What is the half-life of economic injustice?
VoxEU Blog/Review Frontiers of economic research Poverty and Income Inequality

What is the half-life of economic injustice?

In this post, David Miles uses versions of a neoclassical growth model to trace through the evolution of economic injustice over time.

Ask yourself this: how did I come to be as well off as I am? Was it partly through the inheritance of wealth that depends, to some extent, on the unjust acquisition of assets in the far past – maybe by an ancestor? Perhaps more likely – or at least more easily assessed – is that you shared in an economic environment that was richer and more successful than it might have been because some resources in the distant past had been acquired unfairly. Perhaps you are Belgian – or perhaps you are from the Democratic Republic of Congo. In either case you might wonder how your standard of living today may even now reflect the lasting impact of assets being looted from the time of King Leopold II in the late 19th century.

How you answer these questions is fundamental to how you view the legitimacy of the distribution of incomes and wealth today. In order to answer them one needs two things: first, a principle to determine what is, and what is not, just acquisition of wealth or a just source of income; second, a means of using that principle to estimate what fraction of wealth and income today is unjust because of past unfairness. In a recent paper I used a principle put forward by Robert Nozick to provide the first of these things and then use calculations based on standard neoclassical models of economic growth to illustrate its implications for the scale of unfairness today. 

The basic idea is straightforward. It rests on a notion that I think has great intuitive appeal, which is this: If I came by something fairly and willingly pass it on to someone else then the recipient holds it fairly; but if I come by some asset unfairly then the income from that asset, and the evolving value of the asset, remains a form of unjust resources. This is Nozick’s principle of justice in acquisition. It means that if some assets have been acquired unjustly the income derived from them is unjust, as is the ownership of assets acquired out of saving from that income. If we can trace through how income from unjust assets is used – and how the value of such assets evolve – we can keep track of what proportion of total assets and total income (GDP) is unjust. Tracing through how incomes and wealth (capital) evolve over time from some start point is exactly what models of economic growth do – the most famous of which remains the celebrated Solow model. 

I use versions of a neoclassical growth model (elaborations on Solow which allow for human capital and different savings rates out of different sorts of income) to trace through the evolution of economic injustice over time. 

To be more specific, the question I address is this: suppose at some point in the (possibly distant) past some proportion of the aggregate stock of productive assets in a country had been acquired unjustly and that subsequently those assets are not returned to their rightful owners. How does the stock of unjustly acquired assets, and the fraction of aggregate income that is unjust, then evolve over time as a result of this initial position? The answer depends on the rate at which such assets depreciate over time and also on what use is made of the income from such assets. Crucial to the question of whether initial injustice fades quickly or persists a long time is what proportion of the income from such assets is saved and used to accumulate new assets which, because they are funded from unjust income, can be considered unjust. Also crucial is whether saving from unjust income is used to accumulate human as well as physical capital.

In assessing the issue of how injustice fades – or does not – I use neoclassical growth models where there is a unique steady state to which the economy converges. But it turns out that this does not mean that half-lives of injustice – the period until the proportion of income or wealth that is unjust falls to half its value at some initial start point – need be low, or even finite.

The tables below show results from various thought experiments. Each table shows how injustice evolves from a start point in the past where a high proportion of productive assets (machines, commercial buildings, houses, agricultural land and so on) had been acquired unjustly. The proportion of total income (GDP) that is a result of using such unjustly acquired assets is shown as well as the proportion of all assets (wealth) that is unjust. The assumption here is that injustice in later periods simply reflects the impact of past injustice due to long-lived assets and to new capital accumulation from saving out of unjust income. I assume total output is produced by a CES production function using capital (either physical capital alone or including human capital as well) and labour. The labour share parameter is set at 0.7. I illustrate by assuming 70% of total initial capital has been unjustly acquired. But what is important in the tables is not so much whether initial unjustly acquired capital is 70% or some much lower figure, but how quickly injustice in wealth and income changes. The half-life is essentially invariant to the assumed initial level of injustice in asset ownership. 

Table 1 shows a simple case – assets are expropriated from some overseas country and brought back to the home country to be used with domestic labour. That boosts the incomes of owners of those assets but also boosts domestic labour incomes and all such income is counted as ‘unjust’. But it fades over time depending on savings rates and depreciation. Making assumptions on overall saving rates (calibrated to generate plausible rates of return on assets based on historical data) and making assumptions on deprecation rates generates half-lives which tend to be quite low – rarely much more than 30 years. 

Table 1 

The conclusion to be drawn from Table 1 is that when capital is expropriated from foreigners and taken back to the expropriator's country, the half-life of the resulting injustice (viewed as the extra income and wealth enjoyed by the expropriator) may be relatively short – a few decades rather than a few centuries. 

But things are different when assets are expropriated from one group by another within the same country. In this case instead of counting as unjust all incomes that depend on such capital (as is done in the calculations above and as may be appropriate when that capital was legitimately the property of people outside the economy who are deprived of its use), it is more appropriate to now count as unjust only the income derived directly from ownership of that capital. That should include capital income subsequently earned on any additions to capital from saving the capital income on initial unjust assets. If the saving rate out of labour income is relatively low and the saving rate out of capital income is much higher (as is plausible for much of history) injustice can persist much longer than suggested by Table 1. Table 2 illustrates.

In fact, there is a surprising result from this standard neoclassical growth model and the Nozick principles of how unjust acquisitions evolve – if the saving rate out of labour income approaches zero injustice never fades at all. But provided there is a non-trivial amount of saving out of labour income, unjust capital will dwindle a great deal in a few generations. 

It is plausible that today such saving in developed economies is non-trivial. Pension arrangements mean that many workers in recent decades automatically make contributions out of labour income; paying off a mortgage out of labour income is another significant (and relatively recent) form of saving. But savings out of labour income in the more distant past may have been very small and Table 2 suggests that then injustice would have had a much longer half-life.

Table 2 

In the past, saving out of labour income may have been low, but the contribution of human capital (acquired skills due to education and training) was probably less important than it has subsequently become. And the rise of the importance of human capital has significance for the propagation of economic injustice. Some part of human capital is likely to be financed out of income that itself comes from ownership of assets that have been acquired (maybe in the distant past) unfairly. The children of the aristocracy and of the robber barons of the past (and of today's oligarchs) typically have had an unusually good education. 

If the share of total labour remuneration that reflects human capital is very high the half-life of injustice can become very long. Using similar assumptions on asset lives as in Table 2 – but adding in human capital and assuming common saving rates out of all income – the half-life of injustice can rise to 100 years and beyond once human capital accounts for 75% or more of total labour income. In fact, one can show a rather remarkable result, which is that as the share of labour remuneration that reflects human capital (rather than innate labour power) rises towards 100%, the effect of injustice in past acquisition of assets may never fade at all.

Is the rate at which injustice stemming from unfair acquisition of assets fades higher or lower now than in the past? Human capital has become more important in modern economies and saving out of labour income is also higher than in the distant past. These two factors pull in different directions, meaning it is not obvious whether the half-life of injustice is higher or lower than in the past. But it is likely that for much of the period up to the early 20th century saving out of labour income for the great majority of the population was very low – probably negligible. If that saving rate out of labour income was negligible, the half-life of injustice would have been very long. Such saving is now not negligible. The fraction of labour remuneration due to human capital would have to rise to nearly one to offset the effect of that so as to keep the half-life of injustice from falling. While human capital now very likely does account for a much higher share of labour income than a hundred and more years ago, its share has not approached 100%, so on balance it seems likely that the half-life of injustice is now lower in developed economies than in the world of 100 or more years ago.

I have set out a framework for thinking through the evolution of one type of injustice and presented illustrative results based on a calibrated version of a standard growth model. In many ways, I have used a narrow conception of just and unjust outcomes. The value of such an exercise does not require that we believe that the only source of injustice is from the past distribution of assets. But knowing how significant is the impact of past unjust outcomes is fundamental to judging the legitimacy of today's outcomes.