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The effects of agent heterogeneity in environmental preferences on social welfare estimates derived by… Rudd, Murray 1998

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THE EFFECTS OF A G E N T HETEROGENEITY IN E N V I R O N M E N T A L PREFERENCES O N SOCIAL W E L F A R E ESTIMATES DERIVED B Y THE CONTINGENT V A L U A T I O N M E T H O D by M U R R A Y R U D D B.Sc, The University of British Columbia, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Agricultural Economics) We accept this thesis as conforming to^iA^pqqirecj standaydV^ THE UNIVERSITY OF BRITISH COLUMBIA January, 1998 © Murray Rudd, 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 fVV'tf IA1TU<7M EconoWcS The University of British Columbia Vancouver, Canada DE-6 (2/88) A B S T R A C T Human society is facing a variety of broad and unique environmental challenges as we enter the twenty-first century. The economic consequences of environmental changes, such as biodiversity loss, depletion of the ozone layer and global climate change, could have very serious impacts on human societies and economies around the globe. Because environmental goods and services are not traded in established markets, economists have developed the contingent valuation method (CVM) as a tool to place economic values on public environmental goods. These values can then be used to compare the relative economic advantages and costs of development projects using cost-benefit analyses or for environmental litigation purposes. One technical issue of particular interest is related to the effects of agent heterogeneity on the accuracy of the social welfare benefits derived using CVM. This study uses an artificial market approach to examine whether valid estimates of aggregate social welfare are obtained using C V M when artificial agent preferences are heterogeneous with regards to environmental goods and whether there are systematic biases away from true welfare as measured by compensating surplus. I found that estimated values of willingness to pay for three levels of environmental change were significantly less than true willingness to pay for the artificial agents. In addition, an analysis of residual errors showed that the use of the log-linear utility function in the empirical utility difference C V M model provided significantly improved, albeit inaccurate, estimates of true willingness to pay compared to the use of the linear utility function. This study has shown that welfare measures estimated suing C V M are unreliable when there is agent heterogeneity, as is the case in the real world, even under simple conditions when agent preferences are specified in accordance with economic theory and there are no technical complications relating to survey design and delivery. ii T A B L E O F C O N T E N T S Abstract ii Table of Contents iii List of Tables v List of Figures vi 1. Introduction 1 1.1 Environmental Change 1 1.2 Problem Statement 3 1.3 Outline of the Thesis 6 2. Background 8 2.1 Ecological Goods and Services 8 2.2 Environmental Valuation and Decision Making 10 2.2.1 Values and Norms 10 2.2.2 Environmental Altruism 12 2.3 Issues Underlying Nonmarket Valuation 14 2.3.1 Neoclassical Economics 15 2.3.2 Cost-Benefit Analysis 23 2.3.3 Contingent Valuation 24 3. Theoretical Considerations 28 3.1 Producer Surplus and Social Welfare 28 3.2 Consumer Demand and Hicksian Welfare Measures 28 3.2.1 Duality Theory and Consumer Demand 29 3.2.2 Hicksian Welfare Measures 31 3.3 Contingent Valuation 33 3.3.1 Alternative CVM Formats 33 3.3.2 The Random Utility Model 34 3.3.3 Welfare Measures in DCCV 38 3.3.4 Confidence Intervals 41 3.4 A Utility Model that Includes Public Goods 42 3.4.1 Model Structure and Terminology 43 3.4.2 Welfare in the Madriaga and McConnell Model 44 iii 4. Methodology 45 4.1 Experimental Design 45 4.1.1 Model Parameterization 47 4.1.2 Bid Vector Calculation 55 4.1.3 Uncertainty 57 4.1.4 Measures of Bias 58 4.2 Estimation Techniques 59 4.2.1 Logit Bid Curve Regression 59 4.2.2 Calculation of Welfare Measures 60 4.2.3 Root Square Error Calculation 61 4.3 Response Surface Estimation 61 5. Results 62 5.1 Regression Results and Welfare Measure Calculation 62 5.2 Proportional Error Analysis 79 6. Conclusions 81 6.1 Accuracy of Welfare Measures 81 6.2 Sources of Bias 83 6.3 Implications for Environmental Valuation 85 References 88 Appendix 97 iv L I S T O F T A B L E S Table 1 - Classification of Economic Goods 8 Table 2 - Parameter Values and Variability 48 Table 3 - Logit Regression Results and Associated Welfare Measures for a 4% Change in Public Good and Two Utility Difference Functional Forms 63 Table 4 - Logit Regression Results and Associated Welfare Measures for a 10% Change in Public Good and Two Utility Difference Functional Forms 67 Table 5 - Logit Regression Results and Associated Welfare Measures for a 16% Change in Public Good and Two Utility Difference Functional Forms 71 Table 6 - Summary of True WTP and Welfare Estimates 76 Table 7 - Tests of Equivalence of Welfare Measures and True WTP 77 Table 8 - Proportional Error Regression Results 79 v L I S T O F F I G U R E S Figure 1 - Compensating and Equivalent Variation [Surplus] Welfare Measures 32 Figure 2 - WTP for a Change in Environmental Amenity 36 Figure 3 - Welfare Measures Derived From the Bid Curve 39 Figure 4 - Random Distribution of Income 49 Figure 5 - Random Distribution of Parameter a 50 Figure 6 - WTP Distribution for Representative Agent with a 16% Change in the Level of the Environmental Amenity (n = 400) 51 Figure 7 - WTP Distribution for Heterogeneous Agents with a 16% Change in the Level of the Environmental Amenity (n = 400) 52 Figure 8 - /} Parameter Distribution for Sample 112 53 Figure 9 - WTP Distribution for Sample 112 with a 16% Change in the Level of the Environmental Amenity (n = 400) 53 Figure 10 - WTP Distribution for Sample 87 with a 16% Change in the Level of the Environmental Amenity (n = 400) 54 Figure 11 - WTP Distribution for Sample 17 with a 16% Change in the Level of the Environmental Amenity (n = 400) 54 Figure 12 - WTP as a Function of the Level of Environmental Change 55 Figure 13 - Distribution of Bid Values in the Normal Distribution 57 Figure 14 - True WTP Distribution for a 4% Increase in Environmental Amenity 64 Figure 15 - Estimated Mean WTP Distribution, 4% Increase in Environmental Amenity and Linear Utility Function 64 Figure 16 - Estimated Median WTP Distribution, 4% Increase in Environmental Amenity and Linear Utility Function 65 Figure 17 - Estimated Mean WTP Distribution, 4% Increase in Environmental Amenity and Log-Linear Utility Function 65 Figure 18 - Estimated Median WTP Distribution, 4% Increase in Environmental Amenity and Log-Linear Utility Function 66 Figure 19 - True WTP Distribution for a 10% Increase in Environmental Amenity 68 vi Figure 20 - Estimated Mean WTP Distribution, 10% Increase in Environmental Amenity and Linear Utility Function Figure 21 - Estimated Median WTP Distribution, 10% Increase in Environmental Amenity and Linear Utility Function Figure 22 - Estimated Mean WTP Distribution, 10% Increase in Environmental Amenity and Log-Linear Utility Function Figure 23 - Estimated Median WTP Distribution, 10% Increase in Environmental Amenity and Log-Linear Utility Function Figure 24 - True WTP Distribution for a 16% Increase in Environmental Amenity Figure 25 - Estimated Mean WTP Distribution, 16% Increase in Environmental Amenity and Linear Utility Function * Figure 26 - Estimated Median WTP Distribution, 16% Increase in Environmental Amenity and Linear Utility Function Figure 27 - Estimated Mean WTP Distribution, 16% Increase in Environmental Amenity and Log-Linear Utility Function Figure 28 - Estimated Median WTP Distribution, 16% Increase in Environmental Amenity and Log-Linear Utility Function Figure 29 - Individual Sample WTP Confidence Interval for Sample 132 C H A P T E R 1 1. I N T R O D U C T I O N 1.1 Environmental Change Environmental change is accelerating as we near the twenty-first century. Increases in human impacts on the environment, caused by both increasing population levels and per capita consumption levels (Ehrlich and Holdren, 1974; Turner et al, 1990), have led to broad and unique environmental challenges such as the loss of biodiversity (Folke et al, 1996; Gowdy, 1997), ozone depletion (Krupa and Kickert, 1989), alteration of the global nitrogen cycle (Vitousek etal, 1997) and global climatic change (Reilly and Anderson, 1992; Fankhauser, 1995; Michener et al, 1997). The potentially serious consequences of these changes on human sustainability and survival has prompted warnings from environmentalists (Meadows et al, 1992; Worldwatch Institute, 1995). Costanza et al (1997) have recently estimated the annual value of global ecosystem services to be USD $33 trillion, compared to world GDP of $18 trillion. Given the potential magnitude of the value of both market and nonmarket environmental goods and services, anthropocentric influences that alter the ability of the ecosphere to provide these services could have major economic impacts. The costs of environmental change to human societies could be enormous, consisting of: direct costs due to environmental damage to property and crops; mitigation costs; indirect costs from deterioration in human health; losses in biodiversity and declines in environmental quality; losses in industrial productivity; and other social adjustment costs. Complicating matters further, some environmental changes, such as "global warming", may be regionally beneficial, improving agricultural and forestry productivity in temperate climates (Mendelsohn etal, 1994; Helms etal, 1997). Page 1 Human impact on the environment also introduces a significant element of environmental risk that may impose high costs on society. Developments in the emerging "science of complexity" are showing that the ongoing development and evolution in complex, adaptive ecological (and economic) systems are fundamentally unpredictable (Arthur, 1989; Novak and May, 1992) and that threshold effects are to be expected (Bak and Chen, 1991; Kauffman, 1993). Once these thresholds are crossed, non-linear ecological processes can rapidly, and irreversibly, change and move a system towards new "attractors" which may, or may not, be stable (Ferriere and Fox, 1995). There are strong reasons for precautionary approaches in a complex, unpredictable world because human activity may cause breakdown of ecosystem resilience as we cross irreversible thresholds (Arrow etal, 1995; Hollings, 1996; Mangel etal, 1996). Human societies have caused serious environmental impacts for centuries and will, obviously, continue to have significant impacts on the environment in the future. Impacts due to increases in human scale will continue until the human population stabilizes, according to the most optimistic United Nation projections, at a projected level of almost 12-billion by the year 2150 (Goodland and Daly, 1996). The scope of human environmental impacts will also increase, possibly beyond sustainable levels, as global income levels rise and per capita consumption rates in developing countries rise towards Western standards. Over a decade ago, it was estimated that human appropriation of primary global productivity already approached 40% (Vitousek et al, 1986). Furthermore, due to the unique time in which we live, in which species extinction may be of the same order of magnitude as during five other major extinction episodes on earth, it is conceivable that the "valuation decisions our species has made in the recent past and will make during the next few decades will determine the fate of life on Earth for the next tens of millions of years" (Gowdy, 1997: p. 25). Page 2 1.2 Problem Statement Much of the modem social decision making process relies on cost-benefit analysis ("CBA"). Applying this decision-making methodology to issues of environmental development, one asks if the economic costs of a development project outweigh the benefits of conservation (or vice-versa). If CBA is embraced as a decision-making methodology, then nonmarket valuation becomes critically important in the calculation of social benefits. Kunin and Lawton (1996, as cited in Bengtsson et al, 1997), discussing the problem of biodiversity loss, succinctly state the general nature of the nonmarket valuation problem: "In a world of scarcity, it is not enough to decry the continual erosion of the earth's biotic niches. If we expect the majority of people to care, we must have some way to evaluate the loss. Just what is a species worth?" Public environmental goods, by definition, are those that have no market impacts and are, therefore, impossible to value using standard economic techniques; the existence value argument is not a choice variable in an individual's utility maximization problem (Boyle and Bishop, 1987). Yet, people clearly value these public environmental goods and services. Ignoring nonmarket values can lead to the significant underestimation of the economic benefits of conservation, a bias towards development in the decision-making process, reduced social welfare and a misallocation of societal resources with systematic relocation of the most damaging industrial and extractive industries to areas of the world most susceptible to environmental damage (Randall, 1993). The valuation of public goods, and nonmarket environmental goods in particular, has been an area of intense research in the last ten years with much of the focus of on the contingent valuation methodology ("CVM") as an instrument for assessing the value of nonmarket goods. Nonmarket valuations derived using C V M are then used in CBA and/or for damage assessment Page 3 purposes in litigation (Arrow et al, 19931; Hausman, 1993; Smith, 1993). Over two thousand C V M studies in over forty countries were undertaken by 1995 (Carson et al, 1995). If one accepts certain assumptions justifying the use of cost-benefit analysis and contingent valuation methodology then it is possible to use C V M to derive measures of individual welfare resulting from changes in the level (i.e., quantity or quality) of environmental amenities and to aggregate these measures into social welfare measures. Nine important assumptions, which are dealt with in more detail in Section 2 of the thesis, are oudined below: 1) welfare is the satisfaction of preferences; 2) limited human cognitive ability and environmental complexity (of the biological and social environment in which humans live) do not preclude an optimization approach to economics; 3) public goods can be considered to be commodities, definable in terms of rates of exchange with other marketable commodities; 4) utility functions for public goods do exist for individuals; 5) willingness-to-pay can serve as an interpersonal, comparable measure of individual welfare; 6) net benefits and net costs of projects can be compared by aggregating private firm and individual welfares; 7) potential Pareto optimality (i.e., use of Kaldor-Hicks criteria), despite imposing costs on some individuals or sectors in a society, is a valid criteria for decision making in CBA; 8) referendum responses in C V M studies accurately reflect an individual's market responses, not attitudes or general opinions, toward public environmental goods and individuals answer hypothetical questions as they would if they actually had to enter a market transaction; and 9) embedding effects, "yea saying", inappropriate sample frames and other technical issues of survey delivery and design do not invalid welfare estimates in C V M studies. A blue-ribbon NOAA panel of Kenneth Arrow, Robert Solow, Edward Learner, Paul Portnoy, Roy Radner and Howard Schuman was formed to evaluate and recommend a variety of specific measures for CVM surveys and valuation procedures. Much of the controversy generated by CVM valuation results from litigation efforts over the Exxon Valdez oil spill: both sides in the mitigation/compensation dispute (i.e., Exxon and U.S. governments) provided significant funding for CVM research to expert witnesses in the litigation. Page 4 Not surprisingly, "the scientific standing of [contingent valuation] methodology remains unresolved" (McFadden and Leonard, 1993: p. 165). Besides the moral and theoretical objections to the use of economic measures as welfare, there are also potentially other more technical issues that might affect the accuracy of contingent valuation. One technical issue of particular interest, and one which has remained unexamined to date, is related to the issue of heterogeneity in the preferences of economic agents and, specifically, the effects of heterogeneity on the accuracy of benefits derived using CVM. Economic analyses at the aggregate level typically make the simplifying assumption that economic agents are homogeneous and that, as a result, individual economic welfare measures can be aggregated to obtain social welfare. In practice there are certainly significant differences in preferences for environmental goods and services within a given population; the conflicts between environmentalists and the forest industry in British Columbia during recent years provides a pointed local example. An obvious research question arises: given the assumptions outlined above, but in the face of heterogeneous agent preferences with regards to environmental goods and services, does the contingent valuation methodology provide valid estimates of aggregate social welfare? In this research, I use artificial agents operating in artificial markets to examine this question. In particular, the null research hypothesis that I test is: are social welfare measurements derived using CVM in an artificially-generated population of agents who are heterogeneous in their degree of "environmental altruism" equal to the "true" social welfare measures for the population? Mathematically, n (1) H ^ J v f ^ w w , <=i Page 5 where w, is the true willingness to pay of each of the « individuals in the artificial society (market), and w is the average estimated willingness to pay calculated using CVM. This thesis is an exploration of the possibility that C V M (and related estimation models and aggregation) results in systematic biases away from the "true" welfare as measured by the compensating or equivalent surplus (van Kooten and Bulte, 1997). I use a direct utility function developed by Madriaga and McConnell (1987) and calculate associated "true" willingness-to-pay of artificially generated economic agents for changes in an environmental amenity. These true WTP values are then compared to WTP estimates derived using the dichotomous choice contingent valuation methodology. The true and estimated values are compared and the proportional error is analyzed as a function of the characteristics of the artificial economic agents, the magnitude of the change in the environmental amenity, the proportion of existence to use values and the functional form used to represent agent utility. Fifty samples of four hundred agents each (200 environmental "altruists" and 200 "non-altruists") are generated for each of three levels of change in an environmental amenity. Each of the agents is presented with a bid amount (i.e., a proposed fee) for the given level of environmental change and the response is recorded based on the difference between the true WTP (compensating surplus) and the bid amount. A total of 6,000 of the 120,000 artificial survey responses (150 samples, 400 agents per sample, 2 utility functional forms) are then chosen randomly for the response surface analysis of proportional error. 1.3 Outline of the Thesis In the second chapter of this thesis, general issues regarding the definition and valuation of environmental goods and services are briefly examined. The underlying assumptions that must be Page 6 accepted for C V M to be viewed as a valid tool for welfare measurement, as outlined above, are discussed in more detail. The third chapter of the thesis examines the theoretical development of Hicksian measures of consumer surplus used in CBA, derivation of Hanemann's (1984) random utility model and welfare measures for the model. The specific model used in this thesis, Madriaga and McConnell's (1987) direct utility model, is then outlined. The fourth chapter of the thesis, on methodology, presents the more practical aspects of model parameterization, bid vector calculation, and model estimation. The results are presented in the fifth chapter and a discussion of the results, and implications regarding the validity of using C V M for nonmarket valuation, are presented in the final chapter of the thesis. Detailed results of the regressions are included in a separate appendix. Page 7 CHAPTER 2 2. BACKGROUND 2.1 Ecological Goods and Services The central problems of environmental use by human societies relate to maintenance of the productive capacity of resource systems and the efficient allocation of resources amongst competing users. Costanza and Daly (1992) discuss the natural resource system in terms of natural capital and natural income. The flow of the ecological goods and services that comprise natural income includes life support and waste assimilation services as well as the provision of consumer goods (Costanza etal, 1997). Resources can be classified into four general types (Ostrom et al, 1994) characterized by their degree of jointness (ease of exclusion) and subtractability (the extent to which one person's consumption reduces the supply of the good to others) as indicated in Table 1. Table 1 - Classification of Economic Goods Degree of Subtractability Low High Ease of Difficult Public Goods Common Pool Resources Exclusion Easy Toll Goods Private Goods Issues of sustainability are "provision" problems while issues of distribution and equity are 'appropriation" problems. Provision problems threaten the on-going ability of natural capital to Page 8 provide ecological services or the very existence of the natural capital itself. Appropriation problems can lead to provision problems through over-allocation of capital to harvesting efforts. The most pressing ecological problems humans face are recent global threats for which there are no precedents. Human impacts are now so pervasive that resource services such as the atmosphere's ability to assimilate human-made pollutants, once far in excess of our capabilities to tax, could well be approaching their physical limits. Most of these problems, caused by increases in human impacts on the environment, can be viewed in terms of public goods (e.g., ozone, the atmosphere, biodiversity, clean water) becoming increasingly scarce; in some ways, public goods are becoming common pool resources in a global commons. Environmental goods and services provide humans with a number of distinct types of use and non-use values (Krutilla, 1967; Boyle and Bishop, 1987; Madriaga and McConnell, 1987; Smith, 1993). Use values include consumptive, non-consumptive and indirect or passive use values. Non-use values are typically thought to include all "existence" values independent of any current use behaviour by an individual and include bequest value, option value, altruism values, and other inherent or intrinsic values (e.g., resulting from an environmental ethic, the inherent value of species, or stewardship values). McConnell (1997) defines existence value as "a person's willingness to pay for the preservation, protection, or enhancement of resources for which he or she has no plans for personal use" (p. 24). More et al (1996) believe that the distinction between use and non-use value is needlessly confusing because the "non-use" values still fulfill functions for humans. They suggest that on-site and off-site use values may effectively include almost all of the values most often referred to as nonmarket. In their terminology, off-site use values are thought to include altruism, bequest and possibly existence values. They note that bequest is just a special category of altruism and that the logical basis for any inherent existence or intrinsic value in the absence of human Page 9 functions is highly questionable. Boyle and Bishop (1987) have also argued that existence value should be restricted to values arising from altruism. 2.2 Environmental Valuation and Decision Making Given the inevitability of ongoing environmental and social change, humans need to be able to evaluate alternative choices regarding the relative degree of environmental development and our interactions with natural ecological systems. Tough questions arise when examining these issues. How do we quantify environmental change? How do we assess what proportion of the changes are caused by human activity rather than being the result of inherent environmental variability? How do we, or should we even attempt to, put a value on the magnitude and variability of environmental changes? How can we value ecosystem resilience when we do not understand how ecosystems function and where thresholds exist? In response to difficult questions like these, the United States National Research Council (1996) has identified the development of improved social science and risk assessment tools as the top priority in environmental R&D in the next quarter century. Specifically, research programs to improve analytical tools for nonmarket valuation and cost-benefit analysis ("CBA") models were identified as requiring immediate and focused attention. 2.2.1 Values and Norms According to one view, "various conceptions of value share the common idea that values are guides to decision-making" (More et al, 1996: p. 398). Human values serve as criteria that people use to make judgments and to specify the relationships between one thing and another. The authors go on to propose that biological, psychological and social systems of behaviour are hierarchically arranged with each of the systems arranged around ultimate functions within the Page 10 individual (i.e., maintenance of long-term health, self actualization, and the preservation of society). Within the social hierarchy, rational values involve standards for truth, moral values involve standards for conduct, aesthetic values involve standards for appreciation, spiritual values involve standards for meaning, and economic values involve standards for choices between goods and services. As a person deals with complex questions relatively high in the social hierarchy, almost all the issues deal exclusively with intangibles. The most basic questions are: Who are we?; How do we relate to the world? More et al (1996) point out that tangible goods and services have value because they allow us to fulfill certain functions and that the market prices of tangible goods help facilitate these choices. A person's willingness-to-pay to preserve wolves in Alaska, for example, may relate more then to helping provide a person an identity (i.e., what kind of a person am I?) rather than any intrinsic value of the wildlife itself. While this view may help conceptually justify the use of economic analyses for nonmarket goods, because all "non-use" values are tied to the fulfillment of human goals and are really just off-site use values, it is important to be clear about what precisely is being valued (i.e., the resource, knowledge of the resource, satisfaction people derive from the resource or satisfaction that people derive from 'doing something to preserve nature' rather than preserving wolves per se). This raises a question of considerable importance in economic cost-benefit analysis: do individuals make decisions based on market values or on a set of wider values based on social norms (i.e., generally accepted common values)? The debate in the literature is intense about this point (Sagoff, 1988; Kahneman and Ritov, 1994; Blarney etal, 1995; Carson, 1995; Carson etal, 1996; Common etal, 1997; Crowards, 1997; Gowdy, 1997). On the one hand is the view that contingent valuation, the primary economic tool that can be used to probe questions of pure public Page 11 good valuation, "actually captures a hodgepodge of market values and broader values and forces them into the indifference curve framework of market exchange" (Gowdy, 1997: p. 27). On the other hand, Carson et al (1996), strong proponents of contingent valuation, argue that market values, derived by consumer choice, are the only values that matter in CBA and that "the answer from economic theory is very clear: it is utility - whatever its source - that matters for total value. Motives are essentially irrelevant and acceptance of consumer sovereignty is one of the most enshrined principals [sic] of economics" (p. 7). Sagoff (1988) proposes that people make choices according to "citizen values" or "consumer values", depending upon the context in which the choice is put forward. This view is consistent with that of More et al (1996), who believe that there are values other than economic values that are important in social decision making. Sen (1977) has previously noted that our concepts of economic rationality are both too weak and strong at the same time because humans have the capacity for maintaining multiple preference scales. Sen also discusses the concept of "commitment" where commitment is a sort of social duty that may conflict directly with personal welfare. Commitment in an environmental context could lead to a "land ethic" or "environmental ethic" which influences decision making, but in a decidedly nonmarket fashion. 2.2.2 Environmental Altruism One possible reason for people holding nonmarket values for environmental goods is altruism (Randall and Stoll, 1983; Boyle and Bishop, 1987; Madriaga and McConnell, 1987; Johansson, 1993; Crowards, 1997; McConnell, 1997). For the purposes of this thesis, it is sufficient to note that: (1) altruism is largely the result of the development of social norms; and (2) there are different kinds of altruism that lead to different results in CBA. Page 12 Simon (1993) provides an informal definition of altruism in which altruistic behaviour can be considered that in which the actions of one person are influenced by expected consequences (pleasure and pain in addition to economic gain or loss) for other persons. Simon suggests that docility, the tendency to depend on suggestions, recommendations and information obtained through social channels at low cost, contributes to human evolutionary fitness. Wealth is not closely linked to fitness in humans but docility is when the complexity of life is such that there are rewards for the acceptance of social influences and, consequently, altruism. Certain conditions encourage altruism and the evolution of cooperation while others discourage it. Altruism is costly for an individual but there is evidence in both biological (Connor, 1995) and social (Huberman and Glance, 1995; Sethi, 1996; Skyrms, 1996) systems that "non-optimizers" (i.e., altruistic individuals) can outperform "optimizers" (i.e., purely selfish and individualistic agents) even in terms of material selfish gains. Reciprocal altruism relies heavily on the ability of agents to recognize their trading partners and to reward cooperation with later cooperation and punish defection; humans are a trading species with a seemingly unique talent to assess their trading partners (Noe and Hammerstein, 1995). On the second point, McConnell (1997) shows that motivations regarding environmental altruism explicitly matter in the calculation of welfare measurements. Two major types of altruism are non-paternalistic, in which an altruist gains utility from increasing the well-being of others, and paternalistic, in which an altruist values the use of a particular resource by others. It has been alleged that altruistic externalities might result in a kind of double counting in CBA (Diamond and Hausman, 1994), but McConnell shows that altruism has no impact on the cost-benefit outcome if the altruism is non-paternalistic and that benefits for paternalistic altruists can legitimately be used in CBA without double counting problems. Page 13 Paternalistic altruism occurs when an altruistic individual values the use of services from a natural resource by other beneficiaries: a paternalistic altruist gets no utility from preserving a resource if nobody actually visits or uses the resource. Certain types of goods (natural resources, education, medical care, food) are often referred to as "merit goods" and are commodities that might be provided by the market but are viewed as not being worth consuming by some individuals. Paternalistic altruists believe that the consumption of these goods is beneficial to others and, hence, derive utility from providing/preserving merit goods. Under the assumption of paternalistic altruism, it is possible for nonmarket values to enter, and alter, social CBA outcomes. For the purposes of this thesis, it will be assumed that we are dealing with a mixed group of paternalistic "environmental altruists" and that nonmarket valuation of an environmental amenity could, therefore, be used to alter the social choice to preserve or restore the natural resource. 2.3 Issues Underlying Nonmarket Valuation Economics is commonly defined as the study of choice under conditions of scarcity, among alternative means to multiple but given ends (Robbins, 1932). The nature of increasing scarcity of environmental services and the implicit trade-offs required for environmental development naturally suggest, to many people, an "economic" approach be taken in environmental policy development. It is useful to examine the theoretical underpinnings of neoclassical economics, cost-benefit analysis and contingent valuation methodology in order to better understand current environmental valuation techniques. Nine issues of importance are briefly outlined below; an emphasis is placed on identifying the key assumptions affecting the validity and 'reasonableness' of using C V M to value nonmarket goods and using the resultant welfare measures in CBA-based social policy decision making. Page 14 2.3.1 Neoclassical Economics i 2.3.1.1 Is Welfare the Satisfaction of Preferences? Each agent is a decision-maker and is assumed to act rationally in neoclassical theory (Gravelle and Rees, 1992). Rationality, which is commonly taken to mean that agents are consistent self-interested utility maximizers, requires that: 1) agents set out all feasible alternatives rejecting any which are not feasible; 2) agents take into account whatever information is available or worth collecting; 3) agents rank each alternative in order of preference in a way that satisfies assumptions of completeness and internal consistency; and 4) agents choose the alternative highest in the ordering according to their preferences. Hausman and McPherson (1996) point out that agents are rational if and only if their preferences may be represented by ordinal utility functions and their choices maximize utility (utility is not some objective function to be maximized but is merely the result, or an index, of economic agents doing what they most prefer). They note that self-interest is not necessarily required to define rationality. Economics, by implicitly laying down conditions about what agent choice and preference ought to satisfy (i.e., self-interest), is much narrower in scope than utility theory and provides a theory of "the causes and consequences of people's economic choices and of the reasons for them" (Hausman and McPherson, 1995: p. 39, emphasis theirs). They go on to assert that it "is only plausible to identify well-being and the satisfaction of preferences if (1) individuals are rational, (2) individuals are self-interested, (3) individuals are well-informed, and (4) individual preferences are not formed or deformed in odd ways" (Hausman and McPherson, 1995: p. 45). The standard view of economics depicts a world in which a person's preferences define what is best for the person (bear in mind the previous quote from Page 15 Carson et al).2 This can clearly lead to undesirable and inefficient social or environmental consequences if preferences are based, for example, on false beliefs.3 For cost-benefit analysis, a fundamental assumption underlying the theoretical CBA framework is that welfare can, indeed, be equated with preference satisfaction. 2.3.1.2 Is Optimization Possible? For many purposes in neoclassical economics it is not necessary to require every individual to act rationally so long as, in aggregate, enough people act rationally to make theories of behaviour about aggregates applicable. Economists can, hence, assume rationality of an average or "representative" agent. This implies neoclassical economic models take the form of optimization problems. By varying the parameters of the decision problem, behavioural relationships such as supply and demand functions can be traced out (Gravelle and Rees, 1992). The ability of economic agents to optimize is, however, severely limited for two main reasons: (1) human cognitive capabilities are limited (summarized in Simon, 1996) and it might be unreasonable to expect accurate C V M responses given the significant cognitive demands placed upon survey respondents (Gregory et al, 1993); and (2) it is impossible to calculate outcomes of any action in complex natural and social environments (Kauffman, 1993). The neoclassical viewpoint is one of "incredibly smart people in unbelievably simple situations" while the alternative viewpoint is one in which we ask "how believably simple people cope with incredibly complex situations" (LeijonhufVud, 1993). "Motives are essentially irrelevant and acceptance of consumer sovereignty is one of the most enshrined principals [sic] of economics". For example, McDaniels et al (1996) found a persistent lack of connection in personal judgments regarding the causes and consequences of global environmental change processes. Without a clear understanding and connection between personal actions, such as the use of CFC-releasing refrigerators and ozone depletion, people Page 16 In the face of cognitive limitations, people use heuristics and adaptive search techniques to "satsifice" in an ever-evolving environment (Boyd and Richardson, 1993; Arthur, 1994; Simon, 1996). Agents search to find a good solution and then use that as a benchmark in the search for better solutions. This is conceptually far different than the opturiization/mathematical prograrrirning approach posited by neoclassical economics. The editors of the journal Organization Science recently called for papers for a special issue on complexity (Organization Science, 1997). They consider a number of findings about complexity science now to be well-established: many dynamic systems do not reach an equilibrium; processes that appear random may actually be chaotic; complex systems can exhibit highly path-dependent behaviour; complex patterns arise from the interactions of agents following simple rules; complex systems resist reductionist analysis; and complex systems tend to exhibit self-organizing behaviour. Complex systems are inherently non-linear and many problems are non-polynomial hard (NP-hard) and some are fundamentally non-computable (Garey and Johnson, 1979). Solution times to NP-hard problems grow exponentially with the size of the problem4 and are impossible to solve in reasonable amounts of time (Westerhoff et al, 1996; Huberman et al, 1997). "Chaos" is also an important phenomena of non-linear systems theory. Neighbouring points in a system exhibiting chaotic behaviour can diverge over time at an exponentially increasing rate. The essential feature of chaos is that, even in a simple non-linear system with full information, prediction of long-term results is none-the-less impossible (Judson, 1994). may make purchase decisions based on the false belief that aerosol sprays are the main source of CFCs and that the problem with aerosols is under control. A search of all possible paths for the game '20 questions' would involve about 17-billion alternatives while for checkers the number rises to 1040 and for chess to 10IM. Page 17 Neoclassical economics uses an equilibrium methodology to derive the supply and demand relationships for commodities. To fully accept CBA and C V M requires accepting the validity of an equilibrium methodology. 2.3.1.3 Are Public Goods Commodities? The two basic agents of microeconomics are consumers (individuals or households) and firms. Firms produce commodities traded in markets while consumers provide firms with labour and consume commodities. Commodities can be either goods or services and each commodity has a price defined in terms of commodity rates of exchange. The market is a mechanism for the exchange of commodities and does, or will, exist for every commodity. According to Gravelle and Rees (1992), " if no market exists for some product then the good in question is not a commodity in the view of microeconomics" (p. 3). ' The lack of established markets for public environmental goods and services is, therefore, a crucial problem that must be addressed if CBA is to be considered a valid decision-making methodology. The typical response of an economist to this problem is to proffer that any good can become a commodity if property rights are clearly assigned and a trading mechanism established. There are objections to the viewpoint that any ecological service can be "commoditized". Many ecologists object because of the unique and critical role of ecological life support services (Bengtsson et al, 1997).5 It has also been argued (Costanza and Daly, 1992) that natural and manufactured capital (i.e., technology) are fundamental complements rather than substitutes and that ecological services cannot be viewed as commodities because they cannot be used in trade-offs Many environmentalists, particularly those associated with the "deep ecology" movement, also object to the very idea of ecological resource being viewed as a commodity for a variety of moral and philosophical reasons which will not be examined here. Page 18 of the sort outlined above by Gravelle and Rees. When basic life support systems break down, there can be no meaningful commodity rates of exchange between ecological services and market goods. A fundamental assumption of CBA and C V M is, however, that any public good can be viewed as an economic commodity and can, therefore, be valued in terms of commodity rates of exchange. 2.3.1.4 Do Utility Functions for Public Goods Exist? Rational economic agents have preference functions that are characterized by features of the six axioms of choice: reflexivity; completeness; transitivity; continuity; non-satiation; and convexity (Deaton and Muellbauer, 1980). The question remains whether individuals have valid utility functions for public environmental goods even if a "market", real or simulated, can be established. Economic assumptions regarding utility functions are stronger than utility theory used in other fields of social science: the existence of a utility function is a much stronger condition than the existence of a preference ordering (Hausman and McPherson, 1996; Common et al, 1997). On one side of the argument, skeptics believe that unfamiliarity with environmental goods and services, preference uncertainty, the phenomena of derived preferences, and risk factors could lead to utility functions that could not validly be used for environmental valuation purposes (Edwards and von Winterfeldt, 1987; Gregory etal, 1993; Diamond and Hausman, 1994; Kahneman andRitov, 1994; Common etal, 1997). Lexicographic utility functions for environmental goods are one such type of utility function for which researchers have found empirical support (Stevens et al, 1994; Spash and Hanley, 1995). Diamond and Hausman (1994) are highly critical, arguing that "contingent valuation is a deeply flawed methodology for measuring nonuse values... This skepticism arises from the belief Page 19 that the internal consistency problems come for an absence of preferences, not a flaw in survey methodology" (p. 62). On the other hand, Carson argues that consumers face the same sort of problems of unfamiliarity and uncertainty for many market goods yet there is no fundamental reason why consumers cannot value these with minimal effort (Carson, 1995, cited in Kramer and Mercer, 1997). If consumers can form preferences and make decisions regarding market purchases, there should be no reason why, when supplied with adequate information in a properly designed contingent valuation survey, they cannot make intelligent choices regarding environmental goods and services as well. Hausman and McPherson (1996) also argue that it is incorrect to base social policy on unreflective preferences that have not been subjected to challenge. Preferences for public goods, that economists infer from people's choices, can respond to arguments and may change after public debate (Cummings et al, 1995). C V M assumes that utility functions for environmental goods and services do exist over the relevant range of interest. 2.3.1.5 Can Welfare be Expressed in Monetary Terms? Economic CBA requires that an individual's welfare be expressed in monetary terms. If welfare is the satisfaction of preferences in economics, and costs and benefits of conservation [development] fall on different individuals in a society, then a question arises as to how the welfare of different individuals, based on the satisfaction of different and heterogeneous preferences, can be compared. Hausman and McPherson (1996) in noting the complications in trying to compare interpersonal welfare, maintain that "regardless of whatconstraints one plaes on which Page 20 preferences count, one must confront the conceptual difficulties involved in comparing the extent to which the preferences of different individuals are satisfied" (p. 87). Neoclassical economics makes an important assumption in this matter and considers willingness-to-pay ("WTP") as the appropriate measure of individual welfare.6 After making this assumption, it is possible to claim that projects whose total benefits exceed their total costs will increase overall societal welfare; the logical basis for cost-benefit analysis is thus established. There are problems with assuming WTP is a valid welfare measure (Hausman and McPherson, 1996). Firstly, people responding to WTP queries may not provide trathful answers for strategic reasons. Secondly, using WTP as a welfare measure is biased against the poor because WTP is a function of both income and preferences. Finally, there is a conceptual problem in equating only money and welfare. Equating welfare, the satisfaction of personal preferences, with money would seem intuitively wrong for many people who believe that satisfaction in life is derived, at least in part, to fulfilling what More et al (1997) identify as ultimate human functions of self actualization and the preservation of society. According to Simon (1993), "that economic actors desire only economic gain is a far stronger assumption than they maximize utility. It is also empirically false" (p. 158). The use of CBA is based on the premise that interpersonal comparisons of welfare can be made using monetary values for welfare measurement. Recall the definition of WTP by McConnell: "a person's willingness to pay for the preservation, protection, or enhancement of resources for which he or she has no plans for personal use". Page 21 2.3.1.6 Can Individual Welfare Measures be Aggregated? If one accepts that WTP can be used as an interpersonal measure of individual welfare, there still remains the issue of the validity of aggregation of individual WTP to derive social welfare measures. Duffield and Patterson (1991) discuss a number of technical aspects of aggregating WTP measures in C V M studies; these will be addressed further in Chapter 3 of this thesis. On a broader level, homogeneity of economic agents is usually assumed at the macro-level in economics. This assumption is basically a mathematical convenience with no strong basis in theory (Kirman, 1992). Vriend (1996), commenting on results originating with Sonnenshein (1972) and Debreu (1974), asserts that "by now we know that it is theoretically impossible to get the necessary characteristics of aggregate demand functions ... In other words, in the aggregate, the assumptions of individual preferences have in general no implications" (p. 279). It can be shown that the "representative agent" of neoclassical economics can take a decision that conflicts with the preference of each individual in an economy under certain circumstances. To ensure that a representative agent truly represents the interests of a collective of individual agents requires economists to either assume that: (1) individuals have identical nomothetic utility functions; or (2) individuals have nomothetic utility functions and that their relative income distribution is fixed and independent of prices. Clearly, neither of these assumptions is realistic or desirable, yet the use of representative agents is at the core of much of macroeconomics, social welfare theory, cost-benefit analysis and environmental economics. Kirman (1992), in a critical article, points out that "this reduction of the behaviour of a group of heterogeneous agents even if they are all themselves utility maximizers, is not simply an analytical convenience as often explained, but is both unjustified and leads to conclusions which are usually nusleading and often wrong" (p. 279). For CBA it is necessary, however, to assume Page 22 that individual welfare measures, as represented by WTP, can be aggregated to derive social costs and benefits of alternative policies or projects. 2.3.2 Cost-Benefit Analysis Social welfare in economic CBA is the summation of consumer and producer surplus. A particular project is deemed desirable using CBA if the sum of all benefits exceed the total costs of the project. Producer surplus is comparatively simple to calculate in economic CBA and is not exarnined in detail in this thesis. The situation with consumer surplus is much more complicated, especially when public environmental goods factor into the cost-benefit equation. 2.3.2.1 Is Cost-Benefit Analysis Fair? An immediate and obvious issue regarding CBA is the question of fairness or justice. As noted previously in Section 2.3.1.5, there are biases against the poor inherent in analyses that equate individual welfare and money. There are also implications for social justice when using the "potential Pareto improvement" criteria in CBA. Distribution issues cannot be avoided in CBA: any project or program is bound to affect different individuals in different ways. The concept of Pareto improvement proposes that an improvement in social welfare can be made if (identifying "better off' with the more thorough satisfaction of personal preferences) one person can be made better off and all other people in a society are at least as well off as a result of a particular development decision. A Pareto optimal state is one in which nobody can be made better off without making somebody worse off. Because true Pareto improvements are rare (any decision will almost certainly adversely affect someone if welfare is defined as the satisfaction of personal preferences), it is more common to examine potential Pareto improvements using the Kaldor-Hicks criteria (see van Kooten and Page 23 Bulte, 1997, for a general discussion of decision criterion in CBA). In this case, the "winners" in a development decision gain such an amount of welfare, in monetary terms, so as to be able to hypothetically compensate the "losers". CBA can, using the Kaldor-Hicks criteria, be limited to summing and comparing costs and benefits of any particular project. In reality there are no obligations of the "winners" to pay any compensation to the "losers". Hausman and McPherson (1996) note that, as a result, "it is easy to see why many are uneasy about using cost-benefit analysis to make social choices. Like the other Pareto criteria, it [potential Pareto criteria] ignores questions of justice, but unlike other Pareto criteria it sanctions policies that make some people worse off' (p. 97). If one uses a different decision-making framework, such as that proposed by John Rawls (1971) for example, where well-being is measured by an index of available primary social goods, decisions made under CBA criteria might be unacceptable. In a Rawlsian framework, the well-being of the poorest of society would be protected in situations where large benefits accruing to the rich would outweigh relatively low costs being imposed on the poor. To accept the results of CBA, one must be prepared to accept the fact that winners will not compensate losers in a development project and that, as a result, CBA sanctions policies that decrease welfare for some individuals and segments of a society. 2.3.3 Contingent Valuation Contingent valuation methodology has become the preferred means for attempting economic valuation of public environmental goods. According to Madriaga and McConnell (1987), contingent valuation is a direct valuation method that asks people either: (1) how much they would be willing to pay for a change in the quantity and/or quality of an environmental amenity; or (2) whether they would accept or reject a proposed fee for a given level of change. Page 24 The 1980 Comprehensive Environmental Response, Compensation, and Liability Act in the United States imposed a legal mandate for the development of resource valuation methods that could be used for damage assessment purposes. A further U.S. court ruling (State of Ohio et al vs. U.S. Department of the Interior et al, 1989) deemed that reliable nonmarket valuation techniques could be used in cases where environmental goods and services could not be determined based on market pricing mechanisms. In cases where there are some linkages between nonmarket and market goods, it is possible to use hedonic price methods (Rosen, 1974) and travel cost methods (Ward and Loomis, 1986; Fletcher et al, 1990) to value environmental goods but, in many cases, even the weak links needed for these indirect methods are absent. C V M has thus been developed to value pure public goods. The essential feature of C V M is that researchers attempt to construct "hypothetical markets" for public environmental goods. Consumers are queried directly about their willingness to pay and the responses are used to develop estimates of aggregate consumer welfare that can be using in CBA. Carson argues (Carson, 1994, cited in Kramer and Mercer, 1997) that "without CV [contingent valuation], a comprehensive benefit-cost analysis involving nonmarket goods with substantial passive/non-use values is impossible" (p. 2). 2.3.3.1 Do Referendum Responses Reflect Market Values? C V M studies now most commonly present referendum style "take-it-or-leave-it" questions to respondents. These are thought to present individuals with choices most like those they would face in making purchase decisions for market goods (Cameron, 1991). Arrow et al (1993) recommended the use of referendum questions in C V M studies in their NOAA Panel report. According to Diamond (1996), "the link between voting and preferences is complex" (p. 337). As noted earlier (Section 2.3.1.4), individuals may not have well-defined utility functions for Page 25 public environmental goods. Environmental goods and services might be a proxy for other derived goods or may be a way in which people help to define their identity. An additional problem arises with the use of referendum format C V M questions: do people respond to referendum questions in a market fashion? It has been proposed that rational individuals will act differently in market and voting situations (Brennan and Lomasky, 1993). According to Common et al (1997), in a referendum: "The perceived cost to the individual of the act of expressing an attitude, or an ethical commitment, is small, as is the probability of securing an instrumental benefit... Where the environmental attribute has symbolic content, as will often be the case where existence value type issues are involved, responses will be dominated by expressive return considerations, rather than by weighing of instrumental costs and benefits" (p. 228). Mainstream economists also have doubts about the validity of C V M referendum responses being indicative of market decisions (McFadden, 1994). There is a large amount of skepticism about the ability of respondents to answer hypothetical questions in the same way in which they would answer if their money was "on the table" (Neill et al, 1994; Cummings et al, 1997). Income constraints, in particular, are often not well considered in C V M responses and responses can vary considerably after respondents are given the chance to reflect upon, and discuss, their initial decisions (Cummings etal, 1995). For C V M results to be valid measures of nonmarket value it must be assumed that the responses to referendum questions are taken in a market context and that expressive attitudes are not the basis for voting behaviour. 2.3.3.2 Do Technical Issues Preclude Accurate Estimation? There are a wide number of "technical" issues relating to survey design and delivery (e.g., embedding effects, starting point bias, geographic scope of the study, "yea saying", etc...) that can Page 26 influence the outcome of a C V M study. Books by Mitchell and Carson (1989), Hausman (1993) and Bjornstad and Kahn (1996) review many of the main controversies and will not be addressed further in this thesis. Arrow et al (1993) provide a broad range of general guidelines to minimize technical problems involved with CVM survey design and delivery. To accept CBA as valid, one must assume that technical issues of survey design and delivery do not adversely influence C V M welfare measurements. Page 27 CHAPTER 3 3. THEORETICAL CONSIDERATIONS In this chapter, I first examine the theoretical basis for C V M and then the derivation of measures of consumer welfare. In the final portion of this section, the Madriaga and McConnell (1987) model of a public environmental good is outlined. 3.1 Producer Surplus and Social Welfare Duality theory (Shephard, 1953; Diewert; 1974, Blackorby et al, 1978) is at the core of modem microeconomic analysis and is well-developed for both producer (McFadden, 1972; Chambers, 1988) and consumer theory (Deaton and Muellbauer, 1980; Just etal, 1982). The measurement of producer surplus is an important aspect of social welfare theory. It is comparatively easier to measure than consumer surplus because the objective function for private firms can usually be assumed to be either profit maximization or cost minimization. Working with either function is straight forward compared to dealing with unobservable consumer utility functions. The theoretical and methodological difficulties of nonmarket valuation lie on the consumer side; producer surplus issues will not be dealt with any further in this thesis. 3.2 Consumer Demand and Hicksian Welfare Measures Hicksian measures of consumer welfare, compensating variation ("CV") 7 and equivalent variation ("EV"), are theoretically correct measures of consumer welfare suitable for use in economic CBA (Johansson, 1993). Marshallian measures of consumer surplus are not suitable social welfare measures because the values of the welfare estimates exhibit path-dependency (Just Page 28 et al, 1982). Hicksian demand functions and surplus measures can be derived using duality theory. 3.2.1 Duality Theory and Consumer Demand Standard duality results can readily be expanded for the theoretical addition of a public environmental good, z, in the consumer's preference function (Johansson, 1993). When confronted with both market and pure public goods, the problem facing a consumer8 is to maximize his or her direct utility subject to the budget constraint: (2) {MaxxU(x, z) | m - px = 0}, where x is a vector of all market goods, [xi, x 2 , x j , p is a vector of market prices, [pi, p2, pn], z is a variable denoting the quality and/or quantity of the environmental amenity, and m is household income. Solving for first order conditions yields: Ux = Xp, where X is the Lagrange multiplier, Ux is a vector of first-order partial derivatives, [dUld\\, dUldxa,dUldxn], and the budget constraint, m-px = 0. The solutions to the maximization problem are the ordinary, or Marshallian, demand functions: (3) x™ = x"(p,m,z) and X = X (p,m,z), where X can be interpreted as the marginal utility of income. Care should be taken not to confuse "CV" (compensating variation) with "CVM" (contingent valuation method) Technically, in consumer theory, the household is the appropriate level at which decision-making regarding consumption is made (Deaton and Muellbauer, 1980). The terms "consumer" and "agent", as applied in this thesis, will be synonymous with "household". Page 29 Substituting the ordinary demand functions back into the objective function yields the indirect utility function (Equation 3). A valid indirect utility function is continuous, twice differentiable, non-decreasing and quasi-concave in the environmental good, z: (3) V = V(p,m,z) = U(x"(p,m,z),z) The dual to the indirect utility function is the expenditure function: (4) e(p, z, U) = {min x px \ U(x, z)zU}=pxc (p, z, u) = V (p, m, z) The theoretically valid expenditure function is also continuous, twice differentiable and non-decreasing but, unlike the indirect utility function, is quasi-concave in z. The Hicksian, or compensated, demand functions can be recovered from the expenditure function using Shephard's Lemma: <5) < = < ( P , z , ! 7 ) = ^ ^ ) v ' d Pi According to the definition of Young et al (1985), dual functions (i.e., indirect utility and expenditure functions), "describe the results of optimizing responses to input and output prices and constraints rather than global responses to input and output quantities as in the corresponding primal functions" (p.3). Thus, using duality theory, it is possible to specify the Hicksian demand functions by inversion of the indirect utility function which is, itself, a function of an observable vector of market prices, consumer income level, and the quality/quantity of environmental amenity; there is no requirement for the quantification of unobservable reference utility levels. In addition, the use of duality theory implies certain restrictions on demand systems (i.e., adding up, homogeneity, and symmetry), permits the use of flexible form indirect utility functions to derive consumer surplus measures, and makes econometric estimation of demand systems relatively simple (Diewert, 1986). Page 30 3.2.2 Hicksian Welfare Measures Under the assumption of the existence of well-behaved indirect utility functions, one can analyze the effects of a shift in the consumer preference function due to changes in the level of the environmental amenity in question and derive theoretically correct values of consumer welfare. For a change in utility from a reference level (denoted with superscript '0') to a target level (denoted with superscript' 1'), the difference in indirect utility is given by: (6) AV = v(p,m,zi)-V{p,m,z°) Two measures of welfare are commonly used in contingent valuation studies.9 Compensating variation, CV, is defined as "the maximum amount of money that can be taken from the household while leaving it just as well off as it was before an improvement in environmental quality. In other words, CV is the willingness to pay..." (Johansson, 1993: p. 25). Equivalent variation, on the other hand, is the "minimum amount of money that must be given to a household to make it as well off as it could have been after an improvement in environmental quality" (Johansson, 1993: p. 26). In terms of indirect utility, CV and EV can be defined, respectively, as: (7) v(p,m-CV,z1) = v(p,m,z°) (8) v(p,m + EV,z°) = v(p,m,zl) Figure 1 illustrates the welfare measures CV and EV. For a shift in utility from V(0) to V(l), caused by an increase in the quality and/or quantity of a public environmental good, a consumer would be willing to pay up to the amount CV as a "fee". The consumer can now reach a higher level of utility for a given level of income, M . Conversely, a change in utility from V(\) to CV and EV should technically be referred to as compensating and equivalent surpluses; most environmental economists tend to follow the terminology of Johansson even though it is not technically correct (van Kooten and Bulte, 1997). EV can also be viewed as compensation demanded, willingness to accept compensation or willingness to sell. Page 31 F(0), due to a decline in environmental quality, would require a payment of amount (EV) to the individual to compensate for the loss. The amount EV would bring the individual back up to the original indifference curve. The difference between EV and CV can be attributed to a difference in property rights with regards to the environmental good; CV is the correct measure of welfare if the individual has rights to environmental level q(0) and utility level V(0) while EV is the correct measure of welfare if the individual has rights to environmental level q(l) and utility level F(l). q(0) • q(1) Quality of Non-Market Environmental Good Figure 1 - Compensating and Equivalent Variation [Surplus] Welfare Measures It is possible to verify that the signs of CV and EV are the same as the signs of the change in the environmental amenity so long as the marginal utility of income is greater than zero. Furthermore, if the environmental amenity is assumed to be a normal good (i.e., WTP increases with increasing income level) then EV will always be greater than or equal to CV (Johansson, 1993). Page 32 Empirical studies have often found EV values to be implausibly larger than CV with EV up to ten times greater than CV (Hanemann, 1991). This discrepancy has raised a number of doubts about the theoretical and methodological soundness of CVM. Arrow et al (1993) recommended that a willingness-to-pay (i.e., CV) referendum style value elicitation format be used for nonmarket valuation purposes rather than using willingness-to-accept formats. A number of explanations have been posited (see Mansfield, 1996, for a further discussion of the problem and possible alternative explanations of the phenomena) but the large differences in observed CV and EV values remains an open question and are not pursued further in this thesis. 3.3 Contingent Valuation 3.3.1 Alternative CVM Formats A number of different C V M formats are possible. The first choice a researcher faces is deciding on an open-ended versus closed-ended format. In the open-ended format, consumers are asked directly for a precise value of WTP for a change in the level of public good or, alternatively, to select amongst a range of values in a "payment card" format. It has been shown that the open-ended approach is susceptible to bias due to strategic behaviour on the part of respondents (McFadden, 1994). As noted above, the NOAA Panel recommends a closed-end or referendum-style format. The referendum format, introduced by Bishop and Heberlein (1979), usually presents respondents with a proposed fee and given level of change in the environmental amenity. The person is then asked, yes or no, whether they would accept the proposed fee. It has been argued that this "take-it-or-Ieave-it" approach bears a much closer resemblance to true market transactions and should, therefore, provide more accurate estimates of consumer welfare (Cameron, 1991). Page 33 The referendum question presented is typically single-bounded: each respondent is presented with one proposed combination of fee and environmental change to which they respond either yes or no. Welfare measures can be derived using Hanemann's (1984) Random Utility Model framework. This approach is known as the dichotomous choice, contingent valuation (DCCV) method. Practical advantages of DCCV over the open-ended approach are documented by Hoehn and Randall (1987), Bowker and Stoll (1988), Boyle and Bishop (1987) and McFadden (1994). It is also possible to use a double-bounded approach in which a follow-up question is asked (Alberini, 1995a). The rationale is to obtain additional information on the shape of the WTP distribution and increase statistical efficiency but the double-bound referendum format is internally inconsistent and can result in substantially different WTP estimates than either open-ended or single-bound formats (McFadden, 1994). The rest of this study focuses on the use of the standard single-bound DCCV. 3.3.2 The Random Utility Model The dichotomous choice contingent valuation method is based on the Random Utility Model (RUM) of Hanemann (1984). In the RUM, economic agents are asked to provide yes or no answers to questions regarding their willingness-to-pay for certain designated changes in the level (quantity or quality) of an environmental amenity. The yes or no responses are recorded and can then be used to fit binary response models, such as the logit and probit, which in turn can be used to trace out the distribution of the unobservable WTP variable and to provide estimates of agent welfare for changes in the level of the environmental amenity under examination (Alberini, 1995b). Page 34 The basic DCCV methodology assumes that the agent knows his or her own utility function. This true preference function can be denoted as U(i, m; sj where;' is a binary choice variable (1 if the respondent is willing to pay the bid amount, 0 otherwise), m is the agent's income level, and s is a vector of other agent characteristics. To an independent observer, indirect utility can be modeled as a random variable consisting of a given parametric probability distribution term with mean V(i,m;s) and a random error component (Park et al, 1991): (9) U{i,m;s) = V(i,m;s) + ei The stochastic error term, is assumed to be an independently and identically distributed random variable with zero mean. When presented with a proposed fee, the "bid amount" A, to be charged for changing environmental quality from reference level to target level, the rational respondent will accept the bid amount if: (10) V{\,m-A;s) + ex >V(0,m;s) +e0 That is, the respondent will accept the bid, indicated by /' = 1, if the utility level after subtracting the fee from household income is greater than the original reference utility level. In Figure 2, the horizontal lines represent income (m) and income less the proposed fee (m -A). For a change in environmental quality, from q(0) to q(l), an individual with preference function U(0) would not accept the opportunity to pay the fee, A, for improving environmental quality because the fee amount is greater than the monetary value that the person puts on the environmental improvement, (c-d). Conversely, an individual with preference function (7(1) would accept the proposed fee because they could pay the fee and still reach a higher level of utility than their original level U(l). The monetary value they place on the improvement, (c-e), is greater than the proposed fee. One can see that they would also be willing to pay for smaller increases in Page 35 environmental quality back to the point f, where their preference function intersects the horizontal line, m -A. > 0) _l a> E o o m m - A 1 b c d U(0) e U(1) • q(0) q(1) Quantity [Quality] of Non-Market Environmental Good Figure 2 - WTP for a Change in Environmental Amenity This relationship can then be placed in a probabilistic framework and used to trace out the bid curve: (11) Vr{Yes} = Vx{v{\,m-A;s) + ex >V(0,m;s) + e0] (12) Pr{Yes} = Fa(AV), where i ^ ( A F ) is a cumulative density function ("cdf') of the respondent's true maximum WTP. A logistic cdf is most commonly used in DCCV studies: (13) PrlYes) = 1—r Page 36 The indirect utility function is most often assumed to be either linear or log-linear in DCCV studies. Sellar et al (1986) have shown that, theoretically, two restrictions must be satisfied to ensure that Hicksian demand functions, h, are positive and downward sloping: A linear utility function satisfies only the first condition but is always upward sloping thereby violating the second condition. The linear specification of the utility function is, therefore, inappropriate for use in theoretically correct DCCV studies. Despite this theoretical shortcoming, it is still commonly used. In this study, two forms of the utility function are used and later compared using proportional error analysis. For the case of a linear utility function (Equation 16), the utility difference can be easily derived (Equation 17): (16) V(i,m;s) = ai+bm (17) Av = a, -a0-bA-a-bA The income term drops out of the utility difference equation when the utility function is of a linear form. Clearly this is unrealistic and in contrast with a priori expectations that income level should be highly influential in the valuation of public environmental goods. A similar process can be followed for the log-linear utility function (but in which the income variable remains): (14) hq<0 (15) (18) V(i,m;s) = ai+B ln(m) (19) Page 37 Substitution of the log-linear utility difference (Equation 19) into the probabilistic framework yields: (20) Pr{x} = 1 l+e M3) where 0 :< 0 < 1, a = (cti - ceo), and /? < 0. Solving for A yields the willingness-to-pay of an agent for a given change, Av, in an environmental amenity: 3.3.3 Welfare Measures in DCCV Two commonly used measures of welfare are the mean WTP and median WTP. Both can be derived from the bid curve (Figure 3). The median is the fee for which the probability of a "yes" vote for a given level of change in the environmental amenity is 50%. The mean WTP can be calculated as the total area under the bid curve. Duffield and Patterson (1991) provide three general criteria that a welfare measure should satisfy: (1) it should be consistent with the theoretical constraints implied by consumer utility theory; (2) it should be statistically efficient; and (3) it should have properties that allow individual welfare measures to be aggregated. Mean WTP is preferred over median WTP as a true measure of consumer welfare on theoretical grounds. The sum of the values of mean WTP of individuals can be aggregated while median WTP cannot, theoretically, be aggregated (Johansson, Kristrom and Maler, 1989). A problem can occur when using mean WTP, however, if the bid curve does not converge to zero: mean WTP can be infinitely high. Cooper and Loomis (1992) observed large decreases in WTP (21) m \ 1 A--— a + ln —-1 P \<P J Page 38 with different levels of truncation, indicating "fat upper tails". Truncation and normalization of the mean is possible (Boyle and Bishop, 1987) but the summation of individual truncated welfare measures may provide a biased estimate of total consumer surplus due to the arbitrary nature of the truncation point. Figure 3 - Welfare Measures Derived From the Bid Curve Duffield and Patterson (1991) claimed that a truncated mean satisfied all three of their criteria but McFadden (1994) shows that the truncated mean is itself a biased estimator of WTP when the WTP distribution is skewed as is typically the case in empirical studies. Parametric models, in particular, often prejudge the shape of the tails and can be inflexible. The problem is worst when there is an element of strategic respondent behaviour (McFadden, 1994). Median WTP would result in a respondent having a 50% probability of accepting the contingent valuation bid amount. For the log-linear case, the logarithmic term drops out of Equation 21 when <f> = 0.5 and results in: Page 39 (22) A = m -a = w log Median WTP (or other specific probability levels) can thus be easily evaluated using the coefficients of a logit regression and income level. A similar process can be followed for the linear utility model with the results: (23) a-In P (24) a P Mean, or expected, WTP values can be calculated by numerical integration of the area under the bid curve. If WTP is assumed to be non-negative10, then the expected value of WTP can be calculated as: (25) E(WTP) = J [l - F{b)]db = w F(b) is the cumulative probability of a no response to the DCCV question and is a function of the bid amount b. For the linear utility function, expected WTP reduces to A=-a/B while a closed-form approximation of mean WTP for the log-linear utility function was developed by Hanemann(1989): (26) E(WTP)*^-:\n(\ + ea) It is quite conceivable that respondents might have a zero or negative WTP for a particular project (Hanemann and Kristrom, 1995). So-called spike or pulse models have been developed to address this problem but will not be dealt with further in this thesis. Page 40 Given the capability of modern econometric software to perform numerical integration quickfy and accurately, there is really no reason why that integration cannot be used directly to calculate mean WTP values. A number of alternative approaches have been proposed for C V M welfare estimation (Cameron, 1988; Kristrom, 1990; Creel and Loomis, 1997). McFadden (1994) outlines the advantages of using parametric, as opposed to non-parametric, models in contingent valuation studies: it is relatively easy to impose preference axioms; experiments are readily combined; and it is possible to extrapolate the calculation of welfare values to different populations than for those estimated. If, however, the parametric representation of preference is not flexible enough to truly describe behaviour, the estimated mean WTP will usually be an inconsistent estimate of true WTP. Cameron (1991) also uses a parametric approach because it allows easy characterization of the shape of indifference curves. This thesis uses a parametric approach to nonmarket valuation. 3.3.4 Confidence Intervals Adamowicz et al (1989) raise the issue of why confidence intervals are important in nonmarket valuation studies and state that in "many instances the welfare measure is to be used in making a resource allocation decision in a benefit-cost framework. The decision maker can be viewed as a statistician who is testing the hypothesis that the true population welfare measure exceeds the cost of the project or policy under consideration"(p. 415). The statistical problems involved in developing confidence intervals for WTP estimates are complex. Estimation of logit model coefficients is done using the maximum likelihood method and leads to estimators that are asymptotically normal and have desirable asymptotic properties (Amemiya, 1981). Parameter estimates calculated using CVM, themselves random variables, are Page 41 then further used to calculate the non-linear random variable welfare measures (Bockstael and Strand, 1987; Parke/o/, 1991). Approximate distributions of the value of estimated mean WTP can be derived from the RUM framework by applying bootstrapping techniques. Park et al (1991) develop WTP confidence intervals using the methods of Krinsky and Robb (1986). While the technique can be "used to establish the empirical distribution of any estimator which is a nonlinear function of the estimated parameters" (Park et al, 1991: p. 66), the technique is computationally intensive requiring at least 1,000 drawings of parameters for the development of confidence intervals for mean WTP and to account for all parameter interactions. Some studies using similar techniques to develop confidence intervals for environmental valuation techniques used substantially higher numbers of random draws: Adamowicz et al (1989) use, for example, five thousand draws to formulate confidence intervals. The development of confidence intervals for mean WTP estimates is viewed as beyond the scope of this thesis but would be the logical next step in the continuation of the exploration of the effects of agent heterogeneity in environmental valuation.11 3.4 A Utility Model that Includes Public Goods Madriaga and McConnell (1987) introduced a model (the " M M Model") that includes a public environmental good with both a pure existence value component and an influence on market The Excel add-in, Crystal Ball, is designed specifically for Monte Carlo simulations but cannot be used directly because, for each drawing of the parameters, a logit regression must be undertaken. This cannot be undertaken directly in Excel but must be accomplished using a separate software package (SHAZAM in this thesis) which is expensive in terms of time (SHAZAM coefficients must be manually copied back into Excel spreadsheets) and required disk storage space. Presumably it would be possible to integrate and automate a Monte Carlo experiment using Crystal Ball, Excel and a statistical package (e.g., SPSS) if appropriate Visual Basic code was written to pass coefficients back and forth between applications. Alternatively, the entire operation could be accomplished using a mathematical software package such as GAUSS. Page 42 goods through a weak complementary link. Practically, the model can be thought of as a water reservoir level model where people have use values for activities such as recreation and non-use values for environmental services such as the provision of habitat for aquatic animals. A decline in reservoir level could, for example, lead to losses in recreational opportunities and, if the level declined beyond some critical level, could adversely affect the survival of aquatic species for which people held existence values. 3.4.1 Model Structure and Terminology The M M model uses a direct utility function to specify consumer preferences: (27) u{x1,x2,R) = axi + \nx2 +b, where x; is a complementary environmental quality-related market good; x2 is a strongly separable composite market good, /7,x, + p2x2= m; and p2 = 1. In this model, parameters a and b control the relative importance of market versus nonmarket goods in the utility function. The parameters are defined using reference and critical baseline levels for the environmental amenity : (28) a = a(R-Rm) (29) b = e**-s-), where R is the level of the resource; Rm is a critical lower maintenance level of the resource; a is a parameter that controls the degree of linkage between the quasi-fixed environmental market good, xi, and environmental quality parameter or, and Bis a parameter that controls the magnitude of existence value within the utility function. Parameter a impacts the size of the use value, while parameter /? controls the contribution of the environmental good to existence value separate from the market good. Page 43 The utility function is quasi-linear, additive in Xi and x2, existence value is strongly separable in the associated indirect utility function, and there is a weak complementary link between the environmental good and a market good, x\, which is a perfect substitute for the environmental good. The RUM requires that the environmental good is essential: theoretical problems occur if consumption is pushed to zero (Cameron, 1991). This implies that the indifference curve intersects the Xi axis. Following the procedure outline in Section 3.1, the Hicksian demand function for the linked good, xu can be calculated as: (30) U-b-\n + 1 a 3.4.2 Welfare in the Madriaga and McConnell Model Using equations (6) and (7), the utility difference for the M M model can be used to calculate agent WTP (Huang and Smith, 1996): (R, - R0 W (31) # 7 7 > = V -lrL + (R.-R^ a ^ - K ) In A-RJ + e Each agent's WTP is expressed as a function of income, price of the complementary market good, and the target and reference levels of the environmental amenity. This value can be calculated each time an agent is presented with a proposed combination of fee and level of environmental change and a yes or no response generated based on comparing this WTP value with the proposed fee. Page 44 CHAPTER 4 4. METHODOLOGY 4.1 Experimental Design The general procedure in this research project is to generate an artificial population of economic agents that can be queried about their willingness-to-pay for changes in an environmental amenity. For the experiment, the direct utility function of equation (27) is employed. Artificial economic agents in a quasi-Monte Carlo experiment are used for a number of specific reasons: • large numbers of responses to WTP queries can be quickly generated allowing for powerful statistical analysis of the causes of C V M proportional error; • there are no technical problems of strategic responses when using artificial agents; • the artificial agents have exact, "true" WTP values for each possible C V M query without any preference uncertainty; and • costs for real surveys of this size would be prohibitive. Parameters were chosen to reflect "reasonable" distributions of true WTP for the agents. After the initial parameterization, the only parameter that was permitted to vary with each "sample" was parameter /?, the parameter that links nonmarket existence value to the preference function of each agent. Each sample consisted of 400 economic agents - 200 each of altruists with a high mean value of p, and 200 non-altruists with a low mean value of j3. The values of /? were randomly assigned to each group using a standard normal distribution. Three levels of environmental change were used in this study: 4%; 10%; and 16%. At the upper level, the income available for agents to spend on environmental preservation was occasionally exceeded. This would not be realistic for real people as budget constraints are Page 45 binding but, given the few cases in the experiment, there was no restriction of the true WTP value to maximum available income. In any event, the largest bid amounts presented to agents were substantially smaller than minimum agent income so any agent that did have a true WTP higher than his/her income would still respond with the same "yes" vote to a referendum question, even if income constraints were imposed in the model (i.e., if WTP for any agent were truncated at that agent's income level, responses used in the logit regression would remain unchanged). Estimated welfare values calculated using the coefficients of the logit analysis would, hence, be unaffected by the occasional agent having a WTP greater than income. Once a sample was assigned a vector of /? parameters (thereby defining a distribution of true WTP values), a bid vector was constructed for the sample. The bid vector, which will be discussed in more detail below, was formulated by assuming the 'true' WTP distribution was normally distributed and setting fee or bid amounts according to this distribution. The true WTP distribution, in reality, was skewed rightward as is typically the case in real studies (Cooper and Loomis, 1992); the normal approximation of the skewed distribution could be viewed as analogous to having the experimenter conduct pre-survey research or focus groups to find an approximate range over which the bid amounts could be distributed. Each agent in a sample was randomly presented with one of six potential bid amounts (extra low, very low, low, high, very high, extra high) and their response, based on their true WTP, was recorded. Each data set, then, consists of 400 responses to a bid amount query: each response is a function of bid amount, income, parameters a and B, the level of environmental change, and whether the agent is an environmental altruist or not. The 400 responses were then used in a logit regression, the coefficients of which could subsequently be used to construct welfare measures for each sample. The calculated welfare measures included the median WTP and mean WTP. The mean was calculated by numerical Page 46 integration over three separate ranges of zero to: (1) the upper bid amount; (2) 125% of the upper bid amount; and (3) 150% of the upper bid amount. Each logit regression was performed twice, once each for assumed linear and log-linear functional forms for utility (Equations 16 and 18). These functions are the ones assumed in empirical analysis and, clearly, not the true utility functions of the artificial agents (Equation 27). Fifty samples for each level of environmental change were drawn using normally distributed values of B for a mixed population of 200 altruist and 200 non-altruists. Thus a total of 60,000 artificial survey responses were generated (400 agents, 150 samples) and welfare measures calculated for them using each of the two assumed functional forms for utility (giving a total of 120,000 data points). After generating the welfare estimates, the average estimated welfare level was compared to the true welfare level for each agent. The proportional error was then calculated and analyzed for sources of bias by regressing a variety of model and agent characterizing factors upon it. 4.1.1 Model Parameterization The goal of choosing parameters for the M M model was to develop the model in such a way that the distribution of true WTP followed a believable partem for samples of artificial agents. A number of parameter values were considered; most were in a similar range as those used by Huang and Smith (1996). The baseline parameters are found in Table 2. Page 47 Table 2 - Parameter Values and Variability AR=4% AR=10% AR=16% Resource Level Critical Level Ro 10.0 10.0 10.0 Maintenance Level R m 15.0 15.0 15.0 Target Level R i 15.6 16.5 17.4 Prices Weak Complement, 0.5 0.5 0.5 Composite good, P2 1.0 1.0 1.0 Income (Normalized) Mean Income 10.0 10.0 10.0 Standard Deviation 10.0% 10.0% 10.0% Linkage Parameters Use Value, a 0.05 0.05 0.05 Standard Deviation 10.0% 10.0% 10.0% Expected Value of a i 0.25 0.25 0.25 Non-Altruist Passive Value, p 0.125 0.125 0.125 Standard Deviation 20.0% 20.0% 20.0% Expected value of b 1.868 1.868 1.868 Altruist Passive Value, p 0.225 0.225 0.225 Standard Deviation 20.0% 20.0% 20.0% Expected value of b 3.080 3.080 3.080 The income parameter was the first one to be fixed. It is distributed as a normal distribution with mean 10.0 and standard deviation 10%: m ~ N(\0.0,1.0). An income distribution was drawn randomly (Figure 4) and was kept fixed throughout the experiment. It should be noted that the ratio of WTP to income in the experiment is large (occasionally greater than 1.0). If household spending on charities, donations and environmental protection is assumed to be strongly separable from other household spending, the magnitude of the ratio is irrelevant. Page 48 Figure 4 also shows a normal curve overlay based on the single random draw of income for this experiment. 50 40 30 20 10 c a> 3 ST i t 0 / Hi \ \ Si Std. Dev= 1.00 Mean = 10.02 N = 400.00 tf/c <*!«. Sr 77 t? Z> •& INCOME Figure 4 - Random Distribution of Income The second parameter to be fixed was the technical link, the parameter a. The distribution was normally distributed with mean of 0.05 and standard deviation of 10%: a ~ #(0.050,0.005). Again, the random distribution drawn initially (Figure 5) was held constant for the balance of the experiment. Page 49 50 40 Std. Dev = .00 Mean = .050 N = 400.00 % % % % \ % % % % % % % % % Alpha (technical link parameter) Figure 5 - Random Distribution of Parameter a The B parameter defines the magnitude of environmental altruism for an agent in the model. Several types of agents can be defined based on the characteristics of the ft distribution. The simplest situation occurs when B is fixed at 0.175, the mean of average values for non-altruists (P=0.125) and altruists (P=0.225). This is referred to as the representative agent. With this constant B and the random normal distributions of income and a, the WTP distribution for the representative agent is shown in Figure 6 for a 16% increase in the quality of the environmental amenity. Similar tight distributions can be observed for other levels of change in environmental quality for the representative agent case. Page 50 Representative Agent WTP Histogram For a 16% Increase in Environmental Amenity 120 u- o Std. Dev = .35 Mean = 4.42 N = 400.00 % % % % % % % % % % % Fee Amount for Improvement (Income = 10) Figure 6 - WTP Distribution for Representative Agent with a 16% Change in the Level of the Environmental Amenity (n = 400) The next step is to introduce random normal distributions of B for altruists and non-altruists. In the model, I assume the distributions for non-altruists to be normally distributed with mean 0.125 and standard deviation 20%: BNA ~ N(0.l25,0.025). Similarly, the distribution for altruists is assumed to have mean 0.225 and standard deviation 20%: BA ~ iv"(0.225,0.045) . Figure 7 shows the distribution of WTP for a 16% increase in environmental quality for a mixed population of altruists and non-altruists when variance is set to zero (i.e., all altruists have PA=0.225 and all non-altruists have PNA=0. 125). These parameters were chosen to ensure niinimal overlap of WTP distributions under baseline model conditions. Page 51 Heterogeneous Agent WTP Histogram For a 16% Increase in Environmental Amenity 601 1 50' (—^  40' r _ , % % % % % % % % % % % Fee Amount for Improvement (Income = 10) Figure 7 - WTP Distribution for Heterogeneous Agents with a 16% Change in the Level of the Environmental Amenity (n = 400) Each sample in the experiment uses a random drawing of B values distributed around appropriate means. For example, the distribution of B for sample 112 (AR=16%) is shown in Figure 8; note the slight skew rightwards because of the larger variance, in absolute terms, of the parameter for altruists. Using the B distribution shown in Figure 8, the true WTP values for each of the 400 agents in the sample can be calculated: the resulting distribution of WTP for sample 112 is shown in Figure 9. There are several points to note: (1) there are WTP values higher than average income; (2) the rightward skew of WTP is typical of the distributions seen in empirical studies; and (3) there are no zero WTP values. Page 52 Sample 112 % % % % % % % % % % Distribution of Beta Coefficient Figure 8 - B Parameter Distribution for Sample 112 Sample 112 True WTP Histogram For a 16% Increase in Environmental Amenity 100i 1 % % % % % % %%\\% Fee Amount for Improvement (Income = 10) Figure 9 - WTP Distribution for Sample 112 with a 16% Change in the Level of the Environmental Amenity (n = 400) Page 53 Figures 10 and 11 show similar patterns for arbitrarily chosen samples 87 (AR=10%) and 17 (AR=4%), respectively. Sample 87 WTP Histogram For a 10% Increase in Environmental Amenity Std. Dev = .99 Mean = 3.22 N = 400.00 '<* % % % % % % % % % % % Fee Amount for Improvement (Income = 10) Figure 10 - WTP Distribution for Sample 87 with a 16% Change in the Level of the Environmental Amenity (n = 400) Sample 17 WTP Histogram For a 4% Increase in Environmental Amenity 801 1 * % % % % % % % % % % Fee Amount for Improvement (Income = 10) Figure 11 - WTP Distribution for Sample 17 with a 16% Change in the Level of the Environmental Amenity (n = 400) Page 54 Finally, Figure 12 shows a plot of true WTP as a function of the magnitude of the change in environmental amenity for a representative agent and the heterogeneous non-altruists and altruists where there is no variance in the distribution of the B parameter (i.e., BA is fixed at 0.225 for all altruists and at BNA = 0.125 for all non-altruists). The relationship is non-linear. $7.50 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% 18.0% 20.0% Change in Environmental Amenity, A R |~ —a— Altruist —*— Representative Agent Non-Altruist ~| Figure 12 - WTP as a Function of the Level of Environmental Change 4.1.2 Bid Vector Calculation There are two basic schools of thought on bid vector design for DCCV studies. On the one hand, Boyle and Bishop (1987), Cooper and Loomis (1992), McFadden (1994) and Elnagheeb and Jordan (1995) advocate schemes in which there a large number of fairly closely spaced bids over a wide range. According to McFadden (1994), mean squared error ("MSE") is inversely proportional to sample size in this situation. A small number of bids, the common practice in field Page 55 studies, results in MSE declining more slowly in relation to overall sample size. Elnagheeb and Jordan (1995) used as many as seventy-three bid values depending on sample size and the coefficient of variation of WTP obtained in pre-survey samples. A second perspective on bid vector design is advocated by Alberini (1995b) and is followed in this thesis. Alberini found that maximum statistical efficiency occurred with only two bid values. This is somewhat impractical given that one has no a priori expectations of appropriate bid range but she also found that there was very little gain in power using designs with more than six to ten total bid points or by placing bid values far out in the tails of the WTP distribution (i.e., where pr{yes} is less than 3%). For each sample in this experiment, a total of six bid values were calculating for presentation in the bid vector. Calculations were made based on the assumption that the WTP distribution could be, for the purposes of estimating the appropriate bid vector range, assumed to be normal (e.g., see the normal curve overlays of the WTP distributions shown in Figures 9, 10 and 11). Conceptually, this could be considered a rough estimate of the WTP distribution derived from a pre-survey sample or focus group. It was decided to set the upper and lower bids at points where 5% of the area under the normal curve was farther out in the tails (i.e., z = ±1.65). The remaining bid values were calculated by dividing the normal distribution into even partitions (Equation 32). The z-values for the bid vector points are shown in Figure 13. All bids were rounded to the nearest 0.05. where Bv is the upper bid limit (0.95), BL is the lower bid limit (0.05) and N is the total number of desired bids. (32) Division = 0.90 = 0.18, N-l 5 Page 56 The distribution of bid partitions within the normal distribution is shown in Figure 13. 50% i 1 Bid 3 Bid 4 PR -3 -2 -1 0 2 3 Standard Deviation Figure 13 - Distribution of Bid Values in the Normal Distribution 4.1.3 Uncertainty There are two types of uncertainty that could be represented in the model. The first type is preference uncertainty (Hanemann and Kristrom, 1995; Li and Mattsson, 1995). When faced with market decisions about unfamiliar environmental goods with which they have little experience, it is conceivable that individuals might not accurately respond to C V M queries. Instead, they respond in such a way that their WTP would, if queried repeatedly, take the form of a distribution that was clustered around the true value of WTP. The second type of uncertainty is the stochastic element of the RUM model and is based on observer uncertainty. In this case, it is presumed that respondents know their WTP precisely and that the stochastic error component is entirely attributable to the observation process. Page 57 An implication of preference uncertainty would be that analysis of bias should be done using root mean square error (see the following section for details as to calculation of "RMSE"). RMSE, in which average deviations from the true WTP for a particular individual are analyzed, is appropriate if an experimenter repeatedly draws parameters for the same agent. Alternatively, if observer uncertainty is assumed, one can conceptualize the random drawing as the independent sampling of B values from a large population of economic agents. In this case, it is logical to examine the proportional error on an individual basis using root squared error ("RSE"). This is the procedure followed in this thesis. If a large population of agents is assumed, from which samples are repeatedly drawn, it would also be most appropriate to draw income and a parameters repeatedly for each sample. This was not done in this thesis for two reasons: (1) mamtaining constant values of income and a over the 150 samples greatly simplified data organization and storage; and (2) it was prudent to leave open the option of changing assumptions regarding preference uncertainty, which requires constant income and a parameters for each agent across the experiment. 4.1.4 Measures of Bias The primary measure of bias used in this study is the proportional root squared error ("PRSE"). This measure, which compares the estimated and true WTP values, is normalized to allow comparisons between agents and to permit PRSE to be used as a dependent variable in an analysis of sources of bias in the experiment. PRSE is calculated as: (33) PRSEi = Page 58 where w(. is the estimated value of willingness-to-pay calculated using the mean WTP at a truncation level of 125% of the upper bid and wt the true WTP for an individual agent. A total of 120,000 values of PRSEjare available for analysis (400 agents per sample, 150 samples, and 2 regressions using different functional forms for agent utility). Huang and Smith (1996) use proportional root mean square error (PTRMSE), the alternative measure of bias appropriate in cases of preference uncertainty. In their analysis, the error measurement is calculated using: where there are m=100 replications of the parameter draw for each of 200 samples and 12 distinct combinations of a and B parameters. 4.2 Estimation Techniques The model parameters were all generated using the Excel random number generator. Vectors of variables to be used in the logit regressions were then saved in text format. These sets of data for the 150 samples were imported into SHAZAM (1993) Version 7.0 where analysis and calculation of welfare measures was undertaken. Coefficients from the SHAZAM regression results were copied back to a spreadsheet for calculation of PRSE. PRSE and related data vectors were then saved and loaded back into SHAZAM for final analysis. 4.2.1 Logit Bid Curve Regression The logit regression uses a maximum likelihood estimation method. Hypothesis testing can be carried out using the likelihood ratio test and related tests. Both the Log-Likelihood and Page 59 (34) Maddala R 2 were used as measures of the goodness of fit of the model. The Maddala R 2 is a variant of the likelihood-ratio test. It compares the value of the log-likelihood function of the model, evaluated at the maximum likelihood estimates, and the value of the model with an intercept but no regressors (Alberini, 1995b). This gives information on the relevance of the variables in the model but not on whether the fitted distribution is close to the true form. In addition, the success of the logit regression was examined on the basis of the percentage of correct yes and no predictions. 4.2.2 Calculation of Welfare Measures Mean and median welfare was calculated for each sample using SHAZAM functions. Median was calculated using equations (22) and (24) for log-linear and linear utility, respectively, and SHAZAM's "Test" function. In addition to calculating the median WTP for each sample, for the arbitrarily chosen sample 132, WTP was calculated for a variety of non-median probabilities. This was accomplished using equations (23) and (25) but setting the probability of a yes response, <l>, to a variety of values other than 0.5. Mean WTP values were calculated using numerical integration and using the Hanemann (1989) approximation (Equation 26).12 Three integration operations were performed for each sample: the first used the upper bid level itself as the upper integration limit; the second set the upper integration limit at 125% of the highest bid level; and the third set the upper integration limit at 150% of the highest bid level. Results from the approximation are not reported but were extremely close to the results obtained via integration. Page 60 4.2.3 Root Square Error Calculation Once the logit regressions were completed in SHAZAM, the coefficients were copied back into Excel spreadsheets. Estimated WTP was then calculated for each agent in the 150 samples using regression parameters and compared to true WTP for the agent. The proportional root square error, PRSE, was calculated using Equation 36. 4.3 Response Surface Estimation For each sample of 400 agents, a total of forty values of PRSE were selected randomly for further analysis; twenty each were selected from the altruist and non-altruist sub-samples. A total of 6,000 responses were used in the response surface regression. Two models were estimated, one of which included a dummy variable for environmental altruists and one which did not. PRSE was specified as a function of income level, the proposed fee, the level of environmental change, the ratio of existence value to use value (p/a), and dummy variables for the log-linear utility specification and environmental altruists. This specification is similar to that used by Huang and Smith (1996) and allows interesting comparisons with their study (which focused on the effects of varying the existence-to-use-value ratio). Page 61 CHAPTER 5 5. RESULTS Complete regression results for all samples are included in the appendix. In this section, I only present summaries from the different regression scenarios (i.e., level of environmental change and utility functional form choice). 5.1 Regression Results and Welfare Measure Calculation Table 3 presents the regression results and calculated welfare measures for a 4% increase in the quality of the environmental amenity. The table shows mean results for both linear and log-linear utility functions being used in the utility difference framework. The logit model fit varied considerably for the linear utility difference form with percentage correct predictions for individual samples ranging from 72.0% to 92.5% and the log-likelihood from -200.8 to -93.3; complete diagnostic statistics and regression results are shown in the Appendix. The regression results and welfare calculations for the log-linear utility difference are very similar to the linear utility case and the calculated values of mean and median are very close to those obtained using the linear utility specification. Diagnostic statistics show the model using the log-linear utility specification provides marginally improved performance compared to the case for the model utilizing a linear utility function. Estimated WTPs are quite close to each other. The median WTP was 1.401 (s.e. = 0.031) for the linear utility case and 1.404 (s.e. = 0.031) when calculated using the log-linear utility difference. Numerical integration resulted in average values of mean WTP ranging from 1.391 to 1.405. The values of mean WTP at 125% and 150% upper bid truncation levels are virtually identical indicating that the bid curve does converge. Page 62 Table 3 Logit Regression Results and Associated Welfare Measures for a 4% Change in Public Good and Two Utility Difference Functional Forms Linear Log-Linear Model Results p -4.297 -44.026 s.e. 0.431 4.383 a 6.018 6.172 s.e. 0.623 0.633 Log-Likelihood -179.3 -172.7 Maddala R2 0.381 0.401 % Correct 77.39% 78.75% WTP Measures Median 1.401 1.404 s.e. 0.031 0.031 Mean (with truncation) 100% Upper Bid 1.391 1.395 125% Upper Bid 1.400 1.404 150% Upper Bid 1.401 1.405 True 1.438 1.438 True WTP was higher than the estimates calculated for the various welfare measures. The differences are more readily observed in Figures 14 through 18. True WTP distribution for the 50 samples (Figure 14) is fairly tightly clustered around mean 1.438. The distributions of mean and median WTP calculated using the linear utility function (Figures 15 and 16, respectively) are shown using the same x-axis scale. They are more widely distributed and with mean and mode lower than the true mean WTP. Figures 17 and 18 show similar patterns for WTPs calculated using the log-linear utility function. Page 63 Distribution of True WTP For a 4% Increase in Environmental Amenity Std. Dev = .02 Mean = 1.438 N = 50.00 1.310 1.350 1.390 1.430 1.470 1.510 1.550 1.330 1.370 1.410 1.450 1.490 1.530 True WTP Figure 14 - True WTP Distribution for a 4% Increase in Environmental Amenity Distribution of Mean WTP Linear Utility Model For a 4% Increase in Environmental Amenity Std. Dev = .03 Mean = 1.391 N = 50.00 1.310 1.350 1.390 1.430 1.470 1.510 1.550 1.330 1.370 1.410 1.450 1.490 1.530 Mean WTP (linear utility model) Figure 15 - Estimated Mean WTP Distribution, 4% Increase in Environmental Amenity and Linear Utility Function Page 64 Distribution of Median WTP Linear Utility Model For a 4% Increase in Environmental Amenity Std. Dev = .04 Mean = 1.401 N = 50.00 1.310 " 1.350 * 1.390 " 1.430 " 1.470 " 1.510 " 1.550 1.330 1.370 1.410 1.450 1.490 1.530 Median WTP (linear utility model) Figure 16 - Estimated Median WTP Distribution, 4% Increase in Environmental Amenity and Linear Utility Function u_ n Distribution of Mean WTP Log Utility Model For a 4% Increase in Environmental Amenity Std. Dev = .03 Mean = 1.395 N = 50.00 .310 1.350 1.390 1.430 1.470 1.510 1.550 1.330 1.370 1.410 1.450 1.490 1.530 Mean WTP (log utility model) Figure 17 - Estimated Mean WTP Distribution, 4% Increase in Environmental Amenity and Log-Linear Utility Function Page 65 Distribution of Median WTP Log Utility Model For a 4% Increase in Environmental Amenity Std. Dev = .03 Mean = 1.405 N = 50.00 1.310 * 1.350 " 1.390 " 1.430 " 1.470 ' 1.510* 1.550 1.330 1.370 1.410 1.450 1.490 1.530 Median WTP (log utility model) Figure 18 - Estimated Median WTP Distribution, 4% Increase in Environmental Amenity and Log-Linear Utility Function Table 4 shows the results for a 10% increase in the quality of the environmental amenity and the use of a linear utility function for empirical estimation. The patterns are very similar to the previous ones observed at the 4% change level. Page 66 Table 4 Logit Regression Results and Associated Welfare Measures for a 1 0 % Change in Public Good and Two Utility Difference Functional Forms Linear Log-Linear Model Results p -1.726 -17.543 s.e. 0.173 1.749 a 5.429 5.530 s.e. 0.567 0.571 Log-Likelihood -182.1 -176.3 Maddala R 2 0.375 0.393 % Correct 76.90% 78.18% WTP Measures Median 3.148 3.160 s.e. 0.076 0.076 Mean (with truncation) 100% Upper Bid 3.125 3.138 125% Upper Bid 3.147 3.159 150% Upper Bid 3.149 3.161 True 3.255 3.255 Figure 19 shows the distribution of true WTP for this scenario, while Figures 20 and 21 show the distributions for calculated mean and median WTP using the linear utility model, respectively. Figures 22 and 23 show the mean and median WTPs calculated using the log-linear utility difference model. The differences appear more distinct than was the case for a 4% increase in environmental quality. Both distributions of estimated WTP show a degree of bimodality. Page 67 Distribution of True Mean WTP For a 10% Increase in Environmental Amenity Std. Dev = .03 Mean = 3.255 N = 50.00 3.015 3.075 3.135 3.195 3.255 3.315 3.375 3.045 3.105 3.165 3.225 3.285 3.345 True WTP Figure 19 - True WTP Distribution for a 10% Increase in Environmental Amenity Distribution of Mean WTP Linear Utility Model For a 10% Increase in Environmental Amenity Std. Dev = .06 Mean = 3.125 N = 50.00 3.015 3.075 3.135 3.195 3.255 3.315 3.375 3.045 3.105 3.165 3.225 3.285 3.345 Mean WTP (Linear Utility Model) Figure 20 - Estimated Mean WTP Distribution, 10% Increase in Environmental Amenity and Linear Utility Function Page 68 Distribution of Median WTP Linear Utility Model For a 10% Increase in Environmental Amenity 3.015 3.075 3.135 3.195 3.255 3.315 3.375 3.045 3.105 3.165 3.225 3.285 3.345 Median WTP (Linear Utility Model) Figure 21 - Estimated Median WTP Distribution, 1 0 % Increase in Environmental Amenity and Linear Utility Function Distribution of Mean WTP Log Utility Model For a 10% Increase in Environmental Amenity Std. Dev = .06 Mean = 3.138 N = 50.00 3.015 3.075 3.135 3.195 3.255 3.315 3.375 3.045 3.105 3.165 3.225 3.285 3.345 Mean WTP (Log Utility Model) Figure 22 - Estimated Mean WTP Distribution, 1 0 % Increase in Environmental Amenity and Log-Linear Utility Function Page 69 Distribution of Median WTP Log Utility Model For a 10% Increase in Environmental Amenity LL 0 Std. Dev = .06 Mean = 3.160 N = 50.00 3.015 3.075 3.135 3.195 3.255 3.315 3.375 3.045 3.105 3.165 3.225 3.285 3.345 Median WTP (Log Utility Model) Figure 23 - Estimated Median WTP Distribution, 10% Increase in Environmental Amenity and Log-Linear Utility Function The results of the regressions for a 16% increase in the level of environmental quality are shown in Table 4. Again, the patterns are very similar to those seen in the previous scenarios with calculated WTP values lower than true WTP values. Diagnostic statistics are very close to those for the 10% change scenarios. Results and patterns for the log-linear utility specification are similar to those for the linear utility function, but model performance is slightly better and estimated WTP values are slightly higher. Estimated welfare measures, however, remain below true WTP. Thus, for 'large' changes in the quality or quantity of an environmental amenity, the DCCV model results in under-estimates of true welfare. Page 70 Table 5 Logit Regression Results and Associated Welfare Measures for a 16% Change in Public Good and Two Utility Difference Functional Forms Linear Log-Linear Model Results p -1.105 -11.087 s.e. 0.110 1.110 a 5.060 5.153 s.e. 0.533 0.538 Log-Likelihood -180.2 -175.4 Maddala R2 0.381 0.395 % Correct 77.01% 78.45% WTP Measures Median 4.642 4.663 s.e. 0.121 0.120 Mean (with truncation) 100% Upper Bid 4.613 4.628 125% Upper Bid 4.644 4.658 150% Upper Bid 4.649 4.662 True 4.835 4.835 Figure 24 shows the distribution of true WTP and is distinctly higher than distributions of mean and median WTP for either linear (Figures 25 and 26, respectively) or log-linear (Figures 27 and 28, respectively) utility specifications. The bimodality evident at the 10% change level is less evident at the 16% level and appears to have been an artifact of the samples at the 10% level. Page 71 Distribution of True Mean WTP For a 16% Increase in Environmental Amenity Std. Dev = .05 Mean = 4.835 N = 50.00 4.488 ^ 4.562 " 4.636 " 4.710 " 4.784 " 4.858 " 4.932 4.525 4.599 4.673 4.747 4.821 4.895 True WTP Figure 24 - True WTP Distribution for a 16% Increase in Environmental Amenity Distribution of Mean WTP Linear Utility Model For a 16% Increase in Environmental Amenity Std. Dev = .08 Mean = 4.613 N = 50.00 4.470 4.550 4.630 4.710 4.790 4.870 4.950 4.510 4.590 4.670 4.750 4.830 4.910 Mean WTP (Linear Utility Model) Figure 25 - Estimated Mean WTP Distribution, 16% Increase in Environmental Amenity and Linear Utility Function Page 72 Distribution of Median WTP Linear Utility Model For a 16% Increase in Environmental Amenity UL 0 Std. Dev = .08 Mean = 4.642 N = 50.00 4.488 4.562 4.636 4.710 4.784 4.858 4.932 4.525 4.599 4.673 4.747 4.821 4.895 Median WTP (Linear Utility Model) Figure 26 - Estimated Median WTP Distribution, 16% Increase in Environmental Amenity and Linear Utility Function Distribution of Mean WTP Log Utility Model For a 16% Increase in Environmental Amenity Std. Dev = .08 Mean = 4.628 N = 50.00 4.488 4.562 4.636 4.710 4.784 4.858 4.932 4.525 4.599 4.673 4.747 4.821 4.895 Mean WTP (Log Utility Model) Figure 27 - Estimated Mean WTP Distribution, 16% Increase in Environmental Amenity and Log-Linear Utility Function Page 73 Distribution of Median WTP Log Utility Model For a 16% Increase in Environmental Amenity U- o Std. Dev = .08 Mean = 4.663 N = 50.00 4.488 4.562 4.636 4.710 4.784 4.858 4.932 4.525 4.599 4.673 4.747 4.821 4.895 Median WTP (Log Utility Model) Figure 28 - Estimated Median WTP Distribution, 16% Increase in Environmental Amenity and Log-Linear Utility Function Finally, using Equation 22, it is possible to calculate the probability of a yes vote in a referendum at probabilities of other than the 50% median level. SHAZAM can calculate standard errors for Equation 22 so, unlike the case for numerically integrated mean WTP estimates, an approximate 95% confidence range for WTP can be calculated (Figure 29). For the arbitrarily chosen sample 132, for a 20% probability of a yes vote in a referendum, for example, the fee would need to be approximately 5.8 to 6.4 to be within the 95% confidence region. Alternatively, the probability of a yes vote for a fee of 6.0 would, for the approximate 95% confidence interval, range from 16% to 27%. Page 74 Sample 132 WTP Range * Pr{Yes} A Pr{Yes} ° Pr{Yes} Proposed Fee Minus 1.96 [s.e.] Plus 1.96 [s.e.] 0 2 4 6 8 10 12 14 Proposed Fee for Increased Environmental Amenity Figure 29 - Individual Sample WTP Confidence Interval for Sample 132 A summary of the true WTP and estimated welfare values for all combinations of levels of environmental change and utility functional forms is provided in Table 6. These average values are calculated for sample sizes of fifty in all cases and standard error is shown in the parentheses below the mean values. The median and mean WTP, calculated using the 125% or 150% truncation levels, are extremely close in value in all six cases. All estimated welfare measures are, however, approximately 2.9% to 3.8% below true WTP. Page 75 Table 6 - Summary of True WTP and Welfare Estimates Environmental Change (%) Utility Form Median Mean (100%) Mean (125%) Mean (150%) True WTP 4% Linear 1.401 1.391 1.400 1.401 1.438 (0.0356) (0.0346) (0.0354) (0.0355) (0.0155) 4% Log-Linear 1.404 1.395 1.404 1.405 1.438 (0.0345) (0.0338) (0.0344) (0.0344) (0.0345) 10% Linear 3.148 3.125 3.147 3.149 3.255 (0.0628) (0.0582) (0.0611) (0.0616) (0.0311) 10% Log-Linear 3.159 3.138 3.159 3.161 3.255 (0.0622) (0.0594) (0.0618) (0.0623) (0.0311) 16% Linear 4.642 4.613 4.644 4.649 4.835 (0.0792) (0.0757) (0.0791) (0.0799) (0.0548) 16% Log-Linear 4.657 4.628 6.658 4.662 4.835 (0.0806) (0.0761) (0.0803) (0.0814) (0.0548) If one assumes the WTP distributions for the fifty samples at each level of environmental change to be asymptotically normally distributed (a dubious assumption for this number of samples and discussed further in Chapter 6), then it is possible to undertake simple single-factor ANOVA and test the null hypotheses that various welfare measures are equal. Table 10 shows the results of these analyses for a variety of groupings. The null hypothesis of interest is listed in the left hand column: abbreviations used include med (average estimated median welfare), 100% (average estimated mean welfare calculated using a truncation point at the upper bid value), 125% (average estimated mean with a truncation at 125% of the upper bid), 150% (average estimated truncation at 150% of the upper bid), and true (average true WTP). Page 76 Table 7 - Tests of Equivalence of Welfare Measures and True WTP Null Hypothesis d.f. 4% Change 10% Change 16% Change Linear Utility Functional Form med= 100%= 125%= 150%=true 4 and 245 15.8*** 41.7*** 72.3*** med= 100%= 125%= 150% 3 and 196 1.0 1.8 2.1* 100%=125%=150% 2 and 147 1.3 2.5* 3.1** 100%=125% 1 and 98 1.7 3.4* 4.1** 100%=150% 1 and 98 2.1 4.2** 5.2** 125%=150% land 98 0.9 0.8 0.8 med=true 1 and 98 45.5*** 116.7*** 198.9*** 100%=true 1 and 98 76.3*** 194.4*** 280.8*** 125%=tme 1 and 98 47.2*** 124.5*** 195.5*** Log-Linear Utility Functional Form med= 100%= 125%= 150%=true 4 and 245 13.8*** 32.9*** 60.9*** med= 100%= 125%= 150% 3 and 196 0.9 1.6 1.9 100%=125%=150% 2 and 147 1.2 2.3 2.8* 100%=125% 1 and 98 1.6 3.0* 3.7* 100%= 150% land 98 1.9 3.8* 4.7** 125%=150% land 98 0.9 0.8 0.8 med=true land 98 39.1*** 94.4*** 166.6*** 100%= true 1 and 98 65.6*** 152.5*** 242.3*** 125%=true 1 and 98 40.9*** 96.3*** 164.6*** Linear median = Log-linear median 1 and 98 0.3 0.8 0.8 Linear 100% = Log-Linear 100% land 98 0.4 1.2 1.0 Linear 125% = Log-Linear 125% 1 and 98 0.3 1.0 0.8 Linear 150% = Log-Linear 150% 1 and 98 0.3 0.9 0.7 *** indicates significance at 1% level; ** at 5% level; and * at 10% level For the case of the linear utility function in the empirical model, the null hypotheses proposed in Equation (1), that true and estimated welfare measures are equal, are strongly rejected when using either median WTP or mean WTP calculated at the 100% or 125% truncation level. The hypothesis that all four welfare estimates, the median and the mean calculated at three different truncation levels, and true WTP are the same is also strongly rejected. The results are essentially the same for the empirical model that uses a log-linear functional form. In that case, the F-statistics are lower than in the linear utility model but still Page 77 highly significant. True WTP for the heterogeneous population of environmental altruists and non-altruists is significantly different from the WTP estimates derived using the DCGV methodology. It appears that neither of the empirical models can adequately account for the relatively simple utility model developed by Madriaga and McConnell (Equation 27). The utility function is additive and non-linear in the level of environmental change. The empirical models do not fully account for the full magnitude of non-market values held by heterogeneous economic agents. The second issue that is addressed by this analysis is the equivalence of welfare estimates. For the linear utility specification, there is no significant differences between welfare measures at low levels of environmental change. As the magnitude of environmental change increases, mean WTP calculated using the upper bid truncation level becomes significantly different from mean WTP calculated using either the 125% or 150% truncation level. At the highest level of environmental change, the null hypothesis of equality of all four welfare measures is also rejected at the 10% significance level. Similar results are also obtained when using the log-linear utility specification: mean WTP calculated using the 100% truncation level is significantly lower than WTP calculated using either the 125% or 150% truncation level. The null hypothesis of the equality of all four welfare measures cannot, however, be rejected when a log-linear utility function is used in the empirical model. The last section of Table 7 provides F-statistics for tests of the equivalence of welfare measures obtained using linear and log-linear utility specifications in the empirical model. Despite providing slightly better performance, based on diagnostic statistics and lower F-statistics, there is no evidence of statistical superiority of the model that uses the log-linear utility specifications based only on the ANOVA analysis of the first part of this experiment. Page 78 5.2 Proportional Error Analysis Table 8 presents the results of the PRSE (Equation 36) regression analysis for the two models. The dependent variable, the proportion root squared error, is significantly influenced by all independent variables in the two models. Although the R 2 values are low for response surface analysis, it can be seen that all coefficients are significant at the 1% level and the F-statistics are highly significant for both models. Table 8 - Proportional Error Regression Results Model 1 [t-ratio] Model 2 [t-ratio] Regression Results Intercept 0.537 21.5 0.471 19.0 Income -0.0273 -11.4 -0.0271 -11.6 Bid Amount 0.0423 22.4 0.0423 22.9 p/a Existence/Use Ratio -0.00813 -4.5 0.0271 9.7 AR -0.870 -12.2 -0.875 -12.5 Log Utility Dummy Altruist Dummy -0.0185 -3.9 -0.0186 -4.0 -0.120 -16.4 R2 0.106 0.145 F-statistic 142.8 168.9 Durbin-Watson 1.94 1.98 Degrees of Freedom 5,994 5,993 In Model 1, it is assumed that environmental altruists cannot be identified within a sample. I that case, proportional error is positively related to bid value and negatively related to income, the existence/use ratio, the level of environmental change, and the log-linear utility dummy. Page 79 That the bid amount and proportional error are positively related is not surprising and is consistent with the results of Huang and Smith (1996). Also consistent with their results are the negative relationship between proportional error and both income and the ratio of existence value to use value. Their rationale for the negative relation between the existence/use value ratio and proportional error relies on the fact that passive-use values come to dominate total WTP as the ratio rises and that, as a consequence, market goods progressively play a less and less important role in respondent valuation. Huang and Smith did not examine the effects on proportional error of different levels of environmental change nor the difference between linear and log-linear utility specifications. I found a strong negative relationship between proportional error and the level of environmental change and a moderate, but highly significant, relationship with the specification of the utility function. Using a log-linear utility function significantly reduced proportional errors in both models. In Model 2,1 made the assumption that environmental altruists could be accurately identified as such and performed the regression with one additional altruist dummy variable. The altruist dummy and the existence/use value ratio were relatively highly correlated at 0.77. The signs and magnitude of income, bid amount, level of environmental change, and the log-linear dummy variable were very stable and corresponded extremely closely with the coefficients from Model 1. The existence/use value ratio, which had been slightly negative and moderately significant, switched sign and became positively related to proportional error. The altruist dummy was highly significant and negatively related to proportional error. Model 2 provided a better overall fit than Model 1 based on R 2 values, F-statistics and t-statistics for coefficients. Page 80 CHAPTER 6 6. CONCLUSIONS 6.1 Accuracy of Welfare Measures The main goal of this thesis was to answer the technical question of whether estimated welfare measures derived using C V M could provide accurate approximations of true welfare values when agent preferences for nonmarket environmental goods and services are heterogeneous within a population. A quasi- Monte Carlo methodology was chosen to explore the question because, as Kling (1988) points out, the "Monte Carlo approach is particularly attractive for researchers interested in examining the reliability of welfare estimation in various circumstances" (P. 339-340). I am calling the methodology used in this thesis "quasi- Monte Carlo" because of the relatively limited number of draws of the B parameter, which controls the relative importance of existence value in the agent's utility function. In this thesis, only fifty draws were conducted at each of three levels of environmental change in the model. Monte Carlo studies typically use many more random draws in order to fully develop accurate confidence intervals for random variables of interest. Park et al (1991) advocate the use of at least 1,000 draws to develop confidence intervals for WTP estimates and base their techniques on the work of Krinsky and Robb (1986). Other researchers working with C V M and other nonmarket valuation Monte Carlo studies have also used large number of draws to develop confidence intervals for their estimators: Creel and Loomis (1997) use 1,000 drawings, Kling (1997) use 2,000 draws, Alberini (1995b) use up to 3,600 samples with 200 replications each, and Adamowicz et al (1989) use 5,000 draws. Rollins et al Page 81 (1997), however, apply the Krinsky and Robb techniques in an empirical study of willingness to pay for improving water infrastructure and develop confidence intervals for samples of only 121 to 523 respondents. Computer experiments can take advantage of the fact that logit regressions use maximum likelihood methods of estimation and return coefficients with desirable asymptotic normal distributions. These random variables can then be used to construct nonlinear WTP estimates which are, themselves, random variables with desirable maximum likelihood properties (Amemiya, 1981). Fifty samples would not normally be regarded as sufficient to form bootstrap confidence intervals because there could still be a variety of nonlinear interactions between logit coefficients that remain unaccounted for. For this reason, the ANOVA results (Table 7) should be interpreted with care. Despite the difficulties with the statistical validity of the comparisons, it is possible to note a strong indication that true and estimated welfare measures are not equal in the study. The test statistics shown in Table 10 indicate that the null hypotheses - that any of the estimated welfare measures (median or the means truncated at various levels) are equal the true welfare values - are strongly rejected. The discrepancies are also visually apparent in the series of figures presented in Chapter 5. A main conclusion of the thesis is that there are strong indications that estimated and true welfare levels are not equal in the population of heterogeneous agents modeled in this experiment. Without further work, however, it is premature to reject the null hypothesis outright due to the potential problems of assuming large sample properties with only fifty samples for each level of environmental change. A logical extension to the work presented in this thesis would be to expand the number of drawings to one thousand or more; this would allow for the development of valid confidence intervals and more reliable examination of the central hypothesis (Equation 1). Page 82 6.2 Sources of Bias The second objective of the thesis was to explore possible sources of bias in the estimation of agent welfare. I used proportional root square error, a measure comparing each individual agent's estimated and true WTP, as the dependent variable in a response surface regression largely following the methodology of Huang and Smith (1996). The regression used a variety of agent and model characteristics as independent variables in order to isolate factors that contribute significantly to bias in the estimated welfare calculations. Like Huang and Smith, I found that increasing agent income led to a reduction in proportional error, while the bid amount and proportional error are positively related. These results are not surprising and can be explained in terms of nonmarket existence values increasing in importance relative to use value in the total value equation. I also found a negative relationship between proportional error and the level of environmental change. The same rationale applies here with proportional error decreasing as existence value becomes relatively more important in the calculation of total WTP. Huang and Smith's main focus was on the relative effect of existence value compared to market use values in the Madriaga and McConnell (1987) water quality model. They were particularly concerned with the widely held view that C V M studies yield larger welfare estimates compared to open-ended surveys and that the upward bias was particularly evident when WTP was dominated by existence values. The results of this thesis confirm the results of Huang and Smith when altruists and non-altruists were assumed to be unidentifiable (Model 1); in both experiments (Huang and Smith, 1996 and Model 1 in this thesis) the implication is that market commodities play less of a role in total WTP as existence values become more important to people and that, as a Page 83 result, there is a counterintuitive negative relationship between proportional error and the ratio of existence to use values. In this thesis, the analysis was extended and the assumption was taken that altruists could be identified. Model 2 then used a dummy variable for environmental altruism and found that the regression performance improved as a result. The sign on the p/ct ratio (i.e., the ratio of existence to use value) switched but the new altruist dummy variable had a strong negative relationship with proportional error. The net result of the coefficient changes is negative: the combination of the dummy and p/ct coefficients total approximately -0.093 in Model 2 compared with the value of -0.008 for p/a only in Model 1. Other coefficients are very stable between the two models. The combination effect in Model 2 is similar in magnitude to that found for the p/a ratio of -0.0364 in Huang and Smith (1996). The result of most interest in the PRSE analysis is the negative relationship between the log-linear utility functional form and proportional error. It is common knowledge that the linear utility function is not theoretically correct (Sellar et al, 1986) but it appears that nobody has yet provided empirical evidence showing bias for the linear utility specification. Even in this study, the (qualified) ANOVA results (Table 10) show no indication of rejection of the null hypotheses of equality of welfare estimates based on linear and log-linear utility specifications. In the PRSE analysis, however, there is a relatively strong and highly significant negative relationship between welfare estimates based on the log-linear utility function and proportional error: using the log-linear functional form to represent utility decreased bias in the experiment. The high explanatory power of a model with almost 6,000 degrees of freedom is probably key to observing this phenomena: Page 84 6.3 Implications for Environmental Valuation In this thesis, I found that, even if one accepts all the necessary assumptions required to use C V M results in economic CBA, however arguable they are, there is still no guarantee that the welfare measures derived using C V M will be accurate if agents within a population have heterogeneous environmental preferences. The results of this study strongly suggest that welfare estimates derived using CVM, when the population is heterogeneous, are inaccurate and/or biased, although it is clear that further confirmation is required using a larger number of samples. One might ask the question as to whether biases are systematic and can, therefore, be subjected to some sort of calibration process. There seems to be no a priori reason to hold this view; this study used what was considered a reasonable utility function and set of parameters to obtain the current results13. But it is conceivable that there could be utility and parameter specifications that result in welfare estimates that are higher, rather than lower, than true WTP, or that result in a more extreme spread between true and estimated WTP values. What is indicated in this particular study is that welfare measures estimated using C V M are suspect even under simple conditions when agent preferences are specified in accordance with economic theory and there are no complications arising from preference uncertainty, embedding effects or any of the other technical problems relating to survey design and delivery. The "bottom line" is that one must decide whether the use of C V M is justified or not. Carson et al (1996) state that "the link between welfare economics and [contingent valuation] is quite direct: contingent valuation offers the potential to trace out the willingness to pay distribution After identifying the utility function to be used in this thesis, I experimented with a number of different parameter values. The goal was to develop a model which showed WTP patterns typical of empirical studies where there is a skew rightwards. There was no "fishing" for a form or parameters that resulted in bias: no logit regressions were undertaken until the final decisions were made on model form and parameter choices. Page 85 for a population of economic agents for a proposed change in a good. If properly executed, CV is a useful tool for benefit-cost analysis" (p. 2). The results of this thesis stand in direct contrast to the dictum of Carson: even if properly executed, it is highly likely that contingent valuation welfare estimates derived using the dichotomous choice contingent valuation methodology are inaccurate and should be used with caution in CBA whenever heterogeneous preferences are held within a population. My results lend support to the view of McFadden and Leonard (1993), albeit for different reasons14, that "the failures are severe enough and central enough to the [contingent valuation] method to render it untrustworthy in its current implementation" (p. 202). This raises a dilemma for conservationists: people clearly value environmental goods and nonmarket valuation can be critical in environmental preservation, but should C V M be advocated as a valuation tool if the welfare estimates put forward by economic analyses are, in the terminology of McFadden (1994), "fundamentally flawed"? Grove-White (1997) notes that a number of British environmental non-governmental organizations have "tended to explore the tactical possibilities of the use of surrogate valuation methods for advancing their aims" (p. 28) and there has been little incentive for environmentalists, historically the principal explicit critics of traditional CBA, to criticize techniques, however flawed, that have been advanced to assist environmental ends. This is a very important question but one beyond the scope of this study. One approach suggested by Common et al (1997) is to accept the fact that C V M results are not market responses of individuals, but that it may still be possible to structure C V M referenda to assess "citizen values" and, therefore, derive social valuations for environmental goods. McFadden and Leonard based their criticisms on the failure of contingent valuation to meet criterion of psychometric robustness (i.e., people do not have stable preferences for environmental goods). Page 86 The philosophical and, increasingly, the technical arguments against the use of C V M are substantial and should be taken seriously by economists involved in issues of environmental valuation and resource management. The contribution of this thesis is that there is further concern regarding CVM. Welfare estimates from the dichotomous choice model commonly employed in C V M are unreliable when there is agent heterogeneity, as is the case in the real world. Page 87 BIBLIOGRAPHY Adamowicz, W. L., J. J. 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Page 96 A P P E N D I X - L O G I T R E G R E S S I O N R E S U L T S A N D W E L F A R E C A L C U L A T I O N S Regression Summary Samples 1 to 50 AR = 4% 1 2 3 4 5 6 7 8 9 10 Linear Utility Model P -4.268 -4.488 -4.621 -4.440 -4.354 -4.247 -5.847 -4.328 -4.106 -4.306 s.e. 0.427 0.452 0.449 0.434 0.441 0.444 0.578 0.430 0.413 0.435 a 5.967 6.233 6.675 6.375 6.158 5.878 7.898 6.132 5.772 5.947 s.e. 0.613 0.649 0.669 0.645 0.642 0.644 0.798 0.630 0.608 0.628 Log-Likelihood -183.7 -184.4 -161.2 -177.4 -189.5 -188.5 -156.7 -173.3 -186.7 -183.8 Maddala R-Square 0.337 0.370 0.440 0.393 0.355 0.349 0.449 0.404 0.358 0.366 % Correct 78.25% 77.00% 81.50% 76.75% 75.00% 74.75% 81.50% 78.25% 78.75% 77.25% Upper Integral Limit Hi 2.150 2.100 2.200 2.100 2.050 2.100 2.050 2.150 2.150 2.100 V Hi 2.688 2.625 2.750 2.625 2.563 2.625 2.563 2.688 2.688 2.625 X Hi 3.225 3.150 3.300 3.150 3.075 3.150 3.075 3.225 3.225 3.150 Welfare Measures Median WTP 1.398 1.389 1.444 1.436 1.414 1.384 1.351 1.417 1.406 1.381 Mean WTP Hi 1.389 1.380 1.438 1.425 1.401 1.374 1.348 1.408 1.395 1.371 V Hi 1.398 1.388 1.444 1.435 1.413 1.384 1.351 1.416 1.405 1.381 X Hi 1.399 1.389 1.445 1.436 1.415 1.385 1.351 1.417 1.406 1.382 True WTP 1.403 1.434 1.46 1.421 1.421 1.43 1.41 1.447 1.433 1.425 Representative Agent Log-Linear Utility Model P -48.502 -45.502 -46.992 -44.462 -41.379 -45.774 -60.380 -44.339 -42.900 -44.050 s.e. 4.745 4.529 4.581 4.322 4.141 4.741 6.012 4.406 4.262 4.352 a 6.824 6.345 6.802 6.383 5.862 6.383 8.171 6.291 6.056 6.099 s.e. 0.684 0.652 0.682 0.640 0.602 0.691 0.831 0.647 0.629 0.627 Log-Likelihood -163.3 -178.8 -154.7 -175.5 -188.4 -177.1 -146.1 -168.8 -180.5 -176.8 Maddala R-Square 0.433 0.387 0.458 0.399 0.359 0.385 0.478 0.417 0.377 0.388 % Correct 80.25% 76.50% 81.50% 77.50% 76.50% 80.75% 82.00% 80.00% 78.50% 76.25% Upper Integral Limit Hi 2.150 2.100 2.200 2.100 2.050 2.100 2.050 2.150 2.150 2.100 V Hi 2.688 2.625 2.750 2.625 2.563 2.625 2.563 2.688 2.688 2.625 X Hi 3.225 3.150 3.300 3.150 3.075 3.150 3.075 3.225 3.225 3.150 Welfare Measures Median WTP 1.409 1.397 1.450 1.438 1.419 1.397 1.355 1.421 1.414 1.387 Mean WTP Hi 1.404 1.389 1.444 1.427 1.402 1.388 1.353 1.413 1.405 1.378 V Hi 1.409 1.397 1.449 1.437 1.417 1.396 1.355 1.421 1.413 1.386 X Hi 1.409 1.397 1.450 1.438 1.419 1.397 1.355 1.421 1.414 1.387 True WTP 1.403 1.434 1.46 1.421 1.421 1.43 1.41 1.447 1.433 1.425 Representative Agent Page 97 Regression Summary Samples 1 to 50 AR = 4% 11 12 13 14 16 16 17 18 19 20 Linear Utility Model P -4.327 -3.934 -4.541 -4.785 -4.092 -4.792 -4.288 -4.115 -4.539 -4.519 s.e. 0.429 0.412 0.454 0.479 0.406 0.401 0.342 0.420 0.455 0.465 a 5.849 5.431 6.296 6.528 5.709 7.009 6.664 5.843 6.464 6.259 s.e. 0.603 0.601 0.643 0.683 0.593 0.590 0.593 0.604 0.661 0.609 Log-Likelihood -182.1 -195.3 -177.2 -175.4 -176.6 -93.3 -94.5 -183.9 -177.6 -191.5 Maddala R-Square 0.375 0.328 0.394 0.394 0.391 0.600 0.599 0.372 0.392 0.345 % Correct 77.00% 74.00% 78.75% 77.50% 76.75% 91.75% 92.50% 76.00% 78.25% 76.25% Upper Integral Limit Hi 2.050 2.150 2.150 2.100 2.200 2.100 2.150 2.150 2.150 2.050 V Hi 2.563 2.688 2.688 2.625 2.750 2.625 2.688 2.688 2.688 2.563 X Hi 3.075 3.225 3.225 3.150 3.300 3.150 3.225 3.225 3.225 3.075 Welfare Measures Median WTP 1.352 1.380 1.386 1.364 1.395 1.463 1.554 1.420 1.424 1.385 Mean WTP Hi 1.341 1.369 1.380 1.358 1.387 1.453 1.537 1.409 1.416 1.375 V Hi 1.351 1.380 1.386 1.364 1.395 1.462 1.553 1.420 1.424 1.384 X Hi 1.352 1.381 1.387 1.364 1.396 1.463 1.554 1.421 1.424 1.385 True WTP 1.414 1.448 1.435 1.419 1.451 1.442 1.431 1.444 1.443 1.426 Representative Agent Log-Linear Utility Model P -47.072 -41.497 -45.323 -50.978 -40.871 -51.770 -46.170 -41.500 -45.077 -43.731 s.e. 4.653 4.347 4.549 5.154 4.057 4.600 3.912 4.243 4.427 4.447 a 6.343 5.731 6.301 6.928 5.698 7.515 7.124 5.874 6.455 6.047 s.e. 0.645 0.629 0.644 0.724 0.590 0.669 0.655 0.604 0.647 0.636 Log-Likelihood -170.8 -187.1 -173.1 -166.5 -172.8 -89.0 -87.1 -179.9 -173.8 -188.6 Maddala R-Square 0.409 0.355 0.401 0.420 0.402 0.609 0.614 0.385 0.404 0.355 % Correct 78.00% 76.75% 81.25% 78.25% 77.75% 92.25% 92.25% 79.00% 78.25% 76.25% Upper Integral Limit Hi 2.050 2.150 2.150 2.100 2.200 2.100 2.150 2.150 2.150 2.050 V Hi 2.563 2.688 2.688 2.625 2.750 2.625 2.688 2.688 2.688 2.563 X Hi 3.075 3.225 3.225 3.150 3.300 3.150 3.225 3.225 3.225 3.075 Welfare Measures Median WTP 1.350 1.383 1.392 1.361 1.396 1.454 1.545 1.418 1.434 1.385 Mean WTP Hi 1.342 1.374 1.386 1.357 1.388 1.447 1.532 1.407 1.426 1.373 V Hi 1.349 1.383 1.392 1.361 1.396 1.454 1.544 1.417 1.434 1.384 X Hi 1.350 1.384 1.393 1.361 1.397 1.454 1.545 1.418 1.434 1.385 True WTP 1.414 1.448 1.435 1.419 1.451 1.442 1.431 1.444 1.443 1.426 Representative Agent Page 98 Regression Summary Samples 1 to 50 AR = 4% 21 22 23 24 25 26 27 28 29 30 Linear Utility Model P -3.778 -4.178 -3.984 -4.425 -3.813 -4.546 -4.298 -4.392 -3.986 -4.381 s.e. 0.388 0.414 0.403 0.466 0.395 0.439 0.435 0.476 0.408 0.426 a 5.280 5.895 5.604 6.118 5.257 6.354 6.214 6.141 5.529 6.116 s.e. 0.570 0.606 0.586 0.668 0.576 0.635 0.645 0.690 0.587 0.612 Log-Likelihood -195.7 -180.1 -191.8 -184.7 -196.3 -165.3 -183.3 -200.8 -197.8 -170.8 Maddala R-Square 0.331 0.382 0.346 0.364 0.326 0.428 0.374 0.314 0.326 0.411 % Correct 75.00% 77.25% 75.00% 78.00% 75.50% 78.50% 79.50% 72.50% 72.00% 78.75% Upper Integral Limit Hi 2.100 2.150 2.150 2.200 2.150 2.150 2.150 2.100 2.050 2.200 V Hi 2.625 2.688 2.688 2.750 2.688 2.688 2.688 2.625 2.563 2.750 X Hi 3.150 3.225 3.225 3.300 3.225 3.225 3.225 3.150 3.075 3.300 Welfare Measures Median WTP 1.398 1.411 1.407 1.383 1.379 1.398 1.446 1.398 1.387 1.396 Mean WTP Hi 1.381 1.401 1.395 1.379 1.366 1.391 1.435 1.388 1.371 1.390 V_Hi 1.397 1.410 1.406 1.383 1.378 1.398 1.445 1.398 1.386 1.396 X Hi 1.399 1.411 1.407 1.383 1.380 1.398 1.446 1.399 1.388 1.396 True WTP 1.438 1.455 1.448 1.466 1.442 1.439 1.449 1.432 1.416 1.455 Representative Agent Log-Linear Utility Model P -39.857 -43.435 -40.935 -48.264 -38.562 -43.486 -41.352 -39.581 -41.638 -49.853 s.e. 3.991 4.232 4.165 4.963 3.959 4.200 4.103 4.212 4.182 4.879 a 5.611 6.140 5.759 6.700 5.304 6.069 5.983 5.521 5.785 7.008 s.e. 0.590 0.617 0.602 0.712 0.571 0.605 0.609 0.607 0.599 0.706 Log-Likelihood -186.4 -169.6 -186.8 -172.5 -189.8 -166.1 -181.1 -201.2 -188.6 -156.8 Maddala R-Square 0.362 0.414 0.363 0.401 0.347 0.425 0.381 0.313 0.356 0.451 % Correct 75.75% 80.00% 76.75% 80.50% 78.50% 81.00% 79.00% 73.00% 74.75% 81.25% Upper Integral Limit Hi 2.100 2.150 2.150 2.200 2.150 2.150 2.150 2.100 2.050 2.200 V Hi 2.625 2.688 2.688 2.750 2.688 2.688 2.688 2.625 2.563 2.750 X Hi 3.150 3.225 3.225 3.300 3.225 3.225 3.225 3.150 3.075 3.300 Welfare Measures Median WTP 1.410 1.416 1.410 1.391 1.378 1.398 1.449 1.400 1.392 1.408 Mean WTP Hi 1.395 1.407 1.398 1.386 1.366 1.390 1.436 1.383 1.377 1.404 V Hi 1.409 1.415 1.409 1.390 1.377 1.398 1.448 1.396 1.390 1.408 X Hi 1.410 1.416 1.409 1.391 1.379 1.398 1.449 1.397 1.392 1.408 True WTP 1.438 1.455 1.448 1.466 1.442 1.439 1.449 1.432 1.416 1.455 Representative Agent Page 99 Regression Summary Samples 1 to SO AR = 4% 31 32 33 34 36 36 37 38 39 40 Linear Utility Model P -4.390 -4.821 -4.538 -4.220 -4.334 -4.457 -4.065 -3.969 -3.719 -4.210 s.e. 0.451 0.489 0.456 0.417 0.448 0.444 0.404 0.399 0.373 0.431 a 5.968 6.791 6.325 5.946 5.906 6.199 5.660 5.475 5.231 5.818 s.e. 0.634 0.716 0.652 0.616 0.645 0.633 0.573 0.578 0.539 0.622 Log-Likelihood -188.8 -165.4 -171.0 -173.2 -187.6 -180.5 -182.0 -183.8 -185.9 -183.1 Maddala R-Square 0.357 0.424 0.409 0.401 0.350 0.383 0.379 0.368 0.367 0.366 % Correct 76.75% 79.25% 77.00% 78.50% 74.25% 78.00% 76.75% 76.25% 75.00% 76.00% Upper Integral Limit Hi 2.100 2.200 2.200 2.200 2.100 2.100 2.150 2.200 2.250 2.150 V Hi 2.625 2.750 2.750 2.750 2.625 2.625 2.688 2.750 2.813 2.688 X Hi 3.150 3.300 3.300 3.300 3.150 3.150 3.225 3.300 3.375 3.225 Welfare Measures Median WTP 1.359 1.409 1.394 1.409 1.363 1.391 1.393 1.379 1.407 1.382 Mean WTP Hi 1.351 1.404 1.389 1.401 1.354 1.382 1.382 1.371 1.397 1.373 V Hi 1.359 1.409 1.394 1.409 1.362 1.390 1.392 1.379 1.407 1.382 X Hi 1.360 1.409 1.394 1.409 1.363 1.391 1.393 1.380 1.408 1.382 True WTP 1.426 1.456 1.464 1.463 1.436 1.421 1.428 1.452 1.458 1.439 Representative Agent Log-Linear Utility Model P -40.977 -50.779 -48.130 -41.757 -45.689 -44.575 -42.850 -39.545 -40.191 -44.928 s.e. 4.155 5.093 4.715 4.103 4.559 4.353 4.293 4.002 3.972 4.499 a 5.578 7.152 6.752 5.898 6.234 6.197 5.945 5.459 5.671 6.220 s.e. 0.583 0.743 0.678 0.607 0.653 0.617 0.600 0.579 0.572 0.648 Log-Likelihood -189.8 -154.7 -158.4 -169.6 -177.8 -174.2 -174.8 -181.4 -173.4 -172.2 Maddala R-Square 0.353 0.454 0.445 0.412 0.381 0.402 0.401 0.375 0.405 0.399 % Correct 76.50% 80.25% 81.50% 78.00% 76.75% 79.75% 79.25% 78.75% 77.75% 78.25% Upper Integral Limit Hi 2.100 2.200 2.200 2.200 2.100 2.100 2.150 2.200 2.250 2.150 V Hi 2.625 2.750 2.750 2.750 2.625 2.625 2.688 2.750 2.813 2.688 X Hi 3.150 3.300 3.300 3.300 3.150 3.150 3.225 3.300 3.375 3.225 Welfare Measures Median WTP 1.363 1.411 1.405 1.415 1.367 1.392 1.389 1.383 1.413 1.387 Mean WTP Hi 1.352 1.407 1.401 1.406 1.359 1.383 1.381 1.374 1.406 1.380 V Hi 1.363 1.410 1.405 1.414 1.366 1.392 1.389 1.383 1.413 1.386 X Hi 1.364 1.411 1.405 1.415 1.367 1.393 1.390 1.384 1.414 1.387 True WTP 1.426 1.456 1.464 1.463 1.436 1.421 1.428 1.452 1.458 1.439 Representative Agent Page 100 Regression Summary Samples 1 to 50 AR = 4% 41 42 43 44 45 46 47 48 49 50 Linear Utility Model P -3.819 -4.160 -4.357 -3.728 -4.488 -4.613 -4.546 -4.087 -3.669 -3.942 s.e. 0.386 0.418 0.442 0.370 0.462 0.481 0.453 0.418 0.363 0.408 a 5.072 5.978 6.023 5.479 6.182 6.450 6.558 5.691 5.039 5.513 s.e. 0.548 0.619 0.631 0.560 0.646 0.676 0.678 0.590 0.525 0.589 Log-Likelihood -185.2 -180.5 -187.2 -186.5 -186.1 -182.5 -178.5 -192.1 -185.0 -198.7 Maddala R-Square 0.352 0.384 0.362 0.365 0.366 0.377 0.388 0.347 0.366 0.323 % Correct 77.75% 77.00% 75.75% 76.75% 74.50% 74.50% 78.75% 75.00% 77.25% 74.50% Upper Integral Limit Hi 2.150 2.150 2.100 2.200 2.100 2.150 2.150 2.100 2.200 2.100 V Hi 2.688 2.688 2.625 2.750 2.625 2.688 2.688 2.625 2.750 2.625 X Hi 3.225 3.225 3.150 3.300 3.150 3.225 3.225 3.150 3.300 3.150 Welfare Measures Median WTP 1.328 1.437 1.382 1.470 1.377 1.398 1.443 1.393 1.374 1.399 Mean WTP Hi 1.318 1.425 1.373 1.454 1.369 1.392 1.434 1.380 1.362 1.384 V Hi 1.328 1.436 1.382 1.468 1.377 1.398 1.442 1.392 1.374 1.398 X Hi 1.329 1.437 1.383 1.470 1.378 1.399 1.443 1.393 1.375 1.399 True WTP 1.426 1.434 1.44 1.457 1.423 1.445 1.443 1.412 1.458 1.429 Representative Agent Log-Linear Utility Model P -40.296 -42.540 -43.247 -41.147 -41.635 -42.058 -41.967 -42.892 -39.050 -41.791 s.e. 4.070 4.293 4.322 4.105 4.214 4.292 4.213 4.361 3.871 4.276 a 5.352 6.116 5.972 6.047 5.763 5.906 6.047 5.988 5.351 5.851 s.e. 0.571 0.632 0.613 0.616 0.592 0.605 0.627 0.614 0.553 0.615 Log-Likelihood -176.1 -174.4 -181.5 -174.7 -187.3 -182.7 -180.2 -182.2 -177.8 -191.8 Maddala R-Square 0.381 0.402 0.380 0.401 0.362 0.376 0.383 0.378 0.388 0.346 % Correct 78.75% 78.00% 78.25% 78.00% 75.25% 76.75% 76.00% 76.25% 77.50% 75.75% Upper Integral Limit Hi 2.150 2.150 2.100 2.200 2.100 2.150 2.150 2.100 2.200 2.100 V Hi 2.688 2.688 2.625 2.750 2.625 2.688 2.688 2.625 2.750 2.625 X Hi 3.225 3.225 3.150 3.300 3.150 3.225 3.225 3.150 3.300 3.150 Welfare Measures Median WTP 1.330 1.440 1.383 1.472 1.386 1.406 1.443 1.398 1.372 1.402 Mean WTP Hi 1.322 1.429 1.373 1.461 1.375 1.397 1.432 1.388 1.364 1.390 V Hi 1.330 1.439 1.383 1.471 1.386 1.406 1.442 1.398 1.372 1.401 X Hi 1.331 1.440 1.384 1.472 1.387 1.407 1.444 1.399 1.373 1.403 True WTP 1.426 1.434 1.44 1.457 1.423 1.445 1.443 1.412 1.458 1.429 Representative Agent Page 101 Regression Summary Samples 1 to 50 AR = 4% Standard Min Mean Max Error Linear Utility Model P -5.847 -4.297 -3.669 s.e. 0.342 0.431 0.578 a 5.039 6.018 7.898 s.e. 0.525 0.623 0.798 Log-Likelihood -200.8 -179.3 -93.3 Maddala R-Square 0.314 0.381 0.600 % Correct 72.00% 77.39% 92.50% Upper Integral Limit Hi 2.050 2.250 V Hi 2.563 2.813 X Hi 3.075 3.375 Welfare Measures Median WTP 1.328 1.401 - 1.554 0.0356 Mean WTP Hi 1.318 1.391 1.537 0.0346 V Hi 1.328 1.400 1.553 0.0354 X Hi 1.329 1.401 1.554 0.0355 True WTP 1.403 1.438 1.466 0.0155 Representative Agent 1.403 Standard Log-Linear Utility Model Min Mean Max Error P -60.380 -44.026 -38.562 s.e. 3.871 4.383 6.012 a 5.304 6.172 8.171 s.e. 0.553 0.633 0.831 Log-Likelihood -201.2 -172.7 -87.1 Maddala R-Square 0.313 0.401 0.614 % Correct 73.00% 78.75% 92.25% Upper Integral Limit Hi 2.050 2.250 V Hi 2.563 2.813 X Hi 3.075 3.375 Welfare Measures Median WTP 1.330 1.404 1.545 0.0345 Mean WTP Hi 1.322 1.395 1.532 0.0338 V Hi 1.330 1.404 1.544 0.0344 X Hi 1.331 1.405 1.545 0.0344 True WTP 1.403 1.438 1.466 0.0155 Representative Agent 1.403 Regression Summary Samples 51 to 100 AR = 10% 51 52 53 54 55 56 57 58 59 60 Linear Utility Model f> -1.595 -1.776 -1.647 -1.662 -1.698 -1.665 -1.863 -1.681 -1.877 -1.903 s.e. 0.154 0.180 0.167 0.169 0.174 0.167 0.189 0.167 0.183 0.193 a 5.215 5.569 5.086 5.240 5.479 5.139 5.782 5.131 5.829 5.816 s.e. 0.516 0.590 0.543 0.558 0.570 0.544 0.603 0.534 0.591 0.605 Log-Likelihood -175.1 -173.8 -181.3 -189.1 -189.9 -185.6 -191.2 -184.9 -172.1 -182.7 Maddala R-Square 0.400 0.400 0.376 0.355 0.353 0.363 0.350 0.364 0.407 0.376 % Correct 79.25% 78.00% 76.75% 76.25% 76.25% 74.00% 76.50% 76.75% 80.50% 77.50% Upper Integral Limit Hi 5.150 5.050 5.200 5.050 4.950 4.950 4.700 4.900 4.850 4.800 V Hi 6.438 6.313 6.500 6.313 6.188 6.188 5.875 6.125 6.063 6.000 X Hi 7.725 7.575 7.800 7.575 7.425 7.425 7.050 7.350 7.275 7.200 Welfare Measures Median WTP 3.271 3.136 3.088 3.153 3.227 3.087 3.103 3.052 3.106 3.056 Mean WTP Hi 3.244 3.120 3.074 3.131 3.199 3.064 3.078 3.030 3.088 3.039 V Hi 3.270 3.136 3.090 3.153 3.226 3.087 3.102 3.052 3.106 3.055 X_Hi 3.274 3.138 3.092 3.156 3.229 3.090 3.105 3.055 3.107 3.056 True WTP 3.304 3.236 3.298 3.281 3.25 3.251 3.179 3.237 3.197 3.191 Representative Agent Log-Linear Utility Model 3 -16.640 -17.697 -15.476 -17.790 -17.348 -16.485 -19.563 -17.603 -18.170 -19.485 s.e. 1.586 1.763 1.566 1.778 1.760 1.641 1.947 1.752 1.773 1.937 a 5.491 5.562 4.796 5.627 5.617 5.121 6.079 5.397 5.683 5.948 s.e. 0.540 0.580 0.514 0.586 0.578 0.538 0.618 0.561 0.579 0.603 Log-Likelihood -168.4 -168.7 -183.4 -177.4 -184.2 -182.5 -181.7 -177.2 -170.8 -174.1 Maddala R-Square 0.420 0.415 0.369 0.391 0.372 0.372 0.380 0.388 0.411 0.402 % Correct 80.75% 79.75% 77.75% 79.50% 76.50% 74.50% 77.00% 78.00% 80.00% 80.00% Upper Integral Limit Hi 5.150 5.050 5.200 5.050 4.950 4.950 4.700 4.900 4.850 4.800 V Hi 6.438 6.313 6.500 6.313 6.188 6.188 5.875 6.125 6.063 6.000 X Hi 7.725 7.575 7.800 7.575 7.425 7.425 7.050 7.350 7.275 7.200 Welfare Measures Median WTP 3.305 3.148 3.104 3.168 3.242 3.111 3.112 3.071 3.133 3.057 Mean WTP Hi 3.280 3.130 3.084 3.150 3.215 3.086 3.091 3.059 3.110 3.042 V Hi 3.304 3.148 3.106 3.168 3.241 3.111 3.111 3.071 3.132 3.057 X Hi 3.307 3.150 3.109 3.169 3.244 3.114 3.113 3.073 3.134 3.059 True WTP 3.304 3.236 3.298 3.281 3.25 3.251 3.179 3.237 3.197 3.191 Representative Agent Page 103 Regression Summary Samples 51 to 100 AR = 10% 61 62 63 64 65 66 67 68 69 70 Linear Utility Model P -1.529 -1.964 -1.679 -1.669 -1.682 -1.521 -1.780 -1.785 -1.780 -1.524 s.e. 0.154 0.195 0.167 0.171 0.166 0.154 0.175 0.181 0.175 0.154 a 4.932 6.104 5.313 5.177 5.160 4.930 5.713 5.580 5.497 4.737 s.e. 0.514 0.638 0.551 0.548 0.532 0.522 0.577 0.591 0.561 0.514 Log-Likelihood -189.3 -160.1 -175.5 -189.2 -182.1 -188.9 -182.1 -185.0 -178.4 -186.8 Maddala R-Square 0.355 0.438 0.397 0.355 0.377 0.357 0.379 0.367 0.388 0.355 % Correct 73.75% 80.75% 78.25% 75.00% 75.75% 76.75% 76.25% 74.75% 78.25% 77.75% Upper Integral Limit Hi 5.100 5.050 5.150 5.000 4.900 5.150 4.850 4.900 4.900 5.150 V Hi 6.375 6.313 6.438 6.250 6.125 6.438 6.063 6.125 6.125 6.438 X Hi 7.650 7.575 7.725 7.500 7.350 7.725 7.275 7.350 7.350 7.725 Welfare Measures Median WTP 3.226 3.108 3.164 3.101 3.068 3.243 3.210 3.126 3.088 3.108 Mean WTP Hi 3.195 3.098 3.146 3.080 3.044 3.212 3.182 3.105 3.068 3.085 V Hi 3.226 3.108 3.165 3.101 3.068 3.242 3.208 3.126 3.088 3.109 X_Hi 3.230 3.109 3.167 3.104 3.071 3.246 3.211 3.128 3.090 3.113 True WTP 3.3 3.255 3.287 3.261 3.231 3.285 3.225 3.236 3.214 3.278 Representative Agent Log-Linear Utility Model P -14.672 -20.207 -16.653 -17.595 -16.973 -16.950 -19.307 -16.842 -19.025 -14.919 s.e. 1.462 1.998 1.656 1.744 1.677 1.697 1.876 1.716 1.874 1.513 a 4.759 6.321 5.255 5.499 5.200 5.496 6.222 5.284 5.889 4.648 s.e. 0.492 0.656 0.541 0.563 0.533 0.570 0.619 0.560 0.597 0.505 Log-Likelihood -187.3 -153.0 -172.4 -177.7 -177.0 -176.1 -169.9 -185.4 -167.3 -186.1 Maddala R-Square 0.362 0.458 0.406 0.391 0.393 0.396 0.415 0.366 0.421 0.358 % Correct 76.25% 82.50% 78.50% 76.25% 77.75% 78.75% 77.25% 74.75% 79.25% 78.50% Upper Integral Limit Hi 5.100 5.050 5.150 5.000 4.900 5.150 4.850 4.900 4.900 5.150 V Hi 6.375 6.313 6.438 6.250 6.125 6.438 6.063 6.125 6.125 6.438 X Hi 7.650 7.575 7.725 7.500 7.350 7.725 7.275 7.350 7.350 7.725 Welfare Measures Median WTP 3.249 3.133 3.160 3.130 3.069 3.248 3.228 3.142 3.100 3.120 Mean WTP Hi 3.211 3.124 3.142 3.112 3.046 3.227 3.206 3.115 3.085 3.094 V Hi 3.248 3.133 3.161 3.130 3.069 3.247 3.227 3.141 3.100 3.122 X Hi 3.254 3.134 3.163 3.132 3.071 3.250 3.228 3.144 3.101 3.126 True WTP 3.3 3.255 3.287 3.261 3.231 3.285 3.225 3.236 3.214 3.278 Representative Agent Page 104 Regression Summary Samples 51 to 100 AR = 10% 71 72 73 74 75 76 77 78 79 80 Linear Utility Model P -1.775 -1.671 -1.812 -1.638 -1.573 -1.789 -1.885 -1.825 -1.629 -1.525 s.e. 0.175 0.165 0.185 0.167 0.154 0.180 0.189 0.183 0.166 0.154 a 5.634 5.425 5.662 5.120 4.991 5.641 6.003 5.640 4.988 4.773 s.e. 0.577 0.550 0.606 0.548 0.512 0.584 0.626 0.593 0.537 0.506 Log-Likelihood -175.0 -178.3 -184.7 -192.0 -178.7 -179.9 -171.8 -177.3 -186.6 -194.9 Maddala R-Square 0.399 0.390 0.365 0.343 0.386 0.385 0.407 0.385 0.360 0.336 % Correct 78.00% 79.00% 75.50% 74.25% 78.00% 77.50% 78.75% 78.25% 77.00% 75.00% Upper Integral Limit Hi 4.950 5.100 4.950 4.850 5.150 5.000 5.000 4.950 5.050 4.950 V Hi 6.188 6.375 6.188 6.063 6.438 6.250 6.250 6.188 6.313 6.188 X Hi 7.425 7.650 7.425 7.275 7.725 7.500 7.500 7.425 7.575 7.425 Welfare Measures Median WTP 3.174 3.247 3.125 3.126 3.172 3.153 3.184 3.090 3.063 3.131 Mean WTP Hi 3.153 3.223 3.107 3.094 3.148 3.135 3.168 3.074 3.043 3.096 V Hi 3.174 3.246 3.125 3.124 3.172 3.153 3.184 3.090 3.064 3.130 X_Hi 3.176 3.249 3.127 3.129 3.176 3.155 3.185 3.092 3.066 3.135 True WTP 3.229 3.282 3.258 3.201 3.282 3.248 3.259 3.238 3.251 3.255 Representative Agent Log-Linear Utility Model P -17.672 -16.582 -17.790 -16.925 -17.311 -16.514 -18.603 -18.832 -16.518 -15.382 s.e. 1.745 1.613 1.823 1.724 1.670 1.646 1.837 1.872 1.654 1.536 a 5.618 5.404 5.574 5.311 5.499 5.217 5.926 5.851 5.070 4.836 s.e. 0.576 0.537 0.599 0.566 0.551 0.534 0.608 0.606 0.536 0.505 Log-Likelihood -172.2 -173.3 -184.0 -184.5 -166.3 -180.4 -169.3 -170.1 -182.0 -189.9 Maddala R-Square 0.407 0.405 0.368 0.368 0.423 0.384 0.414 0.407 0.374 0.352 % Correct 79.25% 79.00% 76.50% 77.25% 79.75% 78.50% 78.25% 79.00% 77.75% 76.50% Upper Integral Limit Hi 4.950 5.100 4.950 4.850 5.150 5.000 5.000 4.950 5.050 4.950 V Hi 6.188 6.375 6.188 6.063 6.438 6.250 6.250 6.188 6.313 6.188 X Hi 7.425 7.650 7.425 7.275 7.725 7.500 7.500 7.425 7.575 7.425 Welfare Measures Median WTP 3.184 3.264 3.138 3.143 3.182 3.164 3.190 3.112 3.074 3.149 Mean WTP Hi 3.161 3.238 3.118 3.114 3.165 3.139 3.174 3.097 3.055 3.114 V Hi 3.183 3.263 3.137 3.141 3.182 3.164 3.190 3.111 3.075 3.147 X Hi 3.185 3.266 3.140 3.145 3.184 3.167 3.191 3.112 3.077 3.153 True WTP 3.229 3.282 3.258 3.201 3.282 3.248 3.259 3.238 3.251 3.255 Representative Agent Page 105 Regression Summary Samples 61 to 100 AR = 10% 81 82 83 84 85 86 87 88 89 90 Linear Utility Model P -1.862 -1.532 -1.792 -1.701 -1.676 -1.842 -1.952 -1.473 -1.968 -1.563 s.e. 0.186 0.156 0.179 0.173 0.165 0.185 0.198 0.148 0.201 0.166 a 5.811 4.838 5.727 5.188 5.326 5.990 6.109 4.761 6.077 4.885 s.e. 0.608 0.514 0.589 0.555 0.548 0.618 0.645 0.491 0.635 0.547 Log-Likelihood -171.6 -191.7 -182.1 -186.6 -179.2 -174.2 -176.0 -195.2 -178.8 -199.6 Maddala R-Square 0.403 0.348 0.379 0.358 0.384 0.402 0.394 0.336 0.387 0.317 % Correct 77.00% 76.25% 78.00% 75.75% 79.00% 80.25% 77.75% 76.50% 77.25% 72.50% Upper Integral Limit Hi 5.050 5.000 5.000 4.900 5.000 5.100 5.050 4.950 4.900 5.000 V Hi 6.313 6.250 6.250 6.125 6.250 6.375 6.313 6.188 6.125 6.250 X_Hi 7.575 7.500 7.500 7.350 7.500 7.650 7.575 7.425 7.350 7.500 Welfare Measures Median WTP 3.120 3.158 3.195 3.049 3.178 3.252 3.130 3.232 3.088 3.125 Mean WTP Hi 3.107 3.125 3.176 3.028 3.153 3.235 3.113 3.186 3.075 3.097 V Hi 3.120 3.157 3.195 3.049 3.177 3.251 3.129 3.229 3.088 3.125 X Hi 3.122 3.162 3.196 3.052 3.180 3.253 3.131 3.237 3.089 3.129 True WTP 3.287 3.259 3.281 3.229 3.265 3.308 3.22 3.241 3.217 3.274 Representative Agent Log-Linear Utility Model P -16.649 -15.112 -19.202 -17.261 -16.846 -19.941 -19.515 -15.399 -20.360 -14.719 s.e. 1.661 1.523 1.906 1.725 1.650 1.990 1.959 1.533 2.055 1.604 a 5.184 4.779 6.160 5.285 5.352 6.495 6.094 4.977 6.292 4.591 s.e. 0.543 0.501 0.627 0.555 0.545 0.665 0.633 0.508 0.647 0.526 Log-Likelihood -178.4 -188.4 -170.5 -180.8 -172.1 -163.5 -167.8 -189.2 -169.1 -202.4 Maddala R-Square 0.383 0.358 0.414 0.376 0.405 0.433 0.419 0.356 0.416 0.307 % Correct 79.25% 75.25% 79.00% 77.50% 80.50% 81.00% 78.75% 75.75% 81.00% 74.25% Upper Integral Limit Hi 5.050 5.000 5.000 4.900 5.000 5.100 5.050 4.950 4.900 5.000 V Hi 6.313 6.250 6.250 6.125 6.250 6.375 6.313 6.188 6.125 6.250 X Hi 7.575 7.500 7.500 7.350 7.500 7.650 7.575 7.425 7.350 7.500 Welfare Measures Median WTP 3.118 3.167 3.213 3.067 3.182 3.262 3.127 3.237 3.095 3.124 Mean WTP Hi 3.098 3.132 3.197 3.046 3.158 3.250 3.111 3.197 3.084 3.089 V Hi 3.119 3.167 3.212 3.067 3.181 3.262 3.127 3.235 3.095 3.124 X Hi 3.121 3.172 3.214 3.069 3.184 3.263 3.128 3.241 3.096 3.130 True WTP 3.287 3.259 3.281 3.229 3.265 3.308 3.22 3.241 3.217 3.274 Representative Agent Page 106 Regression Summary Samples 51 to 100 AR = 10% 91 92 93 94 95 96 97 98 99 100 Linear Utility Model B -2.163 -1.639 -1.541 -1.673 -1.690 -1.931 -1.538 -1.920 -1.713 -1.751 s.e. 0.216 0.171 0.157 0.162 0.171 0.193 0.153 0.189 0.176 0.175 a 6.551 5.340 4.971 5.278 5.417 6.028 4.896 6.017 5.368 5.574 s.e. 0.675 0.582 0.516 0.534 0.560 0.622 0.516 0.616 0.562 0.575 Log-Likelihood -162.6 -190.8 -192.8 -173.8 -184.5 -172.5 -184.0 -167.3 -197.2 -183.4 Maddala R-Square 0.434 0.348 0.344 0.401 0.371 0.406 0.367 0.421 0.330 0.373 % Correct 78.50% 75.00% 73.25% 77.75% 77.25% 77.00% 75.50% 80.00% 74.00% 76.25% Upper Integral Limit Hi 4.900 5.100 5.000 5.000 5.050 5.000 5.100 5.100 4.850 4.950 V Hi 6.125 6.375 6.250 6.250 6.313 6.250 6.375 6.375 6.063 6.188 X Hi 7.350 7.650 7.500 7.500 7.575 7.500 7.650 7.650 7.275 7.425 Welfare Measures Median WTP 3.029 3.258 3.266 3.157 3.205 3.122 3.184 3.134 3.133 3.183 Mean WTP Hi 3.022 3.232 3.190 3.132 3.182 3.110 3.155 3.124 3.106 3.160 V Hi 3.029 3.258 3.225 3.155 3.204 3.122 3.184 3.134 3.132 3.182 X Hi 3.030 3.261 3.230 3.158 3.207 3.123 3.188 3.135 3.136 3.184 True WTP 3.221 3.311 3.261 3.262 3.271 3.275 3.285 3.269 3.236 3.262 Representative Agent Log-Linear Utility Model P -23.267 -15.482 -16.643 -17.430 -16.985 -20.063 -16.399 -19.273 -17.968 -17.102 s.e. 2.321 1.608 1.655 1.678 1.672 2.034 1.613 1.897 1.838 1.678 a 7.050 5.031 5.357 5.508 5.476 6.270 5.234 6.061 5.632 5.479 s.e. 0.719 0.543 0.539 0.553 0.551 0.654 0.540 0.618 0.585 0.556 Log-Likelihood -149.5 -190.2 -181.2 -165.7 -177.2 -167.8 -172.8 -163.5 -190.1 -180.8 Maddala R-Square 0.470 0.350 0.381 0.424 0.394 0.420 0.402 0.432 0.353 0.381 % Correct 81.00% 75.00% 76.25% 80.25% 77.00% 78.75% 78.50% 81.50% 74.25% 78.75% Upper Integral Limit Hi 4.900 5.100 5.000 5.000 5.050 5.000 5.100 5.100 4.850 4.950 V Hi 6.125 6.375 6.250 6.250 6.313 6.250 6.375 6.375 6.063 6.188 X Hi 7.350 7.650 7.500 7.500 7.575 7.500 7.650 7.650 7.275 7.425 Welfare Measures Median WTP 3.035 3.255 3.224 3.165 3.229 3.130 3.197 3.149 3.139 3.209 Mean WTP Hi 3.029 3.226 3.196 3.144 3.205 3.119 3.173 3.139 3.116 3.182 V Hi 3.035 3.254 3.223 3.164 3.228 3.130 3.196 3.149 3.138 3.207 X Hi 3.035 3.258 3.226 3.167 3.231 3.131 3.199 3.150 3.140 3.210 True WTP 3.221 3.311 3.261 3.262 3.271 3.275 3.285 3.269 3.236 3.262 Representative Agent Page 107 Regression Summary Samples 51 to 100 AR = 10% Standard Min Mean Max Error Linear Utility Model P -2.163 -1.726 -1.473 s.e. 0.148 0.173 0.216 a 4.737 5.429 6.551 s.e. 0.491 0.567 0.675 Log-Likelihood -199.6 -182.1 -160.1 Maddala R-Square 0.317 0.375 0.438 % Correct 72.50% 76.90% 80.75% Upper Integral Limit Hi 4.700 5.200 V Hi 5.875 6.500 X Hi 7.050 7.800 Welfare Measures Median WTP 3.029 3.148 3.271 0.0628 Mean WTP Hi 3.022 3.125 3.244 0.0582 V Hi 3.029 3.147 3.270 0.0611 X Hi 3.030 3.149 3.274 0.0616 True WTP 3.179 3.255 3.311 0.0311 Representative Agent 3.168 Log-Linear Utility Model Min Mean Max P -23.267 -17.543 -14.672 s.e. 1.462 1.749 2.321 a 4.591 5.530 7.050 s.e. 0.492 0.571 0.719 Log-Likelihood -202.4 -176.3 -149.5 Maddala R-Square 0.307 0.393 0.470 % Correct 74.25% 78.18% 82.50% Upper Integral Limit Hi 4.700 5.200 V Hi 5.875 6.500 X Hi 7.050 7.800 Welfare Measures Median WTP 3.035 3.159 3.305 0.0622 Mean WTP Hi 3.029 3.138 3.280 0.0594 V Hi 3.035 3.159 3.304 0.0618 X Hi 3.035 3.161 3.307 0.0623 True WTP 3.179 3.255 3.311 0.0311 Representative Agent 3.168 Page 108 Regression Summary Samples 101 to 150 AR = 16% Sample Number 101 102 103 104 105 106 107 108 109 110 Linear Utility Model P -1.095 -0.950 -1.248 -1.214 -1.294 -1.125 -1.255 -1.280 -1.063 -1.067 s.e. 0.109 0.096 0.122 0.122 0.131 0.112 0.124 0.129 0.107 0.108 a 5.128 4.355 5.795 5.690 5.972 5.225 5.680 6.107 4.850 4.791 s.e. 0.522 0.460 0.590 0.590 0.624 0.535 0.583 0.626 0.515 0.526 Log-Likelihood -187.0 -191.6 -163.0 -174.1 -167.7 -175.8 -173.6 -173.5 -185.1 -182.1 Maddala R-Square 0.363 0.348 0.433 0.402 0.422 0.397 0.403 0.405 0.366 0.363 % Correct 77.50% 74.00% 81.00% 78.00% 79.25% 78.25% 79.00% 78.25% 77.75% 77.75% Upper Integral Limit Hi 7.500 7.650 7.550 7.600 7.500 7.650 7.200 7.400 7.550 7.500 V Hi 9.375 9.563 9.438 9.500 9.375 9.563 9.000 9.250 9.438 9.375 X Hi 11.250 11.475 11.325 11.400 11.250 11.475 10.800 11.100 11.325 11.250 Welfare Measures Median WTP 4.683 4.585 4.645 4.685 4.616 4.644 4.527 4.771 4.564 4.490 Mean WTP Hi 4.641 4.542 4.626 4.665 4.600 4.619 4.502 4.747 4.532 4.461 V_Hi 4.682 4.589 4.645 4.686 4.617 4.645 4.527 4.771 4.566 4.493 X_Hi 4.688 4.597 4.647 4.688 4.618 4.648 4.529 4.773 4.570 4.497 True Agent WTP 4.780 4.821 4.809 4.816 4.751 4.830 4.690 4.795 4.794 4.754 Representative Agent Log-Linear Utility Model P -10.618 -9.589 -12.072 -11.795 -12.730 -11.365 -13.177 -12.339 -10.500 -10.967 s.e. 1.051 0.964 1.195 1.169 1.275 1.131 1.301 1.231 1.067 1.108 a 4.965 4.409 5.592 5.557 5.904 5.313 5.981 5.882 4.768 4.923 s.e. 0.502 0.462 0.571 0.570 0.611 0.545 0.613 0.598 0.505 0.534 Log-Likelihood -186.8 -188.7 -161.2 -172.8 -164.6 -171.5 -167.6 -174.1 -182.9 -177.0 Maddala R-Square 0.364 0.357 0.438 0.406 0.431 0.410 0.421 0.403 0.373 0.379 % Correct 77.00% 74.00% 80.75% 78.00% 79.75% 78.50% 78.50% 78.00% 79.25% 80.25% Upper Integral Limit Hi 7.500 7.650 7.550 7.600 7.500 7.650 7.200 7.400 7.550 7.500 V Hi 9.375 9.563 9.438 9.500 9.375 9.563 9.000 9.250 9.438 9.375 X Hi 11.250 11.475 11.325 11.400 11.250 11.475 10.800 11.100 11.325 11.250 Welfare Measures Median WTP 4.683 4.605 4.639 4.718 4.645 4.682 4.545 4.774 4.548 4.496 Mean WTP Hi 4.635 4.563 4.618 4.693 4.626 4.657 4.525 4.745 4.516 4.469 V Hi 4.682 4.609 4.639 4.718 4.645 4.683 4.545 4.773 4.550 4.498 X Hi 4.688 4.616 4.642 4.721 4.647 4.686 4.546 4.776 4.555 4.502 True Agent WTP 4.78 4.821 4.809 4.816 4.751 4.83 4.69 4.795 4.794 4.754 Representative Agent Page 109 Regression Summary Samples 101 to 150 AR = 16% Sample Number 111 112 113 114 115 116 117 118 119 120 Linear Utility Model a -1.130 -1.227 -1.101 -1.101 -1.095 -1.023 -1.217 -0.953 -1.238 -1.012 s.e. 0.113 0.119 0.108 0.110 0.113 0.107 0.120 0.098 0.123 0.103 a 5.212 5.678 5.014 5.131 5.148 4.667 5.509 4.579 5.758 4.729 s.e. 0.538 0.571 0.517 0.541 0.554 0.510 0.572 0.492 0.588 0.511 Log-Likelihood -172.4 -164.4 -174.3 -175.2 -189.9 -200.8 -166.5 -196.2 -172.2 -186.2 Maddala R-Square 0.408 0.430 0.398 0.397 0.352 0.317 0.412 0.331 0.409 0.361 % Correct 80.00% 79.75% 79.25% 78.00% 74.50% 72.75% 79.25% 76.75% 78.50% 77.50% Upper Integral Limit Hi 7.800 7.550 7.500 7.800 7.600 7.400 7.700 7.650 7.450 7.900 V Hi 9.750 9.438 9.375 9.750 9.500 9.250 9.625 9.563 9.313 9.875 X Hi 11.700 11.325 11.250 11.700 11.400 11.100 11.550 11.475 11.175 11.850 Welfare Measures Median WTP 4.611 4.627 4.553 4.659 4.702 4.563 4.525 4.805 4.650 4.671 Mean WTP Hi 4.592 4.607 4.524 4.636 4.670 4.520 4.511 4.748 4.627 4.642 V Hi 4.613 4.627 4.554 4.661 4.702 4.564 4.527 4.805 4.650 4.674 X Hi 4.615 4.629 4.558 4.664 4.706 4.571 4.528 4.814 4.652 4.678 True Agent WTP 4.837 4.795 4.797 4.859 4.873 4.750 4.854 4.833 4.778 4.909 Representative Agent Log-Linear Utility Model P -11.387 -11.703 -11.953 -11.146 -10.558 -11.590 -12.435 -9.680 -13.441 -10.070 s.e. 1.129 1.124 1.181 1.099 1.086 1.189 1.229 0.991 1.337 1.011 a 5.265 5.428 5.444 5.207 4.979 5.306 5.637 4.684 6.277 4.722 s.e. 0.536 0.544 0.561 0.540 0.534 0.561 0.585 0.503 0.641 0.502 Log-Likelihood -167.5 -164.3 -165.7 -168.3 -189.2 -185.8 -162.0 -192.4 -162.3 -181.5 Maddala R-Square 0.422 0.430 0.423 0.418 0.355 0.366 0.433 0.344 0.437 0.376 % Correct 81.25% 80.50% 81.50% 79.00% 77.25% 76.75% 82.75% 75.75% 81.25% 80.25% Upper Integral Limit Hi 7.800 7.550 7.500 7.800 7.600 7.400 7.700 7.650 7.450 7.900 V Hi 9.750 9.438 9.375 9.750 9.500 9.250 9.625 9.563 9.313 9.875 X Hi 11.700 11.325 11.250 11.700 11.400 11.100 11.550 11.475 11.175 11.850 Welfare Measures Median WTP 4.631 4.645 4.561 4.679 4.723 4.585 4.540 4.847 4.677 4.696 • Mean WTP Hi 4.612 4.621 4.540 4.656 4.685 4.557 4.527 4.789 4.660 4.666 V Hi 4.633 4.646 4.562 4.680 4.723 4.585 4.541 4.845 4.677 4.700 X Hi 4.635 4.649 4.564 4.683 4.729 4.590 4.542 4.854 4.678 4.704 True Agent WTP 4.837 4.795 4.797 4.859 4.873 4.75 4.854 4.833 4.778 4.909 Representative Agent Page 110 Regression Summary Samples 101 to 150 AR=16% Sample Number 121 122 123 124 125 126 127 128 129 130 Linear Utility Model P -1.031 -1.276 -0.905 -1.126 -1.042 -0.989 -1.107 -0.884 -1.046 -0.997 s.e. 0.105 0.126 0.093 0.115 0.102 0.106 0.110 0.091 0.107 0.100 a 4.871 5.997 4.238 5.218 5.051 4.645 5.013 4.083 4.795 4.678 s.e. 0.514 0.604 0.452 0.557 0.516 0.513 0.528 0.455 0.521 0.496 Log-Likelihood -185.9 -164.6 -197.2 -183.5 -174.0 -200.8 -175.7 -197.1 -182.0 -182.0 Maddala R-Square 0.367 0.431 0.329 0.371 0.403 0.318 0.394 0.323 0.372 0.377 % Correct 76.50% 77.75% 73.50% 77.00% 79.75% 72.75% 76.50% 73.00% 76.00% 75.25% Upper Integral Limit Hi 7.750 7.500 7.750 7.550 8.000 7.650 7.750 7.900 7.900 7.850 V Hi 9.688 9.375 9.688 9.438 10.000 9.563 9.688 9.875 9.875 9.813 X Hi 11.625 11.250 11.625 11.325 12.000 11.475 11.625 11.850 11.850 11.775 Welfare Measures Median WTP 4.726 4.700 4.681 4.634 4.846 4.695 4.529 4.620 4.586 4.692 Mean WTP Hi 4.691 4.680 4.630 4.606 4.817 4.652 4.510 4.578 4.565 4.659 V Hi 4.727 4.699 4.685 4.635 4.849 4.697 4.532 4.628 4.590 4.695 X Hi 4.732 4.700 4.695 4.638 4.852 4.704 4.534 4.637 4.594 4.701 True Agent WTP 4.817 4.808 4.885 4.829 4.927 4.849 4.831 4.931 4.937 4.901 Representative Agent Log-Linear Utility Model P -10.573 -13.147 -9.250 -11.030 -10.219 -8.990 -10.822 -8.953 -10.472 -10.747 s.e. 1.065 1.284 0.931 1.100 0.997 0.980 1.077 0.930 1.058 1.080 a 4.988 6.178 4.340 5.129 4.957 4.219 4.903 4.132 4.808 5.032 s.e. 0.516 0.616 0.452 0.534 0.502 0.472 0.516 0.459 0.516 0.531 Log-Likelihood -178.4 -160.3 -189.6 -180.2 -172.9 -203.5 -175.7 -194.7 -177.5 -174.5 Maddala R-Square 0.360 0.443 0.354 0.382 0.407 0.308 0.394 0.331 0.386 0.400 % Correct 78.50% 79.25% 74.75% 80.50% 77.75% 73.25% 80.00% 74.00% 79.50% 78.25% Upper Integral Limit Hi 7.750 7.500 7.750 7.550 8.000 7.650 7.750 7.900 7.900 7.850 V Hi 9.688 9.375 9.688 9.438 10.000 9.563 9.688 9.875 9.875 9.813 X Hi 11.625 11.250 11.625 11.325 12.000 11.475 11.625 11.850 11.850 11.775 Welfare Measures Median WTP 4.725 4.706 4.699 4.657 4.858 4.700 4.538 4.622 4.598 4.689 Mean WTP Hi 4.693 4.689 4.650 4.626 4.826 4.640 4.516 4.582 4.576 4.664 V Hi 4.726 4.706 4.702 4.658 4.859 4.702 4.541 4.630 4.602 4.691 X Hi 4.730 4.708 4.711 4.662 4.864 4.714 4.544 4.638 4.606 4.695 True Agent WTP 4.817 4.808 4.885 4.829 4.927 4.849 4.831 4.931 4.937 4.901 Representative Agent Page 111 Regression Summary Samples 101 to 150 AR=16% Sample Number 131 132 133 134 135 136 137 138 139 140 Linear Utility Model B -0.977 -1.066 -1.078 -1.046 -1.105 -0.887 -1.158 -1.270 -1.070 -1.044 s.e. 0.100 0.105 0.108 0.106 0.114 0.089 0.115 0.126 0.109 0.101 a 4.568 5.005 4.940 4.796 5.196 4.224 5.364 5.898 4.952 4.775 s.e. 0.492 0.526 0.523 0.520 0.551 0.461 0.564 0.607 0.543 0.495 Log-Likelihood -185.3 -166.4 -179.2 -185.3 -180.3 -194.0 -171.6 -154.2 -191.2 -175.3 Maddala R-Square 0.366 0.421 0.384 0.359 0.384 0.336 0.407 0.457 0.347 0.393 % Correct 76.75% 82.00% 75.75% 75.00% 74.00% 75.00% 77.75% 80.75% 75.25% 78.25% Upper Integral Limit Hi 8.000 8.100 7.650 7.650 7.800 7.750 7.850 7.900 7.400 7.650 V Hi 10.000 10.125 9.563 9.563 9.750 9.688 9.813 9.875 9.250 9.563 X Hi 12.000 12.150 11.475 11.475 11.700 11.625 11.775 11.850 11.100 11.475 Welfare Measures Median WTP 4.676 4.696 4.583 4.583 4.703 4.761 4.634 4.643 4.630 4.576 Mean WTP Hi 4.648 4.678 4.556 4.553 4.679 4.700 4.617 4.632 4.589 4.546 V Hi 4.681 4.700 4.585 4.586 4.704 4.763 4.636 4.644 4.629 4.579 X Hi 4.686 4.702 4.589 4.590 4.707 4.774 4.638 4.645 4.635 4.583 True Agent WTP 4.926 4.918 4.809 4.876 4.849 4.880 4.885 4.878 4.768 4.800 Representative Agent Log-Linear Utility Model B -10.046 -10.410 -11.375 -10.226 -11.666 -8.741 -12.407 -12.626 -10.433 -10.807 s.e. 1.017 1.021 1.126 1.044 1.181 0.893 1.238 1.241 1.058 1.060 a 4.708 4.884 5.250 4.694 5.479 4.155 5.735 5.870 4.821 4.952 s.e. 0.502 0.509 0.549 0.511 0.568 0.457 0.599 0.597 0.516 0.514 Log-Likelihood -179.4 -165.4 -169.3 -183.2 -172.4 -195.1 -161.9 -150.4 -190.0 -170.3 Maddala R-Square 0.385 0.424 0.413 0.365 0.408 0.332 0.435 0.467 0.351 0.409 % Correct 77.50% 82.75% 76.75% 76.75% 75.75% 75.00% 81.00% 82.25% 76.25% 80.50% Upper Integral Limit Hi 8.000 8.100 7.650 7.650 7.800 7.750 7.850 7.900 7.400 7.650 V Hi 10.000 10.125 9.563 9.563 9.750 9.688 9.813 9.875 9.250 9.563 X Hi 12.000 12.150 11.475 11.475 11.700 11.625 11.775 11.850 11.100 11.475 Welfare Measures Median WTP 4.694 4.699 4.622 4.597 4.704 4.761 4.630 4.657 4.628 4.589 #REF! Mean WTP #REF! Hi 4.667 4.679 4.599 4.564 4.685 4.697 4.617 4.646 4.584 4.562 V Hi 4.698 4.703 4.623 4.600 4.705 4.763 4.631 4.657 4.628 4.591 X Hi 4.702 4.706 4.626 4.605 4.707 4.776 4.632 4.659 4.635 4.595 True Agent WTP 4.926 4.918 4.809 4.876 4.849 4.88 4.885 4.878 4.768 4.8 Representative Agent Page 112 Regression Summary Samples 101 to 150 AR = 16% Sample Number 141 142 143 144 145 146 147 148 149 150 Linear Utility Model P -0.857 -1.105 -1.058 -1.806 -1.136 -1.171 -1.117 -1.037 -1.063 -1.089 s.e. 0.088 0.111 0.108 0.109 0.112 0.115 0.113 0.107 0.110 0.106 a 4.099 5.102 4.826 5.154 5.289 5.311 5.157 4.839 4.786 5.093 s.e. 0.452 0.533 0.521 0.530 0.557 0.546 0.533 0.520 0.522 0.513 Log-Likelihood -196.9 -176.4 -189.3 -185.4 -171.3 -169.8 -175.6 -188.0 -188.8 -173.5 Maddala R-Square 0.328 0.394 0.353 0.368 0.405 0.415 0.399 0.358 0.352 0.405 % Correct 74.50% 77.75% 74.25% 75.50% 78.00% 78.50% 77.75% 76.00% 74.50% 78.00% Upper Integral Limit Hi 8.050 7.800 7.650 7.500 7.550 7.550 7.850 7.650 7.500 7.750 V Hi 10.063 9.750 9.563 9.375 9.438 9.438 9.813 9.563 9.375 9.688 X Hi 12.075 11.700 11.475 11.250 11.325 11.325 11.775 11.475 11.250 11.625 Welfare Measures Median WTP 4.783 4.616 4.560 4.745 4.656 4.535 4.618 4.665 4.500 4.676 Mean WTP Hi 4.733 4.595 4.533 4.705 4.628 4.515 4.599 4.630 4.470 4.649 V Hi 4.790 4.619 4.563 4.745 4.656 4.537 4.620 4.667 4.503 4.677 X Hi 4.800 4.621 4.567 4.750 4.660 4.539 4.623 4.672 4.507 4.681 True Agent WTP 4.907 4.863 4.846 4.786 4.809 4.793 4.873 4.821 4.764 4.842 Representative Agent Log-Linear Utility Model P -8.826 -11.739 -11.460 -11.395 -11.955 -11.848 -12.202 -10.311 -11.005 -11.574 s.e. 0.902 1.181 1.161 1.143 1.188 1.164 1.233 1.028 1.137 1.114 a 4.235 5.450 5.252 5.401 5.585 5.376 5.638 4.832 4.949 5.439 s.e. 0.461 0.570 0.558 0.550 0.589 0.552 0.583 0.503 0.537 0.541 Log-Likelihood -191.3 -168.3 -177.7 -175.2 -164.0 -165.7 -165.7 -183.1 -183.2 -163.3 Maddala R-Square 0.347 0.418 0.389 0.400 0.427 0.427 0.428 0.374 0.370 0.434 % Correct 75.50% 79.75% 76.50% 76.75% 80.75% 79.50% 77.25% 76.75% 78.50% 80.50% Upper Integral Limit Hi 8.050 7.800 7.650 7.500 7.550 7.550 7.850 7.650 7.500 7.750 V Hi 10.063 9.750 9.563 9.375 9.438 9.438 9.813 9.563 9.375 9.688 X Hi 12.075 11.700 11.475 11.250 11.325 11.325 11.775 11.475 11.250 11.625 Welfare Measures Median WTP 4.805 4.650 4.590 4.747 4.679 4.544 4.627 4.693 4.504 4.707 Mean WTP Hi 4.758 4.633 4.568 4.714 4.655 4.524 4.614 4.656 4.477 4.685 V Hi 4.811 4.651 4.591 4.747 4.679 4.545 4.629 4.694 4.506 4.708 X Hi 4.820 4.654 4.594 4.751 4.682 4.548 4.630 4.700 4.510 4.710 True Agent WTP 4.907 4.863 4.846 4.786 4.809 4.793 4.873 4.821 4.764 4.842 Representative Agent Page 113 Regression Summary Samples 101 to 150 AR = 16% Standard Sample Number Min Mean Max Error Linear Utility Model P -1.806 -1.105 -0.857 s.e. 0.088 0.110 0.131 a 4.083 5.060 6.107 s.e. 0.452 0.533 0.626 Log-Likelihood -200.8 -180.2 -154.2 Maddala R-Square 0.317 0.381 0.457 % Correct 72.75% 77.01% 82.00% Upper Integral Limit Hi 7.200 8.100 V Hi 9.000 10.125 X Hi 10.800 12.150 Welfare Measures Median WTP 4.490 4.642 4.846 0.0792 Mean WTP Hi 4.461 4.613 4.817 0.0757 V Hi 4.493 4.644 4.849 0.0791 X Hi 4.497 4.649 4.852 0.0799 True Agent WTP 4.690 4.835 4.937 0.0548 Representative Agent 4.671 Log-Linear Utility Model P -13.441 -11.087 -8.741 s.e. 0.893 1.110 1.337 a 4.132 5.153 6.277 s.e. 0.452 0.538 0.641 Log-Likelihood -203.5 -175.4 -150.4 Maddala R-Square 0.308 0.395 0.467 % Correct 73.25% 78.45% 82.75% Upper Integral Limit Hi 7.200 8.100 V Hi 9.000 10.125 X Hi 10.800 12.150 Welfare Measures Median WTP 4.496 4.657 4.858 0.0806 Mean WTP Hi 4.469 4.628 4.826 0.0761 V Hi 4.498 4.658 4.859 0.0803 X Hi 4.502 4.662 4.864 0.0814 True Agent WTP 4.690 4.835 4.937 0.0548 Representative Agent 4.671 Page 114 

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