In praise of inequality: public good provision and income distribution
Introduction
A surprising result in the analysis of private contributions to public goods is that total contributions are unaffected by any reallocation of income amongst consumers that leaves the set of contributors (those giving some positive amount) unchanged. This result was first established by Warr (1983)and extended by Bergstrom et al. (1986). That a transfer of income between consumers induces those receiving income to raise their contributions to the public good and that this increase is offset by a reduction in contributions of those losing income is unremarkable. What is striking is that the result that the two changes exactly offset each other is not special to particular utility functions but is a general consequence of the marginality conditions arising from the Nash equilibrium of the contribution game.
The purpose of this paper is to examine some policy-related implications of this result. It is first noted that if the consumers have the same preferences, private contributions to a public good result in the equalisation of utilities across consumers. Phrased another way, even if income is unequally distributed amongst consumers, the consequence of public goods being privately provided is that all consumers have the same utility level at equilibrium. The simple logic behind why this is so is that contributions to the public good increase with income. Hence, those with low incomes benefit from the provision of those whose incomes are higher. The marginality conditions arising from individual optimisation then ensure that since all consume the same quantity of public good, they can only be in equilibrium if they consume the same quantity of private good. Utilities must then be equal.
Both the equalisation of utility result and the independence of total public good provision from the distribution of income result from the fact that both are functions of the sum of income in society. This has the further implication that an extra dollar of income for consumer h raises h's utility by the same amount as an extra dollar to consumer j raises h's utility. That is, consumers are indifferent as to who should receive any extra income that society may generate.
The result that utilities are equalised seems to suggest that as long as the social welfare function is symmetric and concave in utilities, income redistribution is redundant as a tool of economic policy. This is not the case because the model also implies that social welfare is not a concave function of incomes. It is shown that increasing income inequality can raise welfare. A welfare increase necessarily arises if the transfer is such that consumers from whom income is taken just cease contributing to the public good. In the context of the economy considered, it is always optimal to create a wealthy consumer who is the sole provider of the public good.
The organisation of the remainder of the paper is as follows. Section 2provides the proof of the utility equalisation result. The gains from increasing inequality are demonstrated in Section 3. The case of non-identical utilities is considered in Section 4. Conclusions and applications are given in Section 5.
Section snippets
Equalisation of welfare
In order to simplify the analysis, it is initially assumed that there are two identical consumers who play a Nash game in private provision of a pure public good.1 It will be clear that the results extend to any number of consumers. The production technology is linear, markets are competitive and units of measurement of the commodities are chosen so that the producer prices of the private and public goods are both one. The
The benefits of inequality
The fact that utilities are equalised at the equilibrium of the private provision game is suggestive of the conclusion that the government need not engage in any redistributive policies. This conclusion is incorrect. In fact, given an initial allocation of income which is sufficiently uniform that all consumers are contributing to the public good, social welfare can be raised by carrying out a redistribution that reduces the income of some consumers to the point where they no longer contribute.
Non-identical preferences
When the consumers have differing utility functions, the utility-equalisation result is obviously not applicable. However, when both consumers contribute the derived utility function of each is still dependent on the sum of incomes. For each consumer the marginal utilities are equated as in (14) so the consumers remain indifferent to income redistributions.
It is also possible to show that to maximise social welfare incomes should be sufficiently unequal that only one of the consumers
Conclusions
A traditional defence of privilege is that it is the only effective way to ensure arts patronage and other great public works. It is not obvious that if equality is valued for itself this argument can be sufficiently strong to justify policies which deliberately “worsen” income distribution. This paper shows that in a natural benchmark model regressive measures are optimal.
It should be stressed that the results are dependent on public goods being provided solely by private contribution.
Acknowledgements
Thanks are due to Richard Cornes, Todd Sandler and an anonymous referee.
References (9)
- et al.
On the private provision of public goods
Journal of Public Economics
(1986) - et al.
The voluntary provision of a pure public good: the case of reduced CFC emissions and the Montreal Protocol
Journal of Public Economics
(1997) The private provision of a pure public good is independent of the distribution of income
Economics Letters
(1983)Dyke maintenance and other stories: some neglected types of public goods
Quarterly Journal of Economics
(1993)
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