Bundling Private Complements to Finance Public Goods

2 Bundling and Weak Complementarity

Economists often make a sharp distinction between private goods, which can be provided by markets, and public goods, usually provided by governments. In many cases, households use a “bundle” of public goods and private goods to enjoy the services they jointly provide. Outdoor recreation is an example of such bundling. To enjoy a recreation trip, a household bundles the publicly provided resource (the scenic vistas and built amenities) and their time along with some private goods and services, such as transportation and gear.

This bundling can be modeled as complementarity between the public and private goods. Of course, two private goods exhibit complementarity when a lower price for one leads to greater demand for the other. For bundles of public and private goods, complementarity can be understood as arising when an increase in the availability of a public good leads to an increase in the demand for the private good and, symmetrically, when a lower price for the private good leads to a higher marginal willingness to pay (WTP) for the public good (Madden 1991).

A particularly important type of complementarity relationship between public and private goods is known as weak complementarity (Mäler 1974; Smith and Banzhaf 2004). Weak complementarity is defined as a situation where the public good has no value to people unless they also consume positive levels of the linked private good (the weak complement). For example, weak complementarity would hold if people do not care about the ecosystem services at a park unless they take trips to it. In other settings, people might not care about local school quality in a neighborhood unless they have housing in the neighborhood; they might not care about local roads unless they drive on them, for example. But the reverse need not hold. The private good’s value does not have to be exclusively associated with the services of the public good. People can value local housing without valuing schools.

When exploiting this connection, economic measures of changes in the public good can capture use values. Thus, a decision to rely on the assumption of weak complementarity to measure the benefits from an increase in the amount of a public good rules out nonuse values (i.e., values households have for the investment even if they never use it in any instrumental sense). Consequently, the assumption also limits “the extent of the market” to the set of users, even though the set of people who would value the investment may be larger. This assumption may be problematic for some policy questions—the preservation of unique resources such as Yellowstone or the Grand Canyon, for example, or investments that protect ecosystems where nonuse values are important. For our purposes, the focus is on funding for deferred maintenance and ongoing operations, not preservation. Although some of these investments may protect ecosystems, many involve more pedestrian objectives like maintaining bathrooms, trails, and campsites, which are linked to recreation use. Even if there are existence or nonuse values, we can say that weak complementarity holds for that portion of households’ total value linked to recreation.

Weak complementarity is useful because it assures that when there is an investment in a publicly provided asset, the change in households’ Hicksian consumer surplus for their consumption of the private good (that is a weak complement) will be equal to its Hicksian WTP for that investment (Mäler 1974). Figure 1 shows the change in consumer surplus for the private weak complement due to this type of investment. It shows the marginal cost of the private good x (e.g., the travel cost of trips to the park) and two Hicksian demand curves for an individual. The lower demand curve shows the quantity of the private good (trips) that would be demanded when income is adjusted to leave the consumer at the same level of utility, when evaluated at a baseline level of environmental or resource quality, q⁰. The upper demand curve shows the corresponding demand at a higher level of environmental quality, q¹. With people more likely to participate in recreation when the quality of the experience is better, the improvement in environmental quality shifts up the demand for trips. We assume this shift occurs only through the complementarity relationship. Chan and Kotchen (2022) also account for the additional desirability of taking trips when they become linked-by a dedicated tax to increases in q, making them avenues to voluntary contributions to a public good.⁷

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Figure 1

Increased Value of the Private Good Caused by Improved Public Good

Weak complementarity guarantees that the shaded area between these lines, up to the level consumed, is the individual’s value (or at least the recreational value) of the improvement in q because u is constant at choke price p*. Consider Figure 2, which shows the marginal cost of providing the public good q against the marginal social value of a little more q. The shaded area in Figure 2 is the (gross) value of the improvement in the public good from q⁰ to q¹. Weak complementarity guarantees that the shaded areas in the two figures are the same for each individual with these preferences. Consequently, the respective sums across individuals of the two areas are the same, and the aggregate value of the investment is equivalent to the change in aggregate consumer surplus for the private weak complement.

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Figure 2

Value of the Improvement in the Public Good

Another consequence of this equivalence is that the value of the environmental services (Figure 2) can actually be captured in the market for the private good (Figure 1) when excludability is possible. For example, to the extent that households only have recreation values for the services provided by environmental resources, they can be excluded from enjoying them because they can be excluded from entering those areas. This bundling creates incentives for private parties to provide environmental services (Heal 2000, 2003; Anderson and Leal 2001; Anderson and Parker 2013). As we discuss in Section 4, it similarly creates the possibility that gear sellers can benefit from a gear tax. When the bundling relationship is not as salient, as when households have nonuse values for the quality improvements associated with investments in a park, households cannot be entirely excluded from enjoying them, making it more difficult for private parties to capture the full value. Nevertheless, as we discuss in Section 5, the presence of partial values may be sufficient to make the case for taxes on some private goods to support improvements in public resources.

In practice, environmental values can be captured in private markets because weak complementarity implies that it is possible to define the increase in the price of the linked private good that exactly compensates for the improvement in the public good, leaving the consumer indifferent to the combined change (Willig 1978; Smith and Banzhaf 2004). Any actual price increases less than this level would leave consumers better off. Thus, instead of measuring value through income adjustments associated with WTP as economists usually do, one can measure value through price adjustments. However, the “right” price adjustment will differ for each consumer.⁸ Figure 3 reproduces the “fanned” indifference curves from Smith and Banzhaf (2004), which display how weak complementarity affects the trade-off between a private good (x) and a numeraire good (Z). The nonmarket good is q. Three indifference curves are displayed. depicts the baseline situation, with the consumer’s optimum at point B. The dashed indifference curve indicates that at a higher level of q (with q⁰< q¹), the consumer obtains higher utility and purchases more of the private weak complement at point E. The indifference curve depicts the combinations of x and Z that leave utility unchanged at the initial level when quality increases. Weak complementarity implies that q has no value to the individual without x, thus the two solid indifference curves intersect at the vertical axis because at that point where x =0.

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Figure 3

Fanned Indifference Curves and Weak Complementarity

Figure 3 illustrates that an increase in the price of the weak complement can be defined that would just compensate for the quality improvement and bring the consumer from to , with equilibrium at point C. This price increase represents the source for the potential revenue gain to suppliers of the private good from the investment. In addition to giving them an incentive to provide the investment themselves, it creates the possibility that they can benefit from it, even if their own sales are taxed to provide the quality improvement that underlies the increase, a point we return to in Section 4.⁹

3. Economics of Gate Fees

In this section, we show that, when trips to public lands are a weak complement to the services provided by the investment in these lands, it is possible to define a fee increase that, when combined with the investment, leaves households indifferent. If the actual fee required to fund the investment is lower than this critical level, households are better off. To a first-order approximation, this criterion is equivalent to observing an increase in the number of trips from the joint effect of the increased fee and the quality improvement associated with the increased investment.

We model gate fees as comparable to a tax on trips. Trips are a convenient activity to analyze in this way because conventional practice in modeling recreation demand assumes their price is set by distance between a household’s home and the site and by per mile time and transportation costs, both of which are plausibly exogenous to the policy. Thus, trips to the site (which we denote by x) can be modeled as having perfectly elastic supply at a pretax price of p_x. The price of a composite of other consumption goods is normalized to 1. Let q be the quality of the site that is visited. Income is fixed and designated by m. Let V( ) represent the indirect utility function for a representative household. Finally, assume that trips are nonrival, so that if lump-sum taxes were available, the optimal gate fee would be zero.¹⁰

Equation [1] provides the indirect utility function expressed in general terms: [1] Here, designates the fact that our analysis begins with a baseline level of utility.

Now consider a policy that jointly imposes an arbitrary entry fee t and provides a change in quality ∆q. To start our discussion, we assume that the costs of the improvement (∆q) may not necessarily be covered by the revenue generated from t. To a first-order approximation, the change in utility is [2] Dividing both sides by the marginal utility of income, V_m, substituting using Roy’s identify for –V_px / V_m=x, and defining the marginal WTP for q by π = V_q/ V_m, we have [3] This expression simply says that utility for this person increases if, and only if, the fee paid, tx, is less than the value of the investment,π∆q. This obviously must hold to a first-order approximation.

The assumption of weak complementarity brings additional insight. For any arbitrary ∆q, weak complementarity guarantees that there exists a t^* for an individual such that ∆V = 0; that is, (Willig 1978; Smith and Banzhaf 2004). Returning to equation [2], this is given by [4] More generally: [5] That is, welfare at the individual level improves if, and only if, for any ∆q, the actual fee (t) is less than the critical value, t*.

Expression [5] can be interpreted more intuitively if we again divide the numerator and denominator of the right-hand term by the marginal utility of income, which, making use of Roy’s identity, yields [6] As , or the marginal WTP for quality per trip, is a commonly estimated parameter in travel cost modeling, this condition is empirically verifiable in principle. It gives us a starting point for determining a neutral tax from the perspective of the representative individual. Thus, if ∆qis accompanied by a gate fee less than t*, the welfare of the representative visitor is increased by the policy.

However, there is another aspect of this relationship. So far, we have considered simultaneous but essentially arbitrary increases in the fee (t) and in the public good, ∆q. In practice, the ability to realize the ∆q depends on the aggregate revenue raised from the fee increase or tNx, where x is the average number of trips per user and N is the number of users. Budget-balanced combinations of t and ∆q are limited by feasible production plans. Suppose the cost of providing ∆q is C(∆q). Then for any ∆q, budget balance requires t = C(∆q)/Nx.¹¹

Substituting this relationship into expression [6], the policy increases aggregate welfare if and only if [7] or [8] That is, the average cost of providing a unit increase in q must be less than the aggregate marginal value of that increase. If one knew π but not C(∆q), one could back out the break-even cost using this formula. In the simplest case, per unit costs of increasing q are a constant p_q, and the test becomes p_q < Nπ.

This test seems to be a straightforward application of the Samuelson rule, comparing the supply price of the public good with the sum of the marginal WTP across the N households assumed to experience it. This result may be surprising, as we are not using a lump-sum tax. At first, it may appear that in taking a first-order approximation, we have simply assumed away deadweight loss in the market for trips. In fact, the result stems from the fact that weak complementarity guarantees that prices and marginal WTP can be compared in the same space. Our approximation merely simplifies things by translating WTP for quality (Figure 2) into a constant WTP per trip, which shows up as a parallel shift in the demand curve for x (Figure 1). With this simplification, condition [8] guarantees that there is no (net) deadweight loss in the private good to worry about, even though we are not using a lump-sum tax.

In Figure 1, an increase in q, taken in isolation, increases the inverse demand curve as shown. As seen in expression [6], our first-order approximation makes use of a constant π∆q/x, or WTP for ∆q per trip. Assuming WTP per trip is constant implies the inverse demand curve for trips shifts up, in parallel, by π∆q/x. By weak complementarity, the resulting increase in consumer surplus (the shaded area) is equal to the value of an increase in q. Just as we can depict the value of ∆qin the demand for trips, we can depict the Hicksian-equivalent price adjustment or gate fee. The fee would shift the cost of trips up by t. Alternatively, it can be viewed as shifting the demand for trips down, in parallel, by t. Thus, we are comparing two vertical shifts in the demand for trips: the shift up from the change in quality and the shift down from the tax. If the tax equaled the vertical shift up in the demand curve from the increase in q, the demand curve would shift back to its original position, so the number of trips would be unchanged and consumer surplus also would be unchanged. This is precisely what would happen if the fee were set at t^*. In Figure 1, the shaded area would equal the gross benefit from the increase in q and the lost consumer from the tax, and the two would offset. Tax revenue would equal t* (the vertical distance between the two curves) × the baseline quantity. In general, if the actual fee required to support ∆q were less, the welfare of users would be higher on net; if it were higher, the welfare of users would be lower on net.

Note that we are asking whether a given policy proposal introducing a bundled change, namely, the fee t that allows a quality improvement of ∆q, is welfare improving. We are not suggesting it is optimal. If lump-sum taxes were available, the policy would involve deadweight loss relative to that first-best alternative, as the number of trips would be even higher in that case. Chan and Kotchen (2022) evaluate alternative taxation schemes from that perspective.

The foregoing analysis suggests, then, that the key question is whether, on net, the demand curve shifts up or down from the joint effect of the excise tax on trips and the quality improvement. As a shift up in the demand curve, with unchanging marginal cost curve, is associated with an increase in the equilibrium quantity of trips, we have the following proposition:

• Proposition 1: Under weak complementarity, to a first-order approximation, welfare increases from a combination of a dedicated gate fee and the associated increase in public good services if and only if the quantity demanded implied by the Hicksian demand for trips increases on net.

Such a test could be implemented with standard travel cost models, which predict the effect of costs on trips, if they can also account for the quality change at stake. Our use of the term “associated” is intended to highlight the fact that the aggregate revenue raised from the gate fee must cover the cost of the increase in services.

Under an additional restriction, known as the Willig (1978) condition, this insight leads to an extremely simple ex post empirical test for a welfare improvement for users. The Willig condition guarantees that the Marshallian variant of the change in consumer surplus for the private weak complement, shown in Figure 1, is also equivalent to the Hicksian consumer surplus in Figure 2. It is equivalent to restricting the price adjustments discussed above to be invariant to income:¹² [9] Under this restriction, we can use a net change in the Marshallian consumer surplus for the private weak complement as the basis for our assessment—and, hence, observed changes in the Marshallian demand. With these assumptions, we thus have the following proposition:

• Proposition 2: Under weak complementarity and the Willig condition, to a first-order approximation, welfare increases for users from a combination of a dedicated gate fee and the associated increases in public good services if and only if the Marshallian quantity demanded for trips increases on net. Thus, ex post, there is no need for estimates of demand, consumer surplus, or costs. If we observe a ceteris paribus increase in trips, we have an increase in welfare.

4. The Gear Tax

The previous argument leads naturally to a strategy for analyzing the gear tax. The difference between the gate fee and the gear tax lies in the fact that (1) we must account for the welfare of sellers as well as consumers; (2) we can no longer assume a perfectly elastic supply as we could in the case of exogenous travel costs, so we must account for market price adjustments; and (3) the assumption of weak complementarity may not be as plausible for some gear in the tax base as for trips. Here, we first account for points (1) and (2), maintaining the assumption of weak complementarity. In Section 5, we relax that assumption as well.

In principle, a gear tax can increase welfare for both consumers and sellers if it results in a large enough quality improvement in the parks. Figure 4 illustrates the basic argument. It is similar to Figure 1, but we have added a supply side that responds to price instead of fixing travel costs at an exogenous level. Let D⁰ represent baseline market demand, as before. Let D^τ represent the demand curve after it is shifted down by the amount of the excise tax on gear, taken in isolation. Let D¹ represent the ex post demand curve after the revenue is used to improve the quality of one or more parks; it shows the net increase in demand from the tax and the quality effects. Here, we show a case where there is a potential Pareto improvement, with demand shifting up on net. As shown in the diagram, consumer surplus and producer surplus increase. After the tax and the increase in park quality, as a result of using tax revenue for this purpose, the equilibrium price increases from P₀ to P₁. Consumer surplus increases by ACEB – P₁EFP₀, which is positive by construction, and producer surplus increases by P₁CFP₀. Thus, the gear tax can be a strict Pareto improvement, not simply a potential Pareto improvement.

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Figure 4

Joint Effects of Gear Tax and Quality Change

Figure 4 captures the basic intuition of our argument but essentially assumes perfect competition. Arguably, perfect competition is an unrealistic assumption, as there are many large gear purveyors with market power. Market power, in turn, raises the question whether equilibrium prices would adjust in a way to lower welfare for one group. In fact, in this subsection, we demonstrate that the basic logic continues to hold under Cournot competition.

Consider the case of J symmetric firms, each with a quadratic cost function for producing a quantity of gear x_j: [10] The firms face a linear inverse market demand curve: [11] where X is the total quantity of gear produced, .

Profits for firm j are [12] Maximizing these profits with respect to x_j, and taking the production of other firms x_k as given, the first-order condition for the firm is to set marginal revenue equal to marginal cost, or [13] Assuming a symmetric equilibrium, we have x_k = X_j, or, for each firm j: [14] [15] The total quantity is J times this value, or [16] Finally, the equilibrium price is given by substituting this equilibrium quantity back into the inverse demand function, : [17]

How does the gear tax affect this equilibrium? As always, we can model an excise tax in isolation by shifting the demand curve down by an amount t, effectively replacing a with a − tin the demand curve and in the equilibrium price and quantity. As the revenue is used to enhance quality of one or more parks, following our analysis in Section 3, we can view this effect, to a first-order approximation, as a parallel shift up in the demand curve for gear. Denote the net effect of these two parallel shifts as da (i.e., the change in the intercept of the demand curve).

At the margin, the net effect of this tax/ quality enhancement on the equilibrium quantity is or [18] This term will have the same sign as da. Thus, if the net effect is to shift demand up, the equilibrium quantity demanded and quantity supplied increases in the Cournot model. By the same token, the change in price is or [19] This price change also is the same sign as da.¹³

Thus, firms are clearly better off: A shift up in the demand curve leads to higher sales and higher prices, which increases profits. Thus, even though they bear some of the burden of the tax, firms benefit on net if the feedback effect of the quality improvement on demand is sufficiently large.

What about consumers? Consider the Marshallian consumer surplus in this market, . Substituting in the above expressions for X^* and , we have [20] If we consider how the consumer surplus changes with a (i.e., ), we have Again, this change in consumer surplus is the same sign as da (assuming an interior solution where the demand curve intersects the supply curve, which occurs where a > γ). Thus, consumer surplus also increases if the demand curve increases.

Moreover, because perfect competition and a single-price monopoly are special cases of Cournot with J = ∞ and J = 1, respectively, the result holds in those limiting cases as well.

Of course, we usually measure consumer welfare with Hicksian surplus, not Marshallian consumer surplus. Since we are using a first-order approximation in terms of price, we are effectively assuming the price/quality relationship is not affected by income (and, as a result, we are implicitly using the Willig condition along with weak complementarity in this argument). Thus, we have the following proposition:

• Proposition 3: Under weak complementarity and the Willig condition, and under Cournot competition, perfect competition, or monopoly pricing, welfare increases for consumers and sellers from a combination of a dedicated gear tax and the implied corresponding increase in public good services if and only if the Marshallian quantity demanded for gear increases on net.

In sum, the test is the same as that from Section 2: To a first-order approximation, our gauge of the effects was given earlier by expression [7] or [8], repeated here: [7] [8] The demand curve shifts down by t and shifts up by the value of q per unit of gear π / x × the change in q that is consistent with the increased revenues, or given our simplifying assumptions, Nxt / p_q. To a first-order approximation, we still have a gauge of whether P_q < πN for gear as well as for trips.¹⁴

5. Extensions

6. Empirical Evidence

Despite the extensive economic literature on outdoor recreation, we know surprisingly little about some of the questions posed in this article. For 60 years, the workhorse model of recreation valuation has been the travel cost model, in which travel costs are meant to be equivalent to the effect of a gate fee.¹⁵ Although well positioned to address some of our questions, to date the travel cost literature has focused on consumer surplus, ignoring the revenue actually collected by the gate fee. It has also tended to focus on questions associated with the valuation of natural amenities, with less emphasis on how the amenities would be maintained or enhanced with any new revenue that resulted from the fees. Ji et al. (2022) take on this issue directly. Von Haefen and Lupi (2022) similarly consider not how revenue might be reinvested to have feedbacks on the demand for sites but how the tax itself might do so through its effects on congestion.

For the gear tax, the empirical challenges are even greater. The central empirical challenge we face is developing estimates for how the improvements in park quality would affect the demand for recreational gear. This effect underlies the shift in demand we describe in Figure 4. Critically, data do not include measures for the activities involved with the gear or the locations the gear was used. Thus, making the connection between park quality and gear is difficult. When park use has been studied in recreation demand studies, the quality features considered are usually not related to those that might be affected by the activities associated with how parks are maintained. As a result, we selected two sets of estimates for the value of national parks and adapted them in an approximate way to develop estimates of the value share, which can be used to implement the logic implied by expression [6].

With some substitutions, we can rewrite expression [6] in percentage terms, making it more useful for use with the available data: [21]

This expression gives us a starting point for determining the magnitude of a neutral tax (i.e., one where the quality increase exactly offsets the negative effect of the increased price[s] due to the taxes on gear). In our setting, we do not know , so we do not know whether the enhancement in the public good can be paid for the revenues raised by the increased gate fees or gear tax. However, we can still find the critical value t* that would leave recreators indifferent. A proportionate tax of that is less than what is implied by the right side of equation [21] would improve welfare for a representative recreating household.

In other words, by comparing the value share for the services associated with park quality as a fraction of income (πq⁰/m) to the share of income spent on gear , we have a gauge of how much the price can increase (due to the tax) for each percentage point increase in quality. For example, if the value share for park quality is about 1% of income and expenditures on gear about 2%, the price increase that would hold WTP constant for a representative individual (with simultaneous improvements in park quality) would be half the percentage increase in quality. Larger value shares for park quality would allow for larger price increases.

We provide some suggestive evidence on the range of price increases for a given improvement in park quality that would be WTP neutral, as we defined this idea in Section 3. Our estimates leverage some diverse sources for measures of the components of equation [21]. They mix data from different household samples and different time periods. As a result, they are best viewed a “collage of estimates” that were collected to match up available information with the task at hand and should be interpreted as illustrative examples of what could be done once there are better data. If the logic associated with our ratio of shares proves useful, implementing it will require a more extensive effort to assemble consistently defined measures for each of the shares.

The first parameter we require is the income share of park quality, πq⁰/m To estimate this parameter, we use revealed preference and stated preference results in the literature. The results for the revealed preference study we selected (Parsons et al. 2021) are in the top panel of the Table 1. The first column provides the names for each of the southwestern parks included in this study. The second column provides the estimate reported in the study for WTP to avoid removing one of the parks from a set that was visited as a group during a vacation trip. The study assumed each visit was part of a planned trip that included visits to multiple sites. The data were collected with on-site surveys in each of the seven parks considered as part of the study. Initial contacts were made to potential survey respondents in the parks as they entered during a specific time window, and these individuals were sent follow-up surveys to collect the specific data used in the analysis. Our results are taken from one of the models reported in Parsons et al. (2021), a standard random utility model where the choice set involved different portfolios of parks to be visited. The model assumes the trip is undertaken, and the choice set is among a group of alternative portfolios. Each alternative is assumed to involve a distinct mix of southwestern parks. This process defined a composite of multiple site choice alternatives. The costs then correspond to the costs associated with altering the mix of sites, given that the respondent has already taken the trip.

The measure we selected for WTP considers a situation where the model is calculating the amount that would be paid to avoid removing each site, one at a time, from the portfolio. Our estimate is the average for each site over all possible combinations considered for the portfolios and for all of the respondents in the sample in 2002$. To arrive at the value share for each site as part of an annual vacation, we added the average trip expenditures for the vacation from the sample ($822) to the incremental WTP as defined in the analysis (in the second column) for the park. We then compared this sum to an estimate of the average respondent’s income. The results, representing our estimates of πq⁰/m for each park, are shown in the third column of Table 1. The numerator is a measure of the contribution of an individual park, as a quality indicator for an overall recreation trip that included visiting multiple parks. We recognize that this interpretation is a stretch. For illustrative purposes, we view this as a “calibration” exercise required because the revealed preference literature does not appear to have better measures for how park maintenance contributes to the quality of a recreation trip composed of visiting southwestern parks. As a result, we pose a thought experiment: if quality declines associated with lack of maintenance would eliminate a park from a recreationist’s planned portfolio, how much would that person pay to support the maintenance that allows it to continue to be part of the composite of sites visited?

The second set of information required for our ratio in equation [21] is the share of income for recreational gear, . We use two sources to estimate the share. The first is an Outdoor Recreation Association (2017)study. The study uses a preexisting, large internet panel to provide a representative sample and asks about expenditures on recreation equipment and participation in 38 activities. Their analysis involved a national survey in 2016 of spending on recreation gear for 38 activities. We selected three nonmotorized activity groups to construct the share measures—camping, hiking, and mountaineering. Camping includes trips that involve sleeping in a tent or in a lodge; hiking was defined as a day hike; and mountaineering corresponded to a composite of three activities: backpacking, mountaineering, and rock or ice climbing. The report did not include household incomes of the respondents. To convert these values to expenditure shares, we use the 2016 median household income as reported by the St. Louis Federal Reserve Bank.¹⁶ These annual shares are given for the three activities in the second row of Table 1.

Our estimated expenditure share for wildlife viewing was taken from a different source.¹⁷ We used the 2016 National Survey of Fishing Hunting and Wildlife Associated Recreation (U.S. Department of the Interior et al. 2016). This survey reports equipment expenditures for those involving trips around the home and away from the home. We considered the total of these expenditures, omitting trip related expenses, for our measure of gear-related expenses. Median household income for those undertaking wildlife viewing away from home was used to compute these ratios. These estimates are also reported in the second row of the table.

The remaining cells in the top panel of Table 1 can then be interpreted as an array, with each row corresponding to one value of πq0 / m and each column corresponding to one estimate for the value of . Each cell in the matrix computes the ratio of the two, as used in equation [21]. All the estimates exceed unity, implying the tax, in percentage terms, can exceed the quality improvement for parks in percentage terms. The analysis does not consider whether the revenue raised would be sufficient to pay for the increase in quality. We consider these estimated ratios as alternative measures for the scaling factor we defined for computing the WTP-neutral price increase for gear. Each provides an alternative estimate, depending on which of the southwestern parks and type of gear measures was considered most plausible, as an example that best fits the logic underlying equation [21]. The measures for wildlife viewing are the smallest but nonetheless imply taxes, in percentage terms, could be 30%-40% larger than the percentage improvement in park quality. If we consider the activity-specific gear measures, where the income shares are smaller, the increases can be as large as 4.5 times the percentage improvement in quality.

The second panel in Table 1 repeats the logic with a different source for the measure of the value share for park quality. We use the Haefele et al. (2020) stated preference analysis of alternative proposals to address the need for added park funding. This study uses a national sample, not one based on on-site sampling as was the case for the revealed preference estimates. The contingent valuation question presents alternative policies to reduce the size of the park system by selling different amounts of parks, presented in terms of acres or sites sold, as a composite including nature-based, historic, and water-based parks with these terms and the total amounts defined in the survey. The nature-based and water-based groups are presented as lost acres (from the sales), implying that portions of the park acres would be sold off, not necessarily individual parks. For the historic category of parks, the amount sold is presented as sites sold.

Three alternative sets of losses, along with an annual income tax increase to avoid the policy, are presented in the first-choice question.¹⁸ Each respondent was asked to indicate their most preferred and their least preferred alternatives. The option of no change in parks is included with the largest required tax increase of the three presented to each respondent. Separate amounts of tax increase were set for the no sales case, ranging from $115 to $600. For the cases with sales, the amounts sold ranged from 20% to 40%. For the maximum case, the amount of the tax increase was always zero. The middle case was a smaller amount of sales, 0%-10%, and smaller tax increases, $15-$100. Fifteen different combinations were considered, varying the percentages sold and tax amounts with the restrictions on the tax amount for the largest amount sold and for the no sale cases imposed. A rank logit estimator was used to estimate the WTP measures.

The estimates developed for each policy alternative are based on using the estimated parameters from the model to compute a marginal WTP and rescaling that measure by the amount (in acres or sites) for the alternative considered. This “unit value” was then the basis for the WTP measures reported in the bottom panel in Table 1. Many questions could be raised about the implicit assumptions in our somewhat ad hoc benefit transfer strategy to develop these estimates. Because of the limited number of available estimates, we simply used their estimates for WTP and median income to provide another source for the measuring the park quality share.

As expected, given the difference in the object of choice used to measure the numerator of our WTP-neutral scaling factor, these ratios are dramatically larger than those based on the Parsons et al. (2021) study of specific southwestern parks. Neither study ideally matches our needs. Taken as a whole, these estimates simply provide an approximate range for what to expect in efforts using current research to estimate the WTP-neutral scaling factor. Given that the scale of the deferred maintenance and associated park quality is large, small overall improvements in the quality of the system are the likely result of any new funding source. Nevertheless, these “rough” estimates suggest a larger price increase (in percentage terms) would be consistent with a WTP-neutral effect, provided they actually raised sufficient revenue to support the quality enhancements. The implication of this assessment is that, based on the logic outlined in Figure 4, taxes that, as a percentage of gear prices, are larger than the park quality increases in percentage terms can still increase welfare for both buyers and sellers.