economics

Yes, r > g. So what?

By N. Gregory Mankiw

Harvard University

November 24, 2014

This essay was prepared the Annual Meeting of the American Economic Association, January 2014. I am grateful to Laurence Ball, Ben Friedman, David Laibson, Lisa Mogilanski, Lawrence Summers, Gabriel Unger, and Matthew Weinzeirl for comments.

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Thomas Piketty’s book Capital in the Twenty-First Century captured the public’s

attention in a way that few books by economists have. Though its best-seller status was a

surprise, probably even to its author, it has the ingredients that foster wide appeal. The book

addresses a pressing issue of the day in a manner that is learned, literary, speculative,

provocative, and fascinating from beginning to end. While largely a work of economic history, it

does not stop there. Piketty ultimately leads the reader to a vision of what the future may hold

and advice about what policymakers should do about it. That vision is a dystopia of continually

increasing economic inequality due to the dynastic accumulation of capital, leading to a policy

recommendation of a steeply progressive global tax on wealth.

Although I admire Piketty and his book, I am not persuaded by his main conclusions. A

chain is only as strong as its weakest links, and several links in Piketty’s chain of argument are

especially fragile. Other aspects of Piketty’s book may well pass the test of time, but the bottom

line—his vision of the future and the consequent policy advice—most likely will not.

The book documents that the rate of return on private capital r exceeds the economy’s

growth rate g, and it argues that this will likely continue to be the case, perhaps by a larger

amount in the future. He boldly calls this fact “the central contradiction of capitalism.” He

reasons that if r > g, the wealth of the capitalist class will grow faster than the incomes of

workers, leading to an “endless inegalitarian spiral.” To someone who views relatively unfettered

capitalism as one of the great achievements of human history and the best way to organize a

society, as I do, these conclusions present a significant challenge.

The first thing to say about Piketty’s logic is that it will seem strange to any economist

trained in the neoclassical theory of economic growth. The condition r > g should be familiar. In

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the textbook Solow growth model, it arrives naturally as a steady-state condition as long as the

economy does not save so much as to push the capital stock beyond the Golden Rule level.

(Phelps 1961) In this model, r > g is not a problem, but r < g could be. If the rate of return is

less than the growth rate, the economy has accumulated an excessive amount of capital. In this

dynamically inefficient situation, all generations can be made better off by reducing the

economy’s saving rate. From this perspective, we should be reassured that we live in a world in

which r > g because it means we have not left any dynamic Pareto improvements unexploited.

There is, moreover, good reason to doubt that r > g leads to the “endless inegalitarian

spiral” that concerns Piketty. Imagine a wealthy person living in an r > g economy who wants to

ensure that he has an endless stream of wealthy descendants. He can pass his wealth on to his

children, but to ensure that his descendants remain wealthy, he faces three obstacles.

First, his heirs will consume some of the wealth they inherit. For this purpose, the

relevant measure of consumption includes not only food, shelter, and riotous living but also

political and philanthropic contributions, which can be sizeable for wealthy families. A plausible

estimate of the marginal propensity to consume out of wealth, based on both theory and

empirical evidence, is about 3 percent. Thus, if wealth earns a rate of return of r, wealth

accumulates at a rate of about r − 3.

Second, as wealth is passed down from generation to generation, it is divided among a

growing number of descendants. (This would not be a problem for the wealthy patron if his

heirs’ mating were perfectly assortative—that is, if they all married someone of equal wealth.

But matters of the heart are rarely so neat.) To get a rough calibration of this effect, suppose

everyone has a typical family of two children, so the number of descendants doubles every

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generation. Because generations are about 35 years apart, the number of descendants grows at a

rate of 2 percent per year. Thus, if family wealth accumulates at a rate of r − 3, wealth per

descendant grows at a rate of r − 5.

Third, many governments impose taxes on both bequests and capital income. In the

United States today, the estate tax rate is 40 percent (above a threshold). In Massachusetts, where

I live, the state imposes an additional estate tax with a top rate of 16 percent. As a result, at the

margin, about half of a family’s wealth is taxed away by the government once every generation.

If we again assume a generation is 35 years, then estate taxation reduces the accumulation of

dynastic wealth by about 2 percent per year. In addition, capital income taxation during a

person’s life reduces capital accumulation even further. This effect is roughly an additional 1

percent per year, making the total drag of taxes about 3 percent per year. Let’s assume, however,

that our dynasty has especially good tax planning and put the total tax effect at only 2 percent.

Thus, taking taxation into account, wealth per descendant grows at a rate of about r − 7.

We can now recalibrate Piketty’s logic taking these three effects into account. Piketty

reasons that resources of the wealthy would grow relative to the labor income if r > g. We can

now see, however, that this condition is not sufficient once consumption, procreation, and

taxation are accounted for. Instead, to obtain the worrisome “endless inegalitarian spiral,” we

would need the return on capital r to exceed the economy’s growth g by at least 7 percentage

points per year.

This scenario is far from what we have experienced. Piketty estimates the real rate of

return to be about 4 or 5 percent, which seems plausible for a typical balanced portfolio.

Meanwhile, the average growth rate of the U.S. economy has been about 3 percent. So Piketty is

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right that r has exceeded g, but it has done so by only about 2 percentage points, not the more

than 7 percentage points necessary for the creation of Piketty’s imagined dystopia.

Moreover, while economists are notoriously bad at predicting the future, especially over

long horizons, it seems unlikely that, looking forward, r will start exceeding g by more than 7

percentage points. If the real return remains stable at 5 percentage points, the economy’s growth

rate would need to become a negative 2 percent. Secular stagnation would not be enough; we

would need secular decline. Alternatively, if future growth is 2 percent per year, the real rate of

return to capital would need to rise from its historical 5 percent to more than 9 percent. That

figure is nowhere near the return that pension and endowment managers are now projecting from

a balanced portfolio of stocks and bonds.

Hence, the forces of consumption, procreation, and taxation are, and will probably

continue to be, sufficient to dilute family wealth over time. As a result, I don’t see it as likely that

the future will be dominated by a few families with large quantities of dynastic wealth, passed

from generation to generation, forever enjoying the life of the rentier.

But suppose I am wrong. Suppose the dynastic accumulation of capital describes the

future, as Pikkety suggests. I would nonetheless remain skeptical of Piketty’s proposal to place

an additional tax on wealth. A simple, standard neoclassical growth model illustrates the problem

with this policy.

Consider an economy composed of two kinds of people—workers and capitalists. Many

workers supply labor inelastically and immediately consume their earnings. A few capitalists

own the capital stock and, because they represent an infinitely-living dynasty, set their

consumption according to the standard model of an optimizing infinitely-lived consumer (as in

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the Ramsey model). Workers and capitalists come together to produce output, using a production

function that experiences labor-augmenting technological progress, and they earn the value of

their marginal product. In addition, following the advice of Piketty, the government imposes a

tax on capital equal to τ per period, the revenue from which is transferred to workers.

To oversimplify a bit, let’s just focus on this economy’s steady state. Using mostly

conventional notation, it is described by the following equations:

(1) cw = w + τk

(2) ck = (r − τ – g)nk

(3) r = f ’(k)

(4) w = f(k) – rk

(5) g = σ(r – τ – ρ)

where cw is consumption of each worker, ck is the consumption of each capitalist, w is the wage,

r is the (before-tax) rate of return on capital, k is the capital stock per worker, n is the number of

workers per capitalist (so nk is the capital stock per capitalist), f(k) is the production function for

output (net of depreciation), g is the rate of labor-augmenting technological change and thus the

steady-state growth rate, σ is the capitalists’ intertemporal elasticity of substitution, and ρ is the

capitalists’ rate of time preference. Equation (1) says that workers consume their wages plus

what is transferred by the government. Equation (2) says that capitalists consume the return on

their capital after paying taxes and saving enough to maintain the steady-state ratio of capital to

effective workers. Equation (3) says that capital earns its marginal product. Equation (4) says

that workers are paid what is left after capital is compensated. Equation (5) is derived from the

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capitalists’ Euler equation; it relates the growth rate of capitalist’s consumption (which is g in

steady state) to the after-tax rate of return.

Because the steady-state return on capital in this economy is r = g/σ + τ + ρ, the condition

r > g arises naturally. A plausible calibration might be g = 2, τ = 2, ρ = 1, and σ = 1, which leads

to r = 5. In this economy, even though r > g, there is no “endless inegalitarian spiral.” Instead,

there is a steady-state level of inequality. (Optimizing capitalists consume enough to prevent

their wealth from growing faster than labor income.) If we assume the number of workers per

capitalist n is large, then capitalists will enjoy a higher standard of living. In this natural case,

cw/ck , the ratio of workers’ consumption to capitalists’ consumption, can be used as a proxy for

inequality. A more egalitarian outcome is then associated with a higher ratio cw/ck.

Now consider the policy question: What level of capital taxation τ should the government

set? Not surprisingly, the answer depends on the objective function.

If policymakers want to maximize the consumption of workers cw subject to equations (1)

through (5) as constraints, they would choose τ = 0. This result of zero capital taxation is familiar

from the optimal tax literature. (Chamley 1985, Judd 1985, and Atkeson, Chari, and Kehoe 1999,

recently reconsidered by Straub and Werning 2014.) In this economy, because capital taxation

reduces capital accumulation, labor productivity, and wages, it is not desirable even from the

standpoint of workers who hold no capital and who get the subsidies that capital taxation would

finance.

By contrast, suppose the government in this economy were a plutocracy, concerned only

about the welfare of the capitalists. In this case, it would choose τ to maximize ck subject to the

above five equations as constraints. The best plutocratic policy is a capital subsidy financed by

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taxes on workers. That is, plutocrats would make τ as negative as it can be. If there is some

minimum subsistence level for workers, the labor tax and capital subsidy would be driven so

high as to push workers’ consumption down to subsistence.

Now consider a government concerned about inequality between workers and capitalists.

In particular, suppose that policymakers want to increase the ratio cw/ck. In this case, a positive

value for the capital tax τ is optimal. Indeed, if maximizing cw/ck is the only goal, then the capital

tax should be as large as it can be. Taxing capital and transferring the proceeds to workers

reduces the steady-state consumption of both workers and capitalists, but it impoverishes the

capitalists at a faster rate. For a standard production function (f ’ > 0 and f ” < 0), a higher

capital tax always raises cw/ck.

Thus, in this simple neoclassical growth model, a positive tax on capital has little to

recommend it if we care only about levels of consumption, but it may look attractive if we are

concerned about disparities. To misquote Winston Churchill: the inherent vice of the free-market

equilibrium is the unequal sharing of blessings; the inherent virtue of capital taxation is the more

equal sharing of miseries.

So far, I have included only one policy instrument—the one recommended by Piketty—

but we can consider others. A better way to pursue equality in this model economy, and I believe

the real economy as well, is a progressive tax on consumption. Such a tax could equalize living

standards between workers and capitalists without distorting the intertemporal margin and

thereby discouraging capital accumulation. Under a progressive consumption tax, the capitalists

would be just as wealthy as they are without it, but they would not fully enjoy the fruits of their

wealth.

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With this model as background, let’s move to the big question: Why should we be

concerned about inequality in wealth? Why should anyone care if some families have

accumulated capital and enjoy the life of the rentier? Piketty writes about such inequality as if we

all innately share his personal distaste for it. But before we embark on policies aimed at reducing

wealth inequality, such as a global tax on capital, it would be useful to explore why this

inequality matters.

One place to look for answers is Occupy Wall Street, the protest movement that drew

attention to growing inequality. This movement was motivated, I believe, by the sense that the

affluence of the financial sector was a threat to other people’s living standards. In the aftermath

of a financial crisis followed by a deep recession, this sentiment was understandable. Yet the

protesters seemed not to object to affluence itself. If they had, Occupy Wall Street would have

been accompanied by Occupy Silicon Valley, Occupy Hollywood, and Occupy Major League

Baseball. From this perspective, the rentier lifestyle of capitalists should not be a concern. As we

have seen, in a standard neoclassical growth model, the owners of capital earn the value of their

marginal contribution to the production process, and their accumulation of capital enhances the

productivity and incomes of workers.

Another possibility is that we object to wealth inequality because it is not fair. Why

should someone be lucky enough to be born into a family of capitalists whereas someone else is

born into a family of workers? The disparity between workers and capitalists is inconsistent with

the ideal of equal opportunity. Yet that ideal conflicts with another—the freedom of parents to

use their resources to help their children. (Fishkin 1984) Moreover, in considering Piketty’s

proposal of a global capital tax, we have to ask: Would you rather be born into a world in which

we are unequal but prosperous or a world in which we are more equal but all less prosperous?

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Even if equal opportunity is a goal, one might still prefer unequal opportunities to be rich over

equal opportunities to be poor.

A final possibility is that wealth inequality is somehow a threat to democracy. Piketty

alludes to this worry throughout his book. I am less concerned. The wealthy includes supporters

of both the right (the Koch brothers, Sheldon Adelson) and the left (George Soros, Tom Steyer),

and despite high levels of inequality, in 2008 and 2012 the United States managed to elect a left-

leaning president committed to increasing taxes on the rich. The fathers of American democracy,

including George Washington, Thomas Jefferson, John Adams, and James Madison, were very

rich men. With estimated net worth (in today’s dollars) ranging from $20 million to $500

million, they were likely all in the top 0.1 percent of the wealth distribution, demonstrating that

the accumulation of capital is perfectly compatible with democratic values. Yet, to the extent that

wealth inequality undermines political ideals, reform of the electoral system is a better solution

than a growth-depressing tax on capital.

My own view—and I recognize that this is a statement of personal political philosophy

more than economics—is that wealth inequality is not a problem in itself. And I do not see

anything objectionable if the economically successful use their good fortune to benefit their

children rather than spending it on themselves. As a society, we should help those at the bottom

of the economic ladder through such policies as a well-functioning educational system and a

robust social safety net (funded with a progressive consumption tax). And we should help people

overcome impediments to saving, thereby allowing more workers to become capitalists. But if

closing the gap between rich and poor lowers everyone’s standard of living, as I believe Piketty’s

global tax on capital would do, I see little appeal to the proposal.

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References

Atkeson, Andrew, V.V. Chari and Patrick J. Kehoe, 1999, “Taxing Capital Income: A Bad Idea,”

Federal Reserve Bank of Minneapolis Quarterly Review 23, 3–17.

Chamley, Christophe. 1986. “Optimal Taxation of Capital Income in General Equilibrium with

Infinite Lives,” Econometrica 54 (3), 607–622.

Fishkin, James S., Justice, Equal Opportunity and the Family, New Haven: Yale University

Press, 1984.

Judd, Kenneth L. 1985. “Redistributive Taxation in a Simple Perfect Foresight Model,” Journal

of Public Economics 28 (1), 59–83.

Phelps, Edmund S. 1961. "The Golden Rule of Capital Accumulation". American Economic

Review 51: 638–643.

Straub, Ludwig, and Iván Werning. 2014. “Positive Long-Run Capital Taxation: Chamley-Judd

Revisited,” MIT Working Paper.