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Quantitative macro models of wealth inequality: A survey

Wealth inequality is back in the spotlight, but its determinants and the saving behaviour generating it are less clear. This column discusses the mechanisms in dynamic quantitative macro models that give rise to wealth inequality. Different mechanisms give rise to similar observed wealth concentrations, but have very different policy implications. A combination of better empirical analysis and richer models is needed to guide policy.

Introduction and some basic facts about wealth inequality

Thomas Piketty’s Capital in the 21st Century renewed interest in wealth inequality, the forces generating it, and the extent to which policy interventions can affect it. But what do we know about the determinants of the observed high wealth concentration and the saving behaviour generating it?

The answer to this question is important for two reasons. First, there is much debate about why some people are wealthy and some people are poor. Second, both the redistributive and aggregate consequences of government taxes and transfer programmes crucially depend on the type and strength of people's saving motives (Holter et al. 2015).  

Dynamic quantitative models of wealth inequality can help us understand and quantify the determinants of the outcomes that we observe in the data and to evaluate the consequences of policy reform. In recent work, I survey how wealth inequality arises in the dynamic quantitative macro models that have been proposed so far (De Nardi 2015).

Key facts about the distribution of wealth have been highlighted in a large number of studies, including Wolff (1992, 1998), Cagetti and De Nardi (2008), and Moritz and Rios-Rull (2015). A striking aspect of the wealth distribution in the US is its degree of concentration. Over the past 30 years or so, for instance, households in the top 1% of the wealth distribution have held about one-third of the total wealth in the economy, and those in the top 5% have held more than half. At the other extreme, more than 10% of households have little or no assets. While there is agreement that the share held by the richest few is very high, the extent to which the shares of the richest have changed over time (and why) is still the subject of some debate (Piketty 2014, Kopczuk 2014, Saez and Zucman 2014, and Bricker et al. 2015).

Basic Bewley models, saving behaviour, and wealth inequality

The workhorse framework used to study wealth inequality is the Bewley (1977) model, which features an incomplete markets environment in which people save to ‘self-insure’ against idiosyncratic earnings shocks to smooth their consumption. Workers who experience a high earnings shock relatively to their permanent income and have relatively low assets save. Workers who experience a relatively high earnings shock but have relatively high assets dissave. In this framework, precautionary savings are the key force driving wealth concentration, but once a buffer stock of wealth is reached, the agents don't save any more (see Carroll 1997). Hence, the saving rate of the wealthy in these models is often negative. In contrast, in the US data, for instance, rich people keep saving at high rates, which explains the emergence and persistence of their very large estates. The basic version of the Bewley model thus fails to generate the high concentration of wealth in the hands of the richest few because it misses the fact that rich people keep saving.

Previous work has uncovered forces that, when introduced into a Bewley model, keep the saving rates of the wealthiest high and thus generate higher wealth concentration in the hands of a small fraction of households. They include heterogeneity in patience, transmission of bequests and human capital across generations, entrepreneurship or rate of return risk, and high earnings risk for top earners.

Stochastic patience heterogeneity

Krusell and Smith (1998) have shown that patience heterogeneity that changes, on average, every generation in a dynasty can generate increased wealth dispersion. Hendricks (2007), however, has shown that in the context of an overlapping-generations model that matches how wealth inequality changes with age in US data, patience heterogeneity can only go so far in explaining the savings of the very wealthy if households face realistic amounts of earnings risk.

Bequests and transmission of human capital across generations

In previous work (De Nardi 2004), I study an overlapping-generation model in which parents and their children are linked by voluntary and accidental bequests and by transmission of earnings ability. In that model, voluntary bequests are generated by ‘warm-glow’ utility of leaving bequests rather than by perfectly altruistic households.  For reasonably calibrated bequest motives, bequests are luxury goods and the bequest motive to save is thus stronger for the richest households who thus leave more wealth to their offspring, who, in turn, tend to do the same. This mechanism generates larger estates that are transmitted across generations. If children also partially inherit the productivity of their parents, more productive parents leave larger bequests to their children, who, in turn, are also more productive than average in the workplace. The presence of a bequest motive also generates lifetime saving profiles that are more consistent with the data in old age (see De Nardi et al. 2010). This mechanism, however, cannot fully explain the savings of the very wealthy.

Entrepreneurship and rate of return risk

In Cagetti and De Nardi (2006), my co-author and I propose a model in which some households are potentially very productive entrepreneurs, but have to accumulate collateral to borrow and grow their firm. In this calibration, at least for a small fraction of people in the population, optimal firm size is large and the entrepreneur is borrowing constrained. Thus, entrepreneurs, even when rich, want to keep saving to grow their firm to be able to borrow more and reap higher returns from capital. This is the mechanism that, in this framework, keeps the rich people's saving rate high and generates a high wealth concentration. Another mechanism that is often associated with entrepreneurship is risky rates of returns of investment. However, in these models, this mechanism works in a similar way to stochastic preference heterogeneity, as they increase the return savings. Benhabib et al. (2011) characterise this mechanism analytically.

Large labour earnings risk for the top earners

Castaneda et al. (2003) adopt a labour earnings process that implies large earnings risk for the top earners, who thus massively save to self-insure and smooth consumption against earnings risk. Their finding underscores the role of the earnings risk faced by households in shaping saving behaviour. It should be noted, however, that in this framework, and in contrast with the rate of return risk story, labour earnings risk is independent from the size of one's wealth and business capital.

Conclusions: Lessons learned and directions for future research

Basic versions of the Bewley model in which households face earnings shocks that are typically parametrically estimated from micro-level data sets miss the saving behaviour of the rich.

Different mechanisms give rise to similar observed wealth concentrations, but these mechanisms can have vastly different policy implications. For instance, models emphasising entrepreneurship usually imply that the adverse responses of savings and economic activity to increased taxation are significant, and especially so if taxation affects the returns to running a business (Kitao 2008, Cagetti and De Nardi 2009, and Lee 2015). In contrast, in a model with high labour earnings risk for the top earners, Kindermann and Krueger (2015) conclude that the optimal marginal income tax rate is close to 90%. The big difference in responses to taxation between these models is due to the fact that entrepreneurs' savings and investments are responsive to their implicit rate of return, net of taxes. In contrast, the very high earner facing a large probability of becoming a very low earner next period is desperate to save to smooth consumption. Hence, even if taxation substantially reduces their current earrings, as long as expected earnings tomorrow are sufficiently low, households will save at a high rate even when taxes are very high.

Similarly, the specific bequest formulation assumed (altruistic or warm glow, for instance) might be quite important in determining the response to taxation. Interestingly, on this point in De Nardi and Yang (2015), we find that three different types of bequest motives give rise to quantitatively similar responses to estate taxation reform as long as each model is calibrated to match the same facts. However, it is likely that this conclusion is not robust to other policy experiments that raise more revenue than the estate tax.

The stark contrast in policy implications stemming from different motivations to save points to the importance of understanding whether, for instance, the risk that the rich face comes from their earnings or from the capital that they have invested in the firm. A combination of better data and empirical analysis, and richer models that include more than just one mechanism generating more wealth concentration will help answering these questions and thus provide better policy recommendations.


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