Sweet and Not-So-Sweet Charity

Abstract

In order to maximize social welfare, governments should subsidize at different rates the private contributions made to different categories of charitable or publicly beneficial civic activity. This prescription stands in contrast to the standard practice of subsidizing such contributions uniformly and suggests the need to evaluate not only that practice but also the diverse initiatives of several governments to subsidize at higher rates certain contributions made to certain categories. The case for differential subsidies is made here with reference to a model in which the government chooses the subsidies and amounts of direct funding, as well as a personal transfer and linear taxes on earnings and consumption. The categories differ in terms of the individuals contributing, the individuals receiving the charitable goods and services, and the effectiveness of direct funding relative to contributions. A linear expenditure system illustrates the restrictive conditions necessary for uniform contribution subsidies being optimal.

Keywords

charitable contributions philanthropy differential subsidies tax expenditures optimal taxation

Introduction

In many jurisdictions, governments subsidize private financial contributions made to certain civic activities that have been officially identified as charitable or publicly beneficial. The subsidies reduce the price to the donor of providing an eligible donee with a certain quantity of financial resources, either by offering the donee a matching grant, or by offering the donor a deduction from taxable income, or a credit against taxes due.

Most governments that apply these subsidies do so uniformly, treating all contributions as the same. In recent decades, however, several national and subnational governments have considered or introduced a range of initiatives to apply the subsidies differentially, setting higher rates on certain contributions to certain categories.¹ The United States is no exception to this. In 2001, for example, President George W. Bush proposed encouraging state governments to provide a credit—in addition to the federal deduction—on contributions to third-sector organizations “addressing poverty and its impact” (Bush 2001, 11). Such encouragement was to have been in the form of legislation that enabled the states to offset the lost revenues using funds from the federal block grants provided under the Temporary Assistance to Needy Families program, legislation that Congress did not pass. Nevertheless, several state governments acting independently have introduced programs to apply higher subsidies on certain contributions. For example, individuals in Arizona have received since 1996 a credit on new contributions to third-sector organizations offering services to low-income residents, and since 2006, a credit on contributions either to public schools for extracurricular events or to private schools for scholarships and bursaries (Arizona Revised Statutes Title 43-1088, 43-1089). In Florida, the contributions of individuals and corporations have attracted since 2006 a matching grant if made to public universities and colleges for bursaries that support students for whom neither parent has a university degree or college diploma (House Bill No. 795, Chap. 2006-73).

In spite of such initiatives, differential contribution subsidies have received scant attention in the public policy literatures that deal with financing the third sector. If acknowledged at all, they are usually viewed as a hypothetical fiscal tool, not an actual one, and then are either ignored or dismissed (Carmichael 2010).² The economics literature on philanthropy is a case in point.³ Within that literature, the theoretical subfield that models the optimal tax treatment of contributions to civic activities typically precludes considering differential subsidies from the outset by assuming all such activities to be the same in terms of the individuals contributing, the goods and services provided, and the individuals receiving those goods and services (e.g., Feldstein 1980; Andreoni 1990; Saez 2004; Diamond 2006). If differential subsidies are mentioned at all, it is not in a central argument or formal model but rather in a concluding remark (Feldstein 1980, 120) or a footnote (Saez 2004, 2664 n.14).

This article responds to this inattention toward differential subsidies. It models the optimal tax treatment of private contributions to identifiable and distinct categories of civic activity, and derives the necessary conditions for the optimal subsidies on contributions being uniform. The goods and services under each category of civic activity are assumed to be financed by a combination of private contributions and government funding. The categories differ in terms of the individuals contributing, the individuals receiving the goods and services, and the effectiveness of direct funding relative to contributions in producing the goods and services. The government chooses its fiscal tools (the subsidies, amounts of direct funding, taxes on earnings and consumption, and a personal transfer) in order to maximize social welfare, defined as the weighted sum of individuals' well-being arising from their consumption.⁴ All taxes and subsidies are assumed to be linear.⁵

What emerges from the model is a theoretical case for differential subsidies that is more general than what might be inferred from the existing literature. As has been demonstrated elsewhere, the optimal subsidy on contributions comprises two components: a Pigovian subsidy that reflects the gain in social welfare resulting from individuals other than the contributor benefiting from the contribution and a shadow tax that reflects the loss in social welfare resulting from the government being restricted to taxes and subsidies that are distortionary.⁶ However, as demonstrated here, only under very restrictive conditions would those two components be uniform, and thus compose a uniform contribution subsidy across all categories of civic activity. In the presence of an optimal linear income tax, these conditions involve the ones needed for optimal commodity taxes being uniform: specifically, individuals differing in their productivity and nonlabor income but having identical preferences that are weakly separable between contributions and leisure and that generate parallel Engel curves. In addition, the conditions require the direct funding in each category to be greater than zero, have the same relative effectiveness as in other categories, and generate the same marginal tax revenue as in other categories. A linear expenditure system illustrates that if these conditions are not satisfied, then differential subsidies would be optimal.

Model

The model has been made syncretic in order to situate it within the existing literature. By depicting contributions as commodities that generate benefits or utility for the contributors, and in considering whether the subsidies on contributions should be uniform or differential, the model draws from papers that derive conditions under which uniform commodity taxation would be optimal, in the presence of an optimal linear income tax (Atkinson 1977; Deaton 1979). By also depicting contributions as commodities that generate benefits for individuals other than for the contributors, it draws from papers that consider the optimal taxation of commodities in the presence of externalities (Sandmo 1975; Cremer, Gahvari, and Ladoux 1998). Finally, by depicting categories of civic activity as receiving not only private contributions but also government funding, it draws from papers that consider the optimal expenditures on public goods in the presence of distortionary taxation (King 1986; Boadway and Keen 1993). Conceptually, the model is close to that of Saez (2004), apart from it assuming distinct and identifiable categories of civic activity. For expositional purposes, however, the model is similar in structure to those of Diamond (1975) and Dixit and Sandmo (1977). This similarity permits consistency in the definitions of the optimal policy variables, including a generalization of the Samuelson rule for government direct funding: each definition makes reference to the marginal efficiency costs and distributional advantages involved, representing the latter by a covariance expression.

The Individual

Let each individual h derive utility according to the following function:

u^{h} = u^{h} (z^{h}, c^{h}, x^{h}, g^{1 h}, \dots, g^{J h}, Γ^{1}, \dots, Γ^{J}), h = 1, \dots, H,

where h can choose all arguments except the last J of them. The chosen arguments are earnings z^h and three forms of commodity: nontaxed consumption c^h , which is the numeraire; consumption x^h ; and contribution g ^jh made to J categories of civic activity.⁷ Contribution g ^jh is the quantity of financial resources going to category j as a result of expenditure by individual h. It exceeds or falls short of that expenditure, as the government subsidizes or taxes the contributions to that category (see note 11). The last J arguments of equation (1) are the levels of goods and services under the J categories. It is assumed that the utility function is both concave and increasing in all arguments except earnings (the supply of labor time is costly in terms of the leisure time foregone). Individuals differ in their productivity as measured by their wage, and hence in their ability to convert labor time into earnings.

Contributions are assumed to improve social welfare by generating two types of benefits: internal and external. By definition, the internal benefits are those that improve the well-being of the contributor alone, as a consequence of his equipping particular categories of civic activity with financial resources. These benefits depend on his personal preferences and circumstances, and thus could be associated with motives that range from the altruistic to the self-promoting. They might involve the so-called warm glow he feels as a consequence of anonymously doing good (Boulding 1962; Andreoni 1990). Alternatively, they might center on his receiving either some quid-pro-quo commemoration, or a share of the goods or services financed by his contribution. They might derive from his affiliating with a purpose or organization he holds in esteem. Alternatively, they might originate from his avoiding the opprobrium or attracting the approval of his peers, his conscience, or God. The internal benefits are represented in the model by g^jh being an argument of equation (1).

By definition, the external benefits improve the well-being of individuals other than of the contributor. They might be associated with other individuals receiving firsthand the goods and services financed in part by his contributions (e.g., criminal rehabilitation), others benefiting vicariously (e.g., knowing that such rehabilitation is available), or still others receiving goods and services as a consequence of the initial ones having been received firsthand (e.g., lower crime rates). The external benefits are represented in the model by Γ^j, the level of goods and services under category j, being an argument of equation (1), regardless of how or by whom it is financed.

It is assumed that Γ^j is financed by a combination of private contributions and government funding according to the relationship:

Γ^{j} = s^{j} G^{j} + \tilde{G}^{j}, j = 1, \dots, J,

where for category j,

G^{j} \equiv \sum_{h} g^{j h} \geq 0

is the sum of contributions and

\tilde{G}^{j} \geq 0

is the amount of direct funding.⁸ The parameter s ^j measures the difference in the effectiveness of direct funding relative to contributions in producing goods and services of a given quantity or quality.⁹ The number of individuals H is assumed to be sufficiently large and their separate contributions sufficiently small such that each chooses earnings, consumption, and contributions taking Γ^j as given.¹⁰

Individual h chooses c^h , x^h , g^jh , and z^h in order to maximize (1a) subject to the constraint:

c^{h} + [1 + t^{x}] x^{h} + \sum_{j} [1 + t^{j}] g^{j h} \leq [1 - t^{z}] z^{h} + {\overline{y}}^{h} + y,

where t ^z and t ^x are, respectively, the taxes on earnings and consumption; t ^j is the subsidy on contributions to category j;

\overline{y}^{h}

is his nonlabor income which is taken to be exogenous and untaxed; and y is a common lump-sum tax or transfer.¹¹ The outcome of these choices can be represented by the indirect utility function:

v^{h} (y, 1 - t^{z}, 1 + t^{x}, 1 + t^{1}, \dots, 1 + t^{J}, Γ^{1}, \dots, Γ^{J}), h = 1, \dots, H .

The total quantities of earnings, consumption, and contributions can be represented by demand functions:

\begin{array}{l} Z (y, 1 - t^{z}, 1 + t^{x}, 1 + t^{1}, \dots, 1 + t^{J}, {\tilde{G}}^{1}, \dots, {\tilde{G}}^{J}) = Z \equiv \sum_{h} z^{h}, \\ X (y, 1 - t^{z}, 1 + t^{x}, 1 + t^{1}, \dots, 1 + t^{J}, {\tilde{G}}^{1}, \dots, {\tilde{G}}^{J}) = X \equiv \sum_{h} x^{h}, \\ G^{j} (y, 1 - t^{z}, 1 + t^{x}, 1 + t^{1}, \dots, 1 + t^{J}, {\tilde{G}}^{1}, \dots, {\tilde{G}}^{J}) = G^{j} \equiv \sum_{h} g^{h j}, j = 1, \dots, J \end{array}

with means

\overline{Z} \equiv Z / H,

\overline{X} \equiv X / H,

and

Â \overline{G}^{j} \equiv G^{j} / H .

The Government

The government chooses the lump-sum tax or transfer, the taxes and subsidies, and the amounts of direct funding in order to maximize an individualistic social welfare function $W (v^{1}, \dots, v^{H})$ , subject to the constraint of a balanced budget.¹² The Lagrangian is as follows:

W (v^{1}, \dots, v^{H}) + λ [t^{z} Z + t^{x} X + \sum_{j} t^{j} G^{j} - y H - \sum_{j} \tilde{G}^{j}],

where λ is the marginal cost of raising government revenue, measured in units of social welfare.¹³

Lump-sum Tax or Transfer

The first-order condition for the optimal choice of the common lump-sum tax or transfer y is as follows:

\sum_{h} W_{h} [v_{y}^{h} + \sum_{j} v_{Γ j}^{h} s^{j} G_{y}^{j}] + λ [t^{z} Z_{y} + t^{x} X_{y} + \sum_{j} t^{j} G_{y}^{j} - H] = 0,

where W_h ≡ ∂W/∂v^h > 0 is the social marginal utility for individual h. Subscripts denote partial derivatives.

One can simplify this first-order condition by introducing versions of three co-coordinating variables found in the literature. First, define Eⁱ as the social marginal utility of the goods and services under a particular category i of civic activity, where

E^{i} \equiv \sum_{h} W_{h} v_{Γ i}^{h}, i \in 1, \dots, J .

This takes into account the number of individuals who receive the goods and services, the extent to which they benefit, and the priority the government places on their well-being.

Second, define τⁱ as the shadow tax on contributions to category i, where

t^{i} \equiv τ^{i} - s^{i} E^{i} / λ, i \in 1, \dots, J .

The shadow tax measures the difference between the subsidy tⁱ and the Pigovian subsidy

- s^{i} E^{i} / λ

which reduces the price of contributions by the value of the social marginal utility they generate.

Third, define $μ^{h}$ as the social marginal utility of income for individual h, where

μ^{h} \equiv W_{h} v_{y}^{h} + λ [t^{z} z_{y}^{h} + t^{x} x_{y}^{h} + \sum_{j} τ^{j} g_{y}^{j h}]

with mean

\overline{μ} \equiv \sum_{h} μ^{h} / H .

The first term on the right side of equation (10) is the change in social welfare attributable to the utility that h would derive from an increment of income. The second term measures the change in social welfare—apart from that which is attributable to the utility of h—made possible by that increment. This includes the effects of not only the government revenues gained or lost by h’s marginal propensity to spend on earnings, consumption, and contributions, but also the goods and services gained or lost by his marginal propensity to contribute. The mean

\overline{μ}

measures the change in social welfare made possible by providing all individuals with the same increment of income. It is assumed that the government, in defining its distributional objectives, places sufficient priority on the well-being of the poor such that if v_y ^h′ > v_y ^h″ , then μ^h′ > μ^h″, for h′, h″ ∈ 1, . . . , H.

Using the co-coordinating variables Eⁱ , τⁱ, μ^h, and $\overline{μ}$ one can rewrite the first-order condition for the optimal choice of y as follows:

λ = \overline{μ} .

Thus, the government, in choosing the optimal lump-sum tax or transfer, sets the social cost of raising an increment of revenue equal to the social utility of providing all individuals with the same portion of that increment.

Taxes and Subsidies

By applying equations (9) and (10), as well as Roy’s identity and the Slutsky decomposition (of the forms $v_{1 + t x}^{h} = - x^{h} v_{y}^{h}$ and $g_{1 + t x}^{i h} = {\hat{g}}_{1 + t x}^{i h} - x^{h} g_{y}^{i h}$ ), one can write the first-order conditions for the optimal taxes on earnings (t^z ) and consumption (t^x ), and the optimal subsidy on contributions to a particular category i (tⁱ ), as follows:

\begin{array}{l} + t^{z} {\hat{Z}}_{1 - t z} + t^{x} {\hat{X}}_{1 - t z} + \sum_{j} τ^{j} {\hat{G}}_{1 - t z}^{j} = - [1 / λ] \sum_{h} μ^{h} z^{h} + Z \\ = - [\frac{Z}{λ}] z [\frac{cov (μ^{h}, z^{h})}{\bar{Z}} + \bar{μ} - λ], \end{array}

11a

\begin{array}{l} - t^{z} {\hat{Z}}_{1 + t x} - t^{x} {\hat{X}}_{1 + t x} - \sum_{j} τ^{j} {\hat{G}}_{1 + t x}^{j} = - [1 / λ] \sum_{h} μ^{h} x^{h} + X \\ = - [\frac{X}{λ}] [\frac{cov (μ^{h}, x^{h})}{\bar{X}} + \bar{μ} - λ], \end{array}

12a

\begin{array}{l} - t^{z} {\hat{Z}}_{1 + t i} - t^{x} {\hat{X}}_{1 + t i} - \sum_{j} τ^{j} {\hat{G}}_{1 + t i}^{j} = - [1 / λ] \sum_{h} μ^{h} g^{i h} + G^{i} \\ = - [\frac{G^{i}}{λ}] [\frac{cov (μ^{h}, g^{i h})}{{\bar{G}}^{i}} + \bar{μ} - λ], i \in 1, \dots, J, \end{array}

13a

where ⁁ denotes compensated demand.

For fixed producer prices, the left sides of equations (11a), (12a), and (13a) represent the marginal efficiency cost of the tax being determined, or the change in the dead-weight loss of the overall tax structure brought about by a small increase in that rate (Tresch 2002).¹⁴ The right sides of the equations represent the marginal distributional effect of the tax, an effect that depends on the relationship between individuals' demand for what is being taxed or subsidized, and their social marginal utility of income. Taking into account the marginal efficiency cost represented on the left side of each equation, and replacing λ with the value from equation (7b), one can rewrite the first-order conditions for t^z , t^x , and τⁱ as follows:

\frac{\partial L}{\partial t^{z}} = - Z [\frac{cov (μ^{h}, z^{h})}{\overline{μ} \overline{Z}}],

11b

\frac{\partial L}{\partial t^{x}} = - X [\frac{cov (μ^{h}, x^{h})}{\overline{μ} \overline{X}}],

12b

\frac{\partial L}{\partial t^{i}} \equiv \frac{\partial L}{\partial τ^{i}} = - G^{i} [\frac{cov (μ^{h}, g^{i h})}{\overline{μ} {\overline{G}}^{i}}], i \in 1, \dots, J,

13b

where L represents the dead-weight loss of the overall tax structure.

The right sides of equations (11b), (12b), and (13b) center on the mean-standardized covariances between individuals' social marginal utility of income μ^h and their demand for earnings z^h , consumption x^h , and contributions g^ih . If one assumes that in absolute amounts the poor earn less, consume less, and contribute less, then, given the priority the government places on their well-being, these covariances will be negative. The greater these negative covariances, the greater the marginal distributional advantage of taxing earnings and consumption, and shadow taxing contributions, and hence the higher the marginal efficiency costs warranted. If one assumes the marginal efficiency costs to be increasing in the tax rates, then greater negative covariances would result in greater optimal values of t^z , t^x , and τⁱ.¹⁵

From equation (9), the optimal subsidy on contributions to category i consists of two components: the optimal shadow tax τⁱ and the Pigovian subsidy $- s^{i} E^{i} / λ$ . In equation (13b), the optimal shadow tax sets the marginal efficiency cost equal to the marginal distributional advantage, as calculated across the contributors who bear the tax. Thus, categories for which the total contributions are less price-sensitive, or for which compensated substitutes have shadow taxes, would warrant a higher shadow tax than categories for which the demands are more price-sensitive, or for which compensated substitutes have shadow subsidies (see note 15). Conversely, categories for which the contributors are predominantly rich (e.g., arts and education) would warrant a higher shadow tax than would categories for which the contributors include more of the poor (e.g., religion and ethnicity). The Pigovian subsidy reduces the price of contributions by the social marginal utility they generate. Thus, categories for which government direct funding is relatively more effective, or for which the service recipients are few, benefit little, or have a low priority placed on their well-being would warrant a smaller Pigovian subsidy than categories for which direct funding is relatively less effective, or for which the recipients are many, benefit much, or have a high priority.¹⁶

Direct Funding

Government direct funding can accompany contributions as a source of finance for the third sector. It is assumed that the government provides funding in order to increase social welfare, whereas individuals make contributions in order to derive internal benefits. Both within and across categories of civic activity, there is the potential for private and government crowding-out and crowding-in. The government, in assessing the social marginal utility of the goods and services under a certain category, considers the level of contributions and their effectiveness. Accordingly, it may reduce the direct funding for categories where contributions are high and effective. On the other hand, the internal benefits individuals derive from contributing to a particular category could take into account the perceived need for the goods and services as measured, say, by the prevailing amount of direct funding. If so, then individuals may reduce their contributions to categories where that funding is high.

By applying equations (9) and (10), as well as the Slutsky decomposition (of the form $x_{\tilde{G} i}^{h} = {\hat{x}}_{\tilde{G} i}^{h} + [v_{Γ i}^{h} / v_{y}^{h}] x_{y}^{h}$ ), one can write the first-order condition for the optimal choice of ${\tilde{G}}^{i}$ , the amount of direct funding to a particular category i, as follows:

1 - t^{z} {\hat{Z}}_{\tilde{G} i} - t^{x} {\hat{X}}_{\tilde{G} i} - \sum_{j} τ^{j} {\hat{G}}_{\tilde{G} i}^{j} \geq [1 / λ] \sum_{h} μ^{h} m^{i h}, i \in 1, \dots, J,

14a

where

m^{i h} \equiv v_{Γ i}^{h} / v_{y}^{h}

is individual h’s marginal willingness to pay for the goods and services under category i, with total

M^{i} \equiv \sum_{h} [v_{Γ i}^{h} / v_{y}^{h}]

and mean

{\overline{M}}^{i} \equiv M^{i} / H

. The possible inequality in equation (14a) results from the constraint that direct funding be nonnegative.

One can rewrite the first-order condition for ${\tilde{G}}^{i}$ as follows:

1 + \frac{\partial L}{\partial {\tilde{G}}^{i}} \geq {M^{i}|}_{F B} + M^{i} [\frac{cov (μ^{h}, m^{i h})}{\overline{μ} {\overline{M}}^{i}}], i \in 1, \dots, J,

14b

where FB identifies the total marginal willingness to pay for the goods and services under category i, given first-best conditions (i.e., individualized lump-sum taxes satisfy the revenue needs of the government; the shadow taxes on contributions, and the taxes on earnings and consumption, all equal zero). The left side of equation (14b) represents the marginal cost of direct funding for category i. This includes the marginal resource cost in units of the numeraire (represented by 1), and the marginal efficiency cost as measured by the change in the dead-weight loss of the overall tax structure. The right side of equation (14b) represents the marginal benefit of direct funding for category i. This includes the sum of the marginal willingness to pay for the goods and services given first-best conditions, and the marginal distributional advantage as measured by the mean-standardized covariance between individuals' social marginal utility of income and their marginal willingness to pay.

Equation (14b) is a second-best generalization of the Samuelson rule for the optimal provision of public goods.¹⁷ Given first-best conditions, the marginal efficiency cost of direct funding would equal zero. The left side of equation (14b) would comprise only the marginal resource cost. Given first-best conditions, individualized lump-sum taxes would satisfy the distributional objectives of the government, and hence the covariance between the social marginal utilities of income and the marginal willingness to pay for the goods and services would equal zero. The right side of equation (14b) would comprise only the sum of the marginal willingness to pay. Given second-best conditions, however, equation (14b) depicts circumstances in which direct funding for category i would be greater or less than with first-best conditions: it would be greater if a distributional advantage outweighs any positive efficiency cost, while it would be less if a negative efficiency cost outweighs any distributional disadvantage.

In supplementing or accommodating contributions, the optimal amounts of direct funding do not necessarily equalize the social marginal utility of the goods and services Eⁱ across all categories. This can be seen by applying equation (8) to rewrite the first-order condition for the optimal choice of ${\tilde{G}}^{i}$ as follows:

E^{i} \leq λ [1 - R_{\tilde{G} i}], i \in 1, \dots, J,

14c

where

R \equiv t^{x} X + t^{z} Z + \sum_{j} τ^{j} G^{j}

is the total revenue from taxes and shadow taxes;

R_{\tilde{G} i}

is the marginal tax revenue generated by direct funding to category i; and the inequality results from the constraint that funding be nonnegative. By combining equations (14c) and (9), one determines for category i the relationship between the optimal amount of direct funding and the optimal subsidy on contributions:

{\tilde{G}}^{i} > (=) 0 \Rightarrow t^{i} = (>) τ^{i} - s^{i} [1 - R_{\tilde{G} i}], i \in 1, \dots, J .

Thus, in the model, the case for differential contributions subsidies is a general one. The optimal subsidies are uniform only if the optimal shadow taxes are uniform and if government direct funding for each category of civic activity is greater than zero, is as effective in producing goods and services as in other categories, and generates the same marginal tax revenue as in other categories. The following section illustrates this conclusion with reference to a linear expenditure system.

Illustration

The preceding section presents a model in which individuals differ in their preferences, productivity, and nonlabor income, as well as in the priority the government places on their well-being. In this section, it is assumed that preferences comply with a linear expenditure system (Stone 1954). This assumption introduces well-known properties of optimal commodity taxation in the presence of an optimal linear income tax. As demonstrated by Atkinson (1977) and generalized by Deaton (1979), if individuals differ in their productivity and nonlabor income but have identical preferences that are weakly separable between commodities and leisure and that generate parallel Engel curves (conditions that are satisfied by a linear expenditure system), then uniform commodity taxation will be optimal, given an optimal linear income tax. Under these conditions, there remain no opportunities to use differential commodity taxation (including the differential shadow taxation of contributions) to distinguish individuals for distributional purposes. Yet, as noted at the end of the preceding section and as illustrated here, even if these conditions hold, the optimal subsidies on contributions can differ across categories of civic activity.

In order to illustrate this, it is assumed that preferences can be represented by the utility function:

u^{h} = {[T - β^{l h} - z^{h} / p^{h}]}^{α^{l h}} {[c^{h} - β^{c h}]}^{α^{c h}} {[x^{h} - β^{x h}]}^{α^{x h}} \prod_{j} {[g^{j h} - β^{j h}]}^{α^{j h}} + \sum_{j} ϕ^{j h} {[Γ^{j}]}^{δ^{j}},

where z^h , c^h , x^h , and g^jh are as previously defined, and where l^h (the variable distinguishing the α and β parameters of the first bracketed expression) is the amount of leisure time demanded by individual h. This equation replaces leisure with earnings, using the relationship:

z^{h} = [T - l^{h}] p^{h},

where T = 24 is the time endowment of all individuals, and p^h is the productivity of h as measured by his wage. It is assumed that individuals are of two types (h = 1, 2) and that there are 50 individuals of each type (H = 100). Type 1 individuals (the rich) have a higher productivity and nonlabor income than do type 2 individuals (the poor).¹⁸ The government’s distributional goals are such that it places a greater priority on the well-being of the poor.¹⁹

The alpha (α) and beta (β) parameters of the utility function are nonnegative.²⁰ The betas (β^lh, β^ch, β^xh, and β^jh) are the committed quantities of leisure, untaxed consumption, taxed consumption, and contributions demanded by h.²¹ The alphas (α^lh, α^ch, α^xh, and α^jh) sum to one and are the proportions in which h allocates the residual income that remains after he has purchased the committed quantities.²² The phi (ϕ^jh) and delta (δ^j) parameters are nonnegative. The phis determine the benefits that h derives from Γ^j, the level of goods and services under category j. The deltas are less than one and determine the rate at which the social marginal utility of those goods and services diminishes.²³

It is assumed that there are four categories of civic activity. For concreteness, these are Arts and Education (category 1), Religion and Ethnicity (category 2), Health and Welfare (category 3), and Public Administration and Infrastructure (category 4). Individuals derive internal benefits from contributing to all categories but the fourth: α^4h = 0, β^4h = 0 for h = 1, 2. The categories can differ on four fronts: individuals' preferences for contributing; individuals' preferences for the goods and services; the effectiveness of direct funding relative to contributions in producing the goods and services; and the marginal tax revenue generated by direct funding. Each set of differences is described in the following.

With respect to the preferences for contributing: it is assumed that all individuals contribute 3 percent of their residual income. If those preferences differ between rich and poor, then the rich contribute proportionately more to Arts and Education, the poor contribute proportionately more to Health and Welfare, and both contribute the same proportion to Religion and Ethnicity.²⁴

With respect to the preferences for the goods and services, it is assumed that if those preferences differ between rich and poor, then the rich most prefer the goods and services from Arts and Education, the poor most prefer those from Health and Welfare, and both least prefer those from Religion and Ethnicity, apart, say, from the goods and services of the particular religious or ethnic groups with which they identify. All individuals equally benefit from Public Administration and Infrastructure which is financed solely by government direct funding.²⁵

With respect to the effectiveness of direct funding relative to contributions in producing the goods and services, it is assumed that direct funding is more effective in all categories: $s^{j} < 0$ for j = 1, 2, 3. If that effectiveness differs, then it is highest for Health and Welfare, and lowest for Religion and Ethnicity: s ³ < s ¹ < s ².²⁶

With respect to the marginal tax revenue generated by direct funding, it is assumed that if this revenue is the same across categories, then it is zero: $R_{\tilde{G} j} = 0$ for j = 1, 2, 3. If this revenue differs, then it is positive for Arts and Education and negative for Health and Welfare: $R_{\tilde{G} 1} > 0$ , $R_{\tilde{G} 3} < 0$ . These differences are assumed to originate from the effects of direct funding on the committed quantity of leisure. For example, directly funded welfare may reduce the amount of labor that individuals offer, whereas directly funded education may increase it.²⁷

Table 1 presents the results of separately introducing the four sets of differences. These results can be compared to the base case of scenario I. For that base case, the necessary conditions for uniform commodity taxes being optimal are satisfied; specifically, individuals have identical preferences for contributing. In addition, the categories of civic activity are the same not only in terms of the preferences for the goods and services but also in terms of the relative effectiveness and the marginal tax revenue of direct funding. Thus, under scenario I, the government cannot differentiate the shadow taxes for distributional purposes. What is more, the effectiveness of direct funding relative to contributions is the same across categories, and the government is able and willing to adjust the amount of direct funding in order to equate across categories the social marginal utility of the goods and services provided: $E^{j} = λ$ for j = 1, 2, 3. Thus, the optimal contribution subsidies—and the shadow taxes and Pigovian subsidies that compose them—are uniform. Note that the optimal commodity taxes (the tax on consumption and the shadow taxes on contributions) are uniform as well: t^x raises the price of consumption by the same percentage that τ^j raises the price of contributions net of the Pigovian subsidy: $t^{x} = τ^{j} / [1 - s^{j} E^{j} / λ]$ for j = 1, 2, 3.²⁸

Table 1.

Results for a Linear Expenditure System, by Scenario

Category	Variable	Scenario I: Uniform Subsidies	Scenario II: Contribution Tastes Differ between Rich and Poor	Scenario III: Service Tastes Differ between Rich and Poor	Scenario IV: Effectiveness of Direct Funding Differs across Categories	Scenario V: Marginal Tax Revenue from Direct Funding Differs across Categories
Arts and education	t ¹	−.6568	−.5608	−.6473	−.6569	−.6328
	τ ¹	.0432	.1392	.0527	.0431	.0460
	−s ¹ E ¹ /λ	−.7000	−.7000	−.7000	−.7000	−.6788
	${\tilde{G}}^{1}$	453.36	421.50	2257.79	448.47	601.77
	Γ ¹	1114.05	1101.18	2832.96	1109.56	1230.28
Religion and ethnicity	t ²	−.6568	−.6569	−.1297	−.7713	−.6570
	τ ²	.0432	.0431	.1301	.0287	.0430
	−s ² E ² /λ	−.7000	−.7000	−.2598	−.8000	−.7000
	${\tilde{G}}^{2}$	453.36	439.35	0	56.25	447.30
	Γ ²	1114.05	1101.18	316.38	1109.56	1110.35
Health and welfare	t ³	−.6568	−.7478	−.6473	−.5425	−.7055
	τ ³	.0432	−.0478	.0527	.0575	.0369
	−s ³ E ³ /λ	−.7000	−.7000	−.7000	−.6000	−.7424
	${\tilde{G}}^{3}$	453.36	481.05	3501.66	654.57	163.41
	Γ ³	1114.05	1101.18	4076.83	1109.56	912.61
Administration and infrastructure	${\tilde{G}}^{4}$ = Γ ⁴	1114.05	1101.18	1181.49	1109.56	1110.35
Taxes and transfer	t^z	.3508	.3507	.3784	.3507	.3503
	t^x	.1438	.1437	.1758	.1437	.1432
	y	80.10	80.56	61.68	80.20	80.47

Note: t ¹, t ², and t ³ are the subsidies on contributions to three categories of civic activity (Arts and Education; Religion and Ethnicity; and Health and Welfare); τ ¹, τ ², and τ ³ are the shadow taxes on contributions; −s ¹ E ¹ /λ, −s ² E ² /λ, and −s ³ E ³ /λ are the Pigovian subsidies on contributions; ${\tilde{G}}^{1}, {\tilde{G}}^{2}, {\tilde{G}}^{3}$ , and ${\tilde{G}}^{4}$ are the amounts of government direct funding; Γ¹, Γ², Γ³, and Γ⁴ = ${\tilde{G}}^{4}$ are the levels of goods and services financed by direct funding and contributions; t^z is the tax on earnings; t^x is the tax on consumption; and y is the personal transfer.

Only under scenario II are the conditions not satisfied for uniform commodity taxes being optimal. Rich and poor have different preferences for two of the commodities: their contributions to Arts and Education and their contributions to Health and Welfare. As a result, the government can pursue its distributional goals in part by choosing a higher shadow tax on contributions to the category preferred by the rich and a shadow subsidy on contributions to the category preferred by the poor. Thus, although the Pigovian subsidies are uniform, the optimal contribution subsidy is lower for Arts and Education and higher for Health and Welfare.

Under scenario III, the categories differ in the preferences of rich and poor for the goods and services: the rich prefer those of Arts and Education, and the poor prefer those of Health and Welfare. Note that such differences do not necessarily lead to differential contribution subsidies; such an outcome depends on the government not being able to adjust its direct funding in order to equate across categories the social marginal utility of the goods and services.²⁹ Here, the government is only able to adjust its funding both for Arts and Education and for Health and Welfare. For Religion and Ethnicity, for which the goods and services are least preferred by rich and poor alike, the government exhausts its ability to raise social marginal utility by lowering its direct funding (it falls to zero). As a result, the Pigovian subsidy remains smaller for Religion and Ethnicity than for either Arts and Education or Health and Welfare. Given uniform shadow taxes, the optimal contribution subsidy remains smaller as well.

Under scenario IV, the categories differ in the effectiveness of direct funding relative to contributions in producing goods and services. The Pigovian subsidies differ accordingly. Given uniform shadow taxes, the optimal contribution subsidy and the reliance on contributions are smallest for Health and Welfare where direct funding is the most effective and are greatest for Religion and Ethnicity where direct funding is the least effective.

Finally, under scenario V, the categories differ in the marginal tax revenue generated by direct funding. Thus, the government is not willing to adjust its direct funding across the categories in order to equate the social marginal utility of the goods and services. The funding is higher and the social marginal utility lower for Arts and Education where the marginal tax revenue is positive; the funding is lower and the social marginal utility higher for Health and Welfare where the marginal tax revenue is negative. As a result, the Pigovian subsidy is lower for category Arts and Education than for Health and Welfare. Given uniform shadow taxes, the optimal contribution subsidy is lower as well.

Summary

In order to maximize social welfare, governments should subsidize at different rates the private contributions made to different categories of charitable or publicly beneficial civic activity. This is argued with reference to a model in which the government chooses the subsidies and amounts of direct funding, as well as a personal transfer and linear taxes on earnings and consumption. The categories differ in terms of the individuals contributing, the individuals receiving the goods and services, and the effectiveness of direct funding relative to contributions.

As made here, the theoretical case for differential subsidies is a general one. The optimal subsidies are smaller on contributions to categories of civic activity for which there is a smaller Pigovian subsidy or a larger shadow tax. The former is associated with direct funding in a particular category either falling to zero, or being relatively effective in producing goods and services, or generating greater marginal tax revenue. The latter is associated either with total contributions being less price-sensitive or with the contributors featuring individuals on whom the government places less priority. Only under restrictive conditions would the Pigovian subsidies and shadow taxes compose uniform subsidies. A linear expenditure system illustrates the effects of relaxing these conditions.

The policy implications of this theoretical case are clear. Relative to a uniform subsidy, differential subsidies could assist the government in financing the third sector more efficiently and more effectively: more efficiently in the sense that a given amount of government spending could generate greater financial resources for the third sector; more effectively in the sense that a given amount of financial resources could generate greater social welfare. To be sure, realizing these desired outcomes would require more than an awareness of their theoretical bases. It would also require developing the broader empirical and institutional understanding needed in order to design and implement a program of differential subsidies having that potential. Accordingly, this article makes a formal and theoretical case for differential subsidies not simply as an end in itself, but rather as an invitation to the researchers and practitioners concerned with financing the third sector to recognize the worth of that broader understanding, and develop it.

Footnotes

The article and the author have benefited from the constructive suggestions of anonymous reviewers, as well as the comments and encouragement offered on earlier drafts by Rose Anne Devlin, Susan Phillips, Stanley Winer, and attendees of the session on charitable giving at the 2006 meetings of the Canadian Economics Association.

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

The author received no financial support for the research, authorship, and/or publication of this article.

Notes

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