As is well-known, Thomas Piketty made some dire predictions for inequality in his book. Roughly speaking, the idea is that low productivity growth will lead to ever-expanding inequality. While Piketty’s empirical contributions are well received (though there are critics), his main conclusions and policy recommendations are based on his theoretical framework, which has been widely criticized. It is thus not an exaggeration to say that the consensus among leading economists is that Piketty’s theory (and consequently his conclusions) are flawed.

So what happens if we study the question of inequality in a standard macroeconomic model? Does inequality increase as growth falls? And if yes, are such increases huge or are they negligible?

These questions are addressed in a working paper by Carroll and Young (2014) (IDEAS page). They build a standard general equilibrium model where households work, consume and save. The model has two types of households: workers and capitalists, the main difference between them being that capitalists can save by investing in capital (and earning returns on it). Workers on the other hand can only save by transferring money from one period to the next, i.e. they don’t earn returns (they actually lose out due to inflation). Other than this, households are identical.

First off, the authors look at a model in which households’ productivity is not uncertain. In other words, there are no idiosyncratic shocks to productivity, which means no one gets fired, promoted, demoted, etc.

Inequality in this model generally increases as growth falls, in line with Piketty’s predictions. There are basically two components to inequality: (1) the capitalist working hours to laborer working hours ratio, and (2) income from capital, and the amount of capital wealth. (1) increases (and thus so does inequality) if capitalists work relatively more, (2) increases if returns to capital increase relative to wages.

There are lots of interactions here. So what exactly happens as growth falls to 0?

- Wages increase with low growth. This is because capital becomes relatively abundant with low growth (as predicted by Piketty). But a relatively abundant capital makes labor a scarce resource and thus wages rise. This is good for laborers, and thus decreases inequality.
- The wealth of capitalists increases as well. This is because more capital is accumulated. This roughly offsets the effects of higher wages.
- Higher wages are not only good for laborers. They also induce capitalists to work more. So actually the first component of inequality, the working hours ratio, increases. This is bad for inequality.

The general conclusion in the model without idiosyncratic risk to productivity is thus that inequality does increase as growth falls. Interestingly, this is due to the fact that capitalists choose to work more to take advantage of higher wages.

Let’s now move on to a more realistic model that includes productivity risk. I.e. people can get promoted, demoted, fired, etc. In this model inequality arises through two channels again: (1) luck, that is when capitalists’ productivity is high they save more and thus accumulate capital; (2) as the amount of capital increases, returns to capital drop and inequality shrinks.

The model, appropriately calibrated, can roughly reproduce patterns of inequality in the US. It actually somewhat overestimates inequality. The Gini coefficient for wealth (not income) distribution (higher = more inequality) is .84 in the model, and .80 in reality. The results from this model is that as growth drops to 0, inequality actually decreases slightly. But the changes are quantitatively small.

Furthermore, when growth drops it takes a long time for the variables in the model to respond to this change. Returns to capital drop, the capital-to-output ratio increases and so does capital’s share in income. These variables take about 100 years to reach their new steady state level. The Gini coefficient drops, but it takes a long time to settle down as seen below (horizontal axis is number of years after growth dropped to 0).

Recall that a lower Gini, which is what we see here overall means lower inequality.

Why do the implications of this model differ from those of Piketty’s? One main reason is returns to capital. Piketty says that capital accumulation will lead to a higher capital stock in the economy, he maintains that despite this returns to capital will somehow not drop. In the model of Carroll and Young, returns to capital do drop as capital becomes abundant, as consistent with economic intuition. This force counteracts the increased capital stock and on net inequality decreases.

How could returns to capital possibly increase or at least stay constant despite more capital in the economy? This is a hard question to answer. Carroll and Young try adding some additional features to their model that may make this happen.

One idea is financial innovation. Suppose capitalists’ wealth is initially exposed to various random shocks, but then these risks decline because of financial innovation (i.e. capitalists can better hedge/insure their positions using various financial instruments). This will reduce incentives to invest in capital for precautionary motives (i.e. to protect oneself from risk). Therefore, the supply of capital (which is determined by how much capitalists invest in it) will decline overall. This will push returns to capital up.

Still, even in such a model the Gini coefficient will be dropping substantially as we transition to zero growth. This is shown below.

What if we also consider capital-biased technical change, i.e. what if robots come and take our jobs? This will generate higher demand for capital. So now, if we have financial innovation and capital-biased technical change, then we have lower supply and higher demand on the capital market. This will surely push returns up. Still, on net (i.e. after accounting for the increased capital stock) returns will decline and inequality will too.

It’s worth mentioning that assuming that workers can also save properly (as opposed to just keeping money) – through saving instruments that are inferior to those of capitalists – does not alter the results.

In conclusion, Piketty’s predictions for ever-increasing inequality are hard to reproduce in a standard model. The main problem is that returns to capital will inevitably decline if capital stock increases. This will hurt capitalists and the net effect on inequality is negative.

There are two takeaways: (1) according to this model, inequality will thus decline as productivity growth falls to zero, and (2) the quantitative effects of productivity growth on inequality are small. Thus even if some other model can make inequality increase – under sensible modeling and parameter choices -, it would still have to deal with the issue that – quantitatively speaking – inequality is just simply not all that responsive to productivity growth.