Depreciating Capital

Brad DeLong attacks Krusell and Smith for using in some of their thought experiments a depreciation rate of 10%, which is probably too high. Fair point but in my post I assumed a depreciation of just 5% and showed that Solow and Piketty give very different predictions about how the K/Y ratio will change with a change in g.

Furthermore, having read DeLong’s comment, I went to the BEA and compared gross and net domestic product which gives capital depreciation as a fraction of GDP of around 15% in recent decades. At a K/Y ratio of 4 that’s a depreciation rate of 3.75%. Similarly, Inklaar and Timmer in constructing capital stocks for the Penn World Tables estimate a depreciation rate for the U.S. of 4.1%. I reran my simple Excel chart with the lower number, 3.75%.

As you can see, the numbers are very similar to earlier and the key point is still that a decrease in g increases K/Y much more in the Piketty model than in the Solow model. Piketty2 Krusell responds to DeLong here making the additional point that their thought experiments show that Piketty’s assumption about savings is implausible at any depreciation rate (see also Hamilton on this point).

First: if the net rate of saving remains positive as the economy’s growth rate falls toward zero, as Piketty assumes in his second fundamental law of capitalism, the gross saving rate in the economy must approach 100%. This observation is a way of illustrating how unreasonable the behavioral assumptions underlying his theory of saving really are.

Second: according to standard, and much more reasonable, saving theory (based either on the standard textbook Solow growth model or on the permanent-income model), the net saving rate must fall with the rate of growth, and become zero when growth is zero.

…These points are key because Piketty’s predictions are all about what happens as growth falls during the 21st century, as he argues it will.

…both of these results hold no matter what the depreciation rate is (so long as it is positive).

The heart of Piketty’s theory is his expression for the capital share of income in the long run, α = r × s/g with the prediction that if g falls the capital share will rise tremendously. This is a good opportunity to summarize some of the recent points about the theory.

There are no contradictions but many a slip ‘twixt the cup and the lip. Namely, will g fall? If g does fall, will K/Y increase? If K/Y increases will capital’s share of income increase? My answers:

Will g fall? Uncertain. Piketty’s forecast is as good as anyone’s. My own view is that at the global level g has been increasing for several centuries and that this will continue, especially because in this century we will see a massive increase in the number of scientists and engineers as China and then India devote increased human capital to the research frontier.

If g does fall, will K/Y increase? Yes, but probably less than Piketty estimates and more in line with Solow.

If K/Y increases will capital’s share of income increase? Uncertain but more likely no than yes. It depends on the elasticity of substitution between K and L and as Rognlie and Summers argue, the elasticity that Piketty needs is higher than current estimates suggest is the case.


Taking Alex's et al.'s argument at face value, the conclusion is that Piketty's theoretical framework is not as solid as some would like us to believe. However, does not his data analysis still find that the share of the pie accruing to the wealthy has been growing and growing? (minus minor mistakes in excel sheets that do not affect the overall result).
Just trying to get it straight in my head without having to read all the "reviews" and retorts which appear at a rate of like 10 per day!

>share of the pie accruing to the wealthy has been growing and growing - See more at:

There are many problems with that statement. Where are you talking about? Some countries have extremes between the rich and very poor, and I think you would find that the rich in those places in nominal dollars would be upper middle class in western democracies. Or are you talking about someone like Gates or Waltons who not too long ago were of middle class upbringing who through innovation and business acumen became wealthy? Or are you talking about the Bronfman family who have seen their family fortune, originally acquired by selling liquor to the US, decrease by half under the stewardship of their latest stupid son.

Or maybe you are talking about the few people who shorted the mortgage financing market and made vast fortunes. Or is it the guys who got bonuses in 2009 pretty much paid for by the tax payer bailout of the financial system?

Piketty doesn't take into consideration the various existing government policies of redistribution in his numbers. And his policy suggestions would make sure there is no inequality by making sure there isn't growth.

> Or are you talking about someone like Gates or Waltons who not too long
> ago were of middle class upbringing who through innovation and business
> acumen became wealthy?

Umm. Possibly you should be informed that:

* The current Waltons were all born ridiculously wealthy, and mostly have done nothing at all helpful to society except hoard money and do damage.

* Their father, Sam Walton, despite his many colorful anecdotes about being a farmer, was raised by a man who gave up on farming and worked for a farm mortgage company, amassing a quite significant amount of money. His 'I started as a poor-but-honest blah blah blah' shtick is a whitewash designed for the rubes. Like you.

* And of course William Gates Jr, of Microsoft fame, was born to William Gates Sr, who was not only wealthy but was born wealthy, and put through law school by his parents. The reason he was in a position to start Microsoft was because of his father's money and connections and the engineering that he was able to steal.

So both of your examples of ALL those AMAZING people who worked their way up from the middle class recently actually came from 'old' money, in that their families became rich in the 1950s or before. (Never mind the fact that both brought, if anything, negative value to society.)

I mean, there ARE some examples of people who worked their way up from middle class to actually do something worthwhile. They are obviously only the really lucky AND talented/hardworking ones, since a thousand very talented and hardworking people die in obscurity for each one who makes it. But the least you could do is use examples of the actual people who did so, instead of depending on your audience being too busy to look up whether you're simply lying to them or not.

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"K/Y ratio"

I'm guessing this ratio represents the degree to which the ultra-rich are f*cking the rest of us over on an ongoing basis, and thus the amount of lube we collectively need.

The KY jelly thing crossed my alleged mind, too.

Anyhow, I think I'm one of the "rest of us", but I have not been f*cked by anybody not even anybody that's ultra-rich.

Can you explain to me how I am being f*cked? And, why I didn't even get a kiss?

There are people who are dying to fuck you over, and Piketty is their front man. adorable. Look at the mindless water carrier for the filthy stinking rich.

I'll let you in on a little secret, Yancey: you'll never have your nose far enough up the asses of the super rich to have them notice you. They don't need you.

The flipside is that you think the Socialist politicians will pay attention to you, either.

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especially because in this century we will see a massive increase in the number of scientists and engineers as China and then India devote increased human capital to the research frontier, and we all know Asians have bigger brains and higher IQs than other races. - See more at:

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Much ado about nothing. As any real scientist or mathematician knows, as opposed to a political scientist or so-called social scientist or economist, when you approach "limits" such as the limit of zero, you get a different phase space where your assumptions no longer hold. So AlexT's 'thought experiment' about what happens to Capital/Output at zero growth is flawed. More likely, the Piketty and Solow equations are best valid away from the zero boundary, and we see that largely they are in agreement there: look at the ratios at 2% growth, largely Piketty and Solow agree. So Piketty is a bit more pessimistic than Solow, but nothing that invalidates Piketty. Time will tell who is right.

But zero growth is not a hypothetical. First quarter this year the economy shrank in the US. Which model falls off the cliff in that real situation?

Exactly. Growth of zero is not an absolute zero. An interest rate effect might behave very differently as it approaches zero. People who understand math should know this.

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@derek, @Willitts, @Mike W - I've not read Piketty, and neither have you, but I would think he makes vague claims about how "low" a rate of growth is needed. I know Piketty specifies a low rate of growth, but I don't think, and would be shocked, if Piketty says zero growth is a necessary and sufficient condition for his hypothesis or law to hold. So 2 percent growth is arguably, even if real, 'low', since historically over three percent is normal or high. So Piketty and Solow 'agree' to a degree. Much ado about nothing as I said.

I'm not sure. Are these models not a steady state abstraction? Assume 2% growth, this happens. In fact, the economy is not like that. Especially since the subject of all this study is a very very small number of people. If the extremely rich are the issue, don't their numbers, the actual count of the people involved border on the statistically insignificant?

So Piketty is trying to establish a pattern with a bunch of statistics, where he doesn't take into consideration one of the wealth building strategies, buying low in slow times. Isn't that what you did, bought property when it was cheap due to a market downturn, which then appreciated to make you wealthy?

So if a model of wealth growth doesn't work when growth is zero, then it is not a model of reality. Wasn't one poster boy of wealth inequality the hedge fund guy who made some multi billion profit shorting the mortgage security market? If your model doesn't take into consideration those events, your model is useless.

And really, aren't we into anecdote being the only data point territory? Unless, and this is my suspicion, rich is defined as over $40k salary, as one of our illustrious socialists admitted in an election campaign a few years ago.

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"...look at the ratios at 2% growth, largely Piketty and Solow agree."

Isn't that what AT is saying...that only at zero growth does the capital ratio increase significantly? And, if so and 2% growth is more likely, doesn't that invalidate Piketty's claim that there will be dire consequences from the increasing wealth of capitalists?

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The point is not what happens at any particular g, even zero. The point is that Piketty is predicting a much bigger increase in the capital stock when g falls, say from .03 to .02 as Alex illustrates, than do other theories like Solow. If the capital stock doesn't increase massively, as Piketty predicts, then the rest of his predictions about increased inequality have much less bite.

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While DeLong was unnecessarily harsh in his post on Tuesday I do (on reflection) think he is correct to see this net/gross savings issue as non-central.

Just as K&S can say "even if you use a lower depreciation rate is doesn't change our underlying results" so could Piketty say "even if you use gross rather than net savings then my result still stay the same: k/y still increases as g falls".

I'm guessing that empirical evidence could be used to show that neither Piketty nor Solow's models hold 100% in reality.

Would it be to build a hybrid Piketty/Solow model (similar to ones derived in the table above) where S is a function of both income and depreciation and the results would (presumably) depend upon the values one gives to depreciation and the way that total depreciation affects gross s? That is: Net s tails off as k increases but never reaches 0.

That would take this issue off the table and allow the discussion to focus on the 3 key theoretical/empirical question asked at the end:

Will g fall?
If g does fall, will K/Y increase?
If g does fall, will K/Y increase?

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"DeLong was unnecessarily harsh"

The above sentence would apply to virtually everything by DeLong I've ever read.

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Cowen, wisely, has returned to phase two of the Piketty criticism, which is the theoretical criticism, as phase three, the data criticism, suffers from what Groucho Marx observed long ago: who you gonna believe, Piketty's critics or your lying eyes. While phase two suffers from its own weaknesses (for most people it's unintelligible), it's better than phase one, in which Piketty's critics didn't deny a growing inequality but argued that it's a good thing, and phase three. Indeed, phase two's weakness, it's unintelligible, is both a weakness and a strength.

Assuming Piketty is wrong, what would you expect critics to write?

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You do know that critics cycling through different (alleged) weaknesses in an argument may also be consistent with that argument being weak, right?

You do know that different people pointing out multiple ways something may be wrong is consistent with it being wrong, right?

My sense is the theory commentary is coming from different people than the data commentary. I don't read Marginal Revolution as taking a particular side as much as reporting on the conversation about the data. The theory side I think Alex clearly has a view.

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"You do know that critics cycling through different (alleged) weaknesses in an argument may also be consistent with that argument being weak, right?"

In political arguments, it's common to see the blogs dart this way and that way like a coordinated school of fish. The classic example was Ezra Klein's left-wing Journolist. That's why people are inherently skeptical when they see "critics cycling through different (alleged) weaknesses in an argument," whether the skepticism is warranted or not.

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To recap:
If you don't understand something it is unintelligible and therefore ___________.

From context, I think the blank is something bad? What exactly do you think fits in the blank?

There are obvious problems with a personal comprehension based test for the validity of an argument. I would submit that if this post was that confusing to you, you probably don't understand Piketty's theoretical argument either.

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Even if an economic arrangement tends to push wealth/ income of the top tier/s, you still need to explain why that's been allowed to happen politically. Picketty seems to begin politics after a fairly long amount of time has been allowed to pass, during which I assume politics has been going on all along. And, in fact, Piketty gives political solutions to this upward movement if you see it as a problem. Therefore, the upward movement can't be an iron law. We need an analysis in terms of political economy, Piketty's move from Economics to Political Economy doesn't work for me because he's really doing Political Economy all along. It's not really an economic theory at all. I hope I made sense.

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I've been asking this question, but nobody has answered it yet:

If the net after-tax rate of return on capital does not fall, then why should net savings fall? In almost any behavioral model, shouldn't net savings be a function of net rate of return?

Piketty assumes in his simulations that the net rate of return will not fall (at least very little) over the range of K/Y values he projects. So it makes sense to assume that net savings will not fall over this range. Assuming constant gross savings, (implying declining net savings) without a fall in returns wouldn't make much sense, as far as I can tell.

Seems to me this assumption that the net rate of return won't fall as K/Y rises is *the* crucial assumption. This is the one that has been critiqued by Rognlie and Summers. It's not his savings assumption that is extreme, it's his return assumption. I don't think Piketty is making any predictions about what would happen to net savings rates if growth falls to zero and we were to blast right through the golden rule to the point where depreciation was eating so much K that net returns are negative, as in the example presented by Jim Hamilton. This is simply outside the range of what he considers in his simulations.

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Does Piketty ever explicitly predict a 20+ K/Y ratio? If not, then while the point is well taken about using net savings, the Solow prediction isn't pretty either, and is still in line with his broader narrative.

On growth, the secular trend has fallen in the first world. World growth will continue to be high in the near term, but once catch-up growth ends it will be back to the frontier secular growth rate for all. Having many more innovative people on earth will increase growth, but the general prediction of a technology plateau seems inevitable at some point. Sure, there are an infinite number of undiscovered innovations out there, but what fraction are really significant for growth rates in the same way trains, movable type and adult literacy were? So its totally plausible that at some point in the distance future the world will settle into a stead state of balanced per-capita growth, with a K/Y ratio much higher than today's.

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Tyler, I don't understand why you believe the capital/output ratio will increase with declining growth. Do you have any theoretical explanation, other than Solow assumed so? Any empirical support?

Piketty is just making the same mistake that Solow made in a more severe way. The only difference being that Solow's assumes gross income to gross output is stable, which has more mildly wrong implications than Piketty's assumption that net income to net output is stable.

Keynes discovered another, rather silly implication of assuming a constant savings rate: public investment stimulus must increase consumption by the same amount times the inverse of the savings rate.

"It follows, therefore, that, if the consumption psychology of the community is such that they will choose to consume, e.g. nine-tenths of an increment of income, then the multiplier k is 10; and the total employment caused by (e.g.) increased public works will be ten times the primary employment provided by the public works themselves, assuming no reduction of investment in other directions."

Granted he qualified this conclusion some, but how ridiculous is that.

Look at the data. Savings rates aren't stable over time, and they at least as often move together with growth as against growth. Savings is one of the most adaptable economic behaviors we have. It always responds to changing conditions. The notion that there is something stubborn about savings behavior is just plain silly.

Oops, please re-direct my question to Alex. Sorry, forgot to look at the byline. Alex, why do you believe capital/output will increase with slowing growth?

Tom, good points.

The assumption of a constant savings rate in Solow is just a simplification that lets you solve the model more easily. You get the same qualitative responses with a utility-maximizing savings rate..The Solow model works pretty well as macro models go. I discuss at greater length in my four lectures on the model in the MRU Development Economics courses

You are right, however, that nothing is written in stone.

But if you want a behavioral model based on maximizing agents, shouldn't you be solving for a *net* savings rate as a function of the *net* return on capital?

My guess is that constant gross savings just "works pretty well" because it accompanies a technology assumption about how r falls as K increases, which means that as capital accumulates, net savings fall even as gross savings remains constant. The problem is, that isn't the assumption that Piketty believes, so why should he use constant gross savings?

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Thanks for the reply, Alex. So if I understand you right your belief that capital/ouput rises when growth slows is based on how the Solow model behaves when the savings rate is assumed to be constant. I'm sorry I don't have the free 25.5 hours for your whole course but if you point to a relavent section and time pointer where you discuss the relationship of growth to the savings rate and/or capital/output I'll check that out.

I have no beef with using or teaching the Solow model so long as it's not oversimplified or misused. Drawing conclusions abou the real-world relationships between growth, savings and capital/output based on how the Soliw model behaves when the savings rate is artificially held constant while other parameters change is however a misuse and dead wrong. Cheers.

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Piketty´s saving rate is not the gross saving rate S but the “net saving rate” = S-(K/Y)d, that is, it is K that depreciates not Y. (This is also why using 10% d is much too big but a reasonable for (K/Y)d ) This means K/Y=(S-(K/Y)d)/g for Piketty´s model and solving for K/Y we get K/Y=S/(g+d) which is the same predictions about how the K/Y ratio will change with a change in g as the Solow model

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