Abstract
We examine dedicated taxes (i.e., taxes on private goods used to finance public good provision) in a game-theoretic model of impure public goods. We show that a dedicated tax can increase or decrease demand for the taxed good. The optimal dedicated tax generally cannot achieve the Pareto-optimal allocation, but it can generate a conditionally efficient equilibrium with comparatively more or less public good provision, depending in part on complementarity or substitutability between the private and public good. We also demonstrate a neutrality result: when individuals can make direct donations, sufficiently low dedicated taxes will not impact equilibrium allocation.
1 Introduction
This article examines potential advantages and disadvantages of financing the provision of a public good by taxing a related private good. In the theoretical literature on financing public goods, common mechanisms that rely on centralized coordination are income or wealth taxes (often lump sum) or subsidies on private provision. Other approaches rely on the benefits principle, which suggests that individuals who benefit more from the public good should pay more for its provision. Toll roads provide a common example. Another example is that visitors to national parks pay more through admission fees.1 There are, however, reasons the direct benefits principle might not be desirable in many contexts, including distributional equity and administrative feasibility.
These concerns often lead to ideas about taxing related goods, a notion we refer to as a dedicated tax. For example, in lieu of monitoring the distance that drivers travel on public roads for purposes of taxation, gasoline taxes are often used to finance transportation infrastructure. Dedicated taxes are also considered in the context of parks and public lands. Rather than charging high admission fees, public lands and parks can be financed to some degree through taxes on related goods, such as gear taxes on outdoor equipment and hunting licenses.2 The intuitive appeal underlying such policies and proposals is that taxing seemingly related goods or services has advantages for financing public goods. In what follows, we provide a theoretical analysis to evaluate this intuition. We develop an approach for examining the positive and normative consequences of using dedicated taxes to finance public goods.
Our analysis is related to the seminal literature in public finance on the optimal supply of public goods when financed through distortionary taxes (e.g., Diamond and Mirrlees 1971; Stiglitz and Dasgupta 1971; Atkinson and Stern 1974). Part of our contribution is to show results in the context of an impure public good model. By explicitly linking the consumption of a private good with provision of a public good, dedicated taxes create an impure public good similar to that analyzed by Cornes and Sandler (1984, 1994, 1996).3 Using this framework, we can show in a direct and transparent way the incentives that dedicated taxes create, their efficiency consequences, and their potential scope for financing the provision of public goods.
We establish four main results. First, we show the conditions under which a dedicated tax can either increase or decrease demand for the taxed good. Second, we derive intuitive conditions showing how the Nash equilibrium under an optimal dedicated tax cannot achieve the Pareto-optimal allocation, except in the limiting case of a single agent, which is equivalent to assuming there is no public good. Part of the reason is that the dedicated tax does not eliminate the free-riding incentive, which affects consumption of the taxed private good and the public good. Third, we show that the equilibrium level of the public good can exceed or fall short of the Pareto-optimal level, depending in part on whether the public good and taxed private good are Hicksian complements or substitutes, respectively. Finally, we show a neutrality result whereby dedicated taxes that are sufficiently low will have no effect on the equilibrium allocation if individuals have the opportunity to simultaneously make voluntary contributions.
2 Private and Public Goods
To establish some preliminary intuition, we begin with a basic utility maximization problem where a representative individual chooses consumption of private goods while taking the level of a public good as exogenously given. In particular, individual i solves
[1]
where 
and 
are private goods with the respective prices, wi is the individual’s wealth, and X3 is a public good that is exogenously provided at level 
.4 We assume for the time being there is no opportunity for individuals to privately provide the public good. This assumption means that the constraint 
is redundant, but we include it in the statement of the problem with an eye toward the generalizations we consider later. In all cases, we will use 
to denote public good provision that is taken as exogenous by the agent in question. Because we are focusing initially on a single individual, we drop superscripts for now. The solution to equation [1] can be written as demand functions 
for goods j = 1,2.
Although the basic setup does not allow individuals to privately provide the public good, we can derive the individual’s marginal willingness to pay (WTP) for X3. Solving the dual of equation [1] yields an expenditure function 
. It follows that the individual’s marginal WTP for the public good, denoted π3, will itself be a function of the exogenous parameters and satisfy
[2]
This expression indicates how marginal WTP is equal to the compensating change in income for a marginal change in the quantity of the public good.
We consider how changes in the exogenously provided level of the public good affects demand for the private goods. That is, we are interested in what determines the sign of 
. While these results are interesting, the approach that we use for showing them helps provide the basis for the methods we use in subsequent sections.
We derive the comparative static results in terms of familiar price and income effects using notations of virtual prices and income.5
The first step is to consider an alternative utility maximization problem where the individual can choose the aggregate level of the public good X3 at a price equal to π3 in addition to the private goods. For the moment, π3 need not equal that defined in equation [2], but the connection will soon become clear. In this case, the individual’s budget constraint can be rewritten as 
, where the right-hand side is the individual’s virtual full income and represents the endowment plus the value of public good spill-ins. Let 
denote full income. By implicitly choosing π3, we can satisfy the following condition for j = 1,2:
[3]
This means that the solution to the “unrestricted” utility maximization problem, written in terms of demand for the private goods, will be identical to that for equation [1]. For clarity, recall that demand functions with a circumflex (or hat) denote solutions to the “restricted” utility maximization problem equation [1], whereas those without the additional notation on the right-hand side of equation [3] are the unrestricted solutions.
Satisfying equation [3] for both private goods also means that demand for the public good will be a knife-edge solution right at the corner such that
[4]
where the uppercase X3 denotes demand for the aggregate level of the public good, which is equal to the exogenously provided level given in equation [1]. It is worth stating that the value of π3 that satisfies equation [3] is equal to the marginal WTP defined in equation [2]. It is also worth pointing out the former is a Marshallian measure and the latter is a Hicksian measure, so the two will diverge with nonmarginal changes from the initial allocation at which they are defined.6
Differentiating equation [3] with respect to 
produces the comparative statics of interest, where marginal changes in the restricted demand functions can be written in terms of changes in the unrestricted demand functions:
[5]
As before, j indexes the good in question. Additional subscripts k = 1,2,3 represent partial derivatives with respect to the (virtual) prices of goods x1, x2, and X3, respectively, and the subscript μ denotes the partial derivative with respect to virtual income. The last equality comes from substituting in the Slutsky equation, and Cjk denotes the compensated (Hicksian) demand response for good j = 1,2,3 with respect to a change in price k = 1,2,3. Notice that this equation expresses the results in standard price and income responses for a familiar (unrestricted) problem.
The only things that remain to be solved are changes in the virtual magnitudes π3 and μ with respect to a change in 
. Following the procedure described by Cornes and Sandler (1996), this is possible using Cramer’s rule and the identifying equations for (1) the budget constraint, and (2) the level of the public good. We provide the details and solutions in Appendix A1, which can be substituted into equation [5] to yield
[6]
In this equation, the term X3μ captures how the individual’s unrestricted demand for aggregate X3 responds to a change in virtual income, while xjμ denotes the income effect on demand for private goods j = 1,2.
Several results follow. If all goods are normal in the usual unrestricted sense (i.e., X3μ > 0 and xjμ > 0 for j = 1,2), the full income effects on demand are not only positive but 1−π3X3μ > 0 because a unit increase in income must be spent on all goods. This means that equation [6] is positive if xj and X3 are Hicksian complements (i.e., Cj3< 0), whereas the sign is indeterminate if the two goods are Hicksian substitutes (i.e., Cj3> 0). With the special case of quasi-linear preferences of the form xk+F(xj,X3), equation [6] is positive or negative if xj and the public good are Hicksian complements or substitutes, respectively.7Together, these results show how demand for a private good changes with a change in the level of an exogenously provided public good, and whether the goods are complements of substitutes plays a critical role.
3 Pareto Efficiency
We consider an economy that consists of n ≥ 2 individuals and solve for the efficient level of the public good. This provides an important point of comparison for our subsequent consideration of a dedicated tax mechanism. We assume the cost of providing the public good is unity, and without loss of generality, we normalize the level of the public good that does not come through private provision to zero. The aggregate level of the public good is thus defined as 
.
Solving for the set of Pareto-optimal allocations is a matter of standard practice in public economics. All the efficient allocations will satisfy the following first-order conditions:
[7]
where for completeness we include the derivation in Appendix A2. The first condition is the well-known Samuelson condition, where the sum of the marginal rates of substitution between the public good and all private goods equals the corresponding price ratios. The second is the standard condition for private goods, where the marginal rates of substitution between goods equals the price ratio for all individuals.
We simplify things further by assuming identical preferences across all individuals and focusing on the symmetric allocation. The symmetry is helpful for our comparisons that follow and is also the allocation that a social planner would choose with equal welfare weights across individuals. With these assumptions, the unique solution will satisfy
[8]
which is a special case of the conditions in equation [7]. To be clear, this implies the same allocation of private goods for each individual, defined as 
, where asterisks denote the solution for the social planner’s problem. It also defines a unique level of the public good:
[9]
We can verify the standard result that lump-sum taxes can be used to implement the Pareto-optimal allocation. With individualized taxes τi, each individual’s utility maximization problem is
and 
with the corresponding first-order condition:
[10]
Setting each individual’s tax such that 
has two implications. The first is that 
by equation [9], 
satisfies each individual’s budget constraint and equation [10], which is equivalent to the second condition in equation [8]. This means that imposing 
for all i implements the Pareto-efficient and symmetric allocation as an equilibrium outcome. Note the possibility that 
can be a subsidy in some cases if an individual’s endowment is sufficiently low. To simplify things even further in what follows, we make the additional assumption of identical endowments w across individuals. In this case, the optimal lump-sum tax is a uniform “head” or “poll” tax that satisfies
The Pareto-optimal allocation, which we have verified can be implemented with lump-sum taxes, will provide a useful benchmark for our analysis.
4 Dedicated Tax
We turn to a dedicated tax mechanism to finance the public good, which we assume applies to good x2 without loss of generality. In particular, we model a tax rate of t2 per unit x2, and all proceeds are used to finance the provision of X3.8 Examples of such dedicated taxes include the gear taxes to fund parks and real estate transfer taxes to fund the acquisition of open space lands. Other examples, which are not based on government provision, are a number of cases where private goods are bundled with contributions to public goods, such as the 1% For the Planet Program.9
Individual Behavior
We begin with a representative individual’s utility maximization problem taking the exogenously given level of the public good 
which now represents the provision of all others, as given. The individual’s problem can be written as
[11]
Assuming an interior solution, the first-order conditions can be combined to
[12]
This has an intuitive interpretation: the ratio of the marginal utilities (benefits) of the two goods equals the price ratio. The difference here from a more typical setup is that x2 is linked to X3 through t2, which defines the relative and constrained quantities and the tax-inclusive price. As shown in the numerator on the left side, consumption of an additional unit of x2 provides the marginal benefit of U2plus t2U3 Suppressing notation for prices and w, we can fully characterize the solution to equation [11] as the function 
. The choice of 
is then defined through the budget constraint.
An interesting feature of the setup in equation [11] is that for any given level of t2, it is a special case of Cornes and Sandler’s (1984, 1994, 1996) impure public good model. This follows because consumption of the taxed private good becomes associated with joint production of the public good. Nevertheless, the comparative static properties of the model will differ because here the price of the jointly produced good and the technology of joint production are not independent parameters, as they are both functions of t2.10
It is interesting and useful to consider how demand for x2 changes with imposition of the dedicated tax. Keep in mind this is not simply a price effect, as an increase in the tax simultaneously provides the public good. We again use virtual prices and income to derive results in terms of familiar price and income effects. In Appendix A3, we derive the following general result:
[13]
where 
, and 
, and the signs of these latter two expressions follow by negative semi-definiteness of the Slutsky matrix.11
Because equation [13] is rather cumbersome, we consider the special case of quasi-linear preferences to illustrate the different possibilities. Suppose preferences take the form x1 + F(x2,X3). Let us simplify even further by considering a marginal increase in the tax from a starting point of t2 = 0. With these simplifications, equation [13] becomes
[14]
and this establishes several results about how demand for x2 will respond to imposition of a dedicated tax t2.
Table 1 summarizes the qualitative results, showing how they depend in part on the individual’s marginal WTP for the public good at the initial allocation.12 Consider the knife-edge case where the individual’s marginal WTP exactly equals the per unit cost of providing the public good (i.e., π3 = 1). If the tax-linked goods are Hicksian complements or substitutes, demand for the private good will increase or decrease, respectively. Notice that an increase in demand for the private good is rather counterintuitive because imposing a tax increases demand. Interestingly, this suggests that sellers of good x2 might actually benefit from the dedicated tax, which is a possibility that Banzhaf and Smith (2022) explore in greater detail.
The same qualitative results occur if π3 is greater (less) than unity, and the linked goods are complements (substitutes). As shown in the cells with question marks in Table 1, ambiguity occurs when there are effects that push in different directions; that is, when the marginal WTP is less (greater) than unity, and the linked goods are complements (substitutes). Recall, however, that we are considering the special case of quasi-linear preferences to illustrate the range of possibilities. In a more general setting, these different possibilities might occur in each case. The primary insight is that imposing a dedicated tax on a private good to provide a public good can affect demand for the private good in what are likely to be unexpected ways.
Nash Equilibrium
We consider a setup where all n individuals in the economy are simultaneously engaged in private provision of the public good through consumption of the private good subject to the dedicated tax. The model’s setup implies that individuals are playing a game for any given level of the dedicated tax. We therefore consider equilibrium existence and uniqueness for any given tax rate.13
We can write each individual’s demand for private provision as 
, which is each individual’s best-response function.14 A Nash equilibrium is a fixed point at the intersection of all n best-response functions. Equivalently, a Nash equilibrium is a set of choices 
for all i that satisfies the first-order condition [12] for all n individuals with 
. Note that, without loss of generality, we are still normalizing the level of the public good that does not come through private provision to zero.
Kotchen (2007) establishes a sufficient condition for equilibrium existence and uniqueness in the general impure public good model, and as noted, the setup here is a special case for a given level of t2. The condition is based on the slope of each individual’s demand for the aggregate level of the public good with respect to the provision of others. In particular, using the notation here, the sufficient condition is
[15]
where 
is an individual’s demand for the aggregate level of the public good, and the bridge between the two equal expressions is based on the identity 
. Equation [15] implies that an increase in spill-ins (i.e., 
must increase demand for the public good and not decrease demand for the untaxed private good x1. This is essentially a normality assumption with respect to full, virtual income. A further implication is that best-response functions have slopes less than zero and greater than −1, and this monotonicity combined with continuity ensures the existence of a unique fixed point (Kotchen 2007).
With identical individuals, a further implication is that the equilibrium will be symmetric, and we denote it simply as 
for each individual, where the superscript N is used to denote a Nash equilibrium quantity. It follows that the aggregate, equilibrium level of public good provision will satisfy 
, which shows how the solution can be written in terms of each individual’s level of private provision or demand for the taxed private good.
Before turning to the optimal dedicated tax, it is helpful to establish an intermediate result. We have shown that demand for x2can be increasing or decreasing in response to imposing a dedicated tax t2. The question that we consider now is whether 
in equation [13] can take either sign while still satisfying the assumption in equation [15]. The reason is that maintaining both possibilities creates some interesting results that we derive in the next section. In general, the answer is yes, which we show in Appendix A4. This means that even imposing the constraint on individual behavior that is sufficient for a unique Nash equilibrium, it is still possible for 
to be either positive or negative.15
5 Optimal Dedicated Tax
We consider the optimal dedicated tax that a social planner would choose. We also compare the resulting allocation with the Pareto-optimalal location defined in Section 3. We thus compare implications of the allocation consistent with the optimal dedicated tax to that which is first-best, regardless of the policy instrument.
It is important to recognize that with any dedicated tax, individuals continue to play a noncooperative Nash game that the planner must take into account when choosing t2. We can write the planner’s objective as
[16]
where a key feature of this statement of the problem is that 
is consistent with the equilibrium that arises given any choice of the dedicated tax level.
The first-order condition that defines the solution can be written as
[17]
To build intuition, first consider the special case where 
, so that equation [17] simplifies to 
. This expression equates the social marginal benefit of greater public good provision for an individual and the cost of forgone consumption of x1 More generally, however, the equilibrium quantity 
will change with a change in the dedicated tax t2.
It is also the case that the sign of 
can be positive or negative, in much the same way that we showed previously how the individual’s demand response in equation [13] can be positive or negative.16 Equation [17] thus shows that the optimal dedicated tax is set where the social marginal benefits and costs are equated after accounting for the net change in quantities due to (1) the direct effect of the change in t2 and (2) the indirect effect on account of changes in equilibrium demand for the taxed private good 
.
Efficiency
We make comparisons between the Pareto-optimal allocation defined in Section 3 and the allocation implied by the optimally set dedicated tax. In particular, the analysis is based on a comparison between the conditions specified in equations [8] and [12], where the latter is evaluated with a dedicated tax that solves equation [16]. The most straightforward way to establish the key result is to begin by assuming the optimal dedicated tax does in fact implement the Pareto-optimal allocation. This means that 
, which by definition satisfies both conditions in equation [8]. Substituting the second condition of equation [8] into equation [12] and rearranging yields 
, and it follows immediately that this equation can match the first condition in equation [8] only if n = 1. This means that the efficient dedicated tax can implement the Pareto-optimal allocation only in the trivial case of n = 1. The simple intuition in this special case is that the planner can set the tax to exactly balance the individual’s preferred consumption of the public good and taxed private good.
More generally, the preceding steps prove a clear result: for n≥ 2, the optimal dedicated tax cannot achieve the Pareto-optimal allocation, which we have shown could arise with lump-sum taxation. The dedicated tax is therefore a second-best policy in cases where there is an actual public good. Intuition follows from at least two observations. The first is that imposing a dedicated tax layers an additional constraint on how the planner chooses the Pareto-optimal allocation, so if the constraint is binding, the dedicated tax cannot be first best. The second is that a dedicated tax does not eliminate free-riding incentives. To show this, we simply note that with a dedicated tax in place, the private marginal benefit of consuming the taxed private good is U2 + t2U3, whereas the greater social marginal benefit is U2 + nt2U3, which is the same only in the special case of n = 1.
Level of Public Good Provision
We turn our focus to implications for the level of public good provision. Just because the dedicated tax is generally a second-best policy does not tell us whether an optimally chosen dedicated tax will implement more or less of the public good compared with the Pareto-optimalal location. Indeed, provision of the public good may in fact be the primary motive for considering a dedicated tax (in contrast to an efficiency objective).
To illustrate these results, it is useful to further simplify the condition defining the optimal dedicated tax. Substituting the equilibrium condition [12] into equation [17] and rearranging yields
[18]
where 
is the elasticity representing the percentage change in each individual’s equilibrium level of demand for the taxed private good, given a percentage change in the tax rate. Notice that the elasticity differs from a standard demand elasticity for two reasons. First, as mentioned, a change in the tax rate is not simply a change in the price because the revenue is used to provide the public good that directly enters the individual utility functions. Second, the elasticity is for an equilibrium response and not simply a demand response, meaning that the change in behavior of all others is taken into account.17 In this respect, 
captures the causal effect of the tax change on behavior and is therefore related to the “policy elasticity” described by Hendren (2016).18
What does the sign of 
imply about the equilibrium level of public good provision compared to the Pareto-optimal level? In Appendix A6, we derive the results summarized in Table 2 under the assumption of strict concavity of the utility function in all three arguments. The table compares the relative magnitudes of the equilibrium quantities for the optimal dedicated tax with those of the Pareto-optimal allocation. The signs in each cell indicate whether the dedicated-tax equilibrium quantity (of the tax private good or the public good) is greater than (+) or less than (-) the first-best quantity. The different columns correspond to different ranges of 
.
To build intuition for these findings, let us begin with the identity 
. Differentiating with respect to t2 and rearranging, it follows that
This means, for example, that a marginal increase in the dedicated tax will increase (decrease) the equilibrium level of the public good depending on whether
(less) than −1.19 When the elasticity is less than −1, we see in Table 2 that the dedicated tax results are at a relatively lower level of the public good. This is intuitive because increasing the tax further would only serve to lower the level of the public good. When the elasticity is greater than zero, the equilibrium level of the public good is greater with the dedicated tax, and this is driven by the way that increasing the tax has the additional effect of increasing demand for the taxed good. Finally, in the intermediate inelastic case, where 
, either result is possible because increasing the tax decreases demand for the taxed private good, yet the quantity of the public good still increases.
6 Generalization with Direct Donations
We have so far assumed that the only way individuals can provide the public good is through consumption of the taxed private good. Here we generalize the model’s setup to allow the possibility for individuals to make a direct contribution to the public good in addition to the taxed private good. The expanded choice set, with multiple channels for public good provision, is similar to that in Kotchen (2005, 2006) and Chan and Kotchen (2014). An example setting where the setup applies is a community with a real estate transfer tax that is used to fund the acquisition of conservation lands, while a land trust is operating with the same objective. This means that individuals are subject to the tax, which provides the public good, while also having the opportunity to make voluntary contributions to the land trust. We consider how this might change the results.
Prior to implementation of a dedicated tax, we can write the individual’s problem as
where the difference here is that the individual can choose x3 directly. This is the standard setup for private provision of a public good (Bergstrom, Blume, and Varian 1986). Substituting the budget constraint into the maximand and choosing x2 and x3, the Kuhn-Tucker first-order conditions are
[19]
where c.s. denotes the complementary slackness condition. Note that the Pareto-optimal allocation with this setup remains the same as defined in Section 3.
An equivalent and useful way to write the individual’s problem is with the implicit choice of the aggregate level of public good provision:
and 
,
where the corresponding first-order conditions are identical to those above. Following convention for the pure public model (see Bergstrom, Blume, and Varian 1986), we can write the solution as 
, where we have suppressed notation for prices. The argument in f (·) is the individual’s full income: wealth plus the value of public good spill-ins. The standard assumption is to assume normality of all goods, in which case 0 < f ′ ≤ 1, where the inequality holds strictly for an interior solution. This guarantees equilibrium existence and uniqueness. This follows because best-response functions 
are continuous and nonincreasing, which gives a unique fixed point for each individual’s direct contribution (reasons identical to those described in Section 4).
The first-order condition establishing the trade-off between x1 and x2, along with the budget constraint, then defines each individual’s choice of the private goods. The equilibrium level of the public good will therefore satisfy 
, where the overbar represents the equilibrium quantity in this new setting with direct donations. Moreover, each individual’s equilibrium choice is denoted 
, and it follows by definition and symmetry that 
.
We now introduce the dedicated tax to this more general setup. The individual’s problem is
which shows the ways to potentially provide the public good: through consumption of x2, through a direct donation x3, or both. In this case, the Kuhn-Tucker first-order conditions are
[20]Notice that only the first condition in equation [20] differs from that in equation [19]. Assuming there is an interior solution, we can substitute the second condition in equation [20] into the first and find that the conditions defining an equilibrium are identical to those in equation [19]. This establishes a neutrality result: implementing a dedicated tax has no effect on the equilibrium if individuals continue to purchase the private goods and make a direct donation after the tax is imposed. The intuition is that provision through the dedicated tax crowds out direct donations one for one.20
To see the crowding out of one’s direct donations more directly, consider how the equilibrium level of the public good at an interior solution will satisfy
. Totally differentiating, and recognizing that 
because of the same conditions in equations [19] and [20], we have 
. Because the equilibrium allocation of all three goods does not change, it must also hold that 
, meaning that 
. This illustrates how direct donations are decreasing in the dedicated tax rate such that the crowding out is one for one. This follows because, given a change in t2 the change in an individual’s contribution to the public good through consumption of the taxed private good is exactly 
.21
A further implication is that we can define the threshold value 
, which is the level of the tax that exactly reduces direct donations to zero and maintains the same equilibrium allocation. We have thus shown complete neutrality for all 
. Things change, however, for dedicated tax levels greater than 
, where an additional increase in the tax rate will render direct donations irrelevant as they are completely crowded out by tax revenues. Thus, for 
, the model with direct donations reduces to the simpler setup in previous sections.
7 Conclusion
This article examines some of the advantages and disadvantages of using dedicated taxes to finance the provision of public goods. We show how the impure public good model provides a useful way for understanding the positive and normative consequences of dedicated taxes. We began by showing how imposing a dedicated tax can, somewhat counterintuitively, increase or decrease demand for the taxed good. We then showed how the optimal dedicated tax generally cannot achieve the Pareto-optimal allocation. It can, however, generate a conditionally efficient equilibrium with comparatively more or less of the public good, depending in part on whether the public good and the taxed private good are Hicksian complements or substitutes, respectively. We also show a neutrality result: when individuals have the opportunity to make direct donations, dedicated taxes that are sufficiently low will have no effect on the equilibrium allocation.
Several of these results may help illuminate the potential political economy and policy implications of dedicated taxes. For example, the possibility that dedicated taxes can stimulate demand for the taxed good suggests suppliers may actually support dedicated taxes (see also Banzhaf and Smith 2022). Indeed, imposing dedicated taxes may provide a mechanism with benefits akin to corporate social responsibility that links public and private goods (Besley and Ghatak 2007), where all producers must meet the same standards rather than having them be met voluntarily. A similar mechanism, albeit voluntary in membership, is the 1% For the Planet initiative. In this case, the dedicated tax is effectively 1% of sales.
The neutrality result also highlights one of the potential risks of dedicated taxes: the crowding out of direct donations. Consider, for example, how federal funding for wildlife and habitat conservation is often disbursed contingent on matching funds from states, as is the case with the Pittman-Robertson and Dingell-Johnson Acts. Although states typically raise matching funds from hunting and fishing licenses, they also rely on contributions from private organizations, with Ducks Unlimited being a frequent and significant partner with state agencies for securing federal money. Our findings point to the potential unintended crowding out that may accompany a state’s greater reliance on dedicated taxes. If contributions from organizations such as Ducks Unlimited are crowded out, the dedicated taxes may have limited or no effect on total conservation funding. Along these lines, Walls and Ashenfarb (2022) also warn of a second dimension of crowding out, whereby dedicated taxes may crowd out allocations from general revenues.
Acknowledgments
We are grateful to Spencer Banzhaf, Daniel Phaneuf, and two anonymous reviewers for comments on earlier drafts that improved the article. We also appreciate the feedback from participants at the Property and Environment Research Center’s workshop “What Price to Play? The Future of Outdoor Recreation and Public Land Funding” in Bozeman, Montana, on November 7-9, 2019, and for which the article was originally written.
Footnotes
Appendix materials are freely available at http://le.uwpress.org and via the links in the electronic version of this article.
↵1 See the articles by Ji et al. (2022) and Lupi, von Haefen, and Cheng (2022) in this issue that provide empirical analyses of user fees for public lakes and beaches, respectively.
↵2 Banzhaf and Smith (2022) and Walls and Ashenfarb (2022), which are also in this issue, provide background on the history of funding for public lands in the United States, including a discussion of excises on hunting and fishing gear and proposals for broader based gear taxes.
↵3 Earlier articles on related topics include Brownlee’s (1961) discussion of funding public goods and services at least partially through sales receipts and subsequent formalizations of the idea by Cicchetti and Smith (1970) and Holtermann (1972).
↵4 The setup builds on the modeling tradition in the context of privately provided pure and impure public goods, which assumes a fixed endowment of wealth and thereby abstracts from the labor/leisure choice that is central to much of the literature on optimal taxation.
↵5 This is the approach used by Cornes and Sandler (1994, 1996) in their study of the impure public good model, but an earlier analysis that uses the same general approach for comparative static analysis of private and rationed goods is found in Madden (1991).
↵6 See Neary and Roberts (1980) for a related discussion.
↵7 These results provide a specific application of the findings in Madden (1991) about the symmetry of substitute and complement relationships between goods based on changes in prices or quantities of a rationed good, which in this case is the level of the public good.
↵8 We continue to assume that individuals are not able to make direct donations to provide the public good, though we relax this assumption in Section 6.
↵9 Business members of 1% For The Planet commit to donating 1% of gross sales to environmental nonprofits; see https://www.onepercentfortheplanet.org/.
↵10 In particular, using the notation in Cornes and Sandler (1994), the relationship between models follows by setting γ = t2 and p = p2 + t2.
↵11 The Slutsky matrix is equivalent to the Hessian of the expenditure function (i.e., the matrix of derivatives of the compensated demand functions), and we assume the weak inequalities implied by negative semi-definiteness hold strictly.
↵12 See Banzhaf and Smith (2022) for a similar set of results, though motivated with a different setup.
↵13 Our examination of the public good equilibrium is one way our analysis differs from that in Banzhaf and Smith (2022). Although they do consider an equilibrium among suppliers, we simplify the supply side of the market by assuming fixed and exogenous prices.
↵14 Each individual’s demand for the aggregate level of the public good follows by definition:
. Furthermore, recall that lowercase variables d. enote individual consumption or provision, the uppercase X3 denotes aggregate provision, and the circumflex denotes the restricted demand function.↵15 The possibility for
is perhaps somewhat less surprising because as described earlier, t2 operates in part like an increase in price. But we have also discussed how it has the additional effect of increasing the level of the public good for a given level of x2, and the sign of the comparative static depends on the signs and relative magnitudes of multiple price and income effects. Given the condition in equation [15], we show in Appendix A4 that if π ≤1 then assuming the two goods are normal and Hicksian substitutes sufficient for . However, to obtain. the opposite sign with both goods still being normal, a necessary condition is for Hicksian complements, where in particular 0 >C22 > C32(=C23) > C33. Intuitively, this corresponds to the case where x2 has a relatively small own-price response, and it aligns with the strong (Hicksian) complements case described by Cornes and Sandler (1996, 267), where crowding-in can occur.↵16 See Appendix A5 for details on the possible relationship between the signs of
and (i.e., an individual’s demand response and the equilibrium demand response that accounts for the response of all other individuals).↵17 In Appendix A5, we show the exact relationship between the equilibrium elasticity nd the demand respo. nse elasticity, denoted In general, we find n. othing that rules out the possibility for the equilibrium elasticity to take either sign, and in most cases, it will have the same sign as the demand response elasticity.
↵18 The linkages between our analysis and that of Hendren (2016) would be an interesting avenue for further investigation.
↵19 Satisfying equation [18] means that for an interior solution, it must hold that the elasticity satisfies .
↵20 Kotchen (2006) shows that group size must be sufficiently small for interior solutions in the presence of an impure public good and the opportunity to make a direct donation. The same result applies with the setup here, which means that the neutrality result depends on a sufficiently small n. With large n, direct donations will be fully crowded out, and the model reverts back to that considered in the previous sections. We believe that analyzing how properties of dedicated taxes with public goods depend on group size is a question worthy of additional research, drawing on some of the results found in Andreoni (1988) and Kotchen (2006).
↵21 A well-known result in the charitable-giving literature is that one-for-one crowding out no longer occurs in the model of impure altruism (Andreoni 1989, 1990). In the setting considered here, however, the crowding effects continue to hold exactly if warm-glow benefits are derived equally from the public good provision that occurs through either directdonations or the dedicated tax. This follows because both alternatives are perfect substitutes with respect to public good provision at interior solutions. Nevertheless, the perfect substitutability will no longer hold if only one form of giving (e.g., direct donations) confers warm glow benefits and the other (e.g., indirect donations via tax payments) does not. In such cases, the crowding effect will be mitigated in ways consistent with that underlying the standard results in the literature.
