More Matt Rognlie on Piketty

From the comments:

Krugman correctly highlights the importance of the elasticity of substitution between capital and labor, but like everyone else (including, apparently, Piketty himself) he misses a subtle but absolutely crucial point.

When economists discuss this elasticity, they generally do so in the context of a gross production function (*not* net of depreciation). In this setting, the elasticity of substitution gives the relationship between the capital-output ratio K/Y and the user cost of capital, which is r+delta, the sum of the relevant real rate of return and the depreciation rate. For instance, if this elasticity is 1.5 and r+delta decreases by a factor of 2, then (moving along the demand curve) K/Y will increase by a factor of 2^(1.5) = 2.8.

Piketty, on the other hand, uses only net concepts, as they are relevant for understanding net income. When he talks about the critical importance of an elasticity of substitution greater than one, he means an elasticity of substitution in the *net* production function. This is a very different concept. In particular, this elasticity gives us the relationship between the capital-output ratio K/Y and the real rate of return r, rather than the full user cost r+delta. This elasticity is lower, by a fraction of r/(r+delta), than the relevant elasticity in the gross production function.

This is no mere quibble. For the US capital stock, the average depreciation rate is a little above delta=5%. Suppose that we take Piketty’s starting point of r=5%. Then r/(r+delta) = 1/2, and the net production function elasticities that matter to Piketty’s argument are only 1/2 of the corresponding elasticities for the gross production function!

Piketty notes in his book that Cobb-Douglas, with an elasticity of one, is the usual benchmark – and then he tries to argue that the actual elasticity is somewhat higher than this benchmark. But the benchmark elasticity of one, as generally understood, is a benchmark for the elasticity in the gross production function – translating into Piketty’s units instead, that’s only 0.5, making Piketty’s proposed >1 elasticity a much more dramatic departure from the benchmark. (Keep in mind that a Cobb-Douglas *net* production function would be a very strange choice of functional form – implying, for instance, that no matter how much capital is used, its gross marginal product is always higher than the depreciation rate. I’ve never seen anyone use it, for good reason.)

Indeed, with this point in mind, the sources cited in support of high elasticities do not necessarily support Piketty’s argument. For instance, in their closely related forthcoming QJE paper, Piketty and Zucman cite Karabarbounis and Neiman (2014) as an example of a paper with an elasticity above 1. But K&N estimate an elasticity in standard units, and their baseline estimate is 1.25! In Piketty’s units, this is just 0.625.

And this:

What does this all mean for the Piketty’s central points – that total capital income rK/Y will increase, and that r-g will grow? His model imposes a constant, exogenous net savings rate ‘s’, which brings him to the “second fundamental law of capitalism”, which is that asymptotically K/Y = s/g. The worry is that as g decreases due to demographics and (possibly) slower per capita growth, this will lead to a very large increase in K/Y. But, of course, this only means an increase in net capital income rK/Y if Piketty’s elasticity of substitution is above 1, or if equivalently the usual elasticity of substitution is above 2. This is already a very high value, and frankly one to be treated with skepticism.

Meanwhile, it is even harder to get growth in r-g, which most readers take to be Piketty’s central point. Suppose that in recent decades, r has been roughly 5% while g has been 2.5%, and suppose that g will ultimately fall to around 1%. In Piketty’s framework, this implies an increase in steady-state K/Y of 2.5. If there is an elasticity of 1 (in Piketty’s units), this implies a decrease in r from 5% to 2%, and thus a *decrease* in the gap r-g from 2.5% to 1%. The point is that with this unit demand elasticity and the exogenous net savings assumption, it is the ratio r/g rather than the difference r-g that is constant, which means that a decline in g leads to a proportionate decline in r-g. (Note that Krugman’s review is ambiguous about this distinction.)

What would we need to obtain even a tiny increase in r-g in this setting – say, of half a percentage point? We would need r to fall from 5% to only 4% while g fell from 2.5% to 1%, increasing r-g from 2.5% to 3%. But given the 2.5-fold increase in K/Y, a decline in r by a factor of only 1/5th implies an elasticity of substitution (in Piketty’s sense) of nearly 4. This implies an elasticity of substitution in the *usual* gross production function sense of nearly 8, not plausible by any stretch of the imagination.

Unless I’m missing something, the formal apparatus in Piketty’s book simply is not capable of generating the results he touts. There are two very simple issues that break it quantitatively – first, the distinction between elasticities of substitution in the gross and net production functions; and second, the fact that as g falls, an extraordinarily high elasticity of substitution is necessary to prevent r from falling along with it and actually compressing the arithmetic gap between r and g. Perhaps there are modifications to the framework that can redeem it, but as it currently stands I’m baffled.

I believe Matt is correct.  I would simply note that diminishing returns to capital — relative to other factors of production — are likely to hold in the long run.  See also these earlier MR comments by Rognlie and Harless.  And here are Piketty’s lecture notes.


all of that flew over my head. I'm probably going to have to spend the weekend reviewing those concepts

Here's the tl;dr:

Commies trying to sneak in the back door aren't honest.

I'm just going to agree with which ever side fits my political preferences.

Jeez, Urstoff. Don't be so transparent about it, like Krugman.

Throw out some equations first!

I’d like to say thanks to Matt. It’s reassuring to have a modelled or empirical refutation even if only in order to invite another refutation … ad infinitum (although Leopold Kronecker did try to stop the trend by ejecting infinity from math altogether). Rumour is Piketty views math as a “juvenile passion divorced from history”. Ainsi c’est humorous he has aroused devout interest of math-oriented socialist economists. The USA leftist intelligentsia is today literally swooning over a handsome French socialist. Could it be anything at all to do with the subliminal influence of NFL? Google ‘marginal revolution NFL’ … et trouver la nourriture pour la pensée. Then google ‘socialism nfl’. Then contemplate one’s navel.

1. this is a very insightful attack, I think, and I hope Piketty does provide a more concrete defense of his assertion that the net elasticity of substitution is greater than one;

2. that said, it is not quite right to say that Piketty's argument deduces from estimates of elasticity. Rather, Piketty is reversing the direction of the argument:

As we discuss in section 7, rising capital-output ratios together with rising capital shares and declining returns to capital imply an elasticity of substitution between labor and capital higher than 1 – consistent with the results obtained by Karabarbounis and Neiman (2013) over the same period of time...

The first basic fact is that capital shares did rise in rich countries during the 1970-2010 period, from about 15%-25% in the 1970s to 25%-35% in the 2000s-2010s, with large variations over time and across countries (Figure 13). However they did not rise as much as national wealth-national income and domestic capital-output ratios, so that the average of return to wealth – which can be computed as rt = ↵t/!t – declined somewhat (Figure 14). Of course, this decline is what one would expect in any model: when there is more capital, the rate of return to capital must go down. The interesting question is whether it falls more or less than the quantity of capital. According to our data it has fallen less, implying a rising capital share.

There are several ways to think about this piece of evidence. One can think of a model with imperfect competition and an increase in the bargaining power of capital (e.g., due to globalization and increasing capital mobility). One can also think of a production function with three factors – capital, high skill labor and low skill labor – where capital is more strongly complementary with skilled than with unskilled labor. With a rise in skills, and possibly with skill-biased technical change, it can easily generate a rising capital share.

Yet another – and more parsimonious – way to obtain the same result is a standard two-factor, CES production function F(K,L) with an elasticity of substitution # > 1. Importantly, the elasticity does not need to be hugely superior to one in order to account for the observed trends. With an elasticity # around 1.2-1.6, a doubling of capital-output ratio ! can lead to a large rise in the capital share ↵. With large changes in !, one can obtain substantial movements
in the capital share with a production function that is only moderately more flexible than the standard Cobb-Douglas function. For instance, with # = 1.5, the capital share rises from ↵ = 28% to ↵ = 36% if the wealth-income ratio jumps from ! = 2.5 to ! = 5, which is roughly what has happened in rich countries since the 1970s. The capital share would reach ↵ = 42% in case further capital accumulation takes place and the wealth-income ratio attains ! = 8. In
case the production function becomes even more flexible over time (say, # = 1.8), the capital share would then be as large as ↵ = 53%. We do not claim that this scenario will necessarily happen.

Our point is simply that it cannot be excluded. Constant capital-output ratios and capital shares are more of a belief than a well-grounded fact...

We stress that our discussion of capital shares and production functions should be viewed as merely exploratory and illustrative. In many ways, it is more difficult to measure capital shares ↵ than wealth-income ratios ...

i.e., the argument is not that the net elasticity of substitution is more than one in the data. The argument is that the net elasticity of substitution being more than one would be consistent with a Solow approach to national accounts trends. K&N are merely invoked for plausibility, not calibration. The amount of hedging that Piketty and Zucman put here suggests that they are aware of the weakness of the argument - I think many, many readers of MR would say an increase in the bargaining power of capital, or ZMP of low skill labour, are more parsimonious! We think in terms of micro nowadays, do we not? Whilst the two approaches are not contradictory, there needs to be some microfounding of the proposed aggregate production function with net elasticity >1 in terms of (say) the alleged skill-biased technological change. Exploratory illustration is not good enough for a crucial part of the argument.

You're right. The mention of Karabarbounis and Neiman (2013) was in order to emphasize that the confusion between net and gross elasticities seems pervasive - to the point where they're discussing a paper that estimates CES with a gross elasticity of 1.25 as a parallel to their own net elasticity. But they're not building their case on it or anything.

The problem is that I'm not exactly sure what their case is. Their only piece of evidence for a net elasticity > 1 seems to be the observation that the net capital share of income has risen alongside the capital/income ratio over the past several decades. This is interesting, but it's hardly conclusive (all kinds of other shifters could have been moving these numbers along the way), and eyeballing two trends is a much sketchier approach to identifying elasticities than the approaches that have been used in the relevant literature (which have virtually never found such large elasticities). Indeed, as I observe below, looking under the hood at their capital income numbers suggests a very different picture.

My better growth theory days are way behind me (back in comp times) but Piketty is fully aware of the gross-net distinction. From his notes: "Above we use net-of-capital-depreciation production function Y=F(K,L) (i.e. Y = net domestic product), and net-of-capital-depreciation savings S=sY. Sometime people use gross production function Y=F(K,L) (i.e. Y = gross domestic product)...".

I'm also not fully sure of the way you defines elasticity of substitution here. In my mind (and Piketty's I think),

EOS=delta log(K/L)/delta log(w/r)=sigma with a CES production function (the outer exponent i sigma/(sigma-1)).

But here you have it as delta (K/Y)/delta(r). There might be some mapping I am missing, but I don't see the scale being equivalent. Can you elaborate?

Well, I know where the earlier comments by Rognlie and Harless are, since I wrote some of them (and have just added another one), but the link doesn't work.

Look at the content of the link, it's funny.

Link for the lazy:

...and now the link in the text works, but it's the wrong link. (JAM's link above is correct.)

Here's a simple sentence to add to the mix: it is hard to get r staying above g in a way that doesn't raise wages significantly, in the medium run at least.

A capitalist would seek higher wages because that is the return on human capital, and a capitalist seeks to increase all capital assets in real terms.

Pillage and plunder free lunch economist seek to destroy the value of capital because that gives the free lunch of something from the labor of others, plus creates a reduced supply of capital that can demand a higher price reaping profits from destruction of capital assets, and restricting the market so more capital won't be built.

When money is flowing into the capital markets, conservatives advocate increased pillage and plunder of capital assets to produce with zero return on real capital assets. If you are growing firewood and seasoning it in the expectations that you can earn a return on your land and labor, conservatives call for ripping out the wood in existing structures and the furniture and selling it for less to stop the competition. Or the equivalent drill baby drill because making land less valuable is the free lunch that kills jobs and keeps labor cheap and thus drives down human capital. And you must admit that a drilling lease on your land will make you a net loser over the following decade - once the economic extraction is done, your land is worthless for drill baby drill so the land as an asset is worth less. Drill baby drill can never create wealth, only reduce it. Or redistribute it from the landowner to the one in control of the drilling. In contrast, putting up a wind farm on the land will "create wealth" by employing human capital as well as converting coal and iron ore through human capital into steel that will still be worth a great deal in five centuries after its been recycled 5-10-500 times. And the wind will still be blowing in five centuries and still generating electricity if the capital depreciation is offset by reinvestment to refresh the capital.

If the pro formas on wind energy are as predictable and hearty as all that capital will be lining up to put wind mills on a piece of property whether it's been drilled out or not. I could even take some of what the oil company paid me to lease my land and invest in the windmills (or the angel/hedge fund) myself.

"Here’s a simple sentence to add to the mix: it is hard to get r staying above g in a way that doesn’t raise wages significantly, in the medium run at least."

Theoretically, the rich could combat that by lobbying for more immigration to increase the supply of labor relative to capital.

This theory would predict that Mark Zuckerberg and Bill Gates would try to get richer by lobbying Congress for more H-1B visas to lower salaries they have to pay to their workers? Has anybody ever heard of Zuckerberg and Gates doing that?

Another theoretical prediction would be that an incredibly rich foreign oligarch who profits exorbitantly on phone calls between illegal aliens in America and their family and friends in Mexico could become the second biggest shareholder in the New York Times and see his financial interests reflected in nonstop demonization of American skeptics about immigration from Mexico.

But that prediction seems like a long shot ...

Now Yglesias and Matt O'Brien are claiming on Twitter that Piketty is only talking about "empirical results." This is comical. I read the Krugman review (which was quite good) and it's obvious the reason people are interested in the book is because of its predictions about the future. Predictions are not "empirical results," they require a model.

Ok, pretty much over my head. I definitely expect lots of criticism of Piketty's main conclusions out of this group.

One question though: doesn't the historical--and much of the recent--data substantiate Piketty's general theory? Putting away the models and your calculators for a minute, does the empirical evidence back the trends about wealth:income and a fairly constant r of 5%? Can we acknowledge the trend and the possibility of its ongoing trajectory, even if the theoretical basis is not airtight (which, as I understand it is true about almost all economic models)?

Well, sure. But the historical record also supports the idea that the vast majority of all humans will be employed in agriculture. Frankly, so does a lot of the recent data. I presume you don't think that we're in an exceptional period and will shortly revert to all being farmers?

Not trying to be facetious -- just pointing out that things do change. Picketty is saying that, contra the recent wisdom, the preeminence of non-inherited wealth is a minor aberration, and not a fundamental technological change. He may be right, but you can't determine that just by saying, "Look at history!"

Of the overwhelming majority of human history, we were all hunter-gatherers.

Piketty also doesn't take into account changes in consumption inequality that lead to a narrowing of the gap in the decades after the 1840s and prior to WW1. Hoffman, et al. (2002) look at long run real inequality in Europe and show that changes in prices produced by the Industrial Revolution tended to favor the rich at first but rapidly shifted from mid century to favoring the poor and lower classes despite fears by Ricardo and others that land owners (the major early capitalists) would swallow all the gains from growth. In fact, labor's share of income rose and measured inequality overstated true inequality because of the price composition effects. Piketty certainly doesn't deal with these issues today or in the future either.

I believe Greg Clark's Farewell to Alms book contains a lot of consumption data from the first century of industrialization that is also not so consistent with the income trajectories in Piketty's charts.

"I would simply note that diminishing returns to capital — relative to other factors of production — are likely to hold in the long run."

If interest rates on money are zero or negative, then returns to capital should be zero or negative as well,

BUT, returns to capital are much higher than market returns are so high they represent economic profit from restrictions in the market. At least profits were a bad thing back before Reagan introduced voodoo economics and made it populat because if provided free lunches.

But I'd say the evidence is pretty clear that capital assets have had net depreciation in the aggregate for the past three decades at least. The infrastructure built before 1980 was paid for before 1980 on the whole, and that which wasn;r was paid off by 1990, but very little of that old capital asset has been replaced even tholong past the time of replacement and refurbishment. And in many cases, the revenue to provide the returns on those assets has been zero or negative, and that has led to raiding the pension assets of workers to pay for band aid fixes of the aging capital assets.

This is not limited to public capital. The airline industry has failed to pay for its capital assets over the past three decades, declaring serial bankruptcy to redistribute wealth to pay for the capital assets that are so critical to the business. Because the return on capital is so low in the air transport industry, the public assets have not been updates and ATC operates using obsolete hardware with multiple generations of capital upgrades abandoned because it would have bankrupted parts of the private sector with required equipment upgrades. The result is no free flight and huge cost burdens on the industry because they operate according to flight paths defined when FDR was president.

The loss of flight 370 is the result of conservative policies that block investment in capital assets. Instead of requiring the airline industry implement real time monitoring of aircraft to allow free flight and also constantly track equipment and human problems, conservatives have blocked the regulations, and then when things like 370 come along, the government borrows money with no revenue to repay it to pay for finding the aircraft and discover the cause, giving the airline shareholders a free lunch of private profit, public losses.

The railroads have not only abandoned 30% of their track assets in the past three decades, the track they still use is in such bad shape its speed is limited to half what it was in 1940.

The road capital assets are trillions in deficit on capital account and for the past decade the revenue from road use fees has not paid for band aid fixes to the capital assets, with massive debt incurred with zero revenue to service the debt put in place to replace the capital assets that failed and that has become functional obsolete to the point of not being useful.

And drill baby drill is pure burning of capital assets leaving nothing to show for the pillage and plunder of natural capital. Before 1980, the nation was devoting a high portion of labor to building capital assets and had a labor shortage. Thus labor saving burning of fossil fuels would support building capital faster. Note that coal plus iron ore produces a capital asset that is more valuable than the coal or ore, an asset that can be refreshed after use depreciates it using electricity from sustainable sources plus labor. But sustainable electricity is labor embodied in wind farms and hydro and other such assets.

Conservatives have advocated the free lunch of capital destruction and decay to cut the return on human capital - wages and benefits - for the past three decades in order to generate huge profits from capital that they are unwilling to pay for buy employing the labor to build more net capital. By decreasing the quantity of capital and blocking anyone else from increasing capital, the existing capital stock repeats profits beyond the fair market price of capital.

Marketplace surveyed the cost of conservative policies in Georgia which closed hospitals in rural Georgia ensuring the residents suffered poorer economic prospects and health, and ensuring industry will not build in those rural counties without hospitals. The conservative policies to "create wealth" destroyed the built capital and human capital of significant parts of once prosperous parts of Georgia and many other States.

I know people usually skip mulp comments, but this one has at least one buried gem

" The loss of flight 370 is the result of conservative policies that block investment in capital assets"

@Michael B Sullivan: "Of the overwhelming majority of human history, we were all hunter-gatherers." Perhaps that is a very relevant fact. Perhaps the fact that hunter-gatherer societies lasted for millions of years, but our industrial civilization is imploding after a couple hundred years is indication our western version of free-market industrial capitalism is not very sustainable.

Or maybe it's a sign that you're fetishizing disaster and have a ridiculous longing for an utterly imagined golden age.

"For the US capital stock, the average depreciation rate is a little above delta=5%"

Piketty likes to cite examples from high and popular culture such as Downton Abbey as an illustration of the high degrees of inequality in the past. The background to the show is that Downton Abbey was physically depreciating and would have become uninhabitable because the Earl couldn't fix it up out of his own capital if the Earl hadn't married a rich American heiress. Fortunately, that marriage turned out happier than the nonfictional loveless marriage of the Duke of Marlborough to Consuelo Vanderbilt that paid for the refurbishment of Blenheim Palace.

So, depreciation outrunning the return on capital proved a huge issue for the hereditary rich in pre-WWI Europe, forcing them into deeply personal sacrifices.

But, do the current Forbes 400 rich spend as much of their wealth on depreciating stuff as did the old rich. Bill Gates spent $44 million on his house, which is a lot, but still most of his wealth seems to be in financial assets that aren't depreciating.

Actually what you were seeing was the tail end of a century long phenomenon in which the share of national income going to landed capital collapsed while labor's share rose. There were many new rich but huge chunks of the old rural aristocracy were brought low over the course of the nineteenth century. This has long been known in the hopistorical literature.

Evidently, Piketty and Zucman's argument for a high net elasticity of substitution is that the net capital income share has risen alongside the capital/income ratio in recent decades. (This is briefly mentioned in the paper, and Zucman talked about it on twitter.)

Honestly, this isn't a very strong argument. There are many ways that this could happen without such a high elasticity of substitution, e.g. technological change with the right factor bias. One comparison of long term trends is hardly enough to overturn an entire literature, most of which uses much savvier empirical approaches and fails to find an elasticity of even close to the same magnitude.

But I'd like to emphasize a point about the disaggregated data - namely, that if you look at the composition of net domestic capital income, by far the most important contributor to the secular increase in share comes from net housing rents. This is true essentially across the board, in all the countries that Piketty and Zucman study (with the odd exception of Germany). (This is easy to see; all the data, a very impressive effort, is available on their website.) In a world with very high elasticities of substitution, this spectacular increase in the housing share would only make sense if the relative rental cost of housing fell.

Did this actually happen? If anything, the situation was the opposite: in the US, the PCE price index for housing has increased by over 35% relative to the mid-70s, yet according to the Piketty-Zucman data, the housing share of income has increased from an average of 4.3% during the 70s to 6.6% today. This suggests a very low elasticity of substitution - as is always the case when shares and prices are moving in the same direction.

I'm not saying that this inference is decisive either, but it's no less legitimate than the P-Z one, and if anything it emphasizes that it's hard to draw unambiguous conclusions by eyeballing a few time trends. (Which is why we ought to be skeptical when someone reasoning along these lines claims an elasticity far out of line with the empirical literature.) In fact, I think it pretty effectively subverts the P-Z argument - it's rather weird to make a big deal about the rise in net capital income when the most important cause of that rise is net rental income, and looking more closely at that component tells a very different story.

See Section III.C in the paper. Piketty and Zucman group land price changes under 'relative price effects' since the actual supply of land is unrelated to saving. But they argue that volume effects are larger that real price effects "in the long run". Their argument is put forth in Section V.B. I am not sure what their thoughts on relative rental cost of housing are, but I have a sense that they put down the housing prices to an income effect.

I think you are treating the identity as a behavioral equation - the price of capital has gone up due to demand. I might totally be missing somerhing but this seems wrong: housing investment goes up when rents go up through rising demad.

This paper strongly contradicts Picketty's return on capital for 1150-1800:

Gregory Clark's "A Farewell to Alms" lingers over the history of land rents (the main form of the return to capital before the Industrial Revolution) in some detail.

For generations people knew that Marx's writings were based on a theory of value that did not work, yet people still considered his writings the rigorous objective proof for why various policies were preferred. Usually these were different people (who ever read Capital?). Clearly, it didn't matter, what mattered was that there was 'science' behind the objective, the new god, science, who defines the pale. Now every redistributionist economist and journalist loves Piketty because it supports their policies, and most importantly is difficult to disprove. That is, to disagree with the argument, you have to understand issues like the elasticity of substitution, so that gets rid of 90% of the flak. Those who wade into the weeds like Rognlie are then told they miss the big picture and specific assumptions weren't important, etc.

This isn't a crazy anomaly, this is modern science. It's not a proof, it's a spread argument.

Sounds about right.

Piketty is comparing the stocks to flows.

Assume a model where there are two types of people - owners of capital, who have no income other than the return on their ownership of capital, and workers, who own no capital, and whose income is derived entirely from current production.

Owners of capital receive, as income, r x C, per year, where r is the average rate of return, and C is the current capital stock. Unless the stock of capital is being increased, this will remain fixed over time, and thus will represent a declining share of P (total production), so long as g (annual rate of increase in total production) is positive. The change in the income of owners of capital will only increase faster than g when the stock of capital increases at a rate c', greater than g.

The increase in the stock of capital can be paid from capital-owners income, which reduces their net income from r x C to (r - c') x C, but with C growing from year to year at a rate of c'. So long as r < 2 x g, the net income of capital-owners cannot sustainably increase faster than the income of workers. (If the increase in capital is paid for by worker income, that increases the class of capital-owners, which is outside the scope of the analysis.)

Therefore, assuming Piketty is correct about the long-term average real rate of return to capital being about 5%, it is imperative that governments ensure that overall production per capita can increase at an average rate of at least 2.5% annually. That's generally sustainable in the industrial world., even if some countries seem incapable of performing that well due to anti-economic policies.

The real rate of return on government bonds maturing in 10 years or more, and on corporate debentures rated A or better, is about 3% p.a. Hence r = 0.03. Piketty's value of 5% requires bearing substantial risk. Last century, the average real return on the NYSE was 6% - 6.5%.

In OECD countries post WWII, the long run average rate of growth of real GDP per capita ranges from 1.6% (Switzerland, New Zealand) to 2.4% (Norway, Australia). Larger growth rates are contaminated with transition dynamics (spectacular cases in point include Korea, Taiwan, Hong Kong and Singapore). Hence when looking at probable steady states, g = 1.6% to 2.4%, so that 2g > r. Since the GFC, g has fallen. But then so has r, unless we let the stock market dominate the calculation of r.

1929-74, wages were roughly 50% of GDI. By 2013, they had fallen to 42%, an all time low.
1929-70, employee compensation rose from 50% to 58% of GDI. By 2013, it had fallen to 52%, the lowest value since 1941. Over the course of two generations, the large increase in fringe benefits during and since WWII, has been paid for workers accepting lower money wages. Fringe benefits have run 10-11% of GDI since the late 1980s.
I submit that these findings do much to explain the growing economic discontent in the USA. I have no economic explanation for this seeming secular decline in labour's share. These calculations implicitly assume that the profits of proprietorships are wholly a return on invested capital, and not a disguised wage. This assumption is unlikely to matter, given that the importance of proprietorships declined 1929-85, and has been stable since.

My formal economics education is 2 semesters (Micro&Macro Introduction) almost 30 years ago in business school, and still I managed to follow Matt's argument, and understand it. This probably says more about his writing skills than my economics. What it says about math savvy Nobel winners is a whole different story.

After reading the latest (serious and professional, not pundit and/or political) reviews of Piketty's book I take note EVERYTHING in it but the graph generating software has been demolished, so now I have only one economic related question left:

What's Amazon's return policy on books?

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