Income inequality vs. standard of living inequality.

Spiegel, Uriel

I. Introduction

The equality presumption states that an individual, such as an economic planner, who divides a resource between other individuals without having information about their needs and tastes, would be inclined to divide the total resources into equal portions.

However, reality consists on the one hand of great inequality (as reflected for example in the Gini measure) in income and wealth, along with a desire to eliminate inequality on the other hand. There is, however, a general consensus in Western society that a certain level of inequality is acceptable and full equality is therefore not pursued.

The question we raise is why society accepts a certain level of inequality. Several explanations can be given. First, people are aware of the differences in skills, needs, and efforts among members of society, and thus do not feel impelled to strive for greater equality, even when some degree of jealousy or dissatisfaction exists. (The idea of Natural Inequality is discussed in Epstein and Spiegel (2001)). A second explanation can be found in the difference between inequality in income distribution and inequality in the standard of living and consumption among social groups. Many wealthy people do not flaunt their economic resources and lead a relatively modest lifestyle. Thus, it is often the case that differences between the poor and rich are not outwardly obvious, which reduces the motivation on the part of the poor to struggle for changes in income distribution that would reduce income inequality.

A third explanation relies on the explicit and implicit altruism of modern society where the wealthy contribute voluntarily to the general welfare of the weaker elements of society. This creates the phenomenon of automatic tools, i.e. an internal mechanism that reduces inequality and creates a sense of gratitude toward the rich amongst the poor. Thus the poor, once again, refrain from fighting for income redistribution.

Another explanation, which lies at the heart of our paper, is based on the internal mechanism of the behavior of the wealthy who by simply investing in their own needs, bring about a decrease in the inequality of living standards, although not necessarily of income distribution. This phenomenon exists when the utility of each individual is positively related to privately consumed products and to public goods (e.g., religious institutes, education, research, medical instruments, public parks, recreation areas and monuments, etc.), which simultaneously increases their utility and the utility of those who do not take part in financing these goods. We use a specific Cobb-Douglas utility function to demonstrate that at certain levels of income inequality, the consumption of pure public goods is not affected by income distribution, but only by total income levels (see Warr (1983) and Bergstrom et al. (1986)). Moreover, at certain levels of income inequality an individual whose share in income has decreased will not experience a reduction in utility as a result of this decrease as long as the income of all individuals is constant (see Itaya et al. (1997)). Conversely, an individual whose income share has increased will not find that his utility is actually affected by the increased income, because the positive change in income is offset by a positive change in the relative financing burden of the public good, thereby a situation of an automatic reduction in the inequality of utility despite the increased inequality of income distribution. The wealthy voluntarily increase their share of the tax burden and thus the welfare distribution (or standard of living) among members of society is not necessarily affected by changes in the distribution of income. From this it is clear that both the poor and the rich will not necessarily object to the changing income distribution.

This state of affairs may be subject to change at greater levels of income inequality or under different circumstances, when inequality in income distribution is translated into terms of utility distribution that differ from those described above.

In some sense our analysis is parallel to that of the recent analysis of Olszewski and Rosenthal (2004). However, in their analysis the public good is financed by taxes imposed on consumers, while the governing agent must use all taxes to purchase units of the public good. No tax revenue can be diverted to the private good provision. Our use is related to what is termed by Olszewski and Rosenthal (2004) "the anarchy of purely voluntary provision of public good", rather than to compulsory tax collection as a means of financing the public good, since taxes generate deadweight losses. The decision to impose a given tax rate or a given level of public good provision is determined, according to Olszewski and Rosenthal (2004), by the "Leviathan governor". In reality the scenario described by the above authors is indeed more typical and common. Nevertheless, we believe that our analysis is relevant and sheds light on the cases mentioned above. Contributions to churches and other religious institutions, donations for research grants or stipends for needy students, contributions to hospitals and other health institutions for new buildings and/or equipment and instruments, donations towards the construction of public parks, recreation areas, and monuments are all well known and popular devices for the wealthy to show their appreciation to the society that has endowed them with riches by voluntarily trying to reduce utility inequalities. This private financing of public goods is welcomed by the government, which encourages these donations by making them tax deductible, and is also welcomed by the poor who benefit from these donations and in general tend to appreciate the donations and respect the donors. Thus we find that many public institutions are financed in part or in whole by voluntary contributions rather than through a compulsory tax system. In truth, charities and many public institutes are today more likely to depend on voluntary contributions rather than on tax-generated government subsidies, and therefore the poor are willing to accept a higher degree of income inequality since this does not necessarily imply a higher degree of utility inequality. This idea is dramatically illustrated by Warren Buffet who in June 2006 announced that he was donating 85% of his fortune (estimated at US$46 billion) to the Bill and Melinda Gates Foundation in addition to the many billions of dollars that Bill Gates himself has contributed.

The above scenarios address the possibility that the degree of income inequality is not necessarily a true and accurate proxy for the degree of utility inequality, and it is only the degree of utility inequality that represents the true gap in the standard of living. As discussed above, this gap is significantly reduced by voluntary contributions on the part of the wealthy towards public needs. These contributions increase the utility of the poor, and narrow the standard of living gap, but are not picked up by the various "objective" measures such as the Gini coefficient of income distribution. This in effect is the background for the following analysis. We develop the model in the next section, followed by some implications and conclusions.

2. The Model

Assume an economy comprised of two consumers (who may be viewed as representative of two types of consumers). Both consumers have identical Cobb-Douglas utility functions, [U.sub.i], such that utility is a positive function of consumption of a private good, [C.sub.i], and a pure public good, G, as follows:

[U.sub.i] = [C.sup.[alpha].sub.i][G.sup.[beta]]

The only difference between consumers is reflected in their incomes where each consumer i (i = 1, 2) has an income [I.sub.i].

Each consumer maximizes utility subject to a budget constraint, i.e., the problem solved by each consumer is,

Max[U.sub.i]([C.sub.i], G) = [C.sup.[alpha].sub.i][G.sup.[beta]] (1)

subject to:

[C.sub.i] + [P.sub.G](G - [G.sub.j]) = [I.sub.i] (2)

Where [G.sub.j], is the contribution of consumer j to the public good, G, where G = [G.sub.i] + [G.sub.j] , i.e., the consumption of the pure public good is the summation of the contributions of both individuals, since a pure public good is defined such that the consumption of one individual does not reduce the consumption of the other. [P.sub.G] is the price paid per unit of the public good. For simplicity we assume that the price of the private product is the numeraire, namely, [P.sub.C] = 1. This approach is used by Atkinson and Stiglitz (1980) which we follow.

First order condition for maximization requires:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

substituting (3) into the budget constraint (2) gives consumer l's contribution to the public good, [G.sub.1], as a reaction to consumer 2's contribution to the public good, [G.sub.2].

Thus from (3) we get:

[G.sub.1] = ([beta]/[alpha] + [beta])[I.sub.1]/[P.sub.G] - ([alpha]/[alpha] + [beta])[G.sub.2] (4)

In the same way we get the reaction of consumer 2 with respect to consumer 1's investment in the public good, [G.sub.1].

[G.sub.2] = ([beta]/[alpha] + [beta])[I.sub.2]/[P.sub.G] - ([alpha]/[alpha] + [beta])[G.sub.1] (5)

The solution to equations (4) and (5) determines the equilibrium levels of [G.sup.[omicron].sub.1] and [G.sup.[omicron].sub.2] as follows (1):

[G.sup.[omicron].sub.1] = ([beta]([alpha] + [beta])/[([alpha] + [beta]).sup.2] - [[alpha].sup.2])[I.sub.1]/[P.sub.G] - ([alpha][beta]/[([alpha] + [beta]).sup.2] - [[alpha].sup.2])[I.sub.2]/[P.sub.G] (6)

[G.sup.[omicron].sub.2] = ([beta]([alpha] + [beta])/[([alpha] + [beta]).sup.2] - [[alpha].sup.2])[I.sub.2]/[P.sub.G] - ([alpha][beta]/[([alpha] + [beta]).sup.2] - [[alpha].sup.2])[I.sub.1]/[P.sub.G] (7)

We now examine the two following cases:

(a) [G.sub.i] > 0 for everyi; (b) [G.sub.1] or [G.sub.2] equal zero, namely, one of the consumers does not contribute to the public good, G.

When [G.sub.1] and [G.sub.2] are both greater than zero, a summation of equations (6) and (7) gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

where [bar.I] is the total income of the two consumers.

The ratio between [G.sub.1] and [G.sub.2] as obtained by dividing equation (6) by equation (7) is:

[G.sup.0.sub.1]/[G.sup.0.sub.2] = ([alpha] + [beta])[I.sub.1] - [alpha][I.sub.2]/ ([alpha] + [beta])[I.sub.2] - [alpha][I.sub.1]

= [beta][I.sub.1] + [alpha]([I.sub.1] - [I.sub.2])/ [beta][I.sub.2] + [alpha]([I.sub.2] - [I.sub.1]) (9)

Now we examine the second case when either [G.sub.1] or [G.sub.2] is equal to zero. (a)

(d) Assume [G.sub.1] = 0 and [G.sub.2] > 0, then by substituting into equation (6), (b)

[G.sub.1] = 0 if: [I.sub.1] < ( [alpha]/[alpha] + [beta])[I.sub.2] (I)

or: [I.sub.2] > ([alpha] + [beta]/[alpha])[I.sub.1] (c)

Symmetrically, [G.sub.2] = 0 and [G.sub.1] > 0 is obtained by substituting in (7),

[G.sub.2] = 0 if: [I.sub.1] > ([alpha] + [beta]/[alpha])[I.sub.2] (II)

From equations (6), (7), (9), and conditions (I) and (II) we get that when [I.sub.1] > [I.sub.2] we still obtain the condition for contribution by both individuals to the public good, namely,

[G.sub.1,] [G.sub.2] > 0 (III)

if: [I.sub.2] < [I.sub.1] < ([alpha] + [beta]/[alpha])[I.sub.2]

and symmetrically, if: [I.sub.2] > [I.sub.1] (III) holds ( i.e., [G.sub.1],

[G.sub.2] < 0) if: [I.sub.1] < [I.sub.2] < ( [alpha] + [beta]/[alpha])[I.sub.1]

We also find that although [I.sub.2] > [I.sub.1] , when the importance of the public good to the utility of the consumer increases, it (i.e., the higher [beta]) encourages consumer 1 (who is poor) to invest a relatively higher share of his low income in the public good. This we get by taking the derivative of [G.sup.[omicron].sub.1] with respect to [beta] (from equation (6)).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

From equations (8), (9), and (10) we can obtain the following conclusions:

(a) In the extreme case of income distribution inequality, the poor customer does not contribute to financing the public good.

(b) In moderate income distribution inequality cases, both consumers (the wealthy and the poor) contribute to the public good. The total expenditure on the public good is independent of income distribution, and is a function only of the summation of total income of all consumers.

(c) In the case where moderate inequality exists and joint income is fixed, the total supply and consumption of pure public good G does not change, but the distribution of consumers' contribution does. The consumer who enjoys a higher income will increase his relative share in financing G while the consumer with the lower income will reduce his share. This implies implicit cooperation, which is a result of each "selfish" consumer knowing the importance of the public good to his own utility. Thus when the other consumer does not contribute sufficiently towards financing the public good due to a lack of resources, the wealthier consumer will "agree" to cover the costs and will thereby increase the poor consumer's utility as well as his own.

(d) As the importance of the public good increases, (i.e.,[beta] [much greater than] [alpha]), the range of inequality in which both consumers cooperate and contribute to the public good grows.

We now wish to find the level of utility gained by the two consumers when their total joint income equals [bar.I] and is distributed unequally between them. From the information above we may draw the conclusion that four separate regions of income distribution must be identified. In each region we will find different equilibrium amounts of the private good consumed by each consumer and different levels of contributions to the public good.

Region 1: I: [I.sub.1] [greater than or equal to] ([alpha] + [beta]/[alpha])[I.sub.2], therefore, [G.sub.2] = 0,

and G = [G.sub.1] = ([beta]/[alpha] + [beta]) [I.sub.1]/[P.sub.G]. In this case [C.sub.2] = [I.sub.2], and [C.sub.1] = ([alpha]/[alpha] + [beta])[I.sub.1].

Region 2: ([alpha] + [beta]/[alpha])[I.sub.2] > [I.sub.1] [greater than or equal to] [I.sub.2], therefore, both consumers contribute to the public good and from equation (8) we get: G = ([beta]/2[alpha] + [beta])[bar.I]/[P.sub.G]. Substitution into equation (3) gives: [C.sub.1] = [C.sub.2] = [alpha]/[beta][P.sub.G] x G = ([alpha]/2[alpha] + [beta]){bar.I}

Region 3: ([alpha] + [beta]/[alpha])[I.sub.1]> [I.sub.2] [greater than or equal to] [I.sub.1]: This case is equivalent to region 2.

Region 4: [I.sub.2] [greater than or equal to] ([alpha] + [beta]/[alpha])[I.sub.1], therefore, [G.sub.1] = 0, and G = [G.sub.2] = ([beta]/[alpha] + [beta])[I.sub.2]/[P.sub.G]. In this case [C.sub.1] = [I.sub.1,] and [C.sub.2] = ([alpha]/[alpha] + [beta])[I.sub.2]

Region 4 is symmetric and opposite in its results to region 1. By substituting equilibrium values of [C.sub.1,] [G.sub.1] and G into the utility function [U.sub.1] we get the equilibrium utility levels, in all four regions as follows:

Region 1: (11) [U.sub.1] = [C.sup.[alpha].sub.1][G.sup.[beta].sub.1] = ([[alpha].sup. [alpha]] + [[beta].sup.[beta]] x [I.sup.([alpha] + [beta]).sub.1]/[([alpha] + [beta]).sup.[alpha] + [beta]] x [P.sup.[beta].sub.G]

for [I.sub.1] > ([alpha] + [beta]/[alpha])[I.sub.2]

Regions 2, 3: (11')

[U.sub.1] = [C.sup.[alpha].sub.1][([G.sub.1] + [G.sub.2]).sup.[beta]] = [[alpha].sup. [alpha]] + [[beta].sup.[beta]] x [([bar.I]).sup.[alpha] + [beta]]/[(2[alpha] + [beta]).sup.[alpha] + [beta]] x [P.sup.[beta].sub.G]

for ([alpha] + [beta]/[alpha])[I.sub.2] > [I.sub.1] > [I.sub.2] or

when ([alpha] + [beta]/[alpha])[I.sub.1] > [I.sub.2] > [I.sub.1]

Region 4:

[U.sub.1] = [C.sup.[alpha].sub.1] x [G.sup.[beta].sub.2] = [I.sup.[alpha].sub.1][([beta]/[alpha] + [beta]).sup.[beta]] x [([bar.I] - [I.sub.1]/[P.sub.G]).sup.[beta]]

for [I.sub.2] > [alpha] + [beta]/[alpha][I.sub.1]

Using comparative statistics analysis we can show that:

In region 1

d[U.sub.1]/d[I.sub.1]|[bar.I] = [[alpha].sup.[alpha]] x [[beta].sup.[beta]][I.sup.[alpha] + [beta] - 1].sub.1]/ [([alpha] + [beta]).sup.[alpha] + [beta] - 1 [P.sup.[beta].sub.G] > 0

In region 4

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

in regions 2 and 3 d[U.sub.1]/d[I.sub.1]|[bar.I] = 0. Thus, in these two regions, when income changes hands between wealthy and poor groups, there is no change in the utility levels of each group member.

3. Welfare Effects of Income Redistribution

A further question we ask is what will happen to the social welfare function, W, assuming the regular utilitarian approach, i.e., W = [U.sub.1] + [U.sub.2,] when income inequality increases, from a moderate inequality where ([alpha] + [beta]/[alpha])[I.sub.i] [greater than or equal to] [I.sub.j] > [I.sub.i] to a large inequality, such that [I.sub.j] > ([alpha] + [beta]/[alpha])[I.sub.i].

We found above that in the moderate range of inequality [U.sub.1] = [U.sub.2] for any redistribution of income in this range. Thus W is constant as long as [bar.I], the total income, is constant.

However, a further increase in the inequality of income distribution does affect the wealthy positively and the poor negatively. The question is therefore what happens to the "total welfare of society", i.e., to the simple summation of utilities?

In order to compute the utility values, let us write the utilities "close" to the border where [[??].sub.1] = ([alpha] + [beta]/[alpha])[[??].sub.2]. At this income distribution [U.sub.1] is given as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

[[??].sub.2] = [C.sup.[alpha].sub.2] x [G.sup.[beta].sub.1] = [[[[??].sub.2]].sup.[alpha]] [[([beta]/ [alpha] + [beta])[[??].sub.1]/[P.sub.G]].sup.[beta]] (11')

When we increase the income, [I.sub.1], of the wealthy person we lower the income, [I.sub.2], of the poor, i.e., d [I.sub.1] = -d [I.sub.2].

Therefore the change in utility of the wealthy person is positive and equal to:

d[U.sub.1]/d[I.sub.1] = [[alpha].sup.[alpha]][[beta].sup.[beta]] x [[??].sup.[alpha] + [beta]-1.sub.1]/ [([alpha] + [beta]).sup.[alpha] + [beta]-1[P.sup.[beta].sub.G] > 0 (12)

The total change in the utility of the poor is negative. On the one hand, his consumption of the private good [C.sub.2] decreases, but the consumption of the public good distributed by the wealthy, [G.sub.1,] increases. This we compute as follows:

Because d [I.sub.1] = d [I.sub.2,] we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

However, because we calculate this value very close to the "border case" we can substitute [[??].sub.2] with the value ([alpha]/[alpha] + [beta])[[??].sub.1] and obtain the following:

d[U.sub.2]/d[I.sub.1] = [[??].sup.[alpha].sub.2][([beta]/[alpha] + [beta]).sup.[beta]] [[??].sup.[beta].sub.1]/ [P.sup.[beta].sub.G](- [alpha]/[I.sub.1])

= ([[alpha].sup.[alpha][beta]/[([alpha] + [beta]).sup.[alpha] + [beta]])[[??].sup.[alpha] + [beta].sub.1] [- [alpha]/[[??].sub.1]] (14)

Thus from (12) and (14) above we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

Q.E.D.

Conclusion: Any positive increase in the income of the wealthy at the expense of the poor, will lead to an increase in the total welfare of society. This is also the most important conclusion of Itaya et al. (1997), p. 292, namely, that inequality raising policies will also raise social welfare.

This conclusion is true at least for small changes in income inequality close to the border case unless the social welfare function is of the Rawlsian-Egalitarian approach.

4. Cooperative vs. Non-cooperative Solutions

Another welfare question related to the conflict between the efficient solution and the equality point of view, is the comparison between the efficient solution of cooperative behavior of consumers namely the Lindhal solution and the solution we develop in the above model of non-cooperative behavior namely the Nash equilibrium.

In the Lindhal solution where all consumers are identical in tastes (utilities) but differ in incomes the cooperative solution is where each individual contributes to cover the price of public good, [P.sub.G], according to his relative income.

If in our example [I.sub.1] + [I.sub.2] = [[??].sup.I] the contribution of individuals 1 and 2 towards coverage of [P.sub.G]:

is ([I.sub.1]/[I.sub.1] + [I.sub.2]) [P.sub.G] and ([I.sub.2]/[I.sub.1] + [I.sub.2])[P.sub.G], respectively. Each individual, i, maximizes his utility: Max [C.sup.[alpha].sub.i][G.sup.[beta]] subject to his new budget constraint:

[C.sub.i] + ([I.sub.i]/I)[P.sub.G]G = [I.sub.i].

From the F.O.C. (first order condition) where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we get [C.sub.i] = [alpha]/[beta]([I.sub.i]/[bar.I]) [P.sub.G]G.

Substituting into the budget constraint we find the optimal value of [G.sub.L] Lindhal equilibrium:

[G.sub.L] = ([beta]/[alpha] + [beta])[bar.I]/[P.sub.G] and [C.sub.i] = ([alpha]/[alpha] + [beta])[I.sub.i].

An immediate conclusion is that the optimal value of the public good level (Lindhal solution) is not affected by income distribution as each individual contributes to the public good proportionally to his income. We compare this equilibrium to a Nash type equilibrium. In the previous section we distinguished between different regions. In Region 2 and 3 we found that: [G.sub.N] = [G.sub.1] + [G.sub.2] = ([beta]/2[alpha] + [beta])[bar.I]/[P.sub.G], where [G.sub.N] is the level of public good at Nash equilibrium. Thus: [G.sub.L] > [G.sub.N].

However in extreme cases of high inequality where one of the individuals does not contribute to the public good at all, the total level of public good G is increasing and is equal to the following:

In Region 1 where [I.sub.1] > ([alpha] + [beta]/[alpha]) [I.sub.2] then [G.sub.N] = [G.sub.1] = ([beta]/[alpha] + [beta])[I.sub.1]/[P.sub.G] and as [I.sub.1] increases and approaches I, [G.sub.N] = [beta] [bar.I]/[alpha] + [beta][P.sub.G], which is equal to the optimal Lindhal solution, [G.sub.L].

[FIGURE 1 OMITTED]

In region 4 where [I.sub.2] > ([alpha] + [beta]/[alpha])[I.sub.1] then [G.sub.N] = [G.sub.2] = ([beta]/[alpha] + [beta])[I.sub.2]/[P.sub.G] and then as [I.sub.2] approaches to [bar.I], [G.sub.N] also approaches [G.sub.L] (see figure 1).

We can conclude from the above discussion that a higher degree of income inequality may lead to higher and more efficient supply of the public good. However, we can also reach a Lindhal equilibrium by a Nash non-cooperative game when all income is allocated to the wealthy and the income of the poor approaches zero. This may be an "efficient" allocation of income but unacceptable (and undesirable politically and ideologically) to society.

5. Concluding Remarks

Our conclusions from the above are clear. In regions that contain large inequalities in income distribution, increasing the income of consumer 1 (who is poor) by reducing the income of consumer 2, will increase the utility of consumer 1 by less than the utility reduction of consumer 2.

In regions of moderate inequality, the utility of consumer 1 (who is poor) will not change as a result of increasing his income share since he will increase his contribution towards the public good while consumer 2 will decrease his contribution, and therefore neither of the utilities will change.

The important implication of this analysis is that extreme inequality in income can bring about a conflict in which those who have a smaller share in income will strive to improve their position at the expense of those who have a larger share. However, in moderate cases of inequality there is no motivation to take such steps since they will bring no change in the distribution of utility (standard of living), and will only result in technical changes in income distribution without real welfare effects. This result implies that society voluntarily corrects a certain degree of what can be considered as "unfair" inequality in income distribution so that compulsory government intervention is not necessarily required. In more extreme cases when income inequality is high, the government might be required to take further measures towards greater equality in income distribution, in order to reduce utility gaps as well as social tensions.

Our analysis predicts that in areas, such as emerging-market countries, that tend to have an unusually high disparity in income distribution we would expect to observe a relatively high level of voluntary investment in public goods. This is of course subject to the specific utility functions of the citizenry and the degree of utility the people derive from public goods in comparison to private goods. Still, even if we can expect certis paribus more private investment in public goods, that does not necessarily mean that more expenditure on public goods by the wealthy would always reduce the utility and standard of living gap for any degree of income inequality.

An additional consideration is the public sector's progressive tax system that at least partially finances expenditures on public goods. According to our results, a compulsory tax system by an economic planner is useless: neither proportional nor progressive taxes have any real impact on utility distribution, especially if tax revenues are spent on collective or pure public goods. Moreover, the "net income" redistribution will be a natural outcome of voluntary contributions of different income groups who contribute their fair share to the public good financing, without government intervention. The progressive tax burden is at times acceptable and agreeable to the society, especially in extreme income inequality cases, while in other situations it is unnecessary, as the society may generate a similarly efficient solution without the necessity of a formal tax system imposed from above.

Another implication is that in the case of a voluntary supply of the public good, there may be a conflict between efficiency and equality. Whenever income inequality reaches a certain (high) level it may lead to a higher level of public good supply, financed by the wealthy. The increase in inequality may lead to a Pareto improvement in the welfare of the whole society, as it reduces the welfare of the poor, by less than the improvement in the welfare of the wealthy. This is true especially in cases where the relative importance of the public good in terms of utility is high.

Appendix

By taking the value of G2 of equation (5) in equation (4) we get the following:

G1 = ([beta]/[alpha] + [beta])[I.sub.1]/[P.sub.G] - [alpha][beta]/[([alpha] + [beta]).sup.2 [I.sub.2]/[P.sub.G] + [[alpha].sup.2]/[([alpha] + [beta]).sup.2] [G.sub.1] (A.1)

therefore

[G.sub.1] [1 - [[alpha].sup.2]/[([alpha] + [beta]).sup.2] = ([beta]/([alpha] + [beta]) [I.sub.1]/[P.sub.G] - [alpha][beta]/[([alpha] + [beta]).sup.2] [I.sub.2]/[P.sub.G] (A.2)

[G.sub.1][[([alpha] + [beta]).sup.2] - [[alpha].sup.2]/[([alpha] + [beta]).sup.2]] = ([beta]/[alpha] + [beta])[I.sub.1]/[P.sub.G] - [alpha][beta]/[([alpha] + [beta]).sup.2] [I.sub.2]/[P.sub.G] (A.3)

thus

[G.sup.[omicron].sub.1] = ([beta]([alpha] + [beta])/[([alpha] + [beta]).sup.2] - [[alpha].sup.2]) [I.sub.1]/[P.sub.G] - ([alpha][beta]/[([alpha] + [beta]).sup.2] - [[alpha].sup.2])[I.sub.2][P.sub.G] (A.4)

In the same way we can solve the optimal value of [G.sup.0.sub.2]

[G.sup.[omicron].sub.2] = ([beta]([alpha] + [beta])/[([alpha] + [beta]).sup.2] - [[alpha].sup.2])[I.sub.2]/[P.sub.G] - ([alpha][beta]/[([alpha] + [beta]).sup.2] - [[alpha].sup.2])[I.sub.1]/[P.sub.G] (A.5)

References

Atkinson, A. B. and Stiglitz, J. E. (1980). Lectures on Public Economics. McGraw-Hill, W. W. Norton.

Bergstrom, T. C., Blume, L., Varian, H., (1986). "On the Private Provision of Public Goods", Journal of Public Economics, 29, pp. 25-49.

Epstein, G., Spiegel, U., (2001). "Natural Inequality Production and Economic Growth" Labour Economics, 8, pp. 463-473.

Olszewski, W., Rosenthal, H., (2004). "Politically Determined Income Inequality and the Provision of Public Goods", Journal of Public Economic Theory, 5, pp. 707-735.

Itaya, J., Meza, D., Myles, G. D., (1997). "In Praise of Inequality: Public Good Provision and Income Distribution", Economics Letters, 57, pp. 289-296.

Warr, P. G., (1983). "The Private Provision of a Public Good is Independent of the Distribution of Income", Economics Letters, 13, pp. 207-211.

(1) See Appendix for proof.

Uriel Spiegel, The Interdisciplinary Department of Social Sciences, Bar-Ilan University, Ramat-Gan, 52900, Israel, email: spiegeu@mail.biu.ac.il and Visiting Associate Professor, Department of Economics, University of Pennsylvania, PA 19014-6297, USA