Why the Rate of Return Exceeds the Growth Rate

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I have been scratching my head over the elementary inequality r>g  for some time. As you probably know, this inequality is central to the “actuarial” mechanism that drives stratification in Piketty’s work. Basically, if the rate of return (net of depreciation) outpaces the growth of the overall economy, the capital-to-output ratio, \beta=K/Y, or equivalently, the wealth-to-income ratio, increases. Since the upper tail of the wealth distribution obeys the Pareto law, this implies relentlessly increasing stratification under fairly normal conditions. In other words, in light of the demographic-actuarial logic, the only thing standing between patrimonial capitalism and us, is war and taxes.

But how can r>g in perpetuity? This looks like an impossibility: If capital income grows at a faster rate than the economy, then the share of capital income in GDP, a, should rise until it is 100%, and then r could not exceed g anymore. I will show how this is a fallacy in the baseline model of economic growth. It is also true of generic models when we replace key exogenous variables in the Harrod-Domar-Solow-Cobb-Douglas model by their micro-theoretic, endogenously arrived at, equilibrium values. For instance, inter-temporal optimization in dynastic models yield the same results as the basic model. But I will not attempt to show that r>g holds under very general conditions. Instead, I will show how this sheds light on global imbalances and the international economy.

In the baseline model, the output Y_{t} given capital K_{t} and effective labor L_{t} is


Assuming zero population growth, effective labor grows at the same rate as productivity, say g. Suppose that the savings rate is s, and the rate of return in global markets is r. Then, the Harrod-Domar-Solow-Cobb-Douglas model implies that in the steady-state,

r\times s = a\times g,

where a is the capital share in national income, which is mathematically determined by the elasticity of substitution of capital and labor. (It is the exponent of capital if one assumes the Cobb-Douglas production function as we have.) Another immediate implication is that the capital-to-income ratio converges to the ratio of the savings rate and the growth rate of the economy. That is,

\beta:=K/Y \longrightarrow s/g,

which is why low growth regimes imply increasing stratification. Where is r hiding? Well, r must satisfy r\times s =a\times g. Under conditions of global capital mobility, it is r is that given exogenously. At the global level,

r= a\times g/s,

in the steady-state. For commonly observed values of the parameters, the implied rate of return, r is reasonable. For instance, for current global ballpark estimates, a=32\%, g=3\%, and s=24\%, which implies that r=4\%; a good ballpark for long-term interest rates. The only way for r to fall below g is if the share of capital in national income is less than the savings rate, i.e., a<s. But this is almost never the case. The share of capital in national income hovers around a third, whereas the global savings rate stays close to a quarter. Also, if r<g, agents’ inter-temporal optimization requires them to borrow as much money as possible; an implausible result. Under fairly normal conditions then, the rate of return exceeds the rate of growth of the economy.

So where did we go wrong in our apparent contradiction? We went wrong in assuming that r>g implied an unbounded share of capital in income. This does not happen because the marginal return to capital falls as capital intensity rises. In the standard model, capital-to-output ratio stabilizes at s/g, and capital’s share in income, a, determines the rate of return of capital. (Capital’s share in income, a is, in turn, determined by the elasticity of substitution between capital and labor). This implies that that the rate of return on capital may easily exceed the rate of growth of productivity, and hence the overall economy, even if capital’s share in national income is held constant!


The thing to understand is this: The savings rates and growth rates of all countries are codetermined. We can take a to be technologically determined outside the model. Also, under perfect capital mobility, the rate of return is the market-clearing price of capital that equates global savings and investment. Suppose that the capital share is constant across countries. Then the ratio of savings rate to growth rate must be equal across countries as well. Of course, the ratios may be different in the short run, but they will tend to their steady-state values over time. That is, savings-to-growth ratios are equal across national economies in equilibrium.

In the note on global imbalances, my claim was that the high-savings strategy of China, Japan, and Germany works through the capital account to suppress the US savings rate. One can use the straightforward model to understand that dynamic mathematically. In what follows, we will basically mine the steady-state equality g/s=r/a.

Consider a bipolar open international economy where both the poles are price-takers in the global capital market. Suppose that the capital share in every national economy is 30%. Let’s compare two scenarios.

For the reference scenario, suppose that the market clearing rate of return is 6%. Then the ratio of growth rate to the savings rate of every national economy must be 20%. (Which implies that all national economies must converge to the capital-income ratio of 500%). Let’s say in equilibrium we have one pole, say the United States, saving at the rate of 20% (and hence growing at 4%); and the other, say China, saving 30% (and thus growing at 6%).

For the second scenario suppose that China follows a high-savings strategy. By a combination of wage suppression and financial repression, it raises its savings rate to 50%. Let’s say this depresses the market-clearing global real interest rate to 4%. The ratio of growth rate to the savings rate of every national economy must now be 12.5%. So China grows at 6.25%. The United States has to lower its savings rate to say 16%. The US thus grows at 2%.

In light of the lowering of trend growth from 4% to 2% at the center, we may use the current account frame-of-reference and see the second, prevailing, scenario as one of secular stagnation. Equivalently, we may use the capital account frame-of-reference and see the second scenario as one of a global savings glut. This is basically what is going on in the world economy (see chart of American and Chinese savings rate below). Ben Bernanke has inaugurated his blog by arguing that global interest rates are low because of a global savings glut; pointing out that central banks cannot control long-term rates (see picture above of the yield curve courtesy of the Grey Lady). Summers responded by defending his secular stagnation thesis. 

Guys, it’s the same thing. And it is the result of the high-savings strategies of US’ trade partners, especially China. Boosting demand in the United States, whether through fiscal or monetary means is not the answer. This is a US foreign economic policy issue. The right question to ask is this: How can the United States persuade China, Germany, and Japan to lower their savings rate? For unless these states pursue alternate strategies to secure their economic interests there is no hope for resolving global imbalances.

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Piketty’s almost universally hailed Capital in the Twenty-First Century is a work of great importance; one that will be widely known decades from now. The reason why this book is so path-breaking — and this tells us a lot about the discipline — is that Piketty brings two hundred years of economic history to bear on the question of inequality. Economists have for too long ignored both economic history and the distribution of wealth and income. Almost every economist worth his salt has written a raving review. The Economist: “The book aims to revolutionize the way people think about the economic history of the past two centuries. It may well manage the feat.” What’s everyone raving about?

The gist of the Piketty’s argument is that there is a natural tendency in capitalism towards ever-greater stratification. Left to itself, the system inexorably concentrates wealth and capital; and that this means that sooner or later inherited wealth comes to dominate lifetime savings of the highest earners. The causal mechanism that drives this process is the difference between the return on capital and the growth rate of the economy. Piketty finds that the return on capital has been remarkably stable over the very long run, at around 4-5%. Similarly, GDP growth has displayed remarkable stability over the very long run, staying close to 1-2% for countries at the technological frontier and a stable population.

This gap of 3% between the growth of wealth and national income means that the ratio of accumulated wealth to average incomes keeps increasing. On the other hand, growth, whether demographic or in per capita incomes, works against increasing concentration. However, the demographic explosion of the nineteenth and twentieth centuries is unlikely to be repeated and per capita incomes only grow quickly in countries playing technological catch-up — no country at the technological frontier not undergoing sustained demographic expansion has ever achieved growth rates higher than 2%. Given that we are almost back at Victorian era levels of capital/income of 600-700%, the wealthy countries are facing a highly inegalitarian future ripe for political instability.

Let me digress here to provide a quick rule of thumb to familiarize the reader with the magnitudes involved. As of writing, there is a rough 10-1/2 rule in the income distribution: 10% of the highest earners take home roughly 1/2 of all income, the top 1% take 1/4, the top 0.1% take 1/8, and the top 0.01% take 1/16. The distribution of (non-home) wealth is twice as concentrated: the top 10% of wealth holders own 3/4 of all assets, the top 1% own 1/2, the 0.1% own 1/4, and the 0.01% own 1/8. Furthermore, due to the fact that larger asset portfolios earn higher returns on average — roughly 12% for portfolios in billions, 10% for hundreds of millions, 8% for tens of millions, and roughly 6% for smaller ones — capital gains (net increase in wealth) are still more concentrated: with the top 0.1% of wealth holders (with assets in excess of $20m) taking nearly 1/2 of all gains.

Top wealth shares

One last point before we return to Piketty. Earned income dominates the upper levels of the income distribution all the up to the 0.1%, at and above this level, income from capital becomes larger than earned income. Since the dynamic is driven by the differential returns to capital and labor, this is the line of demarcation that one needs to pay attention to. Loosely speaking, a rentier society is one where ‘the weight of the past’ is so strong that even the highest income earners cannot match the unearned incomes of the upper classes, whose inherited wealth allows them a lifestyle that is simply unachievable to those who did not win the birth lottery. That is, Jane Austen’s world.

Piketty’s historical narrative is straightforward. Eighteenth and nineteenth century Europe was a rentier society that grew steadily more stratified. Demographically exploding America, with its virtually free land, was very slow to catch up (modulo the plantation south). It was only at the turn of the century that the United States became half as stratified as Europe; although, by then, it too had become a rentier society. On the eve of the Great War, Europe attained a yet to be outdone peak of both capital/income ratio (700%) as well the highest concentration of wealth. In 1910, the top 1% of British wealth holders owned 70% of national wealth, while the top 0.1% held 40%. The continent was only slightly more egalitarian, with the top 1% controlling just 60%, whereas America’s wealthy lagged behind at 40%.

The period 1914-1945 saw a virtual ‘euthanasia of the rentiers’ as the world wars, depression, hyperinflation, and exaction by the state decimated accumulated fortunes. This process was much more pronounced on the continent which bore the brunt of the physical destruction, debt defaults, and hyperinflation. The Anglo-Saxon countries survived relatively unscathed, but their response to the Great Depression has similar, if attenuated, effects on accumulated wealth. In Piketty’s telling, the rapid growth rates of the post-war period along with anti-capital policies meant both that the return on capital was lower than the rate of growth and asset prices remained depressed. After 1980, return of low growth and a sustained asset price recovery due to capital friendly policies led to a rebound in wealth inequality.

Due to the inherent slowness of the process, it took nearly three decades for the wealthy countries to return to the distribution last seen in the nineteenth century. If there is no major shock or dramatic shift in policies, by 2030, Europe will surpass the levels of concentration attained around 1910. This process will be slower in the United States due to expected population growth of 1%. US wealth stratification is expected to reach pre-World War I European levels only by 2050. Parenthetically, Piketty explains the rise in earned income inequality as being a response to the lowering of tax rates. Skyrocketing executive compensation is responsible for two-thirds of the increase. He reckons that lower taxes on high incomes mean that executives face strong incentives to bargain for fat paychecks.

Piketty’s proposal to thwart this relentless march towards rentierdom is a progressive global tax on capital. That is, an annual tax of a few percent on great wealth facilitated by international data sharing which is already technologically feasible. “One might imagine a rate of 0 percent for net assets below 1 million euros, 1 percent between 1 and 5 million, and 2 percent above 5 million. Or one might prefer a much more steeply progressive tax on the largest fortunes (for instance, a rate of 5 or 10 percent on assets above 1 billion euros).” The benefits include not just a levelling of the playing field, but also a “more just and transparent international tax system.” Although it would be optimal, Piketty thinks it is “utopian,” owing to the political power of plutocrats.


The motor behind the natural law towards ever greater concentration of wealth is the difference between the rate of return on capital and the rate of growth of income. How can the rate of return on capital exceed GDP growth in perpetuity? Piketty has demonstrated that this is what obtained. But why?

Controlling for demographic factors, the growth of national income in the very long run is determined almost solely by growth in productivity. Why has this been more or less constant at 1% over the past two centuries? Why the same rate in the early nineteenth century and the late twentieth? GDP estimates only began during the Great Depression. Perhaps earlier figures are more extrapolations than estimates? It’s hard to believe that American growth rates were constant from the Early Republic through the railroad revolution and the rapid industrialization of the late nineteenth century that forged the greatest industrial power in history. Same goes for Britain. Shouldn’t we expect growth rates in Britain to be substantially higher between 1820-1850??

The near constant 4-5% rate of return on capital is similarly inexplicable. The stability of the rate of return on capital in the nineteenth century is stunning. Piketty talks about how Balzac and Austen move freely between describing a gentleman as being worth £250,000, or equivalently as one with £10,000 in annual income. Why should this be the case? During the long nineteenth century (1815-1914), there was zero inflation in the center countries. Moreover, the bulk of the financial assets were held in government bonds, which paid 4-5%. But landed estates derived their income from agriculture. Even if the former were stabilized by the prudent fiscal management of the Crown, why the latter? In particular, there must’ve been considerable long term movements up and down in the terms of trade between industry and agriculture. Why then were the incomes from landed estates so stable?

Piketty has ignored the long-term effects of monetary regimes. Dumenil and Levy demonstrated the key role played by real interest rates in the distribution of surplus between labor and capital since the end of World War II. The long-term empirical stability of the rate of return on capital is an epiphenomena. Perhaps one that is an artifact of monetary regimes rather than a constant of nature. If this is the case, then — due to the absolute stability of the monetary regime between Waterloo and World War I — the entire century would count as a single observation. Economic historians who are familiar with the rates since the sixteenth century need to weigh in here. Unfortunately for us all, Braudel is long dead.

My personal feeling is that we can’t get to the bottom of this without a solid explanation of these remarkable regularities. There is no such thing as an empirical law — Piketty himself attacks Kuznets for extrapolating from limited data. Although he has compiled an impressive dataset, the forecast for the twenty-first century may have greater variance than he concedes. For now however, we must treat Piketty’s projections as the baseline scenario. Given that variance works both ways, this is already pretty scary.

This is by no means the last you have heard of Piketty on these pages. We’re sure to be talking about this for a long time.