Public Sector Crowding Out of Private Provision of Public Goods:
The Influence of Transactions Costs
Dennis Coates
Department of Economics
UMBC
5401 Wilkins Ave.
Baltimore, MD 21228-5398
Draft:
I benefitted greatly from the comments of Tom Bogart, Art Benavie, Jim Friedman and, especially, Todd Sandler.
Abstract
Whether and to what extent public sector provision of public goods, financed via compulsory taxation, has crowded out voluntary provision has been a topic of intense theoretical and empirical interest since the early 1980s. Warr (1982, 1983) and Roberts (1984) each presented models in which one additional dollar of taxes used to finance an additional dollar of public services would crowd out exactly one dollar of private contributions to the public good. This extreme neutrality result came under fire almost immediately as Cornes and Sandler (1984) showed that crowding out would be less than perfect if contributions to the public good produced both a private benefit to the donor and a public benefit to the community. Schiff (1985) demonstrated a similar result given that public sector provision and own donations are not perfect substitutes. Cornes and Sandler (1984) and Schiff (1985) each show that when donations serve multiple purposes crowding in may occur.
Andreoni (1989) and Steinberg (1987) also present models in which donors derive satisfaction from the act of donating. They find that crowding out is not complete. In fact, Andreoni concludes that "pure altruism is both necessary and sufficient for neutrality." Bernheim (1986) went even farther, demonstrating that perfect crowding out, or policy neutrality, could be derived from a very general model, including one with "distortionary taxation", so long as altruism was of the pure sort. That is, given donors do not derive utility from the act of donating per se, but only from the quantity of the public service available, then Bernheim's result indicates that every public policy initiative can and will be undone by the reactions of private citizens. He argues that such a result is highly questionable. Consequently, researchers should focus more attention on models which account for deriving satisfaction from the act of giving.
Bergstrom, Blume and Varian (1986) note that in the Warr (1982, 1983) and Roberts (1984) models everyone is a donor, that taxes do not fall on non-donors. They develop an approach which accounts for donors and non-donors and changes in the size of the donating group. With their model, they show that public provision will tend to increase total public goods supply whenever someindividuals' coerced donations (taxes) toward the public good provision are in excess of their voluntary contribution levels. Hence, even if donors reduce their donations to zero, the total supply of the public good still will rise.
The models described so far lack realism in that there is typically only one public good to which the individual may contribute or the marginal price of donations is the same across all public goods. Moreover, the public services offered by government are either perfect substitutes for those offered by private charities or donations are a good in themselves. Only Schiff (1985) addresses the possibility that the publicly provided service and the privately provided service are complements. He follows Rose-Ackerman (1981) in allowing this possibility. In deed, Rose-Ackerman (1981) includes far greater realism in her model by allowing for multiple charities with disparate clienteles, motivations, and costs than is standard in the literature. In this context, she finds that increased public provision may generate increased private donations as well as decreased private donations depending on the type of services offered by the public sector and the private sector. Specifically, she argues that private charities might evolve from one type service, say social welfare, to another like legal services or advocacy, to help the clientele in its dealings with the government bureaucracy. In this case, the goods provided are complements not substitutes.
A large body of empirical literature suggests that actual crowding out is not complete, and much indicates crowding in occurs. Much of this literature is reviewed by Steinberg (1991). Abrams and Schmitz (1978, 1984), Schiff (1985) and Kingma (1989) each provide evidence of crowding out. Abrams and Schmitz (1979, 1984) estimate elasticities of contributions with respect to government spending between -.18 and -.30; that is, a ten percent increase in government spending for social welfare or poverty relief induces between a 1.8 percent and a 3 percent reduction in private donations for those purposes. Schiff (1985) finds far larger elasticities, typically greater than one in absolute value, but these elasticities vary both by purpose of the expenditures and by the source. More recently, Kingma (1989) estimates, in a sample of donors to public radio stations in the United States, that an additional dollar of governmental support reduces private donations by about $.15. Reece (1979), Posnett and Sandler (1989), Khanna, Posnett and Sandler (1995) and Callen (1994) find little evidence of any crowding out. In fact, Posnett and Sandler (1989), Khanna, Posnett and Sandler (1995) and Schiff (1985) each find support for crowding in of private contributions by public spending.
A close reading of the existing literature raises two questions. First, do the empirical results provide compelling evidence to reject the pure altruism model of Warr (1982, 1983) and Roberts (1984) or to accept the impure altruism/joint products models? Said differently, is there a reasonable alternative specification of the problem that implies less than complete crowding out without depending on the presence of impure altruism? Second, given an alternative model exists that meets the criteria of the first question, how would the results of Bergstrom, Blume, and Varian be altered within this new framework?
Section I addresses the first question. Bergstrom, Blume and Varian (1986) admit that the results of their model may need to be modified because they "have not incorporated the inefficiencies of administration, the informational imperfections inherent in democratic decision-making, and the excess burden involved in collecting taxes." This section shows that introduction of transactions costs, administrative inefficiencies as used by Bergstrom, et. al., into Warr's (1983) pareto optimal redistribution model implies that less than perfect crowding out is possible without resorting to impure altruism. Moreover, perfect crowding out is possible in the presence of impure altruism. Consequently, empirical results in the literature are not conclusive evidence for or against the pure altruism model. Section II demonstrates the same results in Andreoni's (1989) more general model of public goods provision with impure altruism.
Section III introduces the transactions costs into the Bergstrom Blume and Varian model. Theupshot is, simply, that even when individuals are taxed more than their initial contribution level the total supply of the public good may fall. The explanation is simple, public sector wastage of resources on program administration and bureaucracy is greater than that of the private sector. Consequently,even though more resources enter the system for providing the public service, a larger share is used in the less efficient sector so fewer public goods are produced. Section III concludes.
I. The Pareto Optimal Redistribution Model
I.1. Basic One Donor One Recipient Model
Andreoni (1989) points out that at the base of the Ricardian equivalence and charitable contributions literatures on crowding out lies a public good. In the text which follows, the focus is on contributions for poor relief, following Warr (1982, 1983). However, in the appendix the role of transactions costs in Andreoni's (1989) more general model are developed. The basic implication, that imperfect crowding out is possible without impure altruism, hold in his model as well.
I.1.A Pareto Optimal Redistribution
In this model there are two individuals, one wealthy and one poor. The wealthy individual gets utility from the consumption of the poor, but the poor consumer derives utility only from her own consumption. Let U(cw,cp) be utility of the wealthy individual, where c denotes consumption and the superscripts indicate the wealthy individual w and the poor p. Each individual i has income yi. Donations by the wealthy are represented by vw and the lump-sum tax imposed on the wealthy by T. The budget constraint of the wealthy person is, therefore, yw = cw + vw + T. Consumption by the poor is given by cp = yp + vw + T. Substitution for cw and cp in the utility function of the wealthy and maximization with respect to vw produces the first-order condition -U1 + U2 = 0, where Ui, i=1,2 is the derivative of the utility function of the wealthy individual with respect to the ithargument.
Total differentiation of the first-order condition and rearrangement produces:
where Uij is the ijth second derivative of the utility function of the wealthy individual and the usual concavity assumptions apply. Ricardian equivalence of public and private provision is obtained by setting dyw=dyp=0 which produces dvw = - dT. In words, a dollar increase in taxes paid by the wealthy individual reduces that individual's contribution to the poor by precisely one dollar (Warr, 1983). The other comparative statics effects show that contributions rise with own wealth and fall with that of the poor. Additionally, a dollar reduction in the income of the wealthy offset by a dollar increase in the income of the poor also reduces the contribution by the wealthy dollar for dollar. That is, redistribution of income has no effect on the final consumption of either consumer (Warr, 1983).
The public good in this model is consumption by the poor individual, each consumer derives satisfaction from consumption by the poor and the benefits of that consumption are non-rival. The technology for producing consumption by the poor individual is linear in public and private sector contributions. In deed, such contributions are perfect substitutes in the production process. If they are not perfect substitutes in production, or if the production processes of the public and private sectors differ, then one would not expect policy ineffectiveness of the sort derived above. The next section develops this idea more fully.
I.1.B. Transactions Costs in the One Donor One Recipient Model
The production of consumption by the poor consumer is assumed to differ between the public and private sectors in a straightforward way. First, for each T dollars of taxes collected from the wealthy individual only %T, %%[0,1), reaches the poor. This simple device allows for leakage in the bucket of transfers from the wealthy individual to the poor individual. Second, a private transfer of one dollar costs the donor % dollars. Previous researchers (Weisbrod and Dominguez 1986; Posnett and Sandler 1989; Khanna, Posnett, and Sandler 1995; Callen 1994) have modeled the price to the donor of a dollar of the public service as p=(1-%)/(1-% -%) where % is the donor's marginal tax rate, % is the share of the recipient organizations total expenditure devoted to fundraising and % is the share of the organization's expenditure going to overhead. The price to the taxpayer could, therefore, be greater than, equal to, or less than one. The analysis below is developed for a lump sum tax, however, so the tax rate % is zero by assumption. This expression for p is therefore 1/(1-% -%) and differs from % only in that p is the average donor price and % is the relevant marginal price assumed to be greater than or equal to one. That is, % % (1, %].
The impact of the transactions costs on the consumption possibilities of the two consumers is seen from the relevant budget constraints. The budget constraint of the wealthy person is yw = cw + %vw + T. Consumption by the poor is given by cp = yp + vw + %T. The upshot of the introduction of these differential costs is that public and private sector funds are no longer perfect substitutes in the production of the public good consumption by the poor.
Defining efficiency as either raising consumption by the poor the most for a given reduction in consumption by the wealthy, or as reducing consumption of the wealthy by the least for a given increase in the consumption of the poor, the public sector is more, less, or equally efficient as the private sector as %% is greater, less or equal to 1. Suppose that cw is reduced by $1 either by raising T by $1 or by raising vw sufficiently for %vw to increase by $1. In the first case, consumption by the poor rises by $%, while in the second it increases by $(1/%). The relationship between 1/% and % indicates the relative efficiency of public versus private poor relief.
After accounting for these changes in the relevant budget constraints, total differentiation of the first-order condition for utility maximization by the donor allows solution for the comparative statics effects of changes in T, yw and yp on the contribution levels of the wealthy. At first blush theresults differ from those above only in the presence of the transactions costs parameters % and %. After further scrutiny, however, the effect of a tax increase on the donations level is now ambiguous even though the model does not include impure altruism. Computing the derivative one obtains:
where dvw/dyp = (%U21-U22)/H<0 and dvw/dyw = (%U11 - U21)/H > 0. The bordered Hessian determinant H = %2U11 + U22 - 2%U21. Clearly, if %=%=1, this collapses to the perfect crowding
out result in the literature and demonstrated in section I.A. Even if %%% but %%=1, perfect crowding out results. Each sector is equally inefficient, (1/%)=%, so the shift in financing between the private and public sector has no impact on the equilibrium. The donor's total expenditure on contributions falls by the exact increase in taxation, %dvw/dT = -1; consumption by the poor is unaffected, dcp/dT= -(1-%%)dvw/dyw=0.
When %% % 1, the public sector attempt to provide the public service may crowd out private donations on either a more or a less than dollar per dollar basis. To see this, rewrite equation (2) as:
Note that the left-hand side of this equality is the change in money contributions, which equals -1 when %%=1. However, if %%%1 then the effect of the tax on contributions depends upon the relative inefficiencies in the public and private sector and the sign and size of the effect of recipient income on the donor's level of contributions. If the public sector is less efficient than the private sector and dvw/dyp is negative, then -(1-%%)dvw/dyp is positive. Consequently, under these circumstances crowding out is less than complete. Note that the theoretical implication of less than perfect crowding out has not required impure altruism or joint products. Nor does the result depend upon %=1; hence, Steinberg's (1986) argument that the marginal cost of private donations is unity has no bearing on the analysis. Moreover, empirical work which finds less than perfect crowding out is not, therefore, necessarily a rejection of the "pure" altruism model nor support for impure altruism models. Said differently, empirical findings of incomplete crowding out can be interpreted as support for the existence of differential trasactions costs between the public and private sectors.
Alternatively, suppose that -(1-%%)dvw/dyp is less than zero. This occurs when either dvw/dyp is positive and the public sector is relatively inefficient, or when dvw/dyp is negative and the public sector is relatively efficient. In either event, crowding out is on more than a dollar for dollar basis; that is, a dollar increase in taxes and public provision induces a reduction in private donations in excess of one dollar.
Finally, note that consumption of the poor might rise after imposition of the tax, which is thepurpose of public sector provision, without worrying about contributing and noncontributing groups as Bergstrom, Blume and Varian (1986) have done. The effect of the tax on consumption by the recipient is dcp/dT= -(1-%%)dvw/dyw. This comparative statics effect is positive if donations are a normal good and the public sector is more efficient than the private sector (%%>1). Simply put, the total supply of the public good rises if resources are transferred from the less efficient to the more efficient producer of public services.
Relative to the perfect crowding out models in the literature then, the presence of transactions costs is potentially important. So far, the simple presence of differential transactions costs has generated imperfect crowding out and the possibility of public provision raising the public goods supply without recourse to the models of impure altruism or joint products. Consequently, Andreoni's (1989) claim that pure altruism is both necessary and sufficient for perfect crowding out is not true.
I.1.C. Impure Altruism and Transactions Costs
Cornes and Sandler (1984), Steinberg (1987) and Andreoni (1989) have modeled impure altruism in which the donor benefits both from the level of public service provision and from the act of giving. The model of section I easily incorporates this possibility by introducing donations vw as an independent source of utility; that is U(cw,cp,vw). Moreover, building transactions costs into the model with impure altruism does not fundamentally alter the results, though it does broaden the possibilities. The comparative statics effect of the tax increase on private donations is again given by equation (2) though now dvw/dyp = (%U21-U22-U32)/H* and dvw/dyw = (%U11 - U21-U31)/H*. The bordered Hessian determinant H* = %2U11 + U22 - 2%U21 + U33 - 2%U13 + 2%U23. Consequently, contrary to the suggestion in Bernheim (1986) and Andreoni (1989) impure altruism is neither necessary nor sufficient for the implication of incomplete crowding out; that is, impure altruism is notnecessarily the reason that every government policy is not neutral. Moreover, this result is not specific to the pareto optimal redistribution framework. The next section shows how the transactions costs affect the Andreoni (1989) model, leading to precisely the same conclusion as here.
II. The Andreoni Model of Public Goods Provision
Andreoni (1989) assumes a utility function U(xi, Y, gi) where x is private good, Y is the total of a general public good or service, and gi is agent i's contribtuion to the public good. The budget constraint is wi= xi + gi + %i where wi is the individuals wealth, and %i is her lump sum tax bill. The total supply of the public good Y = %j=1Ngj + %j=1N%j where N is the group size. The introduction of transactions costs into this model is straightforward. First, in the consumer budget constraint % appears before the gi: wi= xi + %gi + %i. Second, the total supply of the public good now must reflect the inefficiency or transactions costs of the public sector: Y = %j=1Ngj + %j=1N%%j. Solving the aggregate supply equation for the voluntary donations of individual i and splitting out her involuntary donations results in: gi = Y - Y-i - %%i, where Y-i=%j%igj + %j%i%%j. Substitution of this expression into the budget constraint, solving for xi, and using that result in the utility function produces the following unconstrained maximization problem:
This equation differs from Andreoni's (1989) equation (2) in the specification of the first and third terms. The first term here involves the tax, which is not the case in Andreoni's model. Additionally, % appears as a coefficient on total public good supply and the supply of the good provided by everyone other than agent i. Finally, the public sector inefficiency parameter % is a coefficient on thetax in the third argument of the utility function here, but is implicitly one in Andreoni's model. These differences arise because Andreoni implicitly assumes that % and % are each 1.
Solving for i's demand for Y one obtains:
Note that in the pure altruism case, in which the second argument of the demand function would not appear, the private sector inefficiency parameter % induces a difference between the derivative of f with respect to wealth w and public goods supplied from all other sources Y-i. The presence of % is important because it alters the derivative dY-i/dwi, which from Andreoni's model is -%i= -fia/(fia + fie), where fia and fie are the derivatives of the demand function with respect to its first (altruistic) and second (egotistic) arguments, respectively. In the current model, dY-i/dwi = -%/% where % is the Andreoni specification.
Solving the model for the comparative statics effects of a tax change on individual i produces the following change in total public goods supply:
where c % (0, 1], c= (1+ %i=1n(1- fia-fie)/(fia+fie))-1 as defined by Andreoni (1989). Note that the comparative statics equation reveals that either crowding out or crowding in may occur depending on the relative efficiencies of the two sectors. Moreover, even if the agent exhibits pure altruism, %=1, imperfect crowding out will occur if %%%1. Finally, if %%=% perfect crowding out occurs even though agent i has impurely altruistic preferences.
III. Bergstrom, Blume and Varian's Nonneutrality Results Qualified
Bergstrom, Blume and Varian (1986) examine the consequences of redistributions which change the set of contributors for levels of public good provision. Starting from an initial equilibrium in which government provision is zero, they allow taxes to be raised for the purpose of providing a public good. In these circumstances their Theorem 6 states:
1) If taxes collected from any individual do not exceed his voluntary contribution to the public good in the absence of government supply, the government's contribution results in an equal reduction in the amount of private contributions.
2) If the government collects some of the taxes that pay for its contribution from non-contributors, then, although private contributions may decrease, the equilibrium total supply of the public good must increase.
3) If the government collects some of the taxes to pay for its contribution by taxing any contributor by more than the amount of his contribution, the equilibrium total supply of public good must increase.
Proof of the theorem makes use of a function constructed from the sum of the inverse demand functions of the contributing group C.
G is the total public goods supply, wi is the wealth of individual i, ti is the tax payment of individual i, and g0 is the governments contribution toward the public good. By definition g0=%i=1Nti, where this sum is over the entire taxpaying population. Hence, g0%%i%Cti. In the situation that the contributing set and the set of taxpayers are identical, the equality is strict. Moreover, F(G, C) = %i=1Nwi. This result implies that the level of public good provision rises with the wealth of the contributors, holding that group constant. In other words, the function F(G,C) is increasing in G.
Proof of the various parts of Bergstrom, Blume and Varian's Theorem 6 is now straightforward. Suppose that all members of society are contributors. In this case, theaggregate wealth of the contributing set is constant, so public goods supply is unchanged by income redistribution. Part 1 of Theorem 6 is equivalent to Warr's (1983) and Cornes and Sandler's (1984) results. Part 2 of the theorem is easily seen from equation (7). The tax payments of the donors are offset by the rise in the public good, but the payments of the non-donors are not. Consequently, F(G',C) = %i%C wi + %i%C ti < %i%C wi = F(G,C); G' is greater than G. The level of the public good is constant if the aggregate wealth of the contributing set is constant, from part 1, but total contributions rise because taxes (forced contributions) are collected from those who do not voluntarily contribute. Part 3 of the theorem simply points out that if the government collects more from any contributor than that person would voluntarily supply, the total supply of the public good must rise.
Suppose instead that g0=%%i=1N ti where %<1. That is, impose the condition that some of the collected tax revenue dissipates before being used for its intended purpose. This assumption is equivalent to forcing the tax and expenditure activities of the government to use (waste) real resources, the administrative inefficiencies mentioned by Bergstrom, Blume and Varian (1986). Write the new function F(G,C;%<1):
Note that F(G',C;%<1) < F(G,C) which implies G' < G. Therefore, for any given tax policy and set of contributors, the dissipation of tax revenues implies a smaller equilibrium public good supply.
Now consider Theorem 6 from Bergstrom, et al. (1986). If everyone is a contributor, the complete crowding out result of part 1 of Theorem 6 no longer holds. Under these circumstances, %i%Cti = 0. Consequently, the second line of equation (8) becomes %i%C wi - (1-%)%i%C ti which is less than %i%C wi. In the process of transferring wealth from private to public hands some of the revenue is lost. The loss of wealth among the contributors (taxes collected) is, therefore, greater than the increase in government provision of the public good; contributor wealth falls as a consequence of the public program. Therefore, total public good supply must be reduced. This is completely analogous to the result shown for the Warr (1983) model in section one. Transactions costs result in partial crowding out without recourse to impure altruism type models.
Using this same style argument it becomes clear that parts 2 and 3 of Bergstrom, Blume and Varian's Theorem 6 no longer are true. In particular, whether total supply of the public good rises, falls or remains constant depends upon how much of the wealth is destroyed or lost in the tax and expenditure process.
Consider equation (8) again. Note that the right hand side is greater than, equal to or less than the pre-program wealth of the contributing group as %%i%Cti is greater than, equal to or less than (1-%)%i%C ti. What this means is that public provision of the public good will cause an increase in total supply if revenues net of wastage raised from non-donors exceed the wastage of donor wealth induced by the program. This is more likely the closer is % to one, (that is, the more efficient is the public sector) the smaller is the set of donors and the larger is the set of non-contributors. On the other hand, total public good supply is more likely to fall the less efficient is the public sector, the larger the set of contributors and thesmaller the set of non-donors.
Part 3 of the theorem fails for a similar reason. Suppose that government taxes a single donor by more than his/her contribution, say a tax of 10 relative to a donation of 9. If dissipation is 20% of revenues then governmental provision is only 8. Hence, dissipation may result in taxation in excess of contributions leading to less provision of the public good.
The analysis of this section has implicitly assumed that the private sector converted one dollar of donations into one dollar of public services. If this is not true, as was assumed above, then the effects of the tax and expenditure policy will depend on which secotr was relatively more efficient in the production of public services. The less efficient is the public sector relative to the private sector, the worse will be the effect on the level of the public service of shifting from voluntary to coerced donations.
Conclusion
This paper has shown that voluntary donations models which do not account for transactions costs are incomplete. Predictions from such models will not incorporate all of the relevant possibilities. In particular, it is not true that impure altruism is "necessary and sufficient" for incomplete crowding out or private provision of public goods by public sector programs. In fact, the analysis shows that impure altruism is neither necessary nor sufficient for impure crowding out to occur. Moreover, taxation of non-donors or taxing donors by more than their voluntary donations levels will not ensure that introduction of the public sector program raises the total supply of the public good.
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