The publication of *Capital in the 21st Century* (Piketty 2014) has put the issue of growth and redistribution back at the centre of the economic debate, both in academic and in policy-oriented discussion.

Almost every commentator has praised Piketty and his coauthors for the painstaking work of data collection on the long-run evolution of income and wealth. Their interpretation of the data, and even more so their predictions of further trends have, understandably, given rise to interesting debates.

The discussion has been framed, explicitly or implicitly, in the language of the standard neoclassical growth model, either with fixed saving behaviour, like in the original Solow (1956) model, or with optimising infinite-lived agents.

In this column I would like to argue that Piketty’s data could also be interpreted in a richer framework, based on Romer’s (1990) model of endogenous technical change.

## A tale of two balanced growth regimes

This is not just for the sake of generality. Discussions on the long-run trends of the growth rate and the rate of interest are more naturally framed in a setting in which both variables are jointly determined. Moreover, and more importantly, one possible interpretation of Piketty’s data is in terms of two regimes of balanced growth, one characterised by higher levels of accumulation of physical capital, the other by stronger accumulation of human capital and ideas. The Romer model provides a language for this interpretation.

Let’s call ‘merit’ a configuration with (*K*/*Y* = 3, *r*=5%, *g*=3%), and ‘rent’ a configuration with (*K*/*Y*=6, *r*=4%, *g*=1.5%), where *Y* is output, *K* is capital, *r* the interest rate, and *g* the growth rate.

Piketty’s data can be (overly) simplified by saying that advanced economies moved from a ‘rent’ configuration in the period 1870–1945 to a ‘merit’ configuration in the period 1945–1980.

Looking at the data for the period 1970–2010 (available for a larger set of countries), the trend since the 1980s seems to be towards a new ‘rent’ period, characterised by a high value of the capital–output ratio *K*/*Y*.^{1}

To interpret this trend, and to make predictions about future tendencies, one needs a theoretical framework.

Piketty (2014) and Piketty and Zucman (2014a, 2014b), base their interpretation on two (by now famous) equations: 1) *α* = *r* *K*/*Y*, and 2) *K*/*Y* = *s*/*g*.

The first is the definition of the capital share of income. The second describes the steady state of a model of capital accumulation (s being the propensity to save).

Using the second equation, Piketty interprets the observed increase in *K*/*Y* since the 1980s as the outcome of lower values of *g*, with little variation in s.

From the first equation, the observed increase in α is compatible with the rise in *K*/*Y* if the rate of return of capital *r* doesn’t decline too much.

## Explaining persistently high rates of return on capital

To justify the label ‘rent’, one should also provide a mechanism explaining why “the past devours the future” (Piketty 2014) in a long-run equilibrium characterised by a high capital–output ratio.

Piketty’s own explanation is based on the fact that, empirically, higher levels of K/Y seem to be associated with higher values for (*r* - *g*), the difference between the interest rate and the growth rate, itself a basic ingredient in dynamic models of wealth inequality (Piketty and Zucman 2014b). Extrapolating into the future, he then argues that, if the trend of decline in g is going to continue, inequality is bound to increase even further, making advanced economies more and more similar to the ‘rentier’ society of 19th-century Europe .

Paul Krugman (2014) illustrates this reasoning in the classical Solow diagram – an exogenous reduction in the growth rate *g*, due to either demographic or technological forces, moves the economy to a new long-run equilibrium characterised by a higher value of *K*/*Y* and a lower value of *r*. What happens to (*r* - *g*) depends on the elasticity of substitution between capital and labour.

As discussed by Krugman, and then in great detail by Krusell and Smith (2014) and Rognlie (2014), predicting ever-increasing values of *K*/*Y* and (*r* - *g*) as *g* declines requires, in the Solow model (and even more in models with optimising savers), implausible parametric restrictions. For empirically relevant ranges of the parameters, no dramatic changes in either the capital–output ratio *K*/*Y* or the share of capital *α* are to be expected. Rising inequality may be a problem, but the mechanism of capital accumulation should not be the focus of policies aiming at reducing it.

## Capital accumulation vs. innovation

An alternative interpretation of the long-run trend from ‘merit’ to ‘rent’ since the 1980s is possible using Romer’s (1990) model, in which the growth rate g and the interest rate *r* are *jointly* determined.

The model can be summarised by two lines in the (*r*, *g*) space. The first line, negatively inclined, describes the production side – a higher interest rate lowers the future profit from innovation and therefore reduces innovation and growth.

The second line summarises the consumption side – growth in consumption depends positively on the difference between the interest rate and the impatience of consumers.

The shift from ‘merit’ to ‘rent’, i.e. a joint reduction in both r and g, requires a downward shift of the line representing production.

**Figure 1**. A graphical summary of Romer’s (1990) endogenous growth model

## Understanding the shift from ‘merit’ to ‘rent’

What could explain this downward shift? The distinction between capital accumulation and innovation, central to Romer’s approach, suggests an answer.

In a set of notes (available at https://sites.google.com/site/eminellisite/research), I explore a variation of the model in which the consumption side is represented by overlapping generations of individuals whose endowment of human capital is subject to idiosyncratic uncertainty influenced by the individual’s effort choice.

This is a way to try to capture a trade-off between pursuing a safer activity of capital accumulation (putting money into a mutual fund, buying a house, investing in mature technologies etc.) and a more risky activity of getting an education and/or working in an innovative firm.

The shift from ‘merit’ to ‘rent’ could then be interpreted as the outcome of institutional changes favouring capital accumulation – subsidies to physical capital accumulation, or a more favourable taxation of capital revenue increase the share of income that does not depend on the choice of effort, thus reducing the incentives to human capital accumulation and innovation, and bringing about a downward shift of the ‘production’ line in the Romer model.

This interpretation, based on an intuition coming from standard dynamic models of saving and effort (Rogerson 1985), doesn’t require particular assumptions about factor substitution, and may be a better representation of Piketty’s own account of a transition towards a ‘rentier’ equilibrium.

## References

Krugman, P (2014), “Notes on Piketty (Wonkish)”, The Conscience of a Liberal, 14 March.

Krusell, P and T Smith (2014), “Is Piketty’s Second Law of Capitalism Fundamental?”, mimeo.

Piketty, T (2014), *Capital in the Twenty-First Century*, Harvard University Press.

Piketty, T and G Zucman (2013), “Rising wealth-to-income ratios, inequality, and growth”, VoxEU.org, 26 September.

Piketty, T and G Zucman (2014a), “Capital is back: wealth to income ratios in rich countries, 1700–2010”, *Quarterly Journal of Economics* 150: 1255–1310.

Piketty, T and G Zucman (2014b), “Wealth and Inheritance in the Long Run”, forthcoming in *Handbook of Income Distribution*, North Holland.

Rogerson, W P (1985), “Repeated Moral Hazard”, *Econometrica* 53: 69–76.

Rognlie, M (2014), “A note on Piketty and diminishing returns to capital”, mimeo.

Romer, P (1990), “Endogenous Technological Change”, *Journal of Political Economy* 98: S71–S102.

Solow, R (1954), “A Contribution to the Theory of Economic Growth”, *Quarterly Journal of Economics* 70: 65–94.

## Footnote

[1] See e.g. Figure 1 in Piketty and Zucman (2014a), also in Piketty and Zucman (2013).