# Willingness to pay for quality improvements: comparative statics and interpretation of contingent valuation results.

I. INTRODUCTION

The increased use of dichotomous choice contingent valuation (CV) data has provided motivation for development of two major models that explain responses to contingent choices. The Hanemann (1984) approach is to specify explicit utility functions and derive functional forms for estimation. The Cameron (1988) approach is to specify a logit or probit (Cameron and James 1987) model for estimation which can then be transformed into a valuation function.(1) While the Hanemann and Cameron approaches are often thought of as competing, McConnell (1990) shows that when the marginal utility of income is constant, the models are linear transforms of one another.(2)

Although McConnell (1990) focuses on dichotomous choice data, the theoretical properties he finds can be used to interpret the results of other types of contingent valuation data. McConnell (1990) focuses on willingness to pay for price changes and investigates briefly two properties of willingness to pay for quality improvements. Since many environmental and natural resource policy issues involve quality changes, such as air and water quality improvements, further investigation of other properties of willingness to pay for quality improvements seems warranted.

The purpose of this paper is to extend the variation function theory of McConnell (1990) for quality improvements by including the effects of changes in own-price, cross-price, and improved quality variables for both resource users and nonusers under an alternative property rights assumption. The variation function model is consistent with discrete choice and continuous valuation function models. The variation function model allows comparisons of contingent valuation results with recreation demand models and identification of substitution and complementarity relationships between trips to natural resource sites. Using an empirical illustration, it is shown how the model can be used to interpret contingent valuation results.

II. WILLINGNESS TO PAY FOR QUALITY IMPROVEMENTS

Suppose consumers have the utility function u(x, q, z), where x = ([x.sub.i]), i = 1,..., n, is a vector of demands for on-site (recreational) use of natural resources, q = ([q.sub.i]) is a vector of natural resource quality characteristics, and z is a composite of all market goods. The expenditure function, m(p, q, u), is found by solving the consumer problem: minimize z + p[prime]x subject to u = u(x, q, z) where p = ([p.sub.i]) is a vector of on-site use prices and z is the numeraire good, [p.sub.z] = 1. The expenditure function measures the minimum amount of money a consumer must spend to achieve a fixed utility level and is increasing in p and u and decreasing in q. Since there are no market prices for the nonmarket goods, the on-site use prices are assumed to be constant marginal costs of use which vary by individual travel distance and time, on-site time, site fees, the costs of variable inputs, etc. In practice, these prices are typically measured as the round-trip travel and time costs.(3)

Willingness to pay is the maximum amount of money consumers would give up in order to enjoy a natural resource quality change. Suppose we wish to value a potential program that would improve water quality in a polluted resource, such as a river, lake, or sound. For simplicity, assume only two natural resources are relevant to the choice problem, [q.sub.1] being the water quality of the polluted resource and [q.sub.2] the water quality of a related natural resource. For concreteness consider a closed-ended question such as: "Would you be willing to pay \$A for a policy with a goal to change water quality from [q.sub.1] to [Mathematical Expression Omitted] for resource 1? The water quality of resource 2 would remain unchanged." The discrete choice question could then be iterated to generate continuous data or transformed using the Cameron (1988) technique to generate a continuous function. Willingness to pay could also result from an open-ended contingent valuation question: "What is the maximum amount of money you would be willing to pay for a policy with a goal to change water quality from [q.sub.1] to [Mathematical Expression Omitted] for resource 1?" A formal definition of the valuation function for the improvement in water quality is

[Mathematical Expression Omitted]

where [WTP.sub.1]([center dot]) is the willingness-to-pay valuation function, [q.sub.1] is a degraded level of water quality and [Mathematical Expression Omitted] is an improved level of water quality, [p.sub.1] is the (on-site use) own-price, and [p.sub.2] is the (on-site use) cross-price of the related resource. Expenditures to maintain the utility level decrease with the increase in quality ([q.sub.1] to [Mathematical Expression Omitted]) so that WT[P.sub.1] [greater than or equal to] 0.

III. COMPARATIVE STATICS FOR RESOURCE USERS

Assume the reference level of utility is [Mathematical Expression Omitted], where y is income and v([center dot]) is the indirect utility function found by solving the problem: maximize u([center dot]) subject to y = z + p[prime]x. Substitution of the indirect function into equation [1] yields the variation function

[Mathematical Expression Omitted]

where [s.sub.1]([center dot]) is the willingness-to-pay variation function. Equation [2] is the equivalent variation function, an alternative assumption about implicit property rights than that made by McConnell (1990).(4) McConnell (1990, 32-33) defines the compensating variation function and finds that willingness to pay is increasing in income for normal goods and decreasing with increases in the degraded ([q.sub.1]) quality level. These two comparative static results are similar for the two property rights assumptions as noted below. However, the assumption employed here facilitates signing and interpretation of own-price and cross-price effects and is more consistent with the following empirical illustration.

Income Effects

The effect of income on the variation function is

[Mathematical Expression Omitted]

where [m.sub.v] = [m.sub.v]([center dot], [q.sub.1]), [Mathematical Expression Omitted], subscripts represent partial derivatives, and all arguments other than quality are suppressed for simplicity. When evaluating at the same quality arguments, [v.sub.y] = 1/[m.sub.v], and(5)

[Mathematical Expression Omitted]

where

[Mathematical Expression Omitted]

and [Mathematical Expression Omitted]. If quality is a normal good the marginal cost of utility will be greater with the degraded quality level and the income effect will be positive. If quality is an inferior good the income effect will be negative. If quality is unrelated to income or if the marginal utility of income is constant, the income effect will be zero. For simplicity, quality is assured to be a normal good for the rest of the paper.

Own-Price Effects

The effect of the own-price on the variation function is

[Mathematical Expression Omitted].

Since [Mathematical Expression Omitted], multiplication by [Mathematical Expression Omitted] yields

[Mathematical Expression Omitted]

which can be expressed as

[Mathematical Expression Omitted]

where [x.sup.h]([center dot]) is the compensated (Hicksian) demand function and, by Roy's identity, [x.sup.m]([center dot]) is the uncompensated (Marshallian) demand function.

Since u = v([center dot], [q.sub.1]) and [x.sup.h][[center dot], v([center dot], [q.sub.1])] = [x.sup.m]([center dot], [q.sub.1]),

[Mathematical Expression Omitted].

Assuming quality is a normal good, the own-price effect will be negative if recreation trips and quality are gross complements.(6) The variation function increases (decreases) with decreases (increases) in the own-price because recreation demand at resource 1, [Mathematical Expression Omitted], is higher with improved quality. If recreation trips and quality are gross substitutes the own-price effect is positive or equal to zero, depending on the size of [Gamma].

Cross-Price Effects

The cross-price effect can be similarly found

[Mathematical Expression Omitted].

This expression can be manipulated as above for own-price effects to yield

[Mathematical Expression Omitted].

Assuming quality is a normal good, if trips to resource 1 and trips to resource 2 are gross substitutes then the cross-price effect is positive or equal to zero, depending on the size of [Gamma]. Increases in the quality of the polluted resource ([q.sub.1] to [Mathematical Expression Omitted]) will increase trips to that resource ([Mathematical Expression Omitted]) and decrease trips to the related resource ([Mathematical Expression Omitted]). If trips to resource 1 and trips to resource 2 are gross complements then the cross-price effect is negative since increases in the quality of resource 1 will increase the demand for trips to resource 1 and resource 2.

Quality Effects

The change in the variation function from a change in the degraded quality level is

[Delta][s.sub.1]/[Delta][q.sub.1] = [m.sub.q1] [less than] 0 [11]

where [absolute value of [m.sub.q1]] = [[Pi].sup.h]([p.sub.1], [p.sub.2], [q.sub.1], [q.sub.2], u) is the inverse Hicksian demand for quality. At any quality level, [[Pi].sup.h] is the marginal willingness to pay for increasing quality.(7) Evaluating [[Pi].sup.h]([center dot], [q.sub.1]) at u = v([center dot])yields the inverse Marshallian demand by duality: [[Pi].sup.h][[center dot], [q.sub.1], v([center dot])] = [[Pi].sup.m]([center dot],[q.sub.1]). The degraded quality effect is negative since expenditures to maintain the reference utility level increase as the quality level worsens.

The change in the variation function from a change in the improved quality level is(8)

[Mathematical Expression Omitted]

which can be rearranged and expressed as

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the inverse Marshallian demand for quality. At any quality level, [[Pi].sup.m] is the marginal willingness to pay for increasing quality. Since [[Pi].sup.m] is positive the improved quality effect is positive.

Comparing the two comparative static quality effects has at least two important implications for contingent valuation models. The inequality, [Mathematical Expression Omitted], should hold with the sufficient conditions [v.sub.qq] [less than] 0 and [v.sub.yq] [greater than] 0. The quality effects also suggest that total willingness to pay increases as the quality change increases.

IV. COMPARATIVE STATICS FOR RESOURCE NONUSERS

Resource nonusers are assumed to currently (with degraded quality) face a price for on-site use which is greater than or equal to their choke price. That is, they face a price that drives recreation demand to zero. Willingness to pay for nonusers is defined as

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the choke price which depends on the quality level. Equation [14] represents a situation where nonusers are not recreation participants even with an increase in quality.

In the previous analysis it was assumed that when water quality improves Marshallian demand increases. It is possible that resource nonusers will enter the recreation market and become users of the resource with the quality improvement. Assuming gross complementarity between quality and recreation demand the following expression for willingness to pay is also plausible for nonusers

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted]. This will hold if the quality improvement is sufficient to increase the Marshallian demand so that the own-price is no longer greater than or equal to the choke price. Equation [15] represents a situation where previous nonparticipants become recreation participants. See Bockstael and McConnell (1993, 1248-49) for further discussion.

Substitution of the relevant reference utility level, [u.sup.*] = v([Mathematical Expression Omitted], [p.sub.2], [Mathematical Expression Omitted], [q.sub.2], y) or [u.sup.*] = v ([p.sub.1], [p.sub.2], [Mathematical Expression Omitted], [q.sub.2], y) respectively, yields the following expressions for the variation function for nonusers

[Mathematical Expression Omitted]

and for nonusers who enter the recreation market and become users

[Mathematical Expression Omitted].

Comparative statistics will differ for nonusers on the arguments where equations [16] and [17] differ from equation [2] above. Since the choke price appears in [16] and [17] for on-site use of resource 1, the own-price effect for nonusers will differ from that of users. However, income, cross-price, and quality effects are as described above for resource users.

Consider first a resource nonuser who does not begin to participate in recreation at resource 1 with the improvement in quality. The own-price effect from equation [16] is

[Mathematical Expression Omitted]

since the choke price does not vary.

From equation [17], the effect of own-price for resource nonusers who become users with the quality improvement is

[Mathematical Expression Omitted].

Multiplication by [Mathematical Expression Omitted] yields

[Mathematical Expression Omitted].

The variation function increases (decreases) with decreases (increases) in the own-price because recreation demand, [Mathematical Expression Omitted], is positive with higher quality. Note that this result is identical to equation [8] for users when [Mathematical Expression Omitted]. The own-price effect for nonusers is a testable hypothesis. The test would indicate whether nonusers stay out of or enter the recreation market with a quality improvement.

V. INTERPRETATION OF CONTINGENT VALUATION RESULTS

These comparative static results can be used to theoretically interpret results of contingent valuation empirical models. Consider a linear variation function

[Mathematical Expression Omitted]

where [[Alpha].sub.i], i = 0,..., 5, are coefficients to be estimated and [Epsilon] is a mean zero error term.(9) The coefficients on [q.sub.1] and [Mathematical Expression Omitted] can be estimated if the quality change that is being valued varies across respondents. If quality is constant across respondents then the constant term will capture the quality change: [Mathematical Expression Omitted]. The coefficients in [21] are theoretically equivalent to the derivatives of the variation function

and

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

[[Alpha].sub.3] = y - , [24]

[[Alpha].sub.4] = -[[Pi].sup.m]([center dot], [q.sub.1]),

and

[Mathematical Expression Omitted].

An empirical application employing contingent valuation data that includes each variable considered above is beyond the scope of this paper. However, as an illustration of the model, an empirical analysis is conducted with data from a telephone survey of 1,033 residents of North Carolina and southeastern Virginia. The data contains measures of willingness to pay for quality improvement in the Albemarle and Pamlico estuarine system in eastern North Carolina, own-price, cross-price, and income variables. The own-price is associated with on-site access to the Albemarle and Pamlico Sounds. The cross-price is associated with access to the Chesapeake Bay.(10) Willingness to pay is a continuous variable.(11)

This data does not have quality measures that vary across respondents, however, it is still useful to discuss the interpretation of the potential coefficient estimates on quality. For instance, the restriction on preferences

-[[Alpha].sub.4] [greater than] [[Alpha].sub.5]/[[Alpha].sub.3] + 1

should hold. If a forecast of willingness to pay outside the range of quality valuation is desired, [[Alpha].sub.5]/([[Alpha].sub.3] + 1) would provide an estimate of the marginal willingness to pay for additional quality levels. For instance if the proposed quality change is measured as the number of waterfowl harvested after wetland habitat restoration and water quality improvement, [[Alpha].sub.5]([[Alpha].sub.3] + 1) would be an estimate of the marginal value of an additional waterfowl. Of course, forecasts outside the quality valuation range must be viewed with caution.

Resource Users

The mean values of the primary variables for the subsample of resource users are [p.sub.1] = \$121, [p.sub.2] = \$143, and y = \$40,325. Construction of the price variables follows the individual travel cost model including the costs of round-trip travel distance and time.(12) Income is the total wage and non-wage household income. The least squares coefficient estimates (t-statistics in parentheses) are [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], and [Mathematical Expression Omitted], F = 9.38 (3 df). Each of the coefficient estimates are significantly different from zero at the .05 level except for the coefficient on cross-price. The expected value of willingness to pay for the quality improvement is \$62 and the average number of recreation trips ([Mathematical Expression Omitted]) is 13.20. The average number of recreational trips to the related natural resource ([Mathematical Expression Omitted]) is 11.94.

These results indicate that quality and recreation demand are gross complements [Mathematical Expression Omitted], trips to the resource sites are gross substitutes [Mathematical Expression Omitted], and quality is a normal good [Mathematical Expression Omitted]. The own-price coefficient can be used to estimate changes in the number of recreation trips. Since [[Alpha].sub.3] + 1 = [Gamma], [Gamma] = 1.001 and substituting mean values and coefficient estimates into equation [22] yields

[Mathematical Expression Omitted]

which can be solved for the average number of trips with a quality improvement: [Mathematical Expression Omitted]. This suggests that about 61 more trips would be taken annually by the sample of users. The number of trips to the related natural resource would change only slightly since the coefficient estimate on cross-price is small and not significantly different from zero. Using equation [23] the coefficient estimates suggest that 1.5 fewer annual trips will be taken to the substitute site by the sample of users.

Resource Nonusers

The mean values of the primary variables for the subsample of resource nonusers are [p.sub.1] = \$146, [p.sub.2] = \$171, and y = \$32,577. The least squares coefficient estimates (t-statistics in parentheses) are [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], and [Mathematical Expression Omitted], F = 25.75 (3 df)]. Each of the coefficient estimates are significantly different from zero at the .05 level. The expected value of willingness to pay for the quality improvement is \$55 and the average number of recreation trips to the related natural resource [Mathematical Expression Omitted] is 7.95.

The regression results for nonusers indicate that quality and recreation demand are gross complements [Mathematical Expression Omitted], the resource sites are gross substitutes [Mathematical Expression Omitted], and quality is a normal good [Mathematical Expression Omitted]. The own-price coefficient indicates that the Marshallian demand for some nonusers will shift with the quality change. Since [[Alpha].sub.3] + 1 = [Gamma], [Gamma] = 1.0005 and substituting mean values and coefficient estimates into equation [22] yields

[Mathematical Expression Omitted]

which can be solved for the average number of trips with a quality improvement: [Mathematical Expression Omitted]. This suggests that about 123 trips would be taken by the sample of nonusers annually after a quality improvement. Substituting mean values and coefficient estimates into equation [23] yields

[Mathematical Expression Omitted]

which suggests a decrease in trips to the related resource to an average of 7.8 or about 112 fewer trips annually for the sample of nonusers.

VI. CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH

This paper has extended the theory underlying the McConnell (1990) variation function for quality improvements in several important ways. First, the effects of the own-price on equivalent variation for both resource users and nonusers are derived. If quality and trips are complements, empirical measures of willingness to pay for quality changes will vary inversely with the own-price for resource users. For resource non-users, willingness to pay will vary inversely with the own-price if the quality improvement leads to resource use (recreation participation). No effect will be found for nonusers if the quality improvement does not lead to resource use. Secondly, the variation function was modeled to include a cross-price term and its potential effects were shown. The cross-price result allows identification of substitution and complementary relationships between trips to natural resource sites. Lastly, the equivalent variation for a quality improvement should increase with increases in the quality change and with income.

These results suggest theoretical validity tests, in addition to testing for effects of income changes on willingness to pay, for contingent valuation models. Contingent valuation researchers can model willingness to pay to depend on own-price, cross-price, and quality variables, in addition to the frequently included income variable. Empirical models that find quality to be a normal good should also, typically, find negative own-price coefficients, positive or negative cross-price coefficients, and degraded quality effects greater in absolute value than improved quality effects. The price results may also be useful in determining market size for aggregation of benefits.

These results also suggest several tests for the convergent validity of contingent valuation models. Since the own-price and cross-price effects can be used to estimate measures of recreational behavior change, they can be compared with results from recreation demand models. The inverse Marshallian demand functions for quality that are derived from the comparative static quality effects can be compared with estimates of the marginal willingness to pay bounded or approximated from revealed preference models using techniques suggested by Neill (1988) and Larson (1992). The contingent valuation model could also be jointly estimated with recreation demand or participation models as convergent validity tests (see Cameron 1992a, 1992b and Cameron and Englin 1991).

Assistant professor, Department of Economics, East Carolina University, Greenville, NC.

The author would like to thank two anonymous referees for suggestions which have significantly improved this paper. Tom Hoban and Bill Clifford are also due thanks for supplying the data in the empirical application.

The research on which this paper is based was financed in part by the U.S. Environmental Protection Agency and the North Carolina Department of Environment, Health, and Natural Resources, through the Albemarle-Pamlico Estuarine Study. Contents of this paper do not necessarily reflect the view and policies of the U.S. EPA or the N.C. DEHNR.

1 Patterson and Duffield (1991) argue that the maximum likelihood estimation procedure emphasized by Cameron (1988) and the logit transformation are computationally equivalent techniques.

2 Sec McConnell (1990) for three conditions when the marginal utility of income is constant. Park and Loomis (1992) provide the first empirical test of the connections between the Hanemann and Cameron approaches. They find that the marginal utility of income is not constant for two of the three conditions, but, the welfare estimates predicted by each model are not significantly different.

3 Randall (1994) argues that travel and time costs systematically understate the price of a recreation trip and is very pessimistic concerning the objective measurement of a trip price due to the heterogeneity of recreation choices. Cameron (1992b) also recognizes that trip prices, as typically measured by travel and time costs, are underestimated but provides hope that the mismeasurement factor can be estimated.

4 The McConnell assumption is that the implicit property rights are associated with the lower quality level. The compensating variation function could be applied to resources such as fish and wildlife populations that can be increased above historical levels through stocking or other types of management programs. For instance improvements in a trophy buck management program or a striped bass program might yield the property rights assumption of McConnell. Environmental pollution, which reduces air and water quality from historical levels, might lead to the property rights assignment of this paper where the higher quality level is associated with the reference level of utility.

5 This result is analogous to the McConnell (1990) income result for the compensating variation.

6 Gross complements and substitutes refer to the Marshallian comparative statics. The gross complementarity between the Marshallian recreation demand and quality can be signed with certain assumptions; see Bockstael and McConnell (1993, 1246). Comparing Hicksian recreation demands with quality changes requires assumptions concerning Hicks complementarity, substitutability, and neutrality. Even after adopting these assumptions, the sign of the price effect is ambiguous since the quality effects on Hicksian recreation demands are ambiguous. See Larson (1992) and Bockstael and McConnell (1993) for a discussion.

7 Several other researchers have investigated this term. Bockstael and McConnell (1993) call the marginal willingness to pay the "virtual price of quality." Their notation is adopted in this paper. Larson (1992) calls the marginal willingness to pay the marginal valuation of the nontraded good with the notation [Mu] = [Pi]. See also Neill (1988).

8 An analogous expression is found by McConnell (1990) for the compensating variation case although the quality effect is for improvements in the degraded quality level. McConnell (1990) does not pursue the effects of changing the level of improved quality on the compensating variation function.

9 Subscripts for individuals are suppressed. This equation could result from least squares or Tobit estimation of open-ended CV data, grouped data estimation of interval CV data (Cameron and Huppert 1988), or the Cameron transformation of logit or probit regression. The linear variation function is common and its interpretation is straightforward. The semi-log and double-log forms are also often estimated. With a bit more manipulation this analysis could also be employed with these other common functional forms.

10 As pointed out by a referee, the choice of related natural resources is problematic due to uncertainty about consumer preferences or collinearity among the price measures. Future CV surveys might include questions which elicit perceived substitute or complementary natural resources from survey respondents. Answers to these questions could guide construction of cross-price variables.

11 After establishing a contingent market through a series of questions designed to describe management alternatives and payment vehicles, respondents are presented with the valuation question: "We already pay for the types of government programs we've just discussed through federal, state, and local taxes. However, government will need more money if water quality and fish and wildlife habitat in the A-P (Albemarle-Pamlico estuarine) system are to be protected. This money would pay for state and local programs to control pollution, monitor water quality, protect habitat, and educate people. The goal would be to make sure water pollution does not get worse and habitat remains the same. Would you and your household be willing to pay \$A, each year, in higher taxes, for these programs, if you knew the money would be used to protect the A-P system?" Respondents were then led through an iterative bidding process to generate continuous willingness-to-pay data. Respondents typically perceive that quality is decreasing in the A-P system so that the management program is perceived to improve quality (Hoban and Clifford 1991). A complete description of the contingent market and data can be found in Whitehead (1992).

12 Several authors have argued that travel costs systematically understate the price of a recreation trip (Parsons 1991; Cameron 1992b; Randall 1994). Parsons (1991) argues that distance to the recreation site is endogeneous to residence location choices and proposes a 2SLS solution to the endogeneity of price. The 2SLS proposal was attempted but did not lead to valid empirical results. This paper will proceed using the operational assumption that travel and time costs are equivalent to trip prices.

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