Income inequality and redistribution

Intuition [Beginner]

Imagine 100 people in a room with $100 total. If everyone has $1, perfect equality. If one person has $99 and the other 99 share $1, extreme inequality. Most real economies fall between these extremes -- but where, exactly?

Income inequality describes how unevenly income is distributed across a population. It exists in every economy, but the degree varies enormously. In the Nordic countries, the richest 10% earn about 5-6 times more than the poorest 10%. In parts of Latin America and sub-Saharan Africa, the ratio can exceed 20 or 30.

Why does it matter? Two reasons. First, extreme inequality may be unfair -- people's incomes depend partly on circumstances they did not choose (family background, health, geography, innate talent). Second, inequality may have economic consequences: it can reduce social mobility, increase political instability, and, according to some economists, reduce overall economic growth.

Measuring inequality requires tools. The simplest is to compare the share of total income going to different groups (e.g., the top 10% versus the bottom 50%). A more comprehensive measure is the Gini coefficient, which summarises the entire income distribution in a single number between 0 (perfect equality) and 1 (perfect inequality).

Governments redistribute income through progressive taxation (higher earners pay a larger share) and transfer payments (welfare, unemployment benefits, social security). The central debate is how much redistribution is appropriate, given the potential trade-off between equity and efficiency.

Visual [Beginner]

Lorenz Curve

  Cumulative % of Income
  100% |                        *
       |                    *
       |                 *  <- Line of equality (45 degrees)
       |              *
       |           *
       |        *  <- Lorenz curve
       |      *      (actual distribution)
       |    *
       |  *
       |*
    0% +-------------------------
       0%                   100%  Cumulative % of Population

  Gini coefficient = A / (A + B)
  where A = area between equality line and Lorenz curve
        B = area below Lorenz curve

  Gini = 0: perfect equality (Lorenz = 45-degree line)
  Gini = 1: perfect inequality (one person has everything)


Redistribution Flow

  Market income --> Tax system --> Government
                       |
  Government --> Transfers --> Low-income households
             --> Public services --> All households

Worked Example [Beginner]

A country with 10 households has the following market incomes (in thousands):

Household	Income	Cumulative Income	Cumulative % Income	Cumulative % Population
1 (poorest)	$5	$5	3.3%	10%
2	$8	$13	8.7%	20%
3	$10	$23	15.3%	30%
4	$12	$35	23.3%	40%
5	$14	$49	32.7%	50%
6	$16	$65	43.3%	60%
7	$18	$83	55.3%	70%
8	$22	$105	70.0%	80%
9	$30	$135	90.0%	90%
10 (richest)	$15	$150	100%	100%

The bottom 50% of households earn 32.7% of total income. The top 10% earn 10.0%. The Lorenz curve bows only slightly below the equality line, indicating moderate inequality.

The Gini coefficient can be approximated from the area between the equality line and the Lorenz curve. A rough calculation gives a Gini of approximately 0.22 -- relatively low by global standards. For comparison: Sweden's Gini is about 0.27, the United States about 0.39, and South Africa about 0.63.

Check Your Understanding [Beginner]

Formal Definition [Intermediate+]

Lorenz curve and Gini coefficient

Let $F(y)$ be the cumulative distribution function of income and $\mu$ be the mean income. The Lorenz curve plots the cumulative share of total income against the cumulative share of population:

$$L(p)=\frac{1}{\mu }{\int }_0^{F^{-1}(p)}ydF(y)$$

where $p$ is the cumulative population share. $L(p)$ maps $[0,1]\to [0,1]$ with $L(0)=0$ and $L(1)=1$.

The Gini coefficient is:

$$G=\frac{A}{A+B}=1-2{\int }_0^{1}L(p)dp$$

Equivalently, $G$ is half the mean absolute difference relative to the mean:

$$G=\frac{{\sum }_{i=1}^n{\sum }_{j=1}^n|y_i-y_j|}{2n^2\bar{y}}$$

Atkinson index

The Atkinson index $A(ϵ)$ incorporates normative judgments about inequality through the inequality aversion parameter $ϵ\ge 0$:

$$A(ϵ)=1-{\left[\frac{1}{n}\sum _{i=1}^n{\left(\frac{y_i}{\bar{y}}\right)}^{1-ϵ}\right]}^{1/(1-ϵ)}$$

for $ϵ\ne 1$. Higher $ϵ$ places more weight on the incomes of the poor. When $ϵ=0$, $A=0$ regardless of distribution (inequality does not matter). As $ϵ\to \infty$, the index reflects only the income of the poorest individual (Rawlsian).

Redistribution and the equity-efficiency trade-off

A standard formulation of the redistribution problem (following Mirrlees, 1971) models the government as choosing a tax schedule $T(y)$ to maximise a social welfare function:

$$\underset{T(\cdot )}{\max }\int W(u_i(c_i,{\ell }_i))dF({\theta }_i)$$

subject to incentive compatibility constraints (individuals choose labour supply ${\ell }_i$ given the tax schedule). Higher redistribution reduces inequality but may reduce labour supply and output. The optimal tax rate balances the marginal social benefit of redistribution against the marginal efficiency cost of distorting work incentives.

Key Concepts [Intermediate+]

• Income inequality: The unequal distribution of income across individuals or households in an economy.
• Wealth inequality: The unequal distribution of net assets. Typically exceeds income inequality.
• Lorenz curve: A graphical representation of income distribution, plotting cumulative income share against cumulative population share.
• Gini coefficient: A summary measure of inequality from 0 (perfect equality) to 1 (perfect inequality). Equals twice the area between the Lorenz curve and the 45-degree line.
• Progressive taxation: A tax system in which the average tax rate increases with income. Higher earners pay a larger share of income in tax.
• Transfer payments: Government payments to individuals (welfare, unemployment benefits, social security) that redistribute income.
• Equity-efficiency trade-off: The idea that redistribution reduces inequality but may reduce economic efficiency by distorting incentives to work, save, and invest.
• Social mobility: The degree to which individuals can move between income groups across generations. High inequality and low mobility often coincide.

Exercise 1. Two countries have the same Gini coefficient of 0.35. Country A has a rapidly growing economy with high social mobility. Country B has stagnant growth and low social mobility. Why might the same Gini tell very different stories about these two societies?

Exercise 2. Suppose a government introduces a flat tax of 20% on all income and redistributes the revenue equally as a lump-sum transfer. Show that this reduces inequality (the post-tax Gini is lower than the pre-tax Gini). Is this system progressive?

Academic Perspectives [Master]

Neoclassical view

Neoclassical economists analyse redistribution through the lens of the equity-efficiency trade-off. The seminal contribution is Mirrlees (1971), which models optimal income taxation as a problem of maximising social welfare subject to incentive compatibility constraints. The key result is that the optimal top marginal tax rate depends on the elasticity of labour supply: if high earners reduce their work effort significantly in response to higher taxes, the optimal tax rate is lower.

Saez (2001) extended the Mirrlees framework, deriving optimal tax formulas expressed in terms of observable elasticities and the income distribution. The Saez formula gives the optimal top marginal tax rate as:

$${\tau }^{\ast }=\frac{1-\bar{g}}{1-\bar{g}+a\cdot e}$$

where $\bar{g}$ is the average social marginal welfare weight of top earners, $a$ is the Pareto parameter of the income distribution, and $e$ is the elasticity of taxable income. This framework supports top marginal rates in the range of 50-80% under plausible parameter values.

Neoclassical economists disagree on the severity of the trade-off. Some (e.g., Feldstein, 1999) emphasise the deadweight loss of taxation. Others (e.g., Diamond and Saez, 2011) argue that the efficiency costs are modest at current tax levels and that significant redistribution is both feasible and desirable.

Keynesian perspective

Keynesians emphasise that inequality can reduce aggregate demand. If marginal propensity to consume falls with income, then transferring income from high earners (who save a large share) to low earners (who spend a larger share) increases total consumption and stimulates economic activity. In this view, redistribution can enhance efficiency, not just equity -- the trade-off may be false during recessions or periods of deficient demand.

Keynesians also point to the macroeconomic instability associated with high inequality. Rajan (2010) argued that rising inequality in the United States led to political pressure for easy credit, contributing to the 2008 financial crisis. Stiglitz (2012) has argued that inequality distorts political institutions, leading to policies that favour the wealthy at the expense of overall economic performance.

Austrian school

Austrian economists are generally opposed to government redistribution, arguing that market outcomes reflect individual choices, productivity, and consumer preferences. Inequality, in the Austrian view, is a natural and desirable feature of a dynamic economy: it reflects differences in talent, effort, risk-taking, and time preference. Entrepreneurs who create value for consumers earn high incomes as a reward; redistributing those incomes reduces the incentive to innovate.

Mises and Hayek argued that government cannot determine a "fair" distribution because value is subjective and knowledge is dispersed. Any redistribution scheme requires a central planner with knowledge that no individual or bureaucracy possesses.

Critics note that this view does not adequately account for the role of inherited wealth, structural barriers, and market power in generating inequality. Market outcomes reflect initial endowments and institutional rules, not purely individual merit.

Marxian perspective

Marxian economists view inequality not as a policy problem to be corrected but as an inherent feature of capitalism. The capitalist mode of production, by paying workers less than the value they produce (extracting surplus value), necessarily generates inequality between the class that owns the means of production and the class that sells its labour. Redistribution through taxation and transfers addresses symptoms but not causes.

Piketty (2014), while not strictly Marxian, provided empirical support for a structural tendency toward inequality in capitalism. His central finding -- that the rate of return on capital ($r$) tends to exceed the rate of economic growth ($g$), causing wealth to concentrate over time -- echoes Marxian concerns about the dynamics of capital accumulation. Piketty's solution (a global wealth tax) is reformist rather than revolutionary, but the analysis of structural inequality aligns with Marxian themes.

Institutional economics

Institutional economists focus on how the "rules of the game" shape the distribution of income. Acemoglu and Robinson (2012) argue that inclusive economic institutions (secure property rights, competitive markets, broad access to opportunities) produce lower inequality and higher growth, while extractive institutions concentrate power and wealth. The distribution of income is not a natural outcome of market forces but a product of political and economic institutions.

Historical Context [Master]

• Pareto (1897): Vilfredo Pareto observed that income distributions across countries and time periods follow a similar power-law pattern (the Pareto distribution). "Pareto's law" suggested that inequality was a stable feature of economies, resistant to policy intervention.
• Kuznets (1955): Simon Kuznets hypothesised an inverted-U relationship between economic development and inequality: inequality rises in early industrialisation (as workers move from low-productivity agriculture to higher-productivity industry) and falls in later stages (as the service sector expands and education spreads). The "Kuznets curve" was influential but has not been supported by late-20th-century data.
• Mirrlees (1971): James Mirrlees developed the theory of optimal income taxation, formalising the equity-efficiency trade-off. He received the Nobel Prize in 1996.
• Atkinson (1970): Anthony Atkinson developed the Atkinson index and pioneered the normative measurement of inequality. His work emphasised that inequality measurement inevitably involves ethical judgments.
• Piketty (2014): Thomas Piketty's Capital in the Twenty-First Century used centuries of tax data from multiple countries to document a long-run trend toward wealth concentration. The book's central claim -- $r>g$ leads to rising inequality -- sparked global debate and brought inequality to the forefront of economic policy discussions.
• World Inequality Report (2018-present): A collaborative effort (Alvaredo, Atkinson, Piketty, Saez, Zucman) providing standardised inequality data for over 100 countries, enabling systematic cross-country and temporal comparisons.

Bibliography [Master]

• Acemoglu, D., & Robinson, J. A. (2012). Why Nations Fail: The Origins of Power, Prosperity, and Poverty. Crown Business.
• Atkinson, A. B. (1970). On the measurement of inequality. Journal of Economic Theory, 2(3), 244-263.
• Diamond, P. A., & Saez, E. (2011). The case for a progressive tax. Journal of Economic Perspectives, 25(4), 165-190.
• Mirrlees, J. A. (1971). An exploration in the theory of optimum income taxation. Review of Economic Studies, 38(2), 175-208.
• Piketty, T. (2014). Capital in the Twenty-First Century. Harvard University Press.
• Rajan, R. G. (2010). Fault Lines: How Hidden Fractures Still Threaten the World Economy. Princeton University Press.
• Saez, E. (2001). Using elasticities to derive optimal income tax rates. Review of Economic Studies, 68(1), 205-229.
• Stiglitz, J. E. (2012). The Price of Inequality. W. W. Norton.