higher rate would increase the producers™ control over his spread risks enough such that he has incentives to scale back his biosecurity (see the tradeoffs described in relation to expression (1.9)). Figure 1.3 A behaviorally -dependent indemnity Finally, consider a behaviorally -independent indemnity. For the sake of comparison , let the indemnity take the form such that it is equal to the equilibrium indemnity earned in the behaviorally -dependent case above. Plugging this indemnity into the producer™s expected profit maximization problem, ( 1.8) is again solved for the best - Figur e 1.4a. Decoupling the indemnity from behavior has lowered the producer™s incentive to 27 Figure 1.2. Also, although RER is unaffected Šthe measure of RER under a behaviorally -independent indemnity is once again given by condition ( 1.7)Šthe magnitude of the strategic complementarities has been reduced (i.e., the best response curve is less steeply sloped along the interval from points A to C relative to Figure 1.2). The reduced complementarities along with reduced biosecurity incentives has expanded the low -effort basin by shifting the expectational T to the right relative to Figure 1.2, increasing the risk of coordination failure. A larger behaviorally -independent indemnity causes the expectational threshold to vanish, as in Figure 1.4b. The only SNE remaining in Figure 1.4b is the low -effort equilibrium A, which is globally stable since biosecurity is a weak strategic complement at this point, and so expe ctations do not matter. Figure 1.4 a) A small and b) large behaviorally -independent indemnity 28 1.6 Optimal Indemnity Design As a final consideration, we explore the social planner™s problem of choosing the optimal indemnity to maximiz e the net economic surplus arising from self -protection. We begin by designing a linear indemnity of the form derived based on the fir st-best optimality conditions. Therefore, we first charac terize the first -best outcome. Our specification of social net economic surplus is based on the simplifying assumptions that the disease has no zoonotic spillovers on human health , that the regulated producers do not have a collective impact on input or output prices, and that society is risk neutral. Also, we continue to assume producers are homogeneous. Given these assumptions, social net economic surplus is simply the aggregate profit to livestock producers: (1.13) cPRNEH,1max . The first term in brackets is the rep resentative producer™s expected profit and the second term in brackets represents the aggregate profits of his N neighbors. For an interior solution, t he opt imal level of biosecurity for the representative producer solves /}{E0, or (1.14) cPP,,. The first LHS term and the RHS term of e quation ( 1.14) are the same as in the producer™s decentralized first order condition ( 1.3) for the case of no ind emnity. The second LHS term (ŒP> 0) accounts for the positive incentives . Intuitively, individual producers do not take account of the positive spillovers generated by their biosecurity. When the decentralized optimality condition ( 1.3) is satisfied as an equality (an interior solution), producers exert too little effort relative to the first -best. The 29 optimal indemnity must therefore give producers an incentive to inter nalize the spillovers arising from the ir coincide with the first -best. Hence, the socially - only weakly greater than the level chosen in the decentraliz ed case . Consider an interior first -best outcome. We can set first -order condition for the case of a decentralized indemnity (equation 1.8) to that of the command optimum (equation 1.14) evaluated at the first -best out come: 19 (1.15) **1,,*PP, * is the first - P P) < 0 is the elasticity of the probability of infection w ith respect to self -protection. The LHS of expression of ( 1.15) represents the positive spi llover effects of biosecurity. The RHS of ( 1.15) is the marginal incenti ve provided by the indemnity. The first -best value * is derived from expression ( 1.15) as * Œ P* P*(1 + *)], where the superscript * means all variables are evaluated a *. *, which is positive, is the aggregate marginal reduction in infection probability from biosecurity. *) > 0. 20 * is unaffected by whether biosecurity is a strate gic complement or substitute. * corrects the inefficiency arising from the spillover effects of one™s self - * is first -best only when this 19 An alternative but equivalent approach involves choosing policy instrument parameters to maximize net economic surplus, condition al on producers™ optimal response to the policy in the decentralized setting (Acocella, Di Bartolomeo and Hughes -Hallett 2013) . 20 * P P = (P P)/P = (P P P contradiction since ( P Œ [ŒP]) is strictly positive given that P P exceeds the absolute value of the marginal impact. 30 rate yields a globally stable SNE at the first -best outcome. Otherwise, in the case of multiple, locally stable equilibria (of which one is the first - * does not guarantee the first -best outcome. The reason is that the linear * does not address an additional source of inefficiency that is generally present in this setting: the problem of coordinating on the first -best equilibrium. This inefficiency can manifest itself when there are multiple decentralized SNE, which may occur when biosecurity is a relatively strong strategic complement. An indemnity with an additional parameter is needed to address this additional source of inefficiency (Tinbergen 1952) . We consider a general indemnity 12 i are parameters to be chosen optimally. The producer™s decentralized first -order condition ( 1.8) may be implicitly solved for the symmetric best -response function conditional on the indemnity, 12)| . The first -best policy parameters are th en chosen to simultaneously solve the following conditions: (1.16a) *21*|,, (1.16b) ,,**0021dd. Condition ( 1.16a) addresses the positive spillovers from biosecurity. Specifically, it states that the best response curve intersects the first - *. This is the sole condition that must be satisfied under traditional notions of optimal indemnity design, and was addressed by our linear indemnity in condition ( 1.15). Condition ( 1.16b) addresses the risk of coordination failure. Specifically, this condition requires the area beneath the producer™s best -response function (the LHS) to be greater than the *. This ensures there are no 31 *, thereby eliminating the risk of coordination failure. 21 Note that (1.16b) is automatically satisfied when biosecurity is a weak strategic complement or a global strategic substitute. Consider first the effect of the optimal indemnity for the general case in which interior equilibria exist prior to the indemnity. There are three cases, as pictured in Figures 1.1aŒc. In each case, the first -best outcome would lie on the 45 curve above and to the right of the high -effort SNE (i.e., point B in figure 1a, or point A in Figures 1.1b or 1.1c). An optimal indemnity will shift the best -response function upwards through the first -best o utcome. In the case of multiple equilibria in Figure 1.1a, the optimal indemnity would al so rotate the best response function clockwise so that it no longer intersects the 45° curve at any point other than the first -best outcome. N ote that condition ( 1.16b) also ensures the new best -response function will intersect the 45 curve from above, gu aranteeing that the solution is globally stable. 22 Finally, c onsider the special case in which the first -best outcome is a corner solution, i.e., B in the no -policy case of Figure 1.1a is a SNE corner solution of (1, 1) that coincides with the first -best optimum. Thus, no indemnity or other instrument is needed under traditional notions of first -best policy design which consider sub -optimal self -protection as the sole source o f inefficiency. However, the first - 21 This condition is the same as that which characterizes first -order stochastic dominance (Mas -Colell, Whinston and Green 1995) . This condition is sufficient to guarantee a unique SNE. To see this, suppose that there exist multiple SNE. This implies the presence of strategic complementarities. In this case, each SNE can be Pareto ranked, with the * must be weakly greater than any decentralized equilibrium, there cannot be a preferred equilibrium above the first - * * if condition (1.16b) is satisfied. 22 For instance, an indemnity of the form 1 2 e value of 1 2 will shift the producer™s best -response function upwards for 2 will increase RER to help eliminate t he risk of coordination failure. Together, these values can be set to accomplish the two objectives defined 3 123)/ Œ1 avoid particularly strong strategic substitute relations that can generate asymmetric equilibria. 32 T. Thus, despite the fact that the first -best outcome is a SNE, the potential exists for producers to equilibrate at a locally stable, sub -optimal outcome (i.e., point A, which is the origin in our model). An indemnity that eliminates the expectational threshold and makes the first -best SNE globally stable can therefore ensure a welfare improvement. Figure 1.5 A best -response function under an op timal indemnity Figure 1.5 depicts the effect of a small, behaviorally -dependent indemnity (i.e., -policy case in Figure 1.2, the first -best optimum (point B) remains an SNE under this indemnity. The indemnity in Figure 1.5 spec ifically addresses the second source of inefficiency described above: it has increased RER and, with it, producer incentives for self - -response function above the 45° line. Here, the first -best outcome B is n ow a unique SNE: by increasing RER, the optimal indemnity provides producers with just enough control over their economic risks so as to eliminate the risk of coordination failure and ensure the long -run sustainability of the first -best outcome. A larger 33 behaviorally -dependent indemnity is unnecessary and may even reduce the SNE level of self -protection and make it unstable (see footnote 7). 1.7 Discussion and Conclusion Understanding that environmental risks are endogenous has had important implications for managing environmental systems (e.g., Archer and Shogren 1996; Kane and Shogren 2000; Treich 2010) . When these risks involve positive spillovers from self -protection, strategic interactions among at -risk individuals may endoge nously affect individual risks. In this chapter , we have attempted to provide insight as to how various strategic relationships endogenously arise and how decision -makers can influence them to enhance disease management. Our analysis of endogenously - and strategically -determined risk has led us to identify a new concept linking these elements: the relative endogeneity of risk , which measures individuals™ ability to tak e control over their own risks. A smaller RER results i n self -protection being strategic complements where coordination failure may occur, whereas a larger RER yields a more stable strategic relation (e.g., weak strategic complements or strategic substitutes) with potentially greater levels of self -protection. Furthermore, disease prevention policies can influence RER , altering strategic relationships and the resulting equilibrium outcomes . Although our numerical analysis is based on a very simple model (i.e., a single period with homogenous agents and non -spat ial interactions) to highlight the link between RER and the strategic interactions between at -risk individuals, we should emphasize that our analytical framework is quite general and could be expanded to incorporate heterogeneous agents interacting via var ious types of spatial disease transmission encompassed by PS (e.g., aerial transmission, direct or indirect contact between herds, etc.). Also, the underlying disease processes may be dynamic, although disease dynamics are not modeled explicitly . Making th e 34 model fully dynamic in this context would primarily involve making the infection probabilities non-stationary, but it would not alter the behavioral response mechanisms assuming that individuals™ behavior is myopic . Myopic behavior makes sense when self -protection efforts do not involve capital accumulation but instead involve the amount of care exerted in each period Š which is t rue of many livestock diseases. Still, the lack of explicit disease dynamics is limiting in the sense that we are unable to see h ow strategic behaviors affect epidemiological dynamics and how these may affect future strategic interactions and , hence , future RER. We leave for future research any extensions to a fully dynamic model, and also to non -myopic behavior for which the strate gic interactions would have clear analogues in differential games. Finally, RER may be of value as a metric. Metrics are commonly used in epidemiology, with R0 (defined as the average number of cases an infected individual generates over the course of its infectious period) being the key metric of interest. R0 typically does not capture economic relationships that may be important in disease transmission (Fenichel et al. 2011) . RER may therefore serve as an economic metric that provides greater insight into both infection risks and the incentives individuals have to self -protect against these risks. We are aware of only one prior study that econometrically estimates the strategic relationships governing self -protection from disease (Kobayashi and Melkonyan 2 011). Future work utilizing structural econometric models may be useful in estimating the technological and economic relationships that determine RER, leading to greater efficacy of disease prevention efforts. 35 CHAPTER 2 ECONOMIC INCENTIVES FOR MANAGING BIOLOGICAL POLLUTION RISKS FROM TRADE 2.1 Introduction The movement of live animals is a key driver behind the spread of infectious livestock diseases (Perry, Grace and Sones 2011) , which can be considered a form of fibiological pollutionfl akin to invasive species (Daszak, Cunningham and Hyatt 2000; Horan et al. 2002) . The economic impact of infectious disease is staggering. For example, the costs of the 2001 foot -and -mouth disease o utbreak in the United Kingdom Šwhich was initially spread throughout Great Britain via animal movements Štotaled over $4 billion (Thompson et al. 2002) . Mechanisms to mitigate the externalities associated with live animal movements have thus gained considera ble attention. Prior literature asserts that trade -related biological pollution externalities arise not from trade per se, but rather from trade of contaminated goods (e.g., Paarlberg and Lee 1998; McAusland and Costello 2004; Mérel and Carter 2008) . This literature acknowledges that it is generally not possible or cost -effective to perfectly identify each contaminated good in trade flows, so trade -based policies comprising tariffs (to internalize expected damages from imported goods) and inspection regime s (to sort out some contaminated goods) are derived to internalize these externalities. All of this prior work implicitly assumes that ( i) individual importers have neither the ability nor the incentive to mitigate externalities from the importation of p otentially contaminated goods and ( ii) social damages from biological pollution are determined from unilateral spillovers (i.e., those who generate social risk are not themselves at risk from others™ actions). Realistically, however, the risk posed by trad e-related biological pollution problems is a fifilterable externalityfl (Shogren and Crocker 1991) : an individual importer can reduce disease 36 spillovers via multiple, private risk management choices. For example, a livestock producer who quarantines newly -imported animals reduces his own disease risks as well as those to neighboring susceptible herds. Producers can also protect others directly via risk -abatement measures such as properly disinfecting shared equipment. Social damages in this context may result from bilateral spillovers (i.e., those who generate social risk are themselves at risk from others™ actions), with strategic interactions arising to the extent that a producer™s risk management measures simultaneously protect himself and neighbors. Simila r issues may arise with biological pollutants other than livestock diseases, such as invasive species that adversely affect the nursery industry (Knowler and Barbier 2005) . The myriad choices involved in generating external risks, combined with stochastici ty and an inability to observe biological pollution at an acceptable cost, suggest the problem of managing trade -related biological pollution risks is analogous to nonpoint source pollution problems (Horan et al. 2002) . In this context, the importation of contaminated goods and private risk management and abatement activities (e.g., inspection effort, biosecurity to prevent spread) can be thought of as inputs to a biological pollution production function. The nonpoint source literature suggests efficient po llution management policies can be based on the biological pollution production function (which serves as a proxy for the externality) or else on the inputs to this function (e.g., Griffin and Bromley 1982) . Prior work on managing risks from trade has inst ead focused on trade -based policies. These can be efficient if trade and surveillance choices are the only inputs to the biological pollution production function. However, trade -based policies are inefficient if biological pollution depends on other, non -trade -based inputs such as importers™ risk abatement effort. 37 This paper explores the design of efficient disease prevention policies when importers can mitigate disease risks to others, thereby generalizing prior work (McAusland and Costello 2004; Mérel and Carter 2008; Paarlberg and Lee 1998) . Specifically, we reframe the problem of managing disease risks from trade as a nonpoint source pollution problem, following the approach of Horan et al. (2002) and Horan and Lupi (2005a; 2005b) . We begin by examining the first -best allocation of disease control efforts in the case of unilateral spillovers. We then compare this allocation with the privately -optimal allocation and derive policy instruments that incentivize first -best disease control efforts. In contrast to prior work, we find the externality extends beyond trade in contaminated goods. We demonstrate that efficient incentive -based mechanisms must target the externality, or each of the choices contributing to the externality, rather than just the trade -rela ted activities. We also show the magnitude of these efficient incentives decreases when producers have greater private risk management incentives and a greater ability to directly protect others. Next, we consider the case when social damages stem from b ilateral disease spillovers between importers. Prior work on trade -related biological pollution has not considered these types of spillovers. In a related study that does not model import decisions, Reeling and Horan (2015) examine bilateral externalities arising from a single biosecurity choice that protects against infection risks from both imports and spread among neighbors. Likewise, Wang and Hennessy (2015) consider the allocation of government and producer effort to prevent and control the spread of d isease within a region. These authors consider bilateral spillovers but do not explore efficient policy design. We extend these analyses by allowing for multiple risk management choices, including import decisions, and by focusing on efficient policy desig n. Within this setting, we find that spillovers between importers may lead to multiple Nash 38 equilibria in the decentralized outcome. These equilibria include one or more sub -optimal equilibria in addition to the first -best equilibrium, and producer™s expec tations about their neighbors™ strategies determine which outcome is pursued. We show that incentives alone may not be sufficient to achieve efficiency; additional command -and -control policies (e.g., regulations) may also be needed alongside incentives (An derson and Francois 1997) . 2.2 Trade and Infection Risks Under Unilateral Spillovers We begin by deriving a simple model of trade and infection risks, which we frame in terms of livestock trade for concreteness. Suppose there exists a region that contains a fi xed number of livestock importers, indexed by i = 1, ... , N, each of whom are price takers operating in competitive input and output markets. Each importer purchases a fixed quantity of live animals, Xi, from outside the region to raise on his farm. Each importer may purchase animals from either a risk -free source known to be devoid of infected animals or from a lower -cost risky source where infected animals are present with a known probability. Denote the share of the animals purchased from the risk -free source as xi [0, 1]. The probability the importer purchases an infected animal is denoted )(iIixP, where 0/iIixP and 0)1(IiP. If any imported animal is infected, then we assume all other animals purchased from the risky source will also become infected during transport. The importer can inspect and quarantine animals purchased from the risky source prior to introducing them into his herd. If an infection is detected, then all imported animals from the risky so urce, (1 Œ xi)Xi, will be culled at per - i to the importer. Denote the probability that infected animals are caught via inspection as )(iCizP, with 0/iCizP, where zi [0, 1] is importer i™s inspection effort. Infect ed animals that are not caught via inspection are assumed to 39 infect the importer™s entire herd, generating a per - i i to the importer. We assume for now that a given importer is not at risk from infection due to neighboring importers™ choices. 23 These additional risks are considered later. Taken together, importer i™s expected losses from disease are ]))(1()1)(()[(),(iiiCiiiiiCiiIiiiiXzPXxzPxPzx i is the importer™s expected losses given that he imports an infected animal, identifies it via inspecti on, i is the importer™s expected losses given that he imports an infected animal but does not detect it, and must therefore cull his entire herd. Given our specification for importer i™s own expected losses, and assuming the importers are risk -neutral, we can define importer i™s private net benefits by (2.1) i(xi, zi, ai) = Bi Œ ci(xi, zi, ai) Œ i(xi, zi). The first right -hand -side (RHS) term, Bi, is the net revenue earned by selling the imported animals. The second RHS term, ci(xi, zi, ai, bi) = w(1 Œ xi)Xihi(zi) + vxiXi, + fi(ai) represents the importer™s costs unrelated to infection. Specifically, the cost of purchasing animals is w(1 Œ xi)Xi h(zi) + vxiXi, where w and v are the exogenous per -unit costs of animals from the risky and risk -free sources, respectively, with w < v. The function hi( hi hi hi(0) = 1 reflects inspection costs. The term )(iiaf is the cost of abatement, where ai [0, 1] is abatement effort to reduce the probability of disease spread off the farm. We assume increasing, convex 23 Spillovers from introduction are unilateral in the sense that the importer™s choices impose damages on society, and not the other way around. Examples of such damages include those a rising from diseases which spread from an importer™s herd into a wildlife reservoir. For instance, bovine tuberculosis (bTb; Mycobacterium bovis ) was introduced with cattle into Australia and New Zealand and quickly became established in feral deer and pos sum populations (Tweddle and Livingstone 1994) . Exposure to bTb, e.g., via hunting and field -dressing infected deer, can result in tuberculosis in humans (Wilkins et al. 2003) . More broadly, the analysis of one -way spillovers is also appropriate for analyz ing risks from other invasive species, e.g., salt cedar ( Tamarix spp. ), an invasive shrub that was introduced via the nursery trade and has become widely established in the Southwestern U.S. (Whitcraft et al. 2007) . 40 abatement costs, fi fi fi(0) = fi(0) = 0. Our assumptions imply ci is linear in xi and increasing and convex in zi and ai ci(xi, zi ai = 0. In what follows, we focus on ci() rather than on its individual components. The profit function is concave in both xi and zi. An infected animal that goes undetected in importer i™s herd may transmit the disease to other, non - As indicated above, the importer can invest in abatement effort, ai, to reduce the probability disease leaves his farm, )(iLiaP, with 0/iLiaP. The probability the disease spreads from importer j™s herd is then (2.2) )()](1)[(),,(jLjjCjjIjjjjjaPzPxPazxe. Let e = (e1, ... , eN). The probability at least one importer transmits the disease to non -importers, so that damages occur, is then given by (2.3) iiiiiDazxeP),,(11)(e, so that expected social damages from disease spread are )()(eeDPD. Social net benefits are defined as the sum of the N importers™ profits less expected social damages. 2.2.1 The Efficient and Privately -Optimal Outcomes under Unilat eral Spillovers We begin by comparing the privately -optimal and efficient outcomes, which clearly indicate the biological pollution externality extends beyond trade in contaminated goods. We then examine how private risk management activities contribute to the externality, with a particular focus on the relative contribution of trade and non -trade related activities. The importer™s private expected profit maximization problem involves choosing xi, zi, and ai to maximize i(). Assuming an interior solution, the importer™s private first order conditions (FOCs) are 41 (2.4) azxvvvcviiiiii,,,0. The FOCs in (2.4) imply the importer™s private marginal benefits from disease mitigation equal his private marginal costs at the privately -optimal solution. The importer™s privately -optimal choice, 0iv, v {x, z, a civi ivi iai = Œci = 0 ( iai = 0), implying 0ia = 0: the individual importer has no incentive to invest in abatement effort since it does not influence his own disease risks. In contrast, the importer™s choices of 0ix and 0iz are weakly positive since these choices do influence his own disease risks. Figure 2. 1 Comparison of privately - and socially -optimal risk mitigation Consider next the efficient outcome, for which the share of risky imports, xi, inspection effort, zi, and risk abatement effort, ai, are chosen to maximize importers™ aggregate expected profits less expected social damages. Assuming for simplicity that imp orters within the region do not have a collective influence on input or output prices, and taking N as fixed (i.e., ignoring the potential for entry and exit), the social planner™s problem is 42 (2.5) )(,,max 1,,eDazxVNiiiiiiazxiii. The first -best choices *ix, *iz, and *ia solve the three FOCs (2.6) .,,;0azxviveDvvcveDvvViiiiiiiiiii The second line of condition (2.6) shows that the marginal net private costs of vi ci vi i vi, optimally equals the marginal expected external benefits from i™s choice of v {x, z, a}, i.e., the marginal avoided social damages stemming from risk mitigation, ŒD eivi]. As ŒD eivi] = Œ PDei eivi ci vi i vi > 0 v {x,z,a } when evaluated at the first -best outcome. This implies importer i™s first -best choice of v, denoted *iv, is greater than his privately -optimal choice, 0iv, which does not account for the marginal expected external benefits of v (Figure 2.1). Comparison of FOCs (2.4) and (2.6) clearly indicates that externalities exist, where the source of the externality created by importer i is the probability that infecti on will be transmitted, ei(). This function can be thought of as a biological pollution production function that measures the external pressures being generated by the importer, and it is the probabilistic analogue of the pollution production functions us ed in more traditional nonpoint source pollution models. Note that this function depends only partly on trade decisions, in contrast to prior work that focuses exclusively on trade decisions involving contaminated goods as the externality (McAusland and Costello 2004; Mérel and Carter 2008) . Note also that the magnitude of the externality is reduced when risks are filterable. To see i 43 = 0 and condition (2.4) bec omes Œcivi = 0 v {x, z, a}, with importer i™s privately optimal choice being 0NDi v= 0 v. The planner™s optimality condition (2.6) becomes (2.7) azxviveDvcvViiiii,,;0. Comparing (2.7) with (2.6) implies that the social optimum occurs at *NDi v < *iv, as illustrated in Figure 2.1. The magnitude of trade -related disease spillovers is smaller when risks are filterable, i.e., *NDi vŒ 0NDi v = *NDi v > *iv Œ 0iv in Figure 2.1. Intuitively, importers™ private incentives for risk mitigation cause them to make choices that reduce the disease risks they face via their own imports. These choices in turn reduce the throughput of disease risks from the importers™ farms to non -importers (i.e., these choices fifilterfl the biological pollution externality). We show in Appendix B that this result holds generally. Hence, the marginal benefits from mitigation beyond privately -optimal levels are reduced, at least for those activ ities for which Œi vi > 0. This finding suggests a need to direct a greater policy focus at the margin on inputs for which individuals have fewer or no private incentives to provide on their own (e.g., risk abatement activities), and to reduce the pol icy focus on inputs where there are greater private incentives to mitigate risk. This result has not arisen in prior work in which trade does not subject individuals to private risks. 2.2.2 Incentives for Efficient Disease Control We now consider how economic in centives might be designed and implemented to eliminate the externalities described above. Incentives for managing disease transmission would ideally be based on the actual emissions of biological pollutants, but there are difficulties with this approach b ecause disease transmission is stochastic and the actual spread of infection to others is 44 generally unobservable (at least, without considerable cost to trace the source of infection; Elbakidze 2007) . Disease control incentives must be based on an alternat ive construct in the face of unobservable emissions. Griffin and Bromley (1982) suggest two alternatives. The first is to base policies on a nonpoint pollution production function that links a firm™s choices to emissions, thereby acting as a performance pr oxy. The analogous construct for biological pollution is the probability of disease transmission, ei(). Such a measure can be thought of as a risk -based performance proxy. The second approach is to target the individual choices or inputs that contribute t o external infection risks through the use of input -based incentives. Both approaches have been analyzed and applied for the case of invasive species (Horan and Lupi 2005a; Horan and Lupi 2005b) . We now examine each in turn. 2.2.2.1 Risk -Based Incentives Suppose a tax ti is levied on the probability of disease transmission, ei, which can be calculated from a predictive model based on observations of the importer™s decisions. 24 The importer™s problem under this risk -based tax is (2.8) ),,(),,(max ,,iiiiiiiiiazxiazxetazxJiii, with FOCs (2.9) azxvvetvvJiiiiiii,,,0. 24 Assuming risk neutrality and perfect traceability, an equivalent approach is to charge importers a fine if an outbreak occurs and the infection can be traced back to their trade decisions. The optimal fine will be larger if traceability is imperfect (i.e., if outbreaks can be traced back to t he original importer with probability less than one). 45 Comparison of (2.9) with the social planner™s FOCs (2.6) indicates that importer i™s optimal choices under the risk -based tax are first -best as long as the tax is set equal to the expected marginal social damages from sp read off his farm, i.e., if, for all i, (2.10) iazxiiiiDt***,, . Intuitively, taxing the probability that disease spreads from the importer™s farm Ša proxy for the true externality of disease transmission Šincentivizes the importer to internalize the consequences of all of his choices on the disease risks to his neighbors. Note also that efficiency can only be attained if the risk -based tax is specific to the importer: the tax would be overdetermined Šand thus at most second -best Šif instead a regulatory authority were to apply a uniform tax rate to all importers. 2.2.2.2 Input -Based Incentives Consider next the outcome under input -based incentives. Specifically, suppose regulators can i, on all animals purchased from the risky source. Suppose a lso that the importer receives per - zi ai for each unit of inspection effort and abatement effort he undertakes on his farm, respectively. 25 The importer™s problem in this case is (2.11) iaiiziiiiiiiiazxiazXxazxJiii)1(),,(max ,, with FOCs (2.12) 0iiiiiXxxJ, 25 The availability of a subsidy for importers™ inspection effort assumes that zi is observable. Alternatively (and equivalently), we could assume that zi is unobservable but that a benevolent regulato ry authority can also provide inspection effort Šsay, zŠin addition to the importer™s own zi. If we assume that zi and z are perfect substitutes, then the aggregate level of inspection effort zi + z chosen by the importer and regulator (perhaps according to a Stackelberg game where the importer chooses his optimal zi given the regulator™s z) will be the same as that chosen using an optimally -derived subsidy. 46 (2.13) 0ziiiizzJ, (2.14) 0aiiiiaaJ. Comparison of (2.12) Œ(2.14) with (2.6) indicates that input -based incentives can achieve izi ai are each set such that (2.15) iXxeDiazxiiiiiii***,,1, (2.16) izeDiazxiiziiii***,,, (2.17) iaeDiazxiiaiiii***,,. The first -best instruments equal the marginal social damage from the externality, D, multiplied eivi for v {z,a } and eivi)/Xi for v = x (i.e., the marginal product of v {x, z, a} in the biological pollution production function). Our earlier results on filterable externalities (see Figure 2.1) suggest the first -best instruments will be smaller the larger are importers™ private incentives for risk mitigation and abatement activities because the magnitude of the externality will be smaller. Note also that the first -best instruments must be applied to each input contributing to ei. These results differ from the literature on trade in goods potentially contaminated with invasive species (e.g., McAusland and Costello 2004; Mérel and Carter 2008; Paarlberg and Lee 1998) in two ways. First, this prior work does not consider non -trade inputs to the bio logical pollution production function . Failure to provide incentives for non -trade inputs (e.g., because these inputs are unobservable) means their levels will be set to zero at the private optimum (i.e., 47 ai = 0), and hence biological pollution control can be at most second -best. It is straightforward to show the second - i zi are the same as those in (2.15) and (2.16), except ai is evaluated at zero. It is unclear whether the second -best incentives are larger or smaller than their first -best count erparts, but these instruments will generally distort trade flows and private incentives for surveillance relative to the first -best outcome. Second, prior work finds uniform incentives to be efficient due to a focus on a single, representative importer . We show efficiency requires the instruments be applied at importer -specific rates since each choice has an importer -specific marginal contribution to ei, which in turn has an importer -specific marginal contribution to D. In the present context, uniform incentives will result in further efficiency losses as importers with a small effect on expected social damages will receive incentives that are too large, and vice versa (Shortle, Abler and Horan 1998) . The nonpoint source pollution literature indicates gai ns can be made from better targeting the design of pollution control incentives (e.g., Babcock et al. 1997; Westra, Easter and Olson 2002) ; the same is likely to be true for biological pollution. 2.3 Trade and Infection Risks under Bilateral Spillovers We now extend the model to consider the case in which disease spillovers are bilateral in the sense that an importing producer™s herd is also at risk of becoming infected via spread from other importers. We show that alternative, command -and -control policies may be needed alongside of economic incentives to achieve efficiency. Suppose importer i™s herd can be infected via spread from any of the N Œ 1 neighboring importers™ herds, should they become infected. Importer i can protect his own herd from this 48 spread of infection by investing in biosecurity effort bi [0, 1].26 We can then write the probability importer i™s herd becomes infected from any of the N Œ 1 neighboring importers as (2.18) ijjjjjiSiiiOiazxebPbP),,(11);(e, where eŒi = (e1, ... , eiŒ1, ei+1, ... , eN) and iSibP is the probability the disease becomes established in i™s herd, conditional on spread from at least one of his neighbors. We assume 0/iiSibbP, so OiP is decreasing in each argument. Importer i™s expected disease losses are (2.19) iOiiiIiiiiiiCiIiiiiCiiiiCiIiPXPXxXPPXPXxPP,)1()1(1)1( i again represents importer i™s expected losses stemming from his own choices. The term iŠwhich does not arise with unilateral sp illovers Šis importer i™s expected losses resulting from others™ choices. The first term in i is the importer i™s expected losses given that he imports an infected animal and identifies it via inspection, but is infected via spread anyway. The second term in i is the importer™s expected losses given that he does not import an infected animal, but is infected via spread anyway. We continue to assume importers can spread the disease to other, non -importing producers, causing expected damage D(e) that are in addition to the damages to neighboring i terms). Taken together, importer i™s expected profits can be written (2.20) );,,(),(),,,();,,,(iiiiiiiiiiiiiiiiiiiibzxzxbazxcBbazxee, 26 We assume the actions that comprise bi are distinct from the actions that comprise ai, which only protect neighboring herds. 49 where ci() takes the same form as in (2.4) except that it now also includes biosecurity costs, which we denote as gi(bi gibi > 0. We can simplify the remaining analysis by rewriting (2.20) as a restricted profit function in bi and ei: );,(iiiRibee (see Appendix B for derivation). This means ei will be treated as a choice in what follows, which is common in pollution problems (e.g., Tietenberg 1985) . 2.3.1 The Efficient and Privately -Optimal Outcomes under Bilateral Externalities The inefficiency ari sing from biological pollution under bilateral spillovers can again be seen by comparing the efficient and privately -optimal outcomes. Importer i™s private problem is to maximize his expected profits, Ri. This problem is solved as a Nash -Cournot game, i.e., bi and ei are chosen to maximize expected profits, taking the externalities generated by other importers ( eŒi) as given. Formally, importer i™s problem is );,(max ,iiiRiebebiie. Assume for now that a unique Nash equilibrium arises as an interior solution, with the privately -optimal choices, 0ib and 0ie, solving 27 (2.21) 0iRib (2.22) .0iRie Conditions (2.21) and (2.22) imply the familiar result that marginal expected net benefits from self -protection and emitting biological pollution, respectively, are zero. 27 Existence of at least one Nash equilibrium follows from a straightforward application of Brouwer™s Fixed Point Theorem (Mas -Colell et al. 1995) . 50 Consider next the social planner™s problem, which is to choose bi and ei i to maxi mize aggregate importer profits less damages to non -importers. Formally, the social planner™s problem is )();,(max ,eeDebiiiiRiiebii. Assuming an interior solution, the first -best choices, *ib and *ie, solve the necessary conditions (2.23) 0iRib (2.24) ,0iDeeijiRjiRi where the term ijiRje/ reflects expected marginal losses to neighboring importers due to spread from importer i™s herd. Comparing (2.23) and (2.24) with (2.21) and (2.22) shows that Šas with the externality to non -importers Šthe externality to importers arises from ei. In contrast, importers will freely choose 0ib at the first -best level whenever policies are implemented to ensure *0iiee. The reason is that bi does not affect the production of biological pollution, and hence no spillovers arise from its under -provision. However, 0ib will be inefficiently large when *0iiee. 2.3.2 Incentives for Efficient Disease Control with Bilateral Externalities Consider how economic incentives might be designed and implemented to eliminate the bilateral externalities described above. Efficiency can be attained through either risk - or input -based incentives, as in the unilatera l case. For conciseness, we derive only the optimal risk -based tax; deriving the optimal input -based incentives provides little additional insight beyond that discussed for the unilateral case. 51 Suppose a tax, ti, is levied on importer i if disease is tra nsmitted from his farm, so that the risk -based tax payment for importer i is tiei. Importer i™s problem under this risk -based tax is iiiiiRiebetebiie;,max ,, with necessary conditions (2.25) 0iRib (2.26) ,0iiRite Comparing (2.26) to (2.24) shows that efficiency can be attained by setting (2.27) iebijijiiiDet****, i, which is similar to the tax rate identified in (2.10) except that now the optimal tax rate incentivizes importers i to internalize the external m arginal damages from his choices to neighboring importers and non -importers. 2.3.3 Efficient Incentives in the Presence of Multiple Equilibria The foregoing assumes the Nash equilibrium of the importer™s problem is unique. This need not be the case in general, as prior work suggests trade -related choices (e.g., import and surveillance decisions) may result in multiple Nash equilibria (Hennessy 2008; Reeling and Horan 2015) . Multiple Nash equilibria arise from non -convexities brought on by strategic complementari ties between producers™ self -protection choices (i.e., an importer™s marginal incentives for self -protection increase with his neighbors™ self -protection). W e show in Appendix B that strategic complementarities between ei and eŒi may exist both before and after implementation of the efficient risk -based tax. 52 The existence of multiple equilibria complicates incentive design due to the potential for coordination failure. Specifically, which equilibrium outcome arises depends on importers™ expectations regar ding their neighbors™ choices. If enough importers believe their neighbors will make choices consistent with the Pareto dominant outcome, which is the efficient equilibrium in the case of an efficient tax (Milgrom and Roberts 1990; Vives 2005) , then the sy stem will converge to this outcome. Coordination failure arises when an insufficient number of importers™ expect neighbors will make efficient choices, so that the system instead equilibrates at a Pareto -dominated equilibrium exhibiting lower risk mitigati on levels. The problem of coordination failure is formalized as follows. Suppose there are multiple roots to the importer™s necessary conditions in ( 2.25) and ( 2.26), resulting in an efficient, Pareto -dominant Nash equilibrium, denoted )),(),...,,((),(***1*1**NNebebeb, and K > 1 Pareto -dominated Nash equilibria, ),()),(),...,,((),(**###1#1##ebebNkNkkkkkebeb for k {1,–, K}. Prices alone cannot generally ensure the system converges to the efficient equilibrium in the presence of multiple equilibria and coordination failure risks (Dasgupt a and Mäler 2003) , which means importers may instead arrive at one of the Pareto -dominated outcomes. Following Anderson and Francois (1997) , the efficient outcome can be ensured through a combination of the efficient risk -based tax in ( 2.27) and command -and -control regulations on ei. Specifically, suppose a regulatory authority can set and enforce a maximum allowable ei, say, i. > *ie.28 Efficiency can then be attained by setting i sufficiently close to *ie , with #ikiee for ##kiikee = 1, ... , K. Importers will then expect their neighbors to make efficient decisions. In 28 In practice, these regulations may be achieved through restrictions on the input s to ei. Specifically, suppose that the regulator can set and enforce regulations designating a minimum share of animals, xi, that can be purchased from the risk -free region. Suppose also the regulator can provide supplemental surveillance, zi, of all animals from the risky region whenever xi i = ei(xi, zi, ai), then these input -based regulations have the same effect as placing a maximum bound on ei. 53 other words, this regulation eliminates importers™ expectations that can lead to coo rdination failure: all Pareto -dominated equilibria are precluded such that the only remaining equilibrium coincides with the efficient outcome. Note also that the regulation i is non -binding in the sense that importers will have private incentives to red uce ei beyond this level. Our results contrast with prior literature in managing trade -related biological pollution spillovers (e.g., McAusland and Costello 2004; Mérel and Carter 2008; Paarlberg and Lee 1998) by showing that incentives alone may not achieve efficiency in the presence of bilateral spillovers. Alternative policy interventions (e.g., regulations) may be needed in addition to price -based incentives to achieve efficiency. This finding also contrast s with the nonpoint source pollution literature (e.g., Griffin and Bromley 1982) , which finds regulations or incentives can be used interchangeably to achieve efficiency. The difference here arises from the fact that importers™ disease risks depend on choi ces made by neighboring importers, leading to strategic interactions that may produce non -convexities and the potential for multiple equilibria; in contrast, the nonpoint literature typically does not assume others™ pollution enters the emitter™s payoff fu nction. 2.4 Discussion and Conclusion Trade -related disease externalities impose considerable economic costs on society, yet the drivers of these externalities are not well understood. Prior work focuses primarily on trade -related decisions (e.g., import and s urveillance choices), which conflates the externality problem with a trade problem. However, other decisions are also relevant. In particular, individual importers™ private risk abatement choices also contribute to disease externalities. Prior work has not considered these choices. 54 We reframe the issue of trade and the spread of an infectious disease as a biological pollution problem, akin to a classic nonpoint source pollution problem. In this context, we show spillovers depend both on trade decisions and on importers™ behaviors and choices once infected animals enter the importing region . Disease spillovers are analogous to those arising from more traditional nonpoint source pollutants, suggesting the spread of disease via animal movements is better viewe d as a pollution problem rather than a trade problem. This finding casts importers as the perpetrators of biological pollution externalities, rather than the victims, suggesting a greater role for disease control and prevention efforts within an importing region. This contrasts with prior work in which the onus is on keeping infected animals and goods from entering the region in the first place (McAusland and Costello 2004; Mérel and Carter 2008; Paarlberg and Lee 1998). We use our approach to derive first -best incentives in the face of unilateral spillovers comprising importer -specific incentives on either ( i) the probability disease spreads from his or her farm or ( ii) on all choices that contribute to the probability of spread. The optimal incentives are smaller the greater are the importer™s private incentives for risk mitigation; hence, policy interventions will have greater marginal benefits when individuals have fewer incentives for private behavior. This result does not arise in prior work that focus es primarily on the choices of a single, representative importer. We also show that multiple equilibria Šand the risk of coordination failure Šmay arise when spillovers are bilateral. Additional command -and -control policies (e.g., regulations) may be needed alongside incentives to avoid coordination failure and suboptimal disease outcomes. Prior work in both the trade and nonpoint source pollution literature does not consider bilateral spillovers, and so policy recommendations do not reflect the potential for coordination failure. 55 Our model of biological pollution is deliberately simple; disease spillovers are viewed as static, with non -spatial interactions between at -risk individuals. This approach is consistent with Šand allows us to glean insights from Šthe t raditional nonpoint source literature (e.g., Griffin and Bromley 1982) . Our results are likely to be relevant in more complex settings, including those where disease spillovers are dynamic. To see this, note that in a dynamic setting the importation of an infected animal and disease transmission from the importer™s herd (the actual externality) occur sequentially in distinct time periods. This might suggest policy intervention is only required to prevent transmission in the latter period. However, the proba bility of transmission at any given time will depend on the stock of infected animals, which in turn depends on trade choices made in a previous period. Disease spillovers therefore depend on both trade choices and private risk abatement choices (e.g., bio security) in a dynamic setting, as in our static model. Optimal policy instruments are derived here under the implicit assumption that non -uniform instruments can be imposed on importers. In reality, significant information asymmetries due to unobservable private disease mitigation choices may preclude the use of non-uniform incentives. International trade agreements that prohibit discriminatory policy instruments are likely to further impede the use of importer -specific tariffs. Biological pollution contr ol will be at most second -best in the face of informational and policy constraints. We leave the extension of second -best instrument design for future work. 56 CHAPTER 3 INTEGRATING POLLUTION MARKETS AC ROSS ENVIRONMENTAL MEDIA: THE CASE OF REACTIVE NITROGEN 3.1 Int roduction An intriguing feature of some environmental markets is the ability to trade across pollutants. This is particularly common in greenhouse gas (GHG) emission markets like California™s Cap -and -Trade Program (CA AB 32) . Under these fimultipollutant markets,fl a regulated polluter that abates emissions of nitrous oxide (N 2O), for example, can sell its corresponding permits to another regulated polluter needing to offset its CO 2 emissions subject to an inter -pollutant excha nge rate. Provisions for multipollutant trading have also existed fo r local air quality markets (e.g., California™s South Coast Air Quality Management District allows new emissions sources to offset their emissions by paying for reductions in emissions of different air pollutants on a case -by-case basis; see Rule 1309 (h) ), although these provisions have rarely, if ever, been exploited (Montero 2001) . Prior work focuses largely on identifying optimal policy parameters for inter -pollutant exchange in greenh ouse gas or air quality trading programs (e.g., Reilly and Richards 1993; Muller and Mendelsohn 2009) or water quality programs (e.g., Hung and Shaw 2005) . Other work has demonstrated that trading across pollutants or other environmental services can incre ase social welfare relative to having separate markets. For example, Montero (2001) uses a simple but general theoretical model to demonstrate that integrated markets can be welfare -enhancing relative to separate markets when abatement cost curves are rela tively steeper than marginal benefit curves. Horan , Shortle , and Shogren (2015) consider the case of a fishery where productivity is degraded by both overfishing and pollution. They show that a market in which 57 fishermen and polluters trade harvest rights for the right to pollute and vice versa can achieve a first -best allocation of fishing and pollution abatement . Consideration of multipollutant trading has Šboth in theory and practice Šbeen restricted to different pollutants that all affect the same resource o r environmental medium. 29 However, recent advances in our understanding of pollutant generation, transport, and fate ha ve increased our knowledge of the linkages between environmental pollutants across environmental media . A striking example involves reacti ve nitrogen (Nr) , a chemical form of nitrogen that is readily converted to different species (i.e., different forms of Nr, such as NOX, N2O, NO 3). A single molecule of Nr released to the environment from disparate activities like agricultural fertilizer use, fossil fuel combustion, or wastewater treatment can travel through multiple environmental media (Galloway et al. 2003) , causing environmental and economic damage in each medium it passes through (Birch et al. 2011) . The pu rpose of this essay is to examine multi -pollutant trading when the pollutants impact different media and when one source (agriculture) is a significant contribut or to the various pollutants and hence affects multiple media. Several features of Nr pollution suggest there may be gains from trading across pollutants (i.e., the different Nr species) and environmental media via integrated markets. First, fluxes of different Nr species like N 2O (a potent GHG with 300 times the global warming potential of CO2) and NO3 (a water pollutant that contributes to eutrophication and hypoxia) are generated as complements from agricultural production choices like nitrogen fertilizer application. Another justification arises from the vast heterogeneity in the costs and benefi ts of abatement of different Nr species. For example, Birch et al. (2011) find the economic damage per unit of Nr emitted in the Chesapeake Bay Watershed can vary by a factor of 100 across the various Nr species , and the 29 Montero™s (2001) model is highly general, but was developed in the context of air pollution trading. 58 marginal abatement costs can vary b y a factor of 20 across Nr sources. This heterogeneity in the costs and benefits of Nr abatement suggests integrated markets may decrease the cost of pollution control by offering regulated polluters additional flexibility in meeting their abatement target s. A final justification for integrated markets lies in the highly mobile nature of Nr, which suggests it may not matter from a biogeochemical perspective in which environmental medium Nr abatement occurs, only that abatement occurs (Galloway et al. 2003) .30 Whether Šand to what degree Šintegrated markets are an improvement over separate markets is an empirical question. Prior work on the design of pollution control policies involving multiple pollutants suggests market design choices , especially trade exchang e ratios defining how pollutants substitute for one another and the level of emissions caps for each pollutant , will determine the gains from a more integrated approach to pollutant tr ading (Woodward 2011; Muller 20 12). However , emissions caps are often not chosen independently of permit trading programs involving agriculture Šlet alone programs that might allow trade across pollutants. For instance, agricultural emissions are generally not regulated unless farmers voluntarily participate in a program that p ays for abatement. In contrast, point sources are generally regulated, such as through National Permit Discharge Elimination System (NPDES) permits for water emissions. Existing and proposed permit markets involving agriculture are designed to allow poin t sources to achieve their permit levels by purchasing credits generated via nonpoint source abatement. Within this setting, it is unlikely that caps would be chosen efficiently absent coordination of the 30 We do not model this feature of Nr emissions for simplicity and because empirical estimates of Nr interconversion rates between species are not well -known, but note only that abatement of Nr fluxes in one environmental medium will ultimately influence fluxes of Nr to other media. 59 distinct regulatory agencies that may manage diffe rent Nr fluxes to different environmental media. 31 Given sub -optimal p ollutant cap s, integrated markets provide an additional policy parameter that may be used to improve the performance of environmental management: an interpollutant exchange rate, or the rate at which abatement of one pollutant can be substituted for abatement of another pollutant. Prior work has attempted to estimate efficient interpollutant exchange rates among a single class of pollutants (e.g., greenhouse gases; Reilly and Richards 1993), but no prior work considers rules for exchanging jointly produced pollutants across different environmental media. The purpose of this essay is to investigate whether integrated markets improve the efficiency of Nr control relative to separate pollutan t- and media -specific markets when permit caps are set independently from the trading rules (e.g., the trade ratio). Using an analytical and numerical model of Nr trading in the Susquehanna River Basin (SRB), the largest source of nutrient inputs to the Ch esapeake Bay Watershed in the Mid -Atlantic U.S., we show that integrated markets improve the efficiency of pollution control relative to separate markets when pollution caps are set arbitrarily. Greater economic performance arises under integrated markets from increased flexibility in meeting pollution targets. The following section develops an analytical model of Nr pollution . The next two sections derive the conditions for efficient Nr abatement and shows only integrated markets that allow multi -pollutant trading can be cost -effective in general. We then develop a numerical 31 A piecemeal approach to environmental regulation is common (Yaffee 1997) . In some cases, a single pollutant is managed by different agencies. For example, NO 3 emissions from point sources in the Unite d States are regulated by the USEPA under the Clean Water Act, but NO 3 emissions from nonpoint sources are typically managed via conservation programs run by the USDA. 60 application of interpollutant trading in the SRB. Nume rical results are discussed, and the final section concludes. 3.2 A Model of Nr Pollution Suppose for simplicity there exist three pollut ing sectors , each of which emit s a different species of Nr. 32 The first sector comprises point source emitters of N 2O to the atmosphere and is referred to heuristically as the industrial sector (indexed by I). The second sector comprises point source emitters of NO 3 to aquatic ecosystems and is referred to heuristically as the wastewater treatment sector (indexed by W). The final sector comprises nonpoint source emitters of both N 2O to the atmosphere and NO 3 to aquatic ecosystems and is referred to heuristically as the agricultur al sector (indexed by A). We assume for simplicity that pollution by all sources is determin istic. 33 Point sources choose their pollution levels directly; d enote N 2O emissions by the industrial sector as eI and NO3 loadings (or emissions that are delivered to a water body) by the wastewater treatment sector as rW. In contrast, the agricultural sector generates both N 2O and NO 3 via the use of a vector of polluting input s, z; agricultural emissions and loadings are thus eA = ge(z) and rA = gr(z), respectively . Moving forward, it will be simpler to work with abatement, rather than pollution levels. Define abatement by the industrial sector as aeI = eI0 Œ eI, where the subscript fi0fl denotes initial emissions prior to abatement. Likewise, wastewater treatment abate ment is arW = rW0 Œ rW, 32 We restrict our focus on NO 3 and N 2O for simplicity, but note that the model can easil y be generalized to accommodate additional Nr species, including NO X and NH 3. 33 More realistically, farmers choose a vector of inputs that influences the distribution of their pollution , since emissions and loadings from nonpoint sources are random functions of weather and other factors (Shortle and Dunn 1986) . We assume for now that farm emissions are deterministic to focus attention on the issue of market integration. We consider an ex tended model with uncertainty later. 61 and agricultural emissions and loadings abatement are aeA = eA0 Œ eA and arA = rA0 Œ rA, respectively. The industrial and wastewater treatment sectors can abate their emissions by investing in pollution control technologies . The in dustrial abatement cost function is )(eIIaC, wh ere 0,IICC. The water treatment sector™s abatement cost function is defined analogously as )(rWWaC. The agricultural sector abates their emissions and loadings by altering input use . We specify the agricultural sector™s abatement cost function as (3.1) )()(min),(0zzzrAeAAaaC subject to )( ),(zzrrAeeAgaga, with CAasA2CA aeA)2 > 0 s {e, r}. We follow Woodward (2011) and assume complementarity in abatement, i.e., 2CAaeAarA < 0. Finally, suppose pollution abatement prevents economic damage that var ies with the aggregate abatement of each pollutant. Let Ee = eI0 Œ aeI + eA0 Œ aeA and Er = rW0 Œ arW + rA0 Œ arA represent the ambient concentration of N 2O and NO 3, respectively. 34 Aggregate abatement of N2O and NO 3 is then ae = Ee0 Œ Ee and ar = Er0 Œ Er, where the subscript fi0fl denotes unregulated pollution (i.e., with zero abatement) . The benefit from abating each species Šexpressed as avoided economic d amage Šis then denoted Be(ae) = De(Ee0) Œ De(Ee) and Br(ar) = Dr(Er0) Œ Dr(Er), where D represents economic damages from ambient pollution . 34 We assume uniform mixing of pollutants for simplicity. Note also that, by defining r as loadings we already account for spatial heterogeneity in NO 3 delivery across sources. 62 3.3 First -Best Nr Control We first consider the efficient , or first -best , allocation of pollution control effort as a baseline for comparison with market outcomes. The efficient outcome is defined as an allocation of pollution control effort that maximizes social net benefits (3. 2) ),()()(max ,,,rAeAArWWeIIrreeaaaaaaCaCaCaBaBVrWeArWeI. Assuming an interior solution, the first -order conditions (FOCs) for problem ( 3.2) are (3.3) eIIBCeV0, (3.4) rWWBCrV0, (3.5) eeAAABaCeV0, (3. 6) rrAAABaCrV0 The FOCs (3.3)Œ(3.6) state the familiar result that, at the first -best abatement levels *eIa, *rWa, *eAa, and *rAa, each agent™s marginal abatement cost s equal the marginal benefit s from abatement. Additional insight can be had by manipulating (3. 3)Œ(3. 6) to yield the following modified equi -marginal condition (3. 7) rrAArWeeAAeIBaCBCBaCBC//, so that the effective marginal cost of abatement Šmeasured by the marginal cost normalized by the marginal avoide d damages from abatement Šis equalized across all sources under the efficient outcome. Note that this assumes the damages caused by each pollutant are fungible: the social planner treats avoided damages from emissions abatement as a perfect substitute for 63 avoided damages from loadings abatement. This contrasts with current approaches to environmental regulation which treat abatement of distinct pollutants (e.g., NO 2 and NO 3) as non-substitutable, even when those pollutants arise from the same source. 3.4 Nr Abat ement under Market Trading Schemes We now consider the outcome under two alternative Nr trading schemes. The first considers the outcome under separate markets for each pollutant and reflects current approaches to Nr management. We then compare this with t he outcome under an integrated market that allows trading across environmental media via multipollutant markets. 3.4.1 Separate NO 3 and N 2O Markets when Caps are Chosen Efficiently Assume N2O emissions and NO 3 loadings are regulated separately by different regulatory agencies that coordinate in setting permit levels, or emissions caps, for each Nr species to promote efficiency. Denote e missions permits by êI, with each permit representing the right to pollute a single unit of N 2O. Likewise, denote loadings permits that cap NO 3 by W. Let the initial emissions permits allocated to the industrial sector be êI0 and the initial loadings permits allocated to the wastewater treatment sector be W0. In contrast, the agricultural sector is endowed with a fixed quantity of offsets corresponding to its unregulated emissions and loadings: êA0 = eA(z0) and A0 = rA(z0), where z0 is their input choice vector prior to abatement . Point sources are regulated such that they must hold one permit or offset for every unit of emissions or loadings they emit .35 In contrast, farmers have an implicit right to pollute, but can voluntarily supply 35 In practice, trades between point and non point sources would be subject to a species - s. Point sources of s s permits from nonpoint sources to offset one unit of their own emissions. The trading ratio reflects imperfect substitutability between point and nonp oint source abatement due to stochastic nonpoint source emissions (Malik et al. 1993). We assume perfect substitutability here to focus our attention on substitution across pollutants, but note that our model could be generalized to account for imperfect s ubstitution. 64 offsets for each pollutant to point source polluters . The price of emissions and loadings permits are denoted pe and pr, respectively . The industrial sector chooses aeI minimize abatement costs, less the cost of purchasing offsets from the agricultural sector . Prior work (e.g., Horan and Shortle 2005) has shown that this problem can be formulated as (3.8) 00‹minIeIIeeIIaeaepaCeI Substi tuting the constraint into the objective function and assuming an interior condition, the FOC for problem ( 3.8) implies (3.9) eIpC. Condition ( 3.9) simply states that, at the optimum, the marginal cost of abatement by the industrial sector (left -hand side [LHS] term) equals the marginal cost of an agricultural offset . The wastewater treatment sector™s problem is analogous to the industrial sector™s ; the cost -minimizing level of abatement satisf ies (3. 10) rWpC. The interpretation of (3. 10) is analogous to that of (3. 9). The agricultural sector ™s problem is to choose aeA and arA to minimize abatement cost less revenue from offset sales : (3.11) .,min,rAreAerAeAAaaapapaaCrAeA The FOC s imply (3.12) eeAApaC, (3. 13) rrAApaC, 65 which state that the marginal abatement cost ( LHS term s in (3. 12) and (3. 13)) equal s the marginal permit revenues for each species ( right -hand side [RHS] term s) at the optimal abatement levels . Conditions (3. 9), (3. 10), (3. 12) and (3. 13), along with the following market -clearing conditions that require post -trade emissions and loadings equal the caps on these pollutants: )()(‹‹0000eAAeIIAIaeaeee and )()(‹‹0000rAArWWAWararrr, form the equilibrium system under separate markets. The separate market equilibrium comprises a vector of prices, peS and prS and pollution abatement allocations, SeIa, SeAa, SrWa, and SrAa, that satisfy this system (where the superscript fi S fl denotes the separate market outcome) . Comparing conditions (3. 9), (3. 10), (3. 12) and (3. 13) with the efficient FOCs in (3. 3)Œ(3. 6) shows the first -best abatement arises when the pollutant caps for emissions and loadings are chosen such that **,|eAeIaaeeBp and **,|rArWaarrBp.36 Specifically, let )()(*0*0*eAAeIIeaeaeE and )()(*0*0*rAArWWrararE be the efficient levels of total emissions and loadings, respectively. Assuming agricultural sources are not initially regulated so that they have implicit initial permit caps of 0Ae and 0Ar, then the efficient permit caps for the emissions and loadings by the industrial and wastewater treatment sectors, respectively, are 0**0‹AeeeEe and 0**0‹ArWrEr. 36 We assume that unregulated emissions by the agricultural sector are sufficiently small that the caps on point sources can, in fact, be chosen such that the market operates efficiently. Generally, the larger are unregulated agricultural secto r emissions, the smaller must be the initial permit allocations to the industrial and wastewater treatment sectors to attain efficiency. 66 3.4.2 An Integrated Nr Market when Caps are Chosen Efficiently Now consider the case in which the market s for each species are integrated so that regulated emitters of either species are allowed to offset their emissions by purchasing permits or offsets representing either N2O or NO 3. We continue to assume the agency can choose permit caps to promote market efficiency. These caps can be set as in non -integrated markets. The first point that must be addressed in designing an integrated Nr market is the commodity to be traded. Without loss of generality, we choose emissions of N 2O as a numeraire. inter -pollutant exchange rate governing trades between emitters of NO3 and N2O. As before, denote the prices of emissions and loadings as pe and pr. The total pollutant capŠdenominated in units of N 2OŠwill then be êI0 + êA0 + [W0 + A0 . The industrial sector ™s problem is again to minimize the cost of abating emissions . In contrast to the separate -market case above, the constraint on the industrial sector ™s emissions can be satisfied by holding permits or offsets for either emissio ns or loadings , as long as the 2O they emit . The wastewater treatment sector ™s problem is analogous to the industrial sector™s problem. The constraint on this sector™s loadings can b e satisfied by holding permits or offsets for either 3 they emit. Agricultural emissions and loadings are only constrained by the offsets that they sell. Prior wo rk on trades involving a trade ratio (e.g., Horan and Shortle 2005) indicate that the following condition must hold in a market equilibrium, (3. 14) rAAeAAWIreeraCaCCCpppp// so that each source is indifferent between purchasing pe rmits and offsets from any other source . Given relation (3. 14), the industrial sector™s problem reduces to problem (3. 8), with t he FOCs 67 satisfy ing condition ( 3.9) above . Analogously, the wastewater treatment sector™s problem reduces to problem (3. 9), with the FOCs being condition (3.10). The farms™ problem reduces to (3. 11), with the FOCs being condition (3. 13). The FOCs (3. 9), (3. 10), (3. 12), and (3. 13), the inter -pollutant exchange rate given by (3. 14), along with the market clearing condition (3.15) rAArWWeAAeIIAWAIararaeaerree00000000‹‹‹‹, form the equilibrium system under an integrated market . Equilibrium comprises a vector of prices, pe** and pr**, and pollution abatement allocations **eIa, **eAa, **rWa and **rAa , that satisfy this system. C omparing c onditions (3. 9), (3. 10), (3. 12), and (3. 13) with the cost -effective FOCs in (3. 3)Œ(3. 6) shows the first -best abatement arises if the permit cap is chosen so that **,|eAeIaaeeBp as before . The optimal value of pr then follows from (3. 15), implying the optimal inter -pollutant exchange rate is the ratio of marginal damages from atmospheric Nr fluxes to the marginal damages from aquatic Nr fluxes, or (3.1 6) ****,,,*/rArWeAeIaaaareBB. This result contrasts with current multi -pollutant markets, which typically determine the inter -pollutant exchange rate according to their physical or chemical qualities. 37 37 For example, different types of GHGs are traded based on their global warming potential, denominated in units of ficar bon equivalents.fl Trades governed by these types of exchange rates are unlikely to be cost -effective as they ignore the economic characteristics of pol lution that vary across pollutant species (Muller 2012; Schmalensee 1993) . An economically -optimal tradin g ratio accounts for these economic characterist ics, as illustrated in (3.15 ). 68 3.4.3 The Relative Efficiency of an Integrated Market when Caps are Exogenous to the Marke t Both separate and i ntegrated markets are first -best under the optimal caps and exchange rates derived above. However, it is unlikely in practice that markets will be designed efficiently because the caps are not determined as a component of the optimal market design . In this case, gains may arise from trading across pollutants since integration offers an additional policy parameter Šthe inter -pollutant exchange rate Šthat can be optimally chosen to attain second -best pollution abatement .38 We derive the sec ond-best inter -pollutant exchange rate the that maximize s social net benefits from abatement, subject to the polluters™ privately optimal abatement choices in the market equilibrium . market equilibrium, t he polluters™ FOCs (3. 9), (3. 10), (3. 12), and (3. 13) can be implicitly solved for the abatement suppl ies as functions aeI(), arW(), aeA(), and arA(). Identifying the sign of the derivative of each of these supply functions requires evaluating a 5 × 5 Hessian matrix, which is analytically intractable. Prior work (Horan and Shortle 2015) indicates the sign of each derivative depends on two effects. A cost effect stems from the final two equalities in equilibrium relation (3.14) implies a larger relative marginal abatement cost, and hence more abatement, in the emissions sector, other things equal (and vice versa for the loadings sector). A quan tity effect arises from condition (3. 16 loadings abatement exchanges for fewer permits of the numeraire commodity (emissions), so that more abatement of loadings is required to produce the same number of emissions permits 38 This is a straightforward consequence of Le Châtelier™s Principle, i.e. that auxiliary constraints (e.g., the constraint that pollutants cannot be exchanged under separate ma rkets) reduce responses to parameter changes, and hence the efficiency of a system at equilibrium (Samuelson 1947). 69 (and vice versa for emissions permits being exchanged for loadings permits). The effects suggest eIa > 0 and eAa > 0, as the cost and quantity effects operate in the same direction. I n contrast, the sign of rWa and rAa are ambiguous due to opposing cost and quantity effects . We will consider outcome s where the price effect dominates (so that rWa, rAa < 0) and where the quantity effect dominates (so that rWa, rAa > 0) below. Substituting the abatement supply functions into the social planner™s objective function (3. 2) and maximizing yields the FOC (3. 17) 0rArAAreAeAAerWWraaCBaaCBaCB. Noting that the marginal cost of abatement each pollutant equals the permit or offset price of each pollutant at the second -best outcome, (3. 17) can be simplified: (3. 17 0rArWIreAeIIeaaCBaaCB. -best inter -pollutant exchange rate (3. 18) rAAeAAWIrIaCaCCCBC//** where ][IeCB is the marginal net benefits from abating emissions and ]/[][rArWeAeIaaaa is the marginal rate of substitution between emissions and loadings . ambiguous in sign otherwise . The second and third equalities in (3.1 7) stem from the market equilibrium conditions repp/, eAAIeaCCp/, an d rAAWraCCp/. 70 The second - ** is the same as the first -best ratio w eAAIeaCCB/, in which case (3. 18) implies rAAWraCCB/. However, a single policy tool is unlikely to be able to correct all the externalities to produce the first -best outcome. This means 0 so that eIBC, and hence the second -best trade ratio will yield rAAWraCCB/. Suppose first that eAAIeaCCB/ increases abatement in each sector) > 0 and the denominators of the two middle terms in (3. 18) indicate that rAAWraCCB/. These results suggest that ***rerIBBBC. It would not be efficient to increase ** *. ** would enhance efficienc y in the emissions sector by reducing the IeCB, this would come at a cost stemming from a n increased difference rWBC. Alternatively, i [loadings] sector) < 0 and the denominators of the two middle terms in (3. 18) indicate that rAAWraCCB/ ** *. Suppose ** < *. Here, increasing ** * would enhance efficiency in the emissions IeCB, but this would come at a cost stemming from an increased difference WrCB. ** *. In ** * would reduce efficiency in the emissions sector by increasing IeCB, while at the same time generating benefits stemming from a smaller difference WrCB. ** * ** * may emerge as an equilibrium outcome 71 eAAIeaCCB/ rAAWraCCB/ by (3.1 8). ** * is ambiguous. ** * ** * increases abatement in both sector s. This increas es efficiency in the loading sector by bringing WC closer to rB. However, this gain is offset by a loss in efficiency from additional abatement in the emissions sector, leading to a greater difference eIBC. I ** * ** * will increase efficiency in the emissions sector by decreasing emissions abatement, reducing the difference eIBC. These gains will be offset by a decrease in l oading abatement, increasing the difference WrCB. rAAWraCCB/. ** *. ** * would increase efficiency in the emissions sector by reducing the difference eIBC. However, these gains would be offset by the resulting increase in abatement by the loading sector that would increas e the difference rWBC. It is unlikely that pollution caps are set efficiently under separate markets . Hence, gains may arise from integration. We now turn to a numerical example of a hypothetical Nr trading program to illustrate the gains from integration in a real -world set ting. 3.5 Numerical Model: Inter -pollutant Trading in the Susquehanna River Basin We now illustrate the theory using an illustrative model of inter -pollutant trading in the Pennsylvania portion of the SRB . Nr pollution in the form of greenhouse gas emissions and water quality loadings poses a major ecological and economic threat to environmental quality in the Chesapeake Bay and surrounding airshed (Birch et al. 2011) . We fo cus on the Pennsylvania portion due to data ava ilability and because Pennsylvania accounts for the greatest share of land 72 area and pollution in the SRB. The SRB features numerous point sources of GHGs and water pollut ion . In addition, Ribaudo and Nickerson (2009) identify several subbasins of the Penns ylvania portion of the SRB as promising for the establishment of point -nonpoint water quality trading markets. Hence, the SRB is an economically interesting example for analyzing the potential for integrated markets. We now provide a brief description of t he simulation. Data for the simulation come primarily from secondary sources. Specifically, data describing aggregate loadings, abatement and marginal abatement costs for point and nonpoint water pollution sources in the SRB are taken from the Chesapeake Bay TMDL (Kaufman et al. 2014). We continue to aggregate all point sources of N loadings into the fiwastewater treatmentfl sector, although our data includes loadings from other types of polluters in addition to wastewater treatment plants. Likewise, we aggr egate nonpoint sources of N loadings into the fiagricultural sector.fl Data describing initial GHG emissions for point sources in the SRB are taken from the US EPA™s flight database. As with the wastewater treatment sector, we aggregate all point sources of emissions into a single fiindustrialfl sector, but note that our data includes emissions from power plants and other regulated point sources. Marginal abatement cost data are taken from RGGI (2014). Finally, agricultural GHG emissions and marginal abatement costs from corn production are estimated following Reeling and Gramig (2012). All sectors™ abatement cost functions are assumed to be cubic in abatement. Specifically, let 3)(eIeIIaaC and 3)(rWrWWaaC be the industrial and wastewater t reatment sectors, where ),(rAeAAaaC = rAeArAeAaaaa33)2/()2/( represent the agricultural sources™ abatement cost 73 agricultural sector emissions and loadings abatement . The marginal damages from emissions are assumed to be constant since GHGs are a globally mixed pollutant. Hence, the damage function for emissions take s the form De(Ee Ee loadings is assumed to take the form 2)(rrrEED is calibrated using data from the Chesapeake TMDL (Kaufman et al. 2014). 3.5.1 Sim ulation Results The numerical model of the SRB is used to simulate abatement costs under separate and integrated markets using calibrated and simulated in Mathematica 7.0 (Wolfram, Inc. 2008). Calibrated parameters can be found in Table 3.1. Consider fir st the efficient outcome as a baseline for comparison with second best abatement allocations. The efficient outcome can arise from the social planner™s problem (3.2), or from efficiently designed integrated or separate pollution markets (i.e., where the in itial emissions and loading caps, êI0 and W0, are chosen to satisfy **,|eAeIaaeeBp and **,|rArWaarrBp, given that nonpoint sources have an implicit right to pollute). Table 3.2 shows the gains from optimally managing emissions and loadings can be significant Šsocial net benefits total nearly $141.5 million for the Pennsylvania portion of the SRB. The efficient inter -pollutant exchange rate is small and suggests the marginal benefits (and hence the marginal costs) from abating loading s are more than 430 times those from abating emissions at the first best outcome. Point sources abate the majority of emissions in the efficient allocation. This contrasts with the 74 Table 3. 1 Simulation Parameters Parameter Description Value Source Marginal cost parameter, industrial sector 0.0021 Reeling and Gramig (2012) Marginal cost parameter, wastewater treatment sector 2.98×10Œ14 Kaufman et al. (2014) Marginal cost parameter, agricultural sector 0.61×10 Œ11 Reeling and Gramig (2012) Marginal cost parameter, agricultural sector 2.43×10 Œ4 Reeling and Gramig (2012) Complementarity parameter , agricultural sector 5×10 Œ6 Assumption Marginal damage from emissions ($ /mtCO 2e) 14 Tol (2005) Marginal damage parameter, loadings 0.08 Kaufman et al. 2014 eI0 Initial industrial emissions (million mtCO2 e) 153.5 US EPA FLIGHT (201 3) eA0 Initial agricultural emissions (million mtCO2 e) 5.54 Reeling and Gramig (2012) rW0 Initial wastewater treatment loadings (thousand mtN) 12 Kaufman et al. (2014) rA0 Initial agricultural loadings (tho usand mtN) 28 Kaufman et al. (2014) loadings sector, in which agriculture abates the majority of pollution. The difference arises due to the agricultural sector™s relatively higher costs of abating emissions, but relatively lower cost of abating loadings. We examine two second -best scenarios in whic h the permit caps for each pollutant have been set sub -optimally since these choices are generally made distinctly in each sector, by different agencies, and they typically do not account for agricultural sources which are not initially regulated and there fore have implicit initial permit caps of 0Ae and 0Ar. Specifically, let the permit caps for the emissions and loadings by the industrial and wastewater treatment sectors, respectively, be *00‹‹eeeee and *00‹‹WrWrr s (for s {e, r}) is a parameter s < 1 (> 1) implies an inefficiently (lax) cap on 75 Table 3. 2 Results from Hypo thetical Pollutant Trading Scenarios, SRB a Social net benefits ($ million ) Emissions abatement Loadings abatement Industry Agriculture WWTb Agriculture First -best 141.5 0.0023 12.52 0.88 0.96 2.86 Separate 2nd -best , e = r = 0.9 c Œ136.22 N/A 25.62 1.80 1.17 3.47 Integrated 2nd -best , e = r = 0.9 72.77 0.0002 7.92 0.56 2.11 6.26 Change from integration 208.99 ŠŠ Œ17.70 Œ1.24 0.94 2.79 Separate 2nd -best , e = r = 1.05 98.89 N/A 5.98 0.42 0.86 2.56 Integrated 2nd -best , e = r = 1.05 103.2 0.001 6.50 0.46 0.68 2.02 Change from integration 4.31 ŠŠ 0.52 0.04 Œ0.18 Œ0.54 a All emissions figures are expressed in millions of metric tons of carbon dioxide equivalents (mtCO 2e). All loadings figures are expressed in thousands of metric tons of nitrogen. -unit increase in emissions (in millions of metric tons of C0 2 equivalents). b WWT = Wastewater treatment sector . c Efficient emissions cap = 140.1 million mtCO 2e; efficient loadings cap = 8 200 mtN. 76 pollution from point sources of s. We continue to assume the agricultural sector has an implicit right to pollute such th at s does not affect the initial allocation of agricultural offsets. Consider first the second -best outcome in which e r = 0.9 such that the permit cap s on point source pollution are set exogenously at 90 percent of their efficient levels. Such an outcome may arise if policymakers implement excessive point source controls since they are unable to regulate nonpoint sources. The social net benefits from pollution abatement decrease dramatically under both market institutions. Consider first the outcome u nder separate markets. The inefficiently strict caps result in greater abatement by all sources relative to the efficient outcome. Excessive abatement drives marginal abatement costs above the marginal benefits from abatement for all sectors. Social net be nefits are negative under separate markets and total nearly $280 million less than the first -best level. In contrast, the effects of excessively strict permit cap s are more muted under integration. Social net benefits decline by nearly half relative to th e first -best outcome but are still positive and large, totaling nearly $73 million. The second -best inter - ** is an order of magnitude smaller than the first -best level, meaning that the effective initial allocation of loadings permit s (given by W0** ) actually increases, despite the decrease in the absolute cap level. This encourages trade (and, hence, abatement) by the loadings sources, and reduces abatement by the emissions sources. Consider finally the second -best outcome in which e r = 1.05. (An emissions cap much beyond this level results in initial permit allocations greater than initial industrial emissions .) Such an outcome may arise if policymakers fail to account for the c omplementarity between agricultural emissions and loadings abatement (Feng and Kling 2005). Social net benefits decline by 28 Œ30 percent under both second -best market outcomes. Under separate 77 markets, excessively large initial permit caps lead to reduced a batement in each market, particularly in the emissions market, where total abatement declines by more than 50 percent from first -best levels. 39 The reduction in abatement is much smaller for the loadings sector. Integration again leads to additional gains relative to separate markets , although t he gains are smaller in this case where the caps have been made less stringent . Integration also results in substitution of emissions abatement for loadings abatement. This is in contrast to the case where caps were overly stringent , as would be expected . Table 3. 3 Pollutant Caps under Each Market Setting Emissions (million mt CO 2e) Loadings (thousand mtN) Effective Loadings (million mt CO 2e)a Industry Agriculture WWT Agriculture WWT Agriculture Efficient 140.1 5.54 8.17 28 0.0023 3.48 11.92 Second -best: e = r = 0.9 126.1 5.54 7.36 28 0.0002 37.45 142.46 e = r = 1.05 147.1 5.54 8.58 28 0.0013 6.72 21.92 a the cap is d enominated in emissions units. This measure only applies to integrated markets. 3.6 Discussion and Conclusion Our results suggest the gains from integrati ng separate pollution markets may be substantial. Despite this, and despite our increasing kno wledge of the linkages between pollutants, little research considers the potential for linking environmental markets across pollutants and environmental media , nor do examples of such markets exist in practice. Alternative approaches for holistic environmental management are being pursued. One example is ficredit bundling ,fl i.e., selling a package of credits , each representing a distinct 39 A 50 percent decrease in abatement from a 5 percent increase in the cap may appear excessive. However, note that abatement and emissions a re different measures, with emissions generally being much larger than abatement. Hence, a small change in allowable emissions generates a relatively large change in abatement. 78 environmental service , as a single commodity (Robertson et al. 2014) . Credit bundling is used in curren t ecosystem service markets (e.g., the Willamette Partnership in Oregon; Deal, Cochran and LaRocco 2012) . However, bundling differs from integration in that it treats environmental services as non -separable. Trading of bundled credits therefore requires se llers to search for buyers interested in a particular bundle of environmental services. Costs related to this search may limit trading activity under credit bundling. In contrast, i ntegrated markets treat environmental services as separable (e.g., the agri cultural sector sells emissions offsets separately from loadings offsets, even though abatement of both pollutants may arise from the same action). This may increase the liquidity of credits or offsets and increase trading, while simultaneously accounting for linkages between pollutants. The extent to which integration enhances market participation is left to future research. Several practical issues may influence the success of integrated markets. First, integration assumes damages from pollution are fung ible , meaning the allocation of damages to different stakeholder groups spread out across the landscape does not matter . However, trading across pollutants and environmental media may result in abatement allocations that vary spatially and/or put different populations at risk . This may be problematic when the effects of pollution are highly localized. The distribution of rents from pollution trading will also change with the flow of permit payments across sectors and/or the landscape. The efficiency and fu nctionality of integrated markets will also depend on several factors not considered he re, including transaction costs and uncertainty about the costs and benefits of pollution abatement. Integrated markets may feature lower transaction costs relative to s eparate markets. Different Nr fluxes affect different environmental media whose protection may fall under the purview of different government agencies. Attaining efficiency under separate caps 79 may be a challenge when doing so requires multiple agencies to coordinate the choice of each pollution cap. In contrast, a single cap need be chosen under an integrated market. There may be an advantage to integrated markets in that they reduce the number of policy parameters that need be negotiated (Weitzman 2014) . Our understanding of the linkages between environmental pollutants is increasing. New approaches will improve the efficiency and sustainability of managing these pollutants. This paper is the first to explore the potential for utilizing integrated markets to manage distinct environmental pollutants across environmental media. That said, much about the dynamics of many pollutants and their implications for efficient market design remains unknown. Further disciplinary and interdisciplinary work is needed to c ombine insights from economics and the natural sciences to more effectively manage linked pollutants. 80 APPENDICES 81 Appendix A Supplementary Information for Chapter 1 In this appendix, we describe the calibration of the numerical model and provide a sensitivity analysis of the numerical model. A.1 Numerical Model Calibration manag ement practices of varied effectiveness. We then use this index, along with the costs of the practices, to construct the cost relation c Chi et al. (2002) calculate the costs of ten biosecurity practices that include various combinations of background checks on purchased livestock, the decision to raise livestock on -farm, and vaccination. Since vaccines for FMD are not commonly used and were not used during the UK FMD epidemic, we ignore all practices involving vaccination. The remaining practices are l isted in Table A1. We rank these practices according to relative effectiveness, assigning a value of 0 to the least effective practice (i.e., purchasing cattle at auction without background checks or vaccination) and 4 to the most effective (closing one™s farm off to introduced animals). Next, we assume the correspondence between practice number and takes the form = 1 Œ eŒpractice# , where practice# corresponds to the ranking in Table A1. This nonlinear relationship is based on the assumption that more 1 when practice # = 4, which corresponds with full investment in biosecurity. (2002) to estimate the cost function c 2. Using OLS, we estimated ln( c 0 1 obtained a valu 0 = 7.1 9, p-value 1 = 1.9 ( p-value = 0.00), with R2 = 1 was not significantly different from 2, we used the value of 2 for as in the text. 40 40 practice# , 82 Next, consider the probability of infection for a given farmer (equation 1 .1, main text). We begin by parameterizing PI() = (1 Œ ). First, since full biosecurity involves closing off one™s farm (so that no animals will be introduced), the maximum achievable reduction in PI()| = 0. We therefo . Green et al. (2006) assume one in ten animals purchased from these markets are infected in modeling the initial spread of FMD from animal movements from livestock markets in northern Eng land. This implies PI biosecurity investment, which we define as conducting background checks on all animals s assumption, we set PI()| 0.68 = 0.1 to derive 3125. Table A. 1 Description of Biosecurity Practices Used in Numerical Example Practice # Farm Access Source of Livestock Background Checks Cost ( £) 0.99 4 Closed N/A N/A 2,446 0.97 3 Open Other producer Yes 1,694 0.90 2 Open Other producer No 1,519 0.68 1 Open Dealer/auction Yes 173 0.00 0 Open Dealer/auction No 0 The infectivity of contact between herds is given by k() = Œ represents the per -unit reduction in infectivity per unit of biosecurity investment. Biosecurity is likely to be only marginally effective in reducing the probability that infection spreads between infected and susceptible herds, especial ly in the absence of vaccination. We therefore assume = 0.15. We assume that the probability that a herd gets infected by a neighboring herd under typical biosecurity (practice 1, Table A1) is k = 0.8. This large probability reflects the 83 larg e transmission risks in Cumbria County due to the high farm density in this county and also the relatively high transmissibility of FMD (Schley et al. 2009) . Then, given our value of v = 0.15, we use the relation k()| = Œ 0.150.68) = 0.8 to s olve for . The number of neighboring producers N is estimated from Brennan et al. (2008) . The authors use network and cluster analysis to identify connections among farms in northwestern England. The authors estimate that indirect contacts occur bet ween 50 of the 56 farms in the 10 km × 10 km area they cover in their analysis. This does not account for connections to farms outside the area of study, so the pool of farms over which these contacts occur may be somewhat larger. We set N = 80 to account for these additional contacts. The effective number of neighboring producers depends on the extent to which the representative producer can protect himself from direct and indirect contacts with these herds. Full biosecurity is not perfectly effective in r educing contact between herds due to aerosol transmission of the FMD virus. Ferguson et al. (2001) utilize another kernel density to approximate the probability that aerial transmission occurs from an infected premises within a given distance. Beyond 2 km, the risk of aerial transmission is negligible. Combining this transmission kernel with herd densi ty data for Cumbria County (Defra 2011) , it was estimated that an average of five herds will be located within a 2 km radius of a given farm, and thus a representative producer could be exposed to aerosol transmission from five neighboring producers, shoul d they become infected. Recalling that the effective number of neighboring producers is N Œ 1)(1 Œ )(1 Œ ), we set Finally, we calculate the representative producer™s co be the value of the herd (assuming all animals within an infected herd are culled) and idling 84 factors of production. There were 6.124 million animals slaughtered in the UK during the outbreak, at a total cost to produce rs of £1.155 billion. This value results in a cost per animal slaughtered of £188.6. We multiply this value by 140, which is the average herd size in Cumbria £26,404. A.2 Sensitivity Analysis We now present the results of a sensitivity analysis (see Table A2) that examines how sensitive the our results are to changes in the model™s parameters. The first column in Table A2 indicates the parameter being considered, as well as the baseli ne value of the indicated parameter. The remaining columns indicate the scenario being considered. We consider three scenarios: (a) the baseline model with no policy, (b) the behaviorally -independent indemnity of (c) the behaviorally -depen dent indemnity of associated with Figure 1.3, not the first -best indemnity). Recall that scenarios (a) and (b) each yield multiple SNE, whereas scenario (c) yielded a unique SNE. Our sensitivity analysis explor es the robustness of these qualitative results. Specifically, for scenarios (a) and (b), we identify the minimum required percentage change in the indicated parameter value, ceteris paribus , that will yield a unique SNE. The opposite is performed for scena rio (c): we identify the minimum required percentage change in the indicated parameter value, ceteris paribus , that will yield multiple SNE. For scenarios (a) and (b), a unique SNE arises when the parameters are changed to sufficiently increase individua ls™ relative control over their spread risks (the opposite arises for our analysis of scenario (c), in which we are trying to produce multiple SNEs). This means N tion 85 wer economic losses means a unique equilibria at a Overall, the results in Table A2 indicate the qualitative results of the main article are insensitive to changes in a single parameter . Qualitative changes are only possible for six of the eight parameters in scenarios (a) and (b), and for only three of the seven variables in scenario (c). Where a qualitative change is possible, the required percentage change in the indicated parameter is substantial. Table A. 2 Sensitivity Analysis Results Parameter a Baseline (a)b Behaviorally -independent indemnity (b) Behaviorally -dependent indemnity (c) -68% -71% -39% -97% -74% Š -60% -73% Š Š Š Š Š Š Š +907% +222% +292% N (80) -70% -78% Š -89% -69% -75% a The initial value of the parameter is in parentheses. b An entry of ŒŒ indicates that multiple equilibria arise for all possible parameter values. 86 Appendix B Supplementary Information for Chapter 2 B.1 Proof that 0**iiNDi vvv There are two cases to consider. First, suppose iiiivveD/)/( v. The proof proceeds in two steps. The first establishes that 0iv < *NDi v < *iv. Consider the first inequality. We know from equations (2.4) and (2.7) that 0iv and *NDi v solve, respectively, (B1) bzxvvcviiii,,, (B2) bzxvvcveDiiii,,. The LHS in ( B2) is greater than the LHS in (B1) by assumption, implying 0iv < *NDi v by convexity of ci. Likewise, from equation (2.6) we know *iv solves (B3) bzxvvcveDviiiiii,,. The LHS in (B3) is greater than the LHS in (B2), implying *NDi v < *iv by convexity of ci. Hence, 0iv < *NDi v < *iv. The second step establishes that *NDi v > *iv Œ 0iv. By convexity of D( i(), and c(), we know 0*0iiivvv (see Figure 2.1, main text). Further, 0*iNDi vv, implying *NDi v > *iv Œ 0iv. For the second case, suppose iiiivveD/)/(, implying 0iv *NDi v (see Figure B1). Importer i™s private marginal costs with and without risks are given by MCi civi + ivi and MCNDi civi, respectively. Differentiating with respect to vi, 87 iNDi iiiiiiiivMCvcvcvvMC222222 i as shown in Figure B1. It must be the case that *0*NDi iivvv by convexity of marginal damages. Figure B. 1 Comparison of privately - and socially -optimal risk mitigation when *0NDi ivv B.2 The Restricted Profit Function and its Properties In this section we derive the restricted profit function );,(iiiRibee and demonstrate that it may be non -convex i n the biological pollution externalities from others, eŒi. The restricted profit function )(Ri is the solution to the problem (B4) );,,,(max ,,iiiiiiazxbazxiiie subject to ),,(iiiiiazxee. The associated Lagrangian is (B5) )],,([);,,,(iiiiiiiiiiiiiazxeebazxLe, 88 i is the Lagrange multiplier. The FOCs from Li can be solved for the response functions );,(iiiibexe, );,(iiiibeze, );,(iiiibeae, and );,(iiiibee. Substituting these responses into the objective function in (B4) yields (B6) );,();),;,(),;,(),;,((iiiRiiiiiiiiiiiiiiiibebbeabezbexeeeee. We now show the after -tax problem, iiRiet)(, may exhibit strategic complementarities Šand hence non -convexities Šresulting in the possibili ty of multiple Nash equilibria. Specifically, strategic complementarities require 0/)(2jiiiRievet, v {e, b} and 0/)(2iiiiRiebet; Vives (2005) ). Relations (B5) and (B6) imply (B7) jijiijiiiRieeeLeeet22)( 0 (B8) 0)(2222jijiijiijiiiRiebebebLebet (by (2.20)) (B9) iiiiiiiRiiiiiRibbeLbeebet222)( 0. iej ibi involves evaluating the determinant of the 5 × 5 bordered Hessian matrix for the Lagrangean in (B5). This is analytically intractable, so we instead use Mathematica 7.0 (Wolfram Research, Inc. 2008) iej ibi numerically for an illustrative numerical example to show that complementarities may exist. For simplicity, we assume homogenous importers, with the functional forms and parameters used in the example summarized in Table B1. The expressions (B7) and (B9) iej = 0.355, ibi = 0.162), implying strategic complementarity. The results are robust to significant perturbations in parameter values. 89 Table B. 1 Functional Forms and Parameter Values used in Illustrative Example Domain Form/Value in Numerical Example Description Units Functions )(iIixP [0, 1] )1()(iiIixxP Probability of importing infected animal from risky region Unitless )(iCizP [0, 1] iiCizzP)( Probability of detecting infection via surveillance Unitless )(iLiaP [0, 1] iiLiaaP1)( Probability of infection leaving importer's herd Unitless )(iSibP [0, 1] )1()(iiSibbP Probability of infection spreading to importer's herd from others Unitless ci(xi, zi, ai, bi) [0, 1] ci(xi, zi, ai, bi) = w(1 Œ xi)X × (1 + hzi2) + vxiX + fa2 + gb2 Costs unrelated to disease $ Parameters N [0, ) 5 Number of neighboring importers Herds [0, 1] 0.9 Probability of importing infected animal if all animals purchased from risky region Unitless [0, 1] 0.5 Probability of detecting infected animal with full surveillance effort Unitless [0, 1] 0.2 % reduction in probability of disease leaving one's herd per unit of abatement effort Unitless [0, 1] 0.9 Probability of disease spreading to one's herd in the absence of biosecurity Unitless [0, 1] 0.9 % reduction in probability of disease spreading to one's herd per unit of biosecurity Unitless X 1 Feedlot capacity Animals 0.6 Losses from culling infected animals before introduction to herd $/animal 0.8 Losses from culling infected animals after introduction to herd $/animal w 0.2 Cost of animals from risky source $/animal 90 Table B1 (con t™d) Domain Form/Value in Numerical Example Description Units v 0.5 Cost of animals from risk -free source $/animal h 1 Surveillance cost parameter Unitless f 1 Abatement cost parameter Unitless g 1 Biosecurity cost parameter Unitless 91 REFERENCES 92 REFERENCE S Acocella, N., G. 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