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Three essays on the management of nonrenewable resources Chapple, Clive 1998

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THREE ESSAYS ON THE MANAGEMENT OF NONRENEWABLE RESOURCES by CLIVE CHAPPLE B.A.Sc. (Honours), University of Surrey, 1986 M.A.Sc., University of British Columbia, 1990 A thesis submitted in partial fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Commerce and Business Administration) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1998 © Clive Chappie, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Cs&A* e= J ^ ^ " " /ipuC> &i/S//OI?l$> /tXt±A"//J. The University of British Columbia Vancouver, Canada Date / o Si>^n lk~ 7f-DE-6 (2/88) The University of British Columbia Abstract THREE ESSAYS ON THE MANAGEMENT OF NONRENEWABLE RESOURCES by Clive Chappie This thesis comprises three essays on the management of nonrenewable resources. Pollution is often associated with the use of nonrenewable resources. Indeed, many of today's most pressing environmental problems are caused by these types of activi-ties. Despite the connection between nonrenewable resource use and environmental degradation, the two issues have been, for the most part, analysed separately by economists. The first paper develops a framework to analyse the effects of a pure-flow externality on the competitive allocation of nonrenewable resources. For com-monly-used specifications of consumer preferences, the competitive allocation is found to be fully optimal for pure-flow externalities exhibiting decreasing marginal disutility. Hence, the paper shows that the presence of a negative externality associ-ated with the use or extraction of a nonrenewable resource need not result in inefficiency. The US 1990 Oil Pollution Act is the most significant attempt yet made by a nation to deal with pollution of its territorial waters. It significantly altered the rights and obli-ii gations of tanker owners operating in US waters, effectively introducing unlimited liability and significantly expanding the definition of spill damages. The second pa-per analyses the effect of the Act on major pelagic oil spills occurring world wide. The hypothesis that the Act had a negligible effect on the number of spills occurring in North America's coastal waters is tested empirically. The results indicate that the Act significantly reduced the number of spills occurring in North American coastal waters and has had no discernible effect on spill frequencies elsewhere. There is a keen and growing interest in the properties of vertical relationships gov-erning the pricing and transfer of intermediate goods. The third paper examines an unusual and commercially-important vertical relationship — the price participation system —which is used extensively in the zinc industry. The paper explores the con-jecture that significant demand uncertainty and risk aversion on the part of zinc smelters might explain why the industry uses the price participation system rather than a more conventional contractual arrangement. The results indicate that these factors do go part way toward explaining why the industry uses the price participa-tion system. iii TABLE OF CONTENTS ABSTRACT II TABLE OF CONTENTS IV LIST OF TABLES VI LIST OF FIGURES Vll ACKNOWLEDGEMENTS VIII INTRODUCTION 1 CHAPTER 1: NONRENEWABLE RESOURCES, EXTERNALITIES, AND PREFERENCES 7 1.1 INTRODUCTION '. : 7 1.2 THE MODEL 13 1.2.1 Competitive Equilibrium 18 1.2.2 Optimal Planning Solution 31 1.3 DISCUSSION 34 1.4 CONCLUSION .' 39 CHAPTER 2: THE 1990 OIL POLLUTION ACT: CONSEQUENCES FOR THE ENVIRONMENT 41 2.1 INTRODUCTION 41 2.2 REVIEW OF THE LITERATURE 49 2.3 ECONOMETRIC SPECIFICATION 50 2.4 DATA 54 2.4.1 Data Sources 55 2.4.2 Construction of Variables 60 2.4.3 Descriptive Statistics 63 iv 2.5 EMPIRICAL RESULTS 65 2.5.1 The Poisson Model 65 2.5.2 Overdispersion and Alternative Model Specifications 71 2.5.3 Regional Heterogeneity 73 2.6 DISCUSSION AND CONCLUSIONS 75 CHAPTER 3: THE ZINC INDUSTRY, PRICE PARTICIPATION, AND PROFITABILITY 77 3.1 INTRODUCTION 77 3.2 THE ZINC INDUSTRY AND PRICE PARTICIPATION 82 3.3 THE MODEL 87 3.3.1 Production When Firms Are Risk Neutral 94 3.3.2 Production When Firms Are Risk Averse 96 3.3.3 Firm, Sector, and Industry Profitability 97 3.4 MODEL CALIBRATION..... 98 3.5 SIMULATION RESULTS AND DISCUSSION 101 3.5.1 Price Participation 102 3.5.2 Standard Input Pricing 108 3.5.3 Price Participation versus Standard Input Pricing 114 3.6 CONCLUSION 127 BIBLIOGRAPHY 130 APPENDIX A ENDOGENOUS DETERMINATION OF INTEREST RATE 135 APPENDIX B AMBIGUOUS EFFECT OF INCREASED LIABILITY ON OIL SPILL FREQUENCIES 137 APPENDIX C A MODEL OF STANDARD INPUT PRICING 145 V LIST OF TABLES Table 2-1: Definition of Variables 59 Table 2-2: Descriptive Statistics 63 Table 2-3: Determinants of Regional Oil Spills (Poisson Regressions) 65 Table 2-4: Poisson Regression, Equation 5 70 Table 2-5: Alternative Econometric Specifications 72 Table 3-1: Parameter Values 100 Table 3-2: Simulation Results Under Price Participation 103 Table 3-3: Simulation Results Under Standard Input Pricing I l l Table 3-4: Price Participation vs. Standard Input Pricing 112 Table 3-5: Choke Price Sensitivity Analysis 119 Table 3-6: Demand Function Slope Sensitivity Analysis 120 Table 3-7: Concentrate Mark-up Sensitivity Analysis 124 vi LIST OF FIGURES Figure 1-1: Competitive Extraction Paths 28 Figure 2-1: Geographic Distribution of Major Pelagic Oil Spills, 1980-1994 57 Figure 2-2: Worldwide Major Pelagic Oil Spills, MMS Database, 1980-1994 64 Figure 3-1: Sequence of Events 93 Figure 3-2: Industry Profit vs. Smelter's Arrow-Pratt Coefficient of Absolute Risk Aversion, r 116 Figure 3-3: Mining and Smelting Sector Profits vs. r Under Price Participation and Standard Input Pricing 116 Figure 3-4: Smelter Utility vs. Zinc Choke Price 121 Figure 3-5: Smelter Utility vs. Concentrate Mark-up 126 vii ACKNOWLEDGEMENTS I would like to thank Jim Brander for sharing his time, his intelligence, and his wis-dom with me over the years. I am also in the debt of Keith Head for his friendship and for his many incisive comments. I would also like to thank Richard Arend, Iain Cockburn, and John Ries for their support and companionship. And a special thanks to Jo, with all my heart, for believing in me, for encouraging me, and, most of all, for being mine. viii Introduction This thesis comprises three essays on the management of nonrenewable resources. The essays share several characteristics. Each examines an issue where a potential or actual market failure is of central importance. Each generates some results that may be of in-terest to policymakers. And each relies heavily on the standard neo-classical economic assumptions of far-sighted rationality, purposeful maximising behaviour, and equilib-rium. Despite these similarities, each essay stands on its own as an analysis of a distinct resource management problem. • * The first paper, "Nonrenewable Resources, Externalities, and Preferences," examines how the presence of a production or consumption externality may affect the use of a nonrenewable resource. In particular, it analyses the effect of a pure-flow negative ex-ternality, such as noise, on the competitive pattern of resource extraction or use. The paper focuses on the question of when, if ever, a negative externality generated during the extraction or use of a nonrenewable resource does not result in an inefficient alloca-tion of the resource. The first paper is strictly theoretical. A simple two-sector, two-factor general equilib-rium model is developed and used to compare the competitive pattern of resource extraction and use in the presence of a negative externality to the welfare-maximising pattern. Explicit functional forms for utility and production functions are used 1 throughout, allowing closed-form solutions to be obtained for variables of interest. When pollution is non-cumulative and exhibits decreasing marginal disutility, competi-tive conditions result in a welfare-maximising allocation of the nonrenewable resource. This is true for a quite general specification of consumer preferences, which includes Cobb-Douglas and logarithmic preferences as special cases. The paper therefore makes the mteresting point that a pollution externality need not result in excessive extraction or use of the nonrenewable resource with which it is associated. The issues addressed in the first paper are significant because many of today's most pressing environmental problems, such as global warming and acid rain, are the direct result of negative externalities generated during the production or use of nonrenewable resources. Furthermore, as the intensity of resource use grows, and with it the magni-tude and importance of these problems, so too does our need to understand the dynamic relationships that exist between externalities and nonrenewable resources. It is only by untangling the complex interactions between these two phenomena that we may be able to successfully solve problems of this sort. The first paper differentiates itself from the existing literature by focusing on the influ-ence of consumer preference structure, by deriving the competitive welfare-maximising extraction paths from fairly general specifications of consumer preferences and produc-tion functions, and by doing so within a general equilibrium framework. The analysis also provides considerably economic structure than any previous analysis of this par-ticular issue. 2 The second paper, "The Oil Pollution Act of 1990: Consequences for the Environment," analyses the effect of the United States' Oil Pollution Act of 1990 on the geographic dis-tribution and frequency of major pelagic oil spills occurring throughout the world. In particular, the hypothesis that the Act had a negligible effect on the number of spills occurring in North America's coastal waters is tested. The second paper is primarily empirical. A production function describing the number of oil spills occurring in a particular geographic region during a particular period is con-structed. It is initially assumed that the spiU-generating function can be described as a Poisson process. More general discrete dependent-variable model specifications are also considered. Regression model parameters are estimated using panel data on the tempo-ral and geographic distribution of major pelagic oil spills occurring worldwide during a 15 year period sparming the introduction of the Act. The results indicate that the Act has significantly reduced the number of spills associated with tanker traffic to and from the North American continent and has had, as of yet, no discernible effect on spill fre-quencies in other parts of the world. The 1990 Oil Pollution Act is undoubtedly the most radical and significant attempt yet made by a single nation to deal directly with pollution of its territorial waters by vessels transporting oil. The Act made sweeping changes to the rights and obligations of tanker owners whose vessels enter or operate in US waters. Many of these changes have been widely criticised, with the introduction of unlimited hability and expansion of the defi-nition of spill damages generating the most controversy. Indeed, it has been argued that far from reducing the number of spills occurring in US waters, the Act's Hability and damage provisions might do just the reverse. More significantly, several intuitively ap-3 pealing mechanisms that might drive this perverse result have been proposed and some supporting anecdotal evidence has been gathered. The primary contribution of this pa-per is adding some solid empirical evidence to this important debate. This paper makes at least three significant advances over previous empirical work ex-arnirting the effects of the 1990 Oil Pollution Act. First, the paper thus far represents the only empirical analysis of the Oil Pollution Act's effect on spill frequencies to explicitly account for geographic heterogeneity. Second, compared to previous analyses, more use is made of the discrete dependent-variable nature of the data, with a wider variety of count model specifications being considered, and a more sophisticated and extensive set of explanatory variables being employed. Third, the paper examines a longer time series of observations than previously available. The third paper, "The Zinc Industry, Price Participation, and Profitability," explores the properties of a somewhat unusual, commercially important, and little studied vertical relationship that is used widely in the non-ferrous metals industries, the so-called price participation system. The work is largely motivated by Brander's (1996) study of the structure and organisation of the Western world's zinc industry. In his paper, Brander convincingly argues that, when firms possess some degree of market power and com-pete in quantities, price participation contracts contain incentives that tend to exacerbate the standard Cournot problem of overproduction and that also tend to make firms react sub-optimally to any demand shocks that the industry might face. The primary goal of this paper is to try to answer the question: Why do some industries chose to use price participation systems when they seem to embody incentive structures 4 that lead to overproduction and so drive down profits? In particular, this paper ex-plores the conjecture that significant uncertainty about the price of refined zinc, and risk aversion on the part of smelters, might go at least some way toward rationalising the zinc industry's use of the price participation system. The results do seem to indicate that risk aversion on the part of smelters and demand uncertainty can go part way toward explaining why the industry might have chosen to use the price participation system, but these factors are unable by themselves to explain why the zinc industry uses the price participation system, at least under market condi-tions prevailing in 1994 (the year for which the model was calibrated). This work is particularly relevant today because the zinc industry is currently consid-ering switching to more conventional pricing contracts. And, if such a change were to occur, it might have significant implications for consumer welfare and international trade. A detailed understanding of the price participation system and its implications is therefore needed if appropriate competition policy governing such a transition is to be formulated and implemented. The paper makes three primary contributions. First, it develops a model of oligopoly under price participation, exphcitly introducing stochastic demand and allowing for the possibility of risk aversion on the part of the output-contiolling firms. In doing so, it significantly expands the limited literature examining this commercially important ver-tical relationship. Second, it identifies the general conditions under which industry participants may have private incentives to use price participation schemes rather than a more conventional standard input pricing contracts. Third, it is the first analysis to ex-5 amine the implications of the zinc industry's proposed move from price participation systems to more conventional pricing contracts. The results indicate that, given current market conditions, such a change would likely improve industry profits, but at the cost of higher consumer prices. Although no firm conclusions about welfare effects are reached, the results do provide some useful insights for policymakers. 6 Chapter 1 Nonrenewable Resources, Externalities, and Preferences 1.1 I N T R O D U C T I O N Resources finite in quantity and lacking any natural process of regeneration pose unique economic problems. Gray (1914) and Hotelling (1931) were the first to analyse such nonrenewable resource problems formally in their pioneering work on the economics of the mine. Yet the issue of exhaustibility received little subsequent attention from the eco-nomic community until the mid 1970s.1 The oil crisis of 1973-74 then forced many people to confront the possibility that some important resources might be substantially depleted in the not too distant future. This sudden realisation precipitated a flurry of activity among some of the leading economic theorists of the time and resulted in many important extensions to Hotelhng's prototypical model. Examples include explicit con-sideration of multiple deposits of variable grade (Shultze, 1974); uncertainty about the availability and characteristics of future substitutes (Hoel, 1978); the possibility of stock augmentation through costly exploration (Pindyck, 1978); and market distortions such 1. Both nonrenewable and renewable resources may be accurately described as exhaustible resources. In this paper, however, the terms exhaustible and exhaustibility are used exclusively in reference to nonrenewable resources. 7 as depletion allowances, monopolies, and externalities (Sweeney, 1977). Considerable effort was also made to incorporate the characteristics of nonrenewable resources into models of optimal economic growth. Much of this work focused on the issue of survival and tried to determine under what circumstances an economy depend-ent on a nonrenewable resource might be able to maintain positive consumption levels in perpetuity. Important early papers devoted to this issue include Solow (1974) and Dasgupta and Heal (1974). As the oil crisis subsided, so did much of the interest in the issue of exhaustibility. But today, as we near the turn of the century, increasing populations, rapid industrialisation, and concomitant environmental degradation are again focusing considerable attention on problems of exhaustibility, this time as a subset of the problems associated with the issue of sustainabiHty.2 Even when compared to the substantial body of work concerning nonrenewable re-sources, the literature on externalities and environmental management is impressively large. Although its roots may be traced to discussions of social and private cost diver-gence by Pigou in the early 1930s, most of the theory has been developed over the last thirty or so years, a period during which concern for the environment has grown sub-stantially, albeit intermittently. It would be difficult and inappropriate to review this extensive literature here. The reader is, however, directed to Baumol and Oates (1988) who offer an excellent synthesis of much of this material. 2. Dasgupta and Heal (1979) present an excellent synthesis of much of the earlier work on the economics of nonrenewable resources, while Lasserre (1991) and Sweeney (1993) provide more recent summaries. 8 Pollution is often associated with either the production or use of nonrenewable re-sources. Indeed, many of today's most pressing environmental problems are direct consequences of these types of activities; for example, the combustion of fossil fuels re-leases large amounts of carbon dioxide into the atmosphere, and accumulation of this greenhouse gas is widely believed to be causing the average global temperature to rise. The physical act of extracting nonrenewable resources can also cause the environment considerable harm; for example, mining activity often destroys wildlife habitat and sce-nic vistas, and sometimes leads to contamination of nearby ground and surface waters by acidic seepage (acid mine drainage). Despite the well-recognised connection between the use of nonrenewable resources and environmental degradation, the two issues have been, for the most part, analysed sepa-rately by economists. This theoretical separation has eased analysis and has helped clarify many of the problems central to each of these important issues. This increased clarity has, however, been purchased at the expense of our understanding of the dy-namic relationships resulting from'the inseparability of these physical processes. Although the literature examining the interaction between nonrenewable and environ-mental resources is relatively small, a wide variety of modelling techniques has been employed, making a brief review difficult; nevertheless, representative papers include Schufze (1974), Sweeney (1977), and Kamien and Schwartz (1982).3 In much of this lit-erature it is often suggested that the interesting feature of the problem is "the dynamic interplay between the exhaustion of natural resources and the i r r e v e r s i b i l i t y [emphasis 3. Kolstad and Krautkraemer (1993) provide a comprehensive review of this literature. 9 added] of environmental damage."4 The implication of this and similar statements ap-pears to be that without pollutants of at least some durability, the problem becomes a trivial extension of the standard static externality problem. If this is correct, a negative externality associated with a nonrenewable resource will result in excessive extraction of the resource relative to the social optimum in the early part of the extraction program. Sweeney (1977) has shown, however, that this may not always be the case, and this pa-per provides further evidence for doubting the validity of this intuitively appealing conclusion. This work differs from the existing literature by focusing on the influence of preference specification and by doing so within a general equilibrium framework. Specifically, it analyses the effect of a pure-flow negative externality, such as noise, on the optimal management of nonrenewable resources. The analysis is carried out within a simple two-factor, two-good general equilibrium framework under the assumption of perfect competition. Explicit functional forms for utility and production functions are used throughout, allowing closed-form solutions to be obtained for variables of interest. This paper derives two main results. First, and most significantly, the paper shows that the presence of a negative externality need not, in principle, cause either inefficiently high or inefficiently fast resource extraction. This contrasts sharply with the static ver-sion of the negative externality problem. As is well known, some form of corrective action, such as a Pigouvian tax, is typically needed to equate the private and social costs of an externahry-generating activity and so ensure an efficient outcome. No such cor-4. See Kolstad and Krautkraemer (1993), page 1227. 10 rective action is required here, however. The paper's second main result is to show that, in a standard dynamic optimisation framework, the resource extraction path depends on the cardinal properties of the utility function, not just its ordinal properties. Both of these results have been anticipated to some extent in the literature. The technical structure of the result concerning the effect of pollution on the extraction path of a non-renewable resource has been obtained by Sweeney (1977). However, Sweeney did not develop the full economic structure provided here, and this led him to underestimate the significance of the effect. More specifically, Sweeney (1977) showed that if the mar-ginal non-internalised pollution cost associated with the extraction of nonrenewable resource grows at the discount rate and the resource is entirely depleted, then internal-ising the externality will not change the competitive extraction path. However, Sweeney then concluded that although such a result was possible, "Non-internalised externalities associated with the extraction or use of depletable resources probably bias markets to-ward current extraction at the expense of future extraction alternatives." In Sweeney's analysis, the extraction path is determined exogenously, while in this paper the com-petitive extraction path of the nonrenewable resource is derived endogenously from fairly standard specifications of consumer preferences and production functions. Con-sidering the model specifications employed, the results indicate that optimality of the competitive outcome may be more likely than Sweeney suggests. One other result in the literature that is suggestive of the result presented here should be mentioned. As is well known, the existence of monopoly within a static framework leads to inefficiency in much the same way as externalities do. Stiglitz (1976) has shown, however, that if demand for a nonrenewable resource is isoelastic, then a monopolist 11 will choose the same resource extraction path as that produced by competitive markets, (i.e., the welfare-maximising extraction path). This paper's second main result—that the cardinal properties of the utility function af-fect the resource extraction path —although known in the dynamic optimisation literature, does not appear to have been widely appreciated in the fields of resource or environmental economics. This result is important because it alerts us to the potential lack of robustness that arises from the use of specific functional forms in the analysis of dynamic resource problems, since even monotonic transformations of the utility func-tions can have important effects on a model's properties. Consequently, ordinally-equivalent preference specifications that would produce identical behaviour within a static framework do not do so within a dynamic framework. In particular, the extrac-tion trajectory of the nonrenewable resource is affected by monotonic transformations of the utility function. The rest of the paper is organised as follows. Section 1.2 specifies the basic model, in-troducing assumptions about the nature of production, resource use, and consumer preferences. Competitive and social planning outcomes are derived and compared; comparative static and comparative dynamic results are presented where appropriate. Section 1.3 summarises and discusses the main results and Section 1.4 considers their implications. 12 1.2 T H E M O D E L Consider a small closed economy of the two-factor, two-sector variety. Factors comprise labour, /, and a nonrenewable resource, x. There is a fixed population of workers, L, and a known quantity of the nonrenewable resource, X. Labour is fully employed and sup-plied inelastically at some wage, w. Each of the economy's two industrial sectors produce one of two generic goods, referred to as manufactures, m, and services, s, re-spectively. Both manufactures and services are fully perishable and cannot be stored. The same is true of the nonrenewable resource. Hence, once extracted, the resource must be used immediately.5 Labour and the nonrenewable resource are imperfect substitutes in the production of manufactures, with firms transforming these inputs according to the following constant-returns production function m = Qxallm-a, 0-1) where 0 is a positive constant indicating the technological efficiency of production, the subscript m references the quantity of labour devoted to manufacturing, and 0 < a < 1. The Cobb-Douglas functional form of (1-1) embodies the assumption that both labour and the nonrenewable resource are essential inputs in the production of manufactures. Production of services also exhibits constant returns to scale, but with labour as the sole input. By a judicious choice of scaling factor, production of services may be described 5. The terms "resource extraction," "resource use," and "resource consumption" are therefore used somewhat interchangeably throughout this paper. 13 by s = ls, (1-2) where the subscript s denotes the quantity of labour devoted to the production of serv-ices. Services are treated as the numeraire good and their price is set to one. Using the full employment assumption, the production of services may be expressed as a function of the total labour supply and employment in the manufacturing sector h=L-lm. (1-3) This small economy is embedded within a much larger world economy. Neither the nonrenewable resource nor the manufactured good can be traded, and labour is inter-nationally immobile. However, the numeraire good, services, can be traded intertemporally at the interest rate r. Consequently, the economy faces the constant ex-ogenous interest rate r. This assumption simplifies the model and does not affect the results of the analysis. In fact, we could alternatively assume that the economy is com-pletely closed and solve for the time path of the momentary interest rate r endogenously. This is done in Appendix A. Extraction of the resource releases a by-product into the environment, which is in some way detrimental to human health. This pollutant is of the pure flow variety—such as noise—and its concentration grows in direct proportion to the rate of resource extrac-tion. By an appropriate choice of scaling factor, these assumptions may be summarised by 14 e = x, (1-4) where e is the ambient level of pollution. Hence, at any particular moment, the ambient level of pollution is equal to the rate of resource extraction. Although this abstracts from the important issues of pollutant accumulation and environmental assimilation, charac-terising pollution in this way is appropriate not only for pure flow externalities, but also for perfectly durable pollutants when the assimilative capacity of the environment ex-ceeds the rate at which pollution is generated. Consumer utility is a function of three variables: (i) consumption of manufactures; (ii) consumption of services; and (iii) the ambient level of pollution. Both manufactures and services are essential, in that both must be consumed for utility to be positive, and the utility derived from a given consumption bundle is a decreasing function of the ambient pollution level. The following function defining instantaneous utility, u, meets these requirements where n is a positive constant which we shall call the pollution sensitivity parameter, 0 < |3 < 1, and 0 < y < 1 • Notice that (1-5) is the Cobb-Douglas sub-utility function embedded within an isoelastic super-utility function. At first glance, (1-5) might appear to be a highly specialised utility function. Although in some sense true, it should be rec-(1-5) (1-6) 15 ognised that (1-5) embodies two other instantaneous utility functions that are commonly employed in the resource economics literature as special cases. When 0 < y < 1, (1-5) is simply a standard Cobb-Douglas utility function, while in the limit as y approaches one, it tends to the logarithmic specification w = f}mm + (l-P)lns-T)lne. (1-7) Consequently, (1-5) represents a less restrictive assumption about consumer preferences than is commonly made. As its name suggests, the pollution sensitivity parameter, r\, indexes the sensitivity of consumers to increases in the ambient level of pollution. The parameter r\ must be posi-tive, and the greater its magnitude, the greater the negative effect of a given amount of pollution on consumer utility. Although the functional form of utility function is quite standard, the negative exponent on the ambient pollution level e embodies the assump-tion that pollution exhibits decreasing marginal disutility; that is, as the environment grows more polluted the negative effect of a little more pollution diminishes. This as-sumption is reasonable for many forms of pollution. Consider noise for example. In a quiet neighbourhood a small increase in traffic noise might reduce utility significantly, while in a busy city such a change is unlikely to reduce utility to any great extent. Litter provides another example. In a virgin forest a small amount of litter could reduce utility significantly, while extra Utter on a dirty city street will likely pass unnoticed. The hypothesis that people experience decreasing marginal disutility as certain types of pollution and environmental degradation worsen is supported by a number of empirical 16 studies. For example, a contingent valuation study of visibility in the Grand Canyon conducted by Shultze et al. (1983) indicates that once an area's natural state has been degraded by pollution, further pollution is of little consequence. Similarly, Bayless (1982) finds that people living in areas with high levels of atmospheric pollution are less willing to pay for improvements in air quality than those living in relatively unpolluted areas. Anderson and Francois (1997) offer a comprehensive survey of the relevant lit-erature. Before proceeding, one formal difficulty related to the interpretation of the instantane-ous utility function (1-5) should be addressed. The Cobb-Douglas form of (1-5) implies that consumer utility in a pollution-free environment (i.e., e = 0) is undefined, and, con-sequently, that preferences are incomplete. Notice, however, that (1-5) also embodies the assumption that both manufactures and services must be consumed for utility to be positive. This ensures, via (1-1) and (1-4), that some extraction of the nonrenewable re-source takes place at all times and that e is always greater than zero. Hence, for all admissible extraction paths, utility is denned and preferences are complete. The parameter y in (1-5) indexes the concavity of the utility function, and as it increases, the utility function grows more concave. For any value of y greater than zero, (1-5) is simply a monotonic transform of the embedded Cobb-Douglas utility function (1-6); therefore, the instantaneous utility functions generated by varying the parameter y are ordinally equivalent and their use in standard static consumer theory would yield iden-tical demand functions and consumer behaviour. However, as will be shown, within a dynamic framework the ordinal properties of the instantaneous utility function are not sufficient to fully characterise consumer behaviour; the cardinal properties of the utility 17 function are also needed. We now consider the characteristics of a competitive economy mcorporating these tech-nological and behavioural assumptions. We first derive the competitive outcome and then compare the resulting intersectoral and intertemporal allocations of labour and nonrenewable resources with those that would be chosen by a welfare-maximising so-cial planner. 1.2.1 Competitive Equil ibrium Ownership of the resource is shared equally among a large number of identical profit-maximising firms facing a constant exogenous interest rate of r. Assuming the cost of resource extraction is negligible, the problem confronting a representative firm is to choose time paths x(t) and lm(t) to maximise 1-a - W, m such that X = -x (1-9) X 0 > X (1-10) X(0) = X 0 , (1-11) where p is the price of manufactures, wm is the wage in the manufacturing sector, X 18 denotes the rate of change of the resource stock with respect to time, and X 0 is the initial aggregate stock of the resource. The wage and the price of manufactures are, of course, treated as exogenous to the firm. This problem may be conveniently analysed using optimal control theory and is char-acterised by the following Hamiltonian H = pQxalm'-a-u>Jm-Xx, (W2) where the costate variable X associated with the dynamic constraint, (1-9), can be inter-preted as the shadow price of the resource.6 The following are among the conditions necessary for an optimum VaQ{lm / xf~a =X (1-13) p(l-a)Q(x/lm)a=wm (1-14) X = rX (1-15) X = -x. (1-16) Interpretation of these conditions is straightforward.7 Equations (1-13) and (1-14) are static efficiency conditions. The former ensures the value marginal product of the re-source is at all times equal to its shadow price, while the latter equates the value 6. See Kamien and Schwartz (1991) for excellent exposition of optimal control theory. 7. Concavity of the Hamiltonian ensures that necessary conditions (1-13) through (1-15) are also sufficient. 19 marginal product of labour to the wage. Equations (1-15) and (1-16) are dynamic effi-ciency conditions. The first indicates that the shadow price of the resource increases exponentially at the interest rate along the entire extraction trajectory X(t) = X(0)exp[rt] (1-17) and the second simply ensures that the dynamic constraint, (1-9), is satisfied. Like manufactures, services are provided by a large number of identical firms, each seeking to maximise the value of its profit stream over an irifirute horizon. But since each service firm has no resource to husband and regards the service-sector wage, ws, and the price of services as given, the problem is reduced to a repeated static optimisa-tion. At each point in time, the representative firm chooses how much labour to employ to maximise its profits subject to (1-2). Remembering that services are the numeraire good and that their price is one, we obtain the familiar first order condition equating wage and price; that is, ws =1. Since labour markets are competitive and labour is freely mobile, both industrial sectors must pay the same wage if both goods are to be produced wm=ws=w = \. (1-18) Firms are owned by a large, constant population of individuals with identical share-holdings and preferences. As previously discussed, individuals gain utility through consumption of manufactures and services, with the utility provided by a given con-sumption bundle decreasing as the ambient level of pollution increases. Since there are 20 many consumers, the purchasing decisions of a particular individual have little effect on aggregate emissions and, consequently, little effect on the level of pollution to which that individual is exposed. It is therefore assumed that consumers do not take account of the pollution-generating aspect of their own consumption decisions; pollution is, in this case, a pure public bad. Given the time path of resource extraction, x(t), and a constant, positive rate of time preference, 8, and the interest rate r, a representative consumer seeks to maximise the discounted value of their utility stream, treating the level of pollution as given; that is, a representative consumer chooses m(f) and s(f) to maximise — r(mh^e-^txp[-St]dt • • (1"19) 1 - y Jo such that I = rl-pm-s (1-20) 1(0) = [ ipBxX"1 + L - lm)exp[-rt]dt (1-21) lim/(f)>0, (1-22) where I represents the wealth of the representative consumer in terms of the numeraire. Equation (1-20) is the dynamic budget constraint; equation (1-21) states that initial wealth is the discounted sum of all future shareholder income and wages; and equation (1-22) simply ensures wealth is at all times non-negative. 21 The current value Hamiltonian associated with this problem is H = [(n^s'-V 1 )l'y ]/(1 -y) + i i ( r l - p m - s ) , C 1" 2 3) where the costate variable | i represents the current marginal utility of wealth. Remem-bering that consumers treat the level of ambient pollution, e, as given, the following necessary conditions are easily derived8 p m M i - Y H s O - » t M ) e - n < i - T ) = p u (1-24) (1 - pjmWi-rtsti-Wa-THg-Ki-r) = u (1-25) A=(5-r)u (1-26) 1 = 7 - / - pm - s . (1_27) Equation (1-24) ensures that the marginal utility of manufactures consumption equals the price of manufactures times the marginal utility of wealth; and equation (1-25) sim-ply ensures that the same holds true for services. The third condition, (1-26), can be integrated with respect to time to yield u(f) = u(0)exp[(8-r)r], (1-28) which shows that along the competitive trajectory the marginal utility of wealth changes 8. Concavity of the Hamiltonian in m and s ensures that necessary conditions (1-24) through (1-27) are also sufficient. 22 at an exponential rate equal to the difference between the rate of time preference and the rate of interest.9 To examine the characteristics of the dynamic equilibrium prevailing in this economy, we assume markets clear. This of course requires that conditions (1-1) and (1-3) hold in aggregate. Equating the marginal utility of wealth, u, in necessary conditions (1-24) and (1-25), and imposing market clearing, we obtain P ( L - / m ) / ( l - P ) = pex«Zj,-a. (1-29) Combining this result with (1-14) and (1-18) yields _ ( l - q ) P L (1-30) m (l-cu3) Since the technological and behavioural parameters a and p are exogenously deter-mined and are assumed constant, (1-30) indicates that the fraction of the total labour force employed in the manufacturing sector does not change along the competitive tra-jectory. This implies, by virtue of (1-3), that employment within the service sector is also constant. Equation (1-30) also indicates that the labour-use trajectory is unaffected by the concavity parameter y. Two further results follow immediately from (1-30): the sectoral employment ratio (i.e., lm/ls) is influenced by neither (i) the total amount of labour available nor (ii) the efficiency of manufacturing production (i.e., the parameter 0). 9. Whether the marginal utility of wealth increases or decreases depends on the relative magnitudes of these 23 We now examine resource extraction within this competitive economy. 13), (1-24), and the market clearing conditions (1-1) and (1-3) yields Combining (1-apeP(l-r)xaP(l-y)-l/Cl-a)p(l-v)(L_/J(l-p)(l-v)e-n(l-y) (1-31) Using (1-17) and (1-28) to substitute for k and u and making use of the fact that e = x, we obtain the following expression implicitly defining the intertemporal path of competi-tive resource extraction ^(ap-nXl-y)-! MO)H(O) ap0P(l-y)zCl-a)P(l-Y)(L_,m)(l-p)(l-. (1-32) exp[5t]. Since the size of the labour force, L, is constant, we know from (1-30) that the expression within the larger bracket is time invariant; therefore, differentiating (1-32) totally with respect to time, we obtain (a(3-r|)(l-y)-l = 5a:/i. (1-33) We now use condition (1-16) to transform this differential equation from one in terms of the rate of resource extraction to one in terms of the resource stock (aB-n)(l-y)-l = 6X/x. (1-34) Solving this equation we obtain the general solution two parameters. The possibility that the rate of time preference and the interest rate are equal is not ruled out. If 8 = r , the marginal utility of wealth is constant along the competitive trajectory. 24 X(f) = A1 + A2 exp - 5 f ( a p - T i ) ( l - y ) - l (1-35) where A-^ and A2 are constants of integration. From (1-11) we know that X(0) = X 0 . We also know that as time tends to infinity the resource stock will asymptotically approach zero lim X(f) = 0. f->oo (1-36) Using these two boundary conditions, we obtain X(f) = X0exp -5f l - ( a p - n ) ( l - y ) (1-37) as the particular solution of (1-35). This relationship describes the intertemporal evolu-tion of the resource stock along the competitive trajectory. An expression describing the rate of resource use may be derived by differentiating (1-37) with respect to time and making use of condition (1-16) x(t) = 5Xn l - ( a p - T ! ) ( l - y ) exp •5f l - ( a p - r , ) ( l - Y ) (1-38) The first thing to notice about this result is that the competitive rate of resource extrac-tion is a function of the pollution sensitivity parameter r\. While consumers do not take the pollution-generating aspects of their own consumption into account when making their purchasing decisions, they do take account of the ambient level of pollution (which 25 they treat as given). It is the magnitude of the parameter n that determines the extent to which ambient level of pollution influences the consumption decision of consumers and, thereby, the resource extraction path. Equation (1-38) also indicates that competitive extraction path is independent of the size of the labour force and the efficiency of the manufacturing technology. Finally, notice that in contrast to the competitive labour-use trajectory described by (1-30), the resource extraction trajectory (1-38) is a function of the concavity parameter y. At this point it is valuable to interpret the parameter combination ccB - n appearing in (1-38). To do so, we substitute the production functions (1-1) and (1-3), and the exter-nality relationship (1-4) into the instantaneous utility function (1-5). This yields consumer utility as a function of resource consumption and labour employment M = [eP(l-Y) x(aP-r,Kl-V) /Cl-a)P(l-y) ( L_ I m )(l-P)(l-Y) j / ( 1 _ y ) (1-39) From this and the restriction placed on y, it is clear that if r\ > aB, the exponent of x is negative. This would imply that aggregate utility is a decreasing function of nonrenew-able resource use. If so, the damage caused by pollution generated during resource extraction or use exceeds the value derived from consumption of the resource as em-bodied in manufactures. In such a case it would be better never to extract the resource. Although possible, such a situation is likely to be uncommon. We therefore restrict our attention to cases where the resource is an asset rather than a liability by the imposing constraint 26 a(3 > r| (1-40) We now consider how the extraction trajectory of the resource changes as consumers grow more sensitive to pollution. Differentiating (1-38) with respect to r| yields dx(t) _ 5X0(1-Y)[5f -1 + (ap - T,)(1 - Y)] [l-(ap-T0(l-Y)] 3 exp -5f l-(ap-r))(l-Y) (1-41) From this we can see that the rate of resource extraction is decreasing in n until time t - [1 - (ap - r|)(l - Y)]/8 and is thereafter increasing in n. As r| increases, implying that consumers have a lower tolerance for pollution, resource extraction is increasingly shifted into the future. This follows directly from the assumption of a positive rate of time preference, 8, since a positive value of this parameter implies future consumption is worth less than present consumption. Therefore, by shifting some resource extraction into the future, concomitant pollution is also shifted. This intertemporal reallocation effectively reduces the negative effect of pollution on consumer welfare by discounting it. Differentiating (1-38) with respect to the concavity parameter Y yields dx(t) _ SX0 (ap - r,)[8f -1 + (ap - r,)(l - Y)] [l-(aP-rO(l-Y)]2 exp -St l-(ap-r,)(l-Y) (1-42) Since ap > r), the rate at which the resource is extracted, x(t), is decreasing in Y until time f = [1 - (ap - r|)(l - Y)]/5 and is increasing in Y thereafter. The effect of varying Y on the resource extraction trajectory is straightforward. As Y increases, and with it the 27 concavity of the instantaneous utility function (1-5), the initial rate of resource extraction falls and extraction is increasingly shifted into the future. In other words, as y grows, consumers increasingly smooth the extraction trajectory by extracting less in the early part of the program and more in the later part of the program. This situation is depicted in Figure 1-1. x • ' Extraction Path Figure 1-1: Competitive Extraction Paths Notice that the isoperimetric condition, (1-16), ensures that at some time t * the extrac-tion trajectory generated by Cobb-Douglas preferences (i.e., y = 0) switches from one of over-extraction to one of under-extraction relative to the trajectory generated by loga-rithmic preferences (i.e., y = 1). 28 The relationship between x(t) and y depicted in Figure 1-1 is the result of the assumption of decreasing marginal utility of resource consumption embodied in utility function (1-5). As the utility function becomes more concave, the marginal utility of resource con-sumption falls and consumers more evenly distribute consumption of the resource across time. This result is not new. In fact, the dependence of the extraction path of a nonrenewable resource on the elasticity of marginal utility has been demonstrated fre-quently (see Dasgupta and Heal (1979), for example). Notice that when y = 1, the competitive extraction trajectory collapses to *(r) = 5X0exp[-5f]. (1-43) Therefore, when preferences are logarithmic, the competitive rate of resource extraction is determined solely by the rate of time preference and the size of the initial stock of the resource. At this point it is useful to recall that the parameter y has no ordinal significance. This can readily be seen by inspecting utility function (1-5) and noticing that changing the value of y simply applies a monotonic transformation to the embedded Cobb-Douglas utility function (1-6). This leads directly to Proposition 1: The extraction path of the nonrenewable resource under com-petitive conditions is a function of the cardinal properties of consumer preferences. 29 Proof : Equation (1-42) shows that the competitive extraction path is affected by the value of y. Since y has no ordinal significance, the proof is complete. • The intuition behind this result is straightforward. The objective function being maxi-mised by the representative consumer is the discounted sum of the utilities obtained at each point in time. Since summing is a cardinal process, it is quite natural to expect the extraction path to be a function of the cardinal properties of consumer's utility function. We now consider the influence of the rate of time preference, 8, on the competitive ex-traction trajectory. Differentiating (1-38) with respect to the rate of time preference yields from which we see that the extraction rate is increasing in 8 until time t = [l-(ap-r|)(l-y)]/8 and is thereafter decreasing in 8. Hence, as the rate of time preference increases (i.e., as consumers increasingly value present consumption over future consumption), extraction is increasingly shifted into the present at the expense of the future. The instantaneous indirect utility function, v, of our representative consumer may be constructed by substituting (1-30) and (1-38) into (1-5) dx(t) _ [ l - ( q p - T i ) ( l - Y ) - 8 f ] X 0 -8f (1-44) 58 [i_(ap-r,)(l-y)]2 exp l-(ap-r,)(l-y)|' 30 v(t) Q P ( I - V ) 1-Y ( l - a ) 6 L l ( 1 - a ) P ( 1 - y ) r ( l - P ) L l - a p 1-ap (1-P)(1-Y) (1-45) SX 0 exp[-St/(l-(aP-ri)(l-Y)] l - (ap-r , ) ( l -Y) (ap-n)(l-y) As one would expect, this result shows that indirect utility is increasing in 9. Thus the greater the technological efficiency of manufactures production, the greater the amount of utility that can be generated from a given quantity of labour and resource inputs. In addition, differentiating the indirect utility function with respect to L shows that v also increases as the size of the labour force grows. This follows from the fact that (1-45) de-fines aggregate utility rather than per capita utility. Finally, combining (1-14) and (1-18) with (1-38), we obtain an expression describing the trajectory of the price of manufactures relative to that of services P(t) = 1 0(1-a) (l-a)PL 1-ap SX0exp[-5t/(l-(ap-ri)(l-Y) l-(ap-r,)(l-Y) (1-46) From this we see that the greater the productivity of the manufacturing sector, the lower the initial price of manufactures. Furthermore, the greater the elasticity of manufactur-ing output with respect to resources (i.e., a), the greater the exponential rate at which the price of manufactures grows. The effect of a on the initial price of manufactures is ambiguous, however. 1.2.2 Optimal Planning Solution We now consider the problem facing a benevolent social planner wishing to maximise 31 consumer welfare. The planner's problem is to choose x(t) and lm(t) to maximise the discounted value of the utility stream of a representative consumer _ L £ m P ( i - v ) s ( i - P ) ( i - r ) e - n ( i - y ) e x p [ _ 8 f ] (1-47) subject to the economy's technological constraints (1-1) and (1-3), the dynamic constraint (1-9), and the non-negativity constraint (1-10). Using these constraints to eUrninate, tn, s, and e from (1-47), the current value Hamiltonian characterising the planner's problem may be written as H = [e p( 1^¥ a p-^ 1- 1 ')4 1- a) | 3( 1- 1')(L-Zm)( 1- 1 5)( 1- 1')]/(l-y)-^, (1-48) where <j) is the shadow price of the resource. This readily yields the following necessary conditions10 (qp -^Q^x^-^-y^i^M^) _ (1-49) ( L _ / m ) ( P - D ( i - r ) " * ( l-a)P(L-/ m ) = ( l -P) / m (1-50) <p = Sep (1-51) X = - x . (1-52) 10. Since we have imposed the constraint ap - r| > 0, concavity of the Hamiltonian is ensured. Necessary conditions (1-49) through (1-52) are therefore also sufficient. 32 mtegrating (1-50) with respect to time yields <K0 = *(0)exp[5f], (1-53) V indicating the shadow price of the resource rises exponentially at the rate of time prefer-ence. Simplifying and rearranging (1-50), we obtain the following expression describing opti-mal employment within the manufacturing sector /m=(l-a)B/(l-aB)L. (1-54) Comparing (1-54) with (1-30) we see that the sectoral and intertemporal allocations of labour chosen by a welfare-maximising social planner are identical to those produced under competitive conditions. But what of the optimal intertemporal allocation of the nonrenewable resource? Sub-stituting (1-53) into (1-49), differentiating the result with respect to time, and making use of condition (1-16) yields [ (ap-n)( l-y ) - l]=x/x. (1-55) This result is familiar. Equation (1-55) is in fact identical to (1-34), the differential equa-tion governing the evolution of the resource stock under competitive conditions. Furthermore, since the social planner faces the same initial and tiansversality conditions as those existing in the competitive economy, the welfare-maximising and competitive 33 extraction trajectories are identical, both being described by (1-38). Thus we have Proposition 2: When extraction or use of a nonrenewable resource generates a pure flow negative externality and consumer preferences are such that pollution exhibits decreasing marginal disutility, the competitive extraction path is wel-fare maximising. Proof: Follows directly from the equality of the resource extraction trajectories (1-34) and (1-55). • In addition, since the competitive extraction path is a function of the concavity parame-ter y, Proposition 2 leads immediately to the following corollary. Corollary 1 : The welfare-maximising extraction path of the nonrenew-able resource is a function of the cardinal properties of the consumer preferences. 1.3 DISCUSSION Within a static framework, the presence of a negative-externality-generating factor of production drives a wedge between the social and private costs of using that factor as an input. This cost differential results in the factor being used excessively compared to its socially optimal level of usage, and some form of corrective action, such as a Pigouvian tax, is typically needed to ensure an efficient outcome. Intuitively, this result should extend in a straightforward manner to a dynamic framework mcorporating nonrenew-able resources. If so, the presence of a negative externality would result in excessive use 34 of the nonrenewable resource in the early part of the program. The main contribution of this paper is to show that this intuition is not always correct, even for quite reasonable and commonly employed model specifications. With an instantaneous utility function of the form described by (1-5), which includes Cobb-Douglas preferences and logarithmic preferences as special cases, the presence of a pure-flow negative externality associated with the extraction of a nonrenewable re-source does not result in a misallocation of resources within a competitive economy; the competitive allocation is in fact welfare-maximising. Under these circumstances, no market failure occurs and no corrective policy intervention is required. The reason for this rather counterintuitive result is as follows. As long as using the non-renewable resource yields some net positive utility, both a welfare-maximising social planner and competitive markets will choose to exhaust the resource eventually. The only question is one of timing; that is, Will the presence of the negative externality result in competitive markets extracting the nonrenewable resource along a sub-optimal in-tertemporal trajectory? It is a well known fact that in the absence of market distortions competitive markets can support the welfare-maximising extraction of a nonrenewable resource. It is also well known that a necessary condition for such welfare-maximising extraction is that the marginal utility of resource use grows at the discount rate along the entire extraction trajectory.11 Indeed, if this condition is not satisfied, then it must be possible to increase utility through an intertemporal reallocation of resource use. 11. Since the possibility of storing the resource is ruled out (except in situ), the resource must be used as it is extracted. 35 Now consider the competitive extraction of a nonrenewable resource that generates a negative externality. Along the competitive trajectory, the marginal utility of resource use e x c l u d i n g the m a r g i n a l d i s u t i l i t y caused by c o n c o m i t a n t p o l l u t i o n must be growing at the discount rate. If not, an intertemporal reallocation of resource use could increase utility. Since the instantaneous utility function, (1-5), is concave in x, the rate at which the re-source is being used must be decreasing at the discount rate. Furthermore, since e = x , the level of pollution must also be decreasing at the discount rate along the competitive trajectory. Finally, since (1-5) is convex in e, as e falls at the discount rate, the marginal disutility associated with the externality must be increasing at the same rate. If so, there is no intertemporal reallocation of pollution that can increase welfare. This implies that the resource is being used at precisely the rate that maximises welfare. This result is not entirely new. Sweeney (1977) has previously shown that if the entire stock of a nonrenewable resource is depleted and if the marginal external cost associated with extraction of the resource grows at the discount rate, then competitive extraction will be welfare-maximising. Sweeney, however, treats the marginal external cost as ex-ogenous and is consequently quite pessimistic about the chances of competitive conditions resulting in socially optimal resource extraction. Indeed, he speculates that although such a result is possible, it is likely to be uncommon. In contrast, in the model presented in this paper, the marginal external cost is determined endogenously. Con-sidering the fairly general specifications of consumer preferences and technology employed, the results obtained would seem to indicate that optimality of the competi-tive outcome may be more likely than Sweeney suggests, at least for those forms of pollution that exhibit decreasing marginal disutility. 36 A further mteresting result is that for the preference specifications considered, the frac-tion of the total labour force employed in each sector, as described by (1-30), is constant, even though labour is a substitute for the nonrenewable resource in the production of manufactures. This implies, of course, that production of manufactures dwindles at an exponential rate along the optimal trajectory, while production of services remains fixed. Condition (1-30) also indicates that the competitive allocation of labour between sectors (i.e., the proportion of the total labour employed in each sector) is influenced by neither the absolute productivity of the manufacturing sector nor the size of the avail-able labour force. Furthermore, (1-38) shows that neither of these factors affect the competitive rate of resource extraction. Finally, it is important to recognise that the instantaneous utility function (1-5) embodies an entire family of ordinally equivalent preference specifications, each of which results in a different resource extraction path. Hence, within a dynamic framework, the extrac-tion path depends upon not only the ordinal properties of the utility function but also its cardinal properties. In summary, this paper identifies a set of conditions that are necessary and sufficient to ensure that competitive markets achieve a welfare-maximising allocation of a nonre-newable resource that generates a negative externality. None of the assumptions used in the model are unreasonable characterisations of reality and similar assumptions are commonly employed in the resource and environmental economics literatures. Never-theless, the model, taken as a whole, embodies a large number of assumptions that interact in a complex manner to generate this unusual result. Consequently, it is quite difficult to know how general the result really is. Future research might involve relax-37 ing some of the model's assumptions to determine the result's level of generality. However, even without such research, it is still possible to speculate about which are likely to be the critical assumptions for obtaining the result. Four assumptions stand out as likely candidates: (i) the assumption that some nonrenewable resource is necessary for the production of manufactures (embodied in equation (1-1)); (ii) the assumption that the manufactures are necessary for utility to be positive (embodied in equation (1-5)); (iii) the assumption that the pollution generated is directly proportional to the amount of resource used (embodied in equation (1-4)); and (iv) the assumption that pollution exhibits decreasing marginal disutility (embodied in equation (1-5)). The first and second of these assumptions together ensure that some amount of the nonrenew-able resource will be used at all times along the competitive trajectory. This in turn ensures that any market distortion, such as the externality, may affect the timing of re-source use, but not the quantity of resource that will be used. This, coupled with the third assumption, ensures that the total amount of pollution generated along the com-petitive trajectory is fixed. Finally, the fourth assumption, the declining marginal disutility of pollution, ensures that as resource use declines asymptotically along the competitive trajectory, and with it the pollution level, the marginal damage it causes rises. The assumption that pollution is non-cumulative is also likely to be important. In real-ity, most forms of pollution are cumulative to some extent; that is, once released into the environment they cause damage over a period of time rather than for just an instant. Therefore, knowing how the analysis will be affected by dropping the non-cumulative assumption is important for judging the general applicability of the main result. Intui-38 tively, it seems likely that a cumulative pollutant will tilt the extraction path toward the future (pollution damage can be reduced by shifting pollution into the future and thereby discounting it.) This tilting of the extraction path toward the future will occur under competitive conditions and under the direction of a welfare-maximising social planner. However, it is not clear if dropping the non-cumulative assumption will cause the competitive and welfare-maximising allocations to diverge. 1.4 C O N C L U S I O N This paper provides a simple dynamic framework with which to analyse the effects of a production externality on the allocation of nonrenewable resources within a competitive economy. For a commonly employed specification of consumer preferences, which in-cludes Cobb-Douglas and logarithmic preferences as special cases, the competitive allocation of labour and the nonrenewable resource is fully optimal for those types of pure-flow externalities exhibiting decreasing marginal disutility. This paper shows that under certain plausible circumstances, the presence of a negative externality associated with the use or extraction of a nonrenewable resource will not result in over-consumption of that resource. Therefore, some form of corrective action, such as a Pig-ouvian tax, is not needed to ensure an efficient outcome. However, no such corrective action is required here. This counterintuitive result is obtained without invoking the assumption of irreversible environmental damage. This paper also shows that the resource extraction path depends upon the cardinal properties of the utility function. Although this result is known in the dynamic pro-gramming literature, it does not seem to be widely appreciated in the fields of either 39 resource or environmental economics. It should be taken as warning about the robust-ness and generality of results based upon specific functional forms, as not even monotonic transformations of these forms will have the same properties. Furthermore, it suggests that the functional forms used to represent consumer preferences within dy-namic policy models are of critical importance, and that as such, the realism of these assumptions should be carefully considered. 40 Chapter 2 The 1990 Oil Pollution Act: Consequences for the Environment 2.1 I N T R O D U C T I O N Although maritime accidents have always been a major concern of tanker owners, the reasons for their concern have changed quite dramatically over the last half-century. Before World War II and for some years thereafter, tanker owners were primarily con-cerned with the financial consequences of damaging a ship or losing a valuable cargo and gave little thought to the environmental damage that might be caused by an oil spill. The reasons for this were quite straightforward. At the time, not only were the seas and oceans thought to be invulnerable and so in no need of protection, but tanker owners faced little or no financial hability for spill damage. Today, however, things are different. Now the environmental costs associated with an oil spill can be as much, if not more, than all other accident costs combined. This change has been brought about by the inception and evolution of regulations governing use of international waters and protecting the rights of coastal states. The. first truly international effort to protect the seas from pollution began in 1948 with 41 the formation of the Intergovernmental Maritime Consultative Organisation, now the International Maritime Organisation (IMO). At first the IMO's anti-pollution efforts were almost exclusively directed at reducing operational spills, such as the discharge of oily wastes from cargo tanks and machinery spaces (Drewrey Shipping Consultants, 1991). It would be another 25 years before a serious attempt was made to regulate acci-dental discharges. It was the grounding of the Torrey Canyon in 1967 off the coast of Cornwall, England that finally forced the international political community to act. The Torrey Canyon released almost 35 million gallons of Kuwaiti crude into the ocean, eventually spoiling hundreds of miles of British and French beaches. At the time, it was the largest oil spill in history. More significantly, it was the first spill to receive extensive media coverage. In the words of M'Gonigle and Zacher (1979), "It was only with the disastrous Torrey Canyon accident that the oil pollution problem ceased being just a minor localised issue or a re-mote statistical trend and became instead a tangible political reality." In fact, the Torrey Canyon disaster was the main motivation for holding the Interna-tional Convention for the Prevention of Pollution from Ships in 1973. The convention, complemented by a 1978 Protocol, resulted in MARPOL 73/78, a set of regulations rep-resenting the international community's first major attempt to tackle the problem of accidental oil spills. MARPOL's solution to the problem was, in essence, to impose more stringent structural requirements on vessels, in the hope of making spills less 42 likely.12 Few changes were made to MARPOL 73/78 between its inception and the early 90s, and those changes that were made mostly concerned issues of implementation rather than modification or expansion of the existing regulatory base. It is only recently, in the wake of the Exxon Valdez disaster in 1989, that the IMO has begun to seriously reconsider its approach to the problem of maritime oil pollution. The largest tanker spill in the history of the United States took place on March 24,1989, when the supertanker Exxon Valdez ran aground in Prince William Sound, Alaska. The impact tore open the ship's hull, spilling almost 11 million gallons of Alaskan crude into the waters of the area's near-pristine wilderness. That spring, oil moved along the Alas-kan coast contammating vast sections of shoreline bordering, among other things, one National Forest, four National Wildlife Refuges, three National Parks, five State Parks, four State Critical Habitat Areas, and one State Game Sanctuary. Oil eventually reached shores as far as 600 miles from the spill site. The media exposed the public, both in the United States and around the world, to disturbing images of the spill's destruction. The public and numerous special interest groups responded by placing the US Government under tremendous political pressure for action. As a result, Congress passed the 1990 Oil Pollution Act on August 18 of the following year. The 1990 Oil Pollution Act (OPA90) is undoubtedly the most radical and significant at-tempt yet made by a single nation to deal directly with pollution of its territorial waters by vessels transporting oil. The main objectives of the Act are to reduce the likelihood of spills occurring and, recognising that spills can never be completely eliminated while 12. Between 1969 and 1974, more than 75 percent of the oil spilled by vessels at sea was discharged as a result of structural failures, groundings, collisions, and other less significant causes such as explosions, breakdowns, 43 tankers operate, to identify those who should be financially liable in the event of a spill and to determine the extent of that uability. To meet these objectives, OPA90 made sweeping changes to the rights and obligations of tanker owners whose vessels enter or operate in US waters. It significantly expanded the potential liability of tanker owners for spills in US waters and introduced unlimited liability under a much broader range of circumstances; it extended the scope of compen-satory spill damages to include a wide range of economic losses, including injury to or loss of natural resources; it requires vessels entering US waters to possess documenta-tion guaranteeing that maximum Uability limits can be met; it requires newly constructed vessels calling at US ports to have double hulls and phases-out the opera-tion of single-hulled vessels in US waters by the year 2015; it stipulates minimum training requirements and maximum working hours for vessel crews; and it requires vessel owners and operators to prepare detailed response plans to deal with "worst-case scenario" spills and to ensure that the necessary tiaining and equipment are provided to make these plans practicable.131415 The Act also made vessel owners solely responsible for spill damages, thereby explicitly excluding the older principle of shared responsibil-ity between cargo owners and tanker owners. The tanker industry's reaction to the Act has been predictably negative for the most and fires (M'Gonigle and Zacher 1979). 13. An owner7s liability in the event of a spill rose from $150 to $1,200 per vessel gross ton, with minimum liabilities of $2 million and $10 million for vessels under and over 3,000 gross tons, respectively. 14. The double hull requirement applies to tankers over 5,000 gross tons; vessels contracted before June 30, 1989 and delivered before January 1,1994 are exempt. 15. The Act calls for the elimination of single-hulled vessels by the year 2010. Single-hulled vessels may, how-ever, operate in US waters until 2015 if they either discharge solely at the Louisiana Offshore Oil Port or lighter (i.e., transfer their cargo to smaller vessels) more than 60 miles offshore. 44 part, with industry representatives labelling it a "hastily concocted set of rules designed to appease public pressure in the aftermath of the [Exxon] Valdez incident" (Drewrey Shipping Consultants Limited 1991). Although nearly every aspect of the Act has been criticised by those in the industry, the introduction of unlimited liability has been by far its most controversial component. Under the Act, if a spill is caused by gross negli-gence, wilful misconduct, or the violation of any federal operating or safety standard, owners and operators face unlimited uability. This contrasts dramatically with the situation prevailing before the Act when, under most circumstances, owners faced a maximum liability of $14 million. mterestingly, some industry representatives have concluded that far from reducing the number of spills occurring in US waters, the Acf s hability provisions might do just the opposite (Price 1991). Their conclusion appears to be based on three main arguments. The first argument focuses on the reallocation of resporisibility for spill damages intro-duced by the Act. Before the Act, responsibility for such damages was shared between tanker owners and cargo owners. However, the Act shifted sole responsibility squarely onto the shoulders of the tanker owners; and by doing so, it is argued, the Act signifi-cantly reduced the incentive for cargo owners to choose the best ships and operators when chartering a tanker. The second argument focuses on the Acf s expansion of the hability of tanker owners for spill damage. One possible effect of that expansion, the desired effect, is that tanker owners will do more to reduce the chances of a spill occurring by investing in spill pre-vention technologies and activities, and by generally taking greater care in day-to-day operations. If is argued, however, that the Act also gives owners a strong incentive to 45 try to shield their assets from liability. To do so, owners might take actions such as re-lying more heavily on debt financing, removing assets further from US jurisdiction, or forming a greater number of limited-nability "one-ship" corporations. Furthermore, those who are most adept at shielding their assets in this way will gain a competitive advantage, since they will presumably face lower finance costs. The third argument, which is closely related to the second, focuses on the differential effect the Acf s hability provisions may have on small and large tanker companies. It is argued that in the event of a spill, small undercapitalised companies will be able to ex-ternalise part of the accident's cost through bankruptcy, while large fully-capitalised companies, such as the oil majors, will have to bear full damages. This will give small firms a competitive advantage over their larger rivals, and will eventually result in as-set-rich companies either reducing the size of their fleets or elunmating them entirely. Taken together these arguments imply that the liability provisions introduced by the Act will encourage and facilitate the entry of small, under-capitalised firms into the tanker industry. The concern with this happening is, of course, that these firms will be able to externalise most, if not all, of the cost associated with a spill through either bank-ruptcy or some similar mechanism; and that this will result in a inefficiently low level of care being taken in tanker operations.16 This type of concern is by no means unique to the tanker industry. This so-called "judgement proof problem" is, in fact, the focus of a small but growing body of work in the law and economics literatures.17 To date little 16. A formal model capturing some elements this argument is presented in the Appendix A. 17. Summers (1983) and Shavell (1986) offer the first formal analyses of the "judgement proof problem". Kraakmen (1984), Easterbrook and Fischel (1985), and Roe(1986) also make major contributions to the devel-46 empirical work has been done on the problem, however. In the few years that have passed since implementation of the Act, a considerable amount of anecdotal evidence has lent support to these arguments. For example, Chev-ron International stated in late 1991 that "Due to concern about the environmental risks associated with deliveries in US waters, shipping into the US and complementary ac-tivities by Chevron International . . . will be reduced significantly." They then promptly scaled back their activities related to deliveries of residual oil to the US (Price 1991). At almost the same time, the "Royal Dutch Shell Group disclosed plans to slash the size of its tanker fleet by almost 50 percent, thereby phasing out its third party trans-portation business" (Price 1991), and Elf Aquitaine announced that it would curtail US oil shipments in vessels it owned or managed. In addition, Ketkar (1995) reports that in 1991 just over 45 percent of all ocean-going tankers trading in the US were owned by single-ship companies, a substantial increase from just under 30 percent in 1980. Not all the available anecdotal evidence supports the "judgement proof" hypothesis, however. In 1993, Gerhard Kurz, Chairman of Oil Companies International Marine Fo-rum and President of Mobil Shipping and Transportation Co., observed "a clear and growing preference for quality tonnage including modern vessels and well-maintained older [tankers] operated by reputable companies" (Knott 1993). The above discussion raises several important questions: Has the 1990 Oil Pollution Act affected the number of oil spills associated with oil tanker operations in US waters? Has opment of the argument. Some empirical support for the conclusions of the argument has been provided by the work of Ringleb and Wiggens (1990). 47 it affected tanker spill rates outside of US waters? Has the Act had a beneficial or detri-mental effect? This paper attempts to answer these questions by conducting an empirical analysis of the temporal and geographic distribution of major pelagic oil spills occurring worldwide during a 15 year period sparming the introduction of the Act. The results of the analysis indicate that the Act has, in fact, significantly reduced the number of spills associated with tanker traffic to and from the North American continent and has had, as of yet, no discernible effect on spill frequencies in other parts of the world. This paper makes at least three advances over previous work. It presents the first em-pirical analysis of the Oil Pollution Act's effect on spill frequencies (i) to explicitly account for geographic heterogeneity in spill frequencies; (ii) to attempt to control for regional differences in tanker traffic flows; and (iii) to make extensive use of the discrete dependent-variable nature of oil spill frequencies by applying a range of count model specifications. This paper also employs a more sophisticated and extensive set of ex-planatory variable than previous analyses. Although Anderson and LaBelle (1994) use the Poisson model in their work on global spill rates, their analysis suffers from several serious deficiencies, the most significant being that their data set ends in 1992, and is therefore too short to clearly discern any effect that the Act may have had. This paper also sheds some light on the general question of what drives the regulatory process. The experience of the tanker industry in the case of the 1990 casts considerable doubt on the general apphcability of the capture hypothesis of regulation, and lends some to support to the alternative public interest and public choice theories. The paper is organised as follows. Section 2.2 gives a brief review of the relevant lit-48 erature. Section 2.3 introduces the statistical models used to explore of the Acf s effect on spill rates and briefly discusses their relative strengths and weaknesses. Section 2.4 provides details of the data used, discusses the construction of the variables, and pres-ents descriptive statistics. Section 2.5 presents the results of the empirical analysis, and Section 2.6 discusses the results and draws conclusions. In addition, Appendix B pres-ents a simple formal model illustrating the ambiguous effect of increased liability on oil spill frequencies. 2.2 REVIEW OF T H E LITERATURE Although this paper constitutes the first formal attempt to identify and quantify the ef-fect of the 1990 Oil Pollution Act on oil spill frequencies, two other recent studies have touched upon the issue. In the first study, Anderson and LaBelle (1994) estimate worldwide tanker oil spill rates per billion barrels (bbl) of oil transported over the peri-ods 1974-1985 and 1974-1992, finding that the rate stayed constant at 1.30 for spills of more than 1,000 bbl and rose slightly from 0.71 to 0.72 for spills of more than 10,000 bbl.18 They also examine spill rates of crude oil from tankers in US coastal waters, con-cluding that over the 1974-1992 period "the annual spill rate did not appear to increase or decrease monotonically." In the second study, the American Petroleum Institute (API) also examines oil spill fre-quencies. Its survey is, however, based solely on US Coast Guard data and therefore restricted to those oil spills occurring in US waters. The API study considers a broader range of spills than does this paper, including not only offshore spills but also those oc-49 curring in bays, harbours, rivers, lakes, and sounds. Although the analysis is restricted to a simple visual inspection of the data, two broad trends in the early 1990s are identi-fied: more small spills, but fewer large tanker spills. 2.3 ECONOMETRIC SPECIFICATION To answer the questions raised in the Introduction, a spill "production function" is es-timated. This function relates the number of spills, S, occurring in a particular geographic region in a given year19 to a vector of inputs, X, and a vector parameters, B. In its most general form, the spill production function can be expressed as Sr(=/(Xr(,B), (2-1) where the subscripts r and t index particular geographic regions and years, respectively. The number of spills occurring in a region must be a non-negative integer, and any model of the spin-generating process should ideally accommodate this restriction; con-sequently, some form of discrete dependent-variable model is called for. Following much of the recent literature dealing with count data, it is initially assumed that spills are generated by a Poisson process.20 This approach is appropriate if the annual number of spills occurring in a region can be thought of as the outcome of a large number of random trials each having some small probability of success. This does not appear to be 18. 1,000 barrels is equal to approximately 42,000 gallons or 136 metric tons. 19. Alternatively, one could chose the total volume of oil spilled in a given region in a given year as the de-pendent variable. However, there are at least two reasons for not doing so. First, much of the time little is known about quantity of oil released in a spill. Second, the damage caused by ten separate 100,000 bbl spills is likely to be much greater than the damage caused by a single 1,000,000 bbl spill. 20. See Henderson and Cockburn 1995, for example. 50 an unreasonable description of the regional spill generating process. According to the Poisson specification, each realisation of the dependent variable is drawn from a Poisson distribution characterised by a single parameter X such that Pr(Sr, = k) = exp(-Xrt )Xkrt /k\ k = 0,1,2 (2-2) where, in this particular case, the dependent variable is the number of spills. As in the standard formulation, the parameter X is assumed an exponential function of the influ-ential variables, X, and the parameters, P Xri =exp(XrfP). (2-3) The exponential form of this equation ensures that the parameter X can only take on non-negative values. Equation (2-3) also embodies the standard assumption that X is a function of a linear combination of X and p. Combining equations (2-2) and (2-3) yields the standard Poisson model £(S r t) = Xrt=exp(Xrfp). (2-4) Although widely employed in econometric studies, the Poisson model is often criticised because it embodies the assumption that the variance and mean of the dependent vari-able are equal. As is well known, this is often a poor assumption as most count data are overdispersed (i.e., their variance exceeds their mean). Several tests have been devel-oped to detect the presence of overdispersion and therefore assess the appropriateness 51 of the Poisson model (see Greene 1990 for examples). As is discussed in Section 2.5.2, two such tests have been applied to the data examined in this paper and both indicate the presence of moderate overdispersion. Fortunately, even if the dependent variable exhibits overdispersion, the Poisson model still yields consistent estimates of the pa-rameters (3. Less fortunately, however, it will typically underestimate their standard errors, resulting in spuriously high levels of significance. Within the context of the Poisson model, overdispersion of the dependent variable is often viewed as evidence that an important explanatory variable, or variables, has been omitted from the equation for X. Hausman, Hall, and Griliches (1984; 1986) have pro-posed using the following model to deal with this type of misspecification21 E(Srt) = X.rt =exp(Xrtp + ert). (2-5) This model is particularly useful if e is a gamma-distributed random variable, in which case the dependent variable follows the negative binomial distribution and maximum likelihood estimates of the parameters P are consistent. Unfortunately, if e follows any other distribution, this attractive consistency result is lost. Gourieroux, Montfont, and Trognon (1984) have proposed an even more general method for dealing with count data displaying overdispersion. They suggest using a 21. Alternatively, one could think of spills as being generated by number of distinct processes, each with its own set explanatory variables and error term. For example, some spills are caused by Acts of God (a storm causing failure of a tanker's superstructure), while some are caused by actions that are somewhat discretionary in nature (a tankerowner's decision to save money by hiring an inexperienced crew), while yet others are sim-ply caused by various forms of human error. In reality, however, there are likely to be strong links between these processes. For example, a decision to employ a poorly trained crew increases the chances of human er-ror, and the decision to reduce, or simply not increase, expenditures on tanker maintenance is likely to increase 52 quasi-generalised pseudo-maximum likelihood estimator that gives consistent estimates of P for error terms drawn from a wide variety of distributions. The Gourieroux, Montfont, and Trognon (GMT) estimator is operationally equivalent to a weighted non-linear least squares estimate of the model Srt =exp(Xr(p) + E r ( . (2-6) Each of the models (2-4) through (2-6) offers certain advantages and disadvantages when dealing with data generated by a poorly understood process such as the one ex-amined here. Because there is no single best model, parameter estimates generated by all three models are reported and compared. Estimates of P derived by fitting model (2-6) using weighted non-linear least squares with robust (White) standard errors are also reported. It seems likely that the chance of a tanker having a spill will vary systematically with the geographic region in which it operates. Possible reasons for this type of variation in-clude different weather conditions or navigational challenges. This type of regional heterogeneity may be exphcitly taken into account within the context of the Poisson model (2-4) by introducing a set of regional dummy variables. By doing so, differences between regions are imphcitfy viewed as parametric shifts of the regression function (a fixed-effects model). An alternative to the fixed effects approach is to view the regional effects as random draws from some underlying distribution that may be estimated. This yields a random the chances of a tanker suffering a structural failure during a storm. For these reasons, as well as more prag-53 effects Poisson model of the following form E ( S r t ) = X . r t =exp(X r t p+u r ) , (2-7) where ur is the time-invariant random disturbance characterising the r* region. With an appropriate specification of the within-group correlation structure, equation (2-7) can be estimated by Generalised Linear Model techniques. The choice between the fixed effects and random effects specifications is not straight-forward. The random effects model is more parsimonious in its use of degrees of freedom, yet this advantage is small when considering only a small number of cross-sectional units, as is the case with the data analysed here. However, implicit in the ran-dom effects specification is the assumption that regional effect is uncorrelated with the other regressors. The fixed effects model does not require this strong assumption. On balance, it would seem that the fixed effects model is more appropriate for the problem at hand. Nevertheless, for the sake of completeness, both fixed and random effects models are estimated and the results compared. 2.4 DATA This section describes the data and the sources from which they are drawn. It also dis-cusses the construction of the variables used in the econometric analysis and presents descriptive statistics. marie ones, it seems appropriate to model spills as generated by a single process. 54-2.4.1 Data Sources Three primary sets of data are used: (i) data on tanker spill incidents drawn from the Worldwide Tanker Spill Database compiled and maintained by the Minerals Manage-ment Service of the US Department of the Interior; (ii) data on the seaborne transport of crude oil and petroleum products drawn from figures published by Maritime Strategies International Ltd. of London, England; and (iii) data on the historical size distribution of the world tanker fleet complied by Shipping Statistics and Market Review published by the Institute of Shipping Economics and Logistics, Bremen, Germany. The Minerals Management Service (MMS) of the US Department of the Interior is re-sponsible for leasing offshore tracts of the US outer continental shelf for oil and gas exploration and production. As part of its effort to assess the risk of oil spills associated with these activities, the MMS maintains a database documenting all major oil spills from tankers and barges occurring throughout the world. To qualify for inclusion in the database, an oil spill must meet three major requirements: the source of the spill must be a vessel carrying either crude oil or a petroleum product as cargo; the spill must be the result of an accident (spills resulting from acts of war are excluded); and the spill must result in the release of at least 1,000 bbl of oil. These requirements ensure the MMS da-tabase includes only a small subset of all oil spills, the last requirement being particularly restrictive, since most spills are in fact quite small.22 Nevertheless, consid-ering only large spills in the current analysis is both appropriate and advantageous for reasons discussed in Section 2.6. 22. For example, in US waters in 1993 only 10 of the 5,678 reported oil spills caused by marine vessels were larger than 10,000 gallons. This includes all oil spills occurring in or reaching navigable waters under US juris-diction, including bays, harbours, rivers, lakes, sounds, and oceans (American Petroleum Institute 1995). 55 Whenever possible, the MMS collects a wide range of information about each spill. This includes commonly available data such as spill date and location, and less frequently reported information such as detailed descriptions of the spill's effect on local flora and fauna. Although the full MMS database is generally unavailable to researchers outside the US Government, summary information has been extracted from the full database and published as the MMS Worldwide Taker Spill Database: An Overview (Anderson and Lear 1994). This publication and its supplementary update (Anderson 1995) pro-vide a comprehensive list of all qualifying tanker spills occurring between 1974 and 1994 (barge spills and spills in inland waterways are excluded).23 The date, location, and size of each spill are reported, along with the vessel name and a description of the type of oil. If possible, spill locations are identified by two pieces of information: the Marsden Square in which the spill occurred and a description of the spill site.24 In many cases, this information is enough to pinpoint the location of the spill to within a few miles; in others, the spill's location cannot be determined more precisely than the area encom-passed by the particular Marsden Square.25 Although the MMS database contains reliable information on spills daring back to 1974, MSI's historical seaborne transport data are. available from only 1980 onwards. The MMS database contains 313 pelagic oil spills for the overlapping 1980-1994 period. The locations of these spills are shown in Figure 2-1. 23. International requirements for the reporting of oil spills were upgraded in 1973. Consequently, data col-lected before this date are not generally released by the MMS. 24. Marsden Squares divide the world into a grid of 10-degree by 10-degree squares of longitude and latitude, with the exception of rectangles at each pole. / 25. The exact area contained within a Marsden Square varies with its location, but the average area contained is roughly 160,000 square miles. 5 6 Comprehensive data on the amount of oil or the number of tankers moving along spe-cific shipping routes are generally unavailable. However, Maritime Strategies International Limited (MSI), one of the foremost private consultancies servicing the shipping and tanker industry, does publish estimates of the quantity of oil transported by sea between major trading regions of the world (Maritime Strategies International Ltd. 1995). MSI generates these estimates from historical country-level data on oil con-sumption, production, exports, and imports. MSI first divides the world into ten major oil-exporting and nine major ou-importing regions and then calculates the amount of crude oil and refined petroleum products shipped between each export-region/import-region pair (separate estimates are provided for crude oil and petroleum products). Hence, the MSI data consist of 180 annual region-to-region oil and product shipment volumes. These data are available for the years 1980 to 1994, inclusive. MSI also publishes information about the size distribution of tankers operating between specific oil exporting and oil importing regions.26 Within the shipping industry, tankers are generally classified according to their dead-weight tonnage (dwt). MSI's particular classification scheme places tankers into one of four size classes: Handymax (10-39 Kdwt); Panamax (40-79 Kdwt); Suezmax (80-149 Kdwt); and Very Large Crude Car-rier (above 150 Kdwt).27 MSI provides estimates of the optimal size class, or range of size classes, of tankers operating between particular oil exporting and oil importing re-gions (for example, Suezmax vessels are optimal for transporting crude between North 26. There is a good but imperfect match between the import-export region combinations for which oil ship-ment data are available and import-export region combinations for which the fleet size distribution data are available. 27. One Kdwt equals 1,000 dwt. 58 Africa and Japan).28 In total, MSI provides data on optimal tanker sizes for 55 economi-cally significant routes for the transport of crude oil and 49 economically significant routes for the transport of oil products. The Institute of Shipping Economics and Logistics (ISL), Bremen, Germany, publishes annual statistics on the size distribution of tankers comprising the world tanker fleet. ISL divides the tanker fleet among a large number of size classes according to dead-weight tonnage and provides data on the total number of vessels in each class. ISL also provides the aggregate dead-weight tonnage of all vessels in each class. Table 2-1: Definition of Variables Variable Definition Units Spills Annual number of qualifying oil spills within a specific Region-Year "spill region" Tankers Measure of tanker traffic density associated with a specific Region-Year "spill region" N A W a t e r s Dummy mdicating observations associated with North Region American oil importing or exporting activities P o s t A c t Dummy mdicating the period after the 1990 Oil Pollution Year Act had been passed NAAcrM Interaction of N A W a t e r s and P o s t A c t dummies Region-Year Time Linear time trend Year Note: (1) NAAct = NAWaters * PostAct 28. The optimal vessel size class for a particular route is primarily determined by the length of the route and any special considerations that may be relevant (e.g., Does the tanker have to pass through the Panama Ca-nal?). 59 2.4.2 Construction of Variables Table 2-1 lists the names of all variables used in the empirical analysis and provides definitions of each. Spills and Tankers are the primary variables. Spills is the annual number of major pelagic oil spills occurring within a particular geographic region, or "spill region." Tankers is a proxy for the amount of tanker traffic experienced by a par-ticular spill region in a given year. Before discussing construction of the variables in detail, it is useful to re-examine Fig-ure 2-1. As previously mentioned, this figure shows the locations of all 313 major pelagic oil spills included in the MMS database for the years 1980 through 1994, inclu-sive. Two important observations can be made: first, the vast majority of spills occur relatively close to land; and second, the spills tend to be concentrated geographically, with regions such as Northern Europe and Atlantic Coast of the US experiencing many spills and regions such as Australasia experiencing relatively few. These two facts, along with the restricted nature of the MSI data, greatly influence the construction of variables. To construct the variable S p i l l s , the world is divided into 11 mutually-exclusive geo-graphic "spill regions": Northern Europe and the former USSR; Latin America and the Caribbean; the Atlantic Coast of North America; the Pacific Coast of North America; Southern Europe, North Africa, and the Eastern Mediterranean; the Red Sea; South and East Africa; West Africa; Southeast Asia; Japan, Korea, and China; the Arabian Gulf and other Asia; and Australasia. The decision to divide the world into these particular spill regions, as opposed to any other possible set of regions, is driven by two considerations: 60 first and foremost, the desire to reflect as closely as possible the natural geographic con-centration of spills readily apparent in Figure 2-1; and second, the need to closely match spill regions with the import/export regions for which MSI oil shipment data are avail-able. Spills occurring relatively close to land are assumed the result of either the oil importing or the oil exporting activities of the country closest to the spill site. Most major oil spills do occur relatively close to land, so there are few cases in which it seems unreasonable to associate a specific spill with a specific country or land mass. Spills for which no rea-sonable association can be made, such as those occurring in the mid-Atlantic, are excluded from the analysis. These spills, however, comprise only a small subset of the MMS spills (23 out of 313). Once appropriate spill regions have been defined, and each spill has been associated with a particular importing or exporting country, the construc-tion of Spills is straightforward. Spills is simply the number of spills occurring in each spill region in each year. Since most major oil spills occur relatively close to land, presumably as tankers either leave or enter coastal waters. If so, it seems reasonable to assume that the number of spills occurring close to a particular section of coast will be an increasing function of the total number of tanker voyages into and out of that region. The variable Tankers is an attempt to quantify this measure of tanker traffic density. As already mentioned, MSI provides data on the optimal size of tankers used to trans-port oil between major oil exporting and importing regions. These data are combined with the tanker fleet size distributions published by ISL and used to estimate the aver-61 1 age dead-weight tonnage of tankers transporting oil between spill regions. With these figures in hand, all that is needed to translate the MSI oil shipment data into estimates of the average number of tanker voyages between regions are data on the utilisation of tanker capacity. Since data of this sort are unavailable, utilisation of tanker capacity is assumed stable and therefore proportional to capacity.29 Tankers is the annual sum of all oil-exporting and on-importing tanker voyages associ-ated with each spill region; for example, if, in a given year, 200 oil-carrying tankers leave a spill region and 300 oil-carrying tankers enter the spill region, then Tankers is 500. In addition to the continuous variables already discussed, four dummy variables are used extensively in the analysis: NAWaters, P o s t A c t , N A A c t , and T i m e . The N A W a t e r s dummy identifies the two regions directly affected by the 1990 Oil Pollution Act (the North American Atlantic Coast Region and the North American Pacific Coast Region). These regions encompass the territorial waters of both the United States and Canada. Separate data on seaborne imports of oil and its derivatives to Canada and the United States were unavailable.30 Hence spills associated with both countries were aggregated and coded as if under the direct influence of OPA90. However, the total number of spills included in the two North American spill regions is generally very similar to the number of spills occurring in US territorial waters There are two reasons for expecting the aggregation of US and Canadian spills to have a 29. Although tankers generally operate at less than 100 percent of capacity, capacity utilisation is generally quite stable over time, especially within vessel size classes (Personal communication with D. Crawley of T.K. Shipping, 1995). 30. Unfortunately, the format of the MSI seaborne transport data prevents any further disaggregation of the data. 62 negligible effect on the analysis: (i) the quantity of oil imported by sea to Canada is small compared with the amount imported by the United States; and (ii) the number of spills occurring in Canadian waters during the 1980 to 1994 period is small compared with the number occurring in US waters. If anything, the aggregation of the Canadian and US spill data should make it more difficult to detect the Acts effect on spill frequen-cies, since Canada did not undergo a similar shift in its legislative regime. The P o s t A c t dummy identifies observations associated with the post-Act period, that is, associated with 1991 or later.31 N A A c t is the interaction of the N A W a t e r s and P o s t A c t dummies, and Time is simply a linear trend. Table 2-2: Descriptive Statistics (165 Observations: 15 Years, 11 Spill Regions) Variable Mean Standard Min Max Deviation Spills 1.7575 1.8843 0 9 Tankers 3,807.8 3,130.9 116.9 12,078.6 N A W a t e r s 0.1818 0.3869 0 1 P o s t A c t 0.2667 0.4436 0 1 N A A c t 0.0485 0.2154 0 1 2.4.3 Descriptive Statistics Table 2-2 presents summary statistics describing the datasheet. Both primary variables, Spills and Tankers, exhibit substantial variation. The mean of Spills is 1.7575, implying an average of just over 19 qualifying oil spills per year during the 1980-1994 period. The 31. Only annual seaborne transport data are available. Consequently, August 18,1990 (i.e., the date the Act was passed) to the end of 1990 is treated as part of the pre-Act period. 63 global spill rate over this period is depicted in Figure 2-2, which shows the distribution of the 313 pelagic spills reported in the MMS database.32 A downward trend in the number of spills occurring is apparent, particularly between the years 1980 and 1990. Figure 2-2: Worldwide Major Pelagic Oil Spills, MMS Database, 1980-1994 40 -i 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 Year Much of the variation of Tankers is generated by cross-sectional (i.e., inter-regional) dif-ferences in tanker traffic densities. This simply reflects the fact that some regions, such as Australasia for example, consistently experience light tanker traffic, while other areas, such as Northern Europe, consistently experience heavy traffic. 32. As discussed in Section 2.4.1, only those MMS spills which occurred relatively close to land and so could be associated with the oil-importing or oil-exporting activities of a particular country or landmass are used in the current study. 64 Interpretation of the mean values of the N A W a t e r s and P o s t A c t dummies is straightfor-ward. The value of the N A W a t e r s mean simply indicates that just over 18 percent (i.e., 2/11) of the observations are associated with North American coastal waters, while the value of the P o s t A c t mean indicates that almost 27 percent (i.e., 4/15) of the observations are associated with the post-Act period. Table 2-3: Determinants of Regional Oil Spills (Poisson Regressions) Equation 1 Equation 2 Equation 3 Equation 4 Equation 5 Intercept -3.6669a -3.6707" -3.6565a -3.4165" -3.4512" (0.5507) (0.5507) (0.5530) (0.5627) (0.5643) ln(Tanfcers) 0.5260a 0.5273a 0.5434a 0.5586" 0.5577" (0.0659) (0.0661) (0.0664) (0.0672) (0.0674) NAWaters -0.0327 -0.0404 -0.0438 0.1439 (0.1455) (0.1459) (0.1456) (0.1512) P o s t A c t -0.6593* -0.1756 0.0490 (0.1569) (0.2239) (0.2288) Time -0.0631" -0.0630" (0.0201) (0.0201) N A A c t ( 4 ) -1.9650" (0.7376) R2 0.205 0.205 0.280 0.313 0.336 Log-likelihood -279.0 -279.0 -268.8 -263.9 -257.7 Notes: (1) Standard errors given in parentheses. (2) Superscript a indicates that the parameter is significantly different from zero at the one percent level. (3) R2 is calculated as the squared correlation between the observed number of spills and the number of spills predicted by the Poisson regression. (4) NAAct = NAWaters * PostAct 2.5 EMPIRICAL RESULTS 2.5.1 The Poisson Model Table 2-3 presents the parameter estimates obtained using five alternative specifications of the Poisson model (2-3). In each specification the primary explanatory variable is 65 \ n ( T a n k e r s ) , the natural log of the regional tanker traffic proxy. Equation 1 contains \ n ( T a n k e r s ) as the sole explanatory variable, while Equations 2 through 5 incrementally add the dummy variables N A W a t e r s , P o s t A c t , T i m e , and N A A c t . Before proceeding, it is useful to highlight some of the implications of using the natural log of Tankers in the Poisson model rather than its level. By entering Tankers in logs, equation (2-3) becomes E(Spills) = K r t = exp(Kln[Tflntersr(] + Mrtcp) (2-8) = Tankers*, exp(Mrtcp) where M is the vector of remaining explanatory variables entered in levels and cp is their associated vector of parameters. It is apparent that entering Tankers in logs implies a direct proportionate relationship between Tankers and Spills modified by a set of mul-tiplicative shift variables. It also implies that the elasticity of Spills with respect to Tankers, K, is constant. Keeping this in mind, the first riving to notice about Table 2-3 is that all five specifica-tions of the Poisson model yield very similar estimates of the elasticity of Spills with respect to Tankers—ranging from 0.5260 to 0.5586—and that all five estimates are signifi-cant at the one percent level. Spills is therefore a concave function of Tankers, and the magnitude of the elasticity implies that as the number of oil-bearing tankers entering or leaving a spill region doubles, the number of spills increases by a factor of about 1.45. Put another way, the more tanker traffic a region experiences, the lower the spill risk an additional tanker represents. 66 There are several possible explanations for this result. Because a considerable portion of the variance of Spills is due to cross-sectional (regional) variation, tanker crews may rightfully view regions having higher traffic densities, such as Northern Europe or the Atlantic Coast of the US, as being particularly hazardous and so may be more diligent when operating in those regions; that is, vessel crews may make a greater effort to avoid spills when they pass though busier, and hence riskier, regions. Another possible ex-planation is that high traffic regions may be better regulated and controlled than low traffic regions; for example, navigational support may be better in the English Channel than off the west coast of Africa. It is also possible that the low observed elasticity is at least in part the result of bias in-troduced by measurement error on the independent variable Tankers. As discussed in Section 2.4.2, Tankers is constructed by combining three other variables, all of which are prone to measurement error. As is well known, such measurement errors may bias the estimated coefficient of the variable toward zero. Comparing Equations 1 and 2, we see that the addition of the N A W a t e r s dummy, which identifies the two North American spill regions, does not add to the explanatory power of the model—R2 remains unchanged.33 It appears, therefore, that when differences in traffic densities are accounted for, spills are no more or no less likely to occur inside the two North American spill regions than elsewhere. Equation 3 adds the P o s t A c t dummy, which identifies spills occurring after the Act's 33. For the Poisson models considered, R2 is calculated as the squared correlation between observed number of spills and the number of spills predicted by the Poisson regression. 67 introduction. The negative and highly significant coefficient associated with this vari-able indicates, ceteris p a r i b u s , that regional spill rates were lower during the four years following the introduction of the Act (1991-1994) than during the previous 11 years. An obvious interpretation of this result is that the Act did in fact push down spill rates. Looking back at Figure 2-2 suggests another explanation, however. Figure 2-2 clearly shows a downward trend in the total number of spills occurring worldwide over the period 1980-1994. This gradual decline may be caused by factors such as improvements in marine technology (better navigational aids or safer tanker designs, for example) or improvements in crew training or vessel management. No matter what the cause, given that regional spill frequencies are steadily declining, it seems possible that the sign and significance of the PostAct dummy is the result of com-paring two adjacent periods experiencing a common decline in spill frequencies. To see if this is in fact the case, a linear time trend, Time, is added to the Poisson model in Equation 4. The magnitude of the coefficient associated with Time (-0.063) is highly significant and has the anticipated negative sign. This result supports the conjecture that with differences in tanker traffic densities taken into account, regional spills rates were declining worldwide between 1980 and 1994. As anticipated, adding the linear time trend causes the significance of the PostAct coefficient to disappear. The dummy variable NAAct—the interaction of the PostAct and N A W a t e r s dummies —is added in Equation 5. It is used to test the hypothesis that the Act caused spill rates asso-ciated with North American tanker traffic to drop relative to spill rates associated with tanker traffic in other regions of the world. The large negative and highly significant 68 coefficient associated with NAAct (-1.965) strongly supports this hypothesis, mdicating that the Act has reduced the number of spills occurring in the North American spill re-gions by roughly 85 percent.34 Several mechanisms could have caused this decline. Among them, one in particular, "vessel shifting," merits comment. Vessel shifting is the geographic redistribution of tankers of differing qualities. If a tanker fleet contains both high- and low-quality vessels, the manager of the fleet may respond to an asymmetric increase in spill liability by shifting the fleet's safer vessels into the areas where hability has grown and, correspondingly, shifting the fleef s riskier vessels into other areas. If such vessel shifting had taken place in response to OPA90, we might expect to see a positive and significant coefficient associated with the P o s t A c t dummy in Equation 5. This would indicate that the frequency of spills had increased outside of the North American regions after the Act had been implemented. Although positive, the P o s t A c t coefficient is both close to zero and insignificant; therefore, al-though we cannot rule out the possibility that some vessel shifting has occurred, we find no evidence of it. If we consider the reason for the significance of the P o s t A c t variable in Equation 3, an obvious question arises: Could the significance of the NAAct dummy be caused by a general decline in the frequency of spills in the North American regions over the 1980-1994 period being greater than the decline in other regions? To investigate this possibil-34. We cannot rule out the possibility that some other event may have significantly affected spills rates at about the same time. The most obvious candidate for such an event is the Gulf War, which began in 1990. The war may have significantly affected on the number of tankers travelling along specific routes in the Middle East. However, the variable Tankers explicitly accounts for such changes in tanker traffic and so should control for any resulting effect on spill rates. It should also be remembered that the MMS database explicitly excludes any spills caused by acts of war. 69 ity Equation 5 is estimated a total of nine times, varying only the year in which the Act is assumed to have been introduced (For example, if we assume the Act was introduced in 1986 instead of 1990, then the P o s t A c t dummy is coded zero for years 1980 through 1986 and one for years 1987 through 1994). Obviously, if we assume the Act was introduced in any year other than 1990, estimate Equation 5, and still obtain similar coefficients and confidence intervals, our confidence in the NAAct result is weakened. Table 2-4 presents the results of this sensitivity analysis. Table 2-4: Poisson Regression, Equation 5 (Sensitivity of Parameter Estimates to Assumed Year of Act Introduction) Year of Intercept In(Tanfcers) N A W a t e r s P o s t A c t T i m e N A A c t V ) Introduction 1983 -3.4407" 0.5603" -0.0236 0.2665 -0.0944" -0.0285 (0.5644) (0.0672) (0.2817) (0.2012) (0.0201) (0.3283) 1984 -3.3607" 0.5600" -0.1425 0.0700 -0.0836" 0.1519 (0.5619) (0.0673) (0.2492) (0.2094) (0.0228) (0.3062) 1985 -3.4147" 0.5587" 0.1325 0.0192 -0.0696" -0.3342 (0.5621) (0.0673) (0.2063) (0.2266) (0.0254) (0.2907) 1986 -3.4852" 0.5599" 0.0891 -0.2946 -0.0387 -0.3015 (0.5645) (0.0673) (0.1905) (0.2430) (0.0273) (0.2948) 1987 -3.4477" 0.5592" 0.1360 -0.0285 -0.0622b -0.4816 (0.5667) (0.0673) (0.1796) (0.2471) (0.0277) (0.3078) 1988 -3.4494" 0.5597" 0.0922 -0.0509 -0.0616b -0.4345 (0.5692) (0.0674) (0.1726) (0.2455) (0.0268) (0.3223) 1989 -3.2791" 0.5566" 0.1217 0.4681 -0.1067" -0.5947" (0.5680) (0.0674) (0.1675) (0.2434) (0.0254) (0.3428) 1990 -3.4676" 0.5583" 0.1066 -0.0799 -0.0572" -0.8681b (0.5656) (0.0673) (0.1575) (0.2314) (0.0223) (0.4302) 1991 -3.4512" 0.5577" 0.1439 0.0490 -0.0630" -1.9650" (0.5643) (0.0674) (0.1512) (0.2288) (0.0201) (0.7376) Notes: (1) Standard errors given in parentheses. (2) Superscripts a, b, and c indicate that the parameter is significantly different from zero at the one, five, and ten percent levels, respectively. . (3) NAAct = NAWaters * PostAct 70 Table 2-4 shows that as the year in which the Act is assumed to be introduced ap-proaches the actual year of introduction (i.e., 1991), the NAAct coefficient gradually changes from being insignificantly different from zero, to being significantly different from zero at the one percent level. Encouragingly, the coefficient becomes significant at the one-percent level exactly when the hypothesised year of introduction is equal to the actual year. It seems likely therefore that the NAAct parameter is picking up variation associated with a regime change occurring at the juncture between the pre-Act and post-Act periods, rather than some other underlying trend in the data. Table 2-4 also shows that the coefficients associated with the variables h \ ( T a n k e r s ) and Time are not very sen-sitive to specification of the Act's year of introduction. To summarise, using the Poisson model we have found that the introduction of OPA90 caused a large and significant reduction in the number of large oil spills occurring close to coast of North America (includes Canada and the US). Furthermore, we have found no evidence that the Act had an effect on the number of spills occurring in other regions of the world. 2.5.2 Overdispersion and Alternative Model Specifications As previously discussed in Section 2.3, within the context of Poisson model overdisper-sion of the dependent variable can lead to spuriously high levels of significance on p estimates. Two econometric tests are used to determine if the regional spill data exam-ined in this paper exhibit overdispersion: the regression method suggested by Hausman, Hall, Griliches (1984) and a likelihood ratio test for overdispersion included 71 in STATA's negative binomial routine.35'36 Both tests indicate the existence of a small degree of overdispersion. This immediately raises the question of whether the Poisson model is an appropriate description of the regional spih-generating process. Table 2-5 addresses this issue by exploring the sensitivity of the results obtained using Equation 5 to the Poisson assumption.37 It does this by presenting estimates of Equation 5 obtained using the alternative statistical models discussed in Section 2.3. Table 2-5: Alternative Econometric Specifications Equation 5 Equation 6 Equation 7 Equation Poisson Negative Nonlinear Least GMT Binomial Squares Intercept -3.4512" -3.5381" -2.9101" -3.4258" (0.5643) (0.6237) (0.7171) (0.5870) \n(Jankers) 0.5577" 0.5700" 0.4888" 0.5588" (0.0674) (0.0755) (0.0828) (0.0722) NAWaters 0.1439 0.1438 0.1517 0.2007 (0.1512) (0.1798) (0.1373) (0.2567) PostAct 0.0490 0.0812 -0.0674 0.3081 (0.2288) (0.2614) (0.2666) (0.3121) Time -0.0630" -0.0657" -0.0545" -0.0771* (0.0201) (0.0236) (0.0237) (0.0316) NAAct® -1.9650" -1.9876" -1.7538b -2.3416" (0.7376) (0.7611) (0.9094) (0.6009) Log-likelihood -257.7 -255.0 -304.0 -228.1 Notes: (1) Standard errors given in parentheses. (2) Superscripts a, b, and c indicate that the parameter is significantly different from zero at the one, five, and ten percent levels, respectively. (3) NAAct = NAWaters * PostAct Table 2-5 shows that the parameter estimates obtained using the negative binomial model (2-4) are very similar to those obtained using the Poisson model. The standard 35. STATA's likelihood ratio test rejected the hypothesis that the data were Poisson distributed at the 2 percent level. 36. STATA (release 4.0) is statistical analysis software published by the Stata Corporation. 37. Equation 5 is duplicated from Table 2-5 to allow easy comparison of results. 72 errors are, however, slightly larger than those calculated using the Poisson model, which is exactly what we would expect if the data are slightly overdispersed and caus-ing the Poisson model to underestimate the size of the confidence intervals. Even so, all the parameters that were significant at the one-percent level when estimated using the Poisson model remain so when estimated using the negative binomial model. Equation 7 presents the coefficients obtained by non-linear least squares estimation of model (2-5) using robust standard errors (estimators of this type are consistent but not efficient). Again, the parameter estimates are quite similar to those obtained using the Poisson model. The primary difference being that the NAAct coefficient drops from being significant at the one percent level to being significant at the five percent level. The other coefficients maintain the same level of significance. Finally, Equation 8 provides the results obtained using the quasi-generalised pseudo-maximum likelihood estimator suggested by Gourieroux, Montfont, and Trognon. As-suming the exp(XB) component of the model is correctly specified, this estimator provides consistent and efficient parameter estimates. Again, the results obtained closely match those obtained using the Poisson model. In this case, however, the mag-nitude of the NAAct parameter increases while its associated standard error shrinks. 2.5.3 Regional Heterogeneity As discussed in Section 2.3, there are strong intuitive reasons for believing there will be region-specific heterogeneity in spill rates. Within the context of the Poisson model, this type of time-invariant regional variation is explicitly accounted for in two ways. First, a random effects model of the type described by equation (2-7) is estimated. The results 73 obtained are shown in Table 2-6 as Equation 9. Again, they closely match those ob-tained by estimating the standard Poisson model (Equation 5). Second, a fixed effects model is estimated by introducing regional dummy variables into standard Poisson model (2-4) and dropping the N A W a t e r s dummy. The results obtained by estimating this model are shown in Table 2-6 as Equation 10. As can be seen, the magnitude and confidence intervals of the NAAct and Time dummies are again very similar to those obtained with Equation 5. However, the coefficient associated with ln(Tanfcers), although quite similar in magnitude to that obtained with the standard Pois-son model, is no longer significant. Furthermore, none of the regional dummies is found to be significant. This result appears to indicate the existence of mulncollinearity between the traffic density proxy l n ( T a n k e r s ) and the regional dummies. To check if this is in fact the case, ln(Tanfers) is dropped from the fixed effects model and the model is then re-estimated. The results obtained are shown in Table 2-6 as Equation 11. As can be seen, all but one of the regional dummies how obtain significance at the 5 percent level or greater. Taken together, the results obtained with the fixed effects and random effects Poisson models indicate that the tanker traffic proxy In(Tanfcers) does a good job of accounting for regional heterogeneity in spills rates and that evidence for explicit consideration of time-invariant cross-sectional variation is weak. 74 2.6 DISCUSSION A N D C O N C L U S I O N S This study examines spills larger than 1,000 bbl (42,000 gallons) only. This restriction requires two comments. First, although OPA90 affects all oil spills occurring in US ter-ritorial waters, the primary objective of the Act is presumably to reduce the number of large oil spills occurring. If so, restricting the analysis to large spills seems appropriate. Second, considering only large spills offers at least one advantage over considering a greater range of spill sizes: the data are likely to be more accurate. This is simply be-cause the larger the spill, the more unlikely it is to avoid detection or to go unreported. The results of the empirical analysis indicate that the 1990 Oil Pollution Act has resulted in a significant decline in the number of major oil spills associated with tanker traffic entering or leaving North American coastal waters. Thus, it would seem that contrary to the expectations of many industry observers, the Act has had its intended effect. The results also indicate that the Act has had, as of yet, no discernible effect on spill frequen-cies outside of North American coastal waters. Therefore, there is no evidence that the decline in spill frequencies in North American waters is simply due to tanker owners responding to the Act by moving their safest tankers and crews to service the US market while simultaneously moving their higher risk vessels to other parts of the world. Hence, the results lend little support to the "vessel shifting" hypothesis. In contrast, the results do seem to indicate that the Act has generated significant local effect. Furthermore, given the speed with which the spill rate in North American waters seems to have fallen in response to the change in the regulatory regime, it is unlikely that the decline is primarily the result of tanker owners phasing out their riskier tankers 75 or upgrading capital. It seems more plausible that tanker owners are simply ensuring that their crews take more care. This paper also sheds some light on the general question of what drives the regulatory process. One of the main theories explaining the evolution of industrial regulation is the "regulatory capture hypothesis." In essence, the capture hypothesis says that firms in-fluence the regulatory process, through lobbying and other similar activities, in ways that ensure that the regulations the industry faces are those that serve its own best inter-ests, rather than the best interests of the public. The tanker industry's experience with the 1990 Oil Pollution Act does not support this hypothesis. It seems clear that the Act was induced as a direct response to pressure exerted by both the general public and various environmental special interest groups in the wake of the Exxon Valdez disaster. Of course, many of the regulations introduced under the Act, such as mandatory double hulling, were being considered for introduction by the regu-latory authority for some time before disaster. But many of the regulations were encountering substantial opposition from the industry at the same time (the aforemen-tioned double hulling being a good example). Indeed, it seems certain that the disaster not only helped ensure the introduction of the regulations but also helped speed up their implementation. Furthermore, the empirical results indicate that the Act had the desired effect of reducing spill frequencies, presumably by forcing tanker owners to in-ternalise certain costs that they had previously chosen to externalise through higher spill frequencies. In conclusion then, this paper tends to lend support to the public interest and public choice theories of regulation, while at the same time undermining the gen-eral apphcability of the capture hypothesis . 76 Chapter 3 The Zinc Industry, Price Participation, and Profitability 3.1 I N T R O D U C T I O N Recently, economists, policy makers, and business practitioners alike have all shown a keen and growing interest in the structure and properties of contractual relationships existing between parties in successive stages of vertical production chains. In contrast to the market for final goods, where a single uniform price is the norm, the types of contract or v e r t i c a l relationships that govern the pricing and transfer of interme-diate goods are often considerably more complex.38 To date research on vertical relationships has been unusually pragmatic in its orientation. Indeed, most work has focussed on the types of contract actually used in practice and on deterrnining the impli-cations of these arrangements for competition in the marketplace. Considerable energy has also been devoted to identifying appropriate policy responses. The vertical relation-ships that have thus far received considerable attention include, among others, quantity-dependent pricing, tying contracts, and royalty systems. However, the range of vertical 77 relationships used in practice is much larger than this short list might indicate/ This paper contributes to the existing literature on vertical relationships by exploring the properties of an unusual and commercially important vertical relationship that is widely used in the non-ferrous metals industries. The relationship in question, often called the price p a r t i c i p a t i o n system, is a mechanism for sharing the profits generated in a final goods market between the producers of the final good and their intermediate good sup-pliers. The following description provides an outline of the system's basic structure:40 An upstream firm supplies an intermediate good to a downstream firm and pays a fee for each unit of the good the downstream firm processes. The fee is initially set to some fraction of the expected price of the final good, but if the price obtained for the final good differs from the expected price, the fee is ad-justed. The fee is increased if the realised price of the final good rises above the expected price and is reduced if the realised price falls below the expected price. The downstream firm produces the final good, sells it for the best available price, and passes the majority of the revenue generated back to the supplier of the intermediate good. From this description, it is clear that the price participation system effectively insulates the downstream firm against some of the risk associated with price uncertainty by passing some of that risk back to the supplier of the intermediate good. Hence, it would 38. For an excellent discussion of the differences between markets for intermediate and final goods see Katz (1989). 39. See Katz (1989) for a more exhaustive list. 40. A more detailed description of the price participation scheme is provided in Section 2. 78 seem, at least a priori, that one of the main functions of the price participation system is to share risk between the producer of the final good and the supplier of the intermediate good. What makes the price participation system particularly mteresting is not only its com-mercial significance, but the fact that if the producers of final good possess some degree of market power the system appears to provide incentives for firms to react sub-optimally to any negative demand shock that they might face. This problem is best il-lustrated by example. Say the producers of the final good were to face a demand shock resulting in downward pressure on the price of their product. Their profit-maximising response would be to cut output to some extent to stop the price falling too far. Of course, the amount by which a particular producer would reduce their output depends on the extent to which the price drop would affect their own profits. It is here that the price participation sys-tem plays a role. By passing some of the negative effect of a price decline back to the producer of the intermediate good, the system ensures that the producers of the final good do not cut their output to the extent that they would if they faced the full effect of the price change. Of course, all producers of the final good face the same incentives and all behave in the same way. The profits of the entire industry (i.e., the aggregate profits of the final good and intermediate good producers) suffer as a result. Of course, a similar problem exists in, and is in fact the very essence of, the standard Cournot oligopoly problem. When a Cournot firm considers increasing production, it considers the negative effect that such a decision would have on its own profitability, 79 but not the effect it would have on the profitability of rival firms. As a result, firms find themselves trapped in a form of the Prisoner's Dilemma, in which all tend to overpro-duce. Of course, without some form of explicit or tacit collusion—which is typically illegal—this problem is unavoidable. Nevertheless, the price participation system seems to make things worse by magnifying each firm's incentive to overproduce. Yet, the fact remains that price participation systems have been used extensively by certain industries for close to a century. The primary goal of this paper is to identify the private incentives of firms to use such contracts. More specifically, this paper explores the possibility that risk aversion on the part of final good producers, and significant un-certainty about the strength of demand for their product, may be able to explain, at least in part, the use of price participation systems. The paper takes the following approach. First, a model of the particular price participa-tion system used by the zinc industry is developed. The reasons for doing this and for not conducting a more general analysis are threefold: first, the zinc industry is a promi-nent example of an important and economically significant industry that uses the price participation system; second, there are strong reasons to suspect that final goods pro-ducers in the zinc industry have some degree of market power; and third, a wealth of information and data on the structure and performance of the zinc industry is readily available.41 Once the model has been developed, it is calibrated with data drawn from the zinc industry. The calibrated model is then used to simulate industry performance under a wide range of assumptions about market structure, demand uncertainty, and 41. In 1994, just over 5.8 million tonnes of zinc were sold in the Western World alone at an average price of $998, and approximately 10 percent of the zinc was exported to the West from former Eastern Bloc countries; 80 risk aversion. Of course, the only way to judge the merit of a particular institutional arrangement such as price participation is to compare its performance relative to that of some practical al-ternative. To make such a comparison, standard input pricing is adopted as a alternative benchmark contract; this is the typical arrangement where downstream firms face uni-form posted prices for the intermediate goods they buy. This arrangement is often held up as the typical form of contract in final good markets (Katz, 1989); however, it is also common in intermediate good markets. More importantly, it is an arrangement that could be easily adopted by the final good and intermediate good producers, at least in principal. This paper is largely motivated by earlier work by Brander (1996) on price participation contracts within the context of the zinc industry. Additional motivation is provided by the fact that after almost a century of using price participation schemes, the zinc indus-try is currently considering swtching to more conventional pricing contracts. If such a change were to occur, it could have far-reaching implications for consumer welfare and international trade. A detailed understanding of the operation of price participation is therefore needed if appropriate competition policy governing such a transition is to be formulated and implemented. This paper makes three primary contributions. First, it extends Brander's (1996) model of oligopoly under price participation by explicitly introducing stochastic demand and by allowing for the possibility of risk aversion on the part of the output-contiolling firms. In doing so, it significantly expands upon previous work exanuning this com-81 mercially important vertical relationship. Second, it identifies the general conditions under which industry participants may have private incentives to use a price participa-tion system rather than a more conventional standard input pricing contract. Third, it is the first analysis to examine the implications of the zinc industry's proposed move from price participation to more conventional pricing contracts. The results indicate that given current market conditions, such a change would likely improve industry profits, but at the cost of higher consumer prices. Although no definitive conclusions about welfare effects are reached, the results do at least provide some guidance for policymak-ers. The rest of the paper is organised as follows. Section 3.2 provides a brief overview of the zinc industry and a more detailed description of the price participation system. Sec-tion 3.3 then develops two models of the price participation system: the first treats downstream firms as risk-neutral profit maximisers; the second examines the more gen-eral case where downstream firms behave as risk-averse utility maximisers. The model is calibrated in Section 3.4 using data drawn from the zinc industry. Simulation results are reported and discussed in Section 3.5, and conclusions are drawn in Section 3.6. 3.2 T H E Z I N C INDUSTRY A N D PRICE PARTICIPATION Zinc is a highly malleable, bluish-white metal, possessing a relatively low melting point and good conductivity. It is easily oxidised when exposed to air, rapidly forming a thin film on its surface that acts as an effective barrier against further oxidation. It has many useful physiochemical properties and is consequently found in a wide range of products including, among others, vitamins, cosmetics, paints, fertilisers, and batteries. 82 Zinc's most economically significant use stems directly from its desirable oxidation characteristics. The process of galvanisation involves coating iron or steel with a thin layer of zinc to protect it against corrosion, either by dipping in molten zinc or by elec-troplating. Once a material has been galvanised, it will not rust as long as the coating's physical integrity is maintained. Other important uses of zinc include alloying with copper to form brass, which is extensively used in the fabrication of medical equipment and ornamental objects; and precision die-casting, particularly in the fabrication of me-chanical parts and accessories used by the auto industry. It is also used extensively by the construction industry as a roofing and guttering sealant. The production of zinc can be thought of as taking place in two physically and economi-cally distinct phases: a mining phase, during which ores are removed from the earth and converted into zinc concentrate; and a smelting phase, during which the zinc con-centrate is transformed into refined zinc.42 Smelters are classified as either integrated smelters or custom smelters depending on their primary source of zinc concentrate. Integrated smelters obtain most of their concentrate from their own mine or mines, while custom smelters rely on independent mines to meet their concentrate requirements. Somewhat surprisingly, both integrated and cus-tom smelters generally enter into similar long-term contracts with mines for concentrate supplies 4 3 Within the integrated sector of the industry, these contracts are essentially 42. To clearly differentiate it from the dissolved zinc contained in concentrate, the final product of smelters— that is the solid zinc suitable for sale on the London Metals Exchange—is referred to as refined zinc rather than simply zinc. 43. Integrated and custom may use the same type of contract for strategic reasons. For example, if smelters are Cournot competitors and integrated smelters were to unilaterally adopt an alternative arrangement that caused them to cut their output, then custom smelters would respond by expanding their output. This potential re-83 transfer pricing agreements. Nevertheless, the similarity of the contracts used in both the integrated and custom sectors of the industry ensures that there is little difference in incentives facing both types of smelter. Presently, about 60 percent of the zinc produced in the Western World is produced by custom smelters, about 35 percent is produced by integrated smelters, and the remaining 5 percent comes from secondary sources, such as recycling (Hamilton and Yates, 1994). Under the terms of a typical price participation contract, the smelter pays for only 85 percent of the zinc contained in the concentrate it, receives from the mine, with the price of this payable zinc being the current price of refined zinc on the London Metals Ex-change (LME). While the smelter pays for only 85 percent of the zinc it receives in concentrate, it typically recovers between 95 and 96 percent. The result is that about 10 percent of the zinc smelters typically receive today is free zinc; that is, it is zinc that smelters recover but do not have to pay for.44 The sale of free zinc is one of the two primary sources of smelter revenue. The other, and far more important source of revenue is a fee the smelter receives from the mine for processing the zinc concentrate into refined zinc. This treatment charge is initially set to some fraction of the zinc basis price, that is, the expected price of refined zinc on the LME over the coming year. The treatment charge is adjusted if the price of zinc differs from sponse might erode the incentives for either integrated or custom smelters to unilaterally switch to a more efficient arrangement (that is, an arrangement that would make all smelters better off if all smelters were to adopt it). 44. The fact that the percentage of payable zinc has not changed even as the percentage of recoverable zinc has risen can be partly explained on the basis of incentive effects. Allowing the smelters to keep the revenues from the sale of all the zinc they recover in excess of the 85 percent that is payable gives smelters a strong incentive to improve recovery efficiencies. Although this might explain a difference between the fraction of zinc that is payable and the fraction that is recoverable, it cannot explain why the payable fraction has not been adjusted at all over the years. 84 the basis price according to an escalator. If the LME price is more than the basis price, the treatment charge is increased, and if the LME price is less than the basis price, the treatment charge is reduced. It is this mechanism for the sharing the costs and benefits of changes in the price of refined zinc between the smelters and mines that gives the price participation system its name. Despite a steadily increasing demand for its product, the zinc industry in the Western World has recently grown concerned about its profitability relative to that of other non-ferrous metal industries. Many factors have contributed to the industry's relatively poor financial performance. They include the increasing efficiency of zinc use; the develop-ment of substitute materials for use in certain traditionally zinc-intensive applications, such as precision die-casting; and the transformation of some former Eastern Bloc coun-tries from net importers of zinc to net exporters. Although most of these factors are beyond the industry's direct control, Brander (1996) has argued that the use of price participation contracts between zinc mines and smelters makes it difficult for the indus-try to effectively respond to external events such as variations in the business cycle. This is important because one of the defining characteristics of the zinc industry is that it faces a highly uncertain price for its final product, refined zinc. For example, over the last 15 years, changes in annual average real price of zinc of 20 percent or more have been common and more extreme swings have occurred. Furthermore, the price of re-fined also exhibits considerable short-term variability, often rising or falling by as much as 10 or 20 percent in the space of a single week (International Lead and Zinc Study Group, 1994). An important aspect of the zinc industry, particularly from the perspective of under-85 standing the imputations of price participation schemes, is the extent to which mines and smelters share control over the quantity of refined zinc produced. The price par-ticipation system has been used by the zinc industry for close to a century, and for much of that time the traditional view within the industry has been that mines exercise more control than smelters over industry activity, with smelters, at least to some degree, act-ing solely as "pass-through" agents for the mines. Today, however, there is good reason to doubt this conclusion. In recent years, independent traders have been playing an in-creasingly important role in meeting the marginal demands of smelters for concentrate. And although most custom smelters typically receive most of their concentrate from particular mines as part of long-term contracts, if the smelters were to decide to expand output, they could simply obtain more concentrate from these independent traders. The smelters' ability to do this implies that they now have considerable independence from mining companies in their production decisions. It is also mteresting to note that under the terms of most long-term contracts, smelters may reduce production below the con-tracted level by simply claiming economic need. Another significant change that has occurred in the industry since price participation contracts were first introduced has been the increasing importance of free zinc as a source of revenue for smelters. When price participation contracts were first negotiated, smelting yields were closer to 85 percent and so all recovered zinc was payable. But over time, as technological progress drove up smelter yields to close to the present 95 or 96 percent, contract negotiations left the fraction of payable zinc unchanged, with the result that 10 percent of the zinc contained in concentrate is now free zinc. In the next section we assume that smelters decide how much refined zinc to produce 86 and that mines supply them with as much concentrate as they need under the terms of the negotiated price participation contract. This assumption reflects the belief that smelters exercise more control than mines over the industry output. It also helps keep the model relatively simple. 3 . 3 T H E M O D E L The smelting sector consists of n smelting firms indexed i = l,...,n, each producing tonnes of refined zinc. The smelting sector's aggregate production of refined zinc, X, is therefore simply All smelters are assumed to be identical. This allows the behaviour of the entire smelt-ing sector and all the firms that comprise it to be fully characterised by examining the actions of a single representative smelter.45-46 The representative smelter—called simply the smelter from this point on—has two sources of revenue: the sale of free zinc and the treatment charge. Revenue raised from the sale of free zinc, F, is just some fraction, co, of the market value of the quantity of re-fined zinc produced 45. For convenience, I use the terms smelter and smelting firm interchangeably throughout this paper. As long as smelting firms are identical, the analysis is equally valid for a smelting sector consisting of single-smelter or multiple-smelter firms. 46. The main results of the analysis are likely to be robust to a more general specification of the model involv-ing heterogeneous firms. 87 F = copx, (3-2) where p is the market price of refined zinc. The smelter is also paid t dollars for proc-essing each tonne of concentrate; therefore, the total revenue raised from the treatment charge, T, is simply where q is the quantity of zinc concentrate processed. The per-unit treatment charge, t, is initially set to some fraction, T , of the expected price of refined zinc, p°, over the coming year. However, if the expected price, called also the basis price, differs from the market price, the treatment charge is adjusted according to the following formula where K is the treatment charge escalator. From this it is clear that if the market price exceeds the basis price (p > p°), the treatment charge is increased; and if the market price falls below the basis price (p < p°), the treatment charge is decreased. Combining (3-3) and (3^ 4) we find that the treatment charge generates revenue T = tq (3-3) r = Tp°+K(p-p°), (3-4) T = xp°q + K(p-p°)q (3-5) when the smelter processes q tonnes of concentrate. 88 At this point, it is useful to explicitly define the relationship between the amount of con-centrate treated by the smelter, q, and the amount of refined zinc produced, x. Holding factors such as smelting technology and concentrate composition constant, the relation-ship may be written as x (3-6) where u is the weight fraction of the zinc in the concentrate and X is the fraction of zinc in the concentrate that can be recovered via smelting.47 We can use this relationship to express the treatment charge revenue as a function of the quantity of refined zinc pro-duced rather than the quantity of concentrate processed T _ T A , K ( P - P V ( 3 _ 7 ) X[i Xp. We now examine the smelter's costs. Under price participation, the smelter does not, in effect, pay for the zinc concentrate it uses.48 It does, however, pay for the countless other factors it uses in the smelting process. Furthermore, it is convenient to represent these other factors by a single aggregate factor, C, and to set its price to one. The variable C then represents both the quantity of the aggregate factor used and its cost. The quantity of the aggregate factor used is, of course, an increasing function of output (i.e., C = C(x), 47. For example, if the weight fraction were 0.5, then one tonne of concentrate would contain 0.5 tonnes of zinc. 48. In reality, the smelter pays the mine the market price of refined zinc for the majority of the zinc it receives in the concentrate. It then process the concentrate, sells the resulting refined zinc at the market price, and re-tains the revenue. Assuming that market prices are stable over the transaction period and that transactions costs are negligible, this arrangement is equivalent to one in which the smelter neither pays for the concentrate it receives nor receives the revenues generated through the sale of the metal it produces. 89 with C > 0). Furthermore, it is assumed that smelting exhibits constant returns to scale, implying C(x) = csx, where cs is both the marginal cost of producing one tonne of re-fined zinc and the quantity of aggregate factor required to do so. Bringing together revenues and costs, the profit of the representative smelter, n, is given by T P ° X K(p-p°)x , (3-8) 7t = apx + —— + ————-—c x The market-clearing price of refined zinc, p, is a decreasing function of smelting sector's aggregate production of zinc, X, and an increasing function of a random variable, e, that indexes the strength of demand. Formally, 5p(X, e) / DX < 0 and 9p(X, e) / <5e > 0. We assume that p is determined according to the following inverse-demand function f«-b(X + Z) + e if a > b ( X + Z ) + e (3-9) P ( ' 8 ) ' 0 if a<b(X + Z) + e where a and b are constants, and Z is the net amount of refined zinc imported from countries in the former Eastern Bloc.49-50 The random variable 6 is discretely distributed 49. Assuming the error term enters the inverse-demand function additively in this way is equivalent to as-suming the choke price is a random variable. This is not the only reasonable way in which uncertainty could be incorporated into the demand function. An alternative specification would have slope of the demand curve, b, as a random variable. Of course, the exact way in which uncertainty enters the demand function could influ-ence the results; however, assuming the choke price is random, as has been done here, is the most common way of characterising uncertainty. Furthermore, the sensitivity analysis carried out on the demand function parameters in Section 3.5.3 indicates that the varying the choke price has a larger effect on the simulation re-sults than does varying the slope of the demand function. 50. When modelling the zinc industry in the Western World, we must take into account the effect of imports from, or exports to, other parts of the world. In reality, the only significant producers of refined zinc outside the control of the western zinc industry are located in the former Eastern Bloc countries. 90 and can realise one of two values, E or E , with equal probability. Thus, E has probabil-ity distribution, g(s), given by |Prob(e = E) = l/2 (3-10) |Prob(E = e) = l/2' where e > e. Thus, if 8 is realised, demand for zinc is relatively strong, but if e is real-ised, demand is relatively weak. It is also assumed that s and E are symmetric, so £(e) = 0.51 Smelters are assumed to be either expected profit maximisers or risk-averse expected utility maximisers. Risk averse smelters are assumed to possesses the following con-stant absolute risk aversion utility function 17(TI) = z - y exp(-ra), (3-11) where n is given by (3-8); z and y are constants, with y > 0; and r is the Arrow-Pratt measure of absolute risk aversion.52-53 The value of the r indicates the smelter's degree 51. I assume e is discretely distributed and can take on one of only two values to keep the model relatively simple. Nevertheless, I believe that the qualitative results of the analysis would not change if I were to adopt a more realistic assumption about the distribution of s. Consider, for example, the likely effect on the smelter's output decision of assuming e uniformly distributed rather than discretely with the same support. With a uniform distribution, there is much greater chance that any realisation of E will be closer to E(e). Consequently, a risk averse smelter would still reduce production below the risk-neutral level, but not to the same extent that it would if it faced the discretely distributed demand. Hence, adopting the more realistic assumption of a uni-formly-distributed E is likely to affect the quantitative, but not the qualitative, results derived from the model. 52. The constant absolute risk aversion utility function has proved to be very popular in both practical and theoretical work. Its function form ensures that r has the following desirable properties: (i) r is larger for a more risk averse smelter than for a less risk averse smelter and (ii) r is unaffected by a arbitrary linear trans-formation of the utility function. See Hey (1978) for a fuller discussion of the utility function's properties and the reasons for its widespread use. 91 of risk aversion, with a larger value of r mdicating a more risk-averse smelter. The smelter's production decision is modelled as a two-stage game, with each stage con-sisting of two distinct phases. Events unfold as follows. In the first phase of Stage 1, mines and smelters take part in contract negotiations during which they select values for the free zinc parameter, co, the treatment charge parameter, x, and the treatment charge escalator, K. These negotiations are beyond the scope of the current analysis, so the val-ues of these parameters are treated as predetermined. Once values for co, x, and K have been determined, the second phase of Stage 1 begins. During this phase, the basis price, p ° , is set to the expected price of zinc in Stage 2; that is, the basis price is set to the price of refined zinc that would prevail in Stage 2 if the random variable e were to realise its expected value of zero. Once the treatment charge parameters, co, x, and K , and the basis price, p°, have been determined, the first phase of Stage 2 begins. At this point, the smelter knows the prob-ability distribution of e but does not what value it will realise; consequently, the smelter maximises either its expected profit £(TC) = ^ 7 I / 2 (3-12) if risk neutral, or its expected utility as defined by its von Neumann-Morgenstern ex-pected utility function 53. As r approaches zero, the behaviour of expected utility maximising smelters facing uncertain demand approaches that of expected profit maximisers. 92 E(U) = £[z-yexp(-m)]/2 E (3-13) if risk averse. Production takes place and the smelter places its output on the market. The second phase of Stage 2 then begins, during which the uncertainty surrounding market demand is resolved (i.e., e is realised) and profits are determined. This sequence of events is represented graphically in Figure 3-1. Figure 3 - 1 : Sequence of Events STAGE 1 Phase 1 Phase 2 STAGE 2 Phase 1 Phase 2 co, x, and K p° set to determined in E[p] bargaining between mines and smelters smelters choose e realised and output levels profits determined The game's equilibrium must be sequentially rational, so the price of zinc that would prevail in Stage 2 if e realised its expected value must equal the basis price chosen in Stage 1. To ensure this, those setting the basis price in Stage 1 must correctly anticipate the behaviour of smelters in Stage 2; that is, they must look forward to Stage 2, solve the smelters' problem, and use the information they gain to determine the appropriate basis price to set. The smelters' production decision must, therefore, be solved by backward induction. In the following two sections, (3.3.1) and (3.3.2), the smelter's problem is solved under the assumption of risk neutrality and then under the assumption of risk aversion, re-93 spectively. The cases are examined separately because the assumption of risk neutrality allows closed-form solutions for the equilibrium output and other variables of interest to be derived, whereas the assumption of risk aversion does not. Under the assumption of risk aversion, only implicit solutions are obtained. 3.3.1 Production: Risk-Neutral Firms If a smelter is risk-neutral, its objective of maximising expected utility is equivalent to that of maximising expected profits. We start by considering the decision facing a rep-resentative smelter in the first phase of Stage 2. At this point the smelter knows the value of co, K , T and also the basis price p°; therefore, taking these parameters and the output of all other smelters as given, the smelter chooses its production level to maxi-mise its expected profit, TC. The smelter's problem is to max XV°X K(v-p°)x s ®px + ——+ r -csx Xu X\x (3-14) subject to the inverse-demand function (3-9). Equivalently, the smelter's problem may be expressed as max co,E[P] + ^ +K ( £ [ p ] - p )X-c*x X\x X\x. (3-15) The first order condition imphcitly defining the smelter's optimal output decision is given by 94 TO0 K(dE[p]/dx + p - p ° ) x . n (3-16) (0(dE[p]dx + p) + - ^ + V m i V ) -cs=0. Since p = a - b ( X + Z) + e, E(e) = 0, and X = ^T" x,, its follows immediately that for any particular smelter dE[p]/dx = - b . (3-17) Also, since all smelters are identical, in equilibrium the symmetry condition X = nx (3-18) must hold, where n is the number of smelters in the industry. Substituting (3-17) into (3-16) and combining the result with (3-18) yields an expression describing the smelter's equilibrium output, x*, as a function of the basis price, p ° , and the other predetermined and exogenous parameters g + bZ xp°-Kp°-XUc5 (3-19) X ~ b(n +1) + b(n + l)(a>k\i + k)' Now, if p° was correctly chosen in Stage 1 and E realised its expected value of zero, then in Stage 2 the market price for refined zinc would be equal to the basis price and no treatment charge adjustment would be necessary. If so 95 p ° = p = a- b ( n x * +Z), (3-20) which implies a - p ° - b Z (3-21) x* = - . nb Equating the right hand sides of equations (3-19) and (3-21) and solving for p° yields o (a-bZ)(a'kii + K) + nX\iCs (3-22) ^ (n + l)co\u + nx + K The basis price, p ° , can now be eliminated from the smelter's production function (3-19). Doing so results in the following expression for the equilibrium output of the repre-sentative smelter in terms of only the predetermined parameters co, T, and K , and the exogenous parameters a, b, Z, n and c5 i r _ ( a - b Z ) ( a > \ \ i + x)-\\ics (3-23) b ( n + + b(nx + K) It is immediately apparent that the smelter's equilibrium output is increasing in the zinc choke price, a, decreasing in the marginal cost of production, cs, and decreasing in the amount of zinc imported from counties in the former Eastern Bloc, Z. 3.3.2 Production: Risk-Averse Firms We again start by considering the decision facing a representative smelter in the first 96 phase of Stage 2. Taking co, K , T, p°, and the output of all other smelters as given, the risk-averse smelter chooses its production level to maximise its expected utility, U. The smelter's problem is to max E(U) = ^ [z - yexp(-rn)]/2 (3-24) subject to the inverse-demand function (3-9). The first order condition implicitly defin-ing the smelter's optimal output decision is given by — ntyexp(-m) dx The non-linearity introduced by the functional form of the utility function precludes explicitly solving for the smelter's equilibrium output, x*. Nevertheless, x* is implic-itly defined by the simultaneous solution of the first order condition (3-25), the linear demand function (3-9), the symmetry condition (3-18), and the sequential rationality constraint (3-20). 3.3.3 Fi rm, Sector, and Industry Profitability Once the smelter's equilibrium output is known, its profit as function of the random variable e can be immediately calculated using (3-8). The profit of the smelting sector, IT, is simply n times the profit of a single smelter, and the profit of the entire zinc in-dustry (i.e., the combined smelting and mining sectors), n, is given by 97 n = npx* -nx*cs -nc m (3-26) where c m is the marginal cost of mining and processing enough ore to produce one tonne of refined zinc. The profit of the mining sector, Um, is simply Before simulations can be carried out using the model of the price participation system developed in the previous section, values of the physical parameters co, X, and u, the contract parameters K and t, the demand parameters a and b, and the marginal costs of mining and smelting, cm and cs, and the utility function parameters, z and y, are re-quired. The model was calibrated using 1994 data drawn from a number of different sources. Estimates of typical values of the physical parameters co, X, and u were taken from Hamilton and Yeats (1994) and Brander (1996). The fraction of zinc recovered from the concentrate via smelting depends on many factors including the presence of trace ele-ments in the concentrate and the particular smelting technology employed. Nevertheless, data suggest that modern smelters commonly recover between 95 or 96 percent of the zinc contained in concentrate; a value 0.95 was therefore adopted for X. The weight fraction of zinc in concentrate also varies, but a fairly typical concentrate contains about 60 percent zinc by weight (i.e., u = 0.6). Also, under the terms of a typical price participation contract, only 85 percent of the zinc contained in the concentrate is r r = n-n s. (3-27) 3.4 MODEL CALIBRATION 98 payable (Hamilton and Yeats, 1994); that is, the smelter must pay for only 85 percent of the zinc it receives from the mine. This, combined with the assumption that smelters typically recover 95 percent of the zinc in the concentrate, implies that 10 percent of zinc in the concentrate is free zinc (i.e., co = 0.1). Values of the price participation parameters K and x were taken from a zinc industry price participation contract that was negotiated and implemented in 1993, but that was still in force in 1994. The contract used is believed to be fairly typical of the contracts that were negotiated during that period. The contract specifies a discrete schedule be-tween the appropriate treatment charge and current the market price of refined zinc. It does, however, also identify the basis price (i.e., the expected price of refined zinc) and the treatment charge to be paid if that price were to be realised. Under the contract, at a basis price of $1,000 per tonne, the treatment charge was approximately 42 percent of the market value of the payable zinc contained in the concentrate. From this and the values of the physical parameters co, X, and ji we can calculate the corresponding value of the treatment charge parameter x: x = 0.42 * (X - co) * u = 0.42 * (0.95 - 0.1) * 0.6 = 0.214. Similarly, the price escalator K was estimated from the schedule to have an approximate value of 0.15. The zinc demand parameters a and b were calculated using 1994 zinc market data taken from Lead and Zinc Statistics (1994) and the elasticity of demand for refined zinc esti-mated by Gupta (1982). In 1994,5.84 million tonnes of refined zinc was purchased in the 99 Western World at an average price of $998.24. Of this, western smelters supplied ap-proximately 90 percent (5.28 million tonnes), while former Eastern Bloc countries supplied the remaining 10 percent (0.55 nullion tonnes). A linear demand function was fitted through the observed the price-quantity combination (5.84 million tonnes, $998.24). Specific estimates of a and b were then obtained by assuming the demand curve had an elasticity of -0.8 (Gupta, 1982) at that point. Doing so yielded estimates of $2,246 and $0,214 per 1000 tonnes for a and b, respectively. Table 3 - 1 : Values of Parameters used in Calibrated Model Variable Description co Fraction of zinc in concentrate recovered via smelting Weight fraction of zinc in concentrate 0.60 Fraction of free zinc 0.10 Treatment charge parameter 0.214 Price escalator 0.15 Linear inverse demand parameter (choke price) 2,246 Linear inverse demand parameter (slope) 0.214 Smelting cost per tonne of zinc metal produced 423 Mining cost per tonne of zinc metal produced 432 Value Units 0.95 $ $/ 103 tonnes $ $ Data on the costs of smelting and nuning in the Western World were taken from Brook Hunt (1994). Average C l smelting and mining costs of $423 per tonne and $432 per 100 tonne were used as estimates of cs and c m , respectively. C l costs include variable costs, but exclude depreciation allowances on fixed capital assets. The only restriction that must be met by the parameters z and y for the Arrow-Pratt con-stant absolute risk aversion utility function to have its desirable properties is that y > 0. For convenience, the parameters are both set to one. Hence, smelter utility can take on a maximum value of one, but is unbounded from below. The values and units of the parameters used in the calibrated model are summarised in Table 3-1. 3.5 S I M U L A T I O N RESULTS A N D DISCUSSION In this section we attempt to identify the circumstances under which the price participa-tion system might be regarded as in some way superior an alternative standard input pricing arrangement. More specifically, we explore the conjecture that the presence of risk aversion on the part of smelters and significant demand uncertainty may go at least some way toward explaining why the price participation system is used by the zinc in-dustry. We first simulate industry behaviour under price participation for a range of model parameterisations and then use the results to make some observations about the general properties of this type of vertical relationship. We also simulate industry be-haviour under the alternative arrangement of standard input pricing and discuss the appropriateness of different assumptions that might be made about pricing of the in-termediate good, zinc concentrate. We compare profits and utilities under the two alternative arrangements, holding smelter preferences and demand uncertainty con-101 stant. Finally, we examine the sensitivity of the simulation results to the values of cer-tain model parameters. 3.5.1 Pr ice P a r t i c i p a t i o n We now examine how the expected utility of smelters is affected by changes in their at-titudes toward risk and changes in the magnitude of demand uncertainty. We also exarnine how the profits of the consolidated zinc industry, the mining sector, and the smelting sector vary as these things change. Since good estimates of the values of the physical parameters X, u, and co and the price participation parameters T and K are avail-able, only the number of smelting firms n, the smelter's Arrow-Pratt measure of absolute risk aversion r, and the upper bound on the demand variable e need be speci-fied before simulations can be conducted.54 Simulations for various combinations of the values of these parameters are conducted. More specifically, the parameters n, r, s are varied over the following ranges: • n (number of smelting firms): 1 to 8 • r (Arrow-Pratt measure of absolute risk aversion55): 0.1 to 10 • e (upper bound on the demand variable e): 0 to 250 These ranges embody a wide range of assumptions about the degree of competition in the smelting sector, attitudes of smelters toward risk, and variability in the strength of 54. Since E = - E , specifying E fixes E. 55. The case where smelters behave as risk-neutral profit maximisers is also examined. Notice that as the Ar-row-Pratt measure of absolute risk aversion r approaches zero, the behaviour of expected utility maximising smelters facing uncertain demand tends to that of expected profit maximisers. 102 zinc demand. The simulation results are reported in Table 3-2, and several trends are immediately apparent. First, as n rises, other things being equal, industry output rises and profits fall. This is, of course, just another example of the standard Cournot result that as the number of firms rises, competition intensifies, driving prices towards mar-ginal cost and reducing firm and industry profits in the process. It should be remembered, however, that the price participation system increases the incentives for smelters to overproduce, even in a world of certainty (see Section 3.1 for a fuller discus-sion of this point). Table 3-2: Simulation Results Under Price Participation Key: n, number of smelting firms; e, value of random variable s when demand is strong; r, smelter7s Arrow-Pratt coefficient of absolute risk aversion; x, smelter's output ('000 tonnes); p& price of refined zinc when demand is strong ($/ tonne); pb, price of refined zinc when demand is weak ($/tonne); E(p), expected price of refined zinc ($/tonne); X, smelting sector7s output ('000 tonnes); E(IT), expected industry profits (million $); E(ns), expected profits of smelting sector (million $); E(n"'), expected profits of mining sector (million $); Ug, smelter utility when demand is strong; l / h smelter utility when demand is weak; E(U), ex-pected smelter utility. 6 r X Ps Pb m X E(TI) E(lf) E(nm) Ub E(U) 0 RN 3,281 1,426 1,426 1,426 3,281 1,873 837 1,037 n/a n/a n/a 62.5 0.1 3,280 1,489 1,364 1,426 3,280 1,874 837 1,037 0.0871 0.0734 0.0802 1 3,272 1,491 1,366 1,428 3,272 1,875 837 1,037 0.5981 0.5337 0.5659 3 3,253 1,494 1,369 1,432 3,253 1,877 839 1,038 0.9353 0.8992 0.9172 5 3,236 1,498 1,373 1,436 3,236 1,879 840 1,039 0.98% 0.9783 0.9840 10 3,202 1,505 1,380 1,443 3,202 1,882 842 1,040 0.9999 0.9995 0.9997 125 0.1 3,277 1,552 1,302 1,427 3,277 1,874 837 1,037 0.0939 0.0665 0.0802 1 3,244 1,559 1,309 1,434 3,244 1,878 839 1,039 0.6272 0.4995 0.5633 3 3,178 1,573 1323 1,448 3,178 1,885 844 1,041 0.9484 0.8773 0.9128 5 3,126 1,584 1334 1,459 3,126 1,889 846 1,042 0.9929 0.9705 0.9817 10 3,058 1,599 1349 1,474 3,058 1,892 849 1,043 0.9999 0.9992 0.9996 250 0.1 3,266 1,679 1,179 1,429 3,266 1,875 838 1,038 0.1072 0.0527 0.0800 1 3,140 1,706 1,206 1,456 3,140 1,888 846 1,042 0.6772 0.4291 0.5531 3 2,944 1,748 1,248 1,498 2,944 1,893 852 1,042 0.9651 0.8267 0.8959 5 2,846 1,769 1,269 1,519 2,846 1,890 852 1,038 0.9961 0.9485 0.9723 10 2,782 1,783 1,283 1,533 2,782 1,886 851 1,035 1.0000 0.9975 0.9987 103 Table 3-2: Simulation Results Under Price Participation (continued) £ r X Pg 0 RN 2,094 1,232 62.5 0.1 2,093 1,295 1 2,090 1,296 3 2,082 1,299 5 2,075 1,302 10 2,059 1,310 125 0.1 2,092 1,358 1 2,079 1,363 3 2,050 1,376 5 2,025 1^ 87 10 1,978 1,406 250 0.1 2,088 1,485 1 2,035 1,507 3 1,938 1,549 5 1,871 1,577 10 1,795 1,610 0 RN 1,215 1,088 62.5 0.1 1,215 1,151 1 1,213 1,152 3 1,211 1,154 5 1,208 1,156 10 1,202 1,161 125 0.1 1,214 1,214 1 1,210 1,218 3 1,200 1,226 5 1,190 1,234 10 1,169 1,252 250 0.1 1,213 1,340 1 1,195 1,356 3 1,158 1,387 5 1,127 1,414 10 1,074 1,459 0 RN 660 998 62.5 0.1 660 1,060 1 660 1,061 3 659 1,062 5 658 1,063 10 657 1,067 125 0.1 660 1,123 1 659 1,125 3 656 1,130 5 653 1,135 10 646 1,148 250 0.1 660 1,249 1 654 1,258 3 643 1,278 5 632 1,296 10 609 1,335 Pb E(P) X 1,232 1,232 4,188 1,170 1,232 4,187 1,171 1,234 4,180 1,174 1,237 4,165 1,177 1,240 4,150 1,185 1,247 4,117 1,108 1,233 4,185 1,113 1,238 4,157 1,126 1,251 4,100 1,137 1,262 4,049 1,156 1,281 3,957 985 1,235 4,176 1,007 1,257 4,070 1,049 1,299 3,875 1,077 1,327 3,742 1,110 1360 3,590 1,088 1,088 4,859 1,026 1,088 4,859 1,027 1,089 4,854 1,029 1,092 4,844 1,031 1,094 4,834 1,036 1,099 4,809 964 1,089 4,857 968 1,093 4,839 976 1,101 4,798 984 1,109 4,760 1,002 1,127 4,677 840 1,090 4,851 856 1,106 4,778 887 1,137 4,630 914 1,164 4,506 959 1,209 4,296 998 998 5,283 935 998 5,282 936 998 5,280 937 1,000 5,274 938 1,001 5,267 942 1,004 5,252 873 998 5,282 875 1,000 5,270 880 1,005 5,246 885 1,010 5,223 898 1,023 5,166 749 999 5,278 758 1,008 5,234 778 1,028 5,141 796 1,046 5,055 835 1,085 4,873 E(n) E(lt) E(lT") 1,579 681 897 1,579 682 897 1,583 683 899 1,590 687 903 1,598 691 907 1,614 699 915 1,580 682 898 1,594 689 905 1,623 704 919 1,647 716 930 1,687 737 950 1,585 685 900 1,637 711 926 1,720 754 966 1,768 779 989 1,813 803 1,010 1,134 459 675 1,134 459 675 1,138 461 677 1,146 465 681 1,154 469 685 1,173 478 695 1,135 460 676 1,150 467 683 1,182 483 699 1,211 497 714 1,273 528 745 1,140 462 678 1,197 490 707 1,307 545 762 1,392 587 804 1,520 652 868 753 271 482 754 271 482 756 273 484 762 276 487 769 278 490 783 286 498 755 272 483 766 277 489 789 289 501 812 300 512 865 326 539 758 273 485 801 294 507 889 338 551 967 376 591 1,122 453 669 E(U) n/a n/a n/a 0.0381 0.0289 0.0335 0.3224 0.2549 0.2886 0.6905 0.5890 0.6398 0.8596 0.7751 0.8174 0.9810 0.9517 0.9664 0.0427 0.0243 0.0335 0.3554 0.2215 0.2884 0.7368 0.5400 0.6384 0.8946 0.7357 0.8151 0.9898 0.9383 0.9641 0.0518 0.0152 0.0335 0.4174 0.1570 0.2872 0.8095 0.4528 0.6312 0.9389 0.6662 0.8026 0.9965 0.9077 0.9521 n/a n/a n/a 0.0141 0.0087 0.0114 0.1330 0.0839 0.1085 0.3502 0.2337 0.2920 0.5148 0.3617 0.4383 0.7698 0.6027 0.6863 0.0169 0.0060 0.0114 0.1577 0.0599 0.1088 0.4086 0.1801 0.2944 0.5900 0.2963 0.4432 0.8430 0.5460 0.6945 0.0223 0.0005 0.0114 0.2063 0.0141 0.1102 0.5152 0.0891 0.3022 0.7122 0.1996 0.4559 0.9260 0.4799 0.7030 n/a n/a n/a 0.0049 0.0019 0.0034 0.0479 0.0189 0.0334 0.1378 0.0568 0.0973 0:2202 0.0945 0.1574 0.3971 0.1878 0.2925 0.0064 0.0004 0.0034 0.0625 0.0047 0.0336 0.1792 0.0187 0.0990 0.2851 0.0385 0.1618 0.5038 0.1082 0.3060 0.0094 -0.0026 0.0034 0.0917 -0.0229 0.0344 0.2604 -0.0496 0.1054 0.4067 -0.0529 0.1769 0.6735 0.0133 0.3434 104 Table 3-2 also reveals that when the number of firms is held constant, the profits of the rruning and smelting sectors, and consequently those of the entire industry, increase monotonically as firms become more risk averse and as the magnitude of demand un-certainty increases. As previously discussed, even when there is no uncertainty about the strength of demand, Cournot competitors, particularly those using a price participa-tion system, tend to overproduce compared to the level that maximises profits. The fact that increasing demand uncertainty or increasing risk aversion results in greater profits then follows immediately from the well known result that risk averse firms tend to re-duce production in the face of demand uncertainty and that such behaviour is independent of market structure.56 It is this output-reducing response to demand un-certainty that moves the industry closer to the profit-maximising level of production, to some extent counteracting the overproduction caused by Cournot competition and ex-acerbated by the price participation system. In summary then, the nature of competition within the smelting sector and the structure of the price participation system tends to drive output above the level that maximises smelting, mining, and industry profits, while risk aversion and demand uncertainty tend to drive output in the opposite direc-tion. It is the competing strength of these two effects that determines how close the industry is to its profit-maximising level of output. Table 3-2 also shows that the expected profits of the smelting sector, the mining sector, and the consolidated industry are increasing functions of demand uncertainty and risk aversion even when the smelting sector behaves as a monopoly (i.e., n = 1). If produc-tion were simply a single-stage decision, such a result would not be possible. In a 56. See Sandmo (1971) and Leland (1972) for classic early discussions of the competitive and monopoly cases, 105 single-stage model, a risk-neutral monopolist (or a risk averse monopolist facing a cer-tain price) would produce at the level that maximises expected profits. This follows directiy from the functional form of the utility function, (3-11), which ensures that the objective of maximising expected utility is equivalent to that of maximising expected profits when decision-makers are risk neutral.57 Consequently, in a single-stage model, any move away from the output level chosen by a risk neutral firm must reduce profits. This is not the case, however, when the output decision takes the form of a two-stage game as it does here. In the second stage of the two-stage game, the monopolist chooses its production level taking the zinc basis price as given. The output level the monopolist chooses does, of course, influence the choice the basis price, but the game structure does not allow the monopolist to control the choice of the basis price.58 If possible, the monopolist would like to commit to resMcting output somewhat in the second stage with the objective of driving up the basis price selected in the first stage. However, since such strategic be-haviour cannot occur, the monopolist tends to produce more zinc than it would if such strategic behaviour were possible. The presence of risk aversion and price uncertainty induces the monopolist to cut pro-duction and moves the monopolisf s output closer to the level it would chose if it could strategically influence the basis price. This of course happens when there is more than a respectively, or Gravelle and Rees (1991), for example, for a more recent summary of the literature. 57. As the Arrow-Pratt measure of absolute risk aversion r approaches zero, the behaviour of expected utility maximising smelters facing uncertain demand tends to that of expected profit maximisers 58 In reality, the basis price is set during the unspecified bargaining process that takes place between the mines and the smelters. Smelters do of course have some influence over the value of the basis price, but the extent of their influence is unknown. 106 single smelter in the market, but the effect is most clearly illustrated in the monopoly case. This result emphasises the importance of accurately describing the way in which the parameters of the price participation contract are selected within the model's game theoretic structure. Another mteresting feature of the simulation results in Table 3-2 is the way in which the equilibrium price of zinc changes as demand uncertainty, risk aversion, and the number of smelting firms varies. Recall from Section 3.4 that the demand curve for zinc was es-timated by fitting a linear function through the average price-quantity combination observed in 1994. (In 1994, the average price of refined zinc was $998.24 and the quan-tity purchased was 5.84 million tonnes.) From Table 3-2 we see that for the simulations to generate an equilibrium price close to the observed price, the smelting industry must be characterised by a relatively high degree of competition; that is p » $998 when n = 8 and smelters are risk neutral. Although the smelter's degree of risk aversion and the magnitude of demand uncertainty also affect the equilibrium price, the effect of market structure (i.e., ri) is more pronounced. Thus, it would seem that in 1994, the year used to calibrate the model, the smelting sector was characterised by a reasonable degree of competition. Table 3-2 also shows that for the parameter values examined the price participation system generates higher profits for the nrining sector than for the smelting sector. Of course, the exact division of total industry profits between the two sectors is a function of n, r, and s, but over the range of parameter values examined the mining sector re-ceives between 67.6 and 54.9 percent. The smelting sector's share of industry profits is 107 increasing in r and E and decreasing in n. It must be remembered, however, that the objective of smelters is to maximise expected utility, not expected profits. The expected utility of smelters can only be compared in a meaningful way when the parameters of the utility function are held constant; therefore, expected utilities are comparable only when the Arrow-Pratt measure of absolute risk aversion, r, is fixed. Keeping this in mind, we see that when r is fixed and competition within the smelting sector is relatively weak (i.e., when n = 1 or 2), the smelter's expected utility decreases as demand uncertainty increases. However, the opposite is true when competition is relatively strong (i.e., when n = 4 or 8). Whether utility rises or falls is determined by the relative strength of two effects. The first effect is that the risk-averse smelters reduce output in response to increasing demand uncertainty. In the majority of cases, this drives up the price of zinc, and the smelter's profits and utility in both the high-demand and low-demand states of the world.59 The second effect is that as demand uncertainty increases and with it the volatility of the zinc price, smelter profits in the high-demand state and those in the low demand state move further apart. Because the smelter is risk averse, utility in the low-demand state drops by more than utility in the high-demand state rises, with the net effect that expected utility falls. 3.5.2 Standard Input Pricing Under standard input pricing, we assume that the mines supply the smelters with as much concentrate as they want at the negotiated concentrate price. Of course, the 59. The only case in which the smelter's expected profits actually fall when the firm cuts back output is when n = 1, r = 10, and e = 250. In this situation the firm cuts output too much and produces less than the profit-maximising amount of zinc. 108 amount of concentrate demanded by the smelters is a function of its price.60 Conse-quently, to calculate outcomes under standard input pricing, we must make an assumption about concentrate pricing. There are many assumptions that could be made. One way to narrow the field, how-ever, is to think about the bargaining process that takes place between the mines and the smelters during which they set the values of the price participation parameters T and K . It is the values of these parameters that determine how the profits earned by the entire industry are divided between the mining and smelting sectors. Consequently, it seems likely that the values of T and K , and therefore the division of industry profits, would be determined by the relative bargaining power of the smelting and mining sectors. Now, if the industry were to use standard input pricing instead of price participation, bar-gaining between the mining and smelting sectors would focus on the price of zinc concentrate rather than the values of T and K . Furthermore, if we assume the relative bargaining power of the two sectors would be the same under the two alternative ar-rangements, we can use the relative profitability of the smelting and mining sectors under price participation as an indicator of what reasonable concentrate pricing under standard input pricing might be. With this in mind, two assumptions that could be made about concentrate pricing are (i) that mines price concentrate above marginal cost by an amount that ensures the mining sector receives the same fraction of total industry profits under standard input pricing as 60. Under standard input pricing, we assume that once the mines and smelters have negotiated a mutually acceptable concentrate price, the supply of concentrate will be completely elastic at that price. Similarly, under price participation, we assume that once the mines and smelters have negotiated a mutually acceptable price participation contract, the supply of concentrate will be completely elastic under the terms of that contract. 109 it does under price participation or (ii) that mines price concentrate above marginal cost by an amount that ensures they earn the same profit per tonne of concentrate under standard input pricing as they earn under price participation. Unfortunately, even though the first assumption has some appeal, it introduces the problem of multiple equilibria. More specifically, if total industry profits are free to vary, more than one concentrate price can support any given ratio of mining-sector-profit to industry-profit under standard input pricing. Of course, the problems intro-duced by multiple equilibria are not insurmountable. Certain rules could be used to choose among the equilibria, for example, always choosing the equilibrium price re-sulting in the highest profits for the entire industry. However, such rules introduce considerable computational difficulties, particularly when, as it is in this case, the num-ber of possible equilibria is high. Consequently, the second assumption—that mines price concentrate above marginal cost by an amount that ensures they earn the same profit per tonne of concentrate produced as they do under price participation—is adopted.61 Simulation results under the assumption of standard input pricing are reported in Ta-ble 3-3. To facilitate comparison of the results generated under the assumption of standard input pricing with those generated under the assumption of price participa-tion, condensed versions of Table 3-2 and Table 3-3 are reproduced together in Table 3-4. 61. Outcomes under this assumption for particular values of n, r, and E the following procedure is used. First, for the values of n, r, and e of interest, the average profit earned by the mines on each of tonne concentrate under the assumption of price participation is calculated. The mines are then assumed to make the same profit on each tonne of concentrate under standard input pricing. 110 Table 3-3: Simulation Results Under Standard Input Pricing Key: n, number of smelting firms; e, value of random variable e when demand is strong; r, smelter7s Arrow-Pratt coefficient of absolute risk aversion; x, smelter's output ('000 tonnes); p& price of refined zinc when demand is strong ($/ tonne); pt, price of refined zinc when demand is weak ($/tonne); E(p), expected price of refined zinc ($/ tonne); X, smelting sector's output ('000 tonnes); E(n), expected industry profits (million $); E(n»), expected profits of smelting sector (million $); E(nm), expected profits of mining sector (million $); Ug, smelter utility when demand is strong; 14 smelter utility when demand is weak; E(U), ex-pected smelter utility. e r X Vz m X E(n) E(lf) E(nm) E(U) 0 RN 431 1,021 1,021 1,021 5,174 1,777 , 1,070 707 n/a n/a n/a 62.5 0.1 431 1,084 959 1,021 5,173 1,776 1,070 706 0.0770 0.0154 0.0462 1 431 1,085 960 1,023 5,166 1,770 1,068 702 0.5540 0.1493 0.3517 3 429 1,089 964 1,026 5,150 1,756 1,063 693 0.9147 0.4118 0.6633 5 428 1,092 967 1,030 5,133 1,744 1,057 687 0.9845 0.6166 0.8005 10 424 1,101 976 1,038 5,094 1,726 1,047 679 0.9998 0.8966 0.9482 125 0.1 431 1,147 897 1,022 5,171 1,774 1,069 705 0.1066 -0.0167 0.0449 1 428 1,153 903 1,028 5,141 1,748 1,060 688 0.6828 -0.1467 0.2681 3 423 1,167 917 1,042 5,077 1,699 1,037 663 0.9722 -0.2528 0.3597 5 418 1,180 930 1,055 5,016 1,669 1,019 650 0.9979 -0.0950 0.4515 10 407 1,209 959 1,084 4,881 1,642 997 645 1.0000 0.6346 0.8173 250 0.1 430 1,274 774 1,024 5,161 1,765 1,066 699 0.1630 -0.0834 0.0398 1 420 1,299 799 1,049 5,045 1,663 1,020 643 0.8423 -0.9646 -0.0611 3 401 1,349 849 1,099 4,810 1,520 935 585 0.9973 -2.7310 -0.8669 5 384 1391 891 1,141 4,612 1,466 896 570 1.0000 -2.6457 -0.8229 10 355 1,467 967 1,217 4,258 1,442 876 566 1.0000 0.1361 0.5680 0 RN 614 1,078 1,078 1,078 4,909 1,871 1,164 707 n/a • n/a n/a 62.5 0.1 613 1,140 1,015 1,078 4,908 1,871 1,165 707 0.0908 0.0333 0.0621 1 612 1,143 1,018 1,080 4,898 1,874 1,169 705 0.6169 0.2934 0.4551 3 609 1,147 1,022 1,085 4,875 1,879 1,178 701 0.9464 0.6662 0.8063 5 607 1,152 1,027 1,090 4,853 1,883 1,185 698 0.9929 0.8530 0.9230 10 600 1,163 1,038 1,101 4,802 1,889 1,194 695 1.0000 0.9856 0.9928 125 0.1 613 1,204 954 1,079 4,904 1,872 1,166 706 0.1185 0.0035 0.0610 1 608 1,212 962 1,087 4,864 1,882 1,183 699 0.7239 0.0685 0.3962 3 597 1,231 981 1,106 4,777 1,892 1,205 687 0.9820 0.3526 0.6673 5 587 1,248 998 1,123 4,698 1,894 1,212 682 0.9990 0.6493 0.8242 10 567 1,282 1,032 1,157 4,538 1,890 1,206 685 1.0000 0.9630 0.9815 250 0.1 611 1,332 832 1,082 4,891 1,876 1,172 703 0.1715 -0.0580 0.0567 1 592 1,365 865 1,115 . 4,734 1,894 1,216 678 0.8587 -0.5071 0.1758 3 554 1,429 929 1,179 4,435 1,864 1,215 649 0.9980 -0.5521 0.2230 5 526 1,478 978 1,228 4,204 1,831 1,188 644 1.0000 0.1285 • 0.5642 10 481 1,554 1,054 1,304 3,851 1,801 1,149 652 1.0000 0.9071 0.9535 0 RN 1,060 1,221 1,221 1,221 4,240 1,551 • 962 589 n/a n/a . n / a 62.5 0.1 1,060 1,284 1,159 1,221 4,238 1,552 963 589 0.1155 0.0674 0.0915 1 1,056 1,287 1,162 1,224 4,223 1,560 971 589 0.7092 0.5071 0.6082 3 1,047 1,294 1,169 1,232 4,188 1,578 989 589 0.9766 0.8872 0.9319 5 1,039 1,301 1,176 1,239 4,156 1,595 1,006 589 0.9982 ' 0.9760 0.9871 10 1,022 1,316 1,191 1,253 4,087 1,629 1,038 .' 591 1.0000 0.9996 0.9998 125 0.1 1,058 1,347 1,097 1,222- 4,233 1,555 966 589 0.1388 0.0427 0.0908 1 1,043 1,361 1,111 1,236 4,171 1,587 998 589 0.7812 0.3793 0.5803 3 1,011 1,387 1,137 1,262 4,046 1,648 1,059 589 0.9908 0.8097 0.9003 5 986 1,409 1,159 1,284 3,942 1,693 1,102 591 0.9997 0.9524 0.9760 10 944 1,445 1,195 1320 3,777 1,756 1,154 602 1.0000 0.9989 0.9995 111 Table 3-3: Simulation Results Under Standard Input Pricing (continued) e r X Pg 250 3 899 1,609 5 840 1,659 10 777 1,713 5 840 1,659 10 777 1,713 0 RN 1,649 1,422 62.5 0.1 1,648 1,485 1 1,638 1,490 3 1,617 1,499 5 1,598 1,507 10 1,564 1,521 125 0.1 1,645 1,549 1 1,606 1,566 3 1,533 1,597 5 1,483 1,618 10 1,425 1,643 250 0.1 1,631 1,680 1 1,490 1,740 3 1,302 1,821 5 1,217 1,857 10 1,158 1,883 Pb E(p) X 1,109 1,359 3,595 1,159 1,409 3361 1,213 1,463 3,107 1,159 1,409 3,361 1,213 1,463 3,107 1,422 1,422 3,299 1360 1,423 3,296 1,365 1,427 3,276 1,374 1,436 3,233 1.382 1,444 3,196 1,396 1,459 . 3,128 1,299 1,424 3,290 1,316 1,441 3,211 1347 1,472 3,066 1368 1,493 2,966 1393 1,518 2,850 1,180 1,430 3,263 1,240 1,490 2,980 1,321 1,571 2,604 1,357 1,607 2,435 1.383 1,633 2,316 E(TQ EfTf) E(n"') 1,811 1,220 592 1,862 1,262 600 1,890 1,262 628 1,862 1,262 600 1,890 1,262 628 1.093 645 448 1.094 645 448 1,102 653 449 1,121 670 450 1,138 687 452 1,179 724 455 1,097 648 448 1,130 679 451 1,198 742 456 1,258 797 461 1371 897 474 1,108 659 449 1,231 773 458 1,437 962 475 1,570 1,079 491 1,729 1,200 529 E(U) 0.9983 0.6181 0.8082 1.0000 0.8783 0.9391 1.0000 0.9922 0.9961 1.0000 0.8783 - 0.9391 1.0000 0.9922 0.9961 n/a n/a n/a 0.1281 0.0914 0.1098 0.7469 0.6189 0.6829 0.9841 0.9465 0.9653 0.9990 0.9927 0.9959 1.0000 1.0000 1.0000 0.1459 0.0727 0.1093 0.7949 0.5422 0.6685 0.9915 0.9149 0.9532 0.9996 0.9851 0.9924 1.0000 0.9998 0.9999 0.1803 0.0350 0.1077 0.8592 0.3754 0.6173 0.9963 0.8158 0.9060 0.9999 0.9447 0.9723 1.0000 0.9967 0.9983 Table 3-4: Price Participation vs. Standard Input Pricing Key: n, number of smelting firms; E , value of random variable E when demand is strong; r, smelter7s Arrow-Pratt coefficient of abso-lute risk aversion; E(p), expected price of refined zinc ($/tonne); X, smelting sector7s output ('000 tonnes); E(n), expected industry profits (million $); E(IT), expected profits of smelting sector (million $); E(lT"), expected profits of mining sector (million $); E(U), ex-pected smelter utility. Panel A: Price Participation Panel B: Standard Input Pricing 8 r E(P) X E(T1) E(if) E(nm) E(U) E(P) X E(n) E(lf) E(rf") E(U) 0 RN 1,426 3,281 1,873 837 1,037 n/a 1,650 2,236 1,777 1,070 707 n/a 62.5 0.1 1,426 3,280 1,874 837 1,037 0.0802 1,650 2,234 1,776 1,070 706 0.1014 1 1,428 3,272 1,875 837 1,037 0.5659 1,654 2,214 1,770 1,068 702 0.6529 3 1,432 3,253 1,877 839 1,038 0.9172 1,663 2,173 1,756 1,063 693 0.9553 5 1,436 3,236 1,879 840 1,039 0.9840 1,670 2,139 1,744 1,057 687 0.9938 10 1,443 3,202 1,882 842 1,040 0.9997 1,681 2,090 1,726 1,047 679 0.9999 125 0.1 1,427 3,277 1,874 837 1,037 0.0802 1,652 2,227 1,774 1,069 705 0.1011 1 1,434 3,244 1,878 839 1,039 0.5633 1,668 2,150 1,748 1,060 688 0.6408 3 1,448 3,178 1,885 844 1,041 0.9128 1,695 2,022 1,699 1,037 663 0.9420 5 1,459 3,126 1,889 846 1,042 0.9817 1,711 1,951 1,669 1,019 650 0.9887 10 1,474 3,058 1,892 849 1,043 0.99% 1,724 1,891 1,642 997 645 0.9997 250 0.1 1,429 3,266 1,875 838 1,038 0.0800 1,657 2,200 1,765 1,066 699 0.0998 1 1,456 3,140 1,888 846 1,042 0.5531 1,714 1,937 1,663 1,020 643 0.5965 3 1,498 2,944 1,893 852 1,042 0.8959 1,774 1,654 1,520 935 585 0.8867 5 1,519 2,846 1,890 852 1,038 0.9723 1,794 1,561 1,466 896 570 0.9594 10 1,533 2,782 1,886 851 1,035 0.9987 1,802 1,522 1,442 876 566 0.9965 112 Table 3-4: Price Participation vs. Standard Input Pricing (continued) Panel A: Price Participation Panel B: Standard Input Pricing 8 r m X E(n) E(lf) EOT") E(U) E(P) X E(n) E(lf) E(n'") E(U) 0 RN 1,232 4,188 1,579 681 897 n/a 1,422 3,299 1,871 1,164 707 n/a 62.5 0.1 1,232 4,187 1,579 682 897 0.0335 1,423 3,296 1,871 1,165 707 0.1098 1 1,234 4,180 1,583 683 899 0.2886 1,427 3,276 1,874 1,169 705 0.6829 3 1,237 4,165 1,590 687 903 0.6398 1,436 3,233 1,879 1,178 701 0.9653 5 1,240 4,150 1,598 691 907 0.8174 1,444 3,196 1,883 1,185 698 0.9959 10 1,247 4,117 1,614 699 915 0.9664 1,459 3,128 1,889 1,194 695 1.0000 125 0.1 1,233 4,185 1,580 682 898 0.0335 1,424 3,290 1,872 1,166 706 0.1093 1 1,238 4,157 1,594 689 905 0.2884 1,441 3,211 1,882 1,183 699 0.6685 3 1,251 4,100 1,623 704 919 0.6384 1,472 3,066 1,892 1,205 687 0.9532 5 1,262 4,049 1,647 716 930 0.8151 1,493 2,966 1,894 1,212 682 0.9924 10 1,281 3,957 1,687 737 950 0.9641 1,518 2,850 1,890 1,206 685 0.9999 250 0.1 1,235 4,176 1,585 685 900 0.0335 1,430 3,263 1,876 1,172 703 0.1077 1 1,257 4,070 1,637 711 926 0.2872 . 1,490 2,980 1,894 1,216 678 0.6173 3 1,299 3,875 1,720 754 966 0.6312 1,571 2,604 1,864 1,215 649 0.9060 5 1,327 3,742 1,768 779 989 0.8026 1,607 2,435 1,831 1,188 644 0.9723 10 1,360 3,590 1,813 803 1,010 0.9521 1,633 2,316 1,801 1,149 652 0.9983 0 RN 1,088 4,859 1,134 459 675 n/a 1,221 4,240 1,551 962 589 n/a 62.5 0.1 1,088 4,859 1,134 459 675 0.0114 . 1,221 4,238 1,552 963 589 0.0915 1 1,089 4,854 1,138 461 677 0.1085 1,224 4,223 1,560 971 589 0.6082 3 1,092 4,844 1,146 465 681 0.2920 1,232 4,188 1,578 989 589 0.9319 5 1,094 4,834 1,154 469 685 0.4383 1,239 4,156 1,595 1,006 589 0.9871 10 1,099 4,809 1,173 478 695 0.6863 1,253 4,087 1,629 1,038 591 0.9998 125 0.1 1,089 4,857 1,135 460 676 0.0114 1,222 4,233 1,555 966 > 589 0.0908 1 1,093 4,839 1,150 467 683 0.1088 1,236 4,171 1,587 998 589 0.5803 3 1,101 4,798 1,182 483 699 0.2944 1,262 4,046 1,648 1,059 589 0.9003 5 1,109 4,760 1,211 497 714 0.4432 1,284 3,942 1,693 1,102 591 0.9760 10 1,127 4,677 1,273 528 745 0.6945 1,320 3,777 1,756 1,154 602 0.9995 250 0.1 1,090 4,851 1,140 462 678 0.0114 1,227 4,212 1,566 977 589 0.0880 1 1,106 4,778 1,197 490 707 0.1102 1,277 3,978 1,678 1,089 589 0.4829 3 1,137 4,630 1,307 545 762 0.3022 1,359 3,595 1,811 1,220 592 0.8082 5 1,164 4,506 1,392 587 804 0.4559 1,409 3,361 1,862 1,262 600 0.9391 10 1,209 4,296 1,520 652 868 0.7030 1,463 3,107 1,890 1,262 628 0.9961 0 RN 998 5,283 753 271 482 n/a 1,078 4,909 1,093 645 448 n/a 62.5 0.1 998 5,282 754 271 482 0.0034 1,078 4,908 1,094 645 448 0.0621 1 998 5,280 756 273 484 0.0334 1,080 4,898 1,102 653 449 0.4551 3 1,000 5,274 762 276 487 0.0973 1,085 4,875 1,121 670 450 0.8063 5 1,001 5,267 769 278 490 0.1574 1,090 4,853 1,138 687 452 0.9230 10 1,004 5,252 783 286 498 0.2925 1,101 4,802 1,179 724 455 0.9928 125 0.1 998 5,282 755 272 483 0.0034 1,079 4,904 1,097 648 448 0.0610 1 1,000 5,270 766 277 489 0.0336 1,087 4,864 1,130 679 451 0.3962 3 1,005 5,246 789 289 501 0.0990 1,106 4,777 1,198 742 456 0.6673 5 1,010 5,223 812 300 512 0.1618 1,123 4,698 1,258 797 461 0.8242 10 1,023 5,166 865 326 539 0.3060 1,157 4,538 1,371 897 474 0.9815 250 0.1 999 5,278 758 273 485 0.0034 1,082 4,891 1,108 659 449 0.0567 1 1,008 5,234 801 294 507 0.0344 1,115 4,734 1,231 773 458 0.1758 3 1,028 5,141 889 338 551 0.1054 1,179 4,435 1,437 962 475 0.2230 5 1,046 5,055 967 376 591 0.1769 1,228 4,204 1,570 1,079 491 0.5642 10 1,085 4,873 1,122 453 669 0.3434 1,304 3,851 1,729 1,200 529 0.9535 113 3.5.3 Price Participation versus Standard Input Pricing We now turn to the question of how the price participation system performs compared to standard input pricing. Comparing Panels A and B of Table 3-4, a few things are im-mediately apparent. For all combinations of parameter values examined62 • the profit of the smelting sector is greater under standard input pricing than under price participation • the profit of the mining sector is greater under price participation than under standard input pricing • the expected utility of a representative smelter is greater under standard input pricing than under price participation, except when the smelting sector sets output as a monopolist • the profits of the entire zinc industry are greater under standard input pricing than under price participation, except in the monopoly case and in the duopoly case when smelters are very risk averse and face considerable demand uncer-tainty (i.e., n = 2, r = 1, and e = 250) Therefore standard input pricing is superior to price participation from the perspective of the smelters and the aggregate zinc industry (i.e., mines and smelters combined), ex-cept in those cases where smelters are highly risk averse, face significant demand uncertainty, and manage to behave collusively. It should also be noted, however, that the profits earned by the mining sector are higher under price participation than under 62. As discussed in Section 3.5.1, the ranges of parameter values examined are thought to cover the combina-tions of demand uncertainty, market structure, and smelter attitudes toward risk that are likely to be encountered in reality; therefore, this not a particularly strong caveat. 114 standard input pricing in all cases. Therefore, if mines were risk-neutral profit maximis-ers, they would prefer price participation. However, we have made no assumption regarding the existence of, or the extent of, risk aversion on the part of the mines. Fur-thermore, there is no obvious reason to believe mat mines or smelters should differ significantly, if at all, in their attitudes toward risk. Consequently, an important ques-tion to ask is, Will the mines prefer price participation if they are risk averse rather than risk neutral? Since expected profits are higher under price participation than under standard input pricing and the variance of mining profits is lower, the answer is clearly yes. In fact, the more risk averse the smelters, the more they will prefer price participa-tion over standard input pricing.63 It seems, therefore, that risk aversion on the part of smelters and demand uncertainty go only part way toward explaining why the industry uses the price participation system. As the magnitude of demand uncertainty and risk aversion grows, the price participa-tion system becomes relatively more attractive; however, the effect is not strong enough to explain the use of the price participation system. For the case where n = 2 and E = 250, Figure 3-2 shows the relationship between industry profits and the risk aver-sion parameter r under both price participation and standard input pricing. Similarly, Figure 3-3 shows the relationship between nrining sector profits smelting profits and the risk aversion parameter r under the two alternative arrangements. 63. Of course, with the existing structure of the model, in which mines have no direct control over the amount of refined zinc being produced, the assumption of risk aversion on the part of mines does affects neither in-dustry output or sectoral profits. If the model were reformulated to allow mines some control over industry production, the conclusion that "the more risk averse the mines, the more the mines prefer price participation to standard input pricing" might no longer hold. 115 Figure 3-2: Industry Profit vs. Smelter's Arrow-Pratt Coefficient of Absolute Risk Aversion, r 2,000 1,900 1,700 1,600 1,500 - • - - Price Participation -Hi—Standard Input Pricing 2 4 6 8 10 Smelter's Arrow-Pratt Coefficient of Absolute Risk Aversion, r Figure 3-3: Mining and Smelting Sector Profits vs. r under Price Participation and Standard Input Pricing 1,500 1,300 1,100 \ 900 700 500 -I -•—Smelting Sector Profits under PP Hi—Smelting Sector Profits under SIP • • • - Mining Sector Profits under PP • * • • Mining Profits Sector under SIP 2 4 6 8 Smelter's Arrow-Pratt Coefficient of Absolute Risk Aversion, r (Key: PP, Price Participation; SIP, Standard Input Pricing) 10 116 As previously discussed, the equilibrium zinc prices generated by the price participation simulations seem to indicate that the behaviour of the smelting sector was likely char-acterised by a fair amount of competition in 1994 (i.e., n > 8). If correct, it would seem that in 1994 the zinc industry was using a contractual arrangement that was suboptimal from the perspectives of all parties involved. This raises the obvious question of what could explain this. One of the more mteresting possibilities is related to the speed with which market in-stitutions adapt to changes in the environments in which they operate. For example, firms are often assumed to adapt rapidly, even instantaneously, to things such as changing costs or demand, and to always behave in ways that maximise expected prof-its or utility, even during these transitions. One possible interpretation of the results presented here is that they are evidence that such transitions may sometimes happen slowly. The price participation system may have been an optimal strategy for the zinc industry some time in the past when things such as the demand for zinc or the nature of competition within the smelting sector were significantly different from what they are today. But as the industry's environment changed and price participation ceased to an optimal arrangement, it continued to be used because of some form of institutional iner-tia. One way to go beyond pure speculation on this issue is to simulate the effects some significant changes that may have affected the zinc industry in the past. Obvious candi-dates include changes in the demand for zinc and changes in the industry's cost structure. Changing the demand function used in the simulations is a particularly good candidate for a sensitivity analysis for at least three reasons. First, the estimate of the own-price 117 elasticity of zinc used to calibrate the linear demand function is likely prone to consider-able error. Second, the demand structure of the zinc industry must have changed considerably since the price participation system used first introduced. And third, the linear demand function used in the model is only an approximation. To make the demand function sensitivity analysis manageable, only a subset of the pre-viously considered combinations of n, r, and e are examined. In particular, the cases where (i) smelters are risk neutral and face no demand uncertainty; (ii) smelters are moderately risk averse (r = 3) and face considerable demand uncertainty (e = 250); and (iii) smelters are highly risk averse (r = 10) and face considerable demand uncer-tainty (e = 250) are considered. These combinations span the range of behavioural and demand assumptions examined in Table 3-2 and Table 3-3. In addition, attention is re-stricted to the n = 2 and n = 8 market structures. The duopoly market structure is examined because, based on the results in Table 3-4, it seems to have the greatest chance (apart from the monopoly case) of providing a situation in which price participation might be superior to standard input pricing. The case where n = 8 is examined because it seems to best represent the true competitive environment facing the industry in 1994, at least based on the equihbrium prices generated by the price participation model. The sensitivity analysis of the effect of the demand function was carried out in two parts. First, the effect of changing the zinc choke price, a, was examined while the slope of the linear function, b, was held constant. Then the effect of changing the slope was examined while the choke price was held constant. Simulations were conducted with a zinc choke price 25 percent above and 25 percent below the base value of 2,246 (i.e., the 118 value of a calculated in Section 3.4 and used in all previous simulations). This repre-sents a parallel shift of the demand curve either toward or away from the origin. Simulations were also carried out with slopes 25 percent larger and 25 percent smaller than the base value of 0.214. The results of the sensitivity analyses on the effects of changing the choke price and of changing the slope of the demand function are reported in Tables 3-5 and 3-6, respectively. Table 3-5: Choke Price Sensitivity Analysis Key: n, number of smelting firms; e, value of random variable e when demand is strong; r, smelter's Arrow-Pratt coefficient of abso-lute risk aversion; a, refined zinc choke price ($/tonne); E(j>), expected price of refined zinc ($/tonne); X, smelting sector's output ('000 tonnes); E(n), expected industry profits (million $); E(ns), expected profits of smelting sector (million $); E(iT"), expected profits of mining sector (million $); E(U), expected smelter utility. n = 2 Panel A: Price Participation Panel B: Standard Input Pricing e r a E(p) X E(IT) E(ir) E(Tf) E(U) E(P) X E(n) E(rf) E(Tf) E(U) 0 RN 1,684 1,077 2,289 508 204 304 n/a 1,181 1,803 587 , 348 240 n/a 2,246 1,232 4,188 1,579 681 897 n/a 1,422 3,299 1,871 1,164 707 n/a 2,807 1,387 6,086 339 1,439 1,800 n/a 1,664 4,794 3,877 2,459 1,418 n/a 250 3 1,684 1,115 2,108 549 226 323 0.2582 1,273 1,371 573 . 363 210 0.4699 2,246 1,299 3,875 1,720 754 966 0.6312 1,571 2,604 1,864 1,215 649 0.9060 2,807 1,477 5,668 , 3,524 1,582 1,942 0.8776 1,845 3,946 3,907 2,555 1,352 0.9955 250 10 1,684 1,171 1,847 •584 247 337 0.6015 1,355 990 495 314 181 0.7411 2,246 1,360 3,590 , 1,813 803 1,010 ,0.9521 1,633 2,316 1,801 1,149 652 0.9983 2,807 1,523 5,450 3,643 1,642 2,001 0.9984 « - 8 1,878 3,793 3,880 2,488 1,393 1.0000 Panel A: Price Participation II KJ Panel B: Standard Input Pricing 8 r a m X E(D) E(rf) E(n™) E(U) X E(TT) E(lf) E(n"') E(U) 0 RN 1,684 949 2,888 271 81 190 n/a 992 2,683 369 193 176 n/a 2,246 998 5,283 . 753- . 271 482 n/a 1,078 4,909 1,093 645 448 n/a 2,807 1,047 7,678 1,471 573 898 n/a 1,163 7,134 2,196 1,362 835 n/a 250 3 1,684 965 2,810 310 101 209 0.0328 1,050 2,416 470 291 180 -0.3142 2,246 1,028 5,141 889 338 551 0.1054 1,179 4,435 1,437 962 475 0.2230 2,807 1,090 7,475 1,757 712 1,045 0.2094 1304 6,477 2,905 2,000 905 0.8405 250 10 1,684 1,000 2,650 383 138 245 0.1206 1,139 1,997 568 383 184 -0.5951 2,246 1,085 4,873 1,122 453 669 0.3434 1,304 3,851 1,729 1,200 529 0.9535 2,807 1,163 7,133 2,199 928 1,271 0.5779 1,429 5,893 3,380 2,330 1,050 0.9999 119 Table 3-6: Demand Function Slope Sensitivity Analysis Key: n, number of smelting firms; e, value of random variable e when demand is strong; r, smelter's Arrow-Pratt coefficient of abso-lute risk aversion; b, slope of linear demand curve for refined zinc ($/'000 tonnes); E(p), expected price of refined zinc ($/tonne); X, smelting sector's output ('000 tonnes); E(n), expected industry profits (million $); E(IT), expected profits of smelting sector (million $); E(nm), expected profits of mining sector (million $); E(U), expected smelter utility. n = 2 Panel A: Price Participation Panel B: Standard Input Pricing s r b m X E(n) E(lf) E ( T f ) E(U) E(P) X E(n) EOT) EOT) E(U) 0 RN 0.161 1,240 5,716 2,202 952 1,249 n/a 1,435 4,503 2,611 1,627 984 n/a 0.214 1,232 4,188 1,579 681 897 n/a 1,422 3,299 1,871 1,164 707 n/a 0.268 1,224 3,270 1,206 519 687 n/a 1,410 2,576 1,429 888 541 n/a 250 3 0.161 1324 5,192 2,437 1,073 1,364 0.7478 1,607 3,429 2,579 1,679 901 0.9572 0.214 1,299 3,875 1,720 754 966 0.6312 1,571 2,604 1,864 1,215 649 0.9060 0.268 1,278 3,066 1,298 567 732 0.5348 1,538 2,096 1,432 931 500 0.8464 250 10 0.161 1,375 4,876 2,536 1,125 1,411 0.9833 1,649 3,171 2,517 1,599 • .917 0.9998 0.214 1,360 3,590 1,813 803 1,010 0.9521 1,633 2,316 1,801 1,149 652 0.9983 0.268 1,342 2,828 1,378 608 769 0.9073 n = 8 1,614 1,812 1,375 882 493 0.9932 Panel A: Price Participation Panel B: Standard Input Pricing 6 r b E(p) X E(n) E(lf) EOT") E(U) m X E(n) E(ns) E(if") E(U) 0 RN 0.161 1,000 7,211 1,047 379 668 n/a 1,082 6,701 1,522 901 621 n/a 0.214 998 5,283 753 271 482 n/a 1,078 4,909 1,093 645 448 n/a 0.268 995 4,126 578 207 371 n/a 1,073 3,834 836 491 345 n/a 250 3 ' 0.161 1,041 6,959 1,293 500 793 0.1475 1,213 5,888 2,106 1,435 671 0.4404 0.214 1,028 5,141 889 338 551 0.1054 1,179 4,435 1,437 962 475 0.2230 0.268 1,019 4,036 662 248 414 0.0802 1,155 3,527 1,059 697 361 0.1261 250 10 0.161 1,110 6,527 1,664 684 981 0.4523 1,338 5,107 2,466 1,699 767 0.9927 0.214 1,085 4,873 1,122 453 669 0.3434 1304 3,851 1,729 1,200 529 0.9535 0.268 1,067 3,857 817 325 492 0.2689 1,273 3,088 1,289 895 394 0.8542 Table 3-5 shows that under both price participation and standard input pricing, as the zinc choke price grows, and with it the demand for zinc, so too do smelter output and the expected price of zinc. This of course pushes up the expected profits for the smelting sector, the mining sector, and the entire industry. More interesting is the result that weaker demand increases the ratio of industry profits under participation to those un-der standard input pricing. This is clearly illustrated in the cases where e = 250 and r = 10 . When n = 2 and the choke price has its base value of 2,246, expected industry 120 profits are slightly higher under price participation than under standard input pricing, that is, $1,812 rrullion compared to $1,800 million. However, when the choke price is reduced by 25 percent to 1,684, expected industry profits under the two systems become $584 million and $495 million, respectively. Although weaker demand significantly re-duces expected industry profits under both systems, it simultaneously increases the relative attractiveness of the price participation system compared to that of standard input pricing, at least from the perspective of maximising expected industry profits. Yet, even with weaker demand, smelters still prefer the standard input pricing ar-rangement because it provides them with higher expected utility. Figure 3-4: Smelter Utility vs. Zinc Choke Price, a 1.0 0.5 xs * 0.0 -0.5 -•-Smelter's E(U) under PP -•-Smelter's E(U) under SIP -1.0 1,500 2,000 2,500 a, Choke Price of Refined Zinc ($/tonne) (Key: PP, Price Participation; SIP, Standard Input Pricing) 3,000 121 This is not the case, however, when the smelting sector is characterised by a greater degree of competition (i.e., n = 8). Looking at the case in Table 3-5, when n = 8, E = 250, and r = 10, and the choke price is 2,246, we see that price participation and standard input pricing yield expected utilities of 0.343 and 0.954, respectively. Clearly, then, under the base case demand conditions, smelters prefer standard input pricing. However, when the choke price is re-duced by 25 percent, the corresponding expected utilities are 0.121 and -0.595 . This situation is depicted graphically in Figure 3-4. Hence, under these weaker demand conditions, smelters prefer price participation. In conclusion, it seems that weaker demand conditions favour price participation relative to standard input pricing. Turning to the results presented in Table 3-6, we see that as the magnitude of the slope of de-mand function, b, grows—corresponding to an increase in the own-price elasticity of demand—the equilibrium zinc price, smelter output, the expected profitability of the mining and smelting sectors, and the expected utility of smelters, all fall. Hence, as the elasticity of demand increases, the market power of the oligopolistic smelters falls and with it their ex-pected profits. Of course, this standard result is a consequence of the assumption of Cournot competition witJun the smelting sector, rather than any particular features of the price partici-pation or standard input pricing arrangements. Although changing the elasticity of demand has a strong effect on the profits earned by smelt-ers and mines under both price participation and standard input pricing, it has little effect on the distribution of industry profits between the two sectors. Under price participation, as b increases, the fraction of the total industry profits going to the smelting sector decreases very shghtly; for example, when n = 2 and smelters are risk neutral, the fraction of total industry 122 profits going to the smelting sector drops from 43.2 percent to 43.1 percent as b is increased by 25 percent over its base value of 0.214. Under standard input pricing there is a non-monotonic relationship between b and the distribution of industry profits between the two sectors. Whether increasing the elasticity of demand results in a rise or fall in the fraction of total industry profits going to the smelting sector depends on the value n, r, and e, and also the initial value of b from which the increase is made. In summary then, there appears to be little that can be said about how a change in the elasticity of demand of zinc might affect the relative attractiveness of the price participation and standard input pricing arrangements to either the mining or smelting sectors of the industry. We now examine how various changes in cost structure might affect the comparison of price participation and standard input pricing. In particular, we examine how changing the con-centrate price affects the simulation results under the two arrangements. So far, we have assumed that mines price concentrate to ensure they make the same profit on each tonne sold under both price participation and standard input pricing. As previously discussed, this is a reasonable assumption if we believe the relative bargaining power of the miriing and smelting sectors would be the same under both arrangements. Nevertheless, it is still an assumption and a fairly strong one. It therefore seems prudent to examine the effect of alternative con-centrate pricing rules. To keep the scale of sensitivity analysis reasonable, we only conduct simulations for certain combinations of r and e, as we did in previous sensitivity analysis examining the effects of changing the values of the demand parameters. Once again we restrict attention to the du-opoly case because, based on the results in Table 3-4, it seems to have the greatest chance of 123 providing a situation in which price participation might be superior to standard input pricing. Table 3-7 reports simulation results under standard input pricing assuming mines price con-centrate at certain arbitrary mark-ups above marginal cost. Mark-ups above marginal cost of zero to 140 percent are considered. Corresponding outcomes under price participation are also reported. Table 3-7: Concentrate Markup Sensitivity Analysis Key: 8, value of random variable s when demand is strong; r, smelter's Arrow-Pratt coefficient of absolute risk aversion; E(p), expected price of refined zinc ($/tonne); X, smelting sector's output ('000 tonnes); E(n), expected industry profits (million $); E(ns), expected profits of smelting sector (million $); E(nm), expected profits of mining sector (million $); E(U), expected smelter utility; MU, percentage mark-up on concentrate above marginal cost. Panel A: Price Participation Panel B: Standard Input Pricing 8 r E(p) X E(n) E(rf) E(lT") E(U) 0 RN 1,232 4,188 1,579 681 897 n/a 250 3 1,299 3,875 1,720 754 966 0.6312 250 10 1,360 3,590 1,813 803 1,010 0.9521 MU E(P) X E(n) E(ir) E(n") E(U) 0% 1,279 3,966 1,683 1,683 0 n/a 20% 1,337 3,697 1,782 1,782 319 n/a 40% 1,395 3,428 1,850 1,850 592 n/a 60% 1,452 3,159 1,886 1,886 819 n/a 80% 1,510 2,890 1,892 1,892 999 n/a 100% 1,567 2,620 1,867 1,867 1,132 n/a 120% 1,625 2,351 1,095 1,095 504 n/a 140% 1,683 2,082 910 910 446 n/a 0% 1,420 3,308 1,870 1,870 0 0.9780 20% 1,473 3,061 1,892 1,892 264 0.9620 40% 1,525 2,817 1,888 1,888 487 0.9374 60% 1,577 2,577 1,860 1,860 668 0.9012 •80% 1,627 2340 1,807 1,807 809 0.8511 100% 1,677 2,107 1,733 1,733 910 0.7851 120% 1,726 1,878 1,636 1,636 974 0.7031 140% 1,774 1,653 932 932 412 0.6067 0% 1,446 3,188 1,884 1,884 0 1.0000 20% 1,503 2,919 1,893 1,893 252 0.9999 40% 1,561 2,651 1,871 1,871 458 0.9997 60% 1,618 2,384 1,819 1,819 618 0.9988 80% 1,675 2,118 1,737 1,737 732 0.9957 100% 1,731 1,856 1,626 1,626 802 0.9863 120% 1,786 1,600 1,489 1,489 830 0.9627 140% 1,838 1,354 894 894 381 0.9122 124 Table 3-7 indicates that when smelters are risk neutral expected industry profits are greater under standard input pricing than under price participation for all mark-ups considered. Also, expected industry profits under standard input pricing actually rise as the mark-up charged by the mines on concentrate increases, at least until the mark-up exceeds 80 percent. The intuition for this result is straightforward. As mines increase the mark-up on concentrate, smelters face higher marginal costs and respond by cutting back production. This moves total industry production of refined zinc closer to the mo-nopoly level and drives up industry profits in the process. Of course, smelter profits fall because they face higher costs, while those of the mining sector rise because they earn more on every tonne of concentrate they sell. But the net result is that expected industry profits rise. However, this cost-induced contraction of output can only enhance profits up to a certain point. If the mark-up becomes too large and the smelters respond by re-ducing output below the monopoly level, industry profits must fall. It would seem that when smelters are risk-neutral, this type of profit-reducing over-contraction of output begins when the mark-up rises above something close to 80 percent. When smelters are risk averse (i.e., r = 3 or 10) and face demand uncertainty, smelter output falls and the equmbrium price of refined zinc rises in response to the larger mark-up as it did in the risk-neutral case. However, once the mark-up rises above 20 percent expected industry profits begin to fall. The decline in expected industry profits begins at a lower mark-up in the risk-averse case than it did in the risk-neutral case be-cause smelters have already contracted their output in response to the demand uncertainty and so are already closer to output level that maximises industry profits. This can be clearly seen by comparing the zero mark-up production level under the as-125 sumption of risk neutrality to the production level under the assumption of moderate risk aversion (i.e., r = 3): 3,966,200 and 3,307,600 tonnes, respectively. Another mteresting feature of the results in Table 3-7 is that when r = 3 and the mark-up is 140 percent, the utility of the smelters is higher under price participation than it is under standard input pricing. This represents a situation in which the price participa-tion system would be preferred to standard input pricing by those in the smelting sector of the industry. However, the expected profits of the mining sector are still lower under price participation than under standard input pricing, so although we can say that smelters would the prefer price participation arrangement, we can not say the same 126 about the mines. (For the case where n = 2, r = 3 and e = 250, Figure 3-5 depicts the relationship between smelter utility and the mark-up on concentrate under standard input pricing. Smelter utility under price participation is also shown.) The general conclusion that can be drawn from Table 3-7 is that the higher the price of concentrate under standard input pricing, the more attractive smelters find the price participation system as an alternative arrangement. This result is not surprising. It simply stems from the fact that under standard input pricing mines can use their control over the price of concentrate as mechanism increasing their own profits at the expense of the smelting sector. 3.6 C O N C L U S I O N In this paper, we develop a model of an important and unusual vertical relationship that has been little studied to date, the price participation system. The model is calibrated using data drawn from zinc industry, one of the most commercially significant indus-tries to use such arrangements. The calibrated model is then used to simulate the zinc industry's performance under a wide range of assumptions about market structure, demand uncertainty, and the degree of risk aversion exhibited by zinc smelters. Finally, performance under price participation is compared to that which might be expected if a more conventional standard input pricing were adopted by the industry. This paper is largely motivated by earlier work by Brander (1996) on price participation contracts within the context of the zinc industry. In his paper, Brander convincingly argues that when firms possess some degree of market power and compete in quanti-127 ties, price participation contracts contain incentives that tend to exacerbate the standard Cournot problem of overproduction. He also points out that these same incentives tend to make firms react sub-optimally to any demand shocks that the industry might face. The primary goal of this paper is to try to answer the question: Why do some industries chose to use price participation systems when they seem to contain incentives that lead to overproduction and drive down profits? In particular, this paper explores the con-jecture that significant uncertainty about the price of refined zinc, and risk aversion on the part of smelters, might go at least some way toward rationalising the zinc industry's use of the price participation system. This conjecture is made because one of the pri-mary functions of the price participation system would seem to be, at least a priori, spreading the risk associated with a highly volatile refined zinc price between the smelt-ers and mines. The somewhat surprising result of this analysis is that demand uncertainty and risk aversion on the part of smelters do not appear able, by themselves, to explain why the zinc industry uses the price participation system, at least under current market condi-tions. Under most, but not all, reasonable parameterisations of the model, the price participation system performs worse, in that it generates either lower profits or lower utility, than a more conventional standard input pricing arrangement. Nevertheless, the analysis does identify certain changes in the industry's commercial environment that make the price participation relatively more attractive. It seems that risk aversion on the part of smelters and demand uncertainty can go part way toward explaining why the industry might have chosen to use the price participation system, since as the magni-tude of demand uncertainty and risk aversion grows, the price participation system 128 becomes relatively more attractive, although still not as attractive as standard input pricing. As previously discussed in Section 3.2, there are good reasons to believe that smelters currently enjoy considerable independence from mines in their production decisions. This may not have always been the case, however. In fact, it seems quite likely that at the turn of the century, when price participation contracts first started being used, smelters had considerably less control over the amount of zinc produced than they do today.64 Of course, the control that smelters currently exert over industry output has been built into the formal model of the price participation system developed in Sec-tion 3.3, and this raises an important question about the model's ability to accurately describe the industry's behaviour in the past. 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(1993) Pigouvian Penalty for Oil Spills. Energy Economics, 15(3), pp. 167-188. KOLSTAD, C. AND J. KRAUTKRAEMER, 1993, "Natural Resource Use and The Environ-ment," in: A. Kneese and J. Sweeney (eds.), Handbook of N a t u r a l Resource and E n e r g y Economics V o l . Ill (Elservier). KRAAKMAN , R.H. (1984) Corporate Liability Strategies and the Costs of Legal Controls. Yale L a w J o u r n a l , 95(5), pp. 857-898. LASSERRE, P., 1991, L o n g - T e r m C o n t r o l of Exhaustible Resources (Harwood Academic Publishers). LELAND, H.E. (1972) The Theory of the Firm Facing Uncertain Demand. A m e r i c a n Eco-n o m i c R e v i e w . 62(3), pp. 278-291. M 'GONIGLE, R.M. & M.W. ZACHER. (1979) Pollution, Politics, and International Law-133 Tankers at Sea. California: University of California Press. Maritime Strategies International Ltd. (1995, Jun) Tanker Market Investment Perform-ance & Outlook. (The Bulk Shipping Planning Service). London, England: Author. McGuRREN, H.J. 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(1992, Jul 27) O i l and Gas Journal, pp. 38-42. 134 Appendix A: Endogenous Determination of Interest Rate We now assume producers and consumers face an endogenous interest-rate time path, r(t), rather than a fixed exogenous rate r. Both producers and consumers take r(t) as given; therefore, necessary conditions (1-13) through (1-16) and (1-24) through (1-27) are identical, except that r is now a function of time rather than a constant. Now, using (1-1), (1-2), (1-3), (1-4) to eliminate m, s, and e from (1-25) and rearranging yields Differentiating this expression logarithmically with respect to time and noting from (1-30) that lm is constant along the equilibrium path yields U = (1 -P)0P(I_Y)/O-a)P(l-r) _i )(1-P)(1-YH j(ap-nXI-Y) (A-l) (l/u = (ap-Ti)(l-Y)i/x (A-2) Combining this result with (1-26) yields 135 ( < X P - T I ) ( 1 - Y ) _ X (A-3) 8-r x From (A-3) and (1-33) we have (gp-ri)(l-Y)_(ap-ri)(l-Y)-l (A-4) 8-r 8 Solving for r yields = 8 (A-5) r "l-(ap-r , ) ( l -Y)' Hence, the interest rate is a positive constant along the competitive trajectory. 136 Appendix B: The Ambiguous Effect of Increased Liability on Oil Spill Frequencies The demand for the services of oil tankers is primarily derived from the demand for crude oil. Also, the cost of tanker services generally accounts for only a small fraction of the cost of crude. Given these two things, the fact that the long-run price elasticity of demand for crude is quite low, even a sharp increase in the price of tanker services is unlikely to have a major effect on the quantity of tanker services demanded. Conse-quently, the demand for tanker services is assumed to be constant and to translate into a requirement for a global tanker fleet corisisting of exactiy N vessels. The global tanker fleet is assumed to consist of two distinct subfleets: a fleet of high-quality vessels and a fleet of low-quality vessels.65 There are a total of NH vessels in the high-quahty fleet and NL vessels in the low-quality fleet, where the subscripts H and L denote high- and low-quality vessels, respectively. The vessels in each fleet are identical except that high-quahty tankers have a a H percent chance of having an oil spill each year, while low-quality tankers have a <xL percent chance, with aL > aH . 137 Each fleet's asset value is simply the aggregate value of the tankers it contains. High-quahty tankers are worth AH and low-quality tankers are worth AL, with AH > AL; therefore, the asset values of the high- and low-quality fleets are N^Ay, and NLAL, re-spectively. In reality, a substantial fraction of the global tanker fleet is owned by large, asset-rich companies such as the oil majors (Exxon and Royal-Dutch Shell being two prominent examples), while the rest is owned by smaller, independent shipping companies with much lower asset values. These small independents sometimes go to great lengths to minimise the exposure of their assets to spill hability risks. For example, a common ploy is for the independent to separate its fleet into a series of legally distinct, limited liability one-ship companies (Ketkar 1995). To capture this aspect of the industry's organisation, albeit in a highly stylised manner, it is assumed that the high-quahty fleet is organised as single corporate unit with assets NHAH, while the low-quality fleet is organised as a series of one-ship companies, each with assets AL. This difference in corporate structure is important because it affects the financial conse-quences of a spill. To see this, one must recognise that if a spill does occur, the maximum cost of the accident to the fleet containing the unlucky vessel is bounded in three ways: first, the loss cannot exceed the total value of the damage caused by the spill, denoted D (including the possibility of punitive damages does not alter the results of the analysis); second, the loss cannot exceed the legally imposed liability limit for compen-65. Of course, in reality no such dichotomy exists; the global tanker fleet comprises a continuous spectrum of vessel qualities. Nevertheless, assuming there are only two types of vessel is the simplest way of formally capturing the idea that tankers do vary in quality. 138 satory spill damages, denoted K; and third, the loss cannot exceed the total asset value of the legal entity responsible for the spill, which is NHAH for a high-quality vessel and AL for a low-quality vessel. Since NHAH > AL, the realised cost of a spill to the high-quality fleet is potentially greater than that of a spill to the low-quality fleet. If a high-quality vessel causes a spill, the cost to the high-quality fleet is denoted SH and the cost to the low-quality fleet is, of course, zero. The same logic applies if a low-quality vessel causes a spill, but with the cost of a spill to the low-quality fleet being denoted SL. For convenience, it is assumed that before the Oil Pollution Act is implemented the value of the damage caused by a spill is greater than the maximum legal liability for spill damages (i.e., D > K) and that the maximum legal liability is exactly equal to the value of a low-quality tanker (i.e., K = AL). It is further assumed that the high-quality fleet functions as single managerial unit, facing operating costs that are an increasing function of fleet size; that is, = C(A^), with C > 0. Marginal operating costs are also assumed to be increasing in fleet size, C" > 0. The low-quality fleet faces the same cost function, but is managed as a series of independent one-ship companies; therefore, the aggregate operating cost of the low-quality fleet, C L , is simply NLC(1). In reality, the global tanker fleet contains many vessels and many owners. Conse-quently, the market for tanker services is assumed to be perfectly competitive.66 The competitive allocation of high- and low-quality tankers can be conveniently derived by solving the problem facing a hypothetical fleet planner given the task of maximising the 66. The assumption of a perfectly competitive tanker market has been made by other workers (for example, see Cockburn and Frank 1992). 139 combined profits of both subfleets given the competitive price of tanker services, P (high- and low-quality tankers provide services of equal value). The planner's problem may be expressed as max II = P ( N H + N L ) - C ( N H ) - N L C ( 1 ) - F H N H - F L N L (B-l) - a H N H S H - a L N L S L subject to N H + N L = N , (B-2) where F H is the cost of renting a high-quality vessel and FL is the cost of renting a low-quality vessel. The last two terms on the right hand side of (B-l) represent the expected cost of oil spills to the high- and low-quality fleets, respectively. Substituting the fleet size constraint (B-2) into (B-l) and solving yields the first order condition = [ttLsL +C(1) + F L]- [ a H S H + C(NH) + F H ] = 0. ( B " 3 ) This relationship imphcitly defines the optimal division of the global fleet between high-and low-quality vessels. An interior solution is assumed to exist and the assumption of increasing marginal operating costs ensures that the second order condition for a maxi-mum is satisfied. Now consider the effect of introducing the 1990 Oil Pollution Acf s hability provisions 140 on the optimal number of high- and low-quality vessels. Under the Act, the maximum legal uability facing a tanker owner in the event of a spill is equal to the full damage caused by the spill, K = D . If the spill is caused by a low-quality vessel, however, the cost of the disaster, S L , is still at most the value of a one-ship company's assets, A L . In contrast, if a high-quality vessel causes the spill, the cost, S H , can now be as much as N H A H (if D > N H A H , the realised cost of the spill to the high-quality fleet is still at most the fleet's total asset value). Hence, introducing the Acts liability provisions is equiva-lent to increasing S H while holding S L constant. The effect of this change on the optimal number of high- and low-quality tankers is easily derived by taking the total differential of (B-3) d N H _ d2n d2n (B-4) d S H ~ dNH8SH'dNH2~ ° H ' ( h ) < This shows that increasing the uability facing tanker owners leads to a decrease in the number of high-quality vessels. Because the demand for tanker services is fixed, this implies an increase in the number of low-quality vessels, and, more importantly, a con-comitant increase in the expected number of spills. One obvious criticism of the previous analysis is that it assumes tanker owners have no control over the probability of a spill occurring. This is, of course, unrealistic. For ex-ample, an owner may choose to exceed the legally-required levels of certain activities aimed at reducing the chance of a spill (e.g., emergency response tiaining of crew). This shortcoming of the model is addressed by making the spill probability, a, a function of each vessel's expenditures on spill prevention activities, w. Of course, the probability of 141 a spill occurring will be decreasing in w, cx'(w) < 0. It also seems safe to assume that there will be diminishing returns to expenditures on spill prevention activities, a"{w) < 0 . Making the spill probability endogenous allows us to dispose of the assumption that there a two physically different types of vessel; now only the corporate structures of the high- and low-quality fleets need differ (the subscripts H and L now indicate only the differing ownership and management structures of the two fleets). The problem facing our hypothetical fleet planner becomes max II = P(NH+NL)-C(NH)-NLC(1)-F(NH+NL) (B-5) NH,NL,wH,wL -a(wH)NHSH-a(wL)NLSL-wHNH-wLNL subject to (A-2), where F is the cost of renting a vessel. Substituting (B-2) into (B-5) and solving yields the following first order conditions ^--[a(wL)SL+C(l) + wL]-[a(wH)SH+C(NH) + wH] = 0 ( B ' 6 ) — - = a(wH)SH+l = 0 v ; dwH and 9n (B-8) — = a(wL)SL+l = 0. v ' dw, Interpretation of these conditions is straightforward. Condition (B-6) ensures that at the 142 optimal point the benefit of having one more high-quality vessel is exactly offset by the cost of having one less low-quality vessel. Condition (B-7) ensures that expenditures on spill prevention activities on high-quality vessels are made such that the marginal return to additional expenditures is zero. And (B-8) is simply the equivalent condition for low-quality vessels. Using conditions (B-6) through (B-8), the implicit function theorem, and Cramer's rule, we can derive relationships describing how the size of the high-quality fleet and expen-ditures on spill prevention activities change when the Uability of the high-quaUty fleet for spiU damages increases. Doing so, we obtain dwi = 0 (B-9) ^ = a"(wH)SHa(wH)a"(wL)SL/\H\ (B"10> and = r'7W..W7,,..Vv'77,,..K. /Iwl ( B - 1 1 ) d^-  C"(NH)a(wH)a"(wH)SL/\H\, where \H\ is the determinant of the Hessian associated with problem (B-5), which must be negative if equations (B-6) through (B-8) are to define a maximum. Equation (B-9) simply tells us that the amount of effort made to prevent spills in the low-quaUty fleet is unaffected by increasing the liabihty facing high-quality vessels. 143 Using (B-10) and the restrictions so far imposed upon the model, we determine that dNH/dSHis negative, which indicates that the number of high-quahty vessels falls. Similarly, (B-ll) indicates that dwH /dSH is positive and so expenditures on spill pre-vention activities on high-quahty vessels rise. The increased hability introduced by the Act therefore results in safer high-quality vessels, but fewer of them. Unfortunately, since the demand for tanker services is fixed, the vacuum created by reducing the num-ber of high-quahty vessels is filled by additional low-quality vessels. Hence, the net effect of Act on the number of spills occurring is determined by the relative strengths of these two compering effects and is therefore ambiguous. 144 Appendix C: A Model of Standard Input Pricing The smelting sector consists of n smelting firms indexed i = l,...,n, each producing x, tonnes of refined zinc. The marginal costs of smelting (i.e., for all factors employed ex-cept concentrate) are constant and equal to cs dollars per tonne of refined zinc produced. Concentrate is obtained from a competitive mining sector at a constant cost of cm dollars for enough concentrate to produce one tonne of refined zinc. The profit function of a representative smelter, TC, is therefore given by n = x(p-cs-cm), (C-l) where the market price of refined zinc, which is function industry output, is given by the inverse demand function (3-9). The utility function of the representative smelter is described by (3-11). The smelter must make its production decision before the random variable e is realised. Therefore, the smelter chooses its production level to maximise its expected utility 145 maxE(LT) = £[z-yexp(-r{;t(p-c s-cm})]/2 (C-2) e taking the output of all other smelters as given. 146 

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