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# An Examination of Utility Consistent Approaches to Modeling Corner Solutions in Recreation Demand

- Abstract
Objectives/Hypothesis: Recreation demand models are used extensively to value existing recreation facilities, as well policies that change the quality attributes and number of recreation sites. However, currently popular modeling frameworks such as those that link a site selection model with a separate participation equation, while providing useful insight into the recreation usage, are hampered by the lack of an underlying utility theoretic framework. The Kuhn-Tucker model, as well as its dual counterpart, provides a promising alternative approach that unifies the site selection and participation decisions in a utility theoretic framework. These models provide a consistent treatment from the behavioral model to the estimating equations. Unfortunately, there has been only limited experience with either of these models to date. The proposed research project is aimed at expanding both our understanding of these models and procedures that will facilitate their use in applied welfare analysis. Approach: The proposed research project will: --Investigate modeling, specification, and econometric issues associated with utility consistent corner solution models in recreation demand. --Compare traditional approaches of modeling recreation demand to the utility consistent methods. --Apply these utility consistent methods to data sets describing recreation use of the Wisconsin Great Lakes Region and Iowa Wetlands. Expected Results: It is anticipated that the research will provide three specific benefits. First, by relaxing both the functional form used to represent individual preferences and the error structure employed to capture variation in preferences, the investigators will expand the range of preferences that the Kuhn-Tucker model can accurately reflect. Second, the investigators' Monte Carlo simulation efforts will provide a better understanding of the tradeoffs between established methods for modeling recreation demand, which rely on standard econometric techniques, and the utility consistent approaches, which present greater (but feasible) challenges in terms of estimation. The simulation exercise will also provide insight into the robustness of Kuhn-Tucker and dual models to changes in modeling assumptions. Finally, while Monte Carlo exercises are useful in examining the performance of a model or estimator over a range of underlying conditions and in a controlled setting, they are often criticized for avoiding real world problems. The application of both the Kuhn-Tucker and dual models to two data sets the will provide insights into the problems encountered in actual valuation settings. The wetland valuation effort, in particular, is of current policy relevance, given proposed federal and state changes to the number and quality of wetlands in the Midwest.
- Metadata
**EPA/NSF ID:**R826615 **Principal Investigators:**Herriges, Joseph A.

Kling, Catherine L.

Phaneuf, Daniel J.**Technical Liaison:****Research Organization:**Iowa State University **Funding Agency/Program:**EPA/ORD/Valuation **Grant Year:**1998 **Project Period:**September 1, 1998 to August 31, 2000 **Cost to Funding Agency:**$134,759 - Project Reports
**Final****Objective:**

Recreation demand models are used extensively to value existing recreation facilities, as well policies that change the quality attributes and number of recreation sites. However, the current popular modeling frameworks, such as those that link a site selection model with a separate participation equation, while providing useful insight into the recreation usage, are hampered by the lack of an underlying utility theoretic framework. The Kuhn-Tucker (KT) model, as well as its dual counterpart, provides a promising alternative approach that unifies the site selection and participation decisions in a utility theoretic framework. These models provide a consistent treatment from the behavioral model to the estimating equations. Unfortunately, there has been only limited experience with either of these models to date. This research project was aimed at expanding both our understanding of these models and procedures that will facilitate their use in applied welfare analysis.**Summary/Accomplishments:**

Progress over the course of this research project can be best divided along the lines of the three initial project objectives.__Objective 1__: Investigate modeling, specification, and econometric issues associated with utility consistent corner solution models in recreation demand.Several lines of research fall within this objective. First, one of the initial concerns regarding the KT model was its complexity. This was not only in terms of the estimation of such models, but also in using them to derive the welfare implications of changing environmental conditions. Prior to this research project, applications of the KT framework were limited to problems involving a small number of commodities (i.e., four or fewer). During the course of this research project, we developed software routines allowing for the estimation of KT models for a much larger number of commodities. Herriges and Phaneuf, for example, estimate a model of recreational wetland usage with 15 sites. More recently, von Haefen, Phaneuf, and Parson have extended this boundary by estimating a 62-site model of the demand for trips to mid-Atlantic beaches by Delaware residents. Phaneuf and Siderelis estimate an eight-site model of coastal paddling trips in eastern North Carolina. This latter paper is of particular practical use in that it not only provides an application that is interest in itself, but also a tutorial on the estimation of KT models, including the requisite GAUSS code.

Second, Phaneuf, Kling, and Herriges generalize the error structure used in the KT model from

*iid*extreme value errors to a generalized extreme value (GEV) error vector. This allows the analyst to specify patterns of correlation among sites in the choice set, much like the nested logit models used in random utility models of site selection. The estimation of KT models with GEV errors is not significantly more complex. However, welfare calculations are made more difficult by this type of error specification. Phaneuf, Kling, and Herriges provide an algorithm for computing the welfare impact of changing site characteristics and availability. The error structures employed with the KT model can be further generalized to allow for more complex patterns of correlations among the available sites using simulation estimation techniques analogous to those employed in the mixed logit models of McFadden and Train (2000). Mixing distributions also can be used to allow for unobserved heterogeneity in consumer preferences. Specifically, the parameters of the assumed utility function can vary across individuals. This random parameters specification is employed in von Haefen, Phaneuf, and Parsons. The welfare estimates obtained from this more general specification are uniformly smaller than those obtained from a fixed coefficients specification.Third, we have extended our analysis of the welfare measures revealed within the KT modeling framework. In standard Random Utility Maximization (or RUM) models, the assumption of weak complementarity is employed to justify the corresponding welfare calculations. Weak complementarity implies that changes to the quality attributes of a site only impact individual visiting that site. While weak complementarity is implicitly imposed in the standard RUM model, its role within the KT framework is less transparent. In Herriges, Kling, and Phaneuf, we examine the components of welfare that can be extracted from KT models, identifying both the components of value that are "revealable" from behavioral data and potential sources of bias in their estimation. Our empirical analysis on Iowa wetland usage indicates that welfare estimates derived from KT models can differ by an order of magnitude, depending upon the assumed source of violations of weak complementarity.

Fourth, in Phaneuf, we have investigated alternative approaches to incorporating the opportunity cost of time within the KT framework, including the use of: (1) a dual constraint model (Bockstael, et al., 1987); (2) full income and full prices via the assumption of weak separability (Larson, 1990); and (3) a conditional demand interpretation (Shaw, et al. 1999), as well as the traditional approach of relying upon a fixed fraction of the wage rate. As one would expect, the competing approaches produce statistically different welfare estimates, requiring a choice by the analyst on the preferred approach. In an empirical application using data on Iowa wetland usage, the dual constraint model from Bockstael, et al. (1987) appears to have the most intuitive appeal, producing welfare estimates for a statewide 20 percent increase in pheasant counts of $22/per season, along with intuitive and statistically significant coefficient estimates.

Fifth, all of the KT models used in our papers employ variations of the Linear Expenditure System (LES) for the consumer's direct utility function. We had initially planned to employ more general functional forms, such as the Tranlog or Almost Ideal Demand systems. Unfortunately, our efforts along these lines were not successful. The fundamental problem in the primal form of the model is that globally convex preferences are required to insure a single crossing point between consumption and nonconsumption, which is needed to specify the likelihood function for use in estimation. This precludes the use of flexible forms in the primal problem. However, we continue to work on the use of more flexible forms in the dual version of the KT model. However, some progress on generalizing the functional form has been made. Von Haefen and Phaneuf provide variations on the basic LES system using repackaging parameters discussed in Willig (1978). Von Haefen, Phaneuf, and Parsons expand this by introducing a translated CES specification that includes the LES model as a special case.

Sixth, Phaneuf and Herriges use the Kuhn-Tucker model to investigate issues in the definition of choice sets in modeling recreation demand. Because of its integrated nature, the KT model is well suited to investigating the impact that alternatives site set definitions may have on both where individuals recreate and the numbers of trips they take. Using data from the Iowa Wetlands Survey, we examined the effects on both parameter estimates and welfare measures of choice sets representing various levels of site aggregation and market scope. As in prior studies based on discrete choice RUM specifications, we find the specification of choice set scope and site aggregation significantly impact welfare measures. For example, even in the relatively homogeneous Prairie Pothole Region of northern Iowa, site aggregation reduces by more than 60 percent of the welfare gain predicted to arise from increasing the pheasant population in the region.

Seventh, and finally, we expand on an idea presented in Crooker and Kling (2000) that suggests that revealed preference data can be combined with stated preference data to produced bounds on the willingness-to-pay for price changes in a recreation effectively. Our insight is to consider the bounds, which contain both RP and SP data, as prior information in the construction of parametric models. Crooker and Phaneuf presented preliminary work for demonstration purposes using the Iowa Wetlands data. The research suggests that these nonparametric bounds may provide a useful basis for model selection.

__Objective 2__: Compare traditional approaches of modeling recreation demand to the utility consistent methods.Two papers provide comparisons between the KT modeling framework and traditional approaches to modeling recreation demand. Herriges, Kling, and Phaneuf provide a comparison between KT models and the traditional linked model of recreation demand. In the linked model, the site selection decision is modeled separately from the decision regarding the overall numbers of trips taken. The paper provides a conceptual comparison between the KT and linked approaches, emphasizing the utility theoretic and integrated nature of the KT model. We also provide a comparison in an empirical setting using data on recreational angling trips to the Wisconsin Great Lakes region. In general, the welfare estimates obtained using the KT model are found to be more stable across empirical specifications than the linked model.

Von Haefen and Phaneuf provide a comparison between the KT modeling approach and count data modeling systems. Again, the paper provides both a comparison of the conceptual underpinnings of the two modeling frameworks and a comparison in an empirical setting, this time using data from the Iowa Wetlands survey. As the paper notes, the conceptual differences between the models stem primarily from what the analyst specifies in each framework. With continuous KT models, the analyst specifies the individual's demand functions up to an unobserved heterogeneity vector, but with count data models, the analyst specifies the expectation of consumer demands. These differences have implications for the development of empirical models. The continuous models use virtual prices to integrate the intensive and extensive margins of choices (i.e., how many trips to take and whether to take any trips at all) within a consistent behavioral framework. In contrast, the count data models give the analyst some flexibility in allowing for additional factors to influence the extensive margin site selection decision that do not enter the intensive margin derived demand decisions. In both frameworks, the preference functions ultimately recovered incorporate all factors that enter the individual's intensive and extensive margins of choice. Welfare estimates derived from the alternative framework, however, have different interpretations. For continuous KT models, the analyst recovers an estimate of the individual's Hicksian consumer surplus arising from changing environmental conditions, whereas for count models, a representative consumer's welfare is recovered. The corresponding empirical analysis found qualitatively different welfare implications arising from the two modeling frameworks. In large part, this appears to stem from how quality attributes are incorporated into each of the models and how preference heterogeneity is considered.

__Objective 3__: Apply these utility consistent methods to data sets describing recreation use of the Wisconsin Great Lakes Region and Iowa Wetlands.While our original proposal had indicated that only two data sets would be used in empirical applications of the KT framework, four data sets in all have been used.

First, the Iowa Wetlands database was used. These studies vary in terms of the specific policy scenario being considered and the specific empirical model being considered, but we consistently find a significant and substantial welfare gain associated with preserving and/or improving wetland regions within the state. For example, Herriges, Kling, and Phaneuf find that a 20 percent increase in the pheasant counts within the state would correspond to a compensating variation of $88 per resident, whereas Phaneuf and Herriges estimate a $321 surplus loss associated with the loss of the Prairie Pothole wetlands.

Second, data from the Wisconsin Great Lakes Region was used in some empirical analyses. Richard Bishop and Audrey Lyke from the University of Wisconsin-Madison collected these data. The surveys provided detailed information on the 1989 angling behavior of Wisconsin fishing-license holders. The analysis in Phaneuf, Kling, and Herriges provides estimates of recreation demand for a variety of policy scenarios associated with changing the stocking practices within the Wisconsin Great Lakes Region and efforts to reduce toxin levels within the lakes. For example, we consider the elimination of programs stocking Lake Trout in Lake Michigan. The loss of lake trout is not found to significantly reduce the welfare of anglers. However, a similar loss of Coho salmon in the region is estimated to reduce welfare by over $300 per season per license holder. A 20 percent reduction in lake toxins is estimated to increase welfare by nearly $110.

Additional empirical analyses of the demand for trips to beaches along the mid-Atlantic region (in von Haefen, Phaneuf, and Parsons) and for coastal paddling trips along the North Carolina shore (in Phaneuf and Siderelis) have been conducted as well using the KT framework. However, as these analyses are still working papers, we do not present specific welfare results in this summary.

**Publications and Presentations:**von Haefen R, Phaneuf DJ. Estimating preferences for outdoor recreation: a comparison of continuous and count data demand systems. Journal of Environmental Economics and Management. 2003, Volume: 45, Number: 3 (MAY), Page: 612-630. Phaneuf DJ, Kling CL, Herriges JA. Estimation and welfare calculations in a generalized corner solution model with an application to recreation demand. The Review of Economics and Statistics 2000;82(1):83-92. Herriges JA, Kling CL, Phaneuf DJ. What's the use? Welfare estimates from revealed preference models when weak complementarity does not hold. Journal of Environmental Economics and Management. **Supplemental Keywords:**

*corners, Kuhn-Tucker KT, wetlands, cost benefit, recreation demand.*, Economic, Social, & Behavioral Science Research Program, Geographic Area, RFA, Scientific Discipline, Ecology and Ecosystems, Economics, Economics & Decision Making, Social Science, State, decision-making, IOWA (IA), Monte Carlo study, Wisconsin (WI), behavior change, behavior model, belief system, corner solution, decision analysis, decision making, econometric analysis, ecosystem valuation, environmental policy, environmental values, non-market valuation, recreational demand, utility consistent approaches - Project Status Reports
For the Year 1999

**Objective:**Recreation demand models are used extensively to value existing recreation facilities, as well as policies that change the quality attributes and number of recreation sites. However, currently popular modeling frameworks, such as those that link a site selection model with a separate participation equation while providing useful insight into the recreation usage, are hampered by the lack of an underlying utility theoretical framework. The Kuhn-Tucker model, as well as its dual counterpart, provides a promising alternative approach that unifies the site selection and participation decisions in a utility theoretical framework. These models provide a consistent treatment from the behavioral model to the estimating equations. Unfortunately, there has been only limited experience with either of these models to date. This research project is aimed at expanding both our understanding of these models and the procedures that will facilitate their use in applied welfare analysis.**Progress Summary:**Progress on this project over the past reporting period can be divided into four areas. First, we have extended the Kuhn-Tucker model in a number of directions, allowing for: (1) a broader range of correlation patterns among the available recreation sites (using GEV errors), (2) a larger number of sites that can feasibly be modeled (15 versus 4 in our earlier work), and (3) more general preference structures. Computer code has been developed in GAUSS to implement both the estimation of these models and the calculation of welfare effects. Second, much of the existing literature relies on the assumption of weak complementarity (either implicitly or explicitly) to justify the welfare estimates derived from recreation demand models. As the Kuhn-Tucker model does not require the imposition of weak complementarity, the question arises as to what types of welfare are revealed within this framework. We have examined the components of welfare that can be extracted from the Kuhn-Tucker models, identifying both the components of value that are "revealable" from behavior data and potential sources of bias in their estimation. Third, we have estimated the Kuhn-Tucker model using data for both the Wisconsin Great Lakes Region and the Iowa Wetlands, providing estimates of the gains and losses to various policy scenarios (detailed in the publications list). For example, a 20 percent reduction in toxins in the Wisconsin Great Lakes region would result in a welfare improvement on the order of $90 per person per season among the population of anglers. Fourth, we have used the Kuhn-Tucker framework to examine issues of choice set definition (e.g., site aggregation and market scope) in modeling recreation demand. We find that significant differences in welfare measures arise from changing the choice set specification.**Future Activities:**Our future activities will center on further pursuing Objectives 1 and 2 set forth in the project proposal.**Objective 1:**Investigate modeling, specification, and econometric issues associated with utility consistent corner solution models in recreation demand. In particular, we plan to consider various flexible functional forms, the use of index number theory to test for and specify sites and their associated prices, and further generalizations of the error structure. We also will investigate ways of better modeling the opportunity cost of time within the Kuhn-Tucker framework.

**Objective 2:**Compare traditional approaches of modeling recreation demand to the utility consistent methods. We already have pursued this objective using data from the Wisconsin Great Lakes region. In the future, we likely will focus on data from the Iowa Wetland Survey, as well as simulated data, for this task.

**Publications and Presentations:**Total Count: 5

__Type____Citation____Journal Searches__Journal Article Phaneuf DJ. A dual approach to modeling corner solutions in recreation demand. Journal of Environmental Economics and Management 1999;37:85-105. *not available*Journal Article Herriges JA, Phaneuf DJ. Choice set definition issues in a Kuhn-Tucker model of recreation demand. Marine Resource Economics 1999;14:343-355. *not available*Journal Article Phaneuf DJ, Kling CL, Herriges JA. Estimation and welfare calculations in a generalized corner solution model with an application to recreation demand. The Review of Economics and Statistics 2000;82(1):83-92. Journal Article Kling CL, Phaneuf DJ. Valuing water quality improvements using revealed preference methods when corner solutions are present. American Journal of Agricultural Economics 1998;80(5):1025-1031. Presentation Herriges JA, Kling CL, Phaneuf DJ. What's the use? Welfare estimates from revealed preference models when weak complementarity does not hold. Presented at the Allied Social Science Association Annual Meeting, Boston, MA, January 6-9, 2000.

**Objective:**Recreation demand models are used extensively to value existing recreation facilities, as well as policies that change the quality attributes and number of recreation sites. However, currently popular modeling frameworks such as those that link a site selection model with a separate participation equation, while providing useful insight into the recreation usage, are hampered by the lack of an underlying utility theoretic framework. The Kuhn-Tucker model, as well as its dual counterpart, provides a promising alternative approach that unifies the site selection and participation decisions in a utility theoretic framework. These models provide a consistent treatment from the behavioral model to the estimating equations. Unfortunately, there has been only limited experience with either of these models to date. This research project is aimed at expanding both our understanding of these models and procedures that will facilitate their use in applied welfare analysis.**Progress Summary:**Progress on this project over the past reporting period can be divided along the lines of objectives 1 and 2 of the project. Several lines of research fall within objective 1, which focuses on model specification and econometric issues associated with utility consistent corner solution models in recreation demand. First, we have extended our analysis of the welfare measures revealed within the Kuhn-Tucker modeling framework. As noted last year, the Kuhn-Tucker model does not require the imposition of weak complementarity, raising both the possibility of rejecting this restriction in estimation and the question as to what components of welfare are revealed by a non-weakly complementarity model. We have examined the components of welfare that can be extracted from KT models, identifying both the components of value that are "revealable" from behavioral data and potential sources of bias in their estimation. Our empirical analysis on Iowa wetland usage indicates that welfare estimates derived from KT models can differ by an order of magnitude depending upon the assumed source of violations of weak complementarity. Second, we have investigated alternative approaches to incorporating the opportunity cost of time within the KT framework, including the use of: (a) a dual constraint model (Bockstael, et al., 1987), (b) full income and full prices via the assumption of weak separability (Larson, 1990), and (c) a conditional demand interpretation (Shaw, et al., 1999), as well as the traditional approach of relying upon a fixed fraction of the wage rate. As one would expect, the competing approaches produce statistically different welfare estimates, requiring a choice by the analyst on the preferred approach. In an empirical application using data on Iowa wetland usage, the dual constraint model from Bockstael, et al. (1987) appears to have the most intuitive appeal, producing welfare estimates for a statewide 20 percent increase in pheasant counts of $22/season, along with intuitive and statistically significant coefficient estimates. Finally, we expand upon an idea presented in Crooker and Kling (2000) (CK) that suggests that revealed preference data can be combined with stated preference data to produce bounds on the willingness to pay for price changes in a recreation good. Our insight is to think about the bounds, which contain both RP and SP data, as prior information in the construction of parametric models. Preliminary work for demonstration purposes has been completed on the Iowa wetlands data. The research suggests that these non-parametric bounds may provide a useful basis for model selection.Objective 2 of the project is to compare traditional approaches of modeling recreation demand to the utility consistent methods. Within this line of research, we have focused our efforts on comparing the KT framework to count data approach employed, for example, by Shonkweiler. Count data models account for the integer nature of recreation demand but address non-consumption from a purely statistical perspective. A second objective in this research is to examine how best to allow for correlations in the unobserved determinants of choice across related sites. The complex structure of both the statistical and economic approach to corners estimation has necessitated the use of relatively simple correlations to date. Recently, Herriges and Phaneuf (2000) have proposed using mixed logit models as a means for allowing more complex cross-site correlation in RUM models. They suggest an error components interpretation that allows complex error structures to be intuitively specified and tractably estimated. We use their insights derived in the context of mixed logit models to allow for correlations in the unobservable determinants of choice for both the economic and statistical specifications of the LES demand system models. The empirical analysis relies upon the Iowa wetland data. Various dimension problems can be specified using this data. The analysis to date suggests that the use of mixed distributions for enhanced error structure makes a significant difference, both in terms of the statistical properties and fit of both models and the magnitude of welfare effects. For example, using the 5-good model Iowa wetland usage, with pheasant counts as an explanatory variable, the simplest independent error KT model provides a seasonal (use-only) welfare measure of $20.33 for a 20 percent increase in counts, while a similar model including correlation terms between sites exhibiting similar characteristics (i.e., riverine wetlands and upland wetlands) produces an estimate of $11.52.

**Future Activities:**Our future activities will center on further pursuing objectives one and two set forth in the project proposal:*Objective 1*. Investigate modeling, specification, and econometric issues associated with utility consistent corner solution models in recreation demand. In particular, we plan to consider various flexible functional forms, the use of index number theory to test for and specify sites and their associated prices, and further generalizations of the error structure.*Objective 2*. Compare traditional approaches of modeling recreation demand to the utility consistent methods. We will continue to pursue this objective using data from the Iowa Wetlands survey, as well as data from an ongoing survey of recreational usage at Clear Lake in north central Iowa.**Publications and Presentations:**Total Count: 8

__Type____Citation____Journal Searches__Journal Article Herriges JA, Phaneuf DJ. Choice set definition issues in a Kuhn-Tucker model of recreation demand. Marine Resource Economics 1999;14:343-355. *not available*Journal Article Phaneuf DJ, Kling CL, Herriges JA. Estimation and welfare calculations in a generalized corner solution model with an application to recreation demand. The Review of Economics and Statistics 2000;82(1):83-92. Journal Article Herriges JA, Kling CL, Phaneuf DJ. What's the use? Welfare estimates from revealed preference models when weak complementarity does not hold. Journal of Environmental Economics and Management (under review, September 2000). Presentation von Haefen RH, Phaneuf DJ. A comparison of economic and statistical approaches to corner solutions with an application to seasonal recreation demand. To be presented at the Allied Social Science Associations? Annual Meeting, New Orleans, LA, January 5-7, 2001 (working paper). *not available*Presentation Crooker J, Phaneuf DJ. Combining nonparametric bounds with parametric models of recreational demand: a Bayesian prior interpretation. Presented at the Heartland Environmental and Resource Economics Workshop, Ames, IA, September 17-18, 2000. *not available*Presentation Crooker J, Phaneuf DJ. Combining nonparametric bounds with parametric models of recreational demand: a Bayesian prior interpretation. Presented at the Western Economics Association Meetings, Vancouver, British Columbia, June 29-July 3, 2000. *not available*Presentation Phaneuf DJ. Kuhn-Tucker models of recreation demand. Presented at the Econometric Society World Congress, Seattle, WA, August 11-16, 2000. *not available*Presentation Phaneuf DJ. Options for including the opportunity cost of time in Kuhn-Tucker models of recreation demand. Presented at the W-133 Western Regional Project Annual Meetings, Kauai, HI, February 26-March 1, 2000 (working paper). *not available*