Why Piketty Is Wrong: Part 1 - The Math

by: Philip Mause


Thomas Piketty's "Capital in the Twenty-First Century" raises important issues.

It argues that returns on capital will eat up ever larger shares of national income.

Piketty's argument is flawed because of assumptions about the return on capital.

He also makes unwarranted assumptions about the elasticity of substitution between capital and labor.

Piketty neglects the fact that under reasonable scenarios, implausibly large levels of savings would be necessary for his projections to materialize.

Thomas Piketty has written an excellent and thought provoking book - for those who have been in a coma the past few weeks, the title is "Capital in the Twenty-First Century". His thesis is in the genre of dystopian futurism. It is a theory under which we all go about our business, work hard, save prudently, floss, clean up after our dogs and avoid fattening foods and - yet - a catastrophe occurs because of the working of inexorable economic laws. Reminiscent of the "population bomb," "peak oil," and "global warming," a newly discovered trend that is unfolding even as we speak will catapult us into a world of misery.

The theory goes like this. Capital tends to grow over time because the return on capital is greater than the rate of growth. Because ownership of capital is concentrated in few hands, much of this return will be reinvested rather than consumed. As more and more capital is reinvested, returns on capital consume a greater and greater percentage of the national income at the expense of wage income, which steadily declines. As time passes, inherited wealth becomes the only path to affluence. Ultimately, we will all inevitably learn what Piketty calls "Vautrin's lesson." At some length (pp. 238-42), Piketty cites Balzac's "Pere Goriot" in which a "shady" character named Vautrin explains to the novel's protagonist that success can be attained only through inheritance and that study, talent and effort are futile. This is especially troubling to me because an uncle warned me once that those who marry for money "wind up earning it."

This series will have three parts. The first will deal with the mathematics and economic dynamics underlying Piketty's thesis. The second will deal with the "spirit of the age" and will discuss Piketty's frequent references to fictional and other sources in an attempt to allow us to appreciate what a more intensely capitalized world will feel like. The third will deal with Piketty's policy prescriptions and will suggest some alternatives.

At the outset, I should say that I find much to agree with in Piketty's work. I think he identifies a phenomenon that is a real danger in societies like Japan and Italy, but is not a plausible danger in the United States. France, alas, is somewhere in between. To the extent a problem has been identified, however, I believe it is more a symptom than a disease, with the fundamental disease being demographic (too few births and failed immigration policy). I think that his work should set off a searching debate, which will hopefully lead to a reordering of tax and regulatory policy away from total dependence on burdens imposed on the employment relationship. That said, I think that the suggestion that inexorable trends inherent to capitalism will propel free market economies toward ruin is fundamentally flawed.

1. The First Fundamental Law of Capitalism

Piketty presents what he describes as the "First Fundamental Law of Capitalism" early in the book. This "law" is that the share of national income going to returns on capital (a) is equal to the rate of return on capital (r) times the capital income ratio (B): a = r x B. Thus, if the value of all capital is $60 trillion and the annual national income is $10 trillion (creating a capital income ratio of 6), then if the rate of return on capital is 5 percent, capital's share of national income will be 30%.

First of all, there is nothing to disagree with here. Indeed, the rule is tautological or definitional like the "rule" that a baseball player's batting average is always the number of hits divided by the number of at bats.

If we multiply both sides of the equation by the national income, then we get an equation that tells us that the income from capital is equal to the value of capital times the rate of return on capital - nothing really revolutionary or remarkable. The form of presentation tends to suggest that the income on capital is somehow "caused" by the amount of capital. In this regard, a bit of further algebraic manipulation transforms the equation into - the value of capital equals the earnings of that capital times 1 over the rate of return on capital leading very quickly to the formula familiar to equity investors: the price equals earnings times the price earnings ratio.

Piketty (I think correctly) uses fair market value rather than original cost or book value to estimate the value of capital in a given economy. By using fair market value, he acknowledges that the total value of capital in an economy can fluctuate significantly from year to year. Investors tend to think of the value of capital as being a function of its ability to generate cash flow rather than viewing cash flow as being the inevitable result of the deployment of more capital. Thus, Piketty's assumption that capital will receive a greater and greater proportion of national income seems questionable. Since the value of capital is generally calculated with reference to cash flow, the argument that capital will become large in relation to national income and that "therefore" a bigger share of national income will be directed toward returns on capital seems exactly backwards.

Still, for reasons explained below, Piketty identifies a tendency for more and more capital to be deployed each year. Over time, if the return on the book value of capital remains constant, this consistent increase in capital deployment inevitably will result in more income going to capital returns. One key problem is the issue of the "return on capital." Arguments have arisen about the impact of increased deployment and therefore availability of capital; more specifically, a strong case can be made that the return on capital will decline as capital is deployed to ever less attractive investments. Piketty addresses this argument, seems to acknowledge its legitimacy but argues that the elasticity of substitution between capital and labor will be greater than 1 in the 21st century, and that therefore, the volume increase in capital will outweigh any decrease in return and will ensure that capital receives a higher percentage of annual income.

This issue is important because, as Piketty points out (p. 237), if the elasticity of substitution between capital and labor is less than one, then an increase in the capital income ratio is actually likely to lead to a decrease in capital's share of annual income. On the other hand, if it is greater than one, the opposite occurs. Piketty spends some time on this issue and argues that the fact that capital's share of income has been increasing in the past 40 or so years demonstrates that it is likely greater than one. I am skeptical because the data is necessarily somewhat sketchy and the result seems to be a bit counterintuitive. This is also one point on which a number of economists have suggested the Piketty is in error.

The assumption that capital can be readily substituted for labor is especially questionable with respect to the enormous capital investment in residential real estate. Stepping back a moment, Piketty acknowledges in the data appendix to his essay "Capital is Back" that "measuring capital is notoriously difficult." He relies on a national account methodology that generally uses fair market value. A huge proportion of privately owned capital consists of owner occupied residences. The "income" generated by such residences is inevitably calculated by attempting to estimate "avoided rent" expense (how much rent did the owner save by virtue of his ownership). There has been a build up of this capital, although its fair market value fluctuates with the vagaries of the real estate market.

The big question is, exactly how will the deployment of more capital in the direction of owner occupied residences eliminate jobs and employment income? In an efficient society, one might imagine the construction of housing near job sites, which would reduce time spent commuting and thereby eliminate jobs for parking attendants, auto repairman and therapists specializing in "road rage" issues. In reality, the tenacious local opposition a developer gets when he tries to build high density housing in one of our prosperous urban areas is worse than the reaction a builder would get if he tried to construct a Center for Transgender Studies in North Waziristan. It is very unlikely that new housing investment will be efficiently targeted to reduce commuting times. Indeed, recent experience suggests just the opposite.

I am not confident that Piketty got this one right. Of course, what is really important for his predictions is not the current elasticity of substitution between capital and labor but what that elasticity will be in future years. And that is inherently uncertain. While I think that this is an issue which deserves further study, I am not really convinced that Piketty has made a persuasive logical or quantitative case that elasticity of substitution is more than one. And, of course, if it isn't, his argument tends to collapse because, by his own admission, the deployment of more capital will actually reduce capital's share of annual income.

The "First Fundamental Law of Capitalism" is really a tautology and doesn't move the ball forward or advance Piketty's thesis at all. It also tends to get the "cash flow"-"valuation" issue bassackwards and misleadingly implies that the deployment of more capital inevitably leads to a proportionate increase in capital's share of national income. The entire thesis rests on a rather shaky assumption concerning the elasticity of substitution between capital and labor.

2. The Second Fundamental Law of Capitalism

The Second Fundamental Law of Capitalism is that the capital/income ratio is (in the long run) equal to the savings rate (s) divided by the growth rate (g) or B = s/g. Put simply, if an economy has a real growth rate of 2 percent and a savings rate of 10 percent, then - over time - the capital income ratio will be 5. Piketty presents an alternate formulation of this rule (footnote at p. 594) as B = s2/g+d where s2 (my notation) is the gross savings rate and d is depreciation.

This law would be a matter of simple arithmetic if capital was calculated based on depreciated original cost or book value. Remember, however, that Piketty (I think correctly) is using fair market value to calculate capital. Thus, in any given year, there can be no guarantee that the market value of capital will have any particular relationship to the amount of capital, which has been deployed in prior years. Indeed, "Mr. Market" is fickle and the fair market value of capital, and therefore, the value of B can vary enormously in the short term as real estate and equity bubbles inflate and pop.

Still, Piketty has a point here. Over a long time, a persistent s/g ratio will drive the level of capital closer to the level suggested by the formula each year, and it is likely that the formula provides a reasonable approximation of long-term average of the B value.

One key problem is that the value of capital, and therefore B, can get very high if r gets low; put another way, stocks can get expensive if the price earnings ratio becomes elevated. At these elevated levels of B, it begins to require a very high savings rate to maintain the ratio.

Using B=s2/g+d and assuming a 2 1/2 depreciation rate (a rate derived by comparing Table 5-3 with Table 5-4 to derive depreciation as percentage of capital) and a 2 percent growth rate, when B equals 6, we would need a gross savings rate of 27 percent just to maintain B. B could get to 6 rather easily if r declined to 4% and capital earned 25% of national income. The table below shows the gross savings rate necessary to sustain B at various rates of return and shares of national income going to capital.

Capital Share of Total Income Capital Return% Capital Income Ratio Necessary Saving Rate
25% 5% 5.00 22.5%
25% 4% 6.25 28.1%
30% 5% 6.00 27.0%
30% 4% 7.50 33.7%
35% 5% 7.00 30.1%
35% 4% 8.75 39.4%
35% 3.5% 10.00 45%

As capital commands a greater percentage of annual income and as the rate of return falls, the capital income ratio must trend much higher and, at higher levels, it requires that a higher and higher percentage of annual income be saved. In this model, the theoretical maximum capital income ratio would be 22.2; at this level, the entire annual income would have to be saved in order to maintain the capital income ratio.

It is important to understand the likely dynamic here. As more capital is invested, the return is likely to decrease simply because higher return opportunities are the first to be exploited. With a lower return, a much higher capital income ratio will be necessary to support a given percentage capital share of annual income. The formula then requires a higher savings rate to support this high capital income ratio. For the United States, some of the savings numbers in the above table are ludicrous. Recent trends are for corporations to disperse earnings to shareholders in the form of dividends or share repurchases rather than to retain earnings, the net result of which is to reduce savings. I am really not worried about the "danger" that Americans will start to save 30% of national income.

On the other hand, some of the entries toward the bottom of the table are not exactly fanciful when we look at countries like Italy or Japan. In the 1980s, when I was running out to the supermarket buying disposable diapers for $27 a bag and would hear commentators on the radio describing how Japan would rule the world because it had a higher savings rate than the United States, I wanted to call in and point out that it is very easy to save money if you are not having any children. Countries with very low birth rates tend to be able to save more money. Ultimately, a country, which entered into a national pact to follow the dictates of Cohle in True Detective and not have any children at all could save a huge amount of money as it descended into oblivion.

Where does this leave us? First of all, a lot depends upon the rate of return. As it declines, the capital income ratio necessary for capital to command a given percentage of the annual income each year gets to be very high and requires high savings rates. Secondly, we have used a 2% growth rate. Piketty argues for 1.5% and this is certainly open to debate. At 1.5% with a 2.5% depreciation rate, you only have to save 4% of total capital each year to achieve stability in the capital income ratio. Again, I think that the countries with ultra low birth rates are likely to have low growth rates so that the problem identified by Piketty will be worse in Japan and Italy.

I really don't think that there is a plausible case that the United States will "save its way into disaster" by saving so much that massive amounts of capital are built up leading to an unacceptable diversion of national income away from wages. This danger may, however, not be fanciful with respect to countries with low growth and have savings rates like Italy and Japan.

3. The Central Contradiction of Capitalism

Piketty describes the Central Contradiction of Capitalism as the fact that the rate of return on capital can be higher for long periods of time than the rate of growth of output - or r>g. He is concerned that this means that wealth accumulated in the past grows faster than the economy as a whole. This in turn leads to the domination of society by that hated group, "rentiers," who live off coupon clippings, rent checks, and dividends contributing nothing to society and devouring and more of its resources year by year.

By r, Piketty means real return on capital and he implies from time to time that the right level is 4-5 percent. As a retiree and author of the "Desperately Seeking Yield" series of articles, I am not quite so sanguine.

First of all, we have - as described above - the fact that a big share of capital is in the form of owner occupied residential housing. The "return" on this large amount of capital is the "avoided rent" enjoyed by owners who save, each year, the rent they would otherwise have had to pay (minus, of course, property taxes, insurance, upkeep and other expenses they would not have had as a renter). It is not hard to find situations in which the fair market value is probably 30, 40 or even 50 times avoided rent - especially in places like San Francisco, New York, London and Paris. And of course, this ignores second homes for which only seasonal avoided rent is appropriate; in many cases, I suspect that upkeep and other costs exceed avoided rent and the net return is zero or less than zero.

The world of true business investments is more complex and doubtlessly provides generally higher returns. However, in many cases, low current returns are accepted on the (often dubious) theory that discounted cash flow from future years should be used in valuation. Thus, much money is invested with the understanding that little or no return is to be earned immediately. Of course, Piketty's formula requires us to review capital's share of income each year so that in these situations, we again have a zero current return on capital.

And then we come to the fact that, in recent years, a huge amount of capital has been directed at "bubbles" at just the wrong time. Money has also been "thrown" at various problems; this seems to be especially popular in the energy sector. The nature of speculative bubbles changes from cycle to cycle, but the trajectory of investor loss seems to be constant. In the 1990s, it was the internet, broadband and competitive telephony. A decade earlier, it was commercial real estate. A decade later, it was residential real estate. Nor is this a recent development. The 19th Century was pockmarked with financial crises created by excessive and redundant investment in railroads.

Piketty might argue that the "new capital" being deployed due to savings each year will be "smart" and will avoid these pitfalls, but recent experience suggests that just the opposite is true. "New money" (depending upon how it is defined) may be especially likely to fly like a moth into the center of the candle of speculative destruction.

I have to catch myself here because I am beginning to hear myself make the argument that the reason Piketty is wrong is that capitalism is inefficient and the investors will waste a good deal of their money, thereby saving the day. That is not what I am saying. Indeed, the bubbles and waves of investment we have seen have served various purposes and have funded (perhaps unwittingly) many worthwhile ventures. The fact that money periodically gets very "easy" has allowed some creative entrepreneurs to "seize the day" and prosper. Just as the tides have sustained the diversity of life along the shores of the oceans by providing a variety of conditions at a variety of times, so does the investment cycle permit the funding of seeds that grow into oak trees and - indeed - sequoias.

I am saying something quite different. The overall real return on all capital including owner occupied residences, second homes, money blown during bubbles, etc., etc., may well be much less than 5 percent. This is especially likely to be true if growth is as slow as Piketty predicts. Calculation of the right number still eludes us (just as we cannot know the elasticity of substitution between capital and labor).

We must then turn to the question of whether the mere fact that r>g is destabilizing as suggested by Piketty. In this regard, I will return to the tables above and add some entries with r levels at 3, 2.5 and 2.25% - all higher than the growth rate of 2% and therefore satisfying the formula.

Capital Share of Total Income Return on Capital Capital Income Ratio Necessary Savings %
25% 3% 7.50 33.8%
25% 2.5% 10 45%
25% 2.25% 11 49.5%
30% 3% 10 45%
30% 2.5% 12 54%
30% 2.25% 13.20 59.4%
35% 3% 11.50 51.8%
35% 2.5% 14.00 63%
35% 2.25% 15.40 69.3%

It is clearly possible for r to be greater than g and still not produce a situation in which it is plausible that returns on capital will gradually eat up the entire national income pie. Most of the results in the above table suggest situations in which it would be impossible for savings to sustain the elevated capital income ratio, and therefore, it would be likely for the capital income ratio to be on the road to decline and for capital's share of annual income to decline as well. What is important, therefore, is not whether capital has a return that is higher than overall growth, but rather how much higher that return is, how that return is likely to change with the deployment of more capital, and how much money is saved.

Piketty has simply not made the case that capitalism contains its own "doomsday machine" that will inexorably cause its demise. His book raises important issues and has led to a useful debate. I am sure more research will lead to a tightening of some of the estimates that are now mere guesses. When I was younger, I never thought I would hear anyone make the argument that the biggest problem for the American economy would be that we could "save" our way into oblivion. But if you live long enough, almost anything is possible.

Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.

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