Does Piketty’s r - g Move with Inequality?

1 At a glance

In “Capital in the 21st Century”, Thomas Piketty argues that income inequality rises when returns to capital (r) exceed the growth rate of the economy (g). This report tests whether this mechanism holds in the UK and USA from 1920 onward, using data on equity returns from the MacroFinance and History Lab, GDP growth per capita from the Maddison Project, and the share of income held by the top 1% of earners from the World Inequality Database. Exploratory analysis shows that inequality is non-stationary and cannot be modeled as a single regime, and the contemporaneous relationship between r-g and inequality is highly unstable over time. However, there is a statistically significant effect in the US when lags are included, but no equivalent result for the UK, a result that is also robust to outliers and stable across sub-periods. Hence, the evidence partially supports Piketty’s hypothesis in the short run for one country, but major external events occurring during the timeline evaluated limit stronger conclusions.

2 Introduction

In his book, “Capital in the 21st Century”, the economist Thomas Piketty has argued that a rise in inequality, defined as the share of income held by the top 1% of earners in the income distribution, is explained by whether the return to capital (r) exceeds the growth rate of GDP per capita (g). The logic behind this effect is that when r exceeds g, wealth holders see their income grow faster than the economy, concentrating an increasing share of total income at the top of the distribution. Thus, the research question is:

Is a higher measure of (r-g) associated with a higher top 1% income share?

The analysis focuses on the short run, using annual data for the UK, US, and Germany stretching back to the late 19th century. But, Germany is ultimately dropped from the formal modelling due to extensive gaps in its mid-century data, narrowing down the scope to the UK and USA from 1920 onward.

3 Data

3.1 Narrowing down the question

The current topic was chosen because it provides a clear economic mechanism to test, connects to a major debate that is relevant to important social issues and provides the opportunity to analyse the data series in many ways.

But, before deciding on this specific topic, I considered a number of other topic choices within economics that involved a specific, testable hypothesis:

1. What are the time lags in which major innovations, such as the invention of the internet or the computer, preceded labor productivity spikes?

This was initially attractive, as it is a highly relevant question currently, given the AI boom. But, it was not chosen as it depends on subjectively choosing when inventions occurred, and national productivity has too many confounding factors to establish a relationship.

2. Is inequality within a country correlated with the degree of conflict or political unrest that it experiences?

This comprises a key question, given the increasing amount of active conflict and social issues surrounding inequality in the world. But, proxying conflict is difficult and subjective across sources, making the analysis much less robust.

3. Has Brexit decreased the Foreign Direct Investment (FDI) of the UK?

Similarly, this is a relevant topic in UK public policy circles. However, the timespan after Brexit is too short, and analysis would need to address overlapping shocks like COVID, which is very difficult to do.

3.2 Final data source

The project combines three source datasets and variables, which all have annual frequencies and are freely available to download online:

Although higher frequency datasets were researched, as they would allow for seasonality analysis, they did not have large numbers of observations, with most stretching back to approximately 2010. Relevant examples here are FRED and the databases of institutions like the World Bank, IMF and OECD. Hence, as the selected datasets in some cases have observations dating back to 1870, they provide more observations, leading to more robust results, that was deemed preferable for the project.

There was also a debate around what specific metrics to use for inequality, capital returns and GDP:

  • For inequality, one can also measure it via the Gini coefficient, or other income shares (top 10%, top 0.1% etc.). Thus, databases including such variables could be relevant. But, the top 1% income share was chosen, as Piketty focuses on this specific share in his analysis and the other metrics had weaker long-run annual coverage.

  • For capital returns, the focus can be placed on broader returns to capital, rather than equity returns alone (as used here). However, equity returns are much better measured, as measuring returns on specific asset classes like housing to incorporate in a broad returns-on-capital index is difficult. Thus, modern estimations for indices on overall returns to capital in the economy are based on significant assumptions. Therefore, return on equity proxies capital returns in the report.

  • For GDP, there is the question of adjusting for Purchasing Power Parities (PPP), which considers the purchasing power of currencies and makes the GDP figures of countries more comparable. But, the project is concerned with the growth rate of GDP, rather than comparing its levels over time, making a PPP adjustment less central.

Therefore, from these datasources, the final annual panel is constructed and then used for the EDA and data analysis sections. The cleaning process keeps the following variables for all 3 countries:

  • GDP per capita growth (g), which is approximated via the logarithm of GDP per capita
  • equity return (r)
  • top 1% income share
  • (r-g), which is computed from the above variables

4 A first look: Germany, UK, USA

4.1 Mapping out gaps in the data

The heatmap provides a direction on missing variables. Germany has a large number of missing variables beginning roughly in the 40s and lasting up until the 80s, which aligns with how it was split in two countries after WW2. The UK and USA have missing variables in similar years, indicating some sort of structural similarity in how data was collected, perhaps due to stronger archival continuity, given a consistent regime being in power (in contrast to Germany).

4.2 A century in three panels

  • GDP per capita growth: Germany exhibits the most extreme volatility, with deep contractions during WW2 and growth exceeding 15% due to post-WW2 reconstruction. The UK shows a notably sharp drop at the very end of the sample, likely due to COVID, which appears more severe than the equivalent US decline. Additionally, the US and Germany are highly volatile during the interwar period (1920s-30s), with more severe Depression-era contractions than the UK. Lastly, all three countries show a reduction in growth volatility after roughly 1950, an effect observed in macroeconomic literature as the “Great Moderation”.

    Return on equity: The UK series is dominated by an enormous spike around 1974–75, where returns exceed 150%, which likely reflects the recovery from the 1973–74 stock market crash. Germany’s equity returns show high volatility in the 1920s–40s, corresponding to the Weimar hyperinflation era and the disruption of WW2. The US series is comparatively stable after 1950, fluctuating within a narrower band than either the UK or Germany. But, across all three countries, returns have become less volatile in the post-war period, though occasional crisis-driven spikes (e.g. due to the 2008 financial crisis) remain visible.

    Top 1% income share: All three countries display a U-shaped pattern, but with important differences. The UK’s trough is the deepest, reflecting aggressive post-war redistribution through high marginal tax rates and the welfare state. The US bottomed out higher, and its subsequent rise has been the steepest, driven by financial sector growth and capital gains concentration. Germany’s trajectory partly differs, as the top 1% share was rising through the 1910s and 1920s, whilst the UK and US shares were roughly flat. This is consistent with the concentration of industrial wealth during the Weimar Republic.

    Hence, these country-specific patterns are consistent with known economic history and support the validity of the assembled dataset.

4.3 Serial dependence

  • The ACF for the 1% income share decreases slowly for all countries, suggesting a trend indicative of non-stationarity. The PACF shows strong persistence at the first lag, reinforcing the likelihood of a unit root. These findings motivate testing formally for stationarity later on.

  • GDP per capita growth and the return on equity show ACF and PACF behaviour that is erratic and mostly very small. Thus, the series are very likely stationary.

4.4 Narrowing down the focus

As a consequence of EDA, the remainder of the report focuses on the UK and US from 1920 onward, as that period for those countries allows the longest-running, continuous analysis. Hence, Germany is excluded due to the data gap in the middle of the 20th century, which would create a highly fragmented sample if it was not excluded.

5 The core test: UK and US from 1920

5.1 Preparing the sample

Post-1920 focused sample used in the main analysis
country start_year end_year n_obs
United Kingdom 1920 2020 100
United States 1920 2020 101

To confirm, this restriction leads to about 100 observations to analyse for each country, providing a long enough period to analyse.

5.2 First impressions from smoothing

Smoothing with a 5-year moving average can allow for an initial, visual analysis on whether there is co-movement in the relevant series. Smoothing does help remove the jagged shape of r-g for both countries, but from simple observation there is no immediate co-movement. This may suggest that there is some lagged relationship. But, before running regressions, it is important to have some diagnostics on the data.

5.3 Formal tests for stationarity and autocorrelation

Diagnostics for autocorrelation and unit-root behaviour
country metric ljung_box_pvalue adf_pvalue
United Kingdom r - g 0.3221 0.0100
United Kingdom Inequality 0.0000 0.4894
United States r - g 0.2056 0.0100
United States Inequality 0.0000 0.9297

These diagnostics tell us the following, at a 5% significance level:

  • UK and US r-g is stationary and has no strong serial autocorrelation

  • UK and US inequality is persistent over time and non-stationary, which is reasonable given its U-shape. This creates an argument to first-difference the variable to address the non-stationarity problem and use Newey-West standard errors to adjust for autocorrelation.

5.4 Searching for regime changes in inequality

Structural break evidence for the inequality series
country supF_pvalue bai_perron_best_n_breaks break_years
United Kingdom 0 0 None
United States 0 0 None

This table evaluates structural instability in inequality. The structural change test based on the F-statistics rejects the null of a single stable regime for both countries, suggesting that the series is not fully constant over time, which is plausible given the major institutional and economic changes across the sample period. However, when breakpoints are selected using the Bai-Perron procedure with BIC, no specific break dates are retained, implying that while instability is present, it is either too gradual, too noisy, or too weakly concentrated at particular dates to justify modelling discrete breaks explicitly. Thus, no adjustment for breakpoints can occur during modelling.

5.5 Outliers in r-g

IQR-based outliers in r - g
country lower_threshold upper_threshold n_outliers outlier_years
United Kingdom -35.085 52.401 2 1974, 1975
United States -36.594 55.886 3 1933, 1937, 2008
IQR-based outliers in inequality
country lower_threshold upper_threshold n_outliers outlier_years
United Kingdom 0.003 0.224 11 1920, 1921, 1922, 1923, 1924, 1925, 1926, 1927, 1928, 1929, 1930
United States 0.046 0.270 0 NA

The outlier table also indicates some useful modelling considerations. Primarily, r-g outliers in the US and UK are years where economic crises have occurred (Great Depression, 2008 crisis, 70s oil shocks), meaning that the regressions are possibly sensitive to these results. Furthermore, inequality outliers for the UK are all lined up in the interwar period. Hence, these outliers imply that full-sample statistics need to be trusted cautiously.

5.6 Consistency of the variables’ relationship

This plot attempts to control for changes in inequality regimes by taking a 10-year rolling correlation between r-g and inequality. However, it does not show a consistent relationship, with the coefficients changing sign repeatedly. Furthermore, the confidence intervals are too wide for a consistently statistically significant relationship. Thus, applying a lag to the relationship becomes even more relevant.

5.7 Finding the right lag

Best distributed-lag horizons selected by BIC
country best_horizon_levels best_bic_levels best_horizon_changes best_bic_changes
United Kingdom 7 488.1467 0 173.3233
United States 0 479.8599 1 249.2766

Before running lagged regressions, it is relevant to determine the optimal lag structure for a multivariate lag regression of the form: inequality ~ year_t + rg_lag_0 + rg_lag_1 + … + rg_lag_K.

This is done via the Bayesian Information Criterion (BIC), where a lower value is more preferable. The measure’s minimization ensures that the addition of model complexity is justified by the improvement in fit it generates. Furthermore, this analysis was conducted with both the levels and first differences of inequality (denoted “changes”) as a y variable, although the latter must be trusted more due to the non-stationarity of the series. Note that the non-stationarity of the series is also controlled by the addition of the trend variable (denoted “year”) in the regression.

Thus, the preferred lag structure is very short in 3 out of 4 cases, which is an unexpected result given the instability of the coefficient in the rolling correlation analysis. This may reflect the fact that the rolling correlation plot used a 10-year window, so any short-run lagged effects within that window are blended into a single pattern rather than appearing as cleanly separated.

5.8 The headline model

Cumulative multiplier: total effect of a 1pp rise in r − g on inequality (joint significance test)
country specification cumulative_estimate cumulative_se wald_stat wald_p_value
United Kingdom Levels (K = 7) -0.3790 0.0657 33.2730 0.0000
United Kingdom Changes (K = 0) 0.0038 0.0021 3.2201 0.0727
United States Levels (K = 0) -0.0070 0.0174 0.1622 0.6872
United States Changes (K = 1) 0.0241 0.0105 5.2110 0.0224

The analysis evaluates individual lag coefficients from regressions with Newey-West standard errors and a joint Wald test of the cumulative multiplier, which captures the total effect of a sustained increase in r − g. Overall, the results reveal a mixed picture for the Piketty hypothesis, with important differences across countries and specifications.

In the first-differenced specification, which is the most reliable given the non-stationarity of inequality, the US model produces a lag-1 coefficient of approximately 0.017 that is statistically significant, with the confidence interval comfortably above zero. The contemporaneous coefficient is also positive, but insignificant. Regardless, the cumulative effect, found by jointly testing the significance of both coefficients, is statistically significant. Thus, a possible interpretation for the result is that a widening of r-g in a given year increases the capital income of wealthy households, but this only translates into a measurably higher income share once those gains are realized, reported in tax data, and reflected in the following year’s inequality statistics.

For the UK, the changes model selects K=0, and the contemporaneous coefficient is positive but small and statistically insignificant, a result confirmed by the joint Wald test. So, there is no meaningful evidence of a short-run link between r-g and inequality changes in the UK. The weaker result may reflect institutional differences, for example the UK’s more progressive post-war tax regime, that may have decreased the transmission of capital returns to top income shares.

The levels specification, although less reliable due to the non-stationarity of inequality, might still be worth interpreting, given the inclusion of the time trend. However, it produces a counterintuitive result for the UK, as the coefficients are negative. This might be a product of spurious correlation, as inequality is U-shaped, so a simple linear time trend cannot control for it. So, the preference for the first-differenced model is reinforced.

Therefore, the strongest finding is a statistically significant one-year lagged effect in the US changes model, providing evidence for a short-run version of Piketty’s mechanism. But, neither country shows evidence of the kind of persistent, cumulative relationship that Piketty’s long-run theory would predict, which is consistent with the instability observed in the rolling correlation analysis above.

5.9 Stress-tests for outlier sensitivity

Previously identified outliers years for r-g may skew the results of the last section. So, quantile regressions at the 25th, 50th and 75th percentile for the inequality change distribution are run, utilizing the same lag structure as before. Thus, the process becomes robust to outliers as the absolute errors are minimised, rather than the squared errors. Note that first-differenced inequality is the dependent variable.

Overall, the estimates across quantiles for the coefficients are very similar to the OLS estimates, showing that outliers have not skewed the above results. Thus, the main analysis and its interpretation become more robust.

5.10 Do the results survive across sub-periods?

To account for the fact that inequality cannot be modeled as a single regime, it is important to observe if the results are also robust across different sub-periods in the sample. However, as doing so requires shrinking down the sample size, reliable inference cannot be drawn conclusively, so differences in magnitude should be treated as suggestive. Note that the same specification is used, with first-differenced inequality as the y variable.

Firstly, for the UK, the statistical insignificance of the coefficients across subsamples reinforces the full-sample result. The same is true for the first lag coefficient for the USA, as there is statistical significance across all subsamples.

For the contemporaneous (lag 0) coefficient for the USA, the picture is more nuanced. While it is insignificant in most subsamples, as in the full sample, it becomes significant in the pre-WW2 years, perhaps because it was a period with weaker redistribution mechanisms. Furthermore, it is notable that when just 7 observations are excluded, corresponding to WW2, the coefficient becomes significant. This may be because the war led to extreme movements of r-g, which are unrelated to the capital accumulation channel that Piketty described, thus producing noise that inflates standard errors when using the full sample.

6 Limitations and suggestions for future work

Several limitations should be kept in mind when interpreting the results:

  • As stated, external forces lead to regime changes, which confounds the evaluation of a longer term relationship. So, future work can look at shorter time periods and data at higher frequencies to evaluate the relationship more cleanly within a regime, for example the interwar period or from the end of WW2 until the collapse of Bretton Woods in the 1970s. Alternatively, more complicated regime switching models can be used.

  • In their current form, the regressions suffer from significant omitted variable bias, due to changing important factors like taxation or inflation. Hence, further analysis should introduce such factors as controls explicitly.

  • In the literature surrounding this relationship, it is often the case that estimates for the broader returns to capital are used, rather than focusing on just equity for the more limited scope of the assignment. Hence, the report’s results should be stress-tested with this more relevant independent variable.

  • Furthermore, in order to detect a long-run relationship reliably, more advanced methods could be used, such as cointegration models, but this analysis does not utilize them as they were outside the scope of the course. Regardless, they are worth testing in further analysis.

  • The analysis assumes that r-g drives inequality, but the opposite is also plausible. For example, rising inequality can shift political power toward deregulation and lower capital taxation, which in turn boosts equity returns and widens r-g. Thus, the estimated coefficients may partly capture feedback from inequality to r-g, rather than the desired mechanism. Hence, instrumental variable approaches could help disentangle the direction of the effect in future work.

7 Conclusion

The project set out to test whether a higher equity-based measure of r-g is associated with higher inequality, measured by the top 1% income share.

This hypothesis was partially supported in the data. Specifically, neither the descriptive plots nor the rolling correlations showed consistent links between the two variables. But, a distributed lag model showed a one-year lagged positive effect with statistical significance for the USA, a result which is insensitive to outliers and sub-periods. The contemporaneous effect for the USA is also positive, and becomes significant once observations during WW2 are removed. The cumulative effect of both lags is also jointly significant. Hence, the results do not reject Piketty’s argument of a relationship between the two, but they do not confirm it, most likely because the relationship between the two variables in this dataset is constantly affected by external forces.