# Bounded Rationality

**All Rich and Equal: The Long-Run Effect of Expansion in Wealth**

Ben O’Neill

University of New South Wales

Email: ben.oneill@hotmail.com

Economists and political philosophers are often concerned with measuring the level of inequality of wealth or income in an economy, and in particular, with the question of whether certain kinds of political systems lead to greater or lesser levels of inequality. Indeed, the issue of inequality is regarded as being sufficiently important that there are textbooks devoted to the subject (e.g., Cowell 2011) and even entire scholarly journals devoted to analysis of the subject (e.g., Journal of Economic Inequality, Journal of Income Distribution).

Notwithstanding this fact, there is widespread confusion about the nature of wealth inequality and the effects of wealth expansion and income mobility over time. One of the most common manifestations of this confusion is regular assertions that equality is increasing and that this is the result of free-market capitalism. It is now common to hear critics of the free market concede that it is conducive to the generation of large amounts of wealth, but claim that this wealth comes with rising inequality.

There is enough of an empirical basis for this to make the story plausible to many people. Comparing static measures of inequality across time there has been rising inequality of incomes in Western countries since approximately after WWII when economies were liberalised to some extent from wartime controls and “progressive” taxation was ameliorated. However, such comparison of static measures is invalid for a number of reasons; it gives a very misleading picture of changes in inequality.

The pseudo-dynamic empirical investigations obtained by comparing static measures of inequality across time fail to account for mobility of incomes over a human lifetime and also fail to account for diminishing marginal returns of wealth. Broadly speaking, as a society becomes wealthier there are two major effects that are both conducive to greater equality (other smaller methodological issues can also affect this; see, e.g., Fields 1987). One is the effect of diminishing marginal returns, which tends to reduce inequality in the attainment of human needs. The other is the effect of greater health and longevity, which affects income mobility in ways which also tend to reduce inequality over time. Analysis of changes in inequality over time can only be done properly by adjusting static measures of inequality to account for these effects.

**Measures of inequality of wealth — the Lorenz curve and Gini coefficient**

Analysis of inequality in wealth usually uses money as the numéraire for calculation of the level of inequality. This is quite a natural measure, insofar as money serves as a proxy for all exchangeable goods in the economy. The inequality in the distribution of wealth or income is often quantified via a transform of the static distribution of that quantity at some point in time. The most common transform is the Lorenz function (Lorenz 1905). In contexts where we are dealing with the distribution of wealth[1] this function tells us the proportion of total wealth held by some specified proportion of the lowest wealth-holders. To obtain a scalar measure of inequality it is common to use the Lorenz curve to determine the Gini coefficient of the distribution (Gastwirth 1972; Dagum 1980). This is a value between zero and one, with zero representing perfect equality of wealth (i.e., everyone has the same amount of wealth) and one representing perfect inequality of wealth (i.e., one person has all the wealth).

The Lorenz curve and Gini coefficient are both invariant to changes in scale, so that the units of measurement of wealth (e.g., dollars, cents, francs, etc.) do not affect the measurement of inequality. This is a desirable property, insofar as these units are arbitrary in a static scenario.[2] However, the measurement of inequality of wealth, measured in money, also means that changes in the average level of wealth over time do not show up in comparisons of wealth inequality across time. When comparisons in the distribution of wealth are made over time it is often the case that the average level of wealth is changing over time (usually increasing) and this does not manifest in any change in the Lorenz curve or Gini coefficient. Even if the average level of wealth doubles or triples over time the distribution of wealth maintains its shape, and this means that there is no change in the measures of inequality over time.

In a static analysis this is entirely desirable — if the distribution of wealth maintains a particular shape (but is subject to changes in scale) then the change in scale should not result in any change in the measured level of inequality. However, as any economist knows, wealth itself is only a means to the end of the satisfaction of human wants. Wealth stands as an adequate proxy for the entire class of exchangeable goods, but as a person’s wealth rises there is diminishing marginal returns to this class of goods, and a consequent substitution to non-exchangeable goods. What is really of interest in the analysis of inequality of wealth is inequality in the satisfactions that are obtained from these varying levels of wealth.

The Lorenz curve and Gini coefficient of raw wealth or income values is pervasive in economic analysis and related political discussion. While somewhat useful in giving a static picture, these measures are often used for more than this. In particular, the Lorenz function and Gini coefficient for raw wealth or income values are often shown changing over time and this is intended to give a picture of the changing level of inequality. The World Bank and other international agencies keep data on the change in these measures over time (World Bank 2014; UNU 2014) and this data is used in analysis of inequality over time (e.g., Biancotti 2006).

Unfortunately, this kind of time comparison can be highly misleading. If the average level of wealth is rising over time then there will be diminishing marginal returns to wealth; the rising level of wealth makes further acquisition of exchangeable goods relatively less valuable than the acquisition of non-exchangeable goods. Use of raw wealth or income values in the Lorenz curve of Gini coefficient ignores this change.

**Measuring inequalities over different times or places**

In cases where we are concerned with a comparison of inequality across two different distributions of wealth (varying by time and/or place) the comparison in raw measures of inequality is not really telling us what we need to know. What is needed is an accounting for the fact that higher levels of wealth come with diminishing marginal returns. This might be accomplished by applying some kind of transformation to the individual wealth or income values to adjust for diminishing returns prior to putting them into inequality measures. Some form of increasing concave transformation with a fixed point at zero could be applied to inflation-adjusted wealth or income values. The concavity of the transformation would ensure that the marginal returns are diminishing, perhaps down to zero.[3] This would yield a measure accounting for diminishing returns to wealth.[4]

There is some recognition of the need to adjust for changing levels of wealth or income. For example, Stanton (2006) advocates adjusting individual income values by taking logarithms prior to calculating inequality measures. (This is one example of an increasing concave transformation that could be applied. It would need to be adjusted slightly to have a fixed point at zero.[5]) This idea is based on a variation of existing methods used in the calculation of the Human Development Index by the United Nations Development Programme. This index uses income data and adjusts per-capita income by taking logarithms. This is explained as an attempt to account for diminishing marginal returns of income (UNDP 1990, p. 13; UNDP 2005, p. 341) but as Stanton rightly points out, it is not sufficient to account for this, owing to the fact that the transformation should be applied to individual values rather than the aggregate (Ibid Stanton, p. 9).

It is easy to illustrate the effects of this kind of adjustment. In Figure 1(a) below we plot the income histogram for a set of randomly generated income values from a log-normal distribution. We consider these raw income values and also consider what happens when we transform them using an appropriate concave function to attempt to account for diminishing marginal returns.[6] The Lorenz curve and associated Gini coefficients for the raw income values and their transformed values are shown in Figure 1(b). It can easily be seen that the transformation yields much greater equality.

**Figure**** 1:** Analysis of inequality for randomly generated income data

The mathematical effect of transforming the individual wealth or income values via a concave function is to make the Lorenz curve closer to the line of perfect equality and reduce the associated Gini coefficient. Intuitively this reflects the fact that when the diminishing marginal returns of wealth are considered, people are more equal than a consideration of their raw wealth or income values would suggest.

**The effect of rising income and wealth**

As wealth rises over time this effect of diminishing marginal returns will yield greater equality even when the shape of the distribution of wealth or income is preserved (i.e., when the inequality in the raw income or wealth values is constant). This is shown in Figure 2 for the same income values. As the income level is scaled upward, with the shape of the distribution remaining the same, the Gini coefficient decreases. So, for example, if the shape of the income distribution shown in Figure 1(a) is preserved but the scale is increased by a factor of (i.e., a multiplicative increase in the income of each individual) the Gini coefficient decreases to the value . Note that this comparison is done *ceteris paribus* since the shape of the income distribution is assumed to be constant over time.

**Figure**** 2:** An example of changes in inequality subject to income growth[7]

If the average wealth and income increases without limit the Gini coefficient of the transformed values approaches zero, meaning that the distribution of the transformed values approaches a state of perfect equality.[8] It is important to understand that this occurs solely due to the scaling — it occurs when the shape of the distribution of the raw values remains the same, so that inequality of the raw values is constant. Intuitively this reflects the fact that every individual in society is becoming extremely wealthy, so that inequalities in wealth manifest in smaller and smaller inequalities in the satisfaction of human needs, with those inequalities eventually being pushed down to zero. In the limit the inequalities in satisfaction become so small that the situation approaches perfect equality, notwithstanding that inequalities in raw wealth and income values are preserved. (One very nice aspect of this kind of analysis is that it is then extremely easy to measure the extent to which the rising level of wealth is responsible for diminishing levels of inequality. This is done by comparing the Gini coefficient of the raw values with the Gini coefficient of the transformed values.)

It is worth stressing that this example is merely illustrative, since there is nothing sacrosanct in the chosen transformation for income or the randomly generated income distribution. The important thing illustrated by the example is the fact that the real level of inequality diminishes as wealth grows, owing to the effect of diminishing marginal returns. The exact rate at which this occurs (shown in Figure 2) depends on the distributional shape of wealth or income, the rate of growth, and the form of the transformation.

Although not dealt with here, other effects of rising wealth should also be considered in dealing with inequality comparisons over time. As wealth rises there is a corresponding increase in health and longevity and this means that the income trajectory for people will tend to be longer. Measurement of inequality should also account for the fact that people in the economy are different ages and are therefore at different points in their income trajectory. These various effects mean that inequality measures which are taken over raw wealth or income values should be regarded as nominal measures. Real measures would require accounting for diminishing marginal returns and age difference in the income trajectory. Most importantly, comparisons of inequality across time should be based on *real* measures which have been adjusted to allow for these issues.

**Conclusion: all rich and equal**

The title of this article alludes to the ultimate consequences of this effect as the average level of wealth and income in an economy grows. If the distribution of wealth remains constant in shape as the average level of wealth grows then each individual in the economy grows rich on an absolute standard. As this occurs there is also a corresponding increase in longevity and therefore a corresponding lengthening of the income trajectory over the age of each person. Even if the shape of the distribution does not remain constant, there will be convergence to inequality due to diminishing marginal returns so long as the changes in the shape of the wealth distribution do not outweigh this effect.

Comparisons of inequality based on nominal measures will give the illusion that inequality is constant in this situation but in reality the level of real inequality is diminishing down to zero. (In more realistic situations where the distributional shape is changing over time the nominal measures may show changes in the level of inequality but these will tend to overestimate the level of inequality more and more over time, and there will still be convergence to perfect equality for all but pathological cases.) Present comparisons of inequality mislead people by failing to account for these effects. They thereby exaggerate present levels of inequality and romanticise past levels of equality.

This has implications in judging institutional frameworks and political philosophies. If it is true that free-market capitalism is conducive to the generation of wealth (as even its critics seem to concede) then it must also be true that *ceteris paribus* this rising wealth will yield real reductions in inequality over time, even when inequalities in the distribution of raw wealth values persists . Indeed, in the limit this will lead to a situation approaching perfect equality. In this sense, the people in such an economy will become *all rich and equal*.

**References**

Biancotti, C. (2006) A polarisation of inequality? The distribution of national Gini coefficients 1970-1996. *Journal of Economic Inequality* **4**, pp. 1-32.

Cowell, F.A. (2011) *Measuring Inequality (Third Edition)*. Oxford University Press: Oxford.

Dagum, C. (1980) The generation and distribution of income, the Lorenz curve and the Gini ratio. *Economie Appliquée* **33**, pp. 327-367.

Diener, E. and Biswas-Diener, R. (2002) Will money increase subjective well-being? A literature review and guide to needed research. *Social Indicators Research* **57(2)**, pp. 119-169.

Diener, E., Sandvik, E., Seidlitz, L. and Diener, M. (1993) The relationship between income and subjective well-being: relative or absolute? *Social Indicators Research ***28(3)**, pp. 195-223.

Easterlin, R.A. (2005) Diminishing marginal utility of income? Caveat emptor. *Social Indicators Research* **70(3)**, pp. 243-255.

Fields, G.S. (1987) Measuring inequality change in an economy with income growth. *Journal of Development Economics* **26(2)**, pp. 357-374.

Gastwirth, J.L. (1972) The estimation of the Lorenz curve and Gini index. *Review of Economics and Statistics* **54(3)**, pp. 306-316.

Kolm, S.C. (1976a) Unequal inequalities I. *Journal of Economic Theory* **12**, pp. 416-442.

Kolm, S.C. (1976b) Unequal inequalities II. *Journal of Economic Theory* **13**, pp. 82-111.

Lorenz, M.O. (1905) Methods for measuring the concentration of wealth. *Journal of the American Statistical Association* **9**, pp. 209-219.

Stanton, E. (2006) *Accounting for inequality: a proposed revision of the Human Development Index*. Working Paper 119, Political Economy Research Institute, University of Massachusetts.

UNDP (1990) *Human Development Report*. United Nations Development Programme: New York.

UNDP (2005) *Human Development Report*. United Nations Development Programme: New York.

UNU-WIDER (2014) *World Income Inequality Database (WIID Version 3.0B)*. United Nations University, September 2014.

Veenhoven, R. (1991) Is happiness relative? *Social Indicators Research* **24(1)**, pp. 1-34.

World Bank (2014) Distribution of income and consumption. *World Development Indicators*, Table 2.9.

[1] To be clear, when we talk about the “distribution” of wealth or income in this article, we are using the term only in the statistical sense. We do not intend to imply or insinuate that wealth or income is actually “distributed” in the sense of being collectively owned and then apportioned out to individuals.

[2] Alternative measures of inequality have been proposed which do not satisfy this scale-invariance property (e.g., Kolm 1976a, 1976b). Nevertheless, scale-invariance is generally regarded to be useful in ensuring that the units of measurement do not affect the inequality measure.

[3] Given a wealth variable it would be sensible to use a second-order differentiable function with *R(0) = o. R'(w) > 0. R” (w) < 0* and *lim _{w→∞} R’ (w)=0*. This captures the requirement of increasing value (i.e., wealth is a good) and diminishing marginal returns with the marginal returns approaching zero in the limit.

[4]Diminishing marginal returns is a core concept in microeconomic theory. When preference relations are strictly convex this yields a utility function which is subject to diminishing marginal returns; Jehle and Reny 2000, p. 19. Nevertheless, “utility” in this context is merely a representation of an underlying ordinal preference. Diminishing marginal returns with respect to wealth and income have also been found empirically in various investigations in “happiness research”; see e.g., Veenhoven 1991, p. 10; Diener *et al* 1993, p. 204; Diener and Biswas-Diener 2002, p. 204; c.f., Easterlin 2005). The latter should be taken with appropriate scepticism since they are based on self-reported happiness levels. We do not wish to venture into the complications of utility theory and interpretation here.

[5] The reason a fixed point at zero is necessary is that the inequality measures we have discussed here apply only to non-negative quantities. Any transformation which gave negative values would make the inequality measures inapplicable. Hence, the “logarithmic” transformation would have to be shifted to give a fixed point at zero, and thereby yield only non-negative values. For example, if the level of wealth is *w ≥ 0* then the transformed value would be *1n (1 + w) ≥ 0*. This would give an increasing concave function with a fixed point at zero.

[6] We let *w _{1},…,w_{n}* be the set of raw income values measured in $1,000s (for our analysis we generated one-thousand of these values). We define the transformed values

*q*which is the same transformation we have previously discussed. This satisfies the suggested properties for such a function.

_{1},…,q_{n}by q_{i}= 1n(1+w_{i})[7] In this figure the Gini coefficient for the value zero on the horizontal axis is the coefficient for the transformed values from the original raw income data. The different values along the horizontal axis represent scale changes in income on a logarithmic scale. The value of negative infinity at the left of the figure reduces the income values arbitrarily low, which means that they are essentially a linear transformation of the raw values. (At this limiting value the Gini coefficient is the same as for the raw income values.)

[8] This follows from the fact that the marginal returns are approaching zero in the limit – i.e., *lim _{w→∞} R’ (w)=0*. This applies more broadly than just in the present example, but we limit ourselves to a single example here.